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1312.4459	050007 2013 S. A. Grigera E. M. Forgan, School of Physics & Astronomy, University of Birmingham, U.K. 050007 We review experimental results, from transport to magnetization measurements, on different graphite samples, from bulk oriented graphite, thin graphite films to transmission electron microscope lamellae, that indicate the existence of granular superconductivity at temperatures above 100 K. The accumulated evidence speaks for a localization of the superconducting phase(s) at certain interfaces embedded in semiconducting crystalline regions with Bernal stacking order. # Invited review: Graphite and its hidden superconductivity P. Esquinazi[inst1] E-mail: [email protected] (25 January 2013; 10 October 2013) ††volume: 5 99 inst1 Division of Superconductivity and Magnetism, Institute for Experimental Physics II, Fakultät für Physik und Geowissenschaften, Universität Leipzig, Linnéstrasse 5, D-04103 Leipzig, Germany. ## 1 Introduction Over the past decade, our interpretation of the magnetic and transport properties of ordered graphite bulk samples has experienced a change respect to the partially accepted general description of their intrinsic properties. The description of graphite one finds in the not-so-old literature tells us that it is a kind of (semi)metal with a finite Fermi energy and carrier (electron plus hole) densities per graphene layer at low temperatures $n_{0}\sim 10^{10}\ldots 10^{12}~{}$cm-2, see e.g., [1, 2, 3]. However, real samples are not necessarily ideal, we mean defect-free, and therefore those carrier densities are not necessarily intrinsic of ideal graphite. The exhaustive experience accumulated in gapless and narrow band semiconductors [4] already indicates us how important defects and impurities (not necessarily magnetic ones but, for example, hydrogen) are in determining some of the measured properties. Therefore, taking experimental data of real samples as intrinsic, without knowing their microstructure and/or defect concentration, was indeed a misleading assumption in the past. This assumption has drastically influenced the description of the band structure of graphite we found nowadays in several books and publications. For example, if graphite has a finite Fermi energy $E_{F}$ (whatever the majority carriers are), as assumed everywhere, up to seven free parameters have to be introduced [2, 5, 6] to describe the apparently ideal band structure of Bernal graphite with the well- known ABAB stacking order of the graphene layers. The impact of well defined two-dimensional interfaces inside graphite samples [7, 8] had not been realized until recent studies of the transport properties as a function of thickness of the graphite sample provided a link to the microstructure of the samples obtained by transmission electron microscope (TEM) studies. We also have to add the sensitivity of the graphite transport properties to very small amount of defects [9]. Those results [7, 9] do not only indicate us that at least a relevant part of the carrier densities measured in graphite is not intrinsic but also that the metallic-like behavior of the electrical resistance does not reflect ideal, defect-free graphite [10]. An anomalous vanishing of the amplitude of the Shubnikov-de Haas (SdH) oscillations decreasing the thickness of the graphite samples was published, more than 10 years ago, [11] without attracting the necessary attention, although those results already suggested that the SdH oscillations are probably not intrinsic of the graphite structure. These results are supported by the absence of SdH oscillations, i.e., no evidence for the existence of a Fermi surface, found recently in bulk oriented samples of high grade and high purity but without internal interfaces [12]. All these results indicate that the internal microstructure of the graphite samples play an important role, a microstructure that was neither characterized nor considered in the discussion of the measured properties of different graphite samples, from highly oriented pyrolytic graphite (HOPG) to Kish or natural graphite, even in nowadays literature [6, 13, 14]. What does this have to do with superconductivity? If we start searching for superconductivity in graphite by measuring the behavior of the electrical resistance ($R$) with temperature ($T$) and magnetic field ($H$), for example, it should be clear that the knowledge of the intrinsic, normal state dependence is needed. Otherwise, we may misleadingly interpret an anomalous behavior due to, for example, the influence of non-percolative, granular superconducting regions embedded in a (normal state) graphite matrix, as intrinsic of the material, clearly missing an interesting aspect of the sample. A reader with expertise in superconductivity might not be convinced that such a mistake could be ever made. However, the ballistic transport characteristics of the graphene layers in ideal graphite with their huge mobility and mean free path [15, 16, 17, 18] provide a high conductivity path in parallel; such that it is not at all straightforward by simple experiments to realize and prove the existence of superconductivity at certain regions in some, not all, graphite samples. One needs indeed to do systematic experiments decreasing the size of the graphite samples (but not too much) to obtain clear evidence for the embedded or “hidden” superconductivity. A note on samples: The internal ordering or mosaicity of the graphite crystalline regions inside commercial HOPG samples is given usually by the grade. For example, the highest ordered pyrolytic graphite samples have is a grade “A”, which means a rocking curve width $\Delta\sim 0.4^{\circ}\pm 0.2^{\circ}$ (“B”, $\Delta\sim 0.8^{\circ}$, etc.). Interestingly, and due to the contribution of two dimensional highly conducting internal interfaces between crystalline regions [7, 10], the highest grade, i.e., smaller rocking curve width, does not always mean that the used sample provides the intrinsic transport properties of ideal graphite. The characterization of the internal structure of usual HOPG samples, as well as the thickness dependence of $R(T)$ to understand the transport and the magnetic properties of graphite, indicate that these two dimensional interfaces are of importance. The existence of rhombohedral inclusions [19, 20] (stacking order ABCABC instead of ABAB of the usual Bernal graphite structure) in HOPG as well as in Kish graphite samples can also have a relationship with the hidden superconductivity in graphite, following the theoretical work in Ref. [21]. According to literature (see e.g., Fig. 2-2 in Ref. [8]), the density of interfaces parallel to the graphene layers in Kish graphite, in regions of several microns length, is notable. Therefore, quantifying the perfection of any graphite sample through the resistivity ratio between 300 K and 4.2 K [8] is not necessarily the best criterion to be used if we are interested on the intrinsic properties of the graphene layers in graphite, because of the high conductivity of the interfaces in parallel to the graphene layers of the sample [10]. Two examples of the interfaces we are referring to can be seen in Fig. 1. On the other hand, commercial HOPG bulk samples are of high purity with average total impurity concentrations below 20 ppm. Especially the existence of magnetic impurities are of importance if the Defect-Induced Magnetism (DIM) is the main research issue. Their concentration remains below a few ppm for high grade HOPG samples [22]. The graphite flakes discussed in this work were obtained by exfoliation of HOPG samples of different batches, by careful mechanical press and rubbing the initial material on a previously cleaned substrate. As substrate, we used p-doped Si with a 150 nm SiN layer on top. We selected the flakes using microscopic and micro-Raman techniques to check their quality. More details on the preparation can be taken from Ref. [7] and other publications cited below. This review is organized as follows. In the next section we discuss the experimental data for $R(T,H)$ from different graphite samples published in the last 12 years and argue that the first hints on unusual superconducting contribution can be already found in those measurements. In section III, we discuss the anomalous hysteresis in the magnetoresistance, a first clear indication for embedded granular superconductivity. Section IV, deals with the Josephson behavior measured in TEM lamellae whereas section V deals with the granular superconducting behavior found in the magnetization of water-treated graphite powder as well as in bulk HOPG samples for fields normal to the interfaces found inside those samples. In the last section, section VI, before the conclusion, we discuss possible origins for the superconducting signals on the basis of earlier and recent experimental and theoretical work. ## 2 The behavior of the resistance vs. temperature at different applied magnetic fields In this section, we discuss the behavior of the resistance $R(T,H)$ of different HOPG samples including Kish graphite. The data we present here were taken from [7, 23, 24] and a quick search in literature demonstrates that these data are reproducible and can be found in different publications, see, e.g., [2, 25, 26, 27]. Figure 2(a) shows the $R(T)$ for different bulk graphite samples of different grades (rocking curve width) and for one sample (HOPG-1) at zero and under a magnetic field applied normal to the main area, i.e., normal to the graphene planes of the sample. This figure reveals a general behavior, namely that the lower the resistivity $\rho$ of the HOPG sample, the more metallic-like its temperature dependence. It is appealing to assume that these characteristics, low $\rho$, low $\Delta$ and the metallic behavior are clear signs for more ideal graphite. Therefore, from the measured $R(T)$, we may conclude that sample HOPG-3 is more ideal than sample HOPG-1 and the latter being more ideal than sample HOPG-2, see Fig. 2(a). This is indeed the usual interpretation found in several reviews in the literature, see e.g., [28, 8]. From a quick look at all the curves in Fig. 2, however, one recognizes a striking similarity between them, although we are comparing different samples with different thickness and some of the curves were measured under a magnetic field applied normal to the graphene layers of the samples; also normal to the interfaces commonly found in some ordered samples [8, 7]. Figure 1: Transmission electron microscope pictures of two different kinds of interfaces and their distribution in HOPG samples. The TEM pictures were taken from two different lamellae, each about 300 nm thick and with the electron beam nearly parallel to the graphene planes of the samples. (a) The interfaces are recognized at the borders of crystalline regions of different gray colors. Taken from [7]. (b) Interfaces found in a HOPG sample used for magnetization measurements (see section V)that reveals hysteretic behavior in field and temperature. Taken from [29]. Let us start discussing the metallic-like behavior of $R(T)$ of sample HOPG-3 in Fig. 2(a). This sample behaves as the HOPG-UC sample shown in (c) at zero magnetic field, having also a maximum at $T\sim 150~{}$K. A “better” metallic character shows the HOPG sample in (b) or the Kish graphite sample in (d), without any maximum in the shown temperature range. Is this metallic-like behavior really intrinsic of ideal graphite? The following experimental evidence does not support such interpretation: First, for samples from the same batch, the metallic character of $R(T)$ vanishes in the whole temperature range when the sample thickness is below $\sim 50~{}$nm [11, 7, 27], see Fig. 2(b). Second, the metallic-like behavior vanishes in the whole temperature range after applying a magnetic field of the order of 1 to 2 kOe, an interesting behavior known as the Metal-Insulator Transition (MIT) [23, 26, 25], see Figs. 2(a), (c) and (d). Note that such magnetic field strength influences mainly the metallic-like region, see e.g., the change of sample HOPG-1 in Fig. 2(a) at zero and at 1 kOe field, an interesting behavior noted first in [30] and interpreted as due to superconducting instabilities. At those field strengths, i.e., $H\sim 1~{}$kOe, the obtained $R(T)$ curves, for samples showing at zero field a metallic-like behavior, resemble the semiconducting-like curves obtained for sample HOPG-2 (Fig. 2(a)) or for samples with small thickness (Fig. 2(b)). At fields higher than a few kOe, the rather large magnetoresistance of graphite starts to play the main role and the $R(T)$ curve increases in the whole temperature range. Finally, all these results added to the existence of well defined interfaces in the metallic-like HOPG samples as well as in Kish graphite, with distances in the $c-$axis direction usually larger than $\sim 30$ nm, indicate that the metallic-like behavior is due to the contribution of these interfaces and it is not intrinsic of the graphene layers of ideal graphite [7, 10]. Therefore, explanations of the MIT based on ideal graphite band models with a large number of free parameters [25, 26] are certainly not the appropriate ones. All the different $R(T)$ curves for different samples shown in Fig. 2 and at zero field can be very well understood assuming the parallel contribution of semiconducting graphite paths in parallel to the one from the highly conducting interfaces [10]. The saturation of the resistance at $T\rightarrow 0$ K is interpreted as due to the finite resistance of the sample surfaces (the free one and the one on the substrate) short circuiting the intrinsic behavior of the bulk graphene layers at low enough temperatures. Figure 2: (a) Normalized resistance vs. temperature for three different HOPG bulk samples. The bottom, metallic-like curve corresponds to the sample HOPG-3, the curves above correspond to HOPG-1 ($H=0$), HOPG-1 ($H=1~{}$kOe), and HOPG-2. The grade and resistivity values are HOPG-1 ($\Delta=1.4^{\circ}$, resistivity at 300 K $\rho(300$K$)=45~{}\mu\Omega$cm), HOPG-2 ($\Delta=1.2^{\circ}$, $\rho(300$K$)=135~{}\mu\Omega$cm) and HOPG-3 ($\Delta=0.5^{\circ}$, $\rho(300$K$)=5~{}\mu\Omega$cm). Taken from [23]. (b) Similar to (a) but for HOPG samples from the same batch but with different size, namely (thickness $\times$ length $\times$ width) L5: $12\pm 3~{}$nm, $27~{}\mu$m, $14~{}\mu$m, L2A: $20\pm 5~{}$nm, $5~{}\mu$m, $10~{}\mu$m, L8A: $13\pm 2~{}$nm, $14~{}\mu$m, $10~{}\mu$m, L8B: $45\pm 5~{}$nm, $3~{}\mu$m, $3~{}\mu$m, L7: $75\pm 5~{}$nm, $17~{}\mu$m, $17~{}\mu$m, HOPG: $17\pm 2~{}\mu$m, $4.4~{}$mm, $1.1~{}$mm, taken from [7]. (c) and (d) Resistance of bulk graphite samples vs. temperature at different applied fields normal to the graphene layers. The sample in (c) is a HOPG bulk sample from Union Carbide of grade A and the sample in (d) is Kish graphite. Taken from [24]. The question is now whether parts of these interfaces hide superconducting regions. It is certainly appealing to suggest that the huge MIT at fields normal to the interfaces (and below $\sim 2~{}$kOe) is related to Josephson- coupled superconducting regions embedded in some of the interfaces. Note that the huge anisotropy of the MIT (parallel fields to the interfaces main plain do not affect the electrical transport) already implies that the regions responsible for the MIT must be laying parallel to the graphene layers [31]. Without the knowledge on the existence of these interfaces, an interpretation of the low field MIT based on the influence of superconductivity has been discussed in detail in the reviews [32, 24]. In those reviews, one can recognize the remarkable similarity between the scaling approaches used to characterize the magnetic-field-induced superconductor-insulator quantum phase transition [33] or of the field-driven MIT in 2D electron (hole) systems [34] and the one obtained for the MIT observed in graphite. As we will see in the next sections, the experimental evidence obtained in the last recent years indicates that granular superconductivity exists within some of those interfaces, indeed. If superconducting patches exist embedded in parts of the interfaces or in other two dimensional regions of the bulk ordered samples, one expects to measure some signs of granular superconductivity, as for example nonlinear $I-V$ curves or hysteresis in the magnetoresistance. However, this is not really observed in bulk large samples. There are at least two reasons for the apparent absence of these expected phenomena. One is the distribution of the input current between the ballistic channels given by the graphene layers [17, 18], the metallic, normal conducting parts of the interfaces and the regions where the superconducting patches exist. In other words, the usual maximum currents used in transport experiments reported in bulk samples may have been small enough so that the current through the superconducting regions remained below the critical Josephson one. The other reason is the experimental voltage sensitivity to measure the possible irreversibility in the magnetoresistance due to the existence of pinned vortices or fluxons. We will see in the next section that part of these problems can be overcome decreasing the sample size; in this way, one obtains the voltage signals from the regions of interest with enough sensitivity. Apart from the large magnetic field sensitivity of the metallic-like resistance measure in bulk graphite samples with interfaces, is there any further hint for the existence of granular superconductivity in those $R(T)$ curves? Yes, this hint is related to the thermally activated function ($\propto\exp(-E_{a}/k_{B}T)$ with $E_{a}$, a sample dependent effective thermal barrier $\sim 30~{}$K) one needs in order to fit the metallic-like contribution below $T\sim 200~{}$K [10]. This function is relevant in spite of only a factor five increase of the resistance between low and high temperatures, see Fig. 2. Skeptical readers can convince themselves about its relevance taking a similar example, as the exponential function used to fit the increase, by a similar factor, of the ultrasonic attenuation with temperature below $T_{c}$ in conventional superconductors. We note that this exponential function has already been used to describe the increasing resistance of bulk graphite samples with temperature and it was speculated to be related to some superconducting-like behavior in graphite [32]. It is clear that this function is not the usual one, one expected for metals or semimetals and that cannot be understood within the usual electron-phonon interaction mechanisms, nor in two dimensions. A similar dependence has been observed in granular AlGe [35], which shows for a particular Al concentration a superconductor-semiconductor transition similar to that reported in Ref. [10] or, after an appropriate scaling in temperature, to some of the curves shown in Fig. 2. The observed thermally activated behavior might be understood on the basis of the Langer-Ambegaokar-McCumber-Halperin (LAMH) model [36, 37] that applies to narrow superconducting channels in which thermal fluctuations can cause phase slips. This interpretation gets further support from the evidence we discuss in the following sections. ## 3 Hysteresis in the magnetoresistance In order to reveal by transport measurements the existence of granular superconductivity in some regions of the graphite samples, we need to increase the sensitivity of the measured voltage to those regions. To achieve this, we decrease the size of the sample enhancing in this way the probability to get some measurable influence of this phenomenon in the voltage. The work in Ref. [38] reported the first observations of an anomalous irreversible behavior in the magnetoresistance (MR) in a few tens of nm thick and several micrometer large multigraphene samples. Hysteresis in the magnetoresistance is a key evidence on the existence of either magnetic order (domains with their walls, for example) or vortices/fluxons and therefore on the existence of superconductivity. Because defects as well as hydrogen can trigger magnetic order in graphite, a first attempt would be to relate the measured hysteresis in the MR with the existence of magnetic order and magnetic domains, for example. However, the data exhibited anomalous hysteresis loops in the MR [38], similar to those observed in granular superconductors with Josephson- coupled grains [39, 40, 41]. The anomalous hysteresis was observed only for magnetic fields perpendicular to the planes, whereas in the parallel to the planes direction, the MR remains negligible. This fact already points out to a remarkable large anisotropic response of the superconducting phase(s) in agreement with the hypothesis that these superconducting regions might be embedded in some of the interfaces found inside some bulk graphite samples [7, 8]. The amplitude of the hysteresis in the MR reported in Ref. [38] vanishes at temperatures $T\sim 10~{}$K, clearly below the temperature at which the resistance shows a maximum, as it is the case for samples HOPG-1 in Fig. 2(a) or sample L2A in Fig. 2(b). It is clear that thermal fluctuations can prevent the establishment of a coherent superconducting state in parts of the sample and therefore zero resistance state is not so simple to be achieved if the superconducting distribution is a mixture of superconducting patches at the interfaces and these are embedded in a multigraphene semiconducting matrix. Moreover, we should take also into account that the voltage electrodes are usually connected at the top surface of the graphite samples picking the voltage difference coming from a non-negligible normal conducting path. One possibility to increase the sensitivity of the measured voltage to the field hysteresis these regions produce is to make a constriction in the middle of the two voltage electrodes, see inset in Fig. 3(a). In this case, we expect a locally narrower distribution of superconducting and normal regions at the constriction such that averaging effects should be less important. Simultaneously, through the constriction the main part of the voltage drop depends mostly on the region at the constriction, see Fig. 2(c) in Ref. [16]. Then, a higher sensitivity to the superconducting paths can be achieved in case they remain at or near the constriction. This idea has been successfully realized in [42] and its main results will be reviewed in this section. Figure 3: (a) Resistance vs. temperature, without constrictions and at zero applied field, for two graphite flakes of size (distance between voltage electrodes $\times$ width $\times$ thickness) for sample 1 (2): $13\times 16\times 0.015~{}(2.6\times 6\times 0.040)$ $\mu$m3. The observed temperature dependence remains for all constrictions widths. The inset shows a scanning electron microscope picture of sample 1 with a constriction width of 4.3 $\mu$m between the two voltage electrodes. The scale bar is 5$~{}\mu$m. (b) Magnetoresistance (MR) vs. applied magnetic field for sample 1 with a $4~{}\mu$m constriction width and at 2 K. The input current was $1~{}\mu$A. Note the clear hysteresis in the MR when the field is swept from $\|H_{\rm max}\|=1000~{}$Oe. The inset shows the difference $\Delta$MR between the curve obtained starting from $H_{\rm max}=+1000~{}$Oe and the return curve measured from $H_{\rm min}=-1000~{}$Oe. (c) The absolute difference between the two MR curves of the hysteresis loop obtained at a fixed magnetic field of $16.6$ Oe for sample 1 without constriction $(\star)$ and for two different constriction widths. The figure also shows the corresponding data for another graphite sample without constrictions $(\blacksquare)$ from [38]. (d) Magnetoresistance measured from a starting maximum field of 1.4 kOe at 10 K for sample 2 with a constriction width of $3~{}\mu$m. Some of the figures and the data were taken from [42]. Let us take two slightly different samples, 1 and 2, with $R(T)-$curves as shown in Fig. 3(a). The aim of the experiment is to study the hysteresis in the MR those samples might show below the temperature at which a maximum in the resistance is measured, in case that maximum is related to the Josephson coupling between superconducting regions. Figure 3(b) shows one example of the anomalous hysteresis loop in the MR. The going down curve (from high, positive to low, negative fields, red arrow), for example, runs below the going up curve (green arrow) in the same quadrant as the field sweep was started, showing a minimum at positive fields of the order of 20 Oe, see also similar curves in Ref. [38]. To present the anomalous behavior clearly, the inset in Fig. 3(b) shows the difference between the two curves. This difference is in clear contrast to the usual hysteresis in superconductors as well as ferromagnets [39, 38], where the minimum (or maximum) in the MR is observed in the opposite field quadrant, and the increasing field resistance curve is usually below the decreasing field one. Figure 3(c) shows the temperature dependence of the difference in the MR between the decreasing and increasing field curves at a fixed magnetic field for sample 1, without and with two constrictions. The results show that the smaller the constriction width, the higher the temperature at which the anomalous hysteresis is observed, decreasing below the sensitivity limit at $T>50$ K for a constriction width of $4~{}\mu$m, whereas the maximum in the $R(T)$ curve is at $\sim 70~{}$K, see Fig. 3(a). The absence of any hysteresis in the MR for sample 2 with a constriction width of $3~{}\mu$m and at $T=10~{}$K indicates that the hysteresis does not come from some artifact due to the used focused ion beam method [42, 43] or due to an artifact in the measurement of the real field applied to the sample. As expected from the $R(T)$ curve, see Fig. 3(a), sample 2 shows the anomalous hysteresis in the MR at lower temperatures than sample 1 [42]. Summarizing this section, the observation of the anomalous hysteresis in the MR – together with the MIT and the relatively large MR at temperatures below the maximum in $R(T)$ – provides already striking hints that granular superconductivity is at work in some regions of these samples. The increase in the temperature region where the hysteresis is observed, decreasing the constriction width, demonstrates the problem of current averaging and voltage sensitivity limits usual experiments with large samples have. Figure 4: (a) Scanning Electron Microscopy (SEM) image of a lamella of 300 nm thickness on a Si/SiN substrate where the yellowish colored areas are the electrodes. A four-point configuration has been prepared with the outer electrodes used to apply current and the inner ones to measure the voltage drop. The $c-$axis runs parallel to the substrate surface and normal to the current direction. (b) Transmission Electron Microscopy (TEM) image of a HOPG lamella. The different brightness corresponds to a different orientation within the $a-b$ plane of the crystalline regions with thickness $>30$ nm. (c) Voltage vs. temperature at different input currents for a lamella of $\sim 800$ nm thickness and with Van der Pauw contact configuration. (d) Current- Voltage characteristics at different temperatures for a lamella of $\sim~{}300$ nm thickness in reduced coordinates, where $R$ is the normal state resistance, $I$ the input current, and $I_{c}$ the critical Josephson current. The continuous curves are fitted to the model proposed in Ref. [44] with $I_{c}(T)$ as the only free parameter. Figures taken from [45]. ## 4 Direct evidence for Josephson behavior in the transport properties of graphite: Measurements in TEM lamellae If the embedded interfaces (or some other quasi two dimensional regions) inside the measured graphite samples have superconducting properties, the best way to check them would be contacting electrodes as near as possible to those interfaces or interface regions and study the behavior as a function of any useful parameter one can take to influence their response. It should be clear that one cannot simply open the graphite sample at the interface and put voltage electrodes at the open surfaces of the interface, simply because it will not remain anymore. A tentative approach to put the contacts as near as possible to an interface has been done in Ref. [46]. Indeed, the observed behavior at low temperatures and as a function of magnetic field appeared to be superconducting-like. A better and appealing evidence for the superconducting behavior embedded in some graphite ordered samples can be obtained by trying to locate the voltage electrodes directly at the inner edges of the interfaces. The work in Ref. [45] prepared TEM lamellae from bulk HOPG samples and using lithography and focused ion beam techniques, current and voltages electrodes at different positions of the samples were prepared. In this way, one tries to contact several of those interface edges simultaneously, as shown in Fig. 4(a). We note, however, that a thin surface layer of disordered graphite exists due to the Ga+ ion irradiation used to cut the lamella from the bulk HOPG sample. This layer has a much larger resistance than the one of the graphene layers or of the interfaces [43] and therefore the input current goes through the lowest resistance path as well as the voltage electrodes pick up the response of the graphite sample with its interfaces. One can see this comparing first the $R(T)$ curves obtained at large enough currents in the lamellae (Fig. 4(c)) with those of graphite samples with top electrodes (Fig. 2). The fact that a zero resistance state (minimum voltage noise $\pm$ 5 nV upon sample) is obtained at low currents with $I-V$ characteristic curves that resemble the one expects for Josephson coupled grains leaves little doubt about the origin of the obtained signals. In the TEM picture of Fig. 4(b), one can see the graphite single crystalline regions (different gray colors) oriented differently between them about the common $c-$axis and having well defined two dimensional interfaces, as high resolution TEM studies revealed [47]. Figure 4(c) shows the voltage vs. temperature measured in a TEM lamella of oriented graphite [45] at different input DC currents, from 100 nA to 10$~{}\mu$A. The clear sharp transition, observed at $\sim 150$ K at the lowest current, shifts to lower temperatures increasing the input DC current. For the largest input currents, the temperature dependence of the resistance of the contacted lamella shows a maximum or follows the intrinsic semiconducting behavior of the graphene layers. This behavior already suggests the existence of high temperature granular superconductivity at some parts of the sample. The study reported in Ref. [45] shows that the transition temperature depends on the prepared sample. This indicates a sample dependent distribution of the superconducting regions and/or some influence of the preparation process or sample size on the superconductivity [47]. We also note that the observed sharp decrease in the measured voltage does not necessarily indicate the critical temperature of the superconducting regions but the temperature below which a percolative granular system shows negligible resistance due to the Josephson coupling at the used input current. Current-voltage characteristic curves at different temperatures and in different lamellae obtained from different HOPG samples have been studied in Ref. [45]. An example of this $I-V$ curves at three temperatures is shown in Fig. 4(d) obtained for a different lamella. The curves follow the expected dependence for a Josephson junction [44] with a temperature dependent critical current, the only free parameter in the fit. Further evidence that speaks for a superconducting origin of the $I-V$ curves is given by the expected detrimental effect of a magnetic field on the superconducting state. This effect can be due to an orbital depairing effect or due to the alignment of the electron spins at much higher fields, in case of singlet coupling. The effect of a magnetic field applied normal and parallel to the interfaces has been studied in detail for thick and thin lamellae in Ref. [45]. Upon sample size (thickness, i.e., width of the graphene planes inside the lamella) the observed effects are from the usual vanishing of the zero resistance state or no effect at all for thin lamellae. A magnetic field of a few kOe applied normal to the interfaces is enough to destroy the Josephson coupling at low temperatures, an effect compatible with the MIT observed in several graphite samples, see section II. Whereas a field applied parallel to them does not influence the $I-V$ curves at all, a fact that speaks for the two dimensionality of the superconducting regions. Nevertheless, the influence of a magnetic field in HOPG samples with interfaces is not as “simple” as in conventional superconductors. For high fields applied normal to the interfaces, the $I-V$ curves show a recovery to the zero resistance state. The observed reentrance appears to be related to the magnetic-field driven reentrance observed at low temperatures in the longitudinal resistance at high enough magnetic fields [48]. This interesting behavior as well as the insensitivity of the $I-V$ curves to magnetic fields in very thin lamellae [45] deserve further studies. We would like to note here that the possible effects of a magnetic field on the superconducting state of quasi two-dimensional superconductors, or in case the coupling does not correspond to a singlet state, are not that clear as in conventional superconductors. For example, results in two different two- dimensional superconductors, including one produced at the interfaces between non superconducting regions [49], show that superconductivity can even be enhanced by a parallel magnetic field. In case the pairing is $p-$type [50], the influence of a magnetic field is expected to be qualitatively different from the conventional, singlet coupling behavior [51, 52] with even an enhancement of the superconducting state at intermediate fields. In case the London penetration depth is much larger than the size of the superconducting regions at the interfaces of our lamellae or if the superconducting coherence length is of the order or larger than the thickness of the lamella, the influence of a magnetic field should be less detrimental. Through these studies, and taking into account that in samples without these interfaces no signature of superconducting or metallic-like behavior has been observed (see also section V) it is appealing to suggest that superconductivity is somewhere hidden at some of those interfaces or interface regions. It should be also clear that not all those interfaces have superconducting regions with similar critical parameters. Those interfaces are formed during the preparation of the HOPG samples based on treatments at very high temperatures ($T>3400^{\circ}$C) and high pressures ($P\sim$10 kg/cm3) and in a non-systematic way [8]. Actually, they are not at all an aim of the production but actually the opposite, they should be avoided in order to enhance the crystal perfection of the bulk HOPG material. It is even possible that, upon the procedure used to control the structure and texture of the graphite sample, the near surface region, for example, can have a different degree of graphitization as inside the bulk HOPG sample [8]. This means that one may obtain different results from different parts of the same HOPG sample. Therefore, disconcerting situations and an apparent lack of reproducibility are preprogrammed in case the research studies are done without taking care of the internal microstructure of the studied samples. ## 5 Magnetization measurements In this section, we present and discuss magnetization measurements done in bulk HOPG samples with and without embedded interfaces and in water treated graphite powders. One of the main problems in interpreting magnetization data for fields applied parallel to the $c-$axis of the graphite structure, i.e., normal to the graphene layers and interfaces, is the need of subtraction of a large diamagnetic background. Due to the small amplitude of the superconducting-like signals in the studied samples, the subtraction of this linear in field background is not so simple, because it is not known with enough certainty to obtain the true field hysteresis after its subtraction. That means that we always have a certain arbitrariness in the shape of the obtained field hysteresis, a situation that will improve with the increase of the amount of material responsible for those superconducting-like signals. The small SQUID signals of interest imply that one should take additional efforts to minimize or rule out possible SQUID artifacts [53, 54, 55]. Therefore, systematic studies of samples of different or equal geometry and magnetic background, with and without interfaces, are necessary. Taking into account: the overall shape of the hysteresis, the slope of the virgin curve at low fields where the subtraction does not affect too much, and the overall experience with ferromagnetic graphite [22, 56], one can rely to a certain extent on the obtained hysteresis shape. Certainly, not only the field hysteresis but also other evidence one gets from magnetization measurements as, e.g., the remanence at zero field as a function of the maximal field applied (see for example measurements for YBa2Cu3O7 in Ref. [57]) and the hysteresis in temperature dependent measurements helps to convince oneself about the existence of some kind of granular superconductivity. The hysteresis between the field cooled (FC) and zero-field cooled (ZFC) curves can help to discern between a superconducting or ferromagnetic-like behavior. The most obvious evidence that speaks against a simple ferromagnetic order of the hysteresis observed as a function of temperature and field is the two dimensionality of the obtained hysteretic signals [29], i.e., the superconducting-like signals are mainly measured for fields normal to the interfaces. This fact is not compatible with any kind of magnetic order including shape or magneto crystalline anisotropy, whatever large they might be. We note that the ferromagnetic response of graphite due to DIM is mostly measured for fields parallel to the graphene layers, parallel to the main area of the samples [22]. ### 5.1 Bulk graphite samples Figure 5: (a) Magnetization of two HOPG bulk samples (HOPG-1 and HOPG-2) after subtraction of a diamagnetic background and of water treated graphite powder (WTGP, right $y-$axis) at 300 K. The HOPG-2 sample shows no hysteresis in contrast to the other two samples. (b) Temperature dependence of the difference between FC and ZFC magnetic moments of the HOPG-1 sample before (b.a.) and after (a.a.) warming the sample up to $\simeq 600~{}$K, at two constant applied fields, 0.5 T (left $y-$axis) and 4 T (right $y$-axis). The field was applied always normal to the interfaces or graphene planes of the samples. Data taken from [29]. Figure 5(a) shows the field hysteresis, after subtraction of the corresponding diamagnetic linear background, at 300 K of two bulk HOPG samples, HOPG-1 and HOPG-2 and a water treated graphite powder (WTGP) (right $y-$axis). A TEM characterization of the internal microstructure of the HOPG-1 sample shows clear evidence for well defined interfaces running parallel to the graphene layers, in contrast to the HOPG-2 sample [29], see Fig. 1(b). These results clearly indicate that the origin of the hysteresis is related to the existence of the interfaces in the HOPG-1 sample. The absence of the hysteresis in the HOPG-2 sample, which has a similar diamagnetic background and overall geometry as the HOPG-1 sample, also indicates that the hysteresis is not due to an obvious SQUID artifact or an artifact in the background subtraction. The field hysteresis is similar to that of WTGP. The narrowing of the hysteresis observed at high fields is expected for granular superconductors [60, 61, 62, 58]. From the hysteresis, as well as measuring the remanent magnetic moment as a function of the applied field [29], one obtains the characteristic Josephson critical fields $h_{c1}^{J}(T)$ and $h_{c2}^{J}(T)$ with values similar to the WTGP [58] and a similar ratio $h_{c2}^{J}(T)/h_{c1}^{J}(T)\sim 3$ [29]. Figure 5(b) shows the magnetic moment hysteresis in temperature (FC minus the ZFC curve) for the HOPG-1 sample as received (b.a.) and after sweeping the temperature up to 500 K (a.a.) [29], at two applied fields. We would like to stress the following features: The hysteresis for the as-received sample starts from the turning point (390 K) and it is positive. The hysteresis in temperature at both applied fields are qualitatively similar, showing a crossing to negative values at low temperatures. Larger ZFC values (smaller in absolute value) than FC ones in the magnetic moment are usually not observed, neither in superconductors nor in ferromagnets and it does appear to be a SQUID artifact [29]. This negative hysteresis in temperature would suggest that the superconducting properties can be enhanced to some extent under a magnetic field, an effect that might be related to the reentrance we have shortly mentioned in section IV. A slight annealing of the HOPG-1 sample of less than one hour at $\sim 500$ K changes drastically the observed hysteresis for both fields (open symbols in Fig. 5(b)). The hysteresis appears to be shifted to lower temperatures but with negative values at high temperatures and high fields. We note that annealing at similar temperatures for several hours produced a decrease in the overall hysteresis observed in WTGP (see supporting information of Ref. [58]). At the state of this research, it is unclear whether pinning properties of vortices and/or of fluxons or the existence of different superconducting phases play a main role in the hysteresis that is observed for fields applied normal to the interfaces. ### 5.2 Water treated graphite powder Figure 6: (a) Field hysteresis at 5 K for a maximum applied field of 40 mT for the water treated graphite powder (S1), the same powder but after pressing it in a pellet with a pressure of $18\pm 5$ MPa (S2) and after pressing it again with a pressure of $60\pm 20$ MPa (S3). The corresponding diamagnetic linear backgrounds were subtracted from the measured data. (b) Difference between the FC and ZFC curve at different applied fields for a water treated graphite powder. Data taken from Ref. [58]. The work of Ref. [58] reports on the magnetic response of WTGPs. The main message of that work is that the WTGP shows a hysteretic behavior in field and temperature compatible with granular superconductivity. As an example, we show in Fig. 6(a) the field hysteresis at 5 K of a WTGP (S1, lose powder without applying significant pressure) and the same WTGP but after pressing it into a pellet with two different pressures (S2,S3). After the diamagnetic background subtraction, the field hysteresis is similar to that obtained for bulk HOPG sample with interfaces, see Fig. 5(a) for similar data but at 300 K. The fact that the hysteresis vanishes after applying pressure to the powder rules out simple SQUID artifacts (the diamagnet background does not diminish after making a pellet from the graphite powder, but the contrary) and also it rules out that the hysteresis is due to a ferromagnetic response due to impurities. Figure 6(b) shows the difference in the magnetic moment between the ZFC and FC curves, as in Fig. 5(b). The behavior of this difference as a function of the applied field appears to be compatible with the one expected for granular superconductors [58]. Note the following features: The hysteresis increases at all $T$ for fields $\mu_{0}H\lesssim 50~{}$mT, showing a maximum near the turning point of 300 K, similar to the HOPG-1 sample in the as-received state, see Fig. 5(b). At fields 0.1 T $\lesssim\mu_{0}H\lesssim 0.2~{}$T the difference decreases at all $T$ and remains rather field independent. At higher fields, however, it increases showing a shift of the crossing point (from negative to positive values) to higher $T$ . This behavior is at odds to the one expected for ferromagnets, even for ferromagnetic nanoparticles [63] as well as for superconductors with a pinning force that decreases with applied field in the shown field range. From the results in [58], and using basic concepts of vortex pinning, we would then conclude that if an upper critical field exists, then it should be clearly larger than 7 T in the temperature range of the figure. In spite of some interesting differences between the behavior obtained for bulk HOPG and WTGP, the similarities already suggest that the water treatment helps to produce a certain amount of interfaces between graphite grains, being the origin for the whole hysteresis. Thermal annealing as well as pressing the WTGP are detrimental indicating that defects and/or hydrogen or oxygen at the interfaces could play an important role in the observed phenomena. ## 6 Discussion Superconductivity in carbon-based systems is a rather old, well recognized fact. This phenomenon was probably first observed in the potassium intercalated graphite $C_{8}K$ [64] back in 1965\. Since then, a considerable amount of studies reported this phenomenon in carbon-based systems, reaching critical temperatures $T_{c}\sim 10$ K in intercalated graphite [65, 66] and above 30 K - though not percolative - in some HOPG samples [59] as well as in doped graphite and amorphous carbon systems [67, 68, 69, 70]. Traces of superconductivity at $T_{c}=65~{}$K have been recently reported in amorphous carbon powder that contained a small amount of sulfur [71]. Superconductivity was found also in carbon nanotubes with $T_{c}=0.55~{}$K [72] and 12 K [73] or possibly even higher critical temperatures [74, 75]. Superconductivity with $T_{c}\sim 4~{}$K in boron-doped diamond [76] and in diamond films with $T_{c}\sim 7~{}$K [77] belong also to the recently published list of carbon- based superconductors. We should note, however, that superconductivity at room temperature in a disordered graphite powder has been already reported in 1974 [78], see also [79], a work that did not attract the necessary attention in the community. Whether quasi two dimensional interfaces play a role in the above mentioned carbon-based superconductors, one can probably rule out only for the intercalated graphite and doped diamond compounds, where the three dimensional superconductivity is characterized by a relatively low critical temperature. We may speculate that the traces of superconductivity found in doped amorphous carbon, disordered or ordered graphite powders may be related to some interfaces between well ordered graphite regions. The experience of the high temperature superconducting oxides already suggests that two dimensionality is advantageous to achieve higher critical temperatures. Apart from the usual transport and magnetization measurements used to characterize the superconducting state, there are scanning tunneling spectroscopy (STS) results obtained on certain disordered regions of a HOPG surface at $T=4.2~{}$K that revealed an apparent energy gap $\sim 100~{}$meV [80]. Although the overall curves resemble a superconducting-like density of states, the authors suggested that the gap originates from charging effects. See further STS results and the discussion in [70]. Theoretical works that deal with superconductivity in graphite as well as in graphene have been published in recent years. For example, $p$-type superconductivity has been predicted to occur in inhomogeneous regions of the graphite structure [50] or $d-$wave high-$T_{c}$ superconductivity [81] based also on resonance valence bonds [82], or at the graphite surface region with rhombohedral stacking due to a topologically protected flat band [83]. For the graphite structure, the experimental evidence obtained in the last years suggests that high temperature superconductivity exists at certain interfaces or interface regions within the usual Bernal structure although the structure of the superconducting regions remains unknown. One can further speculate that due to the high carrier concentration that can be localized at those interfaces, they should be predestined to play a role in triggering superconductivity. Following a BCS approach in two dimensions (with anisotropy), for example, a critical temperature $T_{c}\sim 60~{}$K has been estimated if the density of conduction electrons per graphene plane increases to $n\sim 10^{14}~{}$cm-2, a density that might be induced by defects and/or hydrogen ad-atoms [84] at the interfaces, or by Li deposition [85]. Further predictions for superconductivity in graphene support the premise that $n>10^{13}~{}$cm-2 in order to reach $T_{c}>1~{}$K [86, 87]. On the other hand, the possibility to have high temperature superconductivity at the surface of or in the rhombohedral graphite phase [83, 21] – a phase that sometimes is found in graphite samples [20, 19] – stimulates further careful studies of these hidden interfaces. In the last years, superconductivity has been found at the interfaces between oxide insulators [88] as well as between metallic and insulating copper oxides with $T_{c}\gtrsim 50~{}$K[89]. Also, interfaces in different Bi bicrystals show superconductivity up to 21 K, although Bi bulk is not a superconductor [90, 91]. Finally, we think that some of the interfaces are also the origin for the metallic-like behavior of graphite samples as well as for the quantum Hall effect (QHE) found in some HOPG samples [48, 92]. Because the existence, density as well as the intrinsic properties of these interfaces depend on sample, we can now understand why the reproducibility of the QHE in bulk HOPG samples is rather poor. ## 7 Conclusion In this review, we have discussed the following experimental evidence: Firstly, the temperature and magnetic field dependence of the electrical resistance of bulk and thin films of graphite samples and its relation with the existence of two dimensional interfaces. Secondly, the Josephson behavior of the current-voltage curves with an apparent zero resistance state at high temperatures in especially made TEM lamellae. Thirdly, the anomalous hysteresis in the magnetoresistance observed in graphite thin samples as well as its enhancement restricting the current path within the sample. Finally, the overall magnetization of bulk graphite samples, with and without interfaces, as well as water treated graphite powders. All this experimental evidence as a whole indicates the existence of superconductivity located at certain interfaces inside graphite samples. Although we cannot rule out other interpretations for some of the observations discussed in this work, the whole evidence suggests that superconductivity should be the origin for all the phenomena discussed here. Clearly, the situation is still highly unsatisfactory because several open questions remain, namely, the characteristics of the superconducting phase(s), from the structure to the main superconducting parameters, as “simple” as the critical temperature and critical fields, the coherence and penetration lengths, etc. It is clear that further studies are necessary in the future but the overall work done until now shows us the way to go. ###### Acknowledgements. The author acknowledges the support provided by the Deutsche Forschungsgemeinschaft under contract DFG ES 86/16-1 and the ESF-Nano under the Graduate School of Natural Sciences “BuildMona”. The results presented in this review were part of the Ph.D. thesis of Heiko Kempa (section II), Srujana Dusari (section III) and Ana Ballestar (section IV) as well as the master thesis of Thomas Scheike (section V) done in the Division of Superconductivity and Magnetism of the Institute for Experimental Physics II of the University of Leipzig. The author thanks Dipl. Kris. Annette Setzer, Dr. José Barzola- Quiquia and Dr. Winfried Böhlmann for their experimental assistance and support. The permanent support as well as the discussions with Nicolás García are gratefully acknowledged. Special thanks go to Yakov Kopelevich with whom we started in the year 1999 and in a rather naive way the research of a new and unexpected world behind graphite. ## Note added in proof Since the submission of this manuscript, some new works related to the subject of this review were published. Tight-binding simulations done in Ref. [93] support the work done in Ref. [21] and found that surface superconductivity is robust for ABC stacked multilayer graphene, even at very low pairing potentials. Through the observation of persistent currents in a graphite filled ring-shaped container immersed in alkanes, the author in Ref. [94] claimed possible room temperature superconductivity. 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1312.4583	incollectioninproceedings ††thanks: On leave from Odessa University. # Uncertainty and Analyticity Vladimir V. Kisil School of Mathematics, University of Leeds, Leeds LS2 9JT, England Web: http://www.maths.leeds.ac.uk/~kisilv/ [email protected] (Date: December 1, 2013) ###### Abstract. We describe a connection between minimal uncertainty states and holomorphy- type conditions on the images of the respective wavelet transforms. The most familiar example is the Fock–Segal–Bargmann transform generated by the Gaussian, however, this also occurs under more general assumptions. ###### Key words and phrases: Quantum mechanics, classical mechanics, Heisenberg commutation relations, observables, Heisenberg group, Fock–Segal–Bargmann space, $\mathrm{SU}(1,1)$, Hardy space ###### 1991 Mathematics Subject Classification: Primary 81P05; Secondary 22E27 ## 1\. Introduction There are two and a half main examples of reproducing kernel spaces of analytic function. One is the Fock–Segal–Bargmann (FSB) space and others (one and a half)—the Bergman and Hardy spaces. The first space is generated by the Heisenberg group [Kisil11c]§ 7.3 [Folland89]§ 1.6, two others—by the group $\mathrm{SU}(1,1)$ [Kisil11c]§ 4.2 (this explains our way of counting). Those spaces have the following properties, which make their study particularly pleasant and fruitful: 1. (i) There is a group, which acts transitively on functions’ domain. 2. (ii) There is a reproducing kernel. 3. (iii) The space consists of holomorphic functions. Furthermore, for FSB space there is the following property: 1. iv. The reproducing kernel is generated by a function, which minimises the uncertainty for coordinate and momentum observables. It is known, that a transformation group is responsible for the appearance of the reproducing kernel [AliAntGaz00]Thm. 8.1.3. This paper shows that the last two properties are equivalent and connected to the group as well. ## 2\. The Uncertainty Relation In quantum mechanics [Folland89]§ 1.1, an observable (self-adjoint operator on a Hilbert space $\mathcal{H}{}$) $A$ produces the expectation value $\bar{A}$ on a state (a unit vector) $\phi\in\mathcal{H}{}$ by $\bar{A}=\left\langle A\phi,\phi\right\rangle$. Then, the dispersion is evaluated as follow: $\Delta_{\phi}^{2}(A)=\left\langle(A-\bar{A})^{2}\phi,\phi\right\rangle=\left\langle(A-\bar{A})\phi,(A-\bar{A})\phi\right\rangle=\left\\|(A-\bar{A})\phi\right\\|^{2}.$ (1) The next theorem links obstructions of exact simultaneous measurements with non-commutativity of observables. ###### Theorem 1 (The Uncertainty relation). If $A$ and $B$ are self-adjoint operators on a Hilbert space $\mathcal{H}{}$, then $\textstyle\left\\|(A-a)u\right\\|\left\\|(B-b)u\right\\|\geq\frac{1}{2}\left\|\left\langle(AB- BA)u,u\right\rangle\right\|,$ (2) for any $u\in\mathcal{H}{}$ from the domains of $AB$ and $BA$ and $a$, $b\in\mathbb{R}{}$. Equality holds precisely when $u$ is a solution of $((A-a)+\mathrm{i}r(B-b))u=0$ for some real $r$. ###### Proof. The proof is well-known [Folland89]§ 1.3, but it is short, instructive and relevant for the following discussion, thus we include it in full. We start from simple algebraic transformations: $\displaystyle\left\langle(AB-BA)u,u\right\rangle$ $\displaystyle=$ $\displaystyle\left\langle(A-a)(B-b)-(B-b)(A-a))u,u\right\rangle$ (3) $\displaystyle=$ $\displaystyle\left\langle(B-b)u,(A-a)u\right\rangle-\left\langle(A-a))u,(B-b)u\right\rangle$ $\displaystyle=$ $\displaystyle 2\mathrm{i}\Im\left\langle(B-b)u,(A-a)u\right\rangle$ Then by the Cauchy–Schwartz inequality: $\textstyle\frac{1}{2}\left\langle(AB- BA)u,u\right\rangle\leq\left\|\left\langle(B-b)u,(A-a)u\right\rangle\right\|\leq\left\\|(B-b)u\right\\|\left\\|(A-a)u\right\\|.$ The equality holds if and only if $(B-b)u$ and $(A-a)u$ are proportional by a _purely imaginary_ scalar. ∎ The famous application of the above theorem is the following fundamental relation in quantum mechanics. Recall [Folland89]§ 1.2, that the one- dimensional Heisenberg group $\mathbb{H}^{1}{}$ consists of points $(s,x,y)\in\mathbb{R}^{3}{}$, with the group law: $\textstyle(s,x,y)(s^{\prime},x^{\prime},y^{\prime})=(s+s^{\prime}+\frac{1}{2}(xy^{\prime}-x^{\prime}y),x+x^{\prime},y+y^{\prime}).$ (4) This is a nilpotent step two Lie group. By the Stone–von Neumann theorem [Folland89]§ 1.5, any infinite-dimensional unitary irreducible representation of $\mathbb{H}^{1}{}$ is unitary equivalent to the Schrödinger representation ${\rho_{\hslash}}$ in $\mathcal{L}_{2}{}(\mathbb{R}{})$ parametrised by the Planck constant $\hslash\in\mathbb{R}{}\setminus\\{0\\}$. A physically consistent form of ${\rho_{\hslash}}$ is [Kisil10a](3.5): $[{\rho_{\hslash}}(s,x,y){f}\,](q)=e^{-2\pi\mathrm{i}\hslash(s+xy/2)-2\pi\mathrm{i}xq}\,{f}(q+{\hslash}y).$ (5) Elements of the Lie algebra $\mathfrak{h}_{1}$, corresponding to the infinitesimal generators $X$ and $Y$ of one-parameters subgroups $(0,t/(2\pi),0)$ and $(0,0,t)$ in $\mathbb{H}^{1}{}$, are represented in (5) by the (unbounded) operators $M$ and $D$ on $\mathcal{L}_{2}{}(\mathbb{R}{})$: $\textstyle M=-\mathrm{i}q,\quad D=\hslash\frac{d}{dq},\quad\text{with the commutator}\quad[M,D]=\mathrm{i}\hslash I.$ (6) In the Schrödinger model of quantum mechanics, $f(q)\in\mathcal{L}_{2}{}(\mathbb{R}{})$ is interpreted as a wave function (a state) of a particle, with $M$ and $D$ are the observables of its coordinate and momentum. ###### Corollary 2 (Heisenberg–Kennard uncertainty relation). For the coordinate $M$ and momentum $D$ observables we have the _Heisenberg–Kennard uncertainty_ relation: $\Delta_{\phi}(M)\cdot\Delta_{\phi}(D)\geq\frac{h}{2}.$ (7) The equality holds if and only if $\phi(q)=e^{-cq^{2}}$, $c\in\mathbb{R}_{+}{}$ is the vacuum state in the Schrödinger model. ###### Proof. The relation follows from the commutator $[M,D]=\mathrm{i}\hslash I$, which, in turn, is the representation of the Lie algebra $\mathfrak{h}_{1}$ of the Heisenberg group. The minimal uncertainty state in the Schrodinger representation is a solution of the differential equation: $(M-\mathrm{i}rD)\phi=0$ for some $r\in\mathbb{R}{}$, or, explicitly: $(M-\mathrm{i}rD)\phi=-\mathrm{i}\left(q+r{\hslash}\frac{d}{dq}\right)\phi(q)=0.$ (8) The solution is the Gaussian $\phi(q)=e^{-cq^{2}}$, $c=\frac{1}{2r\hslash}$. For $c>0$, this function is in the state space $\mathcal{L}_{2}{}(\mathbb{R}{})$. ∎ It is common to say that the Gaussian $\phi(q)=e^{-cq^{2}}$ represents the ground state, which minimises the uncertainty of coordinate and momentum. ## 3\. Wavelet transform and analyticity ### 3.1. Induced wavelet transform The following object is common in quantum mechanics [Kisil02e], signal processing, harmonic analysis [Kisil12d], operator theory [Kisil12b, Kisil13a] and many other areas [Kisil11c]. Therefore, it has various names [AliAntGaz00]: coherent states, wavelets, matrix coefficients, etc. In the most fundamental situation [AliAntGaz00]Ch. 8, we start from an irreducible unitary representation ${\rho}$ of a Lie group $G$ in a Hilbert space $\mathcal{H}{}$. For a vector $f\in\mathcal{H}{}$ (called mother wavelet, vacuum state, etc.), we define the map $\mathcal{W}_{f}$ from $\mathcal{H}{}$ to a space of functions on $G$ by: $[\mathcal{W}_{f}v](g)=\tilde{v}(g):=\left\langle v,{\rho}(g)f\right\rangle.$ (9) Under the above assumptions, $\tilde{v}(g)$ is a bounded continuous function on $G$. The map $\mathcal{W}_{f}$ intertwines ${\rho}(g)$ with the left shifts on $G$: $\mathcal{W}_{f}\circ{\rho}(g)=\Lambda(g)\circ\mathcal{W}_{f},\qquad\text{ where }\Lambda(g):\tilde{v}(g^{\prime})\mapsto\tilde{v}(g^{-1}g^{\prime}).$ (10) Thus, the image $\mathcal{W}_{f}\mathcal{H}{}$ is invariant under the left shifts on $G$. If ${\rho}$ is square integrable and $f$ is admissible [AliAntGaz00]§ 8.1, then $\tilde{v}(g)$ is square-integrable with respect to the Haar measure on $G$. At this point, none of admissible vectors has an advantage over others. It is common [Kisil11c]§ 5.1, that there exists a closed subgroup $H\subset G$ and a respective $f\in\mathcal{H}{}$ such that ${\rho}(h)f=\chi(h)f$ for some character $\chi$ of $H$. In this case, it is enough to know values of $\tilde{v}(\mathsf{s}(x))$, for any continuous section $\mathsf{s}$ from the homogeneous space $X=G/H$ to $G$. The map $v\mapsto\tilde{v}(x)=\tilde{v}(\mathsf{s}(x))$ intertwines ${\rho}$ with the representation ${\rho_{\chi}}$ in a certain function space on $X$ induced by the character $\chi$ of $H$ [Kirillov76]§ 13.2. We call the map $\mathcal{W}_{f}:v\mapsto\tilde{v}(x)$ the _induced wavelet transform_ [Kisil11c]§ 5.1. For example, if $G=\mathbb{H}^{1}{}$, $H=\\{(s,0,0)\in\mathbb{H}^{1}{}:\ s\in\mathbb{R}{}\\}$ and its character $\chi_{\hslash}(s,0,0)=e^{2\pi\mathrm{i}\hslash s}$, then any vector $f\in\mathcal{L}_{2}{}(\mathbb{R}{})$ satisfies ${\rho_{\hslash}}(s,0,0)f=\chi_{\hslash}(s)f$ for the representation (5). Thus, we still do not have a reason to prefer any admissible vector to others. ### 3.2. Right shifts and analyticity To discover some preferable mother wavelets, we use the following a general result from [Kisil11c]§ 5. Let $G$ be a locally compact group and ${\rho}$ be its representation in a Hilbert space $\mathcal{H}{}$. Let $[\mathcal{W}_{f}v](g)=\left\langle v,{\rho}(g)f\right\rangle$ be the wavelet transform defined by a vacuum state $f\in\mathcal{H}{}$. Then, the right shift $R(g):[\mathcal{W}_{f}v](g^{\prime})\mapsto[\mathcal{W}_{f}v](g^{\prime}g)$ for $g\in G$ coincides with the wavelet transform $[\mathcal{W}_{f_{g}}v](g^{\prime})=\left\langle v,{\rho}(g^{\prime})f_{g}\right\rangle$ defined by the vacuum state $f_{g}={\rho}(g)f$. In other words, the covariant transform intertwines right shifts on the group $G$ with the associated action ${\rho}$ on vacuum states, cf. (10): $R(g)\circ\mathcal{W}_{f}=\mathcal{W}_{{\rho}(g)f}.$ (11) Although, the above observation is almost trivial, applications of the following corollary are not. ###### Corollary 3 (Analyticity of the wavelet transform, [Kisil11c]§ 5). Let $G$ be a group and $dg$ be a measure on $G$. Let ${\rho}$ be a unitary representation of $G$, which can be extended by integration to a vector space $V$ of functions or distributions on $G$. Let a mother wavelet $f\in\mathcal{H}{}$ satisfy the equation $\int_{G}a(g)\,{\rho}(g)f\,dg=0,$ for a fixed distribution $a(g)\in V$. Then any wavelet transform $\tilde{v}(g)=\left\langle v,{\rho}(g)f\right\rangle$ obeys the condition: $D\tilde{v}=0,\qquad\text{where}\quad D=\int_{G}\bar{a}(g)\,R(g)\,dg,$ (12) with $R$ being the right regular representation of $G$. Some applications (including discrete one) produced by the $ax+b$ group can be found in [Kisil12d]§ 6. We turn to the Heisenberg group now. ###### Example 4 (Gaussian and FSB transform). The Gaussian $\phi(x)=e^{-cq^{2}/2}$ is a null-solution of the operator $\hslash cM-\mathrm{i}D$. For the centre $Z=\\{(s,0,0):\ s\in\mathbb{R}{}\\}\subset\mathbb{H}^{1}{}$, we define the section $\mathsf{s}:\mathbb{H}^{1}{}/Z\rightarrow\mathbb{H}^{1}{}$ by $\mathsf{s}(x,y)=(0,x,y)$. Then, the corresponding induced wavelet transform is: $\tilde{v}(x,y)=\left\langle v,{\rho}(\mathsf{s}(x,y))f\right\rangle=\int_{\mathbb{R}{}}v(q)\,e^{\pi\mathrm{i}\hslash xy-2\pi\mathrm{i}xq}\,e^{-c(q+{\hslash}y)^{2}/2}\,dq.$ (13) The infinitesimal generators $X$ and $Y$ of one-parameters subgroups $(0,t/(2\pi),0)$ and $(0,0,t)$ are represented through the right shift in (4) by $\textstyle R_{}(X)=-\frac{1}{4\pi}y\partial_{s}+\frac{1}{2\pi}\partial_{x},\quad R_{}(Y)=\frac{1}{2}x\partial_{s}+\partial_{y}.$ For the representation induced by the character $\chi_{\hslash}(s,0,0)=e^{2\pi\mathrm{i}\hslash s}$ we have $\partial_{s}=2\pi\mathrm{i}\hslash I$. Cor. 3 ensures that the operator $\hslash c\cdot R_{}(X)+\mathrm{i}\cdot R_{}(Y)=-\frac{\hslash}{2}(2\pi x+{\mathrm{i}\hslash c}y)+\frac{\hslash c}{2\pi}\partial_{x}+\mathrm{i}\partial_{y}$ (14) annihilate any $\tilde{v}(x,y)$ from (13). The integral (13) is known as Fock–Segal–Bargmann (FSB) transform and in the most common case the values $\hslash=1$ and $c=2\pi$ are used. For these, operator (14) becomes $-\pi(x+\mathrm{i}y)+(\partial_{x}+\mathrm{i}\partial_{y})=-\pi z+2\partial_{\bar{z}}$ with $z=x+\mathrm{i}y$. Then the function $V(z)=e^{\pi z\bar{z}/2}\,\tilde{v}(z)=e^{\pi(x^{2}+y^{2})/2}\,\tilde{v}(x,y)$ satisfies the Cauchy–Riemann equation $\partial_{\bar{z}}V(z)=0$. This example shows, that the Gaussian is a preferred vacuum state (as producing analytic functions through FSB transform) exactly for the same reason as being the minimal uncertainty state: the both are derived from the identity $(\hslash cM+\mathrm{i}D)e^{-cq^{2}/2}=0$. ### 3.3. Uncertainty and analyticity The main result of this paper is a generalisation of the previous observation, which bridges together Cor. 3 and Thm. 1. Let $G$, $H$, ${\rho}$ and $\mathcal{H}{}$ be as before. Assume, that the homogeneous space $X=G/H$ has a (quasi-)invariant measure $d\mu(x)$ [Kirillov76]§ 13.2. Then, for a function (or a suitable distribution) $k$ on $X$ we can define the integrated representation: ${\rho}(k)=\int_{X}k(x){\rho}(\mathsf{s}(x))\,d\mu(x),$ (15) which is (possibly, unbounded) operators on (possibly, dense subspace of) $\mathcal{H}{}$. In particular, $R(k)$ denotes the integrated right shifts, for $H=\\{e\\}$. ###### Theorem 5. Let $k_{1}$ and $k_{2}$ be two distributions on $X$ with the respective integrated representations ${\rho}(k_{1})$ and ${\rho}(k_{2})$. The following are equivalent: 1. (i) A vector $f\in\mathcal{H}{}$ satisfies the identity $\Delta_{f}({\rho}(k_{1}))\cdot\Delta_{f}({\rho}(k_{2}))=\left\|\left\langle[{\rho}(k_{1}),{\rho}(k_{1})]f,f\right\rangle\right\|.$ 2. (ii) The image of the wavelet transform $\mathcal{W}_{f}:v\mapsto\tilde{v}(g)=\left\langle v,{\rho}(g)f\right\rangle$ consists of functions satisfying the equation $R(k_{1}+\mathrm{i}rk_{2})\tilde{v}=0$ for some $r\in\mathbb{R}{}$, where $R$ is the integrated form (15) of the right regular representation on $G$. ###### Proof. This is an immediate consequence of a combination of Thm. 1 and Cor. 3. ∎ Example 4 is a particular case of this theorem with $k_{1}(x,y)=\delta^{\prime}_{x}(x,y)$ and $k_{2}(x,y)=\delta^{\prime}_{y}(x,y)$ (partial derivatives of the delta function), which represent vectors $X$ and $Y$ from the Lie algebra $\mathfrak{h}_{1}$. The next example will be of this type as well. ### 3.4. Hardy space Let $\mathrm{SU}(1,1)$ be the group of $2\times 2$ complex matrices of the form $\begin{pmatrix}\alpha&\beta\\\ \bar{\beta}&\bar{\alpha}\end{pmatrix}$ with the unit determinant $\left\|\alpha\right\|^{2}-\left\|\beta\right\|^{2}=1$. A standard basis in the Lie algebra $\mathfrak{su}_{1,1}$ is $A=\frac{1}{2}\begin{pmatrix}0&-\mathrm{i}\\\ \mathrm{i}&0\end{pmatrix},\quad B=\frac{1}{2}\begin{pmatrix}0&1\\\ 1&0\end{pmatrix},\quad Z=\begin{pmatrix}\mathrm{i}&0\\\ 0&-\mathrm{i}\end{pmatrix}.$ The respective one-dimensional subgroups consist of matrices: $e^{tA}=\begin{pmatrix}\cosh\frac{t}{2}&-\mathrm{i}\sinh\frac{t}{2}\\\ \mathrm{i}\sinh\frac{t}{2}&\cosh\frac{t}{2}\end{pmatrix},\ e^{tB}=\begin{pmatrix}\cosh\frac{t}{2}&\sinh\frac{t}{2}\\\ \sinh\frac{t}{2}&\cosh\frac{t}{2}\end{pmatrix},\ e^{tZ}=\begin{pmatrix}e^{\mathrm{i}t}&0\\\ 0&e^{-\mathrm{i}t}\end{pmatrix}.$ The last subgroup—the maximal compact subgroup of $\mathrm{SU}(1,1)$—is usually denoted by $K$. The commutators of the $\mathfrak{su}_{1,1}$ basis elements are $[Z,A]=2B,\qquad[Z,B]=-2A,\qquad[A,B]=-\frac{1}{2}Z.$ (16) Let $\mathbb{T}{}$ denote the unit circle in $\mathbb{C}{}$ with the rotation- invariant measure. The mock discrete representation of $\mathrm{SU}(1,1)$ [Lang85]§ VI.6 acts on $\mathcal{L}_{2}{}(\mathbb{T}{})$ by unitary transformations $[{\rho_{1}}(g)f](z)=\frac{1}{(\bar{\beta}z+\bar{\alpha})}\,f\left(\frac{{\alpha}z+\beta}{\bar{\beta}z+\bar{\alpha}}\right),\qquad g^{-1}=\begin{pmatrix}\alpha&\beta\\\ \bar{\beta}&\bar{\alpha}\end{pmatrix}.$ (17) The respective derived representation ${\rho_{1}}$ of the $\mathfrak{su}_{1,1}$ basis is: $\textstyle{\rho^{A}_{1}}=\frac{\mathrm{i}}{2}(z+(z^{2}+1)\partial_{z}),\quad{\rho^{B}_{1}}=\frac{1}{2}(z+(z^{2}-1)\partial_{z}),\quad{\rho^{Z}_{1}}=-\mathrm{i}I-2\mathrm{i}z\partial_{z}.$ Thus, ${\rho^{B+\mathrm{i}A}_{1}}=-\partial_{z}$ and the function $f_{+}(z)\equiv 1$ satisfies ${\rho^{B+\mathrm{i}A}_{1}}f_{+}=0$. Recalling the commutator $[A,B]=-\frac{1}{2}Z$ we note that ${\rho_{1}}(e^{tZ})f_{+}=e^{\mathrm{i}t}f_{+}$. Therefore, there is the following identity for dispersions on this state: $\textstyle\Delta_{f_{+}}({\rho^{A}_{1}})\cdot\Delta_{f_{+}}({\rho^{B}_{1}})=\frac{1}{2},$ with the minimal value of uncertainty among all eigenvectors of the operator ${\rho_{1}}(e^{tZ})$. Furthermore, the vacuum state $f_{+}$ generates the induced wavelet transform for the subgroup $K=\\{e^{tZ}\,\mid\,t\in\mathbb{R}{}\\}$. We identify $\mathrm{SU}(1,1)/K$ with the open unit disk $D=\\{w\in\mathbb{C}{}\,\mid\,\left\|w\right\|<1\\}$ [Kisil11c]§ 5.5 [Kisil13a]. The map $\mathsf{s}:\mathrm{SU}(1,1)/K\rightarrow\mathrm{SU}(1,1)$ is defined as $\mathsf{s}(w)=\frac{1}{\sqrt{1-\left\|w\right\|^{2}}}\begin{pmatrix}1&w\\\ \bar{w}&1\end{pmatrix}$. Then, the induced wavelet transform is: $\displaystyle\tilde{v}(w)=\left\langle v,{\rho_{1}}(\mathsf{s}(w))f_{+}\right\rangle$ $\displaystyle=\frac{1}{2\pi\sqrt{1-\left\|w\right\|^{2}}}\int_{\mathbb{T}{}}\frac{v(e^{\mathrm{i}\theta})\,d\theta}{1-we^{-\mathrm{i}\theta}}$ $\displaystyle=\frac{1}{2\pi\mathrm{i}\sqrt{1-\left\|w\right\|^{2}}}\int_{\mathbb{T}{}}\frac{v(e^{\mathrm{i}\theta})\,de^{\mathrm{i}\theta}}{e^{\mathrm{i}\theta}-w}.$ Clearly, this is the Cauchy integral up to the factor $\frac{1}{\sqrt{1-\left\|w\right\|^{2}}}$, which presents the conformal metric on the unit disk. Similarly, we can consider the operator ${\rho^{B-\mathrm{i}A}_{1}}=z+z^{2}\partial_{z}$ and the function $f_{-}(z)=\frac{1}{z}$ simultaneously solving the equations ${\rho^{B-\mathrm{i}A}_{1}}f_{-}=0$ and ${\rho_{1}}(e^{tZ})f_{-}=e^{-\mathrm{i}t}f_{-}$. It produces the integral with the conjugated Cauchy kernel. Finally, we can calculate the operator (12) annihilating the image of the wavelet transform. In the coordinates $(w,t)\in(\mathrm{SU}(1,1)/K)\times K$, the restriction to the induced subrepresentation is, cf. [Lang85]§ IX.5: $\displaystyle\mathfrak{L}^{B-\mathrm{i}A}{}=e^{2\mathrm{i}t}(-\frac{1}{2}w+(1-\left\|w\right\|^{2})\partial_{\bar{w}}).$ Furthermore, if $\mathfrak{L}^{B-\mathrm{i}A}{}\tilde{v}(w)=0$, then $\partial_{\bar{w}}(\sqrt{1-w\bar{w}}\cdot\tilde{v}(w))=0$. That is, $V(w)=\sqrt{1-w\bar{w}}\cdot\tilde{v}(w)$ is a holomorphic function on the unit disk. Similarly, we can treat representations of $\mathrm{SU}(1,1)$ in the space of square integrable functions on the unit disk. The irreducible components of this representation are isometrically isomorphic [Kisil11c]*§ 4–5 to the weighted Bergman spaces of (purely poly-)analytic functions on the unit, cf. [Vasilevski99]. ## References	arxiv-papers	2013-12-16T22:38:43	2024-09-04T02:49:55.530641	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Vladimir V. Kisil", "submitter": "Vladimir V Kisil", "url": "https://arxiv.org/abs/1312.4583" }
1312.4588	# Interface magnetic moments enhancement of FePt-L10/MgO(001): an $ab$ $initio$ study R. Cuadrado Department of Physics, University of York, York YO10 5DD, United Kingdom R. W. Chantrell Department of Physics, University of York, York YO10 5DD, United Kingdom ###### Abstract The interface between FePt–L10 and MgO(001) alloys has been studied using density functional calculations. Because the stacking of the face-centered tetragonal L10 phase is formed by alternating Fe and Pt planes, both the Fe- and Pt-terminated contact layers were studied. Furthermore, due to the large mismatch between the in-plane lattice constants of both systems, we have chosen some common $a$ values for both alloys in order to explore in detail the adsorption energy, the electronic structure and the interface magnetism. The adsorption energy has been calculated by subtracting the energy of clean FePt and MgO alloys from the total energy. The preferred adsorbed geometric sites for Fe/Pt atoms are when they lie on $top$ of the O species, having a smaller adsorption energy for the remaining positions. We found that expanding the MgO lattice enhances the magnetic moment of the Fe species but the Pt moments remain almost constant. ## I Introduction The face-centered tetragonal (fct) L10 phase of the 3d–5d binary based alloys such as FePt have recently been the subject of much attention because of their potential applications for the fabrication of ultrahigh density data recording media weller_para . These alloys present high values of the magnetocrystalline anisotropy (MAE) constants (7$\times$107 ergs/cm3) Pirama ; ivanov along the $c$–axis, a preferred orientation direction (easy axis). These high values of the anisotropy are necessary to overcome the superparamagnetic limit Yan in order to avoid the loss of recorded information. There are several methods to achieve the desired ferromagnetic structures such as the alternating monatomic layer deposition of Fe and Pt Shima or alternatively the room temperature deposition of disordered fcc FePt on an underlayer followed by annealing at around 600 ∘C to induce a phase transformation from fcc to fct L10 stacking. Extensive studies such as the effect of the alloying composition and growing FePt films on various underlayers, for instance MgO, Chen PtMn, Chiang or Si, Wu among others, have been carried out to optimize the microstructure and the magnetic properties as well as decreasing the processing temperature Xu . For practical purposes there are still some challenges, for example the ordered FePt grains with perpendicular (001) crystallographic orientation have to be magnetically decoupled from each other for which some materials are added in the fabrication process (see Ref. Peng and references therein). The use of MgO as an underlayer has some practical drawbacks such as its elevated costs, but the MgO single crystalline substrate promotes the out-of-plane anisotropy in contrast to other substrates which tend to promote in-plane anisotropy. Perumal Additionally, because the MgO lattice parameter is larger than that of FePt, it can easily promote the $c_{FePt}$ to remain perpendicular during the growing process, thus reducing the in-plane variants. From a theoretical point of view the transition metal (TM) binary compounds have been extensively studied in the past decade in their bulk phases, slabs, Chepulskii2012 gas phases, Cuadrado ; antoniak ; rollman and interfaces Zhu1 , however only a few theoretical studies based on FePt–L10/MgO(001) interfaces have been carried out Zhu1 ; Zhu2 . In this work we scan different configurations for this interface in a systematic investigation of the potential adsorption geometries, namely, the Fe-/Pt-termination, different atomic adsorption sites and the influence within the electronic structure of changes in the MgO(001) lattice constant, $a_{MgO}$. The paper is structured as follows: The employed theoretical tools are explained briefly in section II. In section III.1 we will summarize the final geometric structures as well as the related adsorption energies. The electronic study is presented in the section III.2 and the magnetic behaviour in III.3. The conclusions and future work are in section IV. ## II Theoretical Methods Figure 1: (Color online) (A) Schematic top view of the four initial adsorption sites for FePt–L10 onto MgO(001) alloy. Only the atoms belonging to the interfaces are shown, i.e., big green and small red spheres for the MgO contact layer and blue spheres on whether Fe- or Pt-terminations is shown. (B) FePt–L10 and MgO unit cells, left and right, respectively. The in-plane lattice constant, $a$, as well as some representative orientations and values have been also depicted in the figure. We have undertaken geometrical, electronic and magnetic structure calculations of the FePt–L10/MgO(001) interface by means of DFT using the SIESTA siesta code. To describe the core electrons we have used fully separable Kleinmann- Bylander kb and norm-conserving pseudopotentials (PP) of the Troulliers- Martins tm type. Our DFT based calculations have been performed within the generalized gradient approximation (GGA) for the exchange correlation (XC) potential following the Perdew, Burke, and Ernzerhof (PBE) version pbe . To address the description of magnetic systems, pseudocore (pc) corrections were used to include in the XC terms not only the valence charge density but also the core charge cc . In order to ease the convergence of three center integrals with the size of the real space grid, $\rho^{c}(r)$ is replaced by a pseudo-core charge density, $\rho^{pc}(r)$, which equals the real core charge density beyond a given radius, $r_{pc}$, while close to the nuclei it becomes a smooth function. The radius $r_{pc}$ should be chosen small enough to ensure that the overlap region between the valence and the core charges is fully taken into account. Based on previous studies of the binary alloys LS-paper , we have chosen for this radius the values of rpc(Fe) = 0.6 Bohrs and rpc(Pt) = 1.0 Bohrs, ensuring that the overlap region between the valence and the core charge is fully taken into account. As the basis set, we have employed double- zeta polarized (DZP) strictly localized numerical atomic orbitals (AO). The confinement energy, $E_{c}$, defined as the energy cost to confine the wave function within a given radius, was set to 100 meV. The so–called electronic temperature –kT in the Fermi-Dirac distribution– was set to 50 meV. In all the cases we ensured convergence of the Brillouin Zone (BZ) integration by considering a k–supercell of (16x16), i.e., 256 k-points. Real space three- center integrals are computed over a three-dimensional grid with a resolution of 900 Ry, a mesh fine enough to ensure convergence of the magnetic properties. The interface system is described by a two-dimensional periodic slab comprising eight MgO(001) plus eight FePt–L10 oriented layers. The L10 structure stacking is a fct phase in which there are alternate planes of Fe and Pt along the (001) direction. Because of this, there arises the possibility of having two kinds of interfaces between FePt–L10 and MgO(001): Fe-terminated (Fe/MgO) and Pt-terminated (Pt/MgO). Furthermore we can see in the Fig. 1A schematically the top views of four possible initial configurations, first and second rows. The small red and big green spheres represent the MgO alloy and the blue represent the first contact plane of the FePt alloy, either Fe or Pt. On the other hand, in the Fig. 1B the unit cells are shown following the same atomic nomenclature as in A but now explicitly the Pt atom presents a bigger size than the Fe ones. Specifically, depending whether the Fe/Pt atoms lie on $top$, $hollow$ or at $bridge$ positions, we have named the configurations as follows: on $top$ of O (Fe/Pt@$top$-O), on $top$ of Mg (Fe/Pt@$top$-Mg), at $hollow$ (Fe/Pt@$hollow$) and in $bridge$ positions (Fe/Pt@$bridge$). Metastable adsorption structures were obtained after relaxing the different proposed models to local minima until forces on atoms were smaller than 0.03 eV/Å. During the minimization process, just two layers of each material were allowed to relax leaving the rest of the atoms in the slab fixed to their bulk positions. The presence of two kinds of atoms in the FePt-L10 phases generates a vertical distortion so that its structure is defined by two quantities, the in-plane lattice parameter, $a$, and the out-of-plane constant, $c$, whose bulk experimental values are $a_{FePt}$=3.86 Å and $c/a$=0.98. The magnesium oxide structure can be described as two inter–penetrating fcc lattices displaced by $a/2(111)$ along the body diagonal of the conventional cube and the bulk experimental value for its lattice parameter is $a_{MgO}$=4.22Å. We observe that the in-plane mismatch between both alloys is $\approx$8.5$\%$. Because of this and in order to scan more geometrical possibilities we have used the four common in-plane lattice values: 4.00Å, 4.05Å, 4.10Å and 4.30Å. The last value corresponds to the optimized lattice constant for the bulk MgO under GGA and it has been taken into account to address how the FePt geometry and magnetic properties change not only with intermediate common $a$ values but also with an $a_{MgO}$ optimized value. According to the changing $a$, the distance between planes will vary too, so it was necessary to optimize the $c$ parameter for each lattice value. The results of both bulk systems and their corresponding out-of-plane distortions are for MgO: c/a = 1.14, 1.12, 1.10, 1.00 and for FePt: c/a = 0.92, 0.90, 0.88, 0.75 for a = 4.00Å, 4.05Å, 4.10Å and 4.30Å, respectively. ## III Results ### III.1 DFT structural relaxations \| \| \| Fe-terminated \| \| \| Pt-terminated \| ---\|---\|---\|---\|---\|---\|---\|--- Site \| a (Å) \| \| Eads \| $z_{I}(Fe)$ \| MMM \| MMNM \| \| \| Eads \| $z_{I}(Pt)$ \| MMM \| MMNM \| @$top$-O \| 4.00 \| \| 0.89 \| 2.23 \| 3.22 \| 0.24 \| \| \| 0.57 \| 2.55 \| 3.26 \| 0.25 \| \| 4.05 \| \| 0.93 \| 2.21 \| 3.26 \| 0.25 \| \| \| 0.60 \| 2.57 \| 3.30 \| 0.26 \| \| 4.10 \| \| 0.97 \| 2.19 \| 3.29 \| 0.25 \| \| \| 0.62 \| 2.56 \| 3.34 \| 0.26 \| \| 4.30 \| \| 1.14 \| 2.16 \| 3.34 \| 0.21 \| \| \| 0.74 \| 2.47 \| 3.39 \| 0.20 \| @$top$-Mg \| 4.00 \| \| 0.19 \| 3.30 \| 3.26 \| 0.25 \| \| \| 0.22 \| 3.16 \| 3.27 \| 0.25 \| \| 4.05 \| \| 0.20 \| 3.47 \| 3.30 \| 0.25 \| \| \| 0.22 \| 3.13 \| 3.30 \| 0.25 \| \| 4.10 \| \| 0.20 \| 3.51 \| 3.33 \| 0.25 \| \| \| 0.23 \| 3.14 \| 3.34 \| 0.26 \| \| 4.30 \| \| 0.22 \| 3.54 \| 3.37 \| 0.21 \| \| \| 0.29 \| 3.07 \| 3.40 \| 0.21 \| @$hollow$ \| 4.30 \| \| 0.57 \| 2.47 \| 3.36 \| 0.21 \| \| \| 0.45 \| 2.68 \| 3.40 \| 0.21 \| @$bridge$ \| 4.30 \| \| 1.13 \| 2.17 \| 3.34 \| 0.21 \| \| \| 0.70 \| 2.51 \| 3.39 \| 0.20 \| Table 1: Adsorption energies, Eads, $z$ heigths between the MgO contact layer and the first FePt-L10 plane, $z_{I}(Fe/Pt)$, average magnetic moments (MM) per atom of the Fe and Pt species of each configuration for Fe-/Pt- terminations, columns 3 to 6 and 7 to 9, respectively. The first two columns represent the four adsorption sites and the common lattice in-plane $a$ values, respectively. Energies are in eV, heights in Å and MM in $\mu_{B}$/at. In the table 1 we present the results of the interface adsorption energies Eads, the perpendicular bond distance $z_{I}$ computed as the $z$ difference of the MgO plane and the Fe/Pt one: $z_{I}$(Ai)=$z_{A_{i}}$–$z_{B_{i}}$ [Ai=Fe,Pt; Bi=Mg,O] and the total magnetic moment (MM) per magnetic/non- magnetic species for all the configurations. Two different contact layers, namely, Fe- and Pt-termination have been taken into account with a common in- plane lattice constant ranging from 4.00 to 4.30Å. The adsorption energies were evaluated after subtracting from the total energy of the FePt+Mg(001) bilayer the energy of the two clean metallic slabs. A general tendency of all @$top$ relaxed structures is that as the in-plane lattice constant expands from 4.00Å to 4.30Å the Eads increases significantly. It is noticeable that for the Fe-terminated configuration on top of O (first four values in the third column of Table 1), the average adsorption energies are larger by 0.35, 0.87 and 0.74 eV compared to those of Pt@$top$-O, Fe@$top$-Mg and Pt@$top$-Mg sites, respectively. Consequently, the bond between the FePt and MgO will be stronger for the Fe@$top$-O, with decreasing stability for the other cases, with Fe@$top$-Mg having the lowest stability. In order to scan more possible accommodation sites for Fe/Pt atoms we also studied Fe/Pt@$hollow$ and Fe/Pt@$bridge$ adsorption configurations (see Fig. 1A) considering only the GGA optimized value of $a_{MgO}$. The Eads values in these cases are between of those of Fe/Pt@$top$-O/-Mg sites, providing a weak chemical bonding for FePt. Regarding the Fe/Pt@$bridge$ case and after relaxing the interfaces, we found that the strong Fe/Pt-O chemical interaction means that the Fe/Pt@$bridge$ atoms move closer to the @$top$-O positions. In fact, the equilibrium position for the Fe/Pt atoms is only $\sim$0.2 Å distant from their @$top$-O positions. We also notice, by inspection of the table, that Eads values are quite similar in both cases. Related to the Eads, the bond distance, $z_{I}$(Ai), will depend on the strength of the bond: the higher the adsorption energies the smaller $z$ distances and stronger bonding will arise. This implies that the Fe/Pt planes will be closer to the MgO contact layer resulting in a complex rearrangement of the charge, changing the final values of the magnetic moments and hence the magnetic behaviour. Except for the case of Fe@$top$-Mg, where Eads remains approximately constant, the interlayer bond distance decrease as $a$ increases. We will discuss in detail in section III.3 the behaviour of the magnetic moments. As a global tendency we can say that the MMM values are augmented as $a$ is increased, by an average amount of $\sim$0.13$\mu_{B}$ and remain almost constant for MMNM with the special characteristic reduction of these values for $a$=4.30Å. ### III.2 Density of states and hybridisation study We display in Fig. 2 the evolution of the spin-resolved density of states (DOS), projected on the Fe/Pt interface atoms, for Fe- and Pt- terminated FePt layers. For both cases results are presented for the @$top$-O and @$top$-Mg adsorption sites. The next FePt–L10 layers after the contact layer (Pt atoms for Fe-termination or Fe for Pt-termination) are not shown here because changes in the electronic states are not significant beyond the interface layer. The three different colours in each graph depict the Fe/Pt DOS of the bilayer (filled turquoise curve), the Fe/Pt for the clean surface (thick solid pink line) and the atomic DOS projection of the atoms in their bulk phase (thin solid black line). Inside each graph the MM values appear on the bottom left corner just for the bilayer configurations together with a positive (negative) value within parenthesis that represents the increase (decrease) of the local MM compared to the Fe/Pt atomic bulk values. Figure 2: (Color online) Density of states (DOS) projected on Fe/Pt atoms of the FePt–L10 interfaces (filled turquoise curve), clean surfaces (thin solid pink line) and the Fe/Pt atoms in their bulk phases (thin solid black line). The Fe-/Pt-terminated configurations are presented in the first and second and in the third and fourth columns, respectively. For each interface termination @$top$-O/-Mg adsorption sites are also shown. The numbers present the MM in $\mu_{B}$/at and in parenthesis their difference with respect to the bulk atomic values. A first inspection of Fig. 2 shows that the DOS at the interface layer is significantly affected for contact with both MgO and vacuum. Further inspection of the DOS for the four rows (O and Mg adsorption sites) regarding both the Fe-/Pt-terminations, shows that, when the Fe/Pt atoms lie @$top$-Mg rather than @$top$-O, the interface Fe/Pt PDOS profile is almost the same as those of clean surfaces either spin-up or spin-down. Consequently the surface states of the FePt alloy will not be altered significantly. As a result, when Fe/Pt are @$top$-Mg the FePt termination behave similarly to a vacuum termination irrespective of the contact with MgO. This confirms the significant difference of $\sim$0.74 eV between the adsorption energies regarding the Mg or O sites (see table 1). Then, as we pointed out in the previous section, the bonding between Fe/Pt and Mg atoms will be weaker than with the O atoms. The abrupt termination of the FePt–L10 alloy in the clean surfaces and in the interface of the bilayer induces within the $d$ electrons a rearrangement which is distinct from that in the bulk phase Although the Fe/Pt coordination will be the same for the whole system, the environment will be modified compared to that of the bulk, given that now the Mg and O atoms are in the former positions of the Fe/Pt species. This is noticeable if we inspect the DOS of the Fe-termination (first two columns). In the bulk case the up-states are located in an energy range of 5 eV, i. e., from -1 to -6 eV for the @$top$-O/-Mg configurations. Despite the fact that the DOS of the interfaces and clean surfaces are quite similar to those of the bulk, their states are located closer to a pronounced peak at 3 eV weakly present in the bulk, leading to a narrowing of both the bilayer and clean surface d-bands. The fact that the spin-up and spin-down black curves have a small shift to lower energy values, compared to the pink and turquoise curves, in the last two columns of Fig. 2 (Pt-termination) can be explained by inspection of the Mulliken population of the bulk FePt. This shows that the bulk atomic species are more charged than the other two systems by an average of $\sim$0.2 e/at. Figure 3: (Color online) Fe and Pt atomic charge difference with respect to their bulk phase counterparts, $\Delta Q_{Fe/Pt}=q_{Fe/Pt}^{bilayer}-q^{bulk}_{Fe/Pt}$, as a function of the in-plane lattice values for the Fe-/Pt-terminated configurations, left and right, respectively. The solid coloured lines depict the evolution of $\Delta Q_{Fe/Pt}$ interface layer @top-O/-Mg adsorption sites and the dashed ones those of the nearest inward layer: Pt for Fe-terminations and Fe for Pt- termination. The analysis of the data shows that the charge transfer between the atoms (orbitals) belonging to an interface is of paramount importance. It gives us a way to understand the hybridisation between atoms and how this influences the bond strength and the magnetic behaviour. In figure 3 it shown that the difference in charge transferred between the magnetic and non–magnetic species is larger compared to the bulk phases. In figure 3A, situating the Fe atoms in the @$top$-O position (blue solid line) increases the amount of charge given to the Pt layer (dashed blue line) and to the first MgO plane by an average of 0.18 e/at. The same situation occurs when the Fe is @$top$-Mg (solid green line) though the effect is smaller. The dispersion ranges from 0.07 e/at up to almost 0.2 e/at for 4.30Å. When the Pt contact layer is @$top$-O (black solid line) it behaves in a similar way to bulk phase so that only for the 4.30Å lattice spacing is the amount of excess charge transferred to the Pt is significant at 0.08 e/at. Similar behavior is shown when comparing the Fe contact configuration in A. The Fe atoms in this case (black dashed line) are responsible for the charge transfer, having almost a constant contribution along all the $a$ values. Finally, the charge transferred to the Pt atoms @$top$-Mg (red solid line) shows very little change with respect to the bulk phase, increasing only slightly for the large $a$ values. The increase is clear however for the Fe atoms: again they lose more charge than in the bulk. In summary, the presence of the MgO changes significantly the FePt-L10 behavior depending on whether the contact layer is Fe or Pt and also where the Fe/Pt atoms are located after relaxation. This implies different kinds of hybridisation between 3$d$ and 5$d$ orbitals of these Fe and Pt atoms respectively. In order to link the MM behaviour with the DOS curves we also observe that in Table 1 the MM values of the Fe atoms increase by 0.13$\mu_{B}/at$ with the in-plane lattice spacing for all the configurations. This small enhancement of the MMM can be observed in the DOS curves noting that the down-state peaks below the Fermi level at $\sim$0.2 eV for $a=$4.00Å move to lie just at the Fermi level for $a=$4.30Å resulting in a deficit of down-states compared to the up-states which are almost constant with $a$. Finally, the DOS projected onto the interface plane of the MgO alloy (not shown here) does not change significantly, there is only some charge transfer among the interface resulting in a very small value of the local MM values (see Sec. III.3). ### III.3 Interface magnetism: Magnetic Moments Figure 4: (Color online) Resolved magnetic moments of Pt and Fe atoms, first and second row, respectively, for both Fe- (left) and Pt-terminations (right) as a function of the in-plane lattice parameter $a$ (in Å). For a common type of symbol the empty symbols represent the MM of the non magnetic species and the filled symbols, those of the Fe atoms. Each kind of color depicts @$top$-O adsorption site (blue squares), @$top$-Mg (green squares), the bulk (red circles) and the clean surface (orange triangles). The crosses and asterisks show the $bridge$ and $hollow$ adsorption sites for 4.30Å, respectively. We provide in Fig. 4 the magnetic moment (MM) values per atom as a function of the in-plane lattice parameter $a$ for all the configurations studied in this work. The global trend of the MM values of the Fe atoms for both Fe-/Pt- terminations (bottom row) is an enhancement of around 0.13$\mu_{B}$/at as the lattice increases from 4.00Å to 4.30Å. Although this behaviour is shared by all the adsorption positions, i.e., @(top-O, top-Mg,hollow,bridge) there is just an increase of 0.06$\mu_{B}$/at in the bulk case (filled red circles) when $a$=4.30Å compared to the smaller lattice considered. The values of the MM of the Pt atoms for both Fe-/Pt-terminations (first row) are enhanced as are those of the Fe atoms when $a$ increases up to $a$=4.15Å. However the MM decreases (increases) for the Fe-terminated alloys (Pt-terminated) for larger $a$. Inspecting the MM values vertically (common lattice) either @top-O or @top-Mg, blue and green filled squares, respectively, we observe that the dispersion between these two adsorption sites is about 0.15$\mu_{B}$/at. It is evident after inspecting table 1 that the lower distance between the Fe/Pt:MgO layers gives the higher adsorption energy. This behaviour plays an important role in the net MM/at values since the charge rearrangement between down and up states of the system promotes the reduction or the increase of the MM/at depending if the adsorption site is @top-O or @top-Mg, respectively. Completing the magnetic study we have added dots to represent the MM values for the Fe/Pt@($hollow$,$bridge$) sites, black asterisks and red crosses, respectively. Their MM values do not have significant changes regarding others with same $a$ value, they are almost the same as those obtained for the Fe/Pt@$top$-O configurations. \| \| \| Fe-terminated \| \| \| Pt-terminated \| ---\|---\|---\|---\|---\|---\|---\|--- Site \| a(Å) \| \| MMMg \| MMO \| \| \| MMMg \| MMO \| @$top$-O \| 4.00 \| \| -0.11 \| 0.03 \| \| \| -0.03 \| 0.03 \| \| 4.05 \| \| -0.12 \| 0.03 \| \| \| -0.03 \| 0.03 \| \| 4.10 \| \| -0.12 \| 0.03 \| \| \| -0.04 \| 0.03 \| \| 4.30 \| \| -0.10 \| 0.04 \| \| \| -0.03 \| 0.04 \| @$top$-Mg \| 4.00 \| \| -0.00 \| 0.02 \| \| \| -0.03 \| 0.02 \| \| 4.05 \| \| -0.02 \| 0.01 \| \| \| -0.03 \| 0.02 \| \| 4.10 \| \| -0.02 \| 0.01 \| \| \| -0.03 \| 0.02 \| \| 4.30 \| \| -0.01 \| 0.01 \| \| \| -0.04 \| 0.03 \| @$hollow$ \| 4.30 \| \| -0.05 \| 0.04 \| \| \| -0.02 \| 0.04 \| @$bridge$ \| 4.30 \| \| -0.10 \| 0.04 \| \| \| -0.03 \| 0.03 \| Table 2: Local magnetic moments (MM) values of the Mg (MMMg) and O (MMO) atoms of the MgO contact plane in $\mu_{B}$. The nomenclature as well as the different adsorption sites and terminations are the same as used throughout this work. Finally we show in Table 2 the site resolved MM values of Mg and O atoms. Note that the Mg species favors the down states while the O sites exhibit the opposite behavior. The most significant difference between the distinct configurations occurs @$top$-O adsorption sites since the MMMg are almost constant, at a value of -0.12 $\mu_{B}$ in comparison with the rest. The O species have an average constant value of $\sim$0.03 $\mu_{B}$. ## IV Conclusions and future work We have carried out a first principles study of the FePt–L10/MgO(001) interface regarding different in-plane lattice constant values and the two possible FePt contact planes due to the L10 stacking. In addition we scanned the magnetic/electronic properties for different adsorption sites for a common Fe/Pt plane, namely, @($top$-O,$top$-Mg,$hollow$,$bridge$). The adsorption energies provide a way to elucidate the preferential adsorption site and lattice constant value. The higher values were obtained for Fe@$top$-O followed by those of the Pt@$top$-O and also when the larger in-plane lattice constant has been taken into account. The stronger chemical bonding then occurs for these configurations having a significant overlap between the $d$-bands and the MgO orbitals. Additionally, we found that as the Eads decreases(increases) the MMM values augment(diminish) however the induced MM within the Pt species is not significantly affected by this variation. The use of GGA as the XC could overestimate the bonding between these materials giving the possibility that from an experimental point of view, the bond distance would be shorter and hence the adsorption energy higher. A useful compliment to the discussion of atomic hybridisation at the end of section III.2 would be a study of the up/down population of the $d$-orbitals. It is however beyond the scope of this work to include such details and instead we have only explored qualitatively the atomic spin-resolved population to understand how the spin charge influences the magnetic behavior. The spin electronic analysis confirms the results of the DOS calculations (Fig. 2). It shows that with increasing lattice constant, $a$, the amount of up (down) charge increases (decreases) leading to an enhancement of the MM values. In addition, figure 4 shows that there is a constant difference of 0.25$\mu_{B}$/at in the MM values for different adsorption positions (see separate blue and green symbols in the Fe-terminated configuration). The lowest adsorption energy value is shown for the Fe@$top$-O and hence this is the most stable configuration. This means that the hybridisation between the MgO and the Fe contact layer is stronger, reducing the MM values and promoting a bigger charge transfer among atoms. For the Pt-terminated configurations, as was pointed out in section III.1, the adsorption energy is smaller and the orbital hybridisation is smoother, therefore the MM values are close to each other for all values of $a$. It has recently been shown fefept that the site-resolved Magnetocrystalline Anisotropy Energy (MAE) of FePt is associated with the Fe sites. This is due to the strong 3d Fe – 5d Pt hybridisation through which the spin-orbit interaction on the Pt atoms is transferred to the electronic states at the Fe sites. Consequently, the complex charge transfer processes at the FePt/MgO interface predicted here might be expected to be reflected in changes in the FePt MAE at the MgO interface. However, the charge transfer effects seem rather localized to the FePt/MgO interface, so the effect might be significant only for ultrathin films. However, this is an interesting possible effect, which is beyond the scope of the current work but certainly worthy of further investigation. ## V Acknowledgments The authors are grateful to Dr T.J Klemmer for helpful discussions. Financial support of the EU Seventh Framework Programme under grant agreement No. 281043, FEMTOSPIN, and Seagate Technology is gratefully acknowledged. ## References * (1) D. Weller and A. Moser, IEEE Trans. Magn., 36, 10 (1999). * (2) N. Piramanayagam J. Appl. Phys. 102, 011301 (2007) * (3) O. A. Ivanov, L. V. Solina, V. A. Demshina and L. M Magat Fiz. Metal Metalloyed 35, 92 (1973). * (4) M. L. Yan, H. Zeng, N. Powers, and D. J. Sellmyer J. Appl. Phys. 91, 8471 (2002) * (5) T. Shima, T. Moriguchi, S. Mitani, and K. Takanashi Appl. Phys. Lett. 80, 288 (2002) * (6) J. S. Chen, B. C. Lim, J. F. Hu, Y. K. Kim, B. Liu, and G. M. Chow Appl. Phys. Lett. 90, 042508 (2007) * (7) C. C. Chiang, C.-H. Lai, and Y. C. Wu Appl. Phys. Lett. 88, 152508 (2006) * (8) Y. C. 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Matter 24, 086005 (2012). * (24) C J Aas, P J Hasnip, R Cuadrado, E M Plotnikova, L Szunyogh, L Udvardi and R W Chantrell Phys Rev B, 88, 174409 (2013).	arxiv-papers	2013-12-16T23:07:26	2024-09-04T02:49:55.537664	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "R Cuadrado and R W Chantrell", "submitter": "Ram\\'on Cuadrado", "url": "https://arxiv.org/abs/1312.4588" }
1312.4641	# From Brans-Dicke theory to Newtonian gravity Sergey Kozyrev Scientific center gravity wave studies Dulkyn e-mail : [email protected] ###### Abstract We present the new interpretation of scalar field for the Brans-Dicke theory. This interpretation is obtained by considering a fixed spacetime structure of manifold. Keywords : scalar-tensor theory, Newtonian physic ## 1 Comparisons with Newtonian gravity The scalar-tensor theory first time was invented by P. Jordan [1] in the 1950’s, and then taken over by C. Brans and R.H. Dicke [2] some years later. In this paper we restrict our discussion to the Brans-Dicke theory [1, 2] that, among of all the alternative theories of classical Einstein’s gravity, is the most studied and hence the best known theory. Scalar-tensor theories of gravity describe the universe as grounded on differentiable arbitrary manifold $M^{4}$ enveloped by a principal bundle formed with isometric representations of a finite continuous Poincaré group. Einstein’s principle of general relativity asserts the invariance under general coordinate transformations of the actions integral grounded on a $M^{4}$ manifold parameterized by variables $x^{\mu},\mu=0,1,2,3$. As it is well known field equation and conservation low of the relativity theory can be obtained from principle of least action. The same principle is the basis of the Brans - Dicke theory 111Units $8\pi G=c=1$ are used throughout the paper. Greek indices range over the coordinates of the 4-manifold and Roman indices over the coordinates of the 3-surfaces. $S={S}_{G}+{S}_{m}$ (1) $S_{G}=\frac{1}{2}\int dx^{4}\sqrt{-g}\left\\{\Phi R-\omega\frac{\Phi^{,\mu}\Phi_{,\mu}}{\Phi}\right\\}+{S}_{m},$ (2) where $R$ is the scalar curvature, $\phi$ is a scalar field, $\omega$ is a dimensionless coupling parameter, and ${S}_{m}$ is an action of ordinary matter (not including the scalar field). An unsatisfactory feature of relativistic theories is that the components of Lagrangian do not have any direct physical interpretation. Note that in abstract manifold, the Ricci scalar and tensor $g_{\mu\nu}$ lose their geometrical meaning that they had in a spacetime and now can be viewed only as a source for the metric. The scalar - tensor theories appears as a theory in which the gravity is described simultaneously by two fields the metric tensor and the scalar fields, the latter being an essential part of the geometrical property of spacetime manifesting its presence in all geometrical phenomena, such as curvature, geodesic motion and so on. On another hand, the Newtonian description of gravitating systems was developed in the 17th century using a scalar potential field and is nowadays a part of most classical mechanics textbooks. It seems reasonable to interpret Newton s absolute space as an absolute Euclidean embedding-space that acts as a container for non-Euclidean geometry. But there may be as well other reasons to contemplate Minkowski space from considerations of scalar gravity. In a gravity theories, a model of spacetime is usually a pair $(M,T)$ where $M$ is $N$-dimensional manifold with suitable topological and differentiable properties and $T$ represent a collection of matter fields on $M$. In approach with fixed spacetime structure, like Newtonian mechanics and special relativity, this model suggest an interpretation of manifold $M$ as independently existing ”container” for the histories of fields and particles. Obviously, the action (2) can be endowed with a structure of a manifold. To obtain the physical interpretation of the scalar field, one may write the action (2) for Minkowski metric in the following manner: $g_{\mu\nu}=\eta_{\mu\nu},$ $R=0$. We need field equations for $\phi,$ so the action (2) for this field must be supplied, $S_{G}=\int dx^{4}\,\eta^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi,$ (3) where $\phi=e^{\Phi}$. Without losing generality, suppose that $\omega=1$. The overall action for the aggregate of N point particles is $S_{m}=-\sum_{i=1}^{n}\int_{-\infty}^{\infty}\\!\\!\\!dt\,\frac{1}{2}m_{i}\dot{q}_{i}^{\mu}\dot{q}_{i}^{\nu}\eta_{\mu\nu}-\sum_{i=1}^{n}\int_{R^{4}}\\!\\!\\!dx\,m_{i}\delta(\textbf{x}-\textbf{q}_{i})\phi.$ (4) where $q_{I}(t)=\\{q_{I}^{\mu}(t)\\},I=1,...,N$ are the trajectory of point particles with mass $m_{I}$. As in Newtonian mechanics, we can consider arbirtary spacelike section $t=const$ given by Euclidean metric. The action (3) provides the following field equations [3]: $\frac{\delta S_{G}}{\delta\phi}=\triangle\phi=0,$ (5) where $\triangle:=\partial_{1}^{2}+\partial_{2}^{2}+\partial_{3}^{2}$ is a three dimensional Laplasian, the gradient of scalar field $\partial\phi$ is taken at the point $q_{I}=\\{t,q_{I}^{\mu}\\}$, where at the time the particle is located. The field equations read $\frac{\delta S}{\delta\phi}=\frac{1}{4\pi G}\triangle\phi-\sum_{i}m_{\delta}(\textbf{x}-\textbf{q}_{i})=0,$ (6) $\frac{\delta S}{\delta q_{i}}=m_{i}\left(\ddot{q}_{i\mu}-\partial_{\mu}\phi\right)=0,$ (7) Then, with this definition, the equation of gravity field have the form $\triangle\phi=\sum_{I}m_{I}\delta(\textbf{x}-\textbf{q}_{I}).$ (8) Thus, we see that the scalar field variables of Brans-Dicke theory play the role of a standard Newtonian potential. The field equation (8) yields the following solution [4] $\phi(t,\textbf{x})=-\sum_{I}\frac{m_{I}}{\|\textbf{x}-\textbf{q}_{I}(t)\|},$ (9) where $\|\textbf{x}-\textbf{q}_{I}\|:=\sqrt{-\eta_{\mu\nu}(x^{\mu}-q_{I}^{\mu})(x^{\nu}-q_{I}^{\nu})}$ (10) Now we consider the model with two particle with masses $m_{I}$ and $m_{J}$. The force $F={F^{\mu}}$, acting on a particle $m_{I}$ by particle $m_{J}$, is $F^{\mu}=m_{I}m_{J}\frac{q_{I}^{\mu}-q_{J}^{\mu}}{\|\textbf{q}_{I}-\textbf{q}_{J}\|^{3}}.$ (11) _It is Newton’s law of gravitation._ ## 2 Discussion In order to make a physical prediction from theory a manifold must be endowed with a structure (metric, connection, curvature …) In contrast with usual formalism of general relativity one can perform a model of spacetime with fixed background geometry, like Newtonian mechanics and special theory of relativity. Then the set up of action implies the imposing the Ricci scalar not as a scalar curvature of spacetime but as a matter source. The standard derivation of the Newtonian gravity as a weak field limit of relativistic theories do not expose the specific feature of Brans-Dicke theory which include Newton s law of gravitation as an exact solution. ## References * [1] P. Jordan, Schwerkraft und Weltall, Vieweg (Braunschweig) 1955\. * [2] C.H.Brans , R. H.Dicke, Phys.Rev.124, 925 (1961). * [3] M.O.Katanaev, Geometrical methods in mathematical physics., arXiv:1311.0733v1 (2013). * [4] V.S.Vladimirov, Uravnenia matematicheskoi fiziki, Nauka, Moskow, 1988\. (Russian)	arxiv-papers	2013-12-17T05:06:37	2024-09-04T02:49:55.549100	{ "license": "Public Domain", "authors": "Sergey Kozyrev", "submitter": "Sergey Kozyrev", "url": "https://arxiv.org/abs/1312.4641" }
1312.4666	# A Functional Approach to Standard Binary Heaps VLADIMIR KOSTYUKOV [email protected] (Dec 2013) ###### Abstract This paper describes a new and purely functional implementation technique of binary heaps. A binary heap is a tree-based data structure that implements priority queue operations (insert, remove, minimum/maximum) and guarantees at worst logarithmic running time for them. Approaches and ideas described in this paper present a simple and asymptotically optimal implementation of immutable binary heap. ## 1 Introduction There are several purely functional implementations of heaps such as _Leftist Heap_ , _Binomial Heap_ (Okasaki, 1999) and _Braun Tree_ (Braun and Rem, 1983), which are definitely the best choice as _priority queues_ in a functional setting. However, there are other heaps around without proper functional implementations. The simplest of them are standard _binary heaps_ , which do not fit well into a functional environment since their reference implementation is based on mutable arrays. This paper presents a new and purely functional implementation technique of binary heaps, with the same asymptotic bounds as in an imperative setting. ## 2 Binary Heaps A _binary heap_ (Williams, 1964) is a data structure that implements priority queue interface and guarantees logarithmic running time for `insert`/`delete` operations and constant time access to `minimum`/`maximum` element. Binary heaps are commonly viewed as binary trees which satisfy two invariants: 1. 1. The _shape_ invariant: the tree is a complete binary tree. 2. 2. The _min-heap_ invariant: each node is less than or equal to each of its children. ## 3 Binary Heap Operations In Scala (Odersky et al., 2004) a binary min heap that holds an integer values might be represented as abstract `Heap` class with two variants: `Branch` and `Leaf`. abstract sealed class Heap { def min: Int def left: Heap def right: Heap def size: Int def height: Int } case class Branch(min: Int, left: Heap, right: Heap, size: Int, height: Int) extends Heap case object Leaf extends Heap { def min: Int = fail("Leaf.min") def left: Heap = fail("Leaf.min") def right: Heap = fail("Leaf.right") def size: Int = 0 def height: Int = 0 } Thus, using pattern matching with case classes, which are actually projections of _Algebraic Data Types_ , `isEmpty` method can be written as def isEmpty: Boolean = this match { case Leaf => true case _ => false } Except for `height` and `size` operations, this signature looks like a functional implementation of _binary search tree_ (Okasaki, 1999). The two new operations are actually accessors to new fields in a heap - its height and size. This additional data should be accessible in constant time to define an efficient and simple _search criteria_ for `insert` and `remove` operations. ## 4 Insertion in O(log n) Insertion into functional binary heap must not violate either of its invariants - neither the shape invariant nor the min-heap invariant. For this purpose two problems should be solved. First, to maintain the shape invariant a new node should be inserted in the first empty spot at the last level of the heap. Second, to maintain the min-heap invariant the inserted node should be _bubbled up_ to the heap root until it becomes greater than its parent. Bubbling up is quite a simple transformation that can be done at each level in constant time. There are two cases depending on whether the violation is at left or right child. In both cases, the violation should be fixed by _swapping_ two nodes - the root node and the child that violates the min-heap invariant. There is also a third case, when it doesn’t violate anything. In this case, a heap should be simply rebuilt with given parameters. In other words, all affected nodes should be copied in order to maintain data structure persistence. More precisely, `bubbleUp` and `insert` operations might be defined as def bubbleUp(x: Int, l: Heap, r: Heap): Heap = (l, r) match { case (Branch(y, lt, rt, _, _), _) if (x > y) => Heap(y, Heap(x, lt, rt), r) case (_, Branch(z, lt, rt, _, _)) if (x > z) => Heap(z, l, Heap(x, lt, rt)) case (_, _) => Heap(x, l, r) } def insert(x: Int): Heap = if (isEmpty) Heap(x) else if (???) bubbleUp(min, left.insert(x), right) else bubbleUp(min, left, right.insert(x)) , where the _smart constructor_ Heap" that creates a new singleton heap is defined as \begin{verbatim} def Heap(x: Int, l: Heap = Leaf, r: Heap = Leaf): Heap = Branch(x, l, r, l.size + r.size + 1, math.max(l.height, r.height) + 1) \end{verbatim} Note that \emph{height} of a heap is defined as max height of its children plus one, while \emph{size} of a heap is defined as sum of its children sizes plus one; and both are calculated only once in a heap constructor. Also, to simplify calculations, suppose that singleton heap’s height is $1$. The last thing to discuss is how to find a proper spot for a new node. This is actually a cornerstone of functional binary heaps. The main idea is based on two definitions of \emph{perfect binary trees}: \emph{math} and \emph{recursive}. Math definition: a perfect binary tree contains \emph{$2^{h + 1} - 1$} nodes, where \emph{$h$} is the height of the tree. Recursive definition: a tree is perfect if its children are perfect trees of the same height. Combining these facts together, one can define \emph{search criteria} which allow to fill a heap level by level from left to right, thereby maintaining the shape invariant. In other words, new nodes should be inserted in a way to make the heap be a perfect tree. This can be simply achieved by following requirements of the recursive definition, using the math definition as an efficient test on tree perfectness. Thus, the search criteria for insertion contain four cases depending on whether the children are perfect trees or not and whether their heights are equal or not. \begin{verbatim} def insert(x: Int): Heap = if (isEmpty) Heap(x) else if (left.size < math.pow(2, left.height) - 1) bubbleUp(min, left.insert(x), right) else if (right.size < math.pow(2, right.height) - 1) bubbleUp(min, left, right.insert(x)) else if (right.height < left.height) bubbleUp(min, left, right.insert(x)) else bubbleUp(min, left.insert(x), right) \end{verbatim} \begin{figure} \hspace{2.6cm} \rput[br](0.58,-1.22){$\alpha$} \pstree[levelsep=20pt,treesep=0.40]{\Tcircle{$x$}} { \psset{ArrowInside=->,arrowscale=2,ArrowInsidePos=0.7} \pstree{\Tp}{\Tfan[linestyle=dashed]} \psset{ArrowInside=none} \Tn \Tcircle[linestyle=none]{$\beta$} } \hspace{1.8cm} \rput[br](0.58,-1.22){$\alpha$} \rput[br](1.94,-1.28){$\beta$} \pstree[levelsep=20pt,treesep=0.40]{\Tcircle{$x$}} { \pstree{\Tp}{\Tfan} \psset{ArrowInside=->,arrowscale=2,ArrowInsidePos=0.7} \pstree{\Tp}{\Tfan[linestyle=dashed]} \psset{ArrowInside=none} } \vspace{1.0cm} \hspace{2.6cm} \rput[br](0.6,-1.44){$\alpha$} \rput[br](1.94,-1.28){$\beta$} \pstree[levelsep=20pt,treesep=0.40]{\Tcircle{$x$}} { \pstree[levelsep=30pt]{\Tp}{\Tfan} \psset{ArrowInside=->,arrowscale=2,ArrowInsidePos=0.7} \pstree{\Tp}{\Tfan} \psset{ArrowInside=none} } \hspace{1.5cm} \rput[br](0.6,-1.44){$\alpha$} \rput[br](1.94,-1.52){$\beta$} \pstree[levelsep=20pt,treesep=0.40]{\Tcircle{$x$}} { \psset{ArrowInside=->,arrowscale=2,ArrowInsidePos=0.7} \pstree[levelsep=30pt]{\Tp}{\Tfan} \psset{ArrowInside=none} \pstree[levelsep=30pt]{\Tp}{\Tfan} } \rput[br](4.6,-0.5){$\|\alpha\| > \|\beta\|$} \rput[br](8.6,-0.5){$\|\alpha\| = \|\beta\|$} \vspace{1cm} \hspace{1.8cm} \psframebox[linewidth=1pt]{ \begin{psmatrix}[rowsep=0.2cm] \trinode{A}{} - perfect tree \hspace{0.2cm} \trinode[linestyle=dashed]{A}{} - not perfect tree \hspace{0.2cm}\\ $\|\alpha\|$ - height of the heap \hspace{0.2cm} \psline[linewidth=1pt,arrowscale=2]{->}(0.3,0.2)(0.6,-0.1) \hspace{0.6cm} - search path direction \end{psmatrix} } \vspace{0.4cm} \caption{Searching for the first empty spot in a heap.} \end{figure} The time complexity of \verb insert” operation is $O(log\;n)$ since it requires to perform bubble up transformations for each node in a search path, and the longest possible path for complete trees is $log\;n$. ## 5 Construction in O(n) Constructing binary heap from unordered input can be done in linear time (Floyd, 1964). Such performance is achieved by algorithm that constructs a complete heap in a bottom-up manner together with fixing all violations of the min-heap invariant. There is only one dangerous case that violates the min- heap invariant - the root node of new heap is greater than its children. This violation can be fixed by _bubbling_ wrongly placed node _down_ to the heap. The `bubbleDown` operation can be written with pattern matching as def bubbleDown(x: Int, l: Heap, r: Heap): Heap = (l, r) match { case (Branch(y, _, _, _, _), Branch(z, lt, rt, _, _)) if (z < y && x > z) => Heap(z, l, bubbleDown(x, lt, rt)) case (Branch(y, lt, rt, _, _), _) if (x > y) => Heap(y, bubbleDown(x, lt, rt), r) case (_, _) => Heap(x, l, r) } [levelsep=35pt,treesep=0.20]$y$ [levelsep=35pt]$x$[linestyle=none]$\alpha$ [linestyle=none]$\beta$ [levelsep=35pt]$z$[linestyle=none]$\gamma$ [linestyle=none]$\delta$ $y<z$$x>y$[levelsep=35pt,treesep=0.20]$x$[levelsep=35pt]$y$[linestyle=none]$\alpha$[linestyle=none]$\beta$[levelsep=35pt]$z$[linestyle=none]$\gamma$[linestyle=none]$\delta$$z<y$$x>z$[levelsep=35pt,treesep=0.20]$z$[levelsep=35pt]$y$[linestyle=none]$\alpha$[linestyle=none]$\beta$[levelsep=35pt]$x$[linestyle=none]$\gamma$[linestyle=none]$\delta$ additional condition for bubbling down Figure 1: Eliminating _min-heap_ invariant violations. The `heapify` operation constructs a _complete_ binary heap in a recursive way by using the ideas of array-based heaps representations - given the index $i$ of a node, the indices of its children - $2i+1$ for left child and $2i+2$ for right child. With inner function it looks like def heapify(a: Array[Int]): Heap = { def loop(i: Int): Heap = if (i < a.length) bubbleDown(a(i), loop(2 i + 1), loop(2 * i + 2)) else Leaf loop(0) } The analysis of these operations is quite tricky. It seems that `heapify`’s running time is $O(n\;log\;n)$, since each call to `bubbleDown` costs $O(log\;n)$ and there are $O(n)$ such calls. This is correct for the upper bound, but it is not asymptotically tight. The thing is that running time of b`bubbleDown` depends on the height of sub-heap it is applied to, and the heights of most sub-heaps are small. Thus, in most cases `bubbleDown` runs in constant time and the total running time of heapify operation is $O(n)$. More detailed analysis can be found at (Cormen et al., 2001). ## 6 Removal in O(log n) Removal is always pain in the neck for most of data structures. However, it is not so bad for binary heaps. There is quite a nice standard algorithm that allows to remove minimum/maximum from the heap in $O(log\;n)$ time. The algorithm joins two phases which maintain both invariants - replacing root node with last inserted one and bubbling it down. Considering that `mergeChildren` is the first phase and `bubbleRootDown` is the second one, `remove` operation might be written as def remove: Heap = if (isEmpty) fail("Empty heap.") else bubbleRootDown(mergeChildren(left, right)) , where `bubbleRootDown` can be defined as wrapper around `bubbleDown` operation that was discussed previously. def bubbleRootDown(h: Heap): Heap = if (h.isEmpty) Leaf else Heap.bubbleDown(h.min, h.left, h.right) The most interesting part of removal is its first phase. Quite a tricky problem should be solved at this phase: replacing the root of the heap with its last inserted node. First of all, it requires finding such node. This is what has already been done in insertion and all one needs to do is change the the search criteria a bit. The search criteria for removal contain four cases as well as for insertion, but it has a different meaning for some of them. First, if both node’s children are empty, then this node is the last inserted one. Second, if both node’s children are perfect trees and the left child is higher than the right one, then the last inserted node is somewhere on the left. Third, it is expected that the last inserted node is somewhere on the right if node’s children are perfect trees of the same height. When the last inserted node is found, it should be _floated_ to the place of the root. This can be done by using _divide and conquer_ algorithm with following ideas. Suppose that the last inserted node of the heap can be recursively floated to the place of its child’s root. Then, depending on whether the child is left or right, its root should be lifted to the place of the heap’s root. Finally, the affected child should be restored. Combining all things together, `mergeChildren` operation might be defined as def mergeChildren(l: Heap, r: Heap): Heap = if (l.isEmpty && r.isEmpty) Leaf else if (l.size < math.pow(2, l.height) - 1) floatLeft(l.min, mergeChildren(l.left, l.right), r) else if (r.size < math.pow(2, r.height) - 1) floatRight(r.min, l, mergeChildren(r.left, r.right)) else if (r.height < l.height) floatLeft(l.min, mergeChildren(l.left, l.right), r) else floatRight(r.min, l, mergeChildren(r.left, r.right)) def floatLeft(x: Int, l: Heap, r: Heap): Heap = l match { case Branch(y, lt, rt, _, _) => Heap(y, Heap(x, lt, rt), r) case _ => Heap(x, l, r) } def floatRight(x: Int, l: Heap, r: Heap): Heap = r match { case Branch(y, lt, rt, _, _) => Heap(y, l, Heap(x, lt, rt)) case _ => Heap(x, l, r) } The remove operation performs two walks along the search path of the heap. First, it searches for the last inserted node and floats it to the place of the root (maintaining the shape invariant). Second, it bubbles new root down (maintaining the min-heap invariant). Thus, keeping in mind that the longest possible path in a complete tree is $log\;n$, the total running time of `remove` is $O(log\;n)$. ## 7 Conclusion Functional setting brings some charm and beauty into data structures implementations. But it is not always possible to design a proper functional implementation that meets performance requirements. Sometimes, it is just close to impossible to convert a RAM-based algorithm into equivalent functional one - to make the mind think in terms of _space_ not _time_. However, it can be done for binary heaps. The suggested implementation technique allows to achieve asymptotically optimal performance along with maintaining data structure persistence. And this is another good example in computer science that combines both elegant abstraction and witty implementation. ## References * [Okasaki, 1999] Okasaki, C. (1999), _Purely functional data structures_ , Cambridge University Press. * [Braun and Rem, 1983] Braun, W. and Rem, M. (1983) A logarithmic implementation of flexible arrays. _Memorandum MR83/4_. Eindhoven University of Technology. * [Williams, 1964] Williams, J. W. J. (1964), Algorithm 232 - Heapsort, _Communications of the ACM_ 7 (6): 347тАУ-348. * [Odersky et al., 2004] Odersky M., Altherr P., Cremet V., Emir B., Maneth S., Micheloud S., Mihaylov N., Schinz M., Stenman E. and Zenger M. (2004), An Overview of the Scala Programming Language, _EPFL Technical Report IC/2004/64_ , EPFL Lausanne. * [Floyd, 1964] Floyd, Robert W., (1964), Algorithm 245 - Treesort 3, _Communications of the ACM_ 7 (12): 701 * [Cormen et al., 2001] Cormen, T. H., Leiserson, C. E. and Rivest, R. L. (2001), _Introduction to Algorithms_ (2nd ed.), Cambridge, Massachusetts: The MIT Press.	arxiv-papers	2013-12-17T07:06:57	2024-09-04T02:49:55.555236	{ "license": "Public Domain", "authors": "Vladimir Kostyukov", "submitter": "Vladimir Kostyukov", "url": "https://arxiv.org/abs/1312.4666" }
1312.4813	# Exclusive processes in proton-proton collisions with the CMS experiment at the LHC ††thanks: Presented at the Low x workshop, May 30 - June 4 2013, Rehovot and Eilat, Israel Laurent [email protected] on behalf of the CMS collaboration Centre for Cosmology, Particle Physics and Phenomenology, Université catholique de Louvain, Chemin du Cyclotron 2, 1348 Louvain-la-Neuve, Belgium ###### Abstract We present the recent measurements of exclusive processes performed in the CMS experiment at the LHC using data collected at a centre of mass energy of 7 TeV. These measurements include the double-pomeron production of photon pairs, the two-photon production of leptons pairs, and the previously undetected two- photon production of $W$ boson pairs. While in case of the two first processes that enables to set limits on production cross-section, in the later case it provides also stringent limits on the anomalous quartic gauge couplings. ## 1 Introduction Central exclusive processes provide an interesting field of study in particle physics. With its excellent performance, the CMS experiment, whose full description can be found elsewhere[1, 2], has managed by now to make a number of significant observations of these processes, hence to probe the Standard model in a unique way. A striking signature is expected for such processes, with two very forward (though undetected) scattered protons and two large rapidity gaps between these protons and the central exclusively produced system. An overview is hereby presented for some of these results, such as the search for central exclusive production of photon pairs, the exclusive two- photon production of lepton pairs, and the achieved limits on anomalous quartic gauge couplings in the two-photon production of $W$ pairs. (a) (b) (c) Figure 1: Various types of exclusive processes involving (a) the double pomeron exchange, (b) the two-photon production, and (c) the photon-pomeron fusion. ## 2 Central-exclusive $\gamma\gamma$ production The search for central exclusive production of two-photon events as represented on Fig. 1(a), constitutes a direct test for the QCD predictions involving the so-called double pomeron exchange. In perturbative calculations this process can be interpreted as a $gg\to\gamma\gamma$ subprocess involving an additional low-momentum gluon exchange enabling the cancellation of the colour flow involved by the interacting gluons. With a data sample corresponding to an integrated luminosity of $36~{}\mathrm{fb}^{-1}$ collected in 2010 by the CMS detector, the limit on the cross-section of central exclusive production of two photons can be quoted for the first time at a centre of mass energy $\sqrt{s}=7~{}\mathrm{TeV}$ after a measurement performed at $\sqrt{s}=1.96~{}\mathrm{TeV}$ by the CDF experiment at the Tevatron[3]. The candidate events are requested to contain two energetic photons with a transverse energy higher than $5.5~{}\mathrm{GeV}$ as measured by the electromagnetic calorimeter within its acceptance (corresponding to a pseudo-rapidity range $\|\eta\|<2.5$). These two photons are furthermore expected to be balanced in their transverse deposited energy and be back-to-back in azimuthal angle. The exclusivity condition is then satisfied when no additional activity (apart from the two photons) is observed in the full detector (thus in the range $\|\eta\|<5.2$). This allows a reduced observation of the proton-dissociative events, when one or both forward scattered proton dissociate in a hadronic system hence additional activity is expected in the forward region of the detector. One of the sources of inefficiency considered in the analysis is due to the relatively high instantaneous luminosity delivered by the LHC machine, whose main drawback is the multiplication of the primary vertices in each event. This phenomenon, the “events pileup”, is expected to lower in a significant way the efficiency of the exclusivity condition with respect to the luminosity. This is shown in Fig. 2 in which the efficiency is evaluated using _zero bias_ events collected parasitically during the data-taking. Figure 2: Exclusivity condition efficiency with respect to the instantaneous luminosity of each protons bunch. This exclusivity efficiency is computed to be the ratio between the number of events passing the selection criteria and the total number of the so-called _zero bias_ events collected in 2010 at $\sqrt{s}=7~{}\mathrm{TeV}$. Figure from [4]. No candidate events are observed with a background expectation of $1.79\pm 0.40$ events arising from the multiple processes involving an inclusive or exclusive production of photons, such as pairs of misidentified photons (from exclusive production of electron pairs) or from the exclusive $\pi^{0}\pi^{0}$ production. An upper limit on the exclusive production cross-section can therefore be set according to this result : $\sigma(E_{T}(\gamma)>5.5~{}\mathrm{GeV},\|\eta(\gamma)\|<2.5)<1.18~{}\mathrm{pb}.$ This limit is then compared to the theoretical prediction from various approaches using different orders in the perturbation theory, and parton densities functions (PDFs). A graphical view of this comparison can be found in Fig. 3, without the most recent prediction [5] computed at the leading order (LO) and the next-to-leading order (NLO), providing a production cross section of $0.180~{}\mathrm{pb}$ for the former using the MSTW08[6] PDF, and $0.039~{}\mathrm{pb}$ for the latter using the MRST99 [7] PDF. These signal predictions are computed using the ExHuMe 1.34[8] event generator implementing the KMR model[9] where the proton-gluon couplings are determined perturbatively. In this model, the two-photon system is then produced by the mean of a quark box, as represented on Fig. 1(a). Figure 3: Comparison of the CMS experimental upper limit on the central diffractive production of photon pairs with the theoretical predictions at leading order (LO) and at the next to leading order (NLO) for different parton density functions. This result shows that the theoretical predictions for this exclusive process, mediated by gluons only, are in good agreement with the observed upper limit on the cross-section. ## 3 Exclusive two-photon production of lepton pairs Two different di-lepton exclusive analyses have been released using the data collected in 2010 at $7~{}\mathrm{TeV}$, namely for the measurements of $\gamma\gamma\to e^{+}e^{-}$[4] and of $\gamma\gamma\to\mu^{+}\mu^{-}$[10], such as represented on Fig. 1(b). The signal for such processes is simulated using the LPAIR [11, 12] Monte Carlo generator developed in the 1980s for the HERA $ep$ collider experiments. It uses the full matrix element calculation for a $pp\to p^{(\ast)}\ell^{+}\ell^{-}p^{(\ast)}$ process, where the photon coupling to a proton is described by the proton electromagnetic form-factors, in case when a fully exclusive production is simulated and both incident protons survived the interaction. In the proton inelastic case the form-factor is replaced by the proton structure functions from HERA fits. The rapidity gap survival probability (due to re-scattering) is not modeled by LPAIR, and it is set to unity in these analyses. The event selection is requiring two leptons which are energy- or momentum- balanced and back-to-back in the transverse plane. This corresponds to a $\left\|p_{T}(\ell^{+})-p_{T}(\ell^{-})\right\|<1~{}\mathrm{GeV}$ as well as an acoplanarity describing the difference in azimuthal angles, $\left\|1-\Delta\phi(\ell^{+},\ell^{-})/\pi\right\|<0.1$. Furthermore, the dimuon analysis requires each of the two muons to carry a transverse momentum larger than $4~{}\mathrm{GeV}$ in the range $\|\eta(\mu)\|<2.1$. In order to reject the exclusive photoproduction of the low-mass resonances such as the $J/\psi$, the $\psi^{\prime}$, and the $\Upsilon(1s,2s,3s)$, an invariant mass cut is applied $m(\mu^{+}\mu^{-})>11.5~{}\mathrm{GeV}$. For the dielectron analysis, a electron-positron pair with a transverse energy deposit in the calorimeters $E_{T}>5.5~{}\mathrm{GeV}$ are selected in the range $\|\eta(e)\|<2.5$. No additonal invariant mass cut is needed in this case. To ensure the exclusivity condition, the dimuon analysis requires no additional tracks within $2~{}\mathrm{mm}$ in the longitudinal direction around the dimuon primary vertex, while the dielectron case rejects all events where any other particles are reconstructed within the full acceptance of the detector, $\|\eta\|<5.2$. This very tight condition, along with a noticeable difference in the collected integrated luminosity, results in large efficiency difference, hence in the number of dimuon candidates (184 events) and the dielectron ones (17 candidates), as represented in Fig. 4. Furthermore, the higher statistics provided by the dimuon channel allows an extraction of the elastic signal contribution as well as the correction to be applied on the theoretical yield predicted for the proton-dissociative part. These corrections are determined from a binned maximum-likelihood fit to the measured $p_{T}(\mu^{+}\mu^{-})$ distribution as represented on Fig. 4(a). (a) (b) Figure 4: Muons (left) and electron (right) pairs transverse momentum distributions for the candidates selected in the two-photon production of leptons pairs. These two results allow to improve the understanding of this purely electromagnetic process, by observing 17 candidates for the dielectron channel and by a measurement of a production cross-section at $\sqrt{s}=7$ TeV for the dimuon channel : $\sigma(pp\to p\mu^{+}\mu^{-}p)=3.38{}^{+0.58}_{-0.55}~{}(\mathrm{stat.})\pm 0.16~{}(\mathrm{syst.})\pm 0.14~{}(\mathrm{lumi.})~{}\mathrm{pb}$ which is consistent with the theoretical prediction. ## 4 Two-photon production of $W$ pairs and limits on anomalous quartic gauge couplings Several processes beyond the Standard model predict an anomalous behaviour of the quartic gauge coupling, such as new gauge bosons production or heavy quarks exchanges[13]. These anomalous quartic gauge couplings (or AQGCs) can be translated into a higher production rate, or discrepancies in the kinematic distributions of multiple final state particles. The LHC experiments have been predicted to be sensitive to such behaviours when involving the two-photon interactions [14, 15]. Hence, a search for these anomalous couplings is performed at high energies in the CMS experiment with the data collected in 2011 at $\sqrt{s}=7$ TeV, using the challenging yet previously unobserved two-photon production of $W^{\pm}$ bosons pairs[16], where both the gauge bosons decay leptonically. The channel of interest for this analysis is the different leptons flavours decay, and especially $e^{\pm}\mu^{\mp}\nu\bar{\nu}$. Indeed, since the high- statistics same-flavour channels are saturated with their main sources of background, the $e^{+}e^{-}\nu\bar{\nu}$ and $\mu^{+}\mu^{-}\nu\bar{\nu}$ final states are set aside in the current analysis. For the former, the dominant sources of background are the inclusive _Drell-Yan_ and the exclusive two-photon productions of $\tau$ leptons pairs as well as the inclusive $W^{+}W^{-}$ production and the leptonic decay of $t\bar{t}$ events. On the other hand, the backgrounds for the later include the inclusive _Drell-Yan_ production of same leptons pairs and exclusive $\gamma\gamma\to\ell^{+}\ell^{-}$ (as seen in section 3), which are predicted to be more than one order of magnitude larger than the exclusive $\gamma\gamma\to W^{+}W^{-}$. The two neutrinos being left undetected, the candidates are to contain two leptons with $p_{T}(\ell)>20$ GeV reconstructed within the full detector acceptance, such as $\|\eta(\ell)\|<2.4$, and matched to one single primary vertex from which no additional tracks originates. Indeed, the high luminosity conditions from the 2011 runs of the LHC giving rise to multiple interactions within the same bunch crossing (the so-called ”pileup”), this condition on the additional tracks is the only one ensuring a sufficient efficiency in the selection. The leptons pair is required to have an invariant mass higher than $20$ GeV, and two kinematic regions can be built to isolate the Standard model or the anomalous quartic gauge couplings search regions. For the former, a transverse momentum $p_{T}(e^{\pm}\mu^{\mp})>30$ GeV is imposed on the leptons pair, while the later is more stringent with a lower cut of $100$ GeV. In order to simulate the theoretical prediction of the Standard model and the anomalous scenarios in these two regions, extra diagrams have to be taken into account in the final computation. These additional processes are the protons single- and double-dissociative cases which cannot be untangled from the purely elastic contribution without the information on the outgoing protons. Hence, an estimation of these two contributions is given by using the high statistics $\mu^{+}\mu^{-}$ final state probed in the same phase space and extracting a ”scale factor” which can then be applied on the $W^{+}W^{-}$ signal : $F=\left.\frac{n(\mu\mu~{}\mathrm{data})-n(\mu\mu~{}\mathrm{background})}{n(\mu\mu~{}\mathrm{elastic})}\right\|_{m(\mu\mu)>160~{}\mathrm{GeV}}=3.23\pm 0.53~{}\mathrm{(stat.+syst.)},$ with the background events defined as the prediction of inclusive _Drell-Yan_ production of muons and taus pairs, and the elastic $\gamma\gamma\to\mu^{+}\mu^{-}$ events number predicted by LPAIR. Given that rescaling in the probed high mass region a theoretical production cross-section is extracted for two kinematic search regions using the CalcHEP [17] generator. The theoretical prediction cross-section for the first region (no acceptance cut, and with the $W^{\pm}$ leptonic decay branching ratio included) is : $\sigma(pp\to p^{(\ast)}W^{+}W^{-}p^{(\ast)}\to p^{(\ast)}\mu^{\pm}e^{\mp}\nu\bar{\nu}p^{(\ast)})=4.0\pm 0.7~{}\mathrm{fb}.$ (a) (b) Figure 5: (a) Events passing the full $\gamma\gamma\to W^{+}W^{-}$ selection, with the leptons pairs’ transverse momentum relaxed. The filled histograms represent the backgrounds while the solid line represents the Standard model prediction of such exclusive two-photon production of $W^{\pm}$ pairs, and the dashed ones are two anomalous quartic gauge couplings examples given as a mean for comparison. (b) One- and two-dimensional limits on the two anomalous quartic gauge couplings parameters according to the CMS upper limits on the $\gamma\gamma\to W^{+}W^{-}$ production cross section. As seen in Fig. 5(a) which depicts the full selected events with the pair transverse momentum cut relaxed, a total of two $\gamma\gamma\to W^{+}W^{-}$ events candidates (displayed in Fig. 6) are observed in this Standard model region. No events are observed in the AQGC search region. An upper limit on the production cross-section is set according to the theoretical predictions. A $95\%$ confidence level interval is given for the Poisson mean for signal events in this window : $\sigma\left(p_{T}(\ell)>20~{}\mathrm{GeV},\|\eta(\ell)\|<2.4,m(e^{\pm}\mu^{\mp})>20~{}\mathrm{GeV},p_{T}(e^{\pm}\mu^{\mp})>100~{}\mathrm{GeV}\right)<1.9~{}\mathrm{fb}.$ Figure 6: Event displays of the two selected Standard model candidates are shown in a zoom over the inner tracking system of CMS. The numerous vertices present on these figures denote the hard events pileup conditions encountered during the 2011 data-taking period. The two parameters controlling these AQGCs can finally be bounded in a tighter way than the previous limits by OPAL[18] and DØ [19]. These 1-and 2-dimensional CMS limits at 95% C.L. are drawn in Fig. 5(b), and are approximately two orders of magnitude more stringent than the best limits obtained at the Tevatron, and at LEP. The one-dimensional bounds are : $\displaystyle\|a^{W}_{0}/\Lambda^{2}\|<1.5\times 10^{-4}~{}\mathrm{GeV}^{-2},\mathrm{~{}and~{}}\|a^{W}_{c}/\Lambda^{2}\|<5.0\times 10^{-4}~{}\mathrm{GeV}^{-2},~{}~{}(\Lambda_{\mathrm{cutoff}}=500~{}\mathrm{GeV})$ $\displaystyle\|a^{W}_{0}/\Lambda^{2}\|<4.0\times 10^{-6}~{}\mathrm{GeV}^{-2},\mathrm{~{}and~{}}\|a^{W}_{c}/\Lambda^{2}\|<1.5\times 10^{-5}~{}\mathrm{GeV}^{-2},~{}~{}(\mathrm{no~{}form~{}factor})\hskip 16.60005pt$ These results either include or not a dipole form factor with a cutoff scale $\Lambda_{\mathrm{cutoff}}=500~{}\mathrm{GeV}$ to avoid the unitarity violation of the anomalous models at high two-photon energies. The case with no form factors is to be taken with care, as they are driven by high-energy two-photon interactions beyond the unitarity bound. With two candidates on an undetected channel, the best limits on these anomalous quartic gauge couplings can be extracted. The limits exceed by two orders of magnitude the previous results, and are competitive with the current CMS analyses of such couplings based on tri-boson production [20]. ## 5 Summary and outlook In this note, several achievements were shown in the experimental search for exclusive processes at the LHC. First, the theoretical predictions for the central-exclusive $\gamma\gamma$ production, mediated by gluons only, are in good agreement with the observed upper limit on the cross-section. This result goes hand in hand with the search for the two-photon production of leptons pairs, which enables to improve the understanding of this purely electromagnetic process with an observation of 17 candidates for the dielectron channel and a measurement of a production cross-section at $\sqrt{s}=7$ TeV consistent with the theoretical prediction for the dimuon channel. Finally, with two candidates on a previously undetected $\gamma\gamma\to W^{+}W^{-}$ process, the best limits on the anomalous quartic gauge couplings can be extracted. These limits, while exceeding two orders of magnitude more stringent results with respect to the previous attempts, are competitive with the current CMS analyses on such couplings. The results presented in this note provide evidence both for the excellent performance of the CMS experiment, and its potential for future measurements of exclusive processes at the LHC. ## References * [1] CMS collaboration, “The CMS experiment at the CERN LHC,” JINST, vol. 3, p. S08004, 2008. * [2] CMS collaboration, “CMS technical design report, volume II: Physics performance,” J. Phys. G, vol. 34, p. 995, 2007. * [3] CDF collaboration, “Observation of Exclusive Gamma Gamma Production in $p\bar{p}$ Collisions at $\sqrt{s}=1.96$ TeV,” Phys.Rev.Lett., vol. 108, p. 081801, 2012. * [4] CMS collaboration, “Search for exclusive or semi-exclusive photon pair production and observation of exclusive and semi-exclusive electron pair production in $pp$ collisions at $\sqrt{s}=7~{}\mathrm{TeV}$,” JHEP, vol. 11, p. 080, 2012. * [5] L. Harland-Lang, V. Khoze, M. Ryskin, and W. Stirling, “The Phenomenology of Central Exclusive Production at Hadron Colliders,” Eur.Phys.J., vol. 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Vermaseren, “Two-photon processes at very high energies,” Nucl. Phys. B, vol. 229, p. 347, 1983. * [13] O. Eboli, M. Gonzalez-Garcia, and S. Lietti, “Bosonic quartic couplings at CERN LHC,” Phys.Rev., vol. D69, p. 095005, 2004. * [14] J. de Favereau de Jeneret, V. Lemaitre, Y. Liu, S. Ovyn, T. Pierzchała, K. Piotrzkowski, X. Rouby, N. Schul, and M. Vander Donckt, “High energy photon interactions at the LHC.” 2009. * [15] T. Pierzchała and K. Piotrzkowski, “Sensitivity to anomalous quartic gauge couplings in photon-photon interactions at the LHC,” Nucl. Phys. Proc. Suppl., vol. 179-180, p. 257, 2008. * [16] CMS collaboration, “Study of exclusive two-photon production of $W^{+}W^{-}$ in $pp$ collisions at $\sqrt{s}=7$ TeV and constraints on anomalous quartic gauge couplings,” JHEP, vol. 1307, p. 116, 2013. * [17] A. Pukhov, “CalcHEP 2.3: MSSM, structure functions, event generation, batchs, and generation of matrix elements for other packages.” 2004\. * [18] OPAL collaboration, “Constraints on anomalous quartic gauge boson couplings from $\nu\bar{\nu}\gamma\gamma$ and $q\bar{q}\gamma\gamma$ events at LEP-2,” Phys. Rev. D, vol. 70, p. 032005, 2004. * [19] DØ collaboration, “Search for anomalous quartic $WW{\gamma}{\gamma}$ couplings in dielectron and missing energy final states in $p\bar{p}$ collisions at $\sqrt{s}=1.96~{}\mathrm{TeV}$.” 2013. * [20] CMS collaboration, “A search for ${WW\gamma}$ and ${WZ\gamma}$ production in $pp$ collisions at $\sqrt{s}=8$ TeV.,” Tech. Rep. CMS-PAS-SMP-13-009, CERN, Geneva, 2013.	arxiv-papers	2013-12-17T15:06:00	2024-09-04T02:49:55.563973	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Laurent Forthomme (for the CMS Collaboration)", "submitter": "Laurent Forthomme", "url": "https://arxiv.org/abs/1312.4813" }
1312.4852	# Identification of Gaussian Process State-Space Models with Particle Stochastic Approximation EM Roger Frigola Fredrik Lindsten Thomas B. Schön Carl E. Rasmussen Dept. of Engineering, University of Cambridge, United Kingdom. Div. of Automatic Control, Linköping University, Sweden. Dept. of Information Technology, Uppsala University, Sweden. ###### Abstract Gaussian process state-space models (GP-SSMs) are a very flexible family of models of nonlinear dynamical systems. They comprise a Bayesian nonparametric representation of the dynamics of the system and additional (hyper-)parameters governing the properties of this nonparametric representation. The Bayesian formalism enables systematic reasoning about the uncertainty in the system dynamics. We present an approach to maximum likelihood identification of the parameters in GP-SSMs, while retaining the full nonparametric description of the dynamics. The method is based on a stochastic approximation version of the EM algorithm that employs recent developments in particle Markov chain Monte Carlo for efficient identification. ###### keywords: System identification, Bayesian, Non-parametric identification, Gaussian processes ## 1 Introduction Inspired by recent developments in robotics and machine learning, we aim at constructing models of nonlinear dynamical systems capable of quantifying the uncertainty in their predictions. To do so, we use the Bayesian system identification formalism whereby degrees of uncertainty and belief are represented using probability distributions (Peterka, 1981). Our goal is to identify models that provide error-bars to any prediction. If the data is informative and the system identification unambiguous, the model will report high confidence and error-bars will be narrow. On the other hand, if predictions are made in operating regimes that were not present in the data used for system identification, we expect the error-bars to be larger. Nonlinear state-space models are a very general and widely used class of dynamical system models. They allow for modeling of systems based on observed input-output data through the use of a latent (unobserved) variable, the _state_ $\mathbf{x}_{t}\in\mathcal{X}\triangleq\mathbb{R}^{n_{x}}$. A discrete-time state-space model (SSM) can be described by $\displaystyle\mathbf{x}_{t+1}$ $\displaystyle=f(\mathbf{x}_{t},\mathbf{u}_{t})+\mathbf{v}_{t},$ (1a) $\displaystyle\mathbf{y}_{t}$ $\displaystyle=g(\mathbf{x}_{t},\mathbf{u}_{t})+\mathbf{e}_{t},$ (1b) where $\mathbf{y}_{t}$ represents the output signal, $\mathbf{u}_{t}$ is the input signal, and $\mathbf{v}_{t}$ and $\mathbf{e}_{t}$ denote i.i.d. noises. The state transition dynamics is described by the nonlinear function $f$ whereas $g$ links the output data at a given time to the latent state and input at that same time. For convenience, in the following we will not explicitly represent the inputs in our formulation. When available, inputs can be straightforwardly added as additional arguments to the functions $f$ and $g$. A common approach to system identification with nonlinear state-space models consists in defining a parametric form for the functions $f$ and $g$ and finding the value of the parameters that minimizes a cost function, e.g. the negative likelihood. Those parametric functions are typically based on detailed prior knowledge about the system, such as the equations of motion of an aircraft, or belong to a class of parameterized generic function approximators, e.g. artificial neural networks (ANNs). In the following, it will be assumed that no detailed prior knowledge of the system is available to create a parametric model that adequately captures the complexity of the dynamical system. As a consequence, we will turn to generic function approximators. Parametrized nonlinear functions such as radial basis functions or other ANNs suffer from both theoretical and practical problems. For instance, a practitioner needs to select a parametric structure for the model, such as the number of layers and the number of neurons per layer in a neural network, which are difficult to choose when little is known about the system at hand. On a theoretical level, fixing the number of parameters effectively bounds the complexity of the functions that can be fitted to the data (Ghahramani, 2012). In order to palliate those problems, we will use Gaussian processes (Rasmussen and Williams, 2006) which provide a practical framework for Bayesian nonparametric nonlinear system identification. Gaussian processes (GPs) can be used to identify nonlinear state-space models by placing GP priors on the unknown functions. This gives rise to the Gaussian Process State-Space Model (GP-SSM) (Turner et al., 2010; Frigola et al., 2013) which will be introduced in Section 2. The GP-SSM is a nonparametric model, though, the GP is in general governed by a (typically) small number of hyper- parameters, effectively rendering the model semiparametric. In this work, the hyper-parameters of the model will be estimated by maximum likelihood, while retaining the full nonparametric richness of the system model. This is accomplished by analytically marginalizing out the nonparametric part of the model and using the particle stochastic approximation EM (PSAEM) algorithm by Lindsten (2013) for estimating the parameters. Prior work on GP-SSMs includes (Turner et al., 2010), which presented an approach to maximum likelihood estimation in GP-SSMs based on analytical approximations and the parameterization of GPs with a pseudo data set. Ko and Fox (2011) proposed an algorithm to learn (i.e. identify) GP-SSMs based on observed data which also used weak labels of the unobserved state trajectory. Frigola et al. (2013) proposed the use of particle Markov chain Monte Carlo to provide a fully Bayesian solution to the identification of GP-SSMs that did not need a pseudo data set or weak labels about unobserved states. However, the fully Bayesian solution requires priors on the model parameters which are unnecessary when seeking a maximum likelihood solution. Approaches for filtering and smoothing using already identified GP-SSMs have also been developed (Deisenroth et al., 2012; Deisenroth and Mohamed, 2012). ## 2 Gaussian Process State-Space Models ### 2.1 Gaussian Processes Whenever there is an unknown function, GPs allow us to perform Bayesian inference directly in the space of functions rather than having to define a parameterized family of functions and perform inference in its parameter space. GPs can be used as priors over functions that encode vague assumptions such as smoothness or stationarity. Those assumptions are often less restrictive than postulating a parametric family of functions. Formally, a GP is defined as a collection of random variables, any finite number of which have a joint Gaussian distribution. A GP $f(\mathbf{x})\in\mathbb{R}$ can be written as $f(\mathbf{x})\sim\mathcal{GP}\big{(}m(\mathbf{x}),k(\mathbf{x},\mathbf{x}^{\prime})\big{)},$ (2) where the mean function $m(\mathbf{x})$ and the covariance function $k(\mathbf{x},\mathbf{x}^{\prime})$ are defined as $\displaystyle m(\mathbf{x})$ $\displaystyle=\mathbb{E}[f(\mathbf{x})],$ (3a) $\displaystyle k(\mathbf{x},\mathbf{x}^{\prime})$ $\displaystyle=\mathbb{E}[(f(\mathbf{x})-m(\mathbf{x}))(f(\mathbf{x}^{\prime})-m(\mathbf{x}^{\prime}))].$ (3b) A finite number of variables from a Gaussian process follow a jointly Gaussian distribution $\begin{bmatrix}f(\mathbf{x}_{1})\\\ f(\mathbf{x}_{2})\\\ \vdots\\\ \end{bmatrix}\sim\mathcal{N}\left(\begin{bmatrix}m(\mathbf{x}_{1})\\\ m(\mathbf{x}_{2})\\\ \vdots\\\ \end{bmatrix},\begin{bmatrix}k(\mathbf{x}_{1},\mathbf{x}_{1})&k(\mathbf{x}_{1},\mathbf{x}_{2})&\\\ k(\mathbf{x}_{2},\mathbf{x}_{1})&k(\mathbf{x}_{2},\mathbf{x}_{2})&\\\ &&\ddots\\\ \end{bmatrix}\right).$ (4) We refer the reader to (Rasmussen and Williams, 2006) for a thorough exposition of GPs. ### 2.2 Gaussian Process State-Space Models In this article we will focus on problems where there is very little information about the nature of the state transition function $f(\mathbf{x}_{t})$ and a GP is used to model it. However, we will consider that more information is available about $g(\mathbf{x}_{t})$ and hence it will be modeled by a parametric function. This is reasonable in many cases where the mapping from states to observations is known, at least up to some parameters. The generative probabilistic model for the GP-SSM is fully specified by $\displaystyle\mathbf{f}_{t+1}\mid\mathbf{x}_{t}$ $\displaystyle\sim\mathcal{GP}\big{(}m_{\boldsymbol{\theta}_{\mathbf{x}}}(\mathbf{x}_{t}),k_{\boldsymbol{\theta}_{\mathbf{x}}}(\mathbf{x}_{t},\mathbf{x}_{t}^{\prime})\big{)},$ (5a) $\displaystyle\mathbf{x}_{t+1}\mid\mathbf{f}_{t+1}$ $\displaystyle\sim\mathcal{N}(\mathbf{x}_{t+1}\mid\mathbf{f}_{t+1},\mathbf{Q}),$ (5b) $\displaystyle\mathbf{y}_{t}\mid\mathbf{x}_{t}$ $\displaystyle\sim p(\mathbf{y}_{t}\mid\mathbf{x}_{t},\boldsymbol{\theta}_{\mathbf{y}}),$ (5c) where $\mathbf{f}_{t+1}=f(\mathbf{x}_{t})$ is the value taken by the state $\mathbf{x}_{t+1}$ after passing through the transition function, but before the application of process noise $\mathbf{v}_{t+1}$. The Gaussian process in (5a) describes the prior distribution over the transition function. The GP is fully specified by its mean function $m_{\boldsymbol{\theta}_{\mathbf{x}}}(\mathbf{x})$ and its covariance function $k_{\boldsymbol{\theta}_{\mathbf{x}}}(\mathbf{x}_{t},\mathbf{x}_{t}^{\prime})$, which are parameterized by the vector of hyper-parameters $\boldsymbol{\theta}_{\mathbf{x}}$. Equation (5b) describes the addition of process noise following a zero-mean Gaussian distribution of covariance $\mathbf{Q}$. We will place no restrictions on the likelihood distribution (5c) which will be parameterized by a finite-dimensional vector $\boldsymbol{\theta}_{\mathbf{y}}$. For notational convenience we group all the (hyper-)parameters into a single vector $\boldsymbol{\theta}=\\{\boldsymbol{\theta}_{\mathbf{x}},\boldsymbol{\theta}_{\mathbf{y}},\mathbf{Q}\\}$. ## 3 Maximum Likelihood in the GP-SSM Maximum likelihood (ML) is a widely used frequentist estimator of the parameters in a statistical model. The ML estimator $\widehat{\boldsymbol{\theta}}^{\text{ML}}$ is defined as the value of the parameters that makes the available observations $\mathbf{y}_{0:T}$ as likely as possible according ot, $\widehat{\boldsymbol{\theta}}^{\text{ML}}=\operatorname{arg\,max}_{\boldsymbol{\theta}}\ p(\mathbf{y}_{0:T}\mid\boldsymbol{\theta}).$ (6) The GP-SSM has two types of latent variables that need to be marginalized (integrated out) in order to compute the likelihood $\displaystyle p(\mathbf{y}_{0:T}\mid\boldsymbol{\theta})=\int p(\mathbf{y}_{0:T},\mathbf{x}_{0:T},\mathbf{f}_{1:T}\mid\boldsymbol{\theta})\ \text{d}\mathbf{x}_{0:T}\,\text{d}\mathbf{f}_{1:T}$ $\displaystyle=\int p(\mathbf{y}_{0:T}\mid\mathbf{x}_{0:T},\boldsymbol{\theta})\left(\int p(\mathbf{x}_{0:T},\mathbf{f}_{1:T}\mid\boldsymbol{\theta})\ \text{d}\mathbf{f}_{1:T}\right)\text{d}\mathbf{x}_{0:T}.$ (7) Following results from Frigola et al. (2013), the latent variables $\mathbf{f}_{1:T}$ can be marginalized analytically. This is equivalent to integrating out the uncertainty in the unknown function $f$ and working directly with a prior over the state trajectories $p(\mathbf{x}_{0:T}\mid\boldsymbol{\theta})$ that encodes the assumptions (e.g. smoothness) of $f$ specified in (5a). The prior over trajectories can be factorized as $\displaystyle p(\mathbf{x}_{0:T}\mid\boldsymbol{\theta})=p(\mathbf{x}_{0}\mid\boldsymbol{\theta})\prod_{t=1}^{T}p(\mathbf{x}_{t}\mid\boldsymbol{\theta},\mathbf{x}_{0:t-1}).$ (8) Using standard expressions for GP prediction, the one-step predictive density is given by $\displaystyle p(\mathbf{x}_{t}\mid\boldsymbol{\theta},\mathbf{x}_{0:t-1})=\mathcal{N}\big{(}\mathbf{x}_{t}\mid\boldsymbol{\mu}_{t}(\mathbf{x}_{0:t-1}),\mathbf{\Sigma}_{t}(\mathbf{x}_{0:t-1})\big{)},$ (9a) where $\displaystyle\boldsymbol{\mu}_{t}(\mathbf{x}_{0:t-1})$ $\displaystyle=\mathbf{m}_{t-1}+\mathbf{K}_{t-1,0:t-2}\widetilde{\mathbf{K}}^{-1}_{0:t-2}\ (\mathbf{x}_{1:t-1}-\mathbf{m}_{0:t-2}),$ (9b) $\displaystyle\mathbf{\Sigma}_{t}(\mathbf{x}_{0:t-1})$ $\displaystyle=\widetilde{\mathbf{K}}_{t-1}-\mathbf{K}_{t-1,0:t-2}\widetilde{\mathbf{K}}^{-1}_{0:t-2}\mathbf{K}_{t-1,0:t-2}^{\top},$ (9c) for $t\geq 2$ and $\boldsymbol{\mu}_{1}(\mathbf{x}_{0})=\mathbf{m}_{0}$, $\mathbf{\Sigma}_{1}(\mathbf{x}_{0})=\widetilde{\mathbf{K}}_{0}$. Here we have defined the mean vector $\mathbf{m}_{0:t-1}\triangleq\begin{bmatrix}m({\mathbf{x}}_{0})^{\top}&\dots&m({\mathbf{x}}_{t-1})^{\top}\end{bmatrix}^{\top}$ and the $(n_{x}t)\times(n_{x}t)$ positive definite matrix $\mathbf{K}_{0:t-1}$ with block entries $[\mathbf{K}_{0:t-1}]_{i,j}=k({\mathbf{x}}_{i-1},{\mathbf{x}}_{j-1})$. These matrices use two sets of indices, as in $\mathbf{K}_{t-1,0:t-2}$, to refer to the off-diagonal blocks of $\mathbf{K}_{0:t-1}$. We also define $\widetilde{\mathbf{K}}_{0:t-1}=\mathbf{K}_{0:t-1}+\mathbf{I}_{t}\otimes\mathbf{Q}$, where $\otimes$ denotes the Kronecker product. Using (8) we can thus write the likelihood (7) as $\displaystyle p(\mathbf{y}_{0:T}\mid\boldsymbol{\theta})=\int p(\mathbf{y}_{0:T}\mid\mathbf{x}_{0:T},\boldsymbol{\theta})p(\mathbf{x}_{0:T}\mid\boldsymbol{\theta})\ \text{d}\mathbf{x}_{0:T}.$ (10) The integration with respect to $\mathbf{x}_{0:T}$, however, is not analytically tractable. This difficulty will be addressed in the subsequent section. A GP-SSM can be seen as a hierarchical probabilistic model which describes a prior over the latent state trajectories $p(\mathbf{x}_{0:T}\mid\boldsymbol{\theta}_{\mathbf{x}},\mathbf{Q})$ and links this prior with the observed data via the likelihood $p(\mathbf{y}_{t}\mid\mathbf{x}_{t},\boldsymbol{\theta}_{\mathbf{y}})$. Direct application of maximum likelihood on $p(\mathbf{y}_{t}\mid\mathbf{x}_{t},\boldsymbol{\theta}_{\mathbf{y}})$ to obtain estimates of the state trajectory and likelihood parameters would invariably result in over-fitting. However, by introducing a prior on the state trajectories111A prior over the state trajectories is not an exclusive feature of GP-SSMs. Linear-Gaussian state-space models, for instance, also describe a prior distribution over state trajectories: $p(\mathbf{x}_{1:T}\mid\mathbf{A},\mathbf{B},\mathbf{Q},\mathbf{x}_{0})$. and marginalizing them as in (10), we obtain the so-called marginal likelihood. Maximization of the marginal likelihood with respect to the parameters results in a procedure known as type II maximum likelihood or empirical Bayes (Bishop, 2006). Empirical Bayes reduces the risk of over-fitting since it automatically incorporates a trade-off between model fit and model complexity, a property often known as Bayesian Occam’s razor (Ghahramani, 2012). ## 4 Particle Stochastic Approximation EM As pointed out above, direct evaluation of the likelihood (10) is not possible for a GP-SSM. However, by viewing the latent states $\mathbf{x}_{0:T}$ as missing data, we are able to evaluate the _complete data_ log-likelihood $\log p(\mathbf{y}_{0:T},\mathbf{x}_{0:T}\mid\boldsymbol{\theta})=\log p(\mathbf{y}_{0:T}\mid\mathbf{x}_{0:T},\boldsymbol{\theta})+\log p(\mathbf{x}_{0:T}\mid\boldsymbol{\theta}),$ (11) by using (5c) and (9). We therefore turn to the Expectation Maximization (EM) algorithm (Dempster et al., 1977). The EM algorithm uses (11) to construct a surrogate cost function for the ML problem, defined as $\displaystyle Q({}$ $\displaystyle\boldsymbol{\theta},\boldsymbol{\theta}^{\prime})=\mathbb{E}_{\boldsymbol{\theta}^{\prime}}[\log p(\mathbf{y}_{0:T},\mathbf{x}_{0:T}\mid\boldsymbol{\theta})\mid y_{0:T}]$ $\displaystyle=\int\log p(\mathbf{y}_{0:T},\mathbf{x}_{0:T}\mid\boldsymbol{\theta})p(\mathbf{x}_{0:T}\mid\mathbf{y}_{0:T},\boldsymbol{\theta}^{\prime})\text{d}\mathbf{x}_{0:T}.$ (12) It is an iterative procedure that maximizes (10) by iterating two steps, expectation (E) and maximization (M), (E) Compute $Q(\boldsymbol{\theta},\boldsymbol{\theta}_{k-1})$. * (M) Compute $\boldsymbol{\theta}_{k}=\operatorname{arg\,max}_{\boldsymbol{\theta}}Q(\boldsymbol{\theta},\boldsymbol{\theta}_{k-1})$. The resulting sequence $\\{\boldsymbol{\theta}_{k}\\}_{k\geq 0}$ will, under weak assumptions, converge to a stationary point of the likelihood $p(\mathbf{y}_{0:T}\mid\boldsymbol{\theta})$. To implement the above procedure we need to compute the integral in (12), which in general is not computationally tractable for a GP-SSM. To deal with this difficulty, we employ a Monte-Carlo-based implementation of the EM algorithm, referred to as PSAEM (Lindsten, 2013). This procedure is a combination of stochastic approximation EM (SAEM) (Delyon et al., 1999) and particle Markov chain Monte Carlo (PMCMC) (Andrieu et al., 2010; Lindsten et al., 2012). As illustrated by Lindsten (2013), PSAEM is a competitive alternative to particle-smoothing-based EM algorithms (e.g. (Schön et al., 2011; Olsson et al., 2008)), as it enjoys better convergence properties and has a much lower computational cost. The method maintains a stochastic approximation of the auxiliary quantity (12), $\widehat{Q}_{k}(\boldsymbol{\theta})\approx Q(\boldsymbol{\theta},\boldsymbol{\theta}_{k-1})$. This approximation is updated according to $\displaystyle\widehat{Q}_{k}(\boldsymbol{\theta})=(1-\gamma_{k})\widehat{Q}_{k-1}(\boldsymbol{\theta})+\gamma_{k}\log p(\mathbf{y}_{0:T},\mathbf{x}_{0:T}[k]\mid\boldsymbol{\theta}).$ (13) Here, $\\{\gamma_{k}\\}_{k\geq 0}$ is a sequence of step sizes, satisfying the usual stochastic approximation conditions: $\sum_{k}\gamma_{k}=\infty$ and $\sum_{k}\gamma_{k}^{2}<\infty$. A typical choice is to take $\gamma_{k}=k^{-p}$ with $p\in\,]0.5,1]$, where a smaller value of $p$ gives a more rapid convergence at the cost of higher variance. In the vanilla SAEM algorithm, $\mathbf{x}_{0:T}[k]$ is a draw from the smoothing distribution $p(\mathbf{x}_{0:T}\mid\mathbf{y}_{0:T},\boldsymbol{\theta}_{k-1})$. In this setting, Delyon et al. (1999) show that using the stochastic approximation (13) instead of (12) in the EM algorithm results in a valid method, i.e. $\\{\boldsymbol{\theta}_{k}\\}_{k\geq 0}$ will still converge to a maximizer of $p(\mathbf{y}_{0:T}\mid\boldsymbol{\theta})$. The PSAEM algorithm is an extension of SAEM, which is useful when it is not possible to sample directly from the joint smoothing distribution. This is indeed the case in our setting. Instead of sampling from the smoothing distribution, the sample trajectory $\mathbf{x}_{0:T}[k]$ in (13) may be drawn from an ergodic Markov kernel, leaving the smoothing distribution invariant. Under suitable conditions on the kernel, this will not violate the validity of SAEM; see (Andrieu and Vihola, 2011; Andrieu et al., 2005). In PSAEM, this Markov kernel on the space of trajectories, denoted as ${P_{\boldsymbol{\theta}}^{N}(\mathbf{x}_{0:T}^{\star}\mid\mathbf{x}_{0:T}^{\prime})}$, is constructed using PMCMC theory. In particular, we use the method by Lindsten et al. (2012), particle Gibbs with ancestor sampling (PGAS). We have previously used PGAS for Bayesian identification of GP-SSMs (Frigola et al., 2013). PGAS is a sequential Monte Carlo method, akin to a standard particle filter (see e.g. (Doucet and Johansen, 2011; Gustafsson, 2010)), but with the difference that one particle at each time point is specified _a priori_. These reference states, denoted as $\mathbf{x}_{0:T}^{\prime}$, can be thought of as guiding the particles of the particle filter to the “correct” regions of the state-space. More formally, as shown by Lindsten et al. (2012), PGAS defines a Markov kernel which leaves the joint smoothing distribution invariant, i.e. for any $\boldsymbol{\theta}$, $\displaystyle\int P_{\boldsymbol{\theta}}^{N}(\mathbf{x}_{0:T}^{\star}\mid\mathbf{x}_{0:T}^{\prime})p(\mathbf{x}_{0:T}^{\prime}\mid\mathbf{y}_{0:T},\boldsymbol{\theta})\text{d}\mathbf{x}_{0:T}^{\prime}=p(\mathbf{x}_{0:T}^{\star}\mid\mathbf{y}_{0:T},\boldsymbol{\theta}).$ (14) The PGAS kernel is indexed by $N$, which is the number of particles used in the underlying particle filter. Note in particular that the desired property (14) holds for any ${N\geq 1}$, i.e. the number of particles only affects the mixing of the Markov kernel. A larger $N$ implies faster mixing, which in turn results in better approximations of the auxiliary quantity (13). However, it has been experienced in practice that the correlation between consecutive trajectories drops of quickly as $N$ increases (Lindsten et al., 2012; Lindsten and Schön, 2013), and for many models a moderate $N$ (e.g. in the range 5–20) is enough to get a rapidly mixing kernel. We refer to (Lindsten, 2013; Lindsten et al., 2012) for details. We conclude by noting that it is possible to generate a sample $\mathbf{x}_{0:T}[k]\sim P_{\boldsymbol{\theta}[k-1]}^{N}(\,\cdot\mid\mathbf{x}_{0:T}[k-1])$ by running a particle-filter-like algorithm. This method is given as Algorithm 1 in (Lindsten et al., 2012) and is described specifically for GP-SSMs in Section 3 of (Frigola et al., 2013). Next, we address the M-step of the EM algorithm. Maximizing the quantity (13) will typically not be possible in closed form. Instead, we make use of a numerical optimization routine implementing a quasi-Newton method (BFGS). Using (11), the gradient of the complete data log-likelihood can be written as $\displaystyle\frac{\partial}{\partial\boldsymbol{\theta}}\log p(\mathbf{y}_{0:T},\mathbf{x}_{0:T}\mid\boldsymbol{\theta})=\sum_{t=0}^{T}\frac{\partial}{\partial\boldsymbol{\theta}}\log p(\mathbf{y}_{t}\mid\mathbf{x}_{t},\boldsymbol{\theta})$ $\displaystyle\hskip 10.00002pt+\sum_{t=1}^{T}\frac{\partial}{\partial\boldsymbol{\theta}}\log p(\mathbf{x}_{t}\mid\mathbf{x}_{0:t-1},\boldsymbol{\theta})+\frac{\partial}{\partial\boldsymbol{\theta}}\log p(\mathbf{x}_{0}\mid\boldsymbol{\theta}),$ (15) where the individual terms can be computed using (5c) and (9), respectively. The resulting PSAEM algorithm for learning of GP-SSMs is summarized in Algorithm 1. Algorithm 1 PSAEM for GP-SSMs 1. 1. Set $\boldsymbol{\theta}_{0}$ and $\mathbf{x}_{0:T}[0]$ arbitrarily. Set $\widehat{Q}_{0}(\boldsymbol{\theta})\equiv 0$. 2. 2. For $k\geq 1$: 1. (a) Simulate $\mathbf{x}_{0:T}[k]\sim P_{\boldsymbol{\theta}[k-1]}^{N}(\,\cdot\mid\mathbf{x}_{0:T}[k-1])$ (run Algorithm 1 in (Lindsten et al., 2012) and set $\mathbf{x}_{0:T}[k]$ to one of the particle trajectories with probabilities given by their importance weights). 2. (b) Update $\widehat{Q}_{k}(\boldsymbol{\theta})$ according to (13). 3. (c) Compute $\boldsymbol{\theta}_{k}=\operatorname{arg\,max}_{\boldsymbol{\theta}}\widehat{Q}_{k}(\boldsymbol{\theta})$. A particular feature of the proposed approach is that it performs smoothing even when the state transition function is not yet explicitly defined. Once samples from the smoothing distribution have been obtained it is then possible to analytically describe the state transition probability density (see (Frigola et al., 2013) for details). This contrasts with the standard procedure where the smoothing distribution is found using a given state transition density. ## 5 Experimental Results In this section we present the results of applying PSAEM to identify various dynamical systems. ### 5.1 Identification of a Linear System Although GP-SSMs are particularly suited to nonlinear system identification, we start by illustrating their behavior when identifying the following linear system $\displaystyle\mathbf{x}_{t+1}$ $\displaystyle=0.8\,\mathbf{x}_{t}+3\,\mathbf{u}_{t}+\mathbf{v}_{t},$ $\displaystyle\mathbf{v}_{t}\sim\mathcal{N}(0,1.5),$ (16a) $\displaystyle\mathbf{y}_{t}$ $\displaystyle=2\,\mathbf{x}_{t}+\mathbf{e}_{t},$ $\displaystyle\mathbf{e}_{t}\sim\mathcal{N}(0,1.5),$ (16b) excited by a periodic input. The GP-SSM can model this linear system by using a linear covariance function for the GP. This covariance function imposes, in a somehow indirect fashion, that the state-transition function in (16a) must be linear. A GP-SSM with linear covariance function is formally equivalent to a linear state-space model where a Gaussian prior is placed over the, unknown to us, parameters ($A=0.8$ and $B=3$) (Rasmussen and Williams, 2006, Section 2.1). The hyper-parameters of the covariance function are equivalent to the variances of a zero-mean prior over $A$ and $B$. Therefore, the application of PSAEM to this particular GP-SSM can be interpreted as finding the hyper- parameters of a Gaussian prior over the parameters of the linear model that maximize the likelihood of the observed data whilst marginalizing over $A$ and $B$. In addition, the likelihood will be simultaneously optimized with respect to the process noise and measurement noise variances ($q$ and $r$ respectively). Figure 1: Convergence of parameters when learning a linear system using a linear covariance function. Figure 2: Linear dynamical system learned using a GP-SSM with linear covariance function. Predictions (a) on training data, and (b) on test data (see text for more details). Figure 1 shows the convergence of the GP hyper-parameters ($l_{x}$ and $l_{u}$) and noise parameters with respect to the PSAEM iteration. In order to judge the quality of the learned GP-SSM we evaluate its predictive performance on the data set used for learning (training set) and on an independent data set generated from the same dynamical system (test set). The GP-SSM can make probabilistic predictions which report the uncertainty arising from the fact that only a finite amount of data is observed. Figure 2 displays the predicted value of $\mathbf{f}_{t+1}-\mathbf{x}_{t}$ versus the true value. Recall that $\mathbf{f}_{t+1}-\mathbf{x}_{t}$ is equivalent to the step taken by the state in one single transition before process noise is added: $f(\mathbf{x}_{t},\mathbf{u}_{t})-\mathbf{x}_{t}$. One standard deviation error bars from the predictive distribution have also been plotted. Perfect predictions would lie on the unit slope line. We note that although the predictions are not perfect, error-bars tend to be large in predictions that are far from the true value and narrower for predictions that are closer to the truth. This is the desired outcome since the goal of the GP- SSM is to represent the uncertainty in its predictions. Figure 3: Convergence of parameters when learning a linear system using a squared exponential covariance function. Figure 4: Linear dynamical system learned using a GP-SSM with squared exponential covariance function. Predictions (a) on training data, and (b) on test data. We now move into a scenario in which the data is still generated by the linear dynamical system in (16) but we pretend that we are not aware of its linearity. In this case, a covariance function able to model nonlinear transition functions is a judicious choice. We use the squared exponential covariance function which imposes the assumption that the state transition function is smooth and infinitely differentiable (Rasmussen and Williams, 2006). Figure 3 shows, for a PSAEM run, the convergence of the covariance function hyper-parameters (length-scales $\lambda_{x}$ and $\lambda_{u}$ and signal variance $\sigma_{f}$) and also the convergence of the noise parameters. The predictive performance on training data and independent test data is presented in Figure 4. Interestingly, in the panel corresponding to training data (a), there is particularly poor prediction that largely underestimates the value of the state transition. However, the variance for this prediction is very high which indicates that the identified model has little confidence in it. In this particular case, the mean of the prediction is 2.5 standard deviations away from the true value of the state transition. ### 5.2 Identification of a Nonlinear System Figure 5: Nonlinear dynamical system with one state and one input. The black mesh represents the ground truth dynamics function and the colored surface is the mean of the identified function. Color is proportional to the standard deviation of the identified function (red represents high uncertainty and blue low uncertainty). Figure 6: State trajectory from a test data set (solid black line). One step ahead predictions made with the identified model are depicted by a dashed line (mean) and a colored interval at $\pm 1$ standard deviation (including process noise). GP-SSMs are particularly powerful for nonlinear system identification when it is not possible to create a parametric model of the system based on detailed knowledge about its dynamics. To illustrate this capability of GP-SSMs we consider the nonlinear dynamical system $\displaystyle\mathbf{x}_{t+1}$ $\displaystyle=a\mathbf{x}_{t}+b\frac{\mathbf{x}_{t}}{1+\mathbf{x}_{t}^{2}}+c\mathbf{u}_{t}+\mathbf{v}_{t},$ $\displaystyle\mathbf{v}_{t}\sim\mathcal{N}(0,q),$ (17a) $\displaystyle\mathbf{y}_{t}$ $\displaystyle=d\mathbf{x}_{t}^{2}+\mathbf{e}_{t},$ $\displaystyle\mathbf{e}_{t}\sim\mathcal{N}(0,r),$ (17b) with parameters $(a,b,c,d,q,r)=(0.5,25,8,0.05,10,1)$ and a known input $\mathbf{u}_{t}=\cos(1.2(t+1))$. One of the challenging properties of this system is that the quadratic measurement function (17b) tends to induce a bimodal distribution in the marginal smoothing distribution. For instance, if we were to consider only one measurement in isolation and $r=0$ we would have $\mathbf{x}_{t}=\pm\sqrt{\frac{\mathbf{y}_{t}}{d}}$. Moreover, the state transition function (17a) exhibits a very sharp gradient in the $\mathbf{x}_{t}$ direction at the origin, but is otherwise parsimonious as $\mathbf{x}_{t}\rightarrow\pm\infty$. Again, we pretend that detailed knowledge about the particular form of (17a) is not available to us. We select a covariance function that consists of a Matérn covariance function in the $\mathbf{x}$ direction and a squared exponential in the $\mathbf{u}$ direction. The Matérn covariance function imposes less smoothness constraints than the squared exponential (Rasmussen and Williams, 2006) and is therefore more suited to model functions that can have sharp transitions. Figure 5 shows the true state transition dynamics function (black mesh) and the identified function as a colored surface. Since the identified function from the GP-SSM comes in the form of a probability distribution over functions, the surface is plotted at $\mathbb{E}[\mathbf{f}^{}\|\mathbf{x}^{},\mathbf{u}^{},\mathbf{y}_{0:T}]$ where the symbol ∗ denotes test points. The standard deviation of $\mathbf{f}^{}$, which represents our uncertainty about the actual value of the function, is depicted by the color of the surface. Figure 6 shows the one step ahead predictive distributions $p(\mathbf{x}_{t+1}^{}\|\mathbf{x}_{t}^{},\mathbf{u}_{t}^{},\mathbf{y}_{0:T})$ on a test data set. ## 6 Conclusions GP-SSMs allow for a high degree of flexibility when addressing the nonlinear system identification problem by making use of Bayesian nonparametric system models. These models enable the incorporation of high-level assumptions, such as smoothness of the transition function, while still being able to capture a wide range of nonlinear dynamical functions. 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In Yee Whye Teh and Mike Titterington, editors, _13th International Conference on Artificial Intelligence and Statistics_ , volume 9 of _W &CP_, pages 868–875, Chia Laguna, Sardinia, Italy, May 13–15 2010\.	arxiv-papers	2013-12-17T16:32:54	2024-09-04T02:49:55.572587	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Roger Frigola, Fredrik Lindsten, Thomas B. Sch\\\"on, Carl E. Rasmussen", "submitter": "Roger Frigola", "url": "https://arxiv.org/abs/1312.4852" }
1312.5013	A Roadmap for Canadian Submillimetre Astronomy _Authors:_ Tracy Webb, Scott Chapman, James Di Francesco, Brenda Matthews, Norm Murray, Douglas Scott, Christine Wilson ###### Contents 1. 1 Executive Summary 2. 2 Submillimetre/millimetre Astronomy 3. 3 SMM in the Context of Canadian LRPs 4. 4 The Submillimetre/Millimetre User Community in Canada 5. 5 The Present SMM Landscape in Canada 1. 5.1 The Atacama Large Millimetre/submillimetre Array 2. 5.2 James Clerk Maxwell Telescope 1. 5.2.1 The JCMT Legacy Survey 2. 5.2.2 PI-led Projects 3. 5.3 Herschel Space Observatory 4. 5.4 Green Bank Telescope and Jansky Very Large Array 5. 5.5 Focused PI Experiments 6. 6 The Future: Commitments and Possibilities 1. 6.1 ALMA 2. 6.2 CCAT 3. 6.3 JCMT 4. 6.4 GBT and JVLA 5. 6.5 SPICA 6. 6.6 Focused PI Experiments 1. 6.6.1 The E and B Experiment 2. 6.6.2 Spider 3. 6.6.3 Super BLAST-Pol 4. 6.6.4 Event Horizon Telescope 7. 6.7 Synergies with Other Facilities 7. 7 A Plan for the Next Decade 1. 7.1 Recommendations 2. 7.2 Funding Considerations 1. 7.2.1 JCMT 2. 7.2.2 CCAT 3. 7.2.3 SPICA and PI Projects ## 1 Executive Summary Submillimetre/millimetre (SMM) wavelengths are a key resource in the study of our cosmic origins. From the Cosmic Microwave Background through starburst galaxies to debris disks, submillimetre observations probe the formation of structure in the universe, on all scales. What was once seen as a niche science with limited technological capabilities has now become a mature field. SMM observations are now a crucial component of essentially all sub-areas of astronomy, and are gathered by a suite of facilities with complementary strengths. Canada holds a long history of excellence and expertise in SMM astronomy, supported not only through continued access to cutting-edge facilities but by strong involvement at the end-to-end science level. The ALMA facility is still under construction but is already the most powerful SMM interferometer in the world, and Canadian scientists are competing strongly for ALMA time through a growing user community (doubling from 23 individuals in Cycle 0 to 55 in Cycle 1). The expected withdrawal of Canada from the James Clerk Maxwell Telescope, and the current lack of funding for its obvious successor CCAT, leaves the Canadian astronomy community at a critical cross-roads. We must continue to invest in SMM facilities at a level that allows Canadians to maintain our position as a world-leader in an important area of science. Historically, Canadians have enjoyed access to the largest, best equipped SMM single dish in the world. Without such continued access, Canadian scientists will lack a key facility for the study of the SMM universe – one fundamental to their future success with ALMA. This document grew from the discussions about the future of SMM astronomy in Canada during a meeting of interested individuals in the SMM community in early 2012. Here we summarize the observational landscape of SMM astronomy in Canada from now to $\sim$2020\. The plans and priorities of the SMM community have been discussed extensively in the reports of the Canadian astronomy Long Range Plan panels. A number of recent changes, particularly the demonstrable success of SCUBA- 2 that led to a resurgence in the JCMT user community (tripling from 14 individuals on PI proposals in semester 11B to 49 in semester 12B), necessitate a revision. To maintain the spirit and primary recommendations of the LRP reports, Canada must remain flexible and ready to respond appropriately to current realities and new opportunities. We argue that because of Canada’s substantial investment in ALMA and numerous PI-led SMM experiments, continued involvement in a large single-dish facility is crucial. In particular, we recommend: (i) an extension of Canadian participation in the JCMT until at least the unique JCMT Legacy Survey program is able to realize the full scientific potential provided by the world-leading SCUBA-2 instrument; (ii) involvement of the entire Canadian community in CCAT, with a large enough share in the partnership that Canadian astronomers can significantly participate at all levels of the facility (decisions, construction, and science). We further recommend continued participation in ALMA development, involvement in many focused PI-led SMM experiments, and partnership in SPICA. We close with an outline of the expected costs of these recommendations and options for funding. ## 2 Submillimetre/millimetre Astronomy Submillimetre/millimetre (SMM) wavelengths encompass the wide range between $\sim$200 $\mu$m and $\sim$10 mm. Over the last three decades, SMM astronomy has grown from a field studying small numbers of near-by objects, to a science capable of undertaking comprehensive investigations of all aspects of the universe, with data quality comparable to what was traditionally only reached with optical wavelengths. The strength and importance of SMM astronomy is extensively outlined elsewhere (such as the SCUBA-2 Legacy Survey Proposals or the CCAT Science Documents), but for completeness we highlight some important aspects here. SMM radiation can be broadly characterized as the light of our cosmic origins. At its lowest frequencies, it probes the very nature of the universe through the primary Cosmic Microwave Background, and the additional imprints of foreground matter in the form of secondary anisotropies (galaxy clusters, gravitational lensing etc). Locally, SMM observations are the work-horses of astronomers seeking to understand the very basics of planet and star formation. In between these two extremes, SMM measurements are diagnostics of the most energetic and dynamic phases of galaxy formation and evolution - the epochs of stellar mass assembly and supermassive black hole growth. As SMM radiation is sandwiched between radio and infrared frequencies, specialized instruments incorporating both optical and radio techniques are required. Moreover, the wavelength range actually observable depends strongly on atmospheric conditions, typically precipitable water vapour content, and some SMM wavelengths are simply inaccessible from the ground from even the driest terrestrial sites. This has resulted in a number of technical challenges which traditionally limited the field, but are now being routinely overcome. SMM measurements can be generally divided into three distinct categories which provide complementary information. We briefly outline these below and highlight the primary scientific uses of each. Continuum Emission: SMM wavelength continuum observations trace extended emission from dust particles within the interstellar medium (ISM). This dust is warmed by nearby hot sources (e.g., young stars) and the interstellar radiation field, and they cool through thermal emission. Therefore, this emission traces the thermal balance in the ISM. In addition, the emission is generally optically thin, so the observed intensity is proportional to the column density and temperature of the emitting dust. Hence, SMM continuum emission is an excellent tracer of mass in the ISM. For example, protostellar or debris disks contain dust warmed by central stellar objects, and these objects glow brightly at SMM wavelengths. Moreover, the dense predecessors to star formation, dense cores, can be well traced by the thermal SMM emission of its dust. SMM continuum emission can also trace galactic structure, such as the dusty spiral arms of galaxies or the interaction zones of colliding galaxies. Moreover, the earliest galaxies are heavily enshrouded by dust, and SMM continuum emission can trace the vigorous star formation rates within high redshift galaxies. Beyond thermal emission from dust, SMM continuum emission can also trace the cosmic microwave background (CMB) and specifically its anisotropies that trace conditions in the early universe. Furthermore, localized deficits or enhancements in the CMB toward galaxy clusters can be caused by the Sunyaev-Zel’dovich Effect, and galaxy cluster masses can be so probed using SMM continuum observations. At the longer SMM wavelengths, i.e., $\geq$ 1 mm, continuum emission may also include significant amounts of free- free radiation from ionized material, e.g., shocked gas in protostellar outflows. Transitions from Molecular or Atomic Gas: Spectral lines are also found at SMM wavelengths, and these provide very important views of galactic and extragalactic environments. For example, the SMM range includes the rest wavelengths of numerous molecular rotational transitions that are excited by conditions found in the ISM (see Figure 1). Since profiles of lines inherently trace velocity structure along a given direction, lines can be used to probe the kinematics and dynamics of gas within the ISM. Moreover, given the strong dependence of transitions on specific excitation conditions, the lines also trace such conditions throughout the ISM. For example, observations of rotational line emission from CO (or its heavier isotopologues) trace the cold components of disks, as well as structure of molecular clouds and their dense cores throughout the Galaxy. Furthermore, molecular lines can trace directly the physical conditions and kinematics of both nearby galaxies and distant objects at high redshift. (In addition, the latter objects can have key atomic lines like the ones of [OI] at 63 $\mu$m, [CII] at 158 $\mu$m, or [NII] at 205 $\mu$m, shifted into the SMM range, allowing expanded probes of their ISMs.) The relative brightnesses of the transitions of several molecular species is a also key probe into the chemistry of the ISM, with the relative abundance of various molecules playing an important role in its thermal balance. Figure 1: Schematic representation of some of the spectral content at SMM wavelengths toward an interstellar cloud (Phillips & Keene 1992). Polarization from Continuum or Line Emission: SMM continuum emission and some lines can also trace magnetic fields in the ISM. The roles magnetic fields play in the dynamical evolution of the ISM are poorly constrained and SMM data remain among the most effective means to trace them. For continuum observations, thermal emission from dust can be measurably (linearly) polarized when magnetic fields preferentially align the orientations of the emitting grains. These data can reveal the magnetic field strengths in the plane of the sky. In addition, polarization of the CMB continuum emission can be used to probe the imprint of gravitational waves during the earliest moments of the universe. For certain line observations, the emission from some molecules (e.g., CCS, CN) can be measurably split and (circularly) polarized due to the Zeeman effect, i.e., the rotational energy levels are subtly shifted in presence of magnetic fields. These data probe the magnetic field strengths along the line of sight. Importantly, SMM continuum and line observations trace denser and more compact structures than are possible to observe using lower-frequency radio observations, e.g., using HI or OH lines. As with any other wavelength regime, SMM astronomy requires different facilities and instruments to achieve different scientific goals. No one instrument or facility can meet the needs of all the lines of inquiry available. Indeed, numerous observatories with diverse instrumentation have been built over the last three decades to exploit the key information residing within SMM wavelengths. Single-dish facilities like the James Clerk Maxwell Telescope (JCMT) and the Green Bank Telescope (GBT) provide the larger-scale view of the SMM universe. (Of special note are airborne and spaceborne observatories like the Balloon-borne Large Aperture Submillimetre Telescope (BLAST), its successor BLAST-Pol, and the Herschel Space Observatory, which trace submm wavelengths unobservable from the ground.) Such facilities, however, are diffraction limited in this wavelength range, and achieving high- resolution images (e.g, subarcsecond) requires interferometric techniques. Such observatories, including now the Atacama Large Millimetre/submillimetre Array (ALMA) and the Karl G. Jansky Very Large Array (JVLA), all serve at present to provide the smaller-scale view. By necessity, however, these data come with intrinsic trade-offs in sensitivity to total flux and larger-scale structures. The complementary nature of single-dish and interferometers underscores the need for access to both types of facility (Indeed, in certain cases, data from each may be combined to provide highly detailed images with information on a wide range of spatial scales.) With over 25 years of access to the JCMT, Canadian astronomers have developed a worldwide reputation as leaders in SMM research. They have recently had access to JCMT, GBT, JVLA, ALMA, BLAST, Planck, and Herschel, enabling high- impact astronomical research at the world’s top SMM observatories. If Canada is to maintain its leadership in astronomical research, however, it cannot do so without continued access to, and support of, the best SMM facilities and instrumentation available worldwide. In the view of the SMM community, this includes BOTH single-dish and interferometric facilities. In this document, we survey the present and future for Canadian SMM facilities, to provide a vision of continued leadership (i.e., a “roadmap”) for Canada in this very important wavelength regime. ## 3 SMM in the Context of Canadian LRPs The Long Range Plan (LRP) series has served Canada well by using community input to outline the priorities and directions of Canadian astronomy for the following 10-15 years. The most recent version, LRP2010, recommended Canadian investment in 13 new facilities, with a total estimated cost of $550M over ten years. The priority rankings of these facilities were organized according to small, medium, and large cost scales, and between space- and ground-based projects. In LRP2010, the highest priority identified for a large-scale, ground-based project was Canadian participation in the optical Thirty Meter Telescope (TMT) project. While final construction budgets are not yet known, TMT is expected to cost at least $\sim$$1.2B, with Canada contributing 25% (or $305M). Once TMT is constructed by the end of the decade, the Square Kilometer Array (SKA) will become Canada’s highest ground-based priority. In the medium-scale category, LRP2010 recommended three projects, at $15M each, one of which – CCAT – is a new SMM facility. The other two include the radio Canadian Hydrogen Intensity Mapping Experiment (CHIME - now funded through a CFI grant) and upgraded instruments on the optical Canada-France-Hawaii Telescope (CFHT). The top two small-scale projects ($<$ $5M) are an arctic-based optical facility and a study for the Next Generation CFHT (ngCFHT). For space-based projects, LRP2010 gave highest priority to Canadian participation in a large-scale dark energy mission of some kind, such as the European Space Agency (ESA) Euclid project, at $100M. Recommended medium- scale, space-based projects included participations in the NASA-ESA International X-ray Observatory (IXO) and the follow-up to Herschel, the Japanese-led far-infrared/submm Space Infra-Red Telescope for Cosmology and Astrophysics (SPICA). LRPs summarize a set of recommendations and guidelines, identified at the beginning of a decade, based on scientific aims. As noted carefully in the LRP2010 report, however, “these are provisional rankings and must be qualified: they are on the basis of science promise and/or long-term potential impact only. All of these projects lack a thorough feasibility study, technical review, and cost analysis…” As new information or opportunities arise, a flexible plan is needed to maximize the scientific return of our current investments. For example, although LRP2010 reiterated phasing out Canadian participation in the JCMT as funding for ALMA ramps up, this recommendation was crucially tied to the then-unknown performance of the flagship SCUBA-2 instrument on JCMT. The report to the LRP2010 panel by the Canadian Astronomical Society’s Ground Based Astronomy Committee (GAC), recommended the transition from the JCMT to ALMA should be done in a timely manner. Since the publication of LRP2010, SCUBA-2 performance has been determined to be of very high quality, and the subsequent response by the Canadian community to use SCUBA-2 has been overwhelmingly positive (see §4.1 below). As such the JCMT remains an extremely relevant facility today. In parallel, Canadian interest in CCAT remains strong, following the recommendation by LRP2010. Indeed, present Canadian participation in CCAT is due to a strong grass-roots effort to raise the initial funds at the university level. The funding source for the minimum contribution to CCAT ($20M - as required by the telescope consortium) has not yet been identified, though the project is moving forward on schedule and predicted to reach first light in 2018. Similarly, SPICA, though targeted for launch in 2022, has not yet been approved by the Japanese Space Agency (JAXA). ## 4 The Submillimetre/Millimetre User Community in Canada The SMM facility user-community is a significant demographic within the larger astronomy community in Canada. It is difficult to objectively and uniquely quantify the size of the community since it encompasses observational astronomers working in multiple wavelength regimes, theorists, and experimentalists building PI-lead instruments. We therefore turn to the user rates of Canada’s forefront facilities as an objective, if incomplete, metric. Table 1 shows the involvement level of Canadian astronomers in ALMA s Cycle 0 (mid-2011) and Cycle 1 (mid-2012) calls for proposals. The fraction of allocated programs led by Canadian PIs rose from 2.7% to 3.1% between Cycle 0 and Cycle 1, but more significantly, the number of Canadian astronomers (i.e., those at Canadian institutions) involved in ALMA proposals worldwide doubled from 23 to 55 individuals, likely arising from the improved capabilities of ALMA from one cycle to the next. Table 1: Canadian ALMA User Statistics Cycle \| Total \| Canadian \| Total \| Allocated to \| Success ---\|---\|---\|---\|---\|--- \| Worldwide \| Lead \| Canadians \| Canadian \| Rate of \| Proposals \| Proposals \| Involved \| PIs \| Canadian PIs 0 \| 924 \| 24 (2.6%) \| 23 \| 3 of 112 (2.7%) \| 3 of 24 (12.5%) 1 \| 1133 \| 26 (2.3%) \| 55 \| 6 of 196 (3.1%) \| 6 of 26 (23%) A comparison of the numbers of unique Canadian users applying for PI time on Canada’s three major off-shore single-aperture facilities shows that the number of JCMT users exhibited significant fluctuation over the four semesters tracked (Table 2). The large increase in unique users in semester 12A can be attributed to the availability of the SCUBA-2 instrument on the telescope. The number of users applying for time in 12B is three times the number for 11B when SCUBA-2 was not offered (and is larger than for the CFHT in the same semester). Note, however, that these numbers do not take into account users of JCMT through the JCMT Legacy Survey who are not involved in PI programs in a given semester. Table 2: Numbers of Unique Canadian Investigators on Proposals to Canada’s Three Major Single-Aperture Off-Shore Facilities Facility \| 11A \| 11B \| 12A \| 12B \| 13A ---\|---\|---\|---\|---\|--- JCMT \| 15 \| 14 \| 31 \| 49 \| 48 CFHT \| 53 \| 39 \| 44 \| 47 \| 54 Gemini \| 48 \| 58 \| 46 \| 71 \| 85 Figure 2: Participants of the NRC (Sub)Millimetre Observing Techniques Summer Workshop held July 2006 in Victoria, BC. ## 5 The Present SMM Landscape in Canada ### 5.1 The Atacama Large Millimetre/submillimetre Array The Atacama Large Millimeter/submillimeter Array (ALMA) is the first billion- dollar ground-based astronomical telescope. It is an SMM interferometer operating on the Atacama desert of northern Chile. ALMA was identified as Canada’s first priority for ground-based facilities in the LRP2000 document, and Canada formally joined the project in 2003 through the North American Partnership for Radio Astronomy (NAPRA). ALMA is presently a collaboration between three partner regions, North America (US, Canada, and Taiwan), Europe (the member states of ESO), and East Asia (Japan and Taiwan, again) and the host country, the Republic of Chile. After years of planning and construction, ALMA is finally nearing completion. Even in its present form, however, ALMA is already the most sensitive SMM interferometer on the planet. It will remain Canada’s most powerful shorter wavelength SMM facility for many years. Figure 3: Four models of ALMA antennas on the Chajnantor plateau. (Credit: ALMA (ESO/NAOJ/NRAO), W. Garnier (ALMA) When complete, ALMA will consist of fifty 12-m antennas, as well as twelve 7-m antennas and four 12-m antennas specifically designed for compact configuration and total power observing, respectively. Each antenna is equipped with a suite of single-pixel receivers that operate over specific wavelength bands (see Table 3). ALMA observations are made with one receiver per telescope, though in principle ALMA could be divided into four subarrays of telescopes using different receivers simultaneously. Received signals are sent to one of two correlators that process the data into continuum data or spectral line data cubes. Table 3: ALMA Receiver Bands Band \| $\lambda\lambda$ (mm) \| Availability ---\|---\|--- 1 \| 6 – 8.5 \| under development 2 \| 3.3 – 4.5 \| under development 3 \| 2.6 – 3.6 \| Cycle 0 onwards 4 \| 1.8 – 2.4 \| Cycle 2 onwards 5 \| 1.4 – 1.8 \| under development 6 \| 1.1 – 1.4 \| Cycle 0 onwards 7 \| 0.8 – 1.1 \| Cycle 0 onwards 8 \| 0.6 – 0.8 \| Cycle 2 onwards 9 \| 0.4 – 0.5 \| Cycle 0 onwards 10 \| 0.3 – 0.4 \| Cycle 2 onwards ALMA has already had two “Early Science” proposal calls to use its Band 3, 6, 7, and 9 receivers. Cycle 0 observations, using a minimum of sixteen 12-m antennas, took place throughout 2012. Cycle 1 observations, using a minimum of thirty-two 12-m antennas (and, if needed, nine 7-m compact configuration antenna and two 12-m total power antennas) began in early 2013. During the Early Science phase, scientific time at the telescope is still being tensioned with ongoing construction, and all observations are only expected on a best- effort basis. Cycle 2 observations are expected to be part of the Full Science phase, using all 66 antennas and Bands 3, 4, 6, 7, 8, 9, and 10, and these will likely begin in 2014. (NB: Bands 1, 2, and 5 are under development.) Through NAPRA, Canadian astronomers have access to the 33.75% of ALMA time available to North American astronomers, with no specific percentage of time defined as Canadian. In comparison, Canada provides 7.25% of the cost of North American ALMA operations. Demand for ALMA time has been extraordinarily high. For Cycle 0 (mid-2011), 924 proposals in total were submitted for $\sim$700 hours of observing time, with 24 led by PIs at a Canadian institution. Only the top 112 proposals ($\sim$12%) received “high priority” status, including three by Canadian PIs. For Cycle 1 (mid-2012), 1133 proposals were submitted for $\sim$800 hours of observing time, with 26 led by Canadian PIs. (Including co-Is, 55 individuals from 14 Canadian institutions were involved in Cycle 1 proposals.) Only the top 196 proposals ($\sim$17%) received “high priority” status, including six by Canadian PIs. These numbers show that Canadians have to date competed successfully for ALMA time at levels just above that of our monetary contribution. Most importantly, over two proposal cycles, the number of successful proposals from PIs at Canadian institutions has increased. Cycle 2 proposals are expected to be due in September 2013. ### 5.2 James Clerk Maxwell Telescope The James Clerk Maxwell Telescope (JCMT), located on the dry summit of Mauna Kea, has been Canada’s primary access to SMM wavelengths for 25 years. At 15-m diameter, JCMT remains the world’s largest present single-dish SMM facility. Historically, the JCMT has been a partnership between the UK (55%), Canada (25%) and the Netherlands (20%). (The University of Hawaii receives 10% of time as the host institution.) The JCMT has been managed by the Joint Astronomy Centre in Hilo, Hawaii together with the United Kingdom InfraRed Telescope (UKIRT). Figure 4: The James Clerk Maxwell Telescope. (Credit: Nik Szymanek) The instrumentation suite at JCMT has made a very significant scientific impact over its lifetime, mostly due to SCUBA, the first submillimetre bolometric “camera.” In the early 2000s when SCUBA was in its prime, JCMT was considered top of its class. JCMT’s instrumentation has all been updated or completely replaced in the last 5-10 years and remains relevant and cutting- edge. The current suite consists of: $\bullet$ SCUBA-2 \- an $\sim$8′$\times$8′-wide 850 $\mu$m and 450 $\mu$m bolometric camera; $\bullet$ HARP \- a $\sim$2′$\times$2′-wide 4 $\times$ 4 325-375 GHz heterodyne receiver array; $\bullet$ RxW \- two single-pixel receivers at 315-375 GHz and 630-710 GHz; and $\bullet$ RxA \- a single-pixel receiver at 211-272 GHz. The fabrication of SCUBA-2 was a collaborative effort between the UK, JAC and several Canadian partners. The Canadian effort involved the warm electronics and data reduction software and was funded largely through a grant from the Canadian Fund for Innovation. Now fully on-line, it produces high-quality continuum images of unprecedented sensitivity at 850 $\mu$m and 450 $\mu$m. HARP uniquely produces stunning wide-field data cubes of similar resolution to SCUBA-2 at 850 $\mu$m. The remaining receivers, RxW and RxA, are older, but they still provide very good scientific return when conditions warrant. Signals received by all three heterodyne JCMT instruments are fed to ACSIS, the digital autocorrelator spectrograph built in Canada by NRC-HIA. Currently, JCMT users access the telescope through the JCMT Legacy Survey (65% of time) and PI time (35%). The strong emphasis on Legacy science is viewed favourably by the wider community. Ample amounts of PI time, however, allow for the follow up of Legacy Survey discoveries, exploration of science not included in the Legacy Surveys, and utilization of SCUBA-2’s ancillary instruments: POL-2, a polarimeter, and FTS-2, a Fourier Transform spectrometer, both of which were built by Canada (Montreal and Lethbridge, respectively). In the following, we explore further both modes of JCMT access. #### 5.2.1 The JCMT Legacy Survey With its latest wide-field instrumentation, HARP and SCUBA-2, the JCMT is able for the first time to conduct very large surveys. Recognizing this opportunity, a new time allocation model was adopted, whereby a majority of UK, Canada, and Dutch observing time was set aside for the large tri-national JCMT Legacy Survey (JLS). Seven components of the JLS were approved in 2005 after extensive peer review, including cosmology (CLS), nearby galaxies (NGS), the Galactic plane (JPS), a spectral line survey of star-forming objects (SLS), nearby star formation (GBS), debris disks around nearby stars (SONS), and an “all-sky” survey (SASSy). JLS HARP observing began in late 2007 and is effectively complete. JLS SCUBA-2 observing began in late 2011 and is ongoing. Canadians are heavily invested in all components of the JLS. Canadian researchers have leadership roles (Co-Principal-Investigators) on all components and the fraction of Canadian participants on each is 20-40%. Most of the JLS time requested was intended for SCUBA-2 observations. In fall 2011, the JLS components needing SCUBA-2 data submitted re-scoped proposals for a new round of peer review, using the actual on-sky performance of SCUBA-2. This process was driven by the limited time available on the telescope, given the impending end of the agreement to operate the JCMT, and not by a lack of scientific interest; completing the peer-reviewed goals of the original JLS would take several times longer than the time now available. The proposals were granted a total of 3490 hours (291 nights) over two years, divided among various weather bands (see Table 4). For comparison, the previous JLS allocation of time with HARP was 962 hours. Table 4: SCUBA-2 Hour Allocations to JLS Components by Weather Band Component \| Band 1 \| Band 2 \| Band 3 \| Band 4 ---\|---\|---\|---\|--- CLS \| 629 \| 695 \| 454 \| GBS \| 70 \| 342 \| \| JPS \| \| \| 450 \| NGLS \| \| 100 \| \| SASSy \| \| \| \| 480 SONS \| \| 135 \| 135 \| #### 5.2.2 PI-led Projects Significant JCMT time remains for PI-led projects. Historically, Canadian use of the JCMT was oversubscribed by factors of $\sim$3 in the early- to mid-2000’s. In the period between SCUBA and SCUBA-2, the oversubscription rate peaked at $\sim$5 in semester 06B but declined to 1.0 in semesters 11A and 11B. (Note, however, that no continuum instrument was available at that time and many members of the Canadian community were engaged in HARP aspects of three JLS components.) Now with access to SCUBA-2, Canadian PI interest in JCMT has returned, though amounts of PI time available are reduced due to the large JLS allocation. Indeed, when the JLS components were re-scoped in fall 2011, some aspects (e.g., certain sky coverages) became impossible to complete and this time was made available back to the community. In semesters 12A-13A, the oversubscription rate was on average 5.9111The oversubscription rates at JCMT for semesters 12A, 12B, and 13A were 5.6, 8.1, and 3.9, respectively.. These numbers clearly indicate the strong interest in SCUBA-2, the other instrumentation, and JCMT in general in PI-led science. Though the JLS was defined to address a wide range of astrophysical phenomena, it is critical for the community to have access to the JCMT for smaller, focused projects. For example, such small projects can respond to scientific developments occurring since the JLS was defined (or re-scoped). Hence, a healthy share of JCMT time should remain for PI-led projects. ### 5.3 Herschel Space Observatory The Herschel Space Observatory deserves special mention in the current Canadian SMM landscape. This ESA-led mission consists of a 3.5-m SMM telescope located at the Sun-Earth L2 point. Herschel was designed to observe SMM wavelength ranges impossible to observe from the ground due to strong atmospheric absorption. Herschel has three instruments: $\bullet$ HIFI \- a heterodyne spectrometer that can observe frequencies over ranges of 480-1250 GHz and 1410-1910 GHz; $\bullet$ SPIRE \- a photometric array with FTS capabilities that can observe at 250 $\mu$m, 350 $\mu$m, and 500 $\mu$m; and $\bullet$ PACS \- a photometric array with grating spectrometer capabilities that can observe at 70 $\mu$m, 100 $\mu$m, and 160 $\mu$m. Canada, through the Canadian Space Agency (CSA), contributed to the construction of the first two of these instruments, through support of efforts at U. Waterloo (M. Fich) and U. Lethbridge (D. Naylor). Herschel was launched successfully by ESA in May 2009 and is expected to run out of cryogens and become inoperable in February-March 2013. Herschel’s time was divided between $\sim$33% Guaranteed Time provided to the consortia that built its three instruments and $\sim$66% Open Time to the world community. Both allocations were further divided into Key Projects requiring $>$100 hours and smaller Regular Projects. Canadians participated in Herschel projects within the Guaranteed Time and Open Time allocations, at both the Key and Regular Project levels. (The former projects were made possible through agreements between CSA and ESA.) Figure 5: The Herschel Space Observatory. (Credit: ESA / AOES Medialab, background: Hubble Space Telescope image (NASA/ESA/STScI) Herschel’s scientific output has been impressive, producing wide far-infrared and submm continuum maps at resolutions similar to those obtained previously from the ground and spectral line data cubes at THz frequencies of unparalleled sensitivity. Exploitation of Herschel data by Canadians is ongoing but all Herschel data will reside for all future users worldwide in raw and pipeline-reduced forms in an ESA-managed archive. In some ways, Herschel data can be used as a pathfinder for future ALMA observations but the two facilities only overlap at the shortest wavelengths ALMA can observe, Bands 9 and 10. Given the relatively small size of Herschel’s aperture, its data cannot be combined directly with ALMA data at overlap wavelengths. No other SMM space-based observatories have been planned worldwide besides the not-yet-approved, Japanese-led SPICA mission slated to launch in 2022 (see §5.5). Figure 6: The NRAO Robert C. Byrd Green Bank Telescope. (Image courtesy of NRAO/AUI) ### 5.4 Green Bank Telescope and Jansky Very Large Array The Robert C. Byrd Green Band Telescope (GBT) and Karl G. Jansky Very Large Array (JVLA) are longer-wavelength SMM/radio facilities operated by NRAO in the United States. The GBT is a 100-m diameter fully steerable single-dish antenna located at Green Bank, West Virginia, equipped with three single-pixel receivers that reach different bands over the longest wavelengths of the SMM range, 3 mm and 7-10 mm, and MUSTANG, a bolometer array that can observe continuum emission at $\sim$3 mm. The Jansky Very Large Array (JVLA) is an expansive retrofit of the Very Large Array interferometer located near Socorro, New Mexico. As with the VLA before, the JVLA consists of twenty-seven 25-m diameter antennas with each antenna equipped with a suite of heterodyne receivers that include wavelengths of 7-10 mm. Indeed, the GBT’s and JVLA’s abilities to observe wavelengths of $\sim$3 mm and 7-10 mm, and Canada’s access to them through the NAPRA, demand their inclusion here. Though the GBT has an impressive aperture, its site is not ideal for mm observations. It can be sensitive when conditions are right but its efficiency is much greater at longer (radio) wavelengths. The JVLA is a big improvement over the older VLA in terms of aperture efficiencies and wavelength coverage. The heart of the JVLA’s profound sensitivity improvement is, however, its new wide-band WIDAR correlator, which was built by NRC-HIA. (This contribution enabled Canada in part to join the ALMA project; see below.) The JVLA site is better than the GBT site for long-wavelength SMM observations. Figure 7: The NRAO Karl G. Jansky Very Large Array. (Image courtesy of NRAO/AUI) The GBT presently operates in Full Science mode. The JVLA is moving from an Early Science phase to a Full Science facility as its vast number of correlator modes are commissioned. Through NAPRA, Canada does not provide funds for the operations of the JVLA or GBT. It has no fixed share of GBT or JVLA time and all Canadian proposals are reviewed in the same manner as all others. At present, the GBT and JVLA provide for Canada premier access to high-resolution continuum and line emission observations at longer SMM wavelengths. (No data about the extent of Canadian participation in recent GBT or JVLA proposal rounds have been compiled for this document.) The future of the GBT is unclear at present; in August 2012, the US National Science Foundation (NSF) AST Portfolio Review Committee (PRC) recommended that in the face of flat or declining NSF budgets, the NSF divest itself from its present support of the GBT so funds can be reallocated to new, high-priority astronomy projects. The JVLA’s future appears more secure, however; it was not targeted for NSF divestment by the PRC. The JVLA is presently the most sensitive facility on the planet over the wavelength range of 7-10 mm (and most radio wavelengths, too). Canada is presently in a partnership with Taiwan, Japan, Chile, and the US to develop Band 1 receivers for ALMA, which will enable $\sim$6-8.5 mm observations from the southern hemisphere at sensitivities moderately improved over those possible with the JVLA. Table 5: Current PI Experiments involving Canadian Researchers facility \| aperture \| location \| focus \| access ---\|---\|---\|---\|--- ACT \| 6m \| Atacama, Chile \| CMB / CMB pol \| closed APEX-SZ \| instrument \| – \| SZ effect/CMB \| closed BLAST \| \| balloon/SP \| CMB/SF \| closed BLAST-Pol \| \| balloon/SP \| SF pol \| closed FTS-2 \| instrument \| JCMT \| broad \| open HIFI \| instrument \| Herschel \| broad \| open POLARBEAR \| 4m \| Atacama, Chile \| CMB pol \| closed1 POL-2 \| instrument \| JCMT \| broad \| open SPT \| 10m \| South Pole \| CMB \| closed2 SCUBA-2 \| instrument \| JCMT \| broad \| open SPIRE-FTS \| instrument \| Herschel \| broad \| open 1 open for non-CMB science 2 closed for now ### 5.5 Focused PI Experiments Though aforementioned facilities are open to the entire Canadian community, some members have access to smaller SMM telescopes that are typically led by PIs and focused on specific experiments. These projects often produce survey- style datasets that address certain goals but these products may be suitable for a host of other unanticipated uses. In Table 5, we outline those focused PI-led SMM facilities that currently include Canadian participation. Unless otherwise noted, these facilities operate as consortia closed to external collaboration, though the data products likely enter the public domain after some period. Figure 8: BLAST and its launch balloon in Antarctica. (Credit: M. Halpern) Figure 9: POLARBEAR (foreground) and ACT. (Credit: UCSD Cosmology) ## 6 The Future: Commitments and Possibilities In this section, we expand on the future prospects for SMM facilities to which Canada has access. For ground-based, single-dish SMM astronomy, Canada has at present access to the JCMT but Canadian support for its operations is planned to cease after September 2014. A new facility, the 25-m diameter CCAT could replace the role of JCMT, but only as early as 2018. Moreover, GBT’s future is also uncertain, though closure plans have yet to be articulated. For interferometric facilities, ALMA’s and JVLA’s futures appear more secure. We reiterate that access to single-dish and interferometric facilities is crucial in the SMM regime, since they provide respectively the larger- and smaller- scale views of phenomena that are sometimes only observable at SMM wavelengths. Moreover, access to single-dish facilities can provide the necessary “ground work” for successful proposals to use the highly oversubscribed interferometers. Of note, no other countries with direct access to a significant shorter wavelength single-dish SMM facility are contemplating its closure as ALMA comes online. (Through ESO, the UK and Netherlands will have access to the 12-m Atacama Pathfinder Experiment (APEX) single-dish telescope.) For space-based SMM astronomy, the successful Herschel mission is almost over as the telescope runs out of cryogens. Though a large and detailed Herschel dataset will be publicly available worldwide, it will always be limited to the current capabilities of Herschel. In this regime, possible successors to Herschel include “Super BLAST-Pol,” a larger aperture follow-up to BLAST-Pol, and SPICA, a clone of the Herschel design (or possibly larger) with a cooled primary mirror and instruments optimized for far-infrared/submm observations. ### 6.1 ALMA ALMA, like the JVLA, is presently finishing its construction and Early Science phases. Its future also appears secure, as no other similar or better facility is planned world-wide. ALMA plans to keep its capabilities modern through a healthy development program. At present, the observatory is considering several near-term possibilities, including adding Band 1 and Band 2 receivers, as well as completing the complement of Band 5 receivers (Table 3). Very long- baseline SMM interferometry with other facilities around the world is also being explored. Though ALMA excels in sensitivity and resolution, it will be limited by its inefficiency in mapping wide fields. Hence, single-dish facilities are needed for circumstances where large-scale data of the SMM universe are needed, or to provide “pathfinder” data for future high resolution data with ALMA. Figure 10: ALMA antennas on the Llano de Chajnantor (Credit: ALMA (ESO/NAOJ/NRAO)) ### 6.2 CCAT CCAT is envisaged as an SMM single-dish facility of 25-m diameter at 5600 m in northern Chile, specifically located on a peak above the plateau where ALMA lies. This site is excellent, having even lower typical precipitable water vapour levels than the adjacent ALMA site and Mauna Kea. Also, the CCAT aperture will be $\sim$2.8$\times$ larger than the surface area of the JCMT and $\sim$2$\times$ the diameter of the 12-m ALMA antennas. These qualities make CCAT an attractive possibility for the future of short-wavelength SMM single-dish astronomy. CCAT was identified as one of the top three medium-scale ground-based observatories in LRP2010, and remains a high priority among the Canadian community. It was further identified by the US NRC Astro2010 survey as the highest priority in a similar medium, ground-based category. Through a recent grass-roots effort, eight Canadian universities (McGill, McMaster, Calgary, Toronto (including CITA, the Department of A&A and the Dunlap Institute), UBC, Waterloo, Dalhousie, Western) secured $550K to join the CCAT consortium as a founding member. This membership status provides a number of tangible benefits such as guaranteed observing time in perpetuity and an active role in the design of the project through instrument contracts and seats on the CCAT Board. The goal of the Canadian consortium is to have Canada eventually join CCAT as a 25% partner, mirroring our investment in the JCMT and ensuring Canadians can make a strong scientific impact in the field. The US partners include Cornell, Caltech, University of Colorado, and AUI and the German partners include the Universität zu Köln and Universität Bonn. The CCAT operational model will be likely driven more by large surveys than smaller PI- led projects, though decisions on survey priorities and wider data access have not yet been made. Figure 11: Wireframe model of CCAT. (Credit: CCAT Consortium) The designs for the first generation of CCAT instruments are modern and ambitious, promising very wide-field continuum and spectroscopic imaging. These instruments will allow wide-field coverage of the southern SMM sky at sensitivities competitive with those of ALMA. Four instruments are now proceeding into an early design phase, after which two are planned to be selected for first light and the other two for installation during early operations. These include: $\bullet$ LWCam \- a bolometric camera at 5 bands from 750 $\mu$m to 2 mm ($\sim$20${}^{\prime}\times$20′); $\bullet$ SWCam \- a bolometric camera at 4 bands from 200 $\mu$m to 670 $\mu$m ($\sim$5′$\times$5′); $\bullet$ X-Spec \- a direct detection wideband survey spectrograph with instantaneous coverage of 195-520 GHz at $R$ = 400-700 ($\sim$20-300 spaxels); and $\bullet$ CHAI \- a dual-frequency band heterodyne focal plane array covering 450 GHz ($\sim$2′$\times$2′) and 830 GHz ($\sim$1′$\times$1′). The estimate for the cost of CCAT construction (including first instruments) is $140M (with first light in $\sim$ 2018). The CCAT telescope itself is presently undergoing a design and development phase funded by the US NSF that will be completed in 2013, and the road contract has been tendered with construction beginning this year (2013). Some private funding for CCAT has been secured by some partners but no further US or Canadian public funding has been yet identified. ### 6.3 JCMT The JCMT partnership in its present form will change in March 2013, when the Netherlands will withdraw support to JCMT. UK and Canadian support is planned to cease at the end of September 2014. The future of the observatory after September 2014 is not clear. A prospectus for new management to take over operations of JCMT will be released in early 2013. If no new management is found, the JCMT will face closure and demolition, with the site returned to its original condition atop Mauna Kea. Note that JCMT support is beginning to erode well ahead of September 2014 as experienced staff move to new stable positions elsewhere. In Canada, the LRP2000 panel strongly advocated support for JCMT cease at the end of the tripartite agreement between the UK, Canada, and the Netherlands to operate the JCMT in 2009, and the funds subsequently freed up be used for ALMA. This idea was originally supported by much of Canada’s SMM community as it was based on the key assumption that the JCMT would have exhausted its scientific potential as ALMA came online. This event, however, has not yet occurred. For example, plans for SCUBA-2 and the JCMT Legacy Surveys were not yet formulated during the period when the LRP2000 panel report was drafted. Throughout the 2000’s SCUBA-2 was developed, in collaboration with Canada (with funding at the $5-10M level from the Canadian Foundation for Innovation (CFI) made to a university consortium). The first SCUBA-2 components usable for observations were available in early 2010, but science usage of the full instrument did not begin until fall 2011. Moreover, the Canadian SMM community at the time of the LRP2000 was smaller than it is today, more than a decade later. It is difficult to assign objective numbers to the increase in community size, but note that four of the seven authors of this document were established as part of the Canadian community after the publication of the LRP2000 report. Figure 12: The James Clerk Maxwell Telescope. (Credit: NRC) With SCUBA-2 in the picture, the agreement to operate JCMT was indeed extended beyond 2009 (to the dates described above). Around the same time, the LRP2010 panel reiterated the earlier LRP2000 report’s recommendation of Canadian withdrawal from JCMT, but defined no specific withdrawal date. For example, they wrote “[w]ithdrawal from the JCMT is expected shortly but with access to both ALMA and the EVLA [now JVLA] Canada is now very well positioned at the forefront of radio/submm astronomy. Canada is also a member of the CCAT consortium and is participating in SKA development.” Since the LRP2010 report, the true on-sky performance of SCUBA-2 has been shown to be excellent and the instrument is still scientifically relevant. For example, its mapping speed is $\sim$200$\times$ that of SCUBA at 850 $\mu$m, and its beam-size at 450 $\mu$m is 4$\times$ smaller than that of Herschel-SPIRE at 500 $\mu$m. SCUBA-2 on the JCMT remains the best facility of its kind in the world. Given the present date for withdrawal from JCMT, the Canadian community will get only three years use of SCUBA-2, its premier instrument, and of course its remaining instrumentation. The JCMT Legacy Survey components were reduced in scale in fall 2011 to $\sim$2 year programs to match the on-sky performance of SCUBA-2 with the historical weather breakdown at JCMT. Since completion assumes the weather holds to statistical norms and no losses of key support staff, there is of course associated risk that even the rescoped JLS may not be fully completed. (The JLS component teams have prioritized their targets in this event.) Moreover, restoring JLS components to their original scopes will not be possible, meaning that the full scientific potential will not be realized. Withdrawal from JCMT in September 2014 will also severely curtail Canadian PI use of SCUBA-2 or HARP, to follow up JLS results, observe regions of the sky not observed by other facilities (e.g., Herschel), or address any new developments that demand submm data. Finally, withdrawal from JCMT will significantly limit use of the two ancillary instruments developed by Canadians, POL-2 and FTS-2, that are currently being commissioned and provide unique data not obtainable at other observatories (i.e., wide-field polarization data and intermediate spectral resolution mapping, respectively). Withdrawal from JCMT will also limit Canadians’ abilities to build on their legacy with Herschel. For example, SCUBA-2 provides information at longer wavelengths (850 $\mu$m) that Herschel could not provide, at resolutions and sensitivities similar to those of Herschel at 250 $\mu$m. In addition, as stated above, SCUBA-2 450 $\mu$m images have a resolution a factor of $\sim$4 higher than that of Herschel at 500 $\mu$m, significantly reducing possible confusion and source blending seen in Herschel 500 $\mu$m data. Though SCUBA-2’s data are more filtered spatially than (Herschel) SPIRE and PACS data, Canadian astronomers have already begun work combining the complementary JCMT and Herschel continuum data to great benefit in more accurately determining the column densities of emitting dust. Moreover, HARP can be used to produce high spectral resolution data of molecular lines that provide key kinematic insights about structures seen in either SCUBA-2 or SPIRE/PACS data (or both). HARP also has proven to be an ancillary instrument to SCUBA-2 itself, as HARP data can provide key information about line emission within the SCUBA-2 850 $\mu$m filter band (mainly from CO) that may be wrongly attributed to continuum flux. Terminating Canada’s involvement in the JCMT in 2014 will also undermine Canada’s investment in ALMA. While ALMA is a forefront facility, its abilities remain finite and in particular it cannot realistically compete with facilities that can observe wide sky fields. JCMT is still the largest single- dish SMM facility on the planet, and single-dish continuum or line pathfinder data, such as those from JCMT, give Canadian ALMA proposals a crucial edge over those of their international peers. We note again that many countries involved in ALMA, e.g., Japan, France, Germany, and Spain have not closed their own short-wavelength SMM single-dish telescopes as ALMA becomes available, in part for this reason. In some cases, the capabilities of these facilities have even been expanded with new instrumentation or upgrades. (Though CCAT could replace JCMT in this role as an ALMA pathfinder, again it will not be available for at least four years after Canada leaves the JCMT partnership.) Finally, we note that Canada’s use of ALMA is small relative to the larger partners in that project, limiting our overall scientific impact as well as the technical and scientific expertise in the SMM regime we can maintain and foster. We believe Canada’s scientific impact with ALMA will be greatly augmented through continued access to a large single-dish SMM observatory. ### 6.4 GBT and JVLA The GBT and JVLA are operated by the National Radio Astronomy Observatory (NRAO) in the US through funding from the NSF to Associated Universities, Inc. (AUI). NRAO also operates ALMA on behalf of North America and the radio- frequency Very Large Baseline Array. Canada has secured access to the NRAO facilities through NAPRA. Figure 13: The Robert C. Byrd Green Bank Telescope. (Image courtesy of NRAO/AUI) The GBT’s future is uncertain given the August 2012 recommendation by the NSF Portfolio Review Committee that NSF divest itself from its commitment to operate the GBT, so that funding can be freed up for facilities to address new astronomical priorities. As of this writing, it is not yet known how NSF, AUI, or NRAO will act on this recommendation, e.g., modifying (reducing) the present operational model for GBT, putting forth the facility for management by new (national or international) partners, or closing it outright. Loss of the GBT (or even loss of open access to it) would be a serious blow to single- dish long-wavelength SMM (and radio) astronomy. The GBT’s surface and imaging capabilities exceed those of its closest competitor, the Effelsberg 100-m Telescope in Germany. Some members of the US (and Canadian) community are actively campaigning to keep open access to the GBT. The JVLA’s future is more secure, as its retrofit and Early Science phase have just been completed. The JVLA is the world’s foremost radio and long- wavelength SMM facility. As with ALMA, the ability to see the large-scale picture is lost with the JVLA but can be restored with data from single-dish facilities. Hence, the loss of the GBT to general use would also impact the scientific potential of the JVLA. Figure 14: The Karl G. Jansky Very Large Array. (Image courtesy of NRAO/AUI) Future developments of the JVLA have not been openly discussed within the US community. ALMA Band 1 receivers on ALMA could exceed the performance of the present 40-50 GHz (Q-band) receivers on JVLA. Further along, the idea of providing mid-scale baselines for the VLBA through a network of single-dish JVLA-like telescopes throughout New Mexico, linking the JVLA with the VLBA, could be resuscitated. (Such a development depends on the future of the VLBA and is beyond the scope of this document.) Eventually, JVLA’s capabilities at radio frequencies $<$10 GHz will be eclipsed by those of the SKA. ### 6.5 SPICA The Space Infrared Telescope for Cosmology and Astrophysics (SPICA) is a planned mission optimized for mid- and far-infrared astronomy with a cryogenically cooled telescope. The baseline telescope is a clone of the 3.5-m Herschel telescope, though other configurations are being considered. To reduce mass, SPICA will be launched at ambient temperature and cooled down in orbit by onboard mechanical coolers with an efficient radiative cooling system. It has been proposed as a Japanese-led JAXA-ESA mission together with extensive international collaboration including Canada, the US, and South Korea. As of 2013, the proposed launch date for SPICA is 2022. Figure 15: The Space Infrared Telescope for Cosmology and Astrophysics (SPICA) concept. Space ghost baby likely not part of final design. (Credit: JAXA) Though optimized for wavelengths mostly shorter than the SMM range, SPICA is of great interest to Canadian astronomers given the success and legacy of the Herschel mission. Through SPICA’s instruments, data will be acquired that are highly complementary to Herschel’s, bridging the presently under-sampled wavelengths between the infrared and SMM regimes also highly impacted by atmospheric absorption. At wavelengths shorter than those covered by Herschel, SPICA data will naturally be of higher resolution than Herschel’s. Moreover, its cooled primary will allow for significant improvements in sensitivity over Herschel. The three baseline SPICA instruments include: $\bullet$ an unnamed mid-infrared (MIR) coronagraph (3.5-27 $\mu$m) with photometric and spectral capabilities ($R$ $\approx$ 200); $\bullet$ an unnamed MIR wide-field camera and high-resolution spectrometer ($R$ $\approx$ 3 $\times$ 104); and $\bullet$ SAFARI \- a far-infrared (30-210 $\mu$m) imaging spectrometer. Canadian astronomers are currently involved in a detector test facility and simulator for SAFARI, primarily through D. Naylor’s group at U. Lethbridge. A fourth instrument, BLISS, a low-resolution ($R$ $\approx$ 500) spectrograph for extragalactic surveys is under design in the US and Canada in consultation with Japan, but based primarily on NASA funding. This instrument is currently exploring several designs and will focus on one of a grating spectrograph, an imaging FTS, or a filter-bank architecture in the coming years. ### 6.6 Focused PI Experiments #### 6.6.1 The E and B Experiment The E and B experiment (EBEX) is a mm-wavelength polarization-sensitive 1.5-m diameter telescope that will be flown by NASA to about 100,000 feet aboard a stratospheric balloon. Detection of polarization from inflationary gravity waves is one of the main science goals of EBEX. EBEX will measure the polarization of the CMB to provide a glimpse of the universe at its very earliest stages. The focal plane of EBEX will be instrumented with 1400 Transition Edge Sensor (TES) detectors, read out with McGill’s Digital Frequency Domain Multiplexer electronics. Polarization sensitivity is achieved with a half-wave plate and polarizing grid. It will flown first on a one day test flight above Texas, before traveling to Antarctica for a $\sim$30 day long duration flight around the South Pole. #### 6.6.2 Spider Spider is a balloon-borne telescope designed to detect the imprint of gravitational waves released in the the first tiny fraction of a second of the universe and thereby imprinted on the CMB. By doing this, Spider will provide insight on the extremely early Universe, and provide a crucial test for models of the early inflation of the Universe. Spider has polarization-sensitive detectors at 100 GHz (3 mm), 150 GHz (2 mm) and 220 GHz (1.4 mm). The first launch expected in summer of 2013. #### 6.6.3 Super BLAST-Pol Super BLAST-Pol is an upgrade of the existing BLAST-Pol telescope (the Balloon-borne Large Aperture Sub-mm Telescope for Polarimetry), which will map polarized dust emission at 250, 350 and 500 $\mu m$ with a resolution of 22′′ at 250 $\mu m$. The telescope will utilize a 2.5 meter aluminum primary mirror, a 28-day hold time cryostat, and a 1,000 detector focal plane array using MKIDs detector technology, which will give Super BLAST-Pol a mapping speed $>$10$\times$ that of BLAST-Pol. The project has funding from NASA, and is being built by the University of Pennsylvania, Northwestern University, Arizona State, NIST, the University of Toronto, Cardiff University and the University of British Columbia. A first science flight is planned for December 2016 from Antarctica: 25$\%$ of the science time will be available for shared risk observing to the astronomy community, making Super BLAST-Pol the first balloon-borne telescope to operate as an observatory. #### 6.6.4 Event Horizon Telescope The Event Horizon Telescope (EHT) is an international consortium to link numerous SMM facilities around the globe into a single very long baseline interferometer. The goal of the EHT is to observe directly 230-450 GHz emission from the immediate environments of supermassive black holes in the centres of our Galaxy and the nearby elliptical galaxy M87. Such observations will probe accretion and jet formation in these highly unusual sites, and test general relativity. Among its many nodes, the EHT will include the JCMT and ALMA. This ambitious project, including participation from the University of Waterloo and the Perimeter Institute of Theoretical Physics, is expected to take a decade to complete due to the need to develop and deploy highly stable frequency standards, new submillimetre dual-polarization receivers, and wide band width VLBI backends and recorders. ### 6.7 Synergies with Other Facilities In this document, we have focused entirely on SMM facilities to which Canada has access. Through its long engagement in JCMT, Canada has built up a considerable community of astronomers who use SMM wavelengths to investigate astrophysical phenomena. Of course, the SMM range is only a small part of the wider electromagnetic spectrum. How do SMM facilities relate to those given high priority (e.g., in LRP2010) at other wavelengths? Optical/infrared telescopes like TMT or Euclid are important for studying the warmer thermal emission from stars (and planets) and galaxies. SMM facilities are used to explore the colder aspects of nature, typically the cold interstellar media of this Galaxy and others. Beyond extinction and interstellar absorption, the cold ISM is generally not probed at optical/infrared wavelengths. The ISM is important, e.g., accounting for mass in spiral galaxies similar to those seen in their stellar components. Similarly, advanced radio telescopes like CHIME and the SKA are important for probing both the ionized components of galaxies through thermal free-free emission and the warmer atomic aspect of the ISM within galaxies through observations of HI. Moreover, such facilities are critical for exploring the signatures of non-thermal processes in the universe, e.g., synchrotron emission. Few molecules strongly emit at the wavelengths generally considered to be “radio” in nature, and thermal dust emission becomes too weak at wavelengths $>$10 mm to be detected. Hence, the SMM regime is in many ways a unique realm where specialized equipment both on the ground and in space are needed to unlock its secrets. Though the optical/infrared and radio facilities prioritized in LRP2010 are very important, attention must be paid to maintain Canada’s leadership in the SMM regime through continued engagement in such facilities on many levels, i.e., beyond just that of ALMA. ## 7 A Plan for the Next Decade ### 7.1 Recommendations We recommend that Canada retain its standing in SMM astronomy by building on its existing strengths and acting on its present and future opportunities. To do so, Canada must retain access to the unique strengths of a single-dish telescope at a level sufficient to maintain and foster scientific and technical expertise. In particular, we present the following recommendations: $\bullet$ Canada must continue to encourage and maintain active participation in ALMA science proposals and in ALMA Development Projects. As a member of the North American ALMA ARC with full access to 33.75% of ALMA time, Canadian PIs need only have strong science cases to achieve high rates of allocated time on ALMA. The results of the first two cycles of ALMA allocations show that Canada is already competing well, obtaining time in excess of our basic monetary contributions. ALMA Development is an ongoing process of furthering the capabilities of ALMA, and Canada has been engaged in Band 1 receiver development for several years. This and other development opportunities should be pursued so that Canada’s science objectives for ALMA are paralleled by hardware and software contributions from Canada as well. $\bullet$ Canada must continue engagement within the JCMT with a smooth transition to CCAT. An extension of our involvement by a minimum of three years (to the end of 2017) will allow for the completion of the Legacy Surveys to their intended (peer-reviewed) level. These surveys remain highly relevant with no comparable facility capable of their execution. Such a timescale is also concurrent with the expected availability of CCAT of 2018. Moreover, the extensive JLS data can continue to involve the community for several years as CCAT ramps up its capabilities. Continued support for JCMT would have the additional benefit of enabling Canadians to follow up discoveries made by the JLS program and from Herschel data. These new observations would be made using the JCMT’s present suite of unique instruments and will realize the scientific potential of the two instruments POL-2 and FTS-2 built with Canadian funding. Access to JCMT will also give Canadians an important edge when devising programs for highly competitive ALMA, for which few pathfinder instruments exist. Finally, continued use of the JCMT will provide a training ground and test-bed facility for CCAT for a growing cohort of involved Canadian students and postdocs. Canadian support for JCMT is presently at the $1M/annum level. Though the future of the partnership of JCMT remains unclear, Canada should be poised to continue engagement at its present level of 25% within whatever consortium may form to take over management of the JCMT. A mechanism for continued Canadian involvement in JCMT must be immediately found. $\bullet$ Canada must move forward with its engagement in CCAT. The minimum level of participation set by the partnership is 10%, but we recommend 25% if Canada is to have a strong voice in the collaboration, a meaningful scientific impact, and the opportunity to maintain and foster expertise. Given its size and location, CCAT will be clearly superior to JCMT and we do not foresee a need for both facilities. Thus, CCAT first-light represents a hard upper-limit on continued Canadian involvement in the JCMT. CCAT will allow Canada to continue building on its heritage in the SMM regime obtained through its engagement in JCMT, and allow for continued advantage for using highly over- subscribed facilities like ALMA. $\bullet$ The JVLA is an impressive facility coming into its own thanks in large part to the Canadian-made WIDAR correlator. We recommend Canadians take advantage of its unique and powerful capabilities in the longer-wavelength millimetre regime. Regarding the GBT, the recent recommendation to NSF that it divest itself from its operations is worrisome in that Canadian access to future single-dish long-wavelength SMM data are threatened. GBT data are impressive on their own, or notably can be used to great effect in combination with JVLA data. The GBT is the top of its class, and no other facility of its type is planned on any timescale. Though Canada has access to GBT or JVLA through the NAPRA agreement, it does not directly fund either facility. How NRAO will proceed on the recommendation to divest from GBT is unknown at this time. We recommend, however, that Canadians stay abreast of potential changes in its operations over the next decade and at the very least be prepared to advocate for continued access. $\bullet$ Canada, specifically the Canadian Space Agency, should leverage its previous successful investment with Herschel to fund Canadian engagement in SPICA. As with Herschel, the CSA should consider funding students and postdoctoral fellows to allow Canadians to exploit fully its investment before the data become widely public. $\bullet$ Though general purpose observatories are extremely useful, sometimes very focused PI-based experiments can provide the answers to pressing problems that arise after larger, open-access observatories and their instruments are built. These nimble experiments can also make great strides with relatively small cost, particularly those based on balloons. Hence, we recommend ongoing funding to focused PI-experiments, like EBEX, Spider, Super BLAST-Pol, POLARBEAR, and ACT through the Canadian Space Agency, CFI, and NSERC. ### 7.2 Funding Considerations #### 7.2.1 JCMT At present, Canada’s involvement in JCMT is provided by the National Research Council of Canada (NRC). NRC is mandated to support Canada’s off-shore astronomy facilities (e.g., Gemini and CFHT) and has begun to fund Canadian participation in ALMA. For Canada to remain engaged in JCMT, the most obvious solution is for NRC to provide new support and continue to manage its operations. Alternatively the Canadian astronomy community must find funding from other national sources. Such a mechanism is not yet clear, but perhaps one where funding from CFI or NSERC managed by ACURA is possible, bridging to future management of CCAT. Current Canadian operating costs of the JCMT are $1.17M per year, and last year’s total operating cost of the JCMT was $4.7M (US). Even considering a stripped down mode of operation for the JCMT of $2M/yr, with Canada continuing to contribute 25% of the costs, then the funding required would be $500k/yr. While not a trivial amount of money it is also not enormous, when one considers the size of the JCMT user base, its scientific impact, and the investment to date. Still, at present there are no appropriate programs available through NSERC or CFI. The initial $500k which secured the Canadian consortium partnership in CCAT was raised through one-time requests to individual universities and institutes, with each contributing $\sim$$50k. While this success confirms that this level of funding is feasible at the university level, this approach is unlikely to possible for continued JCMT funding, as it is not a new initiative. #### 7.2.2 CCAT The Canadian CCAT Consortium plans to submit a proposal for funding to the Canadian Foundation for Innovation (CFI) this year. The CCAT consortium, including Canadian universities and institutes, are actively exploring the potential for private donations. The required funds for construction ($35M) are within reach of a single, or a small number of private donors. The ongoing costs of operation of CCAT, however, are unlikely to be funded through these mechanisms. We note that the expertise of NRC in managing telescopes and providing archival resources makes it a natural partner for the CCAT project. We recommend that the management and operations structure of CCAT be finalized soon, with some clear guidance from Canadian funding agencies for continued and stable means to fund CCAT operations. #### 7.2.3 SPICA and PI Projects The Canadian Space Agency (CSA) is the primary means of funding involvement in new space missions such as SPICA. They also support various smaller programs related to balloon missions, which are relevant and useful for the PI-types of projects in particular. Over the past decade, CSA has provided significant funding resources, including roughly $100M for our participation in JWST, Herschel, and Planck, as well as $1-2M of funding in support of data analysis associated with Herschel, Planck, FUSE, and other operating missions to individual university researchers. CSA thus has the potential in the long run to provide major new funding for space- and balloon-based submillimetre facilities. The CSA’s budget is currently very tight and informal feedback suggests they have no money for new astronomy projects at this time. Budget situations can change dramatically, however, with new directions and funding coming from the federal budget process. The community at large will need to continue to put pressure on the CSA and at the political level to lobby for additional funding for science programs in the CSA’s budget. Acknowledgements: The authors thank Laura Fissel for providing information on Super BLAST-Pol, David Naylor for providing information on SPICA, Gerald Moriarty-Scheiven for providing the ALMA user numbers, Dennis Crabtree for providing the JCMT, CFHT, and Gemini user statistics, and Tom Phillips for providing an updated version of Figure 1. We also thank Matthijs van der Wiel and Gary Davis for pointing out small errors in a previous version of this document.	arxiv-papers	2013-12-18T01:00:14	2024-09-04T02:49:55.585055	{ "license": "Public Domain", "authors": "Tracy Webb (1), Scott Chapman (2), James Di Francesco (3), Brenda\n Matthews (3), Norm Murray (4), Douglas Scott (5), and Christine Wilson (6)\n ((1) McGill University, (2) Dalhousie University, (3) National Research\n Council of Canada, (4) Canadian Institute of Theoretical Astrophysics, (5)\n University of British Columbia, (6) McMaster University)", "submitter": "James Di Francesco", "url": "https://arxiv.org/abs/1312.5013" }
1312.5029	# Hrushovski’s Algorithm for Computing the Galois Group of a Linear Differential Equation Ruyong [email protected]. This work is partially supported by a National Key Basic Research Project of China (2011CB302400) and by a grant from NSFC (60821002). KLMM, AMSS, Chinese Academy of Sciences, Beijing 100190, China ###### Abstract We present a detailed and simplified version of Hrushovski’s algorithm that determines the Galois group of a linear differential equation. There are three major ingredients in this algorithm. The first is to look for a degree bound for proto-Galois groups, which enables one to compute one of them. The second is to determine the identity component of the Galois group that is the pullback of a torus to the proto-Galois group. The third is to recover the Galois group from its identity component and a finite Galois group. ## 1 Introduction In [6], Hrushovski developed an algorithm to compute the Galois groups for general linear differential equations. To the best of my knowledge, this is the first algorithm that works for all linear differential equations with rational function coefficients. Before Hrushovski’s results, the known algorithms were only valid for linear differential equations of special types, for instance, low order or completely reducible equations. The algorithm due to Kovacic ([7]) deals with the second order equations. In [12], the authors determined the structural properties of the Galois groups of second and third order linear differential equations. In many cases these properties can be used to determine the Galois groups. In [2], the authors gave an algorithm to compute the Galois group of linear differential equations that are completely reducible. The reader is referred to [11, 15] for the survey of algorithmic aspects of Galois groups and the references given there for more results. In particular, in [11], the author gave a clear explanation of the method based on Tannakian philosophy and introduced the various techniques that were used in the known algorithms. Throughout the paper, $C$ denotes an algebraically closed field of characteristic zero and $k=C(t)$ is the differential field with the usual derivation $\delta=\frac{d}{dt}$. The algebraic closure of $k$ is denoted by $\bar{k}$. Linear differential equations we consider here will be of the matrix form: $\delta(Y)=AY,$ (1) where $Y$ is a vector with $n$ unknowns and $A$ is an $n\times n$ matrix with entries in $k$. Denote the Picard-Vessiot extension field of $k$ for (1) by $K$ and the solution space of (1) by $V$. Then the Galois group of (1) over $k$, denoted by ${\mbox{\rm Gal}}(K/k)$, are defined as the group of differential automorphisms of $K$ that keep all elements of $k$ fixed. For brevity, we usually use ${\mathcal{G}}$ to denote this group. ${\mathcal{G}}$ is a subgroup of ${\rm GL}(V)$. A matrix in ${\rm GL}_{n}(K)$ whose columns form a basis of $V$ is called a fundamental matrix of (1). Elements of $V^{n}$ are vectors with $n^{2}$ coordinates. For the sake of convenience, elements of $V^{n}$ are also written in the matrix form. In such a case, $V^{n}=\\{Fh\|h\in{\rm Mat}_{n}(C)\\}$, where $F$ is a fundamental matrix of (1). Set $V^{n}_{inv}=\\{Fh\|h\in{\rm GL}_{n}(C)\\}.$ Then $V_{inv}^{n}$ is an open subset of $V^{n}$. Note that if $F=I_{n}$, then $V^{n}_{inv}={\rm GL}_{n}(C)$. In this paper, we will always use $X$ to denote the $n\times n$ matrix whose entries are indeterminates $x_{i,j}$. Without any possible ambiguity, we will also use $X$ to denote the set of indeterminates. Let $Z$ be a subset of $V^{n}_{inv}$. $Z$ is said to be a Zariski closed subset of $V_{inv}^{n}$ if there are polynomials $P_{1}(X),\cdots,P_{m}(X)$ such that $Z={\mbox{\rm Zero}}(P_{1}(X),\cdots,P_{m}(X))\cap V_{inv}^{n}$. In this case, we also say that $Z$ is defined by $P_{1}(X),\cdots,P_{m}(X)$. We will use $N_{d}(V_{inv}^{n})$ to denote the set of all subsets of $V_{inv}^{n}$, which are defined by finitely many polynomials with degree not greater than $d$. Note that here we have no requirement for the coefficients of these polynomials. Suppose that $\tilde{k}$ is an extension field of $k$ and $Z\subseteq V_{inv}^{n}$. If $Z$ can be defined by polynomials with coefficients in $\tilde{k}$, then $Z$ is said to be $\tilde{k}$-definable. The stabilizer of $Z$, denoted by ${\rm stab}(Z)$, is defined as the subgroup of ${\rm GL}(V)$ whose elements keep $Z$ unchange. Let $Z\in N_{d}(V_{inv}^{n})$. In case that we emphasize the degree $d$ of defining polynomials, we will also say “$Z$ is bounded by $d$”. Let $F$ be a fundamental matrix. For any $\sigma\in{\rm GL}(V)$, there exists $[\sigma]\in{\rm GL}_{n}(C)$ such that $\sigma(F)=F[\sigma]$. The map $\phi_{F}:{\rm GL}(V)\rightarrow{\rm GL}_{n}(C)$, given by $\phi_{F}(\sigma)=[\sigma]$, is a group isomorphism and $\phi_{F}({\mathcal{G}})$ is an algebraic subgroup of ${\rm GL}_{n}(C)$. Let ${\mathcal{H}}$ be a subgroup of ${\rm GL}_{n}(V)$. For ease of notations, we use ${\mathcal{H}}_{F}$ to denote $\phi_{F}({\mathcal{H}})$. ${\mathcal{H}}$ is said to be an algebraic subgroup if so is ${\mathcal{H}}_{F}$. Assume that ${\mathcal{H}}$ is an algebraic subgroup. Then ${\mathcal{H}}^{\circ},{\mathcal{H}}^{t}$ are used to denote the pre-images of ${\mathcal{H}}_{F}^{\circ}$ and ${\mathcal{H}}_{F}^{t}$, where ${\mathcal{H}}_{F}^{\circ}$ denotes the identity component of ${\mathcal{H}}_{F}$ and ${\mathcal{H}}_{F}^{t}$ is the intersection of kernels of all characters of ${\mathcal{H}}_{F}$. ${\mathcal{H}}$ is said to be bounded by $d$ if so is ${\mathcal{H}}_{F}$. The key point of Hrushovski’s algorithm is that one can compute an integer $\tilde{d}$ such that there is an algebraic subgroup ${\mathcal{H}}$ (or ${\mathcal{H}}_{F}$) of ${\rm GL}(V)$ bounded by $\tilde{d}$ satisfying $():({\mathcal{H}}^{\circ})^{t}\unlhd{\mathcal{G}}^{\circ}\leq{\mathcal{G}}\leq{\mathcal{H}},\,\,\mbox{or}\,\,\,\,({\mathcal{H}}_{F}^{\circ})^{t}\unlhd{\mathcal{G}}_{F}^{\circ}\leq{\mathcal{G}}_{F}\leq{\mathcal{H}}_{F}.$ For simplicity of presentation, we introduce the following notion. ###### Definition 1.1 The algebraic group ${\mathcal{H}}$ $($or ${\mathcal{H}}_{F}$$)$ in $()$ is called a proto-Galois group of (1). Roughly speaking, Hrushovski’s approach includes the following steps. * $(S1)$ (proto-Galois groups). One can compute an integer $\tilde{d}$ only depending on $n$ such that there is a proto-Galois group of (1), which is bounded by $\tilde{d}$. Let ${\mathcal{H}}$ be the intersection of the stabilizers of $k$-definable elements of $N_{\tilde{d}}(V^{n}_{inv})$. Then ${\mathcal{H}}$ is a desired proto-Galois group of (1). * $(S2)$ (The toric part). Compute ${\mathcal{H}}_{F}^{\circ}$ and let $\chi_{1},\cdots,\chi_{l}$ be generators of the character group of ${\mathcal{H}}_{F}^{\circ}$. Then the map $\varphi=(\chi_{1},\cdots,\chi_{l})$ is a morphism from ${\mathcal{H}}_{F}^{\circ}$ to $(C^{})^{l}$. $\varphi({\mathcal{G}}_{F}^{\circ})$ is the Galois group of some exponential extension $E$ of $\tilde{k}$ over $\tilde{k}$, where $\tilde{k}$ is an algebraic extension of $k$. One can find $E$ by computing the hyperexponential solutions of some symmetric power system of (1). Pulling $\varphi({\mathcal{G}}_{F}^{\circ})$ back to ${\mathcal{H}}_{F}^{\circ}$, one gets ${\mathcal{G}}_{F}^{\circ}$. $(S3)$ (The finite part). Find a finite Galois extension $k_{G}$ of $k$ and a $k_{G}$-definable subset $Z$ of $V^{n}_{inv}$ such that ${\mathcal{G}}^{\circ}={\rm stab}(Z)$. Let $Z_{1}=Z,Z_{2},\cdots,Z_{m}$ be the orbit of $Z$ under the action of ${\mbox{\rm Gal}}(k_{G}/k)$. Then ${\mathcal{G}}=\cup_{i=1}^{m}\\{\sigma\in{\rm GL}(V)\|\sigma(Z)=Z_{i}\\}$. We follow Hrushovski’s approach, but take out the logical language and elaborate the details of the proofs that were only sketched in his paper. We hope this will be helpful for the reader to understand Hrushovski’s approach. Meanwhile, we simplify the first step of his approach. In spite of calculating all $k$-definable elements, we only need compute one $k$-definable element to obtain a proto-Galois group. Hrushovski showed in the part iii@ of [6] how to compute the integer $\tilde{d}$ as claimed in (S1). As well as providing the detailed explanations of his proofs, we present an explicit estimate of the integer $\tilde{d}$. The paper is organized as follows. In Appendix A, we describe a method to find a bound for $k$-definable elements of $N_{d}(V^{n}_{inv})$. In Appendix B, an explicit estimate of the integer $\tilde{d}$ that bounds the proto-Galois groups is presented. These bounds will guarantee the termination of the algorithm. In Sections 2 and 3, we show how to compute a proto-Galois group and then the Galois group. Some computation details are omitted in these sections and will be completed in Section 4. Acknowledgements. Special thanks go to Michael F. Singer for his numerous significant suggestions that improve the paper a lot. In preparing the paper, the author was invited by Michael F. Singer to visit North Carolina State University for two weeks. The author thanks him for the invitation and financial support. Many thanks also go to Shaoshi Chen, Ziming Li and Daniel Rettstadt for their valuable discussions. ## 2 proto-Galois groups This section will be devoted to finding a proto-Galois group of (1). Let $d\in{\mathbb{Z}}_{\geq 0}\cup\\{\infty\\}$ and $I\subseteq k[x_{1,1},\cdots,x_{n,n}]$. $I_{\leq d}$ denotes the set of polynomials in $I$ with degree not greater than $d$. Set $I_{F,d}=\left\\{P(X)\in k[x_{1,1},\cdots,x_{n,n}]_{\leq d}\,\,\left\|\,\,P(F)=0\right.\right\\}\,\,\mbox{and}\,\,Z_{F,d}={\rm Zero}(I_{F,d})\bigcap V_{inv}^{n}$ (2) When $d=\infty$, we use $I_{F}$ and $Z_{F}$ for short. $I_{F}$ consists of algebraic relations among entries of the fundamental matrix $F$. The Galois group of (1) is considered as the subgroup of ${\rm GL}(V)$ that preserves $I_{F}$. Precisely, ${\mathcal{G}}=\\{\sigma\in{\rm GL}(V)\,\,\|\,\,\sigma(Z_{F})=Z_{F}\\}.$ Hence once $I_{F}$ is computed, ${\mathcal{G}}$ will be determined. In general, it is hard to calculate $I_{F}$. While given a nonnegative integer $d$, the results in Appendix A enable us to compute $I_{F,d}$ (see Section 4.1). Moreover, we shall show that if $d$ is large enough, then ${\rm stab}(Z_{F,d})$ will be a proto-Galois group of (1). Corollary B.15 in Appendix B tells us how large the integer $d$ is enough. Let us start with two lemmas. ###### Lemma 2.1 Assume that $U$ is a $C$-definable Zariski closed subset of ${\rm GL}_{n}(C)$ such that ${\mathcal{G}}\subseteq{\rm stab}(FU)$. Then $FU$ is a $k$-definable Zariski closed subset of $V^{n}_{inv}$. Moreover if $U$ is bounded by $d$ then so is $FU$. * Proof. Assume that $U$ is bounded by $d$. Let $J=\left\\{P(X)\in C[x_{1,1},x_{1,2},\cdots,x_{n,n}]_{\leq d}\,\,\|\,\,\forall\,\,u\in U,P(u)=0\right\\}.$ Since $U$ is bounded by $d$, $U={\rm Zero}(J)\cap{\rm GL}_{n}(C)$. Let $\tilde{I}$ be the ideal in $K[x_{1,1},x_{1,2},\cdots,x_{n,n}]$ generated by $\\{P(F^{-1}X)\|P(X)\in J\\}$. Set $\tilde{I}_{\leq d}=\tilde{I}\cap K[x_{1,1},x_{1,2},\cdots,x_{n,n}]_{\leq d}.$ Then $\\{P(F^{-1}X)\|P(X)\in J\\}\subseteq\tilde{I}_{\leq d}$ and $FU={\rm Zero}(\tilde{I}_{\leq d})\cap V^{n}_{inv}$. Moreover, assume that $P(X)$ is a polynomial in $K[x_{1,1},x_{1,2},\cdots,x_{n,n}]_{\leq d}$ satisfying that $P(Fu)=0$ for any $u\in U$. Then one can easily see that $P(FX)$ is a $K$-linear combination of finitely many elements in $J$. Therefore $P(X)\in\tilde{I}_{\leq d}$. Now let $I_{\leq d}=\tilde{I}_{\leq d}\cap k[x_{1,1},x_{1,2},\cdots,x_{n,n}].$ We will show that $I_{\leq d}$ generates $\tilde{I}$. For this, it suffices to prove that $I_{\leq d}$ generates $\tilde{I}_{\leq d}$, since $\tilde{I}_{\leq d}$ generates $\tilde{I}$. We will use the similar argument as in (p.23, [16]) to prove this. Use $\langle I_{\leq d}\rangle$ to denote the ideal of $K[x_{1,1},x_{1,2},\cdots,x_{n,n}]$ generated by $I_{\leq d}$. Assume that $\tilde{I}_{\leq d}$ is not a subset of $\langle I_{\leq d}\rangle$. Pick $Q(X)\in\tilde{I}_{\leq d}\setminus\langle I_{\leq d}\rangle$ such that ${\rm num}(Q(X))$, the number of the monomials of $Q(X)$, is minimal. If ${\rm num}(f)=1$, it is clear that $Q(X)\in\langle I_{\leq d}\rangle$. Hence ${\rm num}(Q(X))>1$. Without loss of generality, we may assume that one of the coefficients of $Q(X)$ equals one and one of them, say $c$, is not in $k$. Let $\sigma\in{\mathcal{G}}$. We use $Q_{\sigma}(X)$ to denote the image of $Q(X)$ by applying $\sigma$ to the coefficients of $Q(X)$. For every $u\in U$, since $Q(\sigma^{-1}(Fu))=0$, $\sigma(Q(\sigma^{-1}(Fu))=Q_{\sigma}(\sigma(\sigma^{-1}(Fu)))=Q_{\sigma}(Fu)=0.$ It implies that $Q_{\sigma}(X)\in\tilde{I}_{\leq d}$ for all $\sigma\in{\mathcal{G}}$. Then the minimality of ${\rm num}(Q(X))$ implies that both $Q(X)-Q_{\sigma}(X)$ and $c^{-1}Q(X)-\sigma(c)^{-1}Q_{\sigma}(X)$ are in $\langle I_{\leq d}\rangle$. Therefore $\forall\,\,\sigma\in{\mathcal{G}},\,\,(\sigma(c)^{-1}-c^{-1})Q(X)\in\langle I_{\leq d}\rangle.$ Since $c\notin k$, there is $\sigma\in{\mathcal{G}}$ such that $\sigma(c)\neq c$. So $Q(X)\in\langle I_{\leq d}\rangle$, a contradiction. Hence $I_{\leq d}$ generates $\tilde{I}_{\leq d}$. $\Box$ The correctness of the statement below is almost obvious. However, since it will be used frequently, we state it as a lemma. ###### Lemma 2.2 Let $H$ be a subgroup of ${\rm GL}_{n}(C)$ and ${\mathcal{N}}={\rm stab}(FH)$. Then ${\mathcal{N}}_{F}=H$. * Proof. One can easily see that ${\mathcal{N}}_{F}\subseteq H$. Assume that $h\in H$. Then there is an element $\sigma_{h}$ in ${\rm GL}(V)$ satisfying that $\sigma_{h}(F)=Fh$. Now for any $h^{\prime}\in H$, $\sigma_{h}(Fh^{\prime})=Fhh^{\prime}\in FH\,\,\mbox{and}\,\,\sigma_{h}(Fh^{-1}h^{\prime})=Fh^{\prime}.$ They imply that $\sigma_{h}\in{\mathcal{N}}$ and then $h\in{\mathcal{N}}_{F}$. This concludes the lemma. $\Box$ ###### Proposition 2.3 Let $\tilde{d}$ be as in Corollary B.15 of Appendix B and ${\mathcal{H}}={\rm stab}(Z_{F,\tilde{d}})$. Then ${\mathcal{H}}$ is a proto-Galois group of (1) and moreover, $Z_{F,\tilde{d}}=F{\mathcal{H}}_{F}$. * Proof. We first show that $Z_{F,\tilde{d}}=F{\mathcal{H}}_{F}$. Assume that $Z_{F,\tilde{d}}=FH$. If $H$ is a group, then one has that ${\mathcal{H}}_{F}=H$ by Lemma 2.2. That is to say, $Z_{F,\tilde{d}}=FH=F{\mathcal{H}}_{F}$. Hence it suffices to show that $H$ is a group. As $F\in Z_{F,\tilde{d}}$, we have that $I_{n}\in H$. Suppose that $h_{1},h_{2}\in H$. For any $P(X)\in I_{F,\tilde{d}}$, the equality $P(Fh_{2})=0$ implies that $P(Xh_{2})$ is an element of $I_{F,\tilde{d}}$. Hence $P(Fh_{1}h_{2})=0$ and then $h_{1}h_{2}\in H$. It remains to prove that for any $h\in H$, $h^{-1}\in H$. As $H$ is closed under the multiplication, $Hh\subseteq H$. Multiplying both sides of $Hh\subseteq H$ by $h$ repeatedly, we have that $\cdots\subseteq Hh^{3}\subseteq Hh^{2}\subseteq Hh\subseteq H.$ It is easy to verify that $H$ is a Zariski closed subset of ${\rm GL}_{n}(C)$ and so is $Hh^{i}$ for all positive integer $i$. The stability of the above sequence indicates that $Hh^{i_{0}+1}=Hh^{i_{0}}$ for some $i_{0}\geq 0$ and therefore $Hh=H$. So $h^{-1}\in H$. Now we prove that ${\mathcal{H}}$ is a proto-Galois group of (1). First of all, as $Z_{F,\tilde{d}}$ is $k$-definable, ${\mathcal{G}}\subseteq{\rm stab}(Z_{F,\tilde{d}})={\mathcal{H}}$. By Corollary B.15, there is a proto- Galois group $\tilde{H}$ of (1) bounded by $\tilde{d}$. Since ${\mathcal{G}}_{F}\subseteq\tilde{H}$, ${\mathcal{G}}\subseteq{\rm stab}(F\tilde{H})$ and then by Lemma 2.1, $F\tilde{H}$ is a $k$-definable element of $N_{\tilde{d}}(V_{inv}^{n})$. Then there are polynomials $Q_{1}(X),\cdots,Q_{s}(X)$ in $k[x_{1,1},x_{1,2},\cdots,x_{n,n}]_{\leq\tilde{d}}$ such that $F\tilde{H}={\rm Zero}(Q_{1}(X),\cdots,Q_{s}(X))\cap V_{inv}^{n}$. It follows from $Q_{i}(F)=0$ that $Q_{i}\in I_{F,\tilde{d}}$ for all $i$ with $1\leq i\leq s$. Hence $F{\mathcal{H}}_{F}=Z_{F,\tilde{d}}\subseteq F\tilde{H}$. This implies that ${\mathcal{H}}_{F}\subseteq\tilde{H}$. Note that $({\mathcal{H}}_{F}^{\circ})^{t}$ is generated by all unipotent elements of ${\mathcal{H}}_{F}^{\circ}$ that are in $\tilde{H}^{\circ}$. Hence $({\mathcal{H}}_{F}^{\circ})^{t}\subseteq(\tilde{H}^{\circ})^{t}\subseteq{\mathcal{G}}_{F}^{\circ}\subseteq{\mathcal{H}}_{F}^{\circ}.$ Then the conclusion follows from the fact that $({\mathcal{H}}_{F}^{\circ})^{t}$ is a normal subgroup of ${\mathcal{H}}_{F}^{\circ}$. $\Box$ ## 3 Recovering Galois groups Throughout this section, $I_{F,\tilde{d}},Z_{F,\tilde{d}}$ and ${\mathcal{H}}$ are as in Proposition 2.3. We will first compute a Zariski closed subset of $Z_{F,\tilde{d}}$ whose stabilizer is ${\mathcal{G}}^{\circ}$. Then using the Galois group of finite extension, we construct $I_{\tilde{F}}$ and then the Galois group ${\mathcal{G}}$, where $I_{\tilde{F}}$ is defined in (2) with some fundamental matrix $\tilde{F}$ and $d=\infty$. Note that ${\mathcal{G}}^{\circ}$ is defined as the pre-image of ${\mathcal{G}}_{F}^{\circ}$ under the map $\phi_{F}$ in Section 1. It is well- known that ${\mathcal{G}}^{\circ}$ is equal to ${\mbox{\rm Gal}}(\bar{k}K/\bar{k})$ where $\bar{k}K$ is the Picard-Vessiot extension field of $\bar{k}$ for (1). ### 3.1 Identity component ${\mathcal{G}}^{\circ}$ Decomposing ${\mathcal{H}}_{F}$ into irreducible components, we obtain its identity component ${\mathcal{H}}_{F}^{\circ}$. The defining equations of ${\mathcal{H}}_{F}^{\circ}$ will lead to a Zariski closed subset $Z_{{\mathcal{H}}^{\circ}}$ of $Z_{F,\tilde{d}}$ such that the stabilizer of $Z_{{\mathcal{H}}^{\circ}}$ is ${\mathcal{H}}^{\circ}$. Let $\chi_{1},\cdots,\chi_{l}$ be generators of $X({\mathcal{H}}_{F}^{\circ})$, where $X({\mathcal{H}}_{F}^{\circ})$ is the group of characters of ${\mathcal{H}}_{F}^{\circ}$. We will show that each character corresponds to a hyperexponential element over $\bar{k}$. Assuming we can find $\chi_{1},\cdots,\chi_{l}$ (and we will show how this can be done), the results in [2] allow us to find algebraic relations among hyperexponential elements associated with $\chi_{1},\cdots,\chi_{l}$. These relations together with $Z_{{\mathcal{H}}^{\circ}}$ produce a Zariski closed subset $Z$ of $Z_{{\mathcal{H}}^{\circ}}$ such that the identity component of ${\rm stab}(Z)$ is ${\mathcal{G}}^{\circ}$. Using the similar argument as constructing $Z_{{\mathcal{H}}^{\circ}}$, we are able to find a Zariski closed subset whose stabilizer is ${{\mathcal{G}}^{\circ}}$. Let $\bar{k}K$ be the Picard-Vessiot extension field of $\bar{k}$ for (1) and $H$ a subgroup of ${\rm GL}_{n}(C)$. For brevity, we will use $H(\bar{k}K)$ to denote ${\rm Zero}(I(H))\cap{\rm GL}_{n}(\bar{k}K)$ where $I(H)$ is the vanishing ideal of $H$ in $C[x_{1,1},x_{1,2},\cdots,x_{n,n}]$. Let ${\mathcal{N}}$ be a subgroup of ${\rm GL}_{n}(V)$. Suppose that $F{\mathcal{N}}_{F}$ is $\bar{k}$-definable and $\Phi$ is the vanishing ideal of $F{\mathcal{N}}_{F}$ in $\bar{k}[x_{1,1},x_{1,2},\cdots,x_{n,n}]$. Let $\gamma$ be an element in ${\rm Zero}(\Phi)\cap{\rm GL}_{n}(\bar{k})$. Then we have the following porposition. ###### Proposition 3.1 * $(a)$ For every $\bar{F}\in F{\mathcal{N}}_{F}$, there is $g_{\bar{F}}\in{\mathcal{N}}_{F}$ such that $\left(\gamma g_{\bar{F}}\right)^{-1}\bar{F}\in{\mathcal{N}}_{F}^{\circ}(\bar{k}K);$ * $(b)$ Let $g_{\bar{F}}$ be an element in ${\mathcal{N}}_{F}$ such that $(a)$ holds for $\bar{F}$ and let $\alpha=\gamma g_{\bar{F}}$. Set $Z_{\alpha}={\rm Zero}\left(\left\\{P(\alpha^{-1}X)\,\,\|\,\,P(X)\in I({\mathcal{N}}_{F}^{\circ})\right\\}\right)\bigcap V_{inv}^{n}.$ Then ${\rm stab}(Z_{\alpha})={\mathcal{N}}^{\circ}$ and $Z_{\alpha}=\bar{F}{\mathcal{N}}_{F}^{\circ}$, which is a Zariski closed subset of $F{\mathcal{N}}_{F}$. * Proof. $(a)$. One can easily verify that ${\rm Zero}(\Phi)\cap{\rm GL}_{n}(\bar{k}K)=F{\mathcal{N}}_{F}(\bar{k}K)$. Then we have that $\bar{F}{\mathcal{N}}_{F}(\bar{k}K)=F{\mathcal{N}}_{F}(\bar{k}K)$ because $\bar{F}\in F{\mathcal{N}}_{F}$. Therefore $\gamma$ is an element of $\bar{F}{\mathcal{N}}_{F}(\bar{k}K)$. Equivalently, $\gamma^{-1}\bar{F}$ is in ${\mathcal{N}}_{F}(\bar{k}K)$. Since ${\mathcal{N}}_{F}$ is $C$-definable, there is $g_{\bar{F}}\in{\mathcal{N}}_{F}$ such that $g_{\bar{F}}^{-1}\gamma^{-1}\bar{F}=\left(\gamma g_{\bar{F}}\right)^{-1}\bar{F}\in{\mathcal{N}}_{F}^{\circ}(\bar{k}K).$ $(b)$. Note that $V_{inv}^{n}=\\{\bar{F}h\,\,\|\,\,h\in{\rm GL}_{n}(C)\\}$. From the definition of $Z_{\alpha}$, for any $\bar{F}h\in Z_{\alpha}$, we have that $\alpha^{-1}\bar{F}h$ belongs to ${\mathcal{N}}_{F}^{\circ}(\bar{k}K)$. It means that $h\in{\mathcal{N}}_{F}^{\circ}$, because $\alpha^{-1}\bar{F}\in{\mathcal{N}}_{F}^{\circ}(\bar{k}K)$ and $h\in{\rm GL}_{n}(C)$. Therefore $Z_{\alpha}\subseteq\bar{F}{\mathcal{N}}_{F}^{\circ}$. It is obvious that $\bar{F}{\mathcal{N}}_{F}^{\circ}$ is a subset of $Z_{\alpha}$. Hence $Z_{\alpha}=\bar{F}{\mathcal{N}}_{F}^{\circ}\subseteq\bar{F}{\mathcal{N}}_{F}=F{\mathcal{N}}_{F}$ that is a Zariski closed subset of $F{\mathcal{N}}_{F}$. Finally, we show that ${\rm stab}(Z_{\alpha})={\mathcal{N}}^{\circ}$. Denote ${\rm stab}(Z_{\alpha})$ by ${\mathcal{W}}$. As $Z_{\alpha}=\bar{F}{\mathcal{N}}_{F}^{\circ}$, Lemma 2.2 implies that ${\mathcal{W}}_{\bar{F}}={\mathcal{N}}_{F}^{\circ}$. From the assumption, $\bar{F}=F\bar{h}$ for some $\bar{h}\in{\mathcal{N}}_{F}$. Hence ${\mathcal{W}}_{F}=\bar{h}{\mathcal{W}}_{\bar{F}}\bar{h}^{-1}=\bar{h}{\mathcal{N}}_{F}^{\circ}\bar{h}^{-1}$. Owing to the normality of ${\mathcal{N}}_{F}^{\circ}$ in ${\mathcal{N}}_{F}$, we have that ${\mathcal{W}}_{F}={\mathcal{N}}_{F}^{\circ}$. Therefore ${\mathcal{W}}={\mathcal{N}}^{\circ}$. $\Box$ ###### Remark 3.2 Propositions 2.3 and 3.1 enable us to compute a Zariski closed subset $Z_{{\mathcal{H}}^{\circ}}$ of $Z_{F,\tilde{d}}$ such that ${\rm stab}(Z_{{\mathcal{H}}^{\circ}})={\mathcal{H}}^{\circ}$. It suffices to compute an element $\alpha$ in ${\rm Zero}(I_{F,\tilde{d}})\cap{\rm GL}_{n}(\bar{k})$ and $\bar{F}\in F{\mathcal{H}}_{F}$ satisfying that $\alpha^{-1}\bar{F}\in{\mathcal{H}}_{F}^{\circ}(\bar{k}K)$. Let $\alpha$ be an element of ${\rm Zero}(I_{F,\tilde{d}})\cap{\rm GL}_{n}(\bar{k})$ and $\bar{F}$ in $F{\mathcal{H}}_{F}$ satisfying that $\alpha^{-1}\bar{F}\in{\mathcal{H}}_{F}^{\circ}(\bar{k}K)$. From the above proposition, we know that such $\alpha$ and $\bar{F}$ exist. ###### Proposition 3.3 Let $\chi:{\mathcal{H}}_{F}^{\circ}\rightarrow C^{}$ be a character of ${\mathcal{H}}_{F}^{\circ}$, which is represented by a polynomial in $C[x_{1,1},x_{1,2},\cdots,x_{n,n},1/\det(X)]$. Then $\chi(\alpha^{-1}\bar{F})$ is a hyperexponential element over $\bar{k}$. Moreover for any $h\in{\mathcal{H}}_{F}^{\circ}$, $\chi(\alpha^{-1}\bar{F}h)=\chi(\alpha^{-1}\bar{F})\chi(h).$ Proof. Since $\chi$ is a character of ${\mathcal{H}}_{F}^{\circ}$, for any $h_{1},h_{2}\in{\mathcal{H}}_{F}^{\circ}(\bar{k}K)$, $\chi(h_{1}h_{2})=\chi(h_{1})\chi(h_{2})$. Note that $\alpha^{-1}\bar{F}\in{\mathcal{H}}_{F}^{\circ}(\bar{k}K)$. For any $h\in{\mathcal{H}}_{F}^{\circ}$, $\chi(\alpha^{-1}\bar{F}h)=\chi(\alpha^{-1}\bar{F})\chi(h)=\chi(\alpha^{-1}\bar{F})\chi(h).$ Suppose that $\sigma\in{\mathcal{G}}^{\circ}$. Then $\sigma(\alpha^{-1}\bar{F})=\alpha^{-1}\bar{F}[\sigma]$ for some $[\sigma]\in{\rm GL}_{n}(C)$. As $\alpha^{-1}\bar{F}$ belongs to ${\mathcal{H}}_{F}^{\circ}(\bar{k}K)$ that is $C$-definable, we have that $\sigma(\alpha^{-1}\bar{F})\in{\mathcal{H}}_{F}^{\circ}(\bar{k}K)$. It follows that $[\sigma]\in{\mathcal{H}}_{F}^{\circ}$. Hence for any $\sigma\in{\mathcal{G}}^{\circ}$, $\sigma\left(\frac{\chi(\alpha^{-1}\bar{F})^{\prime}}{\chi(\alpha^{-1}\bar{F})}\right)=\frac{\chi(\alpha^{-1}\bar{F}[\sigma])^{\prime}}{\chi(\alpha^{-1}\bar{F}[\sigma])}=\frac{\chi(\alpha^{-1}\bar{F})^{\prime}\chi([\sigma])}{\chi(\alpha^{-1}\bar{F})\chi([\sigma])}=\frac{\chi(\alpha^{-1}\bar{F})^{\prime}}{\chi(\alpha^{-1}\bar{F})}.$ Thus $\frac{\chi(\alpha^{-1}\bar{F})^{\prime}}{\chi(\alpha^{-1}\bar{F})}\in\bar{k}$. $\Box$ Suppose that $\chi_{1},\cdots,\chi_{l}$ are the generators of $X({\mathcal{H}}_{F}^{\circ})$, all of which are nontrivial and represented by polynomials in $C[x_{1,1},x_{1,2},\cdots,x_{n,n}]$. Then each character $\chi_{i}$ corresponds to a hyperexponential element $\chi_{i}(\alpha^{-1}\bar{F})$, denoted by $h_{i}$. Let $v_{i}=h_{i}^{\prime}/h_{i}$ for all $i$ with $1\leq i\leq l$, and $E=\bar{k}(h_{1},\cdots,h_{l})$ that is the Picard-Vessiot extension of $\bar{k}$ for the equations $\delta(Y)=\hbox{\rm diag}(v_{1},\cdots,v_{l})Y.$ Note that $E$ is a subfield of $\bar{k}K$. Let ${\mathbf{h}}=(h_{1},\cdots,h_{l})$. Then ${\mathbf{h}}$ is a fundamental matrix of the above equations and ${\mbox{\rm Gal}}(E/\bar{k})$ can naturally be embedded into $(C^{})^{l}$. Denote the image of ${\mbox{\rm Gal}}(E/\bar{k})$ by $T$ under this embedding. That is to say, $T=\left\\{\left.(c_{1},\cdots,c_{l})^{T}\in(C^{})^{l}\,\,\right\|\,\,\exists\,\,\sigma\in{\mbox{\rm Gal}}(E/\bar{k})\,\,s.t.\,\,\sigma(h_{i})=c_{i}h_{i},i=1,\cdots,l\right\\}.$ In [2], the authors show that when $C$ is an algebraically closed computable field, given $v_{1},\cdots,v_{l}$, one can compute a set of elements $S=\\{h_{\eta_{1}},\cdots,h_{\eta_{r}}\\}\subseteq\\{h_{1},\cdots,h_{l}\\}$ such that * $(i)$ $h_{\eta_{1}},\cdots,h_{\eta_{r}}$ are algebraically independent over $C$; * $(ii)$ for each $j\in\\{1,\cdots,l\\}$, there are an element $f_{j}\in\bar{k}$ and integers $m_{j},m_{i,j},m_{j}\neq 0$ satisfying $h_{j}^{m_{j}}=f_{j}\prod_{i=1}^{r}h_{\eta_{i}}^{m_{i,j}}$ if $S$ is nonempty, or $h_{j}^{m_{j}}=f_{j}$ if $S$ is empty. The equalities in $(ii)$ generate almost all algebraic relations among $h_{1},\cdots,h_{l}$. Due to the proof of Proposition 2.5 in [2], the set $\\{y_{j}^{m_{j}}-\prod_{i=1}^{r}y_{\eta_{i}}^{m_{i,j}},j=1,2,\cdots,l\\}$ defines an algebraic subgroup of $(C^{})^{l}$, whose identity component is equal to $T$. Let $\varphi=(\chi_{1},\cdots,\chi_{l})$. Then $\varphi$ is a surjective morphism from ${\mathcal{H}}_{F}^{\circ}$ to $(C^{})^{l}$, for all $\chi_{i}$ are nontrivial. As $\bar{F}\in F{\mathcal{H}}_{F}$, it follows from the normality of ${\mathcal{H}}_{F}^{\circ}$ in ${\mathcal{H}}_{F}$ that ${\mathcal{H}}_{\bar{F}}^{\circ}={\mathcal{H}}_{F}^{\circ}$. Thus ${\mathcal{G}}_{\bar{F}}^{\circ}\subseteq{\mathcal{H}}_{\bar{F}}^{\circ}={\mathcal{H}}_{F}^{\circ}$. Moreover, we have ###### Lemma 3.4 $\varphi({\mathcal{G}}_{\bar{F}}^{\circ})=T.$ * Proof. Note that ${\mathcal{G}}^{\circ}={\mbox{\rm Gal}}(\bar{k}K/\bar{k})$ and $E\subseteq\bar{k}K$ that is a Picard-Vessiot extension field. By the Galois theory, the map $\psi:{\mathcal{G}}^{\circ}\rightarrow{\mbox{\rm Gal}}(E/\bar{k})$ given by $\psi(\sigma)=\sigma\|_{E}$ for any $\sigma\in{\mathcal{G}}^{\circ}$ is surjective. For any $[\sigma]\in{\mathcal{G}}_{\bar{F}}^{\circ}$, there is $\sigma\in{\mathcal{G}}^{\circ}$ such that $\sigma(\bar{F})=\bar{F}[\sigma]$ and then $\displaystyle\psi(\sigma)({\mathbf{h}})$ $\displaystyle=\sigma({\mathbf{h}})=(\sigma(h_{1}),\cdots,\sigma(h_{l}))=(\chi_{1}(\alpha^{-1}\bar{F}[\sigma]),\cdots,\chi_{l}(\alpha^{-1}\bar{F}[\sigma]))$ $\displaystyle=(\chi_{1}(\alpha^{-1}\bar{F})\chi_{1}([\sigma]),\cdots,\chi_{l}(\alpha^{-1}\bar{F})\chi_{l}([\sigma]))=(\chi_{1}([\sigma])h_{1},\cdots,\chi_{l}([\sigma])h_{l}).$ By the definition of $T$, we have that $\varphi([\sigma])=(\chi_{1}([\sigma]),\cdots,\chi_{l}([\sigma]))\in T$. Now assume that $(c_{1},\cdots,c_{l})^{T}\in T$. Then there is $\sigma\in{\mbox{\rm Gal}}(E/\bar{k})$ such that $\sigma(h_{j})=c_{j}h_{j}$ for all $j$ with $1\leq j\leq l$. Due to the surjectivity of $\psi$, there is $\hat{\sigma}\in{\mathcal{G}}^{\circ}$ such that $\psi(\hat{\sigma})=\sigma$. Assume that $\hat{\sigma}(\bar{F})=\bar{F}[\hat{\sigma}]$ for some $[\hat{\sigma}]\in{\mathcal{G}}_{\bar{F}}^{\circ}$. It follows that $c_{j}=\frac{\sigma(h_{j})}{h_{j}}=\frac{\hat{\sigma}(h_{j})}{h_{j}}=\frac{\chi_{j}(\alpha^{-1}\bar{F}[\hat{\sigma}])}{\chi_{j}(\alpha^{-1}\bar{F})}=\frac{\chi_{j}(\alpha^{-1}\bar{F})\chi_{j}([\hat{\sigma}])}{\chi_{j}(\alpha^{-1}\bar{F})}=\chi_{j}([\hat{\sigma}]),\,\,j=1,\cdots,l.$ In other words, $(c_{1},\cdots,c_{l})^{T}$ is the image of $[\hat{\sigma}]$ under the morphism $\varphi$. So $\varphi({\mathcal{G}}_{\bar{F}}^{\circ})=T$. $\Box$ Set $J=\\{P(\alpha^{-1}X)\,\,\|\,\,P(X)\in I({\mathcal{H}}_{F}^{\circ})\\}\bigcup\left\\{\chi_{j}(\alpha^{-1}X)^{m_{j}}-f_{j}\prod_{i=1}^{r}\chi_{\eta_{i}}(\alpha^{-1}X)^{m_{i,j}},j=1,\cdots,l\right\\}.$ Let $Z_{J}={\mbox{\rm Zero}}(J)\cap V^{n}_{inv}$ and $\bar{{\mathcal{H}}}={\rm stab}(Z_{J})$. ###### Proposition 3.5 $\bar{{\mathcal{H}}}^{\circ}={\mathcal{G}}^{\circ}$. * Proof. Assume that $Z_{J}=\bar{F}\bar{H}$ where $\bar{H}\subseteq{\rm GL}_{n}(C)$. Recall that $\alpha^{-1}\bar{F}\in{\mathcal{H}}_{F}^{\circ}(\bar{k}K)$. It is easy to verify that $\bar{H}={\mathcal{H}}_{F}^{\circ}\bigcap{\mbox{\rm Zero}}\left(\left\\{\chi_{j}^{m_{j}}-\prod_{i=1}^{r}\chi_{\eta_{i}}^{m_{i,j}},\,\,j=1,2,\cdots,l\right\\}\right).$ Therefore $\bar{H}$ is a group and then by Lemma 2.2, $\bar{{\mathcal{H}}}_{\bar{F}}=\bar{H}$. To prove $\bar{{\mathcal{H}}}^{\circ}={\mathcal{G}}^{\circ}$, it suffices to show that $\bar{H}^{\circ}={\mathcal{G}}_{\bar{F}}^{\circ}$. Since $\varphi=(\chi_{1},\cdots,\chi_{l})$ is surjective, $\varphi(\bar{H})=(C^{})^{l}\bigcap{\mbox{\rm Zero}}\left(\left\\{y_{j}^{m_{j}}-\prod_{i=1}^{r}y_{\eta_{i}}^{m_{i,j}},j=1,2,\cdots,l\right\\}\right).$ The discussion before Lemma 3.4 indicates that the identity component of $\varphi(\bar{H})$ is equal to $T$. Note that $\ker(\varphi)=({\mathcal{H}}^{\circ}_{F})^{t}=({\mathcal{H}}_{\bar{F}}^{\circ})^{t}$. Then we have $\ker(\varphi)\subseteq{\mathcal{G}}_{\bar{F}}^{\circ}$, since ${\mathcal{H}}$ is a proto-Galois group. As $Z_{J}$ is $\bar{k}$-definable, ${\mathcal{G}}^{\circ}\subseteq\bar{{\mathcal{H}}}$ and then ${\mathcal{G}}_{\bar{F}}^{\circ}\subseteq\bar{{\mathcal{H}}}_{\bar{F}}=\bar{H}$. Now we have $[\bar{H}:{\mathcal{G}}_{\bar{F}}^{\circ}]=\left[\bar{H}/\ker(\varphi):{\mathcal{G}}_{\bar{F}}^{\circ}/\ker(\varphi)\right]=[\varphi(\bar{H}):\varphi({\mathcal{G}}_{\bar{F}}^{\circ})].$ Owing to Lemma 3.4, $\varphi({\mathcal{G}}_{\bar{F}}^{\circ})$ is equal to $T$ that is the identity component of $\varphi(\bar{H})$. Hence $[\bar{H}:{\mathcal{G}}_{\bar{F}}^{\circ}]$ is finite. Thus $\bar{H}^{\circ}={\mathcal{G}}_{\bar{F}}^{\circ}$, which completes the proof. $\Box$ Due to Proposition 3.1, there are $\beta\in{\rm Zero}(J)\cap{\rm GL}_{n}(\bar{k})\,\,\mbox{and}\,\,\tilde{F}\in\bar{F}\bar{{\mathcal{H}}}_{\bar{F}}$ (3) such that $\beta^{-1}\tilde{F}\in{\mathcal{G}}_{\bar{F}}^{\circ}(\bar{k}K)$. Let $Z_{\beta}={\rm Zero}\left(\left\\{P(\beta^{-1}X)\,\,\|\,\,P(X)\in I({\mathcal{G}}_{\bar{F}}^{\circ})\right\\}\right)\cap V_{inv}^{n}.$ By Proposition 3.1 again, we have that $Z_{\beta}=\tilde{F}{\mathcal{G}}_{\bar{F}}^{\circ}$ that is a Zariski closed subset of $\bar{F}\bar{{\mathcal{H}}}_{\bar{F}}$ and ${\rm stab}(Z_{\beta})={\mathcal{G}}^{\circ}$. Furthermore, by Lemma 2.2, ${\mathcal{G}}_{\tilde{F}}^{\circ}={\mathcal{G}}_{\bar{F}}^{\circ}$. ### 3.2 Galois group ${\mathcal{G}}$ Let $\beta$ and $\tilde{F}$ be as in (3). Let $k_{G}$ be the Galois closure of $k(\beta^{-1})$, where $k(\beta^{-1})$ denotes the extension field of $k$ by joining the entries of $\beta^{-1}$. For any $\tau\in{\mbox{\rm Gal}}(k_{G}/k)$, set $J_{\tau(\beta)}=\left\langle\left\\{P(\tau(\beta)^{-1}X)\,\,\|\,\,P(X)\in I({\mathcal{G}}_{\tilde{F}}^{\circ})\right\\}\right\rangle$ where $\langle\rangle$ denotes the ideal in $k_{G}[x_{1,1},x_{1,2},\cdots,x_{n,n}]$ generated by $$. ###### Proposition 3.6 Let $I_{\tilde{F}}$ be defined in (2) with $F=\tilde{F}$ and $d=\infty$ . Then $I_{\tilde{F}}=\left(\bigcap_{\tau\in{\mbox{\rm Gal}}(k_{G}/k)}J_{\tau(\beta)}\right)\bigcap k[x_{1,1},x_{1,2},\cdots,x_{n,n}].$ Proof. Denote the ideal in the righthand side of the above equality by $\Phi$. Suppose that $P(X)\in\Phi$. Then $P(X)\in J_{\beta}$. From the previous subsection, $\tilde{F}\in Z_{\beta}$. So $P(\tilde{F})=0$. That is to say, $P(X)\in I_{\tilde{F}}$. Thus $\Phi\subseteq I_{\tilde{F}}$. Conversely, suppose that $P(X)\in I_{\tilde{F}}$. Then $P(\tilde{F})=0$. Applying ${\mathcal{G}}^{\circ}$ to it, we obtain that $P(\tilde{F}g)=0$ for every $g\in{\mathcal{G}}_{\tilde{F}}^{\circ}$ and furthermore the polynomial $P(\tilde{F}X)$ vanishes on ${\mathcal{G}}_{\tilde{F}}^{\circ}(\bar{k}K)$. Since $\beta^{-1}\tilde{F}\in{\mathcal{G}}_{\tilde{F}}^{\circ}(\bar{k}K)$, $\tilde{F}{\mathcal{G}}_{\tilde{F}}^{\circ}(\bar{k}K)=\beta{\mathcal{G}}_{\tilde{F}}^{\circ}(\bar{k}K)$. It implies that $P(\beta X)$ vanishes on ${\mathcal{G}}_{\tilde{F}}^{\circ}(\bar{k}K)$. Consequently, $P(\beta X)$ belongs to $\langle I({\mathcal{G}}_{\tilde{F}}^{\circ})\rangle$ and then $P(X)\in J_{\beta}$. Because all coefficients of $P(X)$ are in $k$, $P(X)\in J_{\tau(\beta)}$ for all $\tau\in{\mbox{\rm Gal}}(k_{G}/k)$. Hence $P(X)\in\Phi$. $\Box$ If $I_{\tilde{F}}$ is computed, then it is easy to verify that ${\mathcal{G}}_{\tilde{F}}=\\{g\in{\rm GL}_{n}(C)\,\,\|\,\,\forall\,\,P(X)\in I_{\tilde{F}},\,\,P(Xg)\in I_{\tilde{F}}\\}.$ In the following, we present another method to compute ${\mathcal{G}}_{\tilde{F}}$ that avoids computing $I_{\tilde{F}}$. From Proposition 3.20 and Theorem 3.11 of [8], there is a Picard-Vessiot extension field, say $\tilde{K}$, of $k$ that contains $k_{G}$ and $K$ as subfields. Then Galois theory implies that if $\tau$ is an element of ${\mbox{\rm Gal}}(k_{G}/k)$ (or ${\mathcal{G}}$), then there is $\rho\in{\mbox{\rm Gal}}(\tilde{K}/k)$ such that the restriction of $\rho$ on $k_{G}$ (or $K$) is equal to $\tau$. Let $\\{Q_{1}(X),\cdots,Q_{\nu}(X)\\}$ be a set of generators of $I({\mathcal{G}}_{\bar{F}}^{\circ})$. Set $\tilde{G}=\bigcup_{\tau\in{\mbox{\rm Gal}}(k_{G}/k)}\left\\{\,\,g\in{\rm GL}_{n}(C)\left\|\,\,Q_{i}(\tau(\beta)^{-1}\tilde{F}g)=0,\,\,\mbox{for all $i=1,\cdots,\nu$}\right.\right\\}.$ Then we have ###### Proposition 3.7 ${\mathcal{G}}_{\tilde{F}}=\tilde{G}$. * Proof. We first prove that ${\mathcal{G}}_{\tilde{F}}\subseteq\tilde{G}$. Assume that $[\sigma]\in{\mathcal{G}}_{\tilde{F}}$. Then there is $\sigma\in{\mathcal{G}}$ such that $\sigma({\tilde{F}})={\tilde{F}}[\sigma]$ and furthermore there is $\rho\in{\mbox{\rm Gal}}(\tilde{K}/k)$ such that $\rho\|_{K}=\sigma$. Let $\tau=\rho\|_{k_{G}}$. Note that $\tau\in{\mbox{\rm Gal}}(k_{G}/k)$. Then for all $i$ with $1\leq i\leq\nu$, $\rho(Q_{i}(\beta^{-1}{\tilde{F}}))=Q_{i}(\tau(\beta)^{-1}\sigma({\tilde{F}}))=Q_{i}({\tau(\beta)^{-1}\tilde{F}}[\sigma])=0.$ This implies that $[\sigma]\in\tilde{G}$. Conversely, assume that $g\in\tilde{G}$. Then there is $\tau\in{\mbox{\rm Gal}}(k_{G}/k)$ such that $Q_{i}(\tau(\beta)^{-1}{\tilde{F}}g)=0$ for all $i$ with $1\leq i\leq\nu$. There is $\rho\in{\mbox{\rm Gal}}(\tilde{K}/k)$ such that $\rho\|_{k_{G}}=\tau^{-1}$. Since $\rho\|_{K}\in{\mathcal{G}}$, there is $h\in{\mathcal{G}}_{\tilde{F}}$ such that $\rho\|_{K}({\tilde{F}})={\tilde{F}}h$. Now we have that $\forall\,\,i=1,\cdots,\nu,\,\,\rho(Q_{i}(\tau(\beta)^{-1}{\tilde{F}}g))=Q_{i}(\beta^{-1}{\tilde{F}}hg)=0.$ Hence $\beta^{-1}{\tilde{F}}hg\in{\mathcal{G}}_{\tilde{F}}^{\circ}(\tilde{K})$, which implies that $hg\in{\mathcal{G}}_{\tilde{F}}^{\circ}$ and thus $g\in{\mathcal{G}}_{\tilde{F}}$. $\Box$ ## 4 Simplified Hrushovski’s Algorithm Now we are ready to present the algorithm. Instead of ${\mathcal{G}}$, we shall compute ${\mathcal{G}}_{\tilde{F}}$ for some fundamental matrix $\tilde{F}$. Throughout this section, $C$ will denote the algebraic closure of a computable field of characteristic zero. ###### Algorithm 4.1 Input: A linear differential equation (1) with coefficients in $C(t)$. Output: the Galois group ${\mathcal{G}}_{\tilde{F}}$. * $(a)$ By Corollary B.15, determine an integer $\tilde{d}$ such that there is a proto-Galois group of (1) bounded by $\tilde{d}$. * $(b)$ Compute a fundamental matrix $F$ (the first finitely many terms of its formal power series expansion at some point). Compute $I_{F,\tilde{d}}$ and then ${\mathcal{H}}_{F}$, where $I_{F,\tilde{d}}$ is defined in (2) and ${\mathcal{H}}$ is the stabilizer of ${\rm Zero}(I_{F,\tilde{d}})\cap V_{inv}^{n}$. * $(c)$ Compute ${\mathcal{H}}_{F}^{\circ}$ and find a zero $\alpha$ of $I_{F,\tilde{d}}$ in ${\rm GL}_{n}(\overline{C(t)})$ and $\bar{F}\in F{\mathcal{H}}_{F}$ such that $\alpha^{-1}\bar{F}$ is an element of ${\mathcal{H}}_{F}^{\circ}(\bar{k}K)$. * $(d)$ Compute generators of $X({\mathcal{H}}_{F}^{\circ})$, say $\chi_{1},\cdots,\chi_{l}$, which are represented by polynomials in $C[x_{1,1},x_{1,2},\cdots,x_{n,n}]$. Denote $\chi_{i}(\alpha^{-1}\bar{F})$ by $h_{i}$ for $i=1,2,\cdots,l$. By Proposition 3.1, $h_{i}$ is hyperexponential over $\overline{C(t)}$. Compute these $h_{i}$. * $(e)$ Using the method in [2], compute algebraic relations for $h_{1},\cdots,h_{l}$, say $\left\\{h_{j}^{m_{j}}-f_{j}\prod_{i=1}^{r}h_{\eta_{i}}^{m_{i,j}},j=1,2,\cdots,l\right\\}$ where $f_{j}\in\overline{C(t)}$ and $h_{\eta_{1}},\cdots,h_{\eta_{r}}$ are algebraically independent over $C(t)$. * $(f)$ Denote a set of generators of $I({\mathcal{H}}_{F}^{\circ})$ by $\\{P_{1}(X),\cdots,P_{\mu}(X)\\}$ and set $J=\\{P_{i}(\alpha^{-1}X),i=1,\cdots,\mu\\}\bigcup\left\\{\chi_{j}^{m_{j}}(\alpha^{-1}X)-f_{j}\prod_{i=1}^{r}\chi_{\eta_{i}}^{m_{i,j}}(\alpha^{-1}X),j=1,\cdots,l\right\\}.$ Compute $\bar{{\mathcal{H}}}={\rm stab}({\rm Zero}(J)\cap V_{inv}^{n})$ and then $\bar{{\mathcal{H}}}^{\circ}$ that is equal to ${\mathcal{G}}^{\circ}$. * $(g)$ As in the step $(c)$, compute ${\mathcal{G}}_{\bar{F}}^{\circ}$ and find $\beta$ in ${\rm Zero}(J)\cap{\rm GL}_{n}(\overline{C(t)})$ and $\tilde{F}\in\bar{F}\bar{{\mathcal{H}}}_{\bar{F}}$ such that $\beta^{-1}\tilde{F}\in{\mathcal{G}}_{\bar{F}}^{\circ}(\bar{k}K)$. * $(h)$ Denote a set of generators of $I({\mathcal{G}}_{\bar{F}}^{\circ})$ by $\\{Q_{1}(X),\cdots,Q_{\nu}(X)\\}$. Let $k_{G}$ be the Galois closure of $C(t)(\beta^{-1})$, where $C(t)(\beta^{-1})$ is the extension field of $C(t)$ by joining the entries of $\beta^{-1}$. Compute ${\mbox{\rm Gal}}(k_{G}/C(t))$ and ${\mathcal{G}}_{\tilde{F}}=\bigcup_{\tau\in{\mbox{\rm Gal}}(k_{G}/C(t))}\left\\{g\in{\rm GL}_{n}(C)\,\,\|\,\,\forall\,\,i=1,\cdots,\nu,\,\,Q_{i}(\tau(\beta)^{-1}\tilde{F}g)=0\right\\}.$ * $(i)$ Return ${\mathcal{G}}_{\tilde{F}}$. The correctness of the algorithm follows from the results in Sections 2 and 3. In the following, we will present several computation details omitted in the previous sections. Generally, it is difficult to find a fundamental matrix of (1). What we can compute is the first finitely many terms of formal power series solutions of (1) at some point of $C$. Let $z$ be a generic point of $C$. Expanding $A$ at $t=z$, we have $A=A_{0}+A_{1}(t-z)+A_{2}(t-z)^{2}+\cdots,\,\,A_{i}\in{\rm Mat}_{n}\left(C\left[z,\frac{1}{q(z)}\right]\right)$ (4) where $q(z)$ is a polynomial in $C[z]$ such that $q(t)$ is the least common multiple of the denominators of the entries of $A$. Using the above expansion, we can compute a formal power series solutions of (1) that has the following form $\Gamma_{z}=I_{n}+D_{1}(t-z)+D_{2}(t-z)^{2}+\cdots,\,\,D_{i}\in{\rm Mat}_{n}\left(C\left[z,\frac{1}{q(z)}\right]\right).$ If $a$ is an element of $C$ that does not vanish $q(z)$, then $\Gamma_{z}$ can be specialized to $\Gamma_{a}$ that is the formal power series expansion of some fundamental matrix of (1) at $t=a$. In this section, we will use $F_{a}$ to denote the fundamental matrix of (1) in ${\rm GL}_{n}(K)$ whose formal power series expansion at $t=a$ is of the form $\Gamma_{a}$. We will begin with a fundamental matrix $F_{a}$ and then may replace it by other fundamental matrix if necessary during the computation process. ### 4.1 Computing $I_{{F_{a}},\tilde{d}}$ and ${\mathcal{H}}_{F_{a}}$ Corollary A.7 says that the coefficients of defining polynomials in $C(t)[x_{1,1},\cdots,x_{n,n}]_{\leq\tilde{d}}$ of $Z_{F,\tilde{d}}$ can be chosen to be rational functions bounded by an integer $\ell$. Without loss of generality, we may assume that all these coefficients are polynomial in $t$ which are bounded by $2\ell$. Let $P_{{\mathbf{c}}}(X)=\sum_{\|\vec{m}\|\leq\tilde{d}}\left(\sum_{0\leq i\leq 2\ell}c_{i,\vec{m}}(t-a)^{i}\right)X^{\vec{m}},\,\,{\mathbf{c}}=(\cdots,c_{i,\vec{m}},\cdots).$ where the $c_{i,\vec{m}}$ are indeterminates. A small modification of Theorem 1 in [1] yields the following theorem that bounds the order of $P_{{\mathbf{c}}}(F_{a})$. ###### Theorem 4.2 One can compute an integer $N$ depending on $A,n,\ell$ such that $P_{\mathbf{c}}(F_{a})=0\,\,\mbox{or}\,\,ord_{t=a}(P_{\mathbf{c}}(F_{a}))\leq N.$ * Proof. Consider $F_{a}$ as a vector with $n^{2}$ entries. Then $F_{a}$ is a solution of the system (${A^{\oplus n}}$). The system (${A^{\oplus n}}$) is defined in Appendix A. Then the theorem follows from Theorem 1 in [1]. $\Box$ The above theorem can be generalized into the case that the coefficients of $P_{{\mathbf{c}}}(X)$ involve algebraic functions. More precisely, let $\gamma$ be an algebraic function with minimal polynomial $Q(x)$ and let $l=\deg(Q(x))$. Assume that ${\mathbf{w}}=(1,\gamma,\gamma^{2},\cdots,\gamma^{l-1})^{T}$. Then it is easy to see that ${\mathbf{w}}$ is a solution of linear differential equations $\delta(Y)=BY$ where $B\in{\rm Mat}_{l}(C(t))$ can be constructed from $Q(x)$. Let $\tilde{P}_{{\mathbf{c}}}(X)=\sum_{\|\vec{m}\|\leq\tilde{d}}\left(\sum_{j=1}^{l}\left(\sum_{0\leq i\leq 2\ell}c_{i,j,\vec{m}}(t-a)^{i}\right)\gamma^{j}\right)X^{\vec{m}},\,\,{\mathbf{c}}=(\cdots,c_{i,j,\vec{m}},\cdots).$ Assume that $t=a$ is a regular point of $\gamma$. Then ###### Corollary 4.3 There is an integer $\tilde{N}$ depending on $A,Q(x),n,\ell$ such that $\tilde{P}_{\mathbf{c}}(F_{a})=0\,\,\mbox{or}\,\,\mbox{\rm ord}_{t=a}\tilde{P}_{{\mathbf{c}}}(F_{a})\leq\tilde{N}.$ * Proof. We only need to consider the system $\delta(Y)=\hbox{\rm diag}(\underbrace{A,\cdots,A}_{n},B)Y.$ Then the corollary follows from the above theorem. $\Box$ Let ${\cal S}$ be the set of the coefficients of the first $N+2$ terms of $P_{{\mathbf{c}}}(F_{a})$. ${\cal S}$ is a linear system in ${\mathbf{c}}$ and $P_{\bar{{\mathbf{c}}}}(F_{a})=0\Leftrightarrow\,\,\mbox{$\bar{{\mathbf{c}}}$ is a solution of ${\cal S}$}.$ So computing $I_{{F_{a}},\tilde{d}}$ is reduced into solving the linear system ${\cal S}$. Assume that $I_{{F_{a}},\tilde{d}}$ is computed. Then we can compute ${\mathcal{H}}_{F_{a}}$ as follows. For any $h\in{\mathcal{H}}_{F_{a}}$, ${F_{a}}h\in Z_{{F_{a}},\tilde{d}}$. It implies that for any $P(X)\in I_{{F_{a}},\tilde{d}}$, $P(F_{a}h)=0$ and so that $P(Xh)\in I_{F_{a},\tilde{d}}$. This induces the defining equations for ${\mathcal{H}}_{F_{a}}$. More precisely, let $P_{1}(X),\cdots,P_{\mu}(X)$ be a $C(t)$-basis of $I_{F_{a},\tilde{d}}$. Let ${\mathbf{x}}=(\cdots,X^{\vec{m}},\cdots)$ be a vector consisting of all monomials in $x_{1,1},\cdots,x_{n,n}$ with degree not greater than $\tilde{d}$, where $X^{\vec{m}}=x_{1,1}^{m_{1,1}}x_{1,2}^{m_{1,2}}\cdots x_{n,n}^{m_{n,n}}$. For any $h\in{\rm GL}_{n}(C)$, there is $[h]\in{\rm GL}_{n^{2}+\tilde{d}\choose\tilde{d}}(C)$ such that $h\cdot{\mathbf{x}}=(\cdots,(Xh)^{\vec{m}},\cdots)={\mathbf{x}}[h].$ For any $i=1,\cdots,\mu$, there is ${\mathbf{p}}_{i}\in C(t)^{n^{2}+\tilde{d}\choose\tilde{d}}$ such that $P_{i}(X)={\mathbf{x}}{\mathbf{p}}_{i}$. Then we have $h\in{\mathcal{H}}_{F_{a}}\Leftrightarrow\forall i,P_{i}(Xh)={\mathbf{x}}[h]{\mathbf{p}}_{i}\in I_{F_{a},\tilde{d}}\Leftrightarrow\forall i,[h]{\mathbf{p}}_{i}\wedge{\mathbf{p}}_{1}\wedge{\mathbf{p}}_{2}\wedge\cdots\wedge{\mathbf{p}}_{\mu}=0.$ This induces the defining equations for ${\mathcal{H}}_{F_{a}}$. ### 4.2 Computing $\alpha$ and $\bar{F}$ Assume that we have calculated $I_{F_{a},\tilde{d}}$ and ${\mathcal{H}}_{F_{a}}$. Decomposing ${\mathcal{H}}_{F_{a}}$ into irreducible components, we obtain ${\mathcal{H}}_{F_{a}}^{\circ}$. Compute a zero $\bar{\alpha}$ of $I_{F_{a},\tilde{d}}$ in ${\rm GL}_{n}(\overline{C(t)})$. Assume that $b\in C$ is a regular point of $\bar{\alpha}^{-1}$ and $q(b)\neq 0$, where $q(z)$ is as in (4). It is well-known that there is $g\in{\rm GL}_{n}(C)$ such that $F_{a}=F_{b}g$. In general, it is difficult to find such $g$, because we can only compute the first finitely many terms of $\Gamma_{a}$ and $\Gamma_{b}$. Note that $\Gamma_{a}$ (resp. $\Gamma_{b}$) is the formal power sereis expansion of $F_{a}$ (resp. $F_{b}$) at $t=a$ (resp. $t=b$). Fortunately, it is easy to find $h\in{\rm GL}_{n}(C)$ such that $F_{b}h$ belongs to $F_{a}{\mathcal{H}}_{F_{a}}$. Such $h$ can be calculated as below. $F_{b}h\in F_{a}{\mathcal{H}}_{F_{a}}$ if and only if for any $P(X)\in I_{F_{a},\tilde{d}}$, $P(F_{b}h)=0$. Using the bound given in Theorem 4.2, the latter conditions derive the defining equations for $h$. Assume that we have found such $h$. Let $\bar{F}=F_{b}h$. Proposition 3.1 tells us that there is $\bar{g}\in{\mathcal{H}}_{F_{a}}$ such that $(\bar{\alpha}\bar{g})^{-1}\bar{F}\in{\mathcal{H}}_{F_{a}}^{\circ}(\bar{k}K)$. Since both $\bar{\alpha}^{-1}$ and $\bar{F}$ can be expanded as formal power series at the point $t=b$, the bound given in Theorem 4.2 allows us to derive the defining equations for $\bar{g}$ from the condition $(\bar{\alpha}\bar{g})^{-1}\bar{F}\in{\mathcal{H}}_{F_{a}}^{\circ}(\bar{k}K)$. Compute such $\bar{g}$ and let $\alpha=\bar{\alpha}\bar{g}$. ### 4.3 Computing $h_{i}$ By Remark B.18 in Appendix B, we can find a set of generators of $X({\mathcal{H}}_{F_{a}}^{\circ})$, say $\chi_{1},\cdots,\chi_{l}$. Assume that $\chi_{1},\cdots,\chi_{l}$ are defined by polynomials of degree not greater than an integer $\kappa$. Denote the monomials in entries of $\alpha^{-1}\bar{F}$ with degree not greater than $\kappa$ by ${\mathbf{m}}_{1},\cdots,{\mathbf{m}}_{m}$. Then each ${\mathbf{m}}_{i}$ satisfies a linear differential operator $L_{i}$ with coefficients in $C(t)(\alpha^{-1})$. Let $L=LCLM(L_{1},\cdots,L_{m})$. For each $i=1,\cdots,l$, $h_{i}=\chi_{i}(\alpha^{-1}\bar{F})$ is a hyperexponential solution of $L$. To compute $h_{i}$, it means to calculate $v_{i}=h_{i}^{\prime}/h_{i}$ where $i=1,\cdots,l$. From [3], one can compute all hyperexponential solutions of $L$ and then the bounds for minimal polynomials of $\bar{h}^{\prime}/\bar{h}$ where $\bar{h}$ is any hyperexponential solution. Using these bounds and Hermite-Padé approximation, one can recover $v_{i}$ from the series expansion of $\chi_{i}(\alpha^{-1}\bar{F})^{\prime}/\chi_{i}(\alpha^{-1}\bar{F})$. ### 4.4 Computing ${\mathcal{G}}_{\tilde{F}}$ The method described in Section 4.2 can be adapted to find $\beta$ and $\tilde{F}$ in Step (g). So far, in Step (g), we obtain $\beta$ and $\tilde{F}$ satisfying that $\beta^{-1}\tilde{F}\in{\mathcal{G}}_{\bar{F}}^{\circ}(\bar{k}K)$, and a set of generators of $I({\mathcal{G}}_{\bar{F}}^{\circ})$, denoted by $\\{Q_{1}(X),\cdots,Q_{\nu}(X)\\}$. Assume that $t=c$ is the point we pick to find $\beta$ and $\tilde{F}$ in Step (g). That is to say, both $\beta^{-1}$ and $\tilde{F}$ can be expanded as formal power series at the point $t=c$. By Corollary 4.3, there is an integer $\tilde{N}$ such that for all $i$ with $1\leq i\leq\nu$ and $\tau\in{\mbox{\rm Gal}}(k_{G}/C(t))$, $Q_{i}(\tau(\beta)^{-1}\tilde{F}{\mathbf{c}})=0\quad\mbox{or}\quad\mbox{\rm ord}_{t=c}(Q_{i}(\tau(\beta)^{-1}\tilde{F}{\mathbf{c}}))\leq\tilde{N}.$ For each $\tau\in{\mbox{\rm Gal}}(k_{G}/C(t))$, let ${\cal S}_{\tau}$ be the set of coefficients of the first $\tilde{N}+2$ terms of formal power series $Q_{1}(\tau(\beta)^{-1}\tilde{F}{\mathbf{c}}),\cdots,Q_{\nu}(\tau(\beta)^{-1}\tilde{F}{\mathbf{c}})$. Then for each $\tau\in{\mbox{\rm Gal}}(k_{G}/k)$, ${\cal S}_{\tau}$ is the set of polynomials in ${\mathbf{c}}$ and for any $g\in{\rm GL}_{n}(C)$, $Q_{i}(\tau(\beta)^{-1}\tilde{F}g)=0$ for all $i$ with $1\leq i\leq\nu$ if and only if $g\in{\rm Zero}({\cal S}_{\tau}).$ Let ${\mathcal{G}}_{\tilde{F}}=\left(\bigcup_{\tau\in{\mbox{\rm Gal}}(k_{G}/k)}{\rm Zero}({\cal S}_{\tau})\right)\bigcap{\rm GL}_{n}(C).$ We then obtain the desired Galois group. ## Appendix A Bounds for $k$-definable elements of $N_{d}(V^{n}_{inv})$ In this appendix, the symbols in the previous sections are used. Let $W$ be a $C$-vector subspace of $V$ of dimension not greater than $n$. Assume that ${\mathbf{w}}_{1},\cdots,{\mathbf{w}}_{m}$ is a basis of $W$. ###### Definition A.1 $W$ is said to be $k$-definable if there is $M\in GL_{n}(k)$ such that $M({\mathbf{w}}_{1},{\mathbf{w}}_{2},\cdots,{\mathbf{w}}_{m})=\begin{pmatrix}\tilde{W}\\\ 0\end{pmatrix}$ (5) where $\tilde{W}$ is an invertible $m\times m$ matrix. We call $M$ the defining matrix of $W$. ###### Remark A.2 There is some $B\in{\mbox{\rm Mat}}_{m}(k)$ such that $\tilde{W}$ is a fundamental matrix of linear differential equations $\delta(Y)=BY$. Clearly, $V$ is a ${\mathcal{G}}$-module. Moreover, we have ###### Proposition A.3 Let $W$ be a $C$-vector subspace of $V$. Then the following are equivalent: 1. $(a)$ $W$ is $k$-definable; 2. $(b)$ $W$ is a ${\mathcal{G}}$-submodule of $V$. * Proof. $(a)\Rightarrow(b)$. Assume that $W$ is $k$-definable and ${\mathbf{w}}_{1},\cdots,{\mathbf{w}}_{m}$ is a $C$-basis of $W$. Then there is an $M\in{\rm GL}_{n}(k)$ such that (5) holds. For any $\sigma\in{\mathcal{G}}$, it follows from Remark A.2 that there is $[\sigma]\in{\rm GL}_{m}(C)$ such that $\sigma(\tilde{W})=\tilde{W}[\sigma]$. Applying $\sigma$ to (5), we have $M(\sigma({\mathbf{w}}_{1}),\sigma({\mathbf{w}}_{2}),\cdots,\sigma({\mathbf{w}}_{m}))=\begin{pmatrix}\tilde{W}\\\ 0\end{pmatrix}[\sigma].$ Then the above equality and (5) deduce that $(\sigma({\mathbf{w}}_{1}),\sigma({\mathbf{w}}_{2}),\cdots,\sigma({\mathbf{w}}_{m}))=({\mathbf{w}}_{1},{\mathbf{w}}_{2},\cdots,{\mathbf{w}}_{m})[\sigma].$ Hence $W$ is a ${\mathcal{G}}$-submodule. $(b)\Rightarrow(a)$. Assume that $W$ is a ${\mathcal{G}}$-submodule and ${\mathbf{w}}_{1},\cdots,{\mathbf{w}}_{m}$ is a $C$-basis of $W$. Then there is an $m\times m$ submatrix of $({\mathbf{w}}_{1},\cdots,{\mathbf{w}}_{m})$ which is invertible. Without loss of generality, assume that the submatrix consisting of the first $m$ rows of the matrix $({\mathbf{w}}_{1},\cdots,{\mathbf{w}}_{m})$ is invertible. Denote this submatrix by $\tilde{W}$ and the left $(n-m)\times m$ submatrix by $\bar{W}$. Let $\bar{M}=\bar{W}\tilde{W}^{-1}$. Assume that $\sigma\in{\mathcal{G}}$. We then have that $\sigma(({\mathbf{w}}_{1},\cdots,{\mathbf{w}}_{m}))=({\mathbf{w}}_{1},\cdots,{\mathbf{w}}_{m})[\sigma]$ for some $[\sigma]\in{\rm GL}_{m}(C)$. Applying $\sigma$ to $\bar{M}$ yields that $\sigma(\bar{M})=\sigma(\bar{W}\tilde{W}^{-1})=\sigma(\bar{W})\sigma(\tilde{W}^{-1})=\bar{W}[\sigma](\tilde{W}[\sigma])^{-1}=\bar{W}\tilde{W}^{-1}=\bar{M},$ which implies that $\sigma(\bar{M})=\bar{M}$ for all $\sigma\in{\mathcal{G}}$. Hence all entries of $\bar{M}$ belong to $k$. Let $M=\begin{pmatrix}I_{m}&0\\\ -\bar{M}&I_{n-m}\end{pmatrix},$ (6) Then $M$ is a matrix such that (5) holds. $\Box$ ###### Definition A.4 Let $W$ be a $k$-definable subspace of $V$ and $m$ an integer. We call $W$ bounded by $m$ if there is $M$ with $\deg(M)\leq m$ such that (5) holds, where $\deg(M)$ is defined as the maximum of the degrees of entries of $M$. ###### Proposition A.5 There is an integer $\ell$ such that all $k$-definable subspaces of $V$ are bounded by $\ell$. * Proof. Evidently, it is enough to prove Proposition A.5 for the fixed dimension $k$-definable subspaces. Let $W$ be a $k$-definable subspace of $V$ with a basis ${\mathbf{w}}_{1},\cdots,{\mathbf{w}}_{m}$. From the proof of Proposition A.3, $M$ can be chosen to satisfy that there is a permutation matrix $P$ such that $MP$ is of the form (6). In other words, $({\mathbf{w}}_{1},{\mathbf{w}}_{2},\cdots,{\mathbf{w}}_{m})=P\begin{pmatrix}I_{m}&0\\\ \bar{M}&I_{n-m}\end{pmatrix}\begin{pmatrix}\tilde{W}\\\ 0\end{pmatrix},$ where $\bar{M}$ is an $(n-m)\times m$ matrix with entries in $k$. If $\deg(\bar{M})$ is not greater than $\ell$, neither is $\deg(M)$. One can easily see that the coordinates of $P{\mathbf{w}}_{1}\wedge P{\mathbf{w}}_{2}\wedge\cdots\wedge P{\mathbf{w}}_{m}$ are the permutation of those of ${\mathbf{w}}_{1}\wedge{\mathbf{w}}_{2}\wedge\cdots\wedge{\mathbf{w}}_{m}$. This implies that we only need to consider the case $P=I_{n}$. An easy calculation yields that $\displaystyle{\mathbf{w}}_{1}\wedge{\mathbf{w}}_{2}\wedge\cdots\wedge{\mathbf{w}}_{m}$ $\displaystyle=\sum_{1\leq i_{1}<i_{2}<\cdots<i_{m}\leq n}c_{i_{1},i_{2},\cdots,i_{m}}{\mathbf{e}}_{i_{1}}\wedge{\mathbf{e}}_{i_{2}}\wedge\cdots\wedge{\mathbf{e}}_{i_{m}}$ $\displaystyle=\sum_{1\leq i_{1}<i_{2}<\cdots<i_{m}\leq n}a_{i_{1},i_{2},\cdots,i_{m}}\det(\tilde{W}){\mathbf{e}}_{i_{1}}\wedge{\mathbf{e}}_{i_{2}}\wedge\cdots\wedge{\mathbf{e}}_{i_{m}},$ where $c_{i_{1},i_{2},\cdots,i_{m}}\in K,a_{i_{1},i_{2},\cdots,i_{m}}\in k$. Note that $\delta(\tilde{W})=B\tilde{W}$ for some $B\in{\rm Mat}_{m}(k)$. Therefore $\delta(\det(\tilde{W}))=\mbox{\rm{Tr}}(B)\det(\tilde{W})$. It implies that the vector $(\cdots,c_{i_{1},i_{2},\cdots,i_{m}},\cdots)$ is a hyperexponential solution over $k$ of the exterior system $\delta(Y)=\left(\wedge^{m}A\right)Y$. The matrix $\wedge^{m}A$ can be constructed from $A$. Assume that $\bar{M}=(\lambda_{i,j}),1\leq i\leq n-m,1\leq j\leq m$. We further have that $a_{12\cdots m}=1,\,\,a_{12\cdots\hat{j}\cdots mi}=\lambda_{i-m,j},\,\,m+1\leq i\leq n,1\leq j\leq m.$ Assume that ${\mathbf{v}}h$ is a hyperexponential solution of $\delta(Y)=\left(\wedge^{m}A\right)Y$ where $h$ is a hyperexponential element over $k$ and ${\mathbf{v}}\in k^{n\choose m}$. Note that if we fix hyperexponential elements $h$, then from [14, 16], there is an integer $\ell$ such that $\deg({\mathbf{v}})\leq\ell/2$. To obtain the entries of $\bar{M}$ from those of ${\mathbf{v}}$, we need to normalize ${\mathbf{v}}$ at some coordinate, for instance when $P=I_{n}$, $a_{12\cdots m}$ is normalized to 1. Hence $\deg(\bar{M})\leq\ell$. $\Box$ Since we restrict ourselves to Zariski closed subsets of $V^{n}_{inv}$, we need to consider the following two new systems. One is the $n$-direct sum of (1): $\delta(Y)=\hbox{\rm diag}(\underbrace{A,A,\cdots,A}_{n})Y=A^{\oplus n}Y.$ (${A^{\oplus n}}$) Then $V^{n}$ is the solution space of (${A^{\oplus n}}$) and $\hbox{\rm diag}(\underbrace{F,F\cdots,F}_{n})$, denoted by $F^{\oplus n}$, is a fundamental matrix. The other is the symmetric power system of (${A^{\oplus n}}$). Define a map $\displaystyle S^{\leq d}:V^{n}$ $\displaystyle\longrightarrow K^{n^{2}+d\choose d}$ $\displaystyle{\mathbf{v}}=(v_{1,1},v_{2,1},\cdots,v_{n,n})$ $\displaystyle\longrightarrow(\cdots,v_{1,1}^{\mu_{1,1}}v_{1,2}^{\mu_{1,2}}\cdots v_{n,n}^{\mu_{n,n}},\cdots,),\,\,\sum_{1\leq i,j\leq n}\mu_{i,j}\leq d.$ Then $S^{\leq d}({\mathbf{v}})$ is a solution of the symmetric power system of (1), denoted by $\delta(Y)=\left({S^{\leq d}A^{\oplus n}}\right)Y.$ (${S^{\leq d}A^{\oplus n}}$) The matrix ${S^{\leq d}A^{\oplus n}}$ can be constructed from $A$ and its entries are in $k$. Denote the solution space of (${S^{\leq d}A^{\oplus n}}$) by ${(V^{n})^{\circledS\leq d}}$. More details about the symmetric power system could be found in (p. 39, [16]) and [14]. Let $Z$ be an element of $N_{d}(V^{n}_{inv})$. Set $W_{Z}={\rm span}_{C}\\{S^{\leq d}({\mathbf{z}}),{\mathbf{z}}\in Z\\}.$ Then $W_{Z}$ is a subspace of ${(V^{n})^{\circledS\leq d}}$. ###### Proposition A.6 Given an element $Z$ of $N_{d}(V^{n}_{inv})$, $Z$ is $k$-definable if and only if $W_{Z}$ is $k$-definable. * Proof. Suppose that $S^{\leq d}({\mathbf{v}}_{1}),\cdots,S^{\leq d}({\mathbf{v}}_{m})$ is a basis of $W_{Z}$. $(\Rightarrow)$ Assume that $\sigma\in{\mathcal{G}}$. Since $Z$ is $k$-definable, $\sigma({\mathbf{v}})\in Z$ for all ${\mathbf{v}}\in Z$. It implies that for each $i=1,\cdots,m$, $\sigma(S^{\leq d}({\mathbf{v}}_{i}))=S^{\leq d}(\sigma({\mathbf{v}}_{i}))\in W_{Z}$. Consequently, $W_{Z}$ is a ${\mathcal{G}}$-module. By Proposition A.3, $W_{Z}$ is $k$-definable. $(\Leftarrow)$ Assume that $W_{Z}$ is $k$-definable. By Proposition A.3, there is $M\in{\rm GL}_{m}(k)$ such that $M(S^{\leq d}({\mathbf{v}}_{1}),\cdots,S^{\leq d}({\mathbf{v}}_{m}))=\begin{pmatrix}\tilde{W}\\\ 0\end{pmatrix}$ where $\tilde{W}$ is an $m\times m$ matrix. Denote the $i$th-row of $M$ by ${\mathbf{m}}_{i}$ for $i=m+1,\cdots,{n^{2}+d\choose d}$. Let ${\mathbf{x}}=(1,x_{1,1},\cdots,x_{1,1}^{\mu_{1,1}}\cdots x_{n,n}^{\mu_{n,n}},\cdots)$ where $\mu_{1,1}+\mu_{1,2}+\cdots+\mu_{n,n}\leq d$. Then $P_{i}(X)={\mathbf{m}}_{i}\cdot{\mathbf{x}}$ is a polynomial of degree at most $d$ with the coefficients in $k$, where $i=m+1,\cdots,{n^{2}+d\choose d}$. One can easily verify that $Z\subseteq{\rm Zero}\left(P_{m}(X),\cdots,P_{n^{2}+d\choose d}(X)\right)\bigcap V_{inv}^{n}.$ Since $Z\in N_{d}(V^{n}_{inv})$, there are $Q_{1}(X),\cdots,Q_{l}(X)$ with degree at most $d$, which define $Z$. By the dimension argument, for each $i=1,\cdots,l$, $Q_{i}(X)$ is a $\bar{k}$-linear combinations of $P_{m}(X),\cdots,P_{n^{2}+d\choose d}(X)$. Therefore the above inclusion relation is actually an equality. $\Box$ The above two propositions indicate the following corollary. ###### Corollary A.7 There is an integer $\ell$ such that for every $k$-definable element $Z$ of $N_{d}(V^{n}_{inv})$, the coefficients of the defining equations of $Z$ can be chosen to be rational functions whose degrees are not greater than $\ell$. * Proof. Let $Z$ be a $k$-definable element of $N_{d}(V^{n}_{inv})$. Then $W_{Z}$ is a $k$-definable subspace of ${(V^{n})^{\circledS\leq d}}$ by Proposition A.6. Moreover, from the proof of Proposition A.6, the coefficients of the defining equations of $Z$ can be chosen to be the entries of the defining matrix of $W_{Z}$. Hence to prove the corollary, it suffices to show that all $k$-definable subspaces of ${(V^{n})^{\circledS\leq d}}$ are bounded by an integer $\ell$. This can be done by applying Proposition A.5 to ${(V^{n})^{\circledS\leq d}}$. $\Box$ ## Appendix B Bounds for proto-Galois groups In this appendix, we shall find an integer $\tilde{d}$ depending on $n$ with the following property. For any algebraic subgroup $G$ of ${\rm GL}_{n}(C)$, there is an algebraic subgroup $H$ of ${\rm GL}_{n}(C)$, which is bounded by $\tilde{d}$, satisfying $():(H^{\circ})^{t}\unlhd G^{\circ}\leq G\leq H.$ Most of results in this section appeared in the part iii@ of [6], where more families of algebraic subgroups of ${\rm GL}_{n}(C)$ that can be uniformly definable are given. Here we only present those we need. At the same tine, we will use the term “bounded by $d$” instead of “uniformly definable”. As mentioned in Introduction, we elaborate the details of the proofs in [6] and present the explicit estimates of the bounds. Meanwhile, we will show how to compute a set of generators of the character group of a given connected algebraic subgroup. The following notation will be used frequently. ###### Notation B.1 Let $H$ be an algebraic subgroup of ${\rm GL}_{n}(C)$ and $S$ an arbitrary subset of $H$. ${\mathcal{F}}$: a family of algebraic subgroups of ${\rm GL}_{n}(C)$; $H_{\mathcal{F}}$: the intersection of all $H^{\prime}\in{\mathcal{F}}$ with $H\subseteq H^{\prime}$; $N_{H}(H^{\prime})$: the normalizer of $H^{\prime}$ in $H$ where $H^{\prime}$ is an algebraic subgroup of $H$; ${\mathcal{F}}_{mt}(H)$: the family of maximal tori of $H$; ${\mathcal{F}}_{imt}(H)$: the family of intersections of maximal tori of $H$; ${\mathcal{F}}_{up}$: the family of subgroups of ${\rm GL}_{n}(C)$ generated by unipotent elements; $X(H)$: the group of characters of $H$; $H^{t}$: the intersection of kernals of all characters of $H$. An algebraic subgroup $H$ of ${\rm GL}_{n}(C)$ is said to be bounded by $d$ if there are polynomials $Q_{1}(X),\cdots,Q_{m}(X)$ in $C[x_{1,1},x_{1,2},\cdots,x_{n,n}]$ with degree not greater than $d$ such that $H={\rm Zero}(Q_{1}(X),\cdots,Q_{m}(X))\bigcap{\rm GL}_{n}(C).$ For a family of algebraic subgroups of ${\rm GL}_{n}(C)$, say ${\mathcal{F}}$, we say ${\mathcal{F}}$ is bounded by $d$, if every element of ${\mathcal{F}}$ is bounded by $d$. For an ideal $I$ in $C[x_{1,1},x_{1,2},\cdots,x_{n,n}]$, $I$ is said to be bounded by $d$ if there exist generators of $I$ whose degrees are not greater than $d$. Throughout this appendix, unless otherwise specified, subgroups always mean algebraic subgroups. ### B.1 Preparation lemmas To achieve the integer $\tilde{d}$, we need the following degree bounds from computational algebraic geometry. More details on these degree bounds can be found in [4, 9, 13]. For the moment, we assume that $I$ is an ideal in $C[x_{1},\cdots,x_{n}]$. Then we have ###### Proposition B.2 Suppose that $I$ is bounded by $d$. Then there is $\gamma(n,d)$ in ${\mathbb{N}}$ such that $I\cap C[x_{1},\cdots,x_{i}]$ is bounded by $\gamma(n,d)$. ###### Remark B.3 By Groebner bases computation, $\gamma(n,d)$ can be chosen as $2\left(\frac{d^{2}}{2}+d\right)^{2^{n-1}}$, which is less than $(d+1)^{2^{n}}$. The reader is referred to [4] for more details. The following several lemmas play the key role in this appendix. ###### Lemma B.4 Let $H$ be a subgroup of ${\rm GL}_{n}(C)$ bounded by $d$. Then there exists a family ${\mathcal{F}}_{ad}(H)$ of subgroups of $H$ bounded by $\max\\{d,n\\}$ such that for any connected subgroup $H^{\prime}$ of $H$, $N_{H}(H^{\prime})\in{\mathcal{F}}_{ad}(H)$. Particularly, ${\mathcal{F}}_{ad}({\rm GL}_{n}(C))$ is bounded by $n$. Proof. Let ${\mathfrak{g}l}(H)$ be the Lie algebra of $H$. Consider the adjoint action of $H$ on ${\mathfrak{g}l}(H)$. Then ${\mathfrak{g}l}(H^{\prime})$ is a subspace of ${\mathfrak{g}l}(H)$ and $N_{H}(H^{\prime})$ is the stabilizer of ${\mathfrak{g}l}(H^{\prime})$ under the adjoint action. Let $B_{1},\cdots,B_{l}$ be a basis of ${\mathfrak{g}l}(H^{\prime})$ and the ${\mathbf{e}}_{i,j}$ a basis of ${\rm Mat}_{n}(C)$. Then for any $h\in N_{H}(H^{\prime})$, there is $g_{h}\in{\rm GL}_{n^{2}}(C)$ such that $h({\mathbf{e}}_{1,1},\cdots,{\mathbf{e}}_{n,n})h^{-1}=({\mathbf{e}}_{1,1},\cdots,{\mathbf{e}}_{n,n})g_{h}$. It is easy to see that the entries of $g_{h}$ are of the form $P_{l,m}(h)/\det(h)$ where the $P_{l,m}(X)$ are polynomials with degree at most $n$. Assume that $B_{s}=({\mathbf{e}}_{1,1},\cdots,{\mathbf{e}}_{n,n}){\mathbf{b}}_{s}$ for $s=1,\cdots,l$ where ${\mathbf{b}}_{s}=(b_{s,i,j})\in C^{n^{2}}$. Then since $hB_{s}h^{-1}\in{\mathfrak{g}l}(H^{\prime})$, there are $a_{s,1},\cdots,a_{s,l}\in C$ such that $hB_{s}h^{-1}=\sum_{\xi}a_{s,\xi}B_{\xi}$. In other words, $hB_{s}h^{-1}=\sum_{i,j}b_{s,i,j}h{\mathbf{e}}_{i,j}h^{-1}=({\mathbf{e}}_{1,1},\cdots,{\mathbf{e}}_{n,n})g_{h}{\mathbf{b}}_{s}=({\mathbf{e}}_{1,1},\cdots,{\mathbf{e}}_{n,n})\sum_{\xi=1}^{l}a_{s,\xi}{\mathbf{b}}_{\xi}.$ That is $g_{h}{\mathbf{b}}_{s}=\sum_{\xi=1}^{l}a_{s,\xi}{\mathbf{b}}_{\xi}.$ The above nonhomogeneous linear equations has solutions if and only if $\mbox{\rm{rank}}({\mathbf{b}}_{1},\cdots,{\mathbf{b}}_{l})=\mbox{\rm{rank}}({\mathbf{b}}_{1},\cdots,{\mathbf{b}}_{l},g_{h}{\mathbf{b}}_{s}).$ This leads to the equations that together with the defining equations of $H$ define $N_{H}(H^{\prime})$. Since the entries of $g_{h}$ are of the form $P_{l,m}(h)/\det(h)$ where the $P_{l,m}(X)$ are polynomials with degree at most $n$, the defining ideal of $N_{H}(H^{\prime})$ is generated by those of $H$ and the polynomials with degrees $\leq n$. Hence ${\mathcal{F}}_{ad}(H)$ is bounded by $\max\\{d,n\\}$. In particular, when $H={\rm GL}_{n}(C)$, ${\mathcal{F}}_{ad}({\rm GL}_{n}(C))$ is bounded by $n$. $\Box$ Let $\\{\tau_{H,\lambda}:H\rightarrow{\rm GL}_{\mu}(C)\|H\in{\mathcal{F}},\lambda\in\Lambda\\}$ be a family of morphisms from elements of ${\mathcal{F}}$ to ${\rm GL}_{\mu}(C)$ where $\mu$ is a positive integer and $\Lambda$ is a set. Assume that $\tau_{H,\lambda}=(P_{i,j}^{H,\lambda}(X)/Q^{H,\lambda}(X))$. We will say that the $\\{\tau_{H,\lambda}\\}$ are bounded by $m$ if $\deg(P^{H,\lambda}_{i,j}(X))\leq m$ and $\deg(Q^{H,\lambda}(X))\leq m$. ###### Lemma B.5 Let $\\{\tau_{H,\lambda}\,\,\|\,\,H\in{\mathcal{F}},\lambda\in\Lambda\\}$ be as above. Assume that ${\mathcal{F}}$ is bounded by $d$ and $\\{\tau_{H,\lambda}\,\,\|\,\,H\in{\mathcal{F}},\lambda\in\Lambda\\}$ is bounded by $m$. Then * $(a)$ $\\{\tau_{H,\lambda}(H)\\}$ is bounded by $(\bar{d}+1)^{2^{\mu^{2}+n^{2}}}$ where $\bar{d}=\max\\{m+1,d\\}$. * $(b)$ if ${\mathcal{F}}^{\prime}$ is a family of subgroups of ${\rm GL}_{\mu}(C)$ bounded by $d^{\prime}$, then $\\{\tau^{-1}_{H,\lambda}\left(H^{\prime}\cap\tau_{H,\lambda}(H)\right)\|H^{\prime}\in{\mathcal{F}}^{\prime},H\in{\mathcal{F}},\lambda\in\Lambda\\}$ is bounded by $\max\\{d,md^{\prime}\\}.$ * Proof. Assume that $H$ is defined by $S_{H}$, a set of polynomials with degree $\leq d$. $(a)$ $\tau_{H,\lambda}(H)$ is defined by $\left\langle Q^{H,\lambda}(X)y_{1,1}-P^{H,\lambda}_{1,1}(X),\cdots,Q^{H,\lambda}(X)y_{\mu,\mu}-P^{H,\lambda}_{\mu,\mu}(X),S_{H}\right\rangle\bigcap C[y_{1,1},y_{1,2},\cdots,y_{\mu,\mu}].$ By Proposition B.2, $\tau_{H,\lambda}(H)$ is bounded by $(\bar{d}+1)^{2^{\mu^{2}+n^{2}}}$ where $\bar{d}=\max\\{m+1,d\\}$. $(b)$ Assume that $H^{\prime}$ is defined by $g_{1}(Y),\cdots,g_{s}(Y)$ where $\deg(g_{i}(Y))\leq d^{\prime}$. Then one can see that $\tau^{-1}_{H,\lambda}\left(H^{\prime}\cap\tau_{H,\lambda}(H)\right)$ is defined by $S_{H},\,\,g_{1}\left(\left(\frac{P^{H,\lambda}_{i,j}(X)}{Q^{H,\lambda}(X)}\right)\right),\cdots,g_{s}\left(\left(\frac{P^{H,\lambda}_{i,j}(X)}{Q^{H,\lambda}(X)}\right)\right).$ Clearing the denominators, we obtain the defining polynomials of $\tau^{-1}_{H,\lambda}\left(H^{\prime}\cap\tau_{H,\lambda}(H)\right)$ in $C[x_{1,1},x_{1,2},\cdots,x_{n,n}]$, whose degrees are not greater than $\max\\{d,md^{\prime}\\}$. $\Box$ Given a non-negative integer $d$, set $d^{}=\max_{i}\left\\{{{n^{2}+d\choose d}\choose i}^{2}\right\\},\,\,n^{}=d^{}d{n^{2}+d\choose d}.$ ###### Proposition B.6 Assume that ${\mathcal{F}}$ is bounded by $d$. Then for any subgroup $H^{\prime}\subseteq{\rm GL}_{n}(C)$ and $H\in{\mathcal{F}}$ with $H\unlhd H^{\prime}$, there is a family of morphisms $\tau_{H^{\prime},H}:H^{\prime}\rightarrow{\rm GL}_{d^{}}(C)$ with $\ker(\tau_{H^{\prime},H})=H$, which are bounded by $n^{}$. Furthermore, if $H^{\prime}$ varies among a family of subgroups of ${\rm GL}_{n}(C)$ bounded by $d^{\prime}$, then $\\{\tau_{H^{\prime},H}(H^{\prime})\|H^{\prime}\in{\mathcal{F}}^{\prime},H\in{\mathcal{F}}\,\,\mbox{with}\,\,H\unlhd H^{\prime}\\}$ is bounded by $(\bar{d}+1)^{2^{(d^{})^{2}+n^{2}}},\,\,\mbox{where}\,\,\bar{d}=\max\\{n^{}+1,d^{\prime}\\}.$ Proof. $C[x_{1,1},x_{1,2},\cdots,x_{n,n}]_{\leq d}$ is a $C$-vector space with dimension $n^{2}+d\choose d$. The group ${\rm GL}_{n}(C)$ acts naturally on $C[x_{1,1},x_{1,2},\cdots,x_{n,n}]_{\leq d}$, which is defined as follows $\forall\,\,g\in{\rm GL}_{n}(C),\,\,P(x)\in C[x_{1,1},x_{1,2},\cdots,x_{n,n}]_{\leq d},\,\,g\cdot P(X)=P(Xg).$ Suppose that $H\in{\mathcal{F}}$. Let $I_{\leq d}(H)=\\{P(X)\in C[x_{1,1},x_{1,2},\cdots,x_{n,n}]_{\leq d}\,\,\|\,\,P(H)=0\\}.$ It is also a $C$-vector space of finite dimension. Since $I_{\leq d}(H)$ defines $H$, $H={\rm stab}(I_{\leq d}(H))$. Let $\nu=\dim_{C}(I_{\leq d}(H))$ and $E=\bigwedge^{\nu}C[x_{1,1},x_{1,2},\cdots,x_{n,n}]_{\leq d}.$ Then $\dim_{C}(E)={{n^{2}+d\choose d}\choose\nu}\,\,\mbox{and}\,\,\bigwedge^{\nu}I_{\leq d}(H)=C{\mathbf{v}}\,\,\mbox{for some ${\mathbf{v}}\in E$}.$ The action of ${\rm GL}_{n}(C)$ on $C[x_{1,1},x_{1,2},\cdots,x_{n,n}]_{\leq d}$ induces an action of ${\rm GL}_{n}(C)$ on $E$. We will still use $\cdot$ to denote this action. It is easy to see that $H={\rm stab}(C{\mathbf{v}})$. Let $U=\oplus U_{\chi}$ where the $\chi$ are characters of $H$ and $U_{\chi}=\\{{\mathbf{u}}\in E\,\,\|\,\,h\cdot{\mathbf{u}}=\chi(h){\mathbf{u}}\\}.$ Note that the above direct sum runs over a finite set. Assume that $U=\oplus_{i=1}^{s}U_{\chi_{i}}$. It is clear that ${\mathbf{v}}\in U_{\chi_{i}}$ for some $i$. Let $H^{\prime}$ be a subgroup of ${\rm GL}_{n}(C)$ satisfying that $H\unlhd H^{\prime}$. Then $U$ is invariant under the action of $H^{\prime}$. Let ${\mathcal{L}}$ be the set of $C$-linear maps from $U$ to $U$ which leave all $U_{\chi_{i}}$ unchanged. Then since $\dim_{C}(U)\leq\dim_{C}(E)$, $\dim_{C}({\mathcal{L}})\leq(\dim_{C}(U))^{2}\leq(\dim_{C}(E))^{2}={{n^{2}+d\choose d}\choose\nu}^{2}.$ Let ${\mathbf{u}}_{1},\cdots,{\mathbf{u}}_{l}$ be a suitable basis of $U$ such that under this basis, each element of ${\mathcal{L}}$ is represented as the matrix $\hbox{\rm diag}(M_{1},\cdots,M_{s})$ where $M_{i}\in{\rm Mat}_{\dim(U_{\chi_{i}})}(C)$. Furthermore every matrix of the form $\hbox{\rm diag}(M_{1},\cdots,M_{s})$ where $M_{i}\in{\rm Mat}_{\dim(U_{\chi_{i}})}(C)$ represents an element of ${\mathcal{L}}$. For any $h^{\prime}\in H^{\prime}$, there is $[h^{\prime}]\in{\rm GL}_{l}(C)$ such that $(h^{\prime}\cdot{\mathbf{u}}_{1},\cdots,h^{\prime}\cdot{\mathbf{u}}_{l})=({\mathbf{u}}_{1},\cdots,{\mathbf{u}}_{l})[h^{\prime}].$ By an easy calculation, the entries of $[h^{\prime}]$ are polynomials in those of $h^{\prime}$ with degree $\leq d\nu$. For any $L\in{\mathcal{L}}$, we will use $L^{{\mathbf{u}}}$ to denote the matrix in ${\rm GL}_{l}(C)$ satisfies that $L(({\mathbf{u}}_{1},{\mathbf{u}}_{2},\cdots,{\mathbf{u}}_{l}))=({\mathbf{u}}_{1},{\mathbf{u}}_{2},\cdots,{\mathbf{u}}_{l})L^{\mathbf{u}}.$ The action of $H^{\prime}$ on $U$ derives an adjoint action of $H^{\prime}$ on ${\mathcal{L}}$ as follows: for any $L\in{\mathcal{L}},h^{\prime}\in H^{\prime}$, $\displaystyle(h^{\prime}\cdot L)(({\mathbf{u}}_{1},\cdots,{\mathbf{u}}_{l}))$ $\displaystyle=h^{\prime}\cdot L(h^{\prime-1}\cdot{\mathbf{u}}_{1},\cdots,h^{\prime-1}\cdot{\mathbf{u}}_{l})=h^{\prime}\cdot L(({\mathbf{u}}_{1},\cdots,{\mathbf{u}}_{l})[h^{\prime}]^{-1})$ $\displaystyle=h^{\prime}\cdot(({\mathbf{u}}_{1},\cdots,{\mathbf{u}}_{l})L^{\mathbf{u}}[h^{\prime}]^{-1})=({\mathbf{u}}_{1},\cdots,{\mathbf{u}}_{l})[h^{\prime}]L^{\mathbf{u}}[h^{\prime}]^{-1}.$ Fix a basis of ${\mathcal{L}}$, say $L_{1},\cdots,L_{m}$, where $m\leq{{n^{2}+d\choose d}\choose\nu}^{2}$. Then the adjoint action induces a morphism from $H^{\prime}$ to ${\rm GL}_{m}(C)$ $\tau_{H^{\prime},H}:H^{\prime}\longrightarrow{\rm GL}_{m}(C),\,\,\tau_{H^{\prime},H}(h^{\prime})=\eta_{h^{\prime}}$ (7) where $\eta_{h^{\prime}}\in{\rm GL}_{m}(C)$ satisfies that $([h^{\prime}]L_{1}^{\mathbf{u}}[h^{\prime}]^{-1},\cdots,[h^{\prime}]L_{m}^{\mathbf{u}}[h^{\prime}]^{-1})=(L_{1}^{\mathbf{u}},\cdots,L_{m}^{\mathbf{u}})\eta_{h^{\prime}}.$ We will show that $\ker(\tau_{H^{\prime},H})=H$. Suppose that $h^{\prime}\in\ker(\tau_{H^{\prime},H})$. Then $\eta_{h^{\prime}}=I_{m}$. In other words, $[h^{\prime}]L^{\mathbf{u}}=L^{\mathbf{u}}[h^{\prime}]$ for all $L\in{\mathcal{L}}$. It implies that $[h^{\prime}]$ is of the following form: $[h^{\prime}]=\hbox{\rm diag}(\underbrace{c_{1},\cdots,c_{1}}_{\dim(V_{\chi_{1}})},\cdots,\underbrace{c_{s},\cdots,c_{s}}_{\dim(V_{\chi_{s}})}).$ Particularly, $h^{\prime}\cdot{\mathbf{v}}=c_{i}{\mathbf{v}}$ for some $i$. Hence $h^{\prime}\in H={\rm stab}(C{\mathbf{v}})$. One can easily see that $H\subseteq\ker(\tau_{H^{\prime},H})$. Hence $H=\ker(\tau_{H^{\prime},H})$. Since $m\leq d^{}$, ${\rm GL}_{m}(C)$ can be naturally embedded into ${\rm GL}_{d^{}}(C)$. Composing this embedding map with $\tau_{H^{\prime},H}$ induces a morphism from $H^{\prime}$ to ${\rm GL}_{d^{}}(C)$ with kernel $H$. We will still denote this morphism by $\tau_{H^{\prime},H}$. An easy calculation yields that $\tau_{H^{\prime},H}(X)=\left(\frac{P_{i,j}^{H^{\prime},H}(X)}{Q^{H^{\prime},H}(X)}\right)$ where $P_{i,j}^{H^{\prime},H}(X),Q^{H^{\prime},H}(X)$ are polynomials in $x_{i,j}$ and $Q^{H^{\prime},H}(h^{\prime})=\det([h^{\prime}])$. Furthermore, the $P_{i,j}^{H^{\prime},H}(h^{\prime})$ are polynomials in the entries of $[h^{\prime}]$ with degree $\leq l$. Since the entries of $[h^{\prime}]$ are polynomials in those of $h^{\prime}$ with degree $\leq d\nu$, the $P_{i,j}^{H^{\prime},H}(h^{\prime})$ and $Q^{H^{\prime},H}(h^{\prime})$ are polynomials in the entries of $h^{\prime}$ with degree $\leq ld\nu$, which is not greater than $n^{}$. This proves that $\\{\tau_{H,\lambda}\,\,\|\,\,H\in{\mathcal{F}},\lambda\in\Lambda\\}$ is bounded by $n^{}$. Finally, by Lemma B.5, $\\{\tau_{H^{\prime},H}(H^{\prime})\|H^{\prime}\in{\mathcal{F}}^{\prime},H\in{\mathcal{F}}\,\,\mbox{with}\,\,H\unlhd H^{\prime}\\}$ is bounded by $(\bar{d}+1)^{2^{(d^{})^{2}+n^{2}}},\,\,\mbox{where}\,\,\bar{d}=\max\\{n^{}+1,d^{\prime}\\}$. $\Box$ ###### Remark B.7 As linear algebraic groups, $\tau_{H^{\prime},H}(H^{\prime})$ is isomorphic to $H^{\prime}/H$. Therefore Proposition B.6 says that $H^{\prime}/H$ can be uniformly embedded into ${\rm GL}_{d^{}}(C)$ and $H^{\prime}/H$ varies among a bounded family if $H^{\prime}$ does. ###### Lemma B.8 ${\mathcal{F}}_{up}$ is bounded by $(2n^{3}+1)^{8^{n^{2}}}$. * Proof. Assume that $H$ is a subgroup generated by unipotent elements. Then by (p.55, Proposition, [5] ), it is the product of at most $2\dim(H)$ one-dimensional unipotent subgroups. From (p. 96, Lemma C, [5]), we know that one-dimension unipotent subgroup is of the form: $I_{n}+{\mathbf{m}}x+\frac{{\mathbf{m}}^{2}}{2}x^{2}+\cdots+\frac{{\mathbf{m}}^{n-1}}{(n-1)!}x^{n-1},\,\,{\mathbf{m}}^{n}=0,\,\,x\in C,$ where ${\mathbf{m}}\in{\rm Mat}_{n}(C)$ and ${\mathbf{m}}^{n}=0$. Hence $H$ has a polynomial parametrized representation $(Y)=\prod_{i=1}^{2\dim(H)}\left(I_{n}+{\mathbf{m}}_{i}x_{i}+\frac{{\mathbf{m}}_{i}^{2}}{2}x_{i}^{2}+\cdots+\frac{{\mathbf{m}}_{i}^{n-1}}{(n-1)!}x_{i}^{n-1}\right).$ Note that $\dim(H)\leq n^{2}$. Then the polynomials in the above system contains at most $3n^{2}$ variables and are of degree not greater than $2n^{3}$. Eliminating all $x_{i}$ in the above system, we obtain the defining ideal of $H$. Then Proposition B.2 yields the desired bound. $\Box$ ###### Lemma B.9 Both ${\mathcal{F}}_{mt}({\rm GL}_{n}(C))$ and ${\mathcal{F}}_{imt}({\rm GL}_{n}(C))$ are bounded by 1. * Proof. Every maximal torus of ${\rm GL}_{n}(C)$ is conjugate to $(C^{})^{n}$. Hence it is equal to the intersection of ${\rm GL}_{n}(C)$ and a linear subspace of ${\rm Mat}_{n}(C)$. Consequently, ${\mathcal{F}}_{mt}({\rm GL}_{n}(C))$ is bounded by 1. As the intersection of linear subspaces of ${\rm Mat}_{n}(C)$ is still linear, any element of ${\mathcal{F}}_{imt}({\rm GL}_{n}(C))$ is the intersection of ${\rm GL}_{n}(C)$ and a linear subspace of ${\rm Mat}_{n}(C)$. So ${\mathcal{F}}_{imt}({\rm GL}_{n}(C))$ is also bounded by 1. $\Box$ ###### Lemma B.10 Assume that $H$ is a connected subgroup of ${\rm GL}_{n}(C)$. Then $H^{t}$ is generated by all unipotent elements of $H$. Proof. Since $H/H^{t}$ is a torus, there are no nontrivial unipotent elements in $H/H^{t}$. Hence all unipotent elements of $H$ are in $H^{t}$. From Lemma 2.1 in [10], $H^{t}$ is generated by $(P,P)$ and $R_{u}(H)$ where $P$ is a Levi factor of $H$ and $R_{u}(H)$ is the unipotent radical of $H$. Moreover $(P,P)$ is semi-simple, so it is generated by unipotent elements. Therefore $H^{t}$ is generated by all unipotent elements of $H$. $\Box$ ### B.2 Main results Let $J(n)$ be a Jordan bound, so that every finite subgroup of ${\rm GL}_{n}(C)$ contains a normal abelian subgroup of index at most $J(n)$. In the following, we will show the main results in this appdendix. Denote $\kappa_{1}=\max_{i}\left\\{{{n^{2}+(2n^{3}+1)^{8^{n^{2}}}\choose n^{2}}\choose i}^{2}\right\\}\quad\mbox{and}\quad\kappa_{2}=\kappa_{1}(2n^{3}+1)^{8^{n^{2}}}{n^{2}+(2n^{3}+1)^{8^{n^{2}}}\choose n^{2}}.$ (8) ###### Proposition B.11 There exists an integer $I(n)$ that is not greater than $J\left(\max_{i}\left\\{{\kappa_{1}^{2}+1\choose i}\right\\}\right)$ and a family ${\mathcal{F}}$ of subgroups of ${\rm GL}_{n}(C)$ bounded by $\kappa_{3}\triangleq\kappa_{2}(\kappa_{1}^{2}+1)\max_{i}\left\\{{\kappa_{1}^{2}+1\choose i}\right\\}$ (9) with the following property. For every subgroup $H$ of ${\rm GL}_{n}(C)$, there is $H^{\prime}\in{\mathcal{F}}$ such that * $(a)$ $H^{\circ}\leq H^{\prime}$. * $(b)$ $H$ normalizes $H^{\prime}$; so $H^{\prime}\unlhd HH^{\prime}\leq{\rm GL}_{n}(C)$. * $(c)$ $[H:H\cap H^{\prime}]=[HH^{\prime}:H^{\prime}]\leq I(n)$. * $(d)$ Every unipotent element of $H^{\prime}$ lies in $H^{\circ}$. We will show the proposition by separating three cases. ###### Lemma B.12 Proposition B.11 is true for finite groups with $I(n)=J(n)$ and ${\mathcal{F}}$ is bounded by 1. * Proof. Assume that $H\subset{\rm GL}_{n}(C)$ is a finite group. Let $\bar{H}$ be a normal abelian subgroup of $H$ with $[H:\bar{H}]\leq J(n)$. As a finite abelian subgroup of ${\rm GL}_{n}(C)$ is diagonalizable, $\bar{H}$ is contained in some maximal tori of ${\rm GL}_{n}(C)$. Let $H^{\prime}$ be the intersection of maximal tori containing $\bar{H}$. Then $H^{\prime}\in{\mathcal{F}}_{imt}({\rm GL}_{n}(C))$. Clearly, $H$ normalizes $H^{\prime}$. Since $\bar{H}\subseteq H\cap H^{\prime}$, $[H:H^{\prime}\cap H]\leq[H:\bar{H}]\leq J(n)$. The only unipotent element of $H^{\prime}$ is the identity. So $(a)-(d)$ hold for $H,H^{\prime}$. The lemma follows from the fact that ${\mathcal{F}}_{imt}({\rm GL}_{n}(C))$ is bounded by $1$. $\Box$ ###### Lemma B.13 Assume that $H$ is a subgroup whose identity component is a torus. Then Proposition B.11 is true for $H$ with $I(n)=J\left(\max_{i}\left\\{{n^{2}+1\choose i}^{2}\right\\}\right)$ and ${\mathcal{F}}$ is bounded by $(n^{2}+1)\max_{i}\left\\{{n^{2}+1\choose i}^{2}\right\\}.$ * Proof. Let $M=(H^{\circ})_{{\mathcal{F}}_{imt}({\rm GL}_{n}(C))}$ and $N=N_{{\rm GL}_{n}(C)}(M)$. It is easy to verify that $H$ normalizes $M$ and thus $H\subseteq N$. Since $M$ lies in the family ${\mathcal{F}}_{imt}({\rm GL}_{n}(C))$ bounded by $1$, Lemma B.4 implies that $N$ lies in a family ${\mathcal{F}}_{ad}({\rm GL}_{n}(C))$ bounded by $n$. Let $\tilde{n}=\max_{i}\\{{n^{2}+1\choose i}^{2}\\}$. By Proposition B.6, there is a morphism $\tau_{N,M}:N\longrightarrow{\rm GL}_{\tilde{n}}(C)$ satisfies that $\ker(\tau_{N,M})=M$ and $\tau_{N,M}$ is bounded by $\tilde{n}(n^{2}+1)$. As $H^{\circ}\subseteq M$, $\tau_{N,M}(H)$ is a finite subgroup of ${\rm GL}_{\tilde{n}}(C)$. From Lemma B.12, there is $\tilde{M}\in{\mathcal{F}}_{imt}({\rm GL}_{\tilde{n}}(C))$ such that $(a)$-$(c)$ hold for $\tau_{N,M}(H),\tilde{M}$ (with $I(\tilde{n})=J(\tilde{n})$). Let $H^{\prime}=\tau_{N,M}^{-1}(\tilde{M}\cap\tau_{N,M}(N))$. Note that ${\mathcal{F}}_{imt}({\rm GL}_{\tilde{n}}(C))$ is bounded by 1. By Lemma B.5, $H^{\prime}$ is bounded by $\tilde{n}(n^{2}+1)$. We will show that the $H^{\prime}$ satisfy $(a)$-$(d)$ with $I(n)=J(\tilde{n})$. It is clear that $H^{\circ}\leq H^{\prime}$. For any $h\in H$, since $T_{M,N}(H)$ normalizes $\tilde{M}$ and $\tau_{N,M}(N)$, $\tau_{N,M}(hH^{\prime}h^{-1})=\tau_{N,M}(h)(\tilde{M}\cap\tau_{N,M}(N))\tau_{N,M}(h)^{-1}=\tilde{M}\cap\tau_{N,M}(N)=\tau_{N,M}(H^{\prime}).$ Therefore $hH^{\prime}h^{-1}\subseteq H^{\prime}$. This indicate that $hH^{\prime}h^{-1}=H^{\prime}$ for any $h\in H$. In other words, $H$ normalizes $H^{\prime}$. This proves $(b)$. Since both $HH^{\prime}$ and $H^{\prime}$ contain $M$, $\displaystyle[HH^{\prime}:H^{\prime}]$ $\displaystyle=[\tau_{N,M}(HH^{\prime}):\tau_{N,M}(H^{\prime})]=[\tau_{N,M}(H)\tau_{N,M}(H^{\prime}):\tau_{N,M}(H^{\prime})]$ $\displaystyle=[\tau_{N,M}(H):\tau_{N,M}(H)\cap\tau_{N,M}(H^{\prime})]=[\tau_{N,M}(H):\tilde{M}\cap\tau_{N,M}(H)]\leq J(\tilde{n}).$ This proves $(c)$. Suppose that $h^{\prime}$ is an unipotent element of $H^{\prime}$. Then $\tau_{N,M}(h^{\prime})$ is an unipotent element of $\tilde{M}$. However $\tilde{M}$ consists of semi-simple elements. Hence $\tau_{N,M}(h^{\prime})=1$. Then $h^{\prime}\in M$. But $M$ is contained in a torus, so $h^{\prime}=1$. This proves $(d)$. $\Box$ Now we are ready to prove Proposition B.11 for the general case. * Proof. Let $U=(H^{\circ})^{t}$. By Lemma B.10, $U$ is generated by unipotent elements. Then it follows from Lemma B.8 that $U$ is bounded by $(2n^{3}+1)^{8^{n^{2}}}$. Let $N=N_{{\rm GL}_{n}(C)}(U)$. Lemma B.4 indicates that $N$ lies in ${\mathcal{F}}_{ad}({\rm GL}_{n}(C))$ that is bounded by $n$. Let $\kappa_{1}$ and $\kappa_{2}$ be as in (8). Using Proposition B.6 again, there is a morphism $\phi_{N,U}:N\longrightarrow{\rm GL}_{\kappa_{1}}(C)$ such that $\ker(\phi_{N,U})=U$ and $\phi_{N,U}$ is bounded by $\kappa_{2}$. We first prove that $H\leq N$. For any $h\in H$ and any character $\chi$ of $H^{\circ}$, $\chi(hXh^{-1})$ is a character of $H^{\circ}$. Hence for any $u\in U$, $\chi(huh^{-1})=1$. So $huh^{-1}\in U$ for any $u\in U$. In other words, $H$ normalizes $U$. So $H\leq N$. As $H^{\circ}/U$ is a torus, $\phi_{N,U}(H)^{\circ}$ is a torus in ${\rm GL}_{\kappa_{1}}(C)$. Lemma B.13 implies that there is $M^{\prime}\leq{\rm GL}_{\kappa_{1}}(C)$ bounded by $(\kappa_{1}^{2}+1)\max_{i}\left\\{{\kappa_{1}^{2}+1\choose i}^{2}\right\\}$ such that $(a)$-$(d)$ hold for $\phi_{N,U}(H),M^{\prime}$ with $I(n)=J\left(\max_{i}\left\\{{\kappa_{1}^{2}+1\choose i}\right\\}\right)$. Let $H^{\prime}=\phi^{-1}_{U}(M^{\prime}\cap\phi_{N,U}(N))$. Then by Lemma B.5, $H^{\prime}$ is bounded by $\kappa_{3}$, where $\kappa_{3}$ is defined in (9). The similar arguments as in the proof of Lemma B.13 implies $(a)$-$(c)$ hold for $H,H^{\prime}$ with $I(n)=J\left(\max_{i}\left\\{{\kappa_{1}^{2}+1\choose i}\right\\}\right)$. Now let us show that $(d)$ holds. Assume that $u$ is an unipotent element of $H^{\prime}$. Then $\phi_{N,U}(u)$ is an unipotent element of $M^{\prime}$. Since $M^{\prime}$ and $\phi_{N,U}(H)^{\circ}$ satisfy $(d)$, $\phi_{N,U}(u)\in\phi_{N,U}(H)^{\circ}$. Whereas $\phi_{N,U}(H)^{\circ}$ is a torus, $\phi_{N,U}(u)=1$. Thus $u\in U\subseteq H^{\circ}$. $\Box$ ###### Proposition B.14 Let $I(n)$ and $\kappa_{3}$ be as in Proposition B.11. Then there exists a family $\tilde{{\mathcal{F}}}$ of subgroups of ${\rm GL}_{n}(C)$ bounded by $\tilde{d}\triangleq(\kappa_{3})^{I(n)-1}$ with the following property. For any subgroup $H$ of ${\rm GL}_{n}(C)$, there exists $\tilde{H}\in\tilde{{\mathcal{F}}}$ such that $H\leq\tilde{H}$, and every unipotent element of $\tilde{H}$ lies in $H^{\circ}$. * Proof. Let ${\mathcal{F}}$ be as in Proposition B.11 and $\tilde{{\mathcal{F}}}=\\{\bar{H}\,\,\|\,\,\exists\,\,M\in{\mathcal{F}},M\unlhd\bar{H},[\bar{H}:M]\leq I(n)\\}.$ Every element of $\tilde{{\mathcal{F}}}$ is the union of at most $I(n)$ cosets of some element in ${\mathcal{F}}$. It is well-known that the union of two varieties is defined by the product of their corresponding defining polynomials. Hence $\tilde{{\mathcal{F}}}$ is bounded by $\tilde{d}$. Assume that $H$ is a subgroup of ${\rm GL}_{n}(C)$. Let $H^{\prime}$ be an element in ${\mathcal{F}}$ such that $(a)$-$(d)$ in Proposition B.11 hold for $H,H^{\prime}$. Let $\tilde{H}=HH^{\prime}$. Then $\tilde{H}\in\tilde{{\mathcal{F}}}$ by Proposition B.11 (c). The unipotent elements of $\tilde{H}$ lie in $\tilde{H}^{\circ}$ and then in $(H^{\prime})^{\circ}$. As the unipotent elements of $H^{\prime}$ lie in $H^{\circ}$, so do the unipotent elements of $\tilde{H}$. $\Box$ ###### Corollary B.15 Let $\tilde{{\mathcal{F}}}$ be the family as in Proposition B.14. Then for any subgroup $H$ of ${\rm GL}_{n}(k)$, there is $\tilde{H}\in\tilde{{\mathcal{F}}}$ such that $(\tilde{H}^{\circ})^{t}\unlhd H^{\circ}\leq H\leq\tilde{H}.$ * Proof. By Proposition B.14, there is $\tilde{H}\in\tilde{{\mathcal{F}}}$ such that $H\leq\tilde{H}$ and the unipotent elements of $\tilde{H}$ lie in $H^{\circ}$. Then the corollary follows from Lemma B.10 and the fact that $(\tilde{H}^{\circ})^{t}$ is normal in $\tilde{H}^{\circ}$. $\Box$ In the following, $H$ is assumed to be connected. Proposition B.6 allows us to bound the degrees of generators of $X(H)$. $X(H)$ can be viewed as a subset of $C[H]$, the coordinate ring of $H$. The morphism $\varphi:H\rightarrow H^{\prime}$ induces a group homomorphism $\varphi^{\circ}:X(H^{\prime})\rightarrow X(H)$. ###### Proposition B.16 Let $\kappa_{2}$ be as in (8). Then there are generators of $X(H)$, which are represented by polynomials bounded by $\kappa_{2}$. * Proof. From Lemmas B.10 and B.8, $H^{t}$ is bounded by $(2n^{3}+1)^{8^{n^{2}}}$. By Proposition B.6, there is a morphism $\tau_{H,H^{t}}:H\rightarrow{\rm GL}_{\kappa_{1}}(C)$ satisfying that $\ker(\tau_{H,H^{t}})=H^{t}$ and $\tau_{H,H^{t}}$ is bounded by $\kappa_{2}$, where $\kappa_{1}$ is as in (8). $\tau_{H,H^{t}}$ is of the from $\left(\frac{P^{H,H^{t}}_{i,j}(X)}{Q^{H,H^{t}}(X)}\right)\quad\mbox{where}\,\,\deg(P^{H,H^{t}}_{i,j}(X))\leq\kappa_{2},\,\,\deg(Q^{H,H^{t}}(X))\leq\kappa_{2}.$ From the proof of Proposition B.6, $Q^{H,H^{t}}(I_{n})=1$ and for any $h,h^{\prime}\in H$, $Q^{H,H^{t}}(hh^{\prime})=\det([hh^{\prime}])=\det([h][h^{\prime}])=\det([h])\det([h^{\prime}])=Q^{H,H^{t}}(h)Q^{H,H^{t}}(h^{\prime}).$ It implies that $Q^{H,H^{t}}(X)\in X(H)$. Notice that $X((C^{})^{\kappa_{1}})$ is generated by the characters $y_{1},\cdots,y_{\kappa_{1}}$ and so is the group of characters of any its subgroup. Since $\tau_{H,H^{t}}(H)$ is a torus in ${\rm GL}_{\kappa_{1}}(C)$, it is conjugate to a subgroup of $(C^{})^{\kappa_{1}}$. So $X(\tau_{H,H^{t}}(H))$ is generated by some linear polynomials. $\tau_{H,H^{t}}$ induces a group homomorphism: $\displaystyle\tau_{H,H^{t}}^{\circ}:X(\tau_{H,H^{t}}(H))$ $\displaystyle\rightarrow X(H)$ $\displaystyle\chi^{\prime}$ $\displaystyle\rightarrow\chi^{\prime}\circ\tau_{H,H^{t}}$ For any $\chi\in X(H)$ and $h\in H$, since $\chi(hh^{\prime})=\chi(h)$ for all $h^{\prime}\in H^{t}$, there is $g\in C[\tau_{H,H^{t}}(H)]$ such that $g\circ\tau_{H,H^{t}}=\chi$. One can verify that $g$ is actually a character of $\tau_{H,H^{t}}(H)$. Therefore $\tau_{H,H^{t}}^{\circ}$ is surjective. Let $L_{1},\cdots,L_{s}$ be linear polynomials, which generate $X(\tau_{H,H^{t}}(H))$. Then $L_{1}\circ\tau_{H,H^{t}},\cdots,L_{s}\circ\tau_{H,H^{t}}$ generate $X(H)$. Since $Q^{H,H^{t}}(X)\in X(H)$, $Q^{H,H^{t}}(X),(L_{1}\circ\tau_{H,H^{t}})Q^{H,H^{t}}(X),\cdots,(L_{s}\circ\tau_{H,H^{t}})Q^{H,H^{t}}(X)$ still generate $X(H)$ and are polynomials bounded by $\kappa_{2}$. $\Box$ Let $P_{1}(X),\cdots,P_{l}(X)$ be in $C[x_{1,1},x_{1,2},\cdots,x_{n,n}]_{\leq\kappa_{2}}$ such that their images constitute a $C$-basis of $C[x_{1,1},x_{1,2},\cdots,x_{n,n}]_{\leq\kappa_{2}}/(I(H))_{\leq\kappa_{2}}$, where $I(H)$ is the set of polynomials in $C[x_{1,1},x_{1,2},\cdots,x_{n,n}]$ that vanishes on $H$. Let $c_{1},\cdots,c_{l}$ be indeterminates and $P_{\mathbf{c}}(X)=\sum_{i=1}^{l}c_{i}P_{i}(X)$, where ${\mathbf{c}}=(c_{1},\cdots,c_{l})$. Then the conditions $P_{\mathbf{c}}(I_{n})=1\,\,\mbox{and}\,\,\forall\,\,h,h^{\prime}\in H,\,\,P_{\mathbf{c}}(h)P_{\mathbf{c}}(h^{\prime})-P_{\mathbf{c}}(hh^{\prime})=0$ induce a system of algebraic equations $J$ for ${\mathbf{c}}$. Moreover, we have the following proposition. ###### Proposition B.17 $\dim(J)=0$ and for each zero $\bar{{\mathbf{c}}}$ of $J$ in $C^{l}$, $P_{\bar{{\mathbf{c}}}}(X)$ is a character of $H$. * Proof. Evidently, for each $\bar{{\mathbf{c}}}\in{\rm Zero}(J)\cap C^{l}$, $P_{\bar{{\mathbf{c}}}}(X)$ is a morphism from $H$ to $C^{}$ and thus a character of $H$. Suppose that $\bar{{\mathbf{c}}},\bar{{\mathbf{c}}}^{\prime}\in{\rm Zero}(J)\cap C^{l}$ and $P_{\bar{{\mathbf{c}}}}(h)=P_{\bar{{\mathbf{c}}}^{\prime}}(h)$ for all $h\in H$. Then it implies that $P_{\bar{{\mathbf{c}}}}(X)-P_{\bar{{\mathbf{c}}}^{\prime}}(X)\in(I(H))_{\leq\kappa_{2}}$. Hence $\sum_{i=1}^{l}(\bar{c}_{i}-\bar{c}_{i}^{\prime})P_{i}(X)\equiv 0\mod(I(H))_{\kappa_{2}}$ where $\bar{{\mathbf{c}}}=(\bar{c_{1}},\cdots,\bar{c}_{l})$ and $\bar{{\mathbf{c}}}^{\prime}=(\bar{c_{1}}^{\prime},\cdots,\bar{c}_{l}^{\prime})$. Since $P_{1}(X),\cdots,P_{l}(X)$ modulo $(I(H))_{\kappa_{2}}$ are linearly independent over $C$. So $\bar{{\mathbf{c}}}=\bar{{\mathbf{c}}}^{\prime}$. That is to say, the map $\varphi:{\rm Zero}(J)\cap C^{l}\rightarrow X(H)$ defined by $\varphi(\bar{{\mathbf{c}}})=P_{\bar{{\mathbf{c}}}}(X)$ is injective. Now assume that $\dim(J)>0$. Then ${\rm Zero}(J)\cap C^{l}$ is an uncountable set and so is $X(H)$. However, it is known that $X(H)$ is a countable set. This contradiction concludes the proposition. $\Box$ ###### Remark B.18 Given a connect subgroup $H$ of ${\rm GL}_{n}(C)$, Propositions B.16 and B.17 allow us to compute a set of generators of $X(H)$ that are represented by polynomials in $C[x_{1,1},x_{1,2},\cdots,x_{n,n}]$. ## References [1] P.D. Bertrand and F. Beukers, Équations Différentielles Linéaires et Majorations de Multiplicités, Ann.Scient. éc. Norm. Sup., 4(18), 181-192, 1985. * [2] Elie Compoint and Michael F. Singer, Computing Galois groups of completely reducible differential equations, J. Symbolic Comput., 11,1-22,1998. * [3] J.H. Davenport and Michael F. Singer, Elementary and Liouvillian solutions of linear differential equations, J. Symbolic Comput., 2, 237-260, 1986. * [4] Thomas W. Dube, The Structure of polynomial ideals and Gröbner bases, SIAM J. Comput., 19(4), 750 C773, 1990. * [5] James E. Humphreys, Linear Algebraic Groups, Springer-Verlag New York, 1981. * [6] Ehud Hrushovski, Computing the Galois group of a linear differential equation, Banach Center Publications, 58, 97-138, 2002. * [7] Jerald J. Kovacic, An algorithm for solving second order linear homogeneous differential equations. J. Symbolic Comput. 2(1), 3 C43, 1986. * [8] Andy R. Magid, Lectures on Differential Galois Theory, University Lecture Series v.7, American Mathematical Society, 1994. * [9] A. Seidenberg, Constructions in algebra, Trans. AMS, 197, 273-313, 1974. * [10] Micael F. Singer, Moduli of linear differential equations on the Riemann sphere with fixed Galois groups, Pacific Journal of Mathematics, 160(2), 343-395, 1993. * [11] Michael F. Singer, Introduction to the Galois theory of linear differential equations, Algebraic Theory of Differential Equations, M.A.H. MacCallum and A.V. Mikhalov, eds., London Mathematical Society Lecture Note Series (no. 357), Cambridge University Press, 1-82, 2009. * [12] Michael F. Singer and F. Ulmer, Galois groups of second and third order linear differential equations, J. Symbolic Comput., 16, 9 - 36, 1993. * [13] L. van den Dries and K. Schmidt, Bounds in the theory of polynomial rings over fields. A nonstandard approach, Inventiones Mathematicae, 76, 77-91, 1984. * [14] Mark van Hoeij and Jacques-Arthur Weil, An algoirthm for computing invariants of differential Galois groups, J. Pure Appl. Algebra, 117/118, 353-379,1997. * [15] Marius van der Put, Galois theory and algorithms for linear differential equations, J. Symbolic Comput., 39, 451 C463, 2005. * [16] Marius van der Put and Michael F. Singer, Galois Theory of Linear Differential Equations, Springer-Verlag, Berlin Heidelberg, 2003.	arxiv-papers	2013-12-18T03:03:18	2024-09-04T02:49:55.599098	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Ruyong Feng", "submitter": "Ruyong Feng", "url": "https://arxiv.org/abs/1312.5029" }
1312.5075	# Derivation of breakup probabilities from experimental elastic backscattering data V.V.Sargsyan1, G.G.Adamian1, N.V.Antonenko1, W. Scheid2, and H.Q.Zhang3 1Joint Institute for Nuclear Research, 141980 Dubna, Russia 2Institut für Theoretische Physik der Justus–Liebig–Universität, D–35392 Giessen, Germany 3China Institute of Atomic Energy, Post Office Box 275, Beijing 102413, China ###### Abstract We suggest simple and useful method to extract breakup probabilities from the experimental elastic backscattering probabilities in the reactions with toughly and weakly bound nuclei. ###### pacs: 25.70.Jj, 24.10.-i, 24.60.-k Key words: sub-barrier capture, neutron transfer, quantum diffusion approach The fusion (capture) dynamics induced by loosely bound radioactive ion beams is being extensively studied Gomes ; EPJSub4 . However, the long-standing question whether fusion (capture) is enhanced or suppressed with these beams has not yet been answered unambiguously. The study of the fusion reactions involving nuclei close to the drip-lines has led to contradictory results. The lack of a clear systematic behavior EPJSub4 ; PRSGomes5 of the breakup probability as a function of the target charge requires additional experimental and theoretical studies. The quasi-elastic backscattering has been suggested EPJSub4 as an alternative to investigate the breakup probability. Since the quasi-elastic experiments are usually not as complex as the fusion (capture) and breakup measurements, they are well suited to survey the breakup probability. In the present report we will show that by employing the experimental elastic backscattering data, one can extract the breakup probabilities of weakly bound nuclei. So, new method for the study of the breakup probability will be suggested. There is a direct relationship between the capture, quasi-elastic scattering and breakup processes, since any losses from the elastic scattering and breakup channels contribute directly to other channels (the conservation of the total reaction flux at given bombarding energy $E_{\rm c.m.}$ and angular momentum $J$): $\displaystyle P_{qe}(E_{\rm c.m.},J)+P_{cap}(E_{\rm c.m.},J)+P_{BU}(E_{\rm c.m.},J)=$ $\displaystyle=P_{el}(E_{\rm c.m.},J)+P_{rest}(E_{\rm c.m.},J)+P_{BU}(E_{\rm c.m.},J)=1,$ (1) where $P_{rest}=P_{cap}+P_{in}+P_{tr}$, $P_{qe}=P_{el}+P_{in}+P_{tr}$ is the quasi-elastic scattering probability, $P_{BU}$ is the breakup probability, and $P_{cap}$ is the capture probability. The quasi-elastic scattering ($P_{qe}$) is the sum of all direct reactions, which include elastic ($P_{el}$), inelastic ($P_{in}$), and a few nucleon transfer ($P_{tr}$) processes. In Eq. (1), we neglect the deep-inelastic collision process, since we are concerned with low energies. Equation (1) can be rewritten as $\displaystyle\frac{P_{el}(E_{\rm c.m.},J)}{1-P_{BU}(E_{\rm c.m.},J)}+\frac{P_{rest}(E_{\rm c.m.},J)}{1-P_{BU}(E_{\rm c.m.},J)}=P_{el}^{noBU}(E_{\rm c.m.},J)+P_{rest}^{noBU}(E_{\rm c.m.},J)=1,$ (2) where $P_{el}^{noBU}(E_{\rm c.m.},J)=\frac{P_{el}(E_{\rm c.m.},J)}{1-P_{BU}(E_{\rm c.m.},J)}$ and $P_{rest}^{noBU}(E_{\rm c.m.},J)=\frac{P_{rest}(E_{\rm c.m.},J)}{1-P_{BU}(E_{\rm c.m.},J)}$ are the elastic scattering and other channels probabilities, respectively, in the absence of the breakup process. From these expressions we obtain the useful formulas $\displaystyle\frac{P_{el}(E_{\rm c.m.},J)}{P_{rest}(E_{\rm c.m.},J)}=\frac{P_{el}^{noBU}(E_{\rm c.m.},J)}{P_{rest}^{noBU}(E_{\rm c.m.},J)}=\frac{P_{el}^{noBU}(E_{\rm c.m.},J)}{1-P_{el}^{noBU}(E_{\rm c.m.},J)}.$ (3) Using Eqs. (1) and (3), one can find the relationship between the breakup and elastic scattering processes: $\displaystyle P_{BU}(E_{\rm c.m.},J)=1-\frac{P_{el}(E_{\rm c.m.},J)}{P_{el}^{noBU}(E_{\rm c.m.},J)}.$ (4) The last equation is the main result of the present report. Note that similar formula $\displaystyle P_{BU}(E_{\rm c.m.},J)=1-\frac{P_{qe}(E_{\rm c.m.},J)}{P_{qe}^{noBU}(E_{\rm c.m.},J)}$ (5) was derived in Ref. EPJSub4 to relate the breakup and quasi-elastic scattering processes. The reflection elastic or quasi-elastic backscattering probability $P_{el,qe}(E_{\rm c.m.},J=0)=d\sigma_{el,qe}/d\sigma_{Ru}$ for bombarding energy $E_{\rm c.m.}$ and angular momentum $J=0$ is given by the ratio of the elastic or quasi-elastic scattering differential cross section $\sigma_{el,qe}$ and Rutherford differential cross section $\sigma_{Ru}$ at 180 degrees Timmers . Employing Eq. (4) or (5) and the experimental elastic or quasi-elastic backscattering data in the reactions with toughly and weakly bound isotopes-projectiles and the same or almost the same compound nucleus, one can extract the breakup probability of the exotic nucleus. For example, using Eq. (4) or (5) at backward angle, the experimental $P_{el,qe}^{noBU}$[4He+AX] of the 4He+AX reaction with toughly bound nuclei (without breakup), and the experimental $P_{el,qe}$[6He+A-2X] of the 6He+A-2X reaction with weakly bound projectile (with breakup), and assuming approximate equality $V_{b}$(4He+AX)$\approx V_{b}$(6He+A-2X) for the Coulomb barriers of very asymmetric systems, one can extract the breakup probability of the 6He: $\displaystyle P_{BU}(E_{\rm c.m.},J=0)=1-\frac{P_{el,qe}(E_{\rm c.m.},J=0)[^{6}{\rm He}+^{A-2}{\rm X}]}{P_{el,qe}^{noBU}(E_{\rm c.m.},J=0)[^{4}{\rm He}+^{A}{\rm X}]}$ (6) or $\displaystyle P_{BU}(E_{\rm c.m.},J=0)=1-\frac{P_{el,qe}(E_{\rm c.m.},J=0)[^{6}{\rm He}+^{A-2}{\rm X}]}{P_{el,qe}^{noBU}(E_{\rm c.m.},J=0)[^{4}{\rm He}+^{A-2}{\rm X}]}.$ (7) Comparing the experimental elastic or quasi-elastic backscattering probabilities in the presence and absence of breakup data in the reaction pairs 6He+68Zn and 4He+68,70Zn, 6He+122Sn and 4He+122,124Sn, 6He+236U and 4He+236,238U, 8He+204Pb and 4He+204,208Pb, 8Li+207Pb and 7Li+207,208Pb, 7Be+207Pb and 10Be+204,207Pb, 9Be+208Pb and 10Be+207,208Pb, 11Be+206Pb and 10Be+206,207Pb, 8B+208Pb and 10B+206,208Pb, 8B+207Pb and 11B+204,207Pb, 9B+208Pb and 11B+206,208Pb, 15C+204Pb and 12C+204,207Pb, 15C+206Pb and 13C+206,208Pb, 15C+207Pb and 14C+207,208Pb, 17F+208Pb and 19F+206,208Pb, leading to the same or almost the same corresponding compound nuclei, one can analyse the role of the breakup channels in the reactions with the light weakly bound projectiles 6,8He, 8Li, 7,9,11Be, 8,9B, 15C, and 17F at energies near and above the Coulomb barrier. One concludes that the elastic or quasi-elastic backscattering technique could be a very useful tool in the study of breakup. The breakup probabilities can be extracted from the elastic or quasi-elastic backscattering probabilities of systems mentioned above. We thank P.R.S. Gomes and A. Lépina-Szily for fruitful discussions and suggestions. This work was supported by NSFC, RFBR, and JINR grants. The IN2P3(France)-JINR(Dubna) and Polish - JINR(Dubna) Cooperation Programmes are gratefully acknowledged. ## References * (1) L.F. Canto, P.R.S. Gomes, R. Donangelo, and M.S. Hussein, Phys. Rep. 424, (2006) 1. * (2) V.V. Sargsyan, G.G. Adamian, N.V. Antonenko, W. Scheid, and H.Q. Zhang, Phys. Rev. C 86, 054610 (2012). * (3) P.R.S. Gomes et al., Phys. Rev. C 84, 014615 (2011); P.R.S. Gomes, J. Lubian, and L.F. Canto, Phys. Rev. C 79, 027606 (2009). * (4) H. Timmers et al., Nucl. Phys. A584, 190 (1995); H.Q. Zhang, F. Yang, C. Lin, Z. Liu, and Y. Hu, Phys. Rev. C 57, R1047 (1998); A.A. Sonzogni et al., Phys. Rev. C 57, 722 (1998); O.A. Capurro et al., Phys. Rev. C 61, 037603 (2000); S. Santra et al., Phys. Rev. C 64, 024602 (2001); R.F. Simões et al., Phys. Lett. B 527, 187 (2002); S. Sinha et al., Phys. Rev. C 64, 024607 (2001); E. Piasecki et al., Phys. Rev. C 65, 054611 (2002).	arxiv-papers	2013-12-18T09:35:37	2024-09-04T02:49:55.614818	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "V.V.Sargsyan, G.G.Adamian, N.V.Antonenko, W. Scheid, and H.Q.Zhang", "submitter": "Vazgen Sargsyan Dr.", "url": "https://arxiv.org/abs/1312.5075" }
1312.5138	# Locating Multiple Ultrasound Targets in Chorus Lei Song Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing, P.R.China Email:[email protected] Yongcai Wang Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing, P.R.China Email: [email protected] ###### Abstract Ranging by Time of Arrival (TOA) of Narrow-band ultrasound (NBU) has been widely used by many locating systems for its characteristics of low cost and high accuracy. However, because it is hard to support code division multiple access in narrowband signal, to track multiple targets, existing NBU-based locating systems generally need to assign exclusive time slot to each target to avoid the signal conflicts. Because the propagation speed of ultrasound is slow in air, dividing exclusive time slots on a single channel causes the location updating rate for each target rather low, leading to unsatisfied tracking performances as the number of targets increases. In this paper, we investigated a new multiple target locating method using NBU, called _UltraChorus_ , which is to locate multiple targets while allowing them sending NBU signals simultaneously, i.e., in chorus mode. It can dramatically increase the location updating rate. In particular, we investigated by both experiments and theoretical analysis on the necessary and sufficient conditions for resolving the conflicts of multiple NBU signals on a single channel, which is referred as _the conditions for chorus ranging and chorus locating_. To tackle the difficulty caused by the anonymity of the measured distances, we further developed _consistent position generation algorithm_ and _probabilistic particle filter algorithm_ to label the distances by sources, to generate reasonable location estimations, and to disambiguate the motion trajectories of the multiple concurrent targets based on the anonymous distance measurements. Extensive evaluations by both simulation and testbed were carried out, which verified the effectiveness of our proposed theories and algorithms. ## I Introduction (a) one target, one receiver (b) two targets have different distances to the receiver (c) two targets have the same distance to the receiver Figure 1: Experiments to test how the separation of NBU peaks are affected by the distances between the transmitters Locating by Time-of-Arrival (ToA) of Narrow band ultrasound provides good positioning accuracy by using very low cost hardware and simple system architecture, which is widely used in many locating system. e.g. ActiveBat[9], Cricket[6]. However, when multiple targets are transmitting the same band ultrasound to a common set of receivers, inevitable conflicts will happen at the receivers if the multiple targets are not appropriately coordinated. Further, because it is hard to code target ID into NBU, even if the NBU signals from multiple targets can be separated at a receiver, the receiver can hardly determine the source (transmitter) of the NBUs, resulting at locating ambiguity. To tackle these multiple target locating problems, existing NBU based locating systems generally rely on the exclusive working mode of the multiple targets, in which each target is assigned an exclusive time slot by TDMA or CSMA scheme to guarantee the NBU signal transmitted from one target is not conflicted to the others. However, because the propagation speed of the ultrasound is rather slow in the air (e.g., 100 ms are needed for ultrasound propagating 34 meters), the time slot for each target’s each transmission has to be long enough to avoid the transmitted NBU being conflicted with the same frequency NBUs from the other targets. Therefore, in exclusive mode[10][6][9], at any time, only NBU from one target is in propagation, which results at low locating updating rate for individual target when the number of target is large. This on one hand limits the locating capacity (number of simultaneously located targets), on the other hand affects the tracking fidelity, especially when the targets are moving quickly. To deal with these problems, in this paper, we investigated the problem of locating multiple NBU targets in chorus mode, which is to locate a set of targets concurrently by allowing them to transmit NBU signals in the same time slot. In this study, we conducted not only theoretical analysis, but also extensive simulations and hardware experiments. In particular, we addressed the difficulties of _chorus ranging_ (measuring TOAs from multiple concurrent targets) and _chorus locating_ (calculating locations for the multiple targets) from following five aspects: 1) We investigated via experiments on the conditions for a receiver to reliably separate the multiple NBUs from multiple concurrent targets. 2) It leads to the geometric conditions on the relationship among the targets to guarantee non-conflict multiple TOA measurements. 3) Since the measured TOAs lack source identity, we present consistent position generation algorithm, which exploits the historical consistence (in terms of the deviation to the historical position of the targets) to label the potential sources (transmitters) for the anonymous distances, and then to generate and to filter the potential positions via evaluating their self-consistence (in terms of the residue of location calculation). 4) By using the generated consistent positions as input, we proposed probabilistic particle filter algorithm to further disambiguate the trajectories of the multiple targets by using the consistence of the moving speeds and accelerations of targets as the evaluation metrics. 5) At last, location based transmission scheduling algorithm was proposed, which schedules the concurrent transmitters for reliable, online multi-target locating in chorus mode. The rest of this paper is organized as follows. Related work and background are introduced in Section II. We introduced the feasibility of chorus ranging in Section III. The conditions for successful multi-target chorus locating are presented in section IV. Techniques for identifying potential sources of TOA and the particle filter algorithms for trajectory disambiguating are presented in Section V. We proposed location-based transmitter scheduling scheme in Section VI. Simulations and experimental results are presented in Section VII. The paper is concluded with remarks in Section VIII. ## II Related Work and Background Ranging by TOA of NBU is a very attractive technique for fine-grained indoor locating due to its high accuracy, low cost, safe-to-user and user- imperceptibility. It can provide positioning accuracy in centimeter level even in 3D space, which makes it very fascinating in may indoor applications. Popular ultrasound TOA-based indoor locating system include Bat[9], Cricket [6], AUITS[10], LOSNUS[7], etc. Popular application scenario include location- based access control [8], location based advertising delivery[4], healthcare etc. Multiple target locating problem has been investigated in existing systems. When using NBU for TOA-based ranging, there is no room for coding the ID of target. Existing approaches let the target send a ultrasound-radio frequency (RF) pair. The RF signal is for synchronization and identification[9][6][10]. Since there are several Media Access Control(MAC) protocol for RF signal, they can be adopted to coordinating target by just extending the length of the time-slot. Another approach is to explore the broadband ultrasound. Compared to the narrowband version, broadband ultrasound requires the transducer [3] to have better frequency response performance. The broadband ultrasound wave can accommodate identity of target to support multiple targets. Furthermore, if the wave is encoded with orthogonal code[1], two waves can be decoded respectively even overlapped. But broadband locating needs high cost transducers, and the signal is more sensitive to the Doppler effects. To the best of our knowledge, very few results have been reported for locating in chorus mode, because the collision problem of NBUs are generally hard to tackle. In this paper, we investigate conditions and algorithms to resolve this challenge. ## III Feasibility of Chorus Ranging At first, we introduce _exclusive mode_ and _chorus mode_ and presents experiments to investigate the conditions for a receiver to successfully detect NBUs from concurrently transmitting targets. ### III-A Exclusive Mode In conventional approach, when there are multiple targets, to avoid conflicting of NBUs, RF+US signals from different targets are scheduled into different time slots (called _exclusive mode_). In each slot one target broadcasts RF+US signals simultaneously, where the RF signal is used to synchronize timers among the target and the receivers. Then the synchronized receivers measure the TOAs of the successive ultrasound wave from the target to estimate their distances to the target and to calculate the target’s location via by least square estimation or trilateration[5]. The exclusive slot assignment can be realized by utilizing the media access control(MAC) protocols of the RF signal, e.g., CSMA, TDMA[6][9][10]. But, because the propagation speed of ultrasound is quite slow in the air (340 m per second), the time-slot for each exclusive target has to be long enough to avoid NBU conflicting to the successively arrived NBU from other targets. For an example, Cricket[6] assigns each target nearly $100ms$ by CSMA protocol. Because $n$ targets need $n$ exclusive slots, the location updating rate of each individual target is only $O(\frac{1}{n})$, which may become unsatisfactory when there are large number of targets. ### III-B Chorus Mode In contrast to the exclusive mode, in chorus mode, we allow multiple targets to broadcast NBUs in the same time slot. A general way is to use a _RF commander_ to broad RF to synchronize the timers of the targets and the receivers, and let the targets to broadcast NBU signals simultaneously and concurrently with the RF. Each receiver detects the mixed ultrasound signals from the multiple targets in its communication range and tries to separate the NBU signals to estimate the TOAs from the targets, and then to determine the locations of the multiple targets. ### III-C Experiments on Multiple NBU signal Detection Detecting TOAs from concurrently transmitted NBU waves at the receiver is the critically first step for chorus ranging, which determines the feasibility of chorus ranging and locating. We conducted experiments using MTS450CA Cricket nodes [6] to investigate the conditions for successfully multiple TOA detection at a receiver. Before carrying out the experiments, we made some modification to the firmware of Cricket node. Firstly, the policy to detect only the first arising edge was canceled, which is originally designed in Cricket to filter out the NLOS (non- line-of-sight) and the echo signals, because the NLOS and echo waves arrive later than the direct path NBU. In the new version, the received wave power is continuously compared to a threshold. When a rising edge (or wavefront) is detected, a TOA event is reported and the comparator state is set to “high”. When the wave power decreases to be lower than the threshold, the comparator state returns to “low” to be ready for detecting the next wavefront. Secondly, we disable the CSMA protocol in target, so that the targets can send ultrasound simultaneously. #### III-C1 Aftershock The first experiment used one receiver and one target. The screen-shot on oscilloscope is shown in Fig. 1(a). The target send $200\mu s$ ultrasound wave. After about $4$ms, this NBU wave arrives at the receiver, which cause a $1$ms shock on the receiver’s sensor. Because the ultrasound is mechanical wave, the shock on the receiver is much longer than the length of the wave sent by the target. This phenomenon is called _aftershock_. When the sensor in the receiver is experiencing an aftershock, the comparator in the sensor is kept in _high state_ , which will block the detection of the newly arrived NBU wavefront. In other word, aftershock will cause loss of TOA measurements at the receiver. Intuitively, the longer is the aftershock, the more frequent is the loss. From the oscilloscope output, we can also see some secondary peaks caused by the echoes. These secondary peaks can be filtered out because their powers are lower than the threshold. After the energy of the aftershock fades below the threshold, the comparator switches to _low state_ , which is ready for detecting the next NBU wave. #### III-C2 Multiple TOA Detections Two targets and one receiver are used in the second experiment, in which the two targets are placed at different distances from the receiver. When the two targets broadcast ultrasound signal simultaneously, the detected waves at the receiver are shown in Fig. 1(b). In this case, the receiver detects two NBU wavefronts successfully (i.e., two TOAs of ultrasound), which is because the separation between the wavefronts is greater than the length of the aftershock, but note that the detected TOAs are anonymous, i.e., the receiver don’t know their targets. In the third experiments, the two targets have the same distance to the receiver, their generated waves at the receiver are overlapped, as shown in Fig. 1(c). In this case, only one TOA is measured at the first arising edge, which is also anonymous. ### III-D Condition on Detecting Multiple TOAs The above experiments showed clearly that whether two successive NBU waves arrived at a receiver can be successfully detected is determined by the time separation between the two waves. If the time separation is longer than the length of the aftershock generated by the first wave, the wavefront of the second wave can be detected, otherwise, the second wavefront will be lost because the comparator is already in high state. Since the length of the aftershock is affected by the received energy of the NBU signal detected at the receiver and by the inertia of the ultrasound transducer of the receiver, it will be better to choose ultrasound transducer with weak inertia to get shorter aftershock to improve the capability of detecting the successive ultrasound pulses. To formulate the impact of the aftershock, let’s denote $L_{max}$ as the longest possible aftershock generated by the strongest signal at the receivers. Let $v_{u}$ be the speed of the ultrasound, then ###### Definition 1 (confident separation distance) We define $\omega=L_{max}v_{u}$ as the confident separation distance between the successively arrived waves for the receiver to successfully detect their TOAs. From triangle inequality, it is easy to verify that if the TOAs from two concurrent targets can be successfully detected by a receiver, the distance between the two targets must be larger than $\omega$. Let’s further take the audible region of the ultrasound into consideration. We assume all the targets have the same broadcasting power, then: ###### Definition 2 (audible range of ultrasound) We define $r$ as the audible range of the ultrasound, which is the propagation distance of the ultrasound from a target before the wave power is lower than the detectable threshold of the receivers. By combining the separation distance and the audible range, we can arrive at the condition for a receiver to successfully detect TOAs from two concurrent targets. ###### Theorem 1 _We consider two targets $a$ and $b$ are at location $\mathbf{x}_{a}$ and $\mathbf{x}_{b}$ respectively, who send NBU waves in the same time slot, a receiver at location $\mathbf{x}_{x}$ can detect the TOAs of the two waves if:_ $\left\\{{\begin{array}[]{{20}{c}}{\left\|{{d_{a,x}}-{d_{b,x}}}\right\|>\omega}\\\ {{d_{a,x}}\leqslant r,{d_{b,x}}\leqslant r}\end{array}}\right.$ (1) where $d_{i,j}$ calculates the distance between $\mathbf{x}_{i}$ and $\mathbf{x}_{j}$. Figure 2: The blind region of $a$ caused by $b$ For the case of multiple targets, to check whether a receiver can detect TOAs from their concurrent transmissions, we can simply sort their distances to the receiver in an ascending order. When the difference between any two adjacent sorted distances are larger than $\omega$, and when the receiver is in their common audible region, the receiver can successfully detect the TOAs of their concurrent ultrasound waves. ### III-E “Blind Region” Impacted by a Concurrent Target Based on Theorem1, let’s now consider in which region will a receiver lose the TOA from a target $a$ when another target $b$ is transmitting concurrently. ###### Definition 3 (Blind region) Blind region of $a$ caused by $b$ is referred to the region in which the receivers cannot capture TOA from $a$, if $a$ and $b$ send wave at the same time. The area of blind region of $a$ caused by $b$ is denoted by $S^{B}_{a\leftarrow b}$. (a) $d_{a,b}>2r$ (b) $2r\\!\\!-\\!\\!\omega\\!\\!\leq\\!\\!d_{a,b}\\!\\!\leq\\!\\!2r$ (c) $\omega\\!\\!<\\!\\!d_{a,b}\\!\\!<\\!\\!2r\\!\\!-\\!\\!\omega$ (d) $0\\!\\!<\\!\\!d_{a,b}\\!\\!\leq\\!\\!\omega$ Figure 3: The grey region stands for blind region, where the rest part in audible-circle is audible region (a) Blind Region by 2 targets (b) by 3 targets (c) by 4 targets (d) by 5 targets (e) by 6 targets Figure 4: Blind-region of target $a$ caused by different number of other targets In Fig.2 the blind region of $a$ caused by $b$ is shown by the gray region, which is characterized by inequality functions: $0<d_{a,x}-d_{b,x}\leq\omega$ and $d_{ax}\leq r$. When the receivers are locating in this region, the NBU wave from $a$ will be hidden in the aftershock of the wave from $b$, so that the receiver cannot detect TOA from $a$. Depending on the distance between $a$ and $b$, i.e., $d_{a,b}$, $S^{B}_{a\leftarrow b}$ changes from 0 to $\frac{\pi r^{2}}{2}$. Figure 3 shows how $S^{B}_{a\leftarrow b}$ changes with $d_{a,b}$, which indicates that $S^{B}_{a\leftarrow b}$ is a function of $d_{a,b}$. Moreover the area of blind-region can be expressed in close form. $S^{B}_{a\leftarrow b}(d_{a,b})\\!\\!=\\!\\!\left\\{\begin{array}[]{ll}0&\textrm{$d_{a,b}>2r$}\\\ r^{2}(\theta-\sin\theta\cos\theta)&\textrm{$2r-\omega\leq d_{a,b}\leq 2r$}\\\ r^{2}(\theta-\sin\theta\cos\theta)-S_{e}&\textrm{$\omega<d_{a,b}<2r-\omega$}\\\ r^{2}(\theta-\sin\theta\cos\theta)&\textrm{$0<d_{a,b}\leq\omega$}\\\ \end{array}\right.$ (2) For clarity of expression, the detailed expansion of $S^{B}_{a\leftarrow b}(d_{a,b})$ can be referred in Appendix. We can just note that it is a monotone decreasing function of $d_{a,b}$. ## IV Conditions for Chorus Locating Now let’s consider the conditions for localizing multiple targets in the chorus mode. It is widely known that in ranging based locating algorithms such as trilateration, three distance measurements from non-collinear beacons are necessarily required for uniquely determining the position of a target. We therefore investigated the condition for obtaining at least three TOA measurements for a target in the chorus mode. Note that for randomly deployed receivers, the probability of three chosen receivers are collinear are small, therefore, the non-colinear constraint is not considered at this stage. ### IV-A How Many TOA Detectable Regions Are Left? We define the _TOA detectable region (TDR)_ of a target as its audible region minus its blind region. Fig.5 shows the blind region caused by one concurrent target. The white region in the audible circle is the TDR region. When multiple concurrent targets are presenting, the left TDR will be further reduced. We denote the TDR of a target $a$ caused by a concurrent target set $T$ as $S^{D}_{a\leftarrow T}$. The area of $S^{D}_{a\leftarrow T}$ will affect the possible number of receivers in it for whatever distributions of the receivers, which determines the number of TOAs that can be obtained for a target. #### IV-A1 Consider Pairwise Separation Among Targets When the number of the concurrent targets is more than 2, the blind region of the target $a$ is the union area of the blind-regions caused by all other targets in set $T$. $S^{B}_{a\leftarrow T}=\cup_{s\in T}S^{B}_{a\leftarrow s}$ (3) As indicated in (2), $S^{B}_{a\leftarrow b}$ is a monotone decreasing function of $d_{a,b}$, therefore, an intuition is that the less are the pair-wise distances among the targets, the larger is the blind region cased by each target. Therefore, we consider $S^{B}_{a\leftarrow T}$ when the pair-wise distances among all concurrent targets are the same, denoted by $d$. Via such a way, we characterize how the inter distances among the targets and their distributions affect the blind region of a particular target. #### IV-A2 Lower Bound of $S^{D}_{a\leftarrow T}$ in Multiple Target Case When all targets have the same pair-wise distance $d$, because the isotropous feature of the audible circle of $a$, the blind region caused by each individual target has the same shape and the same size. By inclusion-exclusion principle, the union area of these blind regions is the largest when the intersection area of the blind regions is the smallest. This case appears when the other targets are geometrically symmetrically distributed around $a$. Fig.4(e) shows the largest union area of the blind regions of $a$ when $\left\|T\right\|=2,3,4,5,6$ respectively. The corresponding TDR area is the lower bound of $S^{D}_{a\leftarrow T}$ for pairwise separation distance $\geq d$ and when the number of concurrent targets in the audible region is known. We omit the expressions of these lower bounds for space limitation. More generally, when there are unknown number of targets are presenting, we can also derive a lower bound of $S^{D}_{a\leftarrow T}$ for given $d$. It is the area of the inscribed circle centered at $a$ with radius $d/2$ in the TDR as shown in Fig. 5. Therefore, the lower bound of $S^{D}_{a\leftarrow T}$ for given pairwise separation $d$ is: $S^{D}_{a\leftarrow T}\geq\pi\left(\frac{d}{2}\right)^{2}$ (4) It is a monotone increasing function of $d$, which means that the larger is the pair-wise separation among the targets, the larger is the area of TDR for each target. Figure 5: Lower bound of blind-region ### IV-B Probability of Having At Least Three Receivers in TDR Based on the lower bound of $S^{D}_{a\leftarrow T}$, for any given distribution of the receivers, we can evaluate the probability and the expectation of at least three receivers in the TDR region of $a$. Note that different formulas can be utilized to estimate the lower bound of $S^{D}_{a\leftarrow T}$ if we know the number of concurrent targets in the audible region and the minimum separation distance $d$. Let’s consider a general case when the receivers are in Poisson distribution, i.e.$P(n_{r}=k)=\frac{\lambda^{k}e^{-\lambda}}{k!}$, where $\lambda$ is the expected number of receivers in a unit area (e.g. 1 $m^{2}$). By substituting the lower bound of $S^{D}_{a\leftarrow T}\geq\pi\left(\frac{d}{2}\right)^{2}$, the probability of at least three receivers are in $S^{D}_{a\leftarrow T}$ can be calculated as: $1-\sum\limits_{i=0}^{2}{p({n_{r}}=i)\geq}1-{e^{-\frac{{\lambda\pi{d^{2}}}}{2}}}\left[{1+\frac{{\lambda\pi{d^{2}}}}{2}+\frac{{{\lambda^{2}}{\pi^{2}}{d^{4}}}}{8}}\right]$ (5) ###### Theorem 2 _When receivers are in Poisson distribution with $\lambda$ expected receivers in a unit area, when the pair-wise separation among targets are larger than $d$, the probability of at least three receivers are presenting in the TDR of a target is lower bounded by_ $1-{e^{-\frac{{\lambda\pi{d^{2}}}}{2}}}\left[{1+\frac{{\lambda\pi{d^{2}}}}{2}+\frac{{{\lambda^{2}}{\pi^{2}}{d^{4}}}}{8}}\right].$ (6) Fig.6 plots the lower bound of $P(n_{r}\geq 3)$ as a function of $d$ and $\lambda$. We can see that for given $\lambda$, the lower bound of at least three receivers presenting in the TDR of a target increases exponentially with $d$. Note that the figure plots only the lower bound. Because the real TDR area can be much larger than the lower bound area of TDR, in real case, the probability of three receivers are in the TDR of a target can be much closer to 1. Figure 6: The lower bound of the probability of at least three receivers are in the TDR of a target as a function of $d$ and $\lambda$ The results in Fig.6 show the strong feasibility of chorus locating. It only needs the targets are well separated and the receivers have enough density for receivers to obtain at least three TOA-based distances for each concurrent target. Figure 7: The diagram of consistent location generation and probabilistic particle filter algorithms to utilize the anonymous distances measured by receivers to locate and disambiguate the tracks of multiple targets ## V Locate Multiple Targets by Anonymous Distances Above analysis shows the feasibility of detecting multiple anonymous TOAs at a receiver in chorus mode. But the receiver don’t know the source (target) of each TOA. To utilize these TOAs to locate the multiple targets, we developed methods to effectively utilize the anonymous distances to locate the multiple targets and to disambiguate their trajectories. We introduce the proposed algorithms in this Section. ### V-A Overview The overview of the proposed techniques are shown in Fig.7, which contain mainly two parts: 1) consistent position generation and 2) probabilistic particle filter for trajectory disaggregation. In the first part, the inputs are the set of anonymous distances measured by the receivers, denoted by $[\mathbf{D}_{1},\cdots,\mathbf{D}_{m}]$, and the coordinates of these receivers, denoted by $[\mathbf{x}_{1},\cdots,\mathbf{x}_{m}],$where $m$ is the number of receivers. The number of distances measured by the $i$th receiver is $\left\|{{\mathbf{D}_{i}}}\right\|={k_{i}}$. #### V-A1 Overview of Consistent Position Generation Since each three distances from non-collinear receivers can generate a position estimation, enumerating the combinations of these anonymous distances will generate a large amount of possible positions, in which most of the positions are wrong. To avoid the pain of finding needless from the sea of large amount of potential positions, we proposed to firstly find the feasible distance groups by historical-consistency, i.e., by utilizing the consistency of distance measurements with the latest location estimations of the targets (which are provided by the particle filter). After this step, the distance groups are utilized to generate a set of potential positions. To further narrow down the potential position set, we proposed self-consistency to evaluate the residue of location calculation of each potential position. Only the top $N_{c}$ potential locations with good self-consistency will retained to be used as input to the particle filter at time $t$. #### V-A2 Overview of Probabilistic Particle Filter The particle filter maintains the positions of $n$ targets at time $t-1$, denoted by $\\{\mathbf{x}_{i}(t-1)\\}$; maintains $l$ most possible tracks for each target up to time $t-1$, denoted by $\\{\mathbf{T}_{i}(1:t-1)\in R^{l(t-1)}\\}$; the probability distribution function (pdf) of each target’s velocity, denoted by $p_{v}(x)$; and the probability distribution function of each target’s acceleration, denoted by $p_{a}(x)$. Then at time $t$, for each target $i$, by connecting its $l$ tracks at time $t-1$ to $N_{c}$ potential positions at time $t$, $lN_{c}$ particles are generated. The velocity ($v_{j}(t),j=1,\cdots,lNc$) and acceleration ($a_{j}(t),j=1,\cdots,lNc$) of each particle are calculated, based on which, the likelihood of the particle $j$ is evaluated by $p_{v}(v_{j}(t))p_{a}(a_{j}(t))$. Then by ranking the likelihoods of the particles, $l$ top particles will be retained for target $i$ at time $t$, which are used to update the location estimation of target $i$ at time $t$, the historical tracks and the pdfs of velocity and acceleration. We introduce key points of the algorithm in following subsections. ### V-B Consistent Potential Position Generation #### V-B1 Historical Consistency To avoid generating a large amount of misleading potential positions by blind combinations of the anonymous distances, we proposed to measure the _historical consistency_ of the distances to label the distances to reasonable sources . The input of this step is the historical positions of the $n$ targets provided by particle filter and the distance set from the receivers. For a target, since the velocity of the target is upper-bounded in the real scenarios, which is denoted by $v_{e}$, its position at time $t$ will be bounded inside a disk centered at its position at $t-1$, with radius $v_{e}$, i.e., $\|\|\mathbf{x}_{i}(t)-\mathbf{x}_{i}(t-1)\|\|\leq v_{e}$ (7) For a receiver $j$, let $d_{{j},i^{(t-1)}}$ represent the distance from it to $\mathbf{x}_{i}(t-1)$. From triangular inequality, for every distance $D_{k}$ measured by receiver $j$ at time $t$, $D_{k}$’s potential source is labeled to target $i$ if: $\|D_{k}-d_{{j},i^{(t-1)}}\|\leq v_{e}$ (8) Then, only the distances with the same source (target) label will be selected to generate potential positions for the targets using trilateration. This step on one hand reduces the computation cost of generating massive possible positions, on the other hand avoids generating the obviously wrong positions. #### V-B2 Self-Consistency We further evaluate the self-consistency of the generated potential positions to further filter out the unreasonable position candidates. Considering a potential position $\mathbf{x}$ calculated by trilateration using $m$ distances $[D_{1},\cdots,D_{m}]$ from receivers at location $\mathbf{x}_{r_{1}},\cdots,\mathbf{x}_{r_{m}}$, the self-consistency of this location is measured by the residue of the location calculation: $S_{x}=\frac{1}{m}\sum_{i=1}^{m}\left(D_{i}-d_{x,r_{i}}\right)^{2}$ (9) where $d_{x,r_{i}}$ is the distance from $\mathbf{x}$ to receiver $\mathbf{x}_{r_{i}}$. Then only top $N_{c}$ potential positions with the best self-consistency performances will be retained as the input for particle filter to be further processed by particle filter at time $t$. ### V-C Probabilistic Particle Filter The particle filter maintains 1) the locations of $n$ targets at $t-1$; 2) $l$ most possible tracks for each target up to time $t-1$, and 3) the probability density functions (pdfs) of each target’s velocity and acceleration. The pdfs of each target’s velocity and acceleration are calculated based on historically velocity and acceleration up to $t-1$. They are utilized to evaluate the likelihood of the generated particles. #### V-C1 Generate and Evaluate Particles For each target, say $i$, by connecting its $l$ ending locations at $t-1$ (in its $l$ tracks) to the $N_{c}$ potential positions at time $t$, $lN_{c}$ particles are generated, each particle represents a potential track. Then we evaluate the likelihood of each particle $k,k=1,\cdots,lN_{c}$ by the following likelihood function: $c_{k}=p_{v}(v_{k}(t))p_{a}(a_{k}(t))$ (10) where $v_{k}(t)$ and $a_{k}(t)$ are calculated on the particle $k$ by: $v_{k}(t)=\|\mathbf{x}_{k}(t)-\mathbf{x}_{k}(t-1)\|,a_{k}(t)=v_{k}(t)-v_{k}(t-1)$ (11) Then the top $l$ particles with best likelihood will be retained for the target for the next step, and $\mathbf{x}(t)$ in the most possible particle will be output as the position estimation at time $t$. The pdfs of velocity and acceleration are updated accordingly. Such a progress will be applied to all the targets, and the algorithm of the probabilistic particle filter is listed in Algorithm 1. Figure 8: Evaluate the cost of each generated particle Algorithm 1 Probability Particle Filter for a Target $i$ 0: $\mathbf{T}_{i}(\\!1\\!:\\!t-1\\!)$ , possible location $\\{\mathbf{x}_{1},$ $\mathbf{x}_{2},\dots,\mathbf{x}_{n_{c}}\\}$. PDF of velocity $p_{v}(\cdot)$ and PDF of acceleration $p_{a}(\cdot)$. 0: Updated $\mathbf{T}_{i}(1:t)$, $p_{v}(\cdot)$ and $p_{a}(\cdot)$, $\mathbf{x}_{i}(t).$ 1: $\\{p_{1},\dots,p_{l\times n_{c}}\\}\leftarrow\mathbf{T}_{i}(\\!1\\!:\\!t-1\\!)\times\\{\mathbf{x}_{1},\dots,\mathbf{x}_{n_{c}}\\}$ // Generate particles by posible locations of tracks at $t-1$ 2: $\\{c_{1},\dots,c_{l\times n_{c}}\\}\leftarrow\mathbf{0}$ 3: for $i=1:l\times n_{c}$ do 4: $v_{k}(t)=\|\mathbf{x}_{k}(t)-\mathbf{x}_{k}(t-1)\|$ 5: $a_{k}(t)=v_{k}(t)-v_{k}(t-1)$ 6: $c_{k}=p_{v}(v_{k}(t))\cdot p_{a}(a_{k}(t))$ 7: end for 8: $\\{\hat{p}_{1},\dots,\hat{p}_{l\times n_{c}}\\}\leftarrow$ sorting $\\{p_{1},\dots,p_{l\times n_{c}}\\}$ by $\\{c_{1},\dots,c_{l\times n_{c}}\\}$ in ascending order 9: $\mathbf{T}_{i}(1:t)\leftarrow\\{\hat{p}_{1},\dots,\hat{p}_{l}\\}$ // preserve the first $l$ sorted particle 10: $p_{a}(\cdot)\leftarrow UpdatePDF(p_{a}(\cdot),\\{v_{1}(t),\dots,v_{l}(t)\\})$ 11: $p_{v}(\cdot)\leftarrow UpdatePDF(p_{v}(\cdot),\\{a_{1}(t),\dots,a_{l}(t)\\})$ 12: $\mathbf{x}_{i}(t)=\hat{p}_{1}$ Complexity of Algorithm1 can be easily verified. ###### Lemma 1 Complexity of algorithm 1 is $O(nN_{c}l\log(N_{c}l))$ ###### Proof: For each target, the most expensive step is to sort the $lN_{c}$ elements, which takes $O(N_{c}l\log(N_{c}l)))$, so the overall complexity for locating the $n$ targets is $O(nN_{c}l\log(N_{c}l)))$. ∎ The probabilistic particle filter provides good flexibility. 1) It supports the trade off between the locating accuracy and the executing time by changing the number of the preserved particles. 2) The likelihood of each Particle is calculated by considering both the velocity and the acceleration, which is online continuously updated, so that it can be suitable even when the targets have variated motion characters. A potential drawback of this particle filter approach is that a target may be lost when it is too close to other targets. When the location candidates of two targets are almost the same, all particles may follow one target and none particle follows the other. Although such kind of target lost happens only in chance, it affects the tracking performance occasionally. We show this problem can be well solved by location based time-slot scheduling, which is discussed in the next section. ## VI Location based Time-slot Scheduling Locating in chorus mode requires concurrent targets have enough pair-wise separation distances, otherwise the receivers cannot detect TOAs from their concurrent waves. Keeping concurrent targets to be spatially well separated is also important for the particle filter to confidentially disambiguate their tracks. In addition, the initial condition of the particle filter needs the initial location estimations to be as accurate as possible to avoid cascading errors. With consideration of these requirements, we designed location based time-slot assignment (LBTA) to appropriately schedule the concurrent transmissions of the targets. In general, LBTA assigns targets which are close to others or with unknown locations to work in exclusive time-slots to avoid conflict. Targets satisfying the separation distance are scheduled to transmit concurrently. At first in LBTA, a confident separation distance $d_{s}$ is calculated by the lower bound of TDR region (6) based on given density of the receivers, i.e., $\lambda$ to guarantee $P(n_{r}\geq 3)$ approaching 1. Then the targets with known locations will be separated into a set of _$d_{s}$ -separated groups_. Each group consists of several targets with the pair-wise distance among targets in the group is at least $d_{s}$. Then an exclusive time slot is assigned to the targets in the same _$d_{s}$ -separated group_. Exclusive slots are also assigned to the targets with unknown locations. Figure 9: An example of LBTA to assign time slots An example of LBTA is shown in Fig. 9, in which, six targets are presenting. We assume the locations of target $\\{1,\dots,5\\}$ are known and the locations of targets $6$ is still unknown. In this case, the targets with known locations are separated into two _$d_{s}$ -seperated groups_. An exclusive time-slot is assigned to every $d_{s}$-seperated group and the target with unknown location. _LBTA_ can help to solve both initialization problem and the risk of missing target in particle filter. At initial state, location of all $n$ targets are unknown. So $n$ time slots are required to locate the $n$ targets. From then on, all $n$ targets share one time slot unless pairwise distances between some targets are less than $d_{s}$. In this case, partition method on the $n$ target is used to separate the targets into $d_{s}$-separated groups. Although finding the minimum number of $d_{s}$-seperated group is NP-hard[2], this problem can be effectively addressed by a greedy approach in practice when the number of targets are limited. We proposed a greedy _DivideClosestTargets_ algorithm to address it. Algorithm 2 DivideClosestTargets 0: $\\{\mathbf{x}_{1},\dots,\mathbf{x}_{n}\\}$ and $d_{s}$ 0: $d_{s}$-seperated group partition, $\mathcal{G}_{1},\dots,\mathcal{G}_{n_{d}}$ 1: $n_{d}\leftarrow 1$, $\rm{tempg}_{1}\leftarrow\\{\mathbf{x}_{1},\dots,\mathbf{x}_{n}\\}$, $\rm{tempg}_{2}=\emptyset$ 2: while $\cup_{i=1}^{n_{d}-1}\mathcal{G}_{i}\neq\\{\mathbf{x}_{1},\dots,\mathbf{x}_{n}\\}$ do 3: while $(\rm{MinPairWiseDis(\rm{temp}\mathcal{G}_{1})}<d_{s})$ do 4: $[i,j]=$ select the closest pair in $\rm{tempg}_{1}$ 5: $\rm{temp}\mathcal{G}_{1}=\rm{temp}\mathcal{G}_{1}\setminus i$, $\rm{temp}\mathcal{G}_{2}=\rm{temp}\mathcal{G}_{2}+i$ 6: end while 7: $\mathcal{G}_{n_{d}}=\rm{temp}\mathcal{G}_{1}$, $n_{d}=n_{d}+1$ 8: $\rm{temp}\mathcal{G}_{1}=\rm{temp}\mathcal{G}_{2}$, $\rm{temp}\mathcal{G}_{2}=\emptyset$ 9: end while The algorithm always selects the closest pair in the current temp group, and put one of them into a new temp group, until all targets in current temp group have pairwise distance larger than $d_{s}$. This temp group will form a $d_{s}$-separated group. Then the algorithm process the new temp group, until all targets are assigned into $d_{s}$-separated groups. ## VII Evaluation Both simulations and experiments were conducted to evaluate the performances of multiple target locating in chorus mode. More specifically, the locating accuracy, efficiency of scheduling and, robustness of chorus locating against noise were evaluated and reported in this section. ### VII-A Simulation Figure 10: Settings of simulation for chorus locating. #### VII-A1 Settings of Simulation We conducted simulation by developing a multi-agent simulator in MATLAB environment. The setting of our simulation scenario is shown in figure 10. The black diamonds stand for receivers, which are deployed in grid of size $2m\times 2m$. The blue stars stand for targets. Motions of targets are identically independent random walk, that each target walks along a line and turns a random angle every 5 seconds. The velocities of the targets are normally distributed, with $\mu=1$ and $\sigma=0.1$. This motion character is close to real action of human in open space. In simulation, we set the number of targets to 10, whose actions are constrained in a box of size $10m\times 10m$. The length of a time slot, i.e., locating updating interval is set to $100ms$. The audible radius, i.e., $r$ of target is set to be $3m$. $\omega$, which the length of the aftershock is set to $0.33m$. The values of $r$, $\delta$ and $\omega$ in above setting are obtained from real values of Cricket [6] locating system. #### VII-A2 Locating accuracy without ranging noise We firstly evaluate the multiple target locating and trajectory disaggregation performances when no ranging noise is incurred, i.e., ranging error is zero. The accuracy for concurrently multiple target tracking is shown in figure 11(a) and 11(b). Fig.11(a) plots the real trajectories and estimated trajectories, which shows that the estimated trajectories coincide well with the real trajectories even trajectories overlap. The corresponding CDF of the locating error is shown in figure 11(b), which shows that more than $90\%$ of the locating error is less than $1cm$. We found that greater than $1cm$ location error appeared when ranges was lost due to aftershock at a receiver resulting at $<3$ TOAs which leads to incorrect location estimation. #### VII-A3 Accuracy vs. $\omega$ vs. time-slots Location accuracy under different $\omega$ is shown in figure 11(c). The CDFs of ranging errors when $\omega$ equals to $33cm,165cm,330cm$ are presented, which are the corresponding cases when the length of the aftershock are $1ms,5ms,10ms$ respectively. Although the accuracy gets worse with growing of $\omega$, $90\%$ of the locating errors in the 3 cases are still very small. We investigated and found that the good locating performances against the variation of $\omega$ were contributed by LBTA. With the growth of the aftershock, LBTA started to assign more time slots to the targets. The slot assignment results are also shown in Figure 11(d), where the average number of concurrent targets located per times-slot are highly dependent on $\omega$. With growing of $\omega$, the number of concurrently located targets per slot drops from 8 to 1.7. In other word, the chorus mode degenerated to the exclusive mode when $\omega$ is large, i.e., when the aftershock is long. #### VII-A4 Accuracy vs. ranging noises Ranging noises are inevitable in ultrasound based locating systems, therefore noise resistance ability of chorus locating was also evaluated. To simulate the effect of ranging noise, positive offset is randomly added to every distance measurement. Offset is distributed from 0 to $l_{o}$ uniformly. The CDFs of locating errors with different $l_{o}(cm)$ is presented in Fig. 11(e), with $l_{o}$ being $1cm$, $5cm$ and $10cm$ respectively. The corresponding $90\%$-error is $1cm$, $10cm$ and $15cm$. Although there are no explicate anti-noise modules, it is shown that chorus locating can work under the impacts of the ranging noises. (a) Real and estimated trace (b) CDF of locating (c) CDF vs. $\omega$ (d) Efficiency vs. $\omega$ (e) CDF vs. Noise Figure 11: Performance evaluation obtained by simulation ### VII-B Testbed experiment We also conduct hardware experiments by using Cricket nodes. 4 nodes were tuned as receivers, which were deployed in an umbrella-type topology. Three nodes were programmed as targets, which were controlled by a sync-node. More specially, every target sends a NBU pulse once it hears the synchronizing signal from the sync-node. The time slot was set to $100ms$. We modified the firmware of cricket, so that each receiver reports all detectable range measurements to a PC via rs232 cable. Chorus locating algorithm was run at the PC end to calculate the locations for the multiple targets. The setting of the test-bed is shown in Fig.12. (a) Receivers (b) Targets Figure 12: Setting of test-bed Fig.13(a) shows the locating accuracy when a target $A$ was attached to a toy train, which ran along a trail at $1m/s$, while two concurrent targets $b$ and $c$ were placed on the ground. The locations of these concurrent targets were tracked by the four receivers. The obtained trajectories of the target on the train are presented in figure 13(a). Since it is difficult to obtain the ground-truth of mobile target. CDF of static targets is presented in Fig. 13(b). It is shown that more than $90\%$ of the locating errors is less than $15cm$. Therefore these simulation and experiment results verified the efficiency of locating multiple targets in chorus mode and the effectiveness of the proposed algorithms. They show that satisfactory accuracy can generally be obtained by locating in chorus mode. (a) Trace of two target (b) Locating CDF of static target Figure 13: Performance evaluation obtained by testbed experiment ## VIII Conclusion We have investigated to locate multiple narrowband ultrasound targets in chorus mode, which is to allow the targets broadcast ultrasound concurrently to improve position updating rate, while disambiguating their locations by algorithms at the receiver end. We investigated the geometric conditions among the targets for confidently separating the NBU waves at the receivers, and the geometrical conditions for obtaining at least three distances for each concurrent target. To deal with the anonymous distance measurements, we present consistent position generation and probabilistic particle filter algorithms to label potential sources for anonymous distances and to disambiguate the trajectories of the multiple concurrent targets. To avoid conflicts of the close by targets and for reliable initialization, we have also developed a location based concurrent transmission scheduling algorithm. Further work includes more flexible wavefront detection technique to improve threshold based detection which is to further shorten the aftershock and to make the detection be more robust to echoes and noises. ## References [1] M. Alloulah and M. Hazas. An efficient cdma core for indoor acoustic position sensing. In Indoor Positioning and Indoor Navigation (IPIN), 2010 International Conference on, pages 1–5, 2010. * [2] A. V. Fishkin. Disk Graphs: A Short Survey. pages 1–5, Jan. 2004. * [3] M. Hazas and A. Hopper. Broadband ultrasonic location systems for improved indoor positioning. IEEE Transactions on Mobile Computing, 5(5):536–547, 2006. * [4] Z. Junhui and W. Yongcai. Pospush: A highly accurate location-based information delivery system. UBICOMM ’09, pages 52–58, 2009. * [5] H. Liu, H. Darabi, P. Banerjee, and J. Liu. Survey of wireless indoor positioning techniques and systems. Systems, Man, and Cybernetics, IEEE Transactions on, 37(6):1067 –1080, nov. 2007. * [6] N. B. Priyantha, A. Chakraborty, and H. Balakrishnan. The Cricket location-support system. In MobiCom ’00: Proceedings of the 6th annual international conference on Mobile computing and networking. ACM Request Permissions, Aug. 2000\. * [7] H. Schweinzer and M. Syafrudin. Losnus: An ultrasonic system enabling high accuracy and secure tdoa locating of numerous devices. In IPIN 2010, pages 1 –8, 2010. * [8] Y. Wang, J. Zhao, and T. Fukushima. Lock: A highly accurate, easy-to-use location-based access control system. In LoCA, volume 5561 of Lecture Notes in Computer Science, pages 254–270. Springer, 2009. * [9] A. Ward, A. Jones, and A. Hopper. A new location technique for the active office. Personal Communications, IEEE, 4(5):42–47, 1997. * [10] J. Zhao and Y. Wang. Autonomous ultrasonic indoor tracking system. In ISPA ’08, pages 532 –539, 2008. ## IX Apendix Parameters in (2) can be expanded as: $\theta=\arccos\frac{d_{a,b}}{2r}$ (12) and $S_{e}=\int_{0}^{y_{\beta}}\left(2\sqrt{r^{2}-y^{2}}-\omega v_{u}\sqrt{1+\frac{y^{2}}{d_{a,b}^{2}-\frac{1}{4}\omega^{2}v_{u}^{2}}}\ \right)dy$ (13) where $y_{\beta}=\frac{b_{h}}{c_{h}}\sqrt{r^{2}-b_{h}^{2}-2a_{h}r}\\\ $ (14) refers to the $y$ coordination of intersection point of hyperbola and circle. $a_{h}=\frac{v_{u}\omega}{2},b_{h}=\frac{d_{a,b}}{2},c_{h}=\sqrt{a_{h}^{2}+b_{h}^{2}}$ (15)	arxiv-papers	2013-12-14T07:47:51	2024-09-04T02:49:55.624488	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Lei Song, Yongcai Wang", "submitter": "Yongcai Wang", "url": "https://arxiv.org/abs/1312.5138" }
1312.5238	# Positive integer powers of symmetric $(0,1)$-Heptadiagonal matrix Murat GUBES Department of Mathematics, Kamil Ozdag Science Faculty, Karamanoglu Mehmetbey University, 70100 Campus, KARAMAN, TURKEY [email protected] and Durmus BOZKURT Department of Mathematics, Faculty of Science, Selçuk University, 42250 Konya, Turkey [email protected] ###### Abstract. In this paper, we derive a general expression for $m$th powers of symmetric $(0,1)$-heptadiagonal matrices with $n=3k$ order, $n\in\mathbb{N}$ $(k=1,2,3,...,n/3)$. ###### Key words and phrases: Heptadiagonal matrix, matrix powers, Chebyshev Polynomials. ###### 2000 Mathematics Subject Classification: 15A18; 15A15; 11B83 ## 1\. Introduction Let us define the $n$-by-$n$ symmetric $(0,1)$-heptadiagonal matrix $H=(h_{ij})$ as below: (1.1) $H=\left\\{\begin{array}[]{l}1,\text{ for\ }\left\|i-j\right\|=3\text{ }\\\ 0,\text{other}\end{array}\right.\,.$ In literature, there are a lot of papers about matrix powers, determinants and inverses (see [2-5], [7-8], [10] and [12-13]). Here, we get a general expression of $m$th powers $(m\in\mathbb{N})$ for symmetric $(0,1)$-heptadiagonal matrix with $n=3k$ $(k=1,2,3,...,n/3)$ orders. It is known that $m$th $(m\in\mathbb{N})$ power of a matrix $H$ is (1.2) $H^{m}=PJ^{m}P^{-1}$ here $P$ is transforming matrix of $H$ and $J$ is jordan form of $H.$ Let we consider the following determinants (1.3) $H_{n}(\alpha)=\left\|\begin{array}[]{ccccccc}\alpha&0&0&1&\cdots&0&0\\\ 0&\alpha&0&0&1&&0\\\ 0&0&\alpha&0&0&\ddots&\vdots\\\ 1&0&0&\alpha&\ddots&\ddots&1\\\ \vdots&1&0&0&\ddots&&0\\\ 0&&\ddots&\ddots&\ddots&\cdots&0\\\ 0&0&\cdots&1&0&\cdots&\alpha\end{array}\right\|$ and (1.4) $\Delta_{n}(\alpha)=\left\|\begin{array}[]{ccccccc}\alpha&1&0&0&\cdots&0&0\\\ 1&\alpha&1&0&\cdots&&0\\\ 0&1&\alpha&1&0&\ddots&\vdots\\\ &0&1&\alpha&\ddots&\ddots&0\\\ \vdots&&0&0&\ddots&\ddots&0\\\ 0&&&\ddots&\ddots&\cdots&1\\\ 0&0&\cdots&&0&1&\alpha\end{array}\right\|\text{.}$ Using the determinant (1.3), we find (1.5) $\left\|\lambda I-H\right\|=H_{n}(\alpha)$ where $\alpha=\lambda\in\mathbb{R}$. Then, from (1.3) and (1.4), we write (1.6) $H_{n}(\alpha)=\left(\Delta_{\frac{n}{3}}(\alpha)\right)^{3}\text{.}$ By using definition of the $\Delta_{n}(\alpha)$ as in [2,3,4], the recurrence relation is obtained as following (1.7) $\Delta_{n}(\alpha)=\alpha\Delta_{n-1}(\alpha)-\Delta_{n-2}(\alpha)$ where $\Delta_{0}(\alpha)=1,\Delta_{1}(\alpha)=\alpha,\Delta_{2}(\alpha)=\alpha^{2}-1$. By solving difference equation (1.7) and substituting the equation into (1.6), we get (1.8) $\Delta_{\frac{n}{3}}(\alpha)=U_{\frac{n}{3}}(\frac{\alpha}{2})\text{ }$ (1.9) $H_{n}(\alpha)=\left(U_{\frac{n}{3}}(\frac{\alpha}{2})\right)^{3}$ where $U_{n}(x)$ is the $n$th degree Chebyshev polynomial of second kind which is defined; (1.10) $U_{n}(x)=\frac{\sin((n+1)\arccos x)}{\sin(\arccos x)},-1\leq x\leq 1\text{ .}$ It’s well known that all the roots of $U_{n}(x)$ are defined as follows shown in [1], [6] (1.11) $x_{nk}=\cos(\frac{k\pi}{n+1}),-1\leq x_{nk}\leq 1\text{ .}$ By considering (1.5), (1.7)-(1.11), we find eigenvalues of the matrix $H$ (1.12) $\lambda_{k}=2\cos(\frac{3k\pi}{n+3}),k=\overline{1,\frac{n}{3}}\text{.}$ ## 2\. Integer powers of $H$ In this part of the paper, we find the matrix $P$ and $P^{-1}$ for the expression $H=PJP^{-1}$. Secondly, we present a general expression of $H^{m}$ for $m\in\mathbb{N}$ . Let we obtain the eigenvectors of matrix $H$ via linear homogeneous system (2.1) $\left(\lambda_{j}I-H\right)x=0$ where $\lambda_{j}$ are eigenvalues of $H$ . We explicitly write down the expression (2.1) as $\displaystyle\lambda_{j}x_{1}-x_{4}$ $\displaystyle=$ $\displaystyle 0$ $\displaystyle\lambda_{j}x_{2}-x_{5}$ $\displaystyle=$ $\displaystyle 0$ (2.2) $\displaystyle\lambda_{j}x_{3}-x_{6}$ $\displaystyle=$ $\displaystyle 0$ $\displaystyle\vdots$ $\displaystyle-x_{n-5}+\lambda_{j}x_{n-2}$ $\displaystyle=$ $\displaystyle 0$ $\displaystyle-x_{n-4}+\lambda_{j}x_{n-1}$ $\displaystyle=$ $\displaystyle 0$ $\displaystyle-x_{n-3}+\lambda_{j}x_{n}$ $\displaystyle=$ $\displaystyle 0$ By solving system of equations (2.2), we get the eigenvectors of matrix $H$; for $j=1,4,7,10,...,n-5,n-2$ and $k=1,2,...,\frac{n}{3}$ (2.3) $x_{jk}=U_{k-1}\left(\frac{\lambda_{\frac{j+2}{3}}}{2}\right)$ for $j=2,5,8,11,...,n-4,n-1$ and $k=1,2,...,\frac{n}{3}$ (2.4) $x_{jk}=U_{k-1}\left(\frac{\lambda_{\frac{j+1}{3}}}{2}\right)$ for $j=3,6,9,12,...,n-3,n$ and $k=1,2,...,\frac{n}{3}$ (2.5) $x_{jk}=U_{k-1}\left(\frac{\lambda_{\frac{j}{3}}}{2}\right)$ Now, we get the expression (1.2). Since, eigenvalues $\lambda_{k}$ $\left(k=1,2,...,\frac{n}{3}\right)$ are multiple, then each eigenvalue corresponds triple jordan cell $\ J_{j}\left(\lambda_{k}\right)$ in the matrix $J$. Thus, we obtain the jordan form of $H$ as $J=diag\left(\lambda_{1},\lambda_{1},\lambda_{1},\lambda_{2},\lambda_{2},\lambda_{2},...,\lambda_{\frac{n-3}{3}},\lambda_{\frac{n-3}{3}},\lambda_{\frac{n-3}{3}},\lambda_{\frac{n}{3}},\lambda_{\frac{n}{3}},\lambda_{\frac{n}{3}}\right)$ Denoting $j$-th column of $P$ by $P_{j}(j=\left(\overline{1,n}\right))$, $P=(P_{1},P_{2},P_{3},...,P_{n})$. Combining (2.3),(2.4) and (2.5), we achieve each column of $P$ as following; (2.6) $P_{j}=\left(\begin{array}[]{c}U_{0}(\frac{\lambda_{\frac{j+2}{3}}}{2})\\\ 0\\\ 0\\\ U_{1}(\frac{\lambda_{\frac{j+2}{3}}}{2})\\\ 0\\\ 0\\\ \vdots\\\ U_{\frac{n-3}{3}}(\frac{\lambda_{\frac{j+2}{3}}}{2})\\\ 0\\\ 0\end{array}\right),j=1,7,...,n-5,P_{j}=\left(\begin{array}[]{c}0\\\ U_{0}(\frac{\lambda_{\frac{j+1}{3}}}{2})\\\ 0\\\ 0\\\ U_{1}(\frac{\lambda_{\frac{j+1}{3}}}{2})\\\ 0\\\ \vdots\\\ 0\\\ U_{\frac{n-3}{3}}(\frac{\lambda_{\frac{j+1}{3}}}{2})\\\ 0\end{array}\right),j=2,8,...,n-4$ (2.7) $P_{j}=\left(\begin{array}[]{c}0\\\ 0\\\ U_{0}(\frac{\lambda_{\frac{j}{3}}}{2})\\\ 0\\\ 0\\\ U_{1}(\frac{\lambda_{\frac{j}{3}}}{2})\\\ \vdots\\\ 0\\\ 0\\\ U_{\frac{n-3}{3}}(\frac{\lambda_{\frac{j}{3}}}{2})\end{array}\right),j=3,9,...,n-3,\ P_{j}=\left(\begin{array}[]{c}U_{\frac{n-3}{3}}(\frac{\lambda_{\frac{j+2}{3}}}{2})\\\ 0\\\ 0\\\ U_{\frac{n-6}{3}}(\frac{\lambda_{\frac{j+2}{3}}}{2})\\\ 0\\\ 0\\\ \vdots\\\ U_{0}(\frac{\lambda_{\frac{j+2}{3}}}{2})\\\ 0\\\ 0\end{array}\right),j=4,10,...,n-2$ $P_{j}=\left(\begin{array}[]{c}0\\\ U_{\frac{n-3}{3}}(\frac{\lambda_{\frac{j+1}{3}}}{2})\\\ 0\\\ 0\\\ U_{\frac{n-6}{3}}(\frac{\lambda_{\frac{j+1}{3}}}{2})\\\ 0\\\ \vdots\\\ 0\\\ U_{0}(\frac{\lambda_{\frac{j+1}{3}}}{2})\\\ 0\end{array}\right),j=5,11,...,n-1,P_{j}=\left(\begin{array}[]{c}0\\\ 0\\\ U_{\frac{n-3}{3}}(\frac{\lambda_{\frac{j}{3}}}{2})\\\ 0\\\ 0\\\ U_{\frac{n-6}{3}}(\frac{\lambda_{\frac{j}{3}}}{2})\\\ \vdots\\\ 0\\\ 0\\\ U_{0}(\frac{\lambda_{\frac{j}{3}}}{2})\end{array}\right),j=6,12,...n$ where $U_{k}(\lambda_{k})$ denote the second kind Chebyshev polynomials. Hence, the transforming matrix is found as below; for $n=3k,(k=1,3,5,...,n/3)$; (2.29) $\displaystyle{\small P}$ $\displaystyle{\small=}$ $\displaystyle\left[\begin{array}[]{ccccccc}U_{0}(\frac{\lambda_{1}}{2})&0&0&U_{\frac{n-3}{3}}(\frac{\lambda_{2}}{2})&0&0&\cdots\\\ 0&U_{0}(\frac{\lambda_{1}}{2})&0&0&U_{\frac{n-3}{3}}(\frac{\lambda_{2}}{2})&0&\cdots\\\ 0&0&U_{0}(\frac{\lambda_{1}}{2})&0&0&U_{\frac{n-3}{3}}(\frac{\lambda_{2}}{2})&\cdots\\\ U_{1}(\frac{\lambda_{1}}{2})&0&0&U_{\frac{n-6}{3}}(\frac{\lambda_{2}}{2})&0&0&\ddots\\\ 0&U_{1}(\frac{\lambda_{1}}{2})&0&0&U_{\frac{n-6}{3}}(\frac{\lambda_{2}}{2})&0&\cdots\\\ 0&0&U_{1}(\frac{\lambda_{1}}{2})&0&0&U_{\frac{n-6}{3}}(\frac{\lambda_{2}}{2})&\cdots\\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots\\\ U_{\frac{n-3}{3}}(\frac{\lambda_{1}}{2})&0&0&U_{0}(\frac{\lambda_{2}}{2})&0&0&\cdots\\\ 0&U_{\frac{n-3}{3}}(\frac{\lambda_{1}}{2})&0&0&U_{0}(\frac{\lambda_{2}}{2})&0&\cdots\\\ 0&0&U_{\frac{n-3}{3}}(\frac{\lambda_{1}}{2})&0&0&U_{0}(\frac{\lambda_{2}}{2})&\cdots\end{array}\right.$ $\displaystyle\left.\begin{array}[]{cccc}\cdots&U_{0}(\frac{\lambda_{\frac{n}{3}}}{2})&0&0\\\ \cdots&0&U_{0}(\frac{\lambda_{\frac{n}{3}}}{2})&0\\\ \cdots&0&0&U_{0}(\frac{\lambda_{\frac{n}{3}}}{2})\\\ \ddots&U_{1}(\frac{\lambda_{\frac{n}{3}}}{2})&0&0\\\ \cdots&0&U_{1}(\frac{\lambda_{\frac{n}{3}}}{2})&0\\\ \cdots&0&0&U_{1}(\frac{\lambda_{\frac{n}{3}}}{2})\\\ \ddots&\vdots&\vdots&\vdots\\\ \cdots&U_{\frac{n-3}{3}}(\frac{\lambda_{\frac{n}{3}}}{2})&0&0\\\ \cdots&0&U_{\frac{n-3}{3}}(\frac{\lambda_{\frac{n}{3}}}{2})&0\\\ \cdots&0&0&U_{\frac{n-3}{3}}(\frac{\lambda_{\frac{n}{3}}}{2})\end{array}\right]$ for $n=3k,(k=2,4,6,...,n/3)$; (2.51) $\displaystyle{\normalsize P}$ $\displaystyle{\normalsize=}$ $\displaystyle\left[\begin{array}[]{ccccccc}U_{0}(\frac{\lambda_{1}}{2})&0&0&U_{\frac{n-3}{3}}(\frac{\lambda_{2}}{2})&0&0&\cdots\\\ 0&U_{0}(\frac{\lambda_{1}}{2})&0&0&U_{\frac{n-3}{3}}(\frac{\lambda_{2}}{2})&0&\cdots\\\ 0&0&U_{0}(\frac{\lambda_{1}}{2})&0&0&U_{\frac{n-3}{3}}(\frac{\lambda_{2}}{2})&\cdots\\\ U_{1}(\frac{\lambda_{1}}{2})&0&0&U_{\frac{n-6}{3}}(\frac{\lambda_{2}}{2})&0&0&\ddots\\\ 0&U_{1}(\frac{\lambda_{1}}{2})&0&0&U_{\frac{n-6}{3}}(\frac{\lambda_{2}}{2})&0&\cdots\\\ 0&0&U_{1}(\frac{\lambda_{1}}{2})&0&0&U_{\frac{n-6}{3}}(\frac{\lambda_{2}}{2})&\cdots\\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots\\\ U_{\frac{n-3}{3}}(\frac{\lambda_{1}}{2})&0&0&U_{0}(\frac{\lambda_{2}}{2})&0&0&\cdots\\\ 0&U_{\frac{n-3}{3}}(\frac{\lambda_{1}}{2})&0&0&U_{0}(\frac{\lambda_{2}}{2})&0&\cdots\\\ 0&0&U_{\frac{n-3}{3}}(\frac{\lambda_{1}}{2})&0&0&U_{0}(\frac{\lambda_{2}}{2})&\cdots\end{array}\right.$ $\displaystyle\left.\begin{array}[]{cccc}\cdots&U_{\frac{n-3}{3}}(\frac{\lambda_{\frac{n}{3}}}{2})&0&0\\\ \cdots&0&U_{\frac{n-3}{3}}(\frac{\lambda_{\frac{n}{3}}}{2})&0\\\ \cdots&0&0&U_{\frac{n-3}{3}}(\frac{\lambda_{\frac{n}{3}}}{2})\\\ \ddots&U_{\frac{n-6}{3}}(\frac{\lambda_{\frac{n}{3}}}{2})&0&0\\\ \cdots&0&U_{\frac{n-6}{3}}(\frac{\lambda_{\frac{n}{3}}}{2})&0\\\ \cdots&0&0&U_{\frac{n-6}{3}}(\frac{\lambda_{\frac{n}{3}}}{2})\\\ \ddots&\vdots&\vdots&\vdots\\\ \cdots&U_{0}(\frac{\lambda_{\frac{n}{3}}}{2})&0&0\\\ \cdots&0&U_{0}(\frac{\lambda_{\frac{n}{3}}}{2})&0\\\ \cdots&0&0&U_{0}(\frac{\lambda_{\frac{n}{3}}}{2})\end{array}\right]\text{.}$ Denoting $j$-th column of $P^{-1}$ by $\rho_{j}(j=\left(\overline{1,n}\right))$. We obtain the $P^{-1}=(\rho_{1},\rho_{2},\rho_{3},...,\rho_{n-2},\rho_{n-1},\rho_{n})$ matrix such that; (2.52) $\rho_{j}=\left(\begin{array}[]{c}h_{1}U_{\frac{j-1}{3}}(\frac{\lambda_{1}}{2})\\\ 0\\\ 0\\\ h_{2}U_{\frac{j-1}{3}}(\frac{\lambda_{2}}{2})\\\ 0\\\ 0\\\ \vdots\\\ h_{\frac{n}{3}}U_{\frac{j-1}{3}}(\frac{\lambda_{\frac{n}{3}}}{2})\\\ 0\\\ 0\end{array}\right),j=1,4,7,...,n-2\text{ , }\rho_{j}=\left(\begin{array}[]{c}0\\\ h_{1}U_{\frac{j-2}{3}}(\frac{\lambda_{1}}{2})\\\ 0\\\ 0\\\ h_{2}U_{\frac{j-2}{3}}(\frac{\lambda_{2}}{2})\\\ 0\\\ \vdots\\\ 0\\\ h_{\frac{n}{3}}U_{\frac{j-2}{3}}(\frac{\lambda_{\frac{n}{3}}}{2})\\\ 0\end{array}\right),j=2,5,8,...,n-1$ (2.53) $\rho_{j}=\left(\begin{array}[]{c}0\\\ 0\\\ h_{1}U_{\frac{j-3}{3}}(\frac{\lambda_{1}}{2})\\\ 0\\\ 0\\\ h_{2}U_{\frac{j-3}{3}}(\frac{\lambda_{2}}{2})\\\ \vdots\\\ 0\\\ 0\\\ h_{\frac{n}{3}}U_{\frac{j-3}{3}}(\frac{\lambda_{\frac{n}{3}}}{2})\end{array}\right),j=3,6,9,...,n-3,n$ From (2.52) and (2.53), the inverse of matrix $P$ can be written as below; (2.75) $\displaystyle P^{-1}$ $\displaystyle=$ $\displaystyle\left[\begin{array}[]{ccccccc}h_{1}U_{0}(\frac{\lambda_{1}}{2})&0&0&h_{1}U_{1}(\frac{\lambda_{1}}{2})&0&0&\cdots\\\ 0&h_{1}U_{0}(\frac{\lambda_{1}}{2})&0&0&h_{1}U_{1}(\frac{\lambda_{1}}{2})&0&\cdots\\\ 0&0&h_{1}U_{0}(\frac{\lambda_{1}}{2})&0&0&h_{1}U_{1}(\frac{\lambda_{1}}{2})&\cdots\\\ h_{2}U_{0}(\frac{\lambda_{2}}{2})&0&0&h_{2}U_{1}(\frac{\lambda_{2}}{2})&0&0&\cdots\\\ 0&h_{2}U_{0}(\frac{\lambda_{2}}{2})&0&0&h_{2}U_{1}(\frac{\lambda_{2}}{2})&0&\cdots\\\ 0&0&h_{2}U_{0}(\frac{\lambda_{2}}{2})&0&0&h_{2}U_{1}(\frac{\lambda_{2}}{2})&\cdots\\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots\\\ h_{\frac{n}{3}}U_{0}(\frac{\lambda_{\frac{n}{3}}}{2})&0&0&h_{\frac{n}{3}}U_{1}(\frac{\lambda_{\frac{n}{3}}}{2})&0&0&\cdots\\\ 0&h_{\frac{n}{3}}U_{0}(\frac{\lambda_{\frac{n}{3}}}{2})&0&0&h_{\frac{n}{3}}U_{1}(\frac{\lambda_{\frac{n}{3}}}{2})&0&\cdots\\\ 0&0&h_{\frac{n}{3}}U_{0}(\frac{\lambda_{\frac{n}{3}}}{2})&0&0&h_{\frac{n}{3}}U_{1}(\frac{\lambda_{\frac{n}{3}}}{2})&\cdots\end{array}\right.$ $\displaystyle\left.\begin{array}[]{cccc}\cdots&h_{1}U_{\frac{n-3}{3}}(\frac{\lambda_{1}}{2})&0&0\\\ \cdots&0&h_{1}U_{\frac{n-3}{3}}(\frac{\lambda_{1}}{2})&0\\\ \cdots&0&0&h_{1}U_{\frac{n-3}{3}}(\frac{\lambda_{1}}{2})\\\ \cdots&h_{2}U_{\frac{n-3}{3}}(\frac{\lambda_{2}}{2})&0&0\\\ \cdots&0&h_{1}U_{\frac{n-3}{3}}(\frac{\lambda_{2}}{2})&0\\\ \cdots&0&0&h_{1}U_{\frac{n-3}{3}}(\frac{\lambda_{2}}{2})\\\ \ddots&\vdots&\vdots&\vdots\\\ \cdots&h_{\frac{n}{3}}U_{\frac{n-3}{3}}(\frac{\lambda_{\frac{n}{3}}}{2})&0&0\\\ \cdots&0&h_{\frac{n}{3}}U_{\frac{n-3}{3}}(\frac{\lambda_{\frac{n}{3}}}{2})&0\\\ \cdots&0&0&h_{\frac{n}{3}}U_{\frac{n-3}{3}}(\frac{\lambda_{\frac{n}{3}}}{2})\end{array}\right]\text{.}$ Here, we obtain the $h_{k}$ related to eigenvalues of $H$ as the similar meaning in [2-4]. (2.76) $h_{k}=\frac{3\left(-1\right)^{k}\left(\lambda_{k}^{2}-4\right)}{2n+6},n=3k\text{ }(k=2,4,6,...,n/3)$ (2.77) $h_{k}=\left\\{\begin{array}[]{c}\frac{3(-1)^{k+1}\lambda_{\frac{n+6k+3}{6}}^{2}}{2n+6}\text{ },\text{ }if\text{ }1\leq k\leq\frac{n-3}{6}\\\ \frac{6(-1)^{k+1}}{n+3}\text{ },\text{ }if\text{ }k=\frac{n+3}{6}\\\ \frac{3(-1)^{k+1}\lambda_{\frac{n-2k+3}{2}}^{2}}{2n+6}\text{ },\text{ }if\text{ }\frac{n+9}{6}\leq k\leq\frac{n}{3}\end{array}\right.,n=3k\ (k=3,5,7,...,n/3)$ By using (2.29), (2.51),(2.75) and (1.2) we obtain general expression as follow; for $n=3k,$ $k=1,3,5,...,n/3,$ $k\in\mathbb{N}$; (2.78) $\displaystyle\left\\{H^{m}\right\\}_{i,j}$ $\displaystyle=$ $\displaystyle\mathop{\displaystyle\sum}\limits_{k=1}^{\frac{n-3}{6}}\left(\lambda_{2k-1}^{m}h_{2k-1}U_{\frac{i-\delta_{ij}}{3}}(\frac{\lambda_{2k-1}}{2})U_{\frac{j-\delta_{ij}}{3}}(\frac{\lambda_{2k-1}}{2})+\lambda_{2k}^{m}h_{2k}U_{\frac{n-\sigma_{ij}}{3}}(\frac{\lambda_{2k}}{2})U_{\frac{j-\delta_{ij}}{3}}(\frac{\lambda_{2k}}{2})\right)$ $\displaystyle+\lambda_{\frac{n}{3}}^{m}h_{\frac{n}{3}}U_{\frac{i-\delta_{ij}}{3}}(\frac{\lambda_{\frac{n}{3}}}{2})U_{\frac{j-\delta_{ij}}{3}}(\frac{\lambda_{\frac{n}{3}}}{2})$ where $h_{k}$ is defined as in (2.77), for $n=3k,k=2,4,6,...,n/3,k\in\mathbb{N};$ (2.79) $\left\\{H^{m}\right\\}_{i,j}=\mathop{\displaystyle\sum}\limits_{k=1}^{\frac{n}{6}}\left(\lambda_{2k-1}^{m}h_{2k-1}U_{\frac{i-\delta_{ij}}{3}}(\frac{\lambda_{2k-1}}{2})U_{\frac{j-\delta_{ij}}{3}}(\frac{\lambda_{2k-1}}{2})+\lambda_{2k}^{m}h_{2k}U_{\frac{n-\sigma_{ij}}{3}}(\frac{\lambda_{2k}}{2})U_{\frac{j-\delta_{ij}}{3}}(\frac{\lambda_{2k}}{2})\right)$ where $h_{k}$ is defined as in (2.76). $\delta_{ij}$ and $\sigma_{ij}$ are expressed as follows each formula; $\delta_{ij}=\left\\{\begin{array}[]{c}1\text{ \ \ \ \ },\text{ \ \ \ \ }i+j\equiv 2\mathop{\mathrm{m}od}(3)\\\ 2\text{ \ \ \ \ },\text{ \ \ \ \ }i+j\equiv 1\mathop{\mathrm{m}od}(3)\\\ 3\text{ \ \ \ \ },\text{ \ \ \ \ }i+j\equiv 0\mathop{\mathrm{m}od}(3)\end{array}\right.$ and $\sigma_{ij}=\left\\{\begin{array}[]{c}i+2\text{ \ \ \ \ },\text{ \ \ \ \ }i+j\equiv 2\mathop{\mathrm{m}od}(3)\\\ i+1\text{ \ \ \ \ },\text{ \ \ \ \ }i+j\equiv 1\mathop{\mathrm{m}od}(3)\\\ i\text{ \ \ \ \ \ \ \ \ \ },\text{ \ \ \ \ }i+j\equiv 0\mathop{\mathrm{m}od}(3)\end{array}\right.\text{.}$ ## 3\. Numerical Considerations Let’s give two numerical examples for the matrix (1.1). For$\ n=6,J=diag(\lambda_{1},\lambda_{1},\lambda_{1},\lambda_{2},\lambda_{2},\lambda_{2})$ and from the general formula (2.79), we obtain the each elements of $\left\\{H^{m}\right\\}_{ij}$ as $h_{11}=h_{22}=h_{33}=\frac{1}{6}(\lambda_{1}^{m}(-1)^{1}(\lambda_{1}^{2}-4)+\lambda_{2}^{m+1}(-1)^{2}(\lambda_{2}^{2}-4))$ $h_{14}=h_{25}=h_{36}=\frac{1}{6}(\lambda_{1}^{m+1}(-1)^{1}(\lambda_{1}^{2}-4)+\lambda_{2}^{m+2}(-1)^{2}(\lambda_{2}^{2}-4))$ $h_{44}=h_{55}=h_{66}=\frac{1}{6}(\lambda_{1}^{m+2}(-1)^{1}(\lambda_{1}^{2}-4)+\lambda_{2}^{m+1}(-1)^{2}(\lambda_{2}^{2}-4))$ $h_{41}=h_{52}=h_{63}=\frac{1}{6}(\lambda_{1}^{m+1}(-1)^{1}(\lambda_{1}^{2}-4)+\lambda_{2}^{m}(-1)^{2}(\lambda_{2}^{2}-4))\text{.}$ where $\lambda_{k}$ is as (1.12) and also other components are zero. For $n=9,$ let $J=diag(\lambda_{1},\lambda_{1},\lambda_{1},\lambda_{2},\lambda_{2},\lambda_{2},\lambda_{3},\lambda_{3},\lambda_{3})$ and from the general formula (2.78), we obtain the each elements of $\left\\{H^{m}\right\\}_{ij}$ as $a_{11}=a_{22}=a_{33}=\frac{1}{8}[(\lambda_{1}^{m}+\lambda_{3}^{m})(-1)^{2}\lambda_{3}^{2}+4\lambda_{2}^{m}(-1)^{2}(\lambda_{2}^{2}-1)]$ $a_{14}=a_{25}=a_{36}=\frac{1}{8}[(\lambda_{1}^{m+1}+\lambda_{3}^{m+1})(-1)^{2}\lambda_{3}^{2}+4\lambda_{2}^{m}(-1)^{2}(\lambda_{2}^{3}-\lambda_{2})]$ $a_{17}=a_{28}=a_{39}=\frac{1}{8}[(\lambda_{1}^{m}(\lambda_{1}^{2}-1)+\lambda_{3}^{m}(\lambda_{3}^{2}-1))(-1)^{2}\lambda_{3}^{2}+4\lambda_{2}^{m}(-1)^{2}(\lambda_{2}^{2}-1)^{2}]$ $a_{41}=a_{52}=a_{63}=\frac{1}{8}[(\lambda_{1}^{m+1}+\lambda_{3}^{m+1})(-1)^{2}\lambda_{3}^{2}+4\lambda_{2}^{m+1}(-1)^{2}]$ $a_{44}=a_{55}=a_{66}=\frac{1}{8}[(\lambda_{1}^{m+2}+\lambda_{3}^{m+2})(-1)^{2}\lambda_{3}^{2}+4\lambda_{2}^{m+2}(-1)^{2}]$ $a_{47}=a_{58}=a_{69}=\frac{1}{8}[(\lambda_{1}^{m}(\lambda_{1}^{3}-\lambda_{1})+\lambda_{3}^{m}(\lambda_{3}^{3}-\lambda_{3}))(-1)^{2}\lambda_{3}^{2}+4\lambda_{2}^{m}(\lambda_{2}^{3}-\lambda_{2})]$ $a_{71}=a_{82}=a_{93}=\frac{1}{8}[(\lambda_{1}^{m}(\lambda_{1}^{2}-1)+\lambda_{3}^{m}(\lambda_{3}^{2}-1))(-1)^{2}\lambda_{3}^{2}+4\lambda_{2}^{m}(-1)^{2}]$ $a_{74}=a_{85}=a_{96}=\frac{1}{8}[(\lambda_{1}^{m}(\lambda_{1}^{3}-\lambda_{1})+\lambda_{3}^{m}(\lambda_{3}^{3}-\lambda_{3}))(-1)^{2}\lambda_{3}^{2}+4\lambda_{2}^{m+1}(-1)^{2}]$ $a_{77}=a_{88}=a_{99}=\frac{1}{8}[(\lambda_{1}^{m}(\lambda_{1}^{2}-1)^{2}+\lambda_{3}^{m}(\lambda_{3}^{2}-1)^{2})(-1)^{2}\lambda_{3}^{2}+4\lambda_{2}^{m}(-1)^{2}(\lambda_{2}^{2}-1)]\text{.}$ where $\lambda_{k}$ is as (1.12) and also other components are zero. ## 4\. Conclusion We investigate a symmetric Heptadiagonal Matrix in terms of positive integer powers and relationship with second kind Chebyshev Polynomials. In this context, we obtain the eigenvalue formula of (1.1) associate with second kind Chebyshev Polynomial roots. Additionally, we obtain the general formula of positive integer powers for (1.1). In the last section of paper, some illustrative examples are given. ## References * [1] J.C. Mason, D. C. Handscomb, Chebyshev Polynomials, CRC Press, Washington, 2003.R. P. Agarwal, Difference Equations and Inequlities, Marcel Dekker, New York, 1992\. * [2] J. Rimas, On computing of arbitrary positive integer powers for one type of symmetric tridiagonal matrices of even order-II, Appl. Math. Comput. 172 (2006) 245–251. * [3] J. Rimas, On computing of arbitrary positive integer powers for one type of symmetric pentadiagonal matrices of even order, Applied Mathematics and Computation 203 (2008) 582–591. * [4] J. Rimas, On computing of arbitrary positive integer powers for one type of symmetric tridiagonal matrices of odd order-II, Appl. Math. Comput. 174 (2006) 676–683. * [5] J. Gutiérrez-Gutiérrez, Positive integer powers of certain tridiagonal matrices, Applied Mathematics and Computation 202 (2008) 133–140. * [6] L. Fox, J.B. Parke, Chebyshev Polynomials in Numerical Analysis, Oxford University Press, London, 1968. * [7] Maryam Shams Solary, Finding eigenvalues for heptadiagonal symmetric Toeplitz matrices, Journal of Mathematical Analysis and Applications, vol 402/2 (2013), pp 719–730. * [8] M. Elouafi and A. Driss Aiat Hadj, On the powers and the inverse of a tridiagonal matrix, Applied Mathematics and Computation 211 (2009) 137–141. * [9] R. P. Agarwal, Difference Equations and Inequlities, Marcel Dekker, New York, 1992\. * [10] R. Witula and D. Slota, Some phenomenon of the powers of certain tridiagonal and asymmetric matrices, Applied Mathematics and Computation 202 (2008) 348-359. * [11] S. N. Elaydi, An Introduction to Difference Equations, Santa Clara University, 1995. * [12] Q.W. Wang, Z.H. He, A system of matrix equations and its applications, Science China Mathematics, 56 (9) (2013) 1795-1820. * [13] T. Sogabe, New algorithms for solving periodic tridiagonal and periodic pentadiagonal linear systems, Applied Mathematics and Computation, vol. 202, no. 2, pp. 850–856, 2008.	arxiv-papers	2013-12-18T17:37:54	2024-09-04T02:49:55.636084	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Murat Gubes and Durmus Bozkurt", "submitter": "Murat Gubes", "url": "https://arxiv.org/abs/1312.5238" }
1312.5355	# Generative NeuroEvolution for Deep Learning Phillip Verbancsics & Josh Harguess Space and Naval Warfare Systems Center – Pacific San Diego, CA 92101 {phillip.verbancsics,joshua.harguess}@navy.mil ###### Abstract An important goal for the machine learning (ML) community is to create approaches that can learn solutions with human-level capability. One domain where humans have held a significant advantage is visual processing. A significant approach to addressing this gap has been machine learning approaches that are inspired from the natural systems, such as artificial neural networks (ANNs), evolutionary computation (EC), and generative and developmental systems (GDS). Research into deep learning has demonstrated that such architectures can achieve performance competitive with humans on some visual tasks; however, these systems have been primarily trained through supervised and unsupervised learning algorithms. Alternatively, research is showing that evolution may have a significant role in the development of visual systems. Thus this paper investigates the role neuro-evolution (NE) can take in deep learning. In particular, the Hypercube-based NeuroEvolution of Augmenting Topologies is a NE approach that can effectively learn large neural structures by training an indirect encoding that compresses the ANN weight pattern as a function of geometry. The results show that HyperNEAT struggles with performing image classification by itself, but can be effective in training a feature extractor that other ML approaches can learn from. Thus NeuroEvolution combined with other ML methods provides an intriguing area of research that can replicate the processes in nature. ## 1 Introduction Evolution has a significant role in creating biological visual systems [1, 2], thus a potential path for training artificial vision as effective as the biological counterparts is neuro-evolution (NE; [3]) . However, NE has been challenged because of the required structure sizes. While the human brain includes trillions of connections [4, 5], traditional NE approaches produce networks significantly smaller in size [3, 6]. Thus an active area of research is evolutionary approaches that address this gap [6]. One significant area of interest is generative and developmental systems (GDS), which are inspired by principles that map genotype to phenotype in nature. Such principles are enabled by reusing genetic information [4, 7]. For example, biological neural networks exhibit several important organizational principles, such as modularity, regularity, and hierarchy [8, 9], which can be exploited through the indirect encoding and enable evolution of complex structures. The GDS approach investigated in this paper is called the Hypercube-based NeuroEvolution of Augmenting Topologies (HyperNEAT; [6, 10, 11]). HyperNEAT trains an indirect encoding, called a _compositional pattern producing network_ (CPPN; [12]) that represents artificial neural network (ANN) connection weights. HyperNEAT has shown promise in simple visual discrimination tasks [13, 14, 15, 16], but has been limited to two or fewer hidden layers. This paper extends the investigation to deeper architectures by applying HyperNEAT to the MNIST benchmark dataset. Interestingly, the results show that HyperNEAT alone has difficulty in classifying the images, but HyperNEAT is effective at training a feature extractor for backward propagation learning. That is, HyperNEAT can learn the ‘deep’ parts of the neural network, while backward propagation can perform the fine tuning to make classifications from the generated features. ## 2 Background The geometry-based methods that underlie the approach are described in this section. ### 2.1 NeuroEvolution of Augmenting Topologies (NEAT) _Neuro-evolution_ (NE; [3]) methods train artificial neural networks (ANNs) through evolutionary algorithms. As opposed to updating weights according to a learning rule, candidate ANNs are evaluated in a task and assigned fitness to allow selection and creation of a new generation of ANNs by mutating and recombining their selected genomes. The NeuroEvolution of Augmenting Topologies (NEAT) algorithm [17] is a popular neuroevolutionary approach that has been proven in a variety of challenging tasks, including particle physics [18, 19], simulated car racing [20], RoboCup Keepaway [21], function approximation [22], and real-time agent evolution [23], among others [17]. NEAT starts with a population of small, simple ANNs that increase their complexity over generations by adding new nodes and connections through mutation. That way, the topology of the network does not need to be known a priori; NEAT searches through increasingly complex networks as it evolves their connection weights to find a suitable level of complexity. The techniques that facilitate evolving a population of diverse and increasingly complex networks are described in detail in Stanley and Miikkulainen [17]; the important concept for the approach in this paper is that NEAT is an evolutionary method that discovers the right topology and weights of a network to maximize performance on a task. The next section reviews the extension of NEAT called HyperNEAT that allows it to effectively train large neural structures. ### 2.2 CPPNs and HyperNEAT Hypercube-based NEAT (HyperNEAT; [6, 11]) is a GDS extension of NEAT that enables effective evolution of high-dimensional ANNs. The effectiveness of the geometry-based learning in HyperNEAT has been demonstrated in multiple domains, such as multi-agent predator prey [24, 25] and RoboCup Keepaway [26]. A full description of HyperNEAT is in Stanley et al. [6]. The main idea in HyperNEAT is that geometric relationships are learned though an indirect encoding that describes how the _weights_ of the ANN can be _generated_ as a function of geometry. Unlike a direct representation, wherein every connection in the ANN is described individually, an indirect representation describes a pattern of parameters without explicitly enumerating each such parameter. That is, information is reused in such an encoding, which is a major focus in the field of GDS from which HyperNEAT originates [27, 28]. Such information reuse allows indirect encoding to search a compressed space. HyperNEAT discovers the _regularities_ in the geometry and learns from them. The indirect encoding in HyperNEAT is called a _compositional pattern producing network_ (CPPN; [12]), which encodes the _weight pattern_ of an ANN [6, 10]. The idea behind CPPNs is that geometric patterns can be encoded by a _composition of functions_ that are chosen to represent common regularities. In this way, a set of simple functions can be composed into more elaborate functions through hierarchical composition.Formally, CPPNs are _functions_ of geometry (i.e. locations in space) that output connectivity patterns for nodes situated in $n$ dimensions. Consider a CPPN that takes four inputs labeled $x_{1}$, $y_{1}$, $x_{2}$, and $y_{2}$; this point in four-dimensional space can _also_ denote the connection between the two-dimensional points $(x_{1},y_{1})$ and $(x_{2},y_{2})$. The output of the CPPN for that input thereby represents the weight of that connection (figure 1). Figure 1: A CPPN Describes Connectivity. A grid of nodes, called the ANN _substrate_ , is assigned coordinates. (1) Every connection between layers in the substrate is queried by the CPPN to determine its weight; the line connecting layers in the substrate represents a sample such connection. (2) For each such query, the CPPN inputs the coordinates of the two endpoints. (3) The weight between them is output by the CPPN. Thus, CPPNs can generate regular patterns of connections. Because the connection weights are produced as a function of their endpoints, the final pattern is produced with _knowledge_ of the domain geometry, which is literally depicted geometrically within the constellation of nodes. Weight patterns produced by a CPPN in this way are called _substrates_ so that they can be verbally distinguished from the CPPN itself. It is important to note that the structure of the substrate is independent of the structure of the CPPN. The substrate is an ANN whose nodes are situated in a coordinate system, while the CPPN defines the connectivity among the nodes of the ANN. The experimenter defines both the location and role (i.e. hidden, input, or output) of each node in the substrate. In summary, HyperNEAT evolves the topology and weights of the CPPN that _encodes_ ANN weight patterns. An extension of HyperNEAT called HyperNEAT with Link Expression Output (HyperNEAT-LEO) was introduced to constrain connectivity with a bias towards modularity [15]. This extension separates the decision of weight magnitude and expression into _two_ different CPPN outputs and seeds the LEO with the concept of locality. The HyperNEAT-LEO variant is shown in Algorithm 1. The next section reviews deep learning that has been applied to image classification. Input: Substrate Configuration Output: Solution CPPN Initialize population of minimal CPPNs with random weights; while _Stopping criteria is not met_ do foreach _CPPN in the population_ do foreach _Possible connection in the substrate_ do Query the CPPN for weight $w$ of connection and LEO expression value $e$; if _$e >0.0$_ then Create connection with a weight $w$; Run the substrate as an ANN in the task domain to ascertain fitness; Reproduce CPPNs according to the NEAT method to produce the next generation; Output the champion CPPN. Algorithm 1 HyperNEAT-LEO Algorithm ### 2.3 Deep Learning in Image Classification Neural networks have experienced a resurgence thanks to breakthroughs in deep learning that have led to state of the art results in a number of challenging domains [29]. In particular, deep learning approaches have achieved remarkable performance in a number of object recognition benchmarks, often achieving the current best performance on these tasks. Such object recognition tasks where deep learning has achieved the best results include the MNIST hand-written digit dataset [30, 31], traffic sign recognition [32], and the ImageNet Large- Scale Visual Recognition Challenge [33]. The challenge for deep learning is how to effectively train such large neural structures. Traditional supervised learning approaches for neural networks, such as backward propagation, face problems that include the “curse of dimensionality” or vanishing gradient. Thus a popular alternative is to perform pre-training on the deep architecture through unsupervised learning. Restricted Boltzmann Machines [34] and auto-encoders [35] are among the common approaches to performing this unsupervised training. This pre-training can either act as an initial starting point for supervised learning of a deep network or a feature extractor from which machine learning approaches can learn [29]. Alternatively, supervised learning performance can be enhanced through the selection of the deep architecture, such as _convolutional neural networks_ (CNNs). Beginning with the Neocognitron [36], these deep architectures have been inspired by proposed models of the human visual cortex [37] and the concept of local receptive fields that take advantage of the input topology. That is, CNNs enforce a particular geometric knowledge by constructing an architecture that learns features based upon locality. Through local receptive fields, shared weights, and sub-sampling [38], CNNs enable backward propagation (supervised learning) to effectively train deep architectures. The next section introduces approaches to deep learning through HyperNEAT. ## 3 Approach: Deep Learning HyperNEAT Although the HyperNEAT method succeeds in a number of challenging tasks [6, 10, 11, 26] by exploiting geometric regularities, it has not yet been applied to tasks where deep learning is showing promise or to deep architectures. Because HyperNEAT learns as a function of domain geometry, it is well-suited towards architectures similar to CNNs. This approach introduces two modifications to HyperNEAT training. The first modification introduces the idea of HyperNEAT as a feature learner. Conventional HyperNEAT trains a CPPN that defines an ANN that is the solution, that is, the produced ANN is applied directly to the task and then the ANN’s performance on that task determines the CPPN’s fitness score. However, this HyperNEAT modification trains an ANN that transforms inputs into features based upon domain geometry and then the features are given to another machine learning approach to solve the problem, the performance of this learned solution then defines the fitness score of the CPPN for HyperNEAT (figure 2). In this way, HyperNEAT acts as a reinforcement learning approach that determines the best features to extract for another machine learning approach to maximize performance on the task. Figure 2: HyperNEAT Feature Learning. To learn features, HyperNEAT trains CPPNs (1) that generate the connectivity for a defined ANN substrate (2). The ANN substrate processes the inputs from a data set to produced a set of features (3). These features are given to another machine learning algorithm (4) that learns to perform the task (e.g. image classification). Machine learning then produces a solution that is evaluated on testing data (5). The performance of the solution on data not seen during training provides the fitness score of the CPPN for HyperNEAT. In this way, HyperNEAT not only discovers better learning features, but also better features for generalization. The second modification introduces alternative architectures to HyperNEAT. Traditionally, HyperNEAT produces the weight pattern for a ANN substrate that is feed forward, fully connected, and containing only sigmoid activation functions. However, the only information HyperNEAT sees about the substrate is the geometric coordinates of the neurons, thus HyperNEAT could be applied to any graph like structure wherein the nodes have coordinates associated with them. Because CNNs have been demonstrably successful in a number of domains, HyperNEAT is extended to include this alternative ANN architecture. Each of these extensions to HyperNEAT is explored in experiments described in the next section. ## 4 Experimental Setup These investigations are conducted on the MNIST dataset, which is a popular benchmark for machine learning because it is a challenging and relevant real world problem. MNIST is a set of $28\times 28$ pixel images of handwritten digits ($0-9$), separated into 60,000 training images and 10,000 testing images. The goal for machine learning is to correctly classify the handwritten digit contained within each image. To this end, we explore HyperNEAT with four combinations of experimental settings on the benchmark MNIST dataset. These settings are traditional HyperNEAT ANN architecture (feed-forward, fully- connection, sigmoid activation functions) or CNN architecture and HyperNEAT training the solution (an ANN for classification) or feature extractor (ANN that transforms images into features). The architecture for HyperNEAT in these experiments is a multi-layer neural network wherein the layers travel along the $z$-axis, each layer consists of a number of features ($f$-axis), and each feature has a constellation of neurons on the $x,y$-plane corresponding to pixel locations. Thus the CPPN represents points in an eight-dimensional Hyper-cube that correspond to connections in the four-dimensional substrate and each neuron is located at a particular ($x,y,f,z$) coordinate. Each layer is presented by a triple ($X,Y,F$), wherein $F$ is the number of features and $X,Y$ are the pixel dimensions. Thus the input layer is ($28,28,1$), because the input image is $28\times 28$ and contains only grayscale pixel values. The traditional HyperNEAT architecture for these experiments is a seven layer neural network with one input, one output, and five hidden layers, represented by the triples ($28,28,1$), ($16,16,3$), ($8,8,3$), ($6,6,8$), ($3,3,8$), ($1,1,100$), ($1,1,64$), and ($1,1,10$). Each layer in the traditional HyperNEAT ANN architecture is fully connected to the adjacent layers and each neuron has a bipolar sigmoid activation function. The CNN architecture to which HyperNEAT is applied replicates the LeNet-5 architecture that was previously applied to the MNIST dataset [38]. To operate as a feature extractor, the above architectures are modified such that the ANN substrate is cut off before the last hidden layer, that is, for the traditional ANN architecture the ($1,1,100$) layer becomes the new output layer and for the CNN architecture the ($1,1,120$) layer becomes the outputs. Thus each image is passed through the ANN substrate architecture to produce an associated feature vector. These feature vectors are then given to backward propagation to train an ANN with an architecture identical to the architecture that was removed from the substrates. The next section presents the results of these HyperNEAT variants. ## 5 Results For each of these experiments, results are averaged over 30 independent runs of 2500 generations with a HyperNEAT population size of 256. The fitness score is the sum of the true positive rate, true negative rate, positive predictive value, negative predictive value, and accuracy for each class plus the fraction correctly classified overall and the inverse of the mean square error from the correct label outputs. Each run randomly selects 300 images, evenly spread across the classes, from the MNIST training set for training. For regular HyperNEAT (i.e. not feature learning), fitness is determined by applying the ANN substrate to the training images. For HyperNEAT feature learning an additional 1000 images are randomly selected (again evenly spread across classes) from the MNIST training set. Backward propagation training is run for 250 epochs on the 300 selected images and then tested on the different set of 1000 images. The testing performance of the backward propagation trained ANN becomes the fitness of the CPPN for HyperNEAT. After evolution completes, the generation champions are evaluated on the MNIST testing set. In the HyperNEAT for feature learning case, the backward propagation trained ANN learns from the full MNIST training set. As seen in figure 3, HyperNEAT by itself quickly plateaus at a particular performance level and only gradually learns, achieving an average fitness score of $4.2$ by the end of training. The accuracy of these trained solutions on the testing data is also not impressive, achieving only a $23.9\%$ correct classifications. By applying HyperNEAT as a feature learner and allowing backward propagation to train, both fitness score during training and classification correctness during testing increase to $5.7$ and $58.4\%$, respectively. It is interesting to note that in both cases, the fitness score and the correct classifications are not completely correlated, that is, improvements in the fitness score can lead to decreases in classification accuracy. Figure 3: HyperNEAT Performance with Traditional ANN Architecture. HyperNEAT can learn to classify images by itself; however, learning quickly plateaus at a low performance and then learning slows, reaching a fitness score of $4.2$ and testing performance of $23.9\%$ correct classifications. By acting as a feature learner, HyperNEAT does not plateau and achieves improved performance over HyperNEAT alone, finishing training with a fitness score of $5.7$ and a testing score of $58.4\%$. Thus HyperNEAT as a feature learner is more promising for than HyperNEAT alone. Changing HyperNEAT to learn the weights of a CNN architecture, rather than the normal ANN architecture of HyperNEAT, does change performance (figure 4). In this case, HyperNEAT by itself plateaus faster at a lower fitness level, achieving an average fitness score of $4.1$ by the end of training, but the accuracy of these trained solutions on the testing data improves to $27.7\%$ correct classifications. On the other hand, feature learning with HyperNEAT on a CNN architecture significantly improves both metrics, achieving a $7.0$ fitness score and a $92.1\%$ correct classifications. An example of the feature maps generated by these HyperNEAT champions can be seen in figure 5. Figure 4: HyperNEAT Performance with CNN Architecture. The CNN architecture has a small effect on HyperNEAT only performance, lowering the fitness performance plateau to $4.1$, but increasing testing performance of $27.7\%$ correctness. Shifting to CNN architecture significantly improves HyperNEAT’s ability as a feature learner, allowing HyperNEAT to find features that achieve a fitness score $7.0$ and a testing score of $92.1\%$. Thus HyperNEAT can learn effective levels of performance given careful selection of the ANN substrate. (a) (b) Figure 5: Visualization of Feature Maps Generated by HyperNEAT. Example feature maps are shown for the digits nine (a) and three (b). Interestingly, there is a distinct pattern along the feature dimension (top to bottom), demonstrating patterns in the geometry of the feature maps. ## 6 Discussion and Future Work Evolution is a significant factor in the creation of biological systems, including the visual cortex [1, 2]. However, biological neural networks are often deep architectures and conventional NeuroEvolution approaches have been challenged in effectively training ANNs order of magnitude smaller than those found in nature. Emerging research from generative and developmental systems has provided an answer in the form of HyperNEAT. HyperNEAT can effectively learn weight patterns for an ANN substrate by training CPPNs, an indirect encoding that computes weights as a function of geometry. The challenge for HyperNEAT is that, by training an indirect encoding, the ability to control precise weights is diminished, thereby creating difficulties in making fine adjustments for tasks such as classification. That is, a change in the indirect encoding will cause changes across the entire weight pattern, when a change to a single weight is needed. This challenge can be addressed by applying HyperNEAT as a feature learner for another ML approach that then makes the fine adjustments for the task, as shown in this paper. Indeed, the results in this paper demonstrate that HyperNEAT successfully learn features to train a backward propagation trained ANN. Interestingly, changing to the convolutional neural network architecture had a significant impact on HyperNEAT’s performance. Two interesting questions arise: (1) Are there more effective architecture choices? and; (2) How do can they be discovered? A path to answering these questions may be an extension of HyperNEAT known as Evolvable Substrate HyperNEAT (ES-HyperNEAT; [39]). In ES- HyperNEAT, the ANN substrate is not defined a priori (except for inputs and outputs); instead, the pattern generated by the CPPN determines the placement neurons in the hidden layers. An exciting implication of this work is that evolution provides a means to learn a representation that is well-suited for the target machine learning approach. Through reinforcement learning, evolution can measure fitness as a function of how well an approach performs on the generated representation. Because the fitness measure does not depend on a particular error signal or gradient, multiple metrics can be incorporated. For example, the results in the paper incorporated several classification characteristics into the fitness function, such as measures of true positives, true negatives, false positives, and false negatives. By combining measures, evolution can explore multiple paths through the different metrics. In addition, the many candidate solutions that evolution generates will perform differently from each other and these solutions may be combined to create an enhanced feature set. Finally, because the fitness measure can be based upon performance of the trained solution on data not seen by ML approach, the discovered features may encourage learning that generalizes. Thus this approach provides anopportunity to encourage representations that enhance generalizability. ## 7 Conclusion This paper investigated deep learning through NeuroEvolution. While NE has been limited in the size of ANNs that could be effectively trained in the past, novel algorithms that operate on indirect encodings, such as HyperNEAT, allow effective evolution of large ANNs. Prior work with HyperNEAT has shown promise in simple vision tasks that operate on a raw visual field, but with non-deep architectures. By itself, HyperNEAT struggles to find ANNs that perform well in image classification; however, HyperNEAT demonstrates an effective ability to act as a feature extractor by being combined with backward propagation. Thus HyperNEAT provides a potentially interesting path for combining reinforcement learning and supervised learning in image classification, as evolution and lifetime learning combine to create the capabilities in biological neural networks. #### Acknowledgments This work was supported and funded by the SSC Pacific Naval Innovative Science and Engineering (NISE) Program. ## References * [1] Q. V. Le, L. A. Isbell, J. Matsumoto, M. 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1312.5451	# On the perturbations on satellites probing General Relativity S.Sargsyan, G.Yegorian, S.Mirzoyan Center for Cosmology and Astrophysics, Alikhanian National Laboratory, Yerevan, Armenia [email protected] ###### Abstract Non-gravitational Yarkovsky-Rubincam effect for LAGEOS and LAGEOS 2 satellites used to probe General Relativity has been revealed by means of the Kolmogorov analysis of their perturbations. We present the method and its efficiency at modeling of generated systems with properties expected at the satellite laser ranging measurements and then at satellite residual data analysis. ## 1 Introduction Two Earth’s satellites, LAser GEOdynamics Satellites (LAGEOS and LAGEOS2), have been used for testing of Lense-Thirring effect predicted by General Relativity, with resulting accuracy of 10% [1, 2]. The recently launched satellite LARES (LAser RElativity Satellite) is aimed for even higher accuracy testing of Lense-Thirring effect [3]. The analysis [4] of the LAGEOS and LAGEOS 2 residual data of their trajectories, i.e. the differences between real measurements and theoretically predicted trajectories based on the Earth’s gravity field’s possibly accurate reconstruction, has been performed using Kolmogorov method [5, 6, 7]. Kolmogorov analysis of the LAGEOS data revealed higher degree of randomness for LAGEOS than for LAGEOS 2 data, which was explained via the Yarkovsky- Rubincam effect, i.e. thermal thrust due to the thermal radiation by the anisotropic thermal heating of a satellites by Solar and Earth’s radiation. Although both satellites have almost identical orbits and internal structure, since LAGEOS has been on the orbit 16 years longer than LAGEOS 2, the effect of thermal thrust is random in the case of LAGEOS with respect to LAGEOS 2. In fact, the thermal drag is a thermal acceleration of a satellite directed along the spin axis of a satellite. However, whereas the orientation of the LAGEOS spin axis was almost chaotic at the time of the orbital analysis,the spin of LAGEOS 2 had a more stable orientation over the same period, since LAGEOS 2 was launched much later than LAGEOS (16 years). This explains the more chaotic nature of the LAGEOS residuals. Below, we represent the method and its application to generated systems with properties expected for the LAGEOS residuals, which then enabled the analysis of the real data. ## 2 The method vs the generated systems Kolmogorov’s stochasticity parameter is defined for $n$ independent values $\\{X_{1},X_{2},\dots,X_{n}\\}$ of a variable $X$, given in increasing order [5, 6, 7]. Two distribution functions are defined as follows: cumulative distribution function is $F(x)=P\\{X\leq x\\}$, while $F_{n}(x)$ is the empirical distribution function $F_{n}(x)=\left\\{\begin{array}[]{rl}0,&X<x_{1}\\\ k/n,&x_{k}\leq X<x_{k+1}\\\ 1,&x_{n}\leq X.\\\ \end{array}\right.$ (1) Then the stochasticity parameter $\lambda_{n}$ is $\lambda_{n}=\sqrt{n}\ \sup_{x}\|F_{n}(x)-F(x)\|\ .$ (2) Kolmogorov’s theorem states that for any continuous $F$ the limit $\lim_{n\to\infty}P\\{\lambda_{n}\leq\lambda\\}=\Phi(\lambda)$ is converging uniformly and independent on $F$ and $\Phi(\lambda)=\sum_{k=-\infty}^{+\infty}\ (-1)^{k}\ e^{-2k^{2}\lambda^{2}}\ ,\ \ \lambda>0\ ,$ (3) where $\Phi(0)=0$. For large enough $n$ and random sequence $x_{n}$ the stochasticity parameter $\lambda_{n}$ has a distribution tending to $\Phi(\lambda)$. If the sequence is not random, the distribution is different, therefore, the $\Phi(\lambda)$ defines the degree of randomness of a sequence [6, 7]. In order to show how this method can be informative for datasets as those of LARES satellites, we represent some results on the study of generated sequences revealing the behavior of $\lambda_{n}$ for certain classes of sequences. In Fig. 1 we represent the results for random and non-random sequences given as $x_{n}=107n\,mod513$ with randomly chosen coefficients; computations have been performed for 7,000 sequences of 10,000 elements each [8]. The abscissa axes denote $\lambda$ and the ordinate ones denote $\Phi(\lambda)$, the dashed line is the empirical distribution function, the solid line is the $\Phi(\lambda)$. From the Fig. 1a one can see that only for random sequences $\Phi(\lambda)$ practically coincide with the empirical distribution function, as follows from Kolmogorov’s theorem. In Fig.1b $\Phi(\lambda)$ is completely different from the empirical distribution function, as the values of $\lambda$ vary between 0.195 and 0.230. Figure 1: The function $\Phi$ vs stochasticity parameter $\lambda$ for the system $x_{n}=107n\,mod513$. Another class of generated sequences included those given in the form $z_{n}=\alpha x_{n}+(1-\alpha)y_{n}$, so that $x_{n}$ are the random sequences and $y_{n}=\frac{an\pmod{b}}{b}$ are the regular ones, $a$ and $b$ are prime numbers which are being fixed in each particular run, and both sequences within the interval $(0,1)$ have uniform distribution [9]. Parameter $\alpha$ is representing the fractions of random and regular sequences, respectively. Fig. 2 shows the dependence of $\chi^{2}$ vs the parameter $\alpha$. Figure 2: The behavior of $\chi^{2}$ vs the parameter $\alpha$ denoting the fraction of random and regular subsequences. These examples illustrate how the method works for sequences with various degree of correlations and randomness. This method has been applied for the analysis of the properties of cosmic microwave background radiation, data on X-ray clusters ([10, 11] and references therein). ## 3 The satellite residuals The data for LAGEOS and LAGEOS 2 which carried sets of laser reflectors, have been collected via the laser ranging from ground based stations during about 11 years, i.e. 4018 days by step of 14 days [1, 2]. The residuals have been obtained using the measured data and the theoretical trajectories obtained using the modeled gravity field of the Earth (Fig.3). Figure 3: The residuals of LAGEOS and LAGEOS 2. Both satellites have almost identical semi-major axes - for LAGEOS it equals to 12270 km and for LAGEOS 2 to 12160 km - with different orbital inclinations. The important difference is their stay time on the Earth’s orbit; LAGEOS and LAGEOS2 have been launched on 4 May 1976 and 23 October 1992, correspondingly. We used Kolmogorov’s method to analyse the residual datasets of both LARES satellites, to obtain their comparative degree of randomness. Despite the fact that both satellites are identical, one finds non-identical behavior of $\Phi$ for LAGEOS and LAGEOS 2 for Gaussian CDF vs the variation of the standard deviation $d\sigma$ [4] (Fig.4). As shows Fig. 4 the residuals of LAGEOS do possess about 10 times higher degree of randomness (chaos) than of LAGEOS 2. Thus, the behavior of $\Phi$ function for two identical satellite residuals enables to reveal Earth-Yarkovsky or Yarkovsky-Rubincam effect [12], i.e. non- gravitational perturbations acting on the satellites. The possibility of accurate estimation of the contribution of such non-gravitational effects is crucial for the basic goal, i.e. highly accurate testing of predictions of the General Relativity and obtaining constraints on its extensions. Figure 4: Kolmogorov function $\Phi$ vs $d\sigma$, LAGEOS (left), LAGEOS 2 (right). We thank I.Ciufolini, V.Gurzadyan, A.Paolozzi for the joint work on LAGEOS data. S.S. thanks G.Meylan for the hospitality in Laboratoire d’Astrophysique, EPFL, Lausanne, during the work on this paper. ## References * [1] Ciufolini I, Pavlis E 2004 Nature 431 958 * [2] Ciufolini I 2007 Nature 449 41 * [3] Ciufolini I et al 2012 Eur. Phys. J. Plus 127 133 * [4] Gurzadyan V G, Ciufolini I, Sargsyan S, Yegorian G, Mirzoyan S, Paolozzi A 2013 Europhys. Lett. 102 60002 * [5] Kolmogorov A N 1933 G.Ist.Ital.Attuar 4 83 * [6] Arnold V I 2008 Uspekhi Mat. Nauk 63 5 * [7] Arnold V I 2009 Trans. Mosc. Math. Soc. 70 31 * [8] Mirzoyan S, Poghosian E 2009 Mod.Phys.Lett 24 3091 * [9] Gurzadyan V G, Ghahramanyan T, Sargsyan S 2011 Europhys.Lett. 95 19001 * [10] Gurzadyan V G et al 2009 A & A 497 343 * [11] Gurzadyan V G et al 2011 A & A 525 L7 * [12] Rubincam D P 1990, J. Geophys. Res. 95 4881	arxiv-papers	2013-12-19T09:18:12	2024-09-04T02:49:55.662511	{ "license": "Public Domain", "authors": "S.Sargsyan, G.Yegorian, S.Mirzoyan", "submitter": "Seda Sargsyan MRS.", "url": "https://arxiv.org/abs/1312.5451" }
1312.5534	# Shaping the distribution of vertical velocities of antihydrogen in GBAR G. Dufour Laboratoire Kastler-Brossel, CNRS, ENS, UPMC, Campus Jussieu, F-75252 Paris, France P. Debu Institut de Recherche sur les lois Fondamentales de l’Univers, CEA-Saclay, F-91191 Gif sur Yvette, France A. Lambrecht Laboratoire Kastler-Brossel, CNRS, ENS, UPMC, Campus Jussieu, F-75252 Paris, France V.V. Nesvizhevsky Institut Max von Laue - Paul Langevin, 6 rue Jules Horowitz, F-38042, Grenoble, France S. Reynaud Laboratoire Kastler-Brossel, CNRS, ENS, UPMC, Campus Jussieu, F-75252 Paris, France A.Yu. Voronin P.N. Lebedev Physical Institute, 53 Leninsky prospect, Ru-117924 Moscow, Russia ###### Abstract GBAR is a project aiming at measuring the free fall acceleration of gravity for antimatter, namely antihydrogen atoms ($\overline{\mathrm{H}}$). Precision of this timing experiment depends crucially on the dispersion of initial vertical velocities of the atoms as well as on the reliable control of their distribution. We propose to use a new method for shaping the distribution of vertical velocities of $\overline{\mathrm{H}}$, which improves these factors simultaneously. The method is based on quantum reflection of elastically and specularly bouncing $\overline{\mathrm{H}}$ with small initial vertical velocity on a bottom mirror disk, and absorption of atoms with large initial vertical velocities on a top rough disk. We estimate statistical and systematic uncertainties, and show that the accuracy for measuring the free fall acceleration $\overline{g}$ of $\overline{\mathrm{H}}$ could be pushed below $10^{-3}$ under realistic experimental conditions. Keywords :Antihydrogen, Gravitation, Quantum reflection PACS : 04.80.Cc, 06.30.Ft, 34.35.+a, 36.10.Gv ## 1 Introduction Gravitational properties of antimatter have never been measured directly. A promising experimental method to do so consists in producing sufficiently cold antihydrogen atoms ($\overline{\mathrm{H}}$) and timing their free fall in the Earth’s gravity field. This approach is being pursued by AEGIS [1], ATHENA- ALPHA [2], ATRAP [3] and GBAR [4] collaborations. In order to get the highest accuracy for measuring the free fall acceleration $\overline{g}$ of $\overline{\mathrm{H}}$, one has to cool atoms down to low temperatures and to measure, or at least to deduce from design and calculations, the initial velocity distribution. We discuss here the method proposed by Walz and Hänsch [5] which is used in the GBAR project to reach very low temperatures : ${\overline{\mathrm{H}}}^{+}$ ions are trapped and cooled down to the lowest quantum state in a Paul trap, and $\overline{\mathrm{H}}$ is then produced by photo-detaching the excess positron. The photo-detachment pulse is the START signal for the free fall timing measurement, while the STOP signal is provided by the annihilation of $\overline{\mathrm{H}}$ atoms on a detection plate placed at a height $H$ below the center of the ion trap. Precision of this measurement depends crucially on the dispersion of vertical velocities before the free fall, which corresponds to the residual kinetic energy of the atoms after the cooling process. The aim of the present paper is to propose a new filtering method to further reduce the initial distribution of vertical velocities and thus improve the accuracy in the measurement of $\overline{g}$. In section 2 we justify our choice of characteristic values for the spatial localization of the initial atomic cloud by considering the spreading of the freely-falling wave-packet of $\overline{\mathrm{H}}$ in the gravitational field. We describe in section 3 the new method for shaping vertical velocities of $\overline{\mathrm{H}}$ in the quasi-classical approximation, and show in section 4 that the improvement of accuracy due to the velocity selection overcomes the degradation associated with the decrease of the statistics. We then present in section 5 a quantum-mechanical description of the experiment in order to validate the quasi-classical estimations of the preceding sections. In section 6 we list possible systematic effects and show that they scale down compared to those in the case of unrestricted free fall of $\overline{\mathrm{H}}$. We then deduce the accuracy which could be reached on the measurement of $\overline{g}$ under realistic experimental conditions. We neglect throughout this paper systematic effects related to the energy- dependent probability of quantum reflection of $\overline{\mathrm{H}}$ from the detection plate [6]. The atomic recoil in the photo-detachment process induces an additional velocity dispersion which is discussed in the last section on systematic effects. ## 2 Spreading of a freely-falling wave-packet In the simplified description presented in the introduction, the initial distribution at time $t=0$ is the lowest quantum state in the Paul trap. This corresponds to a Gaussian wave-packet with vertical velocity dispersion $\upsilon$ and vertical position dispersion $\zeta$ reaching the minimum in the Heisenberg uncertainty relation: $\displaystyle m\upsilon\zeta=\frac{\hbar}{2}$ (2.1) where $\hbar$ is the reduced Planck constant and $m$ the inertial mass of $\overline{\mathrm{H}}$. After their release from the trap at time $t=0$, atoms start falling freely in the Earth’s gravity field until they reach the detection plate placed at a height $H$ below the center of the trap. The time of fall is measured as the delay $t$ from their release to their annihilation on the detection plate. The acceleration of gravity $\overline{g}$ for antihydrogen is then deduced from the distribution of these fall times. This acceleration $\overline{g}$ for antihydrogen is related to the analog quantity $g$ defined for hydrogen by $\overline{g}=Mg/m$, where $M$ is the gravitational mass of $\overline{\mathrm{H}}$. We now discuss the distribution of free fall times, assuming for simplicity that this distribution is determined by initial dispersions of vertical velocity and position (other sources of uncertainty negligible). If the initial quantum state is poorly localized (large values of $\zeta$) then the spread of the fall times is too large because of the initial dispersion of height. In the opposite case where the wave-packet is too localized (small values of $\zeta$) then the spread of the fall times is too large because of initial dispersion of vertical velocity. An optimum localization of the initial quantum state should be found as a compromise between these two limiting cases. As the variations of position and velocity are uncorrelated in the initial wave-packet, a classical calculation gives the relative spread $\Delta t$ of the free fall times arising from both effects: $\displaystyle\frac{\Delta t}{t_{H}}=$ $\displaystyle\sqrt{{\left(\frac{\zeta}{2H}\right)}^{2}+{\left(\frac{\upsilon}{v_{H}}\right)}^{2}}$ (2.2) $\displaystyle=$ $\displaystyle\sqrt{{\left(\frac{\zeta}{2H}\right)}^{2}+{\left(\frac{\hbar}{2mv_{H}\zeta}\right)}^{2}}~{}.$ (2.3) The second of these relations uses (2.1) while the first one is valid even when $\upsilon$ and $\zeta$ do not reach the minimum in Heisenberg uncertainty relation. We have defined $t_{H}=\sqrt{2H/\overline{g}}$ and $v_{H}=\sqrt{2\overline{g}H}$ as the free fall time and velocity for a free fall height $H$ with zero initial velocity. The optimum size of the initial state, which minimizes $\Delta t$ in (2.3), is: $\displaystyle{\zeta}_{{opt}}=\sqrt{\frac{{\hbar H}}{mv_{H}}}~{}.$ (2.4) It leads to an optimum resolution for the free fall measurement: $\displaystyle{\left(\frac{\Delta t}{t_{H}}\right)}_{{opt}}=\sqrt{\frac{\hbar}{2mv_{H}H}}~{}.$ (2.5) The larger the product $mv_{H}H$ with respect to $\hbar/2$, the better this optimal resolution is. Better precisions are also obtained by increasing the fall height with the characteristics of the trap kept fixed. However, the setup size is limited by practical arguments involving price and space considerations. Note that equation (2.5) is translated in an uncertainty twice larger on the acceleration of gravity $\displaystyle\frac{\Delta\overline{g}}{\overline{g}}=2\frac{\Delta t}{t}$ (2.6) in the simple derivation presented here (a detailed analysis based on Monte- Carlo simulations is given in [4]). With the typical numbers used for the design [4] of the GBAR experiment ($H=0.3$ m so that $v_{H}\approx 2.4$ m/s if $\overline{g}\approx g$), one obtains $\zeta_{opt}\approx 88$ $\mu$m and $\left(\Delta t/t_{H}\right)_{opt}\approx 2.1{\times}{10}^{-4}$. If this optimum operation could be experimentally realized, the accuracy would reach $\left(\Delta\overline{g}/\overline{g}\right)_{opt}\approx 4.2{\times}{10}^{-4}$ for each detection of an annihilation event. With a total number of events $N_{\mathrm{tot}}\approx 2.6{\times}{10}^{4}$, calculated for a typical measuring time of 1 month and an average production rate of 1 ultracold $\overline{\mathrm{H}}$ atoms per period of 100 s, this would lead to the resolution after one month: $\displaystyle\left(\frac{\Delta\overline{g}}{\overline{g}\sqrt{N_{\mathrm{tot}}}}\right)_{opt}=\sqrt{\frac{2\hbar}{mv_{H}HN_{\mathrm{tot}}}}\approx 2.6\times 10^{-6}~{}.$ (2.7) We have assumed there were no large systematic effect. However, the size of the initial cloud used in the design of the GBAR experiment is far from this optimum. The Paul trap is characterized by its oscillation frequency $\omega$ which fixes the velocity and position dispersions in the ground state: $\displaystyle\zeta=\sqrt{\hbar/2m\omega}\quad,\quad\upsilon=\sqrt{\hbar\omega/2m}~{}.$ (2.8) The mean kinetic energy in the ground state is then $m\upsilon^{2}/2=\hbar\omega/4$. Therefore the range of trap frequencies that can be used is limited by the residual kinetic energy of the atoms after cooling. In GBAR, the considered frequency range is 0.1 MHz $<\omega/2\pi<$ 1 MHz, so that one gets 0.22 $\mu$m $>\zeta>$ 0.07 $\mu$m and 0.14 m/s $<\upsilon<$ 0.44 m/s. This means that the initial cloud is smaller than the optimum by about 3 orders of magnitude. The resolution is thus limited by the dispersion of initial velocity: $\displaystyle\frac{\Delta\overline{g}}{\overline{g}\sqrt{N_{\mathrm{tot}}}}\approx\frac{2\upsilon}{v_{H}\sqrt{N_{\mathrm{tot}}}}~{}.$ (2.9) As it is not experimentally feasible to further cool down the ions to reach the optimum size of the initial cloud, we propose in this paper to select the initial vertical velocity of the atoms. This will improve the resolution after each annihilation event by a factor scaling as the reduced velocity range $\Delta v/\upsilon$. The statistics is reduced by a factor scaling as $\sqrt{N/N_{\mathrm{tot}}}{\propto}\sqrt{\Delta v/\upsilon}$ (see equation (3.1)) so that an overall improvement is expected. Also systematic uncertainties will decrease dramatically. The description of the shaping device and the evaluation of its performance are discussed in more details in the next sections. ## 3 Shaping the distribution of vertical velocities of $\overline{\mathrm{H}}$ in GBAR The current design for GBAR is a classical free-fall experiment which aims at an accuracy of the order of 1% [4]. With a quantum detection technique, one could get significantly higher precision, in analogy to spectroscopy [7] or interferometry [8] of near-surface quantum states [9] of ultracold neutrons (UCNs) [10, 11]. However these techniques require high energy resolution and sufficient statistics [12, 13]. The method that we propose in this paper is an intermediate step in this direction which is less precise than the full quantum detection technique but allows for better statistics and simpler design. This method is analogous to the one used in the experiment on the observation of gravitational quantum states of ultra-cold neutrons [7, 14, 15, 16]. The distribution of initial vertical velocities is shaped by selecting the atoms passing through a shaping device consisting of two disks. A scheme of principle of the shaping device where all useful quantities are defined is shown in figure 1. In the sequel of this section, a simple analysis of the problem is presented in terms of quasi-classical arguments, to be confirmed in the next sections. A more complete quantum-mechanical description is also available in papers devoted to ultra-cold neutrons [15, 17, 18, 19, 20]. Figure 1: A scheme of principle of the proposed shaping device: an $\overline{\mathrm{H}}$ atom is released from the Paul trap (central spot) and it bounces a few times on the mirror surface of the bottom disk (arrows); if it scatters on the rough top surface, it annihilates (lightnings); otherwise, it escapes from the aperture between the two disks, and falls to the detection plate where it annihilates (lightning on the detection plate). $R$ is the radius of the bottom and top disks, $r$ is the radius of central openings in the disks, $h$ is the distance between the top surface of the bottom disk and the bottom surface of the top disk, $H$ is the distance between the top of the detection plate and the top of the bottom disk, $L$ is the horizontal distance between the initial spot and the detection point. In the zone between the two disks, atoms with sufficiently small vertical velocities bounce on the bottom mirror disk due to the high efficiency of quantum reflection in the Casimir-Polder potential [6]. If the top surface of the mirror disk is flat, smooth and horizontal, the horizontal velocity component as well as the total energy of the vertical motion do not change and atoms thus pass through the shaping device with high probability. This last statement would be precisely valid for ideal quantum reflection from the mirror surface; otherwise corresponding corrections have to be taken into account (more discussions below). On the other hand, atoms with large vertical velocities rise in the Earth’s gravity field to the height of the rough surface of the top disk and scatter non-specularly on this surface. As this scattering mixes horizontal and vertical velocity components, it leads to rapid loss of scattered $\overline{\mathrm{H}}$ through annihilation on the top or bottom disk. A few remarks are useful at this point: 1) the shaping device has to be coupled with the Paul trap (not shown on the figure); this point is not discussed in this paper except for the role of the openings left in the center of the disks for operating the Paul trap; note that the disks may consist of several sectors not covering the complete $2\pi$ horizontal angle in order to include the Paul trap in the overall design; 2) annihilation events are supposed to be detected with position-sensitive and time-resolving detectors; this will allow one to account for the time spent in the shaping device (see below); 3) due to the cylindrical symmetry of the device, all atoms with small enough vertical velocity components and any value and direction of the horizontal velocity component can pass through it with high probability. In order to describe the operation of the shaping device, we follow possible classical trajectories of atoms from the initial point where they are released to the points where they annihilate. As the size of the initial spot (discussed in section 2) is much smaller than any other characteristic size of the shaping device, it plays no role in the following. We suppose the initial spot of atoms to be placed at the height $H$ of the top surface of the bottom disk (origin for altitude placed at the detection plate). In a first step, we let the radius $r$ of the central opening tend to zero and the radius $R$ of the disk tend to infinity. Disregarding the losses due to imperfect quantum reflection on the mirror disk, we obtain the fraction of atoms going through the angular acceptance of the shaping device as: $\displaystyle\frac{N}{N_{\mathrm{tot}}}\approx\frac{\Delta v}{\upsilon}\sqrt{\frac{1}{2\pi}}$ (3.1) where $\upsilon$ is the standard deviation of the Gaussian distribution of vertical velocities and $\Delta v$ the range of vertical velocities fitting the aperture of the shaping device. With the geometry sketched in figure 1, the latter corresponds to atoms with vertical velocities $0<v<\Delta v$ with $\Delta v$ deduced from the energy needed to rise the height from $H$ to $H+h$ in the gravity field: $\displaystyle\Delta v=\sqrt{2\overline{g}h}~{}.$ (3.2) Note that the fraction of atoms going through the angular acceptance of the shaping device changes as a function of the height of the initial spot above the mirror disk as well as a function of the radius $r$; therefore equation (3.1) has to be modified for other positions of the spot. Also equation (3.1) has been written in the limit of a good velocity selection $\Delta v<\upsilon$, which entails through (3.2) that $h$ has a maximum value $h_{\max}$: $\displaystyle h<h_{\max}=\frac{{\upsilon}^{2}}{2\overline{g}}~{}.$ (3.3) With the GBAR numbers considered above, the maximum value $h_{\max}$ lies in the interval 1-10 mm. If this condition is not obeyed, equation (3.1) has to be replaced by the appropriate integral. We now take into account the finite values of the radii of the central openings $r$ and of the disks $R$. In order to do it properly, we have to consider the shape of the angular distribution of initial velocities. The operation of the Paul trap may indeed require anisotropy to be introduced between horizontal and vertical directions. This can be described by a ratio $\varepsilon$ between frequencies of operation in horizontal and vertical directions ${\omega}_{\mathrm{hor}}={\varepsilon\omega}$ ($\omega$ is the frequency already introduced for operation of the vertical trap). This ratio should be in the interval $2<\varepsilon<4$ for a proper operation of the Paul trap [21]. Using the same reasoning as in the preceding section, we deduce that the horizontal dispersions are $\displaystyle\upsilon_{\mathrm{hor}}=\upsilon\sqrt{\varepsilon}~{},\quad\zeta_{\mathrm{hor}}=\zeta/\sqrt{\varepsilon}~{}.$ (3.4) where $\upsilon$ and $\zeta$ are the dispersions already introduced for operation of the vertical trap. We can now discuss the role of the finite radius $r$ of the central openings. We want to avoid extra loss of statistics at the entrance of the device, and thus choose $r$ small enough so that the angular divergence there fits the angular acceptance of the shaping device: $\frac{h}{r}>\frac{\Delta v}{\upsilon\sqrt{\varepsilon}}~{},\quad r<r_{\max}=\frac{\upsilon\sqrt{\varepsilon h}}{\sqrt{2\overline{g}}}=\sqrt{\varepsilon hh_{\max}}~{}.$ (3.5) To write these relations, we have neglected the effect of gravity on the short distance $r$ and used the value in (3.4) of the root-mean-square (rms) dispersion of horizontal velocity. We then consider the role of the finite radius $R$ of the disk, using the following classical arguments. We want to produce an efficient loss of atoms having too large velocities with respect to the designed angular acceptance of the shaping velocity, and thus choose $R$ large enough so those atoms efficiently touch the top disk. Saying that they touch it at least once, this implies that the time $T$ they spend in the zone between the disks is about two times larger than the time $t_{h}=\sqrt{2h/\overline{g}}$ corresponding to a free fall on a height $h$: $\displaystyle T=\frac{R}{\upsilon\sqrt{\varepsilon}}>2t_{h}=2\sqrt{\frac{2h}{\overline{g}}}$ $\displaystyle\quad\Rightarrow\quad R>{R}_{\min}=\frac{4\upsilon\sqrt{{\varepsilon h}}}{\sqrt{2\overline{g}}}=4r_{\max}~{}.$ (3.6) Again, we have used the dispersion (3.4) of horizontal velocity to calculate the time $T$ spent in the shaping device for an atom with the rms velocity. Of course, the time $T$ depends on the actual horizontal velocity (not its rms value) so that a value larger than that calculated in (3.6) is required to produce an effective shaping for the whole distribution. We want also to stress that the time $T$ appears as a systematical delay in the free fall timing experiment so that its knowledge is crucial for accuracy. Here the fact that annihilation event detectors are position-sensitive is important. Measuring the horizontal distance $L$ between the initial spot and the detection point indeed gives the actual horizontal velocity of the atom $L/T_{\mathrm{tot}}$ with $T_{\mathrm{tot}}$ the time between escape from the trap and annihilation on the detector and allows one to correct the timing measurement for the time spent in the shaping device $T=RT_{\mathrm{tot}}/L$. At the exit of the shaping device, the height lies in the interval $\left[H,H+h\right]$ while the vertical velocity lies in the interval $\left[-\Delta v,+\Delta v\right]$. As discussed in the next section, this affects the resolution of the timing measurement in the same manner as the dispersion of velocities did affect the free fall measurement discussed in section 2. In order to optimize the various parameters, in particular the value of the radius $R$, we have to simulate the whole experiment, that is the photo-detachment, the passage through the shaping device, the free fall from its output slit to the detection plate, the timing of annihilation events, and the correction from the time spent in the device. In the present paper, we use simpler arguments to estimate the resulting accuracy of the measurement. ## 4 Estimation of statistical uncertainty At this point, we have all information needed to give a simple estimation of the statistical accuracy in this experiment. To this aim, we use the analogy with the free fall timing measurement to write the relative spread of the free fall times as (compare with (2.2)): $\displaystyle\frac{\Delta t}{t_{H}}=\sqrt{\alpha{\left(\frac{h}{2H}\right)}^{2}+\beta{\left(\frac{\Delta v}{v_{H}}\right)}^{2}}$ (4.1) $\alpha$ and $\beta$ are dimensionless numbers smaller than unity describing the shapes of position and velocity distributions at the output slit of the shaping device. For simplicity, we have supposed that these distributions are uncorrelated and we have considered that the correction for the time $T$ spent in the shaper has been done. As $\Delta v=\sqrt{2\overline{g}h}$ and $v_{H}=\sqrt{2\overline{g}H}$ with $h{\ll}H$, it follows that the relative spread $\left(\Delta t/t_{H}\right)$ is dominated by the effect of velocity dispersion and can be written as: $\displaystyle\frac{\Delta t}{t_{H}}\approx\sqrt{\frac{{\beta h}}{H}}~{}.$ (4.2) This corresponds to an accuracy $\left(\Delta\overline{g}/\overline{g}\right)\approx 2\sqrt{{\beta h}/H}$ for each detection of an annihilation event. We then obtain the resolution after one month of measurement, taking into account that the number of events is reduced by the velocity selection (compare with (3.1)): $\displaystyle\frac{\Delta\overline{g}}{\overline{g}\sqrt{N}}=2\sqrt{\frac{\beta h}{H}}\sqrt{\frac{\upsilon\sqrt{2\pi}}{N_{\mathrm{tot}}\Delta v}}=2\\!\left(\frac{{\pi h}{\beta}^{2}{\upsilon}^{2}}{\overline{g}{H}^{2}N_{\mathrm{tot}}^{2}}\right)^{1/4}.$ (4.3) It is instructive to compare this resolution with the analogous result obtained without the velocity selection mechanism. The improvement is described by the ratio of (4.3) to (2.9): $\displaystyle 2\left(\frac{{\pi h}{\beta}^{2}{\upsilon}^{2}}{\overline{g}{H}^{2}N_{\mathrm{tot}}^{2}}\right)^{1/4}\left(\frac{2\upsilon}{v_{H}\sqrt{N_{\mathrm{tot}}}}\right)^{-1}=\left(\frac{2{\pi h}{\beta}^{2}}{h_{\max}}\right)^{1/4}~{}.$ (4.4) The best accuracy is therefore achieved for smaller slit sizes. We take the value $H=0.3$ m chosen in the current design of GBAR, the worst case of $\beta=1$ and a velocity dispersion $\upsilon=$ 0.44 m/s and discuss three cases corresponding to decreasing values of $h$: 1. 1. Equation (4.4) shows that $h$ should be smaller than $\approx h_{\max}/2\pi$ for the shaping device to improve the resolution of the experiment. We choose as an example $h=1$ mm, so that the statistics is $N\approx 3.3\times 10^{3}$. The opening radius has to be smaller than $r_{\max}\simeq 3.2\sqrt{\varepsilon}$ mm and the disk radius should be larger than $R_{\min}\simeq 13\sqrt{\varepsilon}$ mm. The statistical accuracy is then $\Delta\overline{g}/\overline{g}\sqrt{N}\approx 2.0\times 10^{-3}$. Note that for a conducting mirror and a maximal vertical velocity $\sqrt{2gh}\approx 0.14$ m/s, the reflection probability for an atom is $78\%$ [6]. To simultaneously improve the resolution and reduce losses from annihilation on the bottom mirror, we move to smaller values of $h$. 2. 2. For $h<50$ $\mu$m, the atom flux through the slit can no longer be evaluated from classical arguments and the quantum behavior of $\overline{\mathrm{H}}$ in the slit between the disks has to be taken into account [7, 14, 15]. At the boundary $h=50$ $\mu$m, the statistics is $N\approx 7.3\times 10^{2}$ and the statistical accuracy is $\Delta\overline{g}/\overline{g}\sqrt{N}\approx 1.0{\times}{10}^{-3}$. The opening radius has to be smaller than $r_{\max}\simeq 0.7\sqrt{\varepsilon}$ mm. Note that the reflection probability for an atom with the maximal velocity $\sqrt{2gh}=3.1\times 10^{-2}$ m/s is 94% for a perfect mirror. 3. 3. For $h<20~{}\mu$m, only atoms in the lowest quantum state can pass through the slit. The reflection probability approaches unity in this case which also corresponds to the highest accuracy for the free fall timing measurement. This quantum limit is analyzed in sections 5.2 and 5.3. The first and second cases provide more comfortable conditions for merging the proposed shaping device and the Paul trap, as well as better statistics. In this discussion, we have disregarded several factors which may decrease statistics (annihilation of $\overline{\mathrm{H}}$ in the bottom disk, non- perfect merging of the angular acceptance of the optical device and the incoming beam of $\overline{\mathrm{H}}$, quantum reflection of $\overline{\mathrm{H}}$ from the reference plate, etc.). These factors have to be evaluated at a later stage. ## 5 Quantum mechanical description We now perform a quantum-mechanical description of the experiment, which will turn out to reproduce the main features and estimations of the quasi-classical treatment given above. ### 5.1 Free fall of a wave-packet We consider the free fall of a pre-formed quantum wave-packet of $\overline{\mathrm{H}}$ in the Earth’s gravity field, and estimate the accuracy of the corresponding time-of-fall measurement. We know that the initial state ${\Psi}_{0}\left(z\right)$ of the wave-packet is a Gaussian function centered in the vertical direction $z$ around the height $H$ of the center of the trap, with the vertical position dispersion given by (2.8): $\displaystyle\Psi_{0}(z)=\left(\frac{m\omega}{\hbar\pi}\right)^{1/4}\exp\left(-\frac{m\omega}{2\hbar}\left(z-H\right)^{2}\right)~{}.$ (5.1) This wave-function is calculated prior to the release, at a time where the gravity is compensated by the trap. After the photo-detachment event, the atom is suddenly released and its state is modified by the free fall in the gravity field. This evolution is given by the propagation equation: $\displaystyle\Psi\left(z,t\right)=\int_{-\infty}^{\infty}G\left(z,z^{\prime},t\right)\Psi_{0}\left(z^{\prime}\right)\mathrm{d}z^{\prime}$ (5.2) where $t$ is the free fall time and $G$ the propagator: $\displaystyle G\left(z,z^{\prime},t\right)=$ $\displaystyle\sqrt{\frac{m}{2i\pi\hbar t}}\exp\left[\frac{im}{2\hbar t}\left(z-z^{\prime}+\frac{\overline{g}t^{2}}{2}\right)^{2}\right]$ $\displaystyle\quad\times\exp\left[\frac{m\overline{g}zt+\frac{1}{6}m\overline{g}^{2}t^{3}}{i\hbar}\right]~{}.$ (5.3) Integrating (5.2) for the initially Gaussian wave-packet (5.1), one gets : $\displaystyle\Psi\left(z,t\right)$ $\displaystyle=\left(\frac{m\omega}{\hbar\pi(1+i\omega t)^{2}}\right)^{1/4}\\!\\!\\!\exp\left[\frac{m\overline{g}zt+\frac{1}{6}m\overline{g}^{2}t^{3}}{i\hbar}\right]$ $\displaystyle\times\exp\left[-\frac{m\omega}{2\hbar(1+i\omega t)}\left(z-H+\frac{\overline{g}t^{2}}{2}\right)^{2}\right]~{}.$ (5.4) Assuming that all atoms annihilate instantaneously when they touch the detection plate at $z=0$, we deduce that the distribution for annihilation times is given by the flux $\mathcal{F}(t)$ of atoms passing through the plane at height $z=0$, that is also the opposite of the current (downward velocities have negative values): $\displaystyle\mathcal{F}(t)$ $\displaystyle=-j(0,t)=-\frac{\hbar}{m}\text{Im}\left(\Psi(0,t)^{}\frac{\partial}{\partial z}\Psi(0,t)\right)~{},$ $\displaystyle=\sqrt{\frac{m\omega^{5}t^{2}}{\hbar\pi(1+\omega^{2}t^{2})^{3}}}\left(H+\frac{\overline{g}t^{2}}{2}+\frac{\overline{g}}{\omega^{2}}\right)$ $\displaystyle\times\exp\left[-\frac{m\omega}{\hbar(1+\omega^{2}t^{2})}\left(\frac{\overline{g}t^{2}}{2}-H\right)^{2}\right]~{}.$ (5.5) This probability distribution is shown in figure 2 for an initially Gaussian wavepacket dropped from 30 cm, in the two cases of an initial size typically expected for the GBAR expected (upper plot) and the optimal size discussed above (lower plot). Figure 2: The distribution of annihilation times of $\overline{\mathrm{H}}$ falling from the height $H=30$ cm; in the upper plot, the initial state is a Gaussian of width $\zeta=70$ nm, a typical value expected in the GBAR experiment; in the lower plot, it is a Gaussian with the optimal width $\zeta_{opt}=88$ $\mu$m. For comparison, the time scale is the same on both graphs. A zoom on the peak is shown in the inset for the lower plot. The optimal case (lower plot) leads to an extremely narrow time distribution, with a peak having the Gaussian shape deduced by expanding at lowest order in $(t-t_{H})$ the distribution (5.5) : $\displaystyle\mathcal{F}(t)\underset{t\approx t_{H}}{\simeq}C\exp\left[-\frac{(t-t_{H})^{2}}{2\Delta t^{2}}\right]~{}.$ (5.6) The width $\Delta t$ of the distribution agrees with the classical result (2.3) : $\displaystyle\Delta t=\sqrt{\frac{\hbar(1+\omega^{2}t_{H}^{2})}{2m\omega\overline{g}^{2}t_{H}^{2}}}=t_{H}\sqrt{{\left(\frac{\zeta}{2H}\right)}^{2}+{\left(\frac{\hbar}{2mv_{H}\zeta}\right)}^{2}}~{}.$ (5.7) The upper plot in figure 2, which corresponds to the typical numbers of the GBAR design, leads to a much broader distribution and shows a deformed shape with respect to a Gaussian distribution. As already discussed, this is a consequence of the large dispersion of initial vertical velocities. ### 5.2 Gravitational quantum states in the shaping device We come now to the discussion of the shaping device in the regime where quantum gravitational states play an important role. The wave-function of the atoms can thus be developed over the basis of eigenstates $\Psi_{n}$ with energies $E_{n}$ in the gravity field, here calculated above a perfectly reflecting mirror [17], $\displaystyle\Psi_{n}(z)=\frac{1}{\sqrt{l}}\frac{\text{Ai}(z/l-\lambda_{n})}{\text{Ai}^{\prime}(-\lambda_{n})}\quad,\quad E_{n}=mgl\lambda_{n}~{}.$ (5.8) The typical scale $l$ of gravitational quantum states is: $\displaystyle l=\left(\frac{\hbar^{2}}{2m^{2}\overline{g}}\right)^{1/3}\approx 5.9~{}\mu\mathrm{m}~{}.$ (5.9) and the quantized energy levels are determined by the zeros of the Airy function Ai: $\displaystyle\text{Ai}(-\lambda_{n})=0$ (5.10) $\displaystyle\lambda_{1}\approx 2.34\;,\;\lambda_{2}\approx 4.09\;,\;\lambda_{3}\approx 5.52\;,\;\ldots$ The high-$n$ states are given by the asymptotic law $\displaystyle\lambda_{n}\underset{n\to\infty}{\approx}\left(\frac{3\pi}{2}\left(n-\frac{1}{4}\right)\right)^{2/3}$ (5.11) Selectivity of the shaping device is based on the sharp dependence of the transmission of eigenstates $\Psi_{n}$ versus the height $h$ of the slit. The detailed formalism in [17] leads to a propagation through the device described by the following propagator: $\displaystyle K(z,z^{\prime},t)=\sum_{n}\Psi_{n}(z)\Psi_{n}(z^{\prime})\exp\left[\frac{(E_{n}-i\Gamma_{n})t}{i\hbar}\right]~{}.$ (5.12) The width $\Gamma_{n}$ of level $n$ becomes large for high values of $n$ [17], as explained by the following qualitative interpretation. When the spatial dispersion $l\lambda_{n}$ of the state $\Psi_{n}$ is smaller than the slit size $h$, the overlap with the absorber is small and the atom has a high probability to pass through the device ($\Gamma_{n}$ small). On the other hand, when $l\lambda_{n}$ is larger than $h$, the overlap of the wave-function with the absorber is significant and atoms have a high probability to be absorbed ($\Gamma_{n}$ large). Figure 3: Transmission of first n=1 and second n=2 gravitational states through a shaping device with a length of 5 cm. As a quantitative illustration, figure 3 shows the probability of transmission for atoms in the two lowest gravitational states $\Psi_{1}$ and $\Psi_{2}$ when the length of the shaping device is $R-r=5$ cm and the roughness amplitude of the top absorber is 1 $\mu$m. A slit size $h=24$ $\mu$m provides 72% transmission probability for the first state but only 0.3% for the second state. This implies that a nearly pure ground state or a superposition of a few lowest gravitational states can be prepared by a suitable choice of the parameters of the shaping device. ### 5.3 Free fall experiment after the velocity shaping The output of the velocity shaping device is a superposition of gravitational quantum states $\Psi_{n}$, determined by the propagator (5.12) calculated for a time $t=(R-r)/v_{\mathrm{hor}}$ for an atomic horizontal velocity $v_{\mathrm{hor}}$. This shaped superposition then falls freely to the detection plate so that the time distribution of annihilation events depends on the properties of the shaped state. We stress again at this point that this supposes that the time $R/v_{\mathrm{hor}}$ spent in the shaping device, and before its entrance, is corrected in the data analysis, $v_{\mathrm{hor}}$ being deduced from the position of the annihilation event. The spatial and velocity dispersions of the state $\Psi_{n}$ can be expressed in terms of $\lambda_{n}$ [22]: $\displaystyle\Delta z_{n}=\frac{2l\lambda_{n}}{3\sqrt{5}}$ $\displaystyle\Delta v_{n}=\frac{\hbar}{ml}\sqrt{\frac{\lambda_{n}}{3}}$ (5.13) In contrast with the case of Gaussian wave-packets discussed above, these dispersions do not reach the minimum in the Heisenberg inequality. Furthermore, $\Delta v_{n}$ and $\Delta z_{n}$ increase simultaneously as functions of $n$. The dispersion of the annihilation time (after correction of the time spent in the device) is thus given for the state $\Psi_{n}$ by (2.2) with $\zeta,\upsilon$ replaced by $\Delta z_{n},\Delta v_{n}$ : $\displaystyle\frac{\Delta t}{t_{H}}=\sqrt{\frac{l^{2}\lambda_{n}^{2}}{45H^{2}}+\frac{l\lambda_{n}}{3H}}{\approx}\sqrt{\frac{\lambda_{n}l}{3H}}$ (5.14) As $l\lambda_{n}\sim h\ll H$, the initial velocity spread still dominates the uncertainty on the annihilation time. It follows that the dispersion of these times is determined by $\Delta v_{n}$ and scales as $\sqrt{\lambda_{n}}$. In order to get an estimate of the dispersions, we suppose that the state in the shaper is an incoherent superposition of the quantum states which fit in the slit. It follows from the arguments in the preceding section that the quantum states which fit in the slit correspond to $n\leq n_{\max}~{},\quad l\lambda_{n_{\max}}\approx h~{}.$ (5.15) We then deduce the dispersion of annihilation times as $\frac{\Delta t}{t_{H}}=\sqrt{\sum_{n}\pi_{n}\frac{l\lambda_{n}}{3H}}~{},$ (5.16) where $\pi_{n}$ is the population in the state $\Psi_{n}$. As the slit size is small compared with the incoming wave-function size, we expect that the states are equally populated among the fitting gravitational quantum states, so that $\pi_{n}\approx 1/{n_{\max}}$ for $n\leq n_{\max}$, $\pi_{n}\approx 0$ otherwise. In the quasi-classical limit where $n_{\max}\gg 1$, we can use the asymptotic expression (5.11) for $\lambda_{n}$ and replace the sum by an integral to find: $\frac{\Delta t}{t_{H}}\approx\sqrt{\frac{h}{5H}}~{}.$ (5.17) This expression scales like the classical result (4.2) with $\beta$ now specified to be $1/5$. The preceding argument disregards the coherence between the components $\Psi_{n}$ in the superposition prepared by the shaping device. This approximation can be justified qualitatively by considering that the effects of coherence are washed out in the averaging associated with free fall propagation as well as horizontal velocity dispersion. However it cannot be considered as exact, and it will have to be confirmed by more precise simulations, to be published in forthcoming papers. Figure 4: The velocity distribution in the ground gravitational state $\Psi_{1}$. Exact quantum calculations can be performed for the special case of an initial state for free fall prepared by the shaper as the ground gravitational state $\Psi_{1}$. The initial velocity distribution, shown in figure 4, has a width $\Delta v\approx 9.5$ mm/s. This is 30 times larger than the optimal velocity spread $\upsilon_{opt}\approx 0.36$ mm/s, but two orders of magnitude smaller than the initial velocity spread in the GBAR experiment. The exact quantum evolution of this initial wave-packet is then obtained by integrating the propagation equations (5.2-5.3). The annihilation time distribution calculated in this manner is shown in figure 5. Its spread is in excellent agreement with the prediction $\Delta t=t_{H}\sqrt{{l\lambda_{1}}/{3H}}\simeq 0.97$ ms deduced from (5.14). As a comparison, this spread was of the order of 45 ms for the free fall measurement performed without velocity shaping. The improvement reflects the velocity selection by the shaping device, which is only partly balanced by the degradation of the statistics (as discussed above). Figure 5: The distribution of arrival times of $\overline{\mathrm{H}}$ falling from the height of 30 cm, assuming that the initial wave-packet has been shaped into the gravitational ground state $\Psi_{1}$. ## 6 Estimation of systematic effects For our proposal to be useful as an improved option of the GBAR measurement, one must ensure that there are no large systematic uncertainties which could contribute at a level comparable to the estimated statistical uncertainty of ${\ 10}^{-3}$. We first examine the additional velocity dispersion caused by the photo- detachment recoil. As discussed in [23], the vertical velocity dispersion due to the absorption of the photon and the positron emission can be kept small ($\sim 0.5$ m/s) by using a horizontal polarized laser beam with an energy tuned at around $\Delta E\approx 10$ $\mu$eV $\approx 0.1$ cm-1 above threshold. The photo-detachment cross section near threshold follows the Wigner law and can be estimated by using the available information in the literature to be $\sigma=6.8\times 10^{-26}(\Delta E/1\text{ cm}^{-1})^{3/2}\approx 2\times 10^{-27}$ m2 [24, 25, 26, 27]. With a $P=1$ W laser beam tuned close the threshold energy $E_{T}=6083$ cm${}^{-1}=0.76$ eV focused on an area $A=10$ $\mu$m $\times 10$ $\mu$m covering the Paul trap center, the photo-detachment rate is $R=\sigma P/AE_{T}=130$ s-1. In GBAR, antihydrogen ions can be produced only every 110 s, the ejection period of the antiproton decelerator at CERN. This time is sufficient to photo-detach the excess positron with high efficiency. The method is to illuminate the ion during a short enough time so as to define the start time with high precision, at a low enough repetition rate so that in case of successful photo-detachment, the free fall is completed before the next laser shot. For example, since the free fall time on 30 cm is only 250 ms, laser shots of 100 $\mu$s duration at a repetition rate of 2 Hz during 100 s allows the start time to be known with enough precision ($4\times 10^{-4}$), it also avoids ambiguity on identifying the successful shot, and leads to a photo- detachment efficiency larger than of 90 %. Since the velocity dispersion induced by the atomic recoil is of the same order as that from the confinement in the Paul trap, one would not gain by trying to get closer to the optimal cloud size. Finally, this effect is equivalent to a slightly warmer antihydrogen cloud, which changes the effective value of the frequency $\omega$ to be used in the calculations, without affecting the principle of the method. A careful analysis of other systematic effects has to be performed in the future, in particular for the following list of possibilities: 1. 1) Uncertainty of shaping/measuring the distribution of vertical velocity components of $\overline{\mathrm{H}}$ within the range of acceptance of the two-disk system; 2. 2) Finite positioning and timing resolution for the detection of annihilation events; 3. 3) Accuracy and reliability for the correction for the time spent in the shaping device; 4. 4) Diffraction of atoms on the mirror edges; 5. 5) Residual electromagnetic effects, and in particular patch effect on mirror surfaces; 6. 6) Defects of mechanical alignments, such as inclinations of the disks and detection plate; 7. 7) Finite precision of production and adjustment of optical elements; 8. 8) Vibrations able to cause parasitic transitions between gravitational quantum states. Monte-Carlo simulations of the experiment are underway; they take into account photo-detachment, coupling of the shaping device with the Paul trap and detector vessel, as well as points 1) and 2). For most of these systematic effects, one may also rely on the experience accumulated in experiments with UCNs [7, 9, 14, 15]. We note that the main systematic uncertainties (in particular 1) are proportional to the ratio $h/H$, and thus decrease strongly when slit heights are decreased. We therefore think that the control of these systematic effects will be improved at small slit heights. ## 7 Conclusion In this paper, we have proposed a new method for shaping vertical velocities of antihydrogen atoms in the timing experiment to be performed by the GBAR collaboration [4]. We have given first estimations of the corresponding statistic uncertainties and listed possible systematic effects. The conclusion of these preliminary estimations, to be confirmed by further analysis, is that the accuracy in the measurement of the free fall acceleration $\overline{g}$ of $\overline{\mathrm{H}}$ atoms could be pushed below 10-3 in realistic experimental conditions. Statistical uncertainties in the experiment are improved for smaller slit heights, which lead to better defined vertical velocities of $\overline{\mathrm{H}}$. This means that a better selection of the range of vertical velocities overweighs the loss in statistics. Systematical uncertainties are expected to decrease even more dramatically for smaller heights of the slit between the two disks in the proposed experimental design. In the optimum experiment where atomic wave-packet is shaped to the lowest quantum state, the effective temperature corresponding to the vertical motion of $\overline{\mathrm{H}}$ is as low as 10 nK. These preliminary estimations have to be confirmed by more complete simulations. We are currently working to develop a fully quantum treatment of the shaping device as well as a complete Monte-Carlo simulations. Let us also mention that an even better accuracy could in principle be obtained by studying interference effects in the time-of-arrival distribution of a coherent superposition of a few lowest-lying gravitational quantum states [12, 13]. ## Acknowledgements The authors thank the ESF Research Networking Programme CASIMIR (casimir- network.org), the GRANIT collaboration and the GBAR collaboration (gbar.in2p3.fr) for providing excellent possibilities for discussions and exchange. ## References [1] A. Kellerbauer, M. Amoretti, A.S. Belov, G. Bonomi, I. Boscolo, R.S. Brusa, M. Büchner, V.M. Byakov, L. Cabaret, C. Canali, C. Carraro, F. Castelli, S. Cialdi, M. de Combarieu, D. Comparat, G. Consolati, N. Djourelov, M. Doser, G. Drobychev, A. Dupasquier, G. Ferrari, P. Forget, L. Formaro, A. Gervasini, M.G. Giammarchi, S.N. Gninenko, G. Gribakin, S.D. Hogan, M. 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1312.5575	, # Effects of the equilibrium model on impurity transport in tokamaks A Skyman, L Fazendeiro, D Tegnered, H Nordman, J Anderson and P Strand Department of Earth and Space Sciences, Chalmers University of Technology, SE-412 96 Gothenburg, Sweden [email protected] [email protected] ###### Abstract Gyrokinetic simulations of ion temperature gradient mode and trapped electron mode driven impurity transport in realistic tokamak geometry are presented and compared with results using simplified geometries. The gyrokinetic results, obtained with the GENE code in both linear and non-linear mode, are compared with data and analysis for a dedicated impurity injection discharge at JET. The impact of several factors on heat and particle transport are discussed, lending special focus to tokamak geometry and rotational shear. To this end, results using $s-\alpha$ and concentric circular equilibria are compared to results with magnetic geometry from a JET-experiment. To further approach experimental conditions, non-linear gyrokinetic simulations are performed with collisions and a carbon background included. The impurity peaking factors, computed by finding local density gradients corresponding to zero particle flux, are discussed. The impurity peaking factors are seen to be reduced by a factor of $\sim 2$ in realistic geometry compared to the simplified geometries, due to a reduction of the convective pinch. It is also seen that collisions reduce the peaking factor for low $Z$ impurities, while increasing it for high charge numbers, which is attributed to a shift in the transport spectra towards higher wave-numbers with the addition of collisions. With the addition of roto-diffusion, an over-all reduction of the peaking factors is observed, but this decrease is not sufficient to explain the flat carbon profiles seen at JET. ###### pacs: 28.52.Av, 52.25.Vy, 52.30.Ex, 52.30.Gz, 52.35.Ra, 52.55.Fa, 52.65.Tt ## 1 Introduction Impurity transport is a matter of crucial relevance for Tokamak fusion plasmas [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32] due to the contribution of impurities to radiation losses and plasma fuel dilution. The impurities can originate either from the sputtering of wall and divertor materials or from deliberate impurity injection in order to reduce power load. The impurities in a tokamak thus cover a large range in charge number $Z$, making it necessary to study the scaling of impurity transport with $Z$. This is even more relevant after the installation of the new ITER-like wall at JET [33], with its beryllium wall and tungsten divertor. In addition, a typical plasma discharge already has such a low particle density, that even a modest amount of impurities can greatly dilute the plasma, thereby reducing the power output. Since most impurity sources are located at the edge of the tokamak, the resulting impurity profile in the core of a given discharge will depend on the balance between diffusive and advective processes. It is well known empirically that shaping of the plasma has a beneficial effect on energy confinement, the Troyon limit and the Greenwald density limit. For example, elongation ($\kappa$) is one of the factors considered in empirical scaling laws (e.g. [34]) of the energy confinement time. Only in recent years, however, has the impact of the geometric configuration on micro- turbulence been rigorously ascertained, e.g. from simulations of the gyrokinetic equations [35, 36, 37, 38, 39]. It has been observed that the heat-flux scales with elongation at fixed average temperature gradients, and that shaping effects enhance residual zonal flows, thus increasing the nonlinear critical temperature gradient [36]. Elongation has been seen to be the dominant effect on micro-turbulence on ion temperature gradient (ITG) and trapped electron (TE) drift-wave instabilities, compared with e.g. triangularity, up–down asymmetry and Dimits shift [38]. However, in spite of these results, the effects of the equilibrium model on impurity transport has not been studied in any detail. Previous work in this area includes [21, 22, 23], in which the impurity transport in a dedicated impurity injection experiment at JET was studied using $s-\alpha$ geometry. It was shown that the profile steepness parameter of the injected impurities was qualitatively reproduced by the simulations. On the other hand, a discrepancy was found for the background carbon profile which was too steep at mid radius compared to the experimental profile. In the present study, an experimental equilibrium is used and compared with results using simpler $s-\alpha$ and circular equilibria [37]. Since the impurity profile is a result of a balance between diffusion and advection, where the main advective term is the inward curvature pinch, a strong dependence on the equilibrium model may be expected. In addition, a more realistic physics description, including the effects of rotational shear on the instabilities and impurity transport (roto-diffusion), is investigated using gyrokinetic simulations. Realistic non-linear gyrokinetic simulations are performed taking into account collisionality, and the presence of a $2\%$C background, consistent with the conditions applying for an impurity injection experiment at JET. The paper is organised as follows. Section 2 describes the gyrokinetic simulations performed, and the fundamental concepts of impurity transport. Section 2.2 describes the particular JET discharge considered. In Section 3, results for the the eigenvalues, turbulent fluctuation levels and transport are presented, along with a discussion on the scaling of the impurity peaking factor with impurity charge $Z$ for the various models considered. A conclusion then follows in Section 4. ## 2 Background ### 2.1 Gyrokinetic model The gyrokinetic simulations in this study were performed using the gyrokinetic turbulence code GENE111http://www.ipp.mpg.de/~fsj/gene/ [40, 41] where a flux–tube domain was considered. This is an Eulerian–type code (i.e., employing a fixed grid in phase space), which allows for arbitrarily long simulation times. Both quasi- and nonlinear simulations including kinetic ions and electrons, and at least one impurity species were performed. All impurities were treated as fully kinetic species with low concentrations. For studying the effects of collisions, we considered the GENE implementation of the Landau–Boltzmann type collision operator [41]. For the simulation domain, all three geometries used a flux tube with periodic boundary conditions in the perpendicular plane. The nonlinear simulations were performed using a $96\times 96\times 32$ grid in the normal, bi-normal, and parallel spatial directions respectively; in the perpendicular and parallel momentum directions, a $12\times 48$ grid was used. For the linear and quasilinear computations, a typical resolution was $12\times 32$ grid points in the normal and parallel directions, with $12\times 64$ grid points in momentum space. To validate the choice of resolutions, convergence tests for mode structure and spectra were performed. The nonlinear simulations were typically run up to $t=600\,a/c_{\text{s}}$ for the experimental geometry scenario, where $a$ represents minor radius and $c_{\text{s}}=\sqrt{T_{e}/m_{i}}$. For the “full scenario,” in which a $2\%$C background and collisions were both present, the simulation time was instead $t\approx 450\,a/c_{\text{s}}$. This shorter time span is due to the much more computationally intensive nature of the second scenario, by a factor of $\sim 2$–$3$. The increased numerical cost was mainly due to collisions, though in part also due to the addition of the $2\%$C background. ### 2.2 Experimental profiles and parameters The physical parameters used in the simulations are presented in table. 1 and were chosen so as to be consistent with the experimental values taken from the JET database for discharge #67330. This was previously analysed in [21], where the $s-\alpha$ geometry model was used. The discharge was part of an impurity- dedicated set of discharges with low MHD activity [32]. In these, extrinsic impurities (Ne, Ar and Ni) were injected via laser ablation and gas injection. Diffusivity $D_{Z}$ and convective velocity $V_{Z}$ were determined by matching spectroscopic data with results obtained from predictive transport codes (see [21] and in particular Fig. 4). Profiles of density and temperature for the discharge are shown in figure 1a. (a) profiles of $T_{i,e}$ and $n_{i}$ (b) profiles of $v_{\text{tor}}$, $q$ and $s$ Figure 1: (colour online) Radial profiles of the background parameters for JET discharge #67730 at $t=47.5\,\mathrm{s}$. The simulations were performed at mid radius ($r/a=0.5$; indicated). For the present study, the parameters used, including the MHD equilibrium (from now on referred to as “experimental geometry”), were taken extracted at mid radius ($r/a=0.5$) at $48\,\mathrm{s}$. The main quantities of interest were elongation $\kappa=1.37$, triangularity $\delta=0.044$, and the fraction of trapped particles $f_{t}=0.55$. The toroidal magnetic field intensity on the axis was $B=3\,\mathrm{T}$, the major radius of the tokamak $R=3\,\mathrm{m}$, electron temperature $T_{e}=1.55\,\mathrm{k}\,\mathrm{eV}$ and plasma $\beta=8\pi n_{e}T_{e}/B^{2}=1.28\times 10^{-3}$, and the normalised gradient of $\beta$ $\alpha=-q^{2}R\mathrm{d}\beta/\mathrm{d}r=0.1026$. The safety factor in the discharge was $q=2.2$ at the considered radius, obtained from EFIT. The collisionality was $\nu_{ei}=0.07\,c_{\text{s}}/R$ ($\nu_{c}=0.28\times 10^{-3}$ in GENE). Effects of purely toroidal rotation were included through the $\boldsymbol{E}\times\boldsymbol{B}$ shearing rate. The normalised gradient scale lengths are defined as $R/L_{n_{j}}=-(R/n_{j})(\mathrm{d}n_{j}/\mathrm{d}r)$ and $R/L_{T_{j}}=-(R/T_{j})(\mathrm{d}T_{j}/\mathrm{d}r)$ where $R$ is the major radius of the tokamak. For the realistic equilibria, the gradients were rescaled to the minor radius with $a=\sqrt{2\Phi_{\text{s}}/B}=1.26\,\mathrm{m}$, where $\Phi_{\text{s}}$ is the toroidal flux at the last closed flux surface. In the considered discharge, the Mach number was $M=0.21$, leading to $\gamma_{\boldsymbol{E}\times\boldsymbol{B}}=-\frac{r}{q}\frac{1}{a}\frac{\partial v_{\text{tor}}}{\partial r}=0.1\,c_{\text{s}}/R$, with the toroidal velocity profile ($v_{\text{tor}}(r)$) as in figure 1b, evaluated at mid radius. Hence, we only consider the effect of flow shear in the limit where the flow is small, neglecting effects of centrifugal and Coriolis forces. These may, however, be important for heavier impurities [9]. Table 1: Parameters for gyrokinetic simulations $T_{i}/T_{e}$: \| $1.02$ ---\|--- $\beta$: \| $1.28\times 10^{-3}$ $s$: \| $0.75$ $q$: \| $2.20$ $\varepsilon=r/R$: \| $0.17$ $\alpha$: \| $0.126$ $\kappa$: \| $1.37$ $\delta$: \| $0.044$ $k_{\theta}\rho_{\text{s}}$: \| $0.2$–$0.6$ $n_{e}$, $n_{i}+Z\,n_{Z}$: \| $1.0\cdot 10^{19}\,\mathrm{m^{-}3}$ $n_{Z}$ _(trace)_ : \| $10^{-6}\cdot 10^{19}\,\mathrm{m^{-}3}$ $Z$: \| $2$–$74$ $R/L_{n_{i,e}}$: \| $2.7$ $R/L_{T_{i}},R/L_{T_{Z}}$: \| $5.6$ $R/L_{T_{e}}$: \| $5.6$ $\nu_{c}$: \| $0$, $0.28\times 10^{-3}$ $\gamma_{\boldsymbol{E}\times\boldsymbol{B}}$: \| $0$–$0.6$ $T_{e}$: \| $1.55\,\mathrm{k}\,\mathrm{eV}$ $B$: \| $3.04\,\mathrm{T}$ ### 2.3 Magnetic equilibrium The gyrokinetic simulations were performed in three different geometries: $s-\alpha$, concentric circular [37] and an MHD equilibrium calculated for the JET-discharge #67330 from EFIT data using the the TRACER code (see [35] for details on the geometry implementation). A cross section of the experimental magnetic equilibrium is shown in figure 2. For the parameters under consideration, $\alpha$ is fairly small ($\alpha=0.1026$), meaning that the $s-\alpha$ and circular model should be near equivalent. The $s-\alpha$ model, however, suffers from inconsistencies of the order $\varepsilon=a/R$ in its standard implementation [35, 37, 38]. Supplementary simulations using circular geometry were therefore performed, in order to better differentiate between effects of magnetic geometry and effects due to the inconsistent $s-\alpha$ metric. Figure 2: Cross section of the magnetic equilibrium, for JET discharge #67730. ### 2.4 Impurity transport For trace impurities, equation (4) can be uniquely written [21] as a linear function of $\nabla n_{Z}$, offset by a convective velocity or “pinch” $V_{Z}$: $\Gamma_{Z}=-D_{Z}\nabla n_{Z}+n_{Z}V_{Z}\Leftrightarrow\frac{R\Gamma_{Z}}{n_{Z}}=D_{Z}\frac{R}{L_{n_{Z}}}+RV_{Z},$ (1) where $D_{Z}$ is the impurity diffusion coefficient, and $V_{Z}$ is the impurity convective velocity, which includes roto-diffusion in the gyrokinetic treatment. Both $D_{Z}$ and $V_{Z}$ are independent of $\nabla n_{Z}$ in the trace impurity limit [16]. In the core of a steady-state plasma with fuelling from the edge (i.e. in the absence of particle sinks or sources in the core), the impurity flux $\Gamma_{Z}$ will go to zero. Since the gradient is a measure of how peaked the impurity density profile is, the gradient of zero impurity flux is referred to as the impurity peaking factor ($PF$). Inserting $R/L_{n_{Z}}=PF$ into the linearised equation (1), it can be seen that the peaking factor quantifies the balance between convective and diffusive impurity transport:222this definition is closely related to the _Péclet number_ [42, 43] $PF=-\frac{R\,V_{Z}}{D_{Z}}.$ (2) Specifically, the sign of the peaking factor is determined by the sign of the pinch, meaning that $PF>0$ is indicative of a net inward impurity pinch, giving a peaked impurity profile. Conversely, if $PF<0$ the net impurity pinch is outward, leading to a hollow impurity profile (also called _flux reversal_). For trace impurities, equation (1) is linear in $R/L_{n_{Z}}$, and the peaking factor is found by calculating the impurity particle flux ($\Gamma_{Z}$) for a range of $R/L_{n_{Z}}$ and solving equation (1) for zero flux. For non-trace species, the $D_{Z}$ and $V_{Z}$ depend on $R/L_{n_{Z}}$, and the peaking factor has to be found by explicitly seeking the gradient yielding zero particle flux. Much of the observed difference between the TE and ITG mode dominated cases can be understood from the convective velocity $V_{Z}$ in equation (1). To lowest order in $Z^{-1}$, the pinch contains two terms that depend on $Z$ [16]: * • thermal diffusion (thermopinch): * – $V_{\nabla T_{Z}}\sim\frac{1}{Z}\frac{R}{L_{T_{Z}}}\omega_{r}$, * – inward for TE mode $\left(V_{\nabla T_{Z}}<0\right)$, outward for ITG mode $\left(V_{\nabla T_{Z}}>0\right)$, * • parallel impurity compression: * – $V_{\parallel,Z}\sim\frac{Z}{A_{Z}}k_{\parallel}^{2}\sim\frac{Z}{A_{Z}q^{2}}\approx\frac{1}{2q^{2}}$, * – outward for TE mode $\left(V_{\parallel_{Z}}>0\right)$, inward for ITG mode $\left(V_{\parallel_{Z}}<0\right)$. Here $1/k_{\parallel}$ represents the wavelength of the parallel structure of the turbulence. Due to the ballooning character of the modes considered, this is proportional to the safety factor ($q$). In addition, the convective velocity contains the curvature pinch, which to lowest order is independent of the impurity charge [21]. The $Z$ dependence in the parallel impurity compression is expected to be weak, since the mass number is approximately $A_{Z}\approx 2Z$ for an impurity species with charge $Z$, and for the high $q$ considered, this term is expected to be small compared to the other pinch contributions. The thermodiffusive contribution, however, can dominate the transport for low $Z$ impurities (such as the Helium ash). For lower charge numbers, the second order correction to the thermal pinch can become important: $V_{T}\sim\left(\omega_{r}\frac{T_{Z}}{T_{e}Z}-\frac{7}{4}\left(\frac{T_{Z}}{T_{e}Z}\right)^{2}\right)\frac{R}{L_{T_{Z}}},$ (3) where the second order term is inward, independent of the mode direction, and finite Larmor-radius effects have been neglected [44, 45, 16, 21]. The direction of the contributions to the pinch are governed mainly by the considered mode’s direction of rotation, which in the considered cases has a different sign for TE and ITG modes [45]. Expanding the convective velocity in (1) into thermal diffusion and “pure convection”, $\Gamma_{Z}=-D_{Z}\nabla n_{Z}+n_{Z}\left(D_{T,Z}\nabla T_{Z}+V_{p,Z}\right),$ (4) their relative contributions to the total peaking factor can be uniquely determined for trace impurities, using the method described in e.g. [46]. Here $V_{p,Z}$ includes the parallel compression and curvature pinches. In the presence of sheared toroidal rotation ($v_{\phi}$), these terms are joined by a roto-diffusive term, which is proportional to $\mathrm{d}v_{\phi}/\mathrm{d}r$. ## 3 Results and discussion ### 3.1 Background turbulence and transport #### 3.1.1 Linear eigenvalue spectra Using GENE, the linear eigenvalues in the studied geometries were calculated at mid-radius of JET L-mode discharge #67330 for a range of different wave numbers. The resulting spectra are compared in figure 3. The results in figure 3a show an overall destabilisation and shift toward higher $k_{\theta}\rho_{\text{s}}$ when moving from $s-\alpha$ to more realistic geometries, consistent with the results reported in [37, 47]. For triangularity of the same order as the experimental value ($\delta=0.044$), the effect on the eigenvalues of $\delta$ was found to be minute, and it is therefore surmised that the elongation is the main factor behind the the observed results. Adding more realism in the form of collisions and a $2\%$ carbon background, in accordance with the “full scenario” considered in the nonlinear runs, both have a stabilising effect, as can be seen in figure 3b. The figure also shows that the stabilising effect is larger for the addition of collisions than for the carbon background, in particular for lower values of $k_{\theta}\rho_{\text{s}}$, where most of the transport normally occurs. In the second figure, the spectra for the ITG mode is supplemented with those of the sub-dominant TE mode, except in the full scenario, where it is completely stabilised, with $\gamma<0$ for the wave-numbers considered. (a) eigenvalue spectra for different geometry models (b) eigenvalue spectra in experimental geometry with added effects Figure 3: (colour online) Growthrate spectra for $s-\alpha$, circular and experimental magnetic equilibrium for an ITG mode dominated case with JET-like parameters (3a), and for the experimental equilibrium with added degrees of realism (3b). For the second figure, _expr._ represents full geometry case with no added effects, in _expr.+C_ a background of $2\%C$ was added, and in _full_ collisions are added. The last case corresponds to the “full-scenario” NL runs in figure 7b. Parameters are presented in table. 1. The effects of sheared toroidal rotation on the ITG growth rate in the circular and experimental equilibrium are shown in figure 4. Both the stabilising perpendicular velocity shear and the destabilising parallel shearing rate are included. For the value of $q$ considered, the mode is destabilised with increasing $\gamma_{\boldsymbol{E}\times\boldsymbol{B}}$. This is because the stabilising component is proportional to $1/q$, wherefore the destabilising effect dominates in the present case [48]. The shearing rate in the considered JET experiment (marked in the figure) is, however, too small to have a significant impact on the mode growth, and is left out of the nonlinear treatment below. Figure 4: (colour online) Scaling of ITG growthrate with $\boldsymbol{E}\times\boldsymbol{B}$ shearing rate. The experimental shearing rate has been indicated (vertical dashed line). #### 3.1.2 Nonlinear transport and fluctuation levels By comparing the transport levels for the different geometries, the difference in the turbulence can be assessed qualitatively. Time series and wave-number spectra of heat and particle flux ($Q$ and $\Gamma$ respectively) for the main ions were produced from nonlinear GENE simulations. The results are presented in figure 5. (a) timeseries of particle fluxes ($\Gamma$) with mean flux indicated (b) timeseries of heat fluxes ($Q$) with mean flux indicated (c) spectra of particle fluxes ($\Gamma$) (d) spectra of heat fluxes ($Q$) Figure 5: (colour online) Timeseries and spectra of particle ($\Gamma$) and heat ($Q$) fluxes for nonlinear GENE simulations for $s-\alpha$ and circular geometry, the experimental equilibrium, and the fully realistic case, with parameters as in table. 1. Spectra show average over radial wavenumbers. The timeseries and spectra have been normalised to the maximum of the corresponding $s-\alpha$ entry. The spectra and time series show a decrease in amplitude consistent with that expected from the linear eigenvalue spectra in figure 3. As can be seen in figures 5a and 5b, the turbulent fluctuation level is increased when moving from $s-\alpha$ to realistic magnetic geometry. The same trend holds true for the spectra in figures 5c and 5d. Simulations using circular geometry [35, 37] place both the spectra and time series between the $s-\alpha$ and experimental case without collisions. This is consistent with the linear eigenvalues in figure 3, and with the results reported in [37]. With the addition of collisions and background carbon, the fluctuation levels are brought down, due to a general reduction of the turbulence, as seen in the the eigenvalue spectra figure 3b. The particle flux ($\Gamma$) is more sensitive to collisions than the heat flux ($Q$), and therefore retains a higher fluctuation level, despite the lower growthrates. In the case with full realism, including collisions and carbon background, the fluctuation levels are lowered to an intermediate level, consistent with higher dissipation from collisions. This is also consistent with the linear eigenvalues presented in figure 3, in which the growthrates for the “full scenario” have an intermediate value between the circular (or $s-\alpha$) and the case where only effects of realistic geometry is considered. As can be seen in figures 5a and 5b, the full case was simulated for a shorter time span, due to its larger computational cost, mainly due to the inclusion of collisions, but also in part due to the $2\%$C background; see Section 2.1. We note that the global confinement time in both L-mode and H-mode plasmas are scaling favourably with elongation, opposite to the trend found here for the core transport. This is likely a result of edge physics not included in the present study [49]. In addition, the zonal flow activity was investigated for each case. However, no significant differences were observed between the considered cases. The NL results can thus be explained qualitatively by the linear physics. ### 3.2 Impurity peaking factors Simulations of impurity transport using a _JET_ -like experimental magnetic equilibrium have been compared to simulations with $s-\alpha$ and circular geometry, using the gyrokinetic code GENE. First, the effects sheared toroidal rotation in the different geometries is investigated The effects on the impurity peaking factors are displayed in figure 6. The peaking factors for several impurity species are shown as a function of $\gamma_{\boldsymbol{E}\times\boldsymbol{B}}$ in the experimental equilibrium. As observed, the peaking factors are reduced by the sheared rotation. This leads to a reversal of the impurity pinch for $\gamma_{\boldsymbol{E}\times\boldsymbol{B}}\gtrsim 0.23$ for all but the lightest impurities ($Z\lesssim 4$) in the experimental equilibrium. In the circular geometry the trends are the same, but the flux reversal for Tungsten ($Z=74$) is shifted to higher rotational sharing rate ($\gamma_{\boldsymbol{E}\times\boldsymbol{B}}\gtrsim 0.28$). This effect, due to roto-diffusion, has been found to be a critical ingredient to include in order to reproduce the Boron profile in ASDEX U [50]. For the considered discharge, however, the shearing rate is small and hence it is neglected in the non-linear analysis. Figure 6: (colour online) Scaling of PF with $\boldsymbol{E}\times\boldsymbol{B}$ shearing rate for different impurity species in circular and shaped equilibrium for wavenumbers near the peak of the corresponding growthrate spectra ($k_{\theta}\rho_{s}=0.3$ and $k_{\theta}\rho_{s}=0.4$ respectively). The experimental shearing rate has been indicated (vertical dashed line). Next, the effects of the geometry on the scaling with impurity charge are investigated. In figure 7a the scaling of the impurity peaking factor ($PF$) with impurity charge ($Z$) is shown, for $s-\alpha$, circular and experimental geometry. Non-linear (NL) gyrokinetic results are compared in figure 7b for added degrees of realism. Error-bars in figure 7b correspond to a conservative standard error estimate of $\pm\sigma$. This was calculated from the NL flux data, where an effective sample size was gauged for the time series and used to estimate the error for the mean flux. This estimate was then propagated through to an error estimate for the peaking factor. The effects on the $PF$-scaling of adding collisions and $2\%$C background, consistent with the considered JET discharge, are shown in figure 7c. For moderate to high impurity charge, weaker scalings of $PF$ with $Z$ were consistently observed when departing from the $s-\alpha$ equilibrium, and the level at which $PF$ saturates for high $Z$ was reduced in the experimental geometry case. The lower $PF$ levels compared to $s-\alpha$ geometry can be attributed to a reduction of the convective pinch due to effects of shaping, but also due to effects of finite inverse aspect ratio ($\varepsilon$) effects, as discussed in Section 2.2. To separate the effects of shaping from the inconsistent $\varepsilon$ effects in the $s-\alpha$ model, simulations using GENE’s circular geometry [37] were performed. Since $\alpha\ll 1$ in the studied discharge, the difference between the results using circular and experimental equilibrium in figure 7a can attributed to shaping effects, mainly due to elongation. Further, a large increase in $PF$ was observed for low $Z$ impurities, most notably He impurities, in circular and experimental geometry. This is due to a reduction of the outward thermopinch in the more realistic equilibrium models, as discussed in Section 3.3. In figure 7b non-linear results using $s-\alpha$ geometry from [21] are compared with NL gyrokinetic simulations using the experimental equilibrium for two different sets of parameters. The first NL set of runs only uses the experimental geometry, whereas the full case also includes a carbon background and collisions, but neglects effects of sheared toroidal rotation (_full_ in figure 3). With the introduction of these two effects, $PF$ is lowered for low $Z$ and increased for high $Z$. As seen in the quasilinear runs, the NL GK results are lower than those predicted by the $s-\alpha$ model, though a qualitative agreement is reached in the full scenario, particularly for high $Z$. When comparing QL and NL results, the former show a more dramatic scaling than the latter and the QL results tend to over-estimate $PF$ for high $Z$, as can be seen in figure 7c. In all models there is, as expected, a saturation of $PF$ in the high $Z$ limit [16, 21, 22], and the observed NL and QL impurity pinches qualitatively agree with the results in [21, 22]. Figure 7c shows the effect of adding collisions and a $2\%$ carbon background to the $PF$ scaling with $Z$, using QL gyrokinetics. The “base case” corresponds to the experimental MHD equilibrium, without any added effects (_expr._ in figure 3). In all cases $k_{\theta}\rho_{\text{s}}=0.4$ was used, since that is the approximate value for the maximum linear growth rate, as shown in figure 3. As with the eigenvalue spectra (figure 3b), it is clear that the addition of collisions to the model has a larger impact than the addition of the $2\%$C background. The addition of the carbon background slightly raises $PF$ for low $Z$, but the addition of collisions lowers it considerably, as well as raising $PF$ for high charge numbers, as seen for the NL runs. These results are consistent with previous observations of the importance of impurity–main ion collisions for core impurity transport [51, 52]. We note that, although a reduction is seen in the peaking for carbon impurities in the realistic case, the flat or hollow profile seen in experiments is not reproduced. Though a further reduction is expected from roto-diffusion, nonlinear simulations of the full scenario with sheared toroidal rotation still show an inward nett transport of the background carbon. (a) comparison of $s-\alpha$, circular and experimental geometries for QL GENE (b) comparison of NL GENE base case and full case (_expr._ and _full_ in figure 3) with results for $s-\alpha$ geometry from [21]; error-bars indicate a standard error of $\pm\sigma$. (c) effects of added realism on QL $Z$-scaling for $k_{\theta}\rho_{s}=0.4$ Figure 7: (colour online) Scalings of impurity peaking factor ($PF$) with impurity charge number ($Z$) from JET discharge #67730 for $s-\alpha$, circular and experimental equilibrium. Parameters as in table. 1. ### 3.3 Contributions to the impurity pinch In order to gain more insight into the results presented in the last section, the contributions to the peaking factor were calculated using the method outlined in [46]. Figure 8 shows the effect of shaping on the thermodiffusive and convective contributions to the impurity peaking factor (equation (4)). It is seen, that the increase in peaking factor observed for lighter impurities is mainly due to the thermopinch, where the inward second order contribution to the thermal pinch comes into dominance over the outward term (equation (3)). This effect becomes more pronounced for lower wave-numbers – especially in the shaped equilibrium – due to the lower mode frequency (figure 3). For higher $Z$, however, the peaking is determined by the balance of the inward convective pinch and the diffusion. This contribution is only weakly dependent on the impurity charge and wave-number, and is reduced substantially in the shaped equilibrium. In figure 9 diffusivities and pinches obtained from NL GENE are compared with with data from [21]. From these results we note that, in contrast to the diffusivity ($D_{Z}$), the convective velocity ($\|V_{Z}\|$) is lower in the realistic equilibrium as compared to the $s-\alpha$ case, despite the corresponding increase in fluctuation levels (figure 5b). This confirms the conclusion that the main reason for the reduced peaking factors obtained in the experimental equilibrium is a reduction of the convective velocity. When comparing the collisionless shaped equilibrium with the full scenario, we note that the diffusivity is decreased, as expected from the observed decrease in fluctuation levels with the addition of collisions. For high $Z$, the reduction in convective velocity is similar in both cases, compared to the $s-\alpha$ equilibrium, leading to higher peaking factors in the full scenario. When comparing the scalings for low ($k_{\theta}\rho_{s}=0.2$) and high ($k_{\theta}\rho_{s}=0.5$) wavenumbers in the case with shaping, it is seen that the thermal diffusion is dominated completely by the higher order terms ($\sim Z^{-2}$) in the low wavenumber case, turning from mostly outward to inward. In the full case, the NL spectrum was seen to shift toward higher wavenumbers, compared to the case with shaping but without collisions and carbon background (figure 5). The difference between the non-linear gyrokinetic results in figure 7b is therefore consistent with the interpretation that higher order terms dominate the thermal diffusion for lower wave numbers, thereby determining the shape of the peaking factor scaling. (a) contributions to the pinch for circular equilibrium (b) contributions to the pinch for experimental equilibrium Figure 8: (colour online) Contributions to the peaking factor ($PF_{k_{\theta}\rho_{s}}^{\text{tot}}$) from thermodiffusion ($PF_{k_{\theta}\rho_{s}}^{\nabla T}$) and pure convection ($PF_{k_{\theta}\rho_{s}}^{\text{conv}}$) as a function of impurity charge ($Z$). Results from QL GENE for wavenumbers as in figure 7, with and without effects of shaping ($\kappa=1.0$, $\kappa=1.37$). Figure 9: (colour online) Scaling of the impurity diffusivity ($D_{Z}$) and total pinch ($RV_{Z}$) with impurity charge for $s-\alpha$ geometry and experimental geometry, with and without collisions and $2\%$C background. Data from NL GENE simulations with standard errors $~{}10$–$20\%$ (omitted for clarity). ## 4 Conclusions The effects of the choice of equilibrium model on impurity transport in ITG/TE mode driven turbulence were studied using gyrokinetic simulations. Results were obtained and contrasted for the experimental MHD equilibrium and the simpler $s-\alpha$ and concentric circular geometries. These results were extended by adding degrees of realism in the physics description, such as a 2% carbon background, collisions, and sheared toridal rotation. The gyrokinetic results, obtained with the GENE code in both quasi- and non-linear mode, were compared with results from a previous study, as well as with expectations from a dedicated impurity injection L-mode discharge at JET. It is found that the different equilibria give qualitatively similar results, but with significant quantitative differences. Linearly, a destabilisation and shift of the growthrate spectrum to higher wave numbers was seen when departing from the simpler geometries, mainly due to elongation. This resulted in larger heat and particle transport, as seen in the nonlinear simulations. The addition of collisions was seen to stabilise the spectrum and reduce transport. The effect of sheared rotation on the mode stability and the impurity transport was studied. A weak destabilisation of the ITG mode with $\boldsymbol{E}\times\boldsymbol{B}$ shear was found. However, this was seen to be a minor contribution for experimentally feasible values of the rotational shearing rate for the considered discharge. The impurity peaking factors ($PF$s), computed by finding local density gradients corresponding to zero particle flux, were derived. The peaking factor was observed to saturate at levels far below neo-classical predictions for high impurity charge numbers. The level of the saturation was considerably lowered when the experimental equilibrium was used, typically by a factor of $\sim 2$. Comparing the diffusivity and convective pinch velocity in the different geometries, it was found that the reduction of $PF$ was mainly due to a reduction of the convective pinch. By decomposing the contributions to the peaking factor into thermodiffusion and pure convection, it was seen that second order contributions to the thermal diffusion became important for low impurity charge. Whereas the first order term is proportional to $\omega_{r}$, and therefore outward for ion temperature gradient modes, the second order contribution to the pinch is always inward, and therefore leads to higher $PF$s. This explains the more pronounced increase in peaking observed for low charge numbers in the shaped equilibria, where the real frequency is lower. Collisions were seen to affect the impurity peaking factors considerably, lowering $PF$ for light impurities, while increasing it for heavy impurities. It was seen, that the addition of collisions and background carbon shifted the nonlinear spectra towards higher wavenumbers, assosciated with higher real frequencies. Therefore, the the difference between the nonlinear results in the experimental equilibrium could be also understood through the second order term in the thermopinch, which was concluded to dominate this pinch contribution for the non-collisional case. However, even combining the effect of collisions with the effect of geometry and rotation in the most realistic simulations, the observed peaking factor for carbon was still too large to explain the rather flat C profile observed in the experiment. One explanation for this descrepancy could lie in the sensitivity of the thermopinch to main ion and impurity temperature, and to the gradients of thereof. A moderate variation in these may bring the background carbon peaking factor down to zero, however, such a treatment is left for future studies. We also note that the treatment of rotation employed in this study neglects centrifugal and Coriolis forces, which may lower the impurity peaking further. ## Acknowledgements The main simulations were performed on resources provided on the Lindgren333See http://www.pdc.kth.se/resources/computers/lindgren/ for details on Lindgren high performance computer, by the Swedish National Infrastructure for Computing (SNIC) at Paralleldatorcentrum (PDC). Additional computations were carried out on resources at Chalmers Centre for Computational Science and Engineering (C3SE)444See http://www.c3se.chalmers.se, also provided by SNIC. 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Fusion 49 085007	arxiv-papers	2013-12-19T15:01:53	2024-09-04T02:49:55.685095	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Andreas Skyman, Luis Fazendeiro, Daniel Tegnered, Hans Nordman, Johan\n Anderson, P\\\"ar Strand", "submitter": "Andreas Skyman", "url": "https://arxiv.org/abs/1312.5575" }
1312.5702	# Complete description of rational points of Diophantine equation $X^{4}+Y^{4}=Z^{4}+W^{4}$ M. A. Reynya Ahva St. 15/14, Haifa, Israel [email protected] ###### Abstract. In this paper we consider Diophantine equation $x^{4}+y^{4}=z^{4}+w^{4}$ (1)We construct some family of cubic curves.We prove that every rational point on Quartica $x^{4}+y^{4}=z^{4}+w^{4}$ can be mapped to a point on some curve of this family. We also prove the opposite: each rational point belonging to our family of curves can be mapped to a rational point on the Quartica. (2) We find the point on our family of curves corresponding to a parametric solution of Leonard Euler. We construct several new parametric solutions of our Quartica, using a parametric solution of Leonard Euler and the algebraic operation on the cubic curves. (3)We present an algorithm to find all rational points on our Quartica. ## 1\. Introduction It is known that the Quartica $X^{4}+Y^{4}=Z^{4}+W^{4}$ has infinitely many integer solutions. Leonard Euler found the first parametric solution. It was proved in the (see [Bi1] by Swinnerton-Dyer that this diophantine equation has infinitely many rational parametric solitions.The following authors have found various parametric solutions: (see [Bi2],[Hr]. The aim of this paper is to give algorithm finding all solutions of this Quartica.In the Section $2$ we show that this Quartica is equivalent to a system of two equations of the third degree in six variables.We prove that every rational point on Quartica can be mapped to a rational point which is solution to this system of two equations, and the opposite: each rational point that is solution to a system of two equations can be mapped to a rational point on the Quartica. Our Quartica has elementary solutions : $(m,n,m,n)$ and $(s,t,-t,s)$. And this means that these elementary solutions are mapped to solutions of a system of equations we constructed. So we got two solutions of our system of equations. Next using these two solutions we find a new solution of our system of equations,as a result we get the expression depending on the 5 variables : $(m,n,r,s,t)$.This expression is the cubic in variables $m$ , $n$ , $r$ with coefficients depending on variables $s$ and $t$.So we construct a family of cubic curves. We prove that this family of cubic curves contains images of all rational points of our Quartica. As an example we take the Quartca point $(59,158,133,134)$ and build cubic curve of our family which contains the image of this point. We also prove the opposite: each rational point belonging to our family of curves is mapped to a rational point on the Quartica. In the section (3) we build several new parametric solutions of our Quartica,using a parametric solution of Leonard Euler. At the beginning we find a point on our family of cubic curves depending on the parameters s and t.Then we find the image of this point on our Quartica.That is, we get a parametric solution to our Quartica. This is the parametric solution of Leonard Euler.In accordance with our costruction every solution on Quartica can be associated to a new solution.We call it the pair solution. Using Mathematika 7 + we find a parametric solution to pair solution of Leonard Euler. Because every point on the Quartica has image on the cubic curve and opposite, and on the cubic curve there exist associative operation we can construct new solutions using solution of Leonard Euler. The process looks like this : 1.We find the image of solution of Leonard Euler on the cubic, let us denote this point $E$ 2.Using the tangent method we build the point $E\cdot E$ on the cubic. 3.We find the image of the point $E\cdot E$ on the Quartica - the new solution. As an example we take the point $E$ \- $(59,158,133,134)$.And we get the point $E\cdot E$ \- $((-8450072351),520471467675,487934246375,59481958899))$, and $((-3535404127283),(-132758926000),(3343735015475),(-2363831080408))$ \- the pair point. Using Mathematika 7 + we find an appropriate parametric solution $E\cdot E$ and its pair. In section 4 we present the algorithm for finding all rational points on the our Quartica. ## 2\. Construction the family of elliptic curves ###### Lemma 2.1. The Quartica $x^{4}+y^{4}=z^{4}+w^{4}$ then, and only then has a rational point (x,y,z,w), if the system equations $\left\\{\begin{array}[]{clrr}a\cdot x^{2}-b\cdot y^{2}=a\cdot z^{2}+b\cdot w^{2},\\\ b\cdot x^{2}+a\cdot y^{2}=-b\cdot z^{2}+a\cdot w^{2}\end{array}\right.$. has the rational point (a,b,x,y,z,w). ###### Proof. To solve the system equations we receive: $a\cdot(x^{2}-z^{2})=b\cdot(y^{2}+w^{2})$, $b\cdot(x^{2}+z^{2})=a\cdot(w^{2}-y^{2})$,and $a/b=(y^{2}+w^{2})/(x^{2}-z^{2})=(x^{2}+z^{2})/(w^{2}-y^{2})$, so $x^{4}+y^{4}=w^{4}+z^{4}$. And opposite: $x^{4}+y^{4}=w^{4}+z^{4}$,$w^{4}-y^{4}=x^{4}-z^{4}$,$(w^{2}-y^{2})\cdot(w^{2}+y^{2})=(x^{2}-z^{2})\cdot(x^{2}+z^{2})$,$(y^{2}+w^{2})/(x^{2}-z^{2})=(x^{2}+z^{2})/(w^{2}-y^{2})$,we will suppose that:$(y^{2}+w^{2})/(x^{2}-z^{2})=(x^{2}+z^{2})/(w^{2}-y^{2})=a/b$, than $(y^{2}+w^{2})/(x^{2}-z^{2})=a/b$,and,$(x^{2}+z^{2})/(w^{2}-y^{2})=a/b$, and we get the initial system equations: $\left\\{\begin{array}[]{clrr}a\cdot x^{2}-b\cdot y^{2}=a\cdot z^{2}+b\cdot w^{2},\\\ b\cdot x^{2}+a\cdot y^{2}=-b\cdot z^{2}+a\cdot w^{2}\end{array}\right.$. ∎ ###### Theorem 2.2. The system of equations: $\left\\{\begin{array}[]{clrr}a\cdot x^{2}-b\cdot y^{2}=a\cdot z^{2}+b\cdot w^{2},\\\ b\cdot x^{2}+a\cdot y^{2}=-b\cdot z^{2}+a\cdot w^{2}\end{array}\right.$. has two parametric solutions : ${x=m,y=n,z=m,w=n,b=0,a=r}$, ${x=s,y=t,z=-t,w=s,b=1,a=(s^{2}+t^{2})/(s^{2}-t^{2})}$ ###### Proof. Points: ${x=m,y=n,z=m,w=n}$ ${x=s,y=t,z=-t,w=s}$ are solutions of Quartica $x^{4}+y^{4}=z^{4}+w^{4}$,therefore for Lemma 2.1 there must be a solution for system of equations: $\left\\{\begin{array}[]{clrr}a\cdot x^{2}-b\cdot y^{2}=a\cdot z^{2}+b\cdot w^{2},\\\ b\cdot x^{2}+a\cdot y^{2}=-b\cdot z^{2}+a\cdot w^{2}\end{array}\right.$. To solve this system for $a$ and $b$ for every point we can to prove the Theorem 2.2. (1) $\left\\{\begin{array}[]{clrr}a\cdot m^{2}-b\cdot n^{2}=a\cdot m^{2}+b\cdot n^{2},\\\ b\cdot m^{2}+a\cdot n^{2}=-b\cdot m^{2}+a\cdot n^{2}\end{array}\right.$. $a=r,b=0$. (2) $\left\\{\begin{array}[]{clrr}a\cdot s^{2}-b\cdot t^{2}=a\cdot t^{2}+b\cdot s^{2},\\\ b\cdot s^{2}+a\cdot t^{2}=-b\cdot t^{2}+a\cdot s^{2}\end{array}\right.$. $a=(s^{2}+t^{2})/(s^{2}-t^{2}),b=1$. ∎ Now we have a system of two equations of the third degree,and two solutions for this system. ###### Theorem 2.3. The system of equations: $\left\\{\begin{array}[]{clrr}a\cdot x^{2}-b\cdot y^{2}=a\cdot z^{2}+b\cdot w^{2},\\\ b\cdot x^{2}+a\cdot y^{2}=-b\cdot z^{2}+a\cdot w^{2}\end{array}\right.$. has parametric solution: $x=(m\cdot g+s),y=(n\cdot g+t),z=(m\cdot g-t),w=(n\cdot g+s),b=(0\cdot g+1),a=(r\cdot g+(s^{2}+t^{2})/(s^{2}-t^{2}))$, where $g=(2\cdot m\cdot s^{2}-2\cdot n\cdot s^{2}+r\cdot s^{3}-r\cdot s^{2}\cdot t+2\cdot m\cdot t^{2}+2\cdot n\cdot t^{2}-r\cdot s\cdot t^{2}+r\cdot t^{3})/(2\cdot(s-t)\cdot(n^{2}-m\cdot r\cdot s-m\cdot r\cdot t))$ from the first equation of the system , and $g=(-2\cdot m\cdot s^{2}+2\cdot n\cdot s^{2}+r\cdot s^{3}+r\cdot s^{2}\cdot t+2\cdot m\cdot t^{2}+2\cdot n\cdot t^{2}-r\cdot s\cdot t^{2}-r\cdot t^{3})/(2\cdot(s+t)\cdot(m^{2}-n\cdot r\cdot s+n\cdot r\cdot t))$ from the second equation of the system ###### Proof. Substitute the expressions: $x=(m\cdot g+s),y=(n\cdot g+t),z=(m\cdot g-t),w=(n\cdot g+s),b=(0\cdot g+1),a=(r\cdot g+(s^{2}+t^{2})/(s^{2}-t^{2}))$, in each of the equations of our system. Because $(x=m,y=n,z=m,w=n,b=0,a=r)$ and $(x=s,y=t,z=-t,w=s,b=1,a=(s^{2}+t^{2})/(s^{2}-t^{2}))$ are solutions to each of the equations ,we receive two linear equations for $g$,because $g=0$ and $g=\infty$ are the roots of each of the equations. If we compute value of $g$ for each of the equations, we get the expressions for $g$ specified in the condition of the theorem. ∎ Now it is clear that condition for the existence of solution for system of equations and for Lemma 2.1 for equation $x^{4}+y^{4}=z^{4}+w^{4}$ is the equality of two expressions for $g$ from Theorema 2.3. Consider the expression resulting from the equality of two expressions for $g$: $(2\cdot m\cdot s^{2}-2\cdot n\cdot s^{2}+r\cdot s^{3}-r\cdot s^{2}\cdot t+2\cdot m\cdot t^{2}+2\cdot n\cdot t^{2}-r\cdot s\cdot t^{2}+r\cdot t^{3})/(2\cdot(s-t)\cdot(n^{2}-m\cdot r\cdot s-m\cdot r\cdot t))$ = $(-2\cdot m\cdot s^{2}+2\cdot n\cdot s^{2}+r\cdot s^{3}+r\cdot s^{2}\cdot t+2\cdot m\cdot t^{2}+2\cdot n\cdot t^{2}-r\cdot s\cdot t^{2}-r\cdot t^{3})/(2\cdot(s+t)\cdot(m^{2}-n\cdot r\cdot s+n\cdot r\cdot t))$, or $(2\cdot m\cdot s^{2}-2\cdot n\cdot s^{2}+r\cdot s^{3}-r\cdot s^{2}\cdot t+2\cdot m\cdot t^{2}+2\cdot n\cdot t^{2}-r\cdot s\cdot t^{2}+r\cdot t^{3})\cdot(2\cdot(s+t)\cdot(m^{2}-n\cdot r\cdot s+n\cdot r\cdot t))$ = $(-2\cdot m\cdot s^{2}+2\cdot n\cdot s^{2}+r\cdot s^{3}+r\cdot s^{2}\cdot t+2\cdot m\cdot t^{2}+2\cdot n\cdot t^{2}-r\cdot s\cdot t^{2}-r\cdot t^{3})\cdot(2\cdot(s-t)\cdot(n^{2}-m\cdot r\cdot s-m\cdot r\cdot t))$. This expression is the cubic in variables $m$ , $n$ , $r$ with coefficients depending on variables $s$ and $t$. ###### Theorem 2.4. For any solution $x$,$y$,$z$,$w$ of the equation $x^{4}+y^{4}=z^{4}+w^{4}$ there are such variable values $m$ , $n$ , $r$ , $s$ , $t$ for the given values $s$ and $t$ variable values $m$ , $n$ , $r$ are the solution of our cubic. ###### Proof. Suppose that ($x$,$y$,$z$,$w$) the rational point of equation, than for Lemma 2.1 there exist $a$ and $b$ so that ($x$,$y$,$z$,$w$,$a$ $b$) the rational point of system equations. Consider the system equations:$x=(m\cdot g+s),y=(n\cdot g+t),z=(m\cdot g-t),w=(n\cdot g+s)$ Suppose that $g=1$,The discriminant of this system is not equal to $0$,therefore $m,n,s,t$ exist. But the points ($m$,$n$,$m$,$n$) and ($s$,$t$,$-t$,$s$) are the solutions of Quartica $x^{4}+y^{4}=z^{4}+w^{4}$,and therefore the points : (${m,n,m,n,0,r}$), (${s,t,-t,s,1,(s^{2}+t^{2})/(s^{2}-t^{2})}$) are the solutions of system of equations.Now in in accordance with our construction the $x,y,z,w,b=1,a=(x^{2}+z^{2})/(w^{2}-y^{2})$ are the solution of our system ,on the other hand because there are such $m,n,s,t$ that $x=(m\cdot g+s),y=(n\cdot g+t),z=(m\cdot g-t),w=(n\cdot g+s)$ for $g=1$, the point $(x=(m\cdot g+s),y=(n\cdot g+t),z=(m\cdot g-t),w=(n\cdot g+s),b=0\cdot g+1,a=r\cdot g+(s^{2}+t^{2})/(s^{2}-t^{2}))$ is the solution of the system if $g=1$ and $r=(x^{2}+z^{2})/(w^{2}-y^{2})-(s^{2}+t^{2})/(s^{2}-t^{2})$,and therefore for any solution $x$,$y$,$z$,$w$ of the equation $x^{4}+y^{4}=z^{4}+w^{4}$ there are such variable values $m$ , $n$ , $r$ , $s$ , $t$ for the given values $s$ and $t$ variable values $m$ , $n$ , $r$ are the solution of our cubic. ∎ EXAMPLE. Now we have solution of Quartica $x^{4}+y^{4}=z^{4}+w^{4}$ : ($x=59$ , $y=158$ , $z=133$ , $w=134$). Show that there is a cubic curve to which this point belongs. Consider the system of equations:$(m\cdot g+s)=59,y=(n\cdot g+t)=158,z=(m\cdot g-t)=133,w=(n\cdot g+s)=134$. Suppose $g=1$,then $m=108,n=183,s=-49,t=-25$. If we substitute into the equation of the cubic the values of $s$ and $t$ we receive: $(-1513\cdot m+888\cdot n+10656\cdot r)\cdot(37\cdot(m^{2}+24\cdot n\cdot r)=(888\cdot m-1513\cdot n+32856\cdot r)\cdot(12\cdot(n^{2}+74\cdot m\cdot r))$ If we substitute in this cubic values $m=108,n=183$ we receive for $r$ equation of second degree ,which must have one rational root of the $g=1$. Since the equation is of second degree there is another rational root corresponding to the new solution. Equation:$72339650100+8604986400\cdot r-1419379200\cdot r^{2}=0$ has two rational roots: $r1=-2797/592,r2=3193/296$ We substitute values for $m,n,s,t$ to Quartica and receive: $(108\cdot g+(-49))^{4}+(183\cdot g+(-25))^{4}-(108\cdot g-(-25))^{4}-(183\cdot g+(-49))^{4}=0$,or $17107200\cdot g-232567200\cdot g^{2}+215460000\cdot g^{3}=0$ This equation has two rational roots :the root $g=1$ corresponding to the point ($x=59$ , $y=158$ , $z=133$ , $w=134$) on the Quartica and the point($m=108,n=183,r=-2797/592$)on the cubic.And the root $g_{1}=3193/296$,corresponding to some new point on the Quartica:($x_{1}=-134413,y_{1}=-34813,z_{1}=111637,w_{1}=-114613$),and the point ($m_{1}=108,n_{1}=183,r_{1}=3193/296$) on the cubic. So picking an arbitrary rational point ($x=59$ , $y=158$ , $z=133$ , $w=134$) on the Quartica we found the cubic to which belong the rational point ($m=108,n=183,r=-2797/592$), where ($m,n,r$) are functions from ($x,y,z,w$) and found one more point on the Quartica ($x_{1}=-134413,y_{1}=-34813,z_{1}=111637,w_{1}=-114613$),corresponding to the point ($m_{1}=108,n_{1}=183,r_{1}=3193/296$) on the same cubic. So the family of cubics is: $(2\cdot m\cdot s^{2}-2\cdot n\cdot s^{2}+r\cdot s^{3}-r\cdot s^{2}\cdot t+2\cdot m\cdot t^{2}+2\cdot n\cdot t^{2}-r\cdot s\cdot t^{2}+r\cdot t^{3})\cdot(2\cdot(s+t)\cdot(m^{2}-n\cdot r\cdot s+n\cdot r\cdot t))$ = $(-2\cdot m\cdot s^{2}+2\cdot n\cdot s^{2}+r\cdot s^{3}+r\cdot s^{2}\cdot t+2\cdot m\cdot t^{2}+2\cdot n\cdot t^{2}-r\cdot s\cdot t^{2}-r\cdot t^{3})\cdot(2\cdot(s-t)\cdot(n^{2}-m\cdot r\cdot s-m\cdot r\cdot t))$. Since each point in our family of cubic curves match point-solution of cubic equations then by Lemma 2 the following is true: every point in our family of cubic curves match point on the surface containing images of all rational points of our Quartica.Since each point in our family of cubic curves match point-solution of cubic equations then Lemma 2 the following is true: every point in our family of cubic curves match point on the Quartica. ## 3\. Construction of parametric solutions of Quartica $X^{4}+Y^{4}=Z^{4}+W^{4}$ Consider the family of cubics constructed in the previous chapter: $(2\cdot m\cdot s^{2}-2\cdot n\cdot s^{2}+r\cdot s^{3}-r\cdot s^{2}\cdot t+2\cdot m\cdot t^{2}+2\cdot n\cdot t^{2}-r\cdot s\cdot t^{2}+r\cdot t^{3})\cdot(2\cdot(s+t)\cdot(m^{2}-n\cdot r\cdot s+n\cdot r\cdot t))$ = $(-2\cdot m\cdot s^{2}+2\cdot n\cdot s^{2}+r\cdot s^{3}+r\cdot s^{2}\cdot t+2\cdot m\cdot t^{2}+2\cdot n\cdot t^{2}-r\cdot s\cdot t^{2}-r\cdot t^{3})\cdot(2\cdot(s-t)\cdot(n^{2}-m\cdot r\cdot s-m\cdot r\cdot t))$. ###### Theorem 3.1. This family of cubics contains the point: ($m=(-((s^{5}-s^{3}\cdot t^{2}-s^{2}\cdot t^{3}+t^{5})/(4\cdot s^{2}\cdot t^{2}))),n=(-((s^{5}-s^{3}\cdot t^{2}+s^{2}\cdot t^{3}-t^{5})/(4\cdot s^{2}\cdot t^{2}))),r=1$) ###### Proof. To prove the theorem it is enough to equate to 0 two expressions in the left and right side of the equation for a family of curves. We get a system of equations: $\left\\{\begin{array}[]{clrr}(2\cdot m\cdot s^{2}-2\cdot n\cdot s^{2}+r\cdot s^{3}-r\cdot s^{2}\cdot t+2\cdot m\cdot t^{2}+2\cdot n\cdot t^{2}-r\cdot s\cdot t^{2}+r\cdot t^{3})=0,\\\ (-2\cdot m\cdot s^{2}+2\cdot n\cdot s^{2}+r\cdot s^{3}+r\cdot s^{2}\cdot t+2\cdot m\cdot t^{2}+2\cdot n\cdot t^{2}-r\cdot s\cdot t^{2}-r\cdot t^{3})=0\end{array}\right.$. Solving this system, provided that the $r=1$,we prove the theorem ∎ Now we have the point on the family of cubics.Now construct a point on Quartica.For this we will solve the equation: $(m\cdot g+s)^{4}+(n\cdot g+t)^{4}-(m\cdot g-t)^{4}-(n\cdot g+s)^{4}=0$ Or substituting the values for $m$ and $n$: $((-((s^{5}-s^{3}\cdot t^{2}-s^{2}\cdot t^{3}+t^{5})/(4\cdot s^{2}\cdot t^{2})))\cdot g+s)^{4}+((-((s^{5}-s^{3}\cdot t^{2}+s^{2}\cdot t^{3}-t^{5})/(4\cdot s^{2}\cdot t^{2})))\cdot g+t)^{4}-((-((s^{5}-s^{3}\cdot t^{2}-s^{2}\cdot t^{3}+t^{5})/(4\cdot s^{2}\cdot t^{2})))\cdot g-t)^{4}-((-((s^{5}-s^{3}\cdot t^{2}+s^{2}\cdot t^{3}-t^{5})/(4\cdot s^{2}\cdot t^{2})))\cdot g+s)^{4}=0$ Using the program Mathematika 7+ we receive: $g=(-((12\cdot s^{4}\cdot t^{4})/((s^{2}-t^{2})\cdot(s^{6}-2\cdot s^{4}\cdot t^{2}-2\cdot s^{2}\cdot t^{4}+t^{6}))))$ And, accordingly: $x=s\cdot(s^{6}+s^{4}\cdot t^{2}-2\cdot s^{2}\cdot t^{4}-3\cdot s\cdot t^{5}+t^{6})$ $y=t\cdot(s^{6}+3\cdot s^{5}\cdot t-2\cdot s^{4}\cdot t^{2}+s^{2}\cdot t^{4}+t^{6})$ $z=-t\cdot(s^{6}-3\cdot s^{5}\cdot t-2\cdot s^{4}\cdot t^{2}+s^{2}\cdot t^{4}+t^{6})$ $w=s\cdot(s^{6}+s^{4}\cdot t^{2}-2\cdot s^{2}\cdot t^{4}+3\cdot s\cdot t^{5}+t^{6})$ This is the solution of Leonard Euler In an example considered above $59,158,133,134$ corresponds to the solution of Leonard Euler for $s=2$,$t=1$: $(x=134,y=133,z=59,w=158)$.However, we got a new solution ($x_{1}=-134413,y_{1}=-34813,z_{1}=111637,w_{1}=-114613$)Call it a pair of the solution of Leonard Euler.To find the solution in parametric form we will carry out the following: Solve the system of equations: $\left\\{\begin{array}[]{clrr}((M+S)=-t\cdot(s^{6}-3\cdot s^{5}\cdot t-2\cdot s^{4}\cdot t^{2}+s^{2}\cdot t^{4}+t^{6}),\\\ $ $(N+T)=s\cdot(s^{6}+s^{4}\cdot t^{2}-2\cdot s^{2}\cdot t^{4}+3\cdot s\cdot t^{5}+t^{6}),\\\ (M-T)==t\cdot(s^{6}+3\cdot$ $s^{5}\cdot t-2\cdot s^{4}\cdot t^{2}+s^{2}\cdot t^{4}+t^{6}),\\\ (N+S)==s\cdot(s^{6}+s^{4}\cdot t^{2}-2\cdot s^{2}\cdot t^{4}-$ $3\cdot s\cdot t^{5}+t^{6})\end{array}\right.$ Solving this system we get: $M=3\cdot(s^{5}\cdot t^{2}+s^{2}\cdot t^{5}),N=(s^{7}+s^{6}\cdot t+s^{5}\cdot t^{2}-2\cdot s^{4}\cdot t^{3}-$ $2\cdot s^{3}\cdot t^{4}+s^{2}\cdot t^{5}+s\cdot t^{6}+t^{7}),S=(-s^{6}\cdot t+2\cdot s^{4}\cdot t^{3}-$ $4\cdot s^{2}\cdot t^{5}-t^{7}),T=(-s^{6}\cdot t+2\cdot s^{4}\cdot t^{3}+2\cdot s^{2}\cdot t^{5}-t^{7})$ Now we solve equation: $(M\cdot g+S)^{4}+(N\cdot g+T)^{4}-(M\cdot g-T)^{4}-(N\cdot+S)^{4}=0$ In accordance with our construction,this equation has the roots: $g_{1}=0,g_{2}=\infty,g_{3}=1$,$g_{3}=1$ \- corresponds to the solution of Euler,and $g_{4}$ we compute with Mathematika 7+: $g_{4}=(-s^{13}+2\cdot s^{12}\cdot t+4\cdot s^{11}\cdot t^{2}-8\cdot s^{10}\cdot t^{3}+8\cdot s^{7}\cdot t^{6}+2\cdot s^{6}\cdot$ $t^{7}-$ $18\cdot s^{5}\cdot t^{8}+18\cdot s^{4}\cdot t^{9}-14\cdot s^{3}\cdot t^{10}+10\cdot s^{2}\cdot t^{11}-s\cdot t^{12}+$ $2\cdot t^{13})/(s^{13}+s^{12}\cdot t-4\cdot s^{11}\cdot t^{2}+14\cdot s^{10}\cdot t^{3}-18\cdot s^{9}\cdot t^{4}-$ $9\cdot s^{8}\cdot t^{5}+28\cdot s^{7}\cdot t^{6}-8\cdot s^{6}\cdot t^{7}-4\cdot s^{3}\cdot t^{10}+5\cdot s^{2}\cdot t^{11}+$ $s\cdot t^{12}+t^{13})$ Substituting the value of $g_{4}$ in each of the expressions:$(M\cdot g_{4}+S),(N\cdot g_{4}+T),(M\cdot g_{4}-T),(N\cdot g_{4}+S)$ we get a new parametric solution : $x=(-t\cdot(s^{18}+3\cdot s^{17}\cdot t-15\cdot s^{16}\cdot t^{2}+15\cdot s^{15}\cdot t^{3}+6\cdot s^{14}\cdot t^{4}-$ $45\cdot s^{13}\cdot t^{5}+82\cdot s^{12}\cdot t^{6}-15\cdot s^{11}\cdot t^{7}-123\cdot s^{10}\cdot t^{8}+$ $171\cdot s^{9}\cdot t^{9}-159\cdot s^{8}\cdot t^{10}+159\cdot s^{7}\cdot t^{11}-98\cdot s^{6}\cdot t^{12}+$ $30\cdot s^{5}\cdot t^{13}-12\cdot s^{4}\cdot t^{14}+3\cdot s^{2}\cdot t^{16}+t^{18}))$ $y=((-s^{19}+s^{18}\cdot t+3\cdot s^{17}\cdot t^{2}+3\cdot s^{16}\cdot t^{3}-21\cdot s^{15}\cdot t^{4}+$ $12\cdot s^{14}\cdot t^{5}+44\cdot s^{13}\cdot t^{6}-86\cdot s^{12}\cdot t^{7}+93\cdot s^{11}\cdot t^{8}-$ $87\cdot s^{10}\cdot t^{9}-3\cdot s^{9}\cdot t^{10}+135\cdot s^{8}\cdot t^{11}-142\cdot s^{7}\cdot t^{12}+$ $100\cdot s^{6}\cdot t^{13}-72\cdot s^{5}\cdot t^{14}+36\cdot s^{4}\cdot t^{15}-12\cdot s^{3}\cdot t^{16}+$ $9\cdot s^{2}\cdot t^{17}-s\cdot t^{18}+t^{19}))$ $z=(t\cdot(s^{18}-3\cdot s^{17}\cdot t+3\cdot s^{16}\cdot t^{2}+21\cdot s^{15}\cdot t^{3}-60\cdot s^{14}\cdot t^{4}+$ $27\cdot s^{13}\cdot t^{5}+58\cdot s^{12}\cdot t^{6}-75\cdot s^{11}\cdot t^{7}+57\cdot s^{10}\cdot t^{8}-$ $63\cdot s^{9}\cdot t^{9}+63\cdot s^{8}\cdot t^{10}-87\cdot s^{7}\cdot t^{11}+100\cdot s^{6}\cdot t^{12}-$ $66\cdot s^{5}\cdot t^{13}+36\cdot s^{4}\cdot t^{14}-18\cdot s^{3}\cdot t^{15}+9\cdot s^{2}\cdot t^{16}+t^{18}))$ $w=((-s^{19}+s^{18}\cdot t+3\cdot s^{17}\cdot t^{2}+3\cdot s^{16}\cdot t^{3}-21\cdot s^{15}\cdot t^{4}+$ $6\cdot s^{14}\cdot t^{5}+44\cdot s^{13}\cdot t^{6}-62\cdot s^{12}\cdot t^{7}-15\cdot s^{11}\cdot t^{8}+$ $129\cdot s^{10}\cdot t^{9}-165\cdot s^{9}\cdot t^{10}+129\cdot s^{8}\cdot t^{11}-88\cdot s^{7}\cdot t^{12}+$ $46\cdot s^{6}\cdot t^{13}-18\cdot s^{5}\cdot t^{14}+6\cdot s^{4}\cdot t^{15}-12\cdot s^{3}\cdot t^{16}+3\cdot s^{2}\cdot$ $t^{17}-s\cdot t^{18}+t^{19}))$ Because every point on the Quartica has image on the cubic curve and opposite, and on the cubic curve there exist an algebraic operation we can construct new solutions,using solution of Leonard Euler. The process looks like this : 1.We find the image of solution of Leonard Euler on the cubic,let us denote this point $E$ 2.Using the tangent method we build the point $E\cdot E$ on the cubic. 3.We find the image of the point $E\cdot E$ on the Quartica - the new solution. Example. We have the ratonal point on the Quartica : $59,158,133,134$ The cubic containg image of this rational point was constructed in the 2 chapter of this paper. This is the cubic : $(-1513\cdot m+888\cdot n+10656\cdot r)\cdot(37\cdot(m^{2}+24\cdot n\cdot r)=(888\cdot m-$ $1513\cdot n+32856\cdot r)\cdot(12\cdot(n^{2}+74\cdot m\cdot r))$ The image point is: $m=108,n=183,r=(-2797/592)$(from chapter 2) Now we can use the tangent method for this cubic curve: We substitude :$m=108$ to $M=(108\cdot g_{1}+k)$ , $n=183$ to $N=(183\cdot g_{1}+1)$ , $r=(-2797/592)$ to $R=((-2797/592)\cdot g_{1}+2)$ : $(-1513\cdot(108\cdot g_{1}+k)+888\cdot(183\cdot g_{1}+1)+10656\cdot((-2797/592)\cdot g_{1}+2))\cdot(37\cdot((108\cdot$ $g_{1}+k)^{2}+24\cdot(183\cdot g_{1}+1)\cdot((-2797/592)\cdot g_{1}+2))=(888\cdot(108\cdot g_{1}+k)-1513\cdot(183\cdot g_{1}+1)$ $+32856\cdot r)\cdot(12\cdot((183\cdot g_{1}+1)^{2}+74\cdot(108\cdot g_{1}+k)\cdot((-2797/592)\cdot g_{1}+2)))$ So we receive: $-((9\cdot g_{1}^{2}\cdot(-20155494924+562635949\cdot k)+$ $72\cdot g_{1}(308932483-21347816\cdot k+570133\cdot k^{2})+$ $4\cdot(-38656812+116715168\cdot k+755688\cdot k^{2}+55981\cdot k^{3}))/($ $3\cdot(-8-130800\cdot g_{1}+34164\cdot g_{1}^{2}-1184\cdot k+2797\cdot g_{1}\cdot k)(672417\cdot g_{1}^{2}-$ $3\cdot g_{1}\cdot(213875+5328\cdot k)-74\cdot(48+k^{2}))))$ We can choose the $k$ so that the coefficient for $g_{1}^{2}$ is equal to $0$: $(-20155494924+562635949\cdot k)=0$ Solving this equation we receive: $k=20155494924/562635949$,and solving equation for $g_{1}$ : $g_{1}=(-3431129689319806/2216545393509777)$ Now we receive values from $M,N,R$: $M=(-97052654280770532/738848464503259)$ , $N=(-208560062584004907/738848464503259)$, $R=6110629743471536675/656097436478893992$. This is the image point on the cubic corresponding the new point on the Quartika. Now we can find this new point,solving the equation: $((-(97052654280770532/738848464503259))\cdot s+(-49))^{4}$ $+((-(208560062584004907/738848464503259))\cdot s+(-25))^{4}$ $-((-(97052654280770532/738848464503259))\cdot s-(-25))^{4}$ $-(-(208560062584004907/738848464503259))\cdot s+(-49))^{4}=0$ This equation has two rational roots : $s_{1}=(-15645116235856509325/187352780702663748309)$,and $s_{2}=(-258596962576140650/711525553297861767)$ This two roots correspond to the two points on the Quartica: $(x_{1}=(-3535404127283),y_{1}=(-132758926000),z_{1}=(3343735015475),w_{1}=(-2363831080408))$ , for $s_{1}$ $(-3535404127283)^{4}+(-132758926000)^{4}=3343735015475^{4}+(-2363831080408)^{4}$ and $(x_{2}=(-8450072351),y_{2}=520471467675,z_{2}=487934246375,w_{2}=359481958899)$,for $s_{2}$ $(-8450072351)^{4}+520471467675^{4}=487934246375^{4}+359481958899^{4}$ Now we return to our cubic : This is the cubic : $(-1513\cdot m+888\cdot n+10656\cdot r)\cdot(37\cdot(m^{2}+24\cdot n\cdot r)=(888\cdot m-1513\cdot n+32856\cdot r)\cdot(12\cdot(n^{2}+74\cdot m\cdot r))$ We substitute the new values of $m=(-97052654280770532/738848464503259)$ , $n=(-208560062584004907/738848464503259)$, and receive the equation of second degree for $r$ : $-5260289575280440614252321193027166875+$ $392359179683252386906081036910885000\cdot r+$ $18514574028136616634982730304244992\cdot r^{2}=0$ This equation has two rational roots : $r_{1}=(-860842465688650025/28219244579737376)$, $r_{2}=6110629743471536675/656097436478893992$. The simple computing show that $r_{1}=(-860842465688650025/28219244579737376)$, correspond to solution $(x_{1}=(-3535404127283),y_{1}=(-132758926000),z_{1}=(3343735015475),w_{1}=(-2363831080408))$ and the $r_{2}=6110629743471536675/656097436478893992$, correspond to solution $(x_{2}=(-8450072351),y_{2}=520471467675,z_{2}=487934246375,w_{2}=359481958899)$ So in accordance with our construction these are the points that we are seeking: $(x_{2}=(-8450072351),y_{2}=520471467675,z_{2}=487934246375,w_{2}=359481958899)$ and $(x_{1}=(-3535404127283),y_{1}=(-132758926000),z_{1}=(3343735015475),w_{1}=(-2363831080408))$ the pair point. Using the program Wolfram Mathematika 7+ we can calculate the parametric solution and its pair parametric solution. If $E$ is the point on the family of curves corresponding to parametric solution of Leonard Euler,the parametric solution corresponding to the point $E\cdot E$ is: $x=(s^{13}+s^{12}\cdot t+2\cdot s^{11}\cdot t^{2}-4\cdot s^{10}\cdot t^{3}-3\cdot s^{9}\cdot t^{4}+3\cdot s^{8}\cdot t^{5}+7\cdot s^{7}\cdot t^{6}+4\cdot s^{6}\cdot t^{7}-12\cdot s^{5}\cdot t^{8}-6\cdot s^{4}\cdot t^{9}+5\cdot s^{3}\cdot t^{10}-s^{2}\cdot t^{11}+s\cdot t^{12}+t^{13})$ $y=(-s^{13}+s^{12}\cdot t+s^{11}\cdot t^{2}+5\cdot s^{10}\cdot t^{3}+6\cdot s^{9}\cdot t^{4}-12\cdot s^{8}\cdot t^{5}-4\cdot s^{7}\cdot t^{6}+7\cdot s^{6}\cdot t^{7}-3\cdot s^{5}\cdot t^{8}-3\cdot s^{4}\cdot t^{9}+4\cdot s^{3}\cdot t^{10}+2\cdot s^{2}\cdot t^{11}-s\cdot t^{12}+t^{13})$ $z=(-(s^{13}+s^{12}\cdot t-s^{11}\cdot t^{2}+5\cdot s^{10}\cdot t^{3}-6\cdot s^{9}\cdot t^{4}-12\cdot s^{8}\cdot t^{5}+4\cdot s^{7}\cdot t^{6}+7\cdot s^{6}\cdot t^{7}+3\cdot s^{5}\cdot t^{8}-3\cdot s^{4}\cdot t^{9}-4\cdot s^{3}\cdot t^{10}+2\cdot s^{2}\cdot t^{11}+s\cdot t^{12}+t^{13}))$ $w=(s^{13}-s^{12}\cdot t+2\cdot s^{11}\cdot t^{2}+4\cdot s^{10}\cdot t^{3}-3\cdot s^{9}\cdot t^{4}-3\cdot s^{8}\cdot t^{5}+7\cdot s^{7}\cdot t^{6}-4\cdot s^{6}\cdot t^{7}-12\cdot s^{5}\cdot t^{8}+6\cdot s^{4}\cdot t^{9}+5\cdot s^{3}\cdot t^{10}+s^{2}\cdot t^{11}+s\cdot t^{12}-t^{13})$ And the pair parametric solution : $x=t\cdot(2\cdot s^{18}-6\cdot s^{17}\cdot t-3\cdot s^{16}\cdot t^{2}-3\cdot s^{15}\cdot t^{3}-9\cdot s^{14}\cdot t^{4}+27\cdot s^{13}\cdot t^{5}+32\cdot s^{12}\cdot t^{6}-33\cdot s^{11}\cdot t^{7}-39\cdot s^{10}\cdot t^{8}+27\cdot s^{9}\cdot t^{9}+21\cdot s^{8}\cdot t^{10}-3\cdot s^{7}\cdot t^{11}+2\cdot s^{6}\cdot t^{12}-6\cdot s^{5}\cdot t^{13}-12\cdot s^{4}\cdot t^{14}+3\cdot s^{2}\cdot t^{16}+2\cdot t^{18})$ $y=-s\cdot(2\cdot s^{18}+3\cdot s^{16}\cdot t^{2}-12\cdot s^{14}\cdot t^{4}+6\cdot s^{13}\cdot t^{5}+2\cdot s^{12}\cdot t^{6}+3\cdot s^{11}\cdot t^{7}+21\cdot s^{10}\cdot t^{8}-27\cdot s^{9}\cdot t^{9}-39\cdot s^{8}\cdot t^{10}+33\cdot s^{7}\cdot t^{11}+32\cdot s^{6}\cdot t^{12}-27\cdot s^{5}\cdot t^{13}-9\cdot s^{4}\cdot t^{14}+3\cdot s^{3}\cdot t^{15}-3\cdot s^{2}\cdot t^{16}+6\cdot s\cdot t^{17}+2\cdot t^{18})$ $z=s\cdot(-2\cdot s^{18}-3\cdot s^{16}\cdot t^{2}+12\cdot s^{14}\cdot t^{4}+6\cdot s^{13}\cdot t^{5}-2\cdot s^{12}\cdot t^{6}+3\cdot s^{11}\cdot t^{7}-21\cdot s^{10}\cdot t^{8}-27\cdot s^{9}\cdot t^{9}+39\cdot s^{8}\cdot t^{10}+33\cdot s^{7}\cdot t^{11}-32\cdot s^{6}\cdot t^{12}-27\cdot s^{5}\cdot t^{13}+9\cdot s^{4}\cdot t^{14}+3\cdot s^{3}\cdot t^{15}+3\cdot s^{2}\cdot t^{16}+6\cdot s\cdot t^{17}-2\cdot t^{18})$ $w=-t\cdot(2\cdot s^{18}+6\cdot s^{17}\cdot t-3\cdot s^{16}\cdot t^{2}+3\cdot s^{15}\cdot t^{3}-9\cdot s^{14}\cdot t^{4}-27\cdot s^{13}\cdot t^{5}+32\cdot s^{12}\cdot t^{6}+33\cdot s^{11}\cdot t^{7}-39\cdot s^{10}\cdot t^{8}-27\cdot s^{9}\cdot t^{9}+21\cdot s^{8}\cdot t^{10}+3\cdot s^{7}\cdot t^{11}+2\cdot s^{6}\cdot t^{12}+6\cdot s^{5}\cdot t^{13}-12\cdot s^{4}\cdot t^{14}+3\cdot s^{2}\cdot t^{16}+2\cdot t^{18})$ ## 4\. The algorithm to compute all rational points of Quartic $X^{4}+Y^{4}=Z^{4}+W^{4}$ Consider the formula of the family of curves constructed in this paper: $(2\cdot m\cdot s^{2}-2\cdot n\cdot s^{2}+r\cdot s^{3}-r\cdot s^{2}\cdot t+2\cdot m\cdot t^{2}+2\cdot n\cdot t^{2}-r\cdot s\cdot t^{2}+r\cdot t^{3})\cdot(2\cdot(s+t)\cdot(m^{2}-n\cdot r\cdot s+n\cdot r\cdot t))$ = $(-2\cdot m\cdot s^{2}+2\cdot n\cdot s^{2}+r\cdot s^{3}+r\cdot s^{2}\cdot t+2\cdot m\cdot t^{2}+2\cdot n\cdot t^{2}-r\cdot s\cdot t^{2}-r\cdot t^{3})\cdot(2\cdot(s-t)\cdot(n^{2}-m\cdot r\cdot s-m\cdot r\cdot t))$. We know that this family contains a point that is the image of solution of Leonard Euler for every $s$ and $t$. We proved that for every rational point from Quartica there exist values $m,n,s,t$ , so that the image of this point belong to some cubic curve from the family of cubic curves. It is easy to see that the formula of family curves contains a variable $r$ only in the first and second degree. ALGORITHM (1)Substituting into the formula of the family of curves all possible combinations of values of the variables $m,n,s,t$ every time we get a second degree equation for $r$.If the discriminant of this equation is a full square we get two pairs of solutions on the family of cubic curves and find two solutions on Quartica. Since the variables $m,n,s,t$ run through all possible integers we cover images of all integer points on Quartica, provided that the discriminant of a quadratic equation for $r$ is full square.If the discriminant equation for $r$ not a full square, we shall now proceed to the second step of the algorithm. (2)If the discriminant equation for $r$ is not a full square we get two conjugate solution on cubic curve in a expansion of the second degree for some numerical values of $s$ and $t$.But this curve also contains the image of Euler’s solution for all values of $s$ and $t$ thus for $s$ and $t$ we selected.So we have three points on the cubic curve: a rational point and two conjugate points of expansion of second degree. We take a line through a rational point and one of the two conjugate points. And get a new point on the cubic curve in expansion of second degree. Our cubic curve contains the point conjugate to the point we constructed. Taking a line through new conjugate points we get a rational point on the curve and then find its image on Quartica. EXAMPLE. Consider the family of cubiks constructed in the previous chapter: $(2\cdot m\cdot s^{2}-2\cdot n\cdot s^{2}+r\cdot s^{3}-r\cdot s^{2}\cdot t+2\cdot m\cdot t^{2}+2\cdot n\cdot t^{2}-r\cdot s\cdot t^{2}+r\cdot t^{3})\cdot(2\cdot(s+t)\cdot(m^{2}-n\cdot r\cdot s+n\cdot r\cdot t))$ = $(-2\cdot m\cdot s^{2}+2\cdot n\cdot s^{2}+r\cdot s^{3}+r\cdot s^{2}\cdot t+2\cdot m\cdot t^{2}+2\cdot n\cdot t^{2}-r\cdot s\cdot t^{2}-r\cdot t^{3})\cdot(2\cdot(s-t)\cdot(n^{2}-m\cdot r\cdot s-m\cdot r\cdot t))$. We substitute:$m=1,n=1,s=1,t=13$,receive equation for $r$:$(r-(13\cdot i/84))\cdot(r+(13\cdot i/84))$,the discriminant equation for $r$ is not a complete square,we receive two solutions of cubic curve in variables $m,n,r$ in a expansion of the second degree :$m=1,n=1,r=(13\cdot i/84)$ and $m=1,n=1,r=-(13\cdot i/84)$.Our first point is:$m_{1}=1,n_{1}=1,r_{1}=(13\cdot i/84)$ To receive the second point we must get the image of Euler solution for $s=1,t=13$,to solve the system of equations: $\left\\{\begin{array}[]{clrr}2\cdot m-2\cdot n+r-r\cdot 13+2\cdot m\cdot 13^{2}+2\cdot n\cdot 13^{2}-r\cdot 13^{2}+r\cdot 13^{3}==0,\\\ -2\cdot m+2\cdot n+r+r\cdot 13+2\cdot m\cdot 13^{2}+2\cdot n\cdot 13^{2}-r\cdot 13^{2}-r\cdot 13^{3}==0\end{array}\right.$. Solving this system we receive the second point on the cubic curve:$m_{2}=(-92232/169),n_{2}=92316/169,r_{2}=1$. Now we look for a new point on the following cubic curve :$(m_{1}\cdot k+m_{2}),(n_{1}\cdot k+n_{2}),(r_{1}\cdot k+r_{2})$ When substituting these expressions into the equation of cubic curve we obtain for $k$ the equation of the first degree: $(69723384192/13-7112448\cdot i)-(733824-9766848\cdot i)\cdot k=0$ Solving this equation we obtain: $k=(7056/169+546\cdot i)$ Next, we calculate a new point on the cubic curve in the expansion $i$: $m_{3}=-504+546\cdot i,n_{3}=588+546\cdot i,r_{3}=(-167/2+84\cdot i/13)$.Obviously that point $m_{3}=-504-546\cdot i,n_{3}=588-546\cdot i,r_{3}=(-167/2-84\cdot i/13)$ also belong to the cubic curve. Taking a line through these two points we get a new rational point on the curve. This is the point : $M=(-2450514024/4855033),N=2851182012/4855033$ , $R=(-810875183/9710066)$ If we now calculate a point on the Quartica for a given point on the curve, we get: $31557461^{4}+2941868^{4}=1827989^{4}+31557968^{4}$ the pair of this point is: $324997193816543^{4}+283678931194359^{4}=329177166160259^{4}+277041948785757^{4}$ ## References * [Bi1] H.P.F.Swinnerton-Dyer, Application of algebraic geometry to number theory ,Number theory institute(Proc.Sympos.Pure Math.,Vol XX,State Univ.New York.Stony Brook.N.Y.,1969)Amer.Math.Soc.,Providence,R.I.,1971 4 (1969) pp.1–52. * [Bi2] R.K.Guy., Sums of Like Powers. Euler’s Conjecture” and ”Some Quartic Equations, Unsolved problems of Number theory,New York: Springer-Verlag, , 1994. 265 (1994) pp. 139-144 and 192-193. * [Hr] G.H.Hardy, Ramanujan Twelve lectures on subjects suggested his life and work,3rd.ed., New York: Chelsea. 13 (1999). * [E] R. L. Ekl, New results in equal sums of like powers, Math. Comput. 67 (1998) 1309–1315. * [LP] L. J. Lander and T. R. Parkin, A counterexample to Euler’s sum of powers conjecture, Math. Comput. 21 (1967) 101–103. * [LPS] L. J. Lander, T. R. Parkin, and J. L. Selfridge, A survey of equal sums of like powers, Math. Comput. 21 (1967) 446–459. * [Wa] B. L. van der Waerden, Algebra, volume 1, 49 paragraph 5.7	arxiv-papers	2013-12-17T19:31:58	2024-09-04T02:49:55.697492	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "M.A.Reynya", "submitter": "Misha Arcadii Reynya", "url": "https://arxiv.org/abs/1312.5702" }
1312.5906	# Invariant metrics for the quaternionic Hardy space Nicola Arcozzi, Giulia Sarfatti Partially supported by the PRIN project Real and Complex Manifolds of the Italian MIUR and by INDAM-GNAMPAPartially supported by INDAM-GNSAGA and by the PRIN project Real and Complex Manifolds of the Italian MIUR ###### Abstract We find Riemannian metrics on the unit ball of the quaternions, which are naturally associated with reproducing kernel Hilbert spaces. We study the metric arising from the Hardy space in detail. We show that, in contrast to the one-complex variable case, no Riemannian metric is invariant under regular self-maps of the quaternionic ball. Key words and phrases: Hardy space on the quaternionic ball; functions of a quaternionic variable; invariant Riemannian metric. Mathematics Subject Classification: 30G35, 46E22, 58B20. Notation. The symbol $\mathbb{H}$ denotes the set of the quaternions $q=x_{0}+x_{1}i+x_{2}j+x_{3}k=\operatorname{Re}(q)+\operatorname{Im}(q)$, with $\operatorname{Re}(q)=x_{0}$ and $\operatorname{Im}(q)=x_{1}i+x_{2}j+x_{3}k$; where the $x_{j}$’s are real numbers and the imaginary units $i,j,k$ are subject to the rules $ij=k,\ jk=i,\ ki=j$ and $i^{2}=j^{2}=k^{2}=-1$. We identify the quaternions $q$ whose imaginary part vanishes, $\operatorname{Im}(q)=0$, with real numbers, $\operatorname{Re}(q)\in\mathbb{R}$; and, similarly, we let $\mathbb{I}=\mathbb{R}i+\mathbb{R}j+\mathbb{R}k$ be the set of the imaginary quaternions. The norm $\|q\|\geq 0$ of $q$ is $\|q\|=\sqrt{\sum_{l=0}^{3}x_{l}^{2}}=(q\overline{q})^{1/2}$, where $\overline{q}=x_{0}-x_{1}i-x_{2}j-x_{3}k$ is the conjugate of $q$. The open unit ball $\mathbb{B}$ in $\mathbb{H}$ contains the quaternions $q$ such that $\|q\|<1$. The boundary of $\mathbb{B}$ in $\mathbb{H}$ is denoted by $\partial\mathbb{B}$. By the symbol $\SS$ we denote the unit sphere of the imaginary quaternions: $q\in\mathbb{I}$ belongs to $\SS$ if $\|q\|=1$. For $I$ in $\mathbb{S}$, the slice $L_{I}=L_{-I}$ in $\mathbb{H}$ contains all quaternions having the form $q=x+yI$, with $x,y$ in $\mathbb{R}$. If $f$ is a real differentiable function on a domain $\Omega\subseteq\mathbb{H}$, we denote its real differential at a point $w\in\Omega$ by the symbol $f_{}[w]$. ## 1 Introduction Let $\mathbb{H}$ be the skew-field of the quaternions. The quaternionic Hardy space $H^{2}(\mathbb{B})$ consists of the formal power series of the quaternionic variable $q$, $f(q)=\sum_{n=0}^{\infty}q^{n}a_{n},$ such that the sequence of quaternions $\\{a_{n}\\}$ satisfies $\\|f\\|_{H^{2}(\mathbb{B})}:=\left(\sum_{n=0}^{\infty}\left\|a_{n}\right\|^{2}\right)^{1/2}<\infty.$ (1) It is easily verified that the series converges to a function $f:\mathbb{B}=\\{q\in\mathbb{H}:\ \|q\|<1\\}\to\mathbb{H}$. The function $f$ is slice regular [13] in the sense of Gentili and Struppa, who developed a version of complex function theory which holds in the quaternionic setting. See the monograph [12] for a detailed account of the theory. The norm (1) can be polarized to obtain an inner product with values in the quaternions, $\left\langle\sum q^{n}a_{n},\sum q^{n}b_{n}\right\rangle_{H^{2}(\mathbb{B})}:=\sum_{n=0}^{\infty}\overline{b_{n}}a_{n}.$ The space $H^{2}(\mathbb{B})$ is a reproducing kernel Hilbert space, in the quaternionic sense: for $w$ in $\mathbb{B}$ and $f$ in $H^{2}(\mathbb{B})$ we have $f(w)=\left\langle f,k_{w}\right\rangle_{H^{2}(\mathbb{B})},\text{ where }k_{w}(q)=k(w,q)=\sum_{n=0}^{\infty}q^{n}\overline{w}^{n}.$ There is a rich interplay between reproducing kernel Hilbert spaces and distance functions. See [3] for an overview and several examples from one- variable holomorphic function space theory. In [9] the connection between metric theory and operator theory is analyzed at a very deep level, and the case of the Hardy space is a model example of that. The seminal article [4] by Aronszajn is still an excellent introduction to the theory of reproducing kernel Hilbert spaces. In this article we are mainly interested in studying metrics on $\mathbb{B}$ which are associated with the function space $H^{2}(\mathbb{B})$. We also provide evidence that the metric properties of the space reflect the behavior of functions in $H^{2}(\mathbb{B})$. The first metric we consider measures the distance between projections of kernel functions in the unit sphere of the Hilbert space $H^{2}(\mathbb{B})$: $\delta(p,q):=\sqrt{1-\left\|\left\langle\frac{k_{q}}{\\|k_{q}\\|_{H^{2}(\mathbb{B})}},\frac{k_{p}}{\\|k_{p}\\|_{H^{2}(\mathbb{B})}}\right\rangle_{H^{2}(\mathbb{B})}\right\|^{2}}.$ (2) In the holomorphic case of $H^{2}(\Delta)$ one obtains this way the pseudo- hyperbolic metric $\delta^{\prime}(z,w)=\left\|\frac{z-w}{1-\overline{w}z}\right\|$. A calculation, see Proposition 4.2 below, gives a formally similar result in the quaternionic case: $\delta(z,w)=\|(1-q\overline{w})^{-}(q-w)\|_{\|_{q=z}},$ for $z,w$ in $\mathbb{B}$. Here, the product $f(q)g(q)$ and the multiplicative inverse $f(q)^{-}$ are not pointwise product and pointwise inverse: they are $$-product and $$-inverse, which are defined so that the usual convolution rule for coefficients of power series’ products holds. See [12], and Section 2 where we summarize some background material on slice regular functions. The infinitesimal version of the pseudo-hyperbolic metric in the complex disc, is the hyperbolic metric in the Riemann-Beltrami-Poincaré disc model: $ds^{2}=\frac{\|dz\|^{2}}{(1-\|z\|^{2})^{2}}$. By infinitesimal version of a distance $\delta_{0}$, we mean the length metric associated with $\delta_{0}$ (see e.g. [15]). The infinitesimal metric associated with $\delta$ is a Riemannian metric $g$ on $\mathbb{B}$. In Theorem 3.2 a formula is produced, which works for a wide class of reproducing kernel quaternionic Hilbert spaces. Here is the special case of the quaternionic Hardy space. ###### Theorem 1.1. (I) The length metric associated with (2) is the Riemannian metric $g$ defined below. For any $w\in\mathbb{B}$, let us identify the tangent space $T_{w}\mathbb{B}$ with $\mathbb{H}$. For any vector $d\in T_{w}\mathbb{B}$, where $w=x+yI_{w}$ lies in $L_{I_{w}}$, we decompose $d=d_{1}+d_{2}$ with $d_{1}$ in $L_{I_{w}}$ and $d_{2}$ in $L_{I_{w}}^{\perp}$, the orthogonal complement of $L_{I_{w}}$ with respect to the Euclidean metric in $\mathbb{H}$. The length of $d$ with respect to $g$ is: $\|d\|_{g(w)}^{2}=\frac{1}{(1-\|w\|^{2})^{2}}\|d_{1}\|^{2}+\frac{1}{\|1-w^{2}\|^{2}}\|d_{2}\|^{2}.$ (3) (II) The isometry group of $g$ is the one generated by the following classes of self-maps of $\mathbb{B}$: (a) regular Möbius transformations of the form $q\mapsto M_{\lambda}(q)=(1-q\lambda)^{-}(q-\lambda)=\frac{q-\lambda}{1-\lambda q},$ with $\lambda$ in $(-1,1)$; * (b) isometries of the sphere of the imaginary units, $q=x+yI\mapsto T_{A}(q)=x+yA(I),$ where $A:\SS\to\SS$ is an isometry of $\,\SS$; * (c) the reflection in the imaginary hyperplane, $q\mapsto R(q)=-\overline{q}.$ In the metric (3), the first, “large” summand is the hyperbolic metric on a slice, while the second “small” summand is peculiarly quaternionic: it measures the variation of a quaternionic Hardy function in the direction orthogonal to the slices. Its small size reflects in quantitative, geometric terms the fact that regular functions are affine in the $\SS$ variable, see [12]. The special rôle of the real axis in the theory of slice regular functions is here reflected in the fact that all isometries of the metric $g$ fix $\mathbb{R}\cap\mathbb{B}$. In particular, contrary to the case of the complex disc, the action is not transitive. Other, more precise, properties of the metric will be stated and proved on route to the proof of Theorem 1.1. We will study geodesics, geodesically complete submanifolds and other geometric properties of the metric. For instance, we will prove that the radius of injectivity is infinite at points of the real diameter of $\mathbb{B}$, and finite elsewhere. The metric has neither positive, nor negative sectional curvature. The proof of Theorem 1.1 is split into two steps. In Theorem 3.2 we will compute the Riemannian metric associated with rather general reproducing kernel quaternionic Hilbert spaces $\mathcal{H}$; that is the length metric associated with the distance function $\delta_{\mathcal{H}}(w,z)=\sqrt{1-\left\|\left\langle\frac{k_{w}}{\\|k_{w}\\|_{\mathcal{H}}},\frac{k_{z}}{\\|k_{z}\\|_{\mathcal{H}}}\right\rangle_{\mathcal{H}}\right\|^{2}}.$ We restrict most of our analysis to spaces of functions defined on symmetric slice domanis in $\mathbb{H}$, with a slice preserving reproducing kernel. Examples are the Hardy space on $\mathbb{B}$ and on the right half-space $\mathbb{H}^{+}=\\{q\in\mathbb{H}\,\|\,\operatorname{Re}(q)>0\\}$ and the Bergman space on $\mathbb{B}$. The classification of the isometries is carried out in Section 4. The Riemannian metric $g$ has a rather small group of isometries, compared with the state of things in the unit disc of the complex plane, or even in the unit ball in several complex variables, with the Bergman-Kobayashi metric. The latter metrics have a transitive group of isometries and, more, the space is isotropic; whereas all isometries of the former have to fix the real line. One might expect that something better is possible. Unfortunately, there is no Riemannian metric on $\mathbb{B}$ which is invariant under regular Möbius functions and which is “democratic” with respect to the sphere of imaginary units. If a geometric invariant for slice regular functions on the quaternionic ball exists, it has to be other than a Riemannian metric. ###### Theorem 1.2. There is no Riemannian metric $m$ on $\mathbb{B}$ having as isometries: * (i) regular Möbius transformations $q\mapsto(1-q\overline{a})^{-}(q-a)u$, with $a$ in $\mathbb{B}$ and $\|u\|=1$; * (ii) maps of the form $q=x+yI\mapsto x+yA(I)$, with $A$ in $O(3)$, the orthogonal group of $\mathbb{R}^{3}$. The proof will be given in Subsection 4.3. We mention here that Bisi and Gentili proved in [5] that the usual Poincaré metric on $\mathbb{B}$ is invariant under classical (non-regular) Möbius maps. A first relationship between the space $H^{2}(\mathbb{B})$ and the metric $g$ concerns the $H^{2}$ norm itself. Let $r\SS^{3}$ be the sphere of radius $0<r<1$ in $\mathbb{B}$, with respect to the usual Euclidean metric; containing quaternions $q=re^{tI}$, with $I$ in $\SS$ and $t$ in $[0,\pi]$. The restriction of $g$ to $r\SS^{3}$ induces a volume form $dVol_{r\mathbb{S}^{3}}$. Let $f$ be in $H^{2}(\mathbb{B})$. Then, $\\|f\\|_{H^{2}(\mathbb{B})}^{2}=\lim_{r\to 1}(1-r^{2})\frac{1}{Vol_{r\mathbb{S}^{3}}(r\mathbb{S}^{3})}\int_{r\mathbb{S}^{3}}\|f\|^{2}dVol_{r\mathbb{S}^{3}},$ a relation similar to the definition of the Hardy norm in the unit disc by means of the Poincaré metric. In Section 5 we use the Caley map $C:q\mapsto(1-q)^{-1}(1+q)$ to write down the metric $g$ in coordinates living in right half-space $\mathbb{H}^{+}:=C(\mathbb{B})=\\{q\in\mathbb{H}\,\|\,\operatorname{Re}(q)>0\\}$. This makes it easier to prove a bilateral estimate for the distance function associated with $g$, Theorem 5.2. As an application, in Theorem 5.4 we further investigate the “rigidity” of the metric $g$, by showing that the only inner functions which are Lipschitz continuous with respect to $g$ have to be be slice preserving. In particular, they have to fix the real diameter of $\mathbb{B}$. A function defined from $\Omega\subseteq\mathbb{H}$ to $\mathbb{H}$ is slice preserving if it maps $L_{I}\cap\Omega$ to $L_{I}$ for all $I$ in $\SS$. We also consider in Subsection 5.1 four equivalent definitions of the Hardy space $H^{2}(\mathbb{H}^{+})$ on $\mathbb{H}^{+}$. First, functions $f$ in $H^{2}(\mathbb{H}^{+})$ might be characterized, pretty much as in the one- dimensional complex case, as inverse Fourier transforms -in the quaternionic sense- of functions $F:[0,\infty)\to\mathbb{H}$ with finite $L^{2}$-norm $\\|F\\|_{L^{2}}=\left(\int_{0}^{\infty}\|F(\zeta)\|^{2}d\zeta\right)^{1/2}.$ Equivalently, $H^{2}(\mathbb{H}^{+})$ is the Hilbert space having reproducing kernel $k_{w}(q)=(q+\overline{w})^{-}$. This second viewpoint has the advantage of relating $H^{2}(\mathbb{H}^{+})$ and $H^{2}(\mathbb{B})$. We show in fact in Proposition 5.1 that the reproducing kernel for $H^{2}(\mathbb{H}^{+})$ is a rescaling of the reproducing kernel for $H^{2}(\mathbb{B})$: $k_{H^{2}(\mathbb{B})}(C^{-1}(w),C^{-1}(z))=\frac{1}{2}(1+z)k_{H^{2}(\mathbb{H}^{+})}(w,z)(1+\overline{w}).$ Here we use the symbols $k_{H^{2}(\mathbb{B})}$ and $k_{H^{2}(\mathbb{H}^{+})}$ for the reproducing kernels on $\mathbb{B}$ and $\mathbb{H}^{+}$, respectively. Hence, third, the functions in $H^{2}(\mathbb{H}^{+})$ might be defined as the rescaled versions of functions in $H^{2}(\mathbb{B})$. Fourth, the norm of $f$ in $H^{2}(\mathbb{H}^{+})$ can be computed as the limit of the integrals of $\|f\|^{2}$ on “horocycles” in $H^{2}(\mathbb{H}^{+})$, when these are endowed with the natural volume form induced by the metric $g$. The space $H^{2}(\mathbb{H}^{+})$ was defined in [2], using a fifth definition, which is shown to give rise to the same reproducing kernel. Our contribution here is mainly relating $H^{2}(\mathbb{H}^{+})$ and the geometry of $\mathbb{H}^{+}$. ## 2 Preliminaries We recall the definition of slice regularity, together with some basic results that hold for this class of functions. We refer to the book [12] for all details and proofs. Let $\mathbb{H}$ denote the four-dimensional (non- commutative) real algebra of quaternions and let $\mathbb{S}$ denote the two- dimensional sphere of imaginary units of $\mathbb{H}$, $\mathbb{S}=\\{q\in\mathbb{H}\,\|\,q^{2}=-1\\}$. One can “slice” the space $\mathbb{H}$ in copies of the complex plane that intersect along the real axis, $\mathbb{H}=\bigcup_{I\in\mathbb{S}}(\mathbb{R}+\mathbb{R}I),\hskip 28.45274pt\mathbb{R}=\bigcap_{I\in\mathbb{S}}(\mathbb{R}+\mathbb{R}I),$ where $L_{I}:=\mathbb{R}+\mathbb{R}I\cong\mathbb{C}$, for any $I\in\mathbb{S}$. Then, each element $q\in\mathbb{H}$ can be expressed as $q=x+yI_{q}$, where $x,y$ are real (if $q\in\mathbb{R}$, then $y=0$) and $I_{q}$ is an imaginary unit. Let $\Omega\subseteq\mathbb{H}$ be a subset of $\mathbb{H}$. For any $I\in\mathbb{S}$, we will denote by $\Omega_{I}$ the intersection $\Omega\cap L_{I}$. We can now recall the definition of slice regular functions, in the sequel simply called regular functions. ###### Definition 2.1. Let $\Omega$ be a domain (open connected subset) in $\mathbb{H}$. A function $f:\Omega\to\mathbb{H}$ is said to be (slice) regular if for any $I\in\mathbb{S}$ the restriction $f_{I}$ of $f$ to $\Omega_{I}$ has continuous partial derivatives and it is such that $\overline{\partial}_{I}f_{I}(x+yI)=\frac{1}{2}\left(\frac{\partial}{\partial x}+I\frac{\partial}{\partial y}\right)f_{I}(x+yI)=0$ for all $x+yI\in\Omega_{I}$. A wide class of examples of regular functions is given by power series with quaternionic coefficients of the form $\sum^{\infty}_{n=0}q^{n}a_{n}$ which converge in open balls centered at the origin. ###### Theorem 2.2. A function $f$ is regular on $B(0,R)=\\{q\in\mathbb{H}\,\|\,\|q\|<R\\}$ if and only if $f$ has a power series expansion $f(q)=\sum^{\infty}_{n=0}q^{n}a_{n}$ converging in $B(0,R)$. For regular functions, it is possible to define an appropriate notion of derivative: ###### Definition 2.3. Let $f$ be a regular function on a domain $\Omega\subseteq\mathbb{H}$. The slice (or Cullen) derivative of $f$ is the regular function defined as $\partial_{c}f(x+yI)=\frac{1}{2}\left(\frac{\partial}{\partial x}-I\frac{\partial}{\partial y}\right)f_{I}(x+yI).$ We will consider domains in certain restricted classes. ###### Definition 2.4. Let $\Omega\subseteq\mathbb{H}$ be a domain. 1. 1. $\Omega$ is called a slice domain if it intersects the real axis and if, for any $I\in\mathbb{S}$, $\Omega_{I}$ is a domain in $L_{I}$. 2. 2. $\Omega$ is called a symmetric domain if for any point $x+yI\in\Omega$, with $x,y\in\mathbb{R}$ and $I\in\mathbb{S}$, the entire two-sphere $x+y\mathbb{S}$ is contained in $\Omega$. The ball $\mathbb{B}$ and the right half-space $\mathbb{H}^{+}=\\{q=x+Iy:\ I\in\SS,\ x>0,\ y\in\mathbb{R}\\}$ are symmetric slice domains. Slice regular functions defined on symmetric slice domains have a peculiar property. ###### Theorem 2.5 (Representation Formula). Let $f$ be a regular function on a symmetric slice domain $\Omega$ and let $x+y\mathbb{S}\subset\Omega$. Then, for any $I,J\in\mathbb{S}$, $f(x+yJ)=\frac{1}{2}[f(x+yI)+f(x-yI)]+J\frac{I}{2}[f(x-yI)-f(x+yI)].$ In particular, there exist $b,c\in\mathbb{H}$ such that $f(x+yJ)=b+Jc$ for any $J\in\mathbb{S}$. When restricted to a sphere of the form $x+y\mathbb{S}$, a regular function is actually affine in the variable $q$. This nice geometric property leads to the following definition ###### Definition 2.6. Let $f$ be a regular function on a symmetric slice domain $\Omega$. The spherical derivative of $f$ is defined as $\partial_{s}f(q)=(q-\overline{q})^{-1}\left(f(q)-f(\overline{q})\right).$ A basic result that establishes a relation between regular functions and holomorphic functions of one complex variable is the following. ###### Lemma 2.7 (Splitting Lemma). Let $f$ be a regular function on a slice domain $\Omega\subseteq\mathbb{H}$. Then for any $I\in\mathbb{S}$ and for any $J\in\mathbb{S}$, $J\perp I$ there exist two holomorphic functions $F,G:\Omega_{I}\to L_{I}$ such that $f(x+yI)=F(x+yI)+G(x+yI)J$ for any $x+yI\in\Omega_{I}$. In general, the pointwise product of functions does not preserve slice regularity. It is possible to introduce a new multiplication operation, which, in the special case of power series, can be defined as follows. ###### Definition 2.8. Let $f(q)=\sum_{n=0}^{\infty}q^{n}a_{n},$ and $g(q)=\sum_{n=0}^{\infty}q^{n}b_{n}$ be regular functions on $B(0,R)$. Their regular product (or $$-product) is $fg(q)=\sum_{n\geq 0}q^{n}\sum_{k=0}^{n}a_{k}b_{n-k},$ regular on $B(0,R)$ as well. The $$-product is related to the standard pointwise product by the following formula. ###### Proposition 2.9. Let $f,g$ be regular functions on a symmetric slice domain $\Omega$. Then $fg(q)=\left\\{\begin{array}[]{l r}0&\text{if $f(q)=0$}\\\ f(q)g(f(q)^{-1}qf(q))&\text{if $f(q)\neq 0$}\end{array}\right.$ The reciprocal $f^{-}$ of a regular function $f$ with respect to the $$-product can be defined. ###### Definition 2.10. Let $f(q)=\sum_{n=0}^{\infty}q^{n}a_{n}$ be a regular function on $B(0,R)$, $f\not\equiv 0$. Its regular reciprocal is $f^{-}(q)=\frac{1}{ff^{c}(q)}f^{c}(q),$ where $f^{c}(q)=\sum_{n=0}^{\infty}q^{n}\overline{a}_{n}$. The function $f^{-}$ is regular on $B(0,R)\setminus\\{q\in B(0,R)\ \|\ ff^{c}(q)=0\\}$ and $ff^{-}=1$ there. For example, in the case of the reproducing kernel for the quaternionic Hardy space $H^{2}(\mathbb{B})$, we have ###### Remark 2.11. The reproducing kernel for $H^{2}(\mathbb{B})$ is $k_{w}(q)=\sum_{n=0}^{\infty}q^{n}\overline{w}^{n}=(1-q\overline{w})^{-}.$ Then we have a natural definition of _regular quotients_ of regular functions, which satisfy ###### Proposition 2.12. Let $f$ and $g$ be regular functions on a symmetric slice domain $\Omega$ and denote by $Z=\\{q\in\Omega\ \|\ ff^{c}(q)=0\\}$. If $T_{f}:\Omega\setminus Z\rightarrow\Omega\setminus$ is defined as $T_{f}(q)=f^{c}(q)^{-1}qf^{c}(q),$ then $f^{-}g(q)=f(T_{f}(q))^{-1}g(T_{f}(q))\quad\text{for every}\quad q\in\Omega\setminus Z_{f^{s}}.$ Important examples of regular quotients that will appear in the sequel are the regular Möbius transformations, of the form $M_{a}(q)=(1-q\overline{a})^{-}(q-a),$ where $a\in\mathbb{B}$, which are regular self-maps of the quaternionic unit ball $\mathbb{B}$. After multiplication on the right by unit-norm quaternions, they are the only self-maps of $\mathbb{B}$ which are regular, with regular inverse. They were introduced by Stoppato in [18]. See also [12]. ## 3 Metrics associated with quaternionic reproducing kernel Hilbert spaces Let $\Omega\subseteq\mathbb{H}$ be a symmetric slice domain and let $\mathcal{H}$ be a reproducing kernel Hilbert space of regular functions on $\Omega$. For the definition and all basic results concerning quaternionic Hilbert spaces see, e.g., [14] and references therein. For the properties we are interested in, the same results hold in quaternion valued Hilbert spaces and complex valued Hilbert spaces, and the proofs are very similar. It is possible to define a metric $\delta_{\mathcal{H}}$ on $\Omega$ in terms of the distance between projections of kernel functions in the unit sphere of the Hilbert space $\mathcal{H}$. Namely, if $k(w,q)=k_{w}(q)$ denotes the reproducing kernel of $\mathcal{H}$, then $\delta_{\mathcal{H}}:\Omega\times\Omega\to\mathbb{R}^{+}$ can be defined as $\delta_{\mathcal{H}}(w,z)=\sqrt{1-\left\|\left\langle\frac{k_{w}}{\\|k_{w}\\|_{\mathcal{H}}},\frac{k_{z}}{\\|k_{z}\\|_{\mathcal{H}}}\right\rangle_{\mathcal{H}}\right\|^{2}}.$ (4) ###### Proposition 3.1. Let $\Omega$ be a symmetric slice domain and let $\mathcal{H}$ be a reproducing kernel Hilbert space of regular functions on $\Omega$. Let $w\in\Omega\cap L_{I_{w}}$ and let $d\in\mathbb{H}$ be such that $w+d\in\Omega$. Consider the decomposition $d=d_{1}+d_{2}$, where $d_{1}\in L_{I_{w}}$ and $d_{2}\in L_{I_{w}}^{\perp}$. Then $\delta^{2}_{\mathcal{H}}(w,w+d)=\frac{\\|k_{w}\\|_{\mathcal{H}}^{2}\left\\|\overline{d_{1}\partial_{c}k_{q}(w)+d_{2}\partial_{s}k_{q}(w)}\right\\|_{\mathcal{H}}^{2}-\left\|\left\langle k_{w},\overline{d_{1}\partial_{c}k_{q}(w)+d_{2}\partial_{s}k_{q}(w)}\right\rangle_{\mathcal{H}}\right\|^{2}}{\\|k_{w}\\|_{\mathcal{H}}^{4}}+O(\|d\|^{2}).$ ###### Proof. Recalling the definition (4) of $\delta_{\mathcal{H}}$, we get $\delta^{2}_{\mathcal{H}}(w,w+d)=\frac{\\|k_{w}\\|^{2}_{\mathcal{H}}\\|k_{w+d}\\|^{2}_{\mathcal{H}}-\left\|\left\langle k_{w},k_{w+d}\right\rangle_{\mathcal{H}}\right\|^{2}}{\\|k_{w}\\|^{2}_{\mathcal{H}}\\|k_{w+d}\\|^{2}_{\mathcal{H}}}.$ (5) We want to have a better description of the numerator of (5). Using the properties of the kernel functions and the fact that regular functions are real analytic functions of $4$ real variables, we can write $k_{w+d}(q)-k_{w}(q)=\overline{k_{q}(w+d)-k_{q}(w)}=\overline{(k_{q})_{}[w](d)}+O(\|d\|^{2})$ where $(k_{q})_{}[w](d)$ denotes the real differential of $k_{q}$ at the point $w$, applied to the vector $d$. We identify here the tangent space $T_{w}\Omega$ with $\mathbb{H}$. Thanks to the decomposition properties of the real differential of regular functions in terms of slice and spherical derivatives, see Remark 8.15 in [12], we have $k_{w+d}(q)-k_{w}(q)=\overline{d_{1}\partial_{c}k_{q}(w)+d_{2}\partial_{s}k_{q}(w)}+O(\|d\|^{2}),$ hence, $\\|k_{w+d}\\|_{\mathcal{H}}^{2}=\\|k_{w}\\|_{\mathcal{H}}^{2}+\left\\|\overline{d_{1}\partial_{c}k_{q}(w)+d_{2}\partial_{s}k_{q}(w)}\right\\|_{\mathcal{H}}^{2}+2\operatorname{Re}\left\langle k_{w},\overline{d_{1}\partial_{c}k_{q}(w)+d_{2}\partial_{s}k_{q}(w)}\right\rangle_{\mathcal{H}}+O(\|d\|^{2})$ and $\displaystyle\left\|\left\langle k_{w},k_{w+d}\right\rangle_{\mathcal{H}}\right\|^{2}=\left\|\\|k_{w}\\|_{\mathcal{H}}^{2}+\left\langle k_{w},\overline{d_{1}\partial_{c}k_{q}(w)+d_{2}\partial_{s}k_{q}(w)}\right\rangle_{\mathcal{H}}\right\|^{2}+O(\|d\|^{2})$ $\displaystyle=\\|k_{w}\\|_{\mathcal{H}}^{4}+\left\|\left\langle k_{w},\overline{d_{1}\partial_{c}k_{q}(w)+d_{2}\partial_{s}k_{q}(w)}\right\rangle_{\mathcal{H}}\right\|^{2}$ $\displaystyle\hskip 170.71652pt+2\\|k_{w}\\|_{\mathcal{H}}^{2}\operatorname{Re}\left\langle k_{w},\overline{d_{1}\partial_{c}k_{q}(w)+d_{2}\partial_{s}k_{q}(w)}\right\rangle_{\mathcal{H}}+O(\|d\|^{2}).$ Therefore $\delta^{2}_{\mathcal{H}}(w,w+d)=\frac{\\|k_{w}\\|_{\mathcal{H}}^{2}\left\\|\overline{d_{1}\partial_{c}k_{q}(w)+d_{2}\partial_{s}k_{q}(w)}\right\\|_{\mathcal{H}}^{2}-\left\|\left\langle k_{w},\overline{d_{1}\partial_{c}k_{q}(w)+d_{2}\partial_{s}k_{q}(w)}\right\rangle_{\mathcal{H}}\right\|^{2}}{\\|k_{w}\\|_{\mathcal{H}}^{4}}+O(\|d\|^{2}).$ ∎ Proposition 3.1 reflects what happens in the complex case, see [16]. In fact the functions $\overline{\partial_{c}k_{q}(w)}$ and $\overline{\partial_{s}k_{q}(w)}$ are regular with respect to the variable $q$, and they reproduce respectively the slice and the spherical derivative of any regular function $f:\Omega\to\mathbb{H}$. In fact, for any $w\in\Omega_{I_{w}}$, if $h\in L_{I_{w}}$, we can write $\displaystyle\partial_{c}f(w)=\lim_{h\to 0,\,h\in L_{I_{w}}}h^{-1}(f(w+h)-f(w))=\lim_{h\to 0,\,h\in L_{I_{w}}}h^{-1}\left(\left\langle f,k_{w+h}\right\rangle_{\mathcal{H}}-\left\langle f,k_{w}\right\rangle_{\mathcal{H}}\right)$ $\displaystyle=\lim_{h\to 0,\,h\in L_{I_{w}}}h^{-1}\left\langle f,\overline{k_{q}(w+h)-k_{q}(w)}\right\rangle_{\mathcal{H}}=\lim_{h\to 0,\,h\in L_{I_{w}}}h^{-1}\left\langle f,\overline{h\partial_{c}k_{q}(w)}\right\rangle_{\mathcal{H}}=\left\langle f,\overline{\partial_{c}k_{q}(w)}\right\rangle_{\mathcal{H}}$ and $\displaystyle\partial_{s}f(w)$ $\displaystyle=(w-\overline{w})^{-1}(f(w)-f(\overline{w}))=(w-\overline{w})^{-1}\left(\langle f,k_{w}\rangle_{\mathcal{H}}-\langle f,k_{\overline{w}}\rangle_{\mathcal{H}}\right)$ $\displaystyle=(w-\overline{w})^{-1}\left\langle f,k_{w}-k_{\overline{w}}\right\rangle_{\mathcal{H}}=(w-\overline{w})^{-1}\left\langle f,\overline{(w-\overline{w})\partial_{s}k_{q}(w)}\right\rangle_{\mathcal{H}}=\left\langle f,\overline{\partial_{s}k_{q}(w)}\right\rangle_{\mathcal{H}}.$ Proposition 3.1 allows us to define a Riemannian metric $g_{\mathcal{H}}$ on the symmetric slice domain $\Omega$. For each point $w\in\Omega$, let us identify the tangent space $T_{w}\Omega$ with $\mathbb{H}=L_{I_{w}}+L_{I_{w}}^{\perp}$. Then the length of a tangent vector $d=d_{1}+d_{2}\in L_{I_{w}}+L_{I_{w}}^{\perp}$ is $\|d\|^{2}_{{g_{\mathcal{H}}}(w)}=\frac{\\|k_{w}\\|_{\mathcal{H}}^{2}\left\\|\overline{d_{1}\partial_{c}k_{q}(w)+d_{2}\partial_{s}k_{q}(w)}\right\\|_{\mathcal{H}}^{2}-\left\|\left\langle k_{w},\overline{d_{1}\partial_{c}k_{q}(w)+d_{2}\partial_{s}k_{q}(w)}\right\rangle_{\mathcal{H}}\right\|^{2}}{\\|k_{w}\\|_{\mathcal{H}}^{4}}.$ (6) ###### Theorem 3.2. Let $\Omega$ be a symmetric slice domain and let $\mathcal{H}$ be a reproducing kernel Hilbert space of regular functions on $\Omega$. Suppose that $k$ is slice preserving: for any $w\in\Omega$ the kernel function $k_{w}$ preserves the slice $L_{I_{w}}$ identified by $w$. Then the length of a tangent vector $d=d_{1}+d_{2}\in L_{I_{w}}+L_{I_{w}}^{\perp}\cong T_{w}\Omega$ with respect to the Riemannian metric $g_{\mathcal{H}}$ associated with $\mathcal{H}$ is given by $\|d\|^{2}_{{g_{\mathcal{H}}}(w)}=\frac{\left(\\|k_{w}\\|_{\mathcal{H}}^{2}\left\\|\overline{\partial_{c}k_{q}(w)}\right\\|_{\mathcal{H}}^{2}-\left\|\overline{\partial_{c}k_{w}(w)}\right\|^{2}\right)}{\\|k_{w}\\|_{\mathcal{H}}^{4}}\|d_{1}\|^{2}+\frac{\left(\\|k_{w}\\|_{\mathcal{H}}^{2}\left\\|\overline{\partial_{s}k_{q}(w)}\right\\|_{\mathcal{H}}^{2}-\left\|\overline{\partial_{s}k_{w}(w)}\right\|^{2}\right)}{\\|k_{w}\\|_{\mathcal{H}}^{4}}\|d_{2}\|^{2}.$ ###### Proof. We begin by working out the numerator in equation (6). We have: $\displaystyle\left\\|\overline{d_{1}\partial_{c}k_{q}(w)+d_{2}\partial_{s}k_{q}(w)}\right\\|_{\mathcal{H}}^{2}=\left\langle\overline{d_{1}\partial_{c}k_{q}(w)+d_{2}\partial_{s}k_{q}(w)},\overline{d_{1}\partial_{c}k_{q}(w)+d_{2}\partial_{s}k_{q}(w)}\right\rangle_{\mathcal{H}}$ $\displaystyle=\left\\|\overline{\partial_{c}k_{q}(w)}\right\\|_{\mathcal{H}}^{2}\|d_{1}\|^{2}+\left\\|\overline{\partial_{s}k_{q}(w)}\right\\|_{\mathcal{H}}^{2}\|d_{2}\|^{2}+2\operatorname{Re}\left\langle\overline{d_{1}\partial_{c}k_{q}(w)},\overline{d_{2}\partial_{s}k_{q}(w)}\right\rangle_{\mathcal{H}}$ and $\displaystyle\left\|\left\langle k_{w},\overline{d_{1}\partial_{c}k_{q}(w)+d_{2}\partial_{s}k_{q}(w)}\right\rangle_{\mathcal{H}}\right\|^{2}=\left\|\overline{d_{1}\partial_{c}k_{w}(w)+d_{2}\partial_{s}k_{w}(w)}\right\|^{2}$ $\displaystyle=\left\|\overline{\partial_{c}k_{w}(w)}\right\|^{2}\|d_{1}\|^{2}+\left\|\overline{\partial_{s}k_{w}(w)}\right\|^{2}\|d_{2}\|^{2}+2\operatorname{Re}\left(\overline{d_{1}\partial_{c}k_{w}(w)}d_{2}\partial_{s}k_{w}(w)\right).$ Hence we are left to prove that both $\operatorname{Re}\left\langle\overline{d_{1}\partial_{c}k_{q}(w)},\overline{d_{2}\partial_{s}k_{q}(w)}\right\rangle_{\mathcal{H}}=\operatorname{Re}\left(d_{2}\left\langle\overline{\partial_{c}k_{q}(w)},\overline{\partial_{s}k_{q}(w)}\right\rangle_{\mathcal{H}}\overline{d_{1}}\right)$ and $\operatorname{Re}\left(\overline{d_{1}\partial_{c}k_{w}(w)}d_{2}\partial_{s}k_{w}(w)\right)=\operatorname{Re}\left(d_{2}\partial_{s}k_{w}(w)\overline{\partial_{c}k_{w}(w)}\,\overline{d_{1}}\right)$ equal zero. Now notice that if $k_{w}$ maps $\Omega_{I_{w}}$ to $L_{I_{w}}$, the same holds true for both $\partial_{c}k_{w}$ and $\partial_{s}k_{w}$. The fact that $d_{1}\in L_{I_{w}}$ and $d_{2}\in L_{I_{w}}^{\perp}$ leads us to conclude. ∎ The hypothesis about kernel functions required in Theorem 3.2 is satisfied by the quaternionic analogues of Hardy and Bergman spaces; see [1, 8]. ## 4 Invariant metrics associated with the Hardy space $H^{2}(\mathbb{B})$ In this section, we turn our attention to the special example of the Hardy space $H^{2}(\mathbb{B})$. We will study the corresponding Riemannian metric $g:=g_{H^{2}(\mathbb{B})}$. Recalling that for any $w$ the kernel function $k_{w}(q)=\sum^{\infty}_{n=0}q^{n}\overline{w}^{n}$ preserves the slice $L_{I_{w}}$, we can directly apply Theorem 3.2 to find the expression of $g$, thus proving the first part of Theorem 1.1. ###### Proposition 4.1. For any $w\in\mathbb{B}$, let us identify the tangent space $T_{w}\mathbb{B}$ with $\mathbb{H}$. For any vector $d\in T_{w}\mathbb{B}$, if $w$ lies in $L_{I_{w}}$ and we decompose $d=d_{1}+d_{2}$ with $d_{1}$ in $L_{I_{w}}$ and $d_{2}$ in $L_{I_{w}}^{\perp}$, then the length of $d$ with respect to $g$ is given by $\|d\|_{g(w)}^{2}=\frac{1}{(1-\|w\|^{2})^{2}}\|d_{1}\|^{2}+\frac{1}{\|1-w^{2}\|^{2}}\|d_{2}\|^{2}.$ (7) ###### Proof. The following equalities can, by their nature, be reduced to simple calculations in the complex plane: $\\|k_{w}\\|_{H^{2}(\mathbb{B})}^{2}=\frac{1}{1-\|w\|^{2}},\quad\left\|\overline{\partial_{c}k_{w}(w)}\right\|^{2}=\frac{\|w\|^{2}}{(1-\|w\|^{2})^{4}},\quad\left\|\overline{\partial_{s}k_{w}(w)}\right\|^{2}=\frac{\|w\|^{2}}{(1-\|w\|^{2})^{2}\|1-w^{2}\|^{2}},$ $\left\\|\overline{\partial_{c}k_{q}(w)}\right\\|_{H^{2}(\mathbb{B})}^{2}=\Big{\\|}\sum_{n\geq 0}n^{2}q^{n}\overline{w}^{n-1}\Big{\\|}_{H^{2}(\mathbb{B})}^{2}=\frac{1+\|w\|^{2}}{(1-\|w\|^{2})^{3}},$ $\left\\|\overline{\partial_{s}k_{q}(w)}\right\\|_{H^{2}(\mathbb{B})}^{2}=\frac{1}{\|w-\overline{w}\|^{2}}\left(\frac{2}{1-\|w\|^{2}}-\frac{1}{1-w^{2}}-\frac{1}{1-\overline{w}^{2}}\right).$ A direct application of Theorem 3.2, then, yields that, with respect to coordinates $(d_{1},d_{2})\in(L_{I_{w}},L_{I_{w}}^{\perp})$, $\|d\|^{2}_{g(w)}=\frac{1}{(1-\|w\|^{2})^{2}}\|d_{1}\|^{2}+\frac{1}{\|1-w^{2}\|^{2}}\|d_{2}\|^{2}.$ ∎ The volume form $dVol_{g}$ associated with the metric $g$ at any point $w=x_{0}+x_{1}i+x_{2}j+x_{3}k\in\mathbb{B}$ is then $dVol_{g}(w)=\frac{dVol_{Euc}(w)}{(1-\|w\|^{2})^{2}\|1-w^{2}\|^{2}},$ where $dVol_{Euc}(w)=dx_{0}dx_{1}dx_{2}dx_{3}$ is the usual Euclidean volume element. ###### Proposition 4.2. Let $\delta:=\delta_{H^{2}(\mathbb{B})}$ be defined as in (4). For any $w,z\in\mathbb{B}$, $\delta(z,w)$ coincides both with the value at $z$ of the regular Möbius transformation $M_{w}$ associated with $w$ and with the vaule at $w$ of the regular Möbius transformation $M_{z}$ associated with $z$, namely $\delta(w,z)=\left\|(1-q\overline{z})^{-}(q-z)\right\|_{\|_{q=w}}=\left\|(1-q\overline{w})^{-}(q-w)\right\|_{\|_{q=z}}.$ ###### Proof. Let $w,z$ be two points in $\mathbb{B}$. By Proposition 2.12, $\|\langle k_{w},k_{z}\rangle_{H^{2}(\mathbb{B})}\|=\|k_{w}(z)\|=\|(1-q\overline{w})^{-}\|_{\|_{q=z}}=\|1-\hat{z}\overline{w}\|^{-1}$ where $\hat{z}=(1-zw)^{-1}z(1-zw)$, which implies $\left\|\left\langle\frac{k_{w}}{\\|k_{w}\\|_{H^{2}(\mathbb{B})}},\frac{k_{z}}{\\|k_{z}\\|_{H^{2}(\mathbb{B})}}\right\rangle_{H^{2}(\mathbb{B})}\right\|^{2}=\|1-\hat{z}\overline{w}\|^{-2}\left(1-\|w\|^{2}\right)\left(1-\|z\|^{2}\right).$ Thus, since $\|\hat{z}\|=\|z\|$, we get $\displaystyle\delta^{2}(w,z)$ $\displaystyle=1-\left\|\left\langle\frac{k_{w}}{\\|k_{w}\\|_{H^{2}(\mathbb{B})}},\frac{k_{z}}{\\|k_{z}\\|_{H^{2}(\mathbb{B})}}\right\rangle_{H^{2}(\mathbb{B})}\right\|^{2}$ $\displaystyle=\left\|1-\hat{z}\overline{w}\right\|^{-2}\left(\left\|1-\hat{z}\overline{w}\right\|^{2}-\left(1-\|w\|^{2}\right)\left(1-\|z\|^{2}\right)\right)$ $\displaystyle=\left\|1-\hat{z}\overline{w}\right\|^{-2}\left(\left(1-\hat{z}\overline{w}\right)\left(1-w\overline{\hat{z}}\right)-\left(1-w\overline{w}\right)\left(1-\hat{z}\overline{\hat{z}}\right)\right)=\left\|1-\hat{z}\overline{w}\right\|^{-2}\left(\hat{z}-w\right)\left(\overline{\hat{z}}-\overline{w}\right)$ $\displaystyle=\left\|1-\hat{z}\overline{w}\right\|^{-2}\left\|\hat{z}-w\right\|^{2}=\left\|\left(1-\hat{z}\overline{w}\right)^{-1}\left(\hat{z}-w\right)\right\|^{2}=\left\|\left(1-q\overline{w}\right)^{-}(q-w)\right\|_{\|_{q=z}}^{2}$ where the last equality follows from Proposition 2.12.∎ The previous relation between the metric $\delta$ (which is the finite version of the metric $g$) and regular Möbius transformations, is not unexpected. In fact, as studied in [6], the real differential $(M_{w})_{}$ of the regular Möbius map $M_{w}$ associated with a point $w\in\mathbb{B}_{I_{w}}$ acts on $L_{I_{w}}$ by right multiplication by $(1-\|w\|^{2})^{-1}$ and on $L_{I_{w}}^{\perp}$ by right multiplication by $(1-\overline{w}^{2})^{-1}$. Looking at equation (7), we see that the coefficients of the metric $g$ at the point $w$ with respect to coordinates $(L_{I_{w}},L_{I_{w}}^{\perp})$ coincide in modulus with the components of $(M_{w})_{}$. Moreover, the fact that $g(w)$ measures vectors in $L_{I_{w}}$ by multiplying their Euclidean length by $\frac{1}{1-\|w\|^{2}}$ means that the restriction of $g$ to a slice $L_{I}$ is the classical Poincaré metric in the unit disc $\mathbb{B}_{I}$. Using spherical coordinates, $\mathbb{B}=\\{re^{tI}\ \|\ r\in[0,1),\ t\in[0,\pi],\ I\in\mathbb{S}\\},$ if $q=re^{tI}$ and we decompose the lenght element $dq=dq_{1}+dq_{2}\in L_{I_{w}}+L_{I_{w}}^{\perp}$, then, since $dI$ is orthogonal to $I$ (because $I$ is unitary) we have $\|d_{1}\|^{2}=dr^{2}+r^{2}dt^{2}$ and $\|d_{2}\|^{2}=r^{2}\sin^{2}t\|dI\|^{2}$ where $\|dI\|$ denotes the usual two-dimensional sphere round metric on $\mathbb{S}\cong\mathbb{S}^{2}$. Therefore we get the expression of the metric tensor $ds^{2}_{g}$ associated with $g$ in spherical coordinates: $ds_{g}^{2}=\frac{dr^{2}+r^{2}dt^{2}}{(1-r^{2})^{2}}+\frac{r^{2}\sin^{2}t\|dI\|^{2}}{(1-r^{2})^{2}+4r^{2}\sin^{2}t}.$ (8) That is, $g$ is a warped product of the hyperbolic metric $g_{hyp}$ on the complex unit disc with the standard round metric $g_{\mathbb{S}}$ on the two- dimensional sphere [17]. ### 4.1 Isometries and geodesics of $(\mathbb{B},g)$ From the expression (7) of $g$, it is clear that three families of functions act isometrically on $(\mathbb{B},g)$: * (a) regular Möbius transformation of the form $q\mapsto M_{\lambda}(q)=(1-q\lambda)^{-}(q-\lambda)=\frac{q-\lambda}{1-q\lambda},$ with $\lambda$ in $(-1,1)$; * (b) isometries of the sphere of imaginary units, which in polar coordinates $r\geq 0,t\in[0,\pi],I\in\mathbb{S}$ read as $q=re^{tI}\mapsto T_{A}(q)=re^{tA(I)},$ where $A:\SS\to\SS$ is an isometry of $\SS$; * (c) the reflection in the imaginary hyperplane, $q\mapsto R(q)=-\overline{q}.$ Our goal is to prove the following classification result, thus proving the second part of Theorem 1.1. ###### Theorem 4.3. The group $\Gamma$ of isometries of $(\mathbb{B},g)$ is generated by maps of type $(a)$, $(b)$ and $(c)$. The proof requires a few steps. To begin with, we identify three classes of totally geodesic submanifolds of $\mathbb{B}$, each one related to a class of isometries. The first family is the one related to isometries of type $(a)$. ###### Lemma 4.4. For any $I\in\mathbb{S}$, the two-dimensional submanifold of $\mathbb{B}$ $\mathbb{B}_{I}=\mathbb{B}\cap L_{I}=\\{re^{tI}\in\mathbb{B}\,\|\,r\in[0,1),t\in[0,2\pi]\\}$ is totally geodesic. In particular, for any $I\in\mathbb{S}$, $\mathbb{B}_{I}$ is an hyperbolic disc. ###### Proof. Fix $I\in\mathbb{S}$ and let $g^{I}_{hyp}$ be the restriction of the metric $g$ to $\mathbb{B}_{I}$, which is just the classical hyperbolic metric in the unit disc. We will show that each geodesic of $(\mathbb{B}_{I},g_{hyp}^{I})$ is still a geodesic of $(\mathbb{B},g)$. Pick two points $w,z$ in $\mathbb{B}_{I}$ and let $\gamma$ be the (hyperbolic) geodesic in $\mathbb{B}_{I}$ joining $w$ with $z$, and $\alpha(\tau)=r(\tau)\left(\cos(t(\tau))+\sin(t(\tau))I(\tau)\right)$ be a parametrized curve which joins $w=\alpha(\tau_{0})$ with $z=\alpha(\tau_{1})$. If $\pi_{I}\left(\alpha\right)$ denotes the piecewise regular curve obtained by projecting $\alpha$ on $\mathbb{B}_{I}$, $\pi_{I}\left(\alpha\right)(\tau)=r(\tau)\left(\cos(t(\tau))+\sin(t(\tau))I\right),$ since $\|dI\|$ is orthogonal to $\mathbb{B}_{I}$, we conclude $\operatorname{length}(\alpha)\geq\operatorname{length}(\pi_{I}\left(\alpha\right))\geq\operatorname{length}(\gamma).$ ∎ The second family of totally geodesic submanifolds is related to isometries of type $(b)$. For any $I\in\mathbb{S}$, we denote by $\mathcal{C}(I)$ the great circle obtained intersecting $\mathbb{S}$ with the plane $L_{I}^{\perp}$. ###### Lemma 4.5. For any $J\in\mathbb{S}$, the three-dimensional submanifold of $\mathbb{B}$ $\mathbb{B}(\mathcal{C}(J))=\\{re^{tI}\in\mathbb{B}\,\|\,r\in[0,1),t\in[0,\pi],I\in\mathcal{C}(J)\\}$ is totally geodesic. ###### Proof. We will prove the statement by showing that the imaginary units identified by all points lying on a same geodesic always belong to the same great circle of $\mathbb{S}$. More precisely, let $\gamma(\tau)=r(\tau)\left(\cos(t(\tau))+\sin(t(\tau))I(\tau)\right)$ be a parametrized geodesic of $(\mathbb{B},g)$ such that $\left\\{\begin{array}[]{l}\gamma(\tau_{0})=x_{0}+y_{0}I_{0}\\\ \gamma^{\prime}(\tau_{0})=v_{0}+w_{0}J_{0}\end{array}\right.$ We want to show that, for any $\tau$, the imaginary unit $I(\tau)$ of $\gamma(\tau)$ belongs to the great circle of $\SS$ identified by $I_{0}$ and $J_{0}$, namely that, for any $\tau$, $I(\tau)\in\mathcal{C}:=\mathcal{C}\left(I_{0}\times J_{0}\right).$ Let $\psi:\mathbb{S}\to\mathbb{S}$ be the reflection of $\mathbb{S}$ with respect to $\mathcal{C}$. Then the curve $\tilde{\gamma}(\tau)=r(\tau)\left(\cos(t(\tau))+\sin(t(\tau))\psi(I(\tau))\right)$ is a geodesic of $(\mathbb{B},g)$ such that $\left\\{\begin{array}[]{l}\tilde{\gamma}(\tau_{0})=\gamma(\tau_{0})\\\ \tilde{\gamma}^{\prime}(\tau_{0})=\gamma^{\prime}(\tau_{0})\end{array}\right.$ since $\psi$ fixes $I_{0}$ and $J_{0}$. By the uniqueness of geodesics with assigned initial conditions, we get that $\tilde{\gamma}(\tau)=\gamma(\tau)$ and hence that $\psi$ fixes $I(\tau)$ for any $\tau$. Therefore we conclude that $I(\tau)\in\mathcal{C}$ for any $\tau\in I$. ∎ The third totally geodesic submanifold is the one related to the last class of isometries, type $(c)$. ###### Lemma 4.6. The three-dimensional submanifold of $\mathbb{B}$ $\mathbb{B}\left(\pi/2\right)=\\{re^{tI}\in\mathbb{B}\,\|\,r\in[0,1),t=\pi/2,I\in\mathbb{S}\\}=\\{rI\,\|\,r\in[0,1),I\in\mathbb{S}\\}$ is totally geodesic. ###### Proof. The statement can be proven following the line of the proof of Lemma 4.5. The ingredients are the fact that the map $R:\mathbb{B}\to\mathbb{B}$, $q\mapsto-\overline{q}$ is an isometry which fixes (punctually) $\mathbb{B}(\pi/2)$, and the uniquness of geodesics with assigned initial conditions. ∎ Considering the intersection of totally geodesic submanifolds of type $\mathbb{B}(C(J))$ with $\mathbb{B}(\pi/2)$ allows us to identify another family of totally geodesic submanifolds of $\mathbb{B}$. ###### Corollary 4.7. Let $\mathcal{C}(J)$ be a great circle in $\mathbb{S}$. Then the two- dimensional submanifold $D\left(\pi/2,\mathcal{C}(J)\right)\subset\mathbb{B}(\pi/2)$, defined as $D\left(\pi/2,\mathcal{C}(J)\right)=\\{re^{tI}\in\mathbb{B}\,\|\,r\in[0,1),t=\pi/2,I\in\mathcal{C}(J)\\}=\\{rI\in\mathbb{B}\,\|\,r\in[0,1),I\in\mathcal{C}(J)\\},$ is totally geodesic. ###### Remark 4.8. Notice that for the two-dimensional submanifold $D\left({\pi}/2,\mathcal{C}(J)\right)$ the following orthogonality relation holds: $D\left({\pi}/2,\mathcal{C}(J)\right)\cap\mathbb{B}_{J}=\\{0\\}\text{ and }T_{0}D\left({\pi}/2,\mathcal{C}(J)\right)=T_{0}\mathbb{B}_{J}^{\perp}.$ Moreover, applying Möbius maps of the form $M_{\lambda}$ to $D\left({\pi}/2,\mathcal{C}(J)\right)$, we can extend the orthogonality relation from the origin to all points in $\mathbb{B}\cap\mathbb{R}$. In this way we obtain a family of totally geodesic submanifolds $D\left(t,\mathcal{C}(J)\right)=M_{\lambda(t)}\left(D\left({\pi}/2,\mathcal{C}(J)\right)\right)$ that, for $t\in[0,\pi]$ and $J\in\mathbb{S}/\\{\pm 1\\}$, defines a foliation of the manifold $\mathbb{B}$. In order to have some understanding of the (global) behavior of the metric $g$, let us investigate some metric properties of the discs of the type $D\left(\frac{\pi}{2},\mathcal{C}(J)\right)$. Since the imaginary units taken into account belong to $\mathcal{C}(J)\cong\mathbb{S}^{1}$, we can change coordinates, setting $I=e^{i\theta}$ and $\|dI\|=d\theta$, so that the metric $g$, on $D\left(\frac{\pi}{2},\mathcal{C}(J)\right)$, reduces to $ds_{D}^{2}=\frac{dr^{2}}{(1-r^{2})^{2}}+\frac{r^{2}d\theta^{2}}{(1+r^{2})^{2}}.$ It is actually convenient to parametrize $D\left(\frac{\pi}{2},\mathcal{C}(J)\right)\subset\mathbb{I}\cong\mathbb{R}^{3}$ as a surface of revolution of the form $(\Phi(\rho),\Psi(\rho)\cos\theta,\Psi(\rho)\sin\theta)$, where $\rho$ is the arc length of the generating curve. Setting $\rho=\rho(r)=\frac{1}{2}\log\frac{1+r}{1-r},$ we get $\frac{dr^{2}}{(1-r^{2})^{2}}=d\rho^{2}\quad\text{and}\quad\frac{r^{2}}{(1+r^{2})^{2}}=\frac{1}{4}\tanh^{2}(2\rho)$ and hence, in coordinates $(\rho,\theta)$, we get that the metric is expressed as $ds_{D}^{2}=d\rho^{2}+\frac{1}{4}\tanh^{2}(2\rho)d\theta^{2}=d\rho^{2}+\Psi^{2}(\rho)d\theta^{2}.$ (9) ###### Remark 4.9. The Gaussian curvature $K$ of the two-dimensional submanifold $D\left(\frac{\pi}{2},\mathcal{C}(J)\right)$ is positive. In fact, see e.g. [11], with respect to coordinates $(\rho,\theta)$ it can be computed as $K=\frac{-\Psi^{\prime\prime}(\rho)}{\Psi(\rho)}$ which is a non-negative quantity since $\Psi(\rho)=\frac{1}{2}\tanh(2\rho)\geq 0$ and $\Psi^{\prime\prime}(\rho)\leq 0$. This in particular implies that the sectional curvature of $(\mathbb{B},g)$ is positive on all sections $D\left(\frac{\pi}{2},\mathcal{C}(J)\right)$, while it is negative on all slices $\mathbb{B}_{I}$. It is possible to study geodesics of $D\left(\frac{\pi}{2},\mathcal{C}(J)\right)$ by means of the Euler-Lagrange equations $\left\\{\begin{array}[]{l}\frac{\partial}{\partial\theta}L=\frac{d}{dt}\frac{\partial}{\partial\dot{\theta}}L\\\ \frac{\partial}{\partial\rho}L=\frac{d}{dt}\frac{\partial}{\partial\dot{\rho}}L\end{array}\right.$ associated with the Lagrangian $L(\rho,\theta,\dot{\rho},\dot{\theta},\tau)=\frac{1}{2}\left(\dot{\rho}^{2}+\frac{\tanh^{2}(2\rho)}{4}\dot{\theta}^{2}\right),$ namely $\left\\{\begin{array}[]{l}0=\frac{d}{dt}\left(\frac{\tanh^{2}(2\rho)}{4}\dot{\theta}\right)\\\ \tanh(2\rho)\frac{1-\tanh^{2}(2\rho)}{4}\dot{\theta}^{2}=\ddot{\rho}.\end{array}\right.$ (10) The first equation in (10) yields $\frac{\tanh^{2}(2\rho)}{4}\dot{\theta}=A,$ for some constant $A$. If $A=0$, we get $\dot{\theta}=0$ and hence the second equation in (10) implies that $\ddot{\rho}=0$. If otherwise $A\neq 0$ we get $\dot{\theta}=\frac{4A}{\tanh^{2}(2\rho)}$ which implies $\|\dot{\theta}\|>4\|A\|$. Therefore all generating curves, with $\dot{\theta}=0$, are geodesics of $D$. Which is not surprising since they correspond to radii $\gamma(r)=re^{\frac{\pi}{2}I}$ for $I\in\mathcal{C}(J)$. The other important fact that arises is that for any point $q\in D\left(\frac{\pi}{2},\mathcal{C}(J)\right)\setminus\\{0\\}$ any geodesic corresponding to $A\neq 0$ intersects in finite time the “radial” geodesic through $q$. This leads to the following result. ###### Lemma 4.10. Let $J\in\mathbb{S}$. For any $q\in D\left(\frac{\pi}{2},\mathcal{C}(J)\right)$ such that $q\neq 0$, the injectivity radius at $q$ is finite. On the other hand, the injectivity radius at $q=0$ is infinite. This important metric property of the point $q=0$ is useful to classify the isometries of $(\mathbb{B},g)$. First of all it tells us that isometries map the real diameter of $\mathbb{B}$ to itself. ###### Lemma 4.11. Let $\Gamma$ be the group of isometries of $(\mathbb{B},g)$. Then, for any $\phi\in\Gamma$, $\phi(\mathbb{B}\cap\mathbb{R})=\mathbb{B}\cap\mathbb{R}$. ###### Proof. Consider first $q=0$. Since the injectivity radius at $q=0$ is infinite, then, for any $\phi\in\Gamma$, the injectivity radius at $\phi(0)$ is infinite as well. By post-composing $\phi$ with a regular Möbius transformation of type $(a)$ $M_{\lambda}$ we can map $0$ to $D\left(\frac{\pi}{2},\mathcal{C}(J)\right)$ (for some $J\in\mathbb{S}$) and hence Lemma 4.10 yields that $M_{\lambda}(\phi(0))=0$. Since $M_{\lambda}$ preserves the real diameter of $\mathbb{B}$, we get that $\phi(0)\in\mathbb{R}$. To conclude, notice that we can map each point of $\mathbb{B}\cap\mathbb{R}$ to $0$ by means of a regular Möbius map of type $(a)$. ∎ We can finally prove the Classification Theorem for isometries of $(\mathbb{B},g)$. ###### Proof of Theorem 4.3. Let $\Phi\in\Gamma$ be an isometry of $(\mathbb{B},g)$. Up to composition with a regular Möbius transformation of type $(a)$ and with the map $R:q\mapsto-\overline{q}$, we can suppose that $\Phi(0)=0$ and that, by Lemma 4.11, $\Phi(\mathbb{B}\cap\mathbb{R}^{+})=\mathbb{B}\cap\mathbb{R}^{+}$. We now show that $\Phi$ fixes $\mathbb{B}(\pi/2)$. Set $\tilde{B}(\pi/2)=\Phi(\mathbb{B}(\pi/2))$. Since $\Phi$ is an isometry, Lemma 4.6 implies that $\tilde{B}(\pi/2)$ is a totally geodesic submanifold of $\mathbb{B}$. Moreover, since $\Phi(0)=0$, since the geodesics starting at $0$ lie on slices, and since, by Lemma 4.4, the slices carry the usual hyperbolic- Poincaré metric: we have that $\Phi$ maps radii $\gamma_{I}(r)=re^{\frac{\pi}{2}I}$ to radii of the form $\Phi(\gamma_{I}(r))=re^{\theta(I)\psi(I)}$ with $\theta(I)\in[0,\pi]$, and $\psi(I)\in\mathbb{S}$. Let us show that $\theta$ is actually constant on $\mathbb{S}$. If $d_{g}$ denotes the distance function on $\mathbb{B}$ associated with $g$, recalling equation (8), on the one hand we have $\displaystyle d_{g}\left(\Phi(\gamma_{I_{1}}(r)),\Phi(\gamma_{I_{2}}(r))\right)$ $\displaystyle=$ $\displaystyle d_{g}\left(\gamma_{I_{1}}(r),\gamma_{I_{2}}(r)\right)=d_{g}\left(re^{\frac{\pi}{2}I_{1}},re^{\frac{\pi}{2}I_{2}}\right)$ (11) $\displaystyle\leq$ $\displaystyle\frac{r}{1+r^{2}}d_{\mathbb{S}}(I_{1},I_{2})\stackrel{{\scriptstyle r\to 1}}{{\longrightarrow}}\frac{1}{2}d_{\mathbb{S}}(I_{1},I_{2}),$ (12) where $d_{\mathbb{S}}$ denotes the usual spherical distance on the unit sphere $\mathbb{S}$. In particular we deduce that $d_{g}\left(\Phi(\gamma_{I_{1}}(r)),\Phi(\gamma_{I_{2}}(r))\right)$ is bounded as a function of $r$. On the other hand, if $\alpha(\tau)=r(\tau)e^{t(\tau)I(\tau)}$ is a parametrized geodesic joining $\alpha(\tau_{1})=\Phi(\gamma_{I_{1}}(r))$ and $\alpha(\tau_{2})=\Phi(\gamma_{I_{2}}(r))$, we have $\displaystyle d_{g}\left(\Phi(\gamma_{I_{1}}(r)),\Phi(\gamma_{I_{2}}(r))\right)=d_{g}\left(re^{\theta({I_{1}})\psi({I_{1}})},re^{\theta({I_{2}})\psi({I_{2}})}\right)=\operatorname{length}\left(\alpha([\tau_{1},\tau_{2}])\right)$ $\displaystyle=\int_{\tau_{1}}^{\tau_{2}}\sqrt{\frac{r^{\prime}(\tau)^{2}+r^{2}t^{\prime}(\tau)^{2}}{1-r(\tau)^{2}}+\frac{r(\tau)^{2}\sin^{2}(t(\tau))I^{\prime}(\tau)^{2}}{(1-r(\tau)^{2})^{2}+4r(\tau)^{2}\sin^{2}(t(\tau))}}d\tau$ $\displaystyle\geq\int_{\tau_{1}}^{\tau_{2}}\sqrt{\frac{r^{\prime}(\tau)^{2}+r(\tau)^{2}t^{\prime}(\tau)^{2}}{1-r(\tau)^{2}}}d\tau\geq d_{hyp}(re^{\theta({I_{1}})I},re^{\theta({I_{2}})I})$ where $I$ is any fixed imaginary unit and $d_{hyp}$ the hyperbolic distance associated to the restriction $g^{I}_{hyp}$ of the metric $g$ to $\mathbb{B}\cap L_{I}$. If, by contradiction, $\theta(I_{1})\neq\theta(I_{2})$, then the distance $d_{hyp}(re^{\theta({I_{1}})I},re^{\theta({I_{2}})I})$ tends to infinity as $r$ goes to $1$, contradicting (11). Then, $\theta(I_{1})=t(\tau_{1})=t(\tau_{2})=\theta(I_{2})$: $\theta$ is constant on $\mathbb{S}$. Therefore we have that $\tilde{B}(\pi/2)$ is ruled by radii of the form $\tilde{\gamma}(r)=re^{t_{0}I}$ for some constant $t_{0}$. If $t_{0}=\pi/2$, then we are done. Suppose then $t_{0}\neq\pi/2$. Since $\tilde{B}(\pi/2)$ and $\mathbb{B}(\pi/2)$ intersect at $0$ and they are three-dimensional submanifolds in $\mathbb{H}$, the intersection $V$ of their respective tangent spaces at $0$ must have dimension $2$ or $3$. Let $v$ be a vector in $V$ and let $r\mapsto re^{Jt}$ be the reparametrized geodesic with initial velocity $v$. The geodesic lies on both $\tilde{B}(\pi/2)$ and $\mathbb{B}(\pi/2)$, hence $t_{0}=t=\pi/2$. (A different proof consists in showing that, if $t_{0}\neq\pi/2$, then $\tilde{B}(\pi/2)$ is not smooth at the origin). The next step is to show that the restriction of $\Phi$ to $\mathbb{B}(\pi/2)$ is an isometry $T_{A}$ of type $(b)$ for some isometry $A$ of the sphere $\mathbb{S}$. We have that $\displaystyle d_{g}\left(\Phi(\gamma_{I_{1}}(r)),\Phi(\gamma_{I_{2}}(r))\right)=d_{g}\left(re^{\frac{\pi}{2}\psi({I_{1}})},re^{\frac{\pi}{2}\psi({I_{2}})}\right).$ (13) We now prove an improvement of (11). ###### Lemma 4.12. $\lim_{r\to 1}d_{g}\left(re^{\frac{\pi}{2}I_{1}},re^{\frac{\pi}{2}I_{2}}\right)=\frac{1}{2}d_{\mathbb{S}}(I_{1},I_{2}).$ ###### Proof of the lemma.. Only the case $I_{1}\neq I_{2}$ is interesting. Let $D\left(\pi/2,\mathcal{C}(J)\right)$ be the two-dimensional manifold introduced in Lemma 4.7 which contains the reparametrized geodesics $r\mapsto re^{\frac{\pi}{2}I_{j}}$, $j=1,2$. The metric $g$ restricted to the totally geodesic surface $D\left(\pi/2,\mathcal{C}(J)\right)$ was discussed earlier in this subsection, where we gave it the expression (9). Since $\Psi^{\prime}(\rho)=1/\cosh(2\rho)\leq 1$, the surface can be isometrically imbedded as a surface $S$ in $\mathbb{R}^{3}$, with parametric equations $(u(s,\theta),v(s,\theta),z(s,\theta))=\chi(s,\theta)$, where: $\begin{cases}u=p(s)\cos(\theta);\crcr v=p(s)\sin(\theta);\crcr z=s.\end{cases}$ Here $s\geq 0$, $\theta\in[-\pi,\pi]$ and $p:[0,+\infty)\to[0,1/2)$ is a smooth, increasing function such that $p(0)=0$ and $\lim_{s\to\infty}p(s)=1/2$. The relationship between $p$ and $\Psi$ is the following: if $\int_{0}^{s}\sqrt{p^{\prime}(\sigma)^{2}+1}d\sigma=\rho,$ then $p(s)=\psi(\rho)$. Now, $r=constant\to 1$ corresponds to $s=constant\to\infty$, and the choice of $I_{1}$ and $I_{2}$ corresponds to a choice of $\theta_{1}$ and $\theta_{2}$. Let $k$ be the metric on the surface. It is elementary that $\lim_{s\to\infty}d_{k}(\chi(s,\theta_{1}),\chi(s,\theta_{2}))=\frac{1}{2}d_{\mathbb{S}^{1}}(\theta_{1},\theta_{2})$ is one-half the distance between $\theta_{1}$ and $\theta_{2}$ on the unit circle, which is the same as one-half the distance between $I_{1}$ and $I_{2}$ in $\SS$. ∎ Equations (11) and (13) together with Lemma 4.12 imply that that $\psi:\mathbb{S}\to\mathbb{S}$ is an isometry of the sphere $\mathbb{S}$, i.e. $\Phi\|_{\mathbb{B}(\pi/2)}=T_{\psi}\|_{\mathbb{B}(\pi/2)}$. In conclusion, $T_{\psi}^{-1}\circ\Phi$ is an isometry that fixes $\mathbb{R}\cap\mathbb{B}$ and $\mathbb{B}(\pi/2)$ and hence its (real) differential at the origin $(T_{\psi}^{-1}\circ\Phi)_{}[0]:T_{0}\mathbb{B}\to T_{0}\mathbb{B}$ is the identity map. Therefore $T_{\psi}^{-1}\circ\Phi$ is the identity map as well and the theorem is proved. ∎ ### 4.2 Relation with the space $H^{2}(\mathbb{B})$ If we restrict the metric $g$ to a three-dimensional sphere $r\mathbb{S}^{3}$ of radius $r$, in spherical coordinates we get $ds^{2}_{r\mathbb{S}^{3}}=\frac{r^{2}}{(1-r^{2})^{2}}dt^{2}+\frac{r^{2}\sin^{2}t}{(1-r^{2})^{2}+4r^{2}\sin^{2}(t)}\|dI\|^{2}$ whose corresponding volume form is $dVol_{r\mathbb{S}^{3}}(re^{tI})=\frac{r^{3}\sin^{2}t}{(1-r^{2})((1-r^{2})^{2}+4r^{2}\sin^{2}(t))}dtdA_{\mathbb{S}}(I)$ where $dA_{\mathbb{S}}$ denotes the area element of the two-dimensional sphere $\mathbb{S}$. This volume form (after a normalization) induces a volume form on the boundary $\mathbb{S}^{3}$ of the unit ball: if $u=e^{sJ}\in\mathbb{S}^{3}$, we have $\displaystyle dVol_{\mathbb{S}^{3}}(u)$ $\displaystyle:=\lim_{r\to 1^{-}}(1-r^{2})dVol_{r\mathbb{S}^{3}}(ru)=\lim_{r\to 1^{-}}\frac{(1-r^{2})r^{3}\sin^{2}s}{(1-r^{2})((1-r^{2})^{2}+4r^{2}\sin^{2}(s))}dtdA_{\mathbb{S}}(I)$ $\displaystyle=\frac{1}{4}dtdA_{\mathbb{S}}(I).$ Notice that $dVol_{\mathbb{S}^{3}}$ is the product of the usual spherical metric on the two-dimensional sphere $\mathbb{S}$ with the metric $dt$ on circles $\mathbb{S}^{3}_{I}$ which appears in the definition of Hardy spaces given in [10]. Moreover in [10] it is proven that any $f\in H^{2}(\mathbb{B})$ has radial limit along almost any radius and hence, denoting (with a slight abuse of notation) the radial limit by $f$ itself, we have $\displaystyle\int_{\mathbb{S}^{3}}\|f(u)\|^{2}dVol_{\mathbb{S}^{3}}(u)$ $\displaystyle=\frac{1}{4}\int_{\mathbb{S}}\int_{0}^{\pi}\|f(e^{tI})\|^{2}dtdA_{\mathbb{S}}(I)=\frac{1}{8}\int_{\mathbb{S}}\int_{0}^{2\pi}\|f(e^{tI})\|^{2}dtdA_{\mathbb{S}}(I)$ $\displaystyle=\frac{1}{8}\int_{\mathbb{S}}\|\|f\|\|^{2}_{H^{2}(\mathbb{B})}dA_{\mathbb{S}}(I)=\frac{\pi}{2}\|\|f\|\|^{2}_{H^{2}(\mathbb{B})}.$ ### 4.3 Proof of Theorem 1.2 We begin by showing that $(\mathbb{B},m)$ has constant negative curvature. Heuristically, a Riemannian metric $m$ satisfying the assumption of the theorem has an isometry group with dimension $dim(\mathbb{B})+dim(\SS^{3})+dim(O(3))=4+3+3=10,$ which is maximal for a four-dimensional Riemannian manifold. Hence, $(\mathbb{B},m)$ has constant curvature. More precisely, we show that the isometry group $\mathcal{I}$ acts transitively on orthonormal frames, a property which is known to imply constant sectional curvature. Given points $a,b$ in $\mathbb{B}$ and orthonormal frames $\\{e_{l}(a):\ l=0,1,2,3\\}$ and $\\{e_{l}(b):\ l=0,1,2,3\\}$ in $T_{a}\mathbb{B}$ and $T_{b}\mathbb{B}$, respectively, we find an isometry $\varphi$ in $\mathcal{I}$ such that its (real) differential $\varphi_{}$ satisfies: $\varphi_{}[a]e_{l}(a)=e_{l}(b)$. We in fact exhibit $\varphi$ mapping $a$ to $0$ and such that $\varphi_{}$ maps the chosen orthonormal frame in $T_{a}\mathbb{B}$ to a fixed orthonormal basis $e_{0}(0),e_{1}(0),e_{2}(0),e_{3}(0)$ of $T_{0}\mathbb{B}$, where $e_{0}(0)$ is the vector tangent to the positive real half-axis. The isometry $M_{a}(q)=(1-q\overline{a})^{-}(q-a)$ maps $a$ to $0$, hence $(M_{a})_{}$ sends $e_{l}(a)$ to a orthonormal frame in $T_{0}\mathbb{B}$ $e^{\prime}_{l}$ for $l=0,\dots,3$. For a suitable choice of $u$ with $\|u\|=1$, the isometry $q\mapsto q\cdot u$ has differential mapping $e^{\prime}_{0}$ to $e_{0}(0)$, and $e^{\prime}_{l}(0)$ to $e^{\prime\prime}_{l}(0)$ ($j=1,2,3$). The isometries $q=x+yI\mapsto T_{A}(q)=x+yA(I)$, $A$ being a fixed element of $O(3)$, all have differentials fixing $e_{0}(0)$. We can find one mapping $e^{\prime\prime}_{l}(0)$ to $e_{l}(0)$ for $l=1,2,3$. The composition of these three isometries is the desired isometry $\varphi$. For $I\in\SS$, consider the subgroup $\mathcal{I}_{I}$ of $\mathcal{I}$ of the isometries fixing the slice $\mathbb{B}_{I}$; which consists of the regular Möbius maps $M_{a}$, with $a$ in $\mathbb{B}_{I}$, and of the maps $q\mapsto q\cdot e^{tI}$. If $\chi_{I}:x+yI\mapsto x+yi$ is the natural bijection from $\mathbb{B}_{I}$ to the unit disc in the complex plane, $\chi_{I}\mathcal{I}_{I}\chi_{I}^{-1}$ identifies $\mathcal{I}_{I}$ with the usual Poincaré group in the complex disc. Hence, the restriction of $m$ to $\mathbb{B}_{I}$ is (isometric to) a constant multiple of the Poincaré metric, which has constant negative curvature. The hyperbolic metric $m$ is realized by the standard Poincaré model on the ball $\mathbb{B}$. The metric $m$ restricted to $\mathbb{B}_{I}$ is realized as $\|d\|^{2}_{m(w)}=\lambda^{2}\frac{\|d\|^{2}}{(1-\|w\|^{2})^{2}}$ (for $w$ in $\mathbb{B}_{I}$ and $d$ in $T_{w}(\mathbb{B}_{I})$), with $\lambda$ which is independent of $I$, since different slices intersect along the real diameter of $\mathbb{B}$. We might set $\lambda=1$. Each slice $\mathbb{B}_{I}$ is totally geodesic, since it is the set of the points fixed by an isometry of the type $x+yJ\mapsto x+yB(J)$, where $B$ is an element of $O(3)$ fixing $\pm I$ and no other element of $\SS$. Then, the radii $r\mapsto ru=\gamma_{u}(r)$ (with $u$ fixed in $\mathbb{H}$, $\|u\|=1$) are (reparametrizations of) geodesics of $(\mathbb{B},m)$, not just of its restriction to a slice, and the distance function on each of them is obtained by integrating $dr/(1-r^{2})$. The three-dimensional spheres $r\mathbb{S}^{3}=\\{q:\ \|q\|=r\\}$ are then metric spheres centered at $0$ for the metric $m$. By Gauss Lemma, they are orthogonal to the curves $\gamma_{u}$. Fix $r$ in $(0,1)$. An argument similar to the one above shows that the subgroup $\mathcal{I}_{0}$ of the isometries fixing $0$ acts transitively on the bundle of the orthogonal frames at points of $r\mathbb{S}^{3}$, hence $r\mathbb{S}^{3}$ is isometric to the usual three- spheres with a multiple of the spherical metric. Since $L_{I}\cap r\mathbb{S}^{3}$ is isometric to a metric one sphere in the Poincaré model of the hyperbolic metric (in dimension two), for each $I$ in $\SS$, $r\mathbb{S}^{3}$ is similarly isometric to a metric three-sphere in the Poincaré model of the hyperbolic metric (in dimension four). But we said that $r\mathbb{S}^{3}$ and $\gamma_{u}$ are orthogonal in their point of intersection; they are both isometric to the corresponding objects in the Poincaré model; the sum of their tangent spaces is the whole tangent space: this shows that the metric $m$ coincides in fact with the Poincaré metric. Concluding, we have shown that the hyperbolic Poincaré metric is invariant under regular Möbius maps, but this contradicts a result of Bisi and Stoppato [6], Remark 5. ## 5 Metric in the right half space $\mathbb{H}^{+}$ Consider the right half space $\mathbb{H}^{+}=\\{q\in\mathbb{H}\,\|\,\operatorname{Re}(q)>0\\}$. The Cayley map $C:\mathbb{B}\to\mathbb{H}^{+}$, $C(q)=(1-q)^{-1}(1+q),$ is a regular bijection from the quaternionic unit ball onto the quaternionic right half space with regular inverse the function $C^{-1}:\mathbb{H}^{+}\to\mathbb{B}$, $C^{-1}(q)=(1+q)^{-1}(q-1).$ The aim of this section is to study the image $(\mathbb{H}^{+},h)$ of $(\mathbb{B},g)$ under the map $C$, where $h$ is the pullback of the metric $g$ by the map $C^{-1}$. In the introduction we labeled $g$ and $h$ by the same letter, since $C$ is by definition an isometry from $(\mathbb{B},g)$ to $(\mathbb{H}^{+},h)$. Let $u\in\mathbb{H}^{+}$ and let $v=v_{1}+v_{2}$ be a tangent vector in $T_{u}\mathbb{H}^{+}\cong L_{I_{u}}+L_{I_{u}}^{\perp}$. The length of $v$ with respect to $h$ is $\|v\|_{h(u)}=\left\|(C^{-1})_{}[u](v)\right\|_{g(C^{-1}(u))}$ where $(C^{-1})_{}[u]$ is the real differential of $C^{-1}$ at the point $u\in\mathbb{H}^{+}$. Recalling the decomposition of the real differential of a regular function in terms of its slice and spherical derivatives, if $v=v_{1}+v_{2}\in L_{I_{u}}+L_{I_{u}}^{\perp}$ we can write $(C^{-1})_{}[u](v)=v_{1}\partial_{c}C^{-1}(u)+v_{2}\partial_{s}C^{-1}(u)=v_{1}\frac{2}{(1+u)^{2}}+v_{2}\frac{2}{\|1+u\|^{2}}.$ Hence $(C^{-1})_{}[u]$ preserves the decomposition $T_{u}\mathbb{H}^{+}=L_{I_{u}}+L_{I_{u}}^{\perp}$ and we get $\displaystyle\|v\|^{2}_{h(u)}$ $\displaystyle=\frac{1}{(1-\|C^{-1}(u)\|^{2})^{2}}\frac{4}{\|1+u\|^{4}}\|v_{1}\|^{2}+\frac{1}{\|1-C^{-1}(u)^{2}\|^{2}}\frac{4}{\|1+u\|^{4}}\|v_{2}\|^{2}$ $\displaystyle=\frac{1}{4\operatorname{Re}(u)^{2}}\|v_{1}\|^{2}+\frac{1}{4\|u\|^{2}}\|v_{2}\|^{2}.$ If $v\in L_{I_{u}}$ then its length is, not surprisingly, the hyperbolic length in the hyperbolic half plane $\mathbb{H}^{+}_{I_{u}}=\\{x+yI_{u}\,\|\,x>0,y\in\mathbb{R}\\}$. Notice that $C$ maps $\mathbb{B}_{I}$ to $\mathbb{H}^{+}_{I}$ for any $I\in\mathbb{S}$ and it maps the totally geodesic submanifold $\mathbb{B}(\pi/2)$ to $\mathbb{H}^{+}(\pi/2):=C\left(\mathbb{B}(\pi/2)\right)=\\{q\in\mathbb{H}^{+}\,\|\,\|q\|=1\\}$ i.e. the right half of the three-dimensional unit sphere $\mathbb{S}^{3}$. Then it is not difficult to verify that the isometry group of $(\mathbb{H}^{+},h)$ is generated by the images under $C$ of isometry of $(\mathbb{B},g)$ of type $(a)$, $(b)$ and $(c)$, (a’) linear maps preserving the positive real half-axis, $q\mapsto q\lambda,$ with $\lambda>0$; * (b’) isometries of the sphere of the imaginary units, which in polar coordinates $r\geq 0$, $t\in[0,\pi/2)$, $I\in\mathbb{S}$ read as $q=re^{tI}\mapsto T_{A}(q)=re^{tA(I)},$ where $A:\SS\to\SS$ is an isometry of $\SS$; * (c’) the inversion in the three-dimensional unit (half) sphere $q\mapsto\frac{1}{\overline{q}}.$ Acting on $\mathbb{H}^{+}(\pi/2)$ by means of isometries of type $(a^{\prime})$ we obtain totally geodesic regions of the form $\\{q\in\mathbb{H}^{+}\,\|\,\|q\|=R\\}$ for $R>0$, that can be sliced in totally geodesic two-dimensional submanifolds, corresponding to submanifolds of type $D(t,C(J))$ in the ball case. In this setting it is possible to introduce horocycles, i.e. hyperplanes of points with constant real part, $H_{c}=\\{q\in\mathbb{H}^{+}\,\|\,\operatorname{Re}(q)=c\\}$ for some constant $c>0$. They deserve the name of horocycles because their intersection with each slice $L_{I}$ is a proper horocycle in the hyperbolic half plane $\mathbb{H}^{+}_{I}$. Isometries of type $(a^{\prime})$ map horocycles one into another. If we restrict the metric $h$ to horocycles we obtain that the length of a vector $v=v_{1}+v_{2}\in T_{u}H_{c}\cong\mathbb{R}I_{u}+L_{I_{u}}^{\perp}$ tangent to the horocycle $H_{c}$ at the point $u\in H_{c}$, can be written as $\|v\|^{2}_{H_{c}}=\frac{1}{4c^{2}}\|v_{1}\|^{2}+\frac{1}{4(c^{2}+\|\operatorname{Im}(u)\|^{2})}\|v_{2}\|^{2}$ and the corresponding volume form at $u=c+x_{1}i+x_{2}j+x_{3}k$ is $dVol_{H_{c}}(u)=\frac{dVol_{Euc}(u)}{8c(c^{2}+\|\operatorname{Im}(u)\|^{2})},$ (since the component in $L_{I_{u}}$ is one-dimensional) where $dVol_{Euc}(u)=dx_{1}dx_{2}dx_{3}$ is the standard Euclidean volume element. If we want to define a (non-degenerate) volume form $dVol_{\partial\mathbb{H}^{+}}$ on the boundary $\partial\mathbb{H}^{+}$ of the quaternionic right half space we can not directly take the limit of $dVol_{H_{c}}$ as $c$ approaches $0$, we need indeed first to normalize it. For any $u\in\partial\mathbb{H}^{+}$, we define $dVol_{\partial\mathbb{H}^{+}}(u)\\!:=\\!\\!\lim_{c\to 0^{+}}c\left(dVol_{H_{c}}(u+c)\right)=\\!\lim_{c\to 0^{+}}\frac{dVol_{Euc}(u+c)}{8(c^{2}+\|\operatorname{Im}(u+c)\|^{2})}=\frac{dVol_{Euc}(u)}{8\|\operatorname{Im}(u)\|^{2}}=\frac{dVol_{Euc}(u)}{8\|u\|^{2}}.$ (14) ### 5.1 Hardy space on $\mathbb{H}^{+}$ We will show that, as in the case of the metric $g$ on $\mathbb{B}$, the invariant metric $h$ on $\mathbb{H}^{+}$ introduced in the previous section and in particular the corresponding volume form (14), is related with the quaternionic Hardy space on the right half space $\mathbb{H}^{+}$. It is possible to define the Hardy space $H^{2}(\mathbb{H}^{+})$ on $\mathbb{H}^{+}$ as the space of regular functions $f:\mathbb{H}^{+}\to\mathbb{H}$ of the form, $f(q)=\int_{0}^{+\infty}e^{-\zeta q}F(\zeta)d\zeta,$ with $F:\mathbb{R}^{+}\to\mathbb{H}$, such that $\|\|f\|\|^{2}_{H^{2}(\mathbb{H}^{+})}:=\int_{0}^{+\infty}\|F(\zeta)\|^{2}d\zeta<+\infty.$ With this definition, the reproducing kernel of $H^{2}(\mathbb{H}^{+})$ is a function $k(q,w)=k_{w}(q)=\int_{0}^{+\infty}e^{-\zeta q}G(\zeta)d\zeta$ where $G:\mathbb{R}^{+}\to\mathbb{H}$ is such that $f(w)=\langle f,k_{w}\rangle_{H^{2}(\mathbb{H}^{+})}=\int_{0}^{+\infty}\overline{G(\zeta)}F(\zeta)d\zeta=\int_{0}^{+\infty}e^{-\zeta w}F(\zeta)d\zeta.$ Hence $G$ has to satisfy $\overline{G(\zeta)}=e^{-\zeta w}$ which implies ${G(\zeta)}=e^{-\zeta\bar{w}}$, i.e. that the kernel function is $k_{w}(q)=\int_{0}^{+\infty}e^{-\zeta q}e^{-\zeta\bar{w}}d\zeta.$ To obtain a closed expression of $k_{w}(q)$, let first $q$ be a (positive) real number. In this case $q$ commutes with all points in $\mathbb{H}^{+}$ and we can write $k_{w}(q)=\int_{0}^{+\infty}e^{-\zeta(q+\bar{w})}d\zeta=\frac{1}{q+\overline{w}}.$ Consider now the function $q\mapsto(q+\overline{w})^{-}$ (here the regular reciprocal is defined with a slight generalization of Definition 2.10, see [12]). This function is regular and it coincides with $q\mapsto(q+\overline{w})^{-1}$ on real numbers. Thanks to the Identity Principle for regular functions, Theorem 1.12 in [12], we obtain that the reproducing kernel is $k_{w}(q)=(w+\overline{q})^{-}=\int_{0}^{+\infty}e^{-\zeta w}e^{-\zeta\bar{q}}d\zeta$. Another way to obtain the reproducing kernel on $H^{2}(\mathbb{H}^{+})$ is the following. ###### Proposition 5.1. Denote by $k_{H^{2}(\mathbb{B})}$ and by $k_{H^{2}(\mathbb{H}^{+})}$ the reproducing kernels of the Hardy space on the unit ball $H^{2}(\mathbb{B})$ and on the right half-space $H^{2}(\mathbb{H}^{+})$ respectively. Let $C:\mathbb{B}\to\mathbb{H}^{+}$ be the Cayley map, $C(q)=(1-q)^{-1}(1+q)$. For any $z,w\in\mathbb{H}^{+}$, the function $k_{H^{2}(\mathbb{B})}(C^{-1}(w),C^{-1}(z))$ is a rescaling of the reproducing kernel of $H^{2}(\mathbb{H}^{+})$: $k_{H^{2}(\mathbb{B})}(C^{-1}(w),C^{-1}(z))=\frac{1}{2}(1+z)k_{H^{2}(\mathbb{H}^{+})}(w,z)(1+\overline{w}).$ ###### Proof. The map $C^{-1}$, having real coefficients, is slice preserving. Hence, we can compose $k_{H^{2}(\mathbb{B})}$ with $C^{-1}$ preserving (left) regularity in the first variable and (right) “anti-regularity” in the second one. We have, then, $\displaystyle k_{H^{2}(\mathbb{B})}(C^{-1}(z),C^{-1}(w))=(1-q\overline{C^{-1}(w)})_{\|_{q=C^{-1}(z)}}^{-}$ $\displaystyle=\left(1-2C^{-1}(z)\operatorname{Re}(C^{-1}(w))+C^{-1}(z)^{2}\|C^{-1}(w)\|^{2}\right)^{-1}\left(1-C^{-1}(z)C^{-1}(w)\right)$ $\displaystyle=(1+z)^{2}\|1+w\|^{2}\left(\|1+w\|^{2}(1+z)^{2}-2(1-z^{2})(1-\|w\|^{2})+(1-z)^{2}\|1-w\|^{2}\right)^{-1}$ $\displaystyle\hskip 113.81102pt\cdot(1+z)^{-1}\left((1+z)(1+w)-(1-z)(1-w)\right)(1+w)^{-1}$ $\displaystyle=(1+z)\frac{1}{4}\left(z^{2}+2z\operatorname{Re}(w)+\|w\|^{2}\right)^{-1}2\left(z+w\right)(1+\overline{w})$ $\displaystyle=\frac{1}{2}(1+z)\left(z+\overline{w}\right)^{-}(1+\overline{w})=\frac{1}{2}(1+z)k_{H^{2}(\mathbb{H}^{+})}(w,z)(1+\overline{w}).$ ∎ Now we want to show that the volume form on $\partial\mathbb{H}^{+}$ obtained in (14) is the natural volume form for the Hardy space $H^{2}(\mathbb{H}^{+})$. In fact, let $f(q)=\int_{0}^{+\infty}e^{-\zeta q}F(\zeta)d\zeta\in H^{2}(\mathbb{H}^{+})$. For any $I\in\mathbb{S}$, we can decompose the function $F$ as $F(\zeta)=F_{1}(\zeta)+F_{2}(\zeta)J$ where $J$ is an imaginary unit orthogonal to $I$ and $F_{1},F_{2}:\mathbb{R}\to L_{I}$. It is possible to prove (see [2]) that functions in $H^{2}(\mathbb{H}^{+})$ have limit at the boundary for almost any point $yI\in\partial\mathbb{H}^{+}=\\{vJ\ \|\ v>0,J\in\mathbb{S}\\}$. If we denote by $dA_{\mathbb{S}}$ the usual surface element of the unit two-dimensional sphere $\mathbb{S}$, thanks to equation (14) and to the orthogonality of $I$ and $J$, we can write $\displaystyle\int_{\partial\mathbb{H}^{+}}\|f(yI)\|^{2}dVol_{\partial\mathbb{H}^{+}}(yI)=\int_{\partial\mathbb{H}^{+}}\|f(yI)\|^{2}\frac{dVol_{Euc}(yI)}{8y^{2}}$ $\displaystyle=\int_{0}^{+\infty}\left(\int_{\mathbb{S}}\left\|\int_{0}^{+\infty}e^{-\zeta yI}F(\zeta)d\zeta\right\|^{2}\frac{y^{2}dA_{\mathbb{S}}(I)}{8y^{2}}\right)dy$ $\displaystyle=\frac{1}{8}\int_{\mathbb{S}}\left(\int_{0}^{+\infty}\left\|\int_{0}^{+\infty}e^{-\zeta yI}\left(F_{1}(\zeta)+F_{2}(\zeta)J\right)d\zeta\right\|^{2}dy\right)dA_{\mathbb{S}}(I)$ $\displaystyle=\frac{1}{8}\int_{\mathbb{S}}\left(\int_{0}^{+\infty}\left\|\int_{0}^{+\infty}e^{-\zeta yI}F_{1}(\zeta)d\zeta+\int_{0}^{+\infty}e^{-\zeta yI}F_{2}(\zeta)Jd\zeta\right\|^{2}dy\right)dA_{\mathbb{S}}(I)$ $\displaystyle=\frac{1}{8}\int_{\mathbb{S}}\left(\int_{0}^{+\infty}\left\|\int_{0}^{+\infty}e^{-\zeta yI}F_{1}(\zeta)d\zeta\right\|^{2}dy+\int_{0}^{+\infty}\left\|\int_{0}^{+\infty}e^{-\zeta yI}F_{2}(\zeta)d\zeta\right\|^{2}dy\right)dA_{\mathbb{S}}(I)$ $\displaystyle=\frac{2\pi}{8}\int_{\mathbb{S}}\left(\int_{0}^{+\infty}\left\|F_{1}(\zeta)\right\|^{2}d\zeta+\int_{0}^{+\infty}\left\|F_{2}(\zeta)\right)\|^{2}d\zeta\right)dA_{\mathbb{S}}(I)$ where the last equality is due to the classical Plancherel Theorem. Therefore, thanks again to the orthogonality of $I$ and $J$, $\int_{\partial\mathbb{H}^{+}}\|f(yI)\|^{2}dVol_{\partial\mathbb{H}^{+}}(yI)=\frac{\pi}{4}\int_{\mathbb{S}}\left(\int_{0}^{+\infty}\left\|F(\zeta)\right\|^{2}d\zeta\right)dA_{\mathbb{S}}(I)=\pi^{2}\|\|f\|\|^{2}_{H^{2}(\mathbb{H}^{+})}.$ ### 5.2 A bilateral estimate for the distance and an application to inner functions In the right half space model it is easier to prove a bilateral estimate for the distance associated with the invariant metric $h$. Fix a imaginary unit $I_{0}$ and define the projection $\pi:x+yI\mapsto x+yI_{0},$ (15) with $x$ real and $y\geq 0$. Let $d_{hyp}$ be hyperbolic distance in $\mathbb{H}_{I_{0}}^{+}=\\{x+yI_{0}:\ x>0,\ y\in\mathbb{R}\\}$: $d_{hyp}$ is the distance associated with the Riemannian metric tensor $ds_{hyp}^{2}=(dx^{2}+dy^{2})/(4x^{2})$. Let now $d_{\SS}$ be the usual spherical distance on the unit two-dimensional sphere $\SS$, associated with the metric tensor $ds^{2}_{\mathbb{S}}$. Then, the metric tensor associated with $h$ can be decomposed as $ds_{h}^{2}=ds^{2}_{hyp}+\frac{y^{2}}{4(x^{2}+y^{2})}ds^{2}_{\mathbb{S}}.$ ###### Theorem 5.2. Let $q_{j}=x_{j}+y_{j}I_{j}$, $j=1,2$, be points in $\mathbb{H}^{+}$: $x_{j}>0$, $y_{j}\geq 0$. The following estimate for the distance function $d_{h}$ associated with the metric $h$ holds: $d_{h}(q_{1},q_{2})\approx d_{hyp}(\pi(q_{1}),\pi(q_{2}))+\min\left\\{\frac{y_{j}}{\|q_{j}\|}:\ j=1,2\right\\}d_{\SS}(I_{1},I_{2}),$ where $\approx$ means that we have a lower and an upper estimate for the right hand side in terms of the left hand side, with multiplicative constants $C_{1},C_{2}$ independent of $q_{1},q_{2}$. ###### Proof. We may suppose that $y_{1}/\|q_{1}\|\leq y_{2}/\|q_{2}\|$. The upper estimate is elementary. Let $\gamma$ be a curve going from $q_{1}$ to $x_{1}+y_{1}I_{2}\in\mathbb{H}_{I_{2}}^{+}$ leaving $x=x_{1}$ and $y=y_{1}$ fixed, and varying the imaginary unit $I$ only. Suppose, more, that $I$ varies along a geodesic on $\SS$, which joins $I_{1}$ and $I_{2}$. Then, $\operatorname{length}(\gamma)=\int_{\gamma}\frac{y_{1}}{2\|q_{1}\|}ds_{\mathbb{S}}=\frac{y_{1}}{2\|q_{1}\|}d_{\SS}(I_{1},I_{2}).$ Let now $\delta$ be a hyperbolic geodesic in $L_{I_{2}}$, joining $x_{1}+y_{1}I_{2}$ and $q_{2}$: $\operatorname{length}(\delta)=d_{hyp}(\pi(q_{1}),\pi(q_{2}))$, which proves the estimate. The lower estimate is more delicate. Let $\gamma$ be a curve in $\mathbb{H}^{+}$ joining $q_{1}$ and $q_{2}$. Then, $\operatorname{length}(\gamma)=\int_{\gamma}ds_{h}\geq\int_{\pi(\gamma)}ds_{hyp}\geq d_{hyp}(\pi(q_{1}),\pi(q_{2})).$ (16) We have then to show that $\int_{\gamma}ds_{h}\gtrsim\frac{y_{1}}{\|q_{1}\|}d_{\SS}(I_{1},I_{2}).$ (17) Since the right hand side of (17) is bounded, and we have already proved (16), it suffices to show that (17) holds when $d_{hyp}(\pi(q_{1}),\pi(q_{2}))\leq 1$. By elementary hyperbolic geometry, see the “sixth model” in [7], and using the fact that dilations $p\mapsto\lambda p$ are isometric for $\lambda>0$, we can assume that $\pi(q_{1})$ and $\pi(q_{2})$ both lie in the square $Q_{n}=\\{x+yI_{0}:\ 1\leq x\leq 2,\ n\leq y\leq n+1\\}\subset L_{I_{0}}$, for some integer $n\geq 0$. Consider now $q_{3}=x_{3}+y_{3}I_{3}$, $y_{3}\geq 0$ the point along $\gamma$ which minimizes $y_{3}/\|q_{3}\|$. We can assume that $\pi(\gamma)$ (hence, $\pi(q_{3})$) is contained in $\tilde{Q}_{n}=\\{z=x+yI_{0}:\ x>0,\ y\geq 0,\ 1/2\leq x\leq 2,\ n-1/2\leq y\leq n+3/2\\}$, otherwise $\operatorname{length}(\gamma)\geq 1$ (which would imply the estimate (17) we are proving). Let $t\geq 0$ be the angle between the positive real half axis $\mathbb{R}^{+}$ and the half line originating at $0$ and passing through $\pi(q_{3})$. For $j=1,2,3$: $t_{j}\approx\sin(t_{j})=y_{j}/\|q_{j}\|.$ We have two cases. Either $y_{3}/\|q_{3}\|\geq 1/2\cdot y_{1}/\|q_{1}\|$, but then we are done because $\int_{\gamma}\frac{y}{2\|q\|}ds^{2}_{\mathbb{S}}\geq\frac{y_{1}}{2\|q_{1}\|}d_{\SS}(I_{1},I_{2}).$ Or $y_{3}/\|q_{3}\|\leq 1/2\cdot y_{1}/\|q_{1}\|$. Then $n=0$, and $\displaystyle\int_{\gamma}ds_{h}$ $\displaystyle\geq$ $\displaystyle\operatorname{length}(\pi(\gamma))\gtrsim\max(\|\pi(z_{1})-\pi(z_{3})\|,\|\pi(z_{2})-\pi(z_{3})\|)\geq\|\pi(z_{1})-\pi(z_{3})\|$ $\displaystyle\gtrsim$ $\displaystyle y_{1}\gtrsim\frac{y_{1}}{\|z_{1}\|}d_{\SS}(I_{1},I_{2}).$ Overall, $\int_{\gamma}ds_{h}\gtrsim\frac{y_{1}}{\|z_{1}\|}d_{\SS}(I_{1},I_{2})$, as wished. ∎ Changing coordinates from the right half plane to the ball, we have the same bilateral estimate in the ball model. ###### Corollary 5.3. Let $d_{g}$ be the invariant distance associated with the metric $g$ in the ball model and let $q_{1},q_{2}$ be points of $\mathbb{B}$. If $\pi$ is defined as in (15), then: $d_{g}(q_{1},q_{2})\approx d_{hyp}(\pi(q_{1}),\pi(q_{2}))+\min\left\\{\frac{y_{j}}{\|1-q_{j}^{2}\|}:\ j=1,2\right\\}d_{\SS}(I_{1},I_{2}).$ A regular function $f:\mathbb{B}\to\mathbb{H}$ is inner if (i) it maps $\mathbb{B}$ into $\mathbb{B}$; (ii) the limit as $r\to 1$ of $f$ along the radius $r\mapsto ru$ exists for $a.e.$ $u$ in $\partial\mathbb{B}$ and it has unitary norm. ###### Theorem 5.4. Let $f:\mathbb{B}\to\mathbb{B}$ be an inner function. Then, $f$ is Lipschitz with respect to the metric $g$ if and only if it is slice preserving. In this case, it is a contraction. It is well known (see [10]) that regular, bounded functions have radial limits along almost all radii, $f(e^{tI}):=\lim_{r\to 1}f(re^{tI})$ exists for $a.e.$ $(t,I)\in[0,\pi]\times\SS$. We start with a Lemma which might have independent interest; for instance, it provides a different route to prove the classification of the isometries for the metric $g$. ###### Lemma 5.5. If $\varphi:\mathbb{B}\to\mathbb{B}$ is Lipschitz with respect to the metric $g$ and $\lim_{r\to 1}\varphi(re^{tI_{1}})=e^{sJ_{1}}\in\partial\mathbb{B}$ (18) exists, with $s\in[0,\pi]$ and $J_{1}\in\SS$; then for each $I_{2}$ in $\SS$, if the limit $\lim_{r\to 1}\varphi(re^{I_{2}t})$ exists, then $\lim_{r\to 1}\varphi(re^{tI_{2}})=e^{sJ_{2}}$ (19) for some $J_{2}$ in $\SS$. The values of $t$ and $s$ in (19) are the same as in (18). ###### Proof. Let $u_{j}=e^{tI_{j}}$, with the same $t\in[0,\pi]$. By Lipschitz continuity, $\displaystyle d(\varphi(ru_{1}),\varphi(ru_{2}))$ $\displaystyle\lesssim$ $\displaystyle d(ru_{1},ru_{2})$ (20) $\displaystyle\approx$ $\displaystyle\frac{rt\|I_{1}-I_{2}\|}{(1-r)+rt}\lesssim\|I_{1}-I_{2}\|$ (21) $\displaystyle\leq$ $\displaystyle 1.$ (22) By the lower estimate in Corollary 5.3 and (20), $d_{hyp}\left(\pi\left(\varphi(ru_{1})\right),\pi\left(\varphi(ru_{2})\right)\right)\lesssim 1.$ But this and elementary hyperbolic geometry imply that, if $\lim_{r\to 1}\varphi(ru_{1})=e^{sJ_{1}}$, then the limit $\lim_{r\to 1}\pi\left(\varphi(ru_{2})\right)=L$ exists and $L=e^{sI_{0}}$ (recall that $\pi:\mathbb{B}\to L_{I_{0}}$). Since $\lim_{r\to 1}\varphi(re^{tI_{2}})=L$ exists by hypothesis and $\pi$ is continuous, it must be $\pi(L)=e^{sI_{0}}$, then $L=e^{sJ_{2}}$ for some $J_{2}$ in $\SS$. ∎ The statement of Lemma 5.5 can be sharpened in several ways. For instance, the Lipschitz assumption might be weakened to a sub-exponential growth assumption. We proceed with the proof of Theorem 5.4. ###### Proof of Theorem 5.4. Being inner, $f$ has boundary limits along radii $r\mapsto re^{tI}$ for $a.e.$ $I$ in $\SS$ and $t$ in $[0,\pi]$. We write for such couples of $(t,I)$: $f(e^{tI}):=\lim_{r\to 1}f(re^{tI})$. We can assume without loss of generality that the limit exists for two antipodal imaginary units $L$ and $-L$, and hence, in view of the Representation Formula 2.5, for any $L\in\mathbb{S}$. If $f$ is regular and Lipschitz with respect to the distance $d_{g}$, thanks on the one hand to the Representation Formula 2.5, on the other hand to Lemma 5.5, we have that for any $I\in\mathbb{S}$ $b(t)+Ic(t)=f(e^{tI})=e^{s(t)J(s,I)}$ where $b(t),c(t)\in\mathbb{H}$, and $s(t)\in[0,2\pi]$ and $J(t,I)\in\mathbb{S}$. Then $\operatorname{Re}(f(e^{tI}))=\operatorname{Re}(f(e^{tL}))$ for any $L\in\mathbb{S}$ and in particular for $L=-I$, which gives $\operatorname{Re}(b(t))-\langle I,c(t)\rangle=\operatorname{Re}(b(t)+Ic(t))=\operatorname{Re}(b(t)-Ic(t))=\operatorname{Re}(b(t))+\langle I,c(t)\rangle$ (where $\langle\cdot,\cdot\rangle$ denotes the standard scalar product in $\mathbb{R}^{4}$). Since $I$ is any imaginary unit, we necessarily have that $c(t)\in\mathbb{R}$. Also, comparing imaginary parts, for any $L_{1},L_{2}\in\mathbb{S}$ we have $\|\operatorname{Im}(f(e^{tL_{1}}))\|=\|\operatorname{Im}(f(e^{tL_{2}}))\|$. Then, if $b=b_{0}+b_{1}K$ with $b_{0},b_{1}\in\mathbb{R}$, $K\in\mathbb{S}$ (omitting the dependence on $t$), when $L_{1}=K$ and $L_{2}=-K$ we get $\|b_{1}+c\|=\|\operatorname{Im}(b_{0}+b_{1}K+cK)\|=\|\operatorname{Im}(f(e^{tK}))\|=\|\operatorname{Im}(f(e^{-tK}))\|=\|\operatorname{Im}(b_{0}+b_{1}K-cK)\|=\|b_{1}-c\|.$ Therefore almost every $t\in[0,\pi]$ belongs to $D\cup E$: $D=\\{t:\ c(t)=0\\},\ E=\\{t:\ b_{1}(t)=0\\}.$ Consider first the case when $t\in D$ holds $a.e.$. Then $f(e^{tI})=b(t)$ for almost every $t$. Since boundary values uniquely identify $f$ (see [10]) and by invariance under rotations of $\SS$, we deduce that $f(re^{tI})=\Phi(r,t),$ for some function $\Phi$. In particular, $f$ can not be open ($dim(f(\mathbb{B}))\leq 2$), hence (see Theorem 7.4 in [12]) it must be constant; thus it is not inner. Then $E$ has positive measure. For $t$ in $E$, $b(t)+Ic(t)=f(e^{tI})=e^{s(t)J(t,I)}$ with $b$ and $c$ real valued, hence $J=I$: $f(e^{tI})=e^{s(t)I}$ (23) for $t$ in $F$. By the Splitting Lemma 2.7, if $J\perp I$ is fixed in $\SS$, then there are holomorphic functions $F,G$ on $\mathbb{B}_{I}$ such that $f(re^{\tau I})=F(re^{\tau I})+G(re^{\tau I})J.$ By (23), $G(e^{tI})=0$ for $t$ in $E$. Since $E$ has positive measure, this implies that $G$ vanishes identically and hence $f(re^{\tau I})=F(re^{\tau I})$ for all $0\leq r<1$ and $0\leq\tau\leq\pi$. That is, $f$ is slice preserving. We have to verify that $f$ is a contraction with respect to the metric $g$, and this can be verified at the infinitesimal level. Let $q$ be a point in a fixed slice $\mathbb{B}\cap L_{I}$. (i) Since $f$ is slice preserving, its restriction to $\mathbb{B}\cap L_{I}$ is an inner function in the one dimensional sense, hence it is a contraction of the Poincaré-hyperbolic metric on $\mathbb{B}\cap L_{I}$. (ii) On the other hand, preserving the slices, $f$ acts isometrically in the $\SS$ variables, with respect to the spherical metric on $\SS$. (iii) Now, the space tangent to $\mathbb{B}\cap L_{I}$ at $q$ and the space tangent to $\operatorname{Re}q+\SS$ at $q$ form an orthogonal decomposition, with respect to the metric $g$, of the space tangent to $\mathbb{B}$. From the expression for $g$ given in (8) and facts (i)-(iii) one easily deduces that $g$ is a contraction. ∎ ## References * [1] D. Alpay, F. Colombo, I. Sabadini, Schur functions and their realizations in the slice hyperholomorphic setting, Integral Equations Operator Theory 72 (2012), 25–289. * [2] D. Alpay, F. Colombo, I. Lewkowicz, I. Sabadini, Realizations of slice hyperholomorphic generalized contractive and positive functions, Preprint arXiv:1310.1035v1 [math.CV] (2013). * [3] N. Arcozzi, R. Rochberg, E. Sawyer, B. D. Wick, Distance functions for reproducing kernel Hilbert spaces, Function spaces in modern analysis, 25–53, Contemp. Math., 547, Amer. Math. Soc., Providence, RI, 2011. * [4] N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. 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Sarfatti, Quaternionic Hardy spaces, Preprint, www.math.unifi.it/users/sarfatti/Hardy.pdf, (2013). * [11] M. P. do Carmo, Differential geometry of curves and surfaces, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1976. * [12] G. Gentili, C. Stoppato, D. C. Struppa, Regular functions of a quaternionic variable, Springer Monographs in Mathematics, Springer, Berlin-Heidelberg, 2013. * [13] G. Gentili, D. C. Struppa, A new theory of regular functions of a quaternionic variable, Adv. Math., 216 (2007), 279–301. * [14] R. Ghiloni, V. Moretti, A. Perotti, Continuous slice functional calculus in quaternionic Hilbert spaces, Rev. Math. Phys. 25 (2013). * [15] M. Gromov, Metric structures for Riemannian and non-Riemannian spaces, Based on the 1981 French original, translated from the French by Sean Michael Bates. Progress in Mathematics, 152, Birkh user Boston, Inc., Boston, MA, 1999. * [16] J. E. McCarthy, Boundary values and Cowen-Douglas curvature, J. Funct. Anal. 137 (1996), 1–18. * [17] B. O’Neill, Semi-Riemannian geometry, with applications to relativity, Pure and Applied Mathematics, 103. Academic Press, Inc., New York, 1983. * [18] C. Stoppato, Regular Moebius transformations of the space of quaternions, Ann. Global Anal. Geom. 39 (2011), 387–401. Nicola Arcozzi Dipartimento di Matematica, Università di Bologna Piazza di Porta San Donato 5, 40126 Bologna, Italy, [email protected] Giulia Sarfatti Dipartimento di Matematica, Università di Bologna Piazza di Porta San Donato 5, 40126 Bologna, Italy, [email protected]	arxiv-papers	2013-12-20T12:06:33	2024-09-04T02:49:55.715651	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Nicola Arcozzi, Giulia Sarfatti", "submitter": "Nicola Arcozzi", "url": "https://arxiv.org/abs/1312.5906" }
1312.5915	# Tightened estimation can improve the key rate of MDI-QKD by more than 100% Yi-Heng Zhou1,Zong-Wen Yu2, and Xiang-Bin Wang1,3111Email Address: [email protected] 1State Key Laboratory of Low Dimensional Quantum Physics, Tsinghua University, Beijing 100084, People s Republic of China 2Data Communication Science and Technology Research Institute, Beijing 100191, China 3 Shandong Academy of Information and Communication Technology, Jinan 250101, People s Republic of China ###### Abstract We present formulas to tightly upper bound the phase-flip errors in decoy state method by using 4 intensities. Our result compressed the bound to about a quarter of known result for MDI-QKD. Based on this, we find that the key rate is improved by more than 100% given weak coherent state sources (WCS), and even more than 200% with the heralded single-photon sources (HSPS). ###### pacs: 03.67.Dd, 42.81.Gs, 03.67.Hk ## I Introduction One of the most fascinating properties of quantum key distribution (QKD) is its unconditional security in theory BB84 ; GRTZ02 . However, most practical devices behave differently form the theoretical models assumed in the security proof. Security for real set-ups of QKD BB84 ; GRTZ02 has become a major problem in this area in the recent years. The insecurity loopholes are mainly due to the imperfect single-photon source and the limited efficiency of the detectors. Fortunately, by using the decoy-state method ILM ; H03 ; wang05 ; LMC05 ; AYKI ; haya ; peng ; wangyang ; rep ; njp , it has been shown that the unconditional security of QKD can still be assured with an imperfect single- photon source PNS1 ; PNS . Besides the source imperfection, the defect in the detectors is another threaten to the security lyderson . To patch up this, several approaches have been proposed. One is the device independent QKD (DI-QKD) ind1 . This technique does not require detailed knowledge of how QKD devices work and can prove security based on the violation of a Bell inequality. Recently, an idea of measurement device independent QKD (MDI-QKD) was proposed based on the idea of entanglement swapping ind3 ; ind2 . There, one can make secure QKD simply by virtual entanglement swapping, i.e., neither Alice and Bob performs any measurement, but they only send out quantum signals to the relay which can be controlled by the un-trusted third party (UTP). After Alice and Bob send out signals, they wait for UTP’s announcement of weather he has obtained the successful detection, and proceed to the standard postprocessing of their sifted data, such as error rate estimation, error correction, and privacy amplification. The only assumption needed in MDI-QKD is that the preparation of the quantum signal sources by Alice and Bob. In practice, in order to obtain a higher key rate or realize a longer distance key distribution, we’d better use laser sources with decoy state method. This has been discussed in Ref. ind2 , and explicit formulas for the practical decoy- state implementation with only three different states was first presented in wangPRA2013 , and then further studied both experimentally tittel1 ; tittel2 ; liuyang and theoreticallywangArxiv ; lopa ; curtty ; Wang3int ; Wang3improve ; Wang3g ; WangModel . In the previous works, the authors considered the effect of finite number of decoy states, but their key rates are notably away form the result obtained with the infinite decoy-state method. The major reason is that the upper bounds of the phase-flip error estimated with these methods are not very tightened. Here in this work, we show how to tightly formulate the upper bound of the phase-flip errors in decoy state method for the regular BB84 protocol and MDI- QKD. Our result compressed the bound to about a quarter of known result for MDI-QKD with WCS, and even about one fifth with HSPS. To achieve the result, we only need 4-intensity decoy state method. Based on this, we find that the key rate is improved by more than 100% with WCS, and even more than 200% with HSPS. ## II Traditional Decoy-state method with only 4 intensities for BB84 protocol In the four-intensity protocol, Alice has four (virtual) sources, the vacuum source $\rho_{0}=\|0\rangle\langle 0\|$ which prepares vacuum pulses, two decoy sources $\rho_{x},\rho_{y}$ which prepare decoy pulses, and the signal source $\rho_{z}$ which prepares signal pulses. In photon-number space, we suppose $\rho_{l}=\sum_{k}a_{k}^{l}\|k\rangle\langle k\|,\quad(l=x,y,z),$ (1) where $\|k\rangle$ is the $k$-photon Fock state, $a_{k}^{l}\geq 0$ for all $k\geq 0$. At each time, Alice will randomly select one of her 4 sources to emit a pulse. For pose-processing, Alice and Bob evaluate the data with the same basis. With the observed total gains and error rates, the final secure key rate can be calculated by the following formula ILM $R=a_{1}^{z}s_{1}[1-H(e_{1})]-S_{z}H(E_{z}),$ (2) where $S_{z}$ and $E_{z}$ denote, respectively, the total gain and error rate of the signal state $\rho_{z}$. $s_{1}$ and $e_{1}$ are, respectively, the fraction and error rate of detection events by Bob that have originated form single-photon pulses emitted by Alice, and $H$ is the binary Shannon entropy. In this paper, we use the capital letter $S(E)$ for known total gains (error rates) and the lowercase letter $s,e$ for unknown variables. In order to estimate the final key rate of this protocol, we need find out the lower bound of the yield $s_{1}$ and the upper bound of the error rate $e_{1}$. In the coming subsection, we devote to estimate the lower bound of $s_{1}$ firstly. ### II.1 The lower bound of the yield $s_{1}$ With given the different sources, Alice randomly chooses quantum channels with different photon-number states. Thus, the total gain with source $\rho_{l}$ can be expressed into the following convex form $S_{l}=\sum_{k\geq 0}a_{k}^{l}s_{k},\quad(l=x,y,z),$ (3) where $s_{k}$ is the yield of an $k$-photon pulse. In order to obtain an effective lower bound of $s_{1}$, we need eliminate the gains associated with the vacuum state from the total gain firstly. Considering this fact, we can rewrite the relations in Eq.(3) into $\tilde{S}_{l}=\sum_{k\geq 1}a_{k}^{l}s_{k},\quad(l=x,y,z),$ (4) where we define $\tilde{S}_{l}=S_{l}-a_{0}^{l}S_{0},\quad(l=x,y,z),$ (5) with $S_{0}$ being the gain of the vacuum source. The lower bound of $s_{1}$ has already been studied wang05 ; LMC05 ; AYKI ; haya . As presented in the previous works, if Alice has 3 different sources $\rho_{o}=\|0\rangle\langle 0\|,\rho_{x},\rho_{y}$, the lower bound of $s_{1}$ can be write into $\underline{s}_{1}(x,y)=\frac{a_{2}^{y}\tilde{S}_{x}-a_{2}^{x}\tilde{S}_{y}}{a_{1}^{x}a_{2}^{y}-a_{1}^{y}a_{2}^{x}},$ (6) under the condition $\frac{a_{k}^{y}}{a_{k}^{x}}\geq\frac{a_{2}^{y}}{a_{2}^{x}}\geq\frac{a_{1}^{y}}{a_{1}^{x}},$ (7) for all $k\geq 2$. It is worth pointing out that the lower bound given by Eq.(6) does not only apply to the weak coherent source, but also to any source as long as it meets the condition in Eq.(7). In this 4-intensity protocol, there are three different no-vacuum sources. In order to get the lower bound of $s_{1}$ directly from Eq.(6), we also need to introduce the following condition $\frac{a_{k}^{z}}{a_{k}^{y}}\geq\frac{a_{2}^{z}}{a_{2}^{y}}\geq\frac{a_{1}^{z}}{a_{1}^{y}},$ (8) for all $k\geq 2$. Then we can obtain some effective lower bounds of $s_{1}$ with Eq.(6) by choosing any two different sources from $\rho_{x},\rho_{y}$ and $\rho_{z}$. After this, we can use the maximum one as the estimation of the lower bound of $s_{1}$ for this 4-intensity protocol $\underline{s}_{1}^{\prime}=\max\\{\underline{s}_{1}(x,y),\underline{s}_{1}(x,z),\underline{s}_{1}(y,z)\\},$ (9) where $\underline{s}_{1}(l,r)$ is just $\underline{s}_{1}(x,y)$ in Eq.(6) with changing $x$ and $y$ into $l$ and $r$ respectively. In order to simplify this expression and derive other main results in this work, we need to define the following function with sources $\rho_{x},\rho_{y}$ and $\rho_{z}$ $\mathcal{G}(i,j,k)=(g_{i}^{x}-g_{j}^{x})(g_{j}^{y}-g_{k}^{y})-(g_{i}^{y}-g_{j}^{y})(g_{j}^{x}-g_{k}^{x}),$ (10) where $g_{m}^{l}=\frac{a_{m}^{l}}{a_{m}^{z}},\quad(m\geq 1;\,l=x,y,z).$ (11) Now, we assume that the states $\rho_{x},\rho_{y}$ and $\rho_{z}$ satisfy the following important condition $\mathcal{G}(i,j,k)\geq 0,$ (12) when $k-j\geq j-i\geq 0$. In Appendix A, we will show that the imperfect sources used in practice such as the weak coherent sources, the heralded source out of the parametric-down conversion, satisfy all the above conditions given by Eqs.(7,8,12). With these conditions presented in Eqs.(7,8,12), we can simplify the lower bound $\underline{s}_{1}^{\prime}$ by $\underline{s}_{1}^{\prime}=\underline{s}_{1}(x,y).$ (13) The detailed proof of this conclusion can be found in Appendix B. ### II.2 The upper bound of the error rate $e_{1}$ In order to estimate the final key rate, we also need the upper bound of the error rate $e_{1}$. In the previous works, the upper bound of $e_{1}$ is obtained by putting the errors with all muti-photon pulses on the error with the single-photon pulse. Explicitly, we can write the upper bound of $e_{1}$ with 3-intensity decoy state method as follows $\overline{e}_{1}=\frac{S_{x}E_{x}-a_{0}^{x}S_{0}E_{0}}{a_{1}^{x}\underline{s}_{1}},$ (14) where $\underline{s}_{1}$ is the lower bound of $s_{1}$, $S_{x}$ and $E_{x}$ are the total gain and error rate of the source $\rho_{x}$ respectively, $S_{0}$ and $E_{0}$ are the total gain and error rate of the vacuum source respectively. With the 3-intensity decoy state method, we can not find out a more better explicit formula to estimate the upper bound of $e_{1}$. In order to get a more tightened upper bound, we need to introduce one more source. This is the main reason for us to consider the 4-intensity decoy state method. Similar to the gain, the error rate can depend on the photon number. Let us denote $e_{k}$ as the error of an $k$-photon pulse. The error rate $E_{l}$ for the source $\rho_{l}(l=x,y,z)$ can be given by $T_{l}=S_{l}E_{l}=\sum_{k\geq 0}a_{k}^{l}s_{k}e_{k},\quad(l=x,y,z).$ (15) If we denote $t_{k}=s_{k}e_{k}$, and $\tilde{T}_{l}=T_{l}-a_{0}^{l}T_{0},\quad(l=x,y,z),$ (16) Eq.(15) can be rewrite into the following equivalent form $\tilde{T}_{l}=\sum_{k\geq 1}a_{k}^{l}t_{k},\quad(l=x,y,z).$ (17) In this 4-intensity decoy state method, there are 3 different no-vacuum sources can be used for Alice. Then we have 3 different relations about $t_{1}$ which are presented in Eq.(17). With these 3 relations, by eliminating the variables $t_{2}$ and $t_{3}$, we obtain the expression of $t_{1}$ as follows $t_{1}=\overline{t}_{1}^{\prime}+\sum_{k\geq 4}f_{t_{1}}(k)t_{k},$ (18) where $\overline{t}_{1}^{\prime}=\frac{a_{1}^{z}a_{2}^{z}a_{3}^{z}}{\mathcal{G}(1,2,3)}\left[(a_{3}^{z}a_{2}^{y}-a_{3}^{y}a_{2}^{z})\tilde{T}_{x}-(a_{3}^{z}a_{2}^{x}-a_{3}^{x}a_{2}^{z})\tilde{T}_{y}+(a_{3}^{y}a_{2}^{x}-a_{3}^{x}a_{2}^{y})\tilde{T}_{z}\right],$ (19) and $f_{t_{1}}(k)=-\frac{\mathcal{G}(2,3,k)}{\mathcal{G}(1,2,3)},\quad(k\geq 4),$ (20) with $\mathcal{G}(1,2,3)$ being defined in Eq.(10). Under the condition presented in Eq.(12), we can easily find out that $f_{t_{1}}(k)\leq 0$ for all $k\geq 4$. Then we can conclude that $\overline{t}_{1}^{\prime}$ given by Eq.(19) is actually a upper bound of $t_{1}$. Then the upper bound of $e_{1}$ can be given by $\overline{e}_{1}^{\prime}=\frac{\overline{t}_{1}^{\prime}}{\underline{s}_{1}^{\prime}}$ (21) where $\underline{s}_{1}^{\prime}$ is the lower bound of $s_{1}$ given by Eq.(13). ### II.3 Numerical Simulation for BB84 protocol Figure 1: (Color online) The ratio of the upper bound of $e_{1}$ between the estimations obtained by using 4-intensity and 3-intensity decoy state methods, i.e., $\overline{e}_{1}^{\prime}/\overline{e}_{1}$, versus the total channel transmission loss. We set $\mu_{1}=0.2$ for decoy state. Figure 2: (Color online) The relative value between the optimal key rate obtained with different methods and the asymptotic limit of the infinite decoy-state method versus the total channel transmission loss. We set $\mu_{1}=0.2$ for decoy states. Figure 3: (Color online) The ratio of the optimal key rate between the estimations obtained by using 4-intensity and 3-intensity decoy state methods versus the total channel transmission loss. We set $\mu_{1}=0.2$ for decoy state. Table 1: List of experimental parameters used in numerical simulations: $e_{0}$ is the error rate of background, $e_{d}$ is the misalignment-error probability; $p_{d}$ is the dark count rate of Bob’s per detector; $\eta_{v}$ is the detection efficiency of Alice’s detector; $p_{dv}$ is the dark count rate of Alice’s detector. $e_{0}$ \| $e_{d}$ \| $p_{d}$ \| $\eta_{v}$ \| $p_{dv}$ ---\|---\|---\|---\|--- 0.5 \| 1.5% \| $3.0\times 10^{-6}$ \| 0.75 \| $1.0\times 10^{-6}$ In this subsection, we will present some numerical simulations to compare the results obtained by using the 3-intensity decoy state method with the results of 4-intensity method for the regular BB84 protocol. As discussed before, we know that the methods presented in this work does not only apply to the weak coherent sources (WCS). Actually, it can be used to estimate the final key rate for any sources that satisfy the condition given by Eq.(7) for the 3-intensity method, and the conditions given by Eqs.(7,8,12) for the 4-intensity method. Below for simplicity, we consider the following two cases. In the first case, we suppose that Alice use WCS. In the second one, we suppose she use the heralded single-photon sources (HSPS) with possion distributions wangArxiv . The Bob’s detectors are identical, i.e., they have the same dark count rate and detection efficiency, and the detection efficiency does not depend on the incoming states. Suppose the overall transmission probability of each photon is $\xi$. In a normal channel, it is common to assume independence between the behaviors of the $n$ photons. Therefore, the transmission efficiency for $n$-photon pulses $\xi_{n}$ is given by $\xi_{n}=1-(1-\xi)^{n}.$ For fair comparison, we use the same parameter values used in UrsinNP2007 for our numerical evaluation. For simplicity, we shall put the detection efficiency to the overall transmittance $\eta=\xi\zeta$. We assume all detectors of Bob have the same detection efficiency $\zeta$ and dark count rate $p_{d}$. In the second case with HSPS, we assume the detector of Alice has the detection efficiency $\eta_{v}$ and dark count rate $p_{dv}$. The values of these parameters are presented in Table 1. With this, the total gains $S_{\mu_{i}}$ and error rates $S_{\mu_{i}}E_{\mu_{i}}$ of Alice’s intensity $\mu_{i}$ ($i=0,1,2$ for 3-intensity method, $i=0,1,2,3$ for 4-intensity method) can be calculated. By using these values, we can estimate the lower bounds of yield $s_{1}$ with Eq.(6) and Eq.(13) for 3-intensity and 4-intensity decoy state methods respectively. Also, we can estimate the upper bounds of error rate $e_{1}$ with Eq.(14) and Eq.(21) for 3-intensity and 4-intensity decoy sate methods respectively. Furthermore, with these parameters, we can estimate the final key rate $R$ of this protocol with Eq.(2). If we fix the density(ies) of the decoy-state source(s) used by Alice, the final key rate will change with Alice taking different intensities for hers signal-state pulses. Here, in order to make a rational and effective comparison, we set the intensities of the decoy source in 3-intensity method and the first decoy source in 4-intensity method are the same and $\mu_{1}=0.2$; let the intensity of the second decoy source in 4-intensity method to be the optimal intensity of signal source in 3-intensity method and assume $\mu_{2}>\mu_{1}$. With these preparations, we can conclude that the lower bounds of $s_{1}$ estimated by using the 3-intensity and 4-intensity decoy state methods are the same, i.e., $\underline{s}_{1}=\underline{s}_{1}^{\prime}$. In order to see more clearly, in Fig.1, we plot the ratio of the upper bound of $e_{1}$ between the estimations obtained by using 4-intensity and 3-intensity decoy state methods, i.e., $\overline{e}_{1}^{\prime}/\overline{e}_{1}$. The relative value between the optimal key rate obtained with different methods and the asymptotic limit of the infinite decoy-state method are shown in Fig.2. In order to clarify the superiority of the 4-intensity decoy state method, we plot the ratio of the optimal key rate between the results obtained by using 4-intensity and 3-intensity decoy state methods in Fig.3. In Fig.1 and Fig.3, the blue dashed lines are obtained with WCS, the red solid lines are obtained with HSPS. In Fig.2, the black dotted line and green dash-dot line are the results obtained by using 3-intensity decoy state method with WCS and HSPS respectively, the blue dashed line and the red solid line are the results obtained by using 4-intensity decoy state with WCS and HSPS respectively. With these 3 figures, we can conclude that the results obtained by using the 4-intensity decoy state method are better than the results of 3-intensity method. But it only have a litter improvement. ## III Tightened formula for decoy-state MDI-QKD with 4 intensities In the protocol, each time a pulse-pair (two-pulse state) is sent to the relay for detection. The relay is controlled by an UTP. The UTP will announce whether the pulse-pair has caused a successful event. Those bits corresponding to successful events will be post-selected and further processed for the final key. Since real set-ups only use imperfect single-photon sources, we need the decoy-state method for security. We assume Alice (Bob) has four sources, $o_{A},x_{A},y_{A},z_{A}$ ($o_{B},x_{B},y_{B},z_{B}$) which can only emit four different states $\rho_{r_{A}}(\rho_{r_{B}})$,$(r=o,x,y,z)$. In the following discussion, we assume $o_{A}$ and $o_{B}$ are two vacuum sources. In photon number space, we have $\rho_{o_{A}}=\|0\rangle\langle 0\|,\rho_{o_{B}}=\|0\rangle\langle 0\|$. For the others, suppose $\rho_{r_{A}}=\sum_{k}a_{k}^{r}\|k\rangle\langle k\|,\rho_{r_{B}}=\sum_{k}b_{k}^{r}\|k\rangle\langle k\|,(r=x,y,z).$ In order to obtain the main results, we also need to introduce the following function $\mathcal{H}(i,j,k)=(h_{i}^{x}-h_{j}^{x})(h_{j}^{y}-h_{k}^{y})-(h_{i}^{y}-h_{j}^{y})(h_{j}^{x}-h_{k}^{x}),$ (22) where $h_{n}^{l}=\frac{b_{n}^{l}}{b_{n}^{z}},\quad(n\geq 1,\,l=x,y,z).$ (23) Now, we assume that the states $\rho_{x_{A(B)}},\rho_{y_{A(B)}}$ and $\rho_{z_{A(B)}}$ satisfy the following important conditions: $\frac{c_{k}^{z}}{c_{k}^{y}}\geq\frac{c_{2}^{z}}{c_{2}^{y}}\geq\frac{c_{1}^{z}}{c_{1}^{y}},\quad\frac{c_{k}^{y}}{c_{k}^{x}}\geq\frac{c_{2}^{y}}{c_{2}^{x}}\geq\frac{c_{1}^{y}}{c_{1}^{x}},\quad(c=a,b),$ (24) for $k\geq 2$, and $\mathcal{G}(i,j,k)\geq 0,\quad\mathcal{H}(i,j,k)\geq 0,$ (25) when $k-j\geq j-i\geq 0$. Similar to $\mathcal{G}$, the imperfect sources used in practice such as the coherent state source, the heralded source out of the parametric-down conversion, satisfy the above restrictions. Given a specific type of source, the above listed different states have different averaged photon numbers (intensities), therefore the states can be obtained by controlling the light intensities. At each time, Alice will randomly select one of her 3 sources to emit a pulse, and so does Bob. The pulse from Alice and the pulse from Bob form a pulse pair and are sent to the un-trusted relay. We regard equivalently that each time a two-pulse source is selected and a pulse pair (one pulse from Alice, one pulse from Bob) is emitted. For post-processing, Alice and Bob evaluate the data sent in two bases separately. The $Z$-basis is used for key generation, while the $X$-basis is used for testing against tampering and the purpose of quantifying the amount of privacy amplification needed. With the observed total gains and error rates, we can calculate the final secure key rate with the following formula ind2 $R=a_{1}^{z}b_{1}^{z}s_{11}^{Z}[1-H(e_{11}^{X})]-S_{z_{A}z_{B}}^{Z}fH(E_{z_{A}z_{B}}^{Z}),$ (26) where $S_{z_{A}z_{B}}^{Z}$ and $E_{z_{A}z_{B}}^{Z}$ denote, respectively, the gain and error rate in the $Z$-basis when both Alice and Bob use $z$-source $\rho_{z_{A}}$ and $\rho_{z_{B}}$; $f$ is the efficiency factor of the error correction method used; $s_{11}^{Z}$ and $e_{11}^{X}$ are the gain and error rate when both Alice and Bob send single-photon states. In this paper, we use capital letter $Z(X)$ for the bases and the lowercase letter $x,y,z$ for the different sources. In order to estimate the final key rate of this protocol, we need find out the lower bound of the yield $s_{11}$ and the upper bound of the error rate $e_{11}$. ### III.1 The lower bound of the yield $s_{11}$ With given the different sources, Alice and Bob randomly choose quantum channels with different photon-number states. Thus, the total gain can be expressed into the following convex form $S_{lr}=\sum_{j,k\geq 0}a_{j}^{l}b_{k}^{r}s_{jk},\quad(l,r=x,y,z),$ (27) when Alice and Bob send pulses with $\rho_{l_{A}}$ and $\rho_{r_{B}}$ respectively. Here and after, we omit the subscripts $A$ and $B$ without causing any ambiguity. It is well-known that, in order to obtain an effective lower bound of $s_{11}$, we need eliminate the gains associated with the vacuum state from the total gain firstly. Considering this fact, we can rewrite the relation in Eq.(27) into $\tilde{S}_{lr}=\sum_{j,k\geq 1}a_{j}^{l}b_{k}^{r}s_{jk},\quad(l,r=x,y,z),$ (28) with $\tilde{S}_{lr}=S_{lr}-a_{0}^{l}S_{0r}-b_{0}^{r}S_{l0}+a_{0}^{l}b_{0}^{r}S_{00},\quad(l,r=x,y,z).$ (29) The lower bound of $s_{11}$ has already been exhaustive studied for 3-intensity decoy state MDI-QKD protocol wangPRA2013 ; wangArxiv ; Wang3int ; Wang3improve ; Wang3g . Until now, the most tightly explicit formula to calculate the lower bound of $s_{11}$ is given in Ref.Wang3int . As presented in Ref.Wang3int , the lower bound of $s_{11}$ with 3 different sources ($o_{A},l_{A},r_{A}$ and $o_{B},l_{B},r_{B}$) used in each side of Alice and Bob can be expressed as $\underline{s}_{11}(l,r)=\frac{(a_{1}^{l}a_{2}^{r}b_{1}^{l}b_{2}^{r}-a_{1}^{r}a_{2}^{l}b_{1}^{r}b_{2}^{l})\tilde{S}_{ll}-b_{1}^{l}b_{2}^{l}(a_{1}^{l}a_{2}^{r}-a_{1}^{r}a_{2}^{l})\tilde{S}_{lr}-a_{1}^{l}a_{2}^{l}(b_{1}^{l}b_{2}^{r}-b_{1}^{r}b_{2}^{l})\tilde{S}_{rl}}{a_{1}^{l}b_{1}^{l}(a_{1}^{l}a_{2}^{r}-a_{1}^{r}a_{2}^{l})(b_{1}^{l}b_{2}^{r}-b_{1}^{r}b_{2}^{l})},$ (30) under the condition $\frac{a_{k}^{r}}{a_{k}^{l}}\geq\frac{a_{2}^{r}}{a_{2}^{l}}\geq\frac{a_{1}^{r}}{a_{1}^{l}},\quad\frac{b_{k}^{r}}{b_{k}^{l}}\geq\frac{b_{2}^{r}}{b_{2}^{l}}\geq\frac{b_{1}^{r}}{b_{1}^{l}},$ for all $k\geq 2$, where $\tilde{S}_{ll},\tilde{S}_{lr},\tilde{S}_{rl}$ are the amended gains defined by Eq.(28). In this 4-intensity protocol, there are 3 no-vacuum sources. We can estimate the effective lower bounds of $s_{11}$ with Eq.(30) by choosing $l$ and $r$ as any two different sources from $x,y,z$. Then we can use the maximum one as the lower bound of $s_{11}$ for this 4-intensity protocol $\underline{s}_{11}^{\prime}=\max\\{\underline{s}_{11}(x,y),\underline{s}_{11}(x,z),\underline{s}_{11}(y,z)\\}.$ (31) Actually, under the assumptions given by Eqs.(24-25), we can simplify the lower bound of $s_{11}$ in Eq.(31) by choosing the lowest two sources at each sides of Alice and Bob, such that $\underline{s}_{11}^{\prime}=\underline{s}_{11}(x,y).$ (32) The detailed proof of this conclusion can be found in appendix B. ### III.2 The upper bound of the error rate $e_{11}$ In order to estimate the final key rate, we also need the upper bound of the error rate $e_{11}$. In previous works, the upper bound of $e_{11}$ is obtained by putting the errors with all multi-photon pairs on the error with the single-photon pair. Explicitly, we can write the upper bound of $e_{11}$ with 3-intensity decoy state method as follows $\overline{e}_{11}=\frac{S_{xx}E_{xx}-a_{0}^{x}S_{0x}E_{0x}-b_{0}^{x}S_{x0}E_{x0}+a_{0}^{x}b_{0}^{x}S_{00}E_{00}}{a_{1}^{x}b_{1}^{x}\underline{s}_{11}},$ (33) where $\underline{s}_{11}$ is the lower bound of $s_{11}$, $S_{lr}$ and $E_{lr}$ are the total gain and error rate when Alice use the source $\rho_{l_{A}}$ and Bob use the source $\rho_{r_{B}}$ respectively. With the numerical results presented in the third subsection of this part, we know that the upper bound obtained with this method is too rough to get an tight estimation of the final key rate comparing with the results obtained by using the infinite-decoy sate method. In order to find out a more tightened upper bound of $e_{11}$, we need introduce one more source in each side of Alice and Bob. This is the main reason for us to consider the 4-intensity decoy state method for MDI-QKD. As expected, we can find out a more tightened upper bound of $e_{11}$ for this protocol. Similar to the total gain, the error rate can be write into the following convex expressions $\tilde{T}_{lr}=\sum_{j,k\geq 1}a_{j}^{l}b_{k}^{r}t_{jk},\quad(l,r=x,y,z),$ (34) where $T_{lr}=E_{lr}S_{lr}$, $t_{jk}=s_{jk}e_{jk}$, and $\tilde{T}_{lr}=T_{lr}-a_{0}^{l}T_{0r}-b_{0}^{r}T_{l0}+a_{0}^{l}b_{0}^{r}T_{00},\quad(l,r=x,y,z).$ (35) In this 4-intensity protocol, there are 3 different no-vacuum sources in each side of Alice and Bob. Then we have 9 different relations about $e_{11}$ which are given by Eq.(34). With these 9 relations, by eliminating the variables $t_{12},t_{21},t_{22},t_{13},t_{23},t_{33},t_{32},t_{31}$, we obtain the expression of $t_{11}$ $t_{11}=\overline{t}_{11}^{\prime}+\sum_{(m,n)\in J_{0}}f_{t_{11}}(m,n)t_{mn},$ (36) where $J_{0}=\\{(m,n)\|m,n\geq 1;m+n\geq 5;(m,n)\neq(2,3);(m,n)\neq(3,3);(m,n)\neq(3,2)\\}$, $\overline{t}_{11}^{\prime}=\frac{(a_{2}^{y}a_{3}^{z}-a_{2}^{z}a_{3}^{y})\mathcal{T}_{b}^{x}-(a_{2}^{x}a_{3}^{z}-a_{2}^{z}a_{3}^{x})\mathcal{T}_{b}^{y}+(a_{2}^{x}a_{3}^{y}-a_{2}^{y}a_{3}^{x})\mathcal{T}_{b}^{z}}{a_{1}^{z}a_{2}^{z}a_{3}^{z}\mathcal{G}(1,2,3)b_{1}^{z}b_{2}^{z}b_{3}^{z}\mathcal{H}(1,2,3)},$ (37) and $f_{t_{11}}(m,n)=-\frac{a_{m}^{z}\mathcal{G}^{\prime}(m)b_{n}^{z}\mathcal{H}^{\prime}(n)}{a_{1}^{z}\mathcal{G}(1,2,3)b_{1}^{z}\mathcal{H}(1,2,3)},$ (38) with $\mathcal{T}_{b}^{l}=(h_{2}^{y}-h_{3}^{y})\tilde{T}_{lx}-(h_{2}^{x}-h_{3}^{x})\tilde{T}_{ly}+(h_{3}^{y}h_{2}^{x}-h_{3}^{x}h_{2}^{y})\tilde{T}_{lz},$ for $l=x,y,z$, and $\displaystyle\mathcal{G}^{\prime}(m)$ $\displaystyle=$ $\displaystyle\left\\{\begin{array}[]{cc}\mathcal{G}(m,2,3),&m=1,2;\\\ \mathcal{G}(2,3,m),&m\geq 3,\end{array}\right.$ $\displaystyle\mathcal{H}^{\prime}(n)$ $\displaystyle=$ $\displaystyle\left\\{\begin{array}[]{cc}\mathcal{H}(n,2,3),&n=1,2;\\\ \mathcal{H}(2,3,n),&n\geq 3.\end{array}\right.$ Here, $h_{k}^{l}$ is defined by Eq.(23), $\mathcal{G}(i,j,k)$ and $\mathcal{H}(i,j,k)$ are defined in Eq.(10) and Eq.(22) respectively. With the conditions presented in Eqs.(24-25), we can prove that $f_{t_{11}}(m,n)\leq 0,$ (41) for all $(m,n)\in J_{0}$. Then we conclude that the expression given by Eq.(37) is actually an upper bound of $t_{11}$. With this, we can estimate the upper bound of $e_{11}$ by the following explicit formula $\overline{e}_{11}^{\prime}=\frac{\overline{t}_{11}^{\prime}}{\underline{s}_{11}^{\prime}},$ (42) where $\underline{s}_{11}^{\prime}$ is the lower bound of $s_{11}$ given in Eq.(32). ### III.3 Numerical Simulation for MDI-QKD Figure 4: (Color online) The estimated values of $e_{11}$ versus the total channel transmission loss for MDI-QKD with WCS and HSPS. We set $\mu_{1}=\nu_{1}=0.1$ for decoy state, and $\mu_{2}=\nu_{2}$. Figure 5: (Color online) The optimal key rate versus the total channel transmission loss using different methods for MDI-QKD with WCS and HSPS. We set $\mu_{1}=\nu_{1}=0.1$ for decoy states, and $\mu_{2}=\nu_{2}$. Figure 6: (Color online) The relative optimal key rate of different methods versus the total channel transmission loss for MDI-QKD with WCS and HSPS. We set $\mu_{1}=\nu_{1}=0.1$ for decoy states, and $\mu_{2}=\nu_{2}$. Figure 7: (Color online) The ratio of the optimal key rates between the estimations obtained by using 4-intensity and 3-intensity decoy state methods versus the total channel transmission loss for MDI-QKD with WCS and HSPS. We set $\mu_{1}=\nu_{1}=0.1$ decoy states, and $\mu_{2}=\nu_{2}$. Figure 8: (Color online) The optimal intensity of the signal states versus the total channel transmission loss using 3-intensity and 4-intensity decoy state methods for MDI-QKD with WCS and HSPS. We set $\mu_{1}=\nu_{1}=0.1$ for decoy states, and $\mu_{2}=\nu_{2}$. Table 2: List of experimental parameters used in numerical simulations: $e_{0}$ is the error rate of background, $e_{d}$ is the misalignment-error probability; $p_{d}$ is the dark count rate of UTP’s per detector; $f$ is the error correction inefficiency; $\eta_{v}$ is the detection efficiency of Alice and Bob’s detector; $p_{dv}$ is the dark count rate of Alice and Bob’s detector. $e_{0}$ \| $e_{d}$ \| $p_{d}$ \| $f$ \| $\eta_{v}$ \| $p_{dv}$ ---\|---\|---\|---\|---\|--- 0.5 \| 1.5% \| $3.0\times 10^{-6}$ \| 1.16 \| 0.75 \| $1.0\times 10^{-6}$ In this section, we will present some numerical simulations to comparing our results with the results obtained by using 3-intensity decoy state method for MDI-QKD Wang3int . As discussed before, we know that the methods presented in this paper does not only apply to the weak coherent sources (WCS). Actually, it can be used to estimate the final key rate for any sources that satisfy the condition given by Eqs.(24-25). Below for simplicity, we consider the following two cases. In the first case, we suppose that Alice and Bob use the WCS. In the second one, we suppose they use the heralded single-photon sources (HSPS) with possion distributions wangArxiv . The UTP locates in the middle of Alice and Bob, and the UTP’s detectors are identical, i.e., they have the same dark count rate and detection efficiency, and their detection efficiency does not depend on the incoming signals. We shall estimate what values would be probably observed for the gains and error rates in the normal cases by the linear models as in wang05 ; ind2 ; WangModel : $\displaystyle\|n\rangle\langle n\|=\sum_{k=0}^{n}C_{n}^{k}\xi^{k}(1-\xi)^{n-k}\|k\rangle\langle k\|$ where $\xi^{k}$ is the transmittance for a distance from Alice to the UTP. For fair comparison, we use the same parameter values used in ind2 for our numerical evaluation, which follow the experiment reported in UrsinNP2007 . For simplicity, we shall put the detection efficiency to the overall transmittance $\eta=\xi^{2}\zeta$. We assume all detectors of UTP have the same detection efficiency $\zeta$ and dark count rate $p_{d}$. In the second case with HPSP, we assume all detectors of Alice and Bob have the same detection efficiency $\eta_{v}$ and dark count rate $p_{dv}$. The values of these parameters are presented in Table 2. With this, by taking the photon- number-cutoff approximation up to 6 photon-number state, the total gains $S_{\mu_{i},\nu_{j}}^{\omega},(\omega=X,Z)$ and error rates $S_{\mu_{i},\nu_{j}}^{\omega}E_{\mu_{i},\nu_{j}}^{\omega},(\omega=X,Z)$ of Alice’s intensity $\mu_{i}$ ($i=0,1,2$ for 3-intensity method, $i=0,1,2,3$ for 4-intensity method) and Bob’s intensity $\nu_{j}$ ($j=0,1,2$ for 3-intensity method, $j=0,1,2,3$ for 4-intensity method) can be calculated. By using these values, we can estimate the lower bounds of yield $s_{11}^{Z}$ with Eq.(30) and Eq.(32) for 3-intensity and 4-intensity decoy state methods respectively. Also, we can estimate the upper bounds of error rate $e_{11}^{X}$ with Eq.(33) and Eq.(42) for these two decoy state methods respectively. Furthermore, with these parameters, we can estimate the final key rate $R$ of this protocol with Eq.(26). If we fix the densities of the decoy-state sources used by Alice and Bob, the final key rate will change with they taking different intensities for their signal-state pulses. Here, in order to make a rational and effective comparison, we set the intensities of the decoy source in 3-intensity method and the first decoy source in 4-intensity method are the same and $\mu_{1}=\nu_{1}=0.1$; let the intensity of the second decoy source in 4-intensity method to be the optimal intensity of signal source in 3-intensity method and assume $\mu_{2}=\nu_{2}>\mu_{1}$ Note . With these preparations, we can conclude that the lower bounds of $s_{11}$ estimated by using the 3-intensity and 4-intensity decoy state methods are the same, i.e., $\underline{s}_{11}=\underline{s}_{11}^{\prime}$. In Fig.4, we plot the upper bound of $e_{11}$ with different methods. The optimal key rates with different methods for WCS and HSPS are shown in the up and down subfigures respectively in Fig.5. To see more clearly, in Fig.6, we plot the relative value between the optimal key rate obtained with different methods and the asymptotic limit of the infinite decoy-state method. In order to clarify the superiority of the 4-intensity decoy state method, we plot the ratio of the optimal key rate between the results obtained by using 4-intensity and 3-intensity decoy state methods in Fig.7. These figures clearly show that our results are better than the pre-existed results. The optimal densities with the optimal key rate versus the total channel transmission loss is given in Fig.8. In Fig.4 and Fig.6, the black dotted line and green dash-dot line are the results obtained by using 3-intensity decoy state method with WCS and HSPS respectively, the blue dashed line and the red solid line are the results obtained by using 4-intensity decoy state with WCS and HSPS respectively, the thick cyan line are the results obtained by using infinite decoy state method. In Fig.5 and Fig.8, the green dotted, the red dashed and the cyan solid lines are the results obtained by using 3-intensity, 4-intensity and the infinite decoy state methods respectively. In Fig.7, the blue dashed lines are obtained with WCS, the red solid lines are obtained with HSPS. ## IV Concluding Remark In conclusion, we show how to tightly formulate the upper bound of the phase- flip errors in decoy state method for the regular BB84 protocol and MDI-QKD. Our result compressed the bound to about a quarter of known result for MDI-QKD with WCS, and even about one fifth with HSPS. To achieve the result, we only need 4-intensity decoy state method. These methods can be applied to the recently proposed protocols with imperfect single-photon source such as the coherent states or the heralded states from the parametric down conversion. Based on this, we find that the key rate is improved by more than 100% with WCS, and even more than 200% with HSPS. Acknowledgement: We acknowledge the support from the 10000-Plan of Shandong province, the National High-Tech Program of China Grants No. 2011AA010800 and No. 2011AA010803 and NSFC Grants No. 11174177 and No. 60725416. Appendix A. Eqs.(7,8,12,24,25) with the imperfect sources used in practice We know that the state emits from a parametric down-conversion (PDC) source is [17,18] $\rho_{l}=\sum_{k}a_{k}^{l}\|k\rangle\langle k\|,$ with $a_{k}^{l}=e^{-l}{l^{k}}/{k!}$ or $a_{k}^{l}={l^{k}}/{(l+1)^{k+1}}$ where $\|k\rangle$ represents an $k$-photon state, $l$ is the intensity (average photon number) of $\rho_{l}$. Firstly, in this appendix, we will prove that the assumptions given by Eqs.(7,8,12) are satisfied by the PDC source. In the 4-intensity protocol, Alice has 3 different no-vacuum sources which are denoted by $\rho_{x},\rho_{y},\rho_{z}$ with $0<x<y<z$. In the case with $a_{k}^{l}=e^{-l}{l^{k}}/{k!}$, we have $\frac{a_{k}^{y}}{a_{k}^{x}}=e^{x-y}\frac{y^{k}}{x^{k}},\quad\frac{a_{k}^{z}}{a_{k}^{y}}=e^{y-z}\frac{z^{k}}{y^{k}},\quad(k\geq 0).$ Then we can easily prove the conclusions in Eqs.(7,8) with $x<y<z$. In order to prove the result presented in Eq.(12), we need the following lemma. Lemma 1. For any two natural number $m,n$ with $m>n\geq 1$, $\mathcal{F}(v)=\frac{1-v^{m}}{1-v^{n}}$ is a monotone increasing function in the domain $v\in(0,1)$. The function $\mathcal{F}(v)=\frac{1-v^{m}}{1-v^{n}}$ can be rewritten into $\displaystyle\mathcal{F}(v)$ $\displaystyle=$ $\displaystyle\frac{\sum_{k=0}^{m-1}v^{k}}{\sum_{k=0}^{n-1}v^{k}}$ $\displaystyle=$ $\displaystyle 1+v^{n}\frac{\sum_{k=0}^{m-n-1}v^{k}}{\sum_{k=0}^{n-1}v^{k}}=1+\frac{\sum_{k=0}^{m-n-1}v^{k}}{\sum_{k=1}^{n}1/v^{k}}.$ This predicts that the function $\mathcal{F}(v)$ is monotone increasing with $m>n\geq 1$ in the domain $v\in(0,1)$. With the definition of $\mathcal{G}(i,j,k)$ in Eq.(10), we have $\displaystyle e^{x+y-2z}\mathcal{G}(i,j,k)$ $\displaystyle=$ $\displaystyle[x_{r}^{i}-x_{r}^{j}][y_{r}^{j}-y_{r}^{k}]-[y_{r}^{i}-y_{r}^{j}][x_{r}^{j}-x_{r}^{k}]$ $\displaystyle=$ $\displaystyle x_{r}^{i}y_{r}^{j}[1-x_{r}^{j-i}][1-y_{r}^{k-j}]-y_{r}^{i}x_{r}^{j}[1-y_{r}^{j-i}][1-x_{r}^{k-j}],$ with $x_{r}=x/z$ and $y_{r}=y/z$. If $x<y<z$, and $k-j\geq j-i\geq 0$, we get $\frac{x_{r}^{i}y_{r}^{j}}{y_{r}^{i}x_{r}^{j}}=\frac{x^{i}}{z^{i}}\frac{y^{j}}{z^{j}}\cdot\frac{z^{i}}{y^{i}}\frac{z^{j}}{x^{j}}=\left(\frac{y}{x}\right)^{j-i}\geq 1,$ and $\displaystyle(1-x_{r}^{j-i})(1-y_{r}^{k-j})-(1-y_{r}^{j-i})(1-x_{r}^{k-j})$ $\displaystyle=$ $\displaystyle(1-x_{r}^{j-i})(1-y_{r}^{j-i})\left(\frac{1-y_{r}^{k-j}}{1-y_{r}^{j-i}}-\frac{1-x_{r}^{k-j}}{1-x_{r}^{j-i}}\right)\geq 0.$ In the last step, we have used Lemma 1. With these relations, we can finish the proof of Eq.(12). In the case with $a_{k}^{l}=l^{k}/(l+1)^{k+1}$, we have $\frac{a_{k}^{y}}{a_{k}^{x}}=\frac{x+1}{y+1}\left(\frac{xy+y}{xy+x}\right)^{k},\quad\frac{a_{k}^{z}}{a_{k}^{y}}=\frac{y+1}{z+1}\left(\frac{yz+z}{yz+y}\right)^{k},$ for all $k\geq 1$. Then we can easily prove the conclusions in Eqs.(7,8) with $x<y<z$. By introducing $\tilde{x}_{r}=\frac{xz+x}{xz+z},\quad\tilde{y}_{r}=\frac{yz+y}{yz+z},$ we find out $\displaystyle\frac{(1+x)(1+y)}{(1+z)^{2}}\mathcal{G}(i,j,k)$ $\displaystyle=$ $\displaystyle[\tilde{x}_{r}^{i}-\tilde{x}_{r}^{j}][\tilde{y}_{r}^{j}-\tilde{y}_{r}^{k}]-[\tilde{y}_{r}^{i}-\tilde{y}_{r}^{j}][\tilde{x}_{r}^{j}-\tilde{x}_{r}^{k}]$ $\displaystyle=$ $\displaystyle\tilde{x}_{r}^{i}\tilde{y}_{r}^{j}[1-\tilde{x}_{r}^{j-i}][1-\tilde{y}_{r}^{k-j}]-\tilde{y}_{r}^{i}\tilde{x}_{r}^{j}[1-\tilde{y}_{r}^{j-i}][1-\tilde{x}_{r}^{k-j}].$ If $x<y<z$, with Lemma 1, we can prove that $\mathcal{G}(i,j,k)\geq 0$ when $k-j\geq j-i\geq 0$. This complete the proof of Eq.(12). Similarly, we can prove that those assumptions in Eqs.(7,8,12) can be fulfilled by the heralded single-photon sources (HSPS) with possion or thermal distributions wangArxiv . When we consider the 4-intensity decoy state method for MDI-QKD, the assumptions presented in Eqs.(24-25) can be fulfilled if Alice and Bob choose PDC sources or HSPS. Appendix B. The derivation of the simplified forms of $s_{1}^{\prime}$ and $s_{11}^{\prime}$ As discussed in section II, the lower bound of $s_{1}$ can be estimated by Eq.(6) when Alice use three different sources $\rho_{o},\rho_{x}$ and $\rho_{y}$. Furthermore, in this case, we can write $s_{1}$ into the following form with $\underline{s}_{1}(x,y)$ $s_{1}=\underline{s}_{1}(x,y)+\sum_{m\geq 3}f_{s_{1}}^{(x,y)}(m)s_{m},$ where $f_{s_{1}}^{(x,y)}(m)=\frac{a_{2}^{x}a_{m}^{y}-a_{2}^{y}a_{m}^{x}}{a_{1}^{x}a_{2}^{y}-a_{1}^{y}a_{2}^{x}},\quad(m\geq 3).$ Similarly, if Alice choose sources $\rho_{o},\rho_{x},\rho_{z}$ and $\rho_{o},\rho_{y},\rho_{z}$, then $s_{1}$ can also be expressed into $\displaystyle s_{1}$ $\displaystyle=$ $\displaystyle\underline{s}_{1}(x,z)+\sum_{m\geq 3}f_{s_{1}}^{(x,z)}(m)s_{m},$ $\displaystyle s_{1}$ $\displaystyle=$ $\displaystyle\underline{s}_{1}(y,z)+\sum_{m\geq 3}f_{s_{1}}^{(y,z)}(m)s_{m},$ with $f_{s_{1}}^{(x,z)}(m)=\frac{a_{2}^{x}a_{m}^{z}-a_{2}^{z}a_{m}^{x}}{a_{1}^{x}a_{2}^{z}-a_{1}^{z}a_{2}^{x}},\quad f_{s_{1}}^{(y,z)}(m)=\frac{a_{2}^{y}a_{m}^{z}-a_{2}^{z}a_{m}^{y}}{a_{1}^{y}a_{2}^{z}-a_{1}^{z}a_{2}^{y}},$ respectively. By calculation, we have $\displaystyle f_{s_{1}}^{(x,y)}(m)-f_{s_{1}}^{(x,z)}(m)$ $\displaystyle=$ $\displaystyle-\frac{a_{2}^{x}a_{1}^{z}a_{2}^{z}a_{3}^{z}\mathcal{G}(1,2,m)}{(a_{1}^{x}a_{2}^{y}-a_{1}^{y}a_{2}^{x})(a_{1}^{x}a_{2}^{z}-a_{1}^{z}a_{2}^{x})},$ $\displaystyle f_{s_{1}}^{(x,z)}(m)-f_{s_{1}}^{(y,z)}(m)$ $\displaystyle=$ $\displaystyle-\frac{a_{2}^{z}a_{1}^{z}a_{2}^{z}a_{3}^{z}\mathcal{G}(1,2,m)}{(a_{1}^{x}a_{2}^{z}-a_{1}^{z}a_{2}^{x})(a_{1}^{y}a_{2}^{z}-a_{1}^{z}a_{2}^{y})}.$ According to the conditions given by Eqs.(7,8,12), we can easily prove that $f_{s_{1}}^{(x,y)}(m)\leq f_{s_{1}}^{(x,z)}(m)\leq f_{s_{1}}^{(y,z)}(m),$ for all $m\geq 3$. So we have $\underline{s}_{1}(x,y)\geq\underline{s}_{1}(x,z)\geq\underline{s}_{1}(y,z).$ This completes the proof of Eq.(13). Now we commit to prove Eq.(32) for MDI-QKD. By choosing any two different no- vacuum sources $\rho_{l_{A(B)}},\rho_{r_{A(B)}}$ form $\rho_{x_{A(B)}},\rho_{y_{A(B)}},\rho_{z_{A(B)}}$, we have $s_{11}=\underline{s}_{11}(l,r)+\sum_{(m,n)\in J_{1}}f_{s_{11}}^{(l,r)}(m,n)s_{mn},$ with $\underline{s}_{11}(l,r)$ being defined in Eq.(30) by replacing $x,y$ with $l,r$ respectively, and $f_{s_{11}}^{(l,r)}(m,n)=\frac{a_{2}^{l}b_{n}^{l}(a_{1}^{l}a_{m}^{r}-a_{1}^{r}a_{m}^{l})(b_{1}^{l}b_{2}^{r}-b_{1}^{r}b_{2}^{l})+a_{m}^{l}b_{1}^{l}(a_{1}^{l}a_{2}^{r}-a_{1}^{r}a_{2}^{l})(b_{2}^{l}b_{n}^{r}-b_{2}^{r}b_{n}^{l})}{a_{1}^{l}b_{1}^{l}(a_{1}^{l}a_{2}^{r}-a_{1}^{r}a_{2}^{l})(b_{1}^{l}b_{2}^{r}-b_{1}^{r}b_{2}^{l})}.$ where $(l,r)\in\\{(x,y),(x,z),(y,z)\\}$, $J_{1}=\\{(m,n)\|m,n\geq 1;m+n\geq 4\\}$. In the coming, we will compare the relations among $f_{s_{11}}^{(x,y)}(m,n),f_{s_{11}}^{(x,y)}(m,n),f_{s_{11}}^{(x,y)}(m,n)$. Firstly, we have $f_{s_{11}}^{(x,y)}(1,n)-f_{s_{11}}^{(x,z)}(1,n)=\frac{-b_{2}^{x}b_{1}^{z}b_{2}^{z}b_{n}^{z}\mathcal{H}(1,2,n)}{(b_{1}^{x}b_{2}^{y}-b_{1}^{y}b_{2}^{x})(b_{1}^{x}b_{2}^{z}-b_{1}^{z}b_{2}^{x})},$ $f_{s_{11}}^{(x,z)}(1,n)-f_{s_{11}}^{(y,z)}(1,n)=\frac{-b_{1}^{z}b_{2}^{z}b_{2}^{z}b_{n}^{z}\mathcal{H}(1,2,n)}{(b_{1}^{x}b_{2}^{y}-b_{1}^{y}b_{2}^{x})(b_{1}^{x}b_{2}^{z}-b_{1}^{z}b_{2}^{x})},$ for all $n\geq 3$. Secondly, we obtain $f_{s_{11}}^{(x,y)}(n,1)-f_{s_{11}}^{(x,z)}(n,1)=\frac{-a_{2}^{x}a_{1}^{z}a_{2}^{z}a_{n}^{z}\mathcal{G}(1,2,n)}{(a_{1}^{x}a_{2}^{y}-a_{1}^{y}a_{2}^{x})(a_{1}^{x}a_{2}^{z}-a_{1}^{z}a_{2}^{x})},$ $f_{s_{11}}^{(x,z)}(n,1)-f_{s_{11}}^{(y,z)}(n,1)=\frac{-a_{1}^{z}a_{2}^{z}a_{2}^{z}a_{n}^{z}\mathcal{G}(1,2,n)}{(a_{1}^{x}a_{2}^{y}-a_{1}^{y}a_{2}^{x})(a_{1}^{x}a_{2}^{z}-a_{1}^{z}a_{2}^{x})},$ for all $n\geq 3$. In the last case, we get $f_{s_{11}}^{(x,y)}(m,n)-f_{s_{11}}^{(x,z)}(m,n)=-\frac{a_{2}^{x}b_{n}^{x}a_{1}^{z}a_{2}^{z}a_{m}^{z}\mathcal{G}(1,2,m)}{b_{1}^{x}(a_{1}^{x}a_{2}^{y}-a_{1}^{y}a_{2}^{x})(a_{1}^{x}a_{2}^{z}-a_{1}^{z}a_{2}^{x})}-\frac{b_{2}^{x}a_{m}^{x}b_{1}^{z}b_{2}^{z}b_{n}^{z}\mathcal{H}(1,2,n)}{a_{1}^{x}(b_{1}^{x}b_{2}^{y}-b_{1}^{y}b_{2}^{x})(b_{1}^{x}b_{2}^{z}-b_{1}^{z}b_{2}^{x})},$ and $f_{s_{11}}^{(x,z)}(m,n)-f_{s_{11}}^{(y,z)}(m,n)\leq-\frac{a_{2}^{y}b_{n}^{y}a_{1}^{z}a_{2}^{z}a_{1}^{z}a_{m}^{z}\mathcal{G}(1,2,m)}{a_{1}^{y}b_{1}^{y}(a_{1}^{x}a_{2}^{z}-a_{1}^{z}a_{2}^{x})(a_{1}^{y}a_{2}^{z}-a_{1}^{z}a_{2}^{y})}-\frac{a_{m}^{y}b_{1}^{z}b_{2}^{z}b_{2}^{z}b_{n}^{z}\mathcal{H}(1,2,n)}{a_{1}^{y}(b_{1}^{x}b_{2}^{z}-b_{1}^{z}b_{2}^{x})(b_{1}^{y}b_{2}^{z}-b_{1}^{z}b_{2}^{y})},$ for all $m,n\geq 2$. In the lase inequality, we have used the assumption presented in Eq.(24). 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Wang, arXiv: 1309.0471v1. * (28) Z.-W. Yu, Y.-H. Zhou, and X.-B. Wang, arXiv: 1309.5886v1. * (29) Q. Wang, and X.-B. Wang, arXiv: 1311.1739v1. * (30) R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, et al., Nat. Phys. 3, 481 (2007). * (31) In the simulations, we set the intensity of the second decoy source in 4-intensity method to be the optimal intensity of signal source in 3-intensity method. In such setting, the key rate of 3-intensity protocol is optimized given the intensity of the weak coherent state to be 0.1 but the key rate of 4-intensity protocol is not optimized. We make such an unfair comparison only in order to clearly demonstrate the advantage of our tightened estimation in the 4-intensity protocol. It is worth to note that, we can actually further improve the key rate of the 4-intensity protocol by taking some weaker intensities for the decoy sources. For example, if we take $\mu_{1}=0.1,\mu_{2}=0.15$ the key rate can be further improved about 10 percent.	arxiv-papers	2013-12-20T12:18:48	2024-09-04T02:49:55.729430	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Y.H. Zhou, Z.W. Yu, and X.B. Wang", "submitter": "Xiang-Bin Wang", "url": "https://arxiv.org/abs/1312.5915" }
1312.6199	# Intriguing properties of neural networks Christian Szegedy Google Inc. &Wojciech Zaremba New York University &Ilya Sutskever Google Inc. &Joan Bruna New York University &Dumitru Erhan Google Inc. &Ian Goodfellow University of Montreal &Rob Fergus New York University Facebook Inc. ###### Abstract Deep neural networks are highly expressive models that have recently achieved state of the art performance on speech and visual recognition tasks. While their expressiveness is the reason they succeed, it also causes them to learn uninterpretable solutions that could have counter-intuitive properties. In this paper we report two such properties. First, we find that there is no distinction between individual high level units and random linear combinations of high level units, according to various methods of unit analysis. It suggests that it is the space, rather than the individual units, that contains the semantic information in the high layers of neural networks. Second, we find that deep neural networks learn input-output mappings that are fairly discontinuous to a significant extent. We can cause the network to misclassify an image by applying a certain hardly perceptible perturbation, which is found by maximizing the network’s prediction error. In addition, the specific nature of these perturbations is not a random artifact of learning: the same perturbation can cause a different network, that was trained on a different subset of the dataset, to misclassify the same input. ## 1 Introduction Deep neural networks are powerful learning models that achieve excellent performance on visual and speech recognition problems [9, 8]. Neural networks achieve high performance because they can express arbitrary computation that consists of a modest number of massively parallel nonlinear steps. But as the resulting computation is automatically discovered by backpropagation via supervised learning, it can be difficult to interpret and can have counter- intuitive properties. In this paper, we discuss two counter-intuitive properties of deep neural networks. The first property is concerned with the semantic meaning of individual units. Previous works [6, 13, 7] analyzed the semantic meaning of various units by finding the set of inputs that maximally activate a given unit. The inspection of individual units makes the implicit assumption that the units of the last feature layer form a distinguished basis which is particularly useful for extracting semantic information. Instead, we show in section 3 that random projections of $\phi(x)$ are semantically indistinguishable from the coordinates of $\phi(x)$. This puts into question the conjecture that neural networks disentangle variation factors across coordinates. Generally, it seems that it is the entire space of activations, rather than the individual units, that contains the bulk of the semantic information. A similar, but even stronger conclusion was reached recently by Mikolov et al. [12] for word representations, where the various directions in the vector space representing the words are shown to give rise to a surprisingly rich semantic encoding of relations and analogies. At the same time, the vector representations are stable up to a rotation of the space, so the individual units of the vector representations are unlikely to contain semantic information. The second property is concerned with the stability of neural networks with respect to small perturbations to their inputs. Consider a state-of-the-art deep neural network that generalizes well on an object recognition task. We expect such network to be robust to small perturbations of its input, because small perturbation cannot change the object category of an image. However, we find that applying an _imperceptible_ non-random perturbation to a test image, it is possible to arbitrarily change the network’s prediction (see figure 5). These perturbations are found by optimizing the input to maximize the prediction error. We term the so perturbed examples “adversarial examples”. It is natural to expect that the precise configuration of the minimal necessary perturbations is a random artifact of the normal variability that arises in different runs of backpropagation learning. Yet, we found that adversarial examples are relatively robust, and are shared by neural networks with varied number of layers, activations or trained on different subsets of the training data. That is, if we use one neural net to generate a set of adversarial examples, we find that these examples are still statistically hard for another neural network even when it was trained with different hyperparameters or, most surprisingly, when it was trained on a different set of examples. These results suggest that the deep neural networks that are learned by backpropagation have nonintuitive characteristics and intrinsic blind spots, whose structure is connected to the data distribution in a non-obvious way. ## 2 Framework Notation We denote by $x\in\mathbb{R}^{m}$ an input image, and $\phi(x)$ activation values of some layer. We first examine properties of the image of $\phi(x)$, and then we search for its blind spots. We perform a number of experiments on a few different networks and three datasets : * • For the MNIST dataset, we used the following architectures [11] * – A simple fully connected network with one or more hidden layers and a Softmax classifier. We refer to this network as “FC”. * – A classifier trained on top of an autoencoder. We refer to this network as “AE”. * • The ImageNet dataset [3]. * – Krizhevsky et. al architecture [9]. We refer to it as “AlexNet”. * • $\sim 10$M image samples from Youtube (see [10]) * – Unsupervised trained network with $\sim$ 1 billion learnable parameters. We refer to it as “QuocNet”. For the MNIST experiments, we use regularization with a weight decay of $\lambda$. Moreover, in some experiments we split the MNIST training dataset into two disjoint datasets $P_{1}$, and $P_{2}$, each with 30000 training cases. ## 3 Units of: $\phi(x)$ Traditional computer vision systems rely on feature extraction: often a single feature is easily interpretable, e.g. a histogram of colors, or quantized local derivatives. This allows one to inspect the individual coordinates of the feature space, and link them back to meaningful variations in the input domain. Similar reasoning was used in previous work that attempted to analyze neural networks that were applied to computer vision problems. These works interpret an activation of a hidden unit as a meaningful feature. They look for input images which maximize the activation value of this single feature [6, 13, 7, 4]. The aforementioned technique can be formally stated as visual inspection of images $x^{\prime}$, which satisfy (or are close to maximum attainable value): $\displaystyle x^{\prime}=\operatorname{arg\,max}_{x\in\mathcal{I}}\langle\phi(x),e_{i}\rangle$ where $\mathcal{I}$ is a held-out set of images from the data distribution that the network was not trained on and $e_{i}$ is the natural basis vector associated with the $i$-th hidden unit. Our experiments show that any random direction $v\in\mathbb{R}^{n}$ gives rise to similarly interpretable semantic properties. More formally, we find that images $x^{\prime}$ are semantically related to each other, for many $x^{\prime}$ such that $\displaystyle x^{\prime}=\operatorname{arg\,max}_{x\in\mathcal{I}}\langle\phi(x),v\rangle$ This suggests that the natural basis is not better than a random basis for inspecting the properties of $\phi(x)$. This puts into question the notion that neural networks disentangle variation factors across coordinates. First, we evaluated the above claim using a convolutional neural network trained on MNIST. We used the MNIST test set for $\mathcal{I}$. Figure 1 shows images that maximize the activations in the natural basis, and Figure 2 shows images that maximize the activation in random directions. In both cases the resulting images share many high-level similarities. Next, we repeated our experiment on an AlexNet, where we used the validation set as $\mathcal{I}$. Figures 3 and 4 compare the natural basis to the random basis on the trained network. The rows appear to be semantically meaningful for both the single unit and the combination of units. (a) Unit sensitive to lower round stroke. (b) Unit sensitive to upper round stroke, or lower straight stroke. (c) Unit senstive to left, upper round stroke. (d) Unit senstive to diagonal straight stroke. Figure 1: An MNIST experiment. The figure shows images that maximize the activation of various units (maximum stimulation in the natural basis direction). Images within each row share semantic properties. (a) Direction sensitive to upper straight stroke, or lower round stroke. (b) Direction sensitive to lower left loop. (c) Direction senstive to round top stroke. (d) Direction sensitive to right, upper round stroke. Figure 2: An MNIST experiment. The figure shows images that maximize the activations in a random direction (maximum stimulation in a random basis). Images within each row share semantic properties. (a) Unit sensitive to white flowers. (b) Unit sensitive to postures. (c) Unit senstive to round, spiky flowers. (d) Unit senstive to round green or yellow objects. Figure 3: Experiment performed on ImageNet. Images stimulating single unit most (maximum stimulation in natural basis direction). Images within each row share many semantic properties. (a) Direction sensitive to white, spread flowers. (b) Direction sensitive to white dogs. (c) Direction sensitive to spread shapes. (d) Direction sensitive to dogs with brown heads. Figure 4: Experiment performed on ImageNet. Images giving rise to maximum activations in a random direction (maximum stimulation in a random basis). Images within each row share many semantic properties. Although such analysis gives insight on the capacity of $\phi$ to generate invariance on a particular subset of the input distribution, it does not explain the behavior on the rest of its domain. We shall see in the next section that $\phi$ has counterintuitive properties in the neighbourhood of almost every point form data distribution. ## 4 Blind Spots in Neural Networks So far, unit-level inspection methods had relatively little utility beyond confirming certain intuitions regarding the complexity of the representations learned by a deep neural network [6, 13, 7, 4]. Global, network level inspection methods _can_ be useful in the context of explaining classification decisions made by a model [1] and can be used to, for instance, identify the parts of the input which led to a correct classification of a given visual input instance (in other words, one can use a trained model for weakly- supervised localization). Such global analyses are useful in that they can make us understand better the input-to-output mapping represented by the trained network. Generally speaking, the output layer unit of a neural network is a highly nonlinear function of its input. When it is trained with the cross-entropy loss (using the Softmax activation function), it represents a conditional distribution of the label given the input (and the training set presented so far). It has been argued [2] that the deep stack of non-linear layers in between the input and the output unit of a neural network are a way for the model to encode a _non-local generalization prior_ over the input space. In other words, it is assumed that is possible for the output unit to assign non- significant (and, presumably, non-epsilon) probabilities to regions of the input space that contain no training examples in their vicinity. Such regions can represent, for instance, the same objects from different viewpoints, which are relatively far (in pixel space), but which share nonetheless both the label and the statistical structure of the original inputs. It is implicit in such arguments that $local$ generalization—in the very proximity of the training examples—works as expected. And that in particular, for a small enough radius $\varepsilon>0$ in the vicinity of a given training input $x$, an $x+r$ satisfying $\|\|r\|\|<\varepsilon$ will get assigned a high probability of the correct class by the model. This kind of smoothness prior is typically valid for computer vision problems. In general, imperceptibly tiny perturbations of a given image do not normally change the underlying class. Our main result is that for deep neural networks, the smoothness assumption that underlies many kernel methods does not hold. Specifically, we show that by using a simple optimization procedure, we are able to find adversarial examples, which are obtained by imperceptibly small perturbations to a correctly classified input image, so that it is no longer classified correctly. In some sense, what we describe is a way to traverse the manifold represented by the network in an efficient way (by optimization) and finding _adversarial examples_ in the input space. The adversarial examples represent low- probability (high-dimensional) “pockets” in the manifold, which are hard to efficiently find by simply randomly sampling the input around a given example. Already, a variety of recent state of the art computer vision models employ input deformations during training for increasing the robustness and convergence speed of the models [9, 13]. These deformations are, however, statistically inefficient, for a given example: they are highly correlated and are drawn from the same distribution throughout the entire training of the model. We propose a scheme to make this process adaptive in a way that exploits the model and its deficiencies in modeling the local space around the training data. We make the connection with hard-negative mining explicitly, as it is close in spirit: hard-negative mining, in computer vision, consists of identifying training set examples (or portions thereof) which are given low probabilities by the model, but which should be high probability instead, cf. [5]. The training set distribution is then changed to emphasize such hard negatives and a further round of model training is performed. As shall be described, the optimization problem proposed in this work can also be used in a constructive way, similar to the hard-negative mining principle. ### 4.1 Formal description We denote by $f:\mathbb{R}^{m}\longrightarrow\\{1\dots k\\}$ a classifier mapping image pixel value vectors to a discrete label set. We also assume that $f$ has an associated continuous loss function denoted by $\textrm{loss}_{f}:\mathbb{R}^{m}\times\\{1\dots k\\}\longrightarrow\mathbb{R}^{+}$. For a given $x\in\mathbb{R}^{m}$ image and target label $l\in\\{1\dots k\\}$, we aim to solve the following box- constrained optimization problem: * • Minimize $\\|r\\|_{2}$ subject to: 1. 1. $f(x+r)=l$ 2. 2. $x+r\in[0,1]^{m}$ The minimizer $r$ might not be unique, but we denote one such $x+r$ for an arbitrarily chosen minimizer by $D(x,l)$. Informally, $x+r$ is the closest image to $x$ classified as $l$ by $f$. Obviously, $D(x,f(x))=f(x)$, so this task is non-trivial only if $f(x)\neq l$. In general, the exact computation of $D(x,l)$ is a hard problem, so we approximate it by using a box-constrained L-BFGS. Concretely, we find an approximation of $D(x,l)$ by performing line- search to find the minimum $c>0$ for which the minimizer $r$ of the following problem satisfies $f(x+r)=l$. * • Minimize $c\|r\|+\textrm{loss}_{f}(x+r,l)$ subject to $x+r\in[0,1]^{m}$ This penalty function method would yield the exact solution for $D(X,l)$ in the case of convex losses, however neural networks are non-convex in general, so we end up with an approximation in this case. ### 4.2 Experimental results Figure 5: Adversarial examples generated for AlexNet [9].(Left) is a correctly predicted sample, (center) difference between correct image, and image predicted incorrectly magnified by 10x (values shifted by 128 and clamped), (right) adversarial example. All images in the right column are predicted to be an “ostrich, Struthio camelus”. Average distortion based on 64 examples is 0.006508. Plase refer to http://goo.gl/huaGPb for full resolution images. The examples are strictly randomly chosen. There is not any postselection involved. Figure 6: Adversarial examples for QuocNet [10]. A binary car classifier was trained on top of the last layer features without fine-tuning. The randomly chosen examples on the left are recognized correctly as cars, while the images in the middle are not recognized. The rightmost column is the magnified absolute value of the difference between the two images. Our “minimimum distortion” function $D$ has the following intriguing properties which we will support by informal evidence and quantitative experiments in this section: 1. 1. For all the networks we studied (MNIST, QuocNet [10], AlexNet [9]), for each sample, we have always managed to generate very close, visually hard to distinguish, adversarial examples that are misclassified by the original network (see figure 5 and http://goo.gl/huaGPb for examples). 2. 2. Cross model generalization: a relatively large fraction of examples will be misclassified by networks trained from scratch with different hyper-parameters (number of layers, regularization or initial weights). 3. 3. Cross training-set generalization a relatively large fraction of examples will be misclassified by networks trained from scratch on a disjoint training set. The above observations suggest that adversarial examples are somewhat universal and not just the results of overfitting to a particular model or to the specific selection of the training set. They also suggest that back- feeding adversarial examples to training might improve generalization of the resulting models. Our preliminary experiments have yielded positive evidence on MNIST to support this hypothesis as well: We have successfully trained a two layer 100-100-10 non-convolutional neural network with a test error below $1.2\%$ by keeping a pool of adversarial examples a random subset of which is continuously replaced by newly generated adversarial examples and which is mixed into the original training set all the time. We used weight decay, but no dropout for this network. For comparison, a network of this size gets to $1.6\%$ errors when regularized by weight decay alone and can be improved to around $1.3\%$ by using carefully applied dropout. A subtle, but essential detail is that we only got improvements by generating adversarial examples for each layer outputs which were used to train all the layers above. The network was trained in an alternating fashion, maintaining and updating a pool of adversarial examples for each layer separately in addition to the original training set. According to our initial observations, adversarial examples for the higher layers seemed to be significantly more useful than those on the input or lower layers. In our future work, we plan to compare these effects in a systematic manner. For space considerations, we just present results for a representative subset (see Table 1) of the MNIST experiments we performed. The results presented here are consistent with those on a larger variety of non-convolutional models. For MNIST, we do not have results for convolutional models yet, but our first qualitative experiments with AlexNet gives us reason to believe that convolutional networks may behave similarly as well. Each of our models were trained with L-BFGS until convergence. The first three models are linear classifiers that work on the pixel level with various weight decay parameters $\lambda$. All our examples use quadratic weight decay on the connection weights: $\textrm{loss}_{decay}=\lambda\sum w_{i}^{2}/k$ added to the total loss, where $k$ is the number of units in the layer. Three of our models are simple linear (softmax) classifier without hidden units (FC10($\lambda$)). One of them, FC10($1$), is trained with extremely high $\lambda=1$ in order to test whether it is still possible to generate adversarial examples in this extreme setting as well.Two other models are a simple sigmoidal neural network with two hidden layers and a classifier. The last model, AE400-10, consists of a single layer sparse autoencoder with sigmoid activations and 400 nodes with a Softmax classifier. This network has been trained until it got very high quality first layer filters and this layer was not fine-tuned. The last column measures the minimum average pixel level distortion necessary to reach $0\%$ accuracy on the training set. The distortion is measure by $\sqrt{\frac{\sum(x_{i}^{\prime}-x_{i})^{2}}{n}}$ between the original $x$ and distorted $x^{\prime}$ images, where $n=784$ is the number of image pixels. The pixel intensities are scaled to be in the range $[0,1]$. In our first experiment, we generated a set of adversarial instances for a given network and fed these examples for each other network to measure the proportion of misclassified instances. The last column shows the average minimum distortion that was necessary to reach 0% accuracy on the whole training set. The experimental results are presented in Table 2. The columns of Table 2 show the error (proportion of misclassified instances) on the so distorted training sets. The last two rows are given for reference showing the error induced when distorting by the given amounts of Gaussian noise. Note that even the noise with stddev 0.1 is greater than the stddev of our adversarial noise for all but one of the models. Figure 7 shows a visualization of the generated adversarial instances for two of the networks used in this experiment The general conclusion is that adversarial examples tend to stay hard even for models trained with different hyperparameters. Although the autoencoder based version seems most resilient to adversarial examples, it is not fully immune either. (a) Even columns: adversarial examples for a linear (FC) classifier (stddev=0.06) (b) Even columns: adversarial examples for a 200-200-10 sigmoid network (stddev=0.063) (c) Randomly distorted samples by Gaussian noise with stddev=1. Accuracy: 51%. Figure 7: Adversarial examples for a randomly chosen subset of MNIST compared with randomly distorted examples. Odd columns correspond to original images, and even columns correspond to distorted counterparts. The adversarial examples generated for the specific model have accuracy 0% for the respective model. Note that while the randomly distorted examples are hardly readable, still they are classified correctly in half of the cases, while the adversarial examples are never classified correctly. Model Name \| Description \| Training error \| Test error \| Av. min. distortion ---\|---\|---\|---\|--- FC10($10^{-4}$) \| Softmax with $\lambda=10^{-4}$ \| 6.7% \| 7.4% \| 0.062 FC10($10^{-2}$) \| Softmax with $\lambda=10^{-2}$ \| 10% \| 9.4% \| 0.1 FC10($1$) \| Softmax with $\lambda=1$ \| 21.2% \| 20% \| 0.14 FC100-100-10 \| Sigmoid network $\lambda=10^{-5},10^{-5},10^{-6}$ \| 0% \| 1.64% \| 0.058 FC200-200-10 \| Sigmoid network $\lambda=10^{-5},10^{-5},10^{-6}$ \| 0% \| 1.54% \| 0.065 AE400-10 \| Autoencoder with Softmax $\lambda=10^{-6}$ \| 0.57% \| 1.9% \| 0.086 Table 1: Tests of the generalization of adversarial instances on MNIST. \| FC10($10^{-4}$) \| FC10($10^{-2}$) \| FC10($1$) \| FC100-100-10 \| FC200-200-10 \| AE400-10 \| Av. distortion ---\|---\|---\|---\|---\|---\|---\|--- FC10($10^{-4}$) \| 100% \| 11.7% \| 22.7% \| 2% \| 3.9% \| 2.7% \| 0.062 FC10($10^{-2}$) \| 87.1% \| 100% \| 35.2% \| 35.9% \| 27.3% \| 9.8% \| 0.1 FC10($1$) \| 71.9% \| 76.2% \| 100% \| 48.1% \| 47% \| 34.4% \| 0.14 FC100-100-10 \| 28.9% \| 13.7% \| 21.1% \| 100% \| 6.6% \| 2% \| 0.058 FC200-200-10 \| 38.2% \| 14% \| 23.8% \| 20.3% \| 100% \| 2.7% \| 0.065 AE400-10 \| 23.4% \| 16% \| 24.8% \| 9.4% \| 6.6% \| 100% \| 0.086 Gaussian noise, stddev=0.1 \| 5.0% \| 10.1% \| 18.3% \| 0% \| 0% \| 0.8% \| 0.1 Gaussian noise, stddev=0.3 \| 15.6% \| 11.3% \| 22.7% \| 5% \| 4.3% \| 3.1% \| 0.3 Table 2: Cross-model generalization of adversarial examples. The columns of the Tables show the error induced by distorted examples fed to the given model. The last column shows average distortion wrt. original training set. Still, this experiment leaves open the question of dependence over the training set. Does the hardness of the generated examples rely solely on the particular choice of our training set as a sample or does this effect generalize even to models trained on completely different training sets? Model \| Error on $P_{1}$ \| Error on $P_{2}$ \| Error on Test \| Min Av. Distortion ---\|---\|---\|---\|--- FC100-100-10: 100-100-10 trained on $P_{1}$ \| 0% \| 2.4% \| 2% \| 0.062 FC123-456-10: 123-456-10 trained on $P_{1}$ \| 0% \| 2.5% \| 2.1% \| 0.059 FC100-100-10’ trained on $P_{2}$ \| 2.3% \| 0% \| 2.1% \| 0.058 Table 3: Models trained to study cross-training-set generalization of the generated adversarial examples. Errors presented in Table correpond to original not-distorted data, to provide a baseline. \| FC100-100-10 \| FC123-456-10 \| FC100-100-10’ ---\|---\|---\|--- Distorted for FC100-100-10 (av. stddev=0.062) \| 100% \| 26.2% \| 5.9% Distorted for FC123-456-10 (av. stddev=0.059) \| 6.25% \| 100% \| 5.1% Distorted for FC100-100-10’ (av. stddev=0.058) \| 8.2% \| 8.2% \| 100% Gaussian noise with stddev=$0.06$ \| 2.2% \| 2.6% \| 2.4% Distorted for FC100-100-10 amplified to stddev=$0.1$ \| 100% \| 98% \| 43% Distorted for FC123-456-10 amplified to stddev=$0.1$ \| 96% \| 100% \| 22% Distorted for FC100-100-10’ amplified to stddev=$0.1$ \| 27% \| 50% \| 100% Gaussian noise with stddev=$0.1$ \| 2.6% \| 2.8% \| 2.7% Table 4: Cross-training-set generalization error rate for the set of adversarial examples generated for different models. The error induced by a random distortion to the same examples is displayed in the last row. To study cross-training-set generalization, we have partitioned the 60000 MNIST training images into two parts $P_{1}$ and $P_{2}$ of size 30000 each and trained three non-convolutional networks with sigmoid activations on them: Two, FC100-100-10 and FC123-456-10, on $P_{1}$ and FC100-100-10 on $P_{2}$. The reason we trained two networks for $P_{1}$ is to study the cumulative effect of changing the hypermarameters and the training sets at the same time. Models FC100-100-10 and FC100-100-10 share the same hyperparameters: both of them are 100-100-10 networks, while FC123-456-10 has different number of hidden units. In this experiment, we were distorting the elements of the test set rather than the training set. Table 3 summarizes the basic facts about these models. After we generate adversarial examples with $100\%$ error rates with minimum distortion for the test set, we feed these examples to the each of the models. The error for each model is displayed in the corresponding column of the upper part of Table 4. In the last experiment, we magnify the effect of our distortion by using the examples $x+0.1\frac{x^{\prime}-x}{\\|x^{\prime}-x\\|_{2}}$ rather than $x^{\prime}$. This magnifies the distortion on average by 40%, from stddev $0.06$ to $0.1$. The so distorted examples are fed back to each of the models and the error rates are displayed in the lower part of Table 4. The intriguing conclusion is that the adversarial examples remain hard for models trained even on a disjoint training set, although their effectiveness decreases considerably. ### 4.3 Spectral Analysis of Unstability The previous section showed examples of deep networks resulting from purely supervised training which are unstable with respect to a peculiar form of small perturbations. Independently of their generalisation properties across networks and training sets, the adversarial examples show that there exist small additive perturbations of the input (in Euclidean sense) that produce large perturbations at the output of the last layer. This section describes a simple procedure to measure and control the additive stability of the network by measuring the spectrum of each rectified layer. Mathematically, if $\phi(x)$ denotes the output of a network of $K$ layers corresponding to input $x$ and trained parameters $W$, we write $\phi(x)=\phi_{K}(\phi_{K-1}(\dots\phi_{1}(x;W_{1});W_{2})\dots;W_{K})~{},$ where $\phi_{k}$ denotes the operator mapping layer $k-1$ to layer $k$. The unstability of $\phi(x)$ can be explained by inspecting the upper Lipschitz constant of each layer $k=1\dots K$, defined as the constant $L_{k}>0$ such that $\forall\,x,\,r~{},~{}\\|\phi_{k}(x;W_{k})-\phi_{k}(x+r;W_{k})\\|\leq L_{k}\\|r\\|~{}.$ The resulting network thus satsifies $\\|\phi(x)-\phi(x+r)\\|\leq L\\|r\\|$, with $L=\prod_{k=1}^{K}L_{k}$. A half-rectified layer (both convolutional or fully connected) is defined by the mapping $\phi_{k}(x;W_{k},b_{k})=\max(0,W_{k}x+b_{k})$. Let $\\|W\\|$ denote the operator norm of $W$ (i.e., its largest singular value). Since the non- linearity $\rho(x)=\max(0,x)$ is contractive, i.e. satisfies $\\|\rho(x)-\rho(x+r)\\|\leq\\|r\\|~{}$ for all $x,r$; it follows that $\\|\phi_{k}(x;W_{k})-\phi_{k}(x+r;W_{k})\\|=\\|\max(0,W_{k}x+b_{k})-\max(0,W_{k}(x+r)+b_{k})\\|\leq\\|W_{k}r\\|\leq\\|W_{k}\\|\\|r\\|~{},$ and hence $L_{k}\leq\\|W_{k}\\|$. On the other hand, a max-pooling layer $\phi_{k}$ is contractive: $\forall\,x\,,\,r\,,~{}\\|\phi_{k}(x)-\phi_{k}(x+r)\\|\leq\\|r\\|~{},$ since its Jacobian is a projection onto a subset of the input coordinates and hence does not expand the gradients. Finally, if $\phi_{k}$ is a contrast- normalization layer $\phi_{k}(x)=\frac{x}{\Big{(}\epsilon+\\|x\\|^{2}\Big{)}^{\gamma}}~{},$ one can verify that $\forall\,x\,,\,r\,,~{}\\|\phi_{k}(x)-\phi_{k}(x+r)\\|\leq\epsilon^{-\gamma}\\|r\\|$ for $\gamma\in[0.5,1]$, which corresponds to most common operating regimes. It results that a conservative measure of the unstability of the network can be obtained by simply computing the operator norm of each fully connected and convolutional layer. The fully connected case is trivial since the norm is directly given by the largest singular value of the fully connected matrix. Let us describe the convolutional case. If $W$ denotes a generic $4$-tensor, implementing a convolutional layer with $C$ input features, $D$ output features, support $N\times N$ and spatial stride $\Delta$, $Wx=\left\\{\sum_{c=1}^{C}x_{c}\star w_{c,d}(n_{1}\Delta,n_{2}\Delta)\,;d=1\,\dots,D\right\\}~{},$ where $x_{c}$ denotes the $c$-th input feature image, and $w_{c,d}$ is the spatial kernel corresponding to input feature $c$ and output feature $d$, by applying Parseval’s formula we obtain that its operator norm is given by $\\|W\\|=\sup_{\xi\in[0,N\Delta^{-1})^{2}}\\|A(\xi)\\|~{},$ (1) where $A(\xi)$ is a $D\times(C\cdot\Delta^{2})$ matrix whose rows are $\forall~{}d=1\dots D~{},~{}A(\xi)_{d}=\Big{(}\Delta^{-2}\widehat{w_{c,d}}(\xi+l\cdot N\cdot\Delta^{-1})\,;\,c=1\dots C\,,\,l=(0\dots\Delta-1)^{2}\Big{)}~{},$ and $\widehat{w_{c,d}}$ is the 2-D Fourier transform of $w_{c,d}$: $\widehat{w_{c,d}}(\xi)=\sum_{u\in[0,N)^{2}}w_{c,d}(u)e^{-2\pi i(u\cdot\xi)/N^{2}}~{}.$ Layer \| Size \| Stride \| Upper bound ---\|---\|---\|--- Conv. $1$ \| $3\times 11\times 11\times 96$ \| $4$ \| $2.75$ Conv. $2$ \| $96\times 5\times 5\times 256$ \| $1$ \| $10$ Conv. $3$ \| $256\times 3\times 3\times 384$ \| $1$ \| $7$ Conv. $4$ \| $384\times 3\times 3\times 384$ \| $1$ \| $7.5$ Conv. $5$ \| $384\times 3\times 3\times 256$ \| $1$ \| $11$ FC. 1 \| $9216\times 4096$ \| N/A \| $3.12$ FC. 2 \| $4096\times 4096$ \| N/A \| $4$ FC. 3 \| $4096\times 1000$ \| N/A \| $4$ Table 5: Frame Bounds of each rectified layer of the network from [9]. Table 5 shows the upper Lipschitz bounds computed from the ImageNet deep convolutional network of [9], using (1). It shows that instabilities can appear as soon as in the first convolutional layer. These results are consistent with the exsitence of blind spots constructed in the previous section, but they don’t attempt to explain why these examples generalize across different hyperparameters or training sets. We emphasize that we compute upper bounds: large bounds do not automatically translate into existence of adversarial examples; however, small bounds guarantee that no such examples can appear. This suggests a simple regularization of the parameters, consisting in penalizing each upper Lipschitz bound, which might help improve the generalisation error of the networks. ## 5 Discussion We demonstrated that deep neural networks have counter-intuitive properties both with respect to the semantic meaning of individual units and with respect to their discontinuities. The existence of the adversarial negatives appears to be in contradiction with the network’s ability to achieve high generalization performance. Indeed, if the network can generalize well, how can it be confused by these adversarial negatives, which are indistinguishable from the regular examples? Possible explanation is that the set of adversarial negatives is of extremely low probability, and thus is never (or rarely) observed in the test set, yet it is dense (much like the rational numbers), and so it is found near every virtually every test case. However, we don’t have a deep understanding of how often adversarial negatives appears, and thus this issue should be addressed in a future research. ## References * [1] David Baehrens, Timon Schroeter, Stefan Harmeling, Motoaki Kawanabe, Katja Hansen, and Klaus-Robert Müller. How to explain individual classification decisions. The Journal of Machine Learning Research, 99:1803–1831, 2010. * [2] Yoshua Bengio. Learning deep architectures for ai. Foundations and trends® in Machine Learning, 2(1):1–127, 2009. * [3] Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. Imagenet: A large-scale hierarchical image database. In Computer Vision and Pattern Recognition, 2009. CVPR 2009. IEEE Conference on, pages 248–255. IEEE, 2009. * [4] Dumitru Erhan, Yoshua Bengio, Aaron Courville, and Pascal Vincent. Visualizing higher-layer features of a deep network. Technical Report 1341, University of Montreal, June 2009. Also presented at the ICML 2009 Workshop on Learning Feature Hierarchies, Montréal, Canada. * [5] Pedro Felzenszwalb, David McAllester, and Deva Ramanan. A discriminatively trained, multiscale, deformable part model. In Computer Vision and Pattern Recognition, 2008. CVPR 2008. IEEE Conference on, pages 1–8. IEEE, 2008. * [6] Ross Girshick, Jeff Donahue, Trevor Darrell, and Jitendra Malik. Rich feature hierarchies for accurate object detection and semantic segmentation. arXiv preprint arXiv:1311.2524, 2013. * [7] Ian Goodfellow, Quoc Le, Andrew Saxe, Honglak Lee, and Andrew Y Ng. Measuring invariances in deep networks. Advances in neural information processing systems, 22:646–654, 2009\. * [8] Geoffrey E. Hinton, Li Deng, Dong Yu, George E. Dahl, Abdel rahman Mohamed, Navdeep Jaitly, Andrew Senior, Vincent Vanhoucke, Patrick Nguyen, Tara N. Sainath, and Brian Kingsbury. Deep neural networks for acoustic modeling in speech recognition: The shared views of four research groups. IEEE Signal Process. Mag., 29(6):82–97, 2012. * [9] Alex Krizhevsky, Ilya Sutskever, and Geoff Hinton. Imagenet classification with deep convolutional neural networks. In Advances in Neural Information Processing Systems 25, pages 1106–1114, 2012. * [10] Quoc V Le, Marc’Aurelio Ranzato, Rajat Monga, Matthieu Devin, Kai Chen, Greg S Corrado, Jeff Dean, and Andrew Y Ng. Building high-level features using large scale unsupervised learning. arXiv preprint arXiv:1112.6209, 2011. * [11] Yann LeCun and Corinna Cortes. The mnist database of handwritten digits, 1998. * [12] Tomas Mikolov, Kai Chen, Greg Corrado, and Jeffrey Dean. Efficient estimation of word representations in vector space. arXiv preprint arXiv:1301.3781, 2013. * [13] Matthew D Zeiler and Rob Fergus. Visualizing and understanding convolutional neural networks. arXiv preprint arXiv:1311.2901, 2013.	arxiv-papers	2013-12-21T03:36:08	2024-09-04T02:49:55.747396	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Christian Szegedy, Wojciech Zaremba, Ilya Sutskever, Joan Bruna,\n Dumitru Erhan, Ian Goodfellow, Rob Fergus", "submitter": "Joan Bruna", "url": "https://arxiv.org/abs/1312.6199" }
1312.6265	# Paley type inequality of the Fourier transform on the Heisenberg group Atef Rahmouni Department of mathematics, King Saudi University, College of Sciences P. O Box 2455 Riyadh 11451, Saudi Arabia. [email protected] ###### Abstract. A paley type inequality for the Fourier transform on $H^{p}(\mathbb{H}^{n}),$ the Hardy space on the Heisenberg group, is obtained for $0<p\leq 1.$ ###### Key words and phrases: Hardy-Littlewood inequality; Heisenberg group. ## 1\. Introduction The study of Hardy spaces has been originated during the 1910’s in the setting of Fourier series and complex analysis in one variable. In 1972, Fefferman and Stein [5] introduced Hardy spaces $H^{p}$ by mean of maximal function $f^{\ast}(x)=\sup_{r>0}\|f\ast\phi_{r}(x)\|$ where $\phi$ belongs to $\mathcal{S}$, the Schwartz space of rapidly decreasing smooth functions satisfying $\int\phi(x)dx=1$. The delation $\phi_{r}$ is given by $\phi_{r}(x)=r^{-n}\phi(x/r).$ We say that a tempered distributions $f\in\mathcal{S}^{\prime}$ is in $H^{p}$ if $f^{\ast}$ is in $L^{p}$. Using the maximal function above, Coifman [4] showed that any $f$ in $H^{p}$ can be represented as a linear combination of atoms, that is $f=\sum_{k=1}^{\infty}\beta_{k}a_{k},\quad\beta_{k}\in\mathbb{C},$ where the $a_{k}$ are atoms and the sum converges in $H^{p}$. Moreover, $\\|f\\|_{H^{p}}\thickapprox\inf\Big{\\{}\sum_{k=1}^{\infty}\|\beta_{k}\|^{p}:\sum_{k=1}^{\infty}\beta_{k}a_{k}\mbox{ is a decomposition of $f$ into\, }atoms\Big{\\}}.$ It has been shown that the study of some analytic problems on $H^{p}(\mathbb{R}^{n})$ is summed up to investigate some properties of these atoms, and therefore the problems become quite simple. In 1980, Taibleson and Weiss [17] gave the definition of molecules belonging to $H^{p},$ and showed that every molecule is in $H^{p}$ with continuous embedding map. By the atomic decomposition and the molecule characterization, the proof of $H^{p}$ boundedness of the operators on Hardy space becomes easier. The theory of $H^{p}$ have been extensively studied in [7] and [6]. In the setting of the euclidian case, Hardy’s inequality for Fourier transform asserts that for all $f\in H^{p}(\mathbb{R}^{n})$ $0<p\leq 1.$ $\int_{\mathbb{R}^{n}}{\|\widehat{f}(\xi)\|^{p}\over\|\xi\|^{n(2-p)}}d\xi\leq\\|f\\|^{p}_{H^{p}{(\mathbb{R}^{n}})},\qquad 0<p\leq 1$ (1.1) where $H^{p}(\mathbb{R}^{n})$ indicates the real Hardy space. Hardy’s type inequality for Fourier transform has been extensively studied in [16]. Kanjin [13] proved Hardy’s inequalities for Hermit and Laguerre expansions for functions in $H^{1}$ and for Hankel transform [12]. In connection with properties of regularity of the spherical means on $\mathbb{C}^{n}$, Thangavelu [18] proved a Hardy’s inequality for special Hermit functions. These standard inequalities for higher dimensional has been studied in [14]. Recently, an extension has been given by [1], the latter establish a Hardy’s type inequality associated with the Hankel transform for over critical exponent $\sigma>\sigma_{0}=2-p.$ We point out here that the result obtained for Hardy’s inequality for the Hankel transform improves the work of Kanjin [12] in which he proved the result for $\sigma_{0}=2-p.$ Although, in [2, 3, 15] extended this form of this inequality to Laguerre hypergroup and its dual. In this paper we are interested in the Heisenberg group $\mathbb{H}^{n}$ is the lie group with underlying manifold $\mathbb{H}^{n}=\mathbb{C}^{n}\times\mathbb{R}$ and multiplication $(z,t).(z^{\prime},t^{\prime})=(z+z^{\prime},t+t^{\prime}+2Im(z.\overline{z^{\prime}}),$ where $\linebreak z=(z_{1},z_{2},...,z_{n})\in\mathbb{C}^{n}.$ If we identify $\mathbb{C}^{n}\times\mathbb{R}$ with $\mathbb{R}^{2n+1}$ by $z_{j}=x_{j}+ix_{j+n},~{}j=1,...,n,$ then the group law can be rewritten as $(x_{1},x_{2},...,x_{2n},t).(y_{1},y_{2},...,y_{2n},t^{\prime})=(x_{1}+y_{2},...,x_{n}+y_{n},t+t^{\prime}-2\sum_{j=1}^{n}(x_{j}y_{j+n}-y_{j}x_{j+n})).$ The reverse element of $u=(z,t)$ is $u^{-1}=(-z,-t)$ and we write the identity of $\mathbb{H}^{n}$ as $0=(0,0).$ Set $X_{j},X_{j+n}$ and $T$ is a basis for the left invariant vector fields on $\mathbb{H}^{n}.$ The corresponding complex vector fields are $Z_{j}=\frac{1}{2}(X_{j}-iX_{j+n})=\frac{\partial}{\partial z_{j}}+i\overline{z}_{j}\frac{\partial}{\partial t},~{}\overline{Z}_{j}=\frac{1}{2}(X_{j}+iX_{j+n})=\frac{\partial}{\partial\overline{z}_{j}}-iz_{j}\frac{\partial}{\partial t},~{}j=1,...,n.$ The Heisenberg group is a connected, simply connected nilpotent Lie group. We define one-parameter dilations on $\mathbb{H}^{n},$ for $R>0,$ by $\rho_{R}(z,t)=(Rz,R^{2}t).$ These dilations are group automorphisms and the Jacobian determinant is $R^{Q},$ where $Q=2n+2$ is the homogeneous dimension of $\mathbb{H}^{n}.$ We will denoted by $f_{\rho}(z,t)=\rho^{-Q}f((z,t)_{\rho})$ the dilated of the function $f$ defined on $\mathbb{H}^{n}.$ A homogeneous norm on $\mathbb{H}^{n}$ is given by $\|(z,t)\|_{\mathbb{H}^{n}}=(\|z\|^{4}+4t^{2})^{1/4},$ With this norm, we define the Heisenberg ball centered at $u=(z,t)$ of radius $r,$ i.e., the set $B(u,r)=\\{v\in\mathbb{H}^{n}:~{}\|uv^{-1}\|_{\mathbb{H}^{n}}<R\\},$ and we denote by $B_{R}=B(0,R)=\\{v\in\mathbb{H}^{n}:~{}\|v\|_{\mathbb{H}^{n}}<R\\}$ the open ball centered at 0, the identity element of $\mathbb{H}^{n},$ with radius $R.$ The volume of the ball $B(u,R)$ is $C_{Q}R^{Q},$ where $C_{Q}$ is the volume of the unit ball $B_{1}.$ The Haar measure $dV$ on $\mathbb{H}^{n}$ coincides with the Lebesgue measure on $\mathbb{C}^{n}\times\mathbb{R}$ which is denoted by $dzd\overline{z}dt.$ Let $J=(j^{1},j^{2},j^{0})\in\mathbb{Z}^{n}_{+}\times\mathbb{Z}^{n}_{+}\times\mathbb{Z}_{+},$ where $\mathbb{Z}_{+}$ the set of all nonnegative integers, we set $h(J)=\|j^{1}\|+\|j^{2}\|+2j^{0},$ where, if $j^{1}=(j^{1}_{1},...,j^{n}_{n}),$ then $\|j^{1}\|=\sum_{k=1}^{n}j^{1}_{k}.$ If $P(z,t)=\sum_{J}a_{J}(z,t)^{J}$ is a polynomial where $(z,t)^{J}=z^{j^{1}}\overline{z}^{j^{2}}t^{j^{0}},$ then we call $\max\\{h(J):a_{J}\neq 0\\}$ the homogeneous degree of $P(z,t).$ The set of all polynomials whose homogeneous degree $\leq s$ is denoted by $\mathcal{P}_{s}.$ Schwartz space on $\mathbb{H}^{n}$ write as $\mathcal{S}(\mathbb{H}^{n}).$ Fix $\lambda>0,$ let $\mathcal{H}_{\lambda}$ be the Bargmann’s space : $\mathcal{H}_{\lambda}=\Big{\\{}F~{}\mbox{holomorphic on}~{}\mathbb{C}^{n}:\\|F\\|^{2}=\big{(}\frac{2\lambda}{\pi}\big{)}^{n}\int_{\mathbb{C}^{n}}\|F(\zeta)\|^{2}e^{-2\lambda\|\zeta\|^{2}}d\zeta<\infty\Big{\\}}.$ Then, $\mathcal{H}_{\lambda}$ is a Hilbert space and the monomials $F_{\alpha,\lambda}(\zeta)=\sqrt{\frac{(2\lambda)^{\|\alpha\|}}{\alpha!}}\zeta^{\alpha},~{}~{}~{}\alpha=(\alpha_{1},\alpha_{2},...,\alpha_{n})\in\mathbb{Z}^{n}_{+}$ form an orthonormal basis for $\mathcal{H}_{\lambda},$ where $\alpha!=\alpha_{1}!\alpha_{2}!...\alpha_{n}!,~{}\|\alpha\|=(\alpha_{1},\alpha_{2},...,\alpha_{n})$ and $\linebreak\zeta^{\alpha}=\zeta^{\alpha_{1}}_{1}\zeta^{\alpha_{2}}_{2}...\zeta^{\alpha_{n}}_{n}.$ Suppose $W_{k,\lambda}$ and $W^{+}_{k,\lambda}$ are the closed operators on $\mathcal{H}_{\lambda}$ such that $\displaystyle W_{k,\lambda}F_{\alpha,\lambda}$ $\displaystyle=$ $\displaystyle(2(\alpha_{k}+1)\lambda)^{1/2}F_{\alpha+e_{k},\lambda},$ $\displaystyle W_{k,\lambda}^{+}F_{\alpha,\lambda}$ $\displaystyle=$ $\displaystyle(2\alpha_{k}\lambda)^{1/2}F_{\alpha- e_{k},\lambda},\qquad\mbox{for}~{}~{}\lambda>0,$ and $\displaystyle W_{k,\lambda}$ $\displaystyle=$ $\displaystyle W_{k,-\lambda}^{+},$ $\displaystyle W_{k,\lambda}^{+}$ $\displaystyle=$ $\displaystyle W_{k,-\lambda},\qquad\mbox{for}~{}~{}\lambda<0,$ where $e_{k}=(0,...,1,...,0)\in\mathbb{Z}^{n}$ with the 1 in the $k$-th position. Then $\prod_{\lambda}(z,t)=exp^{i\lambda t}exp^{(-z.W_{\lambda}+\overline{z}.W_{\lambda}^{+})}$ is an irreducible unitary representation of $\mathbb{H}^{n}$ on $\mathcal{H}_{\lambda},$ where $z.W_{\lambda}=\sum_{k=1}^{n}z_{k}.W_{k,\lambda}.$ The group Fourier transform of $f\in L^{1}(\mathbb{H}^{n})\cap L^{2}(\mathbb{H}^{n})$ is an operator-valued function defined by $\mathcal{F}(f)(\lambda)=\int_{\mathbb{H}^{n}}f(z,t)\prod_{\lambda}(z,t)dV.$ (1.2) Obviously, $\\|\mathcal{F}(f)(\lambda)\\|\leq\\|f\\|_{L^{1}}.$ Here, $\\|-\\|$ denotes the operator norm. Similar as in $\mathbb{R}^{n},$ for $f\in L^{1}(\mathbb{H}^{n})\cap L^{2}(\mathbb{H}^{n}),$ we have the following Plancherel and inversion formulas : $\\|f\\|_{2}^{2}=\frac{2^{n-1}}{\pi^{n+1}}\int_{\mathbb{R}}\\|\mathcal{F}(f)(\lambda)\\|^{2}_{HS}\|\lambda\|^{n}d\lambda,\qquad f\in L^{1}(\mathbb{H}^{n})\cap L^{2}(\mathbb{H}^{n}),$ (1.3) $\int_{\mathbb{R}}tr\Big{(}\prod^{}_{\lambda}(z,t)\mathcal{F}(f)(\lambda)\Big{)}\|\lambda\|^{n}d\lambda=\frac{(2\pi)^{n+1}}{4^{n}}f(u)$ (1.4) where $tr$ is the canonical semifinite trace and $\\|-\\|_{HS}$ denotes the Hilbert-Schmidt norm. For $(\lambda,m,\alpha)\in\mathbb{R}^{}\times\mathbb{Z}^{n}\times\mathbb{Z}^{n}_{+},$ where $\mathbb{R}^{}=\mathbb{R}\backslash\\{0\\},$ we use the notations \| $m_{i}^{+}=\max\\{m_{i},0\\},\qquad$ \| $m_{i}^{-}=-\min\\{m_{i},0\\},$ ---\|---\|--- \| $m^{+}=(m_{1}^{+},m_{2}^{+},...,m_{n}^{+})\qquad$ \| $m^{-}=(m_{1}^{-},m_{2}^{-},...,m_{n}^{-}).$ The partial isometry operator $W^{m}_{\alpha}(\lambda)$ on $\mathcal{H}_{\|\lambda\|}$ by \| $W_{k,\alpha}(\lambda)F_{\beta,\lambda}=(-1)^{\|m^{+}\|}\delta_{\alpha+m^{+},\beta}F_{\alpha+m^{-},\lambda},$ \| $\qquad\mbox{for}~{}~{}\lambda>0;$ ---\|---\|--- \| $W_{\alpha}^{m}(\lambda)=[W_{\alpha}^{m}(-\lambda)]^{},$ \| $\qquad\mbox{for}~{}~{}\lambda<0.$ Thus $\\{W^{m}_{\alpha}(\lambda):m\in\mathbb{Z}^{n},\alpha\in\mathbb{Z}^{n}\\}$ is an orthonormal basis for the Hilbert-Schmidt operators on $\mathcal{H}_{\|\lambda\|}.$ Given a function $f\in L^{2}(\mathbb{H}^{n})$ such that $f(z,t)=\sum_{m,\alpha}f_{m}(r_{1},...,r_{n},t)e^{i(m_{1}\theta_{1}+...+m_{n}\theta_{n})},\qquad\mbox{where}\qquad z_{j}=r_{j}e^{i\theta_{j}},$ then $\mathcal{F}(f)(\lambda)=\sum_{m,\alpha}R_{f}(\lambda,m,\alpha)W^{m}_{\alpha}(\lambda),$ where $R_{f}(\lambda,m,\alpha)=\int_{\mathbb{H}^{n}}f_{m}(r_{1},...,r_{n},t)e^{i\lambda t}\ell_{\alpha_{1}}^{\|m_{1}\|}(2\|\lambda\|r_{1}^{2})...\ell_{\alpha_{n}}^{\|m_{n}\|}(2\|\lambda\|r_{n}^{2})dV,$ and $\ell_{\alpha}^{\|m\|}$ is the Larguerre function of type $\|m\|$ and degree $\|\alpha\|.$ Let $P$ be a polynomial in $z_{j},\overline{z}_{j},t$ on $\mathbb{H}^{n},$ and we define the difference-differential operator $\Delta_{P}$ acting on the Fourier transform of $f\in L^{1}\cap L^{2}(\mathbb{H}^{n})$ by $\Delta_{P}\Big{(}\sum_{m,\alpha}R_{f}(\lambda,m,\alpha)W_{\alpha}^{m}(\lambda)\Big{)}=\sum_{m,\alpha}R_{Pf}(\lambda,m,\alpha)W_{\alpha}^{m}(\lambda),$ namely, $\Delta_{P}\mathcal{F}(f)(\lambda)=\widehat{P(.)f(.)}(\lambda).$ In [9] and [10], the authors gave the explicit expressions for $\Delta_{t},\Delta_{z_{j}}$ and $\Delta_{\overline{z}_{j}}.$ For convenience, we shall write $\Delta^{J}_{(z,t)}=\Delta^{J}.$ The paper is organized as follows. In the Second section we give an appropriate definition of atoms and investigate the atoms characterization of Hardy spaces $H^{p}(\mathbb{H}^{n})$ for $0<p\leq 1.$ In the last section we state and prove our main result: ###### Theorem 1.1. Let $0<p\leq 1,$ and $s\geq J=[Q({1/p}-1)],$ the greatest integer not exceeding $\linebreak Q({1/p}-1).$ Then for any $f\in H^{p}(\mathbb{H}^{n})$ the Fourier transform of $f$ satisfies the following Hardy’s type inequality $\int_{\mathbb{R}}\frac{\\|\mathcal{F}(f)(\lambda)\\|_{HS}^{p}}{\big{(}(2\|\alpha\|+n)\|\lambda\|\big{)}^{\sigma}}\|\lambda\|^{n}d\lambda\leq C(p,n)\\|f\\|^{p}_{H^{p}(\mathbb{H}^{n})},$ (1.5) provided that $\frac{Q}{2}(2-p)\leq\sigma<\frac{Q}{2}+p(\frac{J+1}{2})$ (1.6) where $C(p,n)$ depend only on $p$ and $n.$ Finally, we mention that $C$ will be always used to denote a suitable positive constant that is not necessarily the same in each occurrence. ## 2\. Atomic decomposition for $H^{p}(\mathbb{H}^{n})$ Now we state the definition of atomic Hardy spaces in the setting of the Heisenberg group $H^{p}(\mathbb{H}^{n})$, $0<p\leq 1$. To this end, we introduce the following kind of atoms, which is closely related to the Haar measure $dV.$ ###### Definition 2.1. Let $0<p\leq 1\leq q\leq\infty,p\neq q,s\in\mathbb{Z}$ and $s\geq J=[Q(1/p-1)].$ (Such an ordered triple $(p,q,s)$ is called admissible). A $(p,q,s)$-atom centered at $x_{0}\in\mathbb{H}^{n}$ is a function $a\in L^{q}(\mathbb{H}^{n}),$ supported on a ball $B(x_{0},R)\subset\mathbb{H}^{n}$ with centre $x_{0}=(z_{0},t_{0})$ and satisfying the following * (i) $\\|a\\|_{L^{q}(\mathbb{H}^{n})}\leq\|B(0,r)\|^{\frac{1}{q}-{1\over p}}$, a.e, * (ii) $\displaystyle\int_{\mathbb{H}^{n}}a(x)P(x)dV(x)=0,$ for every $P\in\mathcal{P}_{s}.$ Here, $(i)$ means that the size condition of atoms, and $(ii)$ is called the cancelation moment condition. A characterization of $H^{p}(\mathbb{H}^{n})$ is included in the following statements. ###### Proposition 2.1. Let $0<p\leq 1.$ If $\\{a_{k}\\}_{k=0}^{\infty}$ is a sequence of $p$-atoms, and $\\{\lambda_{k}\\}_{k=0}^{\infty}$ is a sequence of complex numbers with $\Big{(}\sum_{k=0}^{\infty}\|\lambda_{k}\|^{p}\Big{)}^{1/p}<\infty,$ then $\sum_{k=0}^{\infty}\lambda_{k}a_{k}$ converges in $H^{p}(\mathbb{H}^{n})$ and $\Big{\\|}\sum_{k}\lambda_{k}a_{k}\Big{\\|}_{H^{p}(\mathbb{H}^{n})}\leq C(p,n)\Big{(}\sum_{k}\|\lambda_{k}\|^{p}\Big{)}^{1/p}.$ Conversely, if $f\in H^{p}(\mathbb{H}^{n})$ there exists a sequence $\\{a_{k}\\}_{k=0}^{\infty}$ of $p$-atoms, and a sequence $\\{\lambda_{k}\\}_{k=0}^{\infty}$ of complex numbers such that $f=\sum_{k}\lambda_{k}a_{k}\qquad\mbox{and}\qquad\Big{(}\sum_{k}\|\lambda_{k}\|^{p}\Big{)}^{1/p}\leq C(p,n)\\|f\\|_{H^{p}(\mathbb{H}^{n})},$ where $C(p,n)$ depends on $p$ and $n.$ ## 3\. Proof of the main result Now we are in a position to give the proof of the main result. First we stat the following proposition which has its own interest. ###### Proposition 3.1. For all $(z,t)\in\mathbb{H}^{n}$ the function $\prod_{\lambda}(z,t)$ satisfies $\prod_{\lambda}(z,t)=\sum_{2k+\ell\leq J}\omega_{k,\ell}(\lambda,n)~{}z^{k}t^{\ell}+R_{\theta}(z,t),\,\,0<\theta<1,$ (3.1) where $R_{\theta}(z,t)=\sum_{2k+\ell=J+1}\frac{(i\lambda t)^{k}}{k!}.\frac{(z.W_{\lambda}-\overline{z}.W^{+}_{\lambda})^{\ell}}{\ell!}.$ (3.2) Here $\omega_{k,\ell}(\lambda,n)$ are functions expressed by mean of $\lambda,n.$ Set $\mathcal{H}_{\|\lambda\|}^{N}$ be the subspace of $\mathcal{H}_{\|\lambda\|}$ spanned by $\\{W^{0}_{\alpha}(\lambda):\|\alpha\|\leq N\\}.$ Remark that (see [9, 11]) $z.W_{\lambda}-\overline{z}.W^{+}_{\lambda}$ is bounded from $\mathcal{H}_{\|\lambda\|}^{N}$ to $\mathcal{H}_{\|\lambda\|}^{N+1}$ and whose bound $<((2\|\alpha\|+n)\|\lambda\|)^{1/2}\|z\|.$ Then $R_{\theta}(z,t)\leq C\sum_{2k+\ell=J+1}\omega_{k,\ell}~{}\big{(}(2\|\alpha\|+n)\|\lambda\|\big{)}^{k+\frac{\ell}{2}}~{}z^{\ell}~{}t^{k}.$ Proof of Theorem 1.1. Let $f=\sum_{k=0}^{\infty}\beta_{k}a_{k}\in H^{p}(\mathbb{H}^{n}),$ being element of $H^{p}(\mathbb{H}^{n})$ where $a_{k}$ are atoms. Since $0<p\leq 1$ it follows $\int_{\mathbb{R}}\frac{\\|\mathcal{F}(f)(\lambda)\\|_{HS}^{p}}{\big{(}(2\|\alpha\|+n)\|\lambda\|\big{)}^{\sigma}}\|\lambda\|^{n}d\lambda\leq C\sum_{k=0}^{\infty}\|\beta_{k}\|^{p}\int_{\mathbb{R}}\frac{\\|\mathcal{F}(a_{k})(\lambda)\\|_{HS}^{p}}{\big{(}(2\|\alpha\|+n)\|\lambda\|\big{)}^{\sigma}}\|\lambda\|^{n}d\lambda.$ In order to prove Theorem 1.1, it is enough to prove, $\int_{\mathbb{R}}\frac{\\|\mathcal{F}(a_{k})(\lambda)\\|_{HS}^{p}}{\big{(}(2\|\alpha\|+n)\|\lambda\|\big{)}^{\sigma}}\|\lambda\|^{n}d\lambda\leq C.$ (3.3) This follows as $f=\sum_{k=0}^{\infty}\beta_{k}a_{k}$ implies $\mathcal{F}(a_{k})(\lambda)^{p}\leq\Big{\|}\sum_{k}\beta_{k}\mathcal{F}(a_{k})(\lambda)\Big{\|}^{p}\leq\sum_{k=0}^{\infty}\|\beta_{k}\|^{p}\|\mathcal{F}(a_{k})(\lambda)\|^{p}$ and hence $\displaystyle\int_{\mathbb{R}}\frac{\\|\mathcal{F}(f)(\lambda)\\|_{HS}^{p}}{\big{(}(2\|\alpha\|+n)\|\lambda\|\big{)}^{\sigma}}\|\lambda\|^{n}d\lambda$ $\displaystyle\leq$ $\displaystyle C\sum_{k=0}^{\infty}\|\beta_{k}\|^{p}\int_{\mathbb{R}}\frac{\\|\mathcal{F}(a_{k})(\lambda)\\|_{HS}^{p}}{\big{(}(2\|\alpha\|+n)\|\lambda\|\big{)}^{\sigma}}\|\lambda\|^{n}d\lambda$ $\displaystyle\leq$ $\displaystyle C\Big{\\{}\sum_{k=0}^{\infty}\|\beta_{k}\|^{p}\Big{\\}}^{1/p}$ $\displaystyle\leq$ $\displaystyle C\\|f\\|_{H^{p}(\mathbb{H}^{n})}.$ Let us now take $\gamma$ an arbitrary nonnegative real number, and decomposing the left hand side of (3.3) as $\displaystyle\int_{\mathbb{R}}\frac{\\|\mathcal{F}(a_{k})(\lambda)\\|_{HS}^{p}}{\big{(}(2\|\alpha\|+n)\|\lambda\|\big{)}^{\sigma}}\|\lambda\|^{n}d\lambda\\!\\!\\!\\!\\!\\!\\!$ $\displaystyle\\!\\!\\!\\!\\!=\\!\\!\\!\\!\\!\\!\\!\\!\\!$ $\displaystyle\\!\\!\\!\int_{0<\|\lambda\|\leq\gamma}\frac{\\|\mathcal{F}(a_{k})(\lambda)\\|_{HS}^{p}}{\big{(}(2\|\alpha\|+n)\|\lambda\|\big{)}^{\sigma}}\|\lambda\|^{n}d\lambda$ $\displaystyle+$ $\displaystyle\int_{\|\lambda\|>\gamma}\frac{\\|\mathcal{F}(a_{k})(\lambda)\\|_{HS}^{p}}{\big{(}(2\|\alpha\|+n)\|\lambda\|\big{)}^{\sigma}}\|\lambda\|^{n}d\lambda$ $\displaystyle\,\,\,\,:=$ $\displaystyle S_{1}+S_{2}.$ To estimate $S_{1}$ we may use Proposition 3.1, and cancelation property of atoms. Hence, by the cancelation property of atom, $\mathcal{F}(a_{k})(\lambda)=\int_{\mathbb{H}^{n}}\Big{[}\sum_{2k+\ell\leq J}\omega_{k,\ell}(\lambda,n)~{}z^{k}t^{\ell}+R_{\theta}(z,t)\Big{]}a(z,t)~{}dV(z,t).$ Now with the help of properties $(i),(ii)$ for $a(p,\infty,s)$-atoms of $H^{p}(\mathbb{H}^{n})$ together with Proposition 3.1, we get $\displaystyle\mathcal{F}(a_{k})(\lambda)$ $\displaystyle\leq$ $\displaystyle C\sum_{2k+\ell=J+1}\omega_{k,\ell}~{}\big{(}(2\|\alpha\|+n)\|\lambda\|\big{)}^{k+\frac{\ell}{2}}\int_{B(o,R)}~{}z^{\ell}~{}t^{k}\|B(0,R)\|^{-\frac{1}{p}}~{}dV(z,t)$ $\displaystyle\leq$ $\displaystyle C~{}\sum_{2k+\ell=J+1}\omega_{k,\ell}~{}R^{Q(1-\frac{1}{p})+2k+\frac{\ell}{2}}~{}\big{(}(2\|\alpha\|+n)\|\lambda\|\big{)}^{k+\frac{\ell}{2}}.$ Integrating with respect to the measure $d\gamma_{n}(\lambda)=\|\lambda\|^{n}d\lambda$ over the domain $0\leq\|\lambda\|\leq\gamma,$ we obtain $\displaystyle S_{1}$ $\displaystyle=$ $\displaystyle\int_{0<\|\lambda\|\leq\gamma}\frac{\\|\mathcal{F}(a_{k})(\lambda)\\|^{p}}{\big{(}(2\|\alpha\|+n)\|\lambda\|\big{)}^{\sigma}}\|\lambda\|^{n}d\lambda$ $\displaystyle\leq$ $\displaystyle C~{}\sum_{2k+\ell=J+1}\omega_{k,\ell}~{}R^{Q(p-1)+p(2k+\frac{\ell}{2})}\int_{0<\|\lambda\|\leq\gamma}\big{(}(2\|\alpha\|+n)\|\lambda\|\big{)}^{p(k+\frac{\ell}{2})-{\sigma}}~{}\|\lambda\|^{n}d\lambda$ $\displaystyle\leq$ $\displaystyle 2C~{}\sum_{\ell=0}^{J+1}~{}\omega_{\ell}~{}R^{Q(p-1)+p(J+1-\frac{\ell}{2})}\int_{0}^{\gamma}\big{(}(2\|\alpha\|+n)\|\lambda\|\big{)}^{p(\frac{J+1}{2})-{\sigma}}~{}\|\lambda\|^{n}d\lambda.$ That is $S_{1}\leq C~{}R^{Q(p-1)+p(J+1-\frac{\ell}{2})}\gamma^{p(\frac{J+1}{2})+{\frac{Q}{2}}-\sigma},~{}~{}\forall\ell=0,1,...,J+1,$ (3.4) provided that $p(\frac{J+1}{2})+\frac{Q}{2}-\sigma>0,$ which follows from the inequality (1.6). Now to estimate $S_{2},$ we may apply Hölder’s inequality for $q=\frac{2}{p}$ and Plancherel formula. Thus, we immediately obtain $\displaystyle S_{2}$ $\displaystyle\leq$ $\displaystyle\Bigg{(}\int_{\mathbb{R}}(\\|\mathcal{F}(a_{k})(\lambda)\\|^{p})^{\frac{2}{p}}\|\lambda\|^{n}d\lambda\Bigg{)}^{\frac{p}{2}}\Bigg{(}\int_{\|\lambda\|>\gamma}\big{(}(2\|\alpha\|+n)\|\lambda\|\big{)}^{\frac{2\sigma}{p-2}}\|\lambda\|^{n}d\lambda\Bigg{)}^{\frac{2-p}{2}}$ $\displaystyle\leq$ $\displaystyle C\\|\mathcal{F}(a_{k})\\|^{p}_{\mathcal{L}^{2}}\Bigg{(}\int_{\|\lambda\|>\gamma}\big{(}(2\|\alpha\|+n)\|\lambda\|\big{)}^{\frac{2\sigma}{p-2}}\|\lambda\|^{n}d\lambda\Bigg{)}^{\frac{2-p}{2}}$ $\displaystyle\leq$ $\displaystyle 2C\\|\mathcal{F}(a_{k})\\|^{p}_{\mathcal{L}^{2}}\Bigg{(}\int_{\gamma}^{\infty}\big{(}(2\|\alpha\|+n)\|\lambda\|\big{)}^{\frac{2\sigma}{p-2}}\|\lambda\|^{n}d\lambda\Bigg{)}^{\frac{2-p}{2}}$ $\displaystyle\leq$ $\displaystyle C\\|\mathcal{F}(a_{k})\\|^{p}_{\mathcal{L}^{2}}\gamma^{{Q\over 4}(2-p)-\sigma}$ provided that ${Q\over 4}(2-p)-\sigma<0,$ which is a consequence of the left hand side of (1.6). Thanks to Plancherel’s formula for Laguerre Fourier transform it follows $\displaystyle\\|\mathcal{F}(a_{k})\\|^{2}_{\mathcal{L}^{2}}=\\|a_{k}\\|_{L^{2}(\mathbb{H}^{n})}^{2}$ $\displaystyle=$ $\displaystyle\int_{\mathbb{H}^{n}}\|a_{k}(z,t)\|^{2}~{}dV(z,t)$ $\displaystyle\leq$ $\displaystyle\|B(0,R)\|^{1-\frac{2}{p}}$ $\displaystyle\leq$ $\displaystyle C~{}R^{-Q(\frac{2-p}{p})}.$ That is $\\|\mathcal{F}(a_{k})\\|^{p}_{\mathcal{L}^{2}}\leq C~{}R^{-\frac{Q}{2}(2-p)},$ and hence, $S_{2}\leq C~{}R^{-\frac{Q}{2}(2-p)}\gamma^{{{Q\over 4}(2-p)}-\sigma}.$ (3.5) However, to prove that $S_{1}+S_{2}\leq C,$ we shall discuss the cases $0<R<1$ and $R\geq 1.$ Hence, in order to deal with the case $0<R<1,$ we need more precise estimates, so we consider the set $\Gamma_{\gamma};$ the collection of all numbers $\gamma$ satisfying $\Gamma_{\gamma}=\Big{\\{}\gamma>0,~{}\frac{\frac{Q}{2}(2-p)}{{\frac{Q}{4}(2-p)}-\sigma}\log(R)\leq\log(\gamma)\leq\frac{Q(1-p)-p(J+1)}{p(\frac{J+1}{2})+{\frac{Q}{2}}-\sigma}\log(R)\Big{\\}}.$ We mention that the collection $\Gamma_{\gamma}$ above is an nonempty set if and only if $\frac{\frac{Q}{2}(2-p)}{\frac{Q}{4}(2-p)-\sigma}\times\frac{p(\frac{J+1}{2})+{\frac{Q}{2}}-\sigma}{Q(1-p)-p(J+1)}\leq 1$ which is a different formulation of the hand side of (1.6), that is $\frac{Q}{2}(2-p)\leq\sigma.$ Now let us choose $\gamma\in\Gamma_{\gamma}$ and using the fact that $\frac{Q}{2}+p\frac{(J+1)}{2}-\sigma>0$ together with the right hand side of (1.6) it follows that $S_{1}\leq C~{}R^{Q(p-1)+p(J+1)}\gamma^{p(\frac{J+1}{2})+{\frac{Q}{2}}-\sigma}.$ (3.6) Also, with the same choose of $\gamma\in\Gamma_{\gamma}$ and under the condition $\frac{Q}{2}(2-p)<\sigma,$ together with the help of the left hand side of (1.6) we obtain $S_{2}\leq C.$ (3.7) Combining (3.6) and (3.7) we obtain $S_{1}+S_{2}\leq C\qquad\mbox{for}\qquad 0<R<1.$ (3.8) Now, to deal with the case $R\geq 1,$ we may take $\gamma=R^{\frac{Q(1-p)-p(J+1)}{p(\frac{J+1}{2})+{\frac{Q}{2}}-\sigma}}$ (3.9) so, using the fact that $R\geq 1,$ we obtain $\gamma\leq R^{\frac{\frac{Q}{2}(2-p)}{\frac{Q}{4}(2-p)-\sigma}}.$ (3.10) which leads to $S_{1}+S_{2}\leq C\qquad\mbox{for}\qquad R\geq 1.$ (3.11) Hence, to prove (3.3), it is enough to combine (3.8) and (3.11). The proof of the main theorem is completed. Acknowledgements. This project was supported by King Saud University, Deanship of Scientific Research, College of Science Research Center. ## References * [1] M. Assal, Hardy’s type inequality associated with the Hankel transform for overcritical exponent, Integr. Transf. Spec. F., (2010), 1-6. * [2] M. Assal and A. Rahmouni, Hardy’s type inequality associated with the Laguerre Fourier transform, Integr. Transf. Spec. F., 24, (2013), 156–163. * [3] M. Assal and A. Rahmouni, An improved Hardy’s inequality associated with the Laguerre Fourier transform, Collect. Math., (2013), 1–11. * [4] R. R. Coifman, A real-variable characterization of $H^{p},$ Studia Math. 51, (1974), 269-274. * [5] C. Fefferman and E. M. Stein, $H^{p}$ spaces of several variables, Acta Math. 129, (1972), 137-193. * [6] G. B. Folland and E. M. Stein, _Hardy Spaces on Homogeneous Groups,_ Princeton University Press, Princeton, NJ, 1982. * [7] J. Garcia-Cuerva and J. Rubio de Francia, _Weighted Norm Inequalities and Related Topics,_ North Holland, 1985. * [8] S. Giulini, _Bernstein and Jackson theorems for th Heisenberg group,_ J. Austral. Math. Soc. (Series A) 38 (1985), 241-254. * [9] H. P. Liu, The group Fourier transforms and mulitipliers of the Hardy spaces on the Heisenberg group, Approx. Theory $\&$ Its Appl., 7 (1991), 106-117. * [10] C. C. Lin, $L^{p}$ multipliers and their $H^{1}-L^{1}$ estimates on the Heisenberg group, Revista Math. Ibero., 11 (1995), 269-308. * [11] C. C. Lin, H$\ddot{o}$rmander’s $H^{p}$ Multiplier theorem for the Heisenberg group, J. London Math. Soc. (3) 67 (2003) 686-700. * [12] Y. Kanjin, On Hardy-Type Inequalities and Hankel Transforms, Monatshefte $f\ddot{u}$r Mathematik, 127, (1999), 311-319. * [13] Y. Kanjin, Hardy’s inequalities for Hermite and Laguerre expansions, Bull. London Math. Soc., 29, (1997), 331-337. * [14] R. Radha and S. Thangavelu, Hardy’s inequalities for Hermite and Laguerre expansions, Proc. Amer. Math. Soc., 132, (12), (2004), 3525-3536. * [15] A. Rahmouni and M. Assal, Hardy’s type inequality for the critical exponent associated with the inverse Laguerre Fourier transform, Integr. Transf. Spec. F., (2013) 1–9. * [16] E. M. Stein, _Harmonic Analysis, real variable Methods, orthogonality and oscillatory integrals,_ Princeton Univ. Press, Princeton, NJ, 1993. * [17] M. H. Taibleson and G. Weiss, The molecular characterization of certain Hardy spaces, Astérisque 77, (1980), Société Math. de France, Paris, 67-149. * [18] S. Thangavelu, On regularity of twisted spherical means and special Hermite expansion, Proc. Ind. Acad. Sci., 103, (1993), 303-320.	arxiv-papers	2013-12-21T15:36:50	2024-09-04T02:49:55.756662	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Rahmouni Atef", "submitter": "Atef Rahmouni", "url": "https://arxiv.org/abs/1312.6265" }
1312.6286	# Description of the lack of compactness in Orlicz spaces and applications Ines Ben Ayed Université de Tunis El Manar, Faculté des Sciences de Tunis, LR03ES04 Équations aux dérivées partielles et applications, 2092 Tunis, Tunisie [email protected] and Mohamed Khalil Zghal Université de Tunis El Manar, Faculté des Sciences de Tunis, LR03ES04 Équations aux dérivées partielles et applications, 2092 Tunis, Tunisie [email protected] ###### Abstract. In this paper, we investigate the lack of compactness of the Sobolev embedding of $H^{1}(\mathbb{R}^{2})$ into the Orlicz space $L^{{\phi}_{p}}(\mathbb{R}^{2})$ associated to the function $\phi_{p}$ defined by $\phi_{p}(s):={\rm{e}^{s^{2}}}-\displaystyle\sum_{k=0}^{p-1}\frac{s^{2k}}{k!}\cdot$ We also undertake the study of a nonlinear wave equation with exponential growth where the Orlicz norm $\\|.\\|_{L^{\phi_{p}}}$ plays a crucial role. This study includes issues of global existence, scattering and qualitative study. ## 1\. Introduction ### 1.1. Critical $2D$ Sobolev embedding It is well known (see for instance [7]) that $H^{1}(\mathbb{R}^{2})$ is continuously embedded in all Lebesgue spaces $L^{q}(\mathbb{R}^{2})$ for $2\leq q<\infty$, but not in $L^{\infty}(\mathbb{R}^{2})$. It is also known that (for more details, we refer the reader to [21]) (1) $H^{1}(\mathbb{R}^{2})\hookrightarrow L^{{\phi}_{p}}(\mathbb{R}^{2}),\quad\forall p\in\mathbb{N}^{},$ where $L^{{\phi}_{p}}(\mathbb{R}^{2})$ denotes the Orlicz space associated to the function (2) $\displaystyle\phi_{p}(s)=\rm{e}^{s^{2}}-\sum_{k=0}^{p-1}\frac{s^{2k}}{k!}\,\cdot$ The embedding (1) is a direct consequence of the following sharp Trudinger- Moser type inequalities (see [1, 20, 22, 26]): ###### Proposition 1.1. (3) $\sup_{\\|u\\|_{H^{1}}\leq 1}\;\;\displaystyle\int_{\mathbb{R}^{2}}\,\left({\rm e}^{4\pi\|u\|^{2}}-1\right)\,dx:=\kappa<\infty,$ and states as follows: (4) $\\|u\\|_{L^{\phi_{p}}}\leq\frac{1}{\sqrt{4\pi}}\\|u\\|_{H^{1}},$ where the norm $\\|.\\|_{L^{{\phi}_{p}}}$ is given by: $\\|u\\|_{L^{\phi_{p}}}=\inf\,\left\\{\,\lambda>0,\int_{\mathbb{R}^{d}}\,\phi_{p}\left(\frac{\|u(x)\|}{\lambda}\right)\;dx\leq\kappa\,\right\\}.$ For our purpose, we shall resort to the following Trudinger-Moser inequality, the proof of which is postponed in the appendix. ###### Proposition 1.2. Let $\alpha\in[0,4\pi[$ and $p$ an integer larger than $1$. There is a constant $c(\alpha,p)$ such that (5) $\displaystyle\int_{\mathbb{R}^{2}}\left({\rm e}^{\alpha\|u(x)\|^{2}}-\displaystyle\sum_{k=0}^{p-1}\frac{\alpha^{k}\|u(x)\|^{2k}}{k!}\right)\,dx\leq c(\alpha,p)\\|u\\|_{L^{2p}(\mathbb{R}^{2})}^{2p},$ for all $u\in H^{1}(\mathbb{R}^{2})$ satisfying $\\|\nabla u\\|_{L^{2}(\mathbb{R}^{2})}\leq 1$. ### 1.2. Development on the lack of compactness of Sobolev embedding in the Orlicz space in the case $p=1$ In [3], [4] and [5], H. Bahouri, M. Majdoub and N. Masmoudi characterized the lack of compactness of $H^{1}(\mathbb{R}^{2})$ into the Orlicz space $L^{\phi_{1}}(\mathbb{R}^{2})$. To state their result in a clear way, let us recall some definitions. ###### Definition 1.3. We shall designate by a scale any sequence $(\alpha_{n})$ of positive real numbers going to infinity, a core any sequence $(x_{n})$ of points in $\mathbb{R}^{2}$ and a profile any function $\psi$ belonging to the set ${{\mathcal{P}}}:=\Big{\\{}\;\psi\in L^{2}(\mathbb{R},{\rm e}^{-2s}ds);\;\;\;\psi^{\prime}\in L^{2}(\mathbb{R}),\;\psi_{\|]-\infty,0]}=0\,\Big{\\}}.$ Given two scales $(\alpha_{n})$, $(\tilde{\alpha}_{n})$, two cores $(x_{n})$, $(\tilde{x}_{n})$ and tow profiles $\psi$, $\tilde{\psi}$, we say that the triplets $\big{(}(\alpha_{n}),(x_{n}),\psi\big{)}$ and $\big{(}(\tilde{\alpha}_{n}),(\tilde{x}_{n}),\tilde{\psi}\big{)}$ are orthogonal if $\mbox{either}\quad\quad\Big{\|}\log\left(\tilde{\alpha}_{n}/{\alpha}_{n}\right)\Big{\|}\to\infty,$ or $\tilde{\alpha}_{n}=\alpha_{n}$ and $-\frac{\log\|x_{n}-\tilde{x}_{n}\|}{\alpha_{n}}\longrightarrow a\geq 0\,\,\mbox{with}\,\,\psi\,\,\mbox{or}\,\,{\tilde{\psi}}\,\,\mbox{null for}\,\,s<a\,.$ ###### Remarks 1.4. • The profiles belong to the Hölder space $C^{\frac{1}{2}}$. Indeed, for any profile $\psi$ and real numbers $s$ and $t$, we have by Cauchy-Schwarz inequality $\|\psi(s)-\psi(t)\|=\left\|\int_{s}^{t}\psi^{\prime}(\tau)\;d\tau\right\|\leq\\|\psi^{\prime}\\|_{L^{2}(\mathbb{R})}\|s-t\|^{\frac{1}{2}}.$ * • Note also that (see [3]) (6) $\frac{\psi(s)}{\sqrt{s}}\rightarrow 0\quad as\quad s\rightarrow 0\quad and\quad as\quad s\rightarrow\infty.$ The asymptotically orthogonal decomposition derived in [4] is formulated in the following terms: ###### Theorem 1.5. Let $(u_{n})$ be a bounded sequence in $H^{1}(\mathbb{R}^{2})$ such that (7) $u_{n}\rightharpoonup 0,$ (8) $\limsup_{n\to\infty}\\|u_{n}\\|_{L^{\phi_{1}}}=A_{0}>0\quad\quad\mbox{and}$ (9) $\lim_{R\to\infty}\;\limsup_{n\to\infty}\,\\|u_{n}\\|_{L^{\phi_{1}}(\|x\|>R)}=0.$ Then, there exist a sequence of scales $({\alpha}_{n}^{(j)})$, a sequence of cores $({x}_{n}^{(j)})$ and a sequence of profiles $(\psi^{(j)})$ such that the triplets $({\alpha}_{n}^{(j)},{x}_{n}^{(j)},\psi^{(j)})$ are pairwise orthogonal and, up to a subsequence extraction, we have for all $\ell\geq 1$, (10) $u_{n}(x)=\displaystyle\sum_{j=1}^{\ell}\,\sqrt{\frac{\alpha_{n}^{(j)}}{2\pi}}\;\psi^{(j)}\left(\frac{-\log\|x-x_{n}^{(j)}\|}{\alpha_{n}^{(j)}}\right)+{\rm r}_{n}^{(\ell)}(x),\quad\limsup_{n\to\infty}\;\\|{\rm r}_{n}^{(\ell)}\\|_{L^{\phi_{1}}}\stackrel{{\scriptstyle\ell\to\infty}}{{\longrightarrow}}0.$ Moreover, we have the following stability estimate (11) $\\|\nabla u_{n}\\|_{L^{2}}^{2}=\displaystyle\sum_{j=1}^{\ell}\,\\|{\psi^{(j)}}^{\prime}\\|_{L^{2}}^{2}+\\|\nabla r_{n}^{(\ell)}\\|_{L^{2}}^{2}+\circ(1),\quad n\to\infty.$ ###### Remarks 1.6. * • It will be useful later on to point out that for any $q\geq 2$, we have (12) $\\|g_{n}^{(j)}\\|_{L^{q}}\stackrel{{\scriptstyle n\rightarrow\infty}}{{\longrightarrow}}0,$ where $g_{n}^{(j)}$ is the elementary concentration involving in Decomposition (10) defined by (13) $g_{n}^{(j)}(x):=\sqrt{\frac{\alpha_{n}^{(j)}}{2\pi}}\;\psi^{(j)}\left(\frac{-\log\|x-x_{n}^{(j)}\|}{\alpha_{n}^{(j)}}\right).$ Since the Lebesgue measure is invariant under translations, we have $\\|g_{n}^{(j)}\\|_{L^{q}}^{q}=(2\pi)^{-\frac{q}{2}}(\alpha_{n}^{(j)})^{\frac{q}{2}}\int_{\mathbb{R}^{2}}\bigg{\|}\psi^{(j)}\bigg{(}-\frac{\log\|x\|}{\alpha_{n}^{(j)}}\bigg{)}\bigg{\|}^{q}dx.$ Performing the change of variable $s=-\frac{\log\|x\|}{\alpha_{n}^{(j)}}$, yields $\\|g_{n}^{(j)}\\|_{L^{q}}^{q}=(2\pi)^{1-\frac{q}{2}}(\alpha_{n}^{(j)})^{\frac{q}{2}+1}\int^{\infty}_{0}\big{\|}\psi^{(j)}(s)\big{\|}^{q}{\rm e}^{-2\alpha_{n}^{(j)}s}\;ds.$ Fix $\varepsilon>0$. Then in view of (6), there exist two real numbers $s_{0}$ and $S_{0}$ such that $0<s_{0}<S_{0}$ and $\left\|\psi^{(j)}(s)\right\|\leq\varepsilon\sqrt{s},\quad\forall\,s\in[0,s_{0}]\cup[S_{0},\infty[.$ This implies, by the change of variable $u=\alpha_{n}^{(j)}s$, that $\displaystyle(\alpha_{n}^{(j)})^{\frac{q}{2}+1}\int_{0}^{s_{0}}\left\|\psi^{(j)}(s)\right\|^{q}{\rm e}^{-2\alpha_{n}^{(j)}s}\;ds$ $\displaystyle\leq$ $\displaystyle{\varepsilon}^{q}\int_{0}^{\alpha_{n}^{(j)}s_{0}}u^{\frac{q}{2}}{\rm e}^{-2u}\;du$ $\displaystyle\leq$ $\displaystyle C_{q}\,\varepsilon^{q}.$ In the same way, we obtain $\displaystyle(\alpha_{n}^{(j)})^{\frac{q}{2}+1}\int_{S_{0}}^{\infty}\left\|\psi^{(j)}(s)\right\|^{q}{\rm e}^{-2\alpha_{n}^{(j)}s}\;ds$ $\displaystyle\leq$ $\displaystyle C_{q}\,\varepsilon^{q}.$ Finally taking advantage of the continuity of $\psi^{(j)}$, we deduce that $\displaystyle(\alpha_{n}^{(j)})^{\frac{q}{2}+1}\int_{s_{0}}^{S_{0}}\left\|\psi^{(j)}(s)\right\|^{q}{\rm e}^{-2\alpha_{n}^{(j)}s}\;ds$ $\displaystyle\lesssim$ $\displaystyle(\alpha_{n}^{(j)})^{\frac{q}{2}+1}\int_{s_{0}}^{S_{0}}{\rm e}^{-2\alpha_{n}^{(j)}s}\;ds$ $\displaystyle\lesssim$ $\displaystyle(\alpha_{n}^{(j)})^{\frac{q}{2}}\left({\rm e}^{-2\alpha_{n}^{(j)}s_{0}}-{\rm e}^{-2\alpha_{n}^{(j)}S_{0}}\right)\stackrel{{\scriptstyle n\rightarrow\infty}}{{\longrightarrow}}0,$ which ends the proof of the assertion (12). * • Recall that it was proved in [5] that $\\|g_{n}^{(j)}\\|_{L^{\phi_{1}}}\stackrel{{\scriptstyle n\rightarrow\infty}}{{\longrightarrow}}\frac{1}{\sqrt{4\pi}}\,\max_{s>0}\;\frac{\|\psi^{(j)}(s)\|}{\sqrt{s}}\,$ and (14) $\big{\\|}\displaystyle\sum_{j=1}^{\ell}\,g_{n}^{(j)}\big{\\|}_{L^{\phi_{1}}}\stackrel{{\scriptstyle n\to\infty}}{{\longrightarrow}}\sup_{1\leq j\leq\ell}\,\left(\lim_{n\to\infty}\,\\|g_{n}^{(j)}\\|_{L^{\phi_{1}}}\right)\,,$ in the case when the scales $(\alpha_{n}^{(j)})_{1\leq j\leq\ell}$ are pairwise orthogonal. Note that Property (14) does not necessarily remain true in the case when we have the same scales and the pairwise orthogonality of the couples $\big{(}(x^{(j)}_{n}),\psi^{(j)}\big{)}$ (see Lemma $3.6$ in [5]). ### 1.3. Study of the lack of compactness of Sobolev embedding in the Orlicz space in the case $p>1$ Our first goal in this paper is to describe the lack of compactness of the Sobolev embedding (1) for $p>1$. Our result states as follows: ###### Theorem 1.7. Let $p>1$ be an integer larger than $1$ and $(u_{n})$ be a bounded sequence in $H^{1}(\mathbb{R}^{2})$ such that (15) $u_{n}\rightharpoonup 0,$ (16) $\limsup_{n\to\infty}\\|u_{n}\\|_{L^{\phi_{p}}}=A_{0}>0\quad\quad\mbox{and}$ (17) $\lim_{R\to\infty}\;\limsup_{n\to\infty}\,\\|u_{n}\\|_{L^{\phi_{p}}(\|x\|>R)}=0.$ Then, there exist a sequence of scales $({\alpha}_{n}^{(j)})$, a sequence of cores $({x}_{n}^{(j)})$ and a sequence of profiles $(\psi^{(j)})$ such that the triplets $({\alpha}_{n}^{(j)},{x}_{n}^{(j)},\psi^{(j)})$ are pairwise orthogonal in the sense of Definition 1.3 and, up to a subsequence extraction, we have for all $\ell\geq 1$, (18) $u_{n}(x)=\displaystyle\sum_{j=1}^{\ell}\,\sqrt{\frac{\alpha_{n}^{(j)}}{2\pi}}\;\psi^{(j)}\left(\frac{-\log\|x-x_{n}^{(j)}\|}{\alpha_{n}^{(j)}}\right)+{\rm r}_{n}^{(\ell)}(x),$ with $\displaystyle\limsup_{n\to\infty}\;\\|{\rm r}_{n}^{(\ell)}\\|_{L^{\phi_{p}}}\stackrel{{\scriptstyle\ell\to\infty}}{{\longrightarrow}}0.$ Moreover, we have the following stability estimate (19) $\\|\nabla u_{n}\\|_{L^{2}}^{2}=\displaystyle\sum_{j=1}^{\ell}\,\\|{\psi^{(j)}}^{\prime}\\|_{L^{2}}^{2}+\\|\nabla r_{n}^{(\ell)}\\|_{L^{2}}^{2}+\circ(1),\quad n\to\infty.$ ###### Remarks 1.8. * • Arguing as in [5], we can easily prove that (20) $\\|g_{n}\\|_{L^{\phi_{p}}}\stackrel{{\scriptstyle n\rightarrow\infty}}{{\longrightarrow}}\frac{1}{\sqrt{4\pi}}\,\max_{s>0}\;\frac{\|\psi(s)\|}{\sqrt{s}},$ where $g_{n}(x):=\sqrt{\frac{\alpha_{n}}{2\pi}}\;\psi\left(\frac{-\log\|x-x_{n}\|}{\alpha_{n}}\right)\cdot$ Indeed setting $L=\displaystyle\liminf_{n\rightarrow\infty}\\|g_{n}\\|_{L^{\phi_{p}}}$, we have for fixed $\varepsilon>0$ and $n$ sufficiently large (up to subsequence extraction) $\displaystyle\int_{\mathbb{R}^{2}}\Big{(}{\rm e}^{\big{\|}\frac{g_{n}(x+x_{n})}{L+\varepsilon}\big{\|}^{2}}-\displaystyle\sum_{k=0}^{p-1}\displaystyle\frac{\|g_{n}(x+x_{n})\|^{2k}}{(L+\varepsilon)^{2k}k!}\Big{)}\,dx\leq\kappa.$ Therefore, $\displaystyle\int_{\mathbb{R}^{2}}\Big{(}{\rm e}^{\big{\|}\frac{g_{n}(x+x_{n})}{L+\varepsilon}\big{\|}^{2}}-1\Big{)}\,dx\lesssim\kappa+\displaystyle\sum_{k=1}^{p-1}\\|g_{n}\\|_{L^{2k}}^{2k},$ which implies in view of (12) that $\displaystyle\int_{\mathbb{R}^{2}}\Big{(}{\rm e}^{\big{\|}\frac{g_{n}(x+x_{n})}{L+\varepsilon}\big{\|}^{2}}-1\Big{)}\,dx=2\pi\displaystyle\int_{0}^{+\infty}\alpha_{n}{\rm e}^{2\alpha_{n}s\Big{[}\frac{1}{4\pi(L+\varepsilon)^{2}}\big{(}\frac{\psi(s)}{\sqrt{s}}\big{)}^{2}-1\Big{]}}\,ds-\pi\lesssim 1.$ Using the fact that $\psi$ is a continuous function, we deduce that $L+\varepsilon\geq\frac{1}{\sqrt{4\pi}}\displaystyle\max_{s>0}\displaystyle\frac{\|\psi(s)\|}{\sqrt{s}},$ which ensures that $L\geq\frac{1}{\sqrt{4\pi}}\displaystyle\max_{s>0}\displaystyle\frac{\|\psi(s)\|}{\sqrt{s}}\cdot$ To end the proof of (20), it suffices to establish that for any $\delta>0$ $\displaystyle\int_{\mathbb{R}^{2}}\Big{(}{\rm e}^{\big{\|}\frac{g_{n}(x+x_{n})}{\lambda}\big{\|}^{2}}-\displaystyle\sum_{k=0}^{p-1}\displaystyle\frac{\|g_{n}(x+x_{n})\|^{2k}}{(\lambda)^{2k}k!}\Big{)}\,dx\stackrel{{\scriptstyle n\rightarrow\infty}}{{\longrightarrow}}0,$ where $\lambda=\frac{1+\delta}{\sqrt{4\pi}}\displaystyle\max_{s>0}\frac{\|\psi(s)\|}{\sqrt{s}}\cdot$ Since $\displaystyle\int_{\mathbb{R}^{2}}\Big{(}{\rm e}^{\big{\|}\frac{g_{n}(x+x_{n})}{\lambda}\big{\|}^{2}}-\displaystyle\sum_{k=0}^{p-1}\displaystyle\frac{\|g_{n}(x+x_{n})\|^{2k}}{(\lambda)^{2k}k!}\Big{)}\,dx\leq\displaystyle\int_{\mathbb{R}^{2}}\Big{(}{\rm e}^{\big{\|}\frac{g_{n}(x+x_{n})}{\lambda}\big{\|}^{2}}-1\Big{)}\,dx,$ the result derives immediately from Proposition $1.15$ in [5], which achieves the proof of the result. * • Applying the same lines of reasoning as in the proof of Proposition 1.19 in [5], we obtain the following result: ###### Proposition 1.9. Let $\big{(}(\alpha_{n}^{(j)}),(x_{n}^{(j)}),\psi^{(j)}\big{)}_{1\leq j\leq\ell}$ be a family of triplets of scales, cores and profiles such that the scales are pairwise orthogonal. Then for any integer $p$ larger than $1$, we have $\Big{\\|}\displaystyle\sum_{j=1}^{\ell}\,g_{n}^{(j)}\Big{\\|}_{L^{\phi_{p}}}\stackrel{{\scriptstyle n\rightarrow\infty}}{{\longrightarrow}}\sup_{1\leq j\leq\ell}\,\left(\lim_{n\to\infty}\,\big{\\|}g_{n}^{(j)}\big{\\|}_{L^{{\phi}_{p}}}\right)\;,$ where the functions $g^{(j)}_{n}$ are defined by (13). As we will see in Section 2, it turns out that the heart of the matter in the proof of Theorem 1.7 is reduced to the following result concerning the radial case: ###### Theorem 1.10. Let $p$ be an integer strictly larger than $1$ and $(u_{n})$ be a bounded sequence in $H_{rad}^{1}(\mathbb{R}^{2})$ such that (21) $u_{n}\rightharpoonup 0\quad\quad\mbox{and}$ (22) $\limsup_{n\to\infty}\\|u_{n}\\|_{L^{\phi_{p}}}=A_{0}>0.$ Then, there exist a sequence of pairwise orthogonal scales $({\alpha}_{n}^{(j)})$ and a sequence of profiles $(\psi^{(j)})$ such that up to a subsequence extraction, we have for all $\ell\geq 1$, (23) $u_{n}(x)=\displaystyle\sum_{j=1}^{\ell}\,\sqrt{\frac{\alpha_{n}^{(j)}}{2\pi}}\;\psi^{(j)}\left(\frac{-\log\|x\|}{\alpha_{n}^{(j)}}\right)+{\rm r}_{n}^{(\ell)}(x),\quad\limsup_{n\to\infty}\;\\|{\rm r}_{n}^{(\ell)}\\|_{L^{\phi_{p}}}\stackrel{{\scriptstyle\ell\to\infty}}{{\longrightarrow}}0.$ Moreover, we have the following stability estimate $\\|\nabla u_{n}\\|_{L^{2}}^{2}=\displaystyle\sum_{j=1}^{\ell}\,\\|{\psi^{(j)}}^{\prime}\\|_{L^{2}}^{2}+\\|\nabla r_{n}^{(\ell)}\\|_{L^{2}}^{2}+\circ(1),\quad n\to\infty.$ ###### Remarks 1.11. * • Compared with the analogous result concerning the Sobolev embedding of $H_{rad}^{1}(\mathbb{R}^{2})$into $L^{\phi_{1}}$ established in [5], the hypothesis of compactness at infinity is not required. This is justified by the fact that $H^{1}_{rad}(\mathbb{R}^{2})$ is compactly embedded in $L^{q}(\mathbb{R}^{2})$ for any $2<q<\infty$ which implies that (24) $\displaystyle\lim_{n\rightarrow\infty}\\|u_{n}\\|_{L^{q}(\mathbb{R}^{2})}=0,\quad\forall\,2<q<\infty.$ * • In view of Proposition 1.9, Theorem 1.10 yields to $\\|u_{n}\\|_{L^{\phi_{p}}}\to\sup_{j\geq 1}\,\left(\lim_{n\to\infty}\,\\|g_{n}^{(j)}\\|_{L^{\phi_{p}}}\right),$ which implies that the first profile in Decomposition (23) can be chosen such that up to extraction (25) $A_{0}:=\displaystyle\limsup_{n\rightarrow\infty}\\|u_{n}\\|_{L^{\phi_{p}}}=\displaystyle\lim_{n\rightarrow\infty}\left\\|\sqrt{\frac{\alpha_{n}^{(1)}}{2\pi}}\psi^{(1)}\left(-\frac{\log\|x\|}{\alpha_{n}^{(1)}}\right)\right\\|_{L^{\phi_{p}}}.$ Note that the description of the lack of compactness in other critical Sobolev embeddings was achieved in [8, 10, 14] and has been at the origin of several prospectus. Among others, one can mention [2, 6, 9, 11, 19]. ### 1.4. Layout of the paper Our paper is organized as follows: in Section 2, we establish the algorithmic construction of the decomposition stated in Theorem 1.7. Then, we study in Section 3 a nonlinear two-dimensional wave equation with the exponential nonlinearity $u\,\phi_{p}(\sqrt{4\pi}u)$. Firstly, we prove the global well- posedness and the scattering in the energy space both in the subcritical and critical cases, and secondly we compare the evolution of this equation with the evolution of the solutions of the free Klein-Gordon equation in the same space. We mention that $C$ will be used to denote a constant which may vary from line to line. We also use $A\lesssim B$ to denote an estimate of the form $A\leq CB$ for some absolute constant $C$ and $A\approx B$ if $A\lesssim B$ and $B\lesssim A$. For simplicity, we shall also still denote by $(u_{n})$ any subsequence of $(u_{n})$ and designate by $\circ(1)$ any sequence which tends to $0$ as $n$ goes to infinity. ## 2\. Proof of Theorem 1.7 ### 2.1. Strategy of the proof The proof of Theorem 1.7 uses in a crucial way capacity arguments and is done in three steps: in the first step, we begin by the study of $u^{\ast}_{n}$ the symmetric decreasing rearrangement of $u_{n}$. This led us to establish Theorem 1.10. In the second step, by a technical process developed in [4], we reduce ourselves to one scale and extract the first core $(x_{n}^{(1)})$ and the first profile $\psi^{(1)}$ which enables us to extract the first element $\sqrt{\frac{\alpha_{n}^{(1)}}{2\pi}}\;\psi^{(1)}\left(\frac{-\log\|x-x_{n}^{(1)}\|}{\alpha_{n}^{(j)}}\right)$. The third step is devoted to the study of the remainder term. If the limit of its Orlicz norm is null we stop the process. If not, we prove that this remainder term satisfies the same properties as the sequence we start with which allows us to extract a second elementary concentration concentrated around a second core $(x_{n}^{(2)})$. Thereafter, we establish the property of orthogonality between the first two elementary concentrations and finally we prove that this process converges. ### 2.2. Proof of Theorem 1.10 The main ingredient in the proof of Theorem 1.10 consists to extract a scale and a profile $\psi$ such that (26) $\\|\psi^{\prime}\\|_{L^{2}(\mathbb{R})}\geq CA_{0},$ where $C$ is a universal constant. To go to this end, let us for a bounded sequence $(u_{n})$ in $H^{1}_{rad}(\mathbb{R}^{2})$ satisfying the assumptions (21) and (22), set $v_{n}(s)=u_{n}({\rm e}^{-s})$. Combining (24) with the following well-known radial estimate: $\|u(r)\|\leq\frac{C}{r^{\frac{1}{p+1}}}\\|u\\|_{L^{2p}}^{\frac{p}{p+1}}\\|\nabla u\\|_{L^{2}}^{\frac{1}{p+1}}$ where $r=\|x\|$, we infer that (27) $\displaystyle\lim_{n\rightarrow\infty}\\|v_{n}\\|_{L^{\infty}(]-\infty,M])}=0,\quad\forall M\in\mathbb{R}.$ This gives rise to the following result: ###### Proposition 2.1. For any $\delta>0$, we have (28) $\sup_{s\geq 0}\left(\Big{\|}\frac{v_{n}(s)}{A_{0}-\delta}\Big{\|}^{2}-s\right)\to\infty,\quad n\to\infty.$ ###### Proof. We proceed by contradiction. If not, there exists $\delta>0$ such that, up to a subsequence extraction (29) $\sup_{s\geq 0,n\in\mathbb{N}}\;\;\left(\Big{\|}\frac{v_{n}(s)}{A_{0}-\delta}\Big{\|}^{2}-s\right)\leq C<\infty.$ On the one hand, thanks to (27) and (29), we get by virtue of Lebesgue theorem $\displaystyle\int_{\|x\|<1}\;\left({\rm e}^{\|\frac{u_{n}(x)}{A_{0}-\delta}\|^{2}}-\displaystyle\sum_{k=0}^{p-1}\frac{\|u_{n}(x)\|^{2k}}{(A_{0}-\delta)^{2k}k!}\right)\,dx$ $\displaystyle\leq$ $\displaystyle\int_{\|x\|<1}\;\left({\rm e}^{\|\frac{u_{n}(x)}{A_{0}-\delta}\|^{2}}-1\right)\,dx$ $\displaystyle\leq$ $\displaystyle 2\pi\,\int_{0}^{\infty}\;\left({\rm e}^{\|\frac{v_{n}(s)}{A_{0}-\delta}\|^{2}}-1\right)\,{\rm e}^{-2s}\,ds\stackrel{{\scriptstyle n\to\infty}}{{\longrightarrow}}0.$ On the other hand, using Property (27) and the simple fact that for any positive real number $M$, there exists a finite constant $C_{M,p}$ such that $\sup_{\|t\|\leq M}\,\left(\frac{{\rm e}^{t^{2}}-\sum_{k=0}^{p-1}\frac{t^{2k}}{k!}}{t^{2p}}\right)<C_{M,p},$ we deduce in view of (24) that $\int_{\|x\|\geq 1}\;\left({\rm e}^{\|\frac{u_{n}(x)}{A_{0}-\delta}\|^{2}}-\displaystyle\sum_{k=0}^{p-1}\frac{\|u_{n}(x)\|^{2k}}{(A_{0}-\delta)^{2k}k!}\right)\,dx\lesssim\\|u_{n}\\|_{L^{2p}}^{2p}\to 0\,.$ Consequently, $\displaystyle\limsup_{n\to\infty}\,\\|u_{n}\\|_{L^{\phi_{p}}}\leq A_{0}-\delta,$ which is in contradiction with Hypothesis (22). ∎ An immediate consequence of the previous proposition is the following corollary whose proof is identical to the proof of Corollaries 2.4 and 2.5 in [5]. ###### Corollary 2.2. Under the above notations, there exists a sequence $(\alpha_{n}^{(1)})$ in $\mathbb{R}_{+}$ tending to infinity such that (30) $4\,\Big{\|}\frac{v_{n}(\alpha_{n}^{(1)})}{A_{0}}\Big{\|}^{2}-\alpha_{n}^{(1)}\stackrel{{\scriptstyle n\to\infty}}{{\longrightarrow}}\infty$ and for $n$ sufficiently large, there exists a positive constant $C$ such that (31) $\frac{A_{0}}{2}\sqrt{\alpha_{n}^{(1)}}\leq\|v_{n}(\alpha_{n}^{(1)})\|\leq C\sqrt{\alpha_{n}^{(1)}}+\circ(1).$ Now, setting $\psi_{n}(y)=\sqrt{\frac{2\pi}{\alpha_{n}^{(1)}}}v_{n}(\alpha_{n}^{(1)}y),$ we obtain along the same lines as in Lemma 2.6 in [5] the following result: ###### Lemma 2.3. Under notations of Corollary 2.2, there exists a profile $\psi^{(1)}\in\mathcal{P}$ such that, up to a subsequence extraction $\psi_{n}^{\prime}\rightharpoonup(\psi^{(1)})^{\prime}\;in\;L^{2}(\mathbb{R})\quad and\quad\\|(\psi^{(1)})^{\prime}\\|_{L^{2}}\geq\sqrt{\frac{\pi}{2}}A_{0}.$ To achieve the proof of Theorem 1.10, let us consider the remainder term (32) $r_{n}^{(1)}(x)=u_{n}(x)-g_{n}^{(1)}(x),$ where $g_{n}^{(1)}(x)=\sqrt{\frac{\alpha_{n}^{(1)}}{2\pi}}\psi^{(1)}\left(\frac{-\log\|x\|}{\alpha_{n}^{(1)}}\right).$ By straightforward computations, we can easily prove that $(r^{(1)}_{n})$ is bounded in $H^{1}_{rad}(\mathbb{R}^{2})$ and satisfies the hypothesis (21) together with the following property: (33) $\displaystyle\lim_{n\rightarrow\infty}\\|\nabla r_{n}^{(1)}\\|_{L^{2}(\mathbb{R}^{2})}^{2}=\underset{n\rightarrow\infty}{\lim}\\|\nabla u_{n}\\|_{L^{2}(\mathbb{R}^{2})}^{2}-\big{\\|}(\psi^{(1)})^{\prime}\big{\\|}_{L^{2}(\mathbb{R})}^{2}.$ Let us now define $A_{1}=\underset{n\rightarrow\infty}{\limsup}\\|r_{n}^{(1)}\\|_{L^{\phi_{p}}}$. If $A_{1}=0$, we stop the process. If not, arguing as above, we prove that there exist a scale $(\alpha_{n}^{(2)})$ satisfying the statement of Corollary 2.2 with $A_{1}$ instead of $A_{0}$ and a profile $\psi^{(2)}$ in $\mathcal{P}$ such that $r_{n}^{(1)}(x)=\sqrt{\frac{\alpha_{n}^{(2)}}{2\pi}}\psi^{(2)}\left(\frac{-\log\|x\|}{\alpha_{n}^{(2)}}\right)+r_{n}^{(2)}(x),$ with $\\|(\psi^{(2)})^{\prime}\\|_{L^{2}}\geq\frac{\sqrt{2\pi}}{2}A_{1}$ and $\displaystyle\lim_{n\rightarrow\infty}\\|\nabla r_{n}^{(2)}\\|_{L^{2}(\mathbb{R}^{2})}^{2}=\underset{n\rightarrow\infty}{\lim}\\|\nabla r_{n}^{(1)}\\|_{L^{2}(\mathbb{R}^{2})}^{2}-\big{\\|}(\psi^{(2)})^{\prime}\big{\\|}_{L^{2}(\mathbb{R})}^{2}.$ Moreover, as in [5] we can show that $(\alpha_{n}^{(1)})$ and $(\alpha_{n}^{(2)})$ are orthogonal. Finally, iterating the process, we get at step $\ell$ $u_{n}(x)=\displaystyle\sum_{j=1}^{\ell}\,\sqrt{\frac{\alpha_{n}^{(j)}}{2\pi}}\;\psi^{(j)}\left(\frac{-\log\|x\|}{\alpha_{n}^{(j)}}\right)+{\rm r}_{n}^{(\ell)}(x),$ with $\limsup_{n\to\infty}\,\\|r^{(\ell)}_{n}\\|_{H^{1}}^{2}\lesssim 1-A_{0}^{2}-A_{1}^{2}-\cdots-A_{\ell-1}^{2}\,,$ which implies that $A_{\ell}\to 0$ as $\ell\to\infty$ and ends the proof of the theorem. ### 2.3. Extraction of the cores and profiles This step is performed as the proof of Theorem 1.16 in [3]. We sketch it here briefly for the convenience of the reader. Let $u_{n}^{\ast}$ be the symmetric decreasing rearrangement of $u_{n}$. Since $u^{\ast}_{n}\in H^{1}_{rad}(\mathbb{R}^{2})$ and satisfies the assumptions of Theorem 1.10, we infer that there exist a sequence $(\alpha_{n}^{(j)})$ of pairwise orthogonal scales and a sequence of profiles $(\varphi^{(j)})$ such that, up to subsequence extraction, $u_{n}^{}(x)=\displaystyle\sum_{j=1}^{\ell}\,\sqrt{\frac{\alpha_{n}^{(j)}}{2\pi}}\;\varphi^{(j)}\left(\frac{-\log\|x\|}{\alpha_{n}^{(j)}}\right)+{\rm r}_{n}^{(\ell)}(x),\quad\limsup_{n\to\infty}\;\\|{\rm r}_{n}^{(\ell)}\\|_{L^{\phi_{p}}}\stackrel{{\scriptstyle\ell\to\infty}}{{\longrightarrow}}0.$ Besides, in view of (25), we can assume that $A_{0}=\underset{n\rightarrow\infty}{\lim}\left\\|\sqrt{\frac{\alpha_{n}^{(1)}}{2\pi}}\varphi^{(1)}\left(-\frac{\log\|x\|}{\alpha_{n}^{(1)}}\right)\right\\|_{L^{{\Phi}_{p}}}.$ Now to extract the cores and profiles, we shall firstly reduce to the case of one scale according to Section 2.3 in [4], where a suitable truncation of $u_{n}$ was introduced. Then assuming that $u_{n}^{}(x)=\sqrt{\frac{\alpha_{n}^{(1)}}{2\pi}}\;\varphi^{(1)}\left(\frac{-\log\|x\|}{\alpha_{n}^{(1)}}\right),$ we apply the strategy developed in Section 2.4 in [4] to extract the cores and the profiles. This approach is based on capacity arguments: to carry out the extraction process of mass concentrations, we prove by contradiction that if the mass responsible for the lack of compactness of the Sobolev embedding in the Orlicz space is scattered, then the energy used would exceed that of the starting sequence. This main point can be formulated in the following terms: ###### Lemma 2.4 ( Lemma 2.5 in [4]). There exist $\delta_{0}>0$ and $N_{1}\in\mathbb{N}$ such that for any $n\geq N_{1}$ there exists $x_{n}$ such that (34) $\frac{\|E_{n}\cap B(x_{n},\rm{e}^{-b\alpha_{n}^{(1)}})\|}{\|E_{n}\|}\geq\delta_{0}A_{0}^{2},$ where $E_{n}:=\\{x\in\mathbb{R}^{2};\|u_{n}(x)\|\geq\sqrt{2\alpha_{n}^{(1)}}(1-\frac{\varepsilon_{0}}{10})A_{0}\\}$ with $0<\varepsilon_{0}<\frac{1}{2}$, $B(x_{n},\rm{e}^{-b\,\alpha_{n}^{(1)}})$ designates the ball of center $x_{n}$ and radius $\rm{e}^{-b\,\alpha_{n}^{(1)}}$ with $b=1-2\varepsilon_{0}$ and $\|.\|$ denotes the Lebesgue measure. Once extracting the first core $(x_{n}^{(1)})$ making use of the previous lemma, we focus on the extraction of the first profile. For that purpose, we consider the sequence $\psi_{n}(y,\theta)=\sqrt{\frac{2\pi}{\alpha_{n}^{(1)}}}v_{n}(\alpha_{n}^{(1)}y,\theta),$ where $v_{n}(s,\theta)=(\tau_{x_{n}^{(1)}}u_{n})(\rm{e}^{-s}\cos\theta,\rm{e}^{-s}\sin\theta)$ and $(x_{n}^{(1)})$ satisfies $\frac{\|E_{n}\cap B(x_{n},\rm{e}^{-(1-2\varepsilon_{0})\alpha_{n}^{(1)}}\|}{\|E_{n}\|}\geq\delta_{0}A_{0}^{2}.$ Taking advantage of the invariance of Lebesgue measure under translations, we deduce that $\displaystyle\\|\nabla u_{n}\\|_{L^{2}}^{2}$ $\displaystyle=$ $\displaystyle\frac{1}{2\pi}\int_{\mathbb{R}}\int_{0}^{2\pi}\|\partial_{y}\psi_{n}(y,\theta)\|^{2}dyd\theta$ $\displaystyle+$ $\displaystyle\frac{\alpha_{n}^{(1)}}{2\pi}\int_{\mathbb{R}}\int_{0}^{2\pi}\|\partial_{\theta}\psi_{n}(y,\theta)\|^{2}dyd\theta.$ Since the scale $\alpha_{n}^{(1)}$ tends to infinity and the sequence $(u_{n})$ is bounded in $H^{1}(\mathbb{R}^{2})$, this implies that up to a subsequence extraction $\partial_{\theta}\psi_{n}\underset{n\rightarrow\infty}{\rightarrow}0$ and $\partial_{y}\psi_{n}\underset{n\rightarrow\infty}{\rightharpoonup}g$ in $L^{2}(\mathbb{R}\times[0,2\pi])$, where $g$ only depends on the variable $y$. Thus introducing the function $\psi^{(1)}(y)=\int_{0}^{y}g(\tau)d\tau,$ we obtain along the same lines as in Proposition 2.8 in [4] the following result: ###### Proposition 2.5. The function $\psi^{(1)}$ belongs to the set of profiles $\mathcal{P}$. Besides for any $y\in\mathbb{R}$, we have (35) $\frac{1}{2\pi}\int^{2\pi}_{0}\psi_{n}(y,\theta)\,d\theta\to\psi^{(1)}(y),$ as $n$ tends to infinity and there exists an absolute constant $C$ such that (36) $\\|{\psi^{(1)}}^{\prime}\\|_{L^{2}}\geq C\,A_{0}.$ ### 2.4. End of the proof To achieve the proof of the theorem, we argue exactly as in Section 2.5 in [4] by iterating the process exposed in the previous section. For that purpose, we set $r_{n}^{(1)}(x)=u_{n}(x)-g_{n}^{(1)}(x),$ where $g_{n}^{(1)}(x)=\sqrt{\frac{\alpha_{n}^{(1)}}{2\pi}}\psi^{(1)}\left(-\frac{\log\|x-x_{n}^{(1)}\|}{\alpha^{(1)}_{n}}\right).$ One can easily check that the sequence $(r_{n}^{(1)})$ weakly converges to $0$ in $H^{1}(\mathbb{R}^{2})$. Moreover, since $\psi^{(1)}_{\|]-\infty,0]}=0$, we have for any $R\geq 1$ (37) $\\|r_{n}^{(1)}\\|_{L^{\Phi_{p}}(\|x-x_{n}^{(1)}\|\geq R)}=\\|u_{n}\\|_{L^{\Phi_{p}}(\|x-x_{n}^{(1)}\|\geq R)}.$ But by assumption, the sequence $(u_{n})$ is compact at infinity in the Orlicz space $L^{\Phi_{p}}$. Thus the core $(x_{n}^{(1)})$ is bounded in $\mathbb{R}^{2}$, which ensures in view of (37) that $(r_{n}^{(1)})$ satisfies the hypothesis of compactness at infinity (17). Finally, taking advantage of the weak convergence of $(\partial_{y}\psi_{n})$ to ${\psi^{(1)}}^{\prime}$ in $L^{2}(y,\theta)$ as $n$ goes to infinity, we get $\underset{n\rightarrow\infty}{\lim}\\|\nabla r_{n}^{(1)}\\|_{L^{2}}^{2}=\underset{n\rightarrow\infty}{\lim}\\|\nabla u_{n}^{(1)}\\|_{L^{2}}^{2}-\\|{\psi^{(1)}}^{\prime}\\|_{L^{2}}^{2}.$ Now, let us define $A_{1}:=\underset{n\rightarrow\infty}{\limsup}\\|r_{n}^{(1)}\\|_{L^{\Phi_{p}}}$. If $A_{1}=0$, we stop the process. If not, knowing that $(r_{n}^{(1)})$ verifies the assumptions of Theorem 1.7, we apply the above reasoning, which gives rise to the existence of a scale $(\alpha_{n}^{(2)})$, a core $(x^{(2)}_{n})$ satisfying the statement of Lemma 2.4 with $A_{1}$ instead of $A_{0}$ and a profile $\psi^{(2)}$ in $\mathcal{P}$ such that $r_{n}^{(1)}(x)=\sqrt{\frac{\alpha_{n}^{(2)}}{2\pi}}\psi^{(2)}\left(-\frac{\log\|x-x_{n}^{(2)}\|}{\alpha^{(2)}_{n}}\right)+r_{n}^{(2)}(x),$ with $\\|{\psi^{(2)}}^{\prime}\\|_{L^{2}}\geq C\,A_{1}$ and $\underset{n\rightarrow\infty}{\lim}\\|\nabla r_{n}^{(2)}\\|_{L^{2}}^{2}=\underset{n\rightarrow\infty}{\lim}\\|\nabla r_{n}^{(1)}\\|_{L^{2}}^{2}-\\|{\psi^{(2)}}^{\prime}\\|_{L^{2}}^{2}.$ Arguing as in [4], we show that the triplets $\big{(}\alpha_{n}^{(1)},x_{n}^{(1)},\psi^{(1)}\big{)}$ and $\big{(}\alpha_{n}^{(2)},x_{n}^{(2)},\psi^{(2)}\big{)}$ are orthogonal in the sense of Definition 1.3 and prove that the process of extraction of the elementary concentration converges. This ends the proof of Decomposition (10). The orthogonality equality (11) derives immediately from Proposition 2.10 in [4]. The proof of Theorem 1.7 is then achieved. ## 3\. Nonlinear wave equation ### 3.1. Statement of the results In this section, we investigate the initial value problem for the following nonlinear wave equation: (41) $\displaystyle\left\\{\begin{array}[]{lll}\square u+u+u\,\left({\rm e}^{4\pi u^{2}}-\displaystyle\sum_{k=0}^{p-1}\frac{(4\pi)^{k}u^{2k}}{k!}\right)=0,\\\ \\\ u(0)=u_{0}\in H^{1}(\mathbb{R}^{2}),\quad\partial_{t}u(0)=u_{1}\in L^{2}(\mathbb{R}^{2}),\end{array}\right.$ where $p\geq 1$ is an integer, $u=u(t,x)$ is a real-valued function of $(t,x)\in\mathbb{R}\times\mathbb{R}^{2}$ and $\square=\partial^{2}_{t}-\Delta$ is the wave operator. Let us recall that in [17, 18], the authors proved the global well-posedness for the Cauchy problem (41) when $p=1$ and the scattering when $p=2$ in the subcritical and critical cases (i.e when the energy is less or equal to some threshold). Note also that in [24, 25], M. Struwe constructed global smooth solutions to (41) with smooth data of arbitrary size in the case $p=1$. Formally, the solutions of the Cauchy problem (41) satisfy the following conservation law: $\displaystyle\quad\quad E_{p}(u,t)$ $\displaystyle:=$ $\displaystyle\\|\partial_{t}u(t)\\|_{L^{2}}^{2}+\\|\nabla u(t)\\|_{L^{2}}^{2}+\frac{1}{4\pi}\left\\|{\rm e}^{4\pi u(t)^{2}}-1-\displaystyle\sum_{k=2}^{p}\frac{(4\pi)^{k}}{k!}u(t)^{2k}\right\\|_{L^{1}}$ $\displaystyle=$ $\displaystyle E_{p}(u,0):=E_{p}^{0}.$ This conducts us, as in [17], to define the notion of criticality in terms of the size of the initial energy $E_{p}^{0}$ with respect to $1$. ###### Definition 3.1. The Cauchy problem (41) is said to be subcritical if $E_{p}^{0}<1.$ It is said to be critical if $E_{p}^{0}=1$ and supercritical if $E_{p}^{0}>1$. We shall prove the following result: ###### Theorem 3.2. Assume that $E_{p}^{0}\leq 1.$ Then the Cauchy problem (41) has a unique global solution $u$ in the space $\mathcal{C}(\mathbb{R},H^{1}(\mathbb{R}^{2}))\cap\mathcal{C}^{1}(\mathbb{R},L^{2}(\mathbb{R}^{2})).$ Moreover, $u\in L^{4}(\mathbb{R},\mathcal{C}^{1/4})$ and scatters. ### 3.2. Technical tools The proof of Theorem 3.2 is based on priori estimates. This requires the control of the nonlinear term (43) $F_{p}(u):=u\,\left({\rm e}^{4\pi u^{2}}-\displaystyle\sum_{k=0}^{p-1}\frac{(4\pi)^{k}u^{2k}}{k!}\right)$ in $L^{1}_{t}(L^{2}_{x})$. To achieve our goal, we will resort to Strichartz estimates for the 2D Klein-Gordon equation. These estimates, proved in [15], state as follows: ###### Proposition 3.3. Let $T>0$ and $(q,r)\in[4,\infty]\times[2,\infty]$ an admissible pair, i.e $\frac{1}{q}+\frac{2}{r}=1.$ Then, (44) $\\|v\\|_{L^{q}([0,T],{\mathrm{B}}^{1}_{r,2}(\mathbb{R}^{2}))}\lesssim\Big{[}\\|v(0)\\|_{H^{1}(\mathbb{R}^{2})}+\\|\partial_{t}v(0)\\|_{L^{2}(\mathbb{R}^{2})}+\\|\square v+v\\|_{L^{1}([0,T],L^{2}(\mathbb{R}^{2}))}\Big{]},$ where ${\mathrm{B}}^{1}_{r,2}(\mathbb{R}^{2})$ stands for the usual inhomogeneous Besov space (see for example [12] or [23] for a detailed exposition on Besov spaces). Noticing that $(q,r)=(4,8/3)$ is an admissible pair and recalling that ${\mathrm{B}}^{1}_{8/3,2}(\mathbb{R}^{2})\hookrightarrow\mathcal{C}^{1/4}(\mathbb{R}^{2}),$ we deduce that (45) $\\|v\\|_{L^{4}([0,T],\mathcal{C}^{1/4}(\mathbb{R}^{2}))}\lesssim\Big{[}\\|v(0)\\|_{H^{1}(\mathbb{R}^{2})}+\\|\partial_{t}v(0)\\|_{L^{2}(\mathbb{R}^{2})}+\\|\square v+v\\|_{L^{1}([0,T],L^{2}(\mathbb{R}^{2}))}\Big{]}.$ To control the nonlinear term $F_{p}(u)$ in $L^{1}_{t}(L^{2}_{x})$, we will make use of the following logarithmic inequalities proved in [16, Theorem 1.3]. ###### Proposition 3.4. For any $\lambda>\frac{2}{\pi}$ and any $0<\mu\leq 1$, a constant $C_{\lambda}>0$ exists such that for any function $u$ in $H^{1}(\mathbb{R}^{2})\cap\mathcal{C}^{1/4}(\mathbb{R}^{2})$, we have (46) $\\|u\\|^{2}_{L^{\infty}}\leq\lambda\\|u\\|_{\mu}^{2}\log\left(C_{\lambda,\mu}+\frac{2\\|u\\|_{{\mathcal{C}}^{1/4}}}{\\|u\\|_{\mu}}\,\right),$ where $\\|u\\|_{\mu}^{2}:=\\|\nabla u\\|_{L^{2}}^{2}+\mu^{2}\\|u\\|_{L^{2}}^{2}$. ### 3.3. Proof of Theorem 3.2 The proof of this result, divided into three steps, is inspired from the proofs of Theorems $1.8$, $1.11$, $1.12$ in [17] and Theorem $1.3$ in [18]. #### 3.3.1. Local existence Let us start by proving the local existence to the Cauchy problem (41) in the case where $\\|\nabla u_{0}\\|_{L^{2}(\mathbb{R}^{2})}<1$. To do so, we use a standard fixed-point argument and introduce for any nonnegative time $T$ the following space: $\mathcal{E}_{T}=\mathcal{C}([0,T],H^{1}(\mathbb{R}^{2}))\cap\mathcal{C}^{1}([0,T],L^{2}(\mathbb{R}^{2}))\cap L^{4}([0,T],\mathcal{C}^{1/4}(\mathbb{R}^{2}))$ endowed with the norm $\\|u\\|_{T}:=\underset{0\leq t\leq T}{\sup}\Big{[}\\|u(t)\\|_{H^{1}}+\\|\partial_{t}u(t)\\|_{L^{2}}\Big{]}+\\|u\\|_{L^{4}([0,T],\mathcal{C}^{1/4})}.$ For a positive time $T$ and a positive real number $\delta$, we denote by $\mathcal{E}_{T}(\delta)$ the ball in the space $\mathcal{E}_{T}$ of radius $\delta$ and centered at the origin. On this ball, we define the map $\Phi$ by $v\longmapsto\Phi(v)=\widetilde{v},$ where $\square\widetilde{v}+\widetilde{v}=-F_{p}(v+v_{0}),\quad\widetilde{v}(0)=\partial_{t}\widetilde{v}(0)=0$ and $v_{0}$ is the solution of the free Klein-Gordon equation $\square v_{0}+v_{0}=0,\quad v_{0}(0)=u_{0},\quad and\quad\partial_{t}v_{0}(0)=u_{1}.$ Now, the goal is to show that if $\delta$ and $T$ are small enough, then the map $\Phi$ is well-defined from $\mathcal{E}_{T}(\delta)$ into itself and it is a contraction. To prove that $\Phi$ is well-defined, it suffices in view of the Strichartz estimates (44) to estimate $F_{p}(v+v_{0})$ in the space $L^{1}([0,T],L^{2}(\mathbb{R}^{2}))$. Arguing as in [17] and using the Hölder inequality and the Sobolev embedding, we obtain for any $\epsilon>0$ $\displaystyle\displaystyle\int_{\mathbb{R}^{2}}\|F_{p}(v+v_{0})\|^{2}\;dx$ $\displaystyle\leq$ $\displaystyle\displaystyle\int_{\mathbb{R}^{2}}\|F_{1}(v+v_{0})\|^{2}\;dx$ $\displaystyle\lesssim$ $\displaystyle\\|v+v_{0}\\|_{H^{1}}^{2}\,{\rm e}^{4\pi\\|v+v_{0}\\|_{L^{\infty}}^{2}}\left\\|{\rm e}^{4\pi(v+v_{0})^{2}}-1\right\\|_{L^{1+\epsilon}}.$ Since $\\|\nabla u_{0}\\|_{L^{2}}<1$, we can choose $\mu>0$ such that $\\|u_{0}\\|_{\mu}<1$. Since $v_{0}$ is continuous in time, there exist a time $T_{0}$ and a constant $0<c<1$ such that for any $t$ in $[0,T_{0}]$ we have $\\|v_{0}(t)\\|_{\mu}\leq c.$ According to Proposition 3.4, we infer that ${\rm e}^{4\pi\\|v+v_{0}\\|_{L^{\infty}}^{2}}\lesssim\left(1+\frac{\\|v+v_{0}\\|_{\mathcal{C}^{1/4}}}{\delta+c}\right)^{8\eta},$ for some $0<\eta<1$. Besides, applying the Trudinger-Moser inequality (5) for $p=1$, the fact that $4\pi(1+\epsilon)(\delta+c)^{2}\longrightarrow 4\pi c<4\pi\quad\mbox{as}\;\epsilon,\,\delta\rightarrow 0\quad\mbox{and}\quad\left\\|\nabla\left(\frac{v+v_{0}}{\delta+c}\right)\right\\|_{L^{2}}\leq 1$ ensures that $\displaystyle\left\\|{\rm e}^{4\pi(v+v_{0})^{2}}-1\right\\|_{L^{1+\epsilon}}^{1+\epsilon}$ $\displaystyle\leq$ $\displaystyle C_{\epsilon}\left\\|{\rm e}^{4\pi(1+\epsilon)(v+v_{0})^{2}}-1\right\\|_{L^{1}}$ $\displaystyle\leq$ $\displaystyle C_{\epsilon,\delta}\\|v+v_{0}\\|_{L^{2}}^{2}$ $\displaystyle\leq$ $\displaystyle C_{\epsilon,\delta}(1+\\|u_{0}\\|_{H^{1}}+\\|u_{1}\\|_{L^{2}})^{2}.$ Therefore, for any $0<T\leq T_{0}$, we obtain that $\\|F_{p}(v+v_{0})\\|_{L^{1}([0,T],L^{2}(\mathbb{R}^{2}))}\lesssim T^{1-\eta}(1+\\|u_{0}\\|_{H^{1}}+\\|u_{1}\\|_{L^{2}})^{4\eta}.$ Now, to prove that $\Phi$ is a contraction (at least for $T$ small), let us consider two elements $v_{1}$ and $v_{2}$ in $\mathcal{E}_{T}(\delta)$. Notice that, for any $\epsilon>0$, $\displaystyle\|F_{p}(v_{1}+v_{0})-F_{p}(v_{2}+v_{0})\|$ $\displaystyle=$ $\displaystyle\|v_{1}-v_{2}\|(1+8\pi\overline{v}^{2})\left({\rm e}^{4\pi\overline{v}^{2}}-\displaystyle\sum_{k=0}^{p-2}\frac{(4\pi)^{k}\overline{v}^{2k}}{k!}\right)$ $\displaystyle\leq$ $\displaystyle C_{\epsilon}\|v_{1}-v_{2}\|\left({\rm e}^{4\pi(1+\epsilon)\overline{v}^{2}}-1\right),$ where $\overline{v}=(1-\theta)(v_{0}+v_{1})+\theta(v_{0}+v_{2}),$ for some $\theta=\theta(t,x)\in[0,1].$ Using a convexity argument, we get $\displaystyle\|F_{p}(v_{1}+v_{0})-F_{p}(v_{2}+v_{0})\|$ $\displaystyle\leq$ $\displaystyle C_{\epsilon}\left\|(v_{1}-v_{2})\left({\rm e}^{4\pi(1+\epsilon)(v_{1}+v_{0})^{2}}-1\right)\right\|$ $\displaystyle+$ $\displaystyle C_{\epsilon}\left\|(v_{1}-v_{2})\left({\rm e}^{4\pi(1+\epsilon)(v_{2}+v_{0})^{2}}-1\right)\right\|.$ This implies, in view of Strichartz estimates (45), that $\displaystyle\\|\Phi(v_{1})-\Phi(v_{2})\\|_{T}$ $\displaystyle\lesssim$ $\displaystyle\\|F_{p}(v_{1}+v_{0})-F_{p}(v_{2}+v_{0})\\|_{L^{1}([0,T],L^{2}(\mathbb{R}^{2}))}$ $\displaystyle\leq$ $\displaystyle C_{\epsilon}\displaystyle\int_{0}^{T}\left\\|(v_{1}-v_{2})\left({\rm e}^{4\pi(1+\epsilon)(v_{1}+v_{0})^{2}}-1\right)\right\\|_{L^{2}}\,dt$ $\displaystyle+$ $\displaystyle C_{\epsilon}\displaystyle\int_{0}^{T}\left\\|(v_{1}-v_{2})\left({\rm e}^{4\pi(1+\epsilon)(v_{2}+v_{0})^{2}}-1\right)\right\\|_{L^{2}}\,dt,$ which leads along the same lines as above to $\displaystyle\\|\Phi(v_{1})-\Phi(v_{2})\\|_{T}$ $\displaystyle\lesssim$ $\displaystyle T^{1-(1+\epsilon)\eta}(1+\\|u_{0}\\|_{H^{1}}+\\|u_{1}\\|_{L^{2}})^{4(1+\epsilon)\eta}\\|v_{1}-v_{2}\\|_{T}.$ If the parameter $\epsilon$ is small enough, then $(1+\epsilon)\eta<1$ and therefore, for $T$ small enough, $\Phi$ is a contraction map. This implies the uniqueness of the solution in $v_{0}+\mathcal{E}_{T}(\delta)$. Now, we shall prove the uniqueness in the energy space. The idea here is to establish that, if $u=v_{0}+v$ is a solution of (41) in $\mathcal{C}([0,T],H^{1}(\mathbb{R}^{2}))\cap\mathcal{C}^{1}([0,T],L^{2}(\mathbb{R}^{2}))$, then necessarily $v\in\mathcal{E}_{T}(\delta)$ at least for $T$ small. Starting from the fact that $v$ satisfies $\square v+v=-F_{p}(v+v_{0}),\quad v(0)=\partial_{t}v(0)=0,$ we are reduced, thanks to the Strichartz estimates (44), to control the term $F_{p}(v+v_{0})$ in the space $L^{1}([0,T],L^{2}(\mathbb{R}^{2})).$ But $\|F_{p}(v+v_{0})\|\leq\|F_{1}(v+v_{0})\|$, which leads to the result arguing exactly as in [17]. #### 3.3.2. Global existence In this section, we shall establish that our solution is global in time both in subcritical and critical cases. Firstly, let us notice that the assumption $E_{p}^{0}\leq 1$ implies that $\\|\nabla u_{0}\\|_{L^{2}(\mathbb{R}^{2})}<1$, which ensures in view of Section 3.3.1 the existence of a unique maximal solution $u$ defined on $[0,T^{})$ where $0<T^{}\leq\infty$ is the lifespan of $u$. We shall proceed by contradiction assuming that $T^{}<\infty$. In the subcritical case, the conservation law (3.1) implies that $\displaystyle\sup_{t\in(0,T^{})}\\|\nabla u(t)\\|_{L^{2}(\mathbb{R}^{2})}<1.$ Let then $0<s<T^{}$ and consider the following Cauchy problem: (47) $\square v+v+F_{p}(v)=0,\quad v(s)=u(s),\quad\mbox{and}\quad\partial_{t}v(s)=\partial_{t}u(s).$ As in the first step of the proof, a fixed-point argument ensures the existence of $\tau>0$ and a unique solution $v$ to (47) on the interval $[s,s+\tau]$. Noticing that $\tau$ does not depend on $s$, we can choose $s$ close to $T^{}$ such that $T^{}-s<\tau$. So, we can prolong the solution $u$ after the time $T^{}$, which is a contradiction. In the critical case, we cannot apply the previous argument because it is possible that the following concentration phenomenon holds: (48) $\displaystyle\limsup_{t\rightarrow T^{}}\\|\nabla u(t)\\|_{L^{2}(\mathbb{R}^{2})}=1.$ In fact, we shall show that (48) cannot hold in this case. To go to this end, we argue as in the proof of Theorem $1.12$ in [17]. Firstly, since the first equation of the Cauchy problem (41) is invariant under time translation, we can assume that $T^{}=0$ and that the initial time is $t=-1$. Similarly to [17, Proposition 4.2, Corollary 4.4], it follows that the maximal solution $u$ satisfies (49) $\displaystyle\limsup_{t\rightarrow 0^{-}}\\|\nabla u(t)\\|_{L^{2}(\mathbb{R}^{2})}=1,$ (50) $\displaystyle\lim_{t\rightarrow 0^{-}}\\|u(t)\\|_{L^{2}(\mathbb{R}^{2})}=0,$ (51) $\displaystyle\lim_{t\rightarrow 0^{-}}\displaystyle\int_{\|x-x^{}\|\leq-t}\|\nabla u(t,x)\|^{2}\;dx=1,\quad\mbox{and}$ (52) $\forall t<0,\quad\displaystyle\int_{\|x-x^{}\|\leq-t}e_{p}(u)(t,x)\;dx=1,$ for some $x^{}\in\mathbb{R}^{2}$, where $e_{p}(u)$ denotes the energy density defined by $e_{p}(u)(t,x):=(\partial_{t}u)^{2}+\|\nabla u\|^{2}+\frac{1}{4\pi}\left({\rm e}^{4\pi u^{2}}-1-\displaystyle\sum_{k=2}^{p}\displaystyle\frac{(4\pi)^{k}u^{2k}}{k!}\right).$ Without loss of generality, we can assume that $x^{}=0$, then multiplying the equation of the problem (41) respectively by $\partial_{t}u$ and $u$, we obtain formally (53) $\partial_{t}e_{p}(u)-div_{x}(2\partial_{t}u\nabla u)=0,$ (54) $\partial_{t}(u\partial_{t}u)-div_{x}(u\nabla u)+\|\nabla u\|^{2}-\|\partial_{t}u\|^{2}+u^{2}{\rm e}^{4\pi u^{2}}-\displaystyle\sum_{k=1}^{p-1}\displaystyle\frac{(4\pi)^{k}u^{2k+2}}{k!}=0.$ Integrating the conservation laws (53) and (54) over the backward truncated cone $K_{S}^{T}:=\Big{\\{}(t,x)\in\mathbb{R}\times\mathbb{R}^{2}\;\mbox{ such that}\;S\leq t\leq T\;\mbox{and}\;\|x\|\leq-t\Big{\\}}$ for $S<T<0$, we get (55) $\displaystyle\int_{B(-T)}e_{p}(u)(T,x)\;dx-\displaystyle\int_{B(-S)}e_{p}(u)(S,x)\;dx$ $=\frac{-1}{\sqrt{2}}\displaystyle\int_{M_{S}^{T}}\left[\left\|\partial_{t}u\frac{x}{\|x\|}+\nabla u\right\|^{2}+\frac{1}{4\pi}\left({\rm e}^{4\pi u^{2}}-1-\displaystyle\sum_{k=2}^{p}\frac{(4\pi)^{k}u^{2k}}{k!}\right)\;dx\,dt\right],$ (56) $\displaystyle\int_{B(-T)}\partial_{t}u(T)u(T)\;dx-\displaystyle\int_{B(-S)}\partial_{t}u(S)u(S)\;dx+\frac{1}{\sqrt{2}}\displaystyle\int_{M_{S}^{T}}\left(\partial_{t}u+\nabla u.\frac{x}{\|x\|}\right)u\;dx\,dt$ $+\displaystyle\int_{K_{S}^{T}}\left(\|\nabla u\|^{2}-\|\partial_{t}u\|^{2}+u^{2}{\rm e}^{4\pi u^{2}}-\displaystyle\sum_{k=1}^{p-1}\displaystyle\frac{(4\pi)^{k}u^{2k+2}}{k!}\right)\;dx\,dt=0,$ where $B(r)$ is the ball centered at $0$ and of radius $r$ and $M_{S}^{T}:=\Big{\\{}(t,x)\in\mathbb{R}\times\mathbb{R}^{2}\;\mbox{ such that}\;S\leq t\leq T\;\mbox{and}\;\|x\|=-t\Big{\\}}.$ According to (52) and (55), we infer that $\displaystyle\int_{M_{S}^{T}}\left[\left\|\partial_{t}u\frac{x}{\|x\|}+\nabla u\right\|^{2}+\frac{1}{4\pi}\left({\rm e}^{4\pi u^{2}}-1-\displaystyle\sum_{k=2}^{p}\frac{(4\pi)^{k}u^{2k}}{k!}\right)\right]\;dx\,dt=0.$ This implies, using (56) and Cauchy-Schwarz inequality, that (57) $\displaystyle\int_{B(-T)}\partial_{t}u(T)u(T)\;dx-\displaystyle\int_{B(-S)}\partial_{t}u(S)u(S)\;dx$ $+\displaystyle\int_{K_{S}^{T}}\left(\|\nabla u\|^{2}-\|\partial_{t}u\|^{2}+u^{2}{\rm e}^{4\pi u^{2}}-\displaystyle\sum_{k=1}^{p-1}\displaystyle\frac{(4\pi)^{k}u^{2k+2}}{k!}\right)\;dx\,dt=0,$ By virtue of Identities (49) and (50) and the conservation law (3.1), it can be seen that (58) $\partial_{t}u(t)\underset{t\rightarrow 0}{\longrightarrow}0\quad\mbox{in}\;L^{2}(\mathbb{R}^{2}),$ which ensures by Cauchy-Schwarz inequality that (59) $\displaystyle\int_{B(-T)}\partial_{t}u(T)u(T)\;dx\rightarrow 0.$ Letting $T\rightarrow 0$ in (57), we deduce from (59) and the fact that $u^{2}{\rm e}^{4\pi u^{2}}-\displaystyle\sum_{k=1}^{p-1}\displaystyle\frac{(4\pi)^{k}u^{2k+2}}{k!}$ is positive (60) $-\displaystyle\int_{B(-S)}\partial_{t}u(S)u(S)\;dx\leq-\displaystyle\int_{K_{S}^{0}}\|\nabla u\|^{2}\,dx\,dt+\displaystyle\int_{K_{S}^{0}}\|\partial_{t}u\|^{2}\,dx\,dt.$ Multiplying Inequality (60) by the positive number $-\frac{1}{S}$, we deduce that (61) $\displaystyle\int_{B(-S)}\partial_{t}u(S)\frac{u(S)}{S}\;dx\leq\frac{1}{S}\displaystyle\int_{K_{S}^{0}}\|\nabla u\|^{2}\;dx\;dt-\frac{1}{S}\displaystyle\int_{K_{S}^{0}}\|\partial_{t}u\|^{2}\;dx\;dt.$ Now, Identity (58) leads to (62) $\displaystyle\lim_{S\rightarrow 0^{-}}\frac{1}{S}\displaystyle\int_{K_{S}^{0}}\|\partial_{t}u\|^{2}\;dx\;dt=0.$ Moreover, using (51), it is clear that (63) $\displaystyle\lim_{S\rightarrow 0^{-}}\frac{1}{S}\displaystyle\int_{K_{S}^{0}}\|\nabla u\|^{2}\;dx\;dt=-1.$ Finally, since $\frac{u(S)}{S}=\frac{1}{S}\int_{0}^{S}\partial_{t}u(\tau)d\tau,$ then $(\frac{u(S)}{S})$ is bounded in $L^{2}(\mathbb{R}^{2})$ and hence (64) $\displaystyle\lim_{S\rightarrow 0^{-}}\displaystyle\int_{B(-S)}\partial_{t}u(S)\frac{u(S)}{S}\;dx=0.$ The identities (62), (63) and (64) yield a contradiction in view of (61). This achieves the proof of the global existence in the critical case. #### 3.3.3. Scattering Our concern now is to prove that, in the subcritical and critical cases, the solution of the equation (41) approaches a solution of a free wave equation when the time goes to infinity. Using the fact that (65) $\|F_{p}(u)\|\leq\|F_{2}(u)\|,\quad\forall p\geq 2,$ we can apply the arguments used in [18]. More precisely, in the subcritical case the key point consists to prove that there exists an increasing function $C:[0,1[\longrightarrow[0,\infty[$ such that for any $0\leq E<1$, any global solution $u$ of the Cauchy problem (41) with $E_{p}(u)\leq E$ satisfies (66) $\\|u\\|_{X(\mathbb{R})}\leq C(E),$ where $X(\mathbb{R})=L^{8}(\mathbb{R},L^{16}(\mathbb{R}^{2}))$. Now, denoting by $E^{}:=\sup\Big{\\{}0\leq E<1;\;\displaystyle\sup_{E_{p}(u)\leq E}\\|u\\|_{X(\mathbb{R})}<\infty\Big{\\}},$ and arguing as in [18, Lemma 4.1], we can show that Inequality (66) is satisfied if $E_{p}(u)$ is small, which implies that $E^{}>0$. Now our goal is to prove that $E^{}=1$. To do so, let us proceed by contradiction and assume that $E^{}<1$. Then, for any $E\in]E^{},1[$ and any $n>0$, there exists a global solution $u$ to (41) such that $E_{p}(u)\leq E$ and $\\|u\\|_{X(\mathbb{R})}>n$. By time translation, one can reduce to (67) $\\|u\\|_{X(]0,\infty[)}>\frac{n}{2}.$ Along the same lines as the proof of Proposition 5.1 in [18], we can show taking advantage of (65) that if $E$ is close enough to $E^{}$, then $n$ cannot be arbitrarily large which yields a contradiction and ends the proof of the result in the subcritical case. The proof of the scattering in the critical case is done as in Section 6 in [18] once we observed Inequality (65). It is based on the notion of concentration radius $r_{\epsilon}(t)$ introduced in [18]. ### 3.4. Qualitative study In this section we shall investigate the feature of solutions of the two- dimensional nonlinear Klein-Gordon equation (41) taking into account the different regimes. As in [5], the approach that we adopt here is the one introduced by P. Gérard in [13] which consists in comparing the evolution of oscillations and concentration effects displayed by sequences of solutions of the nonlinear Klein-Gordon equation (41) and solutions of the free linear Klein-Gordon equation. (68) $\displaystyle\square v+v=0.$ More precisely, let $(\varphi_{n},\psi_{n})$ be a sequence of data in $H^{1}\times L^{2}$ supported in some fixed ball and satisfying (69) $\varphi_{n}\rightharpoonup 0\quad\mbox{in}\;H^{1},\quad\psi_{n}\rightharpoonup 0\quad\mbox{in}\;L^{2},$ such that (70) $E_{p}^{n}\leq 1,\quad n\in\mathbb{N}$ where $E_{p}^{n}$ stands for the energy of $(\varphi_{n},\psi_{n})$ given by $E_{p}^{n}=\\|\psi_{n}\\|_{L^{2}}^{2}+\\|\nabla\varphi_{n}\\|_{L^{2}}^{2}+\frac{1}{4\pi}\;\Big{\\|}{\rm e}^{4\pi\varphi_{n}^{2}}-1-\displaystyle\sum_{k=2}^{p}\frac{(4\pi)^{k}}{k!}\varphi_{n}^{2k}\Big{\\|}_{L^{1}},$ and let us consider $(u_{n})$ and $(v_{n})$ the sequences of finite energy solutions of (41) and (68) such that $(u_{n},\partial_{t}u_{n})(0)=(v_{n},\partial_{t}v_{n})(0)=(\varphi_{n},\psi_{n}).$ Arguing as in [13], the notion of linearizability is defined as follows: ###### Definition 3.5. Let $T$ be a positive time. We shall say that the sequence $(u_{n})$ is linearizable on $[0,T]$, if $\displaystyle\sup_{t\in[0,T]}E_{c}(u_{n}-v_{n},t)\longrightarrow 0\quad\mbox{as}\quad n\rightarrow\infty,$ where $E_{c}(w,t)$ denotes the kinetic energy defined by: $E_{c}(w,t)=\displaystyle\int_{{\mathbb{R}}^{2}}\left[\|\partial_{t}w\|^{2}+\|\nabla_{x}w\|^{2}+\|w\|^{2}\right](t,x)\;dx.$ For any time slab $I\subset\mathbb{R}$, we shall denote $\\|v\\|_{\mbox{\tiny ST}(I)}:=\sup_{(q,r)\;\mbox{\tiny admissible}}\;\\|v\\|_{L^{q}(I;{\mathrm{B}}^{1}_{r,2}(\mathbb{R}^{2}))}\,.$ By interpolation argument, this Strichartz norm is equivalent to $\\|v\\|_{L^{\infty}(I;H^{1}(\mathbb{R}^{2}))}+\\|v\\|_{L^{4}(I;{\mathrm{B}}^{1}_{8/3,2}(\mathbb{R}^{2}))}\,.$ As ${\mathrm{B}}^{1}_{r,2}(\mathbb{R}^{2})\hookrightarrow L^{p}(\mathbb{R}^{2})$ for all $r\leq p<\infty$ (and $r\leq p\leq\infty$ if $r>2$), it follows that (71) $\\|v\\|_{L^{q}(I;L^{p})}\lesssim\\|v\\|_{\mbox{\tiny ST}(I)},\quad\frac{1}{q}+\frac{2}{p}\leq 1\,.$ As in [5], in the subcritical case, i.e $\displaystyle\limsup_{n\rightarrow\infty}\;E_{p}^{n}<1$, the nonlinearity does not induce any effect on the behavior of the solutions. But, in the critical case i.e $\displaystyle\limsup_{n\rightarrow\infty}\;E_{p}^{n}=1$, it turns out that a nonlinear effect can be produced. More precisely, we have the following result: ###### Theorem 3.6. Let $T$ a strictly positive time. Then 1. (1) If $\underset{n\rightarrow\infty}{\limsup}\,E_{p}^{n}<1$, the sequence $(u_{n})$ is linearizable on $[0,T]$. 2. (2) If $\underset{n\rightarrow\infty}{\limsup}\,E_{p}^{n}=1$, the sequence $(u_{n})$ is linearizable on $[0,T]$ provided that the sequence $(v_{n})$ satisfies (72) $\displaystyle\limsup_{n\to\infty}\;\\|v_{n}\\|_{L^{\infty}([0,T];{L^{\Phi_{p}}})}<\frac{1}{\sqrt{4\pi}}\cdot$ ###### Proof. The proof of Theorem 3.6 is similar to the one of Theorems 3.3 and 3.5 in [5]. Denoting by $w_{n}=u_{n}-v_{n}$, it is clear that $w_{n}$ is the solution of the nonlinear wave equation $\square w_{n}+w_{n}=-F_{p}(u_{n})$ with null Cauchy data. Under energy estimate, we obtain $\\|w_{n}\\|_{T}\lesssim\\|F_{p}(u_{n})\\|_{L^{1}([0,T],L^{2}(\mathbb{R}^{2}))},$ where $\\|w_{n}\\|^{2}_{T}\buildrel\hbox{\footnotesize def}\over{=}\sup_{t\in[0,T]}E_{c}(w_{n},t)$. Therefore, it suffices to prove in the subcritical and critical cases that (73) $\\|F_{p}(u_{n})\\|_{L^{1}([0,T],L^{2}(\mathbb{R}^{2}))}\longrightarrow 0\quad\mbox{as}\quad n\rightarrow\infty.$ Let us begin by the subcritical case. Our goal is to prove that the nonlinear term does not affect the behavior of the solutions. By hypothesis, there exists some nonnegative real $\rho$ such that $\displaystyle\limsup_{n\rightarrow\infty}E_{p}^{n}=1-\rho$. The main point for the proof is based on the following lemma, the proof of which is similar to the proof of Lemma 3.16 in [5] once we observed Inequality (65). ###### Lemma 3.7. For every $T>0$ and $E^{0}_{p}<1$, there exists a constant $C(T,E^{0}_{p})$, such that every solution $u$ of the nonlinear Klein-Gordon equation (41) of energy $E_{p}(u)\leq E^{0}_{p}$, satisfies (74) $\displaystyle\\|u\\|_{L^{4}([0,T];{{\mathcal{C}}}^{1/4})}\leq C(T,E^{0}_{p}).$ Now to establish (73), it suffices to prove that the sequence $(F_{p}(u_{n}))$ is bounded in $L^{1+\epsilon}([0,T],L^{2+\epsilon}(\mathbb{R}^{2}))$ for some nonnegative $\epsilon$ and converges to $0$ in measure in $[0,T]\times\mathbb{R}^{2}$. This can be done exactly as in [5] using the fact that $\|F_{p}(u_{n})\|\leq\|F_{1}(u_{n})\|$. Let us now prove (73) in the critical case. For that purpose, let $T>0$ and assume that (75) $L:=\limsup_{n\to\infty}\;\\|v_{n}\\|_{L^{\infty}([0,T];{L^{\Phi_{p}}})}<\frac{1}{\sqrt{4\pi}}\cdot$ Applying Taylor’s formula, we obtain $F_{p}(u_{n})=F_{p}(v_{n}+w_{n})=F_{p}(v_{n})+F_{p}^{\prime}(v_{n})\,w_{n}+\frac{1}{2}\;F_{p}^{\prime\prime}(v_{n}+\theta_{n}\,w_{n})\,w_{n}^{2},$ for some $0\leq\theta_{n}\leq 1$. Strichartz estimates (44) yields $\\|w_{n}\\|_{\mbox{\tiny ST}([0,T])}\lesssim I_{n}+J_{n}+K_{n},$ where $\displaystyle I_{n}$ $\displaystyle=$ $\displaystyle\\|F_{p}(v_{n})\\|_{L^{1}([0,T];L^{2}(\mathbb{R}^{2}))},$ $\displaystyle J_{n}$ $\displaystyle=$ $\displaystyle\\|F_{p}^{\prime}(v_{n})\,w_{n}\\|_{L^{1}([0,T];L^{2}(\mathbb{R}^{2}))},\quad\mbox{and}$ $\displaystyle K_{n}$ $\displaystyle=$ $\displaystyle\\|F_{p}^{\prime\prime}(v_{n}+\theta_{n}\,w_{n})\,w_{n}^{2}\\|_{L^{1}([0,T];L^{2}(\mathbb{R}^{2}))}.$ As in [5], we have $\displaystyle I_{n}$ $\displaystyle\underset{n\rightarrow\infty}{\longrightarrow}$ $\displaystyle 0\quad\mbox{and}$ $\displaystyle J_{n}$ $\displaystyle\leq$ $\displaystyle\varepsilon_{n}\\|w_{n}\\|_{ST([0,T])},$ where $\varepsilon_{n}\rightarrow 0$. Besides, provided that (76) $\limsup_{n\to\infty}\,\\|w_{n}\\|_{L^{\infty}([0,T];H^{1})}\leq\frac{1-L\,\sqrt{4\pi}}{2},$ we get $K_{n}\leq\varepsilon_{n}\\|w_{n}\\|_{ST([0,T])}^{2},\quad\varepsilon_{n}\rightarrow 0.$ Since $\\|w_{n}\\|_{ST([0,T])}\lesssim I_{n}+\varepsilon_{n}\\|w_{n}\\|_{ST([0,T])}^{2}$, wet obtain by bootstrap argument $\\|w_{n}\\|_{ST([0,T])}\lesssim\varepsilon_{n},$ which ends the proof of the result. ∎ ## 4\. Appendix: Proof of Proposition 1.2 The proof uses in a crucial way the rearrangement of functions (for a complete presentation and more details, we refer the reader to [20]). By virtue of density arguments and the fact that for any function $f\in H^{1}(\mathbb{R}^{2})$ and $f^{}$ the rearrangement of f, we have $\displaystyle\\|\nabla f\\|_{L^{2}}$ $\displaystyle\geq$ $\displaystyle\\|\nabla f^{}\\|_{L^{2}},$ $\displaystyle\\|f\\|_{L^{p}}$ $\displaystyle=$ $\displaystyle\\|f^{}\\|_{L^{p}},$ $\displaystyle\\|f\\|_{L^{\phi_{p}}}$ $\displaystyle=$ $\displaystyle\\|f^{}\\|_{L^{\phi_{p}}}\,,$ one can reduce to the case of a nonnegative radially symmetric and non- increasing function $u$ belonging to ${{\mathcal{D}}}(\mathbb{R}^{2})$. With this choice, let us introduce the function $w(t)=(4\pi)^{\frac{1}{2}}u(\|x\|),\quad\mbox{where}\quad\|x\|={\rm e}^{-\frac{t}{2}}.$ It is then obvious that the functions $w(t)$ and $w^{\prime}(t)$ are nonnegative and satisfy $\displaystyle\int_{\mathbb{R}^{2}}\|\nabla u(x)\|^{2}\,dx$ $\displaystyle=$ $\displaystyle\int_{-\infty}^{+\infty}\|{w}^{\prime}(t)\|^{2}\,dt,$ $\displaystyle\int_{\mathbb{R}^{2}}\|u(x)\|^{2p}\,dx$ $\displaystyle=$ $\displaystyle\frac{1}{4^{p}\,\pi^{p-1}}\int_{-\infty}^{+\infty}\|w(t)\|^{2p}\leavevmode\nobreak\ {\rm e}^{-t}\,dt,$ $\displaystyle\displaystyle\int_{\mathbb{R}^{2}}\left({\rm e}^{\alpha\|u(x)\|^{2}}-\sum_{k=0}^{p-1}\frac{\alpha^{k}\|u(x)\|^{2k}}{k!}\right)dx$ $\displaystyle=$ $\displaystyle\pi\displaystyle\int_{-\infty}^{+\infty}\left({\rm e}^{\frac{\alpha}{4\pi}\|w(t)\|^{2}}-\sum_{k=0}^{p-1}\frac{\alpha^{k}\|w(t)\|^{2k}}{(4\pi)^{k}k!}\right){\rm e}^{-t}\,dt.$ So we are reduced to prove that for any $\beta\in[0,1[$, there exists $C_{\beta}\geq 0$ so that $\displaystyle\int_{-\infty}^{+\infty}\left({\rm e}^{\beta\|w(t)\|^{2}}-\sum_{k=0}^{p-1}\frac{\beta^{k}\|w(t)\|^{2k}}{k!}\right){\rm e}^{-t}dt\leq C({\beta,p})\displaystyle\int_{-\infty}^{+\infty}\|w(t)\|^{2p}{\rm e}^{-t}\,dt,\quad\forall\,\beta\in[0,1[,$ when $\displaystyle\int_{-\infty}^{+\infty}\|{w}^{\prime}(t)\|^{2}dt\leq 1.$ For that purpose, let us set $T_{0}=\sup\bigg{\\{}{t\in\mathbb{R},\leavevmode\nobreak\ w(t)\leq 1\bigg{\\}}}.$ The existence of a real number $t_{0}$ such that $w(t_{0})=0$ ensures that the set $\bigg{\\{}{t\in\mathbb{R},\leavevmode\nobreak\ w(t)\leq 1\bigg{\\}}}$ is non empty. Then $T_{0}\in]-\infty,+\infty].$ Knowing that $w$ is nonnegative and increasing function, we deduce that $w:]-\infty,T_{0}]\longrightarrow[0,1].$ Therefore, observing that $\displaystyle{\rm e}^{s}-\sum_{k=0}^{p-1}\frac{s^{k}}{k!}\leq c_{p}\,s^{p}\,{\rm e}^{s}$ for any nonnegative real $s$, we obtain $\displaystyle\int_{-\infty}^{T_{0}}\left({\rm e}^{\beta\|w(t)\|^{2}}-\sum_{k=0}^{p-1}\frac{\beta^{k}\|w(t)\|^{2k}}{k!}\right){\rm e}^{-t}dt\leq c_{p}\,\beta^{p}\,{\rm e}^{\beta}\displaystyle\int_{-\infty}^{T_{0}}\|w(t)\|^{2p}{\rm e}^{-t}dt.$ To estimate the integral on $[T_{0},+\infty[$, let us first notice that in view of the definition of $T_{0}$, we have for all $t\geq T_{0}$ $\displaystyle w(t)$ $\displaystyle=$ $\displaystyle w(T_{0})+\int_{T_{0}}^{t}{w}^{\prime}(\tau)d\tau$ $\displaystyle\leq$ $\displaystyle w(T_{0})+(t-T_{0})^{\frac{1}{2}}\left(\int_{T_{0}}^{+\infty}{w^{\prime}}(\tau)^{2}d\tau\right)^{\frac{1}{2}}$ $\displaystyle\leq$ $\displaystyle 1+(t-T_{0})^{\frac{1}{2}}.$ Thus, using the fact that for any $\varepsilon>0$ and any $s\geq 0$, we have $(1+s^{\frac{1}{2}})^{2}\leq(1+\varepsilon)s+1+\frac{1}{\varepsilon}=(1+\varepsilon)s+C_{\varepsilon},$ we infer that for for any $\varepsilon>0$ and all $t\geq T_{0}$ (77) $\|w(t)\|^{2}\leq(1+\varepsilon)(t-T_{0})+C_{\varepsilon}.$ Now $\beta$ being fixed in $[0,1[$, let us choose $\varepsilon>0$ so that $\beta(1+\varepsilon)<1$. Then by virtue of (77) $\displaystyle\int_{T_{0}}^{+\infty}\Big{(}{\rm e}^{\beta\|w(t)\|^{2}}-\sum_{k=0}^{p-1}\frac{\beta^{k}\|w(t)\|^{2k}}{k!}\Big{)}{\rm e}^{-t}\,dt$ $\displaystyle\leq$ $\displaystyle\int_{T_{0}}^{+\infty}{\rm e}^{\beta\|w(t)\|^{2}}{\rm e}^{-t}\,dt$ $\displaystyle\leq$ $\displaystyle\frac{{\rm e}^{\beta C_{\varepsilon}-T_{0}}}{1-\beta(1+\varepsilon)}\cdot$ But ${\rm e}^{-T_{0}}=\int_{T_{0}}^{+\infty}{\rm e}^{-t}\,dt\leq\int_{T_{0}}^{+\infty}\|w(t)\|^{2p}\,{\rm e}^{-t}\,dt,$ which gives rise to $\int_{T_{0}}^{+\infty}\Big{(}{\rm e}^{\beta\|w(t)\|^{2}}-\sum_{k=0}^{p-1}\frac{\beta^{k}\|w(t)\|^{2k}}{k!}\Big{)}{\rm e}^{-t}dt\leq\frac{{\rm e}^{\beta C_{\varepsilon}}}{1-\beta(1+\varepsilon)}\int_{T_{0}}^{\infty}\|w(t)\|^{2p}{\rm e}^{-t}\,dt.$ Choosing $C({\beta,p})=\max\Big{(}c_{p}{\rm e}^{\beta}\beta^{p},\displaystyle\frac{{\rm e}^{\beta C_{\varepsilon}}}{1-\beta(1+\varepsilon)}\Big{)}$ ends the proof of the proposition. ## References * [1] S. Adachi and K. 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Struwe, A Super-Critical Nonlinear Wave Equation in 2 Space Dimensions, Milan J. Math. (2011), 129–143. * [25] M. Struwe, Global well-posedness of the Cauchy problem for a super-critical nonlinear wave equation in two space dimensions, Math. Ann. (2011), 707–719. * [26] N.S. Trudinger, On imbedding into Orlicz spaces and some applications, J. Math. Mech. 17 (1967), 473–484.	arxiv-papers	2013-12-21T18:26:29	2024-09-04T02:49:55.764662	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Ines Ben Ayed and Mohamed Khalil Zghal", "submitter": "Majdoub Mohamed", "url": "https://arxiv.org/abs/1312.6286" }
1312.6350	Sparse Portfolio Selection via Quasi-Norm Regularization Caihua Chen111International Center of Management Science and Engineering, School of Management and Engineering, Nanjing Univeristy, China. This author is partially supported by the Natural Science Foundation of Jiangsu Province BK20130550 and the Natural Science Foundation of China NSFC-71271112. Email: [email protected]. , Xindan Li222International Center of Management Science and Engineering, School of Management and Engineering, Nanjing Univeristy, China. This author is partially supported by the Natural Science Foundation of China NSFC-70932003. Email: [email protected]., Caleb Tolman333Department of Management Science and Engineering, School of Engineering, Stanford University, USA. Email: [email protected]. This author is partially supported by AFOSR Grant FA9550-12-1-0396., Suyang Wang444International Center of Management Science and Engineering, School of Management and Engineering, Nanjing Univeristy, China; and Department of Management Science and Engineering, School of Engineering, Stanford University, USA. This author is supported by CSC. Email: [email protected]., Yinyu Ye555Department of Management Science and Engineering, School of Engineering, Stanford University, USA; and International Center of Management Science and Engineering, School of Management and Engineering, Nanjing University. Email: [email protected]. This author is partially supported by AFOSR Grant FA9550-12-1-0396. December 16, 2013. Abstract In this paper, we propose $\ell_{p}$-norm regularized models to seek near-optimal sparse portfolios. These sparse solutions reduce the complexity of portfolio implementation and management. Theoretical results are established to guarantee the sparsity of the second-order KKT points of the $\ell_{p}$-norm regularized models. More interestingly, we present a theory that relates sparsity of the KKT points with Projected correlation and Projected Sharpe ratio. We also design an interior point algorithm to obtain an approximate second-order KKT solution of the $\ell_{p}$-norm models in polynomial time with a fixed error tolerance, and then test our $\ell_{p}$-norm modes on S&P 500 (2008-2012) data and international market data. The computational results illustrate that the $\ell_{p}$-norm regularized models can generate portfolios of any desired sparsity with portfolio variance and portfolio return comparable to those of the unregularized Markowitz model with cardinality constraint. Our analysis of a _combined_ model lead us to conclude that sparsity is not _directly_ related to overfitting at all. Instead, we find that sparsity moderates overfitting only _indirectly_. A combined $\ell_{1}$-$\ell_{p}$ model shows that the proper choose of leverage, which is the amount of additional buying-power generated by selling short can mitigate overfitting; A combined $\ell_{2}$-$\ell_{p}$ model is able to produce extremely high performing portfolios that exceeded the 1/N strategy and all $\ell_{1}$ and $\ell_{2}$ regularized portfolios. Keywords: Markowitz model, sparse portfolio management, $\ell_{p}$-norm regularization, optimality condition, Sharpe ratio. AMS Subject Classifications: 90B50, 90C90,91G10 ## 1 Introduction The origin of modern portfolio theory can be traced back to the early 1950’s, beginning with Markowitz’s work [Markowitz(1952)] on mean-variance formulation. Given a basket of securities, the Markowitz model seeks to find the optimal asset allocation of the portfolio by minimizing the estimated variance with an expected return above a specified level. Although the Markowitz mean-variance model captures the most two essential aspects in portfolio management—risk and return, it is not trivial to implement the model directly in the real world. One of the most critical challenge is the overfitting problem. Overfitting arises from the inability to perfectly estimate the mean and covariance of real-world objects. In fact, due to high dimensionality and non-normal distribution of the unknown variable, these estimates are especially inaccurate for stock data. Indeed, [Merton(1980)] shows that most of the difficulty lies on the mean estimate. Moreover, [DeMiguel et al.(2009)] show that in order to estimate the expected return of portfolio of 25 stocks with satisfactorily low error, one would need on the order of 3000 months of data, which is both extremely difficult to acquire and too long for the model to obey the time-invariance assumptions. The Markowitz model does nothing to prevent the overfitting that comes from mis-estimation, and thus performs poorly across most out-of-sample metrics. For example, [DeMiguel et al.(2009b)] evaluate the out-of-sample performance of the mean-variance model and find that none of algorithms to compute the solution of the Markowitz model consistently outperforms the naive ${1/N}$ (equal amounts of every stock) portfolio. To alleviate the overfitting, several variants of the Markowitz model with regularizers/additional constraints have been proposed in the literature. The modifications can be viewed as adding a prior belief on the true yet unknown return distributions (as suggested by [Merton(1980)]). In [Jagannathan and Ma(2003)], the authors impose a non-shortsale constraint to the mean-variance formulation despite the fact that leading theory speaks against this constraint. Surprisingly, the “wrong” constraint helps the model to find solution with better out-of-sample performance. More recently, [Brodie et al.(2009)] and [Rosenbaum and Tsybakov(2010)] succeed in applying the $\ell_{1}$-norm technique to the Markowitz model to obtain sparse portfolios with higher Sharpe ratio and stability than the naive $1/N$ rule. By adding a norm ball constraint to the portfolio-weight vector, [DeMiguel et al.(2009)] provide a general framework for determining the optimal portfolio. The computational results demonstrate that the norm ball constrained portfolios typically achieve lower out-of-sample variance and higher out-of-sample Sharpe ratio than the proposed strategies in [Jagannathan and Ma(2003)], the naive $1/N$ portfolio and many others in the literature. Meanwhile, the optimal portfolio of Markowitz’s classical model often holds a huge number of assets and some assets admit extremely small weights. Such a solution, however, is not attainable in most situations of the real market. Due to physical, political and economical constraints, investors would be willing to sacrifice a small degree of performance for a more manageable sparse portfolio (see [Shefrin and Statman(2000), Boyle et al.(2012), Guidolin and Rinaldi(2013)] and references therein). An illustrative example comes from the most successful investor of the 20th century, Warren Buffet, who advocates investing in a few familiar stocks, which is also supported by the early work of Keynes (see [Moggridge(1983)]). A popular way to construct the sparse portfolio is via the cardinality constrained portfolio selection (CCPS) model ([Bertsimas and Shioda(2009), Cesarone et al.(2009), Maringer and Kellerer(2003)]) , i.e., choose a specified number of assets to form an efficient portfolio. Unfortunately, the inherent combinatorial property makes the cardinality constrained problem NP- hard generally and hence computationally intractable. By relaxing the hard cardinality constraint, many heuristic methods [Bienstock(1996), Chang et al.(2000)] have been proposed to solve the CCPS. Very recently, by relaxing the objective function as some separable functions, [Gao and Li(2013)] obtain a cardinality constrained relaxation of CCPS with closed-form solution. The new relaxation combined with a branch-and-bound algorithm (Bnb) yields a highly efficient solver, which outperforms CPLEX significantly. The main objective of our paper is to propose a novel and non-CCPS portfolio strategy with complete flexibility in choosing sparsity while still maintaining satisfactory out-of-sample performance. Here, we discuss a new regularization of Markowitz’s portfolio construction both with and without the shortsale constraint. To accomplish this objective, we turn to the $\ell_{p}$-norm ($0<p<1$) regularization which recently attracts a growing interest from the optimization community due to its important role in inducing sparsity. Theoretical and empirical results indicate that the $\ell_{p}$-norm regularization ([Chartrand(2007), Xu et al.(2009), Ji et al.(2013), Saab et al.(2008)]) could have better stability and sparsity than the traditional $\ell_{1}$-norm regularization. In this work, we take a step to study the theoretical and computational performance of the $\ell_{p}$-norm regularized portfolio optimization problem in the framework of the Markowitz model. The contributions of our paper include (i) a novel portfolio strategy to produce 50%–95% more sparse portfolios with competitive out-of-sample performance compared with the Markowitz model and the $\ell_{1}$-norm model; (ii) a polynomial time interior point algorithm to compute the second-order KKT solutions of our $\ell_{p}$-norm models; iii) an extension of the modern portfolio theory that relates sparsity to “Projected correlation” and “Projected Sharpe ratio”; (iv) an “efficient frontier” outlining the optimal tradeoff between sparsity and expected return and variance. The remainder of this paper is organized as follows. In Section 2, we review some relevant portfolio models in the literature and present our $\ell_{p}$-norm regularized formulations for sparse portfolio selection with/without shorting constraints. In Section 3, we develop the $\ell_{p}$-norm regularization portfolio theory with financial interpretation, and design a fast interior point algorithm to compute the KKT points of our regularized models in polynomial time. We also construct some toy examples to show the intuition of our portfolio theory. Section 4 is devoted to the computational results of the regularized models and comparison between different models, which show our portfolio strategies have high sparsity but still maintain out-of-sample performance. Section 5 concludes our work and provides a possible application of our research. All proofs of the propositions can be found in the Appendix I and the details of our interior point algorithm are described in the Appendix II. ## 2 The Related Models Given a portfolio consisting of $n$ stocks. The Markowitz mean-variance portfolio is the solution of the following constrained optimization problem $\begin{array}[]{rl}\min&\displaystyle\,\frac{1}{2}\,x^{T}Qx\\\\[8.5359pt] \mbox{s.t.}&e^{T}x=1,\\\\[2.84544pt] &m^{T}x\geq m_{0},\end{array}$ (2.1) where $Q\in\Re^{n\times n}$ is the estimated covariance matrix of the portfolio, $m\in\Re^{n}$ is the estimated return vector, $m_{0}\in\Re$ is a specific return level, and $e$ is the vector of all ones with a matching dimension. Note also that, if the non-shortsale constraint $x\geq 0$ is added to (2.1), the resulting model is the formulation of the shorting-prohibited Markowitz model. Assume the optimal Lagrangian multiplier associated with the mean constraint is known as $\phi$. Then we can recast the Markowitz model without (with) no-shorting constraint as a linear equality constrained optimization problem $\begin{array}[]{rl}\min&\displaystyle\,\frac{1}{2}\,x^{T}Qx-c^{T}x\\\\[8.5359pt] \mbox{s.t.}&e^{T}x=1,\\\\[2.84544pt] &(x\geq 0),\end{array}$ (2.2) where $c=\phi m$. [Brodie et al.(2009)] discuss the $\ell_{1}$-norm regularized Markowitz model $\begin{array}[]{rl}\min&\displaystyle\,\frac{1}{2}\,x^{T}Qx+\rho\\|x\\|_{1}\\\\[8.5359pt] \mbox{s.t.}&e^{T}x=1,\\\\[2.84544pt] &m^{T}x=m_{0}.\end{array}$ (2.3) Here the $\ell_{1}$-norm of a vector $x\in\Re^{n}$ is defined by $\\|x\\|_{1}:=\sum_{i=1}^{n}\|x_{i}\|$ and $\rho$ is a positive penalty parameter. Sparse portfolios can be obtained by solving (2.3) with increasing values of $\rho$. The $\ell_{1}$-norm, however, cannot be effective in conjunction with the no-shorting constraint, and thus it cannot induce sparsity beyond the sparsity of the no-shorting Markowitz portfolio. This fact can be explained as follows: let $x^{+}$ and $-x^{-}$ denote the positive and negative entries of $x$, respectively. Then, in order to satisfy the budget constraint, we must have: $e^{T}x^{+}=e^{T}x^{-}+1.$ Since $\\|x\\|_{1}=e^{T}x^{+}+e^{T}x^{-}$, we also have that $\\|x\\|_{1}=2e^{T}x^{-}+1$. Thus, adding $\\|x\\|_{1}$ into the objective penalizes shorting activity the sum of the absolute negative entries in $x$ and thus has less effect as a penalty on sparsity. Such a gap motivates us to study the following concave $\ell_{p}$-norm $\,(0<p<1)$ regularization of the no-shorting mean-variance model $\begin{array}[]{rl}\min&\displaystyle\frac{1}{2}\,\,x^{T}Qx-c^{T}x+\lambda\\|x\\|^{p}_{p}\\\\[8.5359pt] \mbox{s.t.}&e^{T}x=1,\\\\[2.84544pt] &x\geq 0,\end{array}$ (2.4) where the $\ell_{p}$-norm of $x\in\Re^{n}$ is defined as $\\|x\\|_{p}=\sqrt[p]{\sum_{j=1}^{n}\|x_{j}\|^{p}}$. And then when $x\geq 0$, $\\|x\\|_{p}^{p}=\sum_{j=1}^{n}x_{j}^{p}$. It is noteworthy that the $\ell_{p}$-norm regularized problem (2.4) can be regarded as a continuous iterative heuristic of the following CCPS problem $\begin{array}[]{rl}\min&\frac{1}{2}x^{T}Qx-c^{T}x\\\\[5.69046pt] \mbox{s.t.}&e^{T}x=1,\\\\[2.84544pt] &\\|x\\|_{0}\leq K,\\\\[2.84544pt] &x\geq 0,\end{array}$ (2.5) where $\\|x\\|_{0}$ represents the number of the nonzero entries of $x$ and $K$ is the chosen limit of stocks to be managed in the portfolio. We also study the portfolio selection problem with the no-shorting constraint removed. Analogues to the above models, we consider the following $\ell_{p}$-norm model $\begin{array}[]{rl}\min&\displaystyle\frac{1}{2}\,\,x^{T}Qx-c^{T}x+\lambda\\|x\\|^{p}_{p}\\\\[8.5359pt] \mbox{s.t.}&e^{T}x=1.\end{array}$ (2.6) Moreover, [DeMiguel et al.(2009)] construct the optimal portfolio with high Sharpe ratio via solving the following the minimum-variance problem subject to a norm ball constraint, i.e., $\begin{array}[]{rl}\min&\displaystyle\,\frac{1}{2}\,x^{T}Qx\\\\[8.5359pt] \mbox{s.t.}&e^{T}x=1,\\\\[2.84544pt] &\\|x\\|\leq\delta,\end{array}$ (2.7) where $\delta$ is a given threshold. Following this work and specifying the general norm as the $\ell_{1}$-norm, we propose the $\ell_{1}$-norm ball constrained the $\ell_{p}$-norm regularized Markowitz model $\begin{array}[]{rl}\min&\frac{1}{2}x^{T}Qx-c^{T}x+\lambda\\|x\\|_{p}^{p}\\\\[5.69046pt] \mbox{s.t.}&e^{T}x=1,\\\\[2.84544pt] &\\|x\\|_{1}\leq\delta.\end{array}$ (2.8) By splitting the vector $x:=x^{+}-x^{-}$, (2.8) can be equivalently written as $\begin{array}[]{rl}\min&\frac{1}{2}(x^{+}-x^{-})^{T}Q(x^{+}-x^{-})-c^{T}(x^{+}-x^{-})+\lambda\\|x^{+}\\|_{p}^{p}+\lambda\\|x^{-}\\|^{p}_{p}\\\\[5.69046pt] \mbox{s.t.}&e^{T}x^{+}-e^{T}x^{-}=1,\\\\[2.84544pt] &e^{T}x^{+}+e^{T}x^{-}\leq\delta,\\\\[2.84544pt] &x^{+}\geq 0,\,x^{-}\geq 0.\end{array}$ (2.9) Besides, we also consider the following $\ell_{2}-\ell_{p}$-norm double regularization Markowitz model which can be seen as a Lagrangian form of (2.7) with a $\ell_{2}$-norm ball $\begin{array}[]{rl}\min&\frac{1}{2}x^{T}Qx-c^{T}x+\lambda\\|x\\|_{p}^{p}+\mu\\|x\\|_{2}^{2}\\\\[5.69046pt] \mbox{s.t.}&e^{T}x=1,\end{array}$ (2.10) as well as its splitting form $\begin{array}[]{rl}\min&\frac{1}{2}(x^{+}-x^{-})^{T}Q(x^{+}-x^{-})-c^{T}(x^{+}-x^{-})+\lambda\\|x^{+}\\|_{p}^{p}+\lambda\\|x^{-}\\|_{p}^{p}+\mu\\|x^{+}-x^{-}\\|_{2}^{2}\\\\[2.84544pt] \mbox{s.t.}&e^{T}x^{+}-e^{T}x^{-}=1,\\\\[2.84544pt] &x^{+}\geq 0,\,x^{-}\geq 0,\end{array}$ (2.11) where $x=x^{+}-x^{-}$. It can be shown later that the regularization models (2.9) and (2.11) always produces a complementary pair $x^{+}$ and $x^{-}$: that is, $x^{+}_{j}x^{-}_{j}=0$ for all $j$. In this paper, we develop theories on the models as well as computation evidences that the models produce sparse portfolios with high out-of-sample Sharpe ratio. ## 3 $\ell_{p}$-norm Regularized Portfolio Theory In this section, we develop theoretical results on the sparsity of the $\ell_{p}$-norm regularized models with toy examples to illustrate the intution and also provide financial interpretation of the theory. Our approach to establish the theoretical results is motivated by the results ([Chen et al(2010)]) in singal processing. For simplicity, hereafter we will fix $p=1/2$. ### 3.1 Bounds of Nonzero Elements of KKT Points First, we develop bounds on the non-zero entries of any KKT solution of the $\ell_{p}$-norm regularized Markowitz model with the non-shortsale constraint $x\geq 0$. ###### Theorem 3.1 Let $\bar{x}$ be any second-order KKT solution of (2.4), that is, a first- order KKT solution that also satisfies the second-order necessary condition, $\bar{P}$ be the support of $\bar{x}$ and $\bar{Q}$ be the corresponding covariance sub-matrix. Furthermore, let $K=\|\bar{P}\|$ and $L_{i}=\bar{Q}_{ii}-\frac{2}{K}(\bar{Q}e)_{i}+\frac{1}{K^{2}}(e^{T}\bar{Q}e),\ i\in\bar{P},$ which are the diagonal entries of the projection of $\bar{Q}$ onto the null space of vector $e$: $\left(I-\frac{1}{K}ee^{T}\right)\bar{Q}\left(I-\frac{1}{K}ee^{T}\right).$ Then it holds that (i) $(K-1)K^{3/2}\leq\frac{4\sum_{i\in\bar{P}}L_{i}}{\lambda}=\frac{4}{\lambda}\Big{[}{\rm tr}(\bar{Q})-\frac{1}{K}e^{T}\bar{Q}e\Big{]}.$ (ii) If $L_{i}=0$ for some $i\in\bar{P}$, then $K=1$ so that $\bar{x}_{i}=1$; otherwise, $\bar{x}_{i}\geq\left(\frac{\lambda(K-1)^{2}}{4L_{i}K^{2}}\right)^{2/3}$ Proof: Please see the proof in the Appendix I. Note that if $\sum_{i\in\bar{P}}L_{i}=0$, the first statement of our theorem implies that $K=1$. This can be explained as follows. $\sum_{i\in\bar{P}}L_{i}=0$ implies the projected $\bar{Q}$ matrix $\left(I-\frac{1}{K}ee^{T}\right)\bar{Q}\left(I-\frac{1}{K}ee^{T}\right)=0.$ Then, $\bar{Q}=\alpha ee^{T}$ for some $\alpha\geq 0$, in which case the portfolio variance $\bar{x}^{T}\bar{Q}\bar{x}=\alpha$ and it is a constant. Thus, the optimal solution of the regularized problem would allocate $100\%$ into the stock with the highest $c_{i}$ or highest return factor. Our theorem also implies that the greater of $\lambda$, the less of $K$. The quantity of $\sum_{i\in\bar{P}}L_{i}$ represents the total diversification coefficient of the set of stocks $i\in\bar{P}$; the smaller of the quantity, the less the size of $\bar{P}$ – the set of selected stocks in the portfolio by the $\ell_{p}$ norm regularized Markowitz model. The second statement provides an even stronger notion: if any $L_{i}=0,\,i\in\bar{P}$, then $K=1$. Basically, it says that investing only into the $i$th stock suffices, since no diversification can help in this case. Note that $L_{i}$ can be interpreted as other stocks’ correlation to stock $i$. If $L_{i}=0$, then other stocks present no diversification to the $i$th stock. Next, we move to the $\ell_{1}$-norm ball constrained $\ell_{p}$-norm regularized Markowitz model and the double regularized model. The following theorems characterize the bound of nonzero elements of any second-order KKT points of problem (2.9) and (2.10). ###### Theorem 3.2 Let $\bar{x}=(\bar{x}^{+},\bar{x}^{-})$ be any second-order KKT solution of problem (2.9) with $\delta>1$, $\bar{P}^{+}$ and $\bar{P}^{-}$ be the support of $\bar{x}^{+}$ and $\bar{x}^{-}$, and $\bar{Q}^{+}$ and $\bar{Q}^{-}$ be the corresponding covariance sub-matrices, respectively. Furthermore, let $K^{+}=\|\bar{P}^{+}\|$ and $K^{-}=\|\bar{P}^{-}\|$, and $L^{j}_{i}=\bar{Q}^{j}_{ii}-\frac{2}{K^{j}}(\bar{Q}^{j}e)_{i}+\frac{1}{(K^{j})^{2}}(e^{T}\bar{Q}^{j}e),\ i\in\bar{P}^{j},\,\,for\,\,j\in\\{+,-\\},$ which are the diagonal entries of the projection of $\bar{Q}^{j}$ onto the null space of vector $e$: $\left(I-\frac{1}{K^{j}}ee^{T}\right)\bar{Q}^{j}\left(I-\frac{1}{K^{j}}ee^{T}\right).$ Then it holds that (i) $\bar{P}^{+}\cap\bar{P}^{-}=\emptyset.$ (ii) $(K^{+}-1)(K^{+})^{3/2}\leq\left({\delta+1\over 2}\right)^{3/2}\frac{4\sum_{i\in\bar{P}^{+}}L_{i}}{\lambda}=\left({\delta+1\over 2}\right)^{3/2}\frac{4}{\lambda}\Big{[}{\rm tr}(\bar{Q}^{+})-\frac{1}{K^{+}}e^{T}\bar{Q}^{+}e\Big{]}$ and $(K^{-}-1)(K^{-})^{3/2}\leq\left({\delta-1\over 2}\right)^{3/2}\frac{4\sum_{i\in\bar{P}^{-}}L_{i}}{\lambda}=\left({\delta-1\over 2}\right)^{3/2}\frac{4}{\lambda}\Big{[}{\rm tr}(\bar{Q}^{-})-\frac{1}{K^{-}}e^{T}\bar{Q}^{-}e\Big{]}.$ (iii) If $L_{i}=0$ for some $i\in\bar{P}^{+}$ (or $i\in\bar{P}^{-}$), then $K^{+}=1$ (or $K^{-}=1$); otherwise, $\bar{x}^{j}_{i}\geq\left(\frac{\lambda(K^{j}-1)^{2}}{4L^{j}_{i}(K^{j})^{2}}\right)^{2/3},\,\,i\in\bar{P}^{j},\,\,{\rm for}\,\,j\in\\{+,-\\}.$ Proof: Please see the proof in the Appendix I. ###### Theorem 3.3 Let $\bar{x}=(\bar{x}^{+},\bar{x}^{-})$ be any second-order KKT solution of (2.11), $\bar{P}^{+}$ and $\bar{P}^{-}$ be the support of $\bar{x}^{+}$ and $\bar{x}^{-}$, and $\bar{P}=\bar{P}^{+}\cup\bar{P}^{-}$. Furthermore, let $\bar{Q}$ be the covariance sub-matrices corresponding to $\bar{P}$, $K=\|\bar{P}\|$, and $L_{i}=\bar{Q}_{ii}+2\mu-\frac{2}{K}(\bar{Q}e)_{i}+\frac{1}{(K)^{2}}(e^{T}\bar{Q}e),\ i\in\bar{P}.$ Then it holds that (i) $\bar{P}^{+}\cap\bar{P}^{-}=\emptyset.$ (ii) If $\\|\bar{x}\\|_{2}\leq\delta$, then $(K-1)K^{3/4}\leq\frac{4\delta^{3/2}\sum_{i\in\bar{P}}L_{i}}{\lambda}=\frac{4\delta^{3/2}}{\lambda}\Big{[}{\rm tr}(\bar{Q})-\frac{1}{K}e^{T}\bar{Q}e\Big{]}.$ (ii) If $L_{i}=0$ for some $i\in\bar{P}$, then $K=1$ so that $\bar{x}_{i}=1$ and $i\in\bar{P}^{+}$; otherwise, $\bar{x}^{j}_{i}\geq\left(\frac{\lambda(K-1)^{2}}{4L_{i}K^{2}}\right)^{2/3},\,\,i\in\bar{P}.$ Proof: Please see the proof in the Appendix I. The theories developed above indicate the importance to compute a second-order KKT solution, rather than just a first-order KKT solution, of the $\ell_{p}$-norm regularized portfolio management problems (2.4) and (2.9). In this paper, we present an interior point algorithm to compute an approximate second KKT point in polynomial time with a fixed error tolerance; see details in the Appendix II. The overall idea of using the interior-point algorithm is to start from a fully supported portfolio $x$ (that is, $x>0$) of every stock in consideration and iteratively eliminate a fraction of stocks at the end of the process. ### 3.2 Characteristics of $L_{i}$ In the theory supporting our model (see Section 3.1), there arose several interesting facts and characteristics to note about the “Projected variances” — $\\{L_{i}\\}$ over the support set of a portfolio selected by the $\ell_{p}$-norm regularized Markowitz models. Given any stock portfolio, with the non-zero portion denoted as $x$, having support $P$ of size $K$ one can rewrite the quantity $L_{i}$ in Theorem 3.1, as follows: $L_{i}=(e^{i}-e^{0})^{T}\bar{Q}(e^{i}-e^{0})={\rm Var}\,[\eta^{T}(e^{i}-e^{0})],$ (3.1) $e_{i}\in R^{K}$ is the vector of all zeros except $1$ at the $i$th position and $e^{0}=\frac{1}{K}e\in R^{k}$. Here $e^{i}$ and $e^{0}$ are the respective distributions obtained by investing 100% in stock $i$ and $\frac{1}{K}$ in each stock of the portfolio $x$, and $\eta$ represents the random return vector of the portfolios. Note that $L_{i}$, $i=1,...,K$, is independent of the entry values of $x$. The difference vector $(e^{i}-e^{0})$ can be viewed as the “cost-neutral portfolio action” that sells an equal amount of everything in the current portfolio and uses all those funds to buy exactly one stock, stock $i$, within the current portfolio. Thus, $L_{i}$ estimates the variance of this action. Let us now consider the feasible and optimal solutions of the Markowitz Model in Lagrangian form: $\begin{array}[]{rl}\min&\frac{1}{2}x^{T}Qx-\phi m^{T}x\\\\[5.69046pt] \mbox{s.t.}&e^{T}x=1,\\\\[2.84544pt] &x\geq 0,\end{array}$ (3.2) where $\phi$ is Lagrangian multiplier associated with the expected return inequality. For any distribution portfolio—the non-zero portion denoted as $x$—one can plot the objective function of moving in a feasible exchange direction $e^{i}-e^{0}$: $\displaystyle f[x+\varepsilon(e^{i}-e^{0})]$ $\displaystyle=\frac{1}{2}[x+\varepsilon(e^{i}-e^{0})]^{T}Q[x+\varepsilon(e^{i}-e^{0})]-\phi m^{T}[x+\varepsilon(e^{i}-e^{0})]$ (3.3) $\displaystyle=f(x)+\varepsilon\,{\rm Cov}\,[x^{T}\eta,(e^{i}-e^{0})^{T}\eta]-\varepsilon\phi(\bar{m}_{i}-\bar{m}_{0})+\frac{1}{2}\varepsilon^{2}L_{i}$ We now consider which stock would increase the variance the least when we remove it from that portfolio $x$. Suppose we remove stock $i$ in the direction $e^{i}-e^{0}$, then we have a new portfolio support $P/\\{i\\}$ with distribution $x^{\prime}=x-\frac{Kx_{i}}{K-1}(e^{i}-e^{0})$. Equation (3.3) would give us the Marginal Costs of Sparsity (3.4) $MCS_{i}=-\frac{K}{K-1}x_{i}[{\rm Cov}(x^{T}\eta,(e^{i}-e^{0})^{T}\eta)-\phi(\bar{m}_{i}-\bar{m}_{0})]+(\frac{K}{K-1})^{2}x_{i}^{2}L_{i}.$ These marginal costs are only _upper-bounds_ on the true costs of sparsity. They do not consider any further improvement that could be made by re- balancing, and thus over-estimate costs. When our current portfolio $x$ is a near-KKT point or local minimizer, we know from the first-order conditions that the first part $[{\rm Cov}\,(x^{T}\eta,(e^{i}-e^{0})^{T}\eta)-\phi(\bar{m}_{i}-\bar{m}_{0})]$ must be near zero and thus the second order term will be a good approximation for the Marginal Cost by itself. Hence, at a near (locally) optimal portfolio $x$, the best candidate for removal can be found by searching for the smallest values of $x_{i}\sqrt{L_{i}}$. Relative Sparsity Cost Index (3.5) ${\rm RSC}_{i}=x_{i}\sqrt{L_{i}}$ Where the smallest non-zero RSC index is the cheapest (on the margin) to eliminate from $x$, and is likely to be the cheapest (absolutely) to remove. Thus the quantity $L_{i}$ can be viewed as measures of elasticity: they indicate how sensitive the objective value is to small cost-neutral changes in $x$; small $L_{i}$ values therefore indicate which stocks could be removed from the portfolio with lowest cost. ### 3.3 The Financial Interpretation of $f^{\prime}(x;e^{i}-e^{0})$ and $\varepsilon_{i}^{}$ The cost-neutral portfolio actions $\big{\\{}e^{i}-e^{0}\,\|\,i\in[1,n]\big{\\}}$ form a basis of the feasible directions, and thus the directional derivatives of the objective along these directions form a method of sensitivity analysis. $f^{\prime}(x,e^{i}-e^{0})={\rm Cov}[x^{T}\eta,(e^{i}-e^{0})^{T}\eta]-\phi(m_{i}-\bar{m})$ (3.6) Where $\bar{m}$ is the average of the expected returns of all stocks in the _support_ of $x$. At an optimal point, these derivatives must be zero. And for small deviations from optimality, these values can be used to approximate any smooth continuous function of the optimal solution. Next, let’s pay more attentions to the optimal step-size along the basic feasible directions. Specifically, given the direction $e^{i}-e^{0}$, the corresponding optimal stepsize $\varepsilon_{i}^{}$ is given by $\varepsilon_{i}^{}=-\frac{f^{\prime}(x;e^{i}-e^{0})}{L_{i}},$ (3.7) which follows directly from (3.3). At any optimal point the directional derivative is zero and thus the optimal step-size is zero; but if we were to consider a small change in the projected gradient, $\varepsilon_{i}^{}$ estimates the changes in optimal solution by taking the direction $e^{i}-e^{0}$. By substituting (3.1) and (3.6) into (3.7), we obtain $\varepsilon_{i}^{}=\phi\frac{m_{i}-\bar{m}_{}}{{\rm Var}\,[(e^{i}-e^{0})^{T}\eta]}-\frac{{\rm Cov}\,[x^{T}\eta,(e^{i}-e^{0})^{T}\eta]}{{\rm Var}\,[(e^{i}-e^{0})^{T}\eta]}.$ The two parts can be easily related to the concepts “Projected correlation” and “Projected Sharpe ratio”, where the Projected correlation is $\bar{\rho}_{i}:=\frac{{\rm Cov}\,[x^{T}\eta,(e^{i}-e^{0})^{T}\eta]}{{\rm Std}\,[x^{T}\eta]{\rm Std}\,[(e^{i}-e^{0})^{T}\eta]},$ (3.8) and the Projected Sharpe ratio is $\bar{S}_{i}=\frac{m_{i}-\bar{m}_{}}{{\rm Std}\,[(e^{i}-e^{0})^{T}\eta]}.$ (3.9) Then the optimal step size can be equivalently written as: $\varepsilon_{i}^{}=\bar{S}_{i}\frac{\phi}{{\rm Std}\,[(e^{i}-e^{0})^{T}\eta]}-\bar{\rho_{i}}\frac{{\rm Std}\,[x^{T}\eta]}{{\rm Std}\,[(e^{i}-e^{0})^{T}\eta]}$ (3.10) It is clear that the optimal step size is sensitive to the inverse of the standard-deviation of the cost-neutral portfolio (inversely), as well as to the current portfolio standard-deviation. The Projected correlation and Projected Sharpe ratio (as well as $\phi$) give the exact coefficients of these relationships. #### 3.3.1 Toy Examples In this section, we illustrate the previous sensitivity analysis by some dummy examples. Consider the first example in Table 1, where the portfolios include three stocks with identically distributed variance yet differing expected returns. The lower returning stock admits a slightly smaller percentage (32.33% vs 34.33%) in the optimal portfolio due to the small reward ($\phi=0.01$) for the expected return. Since the RSC of stock 1 attains the minimum cost of the three stocks, according to our sensitive analysis, the investor would intuitively decrease the investment in the first stock further and thus remove the first stock from the basis to form a sparse portfolio (with the increasing of $\lambda$ ). Direct calculation also shows that this is the lowest cost stock to remove. Table 1: Toy example 1 Mean \| Variance \| $x^{}(\phi=0.01)$ \| $L_{i}$ \| OK to drop \| RSC ---\|---\|---\|---\|---\|--- $\begin{bmatrix}1\\\ 2\\\ 3\\\ \end{bmatrix}$ \| $\begin{bmatrix}2&1&1\\\ 1&2&1\\\ 1&1&2\\\ \end{bmatrix}$ \| $\begin{bmatrix}0.3233\\\ 0.3333\\\ 0.3433\\\ \end{bmatrix}$ \| $\begin{bmatrix}0.6667\\\ 0.6667\\\ 0.6667\\\ \end{bmatrix}$ \| $\begin{bmatrix}Yes\\\ No\\\ No\\\ \end{bmatrix}$ \| $\begin{bmatrix}0.2640\\\ 0.2722\\\ 0.2803\\\ \end{bmatrix}$ Consider the portfolio in Table 2, where two stocks are positively correlated yet a third stock is independent; all the stocks share a common mean and variance. The large value of $L_{1}$ (see the MCS equation in (3.4)) suggests that the first stock may not be a good candidate to be removed, which can seen clearly by comparing the variances of the portfolios with two stocks. Table 2: Toy example 2 Mean \| Variance \| $x^{}(\phi=0.01)$ \| $L_{i}$ \| OK to drop \| RSC ---\|---\|---\|---\|---\|--- $\begin{bmatrix}0\\\ 0\\\ 0\\\ \end{bmatrix}$ \| $\begin{bmatrix}2&0&0\\\ 0&2&1\\\ 0&1&2\\\ \end{bmatrix}$ \| $\begin{bmatrix}0.4355\\\ 0.2823\\\ 0.2823\\\ \end{bmatrix}$ \| $\begin{bmatrix}1.5556\\\ 0.8889\\\ 0.8889\\\ \end{bmatrix}$ \| $\begin{bmatrix}No\\\ Yes\\\ Yes\\\ \end{bmatrix}$ \| $\begin{bmatrix}0.5431\\\ 0.2661\\\ 0.2661\\\ \end{bmatrix}$ Table 3 lists the portfolio consisting of three stocks, where the third stock is actually a zero-cost mutual fund—one that simply invest equally in the first and second stocks. This third stock creates redundancy and thus _infinitely_ many optimal solutions are possible (we have shown one arbitrarily). If we were to drop either the second stock or the third (but not both) from the portfolio, then we would still be able to attain the same optimal objective (75%-25% mix of Stock 1 and Stock 2 respectively for this small $\phi$, and a more balanced mix larger $\phi$). Moreover, we see that $L_{3}=0$, and this fact correctly predicts that there exists a strictly sparser optimal portfolio. Table 3: Toy example 3 Mean \| Variance \| $x^{}(\phi=0.01)$ \| $L_{i}$ \| OK to drop \| RSC ---\|---\|---\|---\|---\|--- $\begin{bmatrix}1\\\ 3\\\ 2\\\ \end{bmatrix}$ \| $\begin{bmatrix}3&1&2\\\ 1&7&4\\\ 2&4&3\\\ \end{bmatrix}$ \| $\begin{bmatrix}0.6875\\\ 0.1925\\\ 0.1200\\\ \end{bmatrix}$ \| $\begin{bmatrix}2\\\ 2\\\ 0\\\ \end{bmatrix}$ \| $\begin{bmatrix}No\\\ BEST\\\ OK\\\ \end{bmatrix}$ \| $\begin{bmatrix}0.9722\\\ 0.2722\\\ 0.00\\\ \end{bmatrix}$ As a last example, consider Table 4, where we have a set of stocks that include two of them with high variance and positive correlation to most other stocks, yet highly negative correlation with each other. These two stocks alone would make an excellent portfolio of size two. Table 4: Toy example 4 mean \| Variance \| $x^{}(\phi=0.01)$ \| $L_{i}$ \| OK to drop \| RSC ---\|---\|---\|---\|---\|--- $\begin{bmatrix}0\\\ 0\\\ 0\\\ 0\\\ \end{bmatrix}$ \| $\begin{bmatrix}8&7&6&6\\\ 7&26&6&0\\\ 6&6&96&-68\\\ 6&0&-68&73\\\ \end{bmatrix}$ \| $\begin{bmatrix}0.2913\\\ 0.1166\\\ 0.2714\\\ 0.3207\end{bmatrix}$ \| $\begin{bmatrix}1.81\\\ 13.81\\\ 83.31\\\ 74.81\end{bmatrix}$ \| $\begin{bmatrix}Yes\\\ No\\\ No\\\ No\end{bmatrix}$ \| $\begin{bmatrix}0.392\\\ 0.433\\\ 2.477\\\ 2.773\end{bmatrix}$ Here we see that the smallest investments in the Markowitz portfolio are not necessarily the stocks to remove (to achieve the best sparse portfolio). The best portfolio with single stock is stock 1. The best portfolio of size 2 contains Stock 3 and 4. The best portfolio of size 3 excludes stock 1. The Relative Sparsity Costs seem to hint at many of those choices. ## 4 Computational Results ### 4.1 Data, Parameters and Models To test the $\ell_{p}$-norm regularized models, we collected historical daily stock price data in S & P 500 index from CRSP Database666We choose this short time-interval due to the need for a large number of intervals and the common belief that the distribution of stock prices fundamentally change shape over decades., which spans from 31/12/2007 to 31/12/2012. We don’t include any company unless it is traded on the market at least 90% of the trading days during the data period, nor do any company not listed on the market for the entire timescale. The total list has 461 companies by 1259 trading days. Since S & P 500 stocks have a high average correlation around 0.4516, for the purpose of testing our model under more uncorrelated data, we further considered a larger dataset that contains 53 commodity ETF daily data from American market, and 236 stocks data of Husheng 300 Index from Chinese market.777This index contains 60% of the market value of stocks listed in Shanghai and Shengzheng Stock Exchange of China. To deal with the mismatch between China and America’s calendars, we set the return of stocks not traded because of holidays on either country to zero. We employ the rolling-window method to evaluate the out-of-sample performance888Taking account into the computational time, we use 36 rolling-windows for No-shorting Constraint case and Shorting-allowed $\ell_{p}$-norm model, $\ell_{1}$-norm ball constrained model, 12 rolling-window for $\ell_{1}$-norm ball constrained $\ell_{p}$-norm regularization model and $\ell_{p}-\ell_{2}$-norm double regularization model, with 500 days and 537 days training window, 21 days and 63 days estimation window respectively. The portfolios obtained from S&P data and International data are named as S&P Portfolio and International Portfolio, respectively. Note that the coefficient $c=\phi m$ in the linear objective term of the regularized models. To solve the $\ell_{p}$-norm Markowitz models, proper $\phi$ values should be be chosen accordingly. To achieve this objective, we first set reasonable values for the minimum target return $m_{0}$, and then calculate the $\phi$-values from the dual variables of the models in constraint form. We use mean, variance and Sharpe Ratio to evaluate the out- of-sample performance, where the Sharpe ratio computed here uses the same method as [DeMiguel et al.(2009)]. ### 4.2 No-shorting Constraint Case In [DeMiguel et al.(2009b)], the authors apply the $\ell_{1}$-norm technique to seek sparse portfolios. The $\ell_{1}$-norm, however, plays no role in the Markowitz model with no-shorting constraints. However, since no-shorting environments and investors exist extensively in the real market, we turn to the $\ell_{p}$-norm regularization to seek portfolios with desired sparsity in this situation. As we will see later, our $\ell_{p}$-norm regularized model (2.4) with no-shorting constraints produces extremely sparse portfolios with comparison to the already sparse Markowitz no-shorting model portfolios. The $\ell_{p}$-norm regularized model is compared with two benchmarks in the framework of Markowitz model with no-shorting constraints. The first one is the Markowitz model without regularization ($\lambda=0$) and the second is the cardinality-constrained portfolio selection (CCPS) model. The global optimal cardinality-constrained portfolios are found by solving the following integer formulation of problem (2.5): $\begin{array}[]{rl}\min&\displaystyle\frac{1}{2}x^{T}Qx-c^{T}x\\\\[5.69046pt] \mbox{s.t.}&e^{T}x=1,\\\\[2.84544pt] &0\leq x\leq d,\\\\[2.84544pt] &e^{T}d\leq K,\\\\[2.84544pt] &d\in\\{0,1\\}^{n}.\end{array}$ #### 4.2.1 In-Sample Performance Table 5 reports the portfolio weight, the mean, the variance and the sparsity of the Markowitz portfolios with the specified return $m_{0}$ ranging from 0.02% to 0.12%. The portfolios range from 19 to 26 stocks, which are about 4.1%-5.6% of the full set. The expected return of each portfolio equals or exceeds the minimum target return. The trend that portfolios with higher target return also have higher estimated variance is clear in the table. Table 6 lists the results of the $\ell_{p}$-norm regularized Markowitz model with $\lambda=5.5e-6$ by our second-order interior point algorithm. Clearly, the resulting portfolios are of low variance and larger sparsity. Specifically, the number of positive position ranges from 3 to 6, which are only 15-25% of the number of stocks in the Markowitz portfolios and 0.5-1.5% of the total number of stocks. We also find that these portfolios have a similar composition to the non-zero unregularized counterparts. The top companies are the same (SO, K, KMB, GIS, AZO) and there is a complete overlap between the unregularized and regularized models: none of the companies in the sparse portfolios were found with 0% stake in the unregularized portfolios. However, Table 5: Results for the Unregularized Markowitz Model: In-Sample Performance. \| $\lambda$ \| 0 ---\|--- $m_{0}$ \| 0.0002 $\phi$ \| 0.00000 Mean \| 0.00029 Variance \| 3.53e-5 Sparsity \| 19 SO \| 0.29541 K \| 0.13471 GIS \| 0.08696 KMB \| 0.07952 PEP \| 0.07762 AZO \| 0.07409 WMT \| 0.0475 MCD \| 0.04266 HSY \| 0.03874 NEM \| 0.03364 CPB \| 0.032 PG \| 0.026 SYY \| 0.01039 FOH \| 0.0093 NFLX \| 0.00565 PPL \| 0.00523 JNJ \| 0.00003 CAG \| 0.00002 ABT \| 0.00002 \| \| $\lambda$ \| 0 ---\|--- $m_{0}$ \| 0.0004 $\phi$ \| 0.00445 Mean \| 0.0004 Variance \| 3.89e-5 Sparsity \| 24 SO \| 0.27974 KMB \| 0.12563 K \| 0.12469 AZO \| 0.0961 GIS \| 0.07693 HSY \| 0.07123 PEP \| 0.06607 WMT \| 0.05964 NEM \| 0.02297 MCD \| 0.01736 AAPL \| 0.01388 CPB \| 0.0132 PG \| 0.01302 CAG \| 0.00545 FOH \| 0.00482 PPL \| 0.00302 DUK \| 0.00246 SYY \| 0.0017 REGN \| 0.00081 ABT \| 0.0007 JNJ \| 0.00003 ORLY \| 0.00002 MNST \| 0.00001 SHW \| 0.00001 \| \| $\lambda$ \| 0 ---\|--- $m_{0}$ \| 0.0006 $\phi$ \| 0.01099 Mean \| 0.0006 Variance \| 3.89e-5 Sparsity \| 26 SO \| 0.25522 KMB \| 0.17225 AZO \| 0.11016 HSY \| 0.10648 K \| 0.09903 GIS \| 0.06227 WMT \| 0.06134 PEP \| 0.0321 AAPL \| 0.02935 REGN \| 0.01786 SHW \| 0.01635 CAG \| 0.01589 ABT \| 0.00896 DUK \| 0.00846 NEM \| 0.00209 THC \| 0.00094 GILD \| 0.00059 PPL \| 0.00005 PG \| 0.00004 ORLY \| 0.00003 MNST \| 0.00003 CPB \| 0.00002 LLY \| 0.00002 BIIB \| 0.00001 RAI \| 0.00001 SYY \| 0.00001 \| $\lambda$ \| 0 ---\|--- $m_{0}$ \| 0.0008 $\phi$ \| 0.01886 Mean \| 0.0008 Variance \| 4.48e-5 Sparsity \| 21 SO \| 0.21501 KMB \| 0.20186 HSY \| 0.13914 AZO \| 0.11891 K \| 0.05701 WMT \| 0.04926 SHW \| 0.04633 AAPL \| 0.03462 REGN \| 0.03401 GIS \| 0.03158 CAG \| 0.02365 BIIB \| 0.01816 DUK \| 0.01522 GILD \| 0.00742 THC \| 0.00495 ABT \| 0.00158 MNST \| 0.00088 ORLY \| 0.00004 LLY \| 0.00002 PEP \| 0.00001 RAI \| 0.00001 \| \| $\lambda$ \| 0 ---\|--- $m_{0}$ \| 0.0010 $\phi$ \| 0.02731 Mean \| 0.001 Variance \| 5.4e-5 Sparsity \| 23 KMB \| 0.22204 HSY \| 0.16869 SO \| 0.15838 AZO \| 0.12639 SHW \| 0.0764 REGN \| 0.05116 BIIB \| 0.04093 AAPL \| 0.03795 WMT \| 0.03061 CAG \| 0.02457 DUK \| 0.02241 GILD \| 0.01254 MNST \| 0.01105 THC \| 0.00928 K \| 0.00699 ORLY \| 0.00008 GIS \| 0.00004 C \| 0.00004 ABT \| 0.00003 HD \| 0.00002 LLY \| 0.00001 RAI \| 0.00001 AMGN \| 0.00001 \| \| $\lambda$ \| 0 ---\|--- $m_{0}$ \| 0.0012 $\phi$ \| 0.03645 Mean \| 0.0012 Variance \| 6.68e-5 Sparsity \| 20 KMB \| 0.22846 HSY \| 0.18915 AZO \| 0.12588 SHW \| 0.10123 SO \| 0.08532 REGN \| 0.06975 BIIB \| 0.05887 AAPL \| 0.03929 DUK \| 0.03077 HD \| 0.0265 GILD \| 0.014 THC \| 0.0135 MNST \| 0.01165 CAG \| 0.00304 C \| 0.00203 WMT \| 0.00019 ORLY \| 0.00002 EXPE \| 0.00002 AMGN \| 0.00001 ABT \| 0.00001 the composition is far from identical as many low-weighted stocks in the unregularized portfolios have large weights in the sparse portfolios. Moreover, Figure 1 shows the number of positive positions versus the regularization parameter $\lambda$ graph of the $\ell_{p}$-norm regularized portfolios. With minor exception, increasing lambda almost always results in a more sparse solution which is consistent with our portfolio theory developed in Section 3. Table 6: Results for $\ell_{p}$-norm Markowitz Regularized Portfolios with $\lambda=5.5e-6$: In-Sample Performance. \| $\lambda$ \| 5.5e-6 ---\|--- $m_{0}$ \| 0.0002 $\phi$ \| 0.00000 Mean \| 0.00025 Variance \| 4.12e-5 Sparsity \| 3 SO \| 0.55327 K \| 0.23209 GIS \| 0.21464 \| \| $\lambda$ \| 5.5e-6 ---\|--- $m_{0}$ \| 0.0004 $\phi$ \| 0.00445 Mean \| 0.00025 Variance \| 4.12e-5 Sparsity \| 3 SO \| 0.55222 K \| 0.22999 GIS \| 0.21779 \| \| $\lambda$ \| 5.5e-6 ---\|--- $m_{0}$ \| 0.0006 $\phi$ \| 0.01099 Mean \| 0.00046 Variance \| 4.05e-5 Sparsity \| 4 SO \| 0.38804 KMB \| 0.31953 K \| 0.16675 HSY \| 0.12568 \| $\lambda$ \| 5.5e-6 ---\|--- $m_{0}$ \| 0.0008 $\phi$ \| 0.01886 Mean \| 0.00061 Variance \| 4.35e-5 Sparsity \| 4 KMB \| 0.35527 SO \| 0.27413 HSY \| 0.20689 AZO \| 0.16371 \| \| $\lambda$ \| 5.5e-6 ---\|--- $m_{0}$ \| 0.0010 $\phi$ \| 0.02731 Mean \| 0.00098 Variance \| 5.76e-5 Sparsity \| 5 KMB \| 0.47565 HSY \| 0.2479 AZO \| 0.19227 REGN \| 0.06476 DUK \| 0.01942 \| \| $\lambda$ \| 5.5e-6 ---\|--- $m_{0}$ \| 0.0012 $\phi$ \| 0.03645 Mean \| 0.00109 Variance \| 6.41e-5 Sparsity \| 6 KMB \| 0.43308 HSY \| 0.25649 AZO \| 0.18254 REGN \| 0.08895 DUK \| 0.02721 THC \| 0.01172 Figure 1: Portfolio Sparsity A comprehensive comparison of computational results between our $\ell_{p}$-norm regularized model and the cardinality constrained portfolio selection (CCPS) model are reported in the Table 7. As can be seen in the table, our regularized $\ell_{p}$-norm performs almost as well as theoretical possible—the difference of the variance estimation between the two models are within 0.2% in all cases and the difference of the mean estimation are within 0.02%. Therefore, compared to the computational intractable cardinality constrained portfolio optimization, our $\ell_{p}$-norm regularized portfolio, which can be obtained in polynomial time, performs almost as well and seeks near optimal sparse portfolios. Table 7: Comparison of Sparsity, Mean and Variance between the $\ell_{p}$-norm Model and CCPS (Daily Return). \| $\ell_{p}$-norm \| CCPS ---\|---\|--- \| $\lambda$ \| Sparsity \| Mean \| Variance \| Sparsity \| Mean \| Variance $m_{0}=0.02\%\quad$ \| 5.0e-7 \| 9 \| 0.05% \| 4.45% \| 9 \| 0.05% \| 4.46% \| 1.0e-6 \| 7 \| 0.03% \| 3.68% \| 7 \| 0.04% \| 3.66% \| 2.0e-6 \| 5 \| 0.03% \| 3.90% \| 5 \| 0.04% \| 3.75% \| 3.5e-6 \| 4 \| 0.02% \| 4.08% \| 4 \| 0.04% \| 3.89% \| 4.5e-6 \| 3 \| 0.02% \| 4.12% \| 3 \| 0.04% \| 4.06% $m_{0}=0.1\%\quad$ \| 5.0e-7 \| 10 \| 0.06% \| 4.57% \| 10 \| 0.05% \| 4.58% \| 1.0e-6 \| 7 \| 0.10% \| 4.90% \| 7 \| 0.10% \| 4.86% \| 2.0e-6 \| 7 \| 0.09% \| 5.27% \| 5 \| 0.09% \| 5.37% \| 3.5e-6 \| 6 \| 0.09% \| 5.18% \| 6 \| 0.09% \| 5.28% \| 4.5e-6 \| 6 \| 0.09% \| 5.18% \| 6 \| 0.09% \| 5.28% #### 4.2.2 Out-of-Sample Performance [Brodie et al.(2009)] show that sparse portfolios are often more robust and thus outperform the portfolios with less sparsity in terms of out-of-sample performance. In their analysis, the no-shorting constraint ($x\geq 0$) is taken as the _most_ extreme sparsity inducing measure. We continued this investigation by taking the no-shorting constraint as the _least_ extreme measure and adding the $\ell_{p}$-norm regularizer onto the objective function. It is interesting to ask whether the sparsest portfolios will outperform other portfolio strategies with less sparsity. Figure 2 and Figure 3 show the out-of-sample portfolio returns and variances obtained by the $\ell_{p}$-norm regularized Markowitz model with $\lambda$ ranging from 5.0e-7 to 5.5e-6 and the CCPS, respectively. From Figure 2, we observe clearly that most of the plots go up slightly and then achieve its maximum, indicating that the portfolios with moderate sparsity (around 10) perform very well, even better than the Markowitz portfolio. However, with the continuously increasing of sparsity, the mean will go down dramatically and thus the regularized portfolios with extreme sparsity perform poorly in the sense of portfolio mean. Figure 3 shows that the variance of the regularized portfolios is increasing with a incremental rate with the increasing sparsity of the portfolios. However, though the highly sparse portfolios performs poorly in the sense of portfolio variance, the intermediate portfolios with about 10 companies suffered a 15-25% increase in variance which is also comparable to the CCPS integer portfolios. Figure 2: Portfolio Returns Figure 3: Portfolio Variances Figure 4: Portfolio Sharpe Ratios Figure 4 shows the out-of-performance Sharpe ratios of our $\ell_{p}$-norm regularized portfolio and the CCPS integer portfolio. Although the Markowitz portfolio (with $\lambda=0$) outperforms our $\ell_{p}$-norm regularized model in terms of the out-of-sample Sharpe ratio, the sparse portfolios may be more implementable due to the transaction costs or logistical limitations reasons. Our results indicate that an intermediate sparse portfolio may get a comparable or at most only 10-20% cost in Sharpe ratio while reducing more construction costs. Also, the $\ell_{p}$-norm regularized approach is competitive with the computationally gigantic integer approach in the sense of out-of-sample performance. ### 4.3 Shorting-Allowed Extension Next we relaxed our constraint to allow the short-selling of stocks. We compare our model (2.6) with the $\ell_{1}$-norm ball constrained portfolios studied by [DeMiguel et al.(2009b)], as the strategy may find sparse portfolios with improved out-of-sample Sharpe ratios. #### 4.3.1 $\ell_{p}$-norm Regularized Model Figure 5 shows that the shorting-allowed Markowitz portfolios behave eccentrically (also see Table 8), with the portfolio including all the stocks no matter the choice of the parameter $\phi$. Meanwhile, our $\ell_{p}$-norm regularized model (2.6) is able to reduce the number of investing stocks drastically. For example, only 22 stocks are involved in the Markowitz regularized portfolio for $\lambda=1e-6$ and $m_{0}=0.06\%$, and thus there is a 95.2% reduction of the portfolio size. The parameter $\lambda$ can be regarded as a server to control the portfolio sparsity. Figure 5: Portfolio Sparsity The out-of-sample results are similar to the shorting-prohibited case. From Table 8, we see that the Sharpe ratio tends to be the highest when $\lambda$ is not too large, and would decrease with the increasing of the parameter $\lambda$. However, even for a significantly small $\lambda$, the regularized portfolios are much more sparse (e.g. 79 versus 461), and of competitive or better performance while compared with the Markowitz portfolio. For larger values of $\lambda$, there is a clear tradeoff between the portfolio sparsity and performance. Table 8: Sharpe Ratio and Sparsity of shorting allowed $\ell_{p}$-norm regularized Model \| $m_{0}=-\infty$ \| $m_{0}=0.02\%$ \| $m_{0}=0.06\%$ \| $m_{0}=0.1\%$ ---\|---\|---\|---\|--- $\lambda$ \| Spa \| SRatio \| Spa \| SRatio \| Spar \| SRatio \| Spar \| SRatio 0 \| 461.0 \| 0.165 \| 461.0 \| 0.161 \| 461.0 \| 0.146 \| 461.0 \| 0.127 5.0e-7 \| 78.1 \| 0.156 \| 58.7 \| 0.161 \| 79.3 \| 0.161 \| 78.2 \| 0.166 1.0e-6 \| 45.1 \| 0.125 \| 27.1 \| 0.120 \| 46.9 \| 0.123 \| 45.2 \| 0.120 2.0e-6 \| 22.9 \| 0.159 \| 13.4 \| 0.159 \| 24.4 \| 0.159 \| 23.4 \| 0.155 2.5e-6 \| 18.4 \| 0.149 \| 11.5 \| 0.152 \| 19.2 \| 0.147 \| 18.8 \| 0.150 3.5e-6 \| 13.4 \| 0.120 \| 7.6 \| 0.120 \| 14.3 \| 0.118 \| 13.4 \| 0.121 4.5e-6 \| 10.9 \| 0.040 \| 6.6 \| 0.036 \| 11.0 \| 0.040 \| 10.8 \| 0.041 5.5e-6 \| 8.5 \| 0.024 \| 4.9 \| 0.023 \| 8.9 \| 0.024 \| 8.6 \| 0.027 Table 9: Sharpe Ratio and Sparsity of $\ell_{1}$-norm ball constrained Markowitz portfoliol \| $m_{0}=-\infty$ \| $m_{0}=0.02\%$ \| $m_{0}=0.06\%$ \| $m_{0}=0.1\%$ ---\|---\|---\|---\|--- $\delta$ \| Spa \| SRatio \| Spa \| SRatio \| Spar \| SRatio \| Spar \| SRatio 1.5 \| 70.5 \| 0.127 \| 70.4 \| 0.109 \| 68.3 \| 0.111 \| 60.9 \| 0.149 2 \| 118.1 \| 0.163 \| 118.6 \| 0.155 \| 116.3 \| 0.177 \| 111.1 \| 0.181 #### 4.3.2 $\ell_{1}$-norm Ball Constrained Model For the purpose of comparison, we also post the results of $\ell_{1}$-norm ball constrained portfolios on the same data set. The $\ell_{1}$-norm ball constrained model considered in this section takes the following form $\begin{array}[]{rl}\min&\frac{1}{2}x^{T}Qx\\\\[5.69046pt] \mbox{s.t.}&e^{T}x=1,\\\\[5.69046pt] &m^{T}x\geq m_{0},\\\\[5.69046pt] &\\|x\\|_{1}\leq\delta,\\\\[5.69046pt] \end{array}$ (4.1) where $\delta\geq 1$. Figure 6 shows the number of nonzero positions versus the threshold parameter $\delta$ of the $\ell_{1}$-norm ball constrained portfolios. It is clear that the sparsity decreases at a fast speed with the increasing of $\delta$. The left-most date points corresponds to the shorting- allowed Markowitz model where the $\ell_{1}$-norm ball constraint is not effective while the right-most data points ($\delta=1$) corresponds to the shorting-prohibited Markowitz mode where the $\ell_{1}$-norm ball constraints takes its most effective role in inducing sparsity. However, even when $\delta=1$, the average sparsity of the portfolio is around 15, which is much more dense than the shorting-allowed portfolio with $\lambda$ smaller than 3.5e-6. Figure 6: Portfolio Sparsity with Different Delta of $\ell_{1}$-Norm Model We can make a detailed comparison between $\ell_{1}$-norm ball constrained model and $\ell_{p}$-norm regularization model by Table 8 and 9. It is easy to see that when $\delta$ equals 1.5 or 2, the out-of-sample performance of $\ell_{p}$-norm models is similar to that of the $\ell_{1}$-norm ball constrained portfolios but the former is much more sparse. However, the performance of the $\ell_{p}$-norm models is surpassed when $\delta$ is increased. In that case, the $\ell_{1}$-norm ball constrained portfolio achieves a better out-of-sample performance with the sacrifice of sparsity, see Figure 7. Moreover, the largest out-of-sample Sharpe Ratio is achieved when $\delta\approx 16$. From this figure, we can also see that the shorting- allowed Markowitz Model (far left points) is better than shorting-prohibited Markowitz Model (far right points). This is consistent with the remark made by [Jagannathan and Ma(2003)] that when daily data is used, shorting-prohibited models perform almost as well. Figure 7: Portfolio Sharpe Ratio with Sparsity ### 4.4 $\ell_{1}$-norm Ball Constrained $\ell_{p}$-norm Regularized Model In the last two sections, we have discussed the computational performance of the $\ell_{1}$-norm ball constrained Markowitz model and the shorting-allowed $\ell_{p}$-norm regularized Markowitz models (2.4) and (2.6) individually. Next, we consider the $\ell_{1}$-norm ball constrained $\ell_{p}$-norm regularized model (2.9) to investigate the relationship between the leverage (characterized by the $\ell_{1}$-norm), the sparsity (induced mostly by the $\ell_{p}$-norm) and the out-of-sample performance. According to the results of the $\ell_{1}$-norm constrained model, we solve our regularized model combined with the $\ell_{1}$-norm constraint with $\delta$ ranging from from 1.5 to 32 and $(\lambda,\phi)$ taking an array of values. This thorough approach are expected to give us a more structured picture of the relationship between the $\ell_{1}$\- and $\ell_{p}$-norms as well as their relationship to the performance. Table 10 reports the out-of-sample computational results for the cases where $\delta$ is taken as 1.5, 2 and 32, and $m_{0}$ is set as $-\infty$ and 0.04%. Since the $\ell_{1}$-norm constrained $\ell_{p}$-norm regularization enjoys the similar trend for different choices of $\delta$, we don’t report the corresponding results for succinctness. From the table, we see clearly that the sparsity, in general, is antagonistic to performance. Thus, there exists a tradeoff between performance and sparsity. Though the performance varies for different values of $\lambda$ , a well performed portfolio can be obtained when $\lambda$ is smaller than 4.5e-06. And when $\lambda$ is not very large, say less then $2e-6$, the sparsity need not come at a high price (of Sharpe ratio) and there are many sparse portfolios with comparable performance to the portfolios found with $\lambda=0$. Also, we find that with the increase of $\lambda$, the leverage of the resulting portfolio decreases significantly. Thus, it seems that the leverage of the portfolio is mostly determined by the choice of $\lambda$. Moreover, we note that there appears to be little cross- effect between sparsity and leverage on performance. Table 10: Sparsity and Sharpe Ratio of the Combined Model and $\ell_{1}$ Norm model for S & P Data with Three Month Estimation Window: Out-of-Sample Performance \| \| $\delta=1.5$ \| $\delta=2$ \| $\delta=32$ ---\|---\|---\|---\|--- model/ $m_{0}$ \| $\lambda$ \| Spar \| Leve \| SRatio \| Spar \| Leve \| SRatio \| Spar \| Leve \| SRatio $\ell_{1}-\ell_{p}/-\infty$ \| 5.0e-7 \| 37.9 \| 1.496 \| 0.074 \| 54.3 \| 1.949 \| 0.159 \| 94.2 \| 3.392 \| 0.214 \| 2.0e-6 \| 16.8 \| 1.340 \| 0.101 \| 23.6 \| 1.526 \| 0.103 \| 29.7 \| 1.806 \| 0.063 \| 4.5e-6 \| 9.3 \| 1.202 \| 0.120 \| 11.9 \| 1.285 \| 0.139 \| 13.8 \| 1.376 \| 0.050 \| 8.0e-6 \| 5.5 \| 1.084 \| -0.094 \| 6.8 \| 1.130 \| -0.026 \| 6.8 \| 1.155 \| -0.138 \| 1.25e-5 \| 4.1 \| 1.049 \| -0.094 \| 4.4 \| 1.063 \| -0.132 \| 4.4 \| 1.068 \| -0.111 $\ell_{1}$-norm/$-\infty$ \| — \| 70.5 \| 1.500 \| 0.234 \| 117.2 \| 2.000 \| 0.195 \| 461 \| 23.571 \| 0.247 $\ell_{1}-\ell_{p}/0.04\%$ \| 5.0e-7 \| 35.1 \| 1.496 \| 0.074 \| 56.6 \| 1.948 \| 0.177 \| 94.4 \| 3.414 \| 0.226 \| 2.0e-6 \| 16.3 \| 1.338 \| 0.114 \| 23.6 \| 1.523 \| 0.115 \| 29.8 \| 1.808 \| 0.086 \| 4.5e-6 \| 9.6 \| 1.197 \| 0.105 \| 12.0 \| 1.284 \| 0.138 \| 13.6 \| 1.378 \| 0.053 \| 8.0e-6 \| 5.6 \| 1.088 \| -0.132 \| 6.8 \| 1.129 \| -0.006 \| 6.7 \| 1.143 \| -0.162 \| 1.25e-5 \| 4.1 \| 1.049 \| -0.098 \| 4.4 \| 1.063 \| -0.133 \| 4.3 \| 1.068 \| -0.115 $\ell_{1}$-norm/$0.04\%$ \| — \| 70.2 \| 1.500 \| 0.201 \| 118.7 \| 2.000 \| 0.207 \| 461.0 \| 23.500 \| 0.234 ### 4.5 $\ell_{2}-\ell_{p}$-norm Double Regularized Model As mentioned in [DeMiguel et al.(2009)], the $\ell_{2}$-norm constraint can be viewed as placing a prior on the 1/N strategy, thus it is reasonable to expect the results close to the 1/N strategy. Yet, most investors would not invest into a portfolio with huge number of stocks, which motivates us to develop a portfolio strategy with less stocks but similar to the 1/N strategy with competitive out-of-sample performance, especially for those passive investors. For this purpose, it is natural to consider the $\ell_{p}$-norm regularization of the $\ell_{2}$-norm constrained Markowitz model $\begin{array}[]{rl}\min&\frac{1}{2}x^{T}Qx+\lambda\\|x\\|_{p}^{p}\\\\[5.69046pt] \mbox{s.t.}&e^{T}x=1,\\\\[5.69046pt] &m^{T}x\geq m_{0},\\\\[5.69046pt] &\\|x\\|_{2}^{2}\leq\delta^{2},\\\\[5.69046pt] \end{array}$ or its Lagranagin version (double regularization Markowitz model (2.10)) to see if we can obtain a portfolio that balances sparsity and uniform prior. The results of the $\ell_{2}-\ell_{p}$ model are shown in Table 11, with different choices of $\lambda$ and $\delta$. The parameters $\mu$ and $\phi$ in the regularized model 2.10 are obtained from the dual variables of problem (4.5) with $\lambda=0$. Seen from the result, the optimal portfolio obtained by the double regularization formulation would include all the stocks in the case that $\lambda=0$ for all values of $\delta$, closely related to the 1/N strategy. Also, the portfolio becomes more sparse with the increasing of $\lambda$ and fixed $\delta$, while more dense with the increasing of $\delta$ and fixed $\lambda$ . This trend shows a tradeoff between $\ell_{p}$-norm regularization and $\ell_{2}$-norm ball constraints. It is also note that the strategy to invest all stocks doesn’t usually perform best in the sense of Sharpe ratio. For example, in the case that $\lambda$=1.25e-5, $\delta$=0.1 and $m_{0}$=0.08%, we can find a portfolio with only 135 stocks yet with a high Sharpe ratio 0.575, which is much better than the Sharpe ratio 0.374 attained with $\lambda=0$. Similar as the observation before, the extremely sparse portfolio often performs poorly showing a tradeoff between sparsity and performance. Also, the most constricting delta ($\delta=0.1$) had the highest performing portfolios, suggesting that the presence of a strong uniform prior on all stocks helps mitigate overfitting due to poor variance/covariance estimates. The out-of-sample performance was _increasing_ in $\lambda$ when $\lambda$ was not too large. These moderately sparse, highly $\ell_{2}$-norm constricted portfolios performed excellently (all had Sharpe Ratio near or above 0.5). Thus the $\ell_{2}$ and $\ell_{p}$ norms appear to exhibit synergy in reducing overfitting. Table 12 lists the out-of-sample computational results of our $\ell_{2}-\ell_{p}$ double regularization model for international data with much more diversity. Compared with the results for S & P data, the overall performance is greatly enhanced, especially for the sparsest portfolios. Very surprisingly, we even find that a portfolio with two stocks perform quite well. And also we see that the cost of sparsity need not be high even for very sparse portfolios if the stock base is favorable. Table 11: Sparsity and Sharpe Ratio of the $\ell_{2}-\ell_{p}$-Norm Double-Regularization Model for S& P Data with Three Month Estimation Window \| \| $\delta=0.1$ \| $\delta=0.2$ \| $\delta=0.3$ \| $\delta=0.4$ ---\|---\|---\|---\|---\|--- \| $\lambda$ \| Spar \| SRatio \| Spar \| SRatio \| Spar \| SRatio \| Spar \| SRatio $m_{0}=0.00\%$ \| 0 \| 461.0 \| 0.409 \| 461.0 \| 0.18 \| 461.0 \| 0.087 \| 461.0 \| 0.035 \| 5.0e-7 \| 321.8 \| 0.431 \| 185.8 \| 0.247 \| 133.0 \| 0.264 \| 116.7 \| 0.214 \| 2.0e-6 \| 236.8 \| 0.414 \| 69.8 \| 0.35 \| 50.3 \| 0.29 \| 33.3 \| 0.231 \| 4.5e-6 \| 164.4 \| 0.504 \| 34.0 \| 0.422 \| 34.3 \| 0.248 \| 15.8 \| 0.14 \| 8.0e-6 \| 166.6 \| 0.498 \| 20.2 \| 0.336 \| 13.5 \| 0.237 \| 8.1 \| 0.096 \| 1.25e-5 \| 105.3 \| 0.536 \| 15.1 \| 0.288 \| 7.6 \| 0.096 \| 5.0 \| -0.083 $m_{0}=0.04\%$ \| 0 \| 461.0 \| 0.389 \| 461.0 \| 0.181 \| 461.0 \| 0.085 \| 461.0 \| 0.046 \| 5.0e-7 \| 321.8 \| 0.405 \| 181.3 \| 0.25 \| 127.0 \| 0.236 \| 111.8 \| 0.233 \| 2.0e-6 \| 233.3 \| 0.385 \| 63.8 \| 0.374 \| 46.2 \| 0.27 \| 36.9 \| 0.251 \| 4.5e-6 \| 189.6 \| 0.414 \| 34.8 \| 0.43 \| 26.5 \| 0.258 \| 22.3 \| 0.147 \| 8.0e-6 \| 188.9 \| 0.52 \| 20.9 \| 0.376 \| 11.3 \| 0.201 \| 8.2 \| 0.122 \| 1.25e-5 \| 132.5 \| 0.553 \| 15.3 \| 0.31 \| 7.8 \| 0.087 \| 4.9 \| -0.082 $m_{0}=0.08\%$ \| 0 \| 460.9 \| 0.374 \| 461.0 \| 0.183 \| 461.0 \| 0.089 \| 461.0 \| 0.052 \| 5.0e-7 \| 343.7 \| 0.376 \| 198.3 \| 0.286 \| 144.7 \| 0.246 \| 120.6 \| 0.243 \| 2.0e-6 \| 254.9 \| 0.366 \| 73.0 \| 0.39 \| 63.4 \| 0.212 \| 32.6 \| 0.222 \| 4.5e-6 \| 179.6 \| 0.379 \| 35.4 \| 0.421 \| 25.6 \| 0.288 \| 27.0 \| 0.212 \| 8.0e-6 \| 165.3 \| 0.444 \| 34.5 \| 0.295 \| 14.2 \| 0.194 \| 8.2 \| 0.114 \| 1.25e-5 \| 134.6 \| 0.575 \| 16.1 \| 0.265 \| 7.8 \| 0.097 \| 5.1 \| -0.079 Table 12: Sparsity and Sharpe Ratio of the $\ell_{2}-\ell_{p}$-Norm Double-Regularization Model for International Data with Three Month Estimation Window \| \| $\delta=0.1$ \| $\delta=0.2$ \| $\delta=0.3$ \| $\delta=0.4$ ---\|---\|---\|---\|---\|--- \| $\lambda$ \| Spar \| SRatio \| Spar \| SRatio \| Spar \| SRatio \| Spar \| SRatio $m_{0}=0.00\%$ \| 0 \| 749.8 \| 0.569 \| 750.0 \| 0.528 \| 749.8 \| 0.509 \| 749.9 \| 0.497 \| 5.0e-7 \| 265.0 \| 0.611 \| 84.8 \| 0.477 \| 60.8 \| 0.443 \| 31.6 \| 0.439 \| 2.0e-6 \| 109.0 \| 0.58 \| 23.4 \| 0.452 \| 14.9 \| 0.439 \| 8.1 \| 0.419 \| 4.5e-6 \| 65.7 \| 0.578 \| 12.3 \| 0.461 \| 5.1 \| 0.417 \| 3.8 \| 0.435 \| 8.0e-6 \| 60.4 \| 0.602 \| 7.67 \| 0.462 \| 3.4 \| 0.415 \| 2.4 \| 0.433 \| 1.25e-5 \| 32.2 \| 0.62 \| 5.4 \| 0.407 \| 2.6 \| 0.424 \| 2.25 \| 0.433 $m_{0}=0.04\%$ \| 0 \| 749.9 \| 0.615 \| 749.9 \| 0.554 \| 750.0 \| 0.524 \| 750.0 \| 0.508 \| 5.0e-7 \| 270.8 \| 0.619 \| 83.8 \| 0.485 \| 48.7 \| 0.445 \| 32.4 \| 0.437 \| 2.0e-6 \| 115.0 \| 0.606 \| 24.5 \| 0.459 \| 12.0 \| 0.441 \| 8.2 \| 0.418 \| 4.5e-6 \| 67.3 \| 0.599 \| 12.58 \| 0.46 \| 5.3 \| 0.416 \| 3.8 \| 0.429 \| 8.0e-6 \| 46.5 \| 0.621 \| 8.0 \| 0.456 \| 3.4 \| 0.413 \| 2.4 \| 0.434 \| 1.25e-5 \| 33.5 \| 0.628 \| 5.5 \| 0.404 \| 2.6 \| 0.423 \| 2.3 \| 0.433 $m_{0}=0.08\%$ \| 0 \| 749.7 \| 0.637 \| 750.0 \| 0.588 \| 749.8 \| 0.544 \| 749.9 \| 0.524 \| 5.0e-7 \| 291.8 \| 0.649 \| 88.2 \| 0.486 \| 49.3 \| 0.446 \| 38.0 \| 0.439 \| 2.0e-6 \| 128.6 \| 0.692 \| 26.3 \| 0.467 \| 12.3 \| 0.434 \| 8.3 \| 0.417 \| 4.5e-6 \| 87.3 \| 0.631 \| 13.1 \| 0.455 \| 5.3 \| 0.414 \| 4.17 \| 0.429 \| 8.0e-6 \| 52.8 \| 0.623 \| 8.5 \| 0.461 \| 3.5 \| 0.41 \| 2.42 \| 0.434 \| 1.25e-5 \| 38.8 \| 0.625 \| 5.8 \| 0.4 \| 2.6 \| 0.4 \| 2.3 \| 0.433 ## 5 Discussions and Conclusions ### 5.1 $\ell_{p}$-norm regularized Dynamic Portfolios A closely related application to our model is the dynamic portfolio selection. Instead of seeking a sparse portfolio, we are looking for a sparse _adjustment_ to an already existing portfolio. Consider the following cardinality constrained optimization model. $\begin{array}[]{rl}\min&\frac{1}{2}x^{T}Qx-c^{T}x\\\\[5.69046pt] \mbox{s.t.}&e^{T}x=1\\\\[2.84544pt] &x\geq 0\\\\[2.84544pt] &\\|x-a\\|_{0}\leq K,\end{array}$ (5.1) Here the $a$-vector is a feasible portfolio ($e^{T}a=1$ and $a\geq 0$), representing the current state of our dynamic portfolio. Similar to the Markowitz model, the dynamic portfolio has found many applications. One is the situation where implementing the portfolio takes a significant amount of time (perhaps we must execute our orders sequentially with long delays in-between) and we wish our first orders to constitute an near-optimal portfolio. Another is the situation where our estimates $Q$ and $c=\phi m$ are themselves varying over time, enough to warrant a re-balancing, yet we still have limits on trading—either due to transaction costs or structural limitations. This model has a non-differentiable point in the middle of the feasible region ($x=a$), but can be reformulated (by substitution: $y=x-a$) to achieve a model very similar to the non-dynamic sparse portfolio model: $\begin{array}[]{rl}\min&\frac{1}{2}y^{T}Qy+Qa^{T}y-c^{T}y\\\ \mbox{s.t.}&e^{T}y=0\\\ &y\geq-a\\\ &\\|y\\|_{0}\leq K,\end{array}$ (5.2) We note that the objective function is still a quadratic function, and that the constraints are also of the same shape. Instead of solving the original model (5.2), we consider the following $p$ norm regularized dynamic Markowitz model $\begin{array}[]{rl}\min&\frac{1}{2}y^{T}Qy+(a^{T}Q-c^{T})y+\lambda\\|y\\|_{p}^{p}\\\\[8.5359pt] \mbox{s.t.}&e^{T}y=0,\\\\[5.69046pt] &y\geq-a.\end{array}$ (5.3) By letting $y=y^{+}-y^{-}$ and using the concavity of $\\|\cdot\\|_{p}^{p}$, we know the regularized model (5.3) can be equivalently written as $\begin{array}[]{rl}\min&\frac{1}{2}(y^{+}-y^{-})^{T}Q(y^{+}-y^{-})+(a^{T}Q-c^{T})(y^{+}-y^{-})+\lambda\\|y^{+}\\|_{p}^{p}+\lambda\\|y^{-}\\|_{p}^{p}\\\\[5.69046pt] \mbox{s.t.}&e^{T}y^{+}-e^{T}y^{-}=0,\\\\[2.84544pt] &y^{+}-y^{-}\geq-a,\\\\[2.84544pt] &y^{+}\geq 0,\,y^{-}\geq 0,\end{array}$ (5.4) which can be further simplified to the following model $\begin{array}[]{rl}\min&\frac{1}{2}(y^{+}-y^{-})^{T}Q(y^{+}-y^{-})+(a^{T}Q-c^{T})(y^{+}-y^{-})+\lambda\\|y^{+}\\|_{p}^{p}+\lambda\\|y^{-}\\|_{p}^{p}\\\\[5.69046pt] \mbox{s.t.}&e^{T}y^{+}-e^{T}y^{-}=0,\\\\[2.84544pt] &y^{+}\geq 0,\,\,\,0\leq y^{-}\leq a,\end{array}$ (5.5) Similar as the non-dynamic $\ell_{p}$\- norm portfolio model, this resulting $\ell_{p}$-norm model can also be solved by the second order interior interior point method. ### 5.2 Conclusions In this paper, we propose an $\ell_{p}$-norm regularized model with/without shortsale constraints to seek near-optimal sparse portfolios to reduce the complexity of portfolio implementation and management. We also study the impact of the $\ell_{1}$ and $\ell_{2}$ norms and their cross-effects on overfitting. Theoretical results is established to guarantee the sparsity of the novel portfolio strategy. Computational evidence also clearly shows that the $\ell_{p}$-norm regularized portfolio is able to choose sparsity with completely flexibility while still maintaining satisfactory out-of-sample performance—comparable to that of the NP cardinality-constrained portfolios. We find that the $\ell_{1}$-norm can be viewed as a prior on the optimal level of portfolio leverage; a small $\ell_{1}$-penalty can improve performance. The $\ell_{1}$ norm greatly reduces the feasible region helping algorithms converge quickly. It also is shown to be synonymous with leverage—a very important financial term and quantity of great theoretical interest. Meanwhile the $\ell_{2}$-norm can be viewed as a prior on the estimated covariances; we find that a large $\ell_{2}$-penalty can greatly improve performance, It also could improve tractability by bounding the feasible region. And $\ell_{2}$-norm and the $\ell_{p}$-norm have positive cross- effects on performance—the combined model consistently portfolios outperformed all others. Generally, when we do not pursue the most sparse portfolio,then the cost of sparsity is low—especially when the original portfolio of stocks is diverse. And our research provides a toolset to evaluate the tradeoffs between sparsity and out-of-sample performance. Our models also importantly provide a theoretical framework. In this framework, sparsity can be studied in relation to leverage, correlation, Sharpe-Ratio and financial theory, where both practical bounds and qualitative insights can be made. ## 6 Appendix ### 6.1 Appendix I: Proofs of the Propositions Proof of Theorem 3.1. Since the second-order necessary condition of (2.4) holds at the point $\bar{x}$, the sub-Hessian matrix of the objective function corresponding to the indices $\bar{P}$ $\bar{Q}-\frac{\lambda}{4}\bar{X}^{-3/2}\succeq 0$ on the null space of $e$. This means the projected Hessian matrix $\left(I-\frac{1}{K}ee^{T}\right)\left(\bar{Q}-\frac{\lambda}{4}\bar{X}^{-3/2}\right)\left(I-\frac{1}{K}ee^{T}\right)$ is positive semidefinite. By direct calculation, we know that the $i$th diagonal entry of the projected Hessian matrix is given by $L_{i}-\frac{\lambda}{4}\left((\bar{x}_{i})^{-3/2}\left(1-\frac{2}{K}\right)+\frac{\sum_{j\in\bar{P}}(\bar{x}_{j})^{-3/2}}{K^{2}}\right)\geq 0,$ (6.1) and also the trace of projected Hessian matrix $\sum_{i\in\bar{P}}L_{i}-\frac{\lambda}{4}\frac{K-1}{K}\sum_{i\in\bar{P}}(\bar{x}_{i})^{-3/2}\geq 0.$ The quantity $\sum_{i\in\bar{P}}(\bar{x}_{i})^{-3/2}$, with $\sum_{i\in\bar{P}}\bar{x}_{i}=1$, achieves its minimum at $\bar{x}_{i}=1/K$ for all $i\in\bar{P}$ with the minimum value $K\cdot K^{3/2}$. Thus, $\frac{\lambda}{4}(K-1)K^{3/2}\leq\sum_{i\in\bar{P}}L_{i},$ or $(K-1)K^{3/2}\leq\frac{4\sum_{i\in\bar{P}}L_{i}}{\lambda},$ which complete the proof of the first claim. Moreover, from (6.1) we have $\frac{\lambda}{4}\left((\bar{x}_{i})^{-3/2}\left(1-\frac{2}{K}\right)+\frac{\sum_{j\in\bar{P}}(\bar{x}_{j})^{-3/2}}{K^{2}}\right)\leq L_{i}.$ Or $\frac{\lambda}{4}\left((\bar{x}_{i})^{-3/2}\left(1-\frac{1}{K}\right)^{2}+\frac{\sum_{j\in\bar{P},j\neq i}(\bar{x}_{j})^{-3/2}}{K^{2}}\right)\leq L_{i},$ which implies $\frac{\lambda}{4}(\bar{x}_{i})^{-3/2}\left(1-\frac{1}{K}\right)^{2}\leq L_{i}.$ (6.2) Hence, if $L_{i}=0$, we must have $K=1$ so that $\bar{x}_{i}$ is the only non- zero entry in $\bar{x}$ and $\bar{x}_{i}=1$. Otherwise, from (6.2), we have the desired second statement in the theorem. Proof of Theorem 3.2.） (i) Assume the contrary that $\bar{P}^{+}\cap\bar{P}^{-}\neq\emptyset$. Then there exists an index $j$ such that $\bar{x}_{j}^{+}>0$ and $\bar{x}_{j}^{-}>0$. Let $\lambda_{1}$ and $\lambda_{2}\,(\leq 0)$ be the optimal Lagrangian multiplier associated with the constraints of (2.9). Since $(x^{+},x^{-})$ is a KKT point of (2.9), it holds that $\left\\{\begin{array}[]{c}\displaystyle\left[Q(\bar{x}^{+}-\bar{x}^{-})\right]_{i}-c_{i}+{\lambda\over 2\sqrt{(\bar{x}^{+})_{i}}}-\lambda_{1}-\lambda_{2}=0\\\ \displaystyle\left[Q(\bar{x}^{-}-\bar{x}^{+})\right]_{i}+c_{i}+{\lambda\over 2\sqrt{(\bar{x}^{-})_{i}}}+\lambda_{1}-\lambda_{2}=0\end{array}\right..$ (6.3) By adding the two equalities above, we have ${\lambda\over 2\sqrt{(\bar{x}^{+})_{i}}}+{\lambda\over 2\sqrt{(\bar{x}^{-})_{i}}}-2\lambda_{2}=0.$ (6.4) However, since $(\bar{x}^{+})_{i}>0,\,\,(\bar{x}^{-})_{i}>0$ and $\lambda_{2}\leq 0$, the equality (6.4) cannot hold. This contradiction shows that $\bar{P}^{+}\cap\bar{P}^{-}\neq\emptyset$. (ii,iii) Since the proof of the remainder parts of this theorem is similar to that of Theorem 1, we omit the details. Proof of Theorem 3.3 .） The proof of this theorem is similar to that of Theorem 1. We omit the details. ### 6.2 Appendix II: Polynomial Time Interior Point Algorithms Most nonlinear optimization solvers can only guarantee to compute a first- order KKT solution. In this section, we extend the interior-point algorithm described in [Bian et al.(2012)] to solve the following generally $\ell_{p}$-norm regularized model $\begin{array}[]{rl}\min&\displaystyle f(x):=\frac{1}{2}\,\,x^{T}Qx-c^{T}x+\lambda\\|x\\|^{p}_{p}\\\\[8.5359pt] \mbox{s.t.}&Ax=b,\\\\[2.84544pt] &x\geq 0,\end{array}$ (6.5) where $A$ is a matrix in $\Re^{p\times n}$, $b$ is a vector in $\Re^{p}$ and the feasible region is strictly feasible. For simplicity, we fix $p={1\over 2}$. Naturally, we would start from an interior-point feasible solution such as the analytical of the feasible set, and let the iterative algorithm to decide which entry goes to zero. This is the basic idea of affine scaling algorithm developed in [Bian et al.(2012)] for regularized nonconvex programming. The algorithm starts from an initial interior-point solution, then follows an interior feasible path and finally converges to either a global minimizer or a second-order KKT solution. At each step, it chooses a new interior point which produces a reduction to the objective function by an affine-scaling trust- region iteration. Specifically, give an interior point $x^{k}$ of the feasible region, the algorithm looks for an objective reduction by a update from $x^{k}$ to $x^{k+1}$. Let $d^{k}$ be a vector in $\Re^{p}$ satisfying $Ad^{k}=0$ and $x^{k+1}:=x^{k}+d^{k}>0$. Using the second Taylor expansion of $f(\cdot)$, we know $f(x^{k+1})\approx f(x^{k})+\frac{1}{2}(d^{k})^{T}\big{(}Q-\frac{\lambda}{4}(X^{k})^{-3/2}\big{)}d^{k}+\big{(}Qx^{k}-c+\frac{\lambda}{2\sqrt{x^{k}}}\big{)}^{T}d^{k},$ where $X^{k}={\rm Diag}(x^{k})$. For given $\varepsilon\in(0,1]$, we solve the ellipsoidal trust-region constrained problem $\begin{array}[]{rl}\min&\displaystyle\frac{1}{2}(d^{k})^{T}\big{(}Q-\frac{\lambda}{4}(X^{k})^{-3/2}\big{)}d^{k}+\big{(}Qx^{k}-c+\frac{\lambda}{2\sqrt{x^{k}}}\big{)}^{T}d^{k}\\\\[8.5359pt] \mbox{s.t.}&Ad^{k}=0,\\\\[5.69046pt] &\\|X_{k}^{-1}d^{k}\\|^{2}\leq\beta^{2}\varepsilon<1,\end{array}$ to obtain the direction $d^{k}$. By letting $\tilde{d^{k}}=X_{k}^{-1}d^{k}$, we can recast the above ellipsoidal trust-region constrained problem above as a ball-constrained quadratic problem $\begin{array}[]{rl}\min&\displaystyle\frac{1}{2}(\tilde{d}^{k})^{T}X^{k}\big{(}Q-\frac{\lambda}{4}(X^{k})^{-3/2}\big{)}X^{k}\tilde{d}^{k}+\big{(}Qx^{k}-c+\frac{\lambda}{2\sqrt{x_{k}}}\big{)}^{T}X^{k}\tilde{d}^{k},\\\\[8.5359pt] \mbox{s.t.}&AX_{k}\tilde{d}^{k}=0,\\\\[2.84544pt] &\\|\tilde{d}^{k}\\|^{2}\leq\beta^{2}\varepsilon.\end{array}$ (6.6) Note that problem (6.6) can be solved efficiently even when it is nonconvex (see [Bian et al.(2012)]). Let $\widetilde{Q}^{k}=X_{k}QX_{k}-\frac{\lambda}{4}\sqrt{X^{k}}$ and $\tilde{c}^{k}=X_{k}(Qx^{k}-c)+\frac{\lambda}{2}\sqrt{x^{k}}$. If $\widetilde{Q}_{k}$ is semidefinite, the solution $\tilde{d}^{k}$ of problem (6.6) satisfies the following necessary and sufficient conditions: $\left\\{\begin{array}[]{l}(\widetilde{Q}^{k}+\mu_{k}I)\tilde{d}^{k}-(AX^{k})^{T}y_{k}=-\tilde{c}^{k},\\\\[2.84544pt] AX^{k}\tilde{d}^{k}=0,\\\\[2.84544pt] \mu_{k}\geq 0,\,\\|\tilde{d}^{k}\\|^{2}\leq\beta^{2}\varepsilon,\,\mu_{k}(\\|\tilde{d}^{k}\\|^{2}-\beta^{2}\varepsilon)=0.\end{array}\right.$ (6.7) In the case that $\widetilde{Q}^{k}$ is indefinite, it holds that $\left\\{\begin{array}[]{l}(\widetilde{Q}^{k}+\mu_{k}I)\tilde{d}^{k}-(AX^{k})^{T}y_{k}=-\tilde{c}^{k},\\\\[2.84544pt] AX_{k}\tilde{d}^{k}=0,\\\\[2.84544pt] \mu_{k}\geq 0,\,N_{k}^{T}\widetilde{Q}^{k}N_{k}+\mu_{k}I\succeq 0,\\\\[2.84544pt] \\|\tilde{d}^{k}\\|=\beta\sqrt{\varepsilon},\end{array}\right.$ (6.8) where $N_{k}$ is an orthogonal basis spanning the space of $X^{k}A^{T}$. To evaluate the performance of the affine scaling method, we need the definitions of $\varepsilon$ scaled first-order and second-order KKT solutions. $x^{}$ is said to be an $\epsilon$ scaled first-order KKT solution of (6.5) if there exists a $y^{}\in\Re^{p}$ such that $\left\\{\begin{array}[]{l}\displaystyle\\|X^{}(Qx^{}-c)+\frac{\lambda}{2}\sqrt{x^{}}-X^{}A^{T}y^{}\\|\leq\epsilon,\\\\[2.84544pt] Ax^{}=b,\\\\[2.84544pt] x^{}\geq 0.\end{array}\right.$ (6.9) Furthermore, if $\displaystyle X^{}QX^{}-\frac{\lambda}{4}\sqrt{X^{}}+\sqrt{\epsilon}I$ is also semidefinite on the null space of $X^{}A^{T}$, we call $x^{}$ an $\epsilon$ scaled second-order KKT solution. If $\varepsilon=0$, the $\varepsilon$ scaled first-order KKT solution reduces to $X^{}(Qx^{}-c)+\frac{\lambda}{2}\sqrt{x^{}}-X^{}A^{T}y^{}=0,$ which is exactly the first-order condition of (6.5). In this case, the $\varepsilon$ scaled second-order condition collapses to $N^{T}X^{}QX^{}N-\frac{\lambda}{4}N^{T}\sqrt{X^{}}N\succeq 0$ (6.10) where $N$ is an orthogonal basis spanning the space of $X^{}A^{T}$. By direct computation, we know (6.10) recovers exactly the second-order optimality condition of problem (6.5). For the convergence analysis of our proposed interior-point algorithm, we make the following standard assumption. For any given $x^{0}\geq 0$ such that $Ax=b$, there exists $R\geq 1$ such that $\sup\\{\\|x\\|_{\infty}:f(x)\leq f(x_{0}),Ax=b,x\geq 0\\}\leq R.$ Under the assumption above, we are able to establish the next theorem showing that the affine scaling is able to obtain either an $\varepsilon$-scaled second-order KKT solution or an $\varepsilon$ global minimizer in polynomial time. ###### Theorem 6.4 Let $\varepsilon\in(0,1]$. There exists a positive number $\tau$ such that the proposed second-order interior point obtains either an $\varepsilon$ scaled second-order KKT solution or $\varepsilon$ global minimizer of (6.5) in no more than $O(\varepsilon^{-3/2})$ iterations provided that $\beta\in(0,\,\tau)$. Proof: With loss of generality, we assume the radius $R=1$ in the assumption. To proceed the proof of this theorem, we first introduce the following Lemma. ###### Lemma 6.1 If $\mu_{k}>\lambda/6\\|\tilde{d}^{k}\\|$ holds for all $k=0,1,2,\ldots$, then the second-order interior point algorithm produces an $\varepsilon$ global minimizer of (6.5) in at most $O(\varepsilon^{-3/2})$ iterations. Proof: By the Taylor expansion of $\sqrt{\cdot}$, it is easily to show that $f(x^{k+1})-f(x^{k})\leq\frac{1}{2}\big{\langle}\tilde{d}^{k},\,\widetilde{Q}^{k}\tilde{d}^{k}\big{\rangle}+\big{\langle}\tilde{c}^{k},\,\tilde{d}^{k}\rangle+\frac{3\lambda}{48}\\|\tilde{d}_{k}\\|^{3}.$ From (6.7) and (6.8), then $\begin{array}[]{rl}f(x^{k+1})-f(x^{k})&\displaystyle\leq\frac{1}{2}\big{\langle}\tilde{d}^{k},\,\widetilde{Q}^{k}\tilde{d}^{k}\big{\rangle}+\big{\langle}-\widetilde{Q}^{k}\tilde{d}^{k}-\mu_{k}\tilde{d}^{k}+(AX_{k})^{T}y_{k},\,\tilde{d}^{k}\rangle+\frac{3\lambda}{48}\\|\tilde{d}^{k}\\|^{3}\\\\[5.69046pt] &\displaystyle=-\frac{1}{2}\tilde{d}^{k}\widetilde{Q}^{k}\tilde{d}^{k}-\mu_{k}\\|\tilde{d}^{k}\\|^{2}+\frac{3\lambda}{48}\\|\tilde{d}^{k}\\|^{3}\\\\[5.69046pt] &\displaystyle=-\frac{1}{2}(v^{k})^{T}(N_{k})^{T}\widetilde{Q}^{k}N_{k}v^{k}-\mu_{k}\\|\tilde{d}^{k}\\|^{2}+\frac{3\lambda}{48}\\|\tilde{d}^{k}\\|^{3}\\\\[5.69046pt] &\displaystyle\leq\frac{\mu_{k}}{2}\\|v^{k}\\|^{2}-\mu_{k}\\|\tilde{d}^{k}\\|^{2}+\frac{\lambda}{48}\\|\tilde{d}^{k}\\|^{3}\\\\[5.69046pt] &\displaystyle=-\frac{\mu_{k}}{2}\\|\tilde{d}^{k}\\|^{2}+\frac{3\lambda}{48}\\|\tilde{d}^{k}\\|^{3}\\\\[5.69046pt] &\displaystyle\leq-\frac{1}{8}\mu_{k}\\|\tilde{d}_{k}\\|^{2},\end{array}$ (6.11) where the second inequality follows from the semidefiniteness of $(N_{k})^{T}(\widetilde{Q}^{k})N_{k}+\mu_{k}I$ and the last inequality comes from the relationship that $\\|\tilde{d}_{k}\\|<6\mu_{k}/\lambda$. Combining (6.11) with the fact that $\\|\tilde{d}^{k}\\|=\beta\sqrt{\varepsilon}$ due to $\mu_{k}>0$, we further have $f(x^{k})-f(x^{0})\leq-\frac{1}{8}\sum_{j=0}^{k-1}\mu_{j}\\|\tilde{d}_{j}\\|^{2}\leq-\frac{\lambda}{48}k\big{(}{\beta^{2}\varepsilon}\big{)}^{3/2}$ and hence the interior-point algorithm produces an $\varepsilon$ global minimizer in $O(\varepsilon^{-\frac{3}{2}})$ iterations. In what follows, we pay more attentions to the case where $\mu_{k}\leq\lambda/6\\|\tilde{d}^{k}\\|$ for some $k$. ###### Lemma 6.2 Let $\beta\leq\min\\{\frac{1}{2},\sqrt{2\over\lambda},\frac{3}{(18\sqrt{2}+2)\lambda}\\}$. If there exists some $k$ such that $\mu_{k}\leq\frac{\lambda}{6}\\|\tilde{d}^{k}\\|$, then $x^{k+1}$ is an $\varepsilon$ second-order KKT solution of (6.5). Proof: (i) We firstly show $x^{k+1}$ is an $\varepsilon$ scaled first order KKT solution when $\beta$ is restricted into the special range. From (6.7) and (6.8), it follows that $-\mu_{k}\tilde{d}^{k}=X^{k}({Q}x^{k+1}-c)-\frac{\lambda}{4}\sqrt{X^{k}}\tilde{d}^{k}+\frac{\lambda\sqrt{x^{k}}}{2}-X^{k}A^{T}y^{k},$ which implies that $Qx^{k+1}-c-A^{T}y^{k}=\frac{\lambda}{4}(X^{k})^{-1/2}\tilde{d}^{k}-\frac{\lambda}{2}(x^{k})^{-1/2}-\mu_{k}(X^{k})^{-1}\tilde{d}^{k}.$ Therefore, we have $\begin{array}[]{rl}&\displaystyle\\|X^{k+1}(Qx^{k+1}-c)+\frac{\lambda\sqrt{x^{k+1}}}{2}-X^{k+1}A^{T}y^{k}\\|\\\\[2.84544pt] =&\displaystyle\\|\frac{\lambda\sqrt{x^{k+1}}}{2}-\frac{\lambda}{2}X^{k+1}(x^{k})^{(-1/2)}+\frac{\lambda}{4}X^{k+1}({X^{k}})^{-1/2}\tilde{d}^{k}-\mu_{k}X^{k+1}(X^{k})^{-1}\tilde{d}^{k}\\|\\\\[7.11317pt] \leq&\displaystyle\\|\frac{\lambda\sqrt{x^{k+1}}}{2}-\frac{\lambda}{2}X^{k+1}(x^{k})^{(-1/2)}+\frac{\lambda}{4}X^{k+1}({X^{k}})^{-1/2}\tilde{d}^{k}\\|+\mu_{k}\\|X^{k+1}(X^{k})^{-1}\tilde{d}^{k}\\|\\\\[7.11317pt] \leq&\displaystyle\frac{\lambda}{2}\\|\sqrt{X^{k}}\\|_{\infty}\\|\sqrt{\tilde{d}^{k}+e}-e-\frac{1}{2}\tilde{d}^{k}+\frac{1}{2}(\tilde{d}^{k})^{2}\\|+\mu_{k}\\|\tilde{d}^{k}\\|(1+\\|\tilde{d}^{k}\\|)\end{array}$ (6.12) Since the condition $\mu_{j}>{\lambda\over 6}\\|\tilde{d}^{j}\\|$ holds for $j=0,1,2\ldots,k-1$, by the proof of Lemma 1, we have $f(x^{k})\leq f(x^{0})$, which together with Assumption 1 implies $\\|x^{k}\\|_{\infty}\leq 1$. Moreover, we know from the proof of Lemma 4 in [Bian et al.(2012)] that $\\|\sqrt{\tilde{d}^{k}+e}-e-\frac{1}{2}\tilde{d}^{k}+\frac{1}{2}({\tilde{d}}^{k})^{2}\\|\leq\frac{1}{2}\\|\tilde{d}^{k}\\|^{2}$ and hence $\begin{array}[]{rl}&\displaystyle\\|(X^{k+1})(Qx^{k+1}-c)+\frac{\lambda\sqrt{x^{k+1}}}{2}-X^{k+1}A^{T}y^{k}\\|\\\\[8.5359pt] \leq&\displaystyle\frac{\lambda}{4}\\|\tilde{d}^{k}\\|^{2}+\frac{3}{2}\mu_{k}\\|\tilde{d}^{k}\\|\leq\frac{\lambda}{2}\\|\tilde{d}^{k}\\|^{2}\leq\varepsilon,\end{array}$ which means $x^{k+1}$ is an $\varepsilon$ scaled first-order KKT solution. (ii) Again from (6.7) and (6.8), we know that $X^{k}QX^{k}-\frac{\lambda}{4}\sqrt{X^{k}}+\mu_{k}I$ is positive semidefinite on the null space that $X^{k}A^{T}$. Let $N_{k}$ be the orthogonal basis of this null space and it therefore holds $N_{k}^{T}(X^{k}QX^{k}-\frac{\lambda}{4}\sqrt{X^{k}})N_{k}\succeq-\mu_{k}I\succeq-\frac{\lambda}{6}\beta\sqrt{\varepsilon}I.$ (6.13) Clearly, $N_{k+1}:=(X^{k+1})^{-1}X^{k}N_{k}$ is a basis of the null space of $X^{k+1}A^{T}$. By simple algebraic computation, we can easily obtain that $\begin{array}[]{rl}&\displaystyle N_{k+1}^{T}\left[X^{k+1}QX^{k+1}-\frac{\lambda}{4}\sqrt{X^{k+1}}+\sqrt{\varepsilon}I\right]N_{k+1}\\\\[11.38092pt] =&\displaystyle N_{k}^{T}(X^{k}QX^{k}-{\lambda\over 4}\sqrt{X^{k}})N_{k}+\sqrt{\varepsilon}N_{k}^{T}\left[X_{k+1}^{-2}(X^{k})^{2}\right]N_{k}\\\\[5.69046pt] &\displaystyle+{\lambda\over 4}N_{k}^{T}\sqrt{X^{k}}\big{[}I-(X^{k})^{3/2}(X^{k+1})^{-3/2}\big{]}N_{k}\\\\[11.38092pt] \displaystyle\succeq&-\frac{\lambda}{6}\beta\sqrt{\varepsilon}I+\sqrt{\varepsilon}N_{k}^{T}(I+D_{k})^{-2}N_{K}+{\lambda\over 4}N_{k}^{T}\sqrt{X^{k}}\big{[}I-(I+D_{k})^{-3/2}\big{]}N_{k}\end{array}$ (6.14) where $D_{k}={\rm Diag}(\tilde{d}^{k})$. Since $\\|\tilde{d}^{k}\\|\leq\beta\sqrt{\varepsilon}\leq{1\over 2}<1$, we know $(I+D_{k})^{-2}\succeq(1+\beta\sqrt{\varepsilon})^{-2}I\,\succeq{1\over 4}I$ (6.15) and $I-(I+D_{k})^{-3/2}\succeq\big{[}1-(1-\beta\sqrt{\varepsilon})^{-3/2}\big{]}I.$ (6.16) Moreover, the mean-value theorem applied to the function $x^{-3/2}$ yields that $1-(1-\beta\sqrt{\varepsilon})^{-3/2}=-{3\over 2}\beta\sqrt{\varepsilon}\theta^{-5/2},$ where $\theta$ is in the open interval $(1-\beta\sqrt{\varepsilon},\,1)$. Note that $\beta\sqrt{\varepsilon}\leq{1\over 2}$, then it holds that $1-(1-\beta\sqrt{\varepsilon})^{-3/2}\geq-6\sqrt{2}\beta\sqrt{\varepsilon}.$ (6.17) By substituting (6.15), (6.16) and (6.17) into (6.14), we immediately get that $N_{k+1}^{T}\left[X^{k+1}QX^{k+1}-\frac{\lambda}{4}\sqrt{X^{k+1}}+\sqrt{\varepsilon}I\right]N_{k+1}\succeq({1\over 4}-{3\sqrt{2}\beta\lambda\over 2}-{\lambda\over 6}\beta)\sqrt{\varepsilon}I\succeq 0.$ Thus $x^{k+1}$ is an $\varepsilon$ scaled second-order KKT solution. ##### According to the above two lemmas, we know the proposed second order interior point obtains either an $\varepsilon$ scaled second KKT solution or $\varepsilon$ global minimizer in no more than $O(\varepsilon^{-3/2})$ iterations provided that $\beta_{k}\leq\min\\{\frac{1}{2},\sqrt{2\over\lambda},\frac{3}{(18\sqrt{2}+2)\lambda}\\}$. This completes the proof of this Theorem. ## References * [Bertsimas and Shioda(2009)] Bertsimas, D., R. Shioda. 2009. 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Science in China Series F-Inf Sci. 52 1–9.	arxiv-papers	2013-12-22T07:07:36	2024-09-04T02:49:55.777799	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Caihua Chen, Xindan Li, Caleb Tolman, Suyang Wang, Yinyu Ye", "submitter": "Caihua Chen", "url": "https://arxiv.org/abs/1312.6350" }
1312.6359	1112010 _Mathematics Subject Classification_ : Primary 30D40; Secondary 51K99 # Normality and boundary behavior of arbitrary and meromorphic functions along simple curves and applications Žarko Pavićević Faculty of Natural Sciences and Mathematics, University of Montenegro, Cetinjski put b.b. 81000 Podgorica, Montenegro [email protected] and Marijan Marković Faculty of Natural Sciences and Mathematics, University of Montenegro, Cetinjski put b.b. 81000 Podgorica, Montenegro [email protected] ###### Abstract. We establish the theorems that give necessary and sufficient conditions for an arbitrary function defined in the unit disk of complex plane in order to has boundary values along classes of equivalencies of simple curves. Our results generalize the well–known theorems on asymptotic and angular boundary behavior of meromorphic functions (Lindölf, Lehto–Virtanen, and Seidel–Walsh type theorems). The results are applied to the study of boundary behavior of meromorphic functions along curves using $P-$sequences, as well as in the proof of the uniqueness theorem similar to Šaginjan’s one. Constructed examples of functions show that the results cannot be improved. ###### Key words and phrases: normal function, normal family of functions, angular limits of analytic functions, simple curves, hyperbolic distance, Fréshet distance ###### Contents 1. 1 Introduction 2. 2 Notations 3. 3 Preliminaries 4. 4 Curvilinear boundary behavior of arbitrary functions 5. 5 Normality of meromorphic functions along simple curves 6. 6 Curvilinear boundary values of meromorphic functions 7. 7 Examples 8. 8 Applications ## 1\. Introduction In this paper we study some problems of the Theory of cluster sets, a theory which is developed in the second half of the twentieth century. It is believed that the first result of this theory were obtained by Sohotsky [49], and independently by Cazorati [6] in 1868th, and Weierstrass [57] in 1876th, which is known in literature as the Theorem on essential singularity of analytic functions (see [48] p. 123). Fundamentals of the Theory of cluster sets are presented in monographs [7], [35], [49], and in the more recent survey paper [29]. The main objects of research in this paper is the asymptotic behavior of meromorphic functions along a simple curve ending in a boundary point of the domain of functions. We emphasize that a very productive area of investigation are domains of the hyperbolic type, i.e., domains where one may define the hyperbolic metric. One of the classical results of The theory of cluster sets related to the asymptotic behavior is the theorem of Lindelöf on angular boundary values of analytic functions [28] (or, see [54]). Further interesting results on the boundary behavior of analytic functions along simple curves were obtained by Seidel [46] and Seidel and Walsh [47] (see also [29]). Lehto and Virtanen’s result from [27], which is a transfer of the results of Lindelöf and Seidel and Walsh to the class of normal meromorphic functions in the unit disc, the class usually denoted by $N$, prompted a further intensive research in the Theory of cluster sets. These investigations were also related to the boundary behavior of functions along sequences of points on the one hand, and on the boundary behavior of harmonic, subharmonic, continuous functions, and normal quasiconformal and equimorphic mappings along (non–)tangential simple curves (see References). While most of these papers concern the boundary properties of functions along simple curves which are at the finite Fréshet distance or finite Hausdorff distance (see eg. [8]), in this paper we define a relation of equivalence in the family of all simple curves in the unit disc which terminate in the same point on the boundary, and study the boundary behavior of functions along classes of equivalence. We also offer an example of two simple curves ending in a point of the boundary of the unit disc which belong to the same equivalence class, such that their Fréshet distance is infinite. This the content of Lemma 3.3. Thus, our results in the paper are generalization of some known results. Namely, using the mentioned relation of equivalence we prove the theorem that give necessary and sufficient conditions for an arbitrary function defined in the unit disk to has a curvilinear boundary value (see Theorem 4.1). This theorem is used in proof of Theorem 4.2, which shows that for an arbitrary function in the unit disk holds an analogue of Theorem 1 in [27] concerning the meromorphic functions. As follows from our Theorem 4.1, the normality along simple curves is a necessary condition for the existence of curvilinear boundary value of functions. In Section 5 we study the normality and boundary behavior of meromorphic functions using the $P-$sequences. We emphasis that the $P-$sequences provide necessary and sufficient conditions for meromorphic functions to be normal (see [16], [38], [17], [14]). Further, in Section 6 we prove theorems that give necessary and sufficient conditions in order that a meromorphic function in the unit disk has a curvilinear boundary value (see Theorems 6.1 and 6.2). These theorems are analogous to the theorems 2, 2’, 4 and 5 in Lehto and Virtanen work [27]. While Theorem 2, 2’, 4 and 5 of Lehto and Virtanen concern the class $N$ of normal meromorphic function in the unit disk, the results of Theorems 6.1 and 6.2 are related to the class of normal meromorphic functions along a simple curve ending in a boundary point of the unit disc. These classes are wider than the class $N$; that will be showed by Examples 7.1 and 7.2 in Section 7. Our results are applied in Section 8 in order to derive Theorem 6.3, which shows that the domain along which there is a single boundary value of meromorphic functions in $N$ from Theorems 2, 2’, 4 and 5 in Lehto and Virtanen work [27] can spread in the case of simple curves which are tangent to the boundary of the unit disc. However, one cannot obtain an extension by using the method of Lehto and Virtanen. Finally, our results are used to show the uniqueness theorem of Šaginjan [52] which is related to the class of boundary analytic functions, and it’s generalization to the class $N$ that was obtained by Gavrilov [18]. We prove a similar result for the class of meromorphic functions in the unit disk that are normal along non–tangential simple curves. ## 2\. Notations By $\mathbb{D}$ we will denote the open unit disk $\\{z:\|z\|<1\\}$ in the complex plane $\mathbb{C}$ and by $\Gamma$ the unit circle $\\{z:\|z\|=1\\}$. Let $d_{ph}(z,w)=\left\|\frac{z-w}{1-z\overline{w}}\right\|\quad\text{and}\quad d_{h}(z,w)=\log\frac{1+d_{ph}(z,w)}{1-d_{ph}(z,w)}$ stands for the pseudo–hyperbolic distance and the hyperbolic distance between $z,\,w\in\mathbb{D}$, respectively. It is well known that $d_{h}$ is a metric in the unit disc, and that $(\mathbb{D},d_{h})$ is the Poincaré disc model for the Lobachevsky geometry. Furthermore, denote by $d_{S}(z,w)=\left\\{\begin{array}[]{ll}\frac{2\|z-w\|}{\sqrt{1+\|z\|^{2}}\sqrt{1+\|w\|^{2}}},&\hbox{$z,\,w\in\mathbb{C}$},\\\ \frac{2}{\sqrt{1+\|z\|^{2}}},&\hbox{$z\in\mathbb{C},\,w=\infty$}.\end{array}\right.$ the spherical metric in the extended complex plane $\overline{\mathbb{C}}=\mathbb{C}\cup\\{\infty\\}$ (Riemann sphere). For $r>0$ we denote by ${D}(r)=\\{\|z\|<r\\}$ the standard open disc in $\mathbb{C}$ with centre in $0$ and radius $r$. For $z\in\mathbb{D}$ let $D_{h}(z,r)=\\{w\in\mathbb{D}:D_{h}(z,w)<r\\}$ be a disc in the hyperbolic metric. Let $D_{S}(w,r),\,w\in\overline{\mathbb{C}}$ denote a disc on the Riemann sphere. For $r^{\prime}\in(0,1)$ the set $D_{ph}(z,r^{\prime})=\\{w\in\mathbb{D}:d_{ph}(z,w)<r^{\prime}\\}$ stands for the pseudo–hyperbolic disc with centre in $z$ and pseudo–hyperbolic radius $r^{\prime}$. In a similar manner one introduces the closed discs in these metrics. It is straightforward to show that (1) $\overline{D}_{h}(z,r^{\prime})=\overline{D}_{ph}(z,r)\quad\text{with}\quad r\in[0,1),\,r^{\prime}=\log\frac{1+r}{1-r}\in[0,\infty).$ The group of all Möbius transforms of $\mathbb{D}$ onto itself (conformal automorphisms of the unit disc) will be denoted by $\mathcal{M}$. A function $f:\mathbb{D}\rightarrow\overline{\mathbb{C}}$ is normal in $\mathbb{D}$ if the family $\\{f\circ\varphi:\varphi\in\mathcal{M}\\}$ is a normal family in the sense of Montel, i.e., if any sequence of this family has a subsequence which is convergent in local topology of $\mathbb{D}$ (uniformly on compact subsets of $\mathbb{D}$). All sequences of functions (or numbers) we mean are convergent in above metrics (if they are convergent). Particulary, the uniform convergence on compact subsets of the disc $\mathbb{D}$ of a sequence of functions $\\{f_{n}:\mathbb{D}\rightarrow\overline{\mathbb{C}}:n\in\mathbb{N}\\}$ to a function $f:\mathbb{D}\rightarrow\overline{\mathbb{C}}$ we mean in the metrics of spaces $(\mathbb{D},d_{ph})$ and $(\overline{\mathbb{C}},d_{S})$, or what is the same, in $(\mathbb{D},d_{h})$ and $(\overline{\mathbb{C}},d_{S})$, as follows from (1). For $w\in\mathbb{D}$ let $\varphi_{w}\in\mathcal{M}$ be defined by $\varphi_{w}(z)=\frac{z+w}{1+z\overline{w}}.$ If $f:\mathbb{D}\rightarrow\overline{\mathbb{C}}$ is any function, we will use the notation $f_{w}$ for $f\circ\varphi_{w}:\mathbb{D}\rightarrow\overline{\mathbb{C}}$, where $\varphi_{w}$ is defined above. In the sequel we will consider the following type of family of functions $\\{f_{n}=f\circ\varphi_{n}\\}$, where $\varphi_{n}=\varphi_{w_{n}}$ and $\\{w_{n}\\}$ is a sequence of points in $\mathbb{D}$ such that $\lim_{n\rightarrow\infty}w_{n}=e^{i\theta}\in\Gamma$. The set $C(f,A,e^{i\theta})=\\{w\in\overline{\mathbb{C}}:\text{there exist a sequence}\,\\{z_{n}\\}\subseteq A,\,\lim_{n\rightarrow\infty}z_{n}=e^{i\theta}\in\Gamma\,\text{such that}\,\lim_{n\rightarrow\infty}f(z_{n})=w\\}$ is the cluster set for the function $f:\mathbb{D}\rightarrow\overline{\mathbb{C}}$ in the point $e^{i\theta}$ along the set $A$ whose closure in $\mathbb{D}\cup\Gamma$ contains $e^{i\theta}$. It may be checked that $C(f,A,e^{i\theta})$ is closed. All curves which appear in the text we mean lie in $\mathbb{D}$, are simple and terminate in a point $e^{i\theta}\in\Gamma$. Let $\gamma$ be a such one curve. The set $\Delta_{r}\gamma=\bigcup_{z\in\gamma}\overline{D}_{ph}(z,r),$ where $r\in[0,1)$, is called a curvilinear angle along the curve $\gamma$ with deflection $r$ and with vertex in $e^{i\theta}$. Particulary, for $r=0$ we have $\Delta_{0}\gamma=\gamma$. Regarding (1), we have $\Delta_{r}\gamma=\bigcup_{z\in\gamma}\overline{D}_{ph}(z,r)=\bigcup_{z\in\gamma}\overline{D}_{h}\left(z,r^{\prime}\right)\quad\text{for all}\quad r\in[0,1).$ For the curvilinear angle $\Delta_{r}\gamma$ we will sometimes use the notation $\Delta_{r^{\prime}}\gamma$. Although this is not fully precise, we believe that a misunderstanding will not occur. ###### Example 2.1. If $\gamma$ is the radius of the disc $\mathbb{D}$ with one endpoint in $e^{i\theta}\in\Gamma$, then the curvilinear angle along $\gamma$ with deflection $r$ is the domain bounded by arcs of two hyper–cycle with endpoints in $e^{i\theta}$ and $e^{-i\theta}$ and by the arc of the circle $\|z\|=r$. That curvilinear angle we call a hyperbolic angle. If $h(\theta,\alpha_{1})$ and $h(\theta,\alpha_{2}),\,-\frac{\pi}{2}<\alpha_{1}<\alpha_{2}<\frac{\pi}{2}$, are chords of the disc $\mathbb{D}$ which with the radius $r_{\theta}$ of $\mathbb{D}$ with one endpoint at $e^{i\theta}$ form angles $\alpha_{1}$ and $\alpha_{2}$, then the sub–domain of $\mathbb{D}$ bounded by these chords and the circle $\left\\{z:\|z-e^{i\theta}\|=r\right\\}$ is the Stolz angle with vertex in $e^{i\theta}$. For any Stolz angle in $\mathbb{D}$ with vertex in $e^{i\theta}$ there exists a hyperbolic angle $\Delta_{r}\gamma$ which is contained in it; we have also the converse: any hyperbolic angle $\Delta_{r}\gamma$ contains a Stolz angle in $\mathbb{D}$ with vertex in $e^{i\theta}$. If $\gamma$ is an arc of the horo–cycle $\\{z:\|z-\frac{e^{i\theta}}{2}\|=\frac{1}{2}\\}$ with endpoints $0$ and $e^{i\theta}$, then the curvilinear angle $\Delta_{r}\gamma$ is the domain bounded by arcs of two horo–cycles which contain $e^{i\theta}$. That curvilinear angle $\Delta_{r}\gamma$ we call the horo–cyclic angle (see [13]). At the end of this section we recall the known definition of the Fréshet distance between two curves. For curves $\gamma_{1}$ and $\gamma_{2}$ the Fréshet distance between them is $d_{F}(\gamma_{1},\gamma_{2})=\inf_{\varphi}\sup_{z\in\gamma_{1}}d_{h}(z,\varphi(z));$ the infimum is taken among all homeomorphisms $\varphi:\gamma_{1}\rightarrow\gamma_{2}$. ## 3\. Preliminaries The following lemma is straightforward and therefore we omit a proof. ###### Lemma 3.1. For all $r\in[0,1)$ and $w\in\mathbb{D}$ we have $\overline{D}_{ph}(w,r)=\varphi_{w}(\overline{D}(r)).$ Thus, $\Delta_{r}\gamma=\bigcup_{w\in\gamma}\overline{D}_{ph}(w,r)=\bigcup_{w\in\gamma}\varphi_{w}(\overline{D}(r)).$ ###### Definition 3.1. Let $\gamma_{1}$ and $\gamma_{2}$ be two curves with a same endpoint in $\Gamma$. If $\gamma_{2}\subseteq\Delta_{r}\gamma_{1}=\bigcup_{w\in\gamma_{1}}\overline{D}_{ph}(w,r)$ for some $r\in(0,1)$, we say that the pseudo–hyperbolic distance between $\gamma_{2}$ and $\gamma_{1}$ is less then $r$. If instead of the pseudo–hyperbolic distance we use the hyperbolic distance, then we say that the hyperbolic distance between $\gamma_{2}$ and $\gamma_{1}$ is less then $r^{\prime}=\log\frac{1+r}{1-r}\in(0,\infty)$ (regarding (1)). In this case we will simply say that the distance between curves if finite (see the following lemma); if this is not the case, we say that the distance is infinite. ###### Example 3.1. Hyper–cycles in $\mathbb{D}$ which terminate in a point $e^{i\theta}\in\Gamma$ are simple curves such that the distance between any of them is finite. The same is true for horo–cycles in $\mathbb{D}$ which terminate in $e^{i\theta}$. However, the distance between any hyper–cycle and any horo–cycle both terminating in $e^{i\theta}$ is infinite. The following definition introduces a relation $\sim$ in the family of all curves in $\mathbb{D}$ which terminate in a same point in $\Gamma$. ###### Definition 3.2. Let $\gamma_{1}$ and $\gamma_{2}$ be curves in the disc $\mathbb{D}$ ending in $e^{i\theta}\in\Gamma$. We write $\gamma_{1}\sim\gamma_{2}$ if there exist $r\in(0,1)$ such that $\gamma_{1}\subseteq\Delta_{r}\gamma_{2}$, i.e., if the distance between $\gamma_{1}$ and $\gamma_{2}$ is finite. ###### Lemma 3.2. The relation $\sim$ is an relation of equivalence in the family of all curves in the disc $\mathbb{D}$ with the same endpoint in $\Gamma$. The class of equivalence for a curve $\gamma$ will be denoted by $[\gamma]$. In order to establish the symmetry property of $\sim$ we will use the following assertion. Let the hyperbolic between $\gamma_{1}$ and $\gamma_{2}$ be less then $r\in(0,\infty)$, then we have: For all $r_{1}\in(0,\infty)$ there exist $r_{2}\in(0,\infty)$ such that $\Delta_{r_{1}}\gamma_{1}\subseteq\Delta_{r_{2}}\gamma_{2}$, and for all $r_{2}\in(0,\infty)$ there exist $r_{1}\in(0,\infty)$ such that $\Delta_{r_{2}}\gamma_{2}\subseteq\Delta_{r_{1}}\gamma_{1}$. ###### Proof. We will firstly prove the assertion. We will proof the first statement, since the second follows immediately from the first one. From Definition 3.1 it follows $\gamma_{1}\subseteq\Delta_{r}\gamma_{2}=\bigcup_{w\in\gamma_{2}}\overline{D}_{h}(w,r)$. Let $r_{2}=r_{1}+r$. We will show $\Delta_{r_{1}}\gamma_{1}\subseteq\Delta_{r_{2}}\gamma_{2}$. Let $z\in\Delta_{r_{1}}\gamma_{1}$. There exists $w_{1}\in\gamma_{1}$ such that $z\in\overline{D}_{h}(w_{1},\gamma_{1})$. Since $\gamma_{1}\subseteq\Delta_{r}\gamma_{2}$, there exist $w_{0}\in\gamma_{2}$ such that $w_{1}\in D_{h}(w_{0},r)$. Using the triangle inequality, we obtain $d_{h}(z,w_{0})\leq d_{h}(z,w_{1})+d_{h}(w_{1},w_{0})<r_{1}+r=r_{2};$ thus $z\in D_{h}(w_{0},r_{2})$, i.e., $z\in\Delta_{r_{2}}\gamma_{2}$. We have proved $\Delta_{r_{1}}\gamma_{1}\subseteq\Delta_{r_{2}}\gamma_{2}$. Let us now establish the that $\sim$ is a relation of equivalence. It is clear that $\gamma\subseteq\Delta_{r}\gamma$ for all $r\in(0,\infty)$, what means that $\sim$ is reflexive. If $\gamma_{1}\sim\gamma_{2}$, then $\gamma_{1}\subseteq\Delta_{r}\gamma_{2}$; from the assertion it follows that there exist $r^{\prime}$ such that $\gamma_{2}\subseteq\Delta_{r^{\prime}}\gamma_{1}$, i.e., $\gamma_{2}\sim\gamma_{1}$. Thus, from $\gamma_{1}\sim\gamma_{2}$ it follows $\gamma_{2}\sim\gamma_{1}$. We have proved that $\sim$ is a symmetry relation. It remains to establish the transitivity of $\sim$. Let $\gamma_{1}\sim\gamma_{2}$ and $\gamma_{2}\sim\gamma_{3}$. It follows $\gamma_{2}\subseteq\Delta_{r_{1}}\gamma_{1}$ and $\gamma_{3}\subseteq\Delta_{r_{2}}\gamma_{2}$ for some $r_{1},\,r_{2}\in(0,\infty)$. As in the assertion, one may prove $\gamma_{3}\subseteq\Delta_{s}\gamma_{1}$ for $s=r_{1}+r_{2}$. Thus, $\gamma_{1}\sim\gamma_{3}$. ∎ ###### Remark 3.1. For any point $e^{i\theta}\in\Gamma$ there exist infinity many classes of equivalence for the relation $\sim$. Namely, if curves $\gamma_{1}$ and $\gamma$ have a different order of contact on the unit circle $\Gamma$ in the point $e^{i\theta}$, then $[\gamma_{1}]\neq[\gamma_{2}]$. For example, all curves in the $\mathbb{D}$ which terminate in $e^{i\theta}$ that are not tangent to $\Gamma$ in $e^{i\theta}$ belong to the same class of equivalences. All horo–cycles tangent on the boundary $\Gamma$ in the point $e^{i\theta}$ belong to the same class of equivalences, etc. ###### Lemma 3.3. If the Fréshet distance between curves $\gamma_{1},\,\gamma_{2}\subseteq\mathbb{D}$ ending in the same point $e^{i\theta}\in\Gamma$ is finite, then $\gamma_{1}\sim\gamma_{2}$. The converse does not hold. ###### Proof. If the Fréshet distance between $\gamma_{1}$ and $\gamma_{2}$ is finite, then from the definition of the Fréshet distance and from Definition 3.2 immediately follows that $\gamma_{1}\sim\gamma_{2}$. Namely, if $d_{F}(\gamma_{1},\gamma_{2})<r$ for some $r\in(0,\infty)$, then for some homeomorphism $\varphi:\gamma_{1}\rightarrow\gamma_{2}$ holds $d(z,\varphi(z))<r$ for all $z\in\gamma_{1}$. This clearly implies $\gamma_{2}\subseteq\Delta_{r}\gamma_{1}$, i.e., $d(\gamma_{1},\gamma_{2})\leq r<\infty$. Thus, in view of Definition 3.2 we have $\gamma_{1}\sim\gamma_{2}$. In order to prove the second statement of this lemma, we will construct an example of two curves $\gamma_{1}$ and $\gamma_{2}$ such that $\gamma_{1}\sim\gamma_{2}$, but the Fréshet distance between $\gamma_{1}$ and $\gamma_{2}$ is infinite. Let $\gamma_{1}$ be a radius od the unit disc $\mathbb{D}$ with one endpoint in $e^{i\theta}$. We will show that for any $r\in(0,\infty)$ there exist a curve $\gamma_{2}\subseteq\Delta_{r}\gamma_{1}$ that the Fréshet distance between $\gamma_{1}$ and $\gamma_{2}$ is not finite. Construction of $\gamma_{2}$. Let us chose points $z_{1},\,w_{1},\,z_{2},\,w_{2},...,z_{n},\,w_{n},...\in\gamma_{1}$ such that $d_{h}(z_{n},z_{n+1})\rightarrow\infty$ as $n\rightarrow\infty,\,d_{h}(z_{n},w_{n+1})\leq 1$ for all $n\geq 1$, and that the order of crossing of $\gamma_{1}$ thought the preceding points is as they appear in the sequence. Regarding the way we selected points $z_{1},\,w_{1},\,z_{2},\,w_{2},...,z_{n},\,w_{n},...$ it follows that $d_{h}(z_{n+1},w_{n})\rightarrow\infty$ as $n\rightarrow\infty$. For $\gamma_{2}$ we will take any curve in $\Delta_{r}\gamma_{1}$ which contains the preceding points in the following order: $z_{1},\,z_{2},\,w_{1},\,z_{3},\,w_{2},\,z_{4},\dots,\,z_{n},\,w_{n-1},\ z_{n+1},\dots$ We will show now that the Fréshet distance between curves $\gamma_{1}$ and $\gamma_{2}$ is not finite. Assume that $\varphi$ is an arbitrary homeomorphism between $\gamma_{2}$ and $\gamma_{1}$. Let $z_{1},\,z_{2},\,z_{3},\dots,\,z_{n},...$ ($w_{1},\,w_{2},\,w_{3},\dots,w_{n},\dots$ which belong to $\gamma_{2}$) be corespondent to $t_{1},\,t_{2},\,t_{3},\dots,t_{n},\dots$ (i.e., $s_{1},\,s_{2},\,s_{3},\dots,s_{n},\dots$ in $\gamma_{1}$) via the homeomorphism $\varphi$. The schedule of crossing of $\gamma_{1}$ thought the preceding points is: $t_{1},\,t_{2},\,s_{1},\,t_{3},\,s_{2},\,t_{4},\dots,t_{n},\ s_{n-1},\,t_{n+1},\,s_{n},\dots,$ i.e., the curve across the point $t_{n+1}$ and then $s_{n}$. Since $d_{h}(z_{n+1},w_{n})\rightarrow\infty$ as $n\rightarrow\infty$, for any $D>0$ there exist an integer $n_{0}\geq 1$ such that for every $n\geq n_{0}$ holds $d_{h}(z_{n+1},w_{n})>2D+1$. However, if the Fréshet distance between $\gamma_{1}$ and $\gamma_{2}$ would be bounded by $D$, then we will have $d_{h}(z_{n+1},\varphi(z_{n+1}))=d_{h}(z_{n+1},t_{n+1})\leq D$ and $d_{h}(w_{n},\varphi(w_{n}))=d_{h}(w_{n},s_{n})\leq D$ for every $n\geq n_{0}$. In view of the relation $d_{h}(z_{n+1},w_{n})>2D+1$ it follows that the curve $\gamma_{1}$ first across the point $s_{n}$ and then $t_{n+1}$. This is the contradiction. ∎ ## 4\. Curvilinear boundary behavior of arbitrary functions ###### Theorem 4.1. Let $f:\mathbb{D}\rightarrow\overline{\mathbb{C}}$ be an arbitrary function, let $\\{w_{n}\\}\subseteq\mathbb{D}$ be a sequence such that $\lim_{n\rightarrow\infty}w_{n}=e^{i\theta}$, and let $c\in\overline{\mathbb{C}}$. The following two conditions are equivalent: 1. (1) the sequence $\\{f_{n}=f\circ\varphi_{n}\\}$, where $\varphi_{n}=\varphi_{w_{n}}$, is convergent in the local topology of $\mathbb{D}$ to the constant function $c$; 2. (2) for any compact subset $K\subset\mathbb{D}$ holds $C\left(f,\bigcup_{n\in\mathbb{N}}\varphi_{n}(K),e^{i\theta}\right)=\\{c\\}.$ ###### Proof. (1) implies (2). Let $K$ be any compact subset of $\mathbb{D}$. Since $\lim_{n\rightarrow\infty}w_{n}=e^{i\theta}$, it follows that $e^{i\theta}$ is the single point od adherence in $\Gamma$ for the set $\bigcup_{n\in\mathbb{N}}\varphi_{w_{n}}(K)$. Since $\\{f_{n}=f\circ\varphi_{w_{n}}\\}$ is uniformly convergent to the constant $c$ on the compact $K$, for every $\varepsilon>0$ we have $d_{S}(f\circ\varphi_{w_{n}}(z),c)<\varepsilon$ for all $z\in K$ if $n\geq n_{0}$, where $n_{0}$ is big enough. Thus, $f\circ\varphi_{w_{n}}(K)\subseteq D_{S}(c,\varepsilon)$ for all $n\geq n_{0}$. It follows $f(\bigcup_{n\geq n_{0}}\varphi_{w_{n}}(K))\subseteq D_{S}(c,\varepsilon)$. In other words, $d_{S}(f(z),c)<\varepsilon$ if $z\in\bigcup_{n\geq n_{0}}\varphi_{w_{n}}(K)$. Now, let $\\{z_{k}\\}\subseteq\bigcup_{n\geq n_{0}}\varphi_{w_{n}}(K)$ be a sequence satisfying $\lim_{k\rightarrow\infty}z_{k}=e^{i\theta}$. We will prove that $\lim_{k\rightarrow\infty}f(z_{k})=c$. For every $k\in\mathbb{N}$ there exist $n_{k}\in\mathbb{N}$ such that $z_{k}\in\varphi_{n_{k}}(K)$. Since $\lim_{k\rightarrow\infty}z_{k}=e^{i\theta}$, we have $n_{k}\rightarrow\infty$ as $k\rightarrow\infty$. If $k$ is big enough, $k\geq k_{0}$, then $n_{k}\geq n_{0}$ and we have $\\{z_{k}:k\geq k_{0}\\}\subseteq\bigcup_{n\geq n_{0}}\varphi_{n}(K)$. Thus, $d_{S}(f(z_{k}),c)<\varepsilon$. In other words $\lim_{k\rightarrow\infty}f(z_{k})=c$. (2) implies (1). Let us prove the contraposition, that is the negation of (1) implies the negation of (2). If the sequence of functions $\\{f_{n}\\}$ does not converge to $c$ in the local topology of $\mathbb{D}$ to $c$, then there exist a compact set $K\subset\mathbb{D}$, a positive number $\varepsilon_{0}$, a subsequence $\\{f_{n_{k}}\\}$, and a sequence $\\{z_{k}\\}\subseteq K$ such that $d_{S}(f_{n_{k}}(z_{k}),c)=d_{S}(f\circ\varphi_{n_{k}}(z_{k}),c)\geq\varepsilon_{0}$ for all $k\in\mathbb{N}$. Denote $u_{k}=\varphi_{n_{k}}(z_{k})$. Then we have $\lim_{k\rightarrow\infty}u_{k}=e^{i\theta},\ \\{u_{k}\\}\subseteq\bigcup_{n\in\mathbb{N}}\varphi_{n}(K)$, and $d_{S}(f(u_{k}),c)\geq\varepsilon_{0}$. This means $C(f,\bigcup_{n\in\mathbb{N}}\varphi_{w_{n}}(K),e^{i\theta})\not\equiv\\{c\\}$, what is negation of (2). ∎ ###### Definition 4.1. A function $f:\mathbb{D}\rightarrow\overline{\mathbb{C}}$ has the $\Delta_{\gamma}-$boundary value $c\in\overline{\mathbb{C}}$ along the curve $\gamma$ which terminates in $e^{i\theta}$ if $C(f,\Delta_{r}\gamma,e^{i\theta})=\\{c\\}$ for all $r\in(0,1)$, i.e., $\bigcup_{r\in(0,1)}C(f,\Delta_{r}\gamma,e^{i\theta})=\\{c\\}.$ ###### Remark 4.1. If a curve $\gamma$ lies in some Stolz angle with vertex in a point $e^{i\theta}\in\Gamma$, then $\Delta_{\gamma}-$boundary value is the same as the ordinary angular boundary value of $f$ in $e^{i\theta}$. In this case the point $e^{i\theta}$ is the Fatou point for the function $f$. If a curve $\gamma\subseteq\mathbb{D}$ is tangent to $\Gamma$ in $e^{i\theta}$ and the order of contact is $1$, t hen $\Delta_{\gamma}-$boundary value for $f$ is the horo–cycle boundary value for $f$ in $e^{i\theta}$. ###### Theorem 4.2. Let $f:\mathbb{D}\rightarrow\overline{\mathbb{C}}$ be any function in the unit disc and let a simple curve $\gamma\subseteq\mathbb{D}$ terminates in a point $e^{i\theta}\in\Gamma$. The following three conditions are equivalent: 1. (1) there exist $\Delta_{\gamma}-$boundary value equal to $c\in\overline{\mathbb{C}}$ in the point $e^{i\theta}$, i.e., $\bigcup_{r\in(0,1)}C(f,\Delta_{r}\gamma,e^{i\theta})=\\{c\\};$ 2. (2) for any sequence $\\{w_{n}\\}\subseteq\gamma$ satisfying $\lim_{n\rightarrow\infty}w_{n}=e^{i\theta}$ the sequence $\\{f_{n}=f\circ\varphi_{w_{n}}\\}$ converges to the constant $c$ in the local topology of $\mathbb{D}$; 3. (3) for any curve $\gamma_{1}\thicksim\gamma$ holds $\lim_{\gamma_{1}\ni z\rightarrow e^{i\theta}}f(z)=c$. Moreover, if there exists $\Delta_{\gamma}-$boundary value for $f$, then it does not depend on the choice of a curve in the class $[\gamma]$. ###### Proof. It follows from Theorem 4.1 that (1) $\Leftrightarrow$ (2). It is clear that (1) implies (3). Let us now prove that (3) implies (1). If (3) holds then $\lim_{\gamma_{1}\ni z\rightarrow e^{i\theta}}f(z)=c$; suppose that (1) is not true. Then for some $r\in(0,1)$ we have $C(f,\Delta_{r}\gamma,e^{i\theta})\not\equiv\\{c\\}$. This means there exist a sequence $\\{z_{n}\\}\subseteq\Delta_{r}\gamma,\ \lim_{n\rightarrow\infty}z_{n}=e^{i\theta}$ such that $\lim_{n\rightarrow\infty}f(z_{n})=a\neq c$ or the previous boundary value does not exist. Points of the sequence $\\{z_{n}\\}$ connect with a curve $\gamma_{1}$ (in any way) such that $\gamma_{1}\subseteq\Delta_{r}\gamma$. Now we have $\lim_{\gamma_{1}\ni z\rightarrow e^{i\theta}}f(z)=c$ or this limit does not exist. Since $\gamma_{1}\thicksim\gamma$, from (3) it follows that our assumption is not correct. Thus, for every $r\in(0,1)$ we have $C(f,\Delta_{r}\gamma,e^{i\theta})=\\{c\\}$. ∎ ###### Definition 4.2. Let $f:\mathbb{D}\rightarrow\overline{\mathbb{C}}$ be any function. If for every curve $\gamma_{1}\in[\gamma]$ we have $\lim_{\gamma_{1}\ni z\rightarrow e^{i\theta}}f(z)=c\in\overline{\mathbb{C}}$, then we say that $c$ is $[\gamma]-$boundary value for $f$, i.e., the boundary value of $f$ along the class $[\gamma]$. From Theorem 4.2 we have ###### Corollary 4.1. Let $f:\mathbb{D}\rightarrow\overline{\mathbb{C}}$ be an arbitrary function and let $\gamma$ be a simple curve which terminates in $\Gamma$. Then the $\Delta_{\gamma}-$boundary value for $f$ exists if and only if there exists $[\gamma]-$boundary value for $f$ and they coincides. ###### Remark 4.2. Since we have infinity many classes of equivalences in the family of curves which terminate in a point $e^{i\theta}$, we can speak about infinity many $\Delta_{\gamma}-$boundary values for $f$ in the point $e^{i\theta}$. ###### Definition 4.3. We say that a function $f:\mathbb{D}\rightarrow\overline{\mathbb{C}}$ is normal along a curve $\gamma$ in $D(r)$ where $r\in(0,1)$ if the family $\\{f_{w}=f\circ\varphi_{w}:w\in\gamma\\}$ is normal in $D(r)$ in the sense of Montel. ###### Definition 4.4. A function $f:\mathbb{D}\rightarrow\overline{\mathbb{C}}$ is normal along a simple curve $\gamma\subseteq\mathbb{D}$ if the family $\\{f_{w}=f\circ\varphi_{w}:w\in\gamma\\}$ is normal in the disc $\mathbb{D}$ in the sense of Montel. ###### Corollary 4.2. If $f:\mathbb{D}\rightarrow\overline{\mathbb{C}}$ has $\Delta_{\gamma}-$boundary value, then $f$ is a normal function along the curve $\gamma$. ###### Proof. If there exists $\Delta_{\gamma}-$boundary value of $f$ equal to $c$, then according to the Theorem 4.2 any subsequence of the family $\\{f_{w}=f\circ\varphi_{w}:w\in\gamma\\}$ is convergent to the constant function $c$. Thus, this family is normal. ∎ ###### Remark 4.3. According to Corollary 4.2 necessary condition for the existence of $\Delta_{\gamma}-$boundary value of a function $f$ in a point $e^{i\theta}\in\Gamma$ is normality of $f$ along the curve $\gamma$. However, normality of the family $\\{f_{w}=f\circ\varphi_{w}:w\in\gamma\\}$ in $\mathbb{D}$ is not a sufficient condition. This shows the example of the meromorphic function from [36], which does not have the radial boundary value in any point of $\Gamma$ and thus it does not have the angular boundary value. This also follows from the example of an elliptic modular function which is a normal analytic function in $\mathbb{D}$ but has the radial boundary value (and thus) only in a countable subset of $\Gamma$. ###### Remark 4.4. Nosiro [36] considered normal meromorphic functions in the disc $\mathbb{D}$ of the first order: a meromorphic function $f$ in the disc $\mathbb{D}$ is normal function of the first order if the family $\\{f\circ\varphi_{w}:w\in\mathbb{D}\\}$ is normal $\mathbb{D}$ and if any boundary function of this family is not a constant. Nosiro proved that normal meromorphic functions of the first order does not poses an angular boundary value. This result follows from from our Theorem 4.1. Our theorem also shows that a normal meromorphic function of the first order does not have $\Delta_{\gamma}-$boundary value in any point of $\Gamma$ for any curve $\gamma$ in disc $\mathbb{D}$ which terminates in $e^{i\theta}\in\Gamma$. Since Theorem 4.1 holds for any function in $\mathbb{D}$ and for $\Delta_{\gamma}-$boundary values, our theorem is a generalization of the result Nosiro. ###### Remark 4.5. Bagemihl and Seidel [4] constructed an analytic function (see Example 2 there) which proves that in Theorem 1 in [4] the condition of normality of the function in $\mathbb{D}$ in order to has a boundary value cannot be removed. Corollary 4.2 also shows that. Corollary 4.2 is a generalization of the result of Bagemihl and Seidel for any function in $\mathbb{D}$ and curvilinear boundary behavior. ###### Remark 4.6. Theorem 4.2 and Corollary 4.2 also show that for the existence of angular boundary values of functions in a point of $\Gamma$ (meromorphic, analytic, harmonic, etc.) one need not assume their normality in the disc $\mathbb{D}$, i.e., it is not necessary to assume that a function is normal with respect to the Möbius group of conformal automorphisms of $\mathbb{D}$ (see [27]), or one need not to assume the condition of normality with respect the hyperbolic or parabolic subgroups or semigroups of the Möbius group (see [19], [30] and [40]). It is enough to assume the condition of normality of the family $\\{f_{w}=f\circ\varphi_{w}:w\in\gamma\\}$ in $\mathbb{D}$, i.e., that $f$ is normal function along the curve $\gamma$ which is not tangent to $\Gamma$. The following theorem is generalization of Theorem 1 in Lehto and Virtanen work [27] for an arbitrary function in the disc $\mathbb{D}$ and for any $\Delta_{\gamma}-$boundary limit. ###### Theorem 4.3. Let $f:\mathbb{D}\rightarrow\overline{\mathbb{C}}$ be any function in the disc $\mathbb{D}$ and let $\gamma\subseteq\mathbb{D}$ be a curve which terminates in $e^{i\theta}\in\Gamma$. Suppose $\lim_{\gamma\ni z\rightarrow e^{i\theta}}f(z)=c\in\overline{\mathbb{C}}$ and assume that $f$ does not have $\Delta_{\gamma}-$boundary limit. Then for every $\varepsilon>0$ there exist two curves $\gamma_{1},\,\gamma_{2}\in[\gamma]$ such that the (pseudo–)hyperbolic distance between $\gamma_{2}$ and $\gamma_{1}$ is less then $\varepsilon$ and such that along $\gamma_{1}$ the function $f$ has the asymptotic boundary value $c$, and along $\gamma_{2}$ does not. ###### Proof. In the proof we will use the hyperbolic metric $d_{h}$. Since $\lim_{\gamma\ni w\rightarrow e^{i\theta}}f(w)=c$ and since $f$ does not have $\Delta_{\gamma}-$boundary limit, from the first part of Theorem 4.1 it follows that $C(f,\Delta_{r_{0}}\gamma,e^{i\theta})\not\equiv\\{c\\}$ for some $r_{0}\in(0,\infty)$. This means that the set $\\{r:C(f,\Delta_{r}\gamma,e^{i\theta})\equiv\\{c\\}:0\leq r<\infty\\}$ is bounded from above by $r_{0}$. Thus, there exist $r_{1}=\sup\\{r:C(f,\Delta_{r}\gamma,e^{i\theta})\equiv\\{c\\}:0\leq r<\infty\\}.$ Let $\varepsilon$ be any positive number. We will consider the following two cases. The case $r_{1}=0$. Then $\gamma=\gamma_{1}$ and according to Theorem 4.2 in the curvilinear angle $\Delta_{\frac{\varepsilon}{2}}\gamma$ there exist a curve $\gamma_{2}$ along which function $f$ does not poses asymptotic value $c$. In view of Definition 3.1 the hyperbolic distance between curves $\gamma_{2}$ and $\gamma_{1}$ is less then $\varepsilon$. The case $r_{1}\in(0,\infty)$. We will denote $r_{1}$ with $r$. Denote by $\gamma_{r}^{+}$ the part of the boundary of curvilinear angle $\Delta_{r}\gamma$ which is above the curve $\gamma$, and with $\gamma_{r}^{-}$ the part of the boundary of $\Delta_{r}\gamma$ which is below the curve $\gamma$. For every $w^{\prime}\in\gamma_{r}^{+}$ there exist $w\in\gamma$ such that $d_{h}(w,w^{\prime})=r$. On the circle (in the metric $d_{h}$) which contains $w$ and $w^{\prime}$ and is orthogonal on $\Gamma$, take the points $u$ which are between $w$ and $w^{\prime}$ and which satisfy $d_{h}(w,u)=r-\frac{\varepsilon}{4}$. Points $u$ make the part of boundary of curvilinear angle $\Delta_{\frac{\varepsilon}{4}}\gamma$ which is on the same side of the curve $\gamma$ as $\gamma_{r}^{+}$. That part of the boundary produces a curve which will be denoted by $\lambda^{\prime}$. Since $\lambda^{\prime}\subseteq\Delta_{r}\gamma$, along $\lambda^{\prime}$ there exists asymptotic value of $f$ equals to $c$. Consider now the curvilinear angle $\Delta_{\frac{\epsilon}{2}}\lambda^{\prime}$. Denote with $\Delta^{+}_{\frac{\epsilon}{2}}\lambda^{\prime}$ the sub–domain of curvilinear angle $\Delta_{\frac{\epsilon}{2}}\lambda^{\prime}$ which is bounded by the curves $\gamma_{r}^{+}$ and ${\lambda^{\prime}}^{+}_{\frac{\varepsilon}{2}}$. Let $r_{2}\in(r_{1},\infty)$. With $(\gamma^{+}_{r_{1}},\gamma^{+}_{r_{2}}]$ denote the domain $\Delta_{r_{2}}^{+}\gamma\setminus\Delta_{r_{1}}^{+}\gamma$, and with $(\gamma^{-}_{r_{1}},\gamma^{-}_{r_{2}}]$ the domain $\Delta_{r_{2}}^{-}\gamma\setminus\Delta_{r_{1}}^{-}\gamma$. In view of the preceding notations, it is not hard to see that $\Delta^{+}_{\frac{\varepsilon}{2}}\lambda^{\prime}\subseteq(\gamma^{+}_{r},\gamma^{+}_{r+\frac{\varepsilon}{4}}]$, what is obvious from the geometric interpretation of these sets. Similarly, $(\gamma^{+}_{r},\gamma^{+}_{r+\frac{\varepsilon}{4}}]\subseteq\Delta^{+}_{\frac{\varepsilon}{2}}\lambda^{\prime}$. Namely, if $z\in(\gamma^{+}_{r},\gamma^{+}_{r+\frac{\varepsilon}{4}}]$, then there exists a disc $D_{h}(w,r+\frac{\epsilon}{4}),\ w\in\gamma$ such that $z\in D_{h}(w,r+\frac{\varepsilon}{4})$. Let $u$ be a point which is in the intersection of the curve $\lambda^{\prime}$ and the hyperbolic half–radius of the disc $D_{h}(w,r+\frac{\varepsilon}{4})$ which contains $z$. Then $z\in D_{h}(u,\frac{\varepsilon}{2})$. Thus $z\in\Delta^{+}_{\frac{\varepsilon}{2}}\lambda^{\prime}$, what implies $(\gamma^{+}_{r},\gamma^{+}_{r+\frac{\varepsilon}{4}}]\subseteq\Delta^{+}_{\frac{\varepsilon}{2}}\lambda^{\prime}$. It follows $\Delta^{+}_{\frac{\varepsilon}{2}}\lambda^{\prime}\subseteq(\gamma^{+}_{r},\gamma^{+}_{r+\frac{\varepsilon}{4}}]$. All we done for $\gamma^{+}_{r}$ may be done also for $\gamma^{-}_{r}$. The resulting curve will be denoted by $\lambda^{\prime\prime}$. In some of the sub–domains $(\gamma^{+}_{r},\gamma^{+}_{r+\frac{\varepsilon}{4}}]$ and $(\gamma^{-}_{r},\gamma^{-}_{r+\frac{\varepsilon}{4}}]$ there exist a sequence of points along which the function $f$ does not have the boundary value, or if there exist, then it is not equals to $c$. This sequence may be connected by a curve $\gamma_{2}$ which lies in the same sub–domain as the sequence. Now we may take $\gamma_{1}=\lambda^{\prime}$ or $\gamma_{1}=\lambda^{\prime\prime}$ what depends on which sub–domain contains the curve $\gamma_{2}$. It is clear that the hyperbolic distance between the curves $\gamma_{2}$ and $\gamma_{1}$ is less then $\varepsilon$. ∎ ###### Theorem 4.4. Let $f:\mathbb{D}\rightarrow\overline{\mathbb{C}}$ be any function in $\mathbb{D}$ and let a curve $\gamma\subseteq\mathbb{D}$ terminates in a point $e^{i\theta}\in\Gamma$. If $C(f,\Delta_{r}\gamma,e^{i\theta})=\\{c\\}$ for some $r\in(0,\infty)$ and if the function $f$ does not have $\Delta_{\gamma}-$boundary value in the point $e^{i\theta}$, then there exist $\gamma_{1}\in[\gamma]$ such that for every $\varepsilon>0$ there exist a curve $\gamma_{\varepsilon}$ at the distance less then $\varepsilon$ from the curve $\gamma_{1}$, such that along one curve $f$ has the asymptotic boundary value $c$, and along the other does not. ###### Proof. That this theorem holds one may see from the proof of Theorem 4.3. One may also chose $\gamma_{1}=\lambda^{\prime}$ and $\gamma_{\varepsilon}=\lambda^{\prime\prime}$. ∎ ## 5\. Normality of meromorphic functions along simple curves With $N$ we denote the class of all normal meromorphic functions in the disk $\mathbb{D}$ (for properties of this class we refer to [27], [29], [34], [45], [4], [16], [14], [26], [55], [32]). For a meromorphic function $f:\mathbb{D}\rightarrow\overline{\mathbb{C}}$ we denote with $f^{\sharp}(z)=\frac{\|f^{\prime}(z)\|}{1+\|f(z)\|^{2}},\quad z\in\mathbb{D}$ the spherical derivate for $f$. The function $f^{\sharp}:\mathbb{D}\rightarrow\mathbb{R}$ is continuous in $\mathbb{D}$. The spherical derivate $f^{\sharp}$ may be used to define the spherical distance between points in the target domain of $f:\mathbb{D}\rightarrow\overline{\mathbb{C}}$. Ostrowski [37] was the first who used the spherical distance, i.e., the spherical derivate in the consideration related with meromorphic functions. Lehto and Virtanen [27] used the Marty criterium (see [22]) in order to derive that a meromorphic function belongs to the class $N$ if and only if $\sup_{z\in\mathbb{D}}(1-\|z\|^{2})f^{\sharp}(z)<\infty.$ By using the Marty criterium for normality of a family of meromorphic functions it is a routine to prove the following ###### Theorem 5.1. Let a curve $\gamma\subseteq\mathbb{D}$ has one endpoint in $e^{i\theta}\in\Gamma$ and let $f:\mathbb{D}\rightarrow\overline{\mathbb{C}}$ be a meromorphic function. For every $r\in(0,1)$ the following conditions are equivalent: 1. (1) $f$ is normal in $D(r)$ along the curve $\gamma$; 2. (2) $\sup_{z\in\Delta_{r}\gamma}(1-\|z\|^{2})f^{\sharp}(z)<\infty.$ From Theorem 5.1 we have ###### Theorem 5.2 (see Theorem 1 in [53]). Let a curve $\gamma\subseteq\mathbb{D}$ terminates in $e^{i\theta}\in\Gamma$ and let $f:\mathbb{D}\rightarrow\overline{\mathbb{C}}$ be a meromorphic function. The following two conditions are equivalent: 1. (1) $f$ is normal in $\mathbb{D}$ along $\gamma$; 2. (2) $\sup_{z\in\Delta_{r}\gamma}(1-\|z\|^{2})f^{\sharp}(z)<\infty$ for all $r\in(0,1)$. ###### Theorem 5.3. A meromorphic function $f:\mathbb{D}\rightarrow\overline{\mathbb{C}}$ is normal along a curve $\gamma$ if and only if it is normal along any curve $\gamma_{1}\in[\gamma]$. ###### Proof. This theorem follows from Theorem 5.2 and Lemma 3.2. ∎ Theorem 5.3 gives an opportunity to introduce a notation of normality of meromorphic functions along classes of simple curves. ###### Definition 5.1. A meromorphic function $f:\mathbb{D}\rightarrow\overline{\mathbb{C}}$ is normal along a class $[\gamma]$ if it is normal along any curve $\gamma_{1}\in[\gamma]$. ###### Remark 5.1. From Theorems 5.2 and 5.3 it follows that the class of normal meromorphic functions in the disc $\mathbb{D}$ investigated by Lehto and Virtanen [27], i.e., functions from $N$, is normal along any simple curve $\gamma$ in the disc $\mathbb{D}$, i.e., it is normal along any class $[\gamma]$ in the disc $\mathbb{D}$. If a curve $\gamma$ lies in some Stolz angle of $\mathbb{D}$ with vertex at $e^{i\theta}$, then the notation of normality along that curve, i.e., along the class $[\gamma]$ of a function $f:\mathbb{D}\rightarrow\overline{\mathbb{C}}$ is equivalent with normality along the hyperbolic semigroup of all Möbius transformations of $\mathbb{D}$ with an attractive point $e^{i\theta}$ ($\pm e^{i\theta}$ are attractors of elements of the semigroup). If a curve belongs to the sub–domain of the disc $\mathbb{D}$ which is bounded by two horocycles which contain $e^{i\theta}$, then the normality of meromorphic functions along the curve $\gamma$, i.e., along the class $[\gamma]$ is the same as the normality along the parabolic semigroup of all Möbius transformations of the disc $\mathbb{D}$ with only one attractive point $e^{i\theta}$. Normality and boundary behavior of meromorphic functions along the hyperbolic and parabolic semigroup and hyperbolic and parabolic subgroup are considered in [19], [30], [40], [16] and [38]. In [16] and [17] Gavrilov considered the normality and boundary behavior of meromorphic function using the notation of $P-$sequences. ###### Definition 5.2 (see [16]). A sequence $\\{z_{n}\\}\subseteq\mathbb{D},\,\lim_{n\rightarrow\infty}\|z_{n}\|=1$ is a $P-$sequence for a meromorphic function $f:\mathbb{D}\rightarrow\overline{\mathbb{C}}$ if for any subsequence $\\{z_{n_{k}}\\}$ and $\varepsilon\in(0,1)$ the function $f$ achieves in the set $\bigcup_{k\in\mathbb{N}}D_{h}(z_{n_{k}},\varepsilon)$ infinity many times all values in $\overline{\mathbb{C}}$ except possibly two. From the definition it follows that any subsequence of $P-$sequence is also a $P-$sequence. Gavrilov (see Theorem 3 in [16]) showed that a meromorphic function $f$ is normal in the disc $\mathbb{D}$, i.e., $f\in N$ is and only if $f$ in $\mathbb{D}$ does not have $P-$sequences. ###### Theorem 5.4 (see Theorem 1 in [17]). If for a sequence $\\{z_{n}\\}\subseteq\mathbb{D},\ \lim_{n\rightarrow\infty}\|z_{n}\|=1$ and a meromorphic function $f:\mathbb{D}\rightarrow\overline{\mathbb{C}}$ hold $\lim_{n\rightarrow\infty}(1-\|z_{n}\|^{2})f^{\sharp}(z)=\infty,$ then $\\{z_{n}\\}$ is a $P-$sequence for $f$. The example of the meromorphic function $f(z)=\exp\left\\{-\exp\left\\{\frac{1}{1-z}\right\\}\right\\}$ from [16] shows that the reverse implication in Theorem 5.4 does not hold. ###### Theorem 5.5 (see Theorem 3 in [17]). A sequence $\\{z_{n}\\}\subseteq\mathbb{D},\,\lim_{n\rightarrow\infty}\|z_{n}\|=1$ is a $P-$sequence for a meromorphic function $f:\mathbb{D}\rightarrow\overline{\mathbb{C}}$ if and only if there exist a sequence of positive numbers $\\{r_{n}\\},\,\lim_{n\rightarrow\infty}r_{n}=0$ such that $\lim_{n\rightarrow\infty}\left\\{\sup_{z\in D_{h}(z_{n},r_{n})}(1-\|z\|^{2})f^{\sharp}(z)\right\\}=\infty.$ ###### Theorem 5.6 (see Theorem 2 and Theorem 5 in [16]). For a meromorphic function $f:\mathbb{D}\rightarrow\overline{\mathbb{C}}$ let $\\{z_{n}\\}\subseteq\mathbb{D},\ \lim_{n\rightarrow\infty}\|z_{n}\|=1$ be a sequence which satisfies $\lim_{n\rightarrow\infty}f(z_{n})=\alpha\in\overline{\mathbb{C}}$. Let $\\{z_{n}^{\prime}\\}$ be a new sequence such that along this one the function $f$ does not poses a limit $\alpha$ and $\lim_{n\rightarrow\infty}d_{h}(z_{n},z_{n}^{\prime})=0$. Then each of $\\{z_{n}\\}$ and $\\{z_{n}^{\prime}\\}$ are $P-$sequences. ###### Theorem 5.7. Let a curve $\gamma\subseteq\mathbb{D}$ terminates in $e^{i\theta}\in\Gamma$. A meromorphic function $f:\mathbb{D}\rightarrow\overline{\mathbb{C}}$ is normal along a curve $\gamma$ in $D(r)$, where $r\in(0,1)$ is fixed, if and only if the function $f$ does not have a $P-$sequenece in $\Delta_{r}\gamma$. One can give a proof of Theorem 5.7 in the similar way as the proof of the following ###### Theorem 5.8 (see Theorem 3 in [53]). A meromorphic function $f:\mathbb{D}\rightarrow\overline{\mathbb{C}}$ is normal along a simple curve $\gamma\subseteq\mathbb{D}$ which terminates in $e^{i\theta}\in\Gamma$ if and only if for all $r\in(0,1)$ the function $f$ does not have a $P-$sequence in $\Delta_{r}\gamma$. From Theorem 5.7 we immediately deduce ###### Theorem 5.9. Let $f:\mathbb{D}\rightarrow\overline{\mathbb{C}}$ be a meromorphic function such that there exits a domain $O$ which contains $e^{i\theta}\in\Gamma$ such that for all $r\in(0,1)$ the function $f$ is bounded in $O\cap\Delta_{r}\gamma$. Then $f$ is normal function along $\gamma$, i.e., $f$ is normal along $[\gamma]$. Altogether, from the results of this section we obtain ###### Proposition 5.1. For a meromorphic function $f:\mathbb{D}\rightarrow\overline{\mathbb{C}}$ and a curve $\gamma$ the following conditions are equivalent: 1. (1) $f$ is normal along $\gamma$; 2. (2) $f$ is normal along $[\gamma]$; 3. (3) for all $r\in(0,1)$ holds $\sup_{z\in\Delta_{r}\gamma}(1-\|z\|^{2})f^{\sharp}(z)<\infty;$ 4. (4) $f$ does not have $P-$sequences in $\Delta_{r}\gamma,\,r\in(0,1)$. This characterization could be stated also on the normality of $f$ along $\gamma$ on $D(r)$ for each $r\in(0,1)$. $P-$sequences, as shows Theorem 4.3 in [16], characterize boundary behavior of meromorphic functions in the unit disc (for example see also [53], [19], [30], [40], [16], [38], [17], [39], [14], [15]). We will use them in the seventh section for the construction of meromorphic functions showing that the results from this paper cannot be improved. ## 6\. Curvilinear boundary values of meromorphic functions ###### Theorem 6.1. Let $f:\mathbb{D}\rightarrow\overline{\mathbb{C}}$ be a meromorphic function, let a curve $\gamma\subseteq\mathbb{D}$ terminates in $e^{i\theta}\in\Gamma$, and let $c\in\overline{\mathbb{C}}$. The following conditions are equivalent: 1. (1) $f$ is normal along $\gamma$ and $\lim_{\gamma\ni z\rightarrow e^{i\theta}}f(z)=c$; 2. (2) $c$ is $\Delta_{\gamma}-$boundary value of $f$. ###### Proof. (1) implies (2): Since $f$ is normal along $\gamma$, for any sequence $\\{w_{n}\\}\subseteq\gamma$ which satisfies $\lim_{n\rightarrow\infty}w_{n}=e^{i\theta}$ there exist a subsequence $\\{w_{n_{k}}\\}$ such that the $\\{f_{n_{k}}(z)=f\circ\varphi_{{n_{k}}}(z)\\}$, where $\varphi_{{n_{k}}}=\varphi_{w_{n_{k}}}$, is convergent to a (meromorphic) function $\psi$ (in the local topology of $\mathbb{D}$). Let $r_{1}\in(0,1)$ and $K=\overline{D}_{ph}(0,r_{1})=\overline{D}(r_{1})=\\{z:\|z\|\leq r_{1}\\}$. For $r\in(0,r_{1})$ let us consider $\gamma\cap\overline{D}_{ph}(w_{n_{k}},r_{1})\setminus D_{ph}(w_{n_{k}},r)$. This set consists of two curves; let $\gamma_{k}$ be one of them and denote $\Gamma_{k}=\varphi_{n_{k}}^{-1}(\gamma_{k})$. For all $m\in\mathbb{N}$ let $\\{z_{k}^{m}\in\Gamma_{k}\\}$ be any sequence that satisfies $\lim_{k\rightarrow\infty}z_{k}^{m}=z_{0}^{m}\in\overline{D}_{ph}(0,r_{1})$. We will show that $\psi(z_{0}^{m})=c$ for all $m\in\mathbb{N}$. Namely, for all $m\in\mathbb{N}$ we have $d_{S}(\psi(z_{0}^{m}),c)\leq d_{S}(\psi(z_{0}^{m}),\psi(z_{k}^{m}))+d_{S}(\psi(z_{k}^{m}),f_{n_{k}}(z_{k}^{m}))+d_{S}(f_{n_{k}}(z_{k}^{m}),c)$ Let $\varepsilon>0$ be any number. Because of continuity of $\psi$ we have $d_{S}(\psi(z_{0}^{m}),\psi(z_{k}^{m}))<\frac{\varepsilon}{3}$, if $k$ is big enough. Since the sequence $\\{f_{n_{k}}\\}$ is convergent to $\psi$ in the local topology of $\mathbb{D}$, we have $d_{S}(\psi(z),f_{n_{k}}(z))<\frac{\varepsilon}{3}$ for all $z\in\overline{D}_{ph}(0,r_{1})$ and if $k$ is big enough. Since $z_{k}^{m}\in\overline{D}_{ph}(0,r)$, we have $d_{S}(\psi(z_{k}^{m}),f_{n_{k}}(z_{k}^{m}))<\frac{\varepsilon}{3}$. Since $z_{k}^{m}\in\Gamma_{k}$, it follows $\varphi_{n_{k}}(z_{k}^{m})=w_{k}^{m}\in\gamma_{k}\subseteq\gamma$ and $\lim_{k\rightarrow\infty}w_{k}^{m}=e^{i\theta}$. Since $c$ is an asymptotic boundary value of $f$ and since $\lim_{k\rightarrow\infty}f_{n_{k}}(w_{k}^{m})=\lim_{k\rightarrow\infty}f\circ\varphi_{n_{k}}(z_{k}^{m})=\lim_{k\rightarrow\infty}f_{n_{k}}(z_{k}^{m})=c$ for big enough $k$, we have $d_{S}(f_{n_{k}}(z_{k}^{m}),c)<\frac{\varepsilon}{3}$ for all $m\in\mathbb{N}$. From the preceding inequalities it follows $d_{S}(\psi(z_{0}^{m}),c)<\varepsilon$ for all $m\in\mathbb{N}$. Since $\varepsilon$ is any positive number, it must be $\psi(z_{0}^{m})=c$ for all $m\in\mathbb{N}$. Since the sequence $\\{z_{0}^{m}\\}$ lies in $\overline{D}_{ph}(0,r_{1})$ and since in this set it has an accumulation point, from the uniqueness theorem, we have $\psi\equiv c$. Thus, it is showed that any convergent sequence (in the local topology of $\mathbb{D}$) of the family $\mathcal{F}_{f,\gamma}=\\{f\circ\varphi_{w}:w\in\gamma\\}$ converges to $c$. We will show now that any sequence of the family $\mathcal{F}_{f,\gamma}$ converges to the constant $c$ in the local topology. Assume contrary, let that there exist a sequence $\\{f_{n}\\}\subseteq\mathcal{F}_{f,\gamma}$, which is not convergent in the local topology to the constant $c$. There exist $\varepsilon>0$ such that for any $k\in\mathbb{N}$ there exist $n_{k}\in\mathbb{N}$ and $z_{n_{k}}\in\overline{D}_{ph}(0,r)$ such that $d_{S}(f_{n_{k}}(z_{n_{k}}),c)\geq\varepsilon$. Since the family $\mathcal{F}_{f,\gamma}$ is normal, the sequence $\\{f_{n_{k}}\\}$ has a subsequence $f_{n_{k_{l}}}$ which is convergent; according to the preceding, it converges to the constant $c$, what is contrary to the assumption $d_{S}(f_{n_{k}}(z_{n_{k}}),c)\geq\varepsilon$. This contradiction shows that any sequence in $\mathcal{F}_{f,\gamma}$ converges in the local topology of $\mathbb{D}$ to the constant $c$. From this and from Theorem 4.1 we have $C(f,\Delta_{r}\gamma,e^{i\theta})=\\{c\\}$ for all $r\in(0,1)$, that is the function $f$ in the point $e^{i\theta}$ has $\Delta_{\gamma}-$boundary value along the curve $\gamma$ equal to $c$. (2) implies (1): Form Theorem 4.1 and condition (2) we have $C(f,\Delta_{r}\gamma,e^{i\theta})=\\{c\\}$ for all $r\in(0,1)$. It follows that any sequence $\\{f_{n}\\}\subseteq\mathcal{F}_{f,\gamma}$ converges to the constant $c$. We infer that $\mathcal{F}_{f,\gamma}$ is normal family in $\mathbb{D}$. Regarding Definition 4.2 this means that $f$ is normal along the curve $\gamma$. From the condition $C(f,\Delta_{r}\gamma,e^{i\theta})=\\{c\\}$ evidently follows $\lim_{\gamma\ni z\rightarrow e^{i\theta}}f(z)=c$. ∎ ###### Remark 6.1. Seidel and Walsh proved (see Theorem 4, p. 199 in [47]): Let $f$ be an analytic function in $\mathbb{D}$ which omits at least two values. Let $\gamma_{1}$ and $\gamma_{2}$ be simple curves in $\mathbb{D}$ which terminate in $1$ with finite Fréshet distance. If $\lim_{\gamma_{1}\ni z\rightarrow 1}f(z)=c\in\overline{\mathbb{C}}$, then also $\lim_{\gamma_{2}\ni z\rightarrow 1}f(z)=c$. This statement remains valid if we assume that $f$ is normal meromorphic function in the disc $\mathbb{D}$ (see Theorem 2.12, p. 131, in [29]). Our Theorem 6.1 shows that the theorem of Seidel and Walsh holds for all curves in the class $[\gamma]$. Lemma 3.3 shows that Theorem 6.1 is a generalization of the previous results. In the similar manner as Theorem 6.1 one can prove the following ###### Theorem 6.2. For a meromorphic function $f:\mathbb{D}\rightarrow\overline{\mathbb{C}},\,r\in(0,1)$ and $c\in\overline{\mathbb{C}}$ the following conditions are equivalent: 1. (1) $f$ is normal in ${D}(r)$ along a curve $\gamma$ and $\lim_{\gamma\ni z\rightarrow e^{i\theta}}f(z)=c$; 2. (2) $C(f,\Delta_{r_{1}}\gamma,e^{i\theta})=c$ for all $r_{1}\in(0,r)$. Form Theorems 5.2, 6.1 and 6.2 we conclude ###### Theorem 6.3. Assume a meromorphic function $f:\mathbb{D}\rightarrow\overline{\mathbb{C}}$ has an asymptotic boundary value along a curve $\gamma$. In order that $f$ has $\Delta_{r}\gamma-$boundary value along the curve $\gamma$ it is necessary and sufficient that $\sup_{z\in\Delta_{r}\gamma}(1-\|z\|^{2})f^{\sharp}(z)<\infty$ for all $r\in(0,1)$, what is equivalent to the condition that $f$ does not contain $P-$sequences in $\Delta_{r}\gamma$. ###### Remark 6.2. Results in Theorems 6.1, 6.2 and 6.3 are generalization of Theorems 2, 2’, 4 and 5 of Lehto and Virtanen [27]. They considered the normal meromorphic functions in $\mathbb{D}$ and its angular boundary values. We consider the normal meromorphic functions along a curve $\gamma$ and $\Delta_{\gamma}-$boundary values. Particulary, if $\gamma$ is not tangent to $\Gamma$, from Theorem 6.1 and 6.3 we derive Theorems 2, 2’, 4 and 5 in [27]. ###### Remark 6.3. Theorem 4.2, Theorem 6.1 and Theorem 6.2 show that each of the following two conditions: 1. (1) $f$ is normal meromorphic function along a curve $\gamma$; 2. (2) a meromorphic function $f$ has an asymptotic boundary value along the curve $\gamma$, is a necessary condition for the existence of $\Delta_{\gamma}-$boundary value ($[\gamma]-$boundary value) of $f$. Taken together, conditions (1) and (2) are necessary and sufficient for the existence of $\Delta_{\gamma}-$boundary value ($[\gamma]-$boundary value) of $f$. ## 7\. Examples The following examples we will construct in the similar way as in [19]. ###### Example 7.1. Let $\gamma$ be a curve terminating in $e^{i\theta}\in\Gamma$. With $\gamma_{r}^{+}$ and $\gamma_{r}^{-}$ we denote the parts of the boundary of the curvilinear angle $\Delta_{r}\gamma,\,r\in(0,1)$ as in the proof of Theorem 4.3. Let $\\{z_{k}\\},\,\lim_{k\rightarrow\infty}z_{k}=e^{i\theta}$ be a sequence of points such that $z_{2m}\in\gamma^{+}_{r_{m}},\,z_{2m-1}\in\gamma^{-}_{r_{m}}$, where $r_{m}\uparrow 1$. Moreover, let $\\{\varepsilon_{k}\\}$ be a sequence of numbers which satisfies: 1. (1) $0<\varepsilon_{k+1}<\varepsilon_{k}$ for all $k\in\mathbb{N}$; 2. (2) $\lim_{k\rightarrow\infty}\varepsilon_{k}=0$; 3. (3) $D_{i}\cap D_{j}$ is the empty set for all $i\neq j$, where $D_{k}=\\{z:\|z-z_{k}\|<\varepsilon_{k}\\}$; 4. (4) $\lim_{k\rightarrow\infty}\sup_{z\in D_{k}}d_{h}(z,z_{k})=0$; 5. (5) $\sum_{k=1}^{\infty}\varepsilon_{k}<\infty$. Let $a_{k}=\varepsilon_{k}^{2}$ for all $k\in\mathbb{N}$ and $f_{0}(z)=\sum_{k=1}^{\infty}a_{k}(z-z_{k})^{-1}$. For fixed $n\in\mathbb{N}$, we have $\left\|\sum_{k\neq n}a_{k}(z-z_{k})^{-1}\right\|\leq\sum_{k=1}^{\infty}\varepsilon_{k}^{2}<\infty.$ It follows that $f_{0}$ is a meromorphic function in the disc $\mathbb{D}$ with poles at $z_{k},\,k\in\mathbb{N}$. Since $\|f_{0}(z_{k}+\varepsilon_{k})\|<\infty$ and $\lim_{k\rightarrow\infty}d_{h}(z_{k},z_{k}+\varepsilon_{k})=0$, from Theorem 5.6 it follows that the sequence $\\{z_{k}\\}$ is a $P-$sequence for $f_{0}$. Since for all $z^{\prime},\,z^{\prime\prime}\in\mathbb{D}\setminus\bigcup_{k\in\mathbb{N}}D_{k}$ holds $\|f_{0}(z^{\prime})-f_{0}(z^{\prime\prime})\|\leq\|z^{\prime}-z^{\prime\prime}\|\sum_{k=1}^{\infty}\varepsilon_{k}$ and since any of sets $\Delta_{r}\gamma,\,r\in(0,1)$ contains a finite number of points from $\\{z_{k}\\}$, it follows that $\limsup_{\Delta_{r}\gamma\ni z\rightarrow e^{i\theta}}\|f_{0}(z)\|=c_{f_{0}}(r)<\infty,\quad r\in(0,1).$ Hence, for all $r\in(0,1)$, the function $f_{0}$ is bounded in $O_{r}\cup\Delta_{r}\gamma$, where $O_{r}=\\{z:\|z-e^{i\theta}\|<1-r\\}$. From Theorem 5.9, it follows that $f_{0}$ is normal in $O_{r}\cup\Delta_{r}\gamma$. Now, from Theorem 5.7 we obtain that $f_{0}$ is normal along the curve $\gamma$ (see [45], p. 35, Montel’s theorem). The way we constructed the function $f_{0}$ shows that any set $\mathcal{A}$ which contains all sets $O_{r}\cap\Delta_{r}\gamma,\,r\in(0,1)$, and any sequence of points $\\{z_{k}\\}$ contains a $P-$sequence of the function $f_{0}$. It is possible to show that there exist vicinities $O_{r},\,r\in(0,1)$ of $e^{i\theta}$ such that $\left(\bigcup_{r\in(0,1)}O_{r}\cap\Delta_{r}\gamma\right)\cup\\{z_{k}:k\in\mathbb{N}\\}\varsubsetneq D\cup O_{e^{i\theta}},$ where $O_{e^{i\theta}}$ is any vicinity of the point $e^{i\theta}$. The preceding facts can be illustrated well if we take for a set $\mathcal{A}$ a horo–cycle which is tangent to $\Gamma$ in the point $e^{i\theta}$ and for the curve $\gamma$ a radius od $\mathbb{D}$ with one endpoint in $e^{i\theta}$. Then the domain $\Delta_{r}\gamma$ is the sub–domain of the disc $\mathbb{D}$ which is bounded by two hyper–cycles which contain $\pm e^{i\theta}$ (see Figure 1). Figure 1. Thus, the function $f_{0}$ shows that in a general case does not exist a set which contains all sets $\Delta_{r}\gamma,\,r\in(0,1)$ such that along that set the function does not poses $P-$sequences. ###### Example 7.2. Let $f_{1}(z)=f_{0}(z)(z-e^{i\theta})$, where $f_{0}$ is the function from the preceding example. Also let $\gamma$ be the curve from the same example. Since for every $\Delta_{r}\gamma,\,r\in(0,1)$ there exist a vicinity $O(r)$ of the point $e^{i\theta}$ such that for every $z\in O(r)\cap\Delta_{r}\gamma$ holds $\|f_{1}(z)\|=\|f_{0}(z)\|\|z-e^{i\theta}\|\leq C(r)\|z-e^{i\theta}\|\rightarrow 0\quad\text{as}\quad z\rightarrow e^{i\theta},$ we have that $0$ is $\Delta_{\gamma}-$boundary value of $f$. On the other side, any $P-$sequence $\\{z_{k}\\}$ for the function $f_{0}$ is a $P-$sequence for $f_{1}$, what may be proved in the same way as for $f_{0}$. That sequence is contained in $\mathcal{A}$. Since $\lim_{k\rightarrow\infty}f_{1}(z_{k})=\infty$, it follows that $\infty,\,0\in C(f,\mathcal{A},e^{i\theta})$. The example of function $f_{1}$ shows that in general case does not exist a set which contains all $\Delta_{r}\gamma,\,r\in(0,1)$ such that along this set the function has the unique cluster point, i.e., Example 7.2 shows that the theorem on the existence of curvilinear boundary values cannot be improved in the direction which means the expansion of sets $\Delta_{r}\gamma,\,r\in(0,1)$. ## 8\. Applications Let $\gamma\subseteq\mathbb{D}$ be a curve which terminates in a point $e^{i\theta}\in\Gamma$ and which is tangent on the cycle $\Gamma$ in that point. Denote by $\Delta_{\alpha,\rho}\gamma,\,\rho\in(0,1),\,\alpha\in(0,\pi)$ the sub–domain of $\mathbb{D}$ bounded by $\gamma$, the chord $h(\theta,\alpha),\,\alpha\in(0,\frac{\pi}{2})$ of $\mathbb{D}$ and by the arc of $D_{\rho}=\\{z:\|z-e^{i\theta}\|=\rho\\},\,\rho\in(0,1)$. Moreover, denote $G^{\theta}_{\gamma,r,\alpha,\rho}=\Delta_{r}\gamma\cup\Delta_{\alpha,\rho}\gamma,\quad r,\,\rho\in(0,1),\,\alpha\in(0,\pi)$ (see Figure 2). It is easy to check that $\Delta_{\alpha,\rho}\gamma\not\subseteq G^{\theta}_{\gamma,r,\alpha,\rho}$ for all $r,\,\rho\in(0,1),\,\alpha\in(0,\pi)$. Figure 2. Lehto and Virtanen (see Remark, p. 53 in [27], or Remark on the page 124 in [29]) showed that a normal meromorphic function $f$ in the disc $\mathbb{D}$ which in a point $e^{i\theta}$ has an asymptotic boundary value $\lim_{\gamma\ni z\rightarrow e^{i\theta}}f(z)=c\in\overline{\mathbb{C}}$ satisfies $C(f,\Delta_{\alpha,\rho}\gamma,e^{i\theta})=\\{c\\}$ for all $\rho\in(0,1)$ and $\alpha\in(0,\pi)$. The following theorem shows that the result of Lehto and Virtanen for the case of a simple curve which is tangent on $\Gamma$ may be improved in the sense that domains $\Delta_{\alpha,\rho}\gamma,\,\rho\in(0,1),\,\alpha\in(0,\pi)$ along which an asymptotic value $c$ exists for a normal meromorphic function in $\mathbb{D}$, may be replaced by the domain $G^{\theta}_{\gamma,r,\alpha,\rho},\,r,\,\rho\in(0,1),\,\alpha\in(0,\pi)$. Namely, we have the following ###### Theorem 8.1. Let $f:\mathbb{D}\rightarrow\overline{\mathbb{C}}$ be a normal meromorphic function in the disc $\mathbb{D}$ i.e., $f\in N$ and let $\lim_{\gamma\ni z\rightarrow e^{i\theta}}f(z)=c\in\overline{\mathbb{C}}$. Then $\bigcup_{r,\,\rho\in(0,1),\,\alpha\in(0,\pi)}C(f,G^{\theta}_{\gamma,r,\alpha,\rho},e^{i\theta})=\\{c\\}.$ ###### Proof. From the mentioned result of Lehto and Virtanen we have $C(f,\Delta_{\alpha,\rho}\gamma,e^{i\theta})=\\{c\\}$ for all $\alpha\in(0,\pi)$ and for all $\rho\in(0,1)$. From Theorem 5.9 it follows $C(f,\Delta_{r}\gamma,e^{i\theta})=\\{c\\}$ for all $r\in(0,1)$. Thus, for all $r,\,\rho\in(0,1)$ and $\alpha\in(0,\pi)$ we have $C(f,G^{\theta}_{\gamma,r,\alpha,\rho},e^{i\theta})=\\{c\\}$; this means that the union of all sets $C(f,G^{\theta}_{\gamma,r,\alpha,\rho},e^{i\theta})$ where $r,\,\rho\in(0,1)\,\alpha\in(0,\pi)$ is equal to $\\{c\\}$. ∎ For simplicity in what follows we will assume that any curve $\gamma$ that appears is a simple curve which connects the center of $\mathbb{D}$ and some point $e^{i\theta}\in\Gamma$ such and any circle $\Gamma_{r}=\\{z:\|z\|=r\\},\ 0<r<1$ intersects in exactly one point. Šaginjan [52] proved the following statement of uniqueness: Let $f(z)$ be an analytic function in the disc $\mathbb{D},\,\|f(z)\|<1,\,z\in\mathbb{D}$, and let $f$ along a curve $\gamma$ satisfies the following estimate (2) $\|f(z)\|\leq\exp\left\\{-\frac{p(1-\|z\|)}{1-\|z\|}\right\\},\quad z\in\gamma,$ where $p(t)$ is a function which arbitrary slow increase to $+\infty$ as $t\rightarrow+0$, then $f(z)\equiv 0$ (see Theorem 2, p. 23, in [52]). The analytic function in $\mathbb{D}$ given by $f(z)=\exp\\{-\frac{1}{1-z}\\}$ shows that the condition cannot be relaxed. Gavrilov proved a theorem which is an analog of the preceding result of Šaginjan: Let $f(z)$ be normal meromorphic in $\mathbb{D}$ and let $\varepsilon$ be any positive number. If $f(z)$ along the radius $\arg z=0$ satisfies the inequality $\|f(z)\|\leq\exp\left\\{-\frac{1}{(1-\|z\|)^{1+\varepsilon}}\right\\},$ then $f(z)\equiv 0$ (see Theorem 1 and 2, pp. 4–6, in [18]). Another results which are generalizations of the theorem of Šaginjan may be found in [11], [25], [56], [9]; see also [23], [24]. Let $P=\\{p(t):t\in(0,b),\,p(t)\uparrow+\infty\ \text{as}\ t\rightarrow 0^{+}\\}$. It is easy to check that if $p(t)\in P$, then $p_{1}(t)=c_{1}p(c_{2}t^{\varepsilon})\in P$, where $c_{1},\,c_{2}$ and $\varepsilon$ are any positive numbers. ###### Lemma 8.1. Let $f:\mathbb{D}\rightarrow\overline{\mathbb{C}}$ be a meromorphic function which is normal along a curve $\gamma$ in $D(r)$ for some $r\in(0,1)$. Furthermore, assume (3) $\|f(z)\|\leq\exp\left\\{-\frac{p(1-\|\varphi(z)\|)}{1-\|\varphi(z)\|}\right\\},\quad z\in\gamma$ where $\varphi:\Delta_{r_{1}}\gamma\rightarrow\mathbb{D},\,r_{1}\in(0,r)$ is a conformal mapping of the curvilinear angle $\Delta_{r_{1}}\gamma$ onto the disk $\mathbb{D}$ such that $\varphi(0)=0$, and let $p(t)$ be an arbitrary function which belong to the class $P$. Then $f(z)\equiv 0$. ###### Proof. From Theorem 6.2 and (3) it follows $C(f,\overline{\Delta_{r_{1}}\gamma},e^{i\theta})=\\{0\\}.$ Without lost of generality, we may assume that $\|f(z)\|<1$ for all $z\in\Delta_{r_{1}}\gamma$. Let $\varphi:\Delta_{r_{1}}\gamma\rightarrow\mathbb{D}$ be a conformal mapping of $\Delta_{r_{1}}\gamma$ onto the disk $\mathbb{D}$. Then $F(\omega)=f\circ\varphi^{-1}(\omega),\,\omega\in\mathbb{D}$ is an analytic function which satisfies $\|F(\omega)\|<1,\,\omega\in\mathbb{D}$. Since $z=\varphi^{-1}(\omega)$, from (3) we obtain $\|f(\varphi^{-1}(\omega))\|\leq\exp\left\\{-\frac{p(1-\|\varphi(\varphi^{-1}(\omega))\|)}{1-\|\varphi(\varphi^{-1}(\omega))\|}\right\\},\quad\omega\in\gamma_{1}=\varphi(\gamma),$ i.e., (4) $\|F(\omega)\|\leq\exp\left\\{-\frac{p(1-\|\omega\|)}{1-\|\omega\|}\right\\},\quad\omega\in\gamma_{1}=\varphi(\gamma),$ where $\gamma_{1}$ is a curve withe endpoint $e^{i\theta}$. From (4) and the theorem of Šaginjan we have $F(w)\equiv 0$ in $\mathbb{D}$. This implies $f(z)\equiv 0$ in $\Delta_{r}\gamma\subseteq\mathbb{D}$ and according to the classical theorem of uniqueness for meromorphic functions we conclude $f(z)\equiv 0$ in $\mathbb{D}$. ∎ For $0<\alpha<\frac{\pi}{2}$ denote $A(e^{i\theta},\alpha,\rho,z)=\\{z\in\mathbb{D}:\|\arg{(e^{i\theta}-z)}\|<\alpha,\,\|e^{i\theta}-z\|<\rho\\},$ where $\rho=\left\\{\begin{array}[]{ll}1,&\hbox{$0<\alpha\leq\frac{\pi}{3}$},\\\ 2\cos\alpha,&\hbox{$\frac{\pi}{3}<\alpha<\frac{\pi}{2}$}.\end{array}\right.$ Then $A(e^{i\theta},\alpha,\rho,z)$ is the Stolz angle in $\mathbb{D}$ with the vertex in $e^{i\theta}$ and with angle $2\alpha$. We have $\alpha\rightarrow\frac{\pi}{2}$ as $\rho\rightarrow 0$. Furthermore, let $[a,b]$ denote a segment in the complex plane with endpoints $a,\,b\in\mathbb{C}$. Denote $\varphi_{1}(z)=-z,\,\varphi_{2}(z)=\rho^{-1}(1+z),\,\varphi_{3}(z)=e^{i\alpha}z,\,\varphi_{4}(z)=z^{\frac{\pi}{2\alpha}}$, let $\varphi_{5}(z)=\frac{1}{2}(z+\frac{1}{z})$ be the function of Zhukovsky, $\varphi_{6}(z)=ze^{-\pi i}$, and $\varphi_{7}(z)=\frac{z-i}{z+i}$. If $w=\varphi_{\alpha}(z)=\varphi_{7}\circ\cdots\circ\varphi_{1}(z)$, then $\varphi_{\alpha}$ is a conformal mapping of the Stolz angle $A(1,\alpha,\rho,z)$ onto $\mathbb{D}$, and $w=\varphi_{\alpha}(z)=1-\frac{4\rho^{\frac{\pi}{2\alpha}}(1-z)^{\frac{\pi}{2\alpha}}}{\left[(1-z)^{\frac{\pi}{2\alpha}}-\rho^{\frac{\pi}{2\alpha}}\right]^{2}+2\rho^{\frac{\pi}{\alpha}}};$ moreover, $\psi(A(e^{i\theta},\alpha,\rho,z))=\mathbb{D},\,\psi(z)=\varphi_{\alpha}(e^{-i\theta}z)$. In the sequel we will simply write $\varphi$ instead of $\varphi_{\alpha}$, where $\alpha\in(0,\frac{\pi}{2})$ is fixed. We have $\varphi^{-1}(A(1,\beta,r,w))\subseteq A(1,\alpha,\rho,z),\,\beta\in(0,\frac{\pi}{2}),\,r\in(0,1),\,\varphi^{-1}([-1,1])=[1-\rho,1],\,\varphi^{-1}(-1)=1-\rho,\,\varphi^{-1}(1)=1$. We formulate our theorems for Stolz angles $A(1,\alpha,\rho,z)$. ###### Lemma 8.2. For all fixed $\alpha,\,\beta\in(0,\frac{\pi}{2})$ there exist constants $m=m(\alpha,\beta)>0$ and $M=M(\alpha,\beta)>0$ such that for all $\omega\in A(1,\beta,\rho,w)$ we have (5) $m(1-\|z\|)^{\frac{\pi}{2\alpha}}\leq 1-\|\omega\|\leq M(1-\|z\|)^{\frac{\pi}{2\alpha}},$ where $z=\varphi^{-1}(\omega)$. ###### Proof. Since for all $z\in A(1,\alpha,\rho,z)$ and $\omega\in A(1,\beta,\rho,w)$ we have $1-\omega=\frac{4\rho^{\frac{\pi}{2\alpha}}(1-z)^{\frac{\pi}{2\alpha}}}{\left[(1-z)^{\frac{\pi}{2\alpha}}-\rho^{\frac{\pi}{2\alpha}}\right]^{2}+2\rho^{\frac{\pi}{\alpha}}}$ and $\|1-\omega\|<\frac{2}{\cos\beta}(1-\|\omega\|)=c(1-\|\omega\|),\quad c=\frac{2}{\cos\beta},$ it follows that for all $\omega\in A(1,\beta,\rho,w)$ and $z=\varphi^{-1}(\omega)$ we have (6) $\begin{split}&\left\|\frac{4\rho^{\frac{\pi}{2\alpha}}(1-z)^{\frac{\pi}{2\alpha}}}{\left[(1-z)^{\frac{\pi}{2\alpha}}-\rho^{\frac{\pi}{2\alpha}}\right]^{2}+2\rho^{\frac{\pi}{\alpha}}}\right\|\frac{\cos\beta}{2}(1-\|z\|)^{\frac{\pi}{2\alpha}}\leq 1-\|\omega\|\leq\\\ &\left\|\frac{4\rho^{\frac{\pi}{2\alpha}}(1-z)^{\frac{\pi}{2\alpha}}}{\left[(1-z)^{\frac{\pi}{2\alpha}}-\rho^{\frac{\pi}{2\alpha}}\right]^{2}+2\rho^{\frac{\pi}{\alpha}}}\right\|\left(\frac{2}{\cos\alpha}\right)^{\frac{\pi}{2\alpha}}(1-\|z\|)^{\frac{\pi}{2\alpha}}.\end{split}$ Since $\alpha$ and $\beta$ are fixed, $r=2\cos\alpha$ is also fixed. Since $\phi(z)=\left\|\frac{4\rho^{\frac{\pi}{2\alpha}}(1-z)^{\frac{\pi}{2\alpha}}}{\left[(1-z)^{\frac{\pi}{2\alpha}}-\rho^{\frac{\pi}{2\alpha}}\right]^{2}+2\rho^{\frac{\pi}{\alpha}}}\right\|$ is a continuous function in $A(1,\alpha,\rho,z)$, the function $f(z)$ on the compact set $\overline{\varphi^{-1}(A(1,\beta,\rho,w))}$ achieves its minimum and maximum; let $\phi_{\rm min}(z)=c_{1}(\alpha,\beta)=c_{1}>0$ and $\phi_{\rm max}(z)=c_{2}(\alpha,\beta)=c_{2}<\infty$. From (6) we obtain that for all $\omega\in A(1,\beta,\rho,w)$ and $z=\varphi^{-1}(\omega)$ holds $m(1-\|z\|)^{\frac{\pi}{2\alpha}}\leq 1-\|\omega\|\leq M(1-\|z\|)^{\frac{\pi}{2\alpha}},$ where we have denoted $m=\frac{\cos\beta}{2}c_{1}>0$ and $M=\left(\frac{2}{\cos\alpha}\right)^{\frac{\pi}{2\alpha}}c_{2}<\infty$. ∎ ###### Theorem 8.2. Let $\gamma$ be a simple curve in the disc $\mathbb{D}$ with one endpoint in $1$ which is not tangent to $\Gamma$. Let $f:\mathbb{D}\rightarrow\overline{\mathbb{C}}$ be a meromorphic function normal in $\mathbb{D}$ along the curve $\gamma$. If for all $\alpha\in(0,\frac{\pi}{2})$ and $p_{1}\in P$ holds (7) $\|f(z)\|\leq\exp\left\\{-\frac{p_{1}(1-\|z\|)}{(1-\|z\|)^{\frac{\pi}{2\alpha}}}\right\\},\quad z\in[\rho,1],$ where $\rho=\left\\{\begin{array}[]{ll}1,&\hbox{$0<\alpha\leq\frac{\pi}{3}$},\\\ 2\cos\alpha,&\hbox{$\frac{\pi}{3}<\alpha<\frac{\pi}{2}$},\end{array}\right.$ then $f(z)\equiv 0$. ###### Proof. From assumptions of this theorem and Theorem 5.3 we obtain that $f$ is normal in $\mathbb{D}$ along the radius of the disc which terminates in $1$, i.e., $f$ is normal in $\mathbb{D}$ along any curve $[a,1],\,0<a<1$. From (7) we have that $0$ is an angular boundary value for $f$. Let $w=\varphi(z)=\frac{4\rho^{\frac{\pi}{2\alpha}}(1-z)^{\frac{\pi}{2\alpha}}}{\left[(1-z)^{\frac{\pi}{2\alpha}}-\rho^{\frac{\pi}{2\alpha}}\right]^{2}+2\rho^{\frac{\pi}{\alpha}}}$ be a conformal mapping which maps the Stolz angle $A(1,\alpha,\rho,z)$ onto the disk $\mathbb{D}$. For any fixed $\beta\in(0,\frac{\pi}{2})$ and $r=1-\rho$ from inequality (5) and Lemma 8.2 we obtain that for all $\omega\in A(1,\beta,\rho,w)$ holds (8) $m(1-\|z\|)^{\frac{\pi}{2\alpha}}\leq 1-\|\omega\|,\quad z=\varphi^{-1}(\omega).$ It follows that this inequality holds for $\omega\in[1-\rho,1]$ and $z=\varphi^{-1}(\omega)\in[\rho,1].$ From (8) we obtain $p(1-\|\varphi(z)\|)\leq p(m(1-\|z\|)^{\frac{\pi}{2\alpha}})$ for all $w\in[1-\rho,1],\,z=\varphi^{-1}(w)\in[\rho,1]$ and $p\in P$. Further, we have $\frac{p(1-\|\varphi(z)\|)}{1-\|\varphi(z)\|}\leq\frac{p(m(1-\|z\|)^{\frac{\pi}{2\alpha}})}{m(1-\|z\|)^{\frac{\pi}{2\alpha}}}.$ Hence (9) $-\frac{p(1-\|\varphi(z)\|)}{1-\|\varphi(z)\|}\geq-\frac{p_{1}(1-\|z\|)}{(1-\|z\|)^{\frac{\pi}{2\alpha}}}$ for $z\in\gamma_{1}=[\rho,1]$, where we have denoted $p_{1}(t)=m^{-1}p(mt^{\frac{\pi}{2\alpha}})\in P$. From (7) and (9) we obtain $\|f(z)\|\leq\exp\left\\{-\frac{p(1-\|\varphi(z)\|)}{1-\|\varphi(z)\|}\right\\},\quad z\in\gamma_{1}=[\rho,1].$ From Lemma 8.1 it follows $f(z)\equiv 0$. ∎ ###### Remark 8.1. From the proof of Theorem 8.2 we see that this theorem holds if instead of normality in $\mathbb{D}$ of $f$ along the curve $\gamma$ we have normality of $f$ along $\gamma_{1}=[a,1]$ in $D_{r}$, where $r,\,a\in(0,1)$. Then in the inequality (7) for the angle $\alpha$ we have $0<\alpha<\frac{\pi}{2}-\arctan\frac{1-r^{2}}{2r}$. Namely, from the preceding conditions it follows that the function $f$, along the domain which is bounded by two horo–cycles that contain $\pm 1$ and $\pm ri$ and the circle $\\{z:\|z-1\|=1\\}$ has a cluster set which contains only $0$. Since for $0<\alpha<\frac{\pi}{2}-\arctan\frac{1-r^{2}}{2r}$ the angle $A(1,\alpha,\rho,z)$ is the subset of this domain, we obtain $C(f,1,A(1,\alpha,\rho,z))=\\{0\\}$, hence a proof in this case goes in the same way as the proof of Theorem 8.2. ###### Theorem 8.3. Let $f:\mathbb{D}\rightarrow\overline{\mathbb{C}}$ be a meromorphic function normal in $D(r)$, where $r\in(0,1)$, along $\gamma_{1}=[a,1],\,0<a<1$. Let, moreover, $\|f(z)\|\leq\exp\left\\{-\frac{1}{(1-\|z\|)^{k}}\right\\},\quad z\in[a,1],$ where $k>\frac{\pi}{2\alpha},\,0<\alpha<\frac{\pi}{2}-\arctan\frac{1-r^{2}}{2r}$. Then $f\equiv 0$. Theorem 8.3 follows immediately from Theorem 8.2 and the inequality $t^{q}<t^{p}$, if $0<t<1$ and $q<p$. ###### Remark 8.2. Theorem 8.3 is actually the result of Gavrilov; see Theorem 2 in [18]. ###### Theorem 8.4. Let $\gamma$ be a curve in $\mathbb{D}$ which terminates in $1$ and which is not tangent to $\Gamma$ in this point. Let $f:\mathbb{D}\rightarrow\overline{\mathbb{C}}$ be a meromorphic function normal in $\mathbb{D}$ along $\gamma$. If for all integers $n\geq 1$ holds (10) $\|f(z)\|\leq\exp\left\\{-\frac{1}{(1-\|z\|)^{1+\frac{1}{n}}}\right\\},\quad z\in[a,1],$ then $f(z)\equiv 0$. ###### Proof. Since $0<\alpha<\frac{\pi}{2}$ and $p_{1},\ p_{2}\in P$ are arbitrary in Theorem 8.2, if we set $\alpha=\frac{n}{2n+1}\pi$ and $p_{1}(t)=t^{-\frac{1}{2n}}$, where $n\geq 1$ is an integer, inequality (7) take the form (10); from Theorem 8.2 we have $f(z)\equiv 0$. ∎ ###### Remark 8.3. 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1312.6394	# Completely bounded Paley projections on anisotropic Sobolev spaces on tori Yanqi QIU ###### Abstract. We study the existence of certain completely bounded Paley projection on the anisotropic Sobolev spaces on tori. Our result should be viewed as a generalization of a similar result obtained by Pełczyński and Wojciechowski in [3]. By a transference method, we obtain similar results on the Sobolev spaces on quantum tori. ## 1\. Introduction Let $S\subset\mathbb{N}^{d}$ be a finite subset, containing the origin and satisfying some saturation conditions. The anisotropic Sobolev space $W^{S}_{1}(\mathbb{T}^{d})$ is defined via the norm $\\|f\\|_{S,1}=\sum_{\gamma\in S}\\|\partial^{\gamma}f\\|_{L_{1}(\mathbb{T}^{d})}.$ In [3], necessary and sufficient conditions on $S$ are given under which there exist the so-called Paley projections on $W^{S}_{1}(\mathbb{T}^{d})$. By the definition, $W^{S}_{1}(\mathbb{T}^{d})$ embeds isometrically in $\ell_{1}^{\|S\|}(L_{1}(\mathbb{T}^{d}))$, it is well-known that on the latter space, there exists a natural operator space structure, and we will equip $W^{S}_{1}(\mathbb{T}^{d})$ with the sub-operator space structure via the above embedding. Following the proofs in [3], we show that under the same conditions on $S$, the projections considered by Pełczyński and Wojciechowski are in fact completely bounded. The complete boundedness of these projections can be applied to obtain similar results on the Sobolev space $W^{S}_{1}(\mathbb{T}_{\theta}^{d})$ associated to the quantum torus $\mathbb{T}^{d}_{\theta}$. ## 2\. Prelininaries Denote by $\mathbb{N}$ the set of non-negative integers. Fix a positive integer $d\geq 1$. The usual scalar product on the Euclidian space $\mathbb{R}^{d}$ is denoted by $\langle\cdot,\cdot\rangle$. We denote by $\mathbb{T}^{d}$ the group $(\mathbb{R}/2\pi\mathbb{Z})^{d}$ equipped with its normalized Haar measure $dx$, it will be identified with the cube $[-\pi,\pi)^{d}$ in a standard way. The dual group of $\mathbb{T}^{d}$ is $\mathbb{Z}^{d}$ such that to each $n\in\mathbb{Z}^{d}$ is assigned the character $\chi_{n}:\mathbb{T}^{d}\to\mathbb{C}$ defined by $\chi_{n}(x)=e^{i\langle x,n\rangle}$. Trigonometric polynomials are complex linear combinations of characters. The set of trigonometric polynomials on $\mathbb{T}^{d}$ is denoted by $\mathscr{P}_{d}$. To each $\gamma=(\gamma(j))\in\mathbb{N}^{d}$, we associate with the partial derivative $\partial^{\gamma}=\frac{\partial^{\|\gamma\|}}{\partial^{\gamma(1)}_{x(1)}\partial^{\gamma(2)}_{x(2)}\dots\partial^{\gamma(d)}_{x(d)}},$ where $\|\gamma\|=\gamma(1)+\gamma(2)+\cdots\gamma(d)$. A smoothness $S$ is a finite subset of $\mathbb{N}^{d}$ which contains the origin 0, and such that: if $\alpha=(\alpha(j))\in S$ then every $\beta=(\beta(j))\in\mathbb{N}$ such that $\beta(j)\leq\alpha(j)$ for $j=1,2,\cdots,d$ belongs to $S$. For each $\gamma\in\mathbb{N}^{d}$, we define the symbol $\sigma_{\gamma}:\mathbb{R}^{d}\to\mathbb{C}$ as the function: if $x=(x(j))\in(\mathbb{R}\setminus 0)^{d}$, then $\sigma_{\gamma}(x)=i^{\|\gamma\|}x^{\gamma}=\prod_{j=1}^{d}(ix(j))^{\gamma(j)},$ otherwise, $\sigma_{\gamma}(x)=0$. The fundamental polynomial of a smoothness $S$ is $Q_{S}=\sum_{\gamma\in S}\|\sigma_{\gamma}\|^{2},$ which is a non-negative function on $\mathbb{R}^{d}$. The Sobolev space $W^{S}_{p}(\mathbb{T}^{d})$ is defined as the completion of $\mathscr{P}_{d}$ with respect to the norm defined as following: if $f\in\mathscr{P}_{d}$, then (1) $\displaystyle\\|f\\|_{S,p}:=\Big{(}\sum_{\gamma\in S}\\|\partial^{\gamma}f\\|_{L_{p}(\mathbb{T}^{d})}^{p}\Big{)}^{1/p}.$ ###### Remark 2.1. The original definition of $\\|f\\|_{S,p}$ in [3] is $\\|f\\|_{S,p}=\Big{(}\int_{\mathbb{T}^{d}}\Big{(}\sum_{\gamma\in S}\|\partial^{\gamma}f(x)\|^{2}\Big{)}^{p/2}dx\Big{)}^{1/p},$ which is equivalent to our definition since $S$ is a finite set. Let $f\in L_{1}(\mathbb{T}^{d})$, its spectrum $\text{spec}(f)$ is $\text{spec}(f):=\\{n\in\mathbb{Z}^{d}:\hat{f}(n)=\int_{\mathbb{T}^{d}}f(x)e^{-i\langle x,n\rangle}dx\neq 0\\}.$ Let $\Lambda\subset\mathbb{Z}^{d}$ be an infinite subset. The projection $P_{\Lambda}:\mathscr{P}_{d}\rightarrow\mathscr{P}_{d}$ is defined by $P_{\Lambda}f=\sum_{n\in\Lambda}\hat{f}(n)e^{i\langle\cdot,n\rangle}$. ###### Definition 2.2. In the above situation, $P_{\Lambda}$ will be called a Paley projection if there is some $K>0$, such that $\\|P_{\Lambda}f\\|_{S,2}\leq K\\|f\\|_{S,1},\text{\quad for all $f\in\mathscr{P}_{d}$,}$ i.e. for all $f\in\mathscr{P}_{d}$, we have $\Big{(}\sum_{n\in\Lambda}Q_{S}(n)\|\hat{f}(n)\|^{2}\Big{)}^{1/2}\leq K\\|f\\|_{S,1}.$ If $P_{\Lambda}$ is a Paley projection, then the natural mapping $W^{S}_{2}(\mathbb{T}^{d})_{\Lambda}\rightarrow W^{S}_{1}(\mathbb{T}^{d})_{\Lambda}$ is an isomorphism. $P_{\Lambda}$ can be uniquely extended to be an projection on $W^{S}_{1}(\mathbb{T}^{d})$, which is still denoted by $P_{\Lambda}:W^{S}_{1}(\mathbb{T}^{d})\rightarrow W^{S}_{1}(\mathbb{T}^{d}).$ For the operator space theory, we refer to the book [5] for a detailed study. Here we recall that the usual $L_{p}$-spaces are equipped with a natural operator space structure (in short o.s.s. For the detail, see e.g.[5] p.178 -p.180). Hence $W^{S}_{1}(\mathbb{T}^{d})$ is an operator space by the embedding $W^{S}_{1}(\mathbb{T}^{d})\subset\ell_{1}^{\|S\|}(L_{1}(\mathbb{T}^{d}))$. We will use the following useful fact: Let $E\subset L_{1}(\Omega,\mu)$ and $F\subset L_{1}(M,\nu)$ be two operator subspaces. Then a linear operator $u:E\rightarrow F$ is completely bounded iff $u\otimes I_{S_{1}}:E(S_{1})\rightarrow F(S_{1})$ is bounded, where $S_{1}$ is the set of trace class operators and $E(S_{1})$ and $F(S_{1})$ are the closures of $E\otimes S_{1}$ and $F\otimes S_{1}$ in $L_{1}(\Omega,\mu;S_{1})$ and $L_{1}(M,\nu;S_{1})$ respectively. Moreover, $\\|u\\|_{cb}=\\|u\otimes I_{S_{1}}\\|.$ Recall that the operator space $C+R$ is a homogeneous Hilbertian operator space, which is determined by the following fact: if $(e_{k})$ is an orthonormal basis of $C+R$ and $(x_{k})$ is a finite sequence in $S_{1}$, then $\\|\sum_{k}x_{k}\otimes e_{k}\\|_{S_{1}[C+R]}=\inf\\{\\|(\sum_{k}y_{k}^{}y_{k})^{1/2}\\|_{S_{1}}+\\|(\sum_{k}z_{k}z_{k}^{})^{1/2}\\|_{S_{1}}\\},$ where the infimum runs over all possible decompositions $x_{k}=y_{k}+z_{k}$. (For the definition of $S_{1}[E]$, see [4]). For convience, we will denote $\|\|\|(x_{k})\|\|\|:=\\|\sum_{k}x_{k}\otimes e_{k}\\|_{S_{1}[C+R]}.$ The following theorem of Lust-Piquard and Pisier will be used in this note. ###### Theorem 2.3. (Lust-Piquard & Pisier) Let $(n_{k})$ be any increasing sequence which is lacunary à la Hadamard, i.e. $\underline{\lim}\frac{n_{k+1}}{n_{k}}>1$. Then there exists $K>0$, such that for any finite sequence $(x_{k})$ in $S_{1}$, we have (2) $\displaystyle\frac{1}{K}\|\|\|(x_{k})\|\|\|\leq\\|\sum_{k}x_{k}e^{in_{k}t}\\|_{L_{1}(\mathbb{T};S_{1})}\leq K\|\|\|(x_{k})\|\|\|.$ ###### Remark 2.4. Under the same condition as in the above theorem, by the equivalence (2), it is easy to see that if $(a_{k})$ is a bounded sequence in $\mathbb{C}$, then $\\|\sum_{k}a_{k}x_{k}e^{in_{k}t}\\|_{L_{1}(\mathbb{T};S_{1})}\lesssim\\|\sum_{k}x_{k}e^{in_{k}t}\\|_{L_{1}(\mathbb{T};S_{1})}.$ If $(a_{k})$ is moreover uniformly separated from 0, i.e. $\inf_{k}\|a_{k}\|>0$, then $\\|\sum_{k}a_{k}x_{k}e^{in_{k}t}\\|_{L_{1}(\mathbb{T};S_{1})}\approx\\|\sum_{k}x_{k}e^{in_{k}t}\\|_{L_{1}(\mathbb{T};S_{1})}.$ ###### Definition 2.5. A smoothness $S\subset\mathbb{N}^{d}$ is said to have Property (O) if there are $\alpha,\beta\in S$ with $\|\alpha\|\not\equiv\|\beta\|\text{ mod 2}$ and $c=(c(j))$ with $c(j)>0$ such that: * (i) $\langle\alpha,c\rangle=\langle\beta,c\rangle=1$ * (ii) $\langle\gamma,c\rangle\leq 1\text{ for all $\gamma\in S$.}$ ###### Remark 2.6. Assume that $S$ has property (O) and let $\alpha,\beta\in S$ be the two points in $S$ as in the definition of property (O). Then there exists a sequence $(n_{k})\subset\mathbb{N}^{d}$ such that (3) $\displaystyle\lim_{k}\inf_{j}n_{k}(j)=\infty$ and (4) $\displaystyle\rho=\min\\{\inf_{k}\frac{\|\sigma_{\alpha}(n_{k})\|}{Q_{S}(n_{k})^{1/2}},\inf_{k}\frac{\|\sigma_{\beta}(n_{k})\|}{Q_{S}(n_{k})^{1/2}}\\}>0.$ For the proof, see Proposition 1.2 in [3]. We end this section by stating the following technical proposition from [3]. ###### Proposition 2.7. (Pełczyński & Wojciechowski ) Let $S\subset\mathbb{N}^{d}$ be a smoothness. Then given $\varepsilon$ with $0<\varepsilon<1$ and $D=1,2,\cdots$ there exists $\rho=\rho(D,\varepsilon)>1$ such that, for every $m,n\in\mathbb{Z}^{d}$, if $\min_{1\leq j\leq d}\|n(j)\|\geq\rho$ and if $\sum_{j=1}^{d}\|n(j)-m(j)\|\leq D$ then $\|1-Q_{S}(n)Q_{S}(m)^{-1}\|<\varepsilon;$ $\sum_{\alpha\in S}\left\|\frac{\|\sigma_{\alpha}(m)\|}{Q_{S}(m)^{1/2}}-\frac{\|\sigma_{\alpha}(n)\|}{Q_{S}(n)^{1/2}}\right\|^{2}<\varepsilon^{2};$ $\sum_{\alpha\in S}\left\|\frac{\sigma_{\alpha}(m)}{Q_{S}(m)^{1/2}}-\frac{\sigma_{\alpha}(n)}{Q_{S}(n)^{1/2}}\right\|^{2}<\varepsilon^{2}.$ ## 3\. Main result ###### Theorem 3.1. If the smoothness $S$ satisfies Property (O), then there exists a completely bounded Paley projection $P_{\Lambda}:W^{S}_{1}(\mathbb{T}^{d})\to W^{S}_{1}(\mathbb{T}^{d})$ associated to some infinite sequence $\Lambda=(n_{k})\subset\mathbb{Z}^{d}$. Moreover, the linear map $\hat{P}:W^{S}_{1}(\mathbb{T}^{d})_{\Lambda}\rightarrow C+R$ defined by $\hat{P}f=\sum_{k=1}^{\infty}Q_{S}(n_{k})^{1/2}\hat{f}(n_{k})e_{k}$ is a complete isomorphism, where $(e_{k})_{k=1}^{\infty}$ is an orthonomal basis of $C+R$. The following lemma will be used in the proof of Theorem 3.1. ###### Lemma 3.2. Assume that $\Sigma\subset\mathbb{Z}^{d}$ is an infinite subset satisfies the conditions $n(1)\geq 1,\text{ $\forall n\in\Sigma\setminus\\{0\\}$}$ and the projection to the first coordinate $\Sigma\rightarrow\mathbb{N}$ defined by $n\mapsto n(1)$ is injective. Assume moreover that $\Lambda=(n_{k})_{k=1}^{\infty}$ is an infinite sequence in $\Sigma$ such that $\inf_{k}\frac{n_{k}(1)}{n_{k-1}(1)}>1.$ Then the natural map $P_{\Sigma,\Lambda}:L_{1}(\mathbb{T}^{d})_{\Sigma}\rightarrow L_{1}(\mathbb{T}^{d})_{\Lambda}$ is completely bounded and $L_{1}(\mathbb{T}^{d})_{\Lambda}$ is completely isomorphic to $C+R$. ###### Proof. We shall prove that the projection $L_{1}(\mathbb{T}^{d};S_{1})_{\Sigma}\rightarrow L_{1}(\mathbb{T}^{d};S_{1})_{\Lambda}$ is bounded. Let $\Gamma$ be the image of the first projection $\Sigma\to\mathbb{N}$. The injectivity of $n\mapsto n(1)$ on $\Sigma$ implies that there is a map $m:\Gamma\to\mathbb{Z}^{d-1}$ such that $n=(n(1),m(n(1)))$ for all $n\in\Sigma$. We write $x\in\mathbb{T}^{d}$ as a pair $x=(t,y)\in\mathbb{T}\times\mathbb{T}^{d-1}$. To each $y\in\mathbb{T}^{d-1}$ and $g\in L_{1}(\mathbb{T}^{d};S_{1})$ we associate with a function $g_{y}:\mathbb{T}\rightarrow S_{1}$ defined by $g_{y}(t)=g(t,y)$. If $g\in L_{1}(\mathbb{T}^{d};S_{1})_{\Sigma}$, then $g(t,y)\sim\sum_{n\in\Sigma}\hat{g}(n)e^{i\langle(t,y),n\rangle}=\sum_{n(1)\in\Gamma}\hat{g}(n)e^{itn(1)}e^{i\langle y,m(n(1))\rangle}.$ This implies that $\text{spec}(g_{y})\subset\Gamma\subset\mathbb{N}$ and $\hat{g_{y}}(n(1))=\hat{g}(n)e^{i\langle y,m(n(1))\rangle}.$ By [2], as operator space, $L_{1}(\mathbb{T})_{\Gamma}$ is completely isomorphic to $C+R$. Hence for any fixed $y\in\mathbb{T}^{d-1}$, $\\|\sum_{n(1)\in\Gamma}\hat{g}(n)e^{itn(1)}e^{i\langle y,m(n(1))\rangle}\\|_{L_{1}(\mathbb{T};S_{1})}\approx\\|\sum_{n(1)\in\Gamma}\hat{g}(n)e^{itn(1)}\\|_{L_{1}(\mathbb{T};S_{1})}.$ It follows that $\displaystyle\\|g\\|_{L_{1}(\mathbb{T}^{d};S_{1})}$ $\displaystyle=$ $\displaystyle\int_{\mathbb{T}^{d-1}}\\|\sum_{n(1)\in\Gamma}\hat{g}(n)e^{itn(1)}e^{i\langle y,m(n(1))\rangle}\\|_{L_{1}(\mathbb{T};S_{1})}dy$ $\displaystyle\approx$ $\displaystyle\\|\sum_{n(1)\in\Gamma}\hat{g}(n)e^{itn(1)}\\|_{L_{1}(\mathbb{T};S_{1})}$ $\displaystyle=$ $\displaystyle\\|\sum_{n(1)\in\Gamma}\hat{g}(n)e^{itn(1)}\\|_{H^{1}(\mathbb{T};S_{1})}.$ Similarly, we have $\\|P_{\Sigma,\Lambda}g\\|_{L_{1}(\mathbb{T}^{d};S_{1})}\approx\\|\sum_{k=1}^{\infty}\hat{g}(n)e^{itn_{k}(1)}\\|_{H^{1}(\mathbb{T};S_{1})}.$ The sequence $(n_{k}(1))_{k=1}^{\infty}$ is lacunary, thus we can apply Corollary 0.4 in [2] to obtain $\\|\sum_{k=1}^{\infty}\hat{g}(n)e^{itn_{k}(1)}\\|_{H^{1}(\mathbb{T};S_{1})}\lesssim\\|\sum_{n(1)\in\Gamma}\hat{g}(n)e^{itn(1)}\\|_{H^{1}(\mathbb{T};S_{1})}.$ Combining the above inequalities, we have $\\|P_{\Sigma,\Lambda}g\\|_{L_{1}(\mathbb{T}^{d};S_{1})}\lesssim\\|g\\|_{L_{1}(\mathbb{T}^{d};S_{1})}.$ This completes the proof that $P_{\Sigma,\Lambda}:L_{1}(\mathbb{T}^{d})_{\Sigma}\to L_{1}(\mathbb{T}^{d})_{\Lambda}$ is completely bounded. The fact that $L_{1}(\mathbb{T}^{d})_{\Lambda}\approx C+R$ is then easy. Indeed, if $h\in L_{1}(\mathbb{T}^{d};S_{1})_{\Lambda}$, then (5) $\displaystyle\\|h\\|_{L_{1}(\mathbb{T}^{d};S_{1})_{\Lambda}}\approx\\|\sum_{k=1}^{\infty}\hat{h}(n)e^{itn_{k}(1)}\\|_{H^{1}(\mathbb{T};S_{1})}\approx\|\|\|(\hat{h}(n))\|\|\|.$ In other words, $L_{1}(\mathbb{T}^{d})_{\Lambda}\simeq C+R$ completely isomorphically. ∎ ###### Remark 3.3. In the situation of Lemma 3.2, the map $\widetilde{P}_{\Sigma,\Lambda}:L_{1}(\mathbb{T}^{d})_{\Sigma}\rightarrow C+R$ defined by $\widetilde{P}_{\Sigma,\Lambda}f=\sum_{k=1}^{\infty}\hat{f}(n_{k})e_{k}$, where $e_{k}$ is an orthonormal basis of $C+R$, is completely bounded. ###### Proof of Theorem 3.1. Our proof follows the proof of Proposition 2.2 in [3]. Let $\alpha,\beta\in S$ and $(n_{k})\in\mathbb{N}^{d}$ be as in Remark 2.6. Since $\|\alpha\|\not\equiv\|\beta\|\text{ mod 2,}$ one can assume that for all $k$, $\text{sign}(\frac{\sigma_{\alpha}(n_{k})}{\sigma_{\beta}(n_{k})})=i^{\|\alpha\|-\|\beta\|}=\tau,$ $\text{sign}(\frac{\sigma_{\alpha}(-n_{k})}{\sigma_{\beta}(-n_{k})})=(-i)^{\|\alpha\|-\|\beta\|}=-\tau.$ Here $\text{sign}(z):=\frac{z}{\|z\|}$ for $z\in\mathbb{C}\setminus 0$. Replacing, if necessary, the sequence $(n_{k})$ by a rapidly increasing subsequence, we can assume without loss of generality that the sequence $(n_{k})$ satisfies the conditions: * (i) $\sum_{r=1}^{k-1}\sum_{j=1}^{d}n_{r}(j)<\min_{j}n_{k}(j)\text{ for $k=2,3,\cdots,$}$ * (ii) $\lim_{k}\frac{\|\sigma_{\alpha}(-n_{k})\|}{\|\sigma_{\beta}(-n_{k})\|}=\lim_{k}\frac{\|\sigma_{\alpha}(n_{k})\|}{\|\sigma_{\beta}(n_{k})\|}=\ell>0,$ * (iii) $\sum_{k=1}^{\infty}\|\sigma_{\alpha}(-n_{k})+\tau\ell\sigma_{\beta}(-n_{k})\|Q_{S}(n_{k})^{-1/2}\\\ =\sum_{k=1}^{\infty}\Big{\|}\|\sigma_{\alpha}(n_{k})\|-\ell\|\sigma_{\beta}(n_{k})\|\Big{\|}Q_{S}(n_{k})^{-1/2}<\frac{1}{2},$ * (iv) $\sum_{k=1}^{\infty}\sum_{m\in B_{k}}\|\sigma_{\alpha}(-m)+\tau\ell\sigma_{\beta}(-m)\|Q_{S}(-m)^{1/2}<1$, where $B_{1}=\\{n_{1}\\}$ and for $k=2,3,\cdots,$ $B_{k}=\Big{\\{}m\in\mathbb{Z}^{d}:\sum_{j=1}^{d}\|m(j)-n_{k}(j)\|\leq\sum_{r=1}^{k-1}\sum_{j=1}^{d}n_{r}(j)\Big{\\}}.$ Notice that item (iv) follows from (iii) , Proposition 2.7 and also the assumption that $(n_{k})$ increase sufficiently fast. Define $M:W^{S}_{1}(\mathbb{T}^{d})\rightarrow L_{1}(\mathbb{T}^{d})$ by $Mf=\partial^{\alpha}f+\tau\ell\partial^{\beta}f-\sum_{k=1}^{\infty}\sum_{m\in B_{k}}(\sigma_{\alpha}(-m)+\tau\ell\sigma_{\beta}(-m))\hat{f}(-m)e^{-i\langle\cdot,m\rangle}.$ Then $M$ is completely bounded. Indeed, consider the map $M\otimes I_{S_{1}}:W^{S}_{1}(\mathbb{T}^{d};S_{1})\rightarrow L_{1}(\mathbb{T}^{d};S_{1})$. If $g\in W^{S}_{1}(\mathbb{T}^{d};S_{1})$, then $\\|\partial^{\alpha}g\\|_{L_{1}(\mathbb{T}^{d};S_{1})}+\\|\partial^{\beta}g\\|_{L_{1}(\mathbb{T}^{d};S_{1})}\leq\\|g\\|_{W^{S}_{1}(\mathbb{T}^{d};S_{1})}.$ Remember that $\sigma_{\gamma}(n)\hat{g}(n)=\int_{\mathbb{T}^{d}}\partial^{\gamma}g(x)e^{-i\langle x,n\rangle}dx$, hence for any $\gamma\in S$, $\\|\sigma_{\gamma}(n)\hat{g}(n)\\|_{S_{1}}\leq\\|\partial^{\gamma}g\\|_{L_{1}(\mathbb{T}^{d};S_{1})}\leq\\|g\\|_{W^{S}_{1}(\mathbb{T}^{d};S_{1})}.$ Hence $Q_{S}(n)^{1/2}\\|\hat{g}(n)\\|_{S_{1}}\leq\|S\|\cdot\\|g\\|_{W^{S}_{1}(\mathbb{T}^{d};S_{1})}.$ Combining with (iv), we have $\displaystyle\\|\sum_{k=1}^{\infty}\sum_{m\in B_{k}}(\sigma_{\alpha}(-m)+\tau\ell\sigma_{\beta}(-m))\hat{g}(-m)e^{-i\langle\cdot,m\rangle}\\|_{L_{1}(\mathbb{T}^{d};S_{1})}$ $\displaystyle\leq$ $\displaystyle\|S\|\cdot\\|g\\|_{W^{S}_{1}(\mathbb{T}^{d};S_{1})}.$ Hence $M\otimes I_{S_{1}}$ is bounded and $M$ is completely bounded. Next, consider the measure $\mu_{R}$ on $\mathbb{T}^{d}$ given by the Riesz product $R(x)=\prod_{k=1}^{\infty}(1+cos\langle x,n_{k}\rangle)=\prod_{k=1}^{\infty}(1+\frac{1}{2}e^{i\langle x,n_{k}\rangle}+\frac{1}{2}e^{-i\langle x,n_{k}\rangle}).$ Then the convolution map $M_{R}:L_{1}(\mathbb{T}^{d})\rightarrow L_{1}(\mathbb{T}^{d})$ defined by $M_{R}f=f\ast\mu_{R}$ is obviously completely contractive. In the definition of this Riesz product, we assume that $\frac{n_{k+1}(1)}{n_{k}(1)}\geq 3$, for all $k=1,2,\cdots$. Notice that we have $\text{spec}(\mu_{R})=\Big{\\{}d_{1}n_{1}+d_{2}n_{2}+\cdots+d_{k}n_{k}:k\in\mathbb{N},d_{1},d_{2},\cdots d_{k}\in\\{-1,0,1\\}\Big{\\}}.$ Claim A: $\text{spec}(\mu_{R})\subset\\{0\\}\cup\bigcup_{k=1}^{\infty}(B_{k}\cup(-B_{k})).$ Indeed, if $m\in\text{spec}(\mu_{R})\setminus\\{0\\}$, then there exist $k\geq 1$ and $d_{1},d_{2},\cdots,d_{k}\in\\{-1,0,1\\}$, such that $d_{k}\neq 0$ and $m=d_{1}n_{1}+d_{2}n_{2}+\cdots+d_{k}m_{k}$. Replacing $m$ by $-m$, if necessary, one may assume that $d_{k}=1$, then $m-n_{k}=\sum_{r=1}^{k-1}d_{r}n_{r}$, it follows that $\sum_{j=1}^{d}\|m(j)-n_{k}(j)\|\leq\sum_{r=1}^{k-1}\sum_{j=1}^{d}n_{r}(j)$, i.e. $m\in B_{k}$. Claim B: The projection to the first coordinate $\text{spec}(\mu_{R})\rightarrow\mathbb{Z}$ is injective. Indeed, if $n,m\in\text{spec}(\mu_{R})$ such that $n(1)=m(1)$, suppose that $n=d_{1}n_{1}+d_{2}n_{2}+\cdots+d_{k}n_{k}$ and $m=d_{1}^{\prime}n_{1}+d_{2}^{\prime}n_{2}+\cdots d_{k^{\prime}}^{\prime}n_{k^{\prime}}$, then by a simple computation (cf. e.g. [1]), we have $k=k^{\prime}$ and $d_{1}=d_{1}^{\prime}$, $d_{2}=d_{2}^{\prime},\cdots,d_{k}=d_{k}^{\prime}$, hence $n=m$. In other words, the projection to the first coordinate $\text{spec}(\mu_{R})\rightarrow\mathbb{Z}$ is injective. Let $\Sigma=\\{0\\}\cup\bigcup_{k=1}^{\infty}B_{k}$. It can be easily checked that the image $\text{Im}(M_{R}M)$ of the composition operator $M_{R}M$ is contained in $L_{1}(\mathbb{T}^{d})_{\Sigma}$. By the definition of $B_{k}$ and condition (i) on the sequence $(n_{k})$ , if $m\in B_{k}$, then $m(1)\geq n_{k}(1)-\sum_{r=1}^{k-1}\sum_{j=1}^{d}n_{r}(j)>0.$ We are now in the situation of Lemma 3.2, thus we obtain a completely bounded projection $P_{\Sigma,\Lambda}:L_{1}(\mathbb{T}^{d})_{\Sigma}\rightarrow L_{1}(\mathbb{T}^{d})_{\Lambda}$. By composition, we obtain the following completely bounded map $P_{\Sigma,\Lambda}M_{R}M:W^{S}_{1}(\mathbb{T}^{d})\rightarrow L_{1}(\mathbb{T}^{d})_{\Lambda}.$ By computation, we have $\displaystyle P_{\Sigma,\Lambda}f=\sum_{k=1}^{\infty}\rho_{k}Q_{S}(n_{k})^{1/2}\hat{f}(n_{k})e^{i\langle\cdot,n_{k}\rangle},$ where $\rho_{k}=\frac{\sigma_{\alpha}(n_{k})+\tau\ell\sigma_{\beta}(n_{k})}{2Q_{S}(n_{k})^{1/2}}$. By (4), $\|\rho_{k}\|=\frac{1}{2}(\|\sigma_{\alpha}(n_{k})\|+\ell\|\sigma_{\beta}(n_{k})\|)Q_{S}(n_{k})^{1/2}\geq\frac{1}{2}\rho(1+\ell).$ On the other hand, it is obvious that $\|\rho_{k}\|\leq\frac{1}{2}(1+\ell)$. Let $g:\mathbb{T}^{d}\rightarrow S_{1}$, then $\displaystyle\\|P_{\Lambda}g\\|_{W^{S}_{1}(\mathbb{T}^{d};S_{1})}$ $\displaystyle=$ $\displaystyle\sum_{\gamma\in S}\\|\sum_{k=1}^{\infty}\sigma_{\gamma}(n_{k})\hat{g}(n_{k})e^{i\langle\cdot,n_{k}\rangle}\\|_{L_{1}(\mathbb{T}^{d};S_{1})}$ By (5) $\displaystyle\approx$ $\displaystyle\sum_{\gamma\in S}\\|\sum_{k=1}^{\infty}\sigma_{\gamma}(n_{k})\hat{g}(n_{k})e^{itn_{k}(1)}\\|_{L_{1}(\mathbb{T};S_{1})}$ By Remark 2 $\displaystyle\lesssim$ $\displaystyle\sum_{\gamma\in S}\\|\sum_{k=1}^{\infty}Q_{S}(n_{k})^{1/2}\hat{g}(n_{k})e^{itn_{k}(1)}\\|_{L_{1}(\mathbb{T};S_{1})}$ By Remark 2 and $\|S\|<\infty$ $\displaystyle\approx$ $\displaystyle\\|\sum_{k=1}^{\infty}\rho_{k}Q_{S}(n_{k})^{1/2}\hat{g}(n_{k})e^{itn_{k}(1)}\\|_{L_{1}(\mathbb{T};S_{1})}$ By (5) $\displaystyle\approx$ $\displaystyle\\|\sum_{k=1}^{\infty}\rho_{k}Q_{S}(n_{k})^{1/2}\hat{g}(n_{k})e^{i\langle\cdot,n_{k}\rangle}\\|_{L_{1}(\mathbb{T}^{d};S_{1})}$ $\displaystyle=$ $\displaystyle\\|(P_{\Sigma,\Lambda}M_{R}M\otimes I_{S_{1}})g\\|_{L_{1}(\mathbb{T}^{d};S_{1})}$ $\displaystyle\lesssim$ $\displaystyle\\|g\\|_{W^{S}_{1}(\mathbb{T}^{d};S_{1})}.$ This completes the proof that $P_{\Lambda}:W^{S}_{1}(\mathbb{T}^{d})\rightarrow W^{S}_{1}(\mathbb{T}^{d})$ is completely bounded. For the second assertion of the theorem, we only need to notice that by Remark 2.6, $\|\sigma_{\alpha}(n_{k})\|\geq\rho Q_{S}(n_{k})^{1/2}$ for all $k$, hence if $g\in W^{S}_{1}(\mathbb{T}^{d};S_{1})_{\Lambda}$, then $\displaystyle\\|g\\|_{W^{S}_{1}(\mathbb{T}^{d};S_{1})}$ $\displaystyle\geq$ $\displaystyle\\|\partial^{\alpha}g\\|_{L_{1}(\mathbb{T}^{d};S_{1})}=\\|\sum_{k=1}^{\infty}\sigma_{\alpha}(n_{k})\hat{g}(n_{k})e^{i\langle\cdot,n_{k}\rangle}\\|_{L_{1}(\mathbb{T}^{d};S_{1})}$ $\displaystyle\gtrsim$ $\displaystyle\\|\sum_{k=1}^{\infty}Q_{S}(n_{k})^{1/2}\hat{g}(n_{k})e^{i\langle\cdot,n_{k}\rangle}\\|_{L_{1}(\mathbb{T}^{d};S_{1})}$ $\displaystyle\approx$ $\displaystyle\\|\sum_{k=1}^{\infty}Q_{S}(n_{k})^{1/2}\hat{g}(n_{k})\otimes e_{k}\\|_{S_{1}[C+R]}.$ ∎ Using Theorem 3.1, then by a classical transference method, we have the following corollary, for the definition of quantum torus $\mathbb{T}_{\theta}^{d}$ and harmonic analysis on it, we refer to the paper [6]. ###### Corollary 3.4. Under the same condition of Theorem 3.1, there exists a completely bounded Paley projection $P_{\Lambda}:W^{S}_{1}(\mathbb{T}^{d}_{\theta})\rightarrow W^{S}_{1}(\mathbb{T}^{d}_{\theta})$ associated to some infinite sequence $\Lambda=(n_{k})\subset\mathbb{Z}^{d}$. ## Acknowledgements The author would like to thank Quanhua Xu for inviting him to Université Franche-Comté and his constant encouragement. ## References * [1] F. R. Keogh. Riesz products. Proc. London Math. Soc. (3), 14a:174–182, 1965. * [2] Françoise Lust-Piquard and Gilles Pisier. Noncommutative Khintchine and Paley inequalities. Ark. Mat., 29(2):241–260, 1991. * [3] A. Pełczyński and M. Wojciechowski. Paley projections on anisotropic Sobolev spaces on tori. Proc. London Math. Soc. (3), 65(2):405–422, 1992. * [4] Gilles Pisier. Non-commutative vector valued $L_{p}$-spaces and completely $p$-summing maps. Astérisque, 247:vi+131, 1998. * [5] Gilles Pisier. Introduction to operator space theory, volume 294 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 2003. * [6] Z. Chen, Q. Xu and Z. Yin. Harmonic analysis on quantum tori. http://arxiv.org/abs/1206.3358	arxiv-papers	2013-12-22T15:51:48	2024-09-04T02:49:55.803867	{ "license": "Public Domain", "authors": "Yanqi Qiu", "submitter": "Yanqi Qiu", "url": "https://arxiv.org/abs/1312.6394" }
1312.6396	# Approximate Dynamical Systems K. R. W. Jones Physics Department, University of Queensland, St Lucia 4072, Brisbane, Australia. (December 1992) ###### Abstract Working notes on setting up approximate dynamical systems and nonlinear eigenvalue problems, here embedded within the theory of complex nonlinear dynamics. Computations parallel those of linear quantum theory except that we use functional methods rather than Hilbert space. ††preprint: UQ Theory: December 1992 ## I Introduction This covers basic concepts behind the theory of approximate dynamical systemsjones , the theory of generalized quantum dynamics, functional methods, theory of propagators, and the basics of nonlinear spectral theory and nonlinear functional analysis. It begins with the key idea of the subject, namely the use of action principles to define, and obtain approximate dynamical systems. Subsequently we see some examples of useful systems of this kind that generate nonlinear equations. ## II Action principles and approximation theory What we really need, to make the business of practical computations efficient, is a larger mathematics which contains both exact and approximate equations of motion — treated in a unified manner. This is best done working from the principle of least action. At the deepest level, physical theories when considered as dynamical systems, derive from action principles. It easy to state these in complete generality, as a shell into which we plug a Lagrangian and turn the crank. Here we concentrate upon equations of motion as “solutions” of the variational problem $\frac{\delta}{\delta x(t)}\int_{t=t_{0}}^{t=t_{f}}L[x,\dot{x},t]\,dt=0.$ (1) The lovely thing about an action principle, when we look at it this way, is that it provides a recipe for constructing new equations of motion that are the result of replacing $L[x,\dot{x},t]$ by some conveniently chosen approximation $L_{\rm app}[x,\dot{x},t]\approx L[x,\dot{x},t]$. If we make sure that our system of mathematics is large enough to capably handle all useful kinds of approximation, then — once we have formalized these in the abstract — we will obtain an entire new system of generalized quantum dynamics. So, the goal is to replace exact action principles by approximate action principles and so obtain entire approximate dynamical systems. Then we look at these, study and classify them, the better to understand their particular merits and deficiencies. ## III Decorrelation as a standard approximation Practical approximations are designed to leave some effect out to make things simple. In quantum theory the one generic effect which makes the theory hard to calculate with, and vizualize, is quantum correlation and quantum entanglement. In the theory of approximate dynamical systems we use some simple tricks to suppress this effect and simplify things. Here is one simple semi–classical example $\langle\frac{\hat{p}^{2}}{2m}+k\hat{x}^{2}\rangle\approx\frac{\langle\hat{p}\rangle^{2}}{2m}+k\langle\hat{x}\rangle^{2},$ (2) where the quantum expectation is replaced by its semi–classical counterpart. A familiar many–body example is the Hartree approximation $\displaystyle\int\psi^{}({\bf x}_{1},{\bf x}_{2})V(\|{\bf x}_{1}-{\bf x}_{2}\|)\psi({\bf x}_{1},{\bf x}_{2})\,d^{3}{\bf x}_{1}d^{3}{\bf x}_{2}\approx$ (3) $\displaystyle\hskip 85.35826pt\int\psi^{}_{1}({\bf x}_{1})\psi^{}_{2}({\bf x}_{2})V(\|{\bf x}_{1}-{\bf x}_{2}\|)\psi_{1}({\bf x}_{1})\psi_{2}({\bf x}_{2})\,d^{3}{\bf x}_{1}d^{3}{\bf x}_{2},$ where $\psi({\bf x}_{1},{\bf x}_{2})$ has been replaced by a factorized pair of wave–functions. In both cases we neglect correlations, or enforce disentanglement, and so modify the degree of the original expression in $\psi$ and $\psi^{}$. It is this modification of degree which is the cause of induced nonlinearity, as we now see with a simple example. ## IV Classical Schrödinger equation The easiest way to express the correspondence between classical and quantum physics is via the Ehrenfest theorem. Starting with the quantum equations: $\frac{d\langle\hat{p}\rangle}{dt}=-\langle H_{x}(\hat{x},\hat{p})\rangle,\;\;\mbox{ and }\;\;\frac{d\langle\hat{x}\rangle}{dt}=+\langle H_{p}(\hat{x},\hat{p})\rangle;$ (4) we introduce the obvious semi–classical approximation: $\langle H_{x}(\hat{x},\hat{p})\rangle\approx H_{x}(\langle\hat{x}\rangle,\langle\hat{p}\rangle),\;\;\mbox{ and }\;\;\langle H_{p}(\hat{x},\hat{p})\rangle\approx H_{p}(\langle\hat{x}\rangle,\langle\hat{p}\rangle);$ (5) and thus obtain the approximate equations: $\frac{d\langle\hat{p}\rangle}{dt}\approx- H_{q}(\langle\hat{x}\rangle,\langle\hat{p}\rangle),\;\;\mbox{ and }\;\;\frac{d\langle\hat{x}\rangle}{dt}\approx+H_{p}(\langle\hat{x}\rangle,\langle\hat{p}\rangle).$ (6) If we now take these as defining a new dynamical system (i.e. we replace $\approx$ by $=$) then our equations reduce to those of Hamilton, $\frac{dP}{dt}=-H_{x}(X,P),\;\;\mbox{ and }\;\;\frac{dX}{dt}=+H_{p}(X,P);$ (7) where we make the obvious identification: $X(t)=\langle\hat{x}\rangle(t)\;\;\mbox{ and }\;\;P(t)=\langle\hat{p}\rangle(t).$ (8) In taking these steps one reduces the quantum problem to a classical problem, in a manner that ignores certain features of the full quantum treatment. Now let us apply this analysis of the Ehrenfest theorem, as a decorrelation approximation, at the general level of the exact quantum action principle $\frac{\delta}{\delta\psi^{}}\int i\hbar\langle\psi\|\frac{d}{dt}\|\psi\rangle-\langle\psi\|\hat{H}(\hat{x},\hat{p})\|\psi\rangle\,dt=0.$ (9) Taking variations with this we obtain $i\hbar\frac{d}{dt}\|\psi\rangle=\hat{H}(\hat{x},\hat{p})\|\psi\rangle,$ (10) as the general equation of motion. However, we could just as well substitute $\langle\psi\|\hat{H}(\hat{x},\hat{p})\|\psi\rangle\approx\langle\psi\|H(\langle\hat{x}\rangle,\langle\hat{p}\rangle)\|\psi\rangle.$ (11) for the energy expectation, and so obtain directly a decorrelated classical wave–equation. There is, however, a minor subtlelty to carrying out this program. In (11) it is not guaranteed that the action principle remains invariant to a re–normalization of $\psi$. Obviously we want to retain that freedom to adjust and preserve normalization. To overcome this difficulty we rescale all coordinate expectations as: $\langle\hat{x}\rangle=\langle\psi\|\hat{x}\|\psi\rangle/n\;\;\mbox{and}\;\;\langle\hat{p}\rangle=\langle\psi\|\hat{p}\|\psi\rangle/n,$ (12) where $n=\langle\psi\|\psi\rangle$. Calculating variational derivatives we find $\frac{\delta\langle\hat{x}\rangle}{\delta\psi^{}}=n^{-1}(\hat{x}-\langle\hat{x}\rangle)\|\psi\rangle,\;\;\mbox{and}\;\;\frac{\delta\langle\hat{p}\rangle}{\delta\psi^{}}=n^{-1}(\hat{p}-\langle\hat{p}\rangle)\|\psi\rangle.$ (13) Invoking now the approximate action principle $\delta\int i\hbar\langle\psi\|\frac{d}{dt}\|\psi\rangle-\langle\psi\|H(\langle\hat{x}\rangle,\langle\hat{p}\rangle)\|\psi\rangle\,dt=0,$ (14) we use the chain rule $\frac{\delta}{\delta\psi^{}}\left[\langle\psi\|H(\langle\hat{x}\rangle,\langle\hat{p}\rangle)\|\psi\rangle\right]=H(\langle\hat{x}\rangle,\langle\hat{p}\rangle)\frac{\delta n}{\delta\psi^{}}+H_{x}(\langle\hat{x}\rangle,\langle\hat{p}\rangle)\frac{\delta\langle\hat{x}\rangle}{\delta\psi^{}}+H_{p}(\langle\hat{x}\rangle,\langle\hat{p}\rangle)\frac{\delta\langle\hat{p}\rangle}{\delta\psi^{}},$ to obtain the approximate equation of motion $i\hbar\frac{d}{dt}\|\psi\rangle=\left\\{H(\langle\hat{x}\rangle,\langle\hat{p}\rangle)\hat{1}+H_{x}(\langle\hat{x}\rangle,\langle\hat{p}\rangle)(\hat{x}-\langle\hat{x}\rangle)+H_{p}(\langle\hat{x}\rangle,\langle\hat{p}\rangle)(\hat{p}-\langle\hat{p}\rangle)\right\\}\|\psi\rangle.$ (15) This is the classical Schrödinger equation, which recovers the Ehrenfest equations of motion in classical form. It propagates wave–packets neglecting dispersion and correlation. The result is that they bounce off barriers and the like just like classical particles. One can construct exact solutions of the above nonlinear integrodifferential equation. To do this we first solve the classical problem to find $X(t)$ and $P(t)$. Next we take any wavefunction $\psi_{0}(x)$ having both position and momentum expectation values equal to zero. Then we form the time–dependent wavefunction $\psi(x,t)=e^{\frac{i}{\hbar}\int_{t_{0}}^{t}L\,d\tau}e^{-iP(t)X(t)/2\hbar}e^{iP(t)x/\hbar}\psi_{0}(x-X(t)),$ (16) where the exact classical action $\int_{t_{0}}^{t}L\,d\tau=\int_{t_{0}}^{t}\left(\frac{P\dot{X}-X\dot{P}}{2}\right)-H(X,P)\,d\tau$ (17) appears as the leading phase factor (showing that the Feynmann–Dirac correspondence is semi–classically exact). This argument can be made constructive, but it is much easier to verify by substitution. Alternatively, given a theory of nonlinear propagators one can set up these equations on a computer and solve them directly to verify this general solution. ## V Physical interpretation When dealing with approximate dynamical systems we must remember that linearity is vital to the Copenhagen interpretation. However, we use nonlinear wave–equations all the time in physics. To interpret them we adopt a computational algorithm viewpoint. We have an exact theory, and quantities that we wish to calculate — e.g. eigenvalues, stationary and time–dependent wavefunctions, expectation values, transition probabilities etc. These we could calculate exactly or approximately. Either way we can apply a physical interpretation that presupposes linearity as an exact property of nature. To the approximately computed, i.e. nonlinearly evolved, physical quantities we apply the Copenhagen interpretation — on the understanding that there is supposed to be an error in our treatment somewhere.111Of course, ultimately the matter of what is correct rests with experiment. For instance, in solving our approximate classical equations we have no need of $\hbar$, nor any specific wavefunction. The solution of the reduced, and thus simplified, problem requires only the initial expectation values. It is thus an approximate method for computing $\langle\hat{x}\rangle(t)$ and $\langle\hat{p}\rangle(t)$. The errors committed are identical, both numerically and conceptually, to those of the familiar Hamiltonian dynamics. Even so, it is pretty useful. One could say the existence of the classical Schrödinger equation, as an excellent semiclassical approximation, explains why classical dynamics fooled us for 300 years! ## References (1) K.R.W. Jones, in prep.	arxiv-papers	2013-12-22T15:59:26	2024-09-04T02:49:55.810533	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "K.R.W. Jones", "submitter": "Kingsley Jones", "url": "https://arxiv.org/abs/1312.6396" }
1312.6431	# Complex short pulse and coupled complex short pulse equations Bao-Feng Feng Department of Mathematics The University of Texas-Pan American Edinburg, TX, 78541-2999, USA ###### Abstract In the present paper, we propose a complex short pulse equation and a coupled complex short equation to describe ultra-short pulse propagation in optical fibers. They are integrable due to the existence of Lax pairs and infinite number of conservation laws. Furthermore, we find their multi-soliton solutions in terms of pfaffians by virtue of Hirota’s bilinear method. One- and two-soliton solutions are investigated in details, showing favorable properties in modeling ulta-short pulses with a few optical cycles. Especially, same as the coupled nonlinear Schrödinger equation, there is an interesting phenomenon of energy redistribution in soliton interactions. It is expected that, for the ultra-short pulses, the complex and coupled complex short pulses equation will play the same roles as the nonlinear Schrödinger equation and coupled nonlinear Schrödinger equation. Keywords: Complex short pulse equation; Coupled complex short pulse equation; Hirota bilinear method; Pfaffian; Envelope soliton; Soliton interaction. ## 1 Introduction The nonlinear Schrödinger (NLS) equation, as one of the universal equations that describe the evolution of slowly varying packets of quasi-monochromatic waves in weakly nonlinear dispersive media, has been very successful in many applications such as nonlinear optics and water waves [1, 2, 3, 4]. The NLS equation is integrable, which can be solved by the inverse scattering transform [5]. However, in the regime of ultra-short pulses where the width of optical pulse is in the order of femtosecond ($10^{-15}$ s), the NLS equation becomes less accurate [6]. Description of ultra-short processes requires a modification of standard slow varying envelope models based on the NLS equation. There are usually two approaches to meet this requirement in the literature. The first one is to add several higher-order dispersive terms to get higher-order NLS equation [2]. The second one is to construct a suitable fit to the frequency- dependent dielectric constant $\epsilon(\omega)$ in the desired spectral range. Several models have been proposed by this approach including the short- pulse (SP) equation [7, 8, 9, 10]. Recently, Schäfer and Wayne derived a so-called short pulse (SP) equation [7] $u_{xt}=u+\frac{1}{6}\left(u^{3}\right)_{xx}\,,$ (1) to describe the propagation of ultra-short optical pulses in nonlinear media. Here, $u=u(x,t)$ is a real-valued function, representing the magnitude of the electric field, the subscripts $t$ and $x$ denote partial differentiation. Apart from the context of nonlinear optics, the SP equation has also been derived as an integrable differential equation associated with pseudospherical surfaces [11]. The SP equation has been shown to be completely integrable [11, 12, 13, 14, 15]. The periodic and soliton solutions of the SP equation were found in [16, 17, 18]. The connection between the SP equation and the sine- Gordon equation through the hodograph transformation was clarified, and then the $N$-soliton solutions including multi-loop and multi-breather ones were given in [19, 20] by using the Hirota bilinear method [21]. The integrable discretization of the SP equation was studied in [22], the geometric interpretation of the SP equation, as well as its integrable discretization, was given in [23]. The higher-order corrections to the SP equation was studied in [24] most recently. Similar to the case of the NLS equation [25], it is necessary to consider its two-component or multi-component generalizations of the SP equation for describing the effects of polarization or anisotropy. As a matter of fact, several integrable coupled short pulse have been proposed in the literature [26, 27, 28, 29, 30, 31]. Most recently, the bi-Hamiltonian structures for the above two-component SP equations were obtained by Brunelli [32]. In the present paper, we propose and study a complex short pulse (CSP) equation $q_{xt}+q+\frac{1}{2}\left(\|q\|^{2}q_{x}\right)_{x}=0\,,$ (2) and its two-component generalization $\displaystyle q_{1,xt}+q_{1}+\frac{1}{2}\left((\|q_{1}\|^{2}+\|q_{2}\|^{2})q_{1,x}\right)_{x}=0\,,$ (3) $\displaystyle q_{2,xt}+q_{2}+\frac{1}{2}\left((\|q_{1}\|^{2}+\|q_{2}\|^{2})q_{2,x}\right)_{x}=0\,.$ (4) As will be revealed in the present paper, both the CSP equation and its two- component generalization are integrable guaranteed by the existence of Lax pairs and infinite number of conservation laws. They have $N$-soliton solutions which can be constructed via Hirota’s bilinear method. The outline of the present paper is organized as follows. In section 2, we derive the CSP equation and coupled complex short pulse (CCSP) equation from the physical context. In section 3, by providing the Lax pairs, the integrability of the proposed two equations are confirmed, and further, the conservation laws, both local and nonlocal ones, are investigated. Then $N$-soliton solutions to both the CSP and CCSP equations are constructed in terms of pfaffians by Hirota’s bilinear method in section 4. In section 5, soliton-interaction for coupled complex short pulse equation is investigated in details, which shows rich phenomena similar to the coupled nonlinear Schrödinger equatoin. In particular, they may undergo either elastic or inelastic collision depending on the initial conditions. For inelastic collisions, there is an energy exchange between solitons, which can allow the generation or vanishing of soliton. The dynamics is more richer in compared with the single component case. The paper is concluded by comments and remarks in section 6. ## 2 The derivation of the complex short pulse and coupled complex short pulse equations In this section, following the procedure in [2, 7], we derive the complex short pulse equation (2) and its two-component generalization that governs the propagation of ultra short pulse packet along optical fibers. ### 2.1 The complex short pulse equation We start with a wave equation for electric field $\nabla^{2}\mathbf{E}-\frac{1}{c^{2}}\mathbf{E}_{tt}=\mu_{0}\mathbf{P}_{tt}\,,$ (5) originated from the Maxwell equation. Here $\mathbf{E}(\mathbf{r},t)$ and $\mathbf{P}(\mathbf{r},t)$ represent the electric field and the induced polarization, respectively, $\mu_{0}$ is the vacuum permeability, $c$ is the speed of light in vacuum. If we assume the local medium response and only the third-order nonlinear effects governed by $\chi^{(3)}$, the induced polarization consists of two parts, $\mathbf{P}(\mathbf{r},t)=\mathbf{P}_{L}(\mathbf{r},t)+\mathbf{P}_{NL}(\mathbf{r},t)$, where the linear part $\mathbf{P}_{L}(\mathbf{r},t)=\epsilon_{0}\int_{-\infty}^{\infty}\chi^{(1)}(t-t^{\prime})\cdot\mathbf{E}(\mathbf{r},t^{\prime})\,dt^{\prime}\,,$ (6) and the nonlinear part $\mathbf{P}_{NL}(\mathbf{r},t)=\epsilon_{0}\int_{-\infty}^{\infty}\chi^{(3)}(t-t_{1},t-t_{2},t-t_{3})\times\mathbf{E}(\mathbf{r},t_{1})\mathbf{E}(\mathbf{r},t_{2})\mathbf{E}(\mathbf{r},t_{3})\,dt_{1}dt_{2}dt_{3}\,.$ (7) Here $\epsilon_{0}$ is the vacuum permittivity and $\chi^{(j)}$ is the $j$th- order susceptibility. Since the nonlinear effects are relatively small in silica fibers, $\mathbf{P}_{NL}$ can be treated as a small perturbation. Therefore, we first consider (5) with $\mathbf{P}_{NL}=0$. Furthermore, we restrict ourselves to the case that the optical pulse maintains its polarization along the optical fiber, and the transverse diffraction term $\Delta_{\perp}\mathbf{E}$ can be neglected. In this case, the electric field can be considered to be one-dimensional and expressed as $\mathbf{E}=\frac{1}{2}\mathbf{e_{1}}\left(E(z,t)+c.c.\right)\,,$ (8) where $\mathbf{e_{1}}$ is a unit vector in the direction of the polarization, $E(z,t)$ is the complex-valued function, and $c.c.$ stands for the complex conjugate. Conducting a Fourier transform on (5) leads to the Helmholtz equation $\tilde{E}_{zz}(z,\omega)+\epsilon(\omega)\frac{\omega^{2}}{c^{2}}\tilde{E}(z,\omega)=0\,,$ (9) where $\tilde{E}(z,\omega)$ is the Fourier transform of $E(z,t)$ defined as $\tilde{E}(z,\omega)=\int_{-\infty}^{\infty}{\ E}(z,t)e^{\mathrm{i}\omega t}\,dt\,,$ (10) $\epsilon(\omega)$ is called the frequency-dependent dielectric constant defined as $\epsilon(\omega)=1+\tilde{\chi}^{(1)}(\omega)\,,$ (11) where $\tilde{\chi}^{(1)}(\omega)$ is the Fourier transform of $\chi^{(1)}(t)$ $\tilde{\chi}^{(1)}(\omega)=\int_{-\infty}^{\infty}\chi^{(1)}(t)e^{\mathrm{i}\omega t}\,dt\,.$ (12) Now we proceed to the consideration of the nonlinear effect. Assuming the nonlinear response is instantaneous so that $P_{NL}$ is given by $P_{NL}(z,t)=\epsilon_{0}\epsilon_{NL}E(z,t)$ [2] where the nonlinear contribution to the dielectric constant is defined as $\epsilon_{NL}=\frac{3}{4}\chi^{(3)}_{xxxx}\|E(z,t)\|^{2}\,.$ (13) In this case, the Helmholtz equation (9) can be modified as $\tilde{E}_{zz}(z,\omega)+\tilde{\epsilon}(\omega)\frac{\omega^{2}}{c^{2}}\tilde{E}(z,\omega)=0\,,$ (14) where $\tilde{\epsilon}(\omega)=1+\tilde{\chi}^{(1)}(\omega)+\epsilon_{NL}\,.$ (15) As pointed out in [7, 3, 8], the Fourier transform $\tilde{\chi}^{(1)}$ can be well approximated by the relation $\tilde{\chi}^{(1)}=\tilde{\chi}_{0}^{(1)}-\tilde{\chi}_{2}^{(1)}\lambda^{2}$ if we consider the propagation of optical pulse with the wavelength between 1600 nm and 3000 nm. It then follows that the linear equation (9) written in Fourier transformed form becomes $\tilde{E}_{zz}+\frac{1+\tilde{\chi}_{0}^{(1)}}{c^{2}}\omega^{2}\tilde{E}-(2\pi)^{2}\tilde{\chi}_{2}^{(1)}\tilde{E}+\epsilon_{NL}\frac{\omega^{2}}{c^{2}}\tilde{E}=0\,.$ (16) Applying the inverse Fourier transform to (16) yields a single nonlinear wave equation $E_{zz}-\frac{1}{c_{1}^{2}}E_{tt}=\frac{1}{c_{2}^{2}}E+\frac{3}{4}\chi^{(3)}_{xxxx}\left(\|E\|^{2}E\right)_{tt}=0\,.$ (17) Similar to [7], we focus on only a right-moving wave packet and make a multiple scales ansatz $E(z,t)=\epsilon E_{0}(\phi,z_{1},z_{2},\cdots)+\epsilon^{2}E_{1}(\phi,z_{1},z_{2},\cdots)+\cdots\,,$ (18) where $\epsilon$ is a small parameter, $\phi$ and $z_{n}$ are the scaled variables defined by $\phi=\frac{t-\frac{x}{c_{1}}}{\epsilon},\quad z_{n}=\epsilon^{n}z\,.$ (19) Substituting (18) with (19) into (17), we obtain the following partial differential equation for $E_{0}$ at the order $O(\epsilon)$: $-\frac{2}{c_{1}}\frac{\partial^{2}E_{0}}{\partial{\phi}\partial{z_{1}}}=\frac{1}{c_{2}^{2}}E_{0}+\frac{3}{4}\chi^{(3)}_{xxxx}\frac{\partial}{\partial{\phi}}\left(\|E_{0}\|^{2}\frac{\partial E_{0}}{\partial{\phi}}\right)\,.$ (20) Finally, by a scale transformation $x=\frac{c_{1}}{2}\phi,\quad t={c_{2}}z_{1},\quad q=\frac{c_{1}\sqrt{6c_{2}\chi^{(3)}_{xxxx}}}{4}E_{0}\,,$ (21) we arrive at the normalized form of the complex short pulse equation (2). ### 2.2 Coupled complex short pulse equation In the previous subsection, a major simplification made in the derivation of the complex short pulse equation is to assume that the polarization is preserved during its propagating inside an optical fiber. However, this is not really the case in practice. For birefringent fibers, two orthogonally polarized modes have to be considered. Therefore, similar to the extension of coupled nonlinear Schrödinger equation from the NLS equation, an extension to a two-component version of the complex short pulse equation (2) is needed to describe the propagation of ultra-short pulse in birefringent fibers. In fact, several generalizations have been proposed for the short pulse equation [26, 27, 28, 29, 30, 31]. Particularly, by taking into account the effects of anisotropy and polarization, Pietrzyk et. al. have derived a general two- component short-pulse equation from the physical context [26]. We follow the approach by Pietrzyk et. al. to derive a two-component complex short pulse equation. However, as shown in subsequent section, the two-component complex short pulse equation admits multi-soliton solutions which reveals richer dynamics in soliton interactions in compared with the real SP equation. We first consider the linear birefringent fiber such that the electric field with an arbitrarily polarized optical fiber can be expressed as $\mathbf{E}=\frac{1}{2}\left(\mathbf{e_{1}}E_{1}(z,t)+\mathbf{e_{2}}E_{2}(z,t)\right)+c.c.\,,$ (22) where $\mathbf{e_{1}}$, $\mathbf{e_{2}}$ are two unit vectors along positive $x$\- and $y$-direction in the transverse plane perpendicular to the optical fiber, respectively, $E_{1}$ and $E_{2}$ are the complex amplitudes of the polarization components correspondingly. Without the presence of nonlinear polarization ($P_{NL}=0$) and the transverse diffraction, the Fourier transform converts (5) into a pair of Helmholtz equations $\tilde{E}_{1,zz}(z,\omega)+\epsilon(\omega)\frac{\omega^{2}}{c^{2}}\tilde{E_{1}}(z,\omega)=0\,,$ (23) $\tilde{E}_{2,zz}(z,\omega)+\epsilon(\omega)\frac{\omega^{2}}{c^{2}}\tilde{E_{2}}(z,\omega)=0\,.$ (24) Same as the scalar case, the frequency-dependent dielectric constant $\epsilon(\omega)=1+\tilde{\chi}^{(1)}(\omega)$, where $\tilde{\chi}^{(1)}$ can be well approximated by the relation $\tilde{\chi}^{(1)}=\tilde{\chi}_{0}^{(1)}-\tilde{\chi}_{2}^{(1)}\lambda^{2}$ for the propagation of optical pulse with the wavelength between 1600 nm and 3000 nm. As indicated in [2], the nonlinear part of the induced polarization $\mathbf{P}_{NL}$ can be written as $\mathbf{P}_{NL}=\frac{1}{2}\left(\mathbf{e_{1}}P_{1}(z,t)+\mathbf{e_{2}}P_{2}(z,t)\right)+c.c.\,,$ (25) where $P_{1}=\frac{3\epsilon_{0}}{4}\chi_{xxxx}^{(3)}\left[\left(\|E_{1}\|^{2}+\frac{2}{3}\|E_{2}\|^{2}\right)E_{1}+\frac{1}{3}(E_{1}^{\ast}E_{2})E_{2}\right]\,,$ (26) $P_{2}=\frac{3\epsilon_{0}}{4}\chi_{xxxx}^{(3)}\left[\left(\|E_{2}\|^{2}+\frac{2}{3}\|E_{1}\|^{2}\right)E_{2}+\frac{1}{3}(E_{2}^{\ast}E_{1})E_{1}\right]\,.$ (27) The last term in Eqs. (26) and (27) leads to the degenerate four-wave mixing. In highly birefringent fibers, the four-wave-mixing term can often be neglected. In this case, we arrive at a coupled nonlinear wave equation $E_{1,zz}-\frac{1}{c_{1}^{2}}E_{1,tt}=\frac{1}{c_{2}^{2}}E_{1}+\frac{3}{4}\chi_{xxxx}^{(3)}\left[\left(\|E_{1}\|^{2}+\frac{2}{3}\|E_{2}\|^{2}\right)E_{1}\right]_{tt}\,,$ (28) $E_{2,zz}-\frac{1}{c_{1}^{2}}E_{2,tt}=\frac{1}{c_{2}^{2}}E_{2}+\frac{3}{4}\chi_{xxxx}^{(3)}\left[\left(\|E_{2}\|^{2}+\frac{2}{3}\|E_{1}\|^{2}\right)E_{2}\right]_{tt}\,.$ (29) Similar to the scalar case, by a multiple scales expansion and an appropriate scaling transformation, a couple complex short pulse equation can be obtained from (28)–(29) $q_{1,xt}+q_{1}+\frac{1}{2}\left((\|q_{1}\|^{2}+\frac{2}{3}\|q_{2}\|^{2})q_{1,x}\right)_{x}=0\,,$ (30) $q_{2,xt}+q_{2}+\frac{1}{2}\left((\|q_{2}\|^{2}+\frac{2}{3}\|q_{1}\|^{2})q_{2,x}\right)_{x}=0\,.$ (31) More generally, we can consider the coupled short pulse equation for elliptically birefringent fibers. In this case, the electric field can be written as $\mathbf{E}=\frac{1}{2}\left(\mathbf{e_{x}}E_{x}(z,t)+\mathbf{e_{y}}E_{y}(z,t)\right)+c.c.\,,$ (32) where $\mathbf{e_{x}}$ and $\mathbf{e_{y}}$ are orthonormal polarization eigenvectors $\mathbf{e_{x}}=\frac{\mathbf{e_{1}}+ir\mathbf{e_{2}}}{\sqrt{1+r^{2}}},\quad\mathbf{e_{y}}=\frac{r\mathbf{e_{1}}-i\mathbf{e_{2}}}{\sqrt{1+r^{2}}}\,.$ (33) The parameter $r$ represents the ellipticity. It is common to introduce the ellipticity angle $\theta$ as $r=\tan(\theta/2)$. The case $\theta=0$ and $\pi/2$ correspond to linearly and circularly birefringent fibers, respectively. Following a procedure similar to the case of linearly birefringent fibers, one can drive the normalized form for the coupled complex short pulse equation $q_{1,xt}+q_{1}+\frac{1}{2}\left((\|q_{1}\|^{2}+B\|q_{2}\|^{2})q_{1,x}\right)_{x}=0\,,$ (34) $q_{2,xt}+q_{2}+\frac{1}{2}\left((\|q_{2}\|^{2}+B\|q_{1}\|^{2})q_{2,x}\right)_{x}=0\,.$ (35) where the parameter $B$ is related to the ellipticity angle $\theta$ as $B=\frac{2+2\sin^{2}\theta}{2+\cos^{2}\theta}\,.$ (36) For a linearly birefringent fiber ($\theta=0$), $B=\frac{2}{3}$, and Eqs. (34)– (35) reduces to Eqs. (30)– (31). For a circularly birefringent fiber ($\theta=\pi/2$), $B=2$. In general, the coupling parameter $B$ depends on the ellipticity angle $\theta$ and can vary from $\frac{2}{3}$ to $2$ for values of $\theta$ in the range from $0$ to $\pi/2$. Note that $B=1$ when $\theta\approx 35^{\circ}$. As discussed in the subsequent section, this case is of particular interest because the coupled system is integrable and admits $N$-soliton solution. ## 3 Lax pairs and conservation laws for the complex and coupled complex short pulse equations ### 3.1 Lax pairs and integrability In [26], a matrix generalization for the SP equation is given based on zero- curvature representation, from which the Lax pairs for several integrable two- component SP equations are explicitly provided. In this subsection, we will show the integrability of the complex short pulse and coupled complex short pulse equations by finding their Lax pairs constructed from another matrix generalization of the SP equation. The Lax pair for the complex short pulse equation (2) can be expressed as $\Psi_{x}=U\Psi,\quad\Psi_{t}=V\Psi\,,$ (37) with $\displaystyle U=\lambda\left(\begin{array}[]{cc}1&q_{x}\\\ q^{}_{x}&-1\end{array}\right),\quad V=\left(\begin{array}[]{cc}-\frac{\lambda}{2}\|q\|^{2}-\frac{1}{4\lambda}&-\frac{\lambda}{2}\|q\|^{2}q_{x}+\frac{q}{2}\\\ -\frac{\lambda}{2}\|q\|^{2}q^{}_{x}-\frac{q^{}}{2}&\frac{\lambda}{2}\|q\|^{2}+\frac{1}{4\lambda}\end{array}\right)\,.$ (38) It can be easily shown that the compatibility condition $U_{t}-V_{x}+[U,\,V]=0$ gives the complex short pulse equation (2). The Lax pair for the coupled complex short pulse equation (3)–(4) is found to be of the form: $\Psi_{x}=U\Psi,\quad\Psi_{t}=V\Psi\,,$ (39) with $U=\lambda\left(\begin{array}[]{cc}I_{2}&Q_{x}\\\ R_{x}&-I_{2}\end{array}\right),\quad V=\left(\begin{array}[]{cc}-\frac{\lambda}{2}QR-\frac{1}{4\lambda}I_{2}&-\frac{\lambda}{2}QRQ_{x}+\frac{1}{2}Q\\\ -\frac{\lambda}{2}RQR_{x}-\frac{1}{2}R&\frac{\lambda}{2}QR+\frac{1}{4\lambda}I_{2}\end{array}\right)\,,$ (40) where $I_{2}$ is a $2\times 2$ identity matrix, $Q$, $R$ are $2\times 2$ matrices defined as $Q=\left(\begin{array}[]{cc}q_{1}&q_{2}\\\ -q_{2}^{\ast}&q_{1}^{\ast}\end{array}\right),\quad R=\left(\begin{array}[]{cc}q_{1}^{\ast}&-q_{2}\\\ q_{2}^{\ast}&q_{1}\end{array}\right)\,.$ (41) Note that $R=Q^{{\dagger}}$, thus, $QR=RQ=(\|q_{1}\|^{2}+\|q_{2}\|^{2})I_{2}\,,$ (42) the compatibility condition $U_{t}-V_{x}+[U,\,V]=0$ for (39) gives the coupled complex short pulse equation (3)–(4). As a matter of fact, the coupled complex short pulse equation can be generalized into a multi-component, or a vector complex short pulse equation $q_{i,xt}+q_{i}+\frac{1}{2}\left(\|\mathbf{q}\|^{2}q_{i,x}\right)_{x}=0\,,\quad i=1,\cdots,n,$ (43) where $\mathbf{q}=(q_{1},q_{2},\cdots,q_{n})$. The integrability of Eq. (43) can be guaranteed by the Lax pair constructed in a similar way as in [33]. $\Psi_{x}=U\Psi,\quad\Psi_{t}=V\Psi\,,$ (44) with $U=\lambda\left(\begin{array}[]{cc}I_{2^{n-1}}&Q_{x}^{(n)}\\\ R_{x}^{(n)}&-I_{2^{n-1}}\end{array}\right),$ $V=\left(\begin{array}[]{cc}-\frac{1}{2}Q^{(n)}R^{(n)}-\frac{1}{4\lambda}I_{2^{n-1}}&-\frac{\lambda}{2}Q^{(n)}R^{(n)}Q_{x}^{(n)}+\frac{1}{2}Q^{(n)}\\\ -\frac{\lambda}{2}R^{(n)}Q^{(n)}R_{x}^{(n)}-\frac{1}{2}R^{(n)}&\frac{1}{2}Q^{(n)}R^{(n)}+\frac{1}{4\lambda}I_{2^{n-1}}\end{array}\right)\,,$ where $I_{2^{n-1}}$ is a $2^{n-1}\times 2^{n-1}$ identity matrix, $Q^{(n)}$ and $R^{(n)}$ are $2^{n-1}\times 2^{n-1}$ matrices can be constructed recursively as follows $Q^{(1)}=q_{1},\quad R^{(1)}=q_{1}^{\ast}\,,$ (45) $Q^{(n+1)}=\left(\begin{array}[]{cc}Q^{(n)}&q_{n+1}I_{2^{n-1}}\\\ -q_{n+1}^{\ast}I_{2^{n-1}}&R^{(n)}\end{array}\right)\,,$ (46) $R^{(n+1)}=\left(\begin{array}[]{cc}R^{(n)}&-q_{n+1}I_{2^{n-1}}\\\ q^{}_{n+1}I_{2^{n-1}}&Q^{(n)}\end{array}\right)\,.$ (47) By the above construction, we have $R^{(n+1)}=(Q^{(n+1)})^{\dagger}$, and further $Q^{(n)}R^{(n)}=R^{(n)}Q^{(n)}=\sum_{i=1}^{n}\|q_{i}\|^{2}I_{2^{n-1}}\,.$ (48) Therefore, the zero curvature condition $U_{t}-V_{x}+[U,\,V]=0$ gives the vector complex coupled short pulse equation (43). ### 3.2 Local and nonlocal conservation laws Following a systematic method developed by in [33, 34, 35, 36], we construct conservation laws for the vector complex short pulse equation, the conservation laws for the complex and coupled short pulse equations can be treated as special cases for $n=1,2$, respectively. To this end, let us rewrite the Lax pair for the vector complex short pulse equation as follows: $\left(\begin{array}[]{c}\Psi_{1}\\\ \Psi_{2}\end{array}\right)_{x}=\left(\begin{array}[]{cc}\lambda I&\lambda Q_{x}\\\ \lambda R_{x}&-\lambda I\end{array}\right)\left(\begin{array}[]{c}\Psi_{1}\\\ \Psi_{2}\end{array}\right)\,,$ (49) $\left(\begin{array}[]{c}\Psi_{1}\\\ \Psi_{2}\end{array}\right)_{t}=\left(\begin{array}[]{cc}-\frac{\lambda}{2}QR-\frac{1}{4\lambda}I_{2}&-\frac{\lambda}{2}QRQ_{x}+\frac{1}{2}Q\\\ -\frac{\lambda}{2}RQR_{x}-\frac{1}{2}R&\frac{\lambda}{2}QR+\frac{1}{4\lambda}I_{2}\end{array}\right)\left(\begin{array}[]{c}\Psi_{1}\\\ \Psi_{2}\end{array}\right)\,.$ (50) Here the size of matrices in the entries of Eqs. (49)–(50) is of $2^{n-1}\times 2^{n-1}$ and is omitted for brevity. If we define $\Gamma\equiv\Psi_{2}\Psi_{1}^{-1}\,$ (51) then we have $2\lambda Q_{x}\Gamma=\lambda Q_{x}R_{x}-Q_{x}((Q_{x})^{-1}\cdot Q_{x}\Gamma)_{x}-\lambda(Q_{x}\Gamma)^{2}\,$ (52) Expanding $Q_{x}\Gamma$ in terms of the spectral parameter $\lambda$ as follows $Q_{x}\Gamma=\sum_{n=0}^{\infty}F_{n}\lambda^{-n}\,,$ (53) and substituting into Eq. (52), we obtain the following relation $2\lambda F_{n}=Q_{x}R_{x}\delta_{n,0}-Q_{x}((Q_{x})^{-1}F_{n-1})_{x}-\sum_{l=0}^{n}F_{l}F_{n-l}.$ (54) The first local conserved density turns out to be $F_{0}=\left(-1+\sqrt{1+\sum\|q_{i,x}\|^{2}}\right)I\,,$ (55) which is associated with a Hamiltonian of $H_{0}=\int\sqrt{1+\|q_{x}\|^{2}}\,dx\,,$ (56) for the complex short pulse equation (2) and $H_{0}=\int\sqrt{1+\|q_{1,x}\|^{2}+\|q_{2,x}\|^{2}}\,dx\,,$ (57) for the coupled complex short pulse equation (3)–(4). Following the procedure in [36], we can find the nonlocal conservation laws for vector complex short pulse equation. To this end, we expand $Q_{x}\Gamma$ as follows $Q_{x}\Gamma=\sum_{n=1}^{\infty}F_{-n}(2\lambda)^{n}\,.$ (58) The first two orders in $\lambda$ yield the following equations $0=Q_{x}R_{x}-Q_{x}\left((Q_{x})^{-1}F_{-1}\right)_{x}\,,$ (59) $2F_{-1}=-Q_{x}\left((Q_{x})^{-1}F_{-2}\right)_{x}\,,$ (60) from which, the first two nonlocal conserved densities can be calculated as $F_{-1}=\frac{1}{2}Q_{x}R\,,$ (61) $F_{-2}=\frac{1}{2}QR\ -\frac{1}{2}\partial_{x}\left(Q\partial_{x}R\right).$ (62) The first one turns out to be a trivial one, the second one accounts for a Hamiltonian $H_{-1}=\frac{1}{2}\int\|q\|^{2}\,dx\,,$ (63) for the complex short pulse equation (2) and $H_{-1}=\frac{1}{2}\int(\|q_{1}\|^{2}+\|q_{2}\|^{2})\,dx\,,$ (64) for the coupled complex short pulse equation (3)–(4). ## 4 Multi-soliton solutions by Hirota’s bilinear method ### 4.1 Bilinear equations and $N$-soliton solution to the complex short pulse equation Proposition 4.1. The complex short pulse equation is derived from the following bilinear equations. $D_{s}D_{y}f\cdot g=fg\,,$ (65) $D^{2}_{s}f\cdot f=\frac{1}{2}\|g\|^{2}\,,$ (66) by dependent variable transformation $q=\frac{g}{f}\,,$ (67) and hodograph transformation $x=y-2(\ln f)_{s}\,,\quad t=-s\,,$ (68) where $D$ is called Hirota $D$-operator defined by $D_{s}^{n}D_{y}^{m}f\cdot g=\left(\frac{\partial}{\partial s}-\frac{\partial}{\partial s^{\prime}}\right)^{n}\left(\frac{\partial}{\partial y}-\frac{\partial}{\partial y^{\prime}}\right)^{m}f(y,s)g(y^{\prime},s^{\prime})\|_{y=y^{\prime},s=s^{\prime}}\,.$ . Proof. Dividing both sides by $f^{2}$, the bilinear equations (65)– (66) can be cast into $\left\\{\begin{array}[]{l}\displaystyle\left(\frac{g}{f}\right)_{sy}+2\frac{g}{f}\left(\ln f\right)_{sy}=\frac{g}{f}\,,\\\\[5.0pt] \displaystyle\left(\ln f\right)_{ss}=\frac{1}{4}\frac{\|g\|^{2}}{f^{2}}\,.\end{array}\right.$ (69) From the hodograph transformation and dependent variable transformation, we then have $\frac{\partial x}{\partial s}=-2(\ln f)_{ss}=-\frac{1}{2}\|q\|^{2}\,,\qquad\frac{\partial x}{\partial y}=1-2(\ln f)_{sy}\,,$ which implies ${\partial_{y}}=\rho^{-1}{\partial_{x}}\,,\qquad{\partial_{s}}=-{\partial_{t}}-\frac{1}{2}\|q\|^{2}{\partial_{x}}\,$ (70) by letting $1-2(\ln f)_{sy}=\rho^{-1}$. Notice that the first equation in (69) can be rewritten as $\left(\frac{g}{f}\right)_{sy}=\left(1-2(\ln f)_{sy}\right)\frac{g}{f}\,,$ (71) or $\rho\left(\frac{g}{f}\right)_{sy}=\frac{g}{f}\,,$ (72) which is converted into $\partial_{x}\left(-\partial_{t}-\frac{1}{2}\|q\|^{2}\partial_{x}\right)q=q\,,$ (73) by using (70). Eq. (73) is nothing but the complex short pulse equation (2). $N$-soliton solution to the bilinear equations (65)–(66) can be expressed by pfaffians similar to the ones for coupled modified KdV equation [37]. To this end, we need to define two sets: $B_{\mu}$ ($\mu=1,2$): $B_{1}=\\{b_{1},b_{2},\cdots,b_{N}\\}$, $B_{2}=\\{b_{N+1},b_{2},\cdots,b_{2N}\\}$, and an index function of $b_{j}$ by $index(b_{j})=\mu$ if $b_{j}\in B_{\mu}$. Theorem 4.2. The pfaffians $\displaystyle f$ $\displaystyle=$ $\displaystyle\mathrm{Pf}(a_{1},\cdots,a_{2N},b_{1},\cdots,b_{2N})\,,$ (74) $\displaystyle g$ $\displaystyle=$ $\displaystyle\mathrm{Pf}(d_{0},\beta_{1},a_{1},\cdots,a_{2N},b_{1},\cdots,b_{2N})\,.$ (75) satisfy the bilinear equations (65)–(66) provided that the elements of the pfaffians are defined by $\mathrm{Pf}(a_{j},a_{k})=\frac{p_{j}-p_{k}}{p_{j}+p_{k}}e^{\eta_{j}+\eta_{k}}\,,\quad\mathrm{Pf}(a_{j},b_{k})=\delta_{j,k}\,,$ (76) $\mathrm{Pf}(b_{j},b_{k})=\frac{1}{4}\frac{\alpha_{j}\alpha_{k}}{p^{-2}_{j}-p^{-2}_{k}}\delta_{\mu+1,\nu}\,,\quad\mathrm{Pf}(d_{l},a_{k})=p_{k}^{l}e^{\eta_{k}}\,,$ (77) $\mathrm{Pf}(b_{j},\beta_{1})=\alpha_{j}\delta_{\mu,1}\,,\quad\mathrm{Pf}(d_{0},b_{j})=\mathrm{Pf}(d_{0},\beta_{1})=\mathrm{Pf}(a_{j},\beta_{1})=0\,.$ (78) Here $\mu=index(b_{j})$, $\nu=index(b_{k})$, $\eta_{j}=p_{j}y+p_{j}^{-1}s$ which satisfying $p_{j+N}=\bar{p}_{j}$, $\alpha_{j+N}=\bar{\alpha}_{j}$, $\bar{p_{j}}$ and $\bar{\alpha}_{j}$ represent the complex conjugates of $p_{j}$ and ${\alpha}_{j}$, respectively. The same notation will be used hereafter. The proof of the Theorem is given in Appendix. Combined with dependent and hodograph transformations (67)–(68), the above pfaffians (74)–(75) give $N$-soliton solution to the complex short pulse equation (2) in parametric form. ### 4.2 One- and two-soliton solutions for the complex short pulse equation In this subsection, we provide one- and two-soliton to the complex short pulse equation (2) and give a detailed analysis for their properties. #### 4.2.1 One-soliton solution Based on (74)–(75), the tau-functions for one-soliton solution ($N=1$) are $\displaystyle f=-1-\frac{1}{4}\frac{\|\alpha_{1}\|^{2}(p_{1}\bar{p}_{1})^{2}}{(p_{1}+\bar{p}_{1})^{2}}e^{\eta_{1}+\bar{\eta}_{1}}\,,$ (79) $g=-\alpha_{1}e^{\eta_{1}}\,.$ (80) Let $p_{1}=p_{1R}+\mathrm{i}p_{1I}$, and we assume $p_{1R}>0$ without loss of generality, then the one-soliton solution can be expressed in the following parametric form $q=\frac{\alpha_{1}}{\|\alpha_{1}\|}\frac{2p_{1R}}{\|p_{1}\|^{2}}e^{\mathrm{i}\eta_{1I}}\mbox{sech}\left(\eta_{1R}+\eta_{10}\right)\,,$ (81) $x=y-\frac{2p_{1R}}{\|p_{1}\|^{2}}\left(\tanh\left(\eta_{1R}+\eta_{10}\right)+1\right)\,,\quad t=-s\,,$ (82) where $\eta_{1R}=p_{1R}y+\frac{p_{1R}}{\|p_{1}\|^{2}}s,\quad\eta_{1I}=p_{1I}y-\frac{p_{1I}}{\|p_{1}\|^{2}}s\,,\quad\eta_{10}=\ln\frac{\|\alpha_{1}\|\|p_{1}\|^{2}}{4p_{1R}}\,.$ (83) Eq. (81) represents an envelope soliton of amplitude $2p_{1R}/\|p_{1}\|^{2}$ and phase $\eta_{1I}$. To analyze the property for the one-soliton solution, we calculate out $\frac{\partial x}{\partial y}=1-\frac{2p^{2}_{1R}}{\|p_{1}\|^{2}}{\mbox{sech}}^{2}(\eta_{1R}+\eta_{10})\,.$ (84) Therefore, $\partial x/\partial y\to 1$ as $y\to\pm\infty$. Moreover, it attains a minimum value of $({p^{2}_{1I}-p^{2}_{1R}})/({p^{2}_{1I}+p^{2}_{1R}})$ at the peak point of envelope soliton where $\eta_{1R}+\eta_{10}=0$. Since ${\partial\|q\|}/{\partial x}=\frac{\partial\|q\|/\partial y}{\partial x/\partial y}$, we can classify this one-soliton solution as follows: * 1. smooth soliton: when $\|p_{1R}\|<\|p_{1I}\|$, ${\partial x}/{\partial y}$ is always positive, which leads to a smooth envelope soliton similar to the envelope soliton for the nonlinear Schrödinger equation. An example with $p_{1}=1+1.5\mathrm{i}$ is illustrated in Fig. 1 (a). * 2. loop soliton: when $\|p_{1R}\|>\|p_{1I}\|$, the minimum value of ${\partial x}/{\partial y}$ at the peak point of the soliton becomes negative. In view of the fact that $\partial x/\partial y\to 1$ as $y\to\pm\infty$, ${\partial x}/{\partial y}$ has two zeros at both sides of the peak of the envelope soliton. Moreover, ${\partial x}/{\partial y}<0$ between these two zeros. This leads to a loop soliton for the envelope of $q$. An example is shown in Fig. (b) with $p_{1}=1+0.5\mathrm{i}$. * 3. cuspon soliton: when $\|p_{1R}\|=\|p_{1I}\|$, ${\partial x}/{\partial y}$ has a minimum value of zero at $\eta_{1R}+\eta_{10}=0$, which makes the derivative of the envelope $\|q\|$ with respect to $x$ going to infinity at the peak point. Thus, we have a cusponed envelope soliton, which is illustrated in Fig. 1 (c) with $p_{1}=1+\mathrm{i}$. (a)(b) (c) Figure 1: Envelope soliton for the complex short pulse equation (2), solid line: $Re(q)$, dashed line: $\|q\|$; (a) smooth soliton with $p_{1}=1+1.5\mathrm{i}$, (b) loop soliotn with $p_{1}=1+0.5\mathrm{i}$, (c) cuspon soliton with $p_{1}=1+\mathrm{i}$. Remark 4.3. The one-soliton solution to the short pulse equation (1) is of loop-type, which lacks physical meaning in the context of nonlinear optics. However, the one-soliton solution to the complex short pulse equation (2) is of breather-type, which allows physical meaning for optical pulse. Remark 4.4. When $\|p_{1R}\|<\|p_{1I}\|$, there is no singularity for one-soliton solution. Moreover, in view of $\eta_{1R}$ associated with the width of envelope soliton and $\eta_{1I}$ associated with the phase, it is obvious that this nonsingular envelope soliton can only contain a few optical cycle. This property coincides with the fact that the complex short pulse equation is derived for the purpose of describing ultra-short pulse propagation. When $\|p_{1R}\|=\|p_{1I}\|$, the soliton becomes cuspon-like one, which agrees with the results in [10] derived from a bidirectional model. #### 4.2.2 Two-soliton solution Based on the $N$-soliton solution of the complex short pulse equation from (74)–(75), the tau-functions for two-soliton solution can be expanded for $N=2$ $\displaystyle f=\mathrm{Pf}(a_{1},a_{2},a_{3},a_{4},b_{1},b_{2},b_{3},b_{4})$ $\displaystyle\quad=1+a_{1\bar{1}}e^{\eta_{1}+\bar{\eta}_{1}}+a_{1\bar{2}}e^{\eta_{1}+\bar{\eta}_{2}}+a_{2\bar{1}}e^{\eta_{2}+\bar{\eta}_{1}}+a_{2\bar{2}}e^{\eta_{2}+\bar{\eta_{2}}}$ $\displaystyle\qquad+\|P_{12}\|^{2}\left(a_{1\bar{1}}a_{2\bar{2}}P_{1\bar{2}}P_{2\bar{1}}-a_{1\bar{2}}a_{2\bar{1}}P_{1\bar{1}}P_{2\bar{2}}\right)e^{\eta_{1}+\eta_{2}+\bar{\eta}_{1}+\bar{\eta}_{2}}\,,$ (85) $\displaystyle g=\mathrm{Pf}(d_{0},\beta_{1},a_{1},a_{2},a_{3},a_{4},b_{1},b_{2},b_{3},b_{4})$ $\displaystyle\quad=\alpha_{1}e^{\eta_{1}}+\alpha_{2}e^{\eta_{2}}+P_{12}\left(\alpha_{1}P_{1\bar{1}}a_{2\bar{1}}-\alpha_{2}P_{2\bar{1}}a_{1\bar{1}}\right)e^{\eta_{1}+\eta_{2}+\bar{\eta}_{1}}$ $\displaystyle\qquad+P_{12}\left(\alpha_{1}P_{1\bar{2}}a_{2\bar{2}}-\alpha_{2}P_{2\bar{2}}a_{1\bar{2}}\right)e^{\eta_{1}+\eta_{2}+\bar{\eta}_{2}}\,,$ (86) where $P_{ij}=\frac{p_{i}-p_{j}}{p_{i}+p_{j}}\,,\quad P_{i\bar{j}}=\frac{p_{i}-\bar{p}_{j}}{p_{i}+\bar{p}_{j}}\,,\quad a_{i\bar{j}}=\frac{\alpha_{i}\bar{\alpha}_{j}(p_{i}\bar{p}_{j})^{2}}{4(p_{i}+\bar{p}_{j})^{2}}\,,$ (87) and $\eta_{j}=p_{j}y+p_{j}^{-1}s$, $\bar{\eta}_{j}=\bar{p}_{j}y+\bar{p}_{j}^{-1}s$. (a)(b) Figure 2: Two-soliton solution to the complex short pulse equation (a) contour plot; (b) profiles at $t=-80$, $80$. To avoid the singularity of the envelope solitons, the conditions $\|p_{1R}\|<\|p_{1I}\|$ and $\|p_{2R}\|<\|p_{2I}\|$ need to be satisfied. When two solitons stay apart, the amplitude of each soliton is of $2\|p_{iR}\|/\|p_{i}\|^{2}$, and the velocity is of $-1/\|p_{i}\|^{2}$ in the $ys$-coordinate system. Therefore, the soliton of larger velocity will catch up with and collide with the soliton of smaller velocity if it is initially located on the left. Furthermore, the collision is elastic, and there is no change in shape and amplitude of solitons except a phase shift. In Fig. 2, we illustrate the contour plot for the collision of two solitons (a), as well as the profiles (b) before and after the collision. The parameters are taken as $\alpha_{1}=\alpha_{2}=1.0$, $p_{1}=1+1.2\mathrm{i}$ and $p_{2}=1+2\mathrm{i}$. Since the velocity of single envelope soliton is $-1/\|p_{i}\|^{2}$ in the $ys$-coordinate system, a bound state can be formed under the condition of $\|p_{1}\|^{2}=\|p_{2}\|^{2}$ if two solitons stay close enough and move with the same velocity. Such a bound state is shown in Fig. 3 for parameters chosen as $\alpha_{1}=\alpha_{2}=1.0$, $p_{1}=1.3+1.8193\mathrm{i}$, $p_{2}=1+2\mathrm{i}$. It is interesting that the envelope of the bound state oscillates periodically as it moves along $x$-axis. (a)(b) Figure 3: Bound state to the complex short pulse equation: (a) 3D plot (b) profiles at $t=-100$, $40$. ### 4.3 Bilinear equations and $N$-soliton solutions to the coupled complex short pulse equation Proposition 4.5. The coupled complex short pulse equation is derived from bilinear equations $D_{s}D_{y}f\cdot g_{i}=fg_{i},\quad i=1,2\,,$ (88) $D^{2}_{s}f\cdot f=\frac{1}{2}\left(\|g_{1}\|^{2}+\|g_{2}\|^{2}\right)\,,$ (89) by dependent variable transformation $q_{1}=\frac{g_{1}}{f},\quad q_{2}=\frac{g_{2}}{f}\,,$ (90) and hodograph transformation $x=y-2(\ln f)_{s}\,,\quad t=-s\,,$ (91) Proof. Dividing both sides of Eqs. (88)–(89) by $f^{2}$, we have $\left(\frac{g_{i}}{f}\right)_{sy}+2\frac{g_{i}}{f}\left(\ln f\right)_{sy}=\frac{g_{i}}{f}\,,$ (92) $\left(\ln f\right)_{ss}=\frac{1}{4}\left(\frac{\|g_{1}\|^{2}}{f^{2}}+\frac{\|g_{2}\|^{2}}{f^{2}}\right)\,.$ (93) From dependent variable and hodograph transformations (90)–(91), we obtain $\frac{\partial x}{\partial s}=-2(\ln f)_{ss}=-\frac{1}{2}\left(\|q_{1}\|^{2}+\|q_{2}\|^{2}\right)\,,\qquad\frac{\partial x}{\partial y}=1-2(\ln f)_{sy}\,,$ which implies ${\partial_{y}}=\rho^{-1}{\partial_{x}}\,,\qquad{\partial_{s}}=-{\partial_{t}}-\frac{1}{2}\left(\|q_{1}\|^{2}+\|q_{2}\|^{2}\right){\partial_{x}}\,$ (94) by letting $1-2(\ln f)_{sy}=\rho^{-1}$. With the use of (94), Eq. (92) can be recast into $\rho\left(\frac{g_{i}}{f}\right)_{sy}=\frac{g_{i}}{f}\,,\quad i=1,2\,,$ (95) which can be further converted into $\partial_{x}\left(-\partial_{t}-\frac{1}{2}(\|q_{1}\|^{2}+\|q_{2}\|^{2})\partial_{x}\right)q_{i}=q_{i}\,,\quad i=1,2\,.$ (96) Eq. (96) is, obviously, equivalent to the coupled complex short pulse equation (3)–(4). $N$-soliton solution for the coupled complex short pulse equation is given in a similar way as the complex short pulse equation by the following theorem. Theorem 4.6. The coupled complex short pulse equation admits the following $N$-soliton solution $q_{i}=\frac{g_{i}}{f},\quad x=y-2(\ln f)_{s}\,,\quad t=-s\,,$ where $f$, $g_{i}$ are pfaffians defined as $\displaystyle f$ $\displaystyle=$ $\displaystyle\mathrm{Pf}(a_{1},\cdots,a_{2N},b_{1},\cdots,b_{2N})\,,$ (97) $\displaystyle g_{i}$ $\displaystyle=$ $\displaystyle\mathrm{Pf}(d_{0},\beta_{i},a_{1},\cdots,a_{2N},b_{1},\cdots,b_{2N})\,,$ (98) and the elements of the pfaffians are determined as $\mathrm{Pf}(a_{j},a_{k})=\frac{p_{j}-p_{k}}{p_{j}+p_{k}}e^{\eta_{j}+\eta_{k}}\,,\quad\mathrm{Pf}(a_{j},b_{k})=\delta_{j,k}\,,$ (99) $\mathrm{Pf}(b_{j},b_{k})=\frac{1}{4}\frac{\sum^{2}_{i=1}\alpha^{(i)}_{j}\alpha^{(i)}_{k}}{p^{-2}_{j}-p^{-2}_{k}}\delta_{\mu+1,\nu}\,,\quad\mathrm{Pf}(d_{l},a_{k})=p_{k}^{l}e^{\eta_{k}}\,,$ (100) $\mathrm{Pf}(b_{j},\beta_{i})=\alpha^{(i)}_{j}\delta_{\mu,i}\,,\quad\mathrm{Pf}(d_{0},b_{j})=\mathrm{Pf}(d_{0},\beta_{i})=\mathrm{Pf}(a_{j},\beta_{i})=0\,.$ (101) Here $\mu=index(b_{j})$, $\nu=index(b_{k})$, $\eta_{j}=p_{j}y+p_{j}^{-1}s+\eta_{j,0}$ which satisfying $p_{j+N}=\bar{p}_{j}$, $\alpha_{j+N}=\bar{\alpha}_{j}$. The proof of the Theorem is given in the Appendix. In the subsequent section, based on the $N$-soliton solution of coupled complex short pulse equation, we will investigate the dynamics of one- and two-solitons in details. Remark 4.7. Through the transformations $x=y-2(\ln f)_{s}\,,\quad t=-s\,,\quad q_{i}=\frac{g_{i}}{f}\,,$ (102) the vector complex short pulse equation (43) can be decomposed into the following bilinear equations $D_{s}D_{y}f\cdot g_{i}=fg_{i},\quad i=1,\cdots,n\,,$ (103) $D^{2}_{s}f\cdot f=\frac{1}{2}\left(\sum^{n}_{i=1}\|g_{i}\|^{2}\right)\,.$ (104) The parametric form of $N$-soliton solution in terms of pfaffians to the vector complex short pulse equation (43) can be given in a very similar from as to to the coupled complex short pulse equation. Here, we omit the details and will report the results later on. ## 5 Dynamics of solitons to the coupled complex short pulse equation ### 5.1 One-soliton solution The tau-functions for one-soliton solution to the coupled complex short pulse equation are obtained from (97)–(98) for $N=1$ $f=-1-\frac{1}{4}\frac{\sum_{i=1}^{2}\|\alpha^{(i)}_{1}\|^{2}(p_{1}\bar{p}_{1})^{2}}{(p_{1}+\bar{p}_{1})^{2}}e^{\eta_{1}+\bar{\eta}_{1}}\,,$ (105) $g_{1}=-\alpha^{(1)}_{1}e^{\eta_{1}}\,,\quad g_{2}=-\alpha^{(2)}_{1}e^{\eta_{1}}\,.$ (106) Let $p_{1}=p_{1R}+\mathrm{i}p_{1I}$, the one-soliton solution can be expressed in the following parametric form $\left(\begin{array}[]{c}q_{1}\\\ q_{2}\end{array}\right)=\left(\begin{array}[]{c}A_{1}\\\ A_{2}\end{array}\right)\frac{2p_{1R}}{\|p_{1}\|^{2}}e^{\mathrm{i}\eta_{1I}}{\mbox{sech}}\left(\eta_{1R}+\eta_{10}\right)\,,$ (107) $x=y-\frac{2p_{1R}}{\|p_{1}\|^{2}}\left(\tanh(\eta_{1R}+\eta_{10})+1\right)\,,\quad t=-s\,,$ (108) where $\eta_{1R}=p_{1R}\left(y+\frac{1}{\|p_{1}\|^{2}}s\right),\quad\eta_{1I}=p_{1I}\left(y-\frac{1}{\|p_{1}\|^{2}}s\right)\,,$ (109) $A_{i}=\frac{\alpha_{1}^{(i)}}{\sqrt{\sum_{i=1}^{2}\|\alpha_{1}^{(i)}\|^{2}}}\,,\quad\eta_{10}=\ln\frac{\sqrt{\sum_{i=1}^{2}\|\alpha_{1}^{(i)}\|^{2}}\|p_{1}\|^{2}}{4\|p_{1R}\|}\,.$ (110) The amplitudes of the single soliton in each component are ${2\|A_{1}\|p_{1R}}/{\|p_{1}\|^{2}}$ and ${2\|A_{2}\|p_{1R}}/{\|p_{1}\|^{2}}$, respectively. Note that $\|A_{1}\|^{2}+\|A_{2}\|^{2}=1$. Same as the analysis for one-soliton solution of complex short pulse equation, if $\|p_{1R}\|<\|p_{1I}\|$, the envelope for one-soliton in each of the component is smooth, whereas, if $\|p_{1R}\|>\|p_{1I}\|$, it becomes a loop (multi-valued) soliton, if $\|p_{1R}\|=\|p_{1I}\|$, it is a cuspon. ### 5.2 Soliton interactions Two-soliton solution for coupled complex short pulse equation is obtained from (97)–(98) for $N=2$. By expanding the pfaffians, the tau-functions for two- soliton solution are expressed by $\displaystyle f=1+e^{\eta_{1}+\bar{\eta}_{1}+r_{1\bar{1}}}+e^{\eta_{1}+\bar{\eta}_{2}+r_{1\bar{2}}}+e^{\eta_{2}+\bar{\eta}_{1}+r_{2\bar{1}}}+e^{\eta_{2}+\bar{\eta}_{2}+r_{2\bar{2}}}$ $\displaystyle\qquad+\|P_{12}\|^{2}\|P_{1\bar{2}}\|^{2}P_{1\bar{1}}P_{2\bar{2}}\left(B_{1\bar{1}}B_{2\bar{2}}-B_{2\bar{1}}B_{1\bar{2}}\right)e^{\eta_{1}+\eta_{2}+\bar{\eta}_{1}+\bar{\eta}_{2}}\,,$ (111) $\displaystyle g_{1}=\alpha^{(1)}_{1}e^{\eta_{1}}+\alpha^{(1)}_{2}e^{\eta_{2}}+P_{12}P_{1\bar{1}}P_{2\bar{1}}\left(\alpha^{(1)}_{2}B_{1\bar{1}}-\alpha^{(1)}_{1}B_{2\bar{1}}\right)e^{\eta_{1}+\eta_{2}+\bar{\eta}_{1}}$ $\displaystyle\qquad+P_{12}P_{1\bar{2}}P_{2\bar{2}}\left(\alpha^{(1)}_{2}B_{1\bar{2}}-\alpha^{(1)}_{1}B_{2\bar{2}}\right)e^{\eta_{1}+\eta_{2}+\bar{\eta}_{2}}\,,$ (112) $\displaystyle g_{2}=\alpha_{1}^{(2)}e^{\eta_{1}}+\alpha_{2}^{(2)}e^{\eta_{2}}+P_{12}P_{1\bar{1}}P_{2\bar{1}}\left(\alpha_{2}^{(2)}B_{1\bar{1}}-\alpha_{1}^{(2)}B_{2\bar{1}}\right)e^{\eta_{1}+\eta_{2}+\bar{\eta}_{1}}$ $\displaystyle\qquad+P_{12}P_{1\bar{2}}P_{2\bar{2}}\left(\alpha_{2}^{(2)}B_{1\bar{2}}-\alpha_{1}^{(2)}B_{2\bar{2}}\right)e^{\eta_{1}+\eta_{2}+\bar{\eta}_{2}}\,,$ (113) where $P_{ij}=\frac{p_{i}-p_{j}}{p_{i}+p_{j}}\,,\quad P_{i\bar{j}}=\frac{p_{i}-\bar{p}_{j}}{p_{i}+\bar{p}_{j}}\,,$ $B_{i\bar{j}}=\frac{\alpha_{i}^{(1)}\bar{\alpha}_{j}^{(1)}+\alpha_{i}^{(2)}\bar{\alpha}_{j}^{(2)}}{4(p_{i}^{-2}-\bar{p}_{j}^{-2})}\,,\quad e^{r_{i\bar{j}}}=\frac{\alpha_{i}^{(1)}\bar{\alpha}_{j}^{(1)}+\alpha_{i}^{(2)}\bar{\alpha}_{j}^{(2)}}{4(p_{i}^{-1}+\bar{p}_{j}^{-1})^{2}}\,.$ and $\eta_{j}=p_{j}y+p_{j}^{-1}s$, $p_{3}=\bar{p}_{1}$, $p_{4}=\bar{p}_{2}$, thus, $\eta_{3}=\bar{\eta}_{1}$, $\eta_{4}=\bar{\eta}_{2}$. Next, we investigate the asymptotic behavior of two-soliton solution. To this end, we assume $p_{1R}>p_{2R}>0$, $p_{1R}/\|p_{1}\|^{2}>p_{2R}/\|p_{2}\|^{2}$ without loss of generality. For the above choice of parameters, we have (i) $\eta_{1R}\approx 0$, $\eta_{2R}\rightarrow\mp\infty$ as $t\rightarrow\mp\infty$ for soliton 1 and (ii) $\eta_{2R}\approx 0$, $\eta_{2R}\rightarrow\pm\infty$ as $t\rightarrow\mp\infty$ for soliton 2\. This leads to the following asymptotic forms for two-soliton solution. (i) Before collision ($t\rightarrow-\infty$) Soliton 1 ($\eta_{1R}\approx 0$, $\eta_{2R}\rightarrow-\infty$): $\displaystyle\left(\begin{array}[]{c}q_{1}\\\ q_{2}\end{array}\right)$ $\displaystyle\rightarrow$ $\displaystyle\left(\begin{array}[]{c}\alpha_{1}^{(1)}\\\ \alpha_{1}^{(2)}\end{array}\right)\frac{e^{\eta_{1}}}{1+e^{\eta_{1}+\bar{\eta}_{1}+r_{1\bar{1}}}}\,,$ (118) $\displaystyle\rightarrow$ $\displaystyle\left(\begin{array}[]{c}A_{1}^{1-}\\\ A_{2}^{1-}\end{array}\right)\frac{2p_{1R}}{\|p_{1}\|^{2}}e^{i\eta_{1I}}{\mbox{sech}}\left(\eta_{1R}+\frac{r_{1\bar{1}}}{2}\right)\,,$ (121) where $\left(\begin{array}[]{c}A_{1}^{1-}\\\ A_{2}^{1-}\end{array}\right)=\left(\begin{array}[]{c}\alpha_{1}^{(1)}\\\ \alpha_{1}^{(2)}\end{array}\right)\frac{1}{\sqrt{\|\alpha_{1}^{(1)}\|^{2}+\|\alpha_{1}^{(2)}\|^{2}}}\,.$ (122) Soliton 2 ($\eta_{2R}\approx 0$, $\eta_{1R}\to\infty$): $\left(\begin{array}[]{c}q_{1}\\\ q_{2}\end{array}\right)\to\left(\begin{array}[]{c}A^{2-}_{1}\\\ A^{2-}_{2}\end{array}\right)\frac{2p_{2R}}{\|p_{2}\|^{2}}e^{i\eta_{2I}}{\mbox{sech}}\left(\eta_{2R}+\frac{r_{1\bar{1}2\bar{2}}-r_{1\bar{1}}}{2}\right)\,,$ (123) where $\left(\begin{array}[]{c}A^{2-}_{1}\\\ A^{2-}_{2}\end{array}\right)=\left(\begin{array}[]{c}e^{r^{(1)}_{1\bar{1}2}}\\\ e^{r^{(2)}_{1\bar{1}2}}\end{array}\right)\frac{e^{-(r_{1\bar{1}2\bar{2}}+r_{1\bar{1}}-r_{2\bar{2}})/{2}}}{\sqrt{\|\alpha^{(1)}_{2}\|^{2}+\|\alpha^{(2)}_{2}\|^{2}}}\,,$ (124) with $e^{r^{(i)}_{1\bar{1}2}}=P_{12}P_{1\bar{1}}P_{2\bar{1}}\left(\alpha^{(i)}_{2}B_{1\bar{1}}-\alpha^{(i)}_{1}B_{2\bar{1}}\right)\,,\quad(i=1,2)$ (125) $e^{r_{1\bar{1}2\bar{2}}}=\|P_{12}\|^{2}\|P_{1\bar{2}}\|^{2}P_{1\bar{1}}P_{2\bar{2}}\left(B_{1\bar{1}}B_{2\bar{2}}-B_{2\bar{1}}B_{1\bar{2}}\right)\,.$ (126) After collision ($t\to\infty$) Soliton 1 ($\eta_{1R}\approx 0$, $\eta_{2R}\to\infty$): $\left(\begin{array}[]{c}q_{1}\\\ q_{2}\end{array}\right)\rightarrow\left(\begin{array}[]{c}A_{1}^{1+}\\\ A_{2}^{1+}\end{array}\right)\frac{2p_{1R}}{\|p_{1}\|^{2}}e^{i\eta_{1I}}{\mbox{sech}}\left(\eta_{2R}+\frac{r_{1\bar{1}2\bar{2}}-r_{2\bar{2}}}{2}\right)\,,$ (127) where $\left(\begin{array}[]{c}A_{1}^{1+}\\\ A_{2}^{1+}\end{array}\right)=\left(\begin{array}[]{c}e^{r_{12\bar{1}}^{(1)}}\\\ e^{r_{12\bar{1}}^{(2)}}\end{array}\right)\frac{e^{-(r_{1\bar{1}2\bar{2}}-r_{1\bar{1}}+r_{2\bar{2}})/{2}}}{\sqrt{\|\alpha_{1}^{(1)}\|^{2}+\|\alpha_{1}^{(2)}\|^{2}}}\,,$ (128) with $e^{r_{12\bar{1}}^{(i)}}=P_{12}P_{1\bar{2}}P_{2\bar{2}}\left(\alpha_{2}^{(i)}B_{1\bar{2}}-\alpha_{1}^{(i)}B_{2\bar{2}}\right)\,,\quad(i=1,2)\,.$ (129) Soliton 2 ($\eta_{2R}\approx 0$, $\eta_{1R}\rightarrow-\infty$): $\left(\begin{array}[]{c}q_{1}\\\ q_{2}\end{array}\right)\rightarrow\left(\begin{array}[]{c}A_{1}^{2+}\\\ A_{2}^{2+}\end{array}\right)\frac{2p_{2R}}{\|p_{2}\|^{2}}e^{i\eta_{2I}}{\mbox{sech}}\left(\eta_{2R}+\frac{r_{2\bar{2}}}{2}\right)\,,$ (130) where $\left(\begin{array}[]{c}A_{1}^{2+}\\\ A_{2}^{2+}\end{array}\right)=\left(\begin{array}[]{c}\alpha_{2}^{(1)}\\\ \alpha_{2}^{(2)}\end{array}\right)\frac{1}{\sqrt{\|\alpha_{2}^{(1)}\|^{2}+\|\alpha_{2}^{(2)}\|^{2}}}\,.$ (131) Similar to the analysis for the CNLS equations [38, 39, 40], the change in the amplitude of each of the solitons in each component can be obtained by introducing the transition matrix $T^{k}_{j}$ by $A_{j}^{k+}=T_{j}^{k}A_{j}^{k-}$, $j,k=1,2$. The elements of transition matrix is obtained from the above asymptotic analysis as $T_{j}^{1}=\left(\frac{P_{12}P_{1\bar{2}}}{\bar{P}_{12}\bar{P}_{1\bar{2}}}\right)^{1/2}\frac{1}{\sqrt{1-\lambda_{1}\lambda_{2}}}\left(1-\lambda_{2}\frac{\alpha_{2}^{(j)}}{\alpha_{1}^{(j)}}\right)\,,\quad j=1,2\,,$ (132) $T_{j}^{2}=\left(\frac{\bar{P}_{12}P_{1\bar{2}}}{P_{12}\bar{P}_{1\bar{2}}}\right)^{1/2}\sqrt{1-\lambda_{1}\lambda_{2}}\left(1-\lambda_{1}\frac{\alpha_{1}^{(j)}}{\alpha_{2}^{(j)}}\right)^{-1}\,,\quad j=1,2\,,$ (133) where $\lambda_{1}=B_{2\bar{1}}/B_{1\bar{1}}$, $\lambda_{2}=B_{1\bar{2}}/B_{2\bar{2}}$. Therefore, in general, there is an exchange of energies between two components of two solitons after the collision. An example is shown in Fig. 4 for the parameters taken as follows $p_{1}=1+1.2\mathrm{i}$, $p_{2}=1+2\mathrm{i}$, $\alpha^{(1)}_{1}=\alpha^{(2)}_{1}=1.0$, $\alpha^{(1)}_{2}=2.0$, $\alpha^{(2)}_{2}=1.0$. (a)(b) (c)(d) Figure 4: Inelastic collision in coupled complex short pulse equation. (a)-(b): contour plot; (c)-(d): profiles before and after the collision. However, only for the special case $\frac{\alpha_{1}^{(1)}}{\alpha_{2}^{(1)}}=\frac{\alpha_{1}^{(2)}}{\alpha_{2}^{(2)}}\,,$ (134) there is no energy exchange between two compoents of solitons after the collision. An example is shown in Fig. 5 for the parameters $p_{1}=1+1.2\mathrm{i}$, $p_{2}=1+2\mathrm{i}$, $\alpha^{(1)}_{1}=\alpha^{(2)}_{1}=1.0$, $\alpha^{(1)}_{2}=\alpha^{(2)}_{2}=1.0$. (a)(b) (c)(d) Figure 5: Elastic collision in coupled complex short pulse equation. It is interesting to note that if we just change the parameters in previous two examples as $\alpha^{(1)}_{2}=0$, $\alpha^{(2)}_{2}=1.0$, the energy of one soliton is concentrated in component $q_{2}$ before the collision. However, component $q_{1}$ gains some energy after the collision. Such an example is shown in Fig. 6. (a)(b) (c)(d) Figure 6: Inelastic collision in coupled complex short pulse equation for $p_{1}=1+1.2{\rm i}$, $p_{2}=1+2{\rm i}$, $\alpha^{(1)}_{1}=\alpha^{(2)}_{1}=1.0$, $\alpha^{(1)}_{2}=0$, $\alpha^{(2)}_{2}=1.0$. (a)-(b): contour plot; (c)-(d): profiles before and after the collision. On the other hand, if we change the parameters as $\alpha^{(1)}_{2}=1.0$, $\alpha^{(2)}_{2}=0$, then the energy of one soliton, which are distributed between two components before the collision is concentrated into one component $q_{2}$ after the collision. The example is shown in Fig. 7. (a)(b) (c)(d) Figure 7: Inelastic collision in coupled complex short pulse equation for $p_{1}=1+1.2\mathrm{i}$, $p_{2}=1+2\mathrm{i}$, $\alpha^{(1)}_{1}=\alpha^{(2)}_{1}=1.0$, $\alpha^{(1)}_{2}=1.0$, $\alpha^{(2)}_{2}=0$. (a)-(b): contour plot; (c)-(d): profiles before and after the collision. ## 6 Concluding Remarks In this paper, we proposed a complex short pulse equation and its two- component generalization. Both of the equations can be used to model the propagation of ultra-short pulses in optical fibers. We have shown their integrability by finding the Lax pairs and infinite numbers of conservation laws. Furthermore, multi-soliton solutions are constructed via Hirota’s bilinear method. In particular, one-soliton solution for the CSP equation is an envelope soliton with a few optical cycles under certain condition, which perfectly match the requirement for the ultra-short pulses. The $N$-solution for complex short pulse equation and its two-component generalization is a benchmark for the study of soliton interactions in ultra-short pulses propagation in optical fibers. It is expected that these analytical solutions can be confirmed from experiments. Similar to our previous results for the integrable discretizations of the short pulse equation [22], how to construct integrable discretizations of the CSP and coupled CSP equations and how to apply them for the numerical simulations is also an interesting topic to be studied. It is obviously beyond the scope of the present paper, we are to report the results on this aspect in a forthcoming paper. ## Appendix Proof of Theorem 4.2 Proof. First we define $(b_{j},\bar{\beta}_{1})=\bar{\alpha}_{j}\delta_{\mu,1}\,,\quad(b_{j},\bar{\beta}_{2})=\bar{\alpha}_{j}\delta_{\mu,2}\,,$ where $index(b_{j})=\mu$ , then from the fact $\mathrm{Pf}(\bar{a}_{j},a_{k})=\mathrm{Pf}(a_{N+j},a_{N+k})\,,\mathrm{Pf}(\bar{b}_{j},b_{k})=\mathrm{Pf}(b_{N+j},b_{N+k})\,,$ we obtain $\bar{f}=f\,,\quad\bar{g}=\mathrm{Pf}(d_{0},\bar{\beta}_{1},a_{1},\cdots,a_{2N},b_{1},\cdots,b_{2N})\,.$ Since $\frac{\partial}{\partial y}\mathrm{Pf}(a_{j},a_{k})=(p_{j}-p_{k})e^{\eta_{j}+\eta_{k}}=\mathrm{Pf}(d_{0},d_{1},a_{j},a_{k})\,,$ $\frac{\partial}{\partial s}\mathrm{Pf}(a_{j},a_{k})=(p^{-1}_{k}-p^{-1}_{j})e^{\eta_{j}+\eta_{k}}=\mathrm{Pf}(d_{-1},d_{0},a_{j},a_{k})\,,$ $\frac{\partial^{2}}{\partial s^{2}}\mathrm{Pf}(a_{j},a_{k})=(p^{-2}_{k}-p^{-2}_{j})e^{\eta_{j}+\eta_{k}}=\mathrm{Pf}(d_{-2},d_{0},a_{j},a_{k})\,,$ $\frac{\partial^{2}}{\partial y\partial s}\mathrm{Pf}(a_{j},a_{k})=(p_{j}p^{-1}_{k}-p_{k}p^{-1}_{j})e^{\eta_{j}+\eta_{k}}=\mathrm{Pf}(d_{-1},d_{1},a_{i},a_{j})\,,$ we then have $\frac{\partial f}{\partial y}=\mathrm{Pf}(d_{0},d_{1},\cdots)\,,$ $\frac{\partial f}{\partial s}=\mathrm{Pf}(d_{-1},d_{0},\cdots)\,,$ $\frac{\partial^{2}f}{\partial s^{2}}=\mathrm{Pf}(d_{-2},d_{0},\cdots)\,,$ $\frac{\partial^{2}f}{\partial y\partial s}=\mathrm{Pf}(d_{-1},d_{1},\cdots)\,.$ Here $\mathrm{Pf}(d_{0},d_{1},a_{1},\cdots,a_{2N},b_{1},\cdots,b_{2N})$ is abbreviated by $\mathrm{Pf}(d_{0},d_{1},\cdots)$, so as other similar pfaffians. Furthermore, it can be shown $\displaystyle\frac{\partial g}{\partial y}=\frac{\partial}{\partial y}\left[\sum_{j=1}^{2N}(-1)^{j}\mathrm{Pf}(d_{0},a_{j})\mathrm{Pf}(\beta_{1},\cdots,\hat{a}_{j},\cdots)\right]$ $\displaystyle=\sum_{j=1}^{2N}(-1)^{j}\left[\left({\partial_{y}}\mathrm{Pf}(d_{0},a_{j})\right)\mathrm{Pf}(\beta_{1},\cdots,\hat{a}_{j},\cdots)+\mathrm{Pf}(d_{0},a_{j}){\partial_{y}}\mathrm{Pf}(\beta_{1},\cdots,\hat{a}_{j},\cdots)\right]$ $\displaystyle=\sum_{j=1}^{2N}(-1)^{j}\left[\mathrm{Pf}(d_{1},a_{j})\mathrm{Pf}(\beta_{1},\cdots,\hat{a}_{j},\cdots)+\mathrm{Pf}(d_{0},a_{j})\mathrm{Pf}(\beta_{1},d_{0},d_{1},\cdots,\hat{a}_{j},\cdots)\right]$ $\displaystyle=\mathrm{Pf}(d_{1},\beta_{1},\cdots)+\mathrm{Pf}(d_{0},\beta_{1},d_{0},d_{1},\cdots)$ $\displaystyle=\mathrm{Pf}(d_{1},\beta_{1},\cdots)\,.$ Here $\hat{a}_{j}$ means that the index $j$ is omitted. Similarly, we can show $\frac{\partial g}{\partial s}=\mathrm{Pf}(d_{-1},\beta_{1},\cdots)\,,$ $\displaystyle\frac{\partial^{2}g}{\partial y\partial s}=\frac{\partial}{\partial y}\left[\sum_{j=1}^{2N}(-1)^{j}\mathrm{Pf}(d_{-1},a_{j})\mathrm{Pf}(\beta_{1},\cdots,\hat{a}_{j},\cdots)\right]$ $\displaystyle=\sum_{j=1}^{2N}(-1)^{j}\left[\left({\partial_{y}}\mathrm{Pf}(d_{-1},a_{j})\right)\mathrm{Pf}(\beta_{1},\cdots,\hat{a}_{j},\cdots)+\mathrm{Pf}(d_{-1},a_{j}){\partial_{y}}\mathrm{Pf}(\beta_{1},\cdots,\hat{a}_{j},\cdots)\right]$ $\displaystyle=\sum_{j=1}^{2N}(-1)^{j}\left[\mathrm{Pf}(d_{0},a_{j})\mathrm{Pf}(\beta_{1},\cdots,\hat{a}_{j},\cdots)+\mathrm{Pf}(d_{-1},a_{j})\mathrm{Pf}(\beta_{1},d_{0},d_{1},\cdots,\hat{a}_{j},\cdots)\right]$ $\displaystyle=\mathrm{Pf}(d_{0},\beta_{1},\cdots)+\mathrm{Pf}(d_{-1},\beta_{1},d_{0},d_{1},\cdots)\,.$ An algebraic identity of pfaffian [21] $\displaystyle\mathrm{Pf}(d_{-1},\beta_{1},d_{0},d_{1},\cdots)\mathrm{Pf}(\cdots)=\mathrm{Pf}(d_{-1},d_{0},\cdots)\mathrm{Pf}(d_{1},\beta_{1},\cdots)$ $\displaystyle\quad-\mathrm{Pf}(d_{-1},d_{1},\cdots)\mathrm{Pf}(d_{0},\beta_{1},\cdots)+\mathrm{Pf}(d_{-1},\beta_{1},\cdots)\mathrm{Pf}(d_{0},d_{1},\cdots)\,,$ implies $({\partial_{s}}{\partial_{y}}g-g)\times f={\partial_{s}}f\times{\partial_{y}}g-{\partial_{s}}{\partial_{y}}f\times g+{\partial_{s}}g\times{\partial_{y}}f\,.$ Therefore, the first bilinear equation is approved. The second bilinear equation can be proved in the same way by Iwao and Hirota [37]. $\displaystyle\frac{\partial^{2}f}{\partial s^{2}}\times 0-\frac{\partial f}{\partial s}\frac{\partial f}{\partial s}$ $\displaystyle=\mathrm{Pf}(d_{-2},d_{0},\cdots)\mathrm{Pf}(d_{0},d_{0},\cdots)-\mathrm{Pf}(d_{-1},d_{0},\cdots)\mathrm{Pf}(d_{-1},d_{0},\cdots)$ $\displaystyle=\sum_{i=1}^{2N}(-1)^{i}\mathrm{Pf}(d_{-2},a_{i})\mathrm{Pf}(d_{0},\cdots,\hat{a}_{i},\cdots)\sum_{j=1}^{2N}(-1)^{j}\mathrm{Pf}(d_{0},a_{j})\mathrm{Pf}(d_{0},\cdots,\hat{a}_{j},\cdots)$ $\displaystyle-\sum_{i=1}^{2N}(-1)^{i}\mathrm{Pf}(d_{-1},a_{i})\mathrm{Pf}(d_{0},\cdots,\hat{a}_{i},\cdots)\sum_{j=1}^{2N}(-1)^{j}\mathrm{Pf}(d_{-1},a_{j})\mathrm{Pf}(d_{0},\cdots,\hat{a}_{j},\cdots)$ $\displaystyle=\sum_{i,j=1}^{2N}(-1)^{i+j}\left[\mathrm{Pf}(d_{-2},a_{i})\mathrm{Pf}(d_{0},a_{j})-\mathrm{Pf}(d_{-1},a_{i})\mathrm{Pf}(d_{-1},a_{j})\right]$ $\displaystyle\quad\times\mathrm{Pf}(d_{0},\cdots,\hat{a}_{i},\cdots)\mathrm{Pf}(d_{0},\cdots,\hat{a}_{j},\cdots)$ $\displaystyle=\sum_{i,j=1}^{2N}(-1)^{i+j+1}\left[p_{i}^{-2}+p_{i}^{-1}p_{j}^{-1}\right]\mathrm{Pf}(a_{i},a_{j})\mathrm{Pf}(d_{0},\cdots,\hat{a}_{i},\cdots)\mathrm{Pf}(d_{0},\cdots,\hat{a}_{j},\cdots)$ The summation over the second term within the bracket vanishes due to the fact that $\displaystyle\sum_{i,j=1}^{2N}(-1)^{i+j+1}p_{i}^{-1}p_{j}^{-1}\mathrm{Pf}(a_{i},a_{j})\mathrm{Pf}(d_{0},\cdots,\hat{a}_{i},\cdots)\mathrm{Pf}(d_{0},\cdots,\hat{a}_{j},\cdots)$ $\displaystyle=\sum_{j,i=1}^{2N}(-1)^{j+i+1}p_{j}^{-1}p_{i}^{-1}\mathrm{Pf}(a_{j},a_{i})\mathrm{Pf}(d_{0},\cdots,\hat{a}_{j},\cdots)\mathrm{Pf}(d_{0},\cdots,\hat{a}_{i},\cdots)$ $\displaystyle=-\sum_{i,j=1}^{2N}(-1)^{i+j+1}p_{i}^{-1}p_{j}^{-1}\mathrm{Pf}(a_{i},a_{j})\mathrm{Pf}(d_{0},\cdots,\hat{a}_{i},\cdots)\mathrm{Pf}(d_{0},\cdots,\hat{a}_{j},\cdots)\,.$ Therefore, $\displaystyle-\frac{\partial f}{\partial s}\frac{\partial f}{\partial s}=\sum_{i,j=1}^{2N}(-1)^{i+j+1}p_{i}^{-2}\mathrm{Pf}(a_{i},a_{j})\mathrm{Pf}(d_{0},\cdots,\hat{a}_{i},\cdots)\mathrm{Pf}(d_{0},\cdots,\hat{a}_{j},\cdots)$ $\displaystyle=\sum_{i=1}^{2N}(-1)^{i+1}p_{i}^{-2}\mathrm{Pf}(d_{0},\cdots,\hat{a}_{i},\cdots)\left[\sum_{j=1}^{2N}(-1)^{j}\mathrm{Pf}(a_{i},a_{j})\mathrm{Pf}(d_{0},\cdots,\hat{a}_{j},\cdots)\right]$ Further, we note that the following identity can be substituted into the term within bracket $\displaystyle\sum_{j=1}^{2N}(-1)^{j}\mathrm{Pf}(a_{i},a_{j})\mathrm{Pf}(d_{0},\cdots,\hat{a}_{j},\cdots)$ $\displaystyle=\mathrm{Pf}(d_{0},a_{i})\mathrm{Pf}(\cdots)+(-1)^{i+1}\mathrm{Pf}(d_{0},\cdots,\hat{b}_{i},\cdots)\,$ which is obtained from the expansion of the following vanishing pfaffian $\mathrm{Pf}(a_{i},d_{0},\cdots)$ on $a_{i}$. Consequently, we have $\displaystyle-\frac{\partial f}{\partial s}\frac{\partial f}{\partial s}=$ $\displaystyle\sum_{i=1}^{2N}(-1)^{i+1}p_{i}^{-2}\mathrm{Pf}(d_{0},\cdots,\hat{a}_{i},\cdots)\left[\mathrm{Pf}(d_{0},a_{i})\mathrm{Pf}(\cdots)+(-1)^{i+1}\mathrm{Pf}(d_{0},\cdots,\hat{b}_{i},\cdots)\right]\,,$ $\displaystyle=-\mathrm{Pf}(\cdots)\mathrm{Pf}(d_{-2},d_{0},\cdots)+\sum_{i=1}^{2N}p_{i}^{-2}\mathrm{Pf}(d_{0},\cdots,\hat{a}_{i},\cdots)\mathrm{Pf}(d_{0},\cdots,\hat{b}_{i},\cdots)\,,$ (A.1) which can be rewritten as $\frac{\partial^{2}f}{\partial s^{2}}f-\frac{\partial f}{\partial s}\frac{\partial f}{\partial s}=\sum_{i=1}^{2N}p_{i}^{-2}\mathrm{Pf}(d_{0},\cdots,\hat{a}_{i},\cdots)\mathrm{Pf}(d_{0},\cdots,\hat{b}_{i},\cdots)\,.$ (A.2) Now, we work on the r.h.s of the second bilinear equation. $\displaystyle\frac{1}{2}\|g\|^{2}=\frac{1}{2}\mathrm{Pf}(d_{0},\beta_{1},\cdots)\mathrm{Pf}(d_{0},\bar{\beta}_{1},\cdots)$ $\displaystyle=\frac{1}{2}\sum_{i,j}^{2N}(-1)^{i+j}\mathrm{Pf}(b_{i},\beta_{1})\mathrm{Pf}(d_{0},\cdots,\hat{b}_{i},\cdots)\mathrm{Pf}(b_{j},\bar{\beta}_{1})\mathrm{Pf}(d_{0},\cdots,\hat{b}_{j},\cdots)$ $\displaystyle=\frac{1}{4}\sum_{i,j}^{2N}(-1)^{i+j}(\alpha_{i}\bar{\alpha}_{j})\mathrm{Pf}(d_{0},\cdots,\hat{b}_{i},\cdots)\mathrm{Pf}(d_{0},\cdots,\hat{b}_{j},\cdots)$ $\displaystyle=\sum_{i,j}^{2N}(-1)^{i+j}\left(p_{i}^{-2}-p_{j}^{-2}\right)\mathrm{Pf}(b_{i},b_{j})\mathrm{Pf}(d_{0},\cdots,\hat{b}_{i},\cdots)\mathrm{Pf}(d_{0},\cdots,\hat{b}_{j},\cdots)$ Next, the expansion of the vanishing pfaffian $\mathrm{Pf}(b_{i},d_{0},\cdots)$ on $b_{i}$ yields $\sum_{j=1}^{2N}(-1)^{i+j}\mathrm{Pf}(b_{i},b_{j})\mathrm{Pf}(d_{0},\cdots,\hat{b}_{j},\cdots)=\mathrm{Pf}(d_{0},\cdots,\hat{a}_{i},\cdots)\,,$ (A.4) which subsequently leads to $\displaystyle\sum_{i,j}^{2N}(-1)^{i+j}p_{i}^{-2}\mathrm{Pf}(b_{i},b_{j})\mathrm{Pf}(d_{0},\cdots,\hat{b}_{i},\cdots)\mathrm{Pf}(d_{0},\cdots,\hat{b}_{j},\cdots)$ $\displaystyle=\sum_{i}^{2N}p_{i}^{-2}\mathrm{Pf}(d_{0},\cdots,\hat{a}_{i},\cdots)\mathrm{Pf}(d_{0},\cdots,\hat{b}_{i},\cdots)\,.$ (A.5) Similarly, we can show that $\displaystyle-\sum_{i,j}^{2N}(-1)^{i+j}p_{j}^{-2}\mathrm{Pf}(b_{i},b_{j})\mathrm{Pf}(d_{0},\cdots,\hat{b}_{i},\cdots)\mathrm{Pf}(d_{0},\cdots,\hat{b}_{j},\cdots)$ $\displaystyle=\sum_{j}^{2N}p_{j}^{-2}\mathrm{Pf}(d_{0},\cdots,\hat{a}_{j},\cdots)\mathrm{Pf}(d_{0},\cdots,\hat{b}_{j},\cdots)\,.$ (A.6) Substituting Eqs. (A.5)–(A.5) into Eq. (Appendix), we arrive at $\frac{1}{2}\|g\|^{2}=2\sum_{i}^{2N}p_{i}^{-2}\mathrm{Pf}(d_{0},\cdots,\hat{a}_{i},\cdots)\mathrm{Pf}(d_{0},\cdots,\hat{b}_{i},\cdots)\,.$ (A.7) Consequently we have $2\frac{\partial^{2}f}{\partial s^{2}}f-2\frac{\partial f}{\partial s}\frac{\partial f}{\partial s}=\frac{1}{2}\|g\|^{2}\,,$ (A.8) which is nothing but the second bilinear equation. Therefore, the proof is complete. The proof of Theorem 4.6 Proof. The proof of the first bilinear equation can be done exactly in the same way as for the complex short pulse equation. In what follows, we prove the second equation by starting from the r.h.s of this equation. Because $\bar{g}_{1}=\mathrm{Pf}(d_{0},\bar{\beta}_{1},a_{1},\cdots,a_{2N},b_{1},\cdots,b_{2N})\,,$ $\bar{g}_{2}=\mathrm{Pf}(d_{0},\bar{\beta}_{2},a_{1},\cdots,a_{2N},b_{1},\cdots,b_{2N})\,,$ the r.h.s of the bilinear equation turns out to be $\displaystyle\frac{1}{2}\left(g_{1}\bar{g}_{1}+g_{2}\bar{g}_{2}\right)$ $\displaystyle=\frac{1}{2}\sum^{2}_{k=1}\sum_{i,j}^{2N}(-1)^{i+j}\mathrm{Pf}(b_{i},\beta_{k})\mathrm{Pf}(d_{0},\cdots,\hat{b}_{i},\cdots)\mathrm{Pf}(b_{j},\bar{\beta}_{k})\mathrm{Pf}(d_{0},\cdots,\hat{b}_{j},\cdots)$ $\displaystyle=\frac{1}{4}\sum_{i,j}^{2N}(-1)^{i+j}\sum^{2}_{k=1}(\alpha^{(k)}_{i}\bar{\alpha}^{(k)}_{j})\mathrm{Pf}(d_{0},\cdots,\hat{b}_{i},\cdots)\mathrm{Pf}(d_{0},\cdots,\hat{b}_{j},\cdots)$ $\displaystyle=\sum_{i,j}^{2N}(-1)^{i+j}\left(p_{i}^{-2}-p_{j}^{-2}\right)\mathrm{Pf}(b_{i},b_{j})\mathrm{Pf}(d_{0},\cdots,\hat{b}_{i},\cdots)\mathrm{Pf}(d_{0},\cdots,\hat{b}_{j},\cdots)$ Similar to the complex short pulse equation, we can show $\frac{1}{2}\left(\|g_{1}\|^{2}+\|g_{2}\|^{2}\right)=2\sum_{i}^{2N}p_{i}^{-2}\mathrm{Pf}(d_{0},\cdots,\hat{a}_{i},\cdots)\mathrm{Pf}(d_{0},\cdots,\hat{b}_{i},\cdots)\,.$ (A.10) Regarding the r.h.s of the bilinear equation, exactly the same as the proof of the Theorem 4.2, we have $\frac{\partial^{2}f}{\partial s^{2}}f-\frac{\partial f}{\partial s}\frac{\partial f}{\partial s}=\sum_{i}^{2N}p_{i}^{-2}\mathrm{Pf}(d_{0},\cdots,\hat{a}_{i},\cdots)\mathrm{Pf}(d_{0},\cdots,\hat{b}_{i},\cdots)\,.$ (A.11) Therefore the second bilinear equation is proved. ## Acknowledgements The author is grateful for the useful discussions with Dr. Yasuhiro Ohta (Kobe University) and Dr. Kenichi Maruno at Waseda University. 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E 67 (2003) 046617.	arxiv-papers	2013-12-22T22:13:12	2024-09-04T02:49:55.817107	{ "license": "Public Domain", "authors": "Bao-Feng Feng", "submitter": "Bao-Feng Feng", "url": "https://arxiv.org/abs/1312.6431" }
1312.6517	# Multiconfiguration Dirac-Hartree-Fock calculations of atomic electric dipole moments of 225Ra, 199Hg, and 171Yb Laima Radžiūtė Gediminas Gaigalas Vilnius University, Institute of Theoretical Physics and Astronomy, A. Goštauto 12, LT-01108, Vilnius, Lithuania Per Jönsson Group for Materials Science and Applied Mathematics, Malmö University, S-20506, Malmö, Sweden Jacek Bieroń Instytut Fizyki imienia Mariana Smoluchowskiego, Uniwersytet Jagielloński, Kraków, Poland ###### Abstract The multiconfiguration Dirac-Hartree-Fock (MCDHF) method has been employed to calculate atomic electric dipole moments (EDM) of 225Ra, 199Hg, and 171Yb. For the calculations of the matrix elements we extended the relativistic atomic structure package GRASP2K Jönsson et al. (2013). The extension includes programs to evaluate matrix elements of $(P,T)$-odd e-N tensor-pseudotensor and pseudoscalar-scalar interactions, the atomic electric dipole interaction, the nuclear Schiff moment, and the interaction of the electron electric dipole moment with nuclear magnetic moments. The interelectronic interactions were accounted for through valence and core-valence electron correlation effects. The electron shell relaxation was included with separately optimised wave functions of opposite parities. ###### pacs: 11.30.Er, 32.10.Dk, 31.15.A-, 24.80.+y ## I Introduction The existence of a non-zero permanent electric dipole moment (EDM) of an elementary particle or a composite system of particles would violate time reversal symmetry (T), as well as the combined charge conjugation and parity symmetry (CP), due to the CPT theorem Khriplovich and Lamoreaux (1997). One of the principal motivations behind the experimental searches of EDMs is to shed light on the observed matter-antimatter asymmetry in the Universe, which in turn is linked to an asymmetry in the Big Bang baryon-antibaryon production. The standard model (SM) of elementary particles cannot explain the matter- antimatter asymmetry in the Universe, as SM predicts sources of CP violation (and of EDMs) several orders of magnitude weaker than those needed to account for the observed baryon numbers. This leads to proliferation of the extensions to the standard model. Some of these extensions predict larger EDMs, sometimes within the reach of current experiments. The experimental searches have not yet detected a non-zero EDM, but they continue to improve the limits on EDMs of individual elementary particles, as well as limits on CP-violating interactions, usually parametrized by the interaction constants $C_{T}$, and $C_{P}$ (see section II for details and the Table II in the reference Griffith et al. (2009) for a summary). These limits constrain the theoretical extensions of the standard model of elementary particles. In recent years these constraints have been set by the measurements of EDMs of neutrons Baker et al. (2006), electrons in a paramagnetic atom (a thallium atom experiment Regan et al. (2002)), electrons in a diamagnetic atom (mercury atom Griffith et al. (2009)), and in TlF and YbF molecules Hudson et al. (2011). The search for EDMs is not restricted to the above species, though — see e.g. Roberts and Marciano (2009). The search for a permanent electric dipole moment of an elementary particle, or a composite system of particles (see Khriplovich and Lamoreaux (1997), or a recent reference Dzuba and Flambaum (2012) for a review), is a challenge, not only for experiments, but also for theories of composite systems. Heavy atoms are excellent examples of composite systems with large EDMs, due to the existence of mechanisms which may induce atomic EDMs several orders of magnitude larger than an intrinsic particle EDM. In the present paper we computed the EDMs of three diamagnetic atoms, 225Ra, 199Hg, and 171Yb. The purpose of the present paper is fourfold. Firstly, we tested the newly developed programs to evaluate matrix elements of $(P,T)$-odd e-N tensor- pseudotensor and pseudoscalar-scalar interactions, the atomic electric dipole interaction, the nuclear Schiff moment, and the interaction of the electron electric dipole moment with nuclear magnetic moments. Secondly, we generated the atomic wave functions in several different approaches, in order to test the dependence of the calculated atomic EDMs on options available in the GRASP2K Jönsson et al. (2013) implementation of the MCDHF method. The approaches depended on the choice of variational energy functional (Average Level versus Optimal Level, with different numbers of optimised levels), the choice of wave functions built on a common orbital set or several separately optimised orbital sets, in the latter case biorthogonal transformations of wave functions had to be applied, as well as on specific methods of one- electron orbital generation. All these approaches are discussed in more detail in section III.2, III.3, III.4 and III.5 and presented in Tables 1, 2, and 3. Thirdly, we sequentially generated several layers of virtual (correlation) orbitals for each of the three elements and observed the effects of electron correlation on atomic EDMs. All valence and core-valence electron correlation effects were included through single and restricted double electron substitutions from core-valence to virtual orbitals. And finally, we provide independently calculated atomic EDMs in 225Ra, 199Hg, and 171Yb, and compare our results with those of other authors. Our results, presented in the Tables 4, 5, 6, and 7, were obtained within the multiconfiguration Dirac-Hartree-Fock (MCDHF) method, using the relativistic atomic structure package GRASP2K Jönsson et al. (2013), which, to the best of our knowledge, and with the exception of one paper Bieroń et al. (2009a) on the Schiff moment in radium, has been employed for the first time in the calculations of matrix elements of $(P,T)$-odd e-N tensor-pseudotensor and pseudoscalar-scalar interactions, nuclear Schiff moment, and interaction of electron electric dipole moment with nuclear magnetic moments. The three atoms 225Ra, 199Hg, and 171Yb, have been chosen on the grounds that they have similar valence shell structure. All these elements are diamagnetic, with closed outer $s$ shell (225Ra $6p^{6}7s^{2}$, 199Hg $5d^{10}6s^{2}$, and 171Yb $4f^{14}6s^{2}$). In the future we will be able extend these calculations to closed-p-valence-shell atoms, as well as to any other, closed- or open-shell system. Our current MCDHF machinery Jönsson et al. (2013) is robust enough to deal with electron correlation effects in arbitrary atomic systems, including the lanthanides and actinides. ## II EDM theory The interactions which mix atomic states of different parities and induce a static electric dipole moment of an atom are quite weak. Therefore an atomic wave function can be expressed as $\displaystyle\widetilde{\Psi}\left(JM_{J}\right)\;$ $\displaystyle=$ $\displaystyle a\Psi\left(\gamma PJM_{J}\right)\;+\;$ (1) $\displaystyle\sum_{i}\;b_{i}\;\Psi\left(\gamma_{i}(-P)J_{i}M_{J_{i}}\right)\,$ where the coefficient $a$ of the dominant contribution can be set to 1. The expansion coefficients of opposite parity ($-P$) admixtures, $b_{i}$, can be found using first order perturbation theory: $\displaystyle b_{i}\;=\;\frac{\left<\Psi\left(\gamma_{i}(-P)J_{i}M_{J_{i}}\right)\|\hat{H}_{int}\|\Psi\left(\gamma PJM_{J}\right)\right>}{E\left(\gamma PJ\right)\;-\;E\left(\gamma_{i}(-P)J_{i}\right)}\,.$ (2) $\hat{H}_{int}$ represents the Hamiltonian of the $(P,T)$-odd interaction, which mixes states of opposite parities. The mixed-parity state of a particular atomic level ${}^{2S+1}L_{J}$ induces a static EDM of an atom: $\displaystyle d_{at}^{int}=\left<\widetilde{\Psi}\left(\gamma JM_{J}\right)\|\hat{D}_{z}\|\widetilde{\Psi}\left(\gamma JM_{J}\right)\right>$ $\displaystyle=$ (3) $\displaystyle\;2\sum_{i}b_{i}\left<\Psi\left(\gamma PJM_{J}\right)\|\hat{D}_{z}\|\Psi\left(\gamma_{i}(-P)J_{i}M_{J_{i}}\right)\right>,$ where $\hat{D}_{z}$ represents the $z$ projection of the electric-dipole moment operator. Eventually an atomic EDM can be written as a sum: $\displaystyle d_{at}^{int}=\;2\sum_{i}\frac{\left<0\|\hat{D}_{z}\|i\right>\left<i\|\hat{H}_{int}\|0\right>}{E_{0}\;-\;E_{i}},$ (4) where $\|0\rangle$ represents the ground state $\|\Psi\left(\gamma PJM_{J}\right>$, with $J=0$ and even parity, and the summation runs over excited states $\|\Psi\left(\gamma_{i}(-P)J_{i}M_{J_{i}}\right>$, with $J_{i}=1$ and odd parity. $E_{0}$ and $E_{i}$ are energies of ground and excited states, respectively. In practice this sum needs to be truncated at some level. Calculations of atomic EDM require evaluation of the matrix element of the static EDM $\left<0\|\hat{D}_{z}\|i\right>$ and the matrix element of the interactions which induced EDM in an atom $\left<i\|\hat{H}_{int}\|0\right>$. The operators associated with the above matrix elements are all one-particle operators. For the general tensor operator $\hat{T}^{k}_{q}$, the matrix element between states of different parity can be expressed by Wigner-Eckart theorem as: $\displaystyle\left<\Psi\left(\gamma PJM_{J}\right)\|\hat{T}_{0}^{k}\|\Psi\left(\gamma_{i}(-P)J_{i}M_{J_{i}}\right)\right>$ $\displaystyle=$ (8) $\displaystyle(-1)^{J-M_{J}}\;\;\sqrt{2J+1}\;\left(\begin{array}[]{ccc}J&k&J_{i}\\\ -M_{J}&0&M_{J_{i}}\end{array}\right)$ $\displaystyle\times\left[\Psi\left(\gamma PJ\right)\\|\hat{T}^{k}\\|\Psi\left(\gamma_{i}(-P)J_{i}\right)\right]\ .$ Expanding the wave functions in configuration state functions (CSFs), $\Phi\left(\gamma PJ\right)$, that are built from one-electron Dirac orbitals, see section III, the reduced matrix elements of $\hat{T}^{k}_{q}$ can be written $\displaystyle\left[\Psi\left(\gamma PJ\right)\\|\hat{T}^{k}\\|\Psi\left(\gamma_{i}(-P)J_{i}\right)\right]$ $\displaystyle=$ (9) $\displaystyle\sum_{r,s}\;c_{r}c_{s}\left[\Phi\left(\gamma_{r}PJ\right)\\|\hat{T}^{k}\\|\Phi\left(\gamma_{s}(-P)J_{i}\right)\right]\,$ where $c_{r}$ and $c_{s}$ are mixing coefficients of CSFs (even and odd parity, respectively). The matrix elements between the CSFs, in turn, can be written as sums of single-particle matrix elements $\displaystyle\left[\Phi\left(\gamma_{r}PJ\right)\\|\hat{T}^{k}\\|\Phi\left(\gamma_{s}(-P)J_{i}\right)\right]$ $\displaystyle=$ (10) $\displaystyle\;\sum_{a,b}\;d^{k}_{ab}(rs)\left[n_{a}\kappa_{a}\\|\hat{t}^{k}\\|n_{b}\kappa_{b}\right].$ In the latter expansion, the $d^{k}_{ab}(rs)$ are known as ‘spin-angular coefficients’ that arise from using Racah’s algebra in the decomposition of the many-electron matrix elements Grant (2007); Gaigalas et al. (1997). The expressions (8), (9) and (10) are general and can be used for any one-particle operator. We consider the following four mechanisms which may induce atomic EDM: tensor- pseudotensor ($\hat{H}_{TPT}$), pseudoscalar-scalar ($\hat{H}_{SPS}$), Schiff moment ($\hat{H}_{SM}$), and electron EDM interaction with nuclear magnetic field ($\hat{H}_{B}$). The interactions, which are all of rank $k=1$, are discussed in more detail in the next sections. In addition the expression for the electric dipole interaction is given. ### II.1 The electric dipole operator The electric-dipole moment operator has the rank $k$=1 in (8), (9), and (10), and the single-particle reduced matrix element $\left[n_{a}\kappa_{a}\\|\hat{t}^{k}\\|n_{b}\kappa_{b}\right]$ in equation (10) can be written as $\displaystyle\left[n_{a}\kappa_{a}\\|\hat{d}^{1}\\|n_{b}\kappa_{b}\right]$ $\displaystyle=$ (11) $\displaystyle-\left[\kappa_{a}\\|C^{1}\\|\kappa_{b}\right]\;\int_{0}^{\infty}\left(P_{a}P_{b}\;+\;Q_{a}Q_{b}\right)\;r\;dr,$ where $P$ and $Q$ are large and small components of the relativistic radial wave functions, respectively. The single-particle angular reduced matrix elements can be expressed as: $\displaystyle\left[\kappa_{a}\\|C^{k}\\|\kappa_{b}\right]$ $\displaystyle=$ (14) $\displaystyle(-1)^{j_{a}+1/2}\;\sqrt{2j_{b}\;+\;1}\left(\begin{array}[]{ccc}j_{a}&k&j_{b}\\\ 1/2&0&-1/2\end{array}\right)\pi\left(l_{a},l_{b},k\right),$ where $\pi\left(l_{a},l_{b},k\right)$ is defined as: $\displaystyle\pi\left(l_{a},l_{b},k\right)$ $\displaystyle=\left\\{\begin{array}[]{ll}1;&\mbox{ if }l_{a}+k+l_{b}\mbox{ even,}\\\ 0;&\mbox{ otherwise. }\end{array}\right.$ (17) ### II.2 Tensor-pseudotensor interaction One of the possible sources of the EDM in diamagnetic atoms is the tensor- pseudotensor (TPT) interaction between electrons and nucleons, violating both parity (P) and time (T)-reversal invariance. It can be expressed as $\displaystyle\hat{H}_{TPT}\;=\;i\sqrt{2}G_{F}C_{T}\;\sum_{j=1}^{N}\;\left(<\bm{{\sigma}}_{A}>\cdot\,\hskip 0.28436pt\bm{\gamma}_{j}\right)\;\rho\left(r_{j}\right)\,.$ (18) $G_{F}$ is the Fermi coupling constant, $A$ is the number of nucleons, $\gamma_{j}$ is the Dirac matrix, and $C_{T}$ is a dimensionless coupling constant of the TPT interaction. $C_{T}$ is equal to zero within the standard model, but it is finite in some theories beyond the standard model of elementary particle physics. According to Dzuba et al Dzuba et al. (2009) $\displaystyle C_{T}<\bm{{\sigma}}_{A}>\;=\;\left<C_{T}^{p}\sum_{p}{\bm{\sigma}_{p}}+C_{T}^{n}\sum_{n}{\bm{\sigma}_{n}}\right>\,,$ (19) where $\left<...\right>$ represents averaging over the nuclear state with the nuclear spin $\bm{I}$. The nuclear charge density distribution $\rho\left(r\right)$ is the normalized to unity two-component Fermi function Dyall et al. (1989) $\rho(r)=\frac{\rho_{0}}{1+e^{(r-b)/a}}$ (20) where $a$ and $b$ depend on the mass of the isotope. The single-particle reduced matrix elements $\left[n_{a}\kappa_{a}\\|\hat{t}^{k}\\|n_{b}\kappa_{b}\right]$ in equation (10) for the tensor - pseudotensor interaction has the form $\displaystyle\left[n_{a}\kappa_{a}\\|\hat{h}_{TPT}^{1}\\|n_{b}\kappa_{b}\right]=\sqrt{2}\;G_{F}\;C_{T}\;<\bm{{\sigma}}_{A}>\;\left[n_{a}\kappa_{a}\\|i\;\hat{\gamma}^{1}\;\rho\left(r\right)\\|n_{b}\kappa_{b}\right]=$ (21) $\displaystyle-\sqrt{2}\;G_{F}\;C_{T}\;<\bm{{\sigma}}_{A}>\left\\{\left[-\kappa_{a}\\|\sigma^{1}\\|\kappa_{b}\right]\;\int_{0}^{\infty}\ P_{b}Q_{a}\;\rho\;dr\right.\left.+\left[\kappa_{a}\\|\sigma^{1}\\|-\kappa_{b}\right]\;\int_{0}^{\infty}\ P_{a}Q_{b}\;\rho\;dr\right\\}\,,$ where the single-particle angular reduced matrix elements can be expressed as: $\displaystyle\left[-\kappa_{a}\\|\sigma^{1}\\|\kappa_{b}\right]$ $\displaystyle=$ $\displaystyle\frac{\left<l_{b}\frac{1}{2}0\frac{1}{2}\|j_{a}\frac{1}{2}\right>\left<l_{b}\frac{1}{2}0\frac{1}{2}\|j_{b}\frac{1}{2}\right>-\left<l_{b}\frac{1}{2}1-\frac{1}{2}\|j_{a}\frac{1}{2}\right>\left<l_{b}\frac{1}{2}1-\frac{1}{2}\|j_{b}\frac{1}{2}\right>}{\left<j_{b}1\frac{1}{2}0\|j_{a}\frac{1}{2}\right>},$ (22) $\displaystyle\left[\kappa_{a}\\|\sigma^{1}\\|-\kappa_{b}\right]$ $\displaystyle=$ $\displaystyle\frac{\left<l_{a}\frac{1}{2}0\frac{1}{2}\|j_{a}\frac{1}{2}\right>\left<l_{a}\frac{1}{2}0\frac{1}{2}\|j_{b}\frac{1}{2}\right>-\left<l_{a}\frac{1}{2}1-\frac{1}{2}\|j_{a}\frac{1}{2}\right>\left<l_{a}\frac{1}{2}1-\frac{1}{2}\|j_{b}\frac{1}{2}\right>}{\left<j_{b}1\frac{1}{2}0\|j_{a}\frac{1}{2}\right>}.$ (23) ### II.3 Pseudoscalar-scalar interaction The interaction Hamiltonian for the pseudoscalar-scalar ($PSS$) interaction between the electrons and the nucleus reads $\displaystyle\hat{H}_{PSS}\;=\;\frac{-G_{F}\;C_{P}}{2\sqrt{2}m_{p}c}\;{\sum_{j=1}^{N}}\;\gamma_{0}\;(\bm{\nabla}_{j}\rho\left(r_{j}\right)<\bm{{\sigma}}_{A}>).$ (24) $C_{P}$ is dimensionless coupling constant of the $PSS$ interaction. Analogously to the $TPT$ interaction, $C_{P}$ constant is zero within the standard model. According to Dzuba et al Dzuba et al. (2009) $\displaystyle C_{P}<\bm{{\sigma}}_{A}>\;=\;\left<C_{P}^{p}\sum_{p}\bm{{\sigma}}_{p}+C_{P}^{n}\sum_{n}\bm{{\sigma}}_{n}\right>\,.$ (25) The single-particle reduced matrix element $\left[n_{a}\kappa_{a}\\|\hat{t}^{k}\\|n_{b}\kappa_{b}\right]$ in the equation (10) for the pseudoscalar-scalar interaction has the form $\displaystyle\left[n_{a}\kappa_{a}\\|\hat{h}_{PSS}^{1}\\|n_{b}\kappa_{b}\right]=-\frac{G_{F}\;C_{P}}{2\sqrt{2}m_{p}c}<\bm{{\sigma}}_{A}>\left[n_{a}\kappa_{a}\\|\gamma_{0}\;\nabla^{1}\rho\left(r\right)\\|n_{b}\kappa_{b}\right]=$ (26) $\displaystyle-\frac{G_{F}\;C_{P}}{2\sqrt{2}m_{p}c}<\bm{{\sigma}}_{A}>\left[\kappa_{a}\\|C^{1}\\|\kappa_{b}\right]\int_{0}^{\infty}\left(P_{a}P_{b}\;-\;Q_{a}Q_{b}\right)\;\frac{d\rho}{dr}\;dr.$ ### II.4 Schiff moment The Hamiltonian of this interaction ($H_{SM}$) can be expressed as: $\displaystyle\hat{H}_{SM}\;=\;\frac{3}{B}\;\sum_{j=1}^{N}\;\left(\bm{S}\cdot\bm{r}_{j}\right)\;\rho\left(r_{j}\right)\,.$ (27) The Schiff moment $\bm{S}$ is directed along the nuclear spin $\bm{I}$ and $\bm{S}\,\equiv\,S\bm{I}/I$, with $S$ being the coupling constant, and $B=\int_{0}^{\infty}\rho(r)r^{4}dr$. The single-particle reduced matrix element $\left[n_{a}\kappa_{a}\\|\hat{t}^{k}\\|n_{b}\kappa_{b}\right]$ in expansion (10) for SM can be factorized into reduced angular matrix element and radial integral $\displaystyle\left[n_{a}\kappa_{a}\\|\hat{h}_{SM}^{1}\\|n_{b}\kappa_{b}\right]=\frac{3}{B}\;S\;\left[n_{a}\kappa_{a}\\|\hat{r}^{1}\;\rho\left(r\right)\\|n_{b}\kappa_{b}\right]=$ (28) $\displaystyle\frac{3}{B}\;S\;\left[\kappa_{a}\\|C^{1}\\|\kappa_{b}\right]\;\int_{0}^{\infty}\left(P_{a}P_{b}\;+\;Q_{a}Q_{b}\right)\;\rho\;r\;dr\,.$ ### II.5 Electron electric dipole moment The operator for the electron EDM interaction with magnetic field of a nucleus can be expressed as: $\displaystyle\hat{H}_{B}\;=\;-id_{e}\;{\sum_{j=1}^{N}}\;(\bm{\gamma}_{j}\;\bm{B}),$ (29) where $d_{e}$ represents the electron electric dipole moment, and $\bm{B}$ the magnetic field of the nucleus. The single-particle reduced matrix element $\left[n_{a}\kappa_{a}\\|\hat{t}^{k}\\|n_{b}\kappa_{b}\right]$ in expansion (10) for operator of electron EDM interaction with magnetic field of a nucleus can be factorized into reduces angular matrix element and radial integral $\displaystyle\left[n_{a}\kappa_{a}\\|h^{el}_{B}\\|n_{b}\kappa_{b}\right]$ $\displaystyle=$ (30) $\displaystyle\frac{d_{e}\mu}{2m_{p}c}\left\\{-3\left[-\kappa_{a}\\|C^{1}\\|-\kappa_{b}\right]\;\int_{R}^{\infty}\frac{Q_{a}P_{b}}{r^{3}}\;dr\right.\;-\;3\left[\kappa_{a}\\|C^{1}\\|\kappa_{b}\right]\int_{R}^{\infty}\frac{P_{a}Q_{b}}{r^{3}}\;dr\;-\;\left[-\kappa_{a}\\|\sigma^{1}\\|\kappa_{b}\right]\;\int_{R}^{\infty}\frac{Q_{a}P_{b}}{r^{3}}\;dr$ $\displaystyle-\;\left[\kappa_{a}\\|\sigma^{1}\\|-\kappa_{b}\right]\int_{R}^{\infty}\frac{P_{a}Q_{b}}{r^{3}}\;dr\;+\;2\left[-\kappa_{a}\\|\sigma^{1}\\|\kappa_{b}\right]\;\int_{0}^{R}\frac{Q_{a}P_{b}}{R^{3}}\;dr\left.\;+\;2\left[\kappa_{a}\\|\sigma^{1}\\|-\kappa_{b}\right]\int_{0}^{R}\frac{P_{a}Q_{b}}{R^{3}}\;dr\right\\},$ where $R$ and $\mu$ represent the nuclear radius and nuclear magnetic moment, respectively. We extended the GRASP2K Jönsson et al. (2013) package for the calculation of the matrix elements (10) and for the calculation of single-particle reduced matrix elements (11), (21), (26), (28), and (30). The extension, presented in this work, includes subroutines for calculation of matrix elements of type $\left<i\|\hat{H}_{int}\|0\right>$ from (4) for tensor-pseudotensor $\hat{H}_{TPT}$, pseudoscalar-scalar $\hat{H}_{PSS}$, Schiff moment $\hat{H}_{SM}$, electron EDM interaction with nuclear magnetic field $\hat{H}_{B}$, and electric dipole moment $\hat{D}_{z}$. ## III MCDHF calculations ### III.1 MCDHF theory We used the MCDHF approach to generate numerical representations of atomic wave functions. An atomic state function (ASF) $\Psi(\gamma PJM_{J})$ is obtained as a linear combination of configuration state functions $\Phi(\gamma_{r}PJM_{J})$, eigenfunctions of the parity $P$, and total angular momentum operators $J^{2}$ and $M_{J}$: $\Psi(\gamma PJM_{J})=\sum_{r}c_{r}\Phi(\gamma_{r}PJM_{J}),$ (31) where $c_{r}$ are configuration mixing coefficients. The multiconfiguration energy functional was based on the Dirac-Coulomb Hamiltonian, given (in a.u.) by $\hat{H}_{DC}=\sum_{j=1}^{N}\Big{(}c\bm{\alpha}_{j}\cdot\bm{p}_{j}+(\beta_{j}-1)c^{2}+V({r_{j}})\Big{)}+\sum_{j<k}^{N}\frac{1}{r_{jk}},$ (32) where $\bm{\alpha}$ and $\beta$ are the Dirac matrices, and $p$ is the momentum operator. The electrostatic electron-nucleus interaction, $V({r_{j}})$, has been generated from a 2-parameter Fermi nuclear charge distribution (20). The effects of the Breit interaction, as well as QED effects, were neglected, since they are expected to be small at the level of accuracy attainable in the present calculations. ### III.2 Energy functionals Several different methods of wave function generation were employed, in order to test the dependence of the calculated atomic EDMs on options available in the GRASP2K Jönsson et al. (2013) implementation of the MCDHF method. One option is related to the variational energy functional in the wave function optimisation procedure. Two general forms of the energy functional are implemented in the GRASP2K Jönsson et al. (2013) package: #### III.2.1 Extended Optimal Level One-electron orbitals based on the Extended Optimal Level (EOL) form are optimised to minimise the energy functional, which is defined through the equation (39) in reference Dyall et al. (1989), where generalised weights (equation (40) in ref. Dyall et al. (1989)) determine a specific atomic state ASF (or a set of ASFs). Consequently, the orbitals in the EOL approach are optimal for a specific atomic state ASF or a set of ASFs. #### III.2.2 Extended Average Level One-electron orbitals based on the Extended Average Level (EAL) form are optimised to minimise the (optionally weighted) sum of energies of all ASFs which may be constructed from a given set of CSFs, so eventually it yields an (optionally weighted) average energy of a set of atomic states. This approach is computationally much cheaper, but usually less accurate than the approach based on the EOL functional. ### III.3 Virtual orbital sets The numerical wave functions were obtained independently for the two parities. The calculations proceeded in two phases. Spectroscopic (occupied) orbitals were obtained in the Dirac-Hartree-Fock approximation. They were kept frozen in all subsequent calculations. Then virtual (correlation) orbitals were generated in several consecutive steps. At each step the virtual set has been extended by one layer of virtual orbitals. A layer is defined as a subset of virtual orbitals, usually with different angular symmetries, optimized simultaneously in one step, and usually frozen in all subsequent steps. In the present paper three or four layers of virtual orbitals of each of the s, p, d, f, g symmetries were generated. At each stage only the outermost layer is optimized and the remaining orbitals (spectroscopic as well as other virtual layers) are kept frozen. Virtual orbitals were generated in an approximation in which all single and restricted double substitutions from valence orbitals and a subset of core orbitals to subsequent layers of virtual orbitals were included. The restriction was applied to double substitutions in such a way that only one electron was substituted from core shells, the other one had to be substituted from the valence shells (i.e. from 7s shell in the case of even parity ground state of radium atom; 7s and 7p shells in the case of odd parity excited states of radium; 6s and 6p in the cases of mercury and ytterbium). Four layers of virtual orbitals were generated for each of the three elements – Ra, Hg, Yb. The combined contribution of the $n=3$ shells to the hyperfine constants of the $7s7p$ ${}^{1}P$ state was evaluated in a previous paper Bieroń and Pyykkö (2005) and found to be negligible, while the combined contribution of the $n=4$ shells was below 1 percent level. Therefore in the present calculations the innermost core orbitals $1s$, $2s$, $2p$, $3s$, $3p$, $3d$ of the radium atom were kept closed for electron substitutions. All other core orbitals, as well as valence orbitals, were subject to electron substitutions. By similar argument, the innermost core orbitals $1s$, $2s$, $2p$ of Hg and Yb were kept closed for electron substitutions. The reader is referred to the papers Bieroń and Pyykkö (2005); Bieroń et al. (2009b) for further details of wave function generation. ### III.4 Non-orthogonal orbital sets The matrix elements of all interactions were calculated between the ground state $ns^{2}$ ($J=0$) and excited states with total angular momentum $J=1$ and opposite parity for 225Ra, 199Hg, and 171Yb. In principle, the optimal wave functions for calculations of EDM matrix elements are obtained in the Extended Optimal Level form (see section III.2.1 above) separately for each parity. The wave functions optimised separately for the ground and excited states are built from independent sets of one-electron orbitals. The two sets are mutually non-orthogonal and they automatically account for relaxation effects involved in calculations of matrix elements between different atomic states Bieroń et al. (2004); Bieroń et al. (2009a). On the other hand, the transition energies obtained from wave functions based separately optimised orbital sets may be less accurate than transition energies obtained from calculations based on a common set of mutually orthogonal one-electron orbitals. The above situation often arises when multiconfiguration expansions are tailored specifically to include only those electron correlation effects that are important for the one-electron expectation values. For one-electron matrix elements involved in the present calculations the dominant contributions arise from single and restricted double substitutions. We have not included the unrestricted double substitutions i.e. the electron correlation effects with dominant contributions to the total energy, as well as higher order substitutions, since their impact on EDMs is indirect and usually small Roberts et al. (2013). We evaluated the effect of the relaxation of the wave functions by performing two parallel sets of calculations based on a common orbital set (orthogonal) and on two separately optimised orbital sets (non-orthogonal), respectively. Table 1 lists the atomic EDM for 225Ra, calculated in several approximations. The first line (denoted 0(DF) in the first column) lists the results obtained with uncorrelated Dirac-Fock wave functions. The following lines provide the results obtained with different numbers (1-4) of virtual orbital layers included in the Virtual Orbital Set (VOS). The number of virtual orbital layers in a given VOS is quoted in the first column. We skipped the ’orthogonal’ calculation with four virtual orbital layers, since the preceding lines show clearly that the effects of non-orthogonality (i.e. the relaxation of wave functions) are of the order of a few percent, up to 11% for the interaction of the electron electric dipole moment with the nuclear magnetic field (eEDM entry in Table 1). The calculation of matrix elements in the non-orthogonal case requires a transformation of one-electron orbitals from which the wave functions of ground and excited states are built. The program BIOTRA2 Jönsson et al. (2013) was applied to transform both wave functions to a biorthonormal form Malmqvist (1986); Olsen et al. (1995) which then permits to use standard Racah algebra in evaluation of matrix elements. Table 1: Contributions to the atomic EDM from TPT, PSS, SM, and electron EDM interactions, calculated for 225Ra, using orthogonal (Orth) and non-orthogonal (Non-O) orbital sets. The number VOS in the first column is the number of virtual orbital layers. Transition energies are experimental. \| TPT \| PSS \| SM \| eEDM ---\|---\|---\|---\|--- VOS \| Orth \| Non-O \| Orth \| Non-O \| Orth \| Non-O \| Orth \| Non-O 0(DF) \| -16.3 \| -15.81 \| -59.7 \| -57.87 \| -6.53 \| -6.32 \| -55.6 \| -46.67 1 \| -14.5 \| -15.51 \| -53.3 \| -57.09 \| -6.28 \| -7.01 \| -48.1 \| -43.69 2 \| -18.8 \| -19.90 \| -69.0 \| -72.95 \| -7.79 \| -8.16 \| -63.5 \| -58.07 3 \| -19.9 \| -20.68 \| -70.3 \| -75.83 \| -8.27 \| -8.59 \| -66.9 \| -60.13 4 \| \| -20.28 \| \| -74.42 \| \| -8.63 \| \| -58.45 ### III.5 Extended Optimal Level calculations The final values of atomic EDMs, presented in the Tables 4, 5, 6, and 7, were obtained with the Extended Optimal Level optimisation procedure described in section III.2.1 above. At each stage of generation of virtual orbital sets, a decision had to be made with respect to the number of atomic levels included in the variational energy functional. Table 2 presents the contributions $d_{at}^{TPT}$ to the atomic EDM of 225Ra from the tensor-pseudotensor interaction (18). The contributions from particular atomic states are listed in subsequent lines. The radial wave functions were optimised within the EOL procedure, with different numbers of EOL levels: 4, 6, 8, 10, or 12 levels, as indicated in the first line of the Table 2. These data were obtained with experimental transition energies quoted from the the NIST Atomic Spectra Database (NIST ASD) NIS . An inspection of the Table 2 (the last line, denoted ’Sum All’) indicates that the $d_{at}^{TPT}$ expectation value becomes stable when eight or more levels are included in the Extended Optimal Level energy functional. Analogous decisions were made for all virtual orbital sets, as well as for the other two elements. The final calculations were made with varying numbers of EOL levels, between 2 levels for uncorrelated Dirac-Fock wave functions, with 6-8 levels in most correlated calculations, and up to 13 levels in one case. ### III.6 Orbital contributions Another interesting conclusion arises from the analysis of contributions of particular one-electron orbitals generated in the EOL optimisation procedure. The analysis presented in the Table 2 was made with only one virtual orbital layer, because the Extended Optimal Level optimisation procedure described in section III.2.1 above becomes unstable with the increasing numbers of virtual layers and of EOL levels. However, already at this level of approximation the dominant contributions come from the singlet $7s7p$ ${}^{1}P$ and triplet $7s7p$ ${}^{3}P$ excited states. The states $7s8p$ ${}^{1}P$ and $7s8p$ ${}^{3}P$, involving $8p$ orbital, contribute 9% and 3%, respectively (and their contributions partially cancel due to different signs). All other states contribute less than one percent each. The following lines present contributions of singlet and triplet states generated by single or double electron substitutions from the reference configuration $7s7p$ to the lowest available orbitals $8s$, $8p$, and $6d$. The line denoted ’Sum s-p’ shows the contributions of the four dominant states generated by single electron substitutions from the reference configuration. The line denoted ’Sum s-d’ shows the sum of entries from the preceding two lines of the $6d7p$ configuration; the line ’s-p+s-d’ shows the sum of all preceding contributions. The next six lines present the contributions of higher lying levels, and the line ’Sum D’ show the sum of the contributions from these six preceding lines. The last line ’Sum All’ shows the total sum of all contributions of all states listed in the preceding lines. We present the partial sums (’s-p’, ’s-d’, ’s-p+s-d’, and ’Sum D’) to show their dependence on the number of EOL levels. The contributions of individual levels are not very stable, and in particular the small contributions may vary significantly, but the partial sums are more stable, and the total sum (’Sum All’) is strongly stabilized by the contributions from the dominant states. It is interesting to make a comparison of Table 2 with Table VI from the reference Latha and Amjith (2013). In reference Latha and Amjith (2013) the contributions from $7s_{1/2}$-$7p_{1/2}$ and $7s_{1/2}$-$8p_{1/2}$ single- particle matrix elements (pairings in their language) are of comparable sizes, -324.468 and -306.133, respectively, while in our calculations the relative sizes of the contributions from $7s_{1/2}$-$8p_{1/2}$, with respect to the contribution from $7s_{1/2}$-$7p_{1/2}$ pairing, are 9% and 3% for singlet and triplet states, respectively. Also, there are differences with respect to the contributions of higher symmetry orbitals. For instance, the contribution from $d_{5/2}$ orbitals is of the order of 4% (see TABLE VII in reference Latha and Amjith (2013)), while in our calculations the contributions from $d_{5/2}$ orbitals are below 1%. It is difficult to explain these differences, but one possible explanation is due to differences in optimisation procedures and radial shapes of one- electron orbitals which resulted from these procedures, as discussed in the section III.5. Different compositions of particular atomic states are likely consequences of differences in radial bases. The authors of the reference Latha and Amjith (2013) used Gaussian basis sets, while in our calculations we use numerical orbitals defined on a grid. We do not have insight into the details of the calculations presented in the reference Latha and Amjith (2013), but their Gaussians are likely to be evenly distributed over the entire configurational space. Different theories use different methods of construction for atomic states. A consequence of these differences is the fact, that comparisons of contributions from particular atomic states or from individual one-electron orbitals are not meaningful. All excited and virtual orbitals generated in our calculations were optimised with multiconfiguration expansions designed for valence and core-valence electron correlation effects, resulting in virtual orbital shapes with maximal overlaps with valence and outer core spectroscopic orbitals. Consequently, the correlation corrections to the wave function are likely to be larger for the lower states included in the Extended Optimal Level procedure. We performed comparison calculations with virtual orbitals generated with three different methods: the Extended Average Level procedure, as described in the section III.2.2; with virtual orbitals generated within the screened hydrogenic approximation; and virtual orbitals from Thomas-Fermi potential. As described in the section III.2.2, one-electron virtual orbitals generated with the EAL functional are optimised to minimise the sum of energies of all states. Hydrogenic and Thomas-Fermi virtual orbitals are not variationally optimized, they just form orthogonal bases. Our comparison calculations indicate, that calculations based on Extended Average Level, hydrogenic, and Thomas-Fermi virtual orbitals converge slower than Extended Optimal Level calculations, and the contributions of higher lying levels are larger, compared to EOL results. Table 2: $d^{TPT}_{at}$contribution to atomic EDM, calculated with the EOL method for 1st VOS, using different numbers of optimized levels and experimental transition energies, in units $\left(10^{-20}C_{T}\left<{\bf\sigma}_{A}\right>\left\|e\right\|\mbox{cm}\right)$, for 225Ra. Numbers in brackets represent powers of 10. Levels \| 4 \| 6 \| 8 \| 10 \| 12 ---\|---\|---\|---\|---\|--- $7s$$7p$ ${}^{3}P$ \| -5.00 \| -4.46 \| -4.63 \| -4.59 \| -4.63 $7s$$7p$ ${}^{1}P$ \| -1.03[1] \| -8.80 \| -8.70 \| -8.69 \| -8.57 $7s$$8p$ ${}^{3}P$ \| \| 0.39 \| 0.30 \| 0.33 \| 0.44 $7s$$8p$ ${}^{1}P$ \| \| -1.12 \| -0.96 \| -1.01 \| -1.24 Sum s-p \| -1.53[1] \| -1.40[1] \| -1.40[1] \| -1.40[1] \| -1.40[1] $6d$$7p$ ${}^{3}D$ \| 2.53[-3] \| -7.72[-4] \| 2.96[-2] \| -9.30[-2] \| -6.91[-2] $6d$$7p$ ${}^{3}P$ \| 1.98[-1] \| -3.08[-2] \| -1.13[-1] \| 3.55[-2] \| 7.34[-2] Sum s-d \| 2.00[-1] \| -3.16[-2] \| -8.33[-2] \| -5.75[-2] \| 4.25[-3] s-p+s-d \| -1.51[1] \| -1.40[1] \| -1.41[1] \| -1.41[1] \| -1.40[1] $6d$$8p$ ${}^{3}D$ \| \| \| -4.79[-2] \| -9.44[-3] \| -3.63[-3] $6d$$8p$ ${}^{3}P$ \| \| \| -1.15[-1] \| -4.90[-2] \| -8.36[-2] $7p$$8s$ ${}^{3}P$ \| \| \| \| -1.96[-2] \| -2.20[-2] $7p$$8s$ ${}^{1}P$ \| \| \| \| -6.02[-3] \| -6.31[-3] $6d$$7p$ ${}^{1}P$ \| \| \| \| \| -5.12[-3] $8s$$8p$ ${}^{3}P$ \| \| \| \| \| 1.50[-3] Sum D \| \| \| -1.63[-1] \| -8.41[-2] \| -1.19[-1] Sum All \| -1.51[1] \| -1.40[1] \| -1.42[1] \| -1.41[1] \| -1.41[1] ### III.7 Transition energies The summation in equation (4) runs over all excited states of appropriate parity and symmetry. The contributions of higher lying levels are gradually decreasing, since they are suppressed both by the energy denominators, as well as by decreasing overlaps of one-electron radial orbitals, entering integrals in the equations: (21), (26), (28), and (30). In numerical calculations they have to be cut off at certain level of accuracy. Except where indicated otherwise, the results presented in the present paper were computed with experimental transition energies in the denominators of the matrix elements in equation (4). The transition energies were calculated from the NIST ASD database NIS and we include levels up to $6d7p~{}^{3}P$ for 225Ra, $6s8p~{}^{1}P$ for 171Yb, and $6s9p~{}^{1}P$ for 199Hg. However, several levels are missing in NIS , so we employed an approach, where those transition energies which were not available, were replaced by the energies calculated with one of the three different methods: (1) using theoretical energies obtained from MCDHF approach; (2) with the energy of the upper level replaced by the energy of the lowest excited state; (3) with the energy of the upper level replaced by the experimental ionisation limit. The choice was made between the above three options in case of each missing level, based on availability of a reliable theoretical energy, or alternatively on the proximity of the lowest excited state or the experimental ionisation limit. To verify this approach we performed test calculations, where all three choices were used together. Table 3 presents the contributions from the tensor- pseudotensor interaction to the atomic EDM of radium isotope 225Ra. Transition energies in Table 3 were taken from: MCDHF RSCF calculation (RSCF), MCDHF RCI calculation (RCI), experimental data (Expt), experimental ionisation limit (ExIL), experimental energy of the lowest excited level (Exp1). The MCDHF RSCF case was a self-consistent-field Extended Optimal Level calculation, with 2, 6, 8 and 6 EOL levels for DF, 1, 2, 3 and 4 VOS, respectively. The MCDHF RCI case was a configuration-interaction calculation with 100 levels included. Their differences indicate the deviation incurred when the number of EOL levels is varied. It should be noted that experimental values of the energies of the $7s7p$ levels were used in all cases in columns ’Expt’, ’ExIL’, and ’Exp1’. The lowest $nsnp$ levels yield the largest contributions to all EDM matrix elements in the present calculations, and their energies are available for all elements in question, therefore replacements were made only for higher lying levels. The number VOS in the first column of Table 3 represents the number of virtual orbital layers. These data indicate the sizes of errors, which may arise from replacing experimental transition energies with experimental ionisation limit (ExIL) or experimental energy of the lowest excited level (Exp1). As can be seen, the deviation is less than 10% in case of radium. The deviations of the data obtained with calculated transition energies are larger, due to the nature of the wavefunctions built from non- orthogonal orbital sets, as explained in the section III.4 above. Table 3: Tensor-pseudotensor interaction contributions to EDM, for 225Ra, in units $\left(10^{-20}C_{T}\left<{\bf\sigma}_{A}\right>\left\|e\right\|\mbox{cm}\right)$, calculated with the EOL method and compared with data from other methods. Transition energies taken from: MCDHF-RSCF calculation (RSCF), experimental data (Expt), MCDHF-RCI calculation (RCI), experimental ionisation limit (ExIL), experimental value of lowest excited level (Exp1). (see text for explanation). The number VOS in the first column is the number of virtual orbital layers. \| 225Ra ---\|--- VOS \| RSCF \| RCI \| Expt \| ExIL \| Exp1 0(DF) \| -18.31 \| -18.31 \| -15.81 \| -15.81 \| -15.81 1 \| -10.37 \| -11.81 \| -15.51 \| -14.70 \| -13.92 2 \| -12.04 \| -12.58 \| -19.90 \| -20.08 \| -20.45 3 \| \| \| -20.68 \| -21.22 \| -22.52 4 \| \| \| -20.28 \| -21.16 \| -22.32 Ref. Dzuba et al. (2009)(DHF) \| \| \| \| \| -3.5 Ref. Dzuba et al. (2009)(CI+MBPT) \| \| \| \| \| -17.6 Ref. Dzuba et al. (2009)(RPA) \| \| \| \| \| -16.7 Ref. Latha and Amjith (2013)(CPHF) \| \| \| \| \| -16.585 ### III.8 Uncertainty estimates Estimates of uncertainty in ab initio calculations are far more difficult than the calculations themselves, particularly in situations, where an atomic property is evaluated, which has not been calculated before within the same approach for any other element. We can indicate possible sources of uncertainties, but their sizes are difficult to estimate. The possible sources of uncertainties are the following. Table 4: Tensor-pseudotensor interaction contributions to EDM, calculated with the EOL method in different virtual sets, in units $\left(10^{-20}C_{T}\left<{\bf\sigma}_{A}\right>\left\|e\right\|\mbox{cm}\right)$, for 225Ra, 199Hg, and 171Yb, compared with data from other methods. \| 225Ra \| 199Hg \| 171Yb ---\|---\|---\|--- VOS \| Ex \| ExJL \| Ex1 \| Ex \| Ex 0(DF) \| -15.81 \| -15.81 \| -15.81 \| -6.15 \| -3.31 1 \| -15.51 \| -14.70 \| -13.92 \| -4.86 \| -1.94 2 \| -19.90 \| -20.08 \| -20.45 \| -5.70 \| -3.71 3 \| -20.68 \| -21.22 \| -22.52 \| -6.10 \| -4.03 4 \| -20.28 \| -21.16 \| -22.32 \| -5.53 \| -4.24 Ref. Dzuba et al. (2009)(DHF) \| -3.5 \| \| \| -2.4 \| -0.70 Ref. Mårtensson-Pendrill (1985)(DHF) \| \| \| \| -2.0 \| Ref. Dzuba et al. (2009)(CI+MBPT) \| -17.6 \| \| \| -5.12 \| -3.70 Ref. Dzuba et al. (2009)(RPA) \| -17.6 \| \| \| -5.89 \| -3.37 Ref. Mårtensson-Pendrill (1985)(RPA) \| \| \| \| -6.0 \| Ref. Latha et al. (2008)(RPA) \| \| \| \| -6.75 \| Ref. Latha and Amjith (2013)(CPHF) \| -16.585 \| \| \| -3.377 \| #### III.8.1 Electron correlation effects In extensive, large-scale calculations the relative accuracy can reach 1-5 percent, depending on the expectation value in question (see eg. Bieroń et al. (2009b); Bieroń and Pyykkö (2005)). An estimate of uncertainty associated with the electron correlation effects can be obtained in several ways. In the limit of very large number of virtual orbital layers an estimate of uncertainty may be related to oscillations of the calculated expectation value plotted as a function of the size of the multiconfiguration expansion Bieroń et al. (2009b). In the present paper an estimate of the uncertainty was based on the differences between the data obtained with the largest two multiconfiguration expansions, represented by 3 and 4 layers of virtual orbitals in Tables 4, 5, 6, and 7. We abstained from extending the virtual sets beyond fourth layer, because there are several other possible sources of uncertainty in the present calculations. An inspection of the Tables indicates that the differences between the last two lines range between 0.47% for the Schiff moment of Ra, and 15.77% for the Schiff moment of Hg (Table 6). We may assume the latter as an estimate of uncertainty associated with the neglected electron correlation effects. Table 5: Pseudoscalar-scalar interaction contributions to EDM, calculated with the EOL method in different virtual sets in units $\left(10^{-23}C_{P}\left<\sigma_{A}\right>\left\|e\right\|\mbox{cm}\right)$ for 225Ra, 199Hg, and 171Yb, compared with data from other methods. VOS \| \| 225Ra \| \| 199Hg \| \| 171Yb ---\|---\|---\|---\|---\|---\|--- 0(DF) \| \| -57.87 \| \| -21.49 \| \| -10.84 1 \| \| -57.09 \| \| -17.16 \| \| -6.31 2 \| \| -72.95 \| \| -19.94 \| \| -12.20 3 \| \| -75.83 \| \| -21.53 \| \| -13.26 4 \| \| -74.42 \| \| -19.45 \| \| -13.94 Ref. Dzuba et al. (2009)(DHF) \| \| -13.0 \| \| -8.7 \| \| -2.4 Ref. Dzuba et al. (2009)(CI+MBPT) \| \| -64.2 \| \| -18.4 \| \| -12.4 Ref. Dzuba et al. (2009)(RPA) \| \| -61.0 \| \| -20.7 \| \| -10.9 #### III.8.2 Wave function relaxation As explained in the section III.4 the effects of wave function relaxation were partially accounted for in the present calculations, by using non-orthogonal orbital sets for the opposite parities. An inspection of Table 1 indicates that the uncertainty which may arise from wave function relaxation effects is of the order of 10%, although this estimate is based on relaxing only the ASF wave function of the ground state on one hand, and the ASF wave functions of all excited states taken together, on the other hand. A more general, albeit far more expensive approach would be to generate separate atomic state functions for the ground state, as well as for each excited state, implying non-orthogonality between all ASFs of both parities. Table 6: Schiff moment contributions to atomic EDM, calculated with the EOL method in different virtual sets, in units $\left\\{10^{-17}[S/(\left\|e\right\|\mbox{fm}^{3})]\left\|e\right\|\mbox{cm}\right\\}$, for 225Ra, 199Hg, and 171Yb, compared with data from other methods. VOS \| \| 225Ra \| \| 199Hg \| \| 171Yb ---\|---\|---\|---\|---\|---\|--- 0(DF) \| \| -6.32 \| \| -2.46 \| \| -1.54 1 \| \| -7.01 \| \| -2.45 \| \| -0.88 2 \| \| -8.16 \| \| -2.23 \| \| -1.83 3 \| \| -8.59 \| \| -2.98 \| \| -2.05 4 \| \| -8.63 \| \| -2.51 \| \| -2.15 Ref. Dzuba et al. (2009)(DHF) \| \| -1.8 \| \| -1.2 \| \| -0.42 Ref. Dzuba et al. (2009)(CI+MBPT) \| \| -8.84 \| \| -2.63 \| \| -2.12 Ref. Dzuba et al. (2009)(RPA) \| \| -8.27 \| \| -2.99 \| \| -1.95 Ref. Dzuba et al. (2002)(RPA) \| \| -8.5 \| \| -2.8 \| \| Ref. Dzuba et al. (2007)(RPA) \| \| \| \| \| \| -1.9 Ref. Latha et al. (2009)(CCSD) \| \| \| \| -5.07 \| \| #### III.8.3 Energy denominators As discussed in the section III.7, the summation in equation (4) runs over all excited states of appropriate parity and symmetry. The NIST Atomic Spectra Database NIS is of course finite, therefore several levels with unknown energies had to be included in the present calculations. The uncertainty which may arise due to replacements described in the section III.7, should not exceed 10% in case of radium atom, and we expect the same order of magnitude in case of ytterbium and mercury. Table 7: Contributions of electron EDM interaction with magnetic field of nucleus, to atomic EDM are calculated with the EOL method in different virtual sets, in units ($d_{e}\times 10^{-4}$), for 225Ra, 199Hg, and 171Yb, compared with data from other methods. VOS \| \| 225Ra \| \| 199Hg \| \| 171Yb ---\|---\|---\|---\|---\|---\|--- 0(DF) \| \| -46.67 \| \| 13.41 \| \| 5.37 1 \| \| -43.69 \| \| 9.58 \| \| 3.17 2 \| \| -58.07 \| \| 12.22 \| \| 5.72 3 \| \| -60.13 \| \| 12.80 \| \| 6.09 4 \| \| -58.45 \| \| 11.45 \| \| 6.44 Ref. Dzuba et al. (2009)(DHF) \| \| -11 \| \| 4.9 \| \| 1.0 Ref. Mårtensson-Pendrill and Öster (1987)(DHF) \| \| \| \| 5.1 \| \| Ref. Dzuba et al. (2009)(CI+MBPT) \| \| -55.7 \| \| 10.7 \| \| 5.45 Ref. Dzuba et al. (2009)(RPA) \| \| -53.3 \| \| 12.3 \| \| 5.05 Ref. Mårtensson-Pendrill and Öster (1987)(RPA) \| \| \| \| 13 \| \| #### III.8.4 Systematic errors The possible sources of systematic errors include: omission of double, triple, and higher order substitutions; the effects of Breit interaction; and QED effects. The calculations of EDMs involve radial integrals of atomic one- electron orbitals, but all these integrals include factors in the integrands, which effectively cut off the integrals outside the nucleus, so the contribution to the integral comes from within or in the vicinity of the nucleus. Therefore an estimate of systematic errors can be made by comparing the EDM calculations with hyperfine structure calculations, where integrand in the form $r^{-2}$ appears in a one-electron integral, which in turn renders the dominant contribution from the first half of the radial orbital oscillation, i.e. near the nucleus. In certain cases in the hyperfine structure calculations the effects of double and triple substitutions can be quite sizeable, of the order of 10-20%, but they often partly cancel and the net deviation is often smaller than 10% Engels (1993); Bieroń et al. (2008). The effects of quadruple and higher order substitutions are negligible. The effects of Breit and QED are usually of the order of 1-2 percent or less for neutral systems. #### III.8.5 Error budget Based on the above estimates, the relative root-mean-square deviation of the present calculations yields $\sigma=25$%. ## IV Final results, discussion, and outlook ### IV.1 Summary Atomic EDMs arising from $(P,T)$-odd tensor-pseudotensor and pseudoscalar- scalar electron-nucleon interactions, nuclear Schiff moment, and interaction of electron electric dipole moment with nuclear magnetic field, are presented in Tables 4, 5, 6, and 7, for 225Ra, 199Hg, and 171Yb. The matrix elements and atomic EDMs were calculated using recently developed programs in the framework of the GRASP2K code Jönsson et al. (2013). One of the objectives of the present calculations was to test these programs. Therefore the results are compared with the data obtained by other methods: random phase aproximations (RPA), many-body perturbation theory and configuration interaction technique (CI+MBPT), coupled-cluster single-double (CCSD), and coupled-perturbed Hartree-Fock (CPHF) theory. These methods are usually more accurate in calculations of properties of closed-shell atoms. An inspection of the Tables indicates that the differences between our results and the data obtained with the RPA methods Dzuba et al. (2009); Latha et al. (2008); Dzuba et al. (2007, 2002); Mårtensson-Pendrill and Öster (1987); Mårtensson-Pendrill (1985) range between 1.5% for the Schiff moment of Ra (Table 6), and 22.1% for the tensor- pseudotensor of Hg (Table 4), all of them within the error bounds estimated in the section III.8.5 above. Despite the reasonable agreement at the level of the correlated calculations, very large differences should be noted at the uncorrelated levels, DF (Dirac- Fock) in our calculations, and DHF (Dirac-Hartree-Fock) in references Dzuba et al. (2009) and Mårtensson-Pendrill and Öster (1987). We used the different symbols to visually differentiate the results obtained with different numerical codes, but the DF and DHF approximations are formally identical within the Dirac-Fock theory, and they should yield similar values, within numerical accuracies of the Dirac-Fock codes. A possible explanation of these large differences may be the fact that in our (DF) calculations the summation in equation (4) runs over only the two lowest excited states, singlet $nsnp$ ${}^{1}P$ and triplet $nsnp$ ${}^{3}P$, which are generated at the Dirac-Fock level of the GRASP2K code Jönsson et al. (2013). On the other hand, in references Dzuba et al. (2009) and Mårtensson-Pendrill and Öster (1987) the summation was probably carried over all excited states, which can be constructed from a suitable set of virtual orbitals. Otherwise we do not have an explanation. Large differences at the level of the correlated calculations should be noted between our results and the data obtained with the CPHF theory Latha and Amjith (2013). The differences are: 18% for TPT of Ra and 39% for TPT of Hg (see Table 4). The largest disagreement appears to be between the result of the present calculations and the value obtained with the CCSD theory Latha et al. (2009) for the Schiff moment of Hg (see Table 6). The difference amounts to 102%. It is difficult to explain some of the abovementioned differences. They may be due to different orbital shapes, orbital contributions, and relaxation effects, discussed in the sections III.6 and III.4, respectively. Another objective of the present calculations was to test the methods of wave function generation, as described in more detail in the section III.1, and of multiconfiguration expansions designed to account for valence and core-valence electron correlation effects. A reasonably good agreement of our results with the data obtained within the RPA and CI+MBPT methods Dzuba et al. (2009); Latha et al. (2008); Dzuba et al. (2007, 2002); Mårtensson-Pendrill and Öster (1987); Mårtensson-Pendrill (1985) seems to indicate that the multconfigurational model employed in the present calculations accounts for the bulk of the electron correlation effects. With adequate computer resources, these calculations may be extended in the future and include also core-core effects. Based on the experiences with other atomic properties, as well as on the present EDM calculations, we expect that the accuracy of the EDM calculations may be improved by a factor of ten, with respect to the current relative root-mean-square deviation of the order of $25$%. ### IV.2 Outlook Several refinements are possible with respect to the methods used in the present paper. To account more accurately for the electron relaxation, separate wave functions for the leading contributors to EDM may be generated. A more general, albeit far more expensive approach would be to generate separate ASFs for the ground state, as well as for each excited state, relaxing orthogonality of the orbital sets between all ASFs of both parities. The expectation values $d_{at}^{int}$ were calculated with theoretical (if reliable), and experimental (if available) transition energies, as explained in the section III.7. In fully correlated calculations theoretical transition energies would have to be evaluated with all single and unrestricted double substitutions. They would be computationally much more expensive than those presented in the present paper, but possible with the currently available massively-parallel computers. Electron correlation effects can also be accounted for using the partitioned correlation function interaction (PCFI) method Verdebout et al. (2013), that allows contributions from single and unrestricted double substitutions deep down in the atomic core to be summed up in a very efficient way. In the near future we will be able to perform fully ab initio calculations for atoms with arbitrary shell structures. We are currently testing the latest version of the GRASP package Jönsson et al. (2013), with angular programs providing full support for arbitrary numbers of electrons in open spdf shells. ###### Acknowledgements. The authors wish to thank the Visby program of the Swedish Institute for a collaborative grant. JB acknowledges the support from the Polish Ministry of Science and Higher Education (MNiSW) in the framework of the scientific grant No. N N202 014140 awarded for the years 2011-2014. The large-scale calculations were carried out with the supercomputer Deszno purchased thanks to the financial support of the European Regional Development Fund in the framework of the Polish Innovation Economy Operational Program (contract no. POIG.02.01.00-12-023/08). ## References * Jönsson et al. (2013) P. Jönsson, G. Gaigalas, J. Bieroń, C. Froese Fischer, and I. Grant, Comput. Phys. Commun. 184, 2197 (2013). * Khriplovich and Lamoreaux (1997) I. B. Khriplovich and S. K. Lamoreaux, _CP Violation Without Strangeness_ (Springer, Berlin, 1997). * Griffith et al. (2009) W. C. 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Acta 86, 429 (1993). * Bieroń et al. (2008) J. Bieroń, C. Froese Fischer, P. Jönsson, and P. Pyykkö, J. Phys. B 41, 115002 (2008). * Verdebout et al. (2013) S. Verdebout, P. Rynkun, P. Jönsson, G. Gaigalas, C. Froese Fischer, and M. Godefroid, J. Phys. B 46, 085003 (2013).	arxiv-papers	2013-12-23T10:57:53	2024-09-04T02:49:55.832194	{ "license": "Public Domain", "authors": "Laima Radziute, Gediminas Gaigalas, Per Jonsson, and Jacek Biero", "submitter": "Gaigalas Gediminas Habil. dr.", "url": "https://arxiv.org/abs/1312.6517" }
1312.6635	# Topic and Sentiment Analysis on OSNs: a Case Study of Advertising Strategies on Twitter Shana Dacres Hamed Haddadi Matthew Purver Cognitive Science Research Group School of Electronic Engineering and Computer Science Queen Mary University of London, UK [email protected] ###### Abstract Social media have substantially altered the way brands and businesses advertise: Online Social Networks provide brands with more versatile and dynamic channels for advertisement than traditional media (e.g., TV and radio). Levels of engagement in such media are usually measured in terms of content adoption (e.g., _likes_ and _retweets_) and sentiment, around a given topic. However, sentiment analysis and topic identification are both non- trivial tasks. In this paper, using data collected from Twitter as a case study, we analyze how engagement and sentiment in promoted content spread over a 10-day period. We find that promoted tweets lead to higher positive sentiment than promoted trends; although promoted trends pay off in response volume. We observe that levels of engagement for the brand and promoted content are highest on the first day of the campaign, and fall considerably thereafter. However, we show that these insights depend on the use of robust machine learning and natural language processing techniques to gather focused, relevant datasets, and to accurately gauge sentiment, rather than relying on the simple keyword- or frequency-based metrics sometimes used in social media research. ###### category: H.3.1 Information Storage and Retrieval Content Analysis and Indexing ###### keywords: Linguistic processing ###### category: I.2.7 Artificial Intelligence Natural Language Processing ###### keywords: Text analysis ###### keywords: Sentiment analysis, Topic analysis, Social Media, Online Advertising, User Engagement ††terms: Algorithms, Measurement ## 1 Introduction Online Social Networks (OSNs) such as Facebook, Twitter, and YouTube have emerged as highly engaging marketing and influence tools, increasingly used by advertisers to promote brand awareness and catalyze word-of-mouth marketing. Researchers have also long recognised the effectiveness of OSNs as a rich source for understanding the spread of information about the real world [20]. For example, Asur et al. [1] analyzed Twitter messages (_tweets_) to predict box-office ratings for newly released movies. Their findings shows that OSNs can be used to make quantitative predictions that outperform those of markets forecasts, by focusing on the sentiment expressed in the tweets. Brands also now recognise the potential of OSNs for gathering market intelligence and insight. In 2012, Twitter announced that 79% of people follow brands to get exclusive content.111http://advertising.twitter.com/2012/05/twitter4brands- event-in-nyc.html This provides the opportunity for brands to participate in real-time conversations to listen to and engage users, respond to complaints and feedback, drive consumer action and broadcast content. Understanding the real engagement of the end users with the brands and their OSN presence has given rise to a number of data analytics, sentiment analysis and social media optimisation startups and academic research projects. However, the techniques required pose a number of challenges and pitfalls often ignored by researchers and analysts, and adopting a particular method naively can lead to problems. Significant progress in Natural Language Processing (NLP) and Machine Learning (ML) has produced models for topic modelling designed for social media [23], and high accuracies in sentiment detection (e.g. [26]), even with the possibility of detecting sarcasm [11]; but care still needs to be taken while using and relying on the relevant tools and techniques straight out of the box [8]. In this study we present a focused case study by examining the content and volume of users’ brand engagement on OSNs to determine the effect of choice of promotion channel on a brand’s influence.222In this work, engagement is defined as adoption of the content by e.g., replying to a tweet, mentioning the brand name, or including the hashtag in a tweet. We are not able to measure external engagement such as sharing content on other OSNs, or clicking on the links in the tweet. We do this by analysing the engagement level of Twitter users, their adoption of brand hashtags, and the sentiment they express, to determine the similarities and differences between two separate advertising strategies on this network: promoted tweets, and promoted trends. We pose a number of questions regarding brands and advertising on OSNs: How does the sentiment for a promotion strategy spread over time? What are the engagement levels for each day of promotion? What is the engagement level (e.g. _retweets_ and _mentions_) for promoted brands and how do these affect the sentiments expressed towards a brand? In order to answer these questions, we use Twitter’s Streaming API service to collect engaged users’ profiles and tweets in regards to promoted influences (tweets and trends) over a busy 10 day shopping period for a selection of brands across different industries. We observe the need to accurately filter the resulting tweets for topical relevance, and compare simple keyword-based methods with a discrimative Machine Learning (ML) approach. We then classify the tweets by sentiment (positive, negative or neutral), and again compare a range of existing methods and tools. We then use this data to establish the driving factors behind the success of promoted influences and differences between advertising strategies. For both tasks, the choice of classification method makes a significant difference, highlighting the care that must be taken when choosing techniques for this kind of analysis. The rest of the paper is organized as follows: In Section 2 we present the some recent related studies. In Section 3 we describe our case study, dataset and its characteristics. In Section 4 we briefly discuss our sentiment analysis and text classification methodology and the challenges which only become apparent upon thorough manual inspection of the data. Section 5 presents our results and the insights gained from our analysis. We conclude the paper and present potential future directions in sentiment and content analysis in Section 6. ## 2 Related Work ### Influence on OSNs Our primary interest in this work is in understanding the factors which govern the effectiveness and influence of campaigns on OSNs. Several recent studies have examined individuals’ influence on OSNs [3], and the effectiveness of online advertising [4, 2], but little attention has been paid to identifying the driving factors behind a brand’s influence on their social audience (although it has been noted that brand names are more important online for some categories [6]). Cheung et al. [5] examined the way information spreads differently within social networks as opposed to word-of-mouth (WOM) broadcasting, by focusing on electronic word-of-mouth (eWOM), showing comprehensiveness and relevance to be the key influences of information adoption. The closest work to ours in understanding brands on Twitter is the study by Jansen et al. [10], who found that 20% of tweets that mentioned a brand expressed a sentiment or opinion concerning that company, product or service. Here, we examine and compare such mentions and sentiments across different promotion strategies available to brands on Twitter, thus specifically investigating advertising effectiveness (see Section 3).333Data availability limits us to effects within OSNs; we cannot determine effects on actual clicks or sales. In a study on the spread of hashtags within Twitter, Romero et al. [24] used over 3 billion tweets 2009-2010 to analyze sources of variation in how the most widely used hashtags spread within its user population. Their results suggested that the mechanism that controls the spread of hashtags related to sports or politics tends to be more persistent than average; repeated exposure to users who use these hashtags affects the probability that a person will eventually use the hashtag more positively than average. However, they only examined hashtags that succeeded in reaching a large number of users. In regards to the focus of promoted influences within Twitter, this raises the question; what distinguishes a promoted item that spreads widely, possibly with positive sentiment, from one that fails to attract attention or is associated with mainly negative sentiment? Our study aims to answer this by examining the sentiment and spread of tweets in relation to brands’ promoted items. ### Analysis Methods Sentiment analysis has been approached across many domains, including products, movie reviews and newspaper articles as well as social media (see e.g [18] for a comprehensive overview). Typically, the methods employed depend either on existing language resources (e.g. sentiment dictionaries or ontologies) or on machine learning from annotated datasets. The former can provide deep insight, but are somewhat inflexible in the face of the non- standard and rapidly changing language used on OSNs, for which few suitable linguistic resources currently exist. The latter are more scalable and can be trained on relevant data (e.g. [14]), but generally depend on large amounts of manual annotation (expensive and often problematic in terms of accuracy) and in some cases the existence of grammatical resources for the language and text domain in question (e.g. [26]). However, some approaches leverage the existence of implicit labelling in the datasets available (_distant supervision_), to avoid the necessity for manual annotation: for example, user ratings provided with movie or product reviews [19, 4]); or author conventions such as emoticons and hashtags on OSNs [7, 17, 21]). Hybrid approaches also exist, e.g. the use of predefined sentiment dictionaries with weights learned from data (e.g. [27]). Identifying the topic of text has also received much attention in NLP research, with methods ranging from the use of existing topic resources or ontologies (e.g. [12]) to unsupervised models for discovery of topics (e.g. [23]). The use of machine learning to detect the relevance (or otherwise) of text to a known topic also has a long history, perhaps most well-known in the form of Naïve Bayes filtering for spam filtering [25]. However, research into OSN behaviour or influence sometimes ignores the spread of sophisticated methods available. Sentiment analysis is often performed based on defined dictionaries (e.g. [28]), and topic identification is often ignored, with datasets filtered purely on keywords or simple Boolean queries. Recently, Goncalves et al. [8] examined the difference in performance across various sentiment analysis approaches on online text, finding significant variations. The effect of these variations in a specific analysis problem is less clear, though: how much does the variation in sophistication (and accuracy) of these methods actually matter? [22] compared statistical and lexicon-based methods and found significant differences at the level of individual messages, although a correlation at the level of their intended analysis (user profiles). Here, we investigate the effect when considering individual advertising campaigns (promoted items). For text relevance, we compare the use of keywords to Naïve Bayes classification via Weka [9]. For sentiment analysis, we examine three existing and freely available tools: the widely-used Data Science Toolkit’s text2sentiment444http://www.datasciencetoolkit.org/ based on a sentiment lexicon [16]; the lexicon-based but data-driven hybrid SentiStrength [27]; and a statistical machine-learning-based approach, Chatterbox’s Sentimental555http://sentimental.co/ (see [21]). ## 3 Data collection We set up a crawler to use the Twitter Streaming API666https://dev.twitter.com/docs/streaming-apis to collect the tweets of interest and all associated metadata (e.g., ID, username, user’s social graph), with details stored in a MySQL database. In this section we briefly describe our dataset and data collection strategy. ### Identifying promoted brands Twitter distinguishes promoted tweets and trends by the use of a _Promoted_ tag. We collected tweets from 11 brands with an active advertising campaign during our study period, across different industry domains, ranging from entertainment to health-care. For each promoted item, the brand names was used to crawl Twitter for tweet data posted in English for a 10 day period. If the promoted item also included a hashtag, the hashtag was also included in the parameters of the crawl’s GET function. This included all tweets that contained keywords such as @BrandName, #BrandName, BrandName, #PromotedHashtag and other brand related terms. These parameter values were selected to keep the dataset both relevant to brand-related tweets, and also manageable for searching purposes. Followers and following information was also tracked on a daily basis for each brand. Industry \| Promotion type \| Brand ---\|---\|--- Electronics \| Promoted tweet \| International CES Promoted tweet \| SONY Promoted trend \| Nintendo UK Travel \| Promoted tweet \| Marriot Entertainment \| Promoted tweet \| BBC One Automobile \| Promoted trend \| Vauxhall Heath Care \| Promoted tweet \| Paints like Me Retail \| Promoted trend \| ASOS Promoted trend \| PespiMax Promoted tweet \| JRebel Telecomms \| Promoted trend \| O2 Network Table 1: Industry sectors and sample brands Details of the selected brands and their promoted type are provided in Table 1. Given that we were interested in promoted items for branding purposes, a range of different brands from different industries were selected. The aim was to include both major, and small brands when selecting promoted items. In addition, a major brand and a small brand enable a comparison of sentiment while weakly controlling for follower count. ### Dataset We identified different industries’ promoted items for 10 day periods between $17^{th}$ December 2012 and $7^{th}$ January 2013. We used non-parallel crawling periods in order to avoid the query limits set by the Twitter API. In total, around 180,000 individual tweets were collected by crawling Twitter continuously, excluding December $21^{st}$ 2012 when there was a 6 hour outage in the crawler API. The crawler collected tweets from around 120,000 different Twitter users engaged in spreading the promoted tweets and trends. Tweets across all topics and with no geographical limits were gathered, as long as they featured the brand’s name/hashtag. When a brand contained more than one directly relevant hashtag, e.g., #Coke and #CocaCola, we included all the relevant hashtags. Twitter users do often repeat their tweets to benefit from repeated exposure. However, in order to remove noise and bias in analysis caused by spam tweets, we removed users who had posted the exact same tweet more than 20 times during our measurement periods, along with their tweets. Twitter users, tweets and tweet timestamps were also cross-analysed to check for spamming accounts. In one case a single user was removed for adding over 8,000 spam tweets to the database. After manual inspection of many tweets and accounts, we are confident that nearly all spam has been removed from our dataset. ## 4 Text Processing & Classification In this section we present the details of our tweet classification (using ML) and sentiment analysis (using existing NLP tools). ### 4.1 Topic Classification One of the major challenges during cleaning the dataset and removing spam was ensuring topic relevance. Our expectation was that this would not be an issue: as in much previous work, our study is looking at all sentiment expressed towards the brands, as long as the tweet matched the parameters of the tweet selection as explained in Section 3. However, whilst sampling tweets for spammers, a general problem surfaced. We found that a keyword-based approach tends to be too broad to accurately identify tweets referring to a particular brand, _O2_ (a UK mobile telecommunications provider and network). Our parameters for collecting tweets for this brand were to match tweets containing O2WhatWouldYouDo and O2 (the hashtag being promoted was #O2WhatWouldYouDo and @O2 is the official brand Twitter handle). Over the 10 day period, 90,000 tweets were collected that matched these keywords. However, examining a random sample of 200 tweets from this dataset showed that over 70% were not referring to the O2 Network brand; many were referring to the “O2 Academy” (a chain of concert venues), the “O2 Arena” (a dome-shaped monstrosity in London), or other senses of ‘O2’ such as oxygen. We also noticed that Twitter users have recently established a new way of using the letter sequence ‘O2’ as a replacement for the letters ‘to’: e.g. “@CokeWave_Thang What Picture You Want Me O2 Put As My BackGround”, “what im goin o2 do o2day”. Experiments with boolean combinations of O2 with other keywords were not successful. A major challenge therefore becomes to filter out non-brand-related tweets automatically: the problem is not trivial, given the variability and unpredictability of language, vocabulary and spelling on Twitter, and the short length of tweets (up to 140 characters); and manual removal of approximately 70% of large datasets is prohibitively labour- intensive. We therefore approached this as a text classification problem and investigated various supervised machine learning approaches using the Weka toolkit [9]. First, we performed a pilot study over a 200-tweet development set to determine a suitable feature representation and classification method; the data was manually labelled as O2-related or otherwise to give a binary decision problem. We tested a variety of classifiers including Naive Bayes, Naive Bayes Multinomial, ID3, IBK and J48 decision trees; features were based on the tweet text using a standard bag-of-words representation (see e.g. [13]) with various scaling methods,777We used Weka’s StringToWordVector filter for text feature extraction and scaling. with the addition of user ID and date of tweet. Given the small size of the dataset, we restricted the feature space to be based on the most common 100 words. We also tested using a simple manual keyword-based filter to remove some common negative instances (using keywords arena, academy, etc) before training (see “manually filtered” results in the figures). Tests were performed using ten-fold cross-validation in order to simulate performance on unseen data. Best performance (overall accuracy) was obtained using only bag-of-words text features, with stopwords removed and a TF-IDF weighting, after manual filtering. The best performing classifiers in cross-validation were J48 and Naive Bayes (NB), with 71% and 91% accuracy respectively. We then compared their performance on a held-out test set: the NB model outperformed the J48 model with 84% accuracy compared to 71% for J48, with training and prediction also noticeably faster for NB (the tree structure of the J48 model made it very slow with larger training sets). Figure 1: NB accuracy with increasing training data. To determine a suitable training set size, we then varied the training set while testing performance on a held-out test dataset of 30 manually labelled tweets. Increasing training set size improved performance (see Figure 1): we tested up to a 2,000-tweet training set; while the curve suggests performance may improve beyond this point, the accuracy on the held-out test set is approaching that on the training set so large improvements are unlikely. The NB classifier trained on 2,000 tweets was therefore used for the experiments below. Figure 2 shows results when tested on a larger, unseen, randomly selected test set of 100 tweets; the version with manual filtering achieves 78% accuracy, 77% recall and 66% precision. Figure 3 gives details of the per- class predictions: without manual filtering, false positives are more common than false negatives (i.e. too much irrelevant data is slipping through); levels are much closer with filtering. Figure 2: Classification results using Naive Bayes. Figure 3: Classification details per class using Naive Bayes. ### 4.2 Sentiment Analysis Having identified tweets with relevant content, we now required a method for sentiment analysis – determining the positive or negative stance of the writer. As discussed in Section 2 above, many methods for sentiment detection exist, with the major distinction being between lexicon-based and machine learning-based approaches. We examined existing tools for Twitter sentiment analysis using both of these approaches in order to determine the most suitable for our data. As a baseline lexicon-based tool we used the freely available Data Science Toolkit888http://www.datasciencetoolkit.org/. The sentiment analyser is based on a sentiment lexicon [16]; we therefore anticipate its coverage to be low but take it to be representative of commonly-used lexicon-based approaches. For a more robust tool for comparison, we examined two alternatives. As a hybrid lexicon/machine-learning tool we chose SentiStrength [27]. This method uses a predetermined list of words commonly associated with negative or positive sentiment, which are given an empirically determined weight (learned from data); new texts are classified by summing the weights of the words they contain. Thelwall et al. [27] report accuracy on Twitter data of 63.7% for positive sentiment and 67.8% for negative when predicting ratings on a 1-5 scale, and accuracies near 95% when predicting a simple binary positive/negative label. However, even though their word lists and weightings are determined for OSN data (including Twitter), this approach may suffer when faced with social text with new words, unexpected spellings and context- dependent language and meaning (see [15]). For a purely ML-based option we used Chatterbox’s Sentimental API,999http://mashape.com/sentimental/sentiment-analysis-for-social-media based on statistical machine learning over large, distantly labelled datasets [21]. This data-based approach means it might be expected to handle slang, errorful or abbreviated text better. Purver & Battersby [21] report accuracies approaching 80% using a similar technique on smaller datasets; Chatterbox report 83.4% accuracy in an independent study.101010See http://content.chatterbox.co/Sentiment\%20Analysis\%20Case\%20Study\%20-\%20Chatterbox\%20and\%20IDL.pdf. Before applying the sentiment analysis tool, and in order to compare the two approaches, a few hundred random tweets were selected from the database and were read and manually labelled for positive or negative sentiment, and both tools were tested on the resulting set. Results showed accuracy below 50% for the lexicon-based Data Science Toolkit, 63% for the hybrid SentiStrength approach, and 84% for the ML-based Chatterbox approach. Error analysis showed one significant source of the latter difference to be sentiment expressed in hashtags (e.g. the negative #shambles), which were detected better by the ML- based approach, presumaby due to their absence from SentiStrength’s predetermined lexicon. We therefore use Chatterbox as the “robust” tool in our experiments below, and compare to the Data Science Toolkit as a purely lexicon-based baseline. ## 5 Results Figure 4: Distribution of promoted tweets volumes over time. Figure 5: Distribution of promoted trends volumes over time. ### Response volume over time To examine the spread of engagement for each promoted item over the 10 day period, we analysed the volume of unique tweets each day in response to each promoted item, then averaged the results across all brands. Figures 4 and 5 display the distribution of this volume in response to _promoted tweets_ (4) and _promoted trends_ (5) per brand. On average, promoted trends led to much higher response volumes. However, the highest percentages of _mentions_ within responses were from promoted tweets, where an average of 18% of tweets each day included an ‘@’ mention to the brand; promoted trends had an average of only 15% mentions per day. This indicates that for a brand to successfully engage users in the content of the promoted item, a promoted tweet is better for this purpose. For example, out of the O2 Network’s $\sim$30,000 tweets, 7,965 included an ‘@’mention to the brand (25%). Results confirmed that the greatest percentage of engagement for a brand’s promoted item takes place on the first day of promotion. On average, 24% of engagements around the promoted item take place on the first day. The effect is most pronounced for _promoted trends_ , with 34% of engagement on average on the first day of promotion, after which the engagement falls dramatically by an average of 25% to 9% by day two and continues to fall thereafter, even if the item is promoted for several days. For _promoted tweets_ , the effect is less pronounced: 19% of the engagement takes place on the first day of promotion, with engagement decreasing by 8% by the second day of promotion. However, it does not continue on a steady decline thereafter, but it rises and falls over the next 8 days, although never again reaching the peak of the first day of promotion. This could be due to the fact that a promoted tweet is usually promoted for several days on Twitter where it occasionally appears at the top of different user’s timeline were users are repeatedly exposed to the item. This finding can be said to conform to Romero et al.’s theory of repeated exposure [24].111111Also see http://advertising.twitter.com/2013/03/Nielsen-Brand-Effect-for-Twitter-How- Promoted-Tweets-impact-brand-metrics.html They found that repeated exposure to a hashtag within Twitter had a significant marginal effect on the probability of adoption of that hashtag. In general, though, these results show that adoption of a promoted item is not a slow gradual shift over several days (as might be assumed) but rather an immediate incline when exposure to the item is new to users. ### Effects on user sentiment The sentiment breakdown for each promoted brand item can be observed in Figures 6 and 7, with Figure 6 showing the results obtained using our chosen machine learning method and Figure 7 those obtained using a keyword-based method (see section 4 above). We observe that in most cases, the percentage of positive sentiment was higher than that of negative and neutral for promoted items. Notable exceptions are the results for two brands, NiveaUK and O2, where neutral and/or negative levels outweigh positive; the ASOS brand also shows little difference between negative and positive levels. However, comparison of the figures that would have been gained using a keyword-based approach (Figure 7) shows misleading results in precisely these interesting cases: apparent positive levels are higher than negative in all cases. Neutral cases also appear much more common; this is due to the low coverage of the keyword lexicon causing large numbers of results with apparently zero sentiment. Use of the more accurate tool (as objectively assessed – see section 4) therefore does appear crucial. Figure 6: Sentiment analysis by brand - machine learning Figure 7: Sentiment analysis by brand - keywords On average, across all brands (promoted tweets and trends), the average percentage of tweets and retweets121212We assume that retweeting users share the same sentiment as the original tweet. which contained a positive sentiment is 50%, that which contained a negative sentiment is 12%, and 38% of tweets had a neutral tone. Figures 8 and 9 then show the distribution of positive and negative sentiments in this response traffic over time. On average, positive sentiment outweighs negative sentiment; on the first day, 49% of the tweets were positive. In general, _promoted tweets_ lead to more positive sentiment and less negative sentiment than _promoted trends_. Figure 8: Positive sentiment distribution over time Figure 9: Negative sentiment distribution over time. In total, 47% of tweets relating to a _promoted tweet_ are positive in sentiment. Day one received the highest percentage of positive sentiment tweets (58%); positive sentiment then continues to dominate over the 10 day period, never falling below 36% of the tweets. Examining _promoted trends_ , we found that, on average, only 37% of tweets relating to a promoted trend contained a positive sentiment. On the first day of promotion, 26% of tweets expressed a negative sentiment, 32% expressed a positive sentiment and 42% expressed no sentiment at all. This shows that Twitter users do not tweet as positively about a promoted trend as they would about a promoted tweet. Instead, a large proportion of tweets relating to a promoted trend contained no emotional words, or if they did, the positive and negative sentiments balanced each other out. They generally contained just the promoted hashtag or generally had an objective, matter-of-fact tone (e.g., - “Get 3G where I live... #O2WhatWouldYouDo”). Taken together with the analysis of engagement volume, these results show that when an item is promoted, the brand and the item get adopted immediately and regarded quite positively by the engaged users. Twitter users welcome the promoted item on Twitter, which has a positive effect on the tweets expressed. The engagement level reduces to an average of 10% of the total tweets on day two, when the item is no longer being promoted, or is no longer seen as “new and interesting”. However, on average, the positive sentiment expressed still outperforms that of negative sentiment and neutral sentiment each day. ### Effect of hashtags on engagement and sentiment Figure 10: Hashtag related engagements for ASOS. Figure 11: Hashtag related engagements for Vauxhall. We then performed two example case studies, using the ASOS and Vauxhall brands, to examine the use of hashtags within promoted items. Figures 10 and 11 show the results. ASOS promoted a trend, #AsosSale, on the $19^{th}$ and $20^{th}$ of December to highlight their Boxing Day sale on the $26^{th}$ of December (day 8 of data collection). Although the promoted hashtag was virtually discarded by day two of data collection, we found that user engagement (use of hashtag, mentions and tweets) for the forthcoming sale continued. This trend is also apparent in Vauxhall’s tweet volumes for their sale which stated on the $27^{th}$ of December (day one of promotion), and ended the day after our 10 day data collection period. The engagement for Vauxhall remained at a consistent level throughout the event (see Figures 5 and 11), despite the rapid drop-off in use of the promoted hashtag. ## 6 Conclusions & Future directions In this paper we present a measurement-driven study of the effects of promoted tweets and trends on Twitter on the engagement level of users, using a number of ML and NLP techniques in order to detect relevant tweets and their sentiments. Our results indicate that use of accurate methods for sentiment analysis, and robust filtering for topical content, is crucial. Given this, we then see that promoted tweets and trends differ considerably in the form of engagement they produce and the overall sentiment associated with them. We found that promoted trends lead to higher engagement volumes than promoted tweets. However, although promoted tweets obtain less engagement than promoted trends, their engagement forms are often more brand inclusive (more direct mentions); and while engagement volumes drop for both forms of promoted items after the first day, this effect is less pronounced for promoted tweets. We also found that although the volume of tweets is highest in promoted trends, they do not lead to the same level of positive sentiment that promoted tweets do. Hence advertisers should carefully assess the trade-offs between high level of engagement, drop-off rate, direct mentions, and positive user sentiment. In the next stage of this study we will investigate the effect of individuals’ influence on the take-up of promoted tweets and trends by their social graph. We will investigate new data at finer granularity (hourly) for events that are time-sensitive, such as major concert ticket sales. This is our first attempt at understanding this space. The advertising campaigns have very different structure and we need to understand these in details. Promoted trends typically stay on the trends list for a day, and promoted tweets are selectively shown to a subset of users for a period of time selected by the advertiser. Without accounting for such nuances, broad statements on the impact of the two forms of advertising are not conclusive. However in this paper we focussed on insights in using sentiment analysis methods and accurate data labelling. We believe our findings could provide a new insight for social network marketing and advertisements strategies, in addition to comparing different methods of classifying and filtering relevant content. ## Acknowledgments The authors thank Chatterbox131313http://chatterbox.co for providing unlimited access to the Sentimental API. ## References * [1] S. Asur and B. A. Huberman. Predicting the future with social media. CoRR, abs/1003.5699, 2010. * [2] T. Blake, C. Nosko, and S. Tadelis. Consumer heterogeneity and paid search effectiveness: A large scale field experiment. 2013\. * [3] M. Cha, H. Haddadi, F. Benevenuto, and K. P. Gummadi. Measuring user influence in twitter: The million follower fallacy. In in ICWSM 10: Proceedings of international AAAI Conference on Weblogs and Social, 2010. * [4] T. Y. Chan, C. Wu, and Y. Xie. Measuring the lifetime value of customers acquired from google search advertising. Marketing Science, 30(5):837–850, Sept. 2011. * [5] C. M. Cheung and D. R. 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In Proceedings of the Conference on Empirical Methods in Natural Language Processing (EMNLP), 2013. To appear. * [27] M. Thelwall, K. Buckley, and G. Paltoglou. Sentiment strength detection for the social web. J. Am. Soc. Inf. Sci. Technol., 63(1):163–173, Dec. 2012. * [28] A. Tumasjan, T. O. Sprenger, P. G. Sandner, and I. M. Welpe. Predicting elections with twitter: What 140 characters reveal about political sentiment. In Proceedings of the Fourth International AAAI Conference on Weblogs and Social Media, 2010.	arxiv-papers	2013-12-23T18:32:06	2024-09-04T02:49:55.847751	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Shana Dacres, Hamed Haddadi, Matthew Purver", "submitter": "Hamed Haddadi", "url": "https://arxiv.org/abs/1312.6635" }
1312.6660	050006 2013 G. B. Mindlin 050006 We describe the operation of a neuronal device which embodies the computational principles of the “paper-and-pencil” machine envisioned by Alan Turing. The network is based on principles of cortical organization. We develop a plausible solution to implement pointers and investigate how neuronal circuits may instantiate the basic operations involved in assigning a value to a variable (i.e., $x=5$), in determining whether two variables have the same value and in retrieving the value of a given variable to be accessible to other nodes of the network. We exemplify the collective function of the network in simplified arithmetic and problem solving (blocks-world) tasks. # A neuronal device for the control of multi-step computations Ariel Zylberberg [equal, lni, liaa, viscog] Luciano Paz E-mail: [email protected] [equal, lni] Pieter R. Roelfsema E-mail: [email protected] [viscog, intne, psydep] Stanislas Dehaene E-mail: [email protected] [collfr, inserm, cea, paris] Mariano Sigman[lni] E-mail: [email protected]: [email protected] (14 June 2013; 9 July 2013) ††volume: 5 99 equal These authors contributed equally to this work lni Laboratory of Integrative Neuroscience, Physics Department, FCEyN UBA and IFIBA, CONICET; Pabellón 1, Ciudad Universitaria, 1428 Buenos Aires, Argentina. liaa Laboratory of Applied Artificial Intelligence, Computer Science Department, FCEyN UBA; Pabellón 1, Ciudad Universitaria, 1428 Buenos Aires, Argentina. viscog Department of Vision and Cognition, Netherlands Institute for Neuroscience, an institute of the Royal Netherlands Academy of Arts and Sciences, Meibergdreef 47, 1105 BA Amsterdam, The Netherlands. intne Department of Integrative Neurophysiology, Center for Neurogenomics and Cognitive Research, VU University, Amsterdam, The Netherlands. psydep Psychiatry Department, Academic Medical Center, Amsterdam, The Netherlands. collfr Collège de France, F-75005 Paris, France. inserm INSERM, U992, Cognitive Neuroimaging Unit, F-91191 Gif/Yvette, France. cea CEA, DSV/I2BM, NeuroSpin Center, F-91191 Gif/Yvette, France. paris University Paris-Sud, Cognitive Neuroimaging Unit, F-91191 Gif/Yvette, France. ## 1 Introduction Consider the task of finding a route in a map. You are likely to start searching the initial and final destinations, identifying possible routes between them, and then selecting the one you think is shorter or more appropriate. This simple example highlights how almost any task we perform is organized in a sequence of processes involving operations which we identify as “atomic” (here search, memory and decision-making). In contrast with the thorough knowledge of the neurophysiology underlying these atomic operations [1, 2, 3], neuroscience is only starting to shed light on how they organize into programs [4, 5, 6]. Partly due to the difficulty of implementing compound tasks in animal models, sequential decision-making has mostly been addressed by the domains of artificial intelligence, cognitive science and psychology [7, 8]. Our goal is to go beyond the available neurophysiological data to show how the brain might sequentialize operations to conform multi-step cognitive programs. We suppose the existence of elementary operations akin to Ullman’s [9] routines, although not limited to the visual domain. Of special relevance to our report is the body of work that has grown out of the seminal work of John Anderson [8]. Building on the notion of “production systems” [10], Anderson and colleagues developed ACT-R as a framework for human cognition [8]. It consists of a general mechanism for selecting productions fueled by sensory, motor, goal and memory modules. The ACT-R framework emphasizes the chained nature of human cognition: at any moment in the execution of a task, information placed in buffers acts as data for the central production system, which feeds-back to these same modules. Despite vast recent progress in our understanding of decision formation in simple perceptual tasks [3], it remains unresolved how the operations required by cognitive architectures may be implemented at the level of single neurons. We address some of the challenges posed by the translation of cognitive architectures to neurons: how neuronal circuits might implement a single operation, how multiple operations are arranged in a sequence, how the output of one operation is made available to the next one. ## 2 Fundamental assumptions and neuronal implementation ### 2.1 The basis for single operations Figure 1: Diffusion processes in single and multi-step tasks. (a) In simple sensory-motor tasks, response selection is mediated by the the accumulation of sensory evidence to a response threshold. (b) In tasks involving multiple steps, there is a parallel competition between a subset of “productions rules” implemented by pools of neurons that accumulate evidence up to a threshold. The selected production can have overt effects (motor actions) as well as covert effects in modifying the state of the memory, after which a new cycle begins. Adapted with permission from Ref. [11]. Insights into the machinery for simple sensory-motor associations come from studies of monkey electrophysiology. Studies of oculomotor decisions —focused primarily on area LIP [12]— have shown that neurons in this area reflect the accumulation of evidence leading to a decision [3]. In a well studied paradigm, monkeys were trained to discriminate the direction of motion of a patch of moving dots and report the direction of motion with an eye-movement response to the target located in the direction of motion [13]. Neurons in LIP which respond with high levels of activity during memory saccades to specific portions of space are recorded during the task. These neurons show ramping activity of their firing rates with a slope that depends on the difficulty of the task, controlled by the proportion of dots that move coherently in one direction [14]. In a reaction time version of the task [13], when monkeys are free to make a saccade any time after the onset of the motion stimuli, the ramping activity continues until a fixed level of activity is reached, signaling the impending saccade (Fig. 1 a). Crucially, the level of this “threshold” does not depend on the difficulty of the task or the time to respond. The emerging picture is that ramping neurons in LIP integrate sensory evidence and trigger a response when activity reaches a threshold. This finding provided strong support for accumulation or race models of decision making which have been previously postulated to explain error rates and reaction times in simple tasks, and match nicely with decision theoretical notions of optimality [15]. While experimental studies have mainly characterized the feedforward flow of information from sensory to motor areas, evidence accumulation is also modulated by contextual and task-related information including prior knowledge about the likelihood and payoff of each alternative [16, 17]. Interestingly, a common currency —the spiking activity of neurons in motor intention areas— may underlie these seemingly unequal influences on decision formation. ### 2.2 Sequencing of multiple operations Brain circuits can integrate sensory evidence over hundreds of milliseconds. This illustrates how the brain decides based on unreliable evidence, averaging over time to clean up the noise. Yet the duration of single accumulation processes is constrained by its analog character, a problem pointed out earlier by von Neumann in his book The computer and the Brain [18]: “in the course of long calculations not only do errors add up but also those committed early in the calculation are amplified by the latter parts of it…”. Modern computers avoid the problem of noise amplification employing digital instead of analog computation. We have suggested that the brain may deal with the amplification of noise by serially chaining individual integration stages, where the changes made by one ramp-to-threshold process represent the initial stage for the next one (Fig. 1 b) [19, 11]. Evidence for the ramp-to-threshold dynamics has been derived from tasks in which the decision reflects a commitment to a motor plan [20, 3]. As others [21], we posit that the ramp-to-threshold process for action-selection is not restricted to motor actions, but may also be a mechanism to select internal actions like the decision of where to attend, what to store in memory, or what question to ask next. Therefore, the activation of a circuit based on sensory and mnemonic evidence is mediated by the accumulation of evidence in areas of the brain which can activate that specific circuit, and which compete with other internal and external actions. Within a single step, the computation proceeds in parallel and settles in a choice. Seriality is the consequence of the competitive selection of internal and external actions that transforms noisy and parallel steps of evidence accumulation into a sequence of discrete changes in the state of the network. These discrete steps clean up the noise and enable the logical flow of the computation. Following the terminology of symbolic cognitive architectures [7, 8], the “ramping” neurons which select the operation to do next are referred as “production” neurons. The competition between productions is driven by inputs from sensory and memory neurons and by the spontaneous activity in the production network. As in single-step decisions [13], the race between productions concludes when neurons encoding one production reach a decision threshold. The neurons which detect the crossing of a threshold by a production also mediate its effects, which is to activate or deactivate other brain circuits. The activated circuits can be motor, but are not restricted to it, producing different effects like changes in the state of working memory (deciding what to remember), activating and broadcasting information which was in a “latent” state (like sensory traces and synaptic memories [22, 23]), or activating peripheral processors capable of performing specific functions (like changing the focus of attention, segregating a figure from its background, or tracing a curve). ### 2.3 Pointers Versatility and flexibility are shared computational virtues of the human brain and of the Turing machine. The simple example of addition (at the root of Turing’s conception) well illustrates what sort of architecture this requires. One can picture the addition of two numbers x and y as displacing y steps from the initial position x. This simple representation of addition as a walk in the number line describes the core connection between movement in space and mathematical operations. It also describes the need of operations that use variables which temporarily bind to values in order to achieve flexibility. In this section we describe how neuronal circuits may instantiate the basic operations involved in assigning a value to a variable (i.e., $x=5$), in determining whether two variables have the same value and in retrieving the value of a given variable. This is in a way a proof of concept, i.e., a way to construct these operations with neurons. We are of course in no position to claim that this instantiation is unique (it is certainly not). However, we have tried to ground it on important principles of neurophsyiology and we believe that this construction raises important aspects and limitations which may generalize to other neuronal constructions of variable assignment, comparison and retrieval mechanisms. Here we introduce the concept of pointers; individual or pool of neurons which can temporarily point to other circuits. When a pointer is active, it facilitates the activation of the circuit to which it is temporarily bound and which is dynamically set during the course of a computation. A pointer “points” to a cortical territory (for instance, to V1). This cortical mantle represents a space of values that a given variable may assume. The cortex is organized in spatial maps representing features (space, orientation, color, 1-D line…) and a pointer can temporarily bind to one of these possible set of values in a way that the activation of the pointer corresponds to the activation of the value and hence functions as a variable (i.e., $x=3$). There are many proposed physiological mechanisms to temporally bind neuronal circuits [24, 25, 26]. A broad class of mechanisms relies on sustained reverberant neuronal activity [26]. A different class relies on small modifications of synaptic activity, which constitute silent memories in latent connections [23]. Here we opt for the second alternative, first because it has a great metabolic advantage allowing to share many memories at very low cost, and more importantly because it separates the processes of variable assignation and variable retrieval. As we describe below in detail, in this architecture the current state of the variable is not broadcast to other areas until it is specifically queried. To specifically implement the binding with neuron-like elements, we follow the classic assumption that when two groups of neurons are active at the same time, a subset of the connection between them is strengthened (Fig. 2). The strengthening of the synapses is bidirectional, and it is responsible for the binding between neuronal populations. To avoid saturation of the connections and to allow for new associations to form, the strength of these connections decays exponentially within a few hundred milliseconds. Specifically, if the connection strength between a pair of populations is $w_{base}$, then when both populations are active the connection strength increases exponentially with a time constant $\tau_{rise}$ to a maximum connection strength of $w_{max}$. When one or both of the populations become inactive then the connection strength decays exponentially back to $w_{base}$ with a time constant of $\tau_{decay}$. Figure 2: Instantiation of variables through plastic synapses. When two neurons become active, the connection between them is rapidly strengthened, forming a transient association which decays within a few hundreds of milliseconds. The value of a pointer can be retrieved by setting the pointer’s domain in a winner-take-all mode and activating the pointer which biases the WTA competition through its strengthened connections. The mechanism described above generates a silent coupling between a pointer and the value to which it points. How is this value recovered? In other words, how can other elements of the program know the value of the variable $X$? The expression of a value stored in silent synapses is achieved by simultaneously activating the pointer circuit and forcing the domain of the variable to a winner-take-all (WTA) mode (Fig. 2). The WTA mode —set by having neurons with self-excitation and cross-inhibition [27]— assures that only one value is retrieved at a time. These neurons make stronger connections with the neurons to which they are bound than with the other neurons. When the network is set in a WTA mode, these connections bias the competition to retrieve the value previously associated with the pointer (Fig. 2). In other words, activation of the pointer by itself is not sufficient to drive synaptic activity to the neuron (or neurons) representing the value to which it points. But it can bias the activation of a specific neuron when co-activated with a tonic activation of the entire network. This architecture is flexible and economic. Value neurons only fire when they are set or retrieved. Memory capacity is constrained by the number of connections and not by the number of neurons. But it also has a caveat. Given that only one variable can be bound to a specific domain at any one time, multiple bindings must be addressed serially. As we show later, this can be accomplished by the sequential firing of production neurons. #### 2.3.1 Compare the value of two variables If two variables $X$ and $Y$ bind to instances in the same domain, it is possible to determine whether the two variables are bound to the same instance, i.e., whether $x=y$. The mechanics of this process is very similar to retrieving the value of a variable. Pointer neurons111In our framework, a pointer can also be a population of neurons that functions as a single pointer. $X$ and $Y$ are co-activated. The equality in the assignment of $X$ and $Y$ can be identified by a coincidence detector. Specifically, this is solved by adjusting the excitability in the value domain in such a way that the simultaneous input on a single value neuron exceeds the threshold but the input of a single pointer does not. This proposed mechanism is very similar to the circuits in the brain stem which —based on coincidence detection of delay lines— encode interaural time difference [28]. This shows the concrete plausibility of generating such dynamic threshold mechanisms that act as coincidence detectors. #### 2.3.2 Assign the value of one variable to another Similarly, to assign the value of $X$ to the variable $Y$ ($Y\leftarrow x$), the value of the variable that is to be copied needs to be retrieved as indicated previously, by activating the variable $X$ and forcing a WTA competition at the variable’s domain. Then, the node coding for variable $Y$ must be activated, which will lead to a reinforcing of the connections between $Y$ and $x$ which will instantiate the new association. ## 3 Concrete implementation of neuronal programs In the previous sections we sketched a set of principles by which brain circuits can control multi-step operations and store temporary information in memory buffers to share it between different operations. Here we demonstrate, as a proof of concept, a neuronal implementation of such circuits in two simple tasks. The first one is a simple arithmetic counter, where the network has to count from an initial number $n_{ini}$ to a final number $n_{end}$, a task that can be seen as the emblematic operation of a Turing device. The second example is a blocks-world problem, a paradigmatic example of multi-step decision making in artificial intelligence [29]. The aim of the first task is to illustrate how the different elements sketched above act in concert to implement neuronal programs. The motivation to implement blocks worlds is to link these ideas to developed notions of visual routines [9, 30, 31]. ### 3.1 Arithmetic counter We designed the network to be generic in the sense of being able to solve any instance of the problem, i.e., any instantiation of $n_{ini}$ and $n_{end}$. We decided to implement a counter, since it constitutes essentially a while loop and hence a basic intermediate description of most flexible computations. In the network, each node is meaningful, and all parameters were set by hand. Of course, understanding how these parameters are adjusted through a learning process is a difficult and important question, but this is left for future work. Each number is represented by a pool of neurons selective to the corresponding numerosity value [32]. A potential area for the neurons belonging to the numerosity domain is the Intra Parietal Sulcus (IPS) [33], where neurons coding for numerosity have been found in both humans and monkeys [34]. In the model, number neurons interact through random lateral inhibitory connections and self-excitation. This allows, as described above, to collapse a broad distribution of number neurons [32] to a pool representing a single number, in a retrieval process during a step of the program. We assume that the newtwork has learned a notion of number proximity and continuity. This was implemented via a transition-network that has asymmetrical connections with the number- network. A given neuron representing the number $n$ excites the transition neuron $n\rightarrow n+1$ population. This in turn excites the neuron that represents the number $n+1$. Again, we do not delve into how this is learned, we assume it as a consolidated mechanism. The numbers-network can be in different modes: it can be quiescent, such that no number is active, or it can be in a winner-take-all mode with only one unit in the active state. Our network implementation of the counter makes use of two variables. The Count variable stores the current count and changes dynamically as the program progresses, after being initialized to $n_{ini}$. The End variable stores the number at which the counting has to stop and is initially set to $n_{end}$. The network behaves basically as a while loop, increasing the value of the Count variable while its value differs from that of the End variable. To increment the count, we modeled a transition-network with units that have asymmetrical connections with the numbers-network. For example, the “$1\rightarrow 2$” node receives excitatory input from the unit coding for number 1 and in turn excites number 2. This network stores knowledge about successor functions, and in order to become active it requires additional input from the production system. As mentioned above, here we do not address how such structure is learned in number representing neurons. Learning to count is an effortful process during childhood [35] by which children learn transition rules between numbers. We postulate that these relations are encoded in structures which resemble horizontal connections in the cortex [36, 37, 38]. In the same way that horizontal connections incorporate transition probabilities of oriented elements in a slow learning process [39, 40], resulting in a Gestalt as a psychological sense of “natural” continuity, we argue that horizontal connections between numerosity neurons can endow the same sense of transition probability and natural continuity in the space of numbers. The successor function can be as an homologous to a matrix of horizontal connections in the array of number neurons. In a way, our description postulates that a certain number of operations are embedded in each domain cortex (orientation selective neurons in V1 for curve tracing, number selective neurons in IPS for arithmetic…). This can be seen as “compiled” routines which are instantiated by local horizontal connections capable of performing operations such as collinear continuity, or “add one”. The program can control which of these operations becomes active at any given step by gating the set of horizontal connections, a process we have referred to as “addressing” the cortex [41]. Just as an example, when older children learn to automatically count every three numbers (1, 4, 7, 10, 13…) we postulate that they have instantiated a new routine (through a slow learning process) capable of establishing the transition matrix of $n\rightarrow n+3$. The repertoire of compiled functions is dynamic and can change with learning [42]. Figure 3: Sketch of the network implementing an arithmetic counter. (a) The network is divided into five sub-networks: productions, memory, pointers (or variables), numbers, and transition-networks. The order in which the productions fire is controlled by the state of the memory network, which is itself controlled by the production system. (b) Dynamics of a subset of neurons in the network. All units are binary except for the production neurons (violet) which gradually accumulate evidence to a threshold. Counting requires a sequence of operations which include changes in the current count, retrieval of successor functions and numeric comparisons. The successive steps of the counting routine are governed by firing of production neurons. The order in which the productions fire is controlled by the content of the memory (Fig. 3). We emphasize that while the production selection process proceeds in parallel —as each production neuron constantly evaluates its input— the selected production strongly inhibits the other production neurons and therefore the evidence accumulated at one step is for the most part lost after a production is selected.222In the absence of external noise (as in the present simulations), only the production with the largest input has higher-than-baseline activity. In Fig. 3 we simulate a network that has to count from numbers 2 to 6. Once the initial and final numbers have been bound to the Count and End variables respectively, the network cycles through six productions. The first production that is selected is the _PrepareNext_ production, whose role is to retrieve the value that results from adding $1$ to the current count. To this end, this production retrieves the current value of the Count variable, and excites the neurons of the transition-network such that the node receiving an input from the retrieved value of Count becomes active (i.e., if 2 is active in the numbers-network, then $2\rightarrow 3$ becomes active in the transition-network). To assure that the retrieved value is remembered for the next step (the actual change of the current count), neurons in the transition-network are endowed with recurrent excitation, and therefore these neurons remain active until explicitly inhibited. The same production also activates a node in the memory-network which excites the _IncrementCount_ production, which is therefore selected next. The role of the next production (_IncrementCount_) is to actually update the current count, changing the binding of the Count variable. The _IncrementCount_ production inhibits all neurons in the number network, to turn it to the quiescent state. Once the network is quiescent, lateral inhibition between number nodes is released and the asymmetrical inputs from the transition-network can activate the number to which it projects. At the same time, the _IncrementCount_ production activates the Count variable, which is then bound to the currently active node in the numbers-network. Notice that this two-step process between the _PrepareNext_ and _IncrementCount_ productions basically re-assigns the value of the Count variable from its initial value $n$ to a new value $n+1$. Using a single production to replace these two (as tried in earlier versions of the simulations) required activating the number and transition neurons at the same time which lead to fast and uncontrolled transitions in the numbers- network. To increase the current count in a controlled manner, we settled for the two-productions solution. The _IncrementCount_ production also activates a memory unit that biases the competition at the next stage, in favor of the _ClearNext_ production. This production shuts up the activity in the transition-network, strongly inhibiting its neurons to compensate for their recurrent excitation. Shutting the activity of these neurons is required at the next step of the routine to avoid changes in the current count when the Count variable is retrieved to be compared with the End variable. After _ClearNext_ , the _RetrieveEnd_ production fires which retrieves the value of the End variable to strengthen the connections between the End variable and the value to which it is bound. This step is required since the strength of the plastic connections decays rapidly, and therefore the instantiation of the variables will be lost if not used or reactivated periodically. Finally, the _CheckEqual_ production is selected to determine if the Count and End variables are equal. If both variables are equal, a node in the memory network is activated which is detected by the _Halt_ production to indicate that the task has been completed; otherwise, the production that is selected next is the _PrepareNext_ production and the production cycle is repeated. In Fig. 3b, we show the dynamics of a subset of neurons for a network that has to count from 2 to 6. With this example we have shown how even a seemingly very easy task such as counting (which can be encoded in up to two lines in virtually any programming language) seems to require a complex set of procedures to coordinate and stabilize all computations, when they are performed by neuronal circuits with slow building of activity and temporal decay. ### 3.2 A world of blocks A natural extension of the numerotopic domain used in the above example is to incorporate problems in which the actor must interact with its environment, and sensory and motor productions ought to be coordinated. Figure 4: Simplified model of the visual system used in the blocks-world simulations. The upper portion shows the different layers and their connections. The early visual area is formed by a first sensory layer of neurons that receive stimulation from the outer world and a second attentional layer with bidirectional connections between the higher color or position tuned areas. The grayed neurons are the ones that present a higher activity. The lower panels show an example of the cuing and search operations. In the latter, a color tuned neuron is stimulated and drives an activity increase in the early visual layer. That later empowers the activity in the position layer concluding the search. The right situation shows the similar cuing operation. The visual system performs a great variety of computations. It can encode a large set of visual features in a parallel feed-forward manner forming its base representation [9, 43, 30, 44], and temporally store these features in a distributed manner [45, 46, 47]. A matrix of lateral connections gated by top- down processing can further detect conjunctions of these feature for object recognition. In analogy with motor routines, the visual system relies on serial chaining of elemental operation [9, 31] to gain computational flexibility. There are many proposals as to which operations are elemental [31], but, as we have discussed above, this list may be fuzzy since the set of elementary operations may be changed by learning [41]. In this framework, atomic operations are those that are encoded in value domains. Here, and for the purpose of implementing a neural circuit capable of solving the blocks worlds, we will focus on a simplified group of three elemental operations: Visual search: the capacity of the system to identify the location of a given feature. Visual cuing: the capacity of the system of highlighting the features that are present at a given location. Shift the processing focus: a method guided by attention to focus the processing of visual features or other computations in a given location. Here we use a simplified representation of the visual system based on previous studies [48, 49, 50]. We assume a hierarchy of two layers of neurons. The first one is tuned to conjunctions of colors and locations in the visual field. The second one has two distinct groups of populations, one with neurons that have large receptive fields that encode color irrespective of their location and another which encodes location independently of color (Fig. 4). The model assumes that neurons in the first layer tuned to a particular color and retinotopic location are reciprocally connected to second layer units that encode the same color or location. Figure 5: Sketch of the network that solves the blocks-world problem. The sensory early visual system receives input from the BW configuration and excites the first attentional layer. The latter is connected to color and position specific areas. The arrows show there is a connection between layers. The individual connections may be excitatory or inhibitory. The connections with the inverted triangle head indicate only excitatory connections exist. This architecture performs visual search of color in a way which resembles the variable assignation described above, through a conjunction mechanism between maps encoding different features. In the model, the color cortex encodes each color redundantly for all positions forming a set of spatial vectors (one for each color). Of course, all these spatial maps selective for a given color can be intermingled in the cortex, as it is also the case with orientation columns which sample all orientations filling space in small receptive fields. If, for instance, a red square is presented in position three, the neuron selective for red (henceforth referred to as in the red map) and with a receptive field in position three will fire. This activation, in turn, propagates to spatial neurons (which are insensitive to color). Thus, if four squares of different colors are presented in positions 1 to 4, the spatial neurons in these positions will fire at the same rate. To search for the spatial position of a red block, the activity of neurons coding for red in the color map must be enhanced. The enhanced activity propagates to the early visual areas which code for conjunctions of color and space, which in turn propagates to the spatial map, highlighting the position where the searched color was present. Spatial selection is triggered by an attention layer which selects the production “attend to red”, addressing the sensory cortex in a way that only locations containing red features will be routed to the spatial neurons. The color of a block at one location can be retrieved by an almost identical mechanism. In this case, the production system sets the attentional network to a given position in space and through conjunction mechanisms (because connections are reciprocal) only the color in the selected position is retrieved. This is a simple device for searching based on the propagation of attentional signals which has been used before in several models (e.g., Ref. [31]) (Fig. 4). Figure 6: Mean firing rate of a subset of neuronal populations involved in the resolution of an instance of BW. The rate is normalized between $0$ (white) and $1$ (black). The horizontal axis represents the ellapsed time in arbitrary units. At the bottom, we show a subset of intermediate BW configurations, aligned to the execution time of the motor commands which lead to these configurations. To bridge these ideas which are well grounded in the visual literature [43, 30, 31] with notions of planning and sequential mental programs, we use this model to implement a solver for a simple set of Blocks-World problems. The blocks-world framework is a paradigmatic artificial intelligence problem that consists of a series of colored blocks piled up on top of a large surface in many columns. The goal is to arrange the blocks according to their color in a given goal configuration. The player can only move one block from the top of any column and place it at the top of another, or on the surface that supports all the blocks. We choose to construct a solver for a restricted blocks-world problem where the surface can only hold 3 columns of blocks and the goal configuration is to arrange them all into one column (that we call the target column).333This restricted problem is equivalent to the Tower of London game [51]. We implement a network with a set of memory and production neurons —analogous to the counter circuit described above— which coordinates a set of visual and motor productions (Fig. 5). The interaction between the memory layer and the production system triggers the execution of elemental visual processes, motor actions and changes in the memory configuration in order to solve any given instance of the problem. To solve this problem, an algorithm needs to be able to find whether a block is in the correct position. For this, it requires, first, a “retrieve color” from a given location function. Normally the location that is intended to be cued is the one that is being attended to. We implement the attended location as a variable population (that we call the processing focus or $PF$ inspired in Ullman’s work [9]), so the “retrieve color” is equal to cue the color in $PF$’s location.444There are works that name $PF$ as Deictic Pointers and suggest that it would be possible to store it also by keeping gaze or even a finger at the relevant location [52]. Second, it must compare the colors in different locations. This can be done by binding the relevant location colors to separate variables and then comparing them in the way described in section 2.3.1 As the goal is to pile all the blocks in the correct order in a given target column, a possible first step towards the goal is to compare the target column with the goal configuration from the bottom to the top. This can be done by chaining several movements of the processing focus with color retrievals and subsequent comparisons. Once the first difference is found, the target column’s upper blocks must be moved away in order to replace the different colored block with the correct one. This process is carried out using several motor productions. Once the target column is free to receive the correct colored block, that color must be searched in the remaining columns. This is done as described earlier in this section. Once found, the $PF$ can be moved there in order to view if there are blocks above it. If there are, motor productions must be chained in order to free the desired block and move it to the target column. After this is done, the program can loop back to comparing the target column with the goal configuration and iteratively solve the problem. Our neuronal implementation chains the productions in a similar way as the one described above and elicits a complex activity pattern (Fig. 6). A detailed explanation of the implementation can be found in the supplementary material [53]. ## 4 Conclusions Here we presented ideas aimed to bridge the gap between the neurophysiology of simple decisions and the control of sequential operations. Our framework proposes a specific set of mechanisms by which multi-step computations can be controlled by neural circuits. Action selection is determined by a parallel competition amongst competing neurons which slowly accumulate sensory and mnemonic evidence until a threshold. Actions are conceived in a broad sense, as they can result in the activation of motor circuits or other brain circuits not directly involved in a movement of the eyes or the limbs. Thresholding the action of the productions results in discrete changes to a meta-stable network. These discrete steps clean up the noise and enable a logical flow of the computation. Comprehending the electrophysiological mechanisms of seriality is hindered by the intrinsic difficulty of training non-human primates in complex multi-step tasks. The ideas presented in this report may serve to guide the experimental search for the mechanisms required to perform tasks involving multiple steps. Neurons integrating evidence towards a threshold should be observed even in the absence of an overt response, for instance in the frontal-eye fields of awake monkeys for the control of attention. Memory neurons should show fast transitions between metastable states, on average every $\sim 100$-$250$ msec, compatible with the mean time between successive productions in ACT-R [8]. As mentioned, we do not address how the productions and the order in which they are executed are learned. There is a vast literature, for instance in reinforcement learning [54, 55, 56] describing how to learn the sequence of actions required to solve a task. Deahene & Changeaux [57] showed how a neuronal network can solve a task similar to the BW that we modelled here, but where the order in which productions fire was controlled by the distance from the game state to the goal. Instead, our aim here was to investigate how the algorithm (the pseudo-code) may be implemented in neuronal circuits —once it has already been learned— from a small set of generic principles. The operation of the proposed neuronal device in a simple arithmetic task and in a neuronal network capable of solving any instance of a restricted Blocks-World domain illustrates the plausibility of our framework for the control of computations involving multiple steps. ###### Acknowledgements. AZ was supported by a fellowship of the Peruilh Foundation, Faculty of Engineering, Universidad de Buenos Aires. LP was supported by a fellowship of the National Research Council of Argentina (CONICET). 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USA 94, 13293 (1997).	arxiv-papers	2013-12-10T13:55:34	2024-09-04T02:49:55.858626	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Ariel D Zylberberg, Luciano Paz, Pieter R Roelfsema, Stanislas\n Dehaene, Mariano Sigman", "submitter": "Mariano Sigman", "url": "https://arxiv.org/abs/1312.6660" }
1312.6680	# Faster all-pairs shortest paths via circuit complexity111This is a preliminary version; comments are welcome. Ryan Williams Stanford University Supported by an Alfred P. Sloan Fellowship, a Microsoft Research Faculty Fellowship, a David Morgenthaler II Faculty Fellowship, and NSF CCF-1212372. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. ###### Abstract We present a new randomized method for computing the min-plus product (a.k.a., tropical product) of two $n\times n$ matrices, yielding a faster algorithm for solving the all-pairs shortest path problem (APSP) in dense $n$-node directed graphs with arbitrary edge weights. On the real RAM, where additions and comparisons of reals are unit cost (but all other operations have typical logarithmic cost), the algorithm runs in time $\frac{n^{3}}{2^{\Omega(\log n)^{1/2}}}$ and is correct with high probability. On the word RAM, the algorithm runs in $n^{3}/2^{\Omega(\log n)^{1/2}}+n^{2+o(1)}\log M$ time for edge weights in $([0,M]\cap{\mathbb{Z}})\cup\\{\infty\\}$. Prior algorithms took either $O(n^{3}/\log^{c}n)$ time for various $c\leq 2$, or $O(M^{\alpha}n^{\beta})$ time for various $\alpha>0$ and $\beta>2$. The new algorithm applies a tool from circuit complexity, namely the Razborov- Smolensky polynomials for approximately representing ${\sf AC}^{0}[p]$ circuits, to efficiently reduce a matrix product over the $(\min,+)$ algebra to a relatively small number of rectangular matrix products over ${\mathbb{F}}_{2}$, each of which are computable using a particularly efficient method due to Coppersmith. We also give a deterministic version of the algorithm running in $n^{3}/2^{\log^{\delta}n}$ time for some $\delta>0$, which utilizes the Yao-Beigel-Tarui translation of ${\sf AC}^{0}[m]$ circuits into “nice” depth-two circuits. ## 1 Introduction The all-pairs shortest path problem (APSP) and its $O(n^{3})$ time solution on $n$-node graphs [Flo62, War62] are standard classics of computer science textbooks. To recall, the input is a weighted adjacency matrix of a graph, and we wish to output a data structure encoding all shortest paths between any pair of vertices—when we query a pair of nodes $(s,t)$, the data structure should reply with the shortest distance from $s$ to $t$ in $\tilde{O}(1)$ time, and a shortest path from $s$ to $t$ in $\tilde{O}(\ell)$ time, where $\ell$ is the number of edges on the path. As the input to the problem may be $\Theta(n^{2}\cdot\log M)$ bits (where $M$ bounds the weights), it is natural to wonder if the $O(n^{3})$ bound is the best one can hope for.222It is not obvious that $o(n^{3})$-size data structures for APSP should even exist! There are $n^{2}$ pairs of nodes and their shortest paths may, in principle, be of average length $\Omega(n)$. However, representations of size $\Theta(n^{2}\log n)$ do exist, such as the “successor matrix” described by Seidel [Sei95]. (In fact, Kerr [Ker70] proved that in a model where only additions and comparisons of numbers are allowed, $\Omega(n^{3})$ operations are required.) Since the 1970s [Mun71, FM71, AHU74] it has been known that the search for faster algorithms for APSP is equivalent to the search for faster algorithms for the min-plus (or max-plus) matrix product (a.k.a. distance product or tropical matrix multiplication), defined as: $(A\star B)[i,j]=\min_{k}(A[i,k]+B[k,j]).$ That is, $\min$ plays the role of addition, and $+$ plays the role of multiplication. A $T(n)$-time algorithm exists for this product if and only if there is an $O(T(n))$-time algorithm for APSP.333Technically speaking, to reconstruct the shortest paths, we also need to compute the product $(A\odot B)[i,j]=\operatorname{arg\,min}_{k}(A[i,k]+B[k,j])$, which returns (for all $i,j$) some $k$ witnessing the minimum $A[i,k]+B[k,j]$. However, all known distance product algorithms (including ours) have this property. Perhaps inspired by the surprising $n^{2.82}$ matrix multiplication algorithm of Strassen [Str69] over _rings_ ,444As $\min$ and $\max$ do not have additive inverses, min-plus algebra and max-plus algebra are not rings, so fast matrix multiplication algorithms do not directly apply to them. Fredman [Fre75] initiated the development of $o(n^{3})$ time algorithms for APSP. He discovered a _non-uniform_ decision tree computing the $n\times n$ min-plus product with depth $O(n^{2.5})$ (but with size $2^{\Theta(n^{2.5})}$). Combining the decision tree with a lookup table technique, he obtained a uniform APSP algorithm running in about $n^{3}/\log^{1/3}n$ time. Since 1975, many subsequent improvements on Fredman’s algorithm have been reported (see Table 1).555There have also been parallel developments in APSP algorithms on sparse graphs [Joh77, FT87, PR05, Pet04, Cha06] and graphs with small integer weights [Rom80, Pan81, Sei95, Tak95, AGM97, SZ99, Zwi02, GS13]. The small- weight algorithms are _pseudopolynomial_ , running in time $O(M^{\alpha}n^{\beta})$ for various $\alpha>0$ and various $\beta$ greater than the (ring) matrix multiplication exponent. However, all these improvements have only saved $\log^{c}n$ factors over Floyd-Warshall: most recently, Chan [Cha07] and Han and Takaoka [HT12] give time bounds of roughly $n^{3}/\log^{2}n$. The consensus appears to be that the known approaches to general APSP may never save more than small ${\text{poly}}(\log n)$ factors in the running time. The methods (including Fredman’s) use substantial preprocessing, lookup tables, and (sometimes) bit tricks, offloading progressively more complex operations into tables such that these operations can then be executed in constant time, speeding up the algorithm. It is open whether such techniques could even lead to an $n^{3}/\log^{3}n$ time Boolean matrix multiplication (with logical OR as addition), a special case of max-plus product. V. Vassilevska Williams and the author [VW10] proved that a large collection of fundamental graph and matrix problems are _subcubic equivalent_ to APSP: Either all these problems are solvable in $n^{3-{\varepsilon}}$ time for some ${\varepsilon}>0$ (a.k.a. “truly subcubic time”), or none of them are. This theory of APSP-hardness has nurtured some pessimism that truly subcubic APSP is possible. Time \| Author(s) \| Year(s) ---\|---\|--- $n^{3}$ \| Floyd [Flo62]/Warshall [War62] \| 1962/1962 $n^{3}/\log^{1/3}n$ \| Fredman [Fre75] \| 1975 $n^{3}/\log^{1/2}n$ \| Dobosiewicz [Dob90]/Takaoka [Tak91] \| 1990/1991 $n^{3}/\log^{5/7}n$ \| Han [Han04] \| 2004 $n^{3}/\log n$ \| Takaoka [Tak04]/Zwick [Zwi04]/Takaoka [Tak05]/Chan [Cha05] \| 2004/2004/2005/2005 $n^{3}/\log^{5/4}n$ \| Han [Han06] \| 2006 $n^{3}/\log^{2}n$ \| Chan [Cha07]/Han-Takaoka [HT12] \| 2007/2012 $n^{3}/2^{\Omega(\log n)^{1/2}}$ \| this paper \| Table 1: Running times for general APSP, omitting poly(log log n) factors. Years are given by the earliest conference/journal publication. (Table adapted from Chan [Cha07].) We counter these doubts with a new algorithm for APSP running faster than $n^{3}/\log^{k}n$ time, for every $k$. ###### Theorem 1.1 On the word RAM, APSP can be solved in $n^{3}/2^{\Omega(\log n)^{1/2}}+n^{2+o(1)}\log M$ time with a Monte Carlo algorithm, on $n$-node graphs with edge weights in $([0,M]\cap{\mathbb{Z}})\cup\\{\infty\\}$. On the real RAM, the $n^{2+o(1)}\log M$ factor may be removed. ###### Remark 1 A similar $n^{2+o(1)}\log M$ time factor would necessarily appear in a complete running time description of all previous algorithms when implemented on a machine that takes bit complexity into account, such as the word RAM—note the input itself requires $\Omega(n^{2}\log M)$ bits to describe in the worst case. In the _real RAM_ model, where additions and comparisons of real numbers given in the input are unit cost, but all other operations have typical cost, the algorithm runs in the “strongly polynomial” bound of $n^{3}/2^{\Omega(\log n)^{1/2}}$ time. Most prior algorithms for the general case of APSP also have implementations in the real RAM. (For an extended discussion, see Section 2 of Zwick [Zwi04].) The key to Theorem 1.1 is a new reduction from min-plus (and max-plus) matrix multiplication to (rectangular) matrix multiplication over ${\mathbb{F}}_{2}$. To the best of our knowledge, all prior reductions from $(\max,+)$ algebra to (the usual) $(+,\times)$ algebra apply the mapping $a\mapsto x^{a}$ for some sufficiently large (or sometimes indeterminate) $x$. Under this mapping, $\max$ “maps to” $+$ and $+$ “maps to” $\times$: $\max\\{a,b\\}$ can be computed by checking the degree of $x^{a}+x^{b}$, and $a+b$ can be computed by $x^{a}\times x^{b}=x^{a+b}$. Although this mapping is extremely natural (indeed, it is the starting point for the field of tropical algebra), the computational difficulty with this reduction is that the sizes of numbers increase exponentially. The new algorithm avoids an exponential blowup by exploiting the fact that $\min$ and addition are _simple_ operations from the point of view of Boolean circuit complexity. Namely, these operations are both in ${\sf AC}^{0}$, i.e., they have circuits of constant-depth and polynomial-size over AND/OR/NOT gates of unbounded fan-in. It follows that min-plus inner products can be computed in ${\sf AC}^{0}$, and the new algorithm manipulates such circuits at the bit- level. (This also means the approach is highly non-black-box, and not subject to lower bounds based on additions and comparisons alone; this is necessary, due to Kerr’s $\Omega(n^{3})$ lower bound.) ${\sf AC}^{0}$ operations are very structured and have many known limitations (starting with [Ajt83, FSS81]). Circuit lower bound techniques often translate algorithmically into nice methods for manipulating circuits. Razborov [Raz87] and Smolensky [Smo87] showed how to randomly reduce size-$s$ and depth-$d$ ${\sf AC}^{0}$ circuits with XOR gates to multivariate polynomials over ${\mathbb{F}}_{2}$ with $2^{(\log s)^{O(d)}}$ monomials and approximate functionality. We show how elements of their reduction can be applied to randomly translate min-plus inner products of $\ell$-length vectors into ${\mathbb{F}}_{2}$ inner products of $n^{0.1}$-length vectors, where $\ell=2^{(\log n)^{\delta}}$ for some $\delta>0$. (The straightforward way of applying this reduction also introduces a ${\text{poly}}(\log M)$ multiplicative factor.) This allows for an efficient reduction from min-plus matrix multiplication of $n\times\ell$ and $\ell\times n$ matrices to a small number of $n\times n^{0.1}$ and $n^{0.1}\times n$ matrix multiplies over ${\mathbb{F}}_{2}$. But such _rectangular_ matrix multiplications can be computed in $n^{2}\cdot{\text{poly}}(\log n)$ arithmetic operations, using a method of Coppersmith [Cop82].666Technically speaking, Coppersmith only proves a bound on the _rank_ of matrix multiplication under these parameters; in prior work [Wil11] the author described at a high level how Coppersmith’s rank bound translates into a full algorithm. An extended exposition of Coppersmith’s algorithm is given in Appendix C. It follows that min-plus matrix multiplication of $n\times\ell$ and $\ell\times n$ matrices is in $n^{2}\cdot{\text{poly}}(\log n)$ time. (There are, of course, many details being glossed over; they will come later.) This algorithm for rectangular min-plus product can be extended to a product of $n\times n$ matrices in a standard way, by partitioning the matrices into $n/\ell$ products of $n\times\ell$ and $\ell\times n$, computing each product separately, then directly comparing the $n/\ell$ minima found for each of the $n^{2}$ entries. All in all, we obtain an algorithm for min-plus matrix product running in $\tilde{O}(n^{3}/\ell+n^{2}\ell\log M)$ time. Since $\ell=2^{(\log n)^{\delta}}\gg\log^{c}n$ for all constants $c$, the ${\text{poly}}(\log n)$ factors can be absorbed into a bound of $\tilde{O}(n^{3}/2^{\Omega(\log n)^{\delta}}+n^{2+o(1)})$.777This $\tilde{O}$ hides not only ${\text{poly}}(\log n)$ but also ${\text{poly}}(\log M)$ factors. By integrating ideas from prior APSP work, the algorithm for rectangular min- plus product can be improved to a strongly polynomial running time, resulting in Theorem 1.1. First, a standard trick in the literature due to Fredman [Fre75] permits us to replace the arbitrary entries in the matrices with $O(\log n)$-bit numbers after only $n^{2+o(1)}$ additions and comparisons of the (real-valued) entries; this trick also helps us avoid translating the additions into ${\sf AC}^{0}$. Then we construct a low-depth AND/XOR/NOT circuit for computing the minima of the quantities produced by Fredman’s trick, using Razborov-Smolensky style arguments to probabilistically translate the circuit into a multivariate polynomial over ${\mathbb{F}}_{2}$ which computes it, with high probability. With care, the polynomial can be built to have relatively few monomials, leading to a better bound. ### 1.1 Applications The running time of the new APSP algorithm can be extended to many other problems; here we illustrate a few. For notational simplicity, let $\ell(n)=\Theta((\log n)^{1/2})$ be such that APSP is in $n^{3}/2^{\ell(n)}$ time, according to Theorem 1.1. It follows from the reductions of [VW10] (and folklore) that: ###### Corollary 1.1 The following problems are all solvable in $n^{3}/2^{\Omega(\ell(n))}$ time on the real RAM. • _Metricity_ : Determine whether an $n\times n$ matrix over ${\mathbb{R}}$ defines a metric space on $n$ points. * • _Minimum weight triangle_ : Given an $n$-node graph with real edge weights, compute $u,v,w$ such that $(u,v),(v,w),(w,u)$ are edges and the sum of edge weights is minimized. * • _Minimum cycle_ : Given an $n$-node graph with real positive edge weights, find a cycle of minimum total edge weight. * • _Second shortest paths_ : Given an $n$-node directed graph with real positive edge weights and two nodes $s$ and $t$, determine the second shortest simple path from $s$ to $t$. * • _Replacement paths_ : Given an $n$-node directed graph with real positive edge weights and a shortest path $P$ from node $s$ to node $t$, determine for each edge $e\in P$ the shortest path from $s$ to $t$ in the graph with $e$ removed. Faster algorithms for some _sparse_ graph problems also follow from Theorem 1.1. An example is that of finding a minimum weight triangle in a sparse graph: ###### Theorem 1.2 For any $m$-edge weighted graph, a minimum weight triangle can be found in $m^{3/2}/2^{\Omega(\ell(m))}$ time. Bremner et al. [BCD+06] show that faster algorithms for $(\min,+)$ matrix product imply faster algorithms for computing the $(\min,+)$ convolution of two vectors $x,y\in({\mathbb{Z}}\cup\\{-\infty\\})^{n}$, which is the vector in $({\mathbb{Z}}\cup\\{-\infty\\})^{n}$ defined as $(x\odot y)[i]=\min_{k=1}^{i}(x[k]+y[i-k]).$ In other words, this is the usual discrete convolution of two vectors in $(\min,+)$ algebra. ###### Corollary 1.2 ([BCD+06]) The $(\min,+)$ convolution of a length-$n$ array is in $n^{2}/2^{\Omega(\ell(n))}$ time. Is it possible that the approach of this paper can be extended to give a “truly subcubic” APSP algorithm, running in $n^{3-{\varepsilon}}$ time for some ${\varepsilon}>0$? If so, we might require an even more efficient way of representing min-plus inner products as inner products over the integers. Very recently, the author discovered a way to efficiently evaluate large depth-two linear threshold circuits [Wil13] on many inputs. The method is general enough that, if min-plus inner product can be efficiently implemented with depth-two threshold circuits, then truly subcubic APSP follows. For instance: ###### Theorem 1.3 Let $M>1$ be an integer. Suppose the $(\min,+)$ inner product of two $n$-vectors with entries in $({\mathbb{Z}}\cap[0,M])\cup\\{\infty\\}$ has polynomial-size depth-two threshold circuits with weights of absolute value at most $2^{{\text{poly}}(\log M)}\cdot 2^{n^{2}}$, constructible in polynomial time. Then for some ${\varepsilon}>0$, APSP is solvable on the word RAM in $n^{3-{\varepsilon}}\cdot{\text{poly}}(\log M)$ time for edge weights in ${\mathbb{Z}}\cap[0,M]$. To phrase it another way, the hypothesis that APSP is _not_ in truly subcubic time implies interesting circuit lower bounds. #### Outline of the rest. In Section 2, we try to provide a relatively succinct exposition of how to solve APSP in less than $n^{3}/\log^{k}n$ time for all $k$, in the case where the edge weights are not too large (e.g., at most ${\text{poly}}(n)$). In Section 3 we prove Theorem 1.1 in full, by expanding considerably on the arguments in Section 2. In Section 4 we illustrate one of the many applications, and consider the possibility of extending our approach to a truly subcubic algorithm for APSP. We conclude in Section 5. ## 2 A relatively short argument for faster APSP We begin with a succinct exposition of a good algorithm for all-pairs shortest paths, at least in the case of reasonable-sized weights. This will illustrate most of the main ideas in the full algorithm. ###### Theorem 2.1 There is a deterministic algorithm for APSP which, for some $\delta>0$, runs in time $\frac{n^{3}\cdot\log M\cdot{\text{poly}}(\log\log M)}{2^{\Omega(\log n)^{\delta}}}$ on $n$-node graphs with edge weights from $([0,M]\cap{\mathbb{Z}})\cup\\{\infty\\}$. To simplify the presentation, we will not be explicit in our choice of $\delta$ here; that lovely torture will be postponed to the next section. Another mildly undesirable property of Theorem 2.1 is that the bound is only meaningful for $M\leq 2^{2^{{\varepsilon}(\log n)^{\delta}}}$ for sufficiently small ${\varepsilon}>0$. So this is not the most general bound one could hope for, but it is effective when the edge weights are in the range $\\{0,1,\ldots,{\text{poly}}(n)\\}$, which is already a difficult case for present algorithms. The $(\log M)^{1+o(1)}$ factor will be eliminated in the next section. Let’s start by showing how ${\sf AC}^{0}$ circuit complexity is relevant to the problem. Define ${\cal W}:=([0,M]\cap{\mathbb{Z}})\cup\\{\infty\\}$; intuitively, ${\cal W}$ represents the space of possible weights. Define the min-plus inner product of vectors $u,v\in{\cal W}^{d}$ to be $(u\star v)=\min_{i}(u[i]+v[i]).$ A min-plus matrix multiplication simply represents a collection of all-pairs min-plus inner products over a set of vectors. ###### Lemma 2.1 Given $u,v\in{\cal W}^{d}$ encoded as $O(d\log M)$-bit strings, $(u\star v)$ is computable with constant-depth AND/OR/NOT circuits of size $(d\log M)^{O(1)}$. That is, the min-plus inner product function is computable in ${\sf AC}^{0}$ for every $d$ and $M$. The proof is relatively straightforward; it is given in Appendix A for completeness. Next, we show that a small ${\sf AC}^{0}$ circuit can be quickly evaluated on all pairs of inputs of one’s choice. The first step is to deterministically reduce ${\sf AC}^{0}$ circuits to depth-two circuits with a symmetric function (a multivariate Boolean function whose value only depends on the sum of the variables) computing the output gate and AND gates of inputs (or their negations) on the second layer. Such circuits are typically called ${\sf SYM}^{+}$ circuits [BT94]. It is known that constant-depth circuits with AND, OR, NOT, and MOD$m$ gates of size $s$ (a.k.a. ${\sf ACC}$ circuits) can be efficiently translated into ${\sf SYM}^{+}$ circuits of size $2^{(\log s)^{c}}$ for some constant $c$ depending on the depth of the circuit and the modulus $m$:888A MOD$m$ gate outputs $1$ if and only if the sum of its input bits is divisible by $m$. ###### Lemma 2.2 ([Yao90, BT94, AG94]) There is an algorithm $A$ and $f:{\mathbb{N}}\times{\mathbb{N}}\rightarrow{\mathbb{N}}$ such that given any size-$s$ depth-$e$ circuit $C$ with AND, OR, and $\text{MOD}m$ gates of unbounded fan-in, $A$ on $C$ runs in $2^{O(\log^{f(e,m)}s)}$ time and outputs an equivalent ${\sf SYM}^{+}$ circuit of $2^{O(\log^{f(e,m)}s)}$ gates. Moreover, given the number of ANDs in the circuit evaluating to $1$, the symmetric function itself can be evaluated in $(\log s)^{O(f(e,m))}$ time. It is easy to see that this translation is really converting circuits into multivariate _polynomials_ over $\\{0,1\\}$: the AND gates represent monomials with coefficients equal to $1$, the sum of these AND gates is a polynomial with $2^{O(\log^{f(e,m)}s)}$ monomials, and the symmetric function represents some efficiently computable function from ${\mathbb{Z}}$ to $\\{0,1\\}$. The second step is to quickly evaluate these polynomials on many chosen inputs, using rectangular matrix multiplication. Specifically, we require the following: ###### Lemma 2.3 (Coppersmith [Cop82]) For all sufficiently large $N$, multiplication of an $N\times N^{.172}$ matrix with an $N^{.172}\times N$ matrix can be done in $O(N^{2}\log^{2}N)$ arithmetic operations.999See Appendix C for a detailed exposition of this algorithm. ###### Theorem 2.2 Let $p$ be a $2k$-variate polynomial over the integers (in its monomial representation) with $m\leq n^{0.1}$ monomials, along with $A,B\subseteq\\{0,1\\}^{k}$ such that $\|A\|=\|B\|=n$. The polynomial $p(a_{1},\ldots,a_{k},b_{1},\ldots,b_{k})$ can be evaluated over all points $(a_{1},\ldots,a_{k},b_{1},\ldots,b_{k})\in A\times B$ in $n^{2}\cdot{\text{poly}}(\log n)$ arithmetic operations. Note that the obvious polynomial evaluation algorithm would require $n^{2.1}$ arithmetic operations. ###### Proof. Think of the polynomial $p$ as being over two sets of variables, $X=\\{x_{1},\ldots,x_{k}\\}$ and $Y=\\{y_{1},\ldots,y_{k}\\}$. First, we construct two matrices $M_{1}\in{\mathbb{Z}}^{n\times m}$ and $M_{2}\in{\mathbb{Z}}^{m\times n}$ as follows. The rows $i$ of $M_{1}$ are indexed by the elements $r_{1},\ldots,r_{\|A\|}\in\\{0,1\\}^{k}$ of $A$, and the columns $j$ are indexed by the monomials $p_{1},\ldots,p_{m}$ of $p$. Let $p_{i}\|_{X}$ denote the monomial $p_{i}$ restricted to the variables $x_{1},\ldots,x_{k}$ (including the coefficient of $p_{i}$), and $p_{i}\|_{Y}$ denote the product of all variables from $y_{1},\ldots,y_{k}$ appearing in $p_{i}$ (here the coefficient of $p_{i}$ is _not_ included). Observe that $p_{i}\|_{X}\cdot p_{i}\|_{Y}=p_{i}$. Define $M_{1}[i,j]:=p_{i}\|_{X}(r_{j})$. The rows of $M_{2}$ are indexed by the monomials of $p$, the columns are indexed by the elements $s_{1},\ldots,s_{\|B\|}\in\\{0,1\\}^{k}$ of $B$, and $M_{2}[i,j]:=p_{j}\|_{Y}(s_{i})$. Observe that $(M_{1}\cdot M_{2})[i,j]=p(r_{i},s_{j})$. Applying Lemma 2.3 for $n\times n^{0.1}$ and $n^{0.1}\times n$ matrices, $M_{1}\cdot M_{2}$ is computable in $n^{2}\cdot{\text{poly}}(\log n)$ operations. ∎ Putting the pieces together, we obtain our “warm-up” APSP algorithm: Proof of Theorem 2.1. Let $A$ and $B$ be $n\times n$ matrices over ${\cal W}=([0,M]\cap{\mathbb{Z}})\cup\\{\infty\\}$. We will show there is a universal $c\geq 1$ such that we can min-plus multiply an arbitrary $n\times d$ matrix $A^{\prime}$ with an arbitrary $d\times n$ matrix $B^{\prime}$ in $n^{2}\cdot{\text{poly}}(\log n,\log\log M)$ time, for $d\leq\frac{2^{(0.1\log n)^{1/c}}}{\log M}$.101010Note that, if $M\geq 2^{2^{(0.1\log n)^{1/c}}}$, the desired running time is trivial to provide. By decomposing the matrix $A$ into a block row of $n/d$ $n\times d$ matrices, and the matrix $B$ into a block column of $n/d$ $d\times n$ matrices, it follows that we can min-plus multiply $n\times n$ and $n\times n$ matrices in time $(n^{3}\cdot\log M\cdot{\text{poly}}(\log\log M))/2^{\Omega(\log n)^{1/c}}.$ So let $A^{\prime}$ and $B^{\prime}$ be $n\times d$ and $d\times n$, respectively. Each row of $A^{\prime}$ and column of $B^{\prime}$ defines a min-plus inner product of two $d$-vectors $u,v\in{\cal W}^{d}$. By Lemma 2.1, there is an ${\sf AC}^{0}$ circuit $C$ of size $(d\log M)^{O(1)}$ computing $(u\star v)$ for all such vectors $u,v$. By Theorem 2.2, that circuit $C$ can be simulated by a polynomial $p:\\{0,1\\}^{O(d\log M)}\rightarrow{\mathbb{Z}}$ of at most $K=2^{(\log(d\log M))^{c}}$ monomials for some integer $c\geq 1$, followed by the efficient evaluation of a function from ${\mathbb{Z}}$ to $\\{0,1\\}$ on the result. For $K\leq n^{0.1}$, Theorem 2.2 applies, and we can therefore compute all pairs of min-plus inner products consisting of rows $A^{\prime}$ and columns of $B^{\prime}$ in time $n^{2}\cdot{\text{poly}}(\log n)$ operations over ${\mathbb{Z}}$, obtaining their min-plus matrix product. But $K\leq n^{0.1}$ precisely when $(\log(d\log M))^{c}\leq 0.1\log n$, i.e., $d\leq\frac{2^{(0.1\log n)^{1/c}}}{\log M}.$ Therefore, we can compute an $n\times\frac{2^{(0.1\log n)^{1/c}}}{\log M}$ and $\frac{2^{(0.1\log n)^{1/c}}}{\log M}\times n$ min-plus matrix product in $n^{2}\cdot{\text{poly}}(\log n)$ arithmetic operations over ${\mathbb{Z}}$. To ensure the final time bound, observe that each coefficient of the polynomial $p$ has bit complexity at most $(\log(d\log M))^{c}\leq(\log n+\log\log M)^{c}\leq{\text{poly}}(\log n,\log\log M)$ (there could be multiple copies of the same AND gate in the ${\sf SYM}^{+}$ circuit), hence the integer output by $p$ has at most ${\text{poly}}(\log n,\log\log M)$ bit complexity as well. Evaluating the symmetric function on each entry takes ${\text{poly}}(\log n,\log\log M)$ time. Hence the aforementioned rectangular min-plus product is in $n^{2}\cdot{\text{poly}}(\log n,\log\log M)$ time, as desired. $\Box$ ## 3 Proof of The Main Theorem In this section, we establish Theorem 1.1. This algorithm will follow the basic outline of Section 2, but we desire a strongly polynomial time bound with a reasonable denominator. To achieve these goals, we incorporate Fredman’s trick into the argument, and we carefully apply the polynomials of Razborov and Smolensky for ${\sf AC}^{0}$ circuits with XOR gates. Here, the final polynomials will be over the field ${\mathbb{F}}_{2}=\\{0,1\\}$ instead of ${\mathbb{Z}}$. Let $A$ be an $n\times d$ matrix with entries from ${\cal W}:=([0,M]\cap{\mathbb{Z}})\cup\\{\infty\\}$, and let $B$ be an $d\times n$ matrix with entries from ${\cal W}$. We wish to compute $C[i,j]=\min_{k=1}^{d}(A[i,k]+B[k,j]).$ First, we can assume without loss of generality that for all $i,j$, there is a unique $k$ achieving the minimum $A[i,k]+B[k,j]$. One way to enforce this is to change all initial $A[i,j]$ entries at the beginning to $A[i,j]\cdot(n+1)+j$, and all $B[i,j]$ entries to $B[i,j]\cdot(n+1)$, prior to sorting. These changes can be made with only $O(\log n)$ additions per entry; e.g., by adding $A[i,j]$ to itself for $O(\log n)$ times. Then, $\min_{k}A[i,k]+B[k,j]$ becomes $\min_{k}(A[i,k]+B[k,j])\cdot(n+1)+k^{\star},$ where $k^{\star}$ is the minimum integer achieving $\min_{k}A[i,k]+B[k,j]$. Next, we encode a trick of Fredman [Fre75] in the computation; his trick is simply that $A[i,k]-A[i,k^{\prime}]\leq B[k^{\prime},j]-B[k,j]\text{~{}if and only if~{}}A[i,k]+B[k,j]\leq A[i,k^{\prime}]+B[k^{\prime},j].$ This subtle trick has been applied in most prior work on faster APSP. It allows us to “prepare” $A$ and $B$ by taking many differences of entries, before making explicit comparisons between entries. Namely, we construct matrices $A^{\prime}$ and $B^{\prime}$ which are $n\times d^{2}$ and $d^{2}\times n$. The columns of $A^{\prime}$ and rows of $B^{\prime}$ are indexed by pairs $(k,k^{\prime})$ from $[d]^{2}$. We define: $A^{\prime}[i,(k,k^{\prime})]:=A[i,k]-A[i,k^{\prime}]\text{~{}and~{}}B^{\prime}[(k,k^{\prime}),j]:=B[k^{\prime},j]-B[k,j].$ Observe that $A^{\prime}[i,(k,k^{\prime})]\leq B^{\prime}[(k,k^{\prime}),j]$ if and only if $A[i,k]+B[k,j]\leq A[i,k^{\prime}]+B[k^{\prime},j]$. For each column $(k,k^{\prime})$ of $A^{\prime}$ and corresponding row $(k,k^{\prime})$ of $B^{\prime}$, sort the $2n$ numbers in the set $S_{(k,k^{\prime})}=\\{A^{\prime}[i,(k,k^{\prime})],B^{\prime}[(k,k^{\prime}),i]~{}\|~{}i=1,\ldots,n\\},$ and replace each $A^{\prime}[i,(k,k^{\prime})]$ and $B[(k,k^{\prime}),j]$ by their rank in the sorted order on $S_{(k,k^{\prime})}$, breaking ties arbitrarily (giving $A$ entries precedence over $B$ entries). Call these new matrices $A^{\prime\prime}$ and $B^{\prime\prime}$. The key properties of this replacement are: 1. 1. All entries of $A^{\prime\prime}$ and $B^{\prime\prime}$ are from the set $\\{1,\ldots,2n\\}$. 2. 2. $A^{\prime\prime}[i,(k,k^{\prime})]\leq B^{\prime\prime}[(k,k^{\prime}),j]$ if and only if $A^{\prime}[i,(k,k^{\prime})]\leq B^{\prime}[(k,k^{\prime}),j]$. That is, the outcomes of all comparisons have been preserved. 3. 3. For every $i,j$, there is a unique $k$ such that $A^{\prime\prime}[i,(k,k^{\prime})]\leq B^{\prime\prime}[(k,k^{\prime}),j]$ for all $k^{\prime}$; this follows from the fact that there is a unique $k$ achieving the minimum $A[i,k]+B[k,j]$. This replacement takes $\tilde{O}(n\cdot d^{2}\cdot\log M)$ time on a word RAM, and $O(n\cdot d^{2}\cdot\log n)$ on the real RAM.111111As observed by Zwick [Zwi04], we do not need to allow for unit cost subtractions in the model; when we wish to compare two quantities $x-y$ and $a-b$ in the above, we simulate this by comparing $x+b$ and $a+y$, as in Fredman’s trick. To determine the $(\min,+)$ product of $A$ and $B$, by the proof of Lemma 2.1 (in the appendix) it suffices to compute for each $i,j=1,\ldots,n$, and $\ell=1,\ldots,\log d$, the logical expression $P(i,j,\ell)=\bigvee_{\begin{subarray}{c}k=1,\ldots,d\\\ \text{$\ell$th bit of $k$ is $1$}\end{subarray}}\bigwedge_{~{}k^{\prime}\in\\{1,\ldots,d\\}}\text{\bf[}A^{\prime\prime}[i,(k,k^{\prime})]\leq B^{\prime\prime}[(k,k^{\prime}),j]\text{\bf]}.$ Here we are using the notation that, for a logical expression $Q$, the expression [$Q$] is either $0$ or $1$, and it is $1$ if and only if $Q$ is true. We claim that $P(i,j,\ell)$ equals the $\ell$th bit of the smallest $k^{\star}$ such that $\min_{k}A[i,k]+B[k,j]=A[i,k^{\star}]+B[k^{\star},j]$. In particular, by construction of $A^{\prime\prime}$, the $\wedge$ in the expression $P(i,j,\ell)$ is true for a given $k^{\star}$ if and only if for all $k^{\prime}$ we have $A[i,k^{\star}]+B[k^{\star},j]\leq A[i,k^{\prime}]+B[k^{\prime},j]$, which is true if and only if $\min_{k^{\prime\prime}=1}^{d}A[i,k^{\prime\prime}]+B[k^{\prime\prime},j]=A[i,k^{\star}]+B[k^{\star},j]$ and $k$ is the smallest such integer (the latter being true due to our sorting constraints). Finally, $P(i,j,\ell)$ is $1$ if and only if the $\ell$th bit of this particular $k^{\star}$ is $1$. This proves the claim. We want to translate $P(i,j,\ell)$ into an expression we can efficiently evaluate arithmetically. We will do several manipulations of $P(i,j,\ell)$ to yield polynomials over ${\mathbb{F}}_{2}$ with a “short” number of monomials. Observe that, since there is always exactly one such $k^{\star}$ for every $i,j$, exactly _one_ of the $\wedge$ expressions in $P(i,j,\ell)$ is true for each fixed $i,j,\ell$. Therefore we can replace the $\vee$ in $P(i,j,\ell)$ with an XOR (also denoted by $\oplus$): $P(i,j,\ell)=\bigoplus_{\begin{subarray}{c}k=1,\ldots,d\\\ \text{$\ell$th bit of $k$ is $1$}\end{subarray}}\bigwedge_{~{}k^{\prime}\in\\{1,\ldots,d\\}}\text{\bf[}A^{\prime\prime}[i,(k,k^{\prime})]\leq B^{\prime\prime}[(k,k^{\prime}),j]\text{\bf]}.$ This is useful because XORs are “cheap” in an ${\mathbb{F}}_{2}$ polynomial, whereas ORs can be expensive. Indeed, an XOR is simply addition over ${\mathbb{F}}_{2}$, while AND (or OR) involves multiplication which can lead to many monomials. In the expression $P$, there are $d$ different ANDs over $d$ comparisons. In order to get a “short” polynomial, we need to reduce the fan-in of the ANDs. Razborov and Smolensky proposed the following construction: for an AND over $d$ variables $y_{1},\ldots,y_{d}$, let $e\geq 1$ be an integer, choose independently and uniformly at random $e\cdot d$ bits $r_{1,1},\ldots,r_{1,d},r_{2,1},\ldots,r_{2,d},~{}\ldots~{},r_{e,1},\ldots,r_{e,d}\in\\{0,1\\}$, and consider the expression $E(y_{1},...,y_{d})=\bigwedge_{i=1}^{e}\left(1+\bigoplus_{j=1}^{d}r_{i,j}\cdot(y_{j}+1)\right),$ where $+$ corresponds to addition modulo $2$. Note that when the $r_{i,j}$ are fixed constants, $E$ is an AND of $e$ XORs of at most $d+1$ variables $y_{j}$ along with possibly the constant $1$. ###### Claim 1 (Razborov [Raz87], Smolensky [Smo87]) For every fixed $(y_{1},...,y_{d})\in\\{0,1\\}^{d}$, $\Pr_{r_{i,j}}[E(y_{1},...,y_{d})=y_{1}\wedge\cdots\wedge y_{d}]\geq 1-1/2^{e}.$ For completeness, we give the simple proof. For a given point $(y_{1},\ldots,y_{d})$, first consider the expression $F_{i}=1+\oplus_{j=1}^{d}r_{i,j}\cdot(y_{j}+1)$. If $y_{1}\wedge\cdots\wedge y_{d}=1$, then $(y_{j}+1)$ is $0$ modulo $2$ for all $j$, and hence $F_{i}=1$ with probability $1$. If $y_{1}\wedge\cdots\wedge y_{d}=0$, then there is a subset $S$ of $y_{j}$’s which are $0$, and hence a subset $S$ of $(y_{j}+1)$’s that are $1$. The probability we choose $r_{i,j}=1$ for an odd number of the $y_{j}$’s in $S$ is at exactly $1/2$. Hence the probability that $F_{i}=0$ in this case is exactly $1/2$. Since $E(y_{1},\ldots,y_{d})=\wedge_{i=1}^{e}F_{i}$, it follows that if $y_{1}\wedge\cdots\wedge y_{d}=1$, then $E=1$ with probability $1$. Since the $r_{i,j}$ are independent, if $y_{1}\wedge\cdots\wedge y_{d}=0$, then the probability is only $1/2^{e}$ that for all $i$ we have $r_{i,j}=1$ for an odd number of $y_{j}=0$. Hence the probability is $1-1/2^{e}$ that some $F_{i}(y_{1},\ldots,y_{d})=0$, completing the proof. Now set $e=2+\log d$, so that $E$ fails on a point $y$ with probability at most $1/(4d)$. Suppose we replace each of the $d$ ANDs in expression $P$ by the expression $E$, yielding: $P^{\prime}(i,j,\ell)=\bigoplus_{\begin{subarray}{c}k=1,\ldots,d\\\ \text{$\ell$th bit of $k$ is $1$}\end{subarray}}E(\text{\bf[}A^{\prime\prime}[i,(k,1)]\leq B^{\prime\prime}[(k,1),j]\text{\bf]},\ldots,\text{\bf[}A^{\prime\prime}[i,(k,k^{\prime})]\leq B^{\prime\prime}[(k,k^{\prime}),j]\text{\bf]}).$ By the union bound, the probability that the (randomly generated) expression $P^{\prime}$ differs from $P$ on a given row $A^{\prime\prime}[i,:]$ and column $B^{\prime\prime}[:,j]$ is at most $1/4$. Next, we open up the $d^{2}$ comparisons in $P$ and simulate them with low- depth circuits. Think of the entries of $A^{\prime\prime}[i,(k,k^{\prime})]$ and $B^{\prime\prime}[(k,k^{\prime}),j]$ as bit strings, each of length $t=1+\log n$. To check whether $a\leq b$ for two $t$-bit strings $a=a_{1},...,a_{t}$ and $b=b_{1},...,b_{t}$ construed as positive integers in $\\{1,\ldots,2^{t}\\}$, we can compute (from Lemma 2.1) $\displaystyle LEQ(a,b)$ $\displaystyle=$ $\displaystyle\left(\bigwedge_{i=1}^{t}(1+a_{i}+b_{i})\right)$ $\displaystyle\oplus\bigoplus_{i=1}^{t}\left((1+a_{i})\wedge b_{i}\wedge\bigwedge_{j=1}^{i-1}(1+a_{j}+b_{j})\right)$ where $+$ again stands for addition modulo $2$. (We can replace the outer $\vee$ with a $\oplus$, because at most one of the $t$ expressions inside of the $\oplus$ can be true for any $a$ and $b$.) The $LEQ$ circuit is an XOR of $t+1$ ANDs of fan-in $\leq t$ of XORs of fan-in at most 3. Applying Claim 1, we replace the ANDs with a randomly chosen expression $E^{\prime}(e_{1},\ldots,e_{t})$, which is an AND of fan-in $e^{\prime}$ (for some parameter $e^{\prime}$ to be determined) of XORs of $\leq t$ fan-in. The new expression $LEQ^{\prime}$ now has the form $\bigoplus_{t+1}\left[\bigwedge_{e^{\prime}}\left[\bigoplus_{\leq t}\left[\text{2 $\oplus$ gates}\right]\right]\right];$ (1) that is, we have an XOR of $t+1$ fan-in, of ANDs of fan-in $e^{\prime}$, of XORs of $\leq t$ fan-in, of XORs of fan-in at most 3. In fact, an anonymous STOC referee pointed out that, by performing additional preprocessing on the matrices $A^{\prime\prime}$ and $B^{\prime\prime}$, we can reduce the $LEQ^{\prime}$ expression further, to have the form $\bigoplus_{t+1}\left[\bigwedge_{e^{\prime}}\left[\text{2 $\oplus$ gates}\right]\right].$ This reduction will be significant enough to yield a better denominator in the running time. (An earlier version of the paper, without the following preprocessing, reported a denominator of $2^{\Omega(\log n/\log\log n)^{1/2}}$.) Each term of the form “$\bigoplus_{\leq t}\left[\text{2 $\oplus$ gates}\right]$” in (1) can be viewed an XOR of three quantities: an XOR of a subset of $O(\log n)$ variables $a_{i}$ (from the matrix $A^{\prime\prime}$), another XOR of a subset of $O(\log n)$ variables $b_{j}$ (from the matrix $B^{\prime\prime}$), and a constant (0 or 1). Given the random choices to construct the expression $E^{\prime}$, we first compute the $(t+1)e^{\prime}$ XORs over just the entries from the matrix $A^{\prime\prime}$ in advance, for all $nd^{2}$ entries in $A^{\prime\prime}$, and separately compute the set of $(t+1)e^{\prime}$ XORs for the $nd^{2}$ entries in $B$, in $\tilde{O}(nd^{2}\cdot(t+1)e^{\prime})$ time. Once precomputed, these XOR values will become the values of variables in our polynomial evaluation later. For each such XOR over an appropriate subset $S$ of the $a_{j}$’s (respectively, some subset $T$ of the $b_{j}$’s), we introduce new variables $a^{\prime}_{S}$ (and $b^{\prime}_{T}$), and from now on we think of evaluating the equivalent polynomial over these new $a^{\prime}_{S}$ and $b^{\prime}_{T}$ variables, which has the form $\bigoplus_{t+1}\left[\bigwedge_{e^{\prime}}\left[\text{2 $\oplus$ gates}\right]\right].$ Combining the two consecutive layers of XOR into one, and applying the distributive law over ${\mathbb{F}}_{2}$ to the AND, $LEQ^{\prime}$ is equivalent to a degree-$e^{\prime}$ polynomial $Q$ over ${\mathbb{F}}_{2}$ with at most $m=(t+1)\cdot 3^{e^{\prime}}$ monomials (an XOR of fan-in at most $m$ of ANDs of fan-in at most $e^{\prime}$). By the union bound, since the original circuit for $LEQ(a,b)$ contains only $t+1$ AND gates, and the probability of error of $E^{\prime}$ is at most $1/2^{e^{\prime}}$, we have that for a fixed pair of strings $(a,b)$, $LEQ(a,b)=LEQ^{\prime}(a,b)$ with probability at least $1-(t+1)/2^{e^{\prime}}$. Recall in the expression $P^{\prime}$, there are $d^{2}$ comparisons, and hence $d^{2}$ copies of the $LEQ$ circuit are needed. Setting $e^{\prime}=3+2\log d+\log t,$ we ensure that, for a given row $i$, column $j$, and $t$ for $P^{\prime}$, $d^{2}$ copies of the $LEQ^{\prime}$ circuit give the same output as $LEQ$ with probability at least $3/4$. Hence we have a polynomial $Q$ in at most $m^{\prime}=(t+1)\cdot 3^{3+2\log d+\log t}$ monomials, each of degree at most $2t$, that can accurately computes all comparisons in $P^{\prime}$ on a given point, with probability at least $3/4$. Plugging $Q$ into the circuit for $P^{\prime}$, the expression $P^{\prime\prime}(i,j,\ell)$ now has the form: $\begin{array}[]{l}\text{An XOR of $\leq d$ fan-in,}\\\ ~{}~{}~{}~{}~{}\text{ANDs of $1+\log d$ fan-in,}\\\ ~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{XORs of $\leq d+1$ fan-in,}\\\ ~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{XORs of $\leq m^{\prime}$ fan-in,}\\\ ~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{ANDs of $e^{\prime}$ variables.}\end{array}$ (The second and third layers are the $E$ circuits; the fourth and fifth layers are the polynomial $Q$ applied to various rows and columns.) Merging the two consecutive layers of XORs into one XOR of fan-in $\leq(d+1)m^{\prime}$, and applying distributivity to the ANDs of $\leq 1+\log d$ fan-in, we obtain a polynomial $Q^{\prime}_{i,j,\ell}$ over ${\mathbb{F}}_{2}$ with a number of monomials at most $\displaystyle d\cdot((d+1)m^{\prime})^{1+\log d}$ $\displaystyle\leq$ $\displaystyle d\cdot((d+1)\cdot(t+1)\cdot 3^{3+2\log d+\log t})^{1+\log d}.$ Further simplifying, this quantity is at most $2^{(1+\log d)\cdot(\log(d+1)+\log(t+1)+(\log 3)(3+2\log d+\log t))}.$ (2) Let $m^{\prime\prime}$ denote the quantity in (2). Provided $m^{\prime\prime}\leq n^{0.1}$, we will be able to apply a rectangular matrix multiplication in the final step. This is equivalent to $\displaystyle\log_{2}(m^{\prime\prime})\leq 0.1\log n.$ (3) Recall $t=1+\log n$, and note that $\log_{2}(m^{\prime\prime})$ expands to a sum of various powers of logs. For $d\geq t$, the dominant term in $\log_{2}(m^{\prime\prime})$ is $2(\log^{2}d)(\log 3)\leq O((\log d)^{2})$. Choosing $d=2^{\delta\cdot(\log n)^{1/2}}$ for sufficiently small $\delta>0$, inequality (3) will be satisfied, and the number $m^{\prime\prime}$ will be less than $n^{0.1}$. Finally, we apply Coppersmith’s rectangular matrix multiplication (Lemma 2.3) to evaluate the polynomial $Q^{\prime}_{i,j,\ell}$ on all $n^{2}$ pairs $(i,j)$ in $n^{2}\cdot{\text{poly}}(\log n)$ time. For a fixed $\ell=1,\ldots,\log d$, the outcome is a matrix product $D_{\ell}$ such that, for every $(i,j)\in[n]^{2}$ and for each $\ell=1,\dots,\log d$, $\displaystyle\Pr[D_{\ell}[i,j]=P(i,j,\ell)]$ $\displaystyle=$ $\displaystyle\Pr[D_{\ell}[i,j]\text{ is the $\ell$th bit of the smallest $k^{\star}$ such that}$ $A[i,k^{\star}]+B[k^{\star},j]=\min_{k}(A[i,k]+B[k,j])$] $\displaystyle\geq$ $\displaystyle 3/4.$ Correct entries for all $i,j$ can be obtained with high probability, using a standard “majority amplification” trick. Let $c$ be an integer parameter to set later. For every $\ell=1,\ldots,\log d$, choose $c\log n$ independent random polynomials $Q^{\prime}_{i,j,\ell}$ according to the above process, and evaluate each one on all $i,j\in[n]^{2}$ using a rectangular matrix product, producing 0-1 matrices $D_{\ell,1},\ldots,D_{\ell,c\log n}$ each of dimension $n\times n$. Let $C_{\ell}[i,j]=MAJ(D_{\ell,1}[i,j],\ldots,D_{\ell,c\log n})[i,j],$ i.e., $C_{\ell}[i,j]$ equals the majority bit of $D_{\ell,1}[i,j],\ldots,D_{\ell,c\log n}[i,j]$. We claim that $C_{\ell}[i,j]$ equals the desired output for all $i,j,\ell$, with high probability. For every $(i,j)\in[n]^{2}$, $\ell\in[\log d]$, and $k=1,\ldots,c\log n$, we have $\Pr[D_{\ell,k}[i,j]=P(i,j,\ell)]\geq 3/4$. Therefore for the random variable $X:=\sum_{k=1}^{c\log n}$ [$D_{\ell,k}[i,j]=P(i,j,\ell)$], we have $E[X]\geq(3c\log n)/4$. In order for the event $MAJ(D_{\ell,1}[i,j],\ldots,D_{\ell,c\log n})\neq P(i,j,\ell)$ to happen, we must have that $X<(c\log n)/2$. Recall that if we have independent random variables $Y_{i}$ that are $0$-$1$ valued with $0<E[Y_{i}]<1$, the random variable $Y:=\sum_{i=1}^{k}Y_{i}$ satisfies the tail bound $\Pr\left[Y<(1-{\varepsilon})E[Y]\right]\leq e^{-{\varepsilon}^{2}E[Y]/2}$ (e.g., in Motwani and Raghavan [MR95], this is Theorem 4.2). Applying this bound, $\displaystyle\Pr[C_{\ell}(i,j)\neq P(i,j,\ell)]$ $\displaystyle=$ $\displaystyle\Pr[MAJ(D_{\ell,1}[i,j],\ldots,D_{\ell,c\log n}[i,j])\neq P(i,j,\ell)]$ $\displaystyle\leq$ $\displaystyle\Pr\left[X<(c\log n)/2\right]\leq\Pr\left[X<(1-1/3)E[X]\right]$ $\displaystyle\leq$ $\displaystyle e^{-(2/3)^{2}E[X]/2}=e^{-4E[X]/18}.$ Set $c=18$. By a union bound over all pairs $(i,j)\in[n]^{2}$ and $\ell\in[\log d]$, $\displaystyle\Pr[\text{There are $i,j,\ell$, }C_{\ell}\neq P(i,j,\ell)]$ $\displaystyle\leq$ $\displaystyle(n^{2}\log d)\cdot e^{-4\log n}\leq(\log d)/n^{2}.$ Set $c=18$. By a union bound over all pairs $(i,j)\in[n]^{2}$ and $\ell\in[\log d]$, $\Pr[\text{there are $i,j,\ell$, }C_{\ell}\neq P(i,j,\ell)]\leq(n^{2}\log d)\cdot e^{-4\log n}\leq(\log d)/n^{2}.$ Therefore for $d=2^{\delta(\log n)^{1/2}}$, the algorithm outputs the min-plus product of an $n\times d$ and $d\times n$ matrix in $n^{2}\cdot{\text{poly}}(\log n)+n\cdot d^{2}\cdot(\log M)$ time, with probability at least $1-(\log n)/n^{2}$. Applying this algorithm to $n/d$ different $n\times d$ and $d\times n$ min- plus products, the min-plus product of two $n\times n$ matrices is computable in time $n^{3}/2^{\Omega(\log n)^{1/2}}$ on the real RAM with probability at least $1-(\log n)/n$, by the union bound. (On the word RAM, there is an extra additive factor of $n^{2+o(1)}\cdot\log M$, for the initial application of Fredman’s trick.) ### 3.1 Derandomizing the algorithm The APSP algorithm can be made deterministic with some loss in the running time, but still asymptotically better than $n^{3}/(\log n)^{k}$ for every $k$. See Appendix B for the proof. ###### Theorem 3.1 There is a $\delta>0$ and a deterministic algorithm for APSP running in $n^{3}/2^{(\log n)^{\delta}}$ time on the real RAM. ## 4 Some Applications All applications referred to the introduction follow straightforwardly from the literature, except for possibly: Reminder of Theorem 1.2 For any $m$-edge weighted graph, a minimum weight triangle can be found in $m^{3/2}/2^{\Omega(\ell(m))}$ time. ###### Proof. We follow the high-degree/low-degree trick of Alon, Yuster, Zwick [AYZ97]. To find a minimum edge-weight triangle with $m$ edges, let $\Delta\in[1,m]$ be a parameter and consider two possible scenarios: 1. 1. _The min-weight triangle contains a node of degree at most $\Delta$._ Here, $O(m\cdot\Delta)$ time suffices to search for the triangle: try all possible edges $\\{u,v\\}$ with $\deg(v)\leq\Delta$, and check if there is a neighbor of $v$ which forms a triangle with $u$, recording the triangle encountered of smallest weight. 2. 2. _The min-weight triangle contains only nodes of degree at least $\Delta$._ Let $N$ be the number of nodes of degree at least $\Delta$; by counting, $N\leq 2m/\Delta$. Searching for a min-weight triangle on these $N$ nodes can be done in $O(N^{3}/2^{\Omega(\ell(N))})$ time, by reduction to $(\min,+)$ matrix multiplication. In particular, one $(\min,+)$ matrix multiply will efficiently compute the weight of the shortest path of two edges from $u$ to $v$, for every pair of nodes $u,v$. We can obtain the minimum weight of any triangle including the edge $\\{u,v\\}$ by adding the two-edge shortest path cost from $u$ to $v$ with the weight of $\\{u,v\\}$. Hence this step takes $O\left(\frac{m^{3}}{\Delta^{3}2^{\Omega(\ell(m/\Delta))}}\right)$ time. To minimize the overall running time, we want $m\cdot\Delta\approx m^{3}/(\Delta^{3}2^{\Omega(\ell(m/\Delta))}).$ For $\Delta=m^{1/2}/2^{\ell(m)}$, the runtime is $O(m^{3/2}/2^{\Omega(\ell(m))})$. ∎ ### 4.1 Towards Truly Subcubic APSP? It seems likely that the basic approach taken in this paper can be extended to discover even faster APSP algorithms. Here we outline one concrete direction to pursue. A ${\sf SYM}\circ{\sf THR}$ circuit is a logical circuit of three layers: the _input layer_ has $n$ Boolean variables, the _middle layer_ contains _linear threshold gates_ with inputs from the input layer, and the _output layer_ is a single gate taking inputs from the middle layer’s outputs and computing a Boolean symmetric function, i.e., the output of the function depends only on the number of true inputs. Every linear threshold gate in the circuit with inputs $y_{1},\ldots,y_{t}$ has its own collection of weights $w_{1},\ldots,w_{t},w_{t+1}\in{\mathbb{Z}}$, such that the gate outputs $1$ if and only if $\sum_{i=1}^{t}w_{i}\cdot y_{i}\geq w_{t+1}$ holds. It is an open frontier in circuit complexity to exhibit explicit functions which are not computable efficiently with ${\sf SYM}\circ{\sf THR}$ circuits. As far as we know, it could be that huge complexity classes like ${\sf EXP}^{\sf NP}$ have ${\sf SYM}\circ{\sf THR}$ circuits with only ${\text{poly}}(n)$ gates. (Allowing exponential weights is crucial: there are lower bounds for depth-two threshold circuits with small weights [HMP+93].) Reminder of Theorem 1.3 Let $M>1$ be an integer. Suppose the $(\min,+)$ inner product of two $n$-vectors with entries in $({\mathbb{Z}}\cap[0,M])\cup\\{\infty\\}$ has polynomial-size ${\sf SYM}\circ{\sf THR}$ circuits with threshold weights of absolute value at most $2^{{\text{poly}}(\log M)}\cdot 2^{n^{2}}$, constructible in polynomial time. Then APSP is solvable on the word RAM in $n^{3-{\varepsilon}}\cdot{\text{poly}}(\log M)$ time for some ${\varepsilon}>0$ for edge weights in ${\mathbb{Z}}\cap[0,M]$. That is, efficient depth-two circuits for $(\min,+)$ inner product would imply a truly subcubic time algorithm for APSP. The proof applies a recent algorithm of the author: ###### Theorem 4.1 ([Wil13]) Given a ${\sf SYM}\circ{\sf THR}$ circuit $C$ with $2k$ inputs and at most $n^{1/12}$ gates with threshold weights of absolute value at most $W_{b}$, and given two sets $A,B\subseteq\\{0,1\\}^{k}$ where $\|A\|=\|B\|=n$, we can evaluate $C$ on all $n^{2}$ points in $A\times B$ using $n^{2}\cdot{\text{poly}}(\log n)+n^{1+1/12}\cdot{\text{poly}}(\log n,\log W_{b})$ time. A similar theorem also holds for depth-two threshold circuits (${\sf THR}\circ{\sf THR}$). Note the obvious algorithm for the above evaluation problem would take at least $\Omega(n^{2+1/12})$ time. Proof of Theorem 1.3. Assuming the hypothesis of the theorem, there is some $k$ such that the $(\min,+)$ inner product of two $d$-vectors with entries in $([0,M]\cap{\mathbb{Z}})\cup\\{\infty\\}$ can be computed with a depth-two linear threshold circuit of at most $(d\cdot\log M)^{k}$ gates. Setting $d=\min\\{1,n^{1/(12k)}/(\log M)^{k}\\}$, the number of gates in the circuit is bounded by $n^{1/24}$. (For sufficiently large $M$, $d$ will be $1$, but in this case a time bound of $n^{3-{\varepsilon}}\cdot{\text{poly}}(\log M)$ for APSP is trivial.) Letting $A$ be the rows of one $n\times d$ matrix $A^{\prime}$, and letting $B$ be the columns of another $d\times n$ matrix $B^{\prime}$, Theorem 4.1 says that we can $(\min,+)$-multiply $A^{\prime}$ and $B^{\prime}$ with entries from $([0,M]\cap{\mathbb{Z}})\cup\\{\infty\\}$ in $n^{2}\cdot{\text{poly}}(\log n,\log M)$ time. To compute the $(\min,+)$-multiplication of two $n\times n$ matrices, we reduce it into $n/d$ multiplies of $n\times d$ and $d\times n$ (as in Theorems 2.1 and 1.1), resulting in an algorithm running in time $O(n^{3-1/(12k)}\cdot(\log M)^{k})$. In a graph with edge weights in ${\mathbb{Z}}\cap[0,M]$, the shortest path between nodes $u$ and $v$ either has length at most $nM$, or it is $\infty$. The above argument shows we can compute min-plus matrix products with entries up to $nM$ in time $n^{3-{\varepsilon}}\cdot{\text{poly}}(\log nM)\leq n^{3-{\varepsilon}^{\prime}}{\text{poly}}(\log M)$, for some ${\varepsilon},{\varepsilon}^{\prime}>0$. Therefore, APSP can be computed in the desired time, since the necessary min-plus matrix products can be performed in the desired time. $\Box$ ## 5 Discussion The method of this paper is generic: the main property of APSP being used is that min-plus inner product and related computations are in ${\sf AC}^{0}$. Other special matrix product operations with “inner product” definable in ${\sf AC}^{0}$ (or even ${\sf ACC}$) are also computable in $n^{3}/2^{(\log n)^{\delta}}$ time, as well. (Note that ${\sf AC}^{0}$ by itself is not enough: one must also be able to reduce inner products on vectors of length $n$ to $\tilde{O}(n/d)$ inner products on vectors of length at most $d^{{\text{poly}}(\log d)}$, as is the case with $(\min,+)$ inner product.) Other fundamental problems have simple algorithms running in time $n^{k}$ for some $k$, and the best known running time is stuck at $n^{k}/\log^{c}n$ for some $c\leq 3$. (The phrase “shaving logs” is often associated with this work.) It would be very interesting to find other basic problems permitting a “clean shave” of all polylog factors from the runtime. Here are a few specific future directions. 1\. Subquadratic 3SUM. Along with APSP, the 3SUM problem is another notorious polynomial-time solvable problem: _given a list of integers, are there three which sum to zero_? For lists of $n$ numbers, an $O(n^{2})$ time algorithm is well-known, and the conjecture that no $n^{1.999}$ time algorithm exists is significant in computational geometry and data structures, with many intriguing consequences [GO95, BHP01, SEO03, Pat10, VW13]. Baran, Demaine, and Patrascu [BDP05] showed that 3SUM is in about $n^{2}/\log^{2}n$ time (omitting ${\text{poly}}(\log\log n)$ factors). Can this be extended to $n^{2}/2^{(\log n)^{\delta}}$ time for some $\delta>0$? It is natural to start with solving Convolution-3SUM, defined by Patrascu [Pat10] as: _given an array $A$ of $n$ integers, are there $i$ and $j$ such that $A[i]+A[j]=A[i+j\pmod{n}]$?_ Although this problem looks superficially easier than 3SUM, Patrascu showed that if Convolution-3SUM is in $n^{2}/(f(n\cdot f(n)))^{2}$ time then 3SUM is in $n^{2}/f(n)$ time. That is, minor improvements for Convolution-3SUM would yield similar improvements for 3SUM. 2\. Subquadratic String Matching. There are many problems involving string matching and alignment which are solvable using dynamic programming in $O(n^{2}/\log n)$ time, on strings of length $n$. A prominent example is computing the _edit distance_ [MP80]. Can edit distance be computed in $n^{2}/2^{(\log n)^{\delta}}$ time? 3\. Practicality? There are two potential impediments to making the approach of this paper work in practice: (1) the translation from ${\sf AC}^{0}[2]$ circuits to polynomials, and (2) Coppersmith’s matrix multiplication algorithm. For case (1), there are no large hidden constants inherent in the Razborov-Smolensky translation, however the expansion of the polynomial as an XOR of ANDs yields a quasi-polynomial blowup. A careful study of alternative translations into polynomials would likely improve this step for practice. For case (2), Coppersmith’s algorithm as described in Appendix C consists of a series of multiplications with Vandermonde and inverse Vandermonde matrices (which are very efficient), along with a recursive step on $2\times 3$ and $3\times 2$ matrices, analogous to Strassen’s famous algorithm. We see no theoretical reason why this algorithm (implemented properly) would perform poorly in practice, given that Strassen’s algorithm can be tuned for practical gains [GG96, CLPT02, DN09, BDLS12, BDH+12]. Nevertheless, it would likely be a substantial engineering challenge to turn the algorithms of this paper into high-performance software. 4\. APSP For Sparse Graphs? Perhaps a similar approach could yield an APSP algorithm for $m$-edge, $n$-node graphs running in $\tilde{O}(mn/2^{(\log n)^{\delta}}+n^{2})$ time, which is open even for undirected, unweighted graphs. (The best known algorithms are due to Chan [Cha06] and take roughly $mn/\log n$ time.) 5\. Truly Subcubic APSP? What other circuit classes can compute $(\min,+)$ inner product and also permit a fast evaluation algorithm on many inputs? This question now appears to be central to the pursuit of truly subcubic ($n^{3-{\varepsilon}}$ time) APSP. Although we observe in the paper that $(\min,+)$ inner product is efficiently computable in ${\sf AC}^{0}$, the usual algebraic $(+,\times)$ inner product is in fact _not_ in ${\sf AC}^{0}$. (Multiplication is not in ${\sf AC}^{0}$, by a reduction from Parity [CSV84].) This raises the intriguing possibility that $(\min,+)$ matrix product (and hence APSP) is not only in truly subcubic time, but could be _easier_ than integer matrix multiplication. A prerequisite to this possibility would be to find new Boolean matrix multiplication algorithms which do not follow the Strassenesque approaches of the last 40+ years. Only minor progress on such algorithms has been recently made [BW09]. #### Acknowledgements. 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Before giving the circuit, let us briefly discuss the encoding of $\infty$. In principle, all we need is that $\infty$ encodes an integer greater than $2M$, and that addition of $\infty$ with any number equals $\infty$ again. With that in mind, we use the following convention: $t=3+\log M$ bits are allocated to encode each number in $\\{0,\ldots,M\\}$ (with two leading zeroes), and $\infty$ is defined as the all-ones string of $t$ bits. The addition of two $t$-bit strings $x$ and $y$ is computable in $t^{O(1)}$ size and constant depth, using a “carry-lookahead” adder [SV84] (with an improved size bound in [CFL85]). Recall that a carry-lookahead adder determines in advance: * • which bits of $x$ and $y$ will “generate” a $1$ when added, by taking the AND of each matching pair of bits from $x$ and $y$, and * • which bits of $x$ and $y$ will “propagate” a $1$ if given a $1$ from the summation of lower bits, by taking the XOR of each matching pair of bits. The “generate” and “propagate” bits can be generated in constant depth. Given them, in parallel we can determine (for all $i$) which bits $i$ of the addition will generate a carry in constant depth, and hence determine the sum in constant depth. (To handle the case of $\infty$, we simply add a side circuit which computes in parallel if one of the inputs is all-$1$s, in which case all output bits are forced to be $1$.) Chandra, Stockmeyer, and Vishkin [CSV84] show how to compute the minimum of a collection of numbers (given as bit strings) in ${\sf AC}^{0}$. For completeness, we give a construction here. First, the comparison of two $t$-bit strings $x$ and $y$ (construed as non-negative integers) is computable in ${\sf AC}^{0}$. Define $LEQ(x,y)=\left(\bigwedge_{i=1}^{t}(1+x_{i}+y_{i})\right)\vee\bigvee_{i=1}^{t}\left((1+x_{i})\wedge y_{i}\wedge\bigwedge_{j=1}^{i-1}(1+x_{j}+y_{j})\right),$ where $+$ stands for addition modulo $2$. The first disjunct is true if and only if $x=y$. For the second disjunct and given $i=1,\ldots,t$, the inner expression is true if and only if the first $i-1$ bits of $x$ and $y$ are equal, the $i$th bit of $x$ is $0$, and the $i$th bit of $y$ is 1. Replacing each $1+a+b$ with $(\neg a\vee b)\wedge(a\vee\neg b)$, we obtain an ${\sf AC}^{0}$ circuit. To show that the minimum of $d$ $t$-bit strings $x_{1},\ldots,x_{d}$ is in ${\sf AC}^{0}$, we first prove it for the case where the minimum is _unique_ (in the final algorithm of Theorem 1.1, this will be the case). The expression $MIN(x_{i}):=\bigwedge_{~{}j\in\\{1,\ldots,d\\}}LEQ(x_{i},x_{j})$ is true if and only if $x_{i}=\min_{j}x_{j}$. When the minimum is unique, the following computes the $\ell$th bit of the minimum $x_{i}$: $MIN_{\ell}(x_{1},\ldots,x_{d}):=\bigvee_{\begin{subarray}{c}i=1,\ldots,d\\\ \text{$\ell$th bit of $i$ is $1$}\end{subarray}}MIN(x_{i}).$ Finally, let us handle the case where there is more than one minimum $x_{i}$. We will compute the minimum $i^{\star}$ such that $MIN(x_{i^{\star}})=1$, then output the $\ell$th bit of $x_{i^{\star}}$. Define $MINBIT(x_{i},i,j)$ to be true if and only if ($MIN(x_{i})$ and the $j$th bit of $i$ is $1$) or $\neg MIN(x_{i})$. It is easy to see that $MINBIT$ is in ${\sf AC}^{0}$. For all $i=1,\ldots,d$, compute the $d$-bit string $f(x_{i}):=MINBIT(x_{i},i,1)\cdots MINBIT(x_{i},i,d)$ in constant depth. The function $f$ maps every non-minimum string $x_{i^{\prime}}$ to the all-ones $d$-bit string, and each minimum string $x_{i}$ to its position $i$ as it appeared in the input. Now, computing the minimum over all $f(x_{i})$ determines the smallest $i^{\star}$ such that $x_{i^{\star}}$ is a minimum; this minimum can be found in constant depth, as observed above. ∎ ## Appendix B Appendix: Derandomizing the APSP algorithm Reminder of Theorem 3.1 There is a $\delta>0$ and a deterministic algorithm for APSP running in $n^{3}/2^{(\log n)^{\delta}}$ time on the real RAM. The proof combines the use of Fredman’s trick in Theorem 1.1 with the deterministic reduction from circuits to polynomials in Theorem 2.1. Take the ${\sf AC}^{0}[2]$ circuit $C$ for computing min-plus inner products on length $d$ vectors, as described in the proof of Theorem 1.1 of Section 3. The circuit $C$ is comprised of circuits $C_{1},\ldots,C_{\log d}$ such that each $C_{\ell}$ takes a bit string of length $k=2d^{2}\log(2n)$ representing two vectors $u,v$ from $\\{1,\ldots,2n\\}^{d}$, and outputs the $\ell$th bit of the smallest $k^{\star}$ such that the min-plus inner product of $u$ and $v$ equals $u[k^{\star}]+v[k^{\star}]$. Applying Lemma 2.2 from Theorem 2.1, we can reduce each $C_{\ell}$ to a ${\sf SYM}$ circuit $D_{\ell}$ of size $2^{\log^{c}k}$ for some constant $c\geq 1$. Then, analogously to Theorem 2.2, we reduce the evaluation of $D_{\ell}$ on inputs of length $k$ to an inner product (over ${\mathbb{Z}}$) of two $0-1$ vectors $u^{\prime},v^{\prime}$ of length $2^{\log^{c}k}$. For every AND gate $g$ in $D$ that is an AND of bits $\\{x_{i_{1}},\ldots,x_{i_{t}},\ldots,y_{j_{1}},\ldots,y_{j_{t}}\\}$, we associate it with a component $g$ in the two vectors; in the $g$th component of $u^{\prime}$ we multiply all the $x_{i_{k}}$ bits owned by $u$, and in the $g$th component of $v^{\prime}$ we multiply all the $y_{j_{k}}$ bits owned by $v$. Therefore we can reduce an $n\times d$ and $d\times n$ min-plus matrix product to a matrix product over the integers, by replacing each row $u$ of the first matrix by a corresponding $u^{\prime}$ of length $\ell=2^{\log^{c}k}=2^{(\log(2d^{2}\log(2n)))^{c}}\leq 2^{(3\log d+2\log\log n)^{c}},$ and replacing each column $v$ of the second matrix by a corresponding $v^{\prime}$ of length $\ell$. When $d$ is small enough to satisfy $2^{(3\log d+2\log\log n)^{c}}\leq n^{0.1}$, this is a reduction from $n\times d$ and $d\times n$ min-plus product to $n\times n^{0.1}$ and $n^{0.1}\times n$ matrix product over ${\mathbb{Z}}$, with matrices containing $0$-$1$ entries. As argued earlier, this implies an $\tilde{O}(n^{3}/d)$ time algorithm for min- plus product. We derive an upper bound on $d$ as follows: $\displaystyle 2^{(3\log d+2\log\log n)^{c}}$ $\displaystyle\leq$ $\displaystyle n^{0.1}$ $\displaystyle\iff 3\log d+2\log\log n$ $\displaystyle\leq$ $\displaystyle(0.1\log n)^{1/c}$ $\displaystyle\iff\log d$ $\displaystyle\leq$ $\displaystyle\frac{(0.1\log n)^{1/c}-2\log\log n}{3},$ hence $d=2^{(0.1\log n)^{1/c}/4}$ suffices for sufficiently large $n$. Examining the proof of Theorem 2.2 shows that we can estimate $c=2^{\Theta(d^{\prime})}$, where $d^{\prime}$ is the depth of the original ${\sf AC}^{0}[2]$ circuit. However, as Beigel and Tarui’s proof also works for the much more expressive class ${\sf ACC}$ (and not just ${\sf AC}^{0}[2]$), we are confident that better estimates on $c$ are possible with a different argument, and hence refrain from calculating an explicit bound here. ## Appendix C Appendix: An exposition of Coppersmith’s algorithm In 1982, Don Coppersmith proved that the rank (that is, the number of essential multiplications) of $N\times N^{0.1}$ and $N^{0.1}\times N$ matrix multiplication is at most $O(N\log^{2}N)$. Prior work has observed that his algorithm can also be used to show that the total number of arithmetic operations for the same matrix multiply is $N\cdot{\text{poly}}(\log N)$. However, the implication is not immediate, and uses specific properties of Coppersmith’s algorithm. Because this result is so essential to this work and another recent circuit lower bound [Wil13], we give a self-contained exposition here. ###### Theorem C.1 (Coppersmith [Cop82]) For all sufficiently large $N$, the rank of $N\times N^{.1}\times\times N$ matrix multiplication is at most $O(N^{2}\log^{2}N)$. We wish to derive the following consequence of Coppersmith’s construction, which has been mentioned in the literature before [SM83, ACPS09, Wil11]: ###### Lemma C.1 For all sufficiently large $N$, and $\alpha\leq.172$, multiplication of an $N\times N^{\alpha}$ matrix with an $N^{\alpha}\times N$ matrix can be done in $N^{2}\cdot{\text{poly}}(\log N)$ arithmetic operations, over any field with $O(2^{{\text{poly}}(\log N)})$ elements. For brevity, we will use the notation “$\ell\times m\times n$ matrix multiply” to refer to the multiplication of $\ell\times m$ and $m\times n$ matrices (hence the above gives an algorithm for $N\times N^{\alpha}\times N$ matrix multiply). Note Lemma C.1 has been “improved” in the sense that the upper bound on $\alpha$ has been increased mildly over the years [Cop97, HP98, KZHP08, Gal12]. However, these later developments only run in $N^{2+o(1)}$ time, not $N^{2}\cdot{\text{poly}}(\log N)$ time (which we require). Our exposition will expand on the informal description given in recent work [Wil11]. First, observe that the implication from Theorem C.1 to Lemma C.1 is not immediate. For example, it could be that Coppersmith’s algorithm is non- uniform, making it difficult to apply. As far as we know, one cannot simply take “constant size” arithmetic circuits implementing the algorithm of Theorem C.1 and recursively apply them. In that case, the ${\text{poly}}(\log N)$ factor in the running time would then become $N^{{\varepsilon}}$ for some constant ${\varepsilon}>0$ (depending on the size of the constant-size circuit). To keep the overhead polylogarithmic, we have to unpack the algorithm and analyze it directly. ### C.1 A short preliminary Coppersmith’s algorithm builds on many other tools from prior matrix multiplication algorithms, many of which can be found in the highly readable book of Pan [Pan84]. Here we will give a very brief tutorial of some of the aspects. #### Bilinear algorithms and trilinear forms. Essentially all methods for matrix multiplication are bilinear (and if not, they can be converted into such algorithms), meaning that they can be expressed in the so-called trilinear form $\sum_{ijk}A_{ik}B_{kj}C_{ji}+p(x)=\sum_{\ell=1}^{5}(\sum_{ij}\alpha_{ij}A_{ij})\cdot(\sum_{ij}\beta_{ij}B_{ij})\cdot(\sum_{ij}\gamma_{ij}C_{ij})$ (4) where $\alpha_{ij}$, $\beta_{ij}$, and $\gamma_{ij}$ are constant-degree polynomials in $x$ over the field, and $p(x)$ is a polynomial with constant coefficient $0$. Such an algorithm can be converted into one with no polynomials and minimal extra overhead (as described in Coppersmith’s paper). Typically one thinks of $A_{ik}$ and $B_{kj}$ as entries in the input matrices, and $C_{ji}$ as indeterminates, so the LHS of (4) corresponds to a polynomial whose $C_{ji}$ coefficient is the $ij$ entry of the matrix product. Note the transpose of the third matrix $C$ corresponds to the final matrix product. To give an explicit example, we assume the reader is familiar with Strassen’s famous method for $2\times 2\times 2$ matrix multiply. Strassen’s algorithm can be expressed in the form of (4) as follows: $\displaystyle\sum_{i,j,k=0,1}A_{ik}B_{kj}C_{ji}$ $\displaystyle=$ $\displaystyle(A_{00}+A_{11})(B_{00}+B_{11})(C_{00}+C_{11})$ $\displaystyle+(A_{10}+A_{11})B_{00}(C_{01}-C_{11})+A_{00}(B_{01}-B_{11})(C_{10}+C_{11})$ $\displaystyle+(A_{10}-A_{00})(B_{00}+B_{01})C_{11}+(A_{00}+A_{01})B_{11}(C_{10}-C_{00})$ $\displaystyle+A_{11}(B_{10}-B_{00})(C_{00}+C_{01})+(A_{01}-A_{11})(B_{10}+B_{11})C_{00}.$ The LHS of (4) and (C.1) represents the trace of the product of three matrices $A$, $B$, and $C$ (where the $ij$ entry of matrix $X$ is $X_{ij}$). It is well known that every bilinear algorithm naturally expresses multiple algorithms through this trace representation. Since $tr(ABC)=tr(BCA)=tr(CAB)=tr((ABC)^{T})=tr((BCA)^{T})=tr((CAB)^{T}),$ if we think of $A$ as a symbolic matrix and consider (4), we obtain a new algorithm for computing a matrix $A$ when given $B$ and $C$. Similarly, we get an algorithm for computing a $B$ when given $A$ and $C$, and analogous statements hold for computing $A^{T}$, $B^{T}$, and $C^{T}$. So the aforementioned algorithm for multiplying a sparse $2\times 3$ and sparse $3\times 2$ yields several other algorithms. #### Schönhage’s decomposition paradigm. Coppersmith’s algorithm follows a specific paradigm introduced by Schönhage [Sch81] which reduces arbitrary matrix products to slightly larger matrix products with “structured nonzeroes.” The general paradigm has the following form. Suppose we wish to multiply two matrices $A^{\prime\prime}$ and $B^{\prime\prime}$. 1. 1. First we preprocess $A^{\prime\prime}$ and $B^{\prime\prime}$ in some efficient way, decomposing $A^{\prime\prime}$ and $B^{\prime\prime}$ into structured matrices $A,A^{\prime},B,B^{\prime}$ so that $A^{\prime\prime}\cdot B^{\prime\prime}=A^{\prime}\cdot A\cdot B\cdot B^{\prime}$. (Note, the dimensions of $A^{\prime}\cdot A$ may differ from $A^{\prime\prime}$, and similarly for $B^{\prime}\cdot B$ and $B^{\prime\prime}$.) The matrices $A$ and $B$ are sparse “partial” matrices directly based on $A^{\prime\prime}$ and $B^{\prime\prime}$, but they have larger dimensions, and only contain nonzeroes in certain structured parts. The matrices $A^{\prime}$ and $B^{\prime}$ are very simple and explicit matrices of scalar constants, chosen independently of $A^{\prime\prime}$ and $B^{\prime\prime}$. (In particular, $A^{\prime}$ and $B^{\prime}$ are Vandermonde-style matrices.) 2. 2. Next, we apply a specialized constant-sized matrix multiplication algorithm in a recursive manner, to multiply the structured $A$ and $B$ essentially optimally. Recall that Strassen’s famous matrix multiplication algorithm has an analogous form: it starts with a seven-multiplication product for $2\times 2\times 2$ matrix multiplication, and recursively applies this to obtain a general algorithm for $2^{M}\times 2^{M}\times 2^{M}$ matrix multiplication. Here, we will use an _optimal_ algorithm for multiplying constant-sized matrices with zeroes in some of the entries; when this algorithm is recursively applied, it can multiply sparse $A$ and $B$ with nonzeroes in certain structured locations. 3. 3. Finally, we postprocess the resulting product $C$ to obtain our desired product $A^{\prime\prime}\cdot B^{\prime\prime}$, by computing $A^{\prime}\cdot C\cdot B^{\prime}$. Using the simple structure of $A^{\prime}$ and $B^{\prime}$, the matrix products $D:=A^{\prime}\cdot C$ and $D\cdot B^{\prime}$ can be performed very efficiently. Our aim is to verify that each step of this process can be efficiently computed, for Coppersmith’s full matrix multiplication algorithm. ### C.2 The algorithm The construction of Coppersmith begins by taking input matrices $A^{\prime\prime}$ of dimensions $2^{4M/5}\times{M\choose 4M/5}2^{4M/5}$ and $B^{\prime\prime}$ of dimensions ${M\choose 4M/5}2^{4M/5}\times 2^{M/5}$ where $M\approx\log N$, and obtains an $O(5^{M}{\text{poly}}(M))$ algorithm for their multiplication. Later, he symmetrizes the construction to get an $N\times N\times N^{\alpha}$ matrix multiply. We will give this starting construction and show how standard techniques can be used to obtain an $N\times N^{\alpha}\times N$ matrix multiply from his basic construction. The multiplication of $A^{\prime\prime}$ and $B^{\prime\prime}$ will be derived from an algorithm which computes the product of $2\times 3$ and $3\times 2$ matrices with zeroes in some entries. In particular the matrices have the form: $\left(\begin{array}[]{ccc}a_{11}&a_{12}&a_{13}\\\ 0&a_{22}&a_{23}\end{array}\right),\left(\begin{array}[]{cc}b_{11}&b_{12}\\\ b_{21}&0\\\ b_{31}&0\end{array}\right),$ and the algorithm is given by the trilinear form $\displaystyle(a_{11}+x^{2}a_{12})(b_{21}+x^{2}b_{11})(c_{11})+(a_{11}+x^{2}a_{13}(b_{31})(c_{11}-xc_{21})+(a_{11}+x^{2}a_{22})(b_{21}-xb_{21})(c_{22})$ (6) $\displaystyle+(a_{11}+x^{2}a_{23})(b_{31}+xb_{12})(c_{12}+xc_{21})-(a_{11})(b_{21}+b_{31})(c_{11}+c_{12})$ $\displaystyle=x^{2}(a_{11}b_{11}c_{11}+a_{11}b_{12}c_{21}+a_{12}b_{21}c_{11}+a_{13}b_{31}c_{11}+a_{22}b_{21}c_{12}+a_{23}b_{31}c_{12})+x^{3}\cdot P(a,b,c,x).$ That is, by performing the five products of the linear forms of $a_{ij}$ and $b_{k\ell}$ on the LHS, and using the $c_{ij}$ to determine how to add and subtract these products to obtain the output $2\times 2$ matrix, we obtain a polynomial in each matrix entry whose $x^{2}$ coefficients yield the final matrix product $c_{ij}$. When the algorithm given by (6) is applied recursively to $2^{M}\times 3^{M}$ and $3^{M}\times 2^{M}$ matrices (analogously to how Strassen’s algorithm is applied to do $2^{M}\times 2^{M}\times 2^{M}$ matrix multiply), we obtain an algorithm that can multiply matrices $A$ and $B$ with dimensions $2^{M}\times 3^{M}$ and $3^{M}\times 2^{M}$, respectively, where $A$ has $O(5^{M})$ nonzeroes, $B$ has $O(4^{M})$ nonzeroes, and these nonzeroes appear in a highly regular pattern (which can be easily deduced). This recursive application of (6) will result in polynomials in $x$ of degree $O(M)$, and additions and multiplications on such polynomials increase the overall time by an $M\cdot{\text{poly}}(\log M)$ factor. Therefore we can multiply these $A$ and $B$ with structured nonzeroes in $O(5^{M}\cdot{\text{poly}}(M))$ field operations. The decomposition of $A^{\prime\prime}$ and $B^{\prime\prime}$ is performed as follows. We choose $A^{\prime}$ and $B^{\prime}$ to have dimensions $2^{4M/5}\times 2^{M}$ and $2^{M}\times 2^{M/5}$, respectively, and such that all $2^{4M/5}\times 2^{4M/5}$ submatrices of $A^{\prime}$ and $2^{M/5}\times 2^{M/5}$ submatrices of $B^{\prime}$ are non-singular. Following Schönhage, we pick $A^{\prime}$ and $B^{\prime}$ to be rectangular Vandermonde matrices: the $i,j$ entry of $A^{\prime}$ is $(\alpha_{j})^{i-1}$, where $\alpha_{1},\alpha_{2},\ldots$ are distinct elements of the field; $B^{\prime}$ is defined analogously. Such matrices have three major advantages: (1) they can be succinctly described (with $O(2^{M})$ field elements), (2) multiplying these matrices with arbitrary vectors can be done extremely efficiently, and (3) inverting an arbitrary square submatrix can be done extremely efficiently. More precisely, $n\times n$ Vandermonde matrices can be multiplied with arbitrary $n$-vectors in $O(n\cdot{\text{poly}}(\log n))$ operations, and computing the inverse of an $n\times n$ Vandermonde matrix can be done in $O(n\cdot{\text{poly}}(\log n))$ operations (for references, see [CKY89, BP94]). In general, operations on Vandermonde matrices, their transposes, their inverses, and the transposes of inverses can be reduced to fast multipoint computations on univariate polynomials. For example, multiplying an $n\times n$ Vandermonde matrix with a vector is equivalent to evaluating a polynomial (with coefficients given by the vector) on the $n$ elements that comprise the Vandermonde matrix, which takes $O(n\log n)$ operations. This translates to $O(n\cdot{\text{poly}}(\log n))$ arithmetic operations. The matrices $A$ and $B$ have dimensions $2^{M}\times 3^{M}$ and $3^{M}\times 2^{M}$, respectively, where $A$ has only $O(5^{M})$ nonzeroes, $B$ has only $O(4^{M})$ nonzeroes, and there is an optimal algorithm for multiplying $2\times 3$ (with 5 nonzeroes) and $3\times 2$ matrices (with 4 nonzeroes) that can be recursively applied to multiply $A$ and $B$ optimally, in $O(5^{M}\cdot{\text{poly}}(M))$ operations. Matrices $A$ and $B$ are constructed as follows: take any one-to-one mapping between the ${M\choose 4M/5}2^{M/5}$ columns of the input $A^{\prime\prime}$ and columns of the sparse $A$ with exactly $2^{4M/5}$ nonzeroes. For these columns $q$ of $A$ with $2^{4M/5}$ nonzeroes, we compute the inverse $A_{q}^{-1}$ of the $2^{4M/5}\times 2^{4M/5}$ minor $A_{q}$ of $A^{\prime}$ with rows corresponding to the nonzeroes in the column, and multiply $A_{q}^{-1}$ with column $q$ (in $2^{4M/5}\cdot{\text{poly}}(M)$ time). After these columns are processed, the rest of $A$ is zeroed out. Then, there is a one-to-one correspondence between columns of $A^{\prime\prime}$ and nonzero columns of $A^{\prime}\cdot A$. Performing a symmetric procedure for $B^{\prime\prime}$ (with the same mapping on rows instead of columns), we can decompose it into $B$ and $B^{\prime}$ such that there is a one-to-one correspondence between rows of $B^{\prime\prime}$ and nonzero rows of $B\cdot B^{\prime}$. It follows that this decomposition takes only $O({M\choose 4M/5}2^{4M/5}\cdot 2^{4M/5}\cdot{\text{poly}}(M))$ time. Since $5^{M}\approx{M\choose 4M/5}4^{4M/5}$ (within ${\text{poly}}(M)$ factors), this quantity is upper bounded by $5^{M}\cdot{\text{poly}}(M)$. After $A$ and $B$ are constructed, the constant-sized algorithm for $2\times 3$ and $3\times 2$ mentioned above can be applied in the usual recursive way to multiply the sparse $A$ and $B$ in $O(5^{M}\cdot{\text{poly}}(M))$ operations; call this matrix $Z$. Because $A^{\prime}$ and $B^{\prime}$ are Vandermonde, the product $A^{\prime}\cdot Z\cdot B^{\prime}$ can be computed in $O(5^{M}\cdot{\text{poly}}(M))$ operations. Hence we have an algorithm for multiplying matrices of dimensions $2^{4M/5}\times{M\choose 4M/5}2^{4M/5}$ and ${M\choose 4M/5}2^{4M/5}\times 2^{M/5}$ that is explicit and takes $5^{M}\cdot{\text{poly}}(M)$ operations. Call the above algorithm Algorithm 1. Observe Algorithm 1 also works when the entries of $A^{\prime\prime}$ and $B^{\prime\prime}$ are themselves matrices over the field. (The running time will surely increase in proportion to the sizes of the underlying matrices, but the bound on the number of operations on the entries remains the same.) Up to this point, we have simulated Coppersmith’s construction completely, and have simply highlighted its efficiency. By exploiting the symmetries of matrix multiplication algorithms in a standard way, we can extract more algorithms from the construction. The trace identity tells us that $tr(ABC)=tr(BCA),$ implying that the expression (6) can also be used to partially multiply a $3^{M}\times 2^{M}$ matrix $B$ with at most $4^{M}$ structured nonzeroes and “full” $2^{M}\times 2^{M}$ matrix $C$ in $5^{M}\cdot{\text{poly}}(M)$ operations, obtaining a $3^{M}\times 2^{M}$ matrix $A^{T}$ with at most $5^{M}$ nonzeroes. In our Algorithm 1, we have a decomposition of $A$ and $B$; in terms of the trace, we can derive: $tr(A^{\prime\prime}B^{\prime\prime}\cdot C^{\prime\prime})=tr(A^{\prime}A\cdot BB^{\prime}\cdot C^{\prime\prime})=tr(B\cdot B^{\prime}C^{\prime\prime}A^{\prime}\cdot A).$ This can be applied to obtain an algorithm for ${M\choose 4M/5}2^{4M/5}\times 2^{M/5}\times 2^{4M/5}$ matrix multiplication, as follows. Given input matrices $B^{\prime\prime}$ and $C^{\prime\prime}$ of the respective dimensions, we decompose $B^{\prime\prime}$ into a $3^{M}\times 2^{M}$ $B$ with $O(4^{M})$ nonzeroes and $2^{M}\times 2^{M/5}$ Vandermonde $B^{\prime}$, as described above. Letting $A^{\prime}$ be a Vandermonde $2^{4M/5}\times 2^{M}$ matrix, we compute the matrix $C:=B^{\prime}\cdot C^{\prime\prime}\cdot A^{\prime}$ in at most $4^{M}\cdot{\text{poly}}(M)$ operations. Noting that $C$ is $2^{M}\times 2^{M}$, we can then multiply $B$ and $C$ in $5^{M}\cdot{\text{poly}}(M)$ operations. This results in a $3^{M}\times 2^{M}$ matrix $A^{T}$ with at most $5^{M}$ nonzeroes. The final output $A^{\prime\prime}$ is obtained by using the one-to-one mapping to extract the appropriate ${M\choose 4M/5}2^{4M/5}$ rows from $A^{T}$, and multiplying each such row by the appropriate inverse minor of $A^{\prime}$ (corresponding to the nonzeroes of that row). This takes at most ${M\choose 4M/5}2^{4M/5}\cdot 2^{M}\cdot{\text{poly}}(M)\leq 5^{M}\cdot{\text{poly}}(M)$ operations. Call this Algorithm 2. From Algorithm 2 we immediately obtain an algorithm for $2^{4M/5}\times 2^{M/5}\times{M\choose 4M/5}2^{4M/5}$ matrix multiplication as well: given input matrices $(C^{\prime\prime})^{T}$ and $(B^{\prime\prime})^{T}$ of te respective dimensions, simply compute $B^{\prime\prime}\cdot C^{\prime\prime}$ using Algorithm 2, and output the transpose of the answer. Call this Algorithm 3. Finally, by “tensoring” Algorithm 2 with Algorithm 3, we derive an algorithm for matrix multiplication with dimensions ${M\choose 4M/5}2^{4M/5}\cdot 2^{4M/5}\times 2^{2M/5}\times{M\choose 4M/5}2^{4M/5}\cdot 2^{4M/5}\geq 5^{M}/M\times 4^{M/5}\times 5^{M}/M.$ That is, we divide the two input matrices of large dimensions into blocks of $2^{4M/5}\times 2^{M/5}$ and $2^{M/5}\times{M\choose 4M/5}2^{4M/5}$ dimensions, respectively. We execute Algorithm 2 on the blocks, and call Algorithm 3 when the product of two blocks is needed. As both Algorithm 2 and Algorithm 3 are explicit and efficient, their “tensorization” inherits these properties. Algorithm 2 uses $5^{M}\cdot{\text{poly}}(M)$ operations, and each operation can take up to $5^{M}\cdot{\text{poly}}(M)$ time (due to calls to Algorithm 3). Therefore, we can perform a $5^{M}\times 4^{2M/5}\times 5^{M}$ matrix multiply over fields with $2^{{\text{poly}}(M)}$ elements, in $5^{2M}\cdot{\text{poly}}(M)$ time. Setting $n=\log(M)/\log(5)$, the algorithm runs in $n^{2}\cdot{\text{poly}}(\log n)$ time for fields with $2^{{\text{poly}}(\log n)}$ elements.	arxiv-papers	2013-12-23T20:59:43	2024-09-04T02:49:55.870720	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Ryan Williams", "submitter": "Ryan Williams", "url": "https://arxiv.org/abs/1312.6680" }
1312.6756	# Multi-dimensional Conversation Analysis across Online Social Networks William Lucia University of Insubria Via Mazzini 5, Varese, Italy [email protected] Cuneyt Gurcan Akcora University of Insubria Via Mazzini 5, Varese, Italy [email protected] Elena Ferrari University of Insubria Via Mazzini 5, Varese, Italy [email protected] ###### Abstract With the advance of the Internet, ordinary users have created multiple personal accounts on online social networks, and interactions among these social network users have recently been tagged with location information. In this work, we observe user interactions across two popular online social networks, Facebook and Twitter, and analyze which factors lead to retweet/like interactions for tweets/posts. In addition to the named entities, lexical errors and expressed sentiments in these data items, we also consider the impact of shared user locations on user interactions. In particular, we show that geolocations of users can greatly affect which social network post/tweet will be liked/ retweeted. We believe that the results of our analysis can help researchers to understand which social network content will have better visibility. ## I Introduction Recent years have seen an increasing number of successful online social networks catering to different social needs on the Internet. From professional social network LinkedIn to personal social network Facebook, online social networks are diversified to meet needs of a growing online population. As a result, an Internet user has several accounts on online social networks, where his/her activities are governed by different motivations. Two of the most successful online social networks, Facebook and Twitter, have grown in recent years to accommodate millions of social network users, and hold millions of personal profiles for the same sets of users. Dynamics of these online social networks are affected by different factors; the directed nature of Twitter enables fast and efficient information propagation, whereas undirected Facebook is still meshed by close familial or regional networks, and users interact in more conservative ways. Despite these differences, both Facebook and Twitter provide means (i.e., like and retweet, respectively) to make another user’s post visible to a larger audience. Analyzing how retweets and likes are employed provides clues in understanding how to spread ideas, disseminate news and propagate influence on a social network. In this paper, we study the problem of how a post is liked/tweeted by social network users across Facebook and Twitter. Rather than external features such as the network structure or user information, we focus on tweets/posts themselves to understand what aspects of tweets/posts help them to get retweeted and liked. Moreover, we analyze how location information influences retweet, comment and like interactions. To this end, we extract the following features from text based social network posts: named entities, lexical errors and sentiments. In named entity recognition, we locate atomic elements of seven categories: people, organizations, locations, date, time, percentage and money. By observing found entities, we show how the functionality (i.e., usage purpose) of online social networks can be mined. In lexical analysis, we locate lexical errors of texts, and classify these errors into ten most frequently committed errors. An analysis of these lexical errors show that social network users make similar mistakes on both sites. Sentiment analysis aims at studying user interactions for different sentiments that are found in social network posts. We investigate the impact of positive/negative/neutral sentiments on conversation patterns. Furthermore, using location information from Twitter bios and Facebook profiles, we show that users’ interaction patterns are heavily dependent on their current locations. Although online social networks have connected millions of users from all over the world, interaction patterns are still location limited, and governed by densely connected networks. With the increasing popularity of online social networks, many studies have been carried on to better understand how humans communicate in a global setting (see e.g., [2, 15]). For instance, a comprehensive analysis has found that online social networks, such as Twitter, can also act as an information source [7] to a high degree. This functionality of Twitter is provided by tweets and conversations that occur among users [5, 12, 13]. In conversational studies, the use of hashtags (i.e., Twitter topics) and addressivity (i.e., @ sign) have been found to increase interactions among Twitter users [4]. However, in such work the studied dimensions are limited to few features, such as hashtags, whereas we consider additional conversational dimensions such as sentiment analysis. Moreover, their analysis is limited to Twitter, whereas we focus also on Facebook posts. From a theoretical point of view, sentiment analysis is a widely studied problem in research work [6, 1], but its impact on conversational interactions has not been well studied. Another dimension of our work is related to geolocation studies. Recent works by Takhteyevet al. [14] and Leetaru et al. [9] have analyzed the geography of Twitter and found that users communicate more often with those closest to them. However, these works are limited to Twitter only. In geolocation research, locations of friends have been studied from a privacy point of view on Facebook [8]. In this work, we come to the similar conclusion that friends/followers share common locations to a high degree, but our focus is different in that we analyze the impact of locations on conversational interactions, by looking at like/retweet locations. The paper is organized as follows. Section II explains our data collection process on Facebook and Twitter. Section III explains our methodology, and discusses the software tools we used. In Section IV, we analyze the four dimensions we considered, whereas Section V shows how considered dimensions affect each other. Finally, Section VI combines our insights with geolocation data, and presents our results in a visual way. ## II Data collection Data sets of this paper have been created by querying Twitter.com and Facebook.com APIs. A Facebook application 111developers.facebook.com was used to query Facebook for user data in an offline manner and store posts from the 2008-2013 period. Through the application, 75 users have given us permission to track data items (i.e., status posts, photos, videos, etc.) along with comments and likes these items receive from other Facebook users. This data crawling allowed us to track conversations of 670K Facebook users. Each considered data item has been posted on Facebook by users, their friends or friends of friends. Similarly, likes/comments on these data items belong to the users, their friends or friends of friends. Facebook status posts can contain pictures and videos. As we cannot analyze contents of these additional data items, we limit our analysis to Facebook comments that consist of textual data only. In what follows, we will refer to Facebook comments as Facebook texts. Our Twitter dataset comes from a crawl between December 2012 and April 2013, and the considered tweets have been posted between 2008 and 2013. By using a Twitter application 222dev.twitter.com, we have stored bio information and last 10 tweets of 11M Twitter users. In what follows, we will use the terms Twitter texts and tweets interchangeably. For geolocation analysis we used bio information on user profiles. Bio information for each user contains a current location field, where unstructured texts can be entered for city/country values, e.g., Boston, MA. We have used Google Geocode API 333developers.google.com/maps/documentation/geocoding/ to convert these location texts into longitude-latitude values. ## III Methodology and tools In order to understand what factors improve the chances of a text getting liked/retweeted by other users, we have considered the following dimensions: named entity recognition, lexical and sentiment analyses. In what follows, we will discuss our methodology for mining each of these dimensions. In given sentences, named entity recognition (NER) [11] locates atomic elements of predefined categories such as names of people, cities, locations, time mentions, and money values. For entity recognition, we used Apache OpenNLP 444http://opennlp.apache.org/ that allows training its classifier with seven different categories. We have used the categories “people, organizations, locations, date, time, percentage and money”. This tool has a precision of 0.8 and a recall of 0.74. Lexical errors are grammar and spelling mistakes/typos found in sentences. These are noun-verb agreement errors, missing words, extra words, wrong words, confusion of similar words, wrong word order, comma errors, and whitespace errors. For Facebook and Twitter texts, we used LanguageTool555http://www.languagetool.org to find lexical errors. These errors are predefined text patterns defined in an XML file. The tool reaches a precision of 93% on the English texts. Sentiment analysis aims at extracting the general sentiment from a given sentence, identifying whether it expresses a positive, negative or neutral emotion. For this purpose, we used the _Sentiment140 API_ , a Sentiment Analysis tool developed by Go et al. [3], which is a recent and widely used tool. The tool is based on a machine learning algorithm for classifying the sentiment of Twitter messages using distant supervision. The training data consists of Twitter messages with emoticons (i.e., pictorial representation of a facial expression), which are used as noisy labels. This type of training data is abundantly available and can be obtained through automated means. The underlying idea is to use emoticons to learn which words co-appear with emoticons, and use this information in machine learning algorithms, such as naive bayes and support vector machines. They show that the tool has an accuracy above 80% when trained with emoticon data [3]. ## IV Mining Dimensions In mining the considered dimensions, we have focused on English language texts. To this end, we have used the Ldig library in Python 666https://github.com/shuyo/ldig, which has a precision of 99.1% in detecting the English language. For Facebook and Twitter datasets, counts of English language texts and dimensional statistics are given in Table I. In columns named entity, error and sentiment, we give the percentages of English texts which contain at least one named entity, lexical error and sentiment tag, respectively. In the following sections, these values will be explained in detail. TABLE I: Dimensional statistics and counts of English texts. Entitity, error and sentiment values are given in percentages. \| Count \| Entity \| Error \| Sentiment ---\|---\|---\|---\|--- Twitter \| 10.6M \| 29.8% \| 81% \| 32% Facebook \| 1M \| 13% \| 69% \| 22% (a) Facebook entities. (b) Twitter entities. Figure 1: Entities and their percentage in Facebook and Twitter posts. ### IV-A Entity Recognition In Table I, percentage of English texts which contain at least one named entity are 29.8% and 13% for Twitter and Facebook texts, respectively. Figure 1 further details the composition of entities in these texts. Along seven different categories, we see that on Facebook, Persons are mentioned in texts 35.14% of the time, whereas this value is 22.66% for tweets. Date and time entities are mentioned twice as much on Twitter, but organizations have similar percentage values. We attribute the difference in date/time values to Twitter users’ high mobility (i.e., mobile phone usage) compared to other social network users [10]. Because of this mobility, tweets are more related to events happening in real time. For example, 1.5% of tweets contain the word today, whereas this value is only 0.02% for Facebook texts. In time entities, we found a similar pattern with tonight appearing in 0.06% of tweets and 0.0012% of Facebook texts, respectively. As seen on Figures 2 and 4, some organizations are frequently mentioned on both Twitter and Facebook, but users on Twitter are more likely to mention a broader variety of organizations, ranging from commercial brands to news agencies. In contrast, the two most frequently mentioned organizations on Facebook are related to basketball, and other organizations are not very frequently mentioned. Figure 2: Most frequently mentioned organizations on Facebook. (a) Facebook sentiment (b) Twitter sentiment Figure 3: Percentage of sentiments in Facebook and Twitter posts. Figure 4: Most frequently mentioned organizations on Twitter. ### IV-B Sentiment Analysis We have found positive sentiment to be more common than negative sentiment on both Facebook and Twitter. Percentages of posts with negative, positive and neutral sentiment tags are shown in Figure 3. Although Facebook friends are more likely to be real life acquaintances than Twitter followers, and Facebook posts are more likely to be directed to a private audience, only 22% of Facebook posts contain a sentiment. On Twitter, tweets are public, but 32% of tweets contain a sentiment, and 14% are positive sentiment tweets. Negative sentiments are expressed less often on both sites, with 8% and 6.85% on Facebook and Twitter, respectively. In Tables II and III we show some post examples with neutral, positive and negative sentiments. As posts on both sites are short texts, sentiments that are expressed in posts are mostly dependent on a limited number of words. For example, in Table III, the first negative tweet has its sentiment expressed by the word ‘lonely’. However, the sentiment tool performs well in labeling sentiments when texts include profanity or other swear words. The following tweets are labeled as: * • Negative: i dont care what others think any more so if you dont like me suck it up and keep your mouth shut or go f*** off. yes im in a bad mood nite * • Positive: here’s to me actually making some f****** money!!!!!!!!! whoo hoo!!!!!!!!!!!!!! * • Neutral: F* I, we all do. TABLE II: Examples of Facebook posts with sentiments. Positive \| I love this kid!! ---\|--- __\| I like Cinebistro too! Cant wait __\| to see the Artist __\| Thanks AJ and SW crew for bringing __\| both you and Thursday back to our shores!! Negative \| No and no!!! __\| I dont think it worth it, it doesn’t __\| give back as I look at it, its ugly __\| and… why do they call it celtic? __\| Of course it is. Their negative attacks on each __\| other alone are bringing some nasty skeletons __\| out of the closet that will hurt whichever __\| Republican becomes the nominee in November. Neutral \| I never heard Shaolin monks went to Chinese __\| university or teach some class in school, so they __\| teach Chinese culture more than kungfu __\| Im waiting on a special phone calll…cake up time __\| They have a show in new Orleans on march 29 TABLE III: Examples of tweets with sentiments. Positive \| watching a snuff movie, so funny, I love these things ---\|--- __\| hey Laura thanks for the invite. __\| I am eating grapes. __\| Good Morning and good Ester to everyone __\| from Turin (Italy)!!! Neutral \| Signing up for twitter __\| Catching up with online things… __\| Eating an apple Negative \| so lonely =( __\| lonely days………when will these lonely __\| days leave me? __\| is bored beyond belief We give the precision and recall values for sentiment detection in Table IV. Overall, precision values for both online social networks are high, whereas the lowest values are obtained for positive recall for Facebook posts (54%) and negative recall for tweets and posts (63% and 64%, respectively). These values have been obtained by asking to a group of three validators to assign a sentiment to messages, given a sample composed by 200 posts and 200 tweets. Finally, we have computed the average values of precision and recall obtained from each validator. TABLE IV: Precision and recall values for Twitter and Facebook sentiments. + and - signs refer to positive and negative sentiments, whereas P. and R. refer to precision and recall, respectively. \| +P. \| +R. \| -P. \| -R. \| Neut. P. \| Neut. R. ---\|---\|---\|---\|---\|---\|--- Twitter \| 88% \| 83% \| 79% \| 63% \| 87% \| 79% Facebook \| 81% \| 54% \| 82% \| 64% \| 75% \| 82% (a) Likes for negative and positive comments. (b) Retweets for negative and positive tweets. Figure 5: The impact of sentiments on like and retweet counts. ### IV-C Lexical Analysis In lexical analysis, we locate spelling or grammar mistakes within individual sentences of a text. To this end, we have replaced @usernames and #hashtags on Twitter with generic words before running the lexical analysis tool, so that site specific features (e.g., hashtag usage) are stripped and the tool can parse sentences without errors. For example, ’Ask @user about #ny’ becomes ’Ask William about New York’. With this transformation, Table I shows that 81% and 69% of Twitter and Facebook texts contain at least one lexical error. These errors are better analyzed in Figures 6 and 7. An interesting error that appears on Twitter but not on Facebook is the absence of a proper verb in a sentence. Overall, errors on both online social networks are very similar; word spelling mistakes and absence of uppercase letters in the beginning of sentences are major errors. Figure 6: Most common lexical errors on Facebook. Figure 7: Most common lexical errors on Twitter. ## V Dimension interplay An interesting part of mining conversational user interactions is finding how different dimensions affect each other or retweet/like counts individually. In this section, we will look at dimension correlations and analyze how these interplays affect likes/retweets of user texts on Facebook and Twitter. In particular, we will analyze the interplay between: i) retweet/like counts and sentiment, ii) sentiment and lexical errors, and iii) lexical errors and likes/retweets. Figure 5 shows the impact of sentiments on retweet and like counts. In the figure, we see that negative texts receive more likes on average on Facebook, whereas on Twitter neutral tweets are retweeted more. Despite similar retweet counts, positive tweets are retweeted more often than the negative tweets. We explain the high like count of negative Facebook posts by expressions of user compassion. Facebook posts, such as “Lexi better not die tonight. :(” (96 likes), receive higher like counts. On Twitter neutral tweets, such as news, are retweeted by many users. Figure 8: Average lexical errors by sentiments in Facebook and Twitter posts. Mining both sentiments and lexical errors allows us to see how user emotions lead to lexical errors due to stress, anger or sadness. Figure 8 shows average lexical errors for positive and negative sentiment texts, for both Facebook and Twitter. Positive tweets have been found to contain more than 2 errors, whereas negative Facebook comments contain 1.8 errors in average. In Tables V and VI we show some text examples with neutral, positive and negative sentiments containing errors. TABLE V: Examples of Facebook posts with sentiments containing errors. Positive \| strawberries would be the best.. oh my how good… ---\|--- __\| and make a chesecake from this… OH yes… Negative \| Cant believe I pay tuition money for this stuff…;-( Neutral \| Dear NBC: Please make your videos __\| viewable in other countries outside America. __\| Sincerly, - The rest of the world TABLE VI: Examples of tweets with sentiments containing errors. Positive \| Oh twitter! Me loves me loves! ---\|--- __\| It’ll lead me to my precioussssss! Negative \| write the wrong words for the wrong thing __\| and make it worse and worse. wahahha Neutral \| looking at a little tiger cat __\| who says hes in my server doing maintenance Figure 9: Average like counts by number of lexical errors. Figure 10: Average retweet counts by number of lexical errors. Lexical errors can also affect the number of times a text is retweeted/liked by other users. In Figures 9 and 10 we show the average value of likes/retweets a text receives for different values of lexical errors. Although there are as many as 15 lexical errors in both figures, number of posts with more than 5 errors are very low on both online social networks. An example with 15 errors is the tweet omg omg omg omg omg omg omg 7 days too goo and i shall be in greece :) also my heads f*** stupid boy y does he always do this 2 me :( x Especially on Figure 9 we see that an increasing number of lexical errors leads to lower like counts on Facebook. Figure 10 shows a similars pattern for Twitter posts, but in this case retweet counts decrease with increasing numbers of lexical errors after the first error. ## VI Influence of Geolocation on Conversations Although some research work [5, 12, 13] have worked on topical conversations on Twitter, a comparative analysis of locations of social network users across multiple online social networks has not been studied yet. In order to analyze the impact of geolocations, we converted textual current locations of Twitter users into geographical longitude-latitude values. From these values, Figure 11 shows the current location of Twitter users on a world map. On this map, red points and green points correspond to users who post a tweet and retweet a posted tweet, respectively. Edges between these two types of users connect them on the map. From the map, we see that a large percentage of retweeted tweets come from the east coast of USA and north Europe. Similarly, most edges are created between these two parts of the world. The absence of China and most of the Russian territories are prominent features on the map. The high concentration of Canadian cities around the northern border of USA is also visible from the map. Another presentation of this location data allows us to measure the distance between two users in miles. A zero distance shows that a user $u$ who retweeted a tweet from user $x$ lives in the city where user $x$ currently lives. By plotting the number of such user pairs for each distinct distance value 777We have put distances into 100 mile buckets. yields Figure 12. In the figure, we see that a big percentage of user pairs have zero distance between them. Numbers of user pairs are shown to decrease with increasing distances. An early anomaly in this trend is the low number of user pairs around 2500 miles. This distance corresponds to the separation between USA and Europe. Figure 11: [Color online] Locations of Twitter users. Edges show retweet behavior among Twitter users. Figure 12: In miles, distances between user pairs who retweet each other’s tweets. Unlike Twitter, Facebook privacy settings are very strict, and users are less willing to share their location information. Due to this shortcoming, we could not repeat the distance experiment on the Facebook dataset. For a similar distance notion on Facebook dataset, we have used the locale 888Locale is the user chosen interface language for Facebook.com, such as EN_US (English in USA), EN_GB (English in Great Britain) field to analyze user proximity. We explain this choice with the empirical observation that users from a country mainly use the official language of the country in the Facebook interface. The main exception to this observation is that English is also widely used by other nationalities. Figure 13 shows locale pairs for Facebook users. In the figure, a locale value $l$ is connected by an edge to another locale $m$ if users from $l$ constitute 30% or more of users who have liked Facebook comments of users from the locale $m$. Although EN_US is connected to a big portion of other locales, most locales have self loops, or they are connected to a small number of other locales. Another relevant feature of the figure is the community of Latin languages connected together, such as it_IT, es_ES, fr_FR, etc.; users of Latin languages interact more often. Figure 13: Like interactions among Facebook users of different locales. Self loops show that most of users from a locale likes comments from users of the same locale. In the previous sections, we have shown how different dimensions of user generated texts affect how many times a particular tweet/comment will be liked or commented. Regardless of these numbers, our geolocation experiments show that users who like/comment a text are more likely to live closer to the owner of the text. ## VII Conclusion In this paper, we have analyzed conversational user interactions on two popular online social networks, Facebook and Twitter. We have found common user behavior in interacting with posts of similar sentiments (i.e., positive or negative). Furthermore, texts on both web sites have been found to exhibit similar lexical errors, but these errors result in differing behaviors in user interactions. Our conversational analysis has been complemented with a location analysis of users on both online social networks. This approach has shown that geolocations of users can greatly affect which social network posts will be liked and retweeted and will have better visibility. From an information propagation point of view, our results can help in choosing seed nodes to disseminate news or advertisements effectively. Furthermore, propagation of any data can greatly benefit from an increased location awareness, because users tend to interact with others who are from the same locations. ## References * [1] Apoorv Agarwal, Boyi Xie, Ilia Vovsha, Owen Rambow, and Rebecca Passonneau. Sentiment analysis of twitter data. In Proceedings of the Workshop on Languages in Social Media, pages 30–38. Association for Computational Linguistics, 2011. * [2] Nicole B Ellison et al. Social network sites: Definition, history, and scholarship. Journal of Computer-Mediated Communication, 13(1):210–230, 2007\. * [3] Alec Go, Richa Bhayani, and Lei Huang. Twitter sentiment classification using distant supervision. CS224N Project Report, Stanford, pages 1–12, 2009. * [4] C Honey and Susan C Herring. Beyond microblogging: Conversation and collaboration via twitter. In System Sciences, 2009. HICSS’09. 42nd Hawaii International Conference on, pages 1–10. IEEE, 2009. * [5] Akshay Java, Xiaodan Song, Tim Finin, and Belle Tseng. Why we twitter: understanding microblogging usage and communities. In Proceedings of the 9th WebKDD and 1st SNA-KDD 2007 workshop on Web mining and social network analysis, pages 56–65. ACM, 2007. * [6] Efthymios Kouloumpis, Theresa Wilson, and Johanna Moore. Twitter sentiment analysis: The good the bad and the omg. In Proceedings of the Fifth International AAAI Conference on Weblogs and Social Media, pages 538–541, 2011. * [7] Haewoon Kwak, Changhyun Lee, Hosung Park, and Sue Moon. What is twitter, a social network or a news media? In Proceedings of the 19th international conference on World wide web, pages 591–600. ACM, 2010. * [8] Sebastian Labitzke, Florian Werling, Jens Mittag, and Hannes Hartenstein. Do online social network friends still threaten my privacy? In Proceedings of the third ACM conference on Data and application security and privacy, pages 13–24. ACM, 2013. * [9] Kalev Leetaru, Shaowen Wang, Guofeng Cao, Anand Padmanabhan, and Eric Shook. Mapping the global twitter heartbeat: The geography of twitter. First Monday, 18(5), 2013. * [10] Amanda Lenhart and Susannah Fox. Twitter and status updating. Pew Internet & American Life Project Washington DC, 2009. * [11] David Nadeau and Satoshi Sekine. A survey of named entity recognition and classification. Lingvisticae Investigationes, 30(1):3–26, 2007. * [12] Meenakshi Nagarajan, Hemant Purohit, and Amit Sheth. A qualitative examination of topical tweet and retweet practices. In International AAAI Conference on Weblogs and Social Media. AAAI, 2010. * [13] Bongwon Suh, Lichan Hong, Peter Pirolli, and Ed H Chi. Want to be retweeted? large scale analytics on factors impacting retweet in twitter network. In Social Computing (SocialCom), 2010 IEEE Second International Conference on, pages 177–184. IEEE, 2010. * [14] Yuri Takhteyev, Anatoliy Gruzd, and Barry Wellman. Geography of twitter networks. Social Networks, 34(1):73–81, 2012. * [15] Shaozhi Ye and S Felix Wu. Measuring message propagation and social influence on twitter. com. In Social informatics, pages 216–231. Springer, 2010.	arxiv-papers	2013-12-24T04:16:11	2024-09-04T02:49:55.886160	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "William Lucia and Cuneyt Gurcan Akcora and Elena Ferrari", "submitter": "Cuneyt Gurcan Akcora", "url": "https://arxiv.org/abs/1312.6756" }
1312.6785	# Spectral properties of Compton inverse radiation: Application of Compton beams Eugene Bulyak1 and Junji Urakawa2 1 NSC KIPT, Kharkov, Ukraine 2 KEK, Tsukuba, Ibaraki, Japan [email protected] ###### Abstract Compton inverse radiation emitted due to backscattering of laser pulses off the relativistic electrons possesses high spectral density and high energy of photons – in hard x–ray up to gamma–ray energies – because of short wavelength of laser radiation as compared with the classical electromagnetic devices such as undulators. In this report, the possibility of such radiation to monochromatization by means of collimation is studied. Two approaches have been considered for the description of the spectral–angular density of Compton radiation based on the classical field theory and on the quantum electrodynamics. As is shown, both descriptions produce similar total spectra. On the contrary, angular distribution of the radiation is different: the classical approach predicted a more narrow radiation cone. Also proposed and estimated is a method of the ‘electronic’ monochromatization based on the electronic subtraction of the two images produced by the electron beams with slightly different energies. A ‘proof–of–principle’ experiment of this method is proposed for the LUXC facility of KEK (Japan). ## 1 Introduction Sources of Compton radiation, in which photons of intense laser pulse scattered off from relativistic electrons, are able to produce bright x–ray beams with narrow bandwidth. The process of Compton scattering can be treated as two–particle elastic scattering: one of the particle is represented by the electron, another one by the photon. Since the laser photon has negligibly small energy as compared with electron’s, the recoil of electron is negligibly small – the photon is scattered off within a narrow cone along electron’s trajectory. Such sources have a substantial potential for applications in different areas of medicine, biology, physics, etc. This potential is emerged not only from its brightness, but tunability and ‘quasimonochromaticity’ of the spectrum. The report highlights spectral properties of Compton sources and presents a potential application of these sources for x–ray angiography, which possesses a substantial advantage over the conventional methods. ## 2 Spectrum of collimated Compton inverse radiation The spectrum of inverse Compton radiation is determined by two relations, (i) kinematic dependence of the energy of a scattered off quantum, $E_{\mathrm{x}}$ upon the crossing angle $\phi$ between the laser photon and the electron, energy of both the electron and the photon, and the scattering angle $\psi$, and (ii) the differential cross section – dependence of probability of scattering the photon at angle $\psi$. Within the small–angle approximation, $\psi\ll 1$ (the crossing angle $\phi=0$ corresponds to the head–on collision), neglected the electron recoil, these relations read ([1]): $E_{\mathrm{x}}=\frac{2\gamma^{2}(1+\cos\phi)E_{\mathrm{las}}}{1+\gamma^{2}\psi^{2}}\;;\qquad\mathrm{d}\sigma=8\pi r^{2}_{0}\frac{\psi\gamma^{2}\left(1+\gamma^{4}\psi^{4}\right)}{\left(1+\gamma^{2}\psi^{2}\right)^{4}}\mathrm{d}\psi\;,$ where $\gamma=E_{\mathrm{e}}/m_{\mathrm{e}}c^{2}$ is the Lorentz factor of the electron, $r_{0}$ is the classical electron radius. Convolution of the cross section with distribution functions of both the electron bunch and the photon pulse taking into account the kinematic relation will produce a real collimated spectrum of the radiation. As it follows from the kinematics, the width of spectrum is determined by the collimation angles $\psi_{i}\leq\psi\leq\psi_{f}$, as is depicted in Fig.2. The spectral–angular density of the collimated Compton radiation for an ideal case (monoenergetic electrons and laser photons with parallel trajectories) is presented in Fig.2. Figure 1: Scheme of the Compton radiation collimation. Figure 2: Spectral-angular density of the Compton radiation. For the vast majority of practical sources the inner collimation angle is zero, $\psi_{i}=0$. We will consider below this particular case of an iris collimator. In general, the spectrum is dependent on the following factors: the collimator opening angle, the energy spread of electrons and photons and the angular spread of their trajectories at the interaction point (IP). In practical cases, the angular spread of photon ‘trajectories’ produces least impact on the spectrum, as it can be seen from the kinematics: Its effect is proportional to the spread squared because IP is usually set up at the laser waist, while the electron spread impact is $\gamma^{2}$ times larger. Also the spread of electron energy within the bunch usually much higher than that of photons. Therefore below we will consider effects of the electron bunch phase volume – spreads of energy and trajectories – upon the collimated spectrum of scattered off laser quanta. ### 2.1 Spread of electrons energy The energy spread dilutes the spectral–angular density curve in Fig.2 along the energy axis. Since the energy spread in bunches of Compton sources is usually small – from a few percents down to fraction of a percent, – the corresponding partial energy spread of x–rays is equal to doubled the electrons’: For the normal (Gaussian) distribution of electrons’ energy in the bunch, the ‘pin-hole’ collimated x–ray beam has the doubled reduced dispersion: $\sigma_{x}/E_{x}\approx 2\sigma_{e}/E_{e}$. (Detailed study on the spectrum for head-on collision is presented in [2].) Analytical collimated spectra for finite energy spread of electrons with parallel trajectories are presented in Fig.4. Figure 3: X-ray spectra for the electron energy spreads $\sigma_{e}\gamma=(0,0.01,0.02,0.03)$ into collimating range $\psi\gamma=(0,0.1)$. Figure 4: X-ray energy spectra for the electron trajectories spreads $\sigma_{e}\gamma=(0.01,0.1,0.5,1.0)$ into collimating range $\psi\gamma=(0,0.1)$. ### 2.2 Angular spread of electrons trajectories The angular spread of electrons’ trajectories induces much more widening into the spectrum since the radiation is emitted within narrow cone along the trajectory. (It should be noted that a quantum emitted at the angle $1/\gamma$ to the electron trajectory has a half of the energy of that emitted along the trajectory.) Also tight focusing of the bunches at the interaction point aimed to gain the yield of x rays, increases the angular spread. A collimated spectrum with account for the angular spread has asymmetric and more complicated shape than that due to the energy spread. It is specified with steep high–energy cutoff (stemmed from the energy conservation law) and long low–energy tail, see Fig.4. Examples of simulated spectra for different collimation opening angles is presented in Fig.5 (the parameters for simulation were taken similar to those of LUCX facility, see [3]). Figure 5: Spectra for collimation $\psi\gamma=0.2,0.4,0.6,0.8,1.0$ (from the narrowest to widest profiles). Angular horizontal and vertical spreads in the bunch are $\gamma\sigma_{x,y}=0.08,0.32$. From the study on the collimated x-ray spectra generated by Compton sources the points are followed: * • The spectrum width is limited by the high energy cutoff, no photons with higher energy. * • The spectrum has the maximum close to the high-energy cutoff. * • High-energy cutoff of the spectrum is steep. * • The collimator opening angle should not be more narrow than the (approximately doubled) angular spread of electron trajectories at IP. More narrow collimation will reduce the spectral maximum but not the width. ## 3 Application of Compton sources for angiography Angiography, a medical x–ray based imaging technique to visualize the inside of blood vessels and organs of the body, uses a radio-opaque contrast agent injected into the blood vessel. The difference between x–ray images without and with the contrast agent in the blood produces a picture of the blood flow. Iodine–based radiocontrast agents are most commonly used in angiography with maximum contrast energy of x–rays just above the K–edge of iodine, 33.17 keV. Hence, the x–rays within a band of $\sim$30–35 keV most effectively produce the images. Other energies are redundant and even detrimental since they cause additional radiation load. The Compton x–ray sources with the tunable energy peak possess certain advantages over the conventional bremsstrahlung sources. In Fig.6, there an ideal Compton spectrum and a bremsstrahlung one are presented that produce equal fluxes within 30–35 keV energy range. Figure 6: Bremstrahlung (70 keV electrons and tungsten radiator) and Compton spectra. the vertical lines limit a useful x-ray energy band, the gray areas beyond show redundant x–ray radiation load from the bremsstrahlung spectrum. As it can be seen from the picture, the conventional source produces much more background x–ray quanta (the gray areas beyond 30–35 keV range). ### 3.1 Simulation of Compton angiography For verification of Compton sources applicability for angiography, a digital model has been created. The models simulates the image of the tissue composed from the skeletal muscle, the bone cortical, blood and iodine components (x–ray attenuation coefficients were taken from NIST [5]). Distribution of the energy of x–ray quanta is simulated with Monte Carlo method according to the collimate ideal Compton spectrum. Distribution of the impinging quanta over the tissue face is uniform, random. A run of simulation on the model reveals applicability of Compton x–ray source for angiography as is presented in Fig.7 (left). The model input data was: x–ray energy ranges 23.3–35 keV; number of quanta in the range $2\times 10^{7}$ (total number over the spectrum $2\times 10^{7}$); the sensor mesh $100\times 100$ pixels (1 mm2 pixels on $10\times 10\,\mbox{cm}^{2}$ tissue); the surface densities of tissue’s substances are as follows (in g/cm2): muscle 5, bone 0.5, blood 0.5, iodine 0.0125. The muscle component is uniform density over the tissue, the bone–like substance is added at the bottom ($2<y<4$ cm), the blood one at the top of the tissue ($7<y<8$ cm), iodine is placed in the left half of the blood tape ($x<5$ cm). Figure 7: X–ray image with $E_{x}^{max}=35\,\mbox{keV}$ (left) and a subtracted ‘30 keV’ minus ‘35 keV’ image (right). To compare the left half of ‘35 keV’ image in Fig.7 (blood with the contrast agent) with the right half (no contrast agent), one can detect location of the iodine. Total flux density for the simulated case is $4\times 10^{5}$ quanta per cm2. ### 3.2 Advanced method of Compton angiography Specific properties of the Compton x–ray radiation spectrum – steep high- energy cutoff with the maximum near it, and strong dependence of the maximal energy upon the energy of electrons, – enable us to propose an advanced method of Compton angiography. The method consists in following. Dissimilar to the conventional angiography where location of the contrast agent is revealed as a result of subtracting the image without the agent from that with it, the both images are taken with presence of the contrast agent but one of them at the maximum energy of the spectrum tuned just below the K–edge of the agent, another – above it. Difference between these images will display the contrast agent location. Simulations of the method have validated it. In Fig.7 (right panel) a subtracted image is presented, the image of the described above tissue at the maximum energy of Compton x–rays of 35 keV (left panel) was subtracted from the image with the maximum energy 30 keV. Since opaqueness of the agent increases with increasing the energy above K–edge while the other components become more transparent, the location of agent in the difference image is lighter than the others (see Fig.7, right panel). Thus, the advanced method is capable to reveal the location of the contrast agent much faster than the conventional Compton angiography method with about the same flux of x–ray quanta. Speed up of the imaging is due to the fact that injection of the contrast agent into the blood vessel will take much more time than switching of the accelerator energy (we believe it will take a fraction of second). ### 3.3 ‘Proof–of–principle’ experiment proposal for LUCX The ‘proof–of–principle’ experiment must verify the basic suggestion for Compton x–ray source: opaqueness of a contrast agent increases with increasing the maximal energy of spectra above the K-edge of contrast agent. Also ability of Compton sources to gather sufficient statistics revealing the location of the contrast agent should be proved. Such an experiment is proposed to conduct at LUCX facility [3, 4]. Since the energy of Compton x–ray photons in this accelerator with YAG laser (1 eV) can not overlap the K–line of iodine, we propose to use bromine as a contrast agent with K-edge at 13.4737 keV, see [5]. We have simulated a ‘proof–of–principle’ experiment for LUCX. Results of the simulation is depicted in Fig.8. The sample of $20\times 20$ cells was irradiated by $2\times 10^{6}$ quanta of each energy, $E_{x}^{(max)}=13,16\,\mbox{keV}$ (the images ‘A’ and ‘B’, respectively). The Compton spectrum was collimated to $1/\gamma$ opening angle. The sample thickness (muscle skeletal) was 0.5 g/cm2 with additional 0.15 g/cm2 in top left corner, bromine thickness was 10 mg/cm2 (bottom left corner). Figure 8: ‘13 keV’ image (A, left), ‘16 keV’ image (B, right) and the subtracted density (A-B, centre). As is follows from the images, one cannot tell from single run (A and B images) which of the shadows induced by bromine or by an additional thickness. The bromine shadow is thinner when the spectrum does not overlap the K–peak, ‘A’ image. The subtracted image, ‘A-B’ clearly indicates which of shadows belong to the additional thickness or to the bromine – the contrast agent is more transparent for lower energy inverse to the regular behavior. ## 4 Summary and conclusion X–ray radiation generated by the Compton sources has the spectrum with the steep high–energy cutoff, which is independent on the collimation opening angle. The specific shape of the spectrum with tunable maximum is able to substantially reduce radiation load for the angiography procedure. We propose a novel angiography procedure consisting in deriving a subtracted image from the two images having been made with different maximum energy of the Compton spectrum, one with the energy below the K–edge of a contrast agent, another – with the maximum energy overlapped it. The carried out simulations proved the suggestion of advantage the Compton subtracted scheme for angiography. Also we proposed a ‘proof–of–principle’ experiment that can be conducted at LUCX facility of KEK employing bromine as a radiocontrast agent. Authors wishing to acknowledge Prof. A. Dovbnya, Drs. P. Gladkikh and V. Skomorokhov for their assistance and helpful discussions. Work is supported by the Photon and Quantum Basic Research Coordinated Development Program by the Ministry of Education, Culture, Sports, Science and Technology ‘Fundamental Technology Development for High Brightness X-ray Source and the Imaging by Compact Accelerator.’ ## References ## References * [1] Bulyak E and Skomorokhov V 2004 Proc. Eur. Part. Accel. Conf. 2004 report THPKF063 * [2] Sun C, Li J, Rusev G, Tonchev A P and Wu Y K 2009 Phys. Rev. ST-AB 12 062801 * [3] Fukuda M, Araki S, Aryshev A, Honda Y, Terunuma N, Urakawa J, Sakaue K and Washio M 2013 Proc. NAPAC’13 report TUPMA01 * [4] Sakaue K, Washio M, Fukuda M, Honda Y, Terunuma N, Urakawa J 2013 Proc. IPAC 2013 report WEPWA017 * [5] http://www.nist.gov/pml/data/xraycoef/	arxiv-papers	2013-12-24T09:44:38	2024-09-04T02:49:55.895185	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Eugene Bulyak and Junji Urakawa", "submitter": "Eugene Bulyak", "url": "https://arxiv.org/abs/1312.6785" }
1312.6850	# A note on 2-local representations of C∗-algebras Antonio M. Peralta Departamento de Análisis Matemático, Universidad de Granada, Facultad de Ciencias 18071, Granada, Spain [email protected] ###### Abstract. We survey the results on linear local and 2-local homomorphisms and zero products preserving operators between C∗-algebras, and we incorporate some new precise observations and results to prove that every bounded linear 2-local homomorphism between C∗-algebras is a homomorphism. Consequently, every linear 2-local ∗-homomorphism between C∗-algebras is a ∗-homomorphism. ###### Key words and phrases: Local homomorphism, local ∗-homomorphism; 2-local homomorphism, 2-local ∗-homomorphism; local representation; 2-local representation ###### 2011 Mathematics Subject Classification: Primary 46L05; 46L40 Author partially supported by the Spanish Ministry of Science and Innovation, D.G.I. project no. MTM2011-23843, and Junta de Andalucía grant FQM375. ## 1\. Introduction Most of the authors agree in acknowledging the papers of R.V. Kadison [24] and D.R. Larson and A.R. Sourour [30] as the pioneering contributions to the theory of local derivations and local automorphisms on Banach algebras, respectively. We recall that a linear mapping $T$ from a Banach algebra $A$ into a Banach algebra $B$ is said to be a _local homomorphism_ if for every $a$ in $A$ there exists a homomorphism $\Phi_{a}:A\to B$, depending on $a$, satisfying $T(a)=\Phi_{a}(a)$. When $A$ and $B$ are C∗-algebras, and for each $a$ in $A$ there exists a ∗-homomorphism $\Phi_{a}:A\to B$, depending on $a$, satisfying $T(a)=\Phi_{a}(a)$, the mapping $T$ is called a _local ∗-homomorphism_. _Local automorphisms_ , _local ∗-automorphisms_, and _local derivations_ are similarly defined. R.V. Kadison proved in [24] that every bounded local derivation on a von Neumann algebra (i.e. a C∗-algebra which is also a dual Banach space) is a derivation. After Kadison’s contribution, a multitude of researchers explored the same problem for general C∗-algebras (see, for example, [1, 4, 21, 31, 44], and [50]). The definitive answer is due to B.E. Johnson [23], who proved that every local derivation from a C∗-algebra $A$ into a Banach $A$-bimodule is a derivation, even if not assumed a priori to be so. Much more recently, local triple derivations on C∗-algebras and JB∗-triples have been studied in [33, 8, 9] and [14]. The knowledge about local homomorphisms and local ∗-homomorphisms between C∗-algebras is less conclusive. D.D. Larson and A.R. Sourour proved in [30] that for an infinite dimensional Banach space $X$, every surjective local automorphism $T$ on the Banach algebra $B(X),$ of all bounded linear operators on $X$, is an automorphism. When $X$ is a separable Hilbert space M. Brešar and P. Šemrl showed that the hypothesis concerning the surjectivity is superfluous (cf. [5]). A related result was established by C. Batty and L. Molnar in [3], where they proved that for a properly infinite von Neumann algebra $\mathcal{M}$, the group, Aut$(\mathcal{M})$, of all ∗-automorphisms on $\mathcal{M}$ is reflexive, i.e. if a linear mappings $T:\mathcal{M}\to\mathcal{M}$ satisfies that for every $a\in\mathcal{M}$, $T(a)$ belongs to the strong-closure of the set $\\{\Phi(a):\phi\in\hbox{Aut}(\mathcal{M})\\}$, then $T$ lies in Aut$(\mathcal{M})$. Furthermore, for each Hilbert space $H$ of dimension $n\geq 3$, the group Aut$(B(H))$ is reflexive. In [41, §2], F. Pop provides an example of a local homomorphism from $M_{2}(\mathbb{C})$ into $M_{4}(\mathbb{C})$ which fails to be multiplicative (cf. Example 3.13). In 1997, P. Šemrl [43] introduces 2-local derivations and 2-local automorphisms in the following sense: Let $A$ be a Banach algebra, a mapping $T:A\to A$ is a _2-local automorphism_ if for every $a,b\in A$ there is an automorphism $T_{a,b}:A\to A,$ depending on $a$ and $b$, such that $T_{a,b}(a)=T(a)$ and $T_{a,b}(b)=T(b)$ (no linearity, surjectivity or continuity of $T$ is assumed). In the just quoted paper, Šemrl proves that for every infinite-dimensional separable Hilbert space $H$, every 2-local automorphism $T:B(H)\to B(H)$ is an automorphism. In [28], S.O. Kim and J.S. Kim show that every surjective 2-local ∗-automorphism on a prime C∗-algebra or on a C∗-algebra such that the identity element is properly infinite is a ∗-automorphism (see [12, 15, 34, 35, 36, 37] and [27] for other related results). A closer look at Šemrl’s paper [43] shows that the connections with the Gleason-Kahane-Żelazko theorem (cf. [19, 25]) didn’t go unnoticed to him. Borrowing a paragraph from [43, Introduction], we notice that Gleason-Kahane- Żelazko theorem can be reformulated in the following sense: every unital linear local homomorphism from a unital complex Banach algebra $A$ into $\mathbb{C}$ is multiplicative (cf. [2]). S. Kowalski and Z. Slodkowski [29] established a 2-local version of the Gleason-Kahane-Żelazko theorem, showing that every 2-local homomorphism $T:A\to\mathbb{C}$ is linear and multiplicative. In order to keep coherence with the terminology employed by P. Šemrl, a mapping $T$ between C∗-algebras $A$ and $B$ is called a _2-local homomorphism_ (respectively, _2-local ∗-homomorphism_) if for every $a,b\in A$ there exists a bounded homomorphism (respectively, a _∗ -homomorphism_) $\Phi_{a,b}:A\to B$, depending on $a$ and $b$, such that $\Phi_{a,b}(a)=T(a)$ and $\Phi_{a,b}(b)=T(b)$. _2-local Jordan homomorphisms_ , _2-local Jordan ∗-homomorphisms_ and _2-local automorphisms_ are defined in a similar fashion. We recall that a linear mapping $\Phi:A\to B$ is said to be a Jordan homomorphism whenever $\Phi(a^{2})=\Phi(a)^{2}$ (equivalently, $\Phi$ preserves the Jordan products of the form $a\circ b:=\frac{1}{2}(ab+ba)$). In 2004, new studies on 2-local linear maps between C∗-algebras were developed by D. Hadwin and J. Li [21] and F. Pop [41], though these papers seem to be mutually disconnected at the publication moment. Hadwin and Li prove that every bounded linear and unital 2-local homomorphism (respectively, 2-local ∗-homomorphism) from a unital C∗-algebra of real rank zero into itself is a homomorphism (respectively, a ∗-homomorphism) [21, Theorem 3.6]. As a consequence, every linear and surjective 2-local ∗-automorphism on a unital C∗-algebra of real rank zero is a ∗-automorphism (cf. [21, Theorem 3.7]). The main contribution of F. Pop in [41] establishes that every bounded linear 2-local homomorphism (respectively, 2-local ∗-homomorphism) from a von Neumann algebra into another C∗-algebra is a homomorphism (respectively, a ∗-homomorphism) [41, Corollary 3.6]. In 2006, J.-H. Liu and N.-C. Wong made their own contribution to the study of not necessarily continuous nor linear 2-local homomorphisms between standard operator algebras on locally convex spaces [32]. We recall that a standard operator algebra $\mathcal{A}$ on a locally convex space $X$, is a subalgebra of $B(X)$ containing the algebra $\mathcal{F}(X)$ of all continuous finite rank operators on $X$. Liu and Wong prove, without assuming linearity, surjectivity or continuity, that every 2-local automorphism of $\mathcal{F}(X)$ is an algebra homomorphism. In case $X$ is a Frechet space with a Schauder basis and $\mathcal{A}$ contains all locally compact operators, it can be concluded that every 2-local automorphism on $\mathcal{A}$ is an automorphism. Furthermore, a 2-local automorphism $\Theta$ of a standard operator algebra $\mathcal{A}$ on a locally convex space $X$ is an algebra homomorphism provided that the range of $\Theta$ contains $\mathcal{F}(X)$, or $\Theta$ is continuous in the weak operator topology (cf. [32]). In the just quoted paper, the authors study the question of when a 2-local automorphism of a C∗-algebra is an automorphism, showing that every linear 2-local automorphism $T$ of a C∗-algebra whose range is a C∗-algebra is an algebra homomorphism. It seems natural to ask whether the above results of Hadwin-Li and Pop remain true for general C∗-algebras. This paper, which has an almost expository aim, combined with new research results, we give a positive answer to this question, showing that every bounded linear 2-local homomorphism between C∗-algebras is a homomorphism, and consequently, every linear 2-local ∗-homomorphism between C∗-algebras is a ∗-homomorphism (Theorem 3.9). In particular, according to the terminology in [41], every 2-local (∗-)representation of a C∗-algebra is a (∗-)representation (Corollary 3.10). In Example 3.14 we present a linear 2-local ∗-automorphism on $M_{2}(\mathbb{C})$ which is not multiplicative. We survey the connections between this problem and the theory of linear zero products preservers developed by J. Alaminos, M. Bresar, J. Extremera and A. Villena in [1]. The novelties in this paper include an independent proof which is not based on the result in [1] together with the precise observations to provide a definitive answer to the whole line of problems on linear preservers on C∗-algebras presented above. Here we make use of techniques developed in the setting of JB∗-triples, the use of compact-Gδ projections in the bidual of a C∗-algebra, and the study of the connections between (linear) 2-local homomorphisms and zero product preserving mappings. Although the results presented here could have been obtained by combining some of the results that we shall review later, the equivalence between bounded linear 2-local homomorphisms and bounded homomorphisms between C∗-algebras has not been explicitly stated before. ## 2\. Techniques of Jordan algebras and JB∗-triples Every C∗-algebra $A$ admits a Jordan product defined by $a\circ b=\frac{1}{2}(ab+ba)$. The Jordan product is commutative but not necessarily associative. Let $B$ be another C∗-algebra. A linear map $T:A\to B$ is said to be a _Jordan homomorphism_ whenever it preserves Jordan products, or equivalently, when $T(a^{2})=T(a)^{2}$, for every $a$. A _Jordan ∗-homomorphism_ is a Jordan homomorphism which maps self-adjoint elements into self-adjoint elements. For each element $a$ in $A$, the symbol $U_{a}$ will denote the linear map $U_{a}:A\to A$ defined by $U_{a}(x):=axa$. Since, for every $a,x\in A$ we have $U_{a}(x)=2(a\circ x)\circ a-a^{2}\circ x$, every Jordan homomorphism $T:A\to B$ satisfies $T(U_{a}(x))=U_{T(a)}(T(x))$. Let $T:A\to B$ be a Jordan homomorphism between C∗-algebras. Since $A^{}$ and $B^{}$ are von Neumann algebras, $T^{}:A^{}\to B^{}$ is weak∗ continuous, and the product of every von Neumann algebra is separately weak∗ continuous (cf. [42, Theorem 1.7.8]), we deduce, via Goldstine’s theorem, that $T^{}:A^{}\to B^{}$ is a Jordan homomorphism. Since the involution of a von Neumann algebra is weak∗ continuous (cf. [42, Theorem 1.7.8]), $T^{*}$ is a Jordan ∗-homomorphism whenever $T$ is a Jordan ∗-homomorphism. There is some benefit in considering a C∗-algebra as an element in the wider class of JB∗-triples. A _JB ∗-triple_ is a complex Banach space $E$ equipped with a triple product $\\{\cdot,\cdot,\cdot\\}:E\times E\times E\rightarrow E$ which is linear and symmetric in the outer variables, conjugate linear in the middle variable and satisfies the following conditions: 1. $(a)$ (Jordan identity) $\\{a,b,\\{x,y,z\\}\\}=\\{\\{a,b,x\\},y,z\\}-\\{x,\\{b,a,y\\},z\\}+\\{x,y,\\{a,b,z\\}\\},$ for $a,b,x,y,z$ in $E$; 2. $(b)$ For each $a\in E$, the mapping $L(a,a):E\rightarrow E,$ $x\mapsto\\{a,a,x\\}$ is an hermitian (linear) operator with non-negative spectrum; 3. $(c)$ $\\|\\{x,x,x\\}\\|=\\|x\\|^{3}$ for all $x\in E$. Every C∗-algebra is a JB∗-triple via the triple product given by $\left\\{x,y,z\right\\}=\frac{1}{2}(xy^{}z+zy^{}x).$ It was shown by Poincar in the early 1900s, that the Riemann mapping theorem fails when the complex plane is replaced by a complex Banach space of higher dimension. Although, a complete holomorphic classification of bounded simply connected domains in arbitrary complex Banach spaces is unattainable, bounded symmetric domains in finite dimensions were studied and classified by E. Cartan [10]. In the setting of complex Banach spaces of arbitrary dimension, W. Kaup proved, in [26], that a complex Banach space is a JB∗-triple if, and only if, its open unit ball is a bounded symmetric domain, and every bounded symmetric domain in a complex Banach space is biholomorphically equivalent to the open unit ball of a JB∗-triple; showing that the category of all bounded symmetric domains with base point is equivalent to the category of JB∗-triples. We refer to monographs [46] and [11] for the basic theory of JB∗-triples and JB∗-algebras. Spectral resolutions of non-normal elements in a C∗-algebra is a completely hopeless goal. However, in every JB∗-triple $E$, the JB∗-subtriple $E_{a}$ generated by a single element $a\in E$ is (isometrically) JB∗-isomorphic to $C_{0}(L)$ for some locally compact Hausdorff space $L\subseteq(0,\\|a\\|]$, such that $L\cup\\{0\\}$ is compact. It is also known that there exists a JB∗-triple isomorphism $\Psi_{a}:E_{a}\to C_{0}(L),$ satisfying $\Psi(a)(t)=t$ $(t\in L)$ (compare [26, Lemma 1.14]). In particular, there exists a unique element $a^{[\frac{1}{3}]}\in E_{a}$ such that $\\{a^{[\frac{1}{3}]},a^{[\frac{1}{3}]},a^{[\frac{1}{3}]}\\}=a.$ When $E=A$ is a C∗-algebra, $a^{[\frac{1}{3}]}(a^{[\frac{1}{3}]})^{}a^{[\frac{1}{3}]}=\\{a^{[\frac{1}{3}]},a^{[\frac{1}{3}]},a^{[\frac{1}{3}]}\\}=a.$ In order to simplify notation, for each element $a$ in a JB∗-triple $E$ we write $a^{[1]}=a$ and $a^{[2n+1]}:=\left\\{a,a^{[2n-1]},a\right\\}$ $(\forall n\in\mathbb{N})$. It is known that JB∗-triples are power associative, that is, $\left\\{a^{[2k-1]},a^{[2l-1]},a^{[2m-1]}\right\\}=a^{[2(k+l+m)-3]},$ for every $k,l,m\in\mathbb{N}$ (cf. [11, Lemma 1.2.10]). ## 3\. Local and 2-local representations of C∗-algebras Let $A$ and $B$ be C∗-algebras. Clearly, every local ∗-homomorphism $T:A\to B$ is automatically continuous and contractive. Indeed, since for each $a\in A$, there exists a ∗-homomorphism $\Phi_{a}:A\to B$ satisfying $T(a)=\Phi_{a}(a)$, we have $\\|T(a)\\|=\\|\Phi_{a}(a)\\|\leq\\|a\\|$. Concerning (local) homomorphisms, many basic questions are still open, like automatic continuity of homomorphisms between C∗-algebras ([13, Question 5.4.D] or [47, Question 1]). The next result summarizes some clear facts about local homomorphisms. ###### Lemma 3.1. Let $A,B$ and $C$ denote C∗-algebras, $T:A\to B$ a local homomorphism (respectively, a local ∗-homomorphism) and $\Phi:B\to C$ a homomorphism (respectively, a ∗-homomorphism), then $\Phi T$ is a local homomorphism (respectively, a local ∗-homomorphism). Every local ∗-homomorphism between C∗-algebras is positive.$\hfill\Box$ By Gelfand theory the set of extreme points in the positive part of the unit ball in the dual, $\mathcal{B}^{}$, of a commutative C∗-algebra $\mathcal{B}$ is precisely the set $X$ of non-zero multiplicative functionals on $\mathcal{B}$, and thus the identification $\mathcal{B}=C_{0}(X)$ was established. Therefore, non-zero homomorphisms from $C_{0}(L)$ into $\mathbb{C}$ identify with those functionals $\delta_{s}:C_{0}(L)\to\mathbb{C}$, $\delta_{s}(f)=f(s),$ where $s$ runs in $L.$ The refinement of the Gleason-Kahane-Żelazko theorem established by Żelazko in [49] asserts that for every complex Banach algebra $\mathcal{B}$ (not necessarily unital nor commutative), every linear selection from the spectrum $\varphi:\mathcal{B}\to\mathbb{C}$ (i.e. $\varphi(a)\in\sigma(a)$, for every $a\in\mathcal{B}$) is multiplicative. Another interesting result, implicitly established by J.P. Kahane and W. Żelazko in [25, Theorem 3], will be applied in the next proposition. ###### Proposition 3.2. Let $L_{1}$ and $L_{2}$ be locally compact Hausdorff spaces and let $T:C_{0}(L_{1})\to C_{0}(L_{2})$ be a local homomorphism. Then, for each $s\in L_{2}$, the mapping $\delta_{s}T:C_{0}(L_{1})\to\mathbb{C}$ is a ∗-homomorphism. In particular, $T$ is a ∗-homomorphism. ###### Proof. Let us assume that $\delta_{s}T\neq 0$. We shall prove that $\delta_{s}T=\delta_{t}$ for a unique $t\in L_{1}$. Since $T$ is a local homomorphism, $\delta_{s}T$ is a local homomorphism. So, given $f\in C_{0}(L_{1})$ there exists a homomorphism $\Phi_{f}:C_{0}(L_{1})\to\mathbb{C}$, and hence an element $t_{f}\in L_{1}$, satisfying $\delta_{s}T(f)=\Phi_{f}(f)=f(t_{f}).$ Now, Theorem 3 in [25] proves that $\delta_{s}T$ is a multiplicative functional. ∎ ###### Corollary 3.3. Let $T:A\to B$ be a local homomorphism between C∗-algebras, where $B$ is commutative. Then $T$ is a Jordan ∗-homomorphism and, consequently, is continuous. ###### Proof. Let $a$ be a self adjoint element in $A$. Considering the C∗-subalgebra, $\mathcal{C},$ generated by $a$, the mapping $T\|_{\mathcal{C}}:\mathcal{C}\to B$ is a local homomorphism between commutative C∗-algebras. Proposition 3.2 assures that $T\|_{\mathcal{C}}$ is a ∗-homomorphism. Therefore $T(a^{2})=T(a)^{2}$ for every $a\in A_{sa}$, and hence $T$ is a Jordan ∗-homomorphism. ∎ When $C(K)$ is replaced with the real C∗-algebra $C(K,\mathbb{R})$, of all real-valued continuous functions on $K$, the corresponding versions of the above results are not, in general, true. For example, the linear operator $T:C([0,1],\mathbb{R})\to\mathbb{R}$, $\displaystyle T(a):=\int_{0}^{1}a(t)dt$ is not multiplicative. However, the mean-value theorem implies that $T$ is a local homomorphism. The existence of bounded linear operators between C∗-algebras which are local homomorphisms and fail to be multiplicative (cf. [41, example in §2]) led F. Pop to focus his attention on 2-local homomorphisms (called 2-local representations by Pop). The just-mentioned counter-example, provided by Pop, is not multiplicative but it is a Jordan homomorphism (see Example 3.13). The latter property is actually satisfied by every linear 2-local homomorphism between C∗-algebras (cf. [32, Lemma 2.1]). ###### Proposition 3.4. Every linear 2-local homomorphism between C∗-algebras is a Jordan homomorphism. Every linear 2-local ∗-homomorphism between C∗-algebras is a Jordan ∗-homomorphism. ###### Proof. When $T$ is a linear 2-local homomorphism, for each $a\in A$, there exists a homomorphism $\Phi_{a,a^{2}}:A\to B$ such that $T(a)=\Phi_{a,a^{2}}(a)$ and $T(a^{2})=\Phi_{a,a^{2}}(a^{2})$. Then, $T(a)^{2}=\Phi_{a,a^{2}}(a)^{2}=\Phi_{a,a^{2}}(a^{2})=T(a^{2}),$ confirming that $T$ is a Jordan homomorphism. ∎ In the setting of von Neumann algebras, the hypothesis in Proposition 3.4 can be relaxed. Indeed, in [41, Proposition 1.4], F. Pop establishes that every bounded linear local homomorphism from a commutative von Neumann algebra into $B(H)$ is multiplicative, and hence a representation. This result applies to get: ###### Corollary 3.5. Let $T:M\to B$ be a bounded linear local homomorphism from a von Neumann algebra into a C∗-algebra. Then $T$ is a Jordan homomorphism. Consequently, every linear local ∗-homomorphism from a von Neumann algebra into a C∗-algebra is a Jordan ∗-homomorphism. ###### Proof. Let $T:M\to B$ be a bounded linear local homomorphism. Making use of the representation theory and Lemma 3.1, we can assume that $B=B(H)$ for a suitable complex Hilbert space $H$. Let $a$ be a self-adjoint element in $M$, and let $\mathcal{C}$ denote the von Neumann subalgebra of $M$ generated by $a$ and $1$. Clearly, $T\|_{\mathcal{C}}:\mathcal{C}\to B(H)$ is a bounded linear local homomorphism. By [41, Proposition 1.4], $T\|_{\mathcal{C}}$ is multiplicative. Therefore, $T(a^{2})=T(a)^{2},$ for every $a\in M_{sa}$. This implies that $T(a\circ b)=T(a)\circ T(b)$ for every $a,b\in M_{sa}$ and hence $T((a+ib)^{2})=T(a^{2}-b^{2}+2ia\circ b)=T(a)^{2}-T(b)^{2}+2iT(a)\circ T(b)=T(a+ib)^{2}$, for every $a,b\in M_{sa}$, which proves the statement. ∎ It seems natural to ask whether the above mentioned results of Hadwin-Li and Pop hold when the domain is a general C∗-algebra. We shall see that the answer is intrinsically related to zero-products preserving operators between C∗-algebras. Let $A$ and $B$ be C∗-algebras. A mapping $f:A\to B$ is said to be _orthogonality preserving_ on a subset $U\subseteq A$ when the implication $a\perp b\Rightarrow f(a)\perp f(b),$ holds for every $a,b\in U$. We recall that elements $a,b$ in $A$ are said to be _orthogonal_ (denoted by $a\perp b$) whenever $ab^{}=b^{}a=0$. When the implication $ab=0\Rightarrow f(a)f(b)=0$ holds for every $a,b\in U$, we shall say that $f$ _preserves zero products_ or is _zero products preserving_ on $U$. In the case $A=U$, we shall simply say that $f$ is _orthogonality preserving_ or that $f$ _preserves zero products_ , respectively. Every homomorphism between C∗-algebras preserves zero products and every ∗-homomorphism is orthogonality preserving. ###### Lemma 3.6. (cf. [32, Lemma 2.1]) Let $T:A\to B$ be a map between C∗-algebras. Suppose $T$ is a 2-local ∗-homomorphism (respectively, a 2-local homomorphism), then $T$ is orthogonality preserving (respectively, zero products preserving). ###### Proof. Given $a,b\in A$ with $a\perp b$, we take a ∗-homomorphism $\Phi_{a,b}:A\to B$ satisfying $T(a)=\Phi_{a,b}(a)$ and $T(b)=\Phi_{a,b}(b)$. Clearly, $T(a)=\Phi_{a,b}(a)\perp\Phi_{a,b}(b)=T(b)$. The other statement follows similarly. ∎ Orthogonality preserving bounded linear maps between C∗-algebras have been completely described in [6, Theorem 17] (see [7] and [18] for completeness). Let $A$ be a C∗-algebra. An element $x$ in the von Neumann algebra $A^{}$ is a _multiplier_ for $A$ if $xA\subseteq A$ and $Ax\subseteq A$. The symbol $M(A)$ will denote the set of all multiplier of $A$ in $A^{}.$ It is known that $M(A)$ is a unital C∗-subalgebra of $A^{}$. Multipliers are uninteresting if the algebra $A$ possesses a unit, because in such a case $M(A)=A$ (see [40, §3.12] for more details). Let $W$ be a von Neumann algebra. For each normal positive functional $\varphi\in W_{}$ the mapping $W\times W\to\mathbb{C}$, $(x,y)_{\varphi}:=\frac{1}{2}\varphi(xy^{}+y^{}x)$ defines a semi-positive sesquilinear form on $W$. The corresponding prehilbertian seminorm on $W$ is defined by $\\|x\\|_{\varphi}:=(x,x)_{\varphi}^{\frac{1}{2}}=\left(\frac{1}{2}\varphi(xx^{}+x^{}x)\right)^{\frac{1}{2}}.$ The _strong ∗ topology_ of $W$ (denoted by $s^{}(W,W_{})$) is the locally convex topology on $W$ defined by all the seminorms $\\|.\\|_{\varphi}$, where $\varphi$ runs in the set of all positive functionals in $W_{}$ (cf. [42, Definition 1.8.7]). It is known that the strong topology of $W$ is compatible with the duality $(W,W_{})$, that is a functional $\psi:W\to\mathbb{C}$ is $s^{}(W,W_{})$ if and only if it is weak∗ continuous (see [42, Corollary 1.8.10]). It is also known, from the above fact together with the Grothedieck- Pisier-Haagerup inequality (cf. [20]), that a linear map between von Neumann algebras is strong∗ continuous if and only if it is weak∗ continuous. We also recall that the product of every von Neumann algebra is jointly strong∗ continuous on bounded sets (see [42, Proposition 1.8.12]). The next result is a subtle variant of [48, Lemma 2.2]. The proof applies techniques of JB∗-triples in a similar fashion to the arguments given in the proofs of [7, Proposition 3.1], [16, Proposition 1.3], and [48, Lemma 2.2]. ###### Proposition 3.7. Let $T:A\to B$ be a bounded linear map between C∗-algebras sending zero products in $A$ to zero products in $B$. Then the restricted map $T^{}\|_{M(A)}:M(A)\to B^{}$ sends zero products in $M(A)$ to zero products in $B^{}$. ###### Proof. We fix $a,b\in M(A)$ with $ab=0$. For each natural, $n$, the odd triple power $a^{[3]}=aa^{}a$, $a^{[2n+1]}=a(a^{[2n-1]})^{}a$, satisfies that $a^{[2n-1]}b=0$. Thus, we deduce that, $\alpha b=0$, for every $\alpha$ in the JB∗-subtriple, $M(A)_{a},$ of $M(A)$ generated by $a$. The same argument shows that (3.1) $\alpha\beta=0$ for every $\alpha\in M(A)_{a}$ and $\beta\in M(A)_{b}$. Consequently, we have $a^{[\frac{1}{3}]}b^{[\frac{1}{3}]}=0.$ Since $M(A)$ is a C∗-subalgebra of $A^{}$, by Goldstine’s Theorem, we can find bounded nets $(x_{\lambda})$ and $(y_{\mu})$ in $A$, converging in the weak∗ topology of $A^{}$ to $a^{[\frac{1}{3}]}$ and $b^{[\frac{1}{3}]}$, respectively. The nets $\left(a^{[\frac{1}{3}]}x_{\lambda}^{}a^{[\frac{1}{3}]}\right)$ and $\left(b^{[\frac{1}{3}]}y_{\mu}^{}b^{[\frac{1}{3}]}\right)$ lie in $A$, and $\left(a^{[\frac{1}{3}]}x_{\lambda}^{}a^{[\frac{1}{3}]}\right)\left(b^{[\frac{1}{3}]}y_{\mu}^{}b^{[\frac{1}{3}]}\right)=0,$ for every $\lambda$ and $\mu$. By hypothesis, $T$ is zero products preserving, and hence, (3.2) $T\left(a^{[\frac{1}{3}]}x_{\lambda}^{}a^{[\frac{1}{3}]}\right)T\left(b^{[\frac{1}{3}]}y_{\mu}^{}b^{[\frac{1}{3}]}\right)=0,$ for every $\lambda$ and $\mu$. Finally, taking weak∗-limits in $\lambda$ and $\mu$, the weak∗ continuity of $T^{}$ and the separate weak∗-continuity of the product of $A^{}$, together with (3.2), give $0=T^{}\left(a^{[\frac{1}{3}]}(a^{[\frac{1}{3}]})^{}a^{[\frac{1}{3}]}\right)T^{*}\left(b^{[\frac{1}{3}]}(b^{[\frac{1}{3}]})^{}b^{[\frac{1}{3}]}\right)=T^{}(a)T^{}(b),$ which completes the proof. ∎ Let $A$ be a C∗-algebra, a projection $p$ in $A^{}$ is called _compact- $G_{\delta}$_ (relative to $A$) whenever there exists a positive, norm-one element $a$ in $A$ such that $p$ coincides with the weak∗-limit (in $A^{}$) of the sequence $(a^{n})_{n}$. Following standard notation, we shall say that $p$ is a _range projection_ when there exists a positive, norm-one element $a\in A$ for which $p$ is the weak∗-limit of the sequence $(a^{\frac{1}{n}})_{n}$. Our next result can be derived from [1, Theorem 4.1] (compare Remark 3.11). To our knowledge, it has never been stated in the form presented here. We also include a new proof which is independent from the arguments in [1]. ###### Theorem 3.8. Let $A$ and $B$ be C∗-algebras with $A$ unital. Let $J:A\to B$ be a bounded Jordan homomorphism preserving zero products. Then $J$ is a homomorphism. ###### Proof. Since $J$ is a Jordan homomorphism, we deduce that $J(1)=e$ is an idempotent in $B$ and $J(a)=J(U_{1}(a))=U_{J(1)}(J(a))=U_{e}(J(a))=eJ(a)e,$ for every $a\in A$. Since $J^{}:A^{}\to B^{}$ is a Jordan homomorphism too, we can actually assure that (3.3) $J^{}(a)=eJ^{}(a)e=eJ^{}(a)=J^{}(a)e,$ for every $a\in A$. Since $J$ preserves zero products, given $a,b\in A$ with $ab=0$, we have $J(ba)=J(ab+ba)=J(a)J(b)+J(b)J(a)=J(b)J(a),$ and consequently (3.4) $J(bza)=J(bz)J(a)=J(b)J(za),$ for every $a,b,z\in A$ with $ab=0$. Let us consider a compact-$G_{\delta}$ projection $p\in A^{}$, that is, there exists a positive, norm-one element $a$ in $A$ such that $\displaystyle p=w^{}-\lim_{n}a^{n}$. We can identify the C∗-subalgebra of $A$ generated by $1$ and $a$ with $C(K),$ where $K\subseteq[0,1]$, $1\in K$, and $a(t)=t$ in the corresponding identification. Let us define two sequences $(y_{n})$ and $(z_{n})$ in $C(K)$ given by $y_{n}(t):=\left\\{\begin{array}[]{ll}1,&\hbox{if $t\in K\cap[0,1-\frac{1}{n}]$;}\\\ -2nt+2n-1,&\hbox{if $t\in K\cap[1-\frac{1}{n},1-\frac{1}{2n}]$;}\\\ 0,&\hbox{if $t\in K\cap[1-\frac{1}{2n},1]$,}\\\ \end{array}\right.$ and $z_{n}(t):=\left\\{\begin{array}[]{ll}0,&\hbox{if $t\in K\cap[0,1-\frac{1}{3n}]$;}\\\ 3nt-3n+1,&\hbox{if $t\in K\cap[1-\frac{1}{3n},1]$}\\\ \end{array}\right..$ It is easy to check that $0\leq y_{n},z_{n}$, $(y_{n})$ is increasing, $(z_{n})$ is decreasing, $y_{n}z_{m}=z_{m}y_{n}=0$ for every $n,m\in\mathbb{N}$, $m\geq n$, $w^{}-\lim_{n}y_{n}=1-p$, and $w^{}-\lim_{n}z_{n}=p$ in $A^{}$. By hypothesis, for every $z$ in $A$, and every $n,m$ in $\mathbb{N}$ with $m\geq n$, we have $J(zz_{m})J(y_{n})=0$, and thus $0=w^{}-\lim_{m\geq n}J(zz_{m})J(y_{n})=J^{*}(zp)J(y_{n}),\ \hbox{ for every }n\in\mathbb{N},$ which implies that $0=w^{}-\lim_{n}J^{}(zp)J(y_{n})=J^{}(zp)J(1-p),$ and hence $J^{}(zp)e=J^{}(zp)J(1)=J^{}(zp)J(p).$ It follows from (3.3) that (3.5) $J^{}(zp)=J^{}(zp)e=J^{}(zp)J^{}(p),$ for every $z\in A^{}$. Applying (3.4) we deduce that $J(y_{n})J(zz_{m})=J(y_{n}zz_{m}),$ and $J(z_{m})J(zy_{n})=J(z_{m}zy_{n}),$ for every $z\in A$, $n,m\in\mathbb{N}$ with $m\geq n$. Taking weak∗-limits in $m,n\to\infty$, we get (3.6) $J^{}(1-p)J^{}(zp)=J^{}((1-p)zp),$ and (3.7) $J^{}(p)J^{}(z(1-p))=J^{}(pz(1-p)),$ for every $z$ in $A$ or in $A^{}.$ Combining (3.6) with (3.3) we get $J^{}(zp)-J^{}(p)J^{}(zp)=J^{}(1)J^{}(zp)-J^{}(p)J^{}(zp)=J^{}(zp)-J^{}(pzp),$ and thus (3.8) $J^{}(p)J^{}(zp)=J^{}(pzp),$ for every $z$ in $A$ or in $A^{}.$ Now, combining (3.7) and (3.8), we deduce that $J^{}(p)J^{}(z)=J^{}(pz),$ for every $z\in A^{}.$ We have therefore proved that (3.9) $J^{}(pz)=J^{}(p)J^{}(z),$ for every $z\in A^{}$ and every compact-$G_{\delta}$ projection $p\in A^{}$. Finally, take an arbitrary self adjoint element $b\in A$ and identify the C∗-subalgebra of $A$ generated by $1$ and $b$ with a $C(K)$-space for a suitable $K\subset[-\\|b\\|,\\|b\\|]$. The property proved in (3.9) shows that $J^{}(pz)=J^{}(p)J^{}(z)$ for every projection $p$ of the form $p=\chi_{{}_{[\alpha,\beta]\cap K}}$ with $[\alpha,\beta]\subseteq[-\\|b\\|,\\|b\\|]$. Having in mind that projections $q\in C(K)^{}\subseteq A^{}$ of the form $q=\chi_{{}_{(\alpha,\beta)\cap K}},$ with $(\alpha,\beta)\subseteq K$ can be approximated in the strong∗ topology of $A^{}$ by sequences of projections $\displaystyle(p_{n})=\left(\chi_{{}_{[\alpha-\frac{1}{n},\beta+\frac{1}{n}]\cap K}}\right)$, we deduce that $J^{}(qz)=J^{}(q)J^{}(z)$ for every such projection $q$ and every $z\in A^{*}$. It is well known that $b$ (regarded as an element in $C(K)\subseteq A$) can be approximated in norm by a finite linear combinations of mutually orthogonal projections of the form $\chi_{{}_{[\alpha,\beta]\cap K}}$ and $\chi_{{}_{(\alpha,\beta)\cap K}}$ with $[\alpha,\beta]\subseteq[-\\|b\\|,\\|b\\|]$ (i.e. steps functions). Therefore, $J(b)J(z)=J(bz)$, for every $z,b\in A$ with $b=b^{}$ and, by linearity, $J$ is a homomorphism. ∎ We can now prove the main result concerning 2-local homomorphisms. ###### Theorem 3.9. Every bounded linear 2-local homomorphism between C∗-algebras is a homomorphism. Every linear 2-local ∗-homomorphism between C∗-algebras is a ∗-homomorphism. ###### Proof. Let $T:A\to B$ be a bounded 2-local homomorphism between C∗-algebras. Proposition 3.4 implies that $T$ is a Jordan homomorphism. By the 2-local property, we deduce, via Lemma 3.6, that $T$ preserves zero products. Proposition 3.7 implies that $T^{}\|_{M(A)}:M(A)\to B^{}$ preserves zero products. Finally, since $T^{}\|_{M(A)}:M(A)\to B^{}$ is a Jordan homomorphism which preserves zero products and $M(A)$ is unital, the above Theorem 3.8 gives the desired statement. ∎ Clearly, Theorems 3.6 and 3.7 in [21] are direct consequences of the above Theorem 3.9. Accordingly to the notation in [41], given a C∗-algebra $A$ and a complex Hilbert space $H$, a bounded linear map $T:A\to B(H)$ is called a _2-local representation_ of $A$ whenever it is a 2-local homomorphism. The next result generalizes [41, Corollary 3.6] to the general setting of C∗-algebras. ###### Corollary 3.10. Let $A$ be a C∗-algebra. Every 2-local representation of a $A$ is a representation.$\hfill\Box$ ###### Remark 3.11. It should be noted here that Theorem 3.8 can be derived from [1, Theorem 4.1]. Indeed, in the just commented result the authors prove that for every unital C∗-algebra $A$, every Banach algebra $B$, and every bounded linear operator $T:A\to B$ preserving zero products, then $T(1)T(xy)=T(x)T(y),$ for all $x,y$ in $A$. Therefore, if $J:A\to B$ is a bounded Jordan homomorphism preserving zero products, by the first part of the argument in the proof of Theorem 3.8, $J(1)=e$ is an idempotent in $B$ and $J(a)=eJ(a)e=eJ(a)=J(a)e,$ for every $a\in A$, and hence $J(xy)=J(1)J(xy)=J(x)J(y),$ for all $x,y$ in $A$. That is, Theorem 3.8 holds when $B$ is a Banach algebra. Proposition 3.7 is needed for the non-unital version of Theorem 3.9 ###### Problem 3.12. Is every (not necessarily linear) 2-local (∗-)homomorphism between C∗-algebras a (∗-)homomorphism? Equivalently, determine whether the hypothesis concerning linearity in Theorem 3.9 is superfluous. As we have commented before, we cannot expect that a local homomorphism between C∗-algebras is a homomorphism (see [41, §2]). We shall take a closer look at the counter-example provided by F. Pop. ###### Example 3.13. We know, from [41, §2], that the mapping $T:M_{2}(\mathbb{C})\to M_{4}(\mathbb{C})$, $T\left(\left(\begin{array}[]{cc}a&b\\\ c&d\\\ \end{array}\right)\right)=\left(\begin{array}[]{cccc}a&0&b&0\\\ 0&a&0&c\\\ c&0&d&0\\\ 0&b&0&d\\\ \end{array}\right),$ is a local homomorphism which is not multiplicative. Is easy to check that the above $T$ is a unital Jordan ∗-homomorphism. We claim that $T$ is not a local ∗-homomorphism. Otherwise, there exits a ∗-homomorphism $\pi=\pi_{\tiny\left(\begin{array}[]{cc}1&1\\\ 2&0\\\ \end{array}\right)}:M_{2}(\mathbb{C})\to M_{4}(\mathbb{C})$ satisfying $\pi\left(\begin{array}[]{cc}1&1\\\ 2&0\\\ \end{array}\right)=T\left(\begin{array}[]{cc}1&1\\\ 2&0\\\ \end{array}\right)=\left(\begin{array}[]{cccc}1&0&1&0\\\ 0&1&0&2\\\ 2&0&0&0\\\ 0&1&0&0\\\ \end{array}\right).$ Therefore $\pi\left(\begin{array}[]{cc}2&2\\\ 2&4\\\ \end{array}\right)=\pi\left(\left(\begin{array}[]{cc}1&1\\\ 2&0\\\ \end{array}\right)\left(\begin{array}[]{cc}1&1\\\ 2&0\\\ \end{array}\right)^{}\right)$ $=\pi\left(\begin{array}[]{cc}1&1\\\ 2&0\\\ \end{array}\right)\pi\left(\begin{array}[]{cc}1&1\\\ 2&0\\\ \end{array}\right)^{}=\left(\begin{array}[]{cccc}2&0&2&0\\\ 0&5&0&1\\\ 2&0&4&0\\\ 0&1&0&1\\\ \end{array}\right),$ $\pi\left(\begin{array}[]{cc}5&1\\\ 1&1\\\ \end{array}\right)=\pi\left(\left(\begin{array}[]{cc}1&1\\\ 2&0\\\ \end{array}\right)^{}\left(\begin{array}[]{cc}1&1\\\ 2&0\\\ \end{array}\right)\right)=\left(\begin{array}[]{cccc}5&0&1&0\\\ 0&2&0&2\\\ 1&0&1&0\\\ 0&2&0&4\\\ \end{array}\right),$ $\pi\left(\begin{array}[]{cc}0&0\\\ -2&4\\\ \end{array}\right)=\left(\begin{array}[]{cccc}0&0&0&0\\\ 0&3&0&-3\\\ -2&0&4&0\\\ 0&-1&0&1\\\ \end{array}\right),$ $\pi\left(\begin{array}[]{cc}-8&0\\\ 0&2\\\ \end{array}\right)=\pi\left(\left(\begin{array}[]{cc}2&2\\\ 2&4\\\ \end{array}\right)-2\left(\begin{array}[]{cc}5&1\\\ 1&1\\\ \end{array}\right)\right)=\left(\begin{array}[]{cccc}-8&0&0&0\\\ 0&1&0&-3\\\ 0&0&2&0\\\ 0&-3&0&-7\\\ \end{array}\right),$ $\pi\left(\begin{array}[]{cc}0&0\\\ 0&20\\\ \end{array}\right)=\pi\left(\left(\begin{array}[]{cc}0&0\\\ -2&4\\\ \end{array}\right)\left(\begin{array}[]{cc}0&0\\\ -2&4\\\ \end{array}\right)^{}\right)=20\left(\begin{array}[]{cccc}0&0&0&0\\\ 0&9/10&0&-3/10\\\ 0&0&1&0\\\ 0&-3/10&0&1/10\\\ \end{array}\right),$ $\pi\left(\begin{array}[]{cc}1&0\\\ 0&0\\\ \end{array}\right)=\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&1/10&0&3/10\\\ 0&0&0&0\\\ 0&3/10&0&9/10\\\ \end{array}\right),$ $\pi\left(\begin{array}[]{cc}0&0\\\ 1&0\\\ \end{array}\right)=\left(\begin{array}[]{cccc}0&0&0&0\\\ 0&3/10&0&9/10\\\ 1&0&0&0\\\ 0&-1/10&0&-3/10\\\ \end{array}\right),$ and $\pi\left(\begin{array}[]{cc}0&1\\\ 0&0\\\ \end{array}\right)=\left(\begin{array}[]{cccc}0&0&1&0\\\ 0&3/10&0&-1/10\\\ 0&0&0&0\\\ 0&9/10&0&-3/10\\\ \end{array}\right),$ which gives $\pi\left(\left(\begin{array}[]{cc}1&1\\\ 0&1\\\ \end{array}\right)\left(\begin{array}[]{cc}1&1\\\ 0&0\\\ \end{array}\right)\right)\neq\pi\left(\left(\begin{array}[]{cc}1&1\\\ 0&1\\\ \end{array}\right)\right)\pi\left(\left(\begin{array}[]{cc}1&1\\\ 0&0\\\ \end{array}\right)\right),$ contradicting that $\pi$ is a homomorphism. Furthermore, the elements $\displaystyle a=\left(\begin{array}[]{cc}1&-1\\\ 1&-1\\\ \end{array}\right)$ and $\displaystyle b=\left(\begin{array}[]{cc}1&2\\\ 1&2\\\ \end{array}\right)$ satisfy $ab=0$ and $T(a)T(b)\neq 0,$ which shows that $T$ does not preserves zero products. It seems natural to ask whether every local ∗-homomorphism between C∗-algebras is multiplicative. We shall see that the answer to this question is, in general, negative. The next example illustrates this fact and provides an easier argument to Pop’s counterexample. ###### Example 3.14. A problem posed by P.R. Halmos in [22, Proposition 159] asks whether every square complex matrix is unitarily equivalent to its transpose. In other words, given $a\in M_{n}(\mathbb{C})$, when does there exist a unitary matrix $u\in M_{n}(\mathbb{C})$ satisfying $u^{}au=a^{t}$? More generally, the problem of deciding whether two given square matrices $a$ and $b$ over the field of complex numbers are unitarily equivalent was positively solved by W. Specht [45] who found a (more or less satisfactory) necessary and sufficient condition for two complex square matrices to be unitarily equivalent. In the setting of $2\times 2$ matrices the conditions are much more simple; F.D. Murnaghan [38] showed that, the traces of $a$, $a^{2}$, and $a^{}a$ form a complete set of invariants to determine when two matrices in $M_{2}(\mathbb{C})$ are unitarily equivalent (i.e. two matrices $a,b\in M_{2}(\mathbb{C})$ are unitarily equivalent if and only if $\hbox{tr}(a)=\hbox{tr}(b)$, $\hbox{tr}(a^{2})=\hbox{tr}(b^{2})$, and $\hbox{tr}(a^{}a)=\hbox{tr}(b^{}b)$). Some years later, C. Pearcy [39] obtained a list of nine conditions to determine when $a,b\in M_{3}(\mathbb{C})$ are unitarily equivalent (see [17] for a recent publication on these topics). 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1312.6856	footnote # Ramification conjecture and Hirzebruch’s property of line arrangements D. Panov and A. Petrunin ###### Abstract The ramification of a polyhedral space is defined as the metric completion of the universal cover of its regular locus. We consider mainly polyhedral spaces of two origins: quotients of Euclidean space by a discrete group of isometries and polyhedral metrics on $\mathbb{C}\mathrm{P}^{2}$ with singularities at a collection of complex lines. In the former case we conjecture that quotient spaces always have a $\mathrm{CAT}[0]$ ramification and prove this in several cases. In the latter case we prove that the ramification is $\mathrm{CAT}[0]$ if the metric on $\mathbb{C}\mathrm{P}^{2}$ is non-negatively curved. We deduce that complex line arrangements in $\mathbb{C}\mathrm{P}^{2}$ studied by Hirzebruch have aspherical complement. ## 1 Introduction The main objects of this article are Euclidean polyhedral spaces and their _ramifications_. The ramification of a polyhedral space is the metric completion of the universal cover of its regular locus. We are interested in the situation when the ramification is $\mathrm{CAT}[0]$. Two classes of polyhedral spaces that will play the most important role are quotients of $\mathbb{R}^{n}$ by discrete isometric actions, and polyhedral Kähler manifolds (i.e., polyhedral manifolds with a complex structure). Quotients of $\mathbb{R}^{m}$ and ramification conjecture. We start with the case of $\mathbb{R}^{m}$ quotients where the ramification space admits an alternative description in terms of arrangements of planes of (real) codimension $2$ (we will call such planes _hyperlines_). Consider a discrete isometric and orientation-preserving action $\Gamma\curvearrowright\mathbb{R}^{m}$. Denote by $\mathcal{L}_{\Gamma}$ the arrangement of all hyperlines which are fixed by at least one non-identical element in $\Gamma$. Define the ramification of $\Gamma\curvearrowright\mathbb{R}^{m}$ (briefly $\mathop{\rm Ram}\nolimits_{\Gamma}$) as the universal cover of $\mathbb{R}^{m}$ branching infinitely along each hyperline in $\mathcal{L}_{\Gamma}$. More precisely, if $\tilde{W}_{\Gamma}$ denotes the universal cover of $W_{\Gamma}=\mathbb{R}^{m}\backslash\left(\bigcup_{\ell\in\mathcal{L}_{\Gamma}}\ell\right)$ equipped with the length metric induced from $\mathbb{R}^{m}$ then $\mathop{\rm Ram}\nolimits_{\Gamma}$ is the metric completion of $\tilde{W}_{\Gamma}$. One of the main motivations of this paper is the following conjecture. 1.1. Ramification conjecture. Let $\Gamma\curvearrowright\mathbb{R}^{m}$ be a properly discontinuous isometric orientation-preserving action. Then 1. a) $\mathop{\rm Ram}\nolimits_{\Gamma}$ is a $\mathrm{CAT}[0]$ space. 2. b) The natural inclusion $\tilde{W}_{\Gamma}\hookrightarrow\mathop{\rm Ram}\nolimits_{\Gamma}$ is a homotopy equivalence. If an action $\Gamma\curvearrowright\mathbb{R}^{m}$ satisfies the Ramification conjecture, we have an immediate corollary. Since $\mathrm{CAT}[0]$ spaces are contactable, $\tilde{W}_{\Gamma}$ is also contractible, and so $W_{\Gamma}$ is aspherical. Ramification conjecture generalizes a conjecture of Allcock [2, Conjecture 1.4] on finite _reflection groups_ (recall that a reflection group is a discrete group generated by a set of reflections of a Euclidean space). Allcock considers the case of the action $\Gamma\curvearrowright\mathbb{C}^{m}$ of a finite reflection group $\Gamma$ that complexifies the orientation reversing action of $\Gamma$ on $\mathbb{R}^{m}$ generated by reflections. Allcock’s conjecture is related to an earlier conjecture of Charney and Davis (see [9, Conjecture 3]) which in turn is motivated by a conjecture of Arnold, Pham and Thom on complex hyperplane arrangements. In the following theorem we collect the partial cases of Ramification conjecture which we can prove. 1.2. Theorem. The Ramification conjecture holds in the following cases: * $(\mathrm{R}^{+})$ If the action $\Gamma\curvearrowright\mathbb{R}^{n}$ is the orientation preserving index two subgroup of a reflection group. * $(\mathbb{Z}_{2})$ If $\Gamma$ is isomorphic to $\mathbb{Z}_{2}^{k}$. * $(\mathbb{R}^{3})$ If $m\leqslant 3$. * $(\mathbb{C}^{2})$ If $m=4$, and the action $\Gamma\curvearrowright\mathbb{R}^{4}$ preserves a complex structure on $\mathbb{R}^{4}$. The most involved case is $(\mathbb{C}^{2})$; it is proved in Section 9 and relies on Theorem 8, which is the main technical result of this paper. The proofs of other cases are simpler. The case $(\mathrm{R}^{+})$ follows from more general Proposition 4. In Section 7 we give two proofs of the case $(\mathbb{R}^{3})$, one is based on Theorem 6 and Zalgaller’s theorem 3 and the other on the case $(\mathrm{R}^{+})$. 1.3. Corollary. Let $S_{3}\curvearrowright\mathbb{C}^{3}$ be the action of symmetric group by permuting coordinates of $\mathbb{C}^{3}$. Then $\mathop{\rm Ram}\nolimits_{S_{3}}$ is a $\mathrm{CAT}[0]$ space. The above corollary is deduced from the $(\mathbb{C}^{2})$-case of Theorem 1 since the action $S_{3}\curvearrowright\mathbb{C}^{3}$ splits as a sum of an action on $\mathbb{C}^{2}$ and a trivial action on $\mathbb{C}^{1}$. This corollary also follows from a result of Charney and Davis in [8]. Polyhedral manifolds and Hirzebruch’s question. Our study of ramifications of polyhedral manifolds sheds some light on a question of Hirzebruch on complex line arrangements in $\mathbb{C}\mathrm{P}^{2}$ asked in [15]. To state this question recall the notion of complex reflection groups and arrangements. A finite complex reflection group is a group $\Gamma$ acting on $\mathbb{C}^{m}$ by complex linear transformation generated by elements that fix a complex hyperplane in $\mathbb{C}^{m}$. The arrangement of complex hyperplanes111i.e., the set of hyperplanes fixed by at least one non-trivial element of $\Gamma$ $\mathcal{L}_{\Gamma}$ in $\mathbb{C}^{m}$ and its projectivization in $\mathbb{C}\mathrm{P}^{m-1}$ are called complex reflection arrangements. 1.4. Hirzebruch’s question, [15]. Let $\mathcal{L}$ be a complex line arrangement in $\mathbb{C}\mathrm{P}^{2}$ consisting of $3{\hskip 0.5pt\cdot\nobreak\hskip 0.5pt}n$ lines such that each line of $\mathcal{L}$ intersect others at exactly $n+1$ points. Is it true that $\mathcal{L}$ is a complex reflection arrangement? The above property will be called Hirzebruch’s property. Hirzebruch noticed that all complex reflection line arrangements in $\mathbb{C}\mathrm{P}^{2}$ satisfy this property. These line arrangements consist of two infinite series and five exceptional examples. The infinite series are called $A_{m}^{0}$ ($m\geqslant 3$) and $A_{m}^{3}$ $(m\geqslant 2)$ and correspond to reflection groups $G(m,m,3)$ and $G(m,p,3)$ ($p<m$) from Shephard–Todd classification. Five exceptional examples correspond to reflection groups $G_{23},G_{24},G_{25},G_{26},G_{27}$. Hirzebruch’s question is still open, but we are able to prove the following. 1.5. Theorem. All line arrangements satisfying Hirzebruch’s property have aspherical complements. Note that if the answer to Hirzebruch’s question were positive, this theorem would follow from the work of Bessis [5]. Bessis finished the proof of the old conjecture stating that complements to finite complex reflection arrangements are aspherical. Namely he proved this statement for the cases of groups $G_{24}$, $G_{27}$, $G_{29}$, $G_{31}$, $G_{33}$ and $G_{34}$. As an immediate corollary of our theorem we get a new geometric proof of Bessis’s theorem for the cases of groups $G_{24}$ and $G_{27}$. Theorem 1 has a generalisation to a larger class of arrangements, described in Corollary 11. Note that on the one hand line arrangements with aspherical complements are quite rear, on the other hand no idea exists at the present of how to classify them. About the proof of Theorem 1. It follows from [18, Corollary 7.8], that for any arrangement satisfying Hirzebruch’s property except the union of three lines, there is a non-negatively curved polyhedral metric on $\mathbb{C}\mathrm{P}^{2}$ with singularities at this arrangement. Hence to prove the theorem it is enough to show that the ramification of this polyhedral metric satisfies conditions a) and b) of Conjecture 1. Let us sketch how this is done. Consider a $3$-dimensional pseudomanifold $\Sigma$ with a piecewise spherical metric. Define the _singular locus_ $\Sigma^{{\star}}$ of $\Sigma$ as the set of points in $\Sigma$ which do not admit a neighborhood isometric to an open domain in the unit $3$-sphere. Then the ramification of $\Sigma$ is defined as the completion of the universal cover $\tilde{\Sigma}^{\circ}$ of the _regular locus_ $\Sigma^{\circ}=\Sigma\backslash\Sigma^{{\star}}$. The obtained space will be denoted as $\mathop{\rm Ram}\nolimits\Sigma$. In Theorem 8 we characterize spherical polyhedral three-manifolds $\Sigma$ admitting an isometric $\mathbb{R}^{1}$-action with geodesic orbits such that $\mathop{\rm Ram}\nolimits\Sigma$ is $\mathrm{CAT}[1]$. The key condition in Theorem 8 is that all points in $\Sigma$ lie sufficiently close to the singular locus. The existence of an $\mathbb{R}^{1}$-action as above on $\Sigma$ is equivalent to the existence of a complex structure on the Euclidean cone over $\Sigma$; see Theorem 3. The latter permits us to apply Theorem 8 in the proof of Theorem 1 since any non-negatively curved polyhedral metric on $\mathbb{C}\mathrm{P}^{2}$ has complex holonomy. It follows then that the ramification of $\mathbb{C}\mathrm{P}^{2}$ is locally $\mathrm{CAT}[0]$ and by an analogue of Hadamard–Cartan theorem it is globally $\mathrm{CAT}[0]$; see Proposition 3. ## 2 More questions and observations Ramification of a polyhedral space. A Euclidean polyhedral space with nonnegative curvature in the sense of Alexandrov has to be a pseudomanifold, possibly with a nonempty boundary. In fact, a stronger statement holds, a Euclidean polyhedral space $\mathcal{P}$ has curvature bounded from below in the sense of Alexandrov if and only if its regular locus $\mathcal{P}^{\circ}$ is connected and convex in $\mathcal{P}$; i.e., any minimizing geodesic between points in $\mathcal{P}^{\circ}$ lies completely in $\mathcal{P}^{\circ}$ (compare [16, Theorem 5]). Recall that the ramification of $\mathcal{P}$ is defined as the completion of the universal cover $\tilde{\mathcal{P}}^{\circ}$ of the _regular locus_ $\mathcal{P}^{\circ}$. Next question is intended to generalise the Ramification conjecture to a wider setting that is not related to group actions. 2.1. Question. Let $\mathcal{P}$ be a Euclidean polyhedral space. Suppose $\mathcal{P}$ has nonnegative curvature in the sense of Alexandrov. What additional conditions should be imposed on $\mathcal{P}$ to guarantee that $\mathop{\rm Ram}\nolimits\mathcal{P}$ is a $\mathrm{CAT}[0]$ space and the inclusion $\tilde{\mathcal{P}}^{\circ}\hookrightarrow\mathop{\rm Ram}\nolimits\mathcal{P}$ is a homotopy equivalence? For a while we thought that no additional condition on $\mathcal{P}$ should be imposed; i.e., $\mathop{\rm Ram}\nolimits\mathcal{P}$ is always a $\mathrm{CAT}[0]$ space. But then we found a counterexample in dimension 4 and higher; see Theorem 10. Nevertheless, Theorem 6 joined with Zalgaler’s Theorem 3 imply that no additional condition is needed if $\mathop{\rm dim}\nolimits\mathcal{P}\leqslant 3$. Theorem 11 also gives an affirmative answer in a particular 4-dimensional case. The latter theorem is used to prove Theorem 1, it also proves [18, Conjecture 8.2]. We don’t know what conditions should be imposed in general if $\mathop{\rm dim}\nolimits\mathcal{P}\geqslant 4$ but would like to formulate a conjecture in one interesting non-trivial case. 2.2. Conjecture. Let $\mathcal{P}$ be a Euclidean polyhedral space with nonnegative curvature in the sense of Alexandrov. Suppose that $\mathcal{P}$ is homeomorphic to $\mathbb{C}\mathrm{P}^{m}$ and its singularities form a complex hyperplane arrangement on $\mathbb{C}\mathrm{P}^{m}$. Then $\mathop{\rm Ram}\nolimits\mathcal{P}$ is $\mathrm{CAT}[0]$ and the inclusion $\tilde{\mathcal{P}}^{\circ}\hookrightarrow\mathop{\rm Ram}\nolimits\mathcal{P}$ is a homotopy equivalence. This conjecture holds for $m=2$ by Theorem 11. Existence of higher-dimensional examples of such polyhedral metrics on $\mathbb{C}\mathrm{P}^{m}$ can be deduced from [11]. Two-convexity of the regular locus. The same argument as in [20] shows that the regular locus $\mathcal{P}^{\circ}$ of a polyhedral space is two-convex, i.e., it satisfies the following property. _Assume $\Delta$ is a flat tetrahedron. Then any locally isometric geodesic immersion in $\mathcal{P}^{\circ}$ of three faces of $\Delta$ which agrees on three common edges can be extended to a locally isometric immersion $\Delta\looparrowright\mathcal{P}^{\circ}$._ From the main result of Alexander, Berg and Bishop in [3], it follows that every simply connected two-convex a flat manifold with a smooth boundary is $\mathrm{CAT}[0]$. Therefore, if one could approximate $(\mathop{\rm Ram}\nolimits\mathcal{P})^{\circ}$ by flat two-convex manifolds with smooth boundary, Alexander–Berg–Bishop theorem would imply that $\mathop{\rm Ram}\nolimits\mathcal{P}\in\nobreak\mathrm{CAT}[0]$. This looks as a nice plan to approach the problem, but it turns out that such a smoothing does not exist even for the action $\mathbb{Z}_{2}^{2}\curvearrowright\mathbb{C}^{2}$ which changes the signs of the coordinates; see the discussion after Proposition 5.3 in [20] or the solution of Problem 42222The numeration might change with time. in [22] for more details. Ramification around a subset. Given a subset $A$ in a metric space $X$, define $\mathop{\rm Ram}\nolimits_{A}X$ as the completion of the universal cover of $X\backslash A$. Then results of Charney and Davis in [8] imply the following: 1. (i) Let $x$, $y$ and $z$ be distinct points in $\SS^{2}$. Then $\mathop{\rm Ram}\nolimits_{\\{x,y,z\\}}\SS^{2}\in\mathrm{CAT}[1]$ if and only if the triangle $[xyz]$ has perimeter $2{\hskip 0.5pt\cdot\nobreak\hskip 0.5pt}\pi$. In particular the points $x$, $y$ and $z$ lie on a great circle of $\SS^{2}$. 2. (ii) Let $X$, $Y$ and $Z$ be disjoint great circles in $\SS^{3}$. Then $\mathop{\rm Ram}\nolimits_{X\cup Y\cup Z}\SS^{3}\in\mathrm{CAT}[1]$ if and only if $X$, $Y$ and $Z$ are fibers of the Hopf fibration $\SS^{3}\to\SS^{2}$ and their images $x,y,z\in\SS^{2}$ satisfy condition (i)333More precisely, the quotient metric on the base $\SS^{2}$ has curvature $4$, so $[xyz]$ should have perimeter $\pi$.. The following two observations give a link between the above results and Question 2. It turns out that if $\mathcal{P}_{n}$ is a sequence of 2-dimensional spherical polyhedral spaces with exactly 3 singular points that approach $\SS^{2}$ in the sense of Gromov–Hausdorff then the limit position of singular points on $\SS^{2}$ satisfies (i). With a bit more work one can show a similar statement holds in the 3-dimensional case. More precisely, let $\mathcal{P}_{n}$ be a sequence of 3-dimensional spherical polyhedral spaces with the singular locus formed by exactly 3 circles. If $\mathcal{P}_{n}$ approaches $\SS^{3}$ in the sense of Gromov–Hausdorff then the limit position of singular locus satisfies (ii). We finish the discussion with one more conjecture. 2.3. Conjecture. Let $\mathcal{H}$ be a complex hyperplane arrangement in $\mathbb{C}^{m}$ . Then $\mathop{\rm Ram}\nolimits_{\mathcal{H}}\mathbb{C}^{m}$ is $\mathrm{CAT}[0]$ if and only if the following condition holds. Let $\ell$ be any complex hyperline444i.e., an affine subspace of complex codimension 2 that belongs to more than one complex hyperplane of $\mathcal{H}$. Then for any complex hyperplane $h\subset\mathbb{C}^{m}$ containing $\ell$ there is a hyperplane $h^{\prime}\in\mathcal{H}$ containing $\ell$ such that the angle between $h^{\prime}$ and $h$ is at most $\frac{\pi}{4}$. Note that all complex reflection hyperplane arrangements satisfy the conditions of this conjecture. The two-dimensional version of this conjecture is Corollary 8, and the “only if” part follows from this corollary. If this conjecture holds then, using the orbi-space Hadamard–Cartan theorem 3 and Allcock’s lemma 3 in the same way as in the proof of Theorem 1, one shows that the inclusion $(\mathop{\rm Ram}\nolimits_{\mathcal{H}}\mathbb{C}^{m})^{\circ}\hookrightarrow(\mathop{\rm Ram}\nolimits_{\mathcal{H}}\mathbb{C}^{m})$ is a homotopy equivalence. Hence this conjecture gives and alternative geometric approach to Bessis’s result [5] on asphericity of complements to complex reflection arrangements. ## 3 Preliminaries Three types of ramifications. Recall that we consider three types of ramifications which are closely related: for group actions, for polyhedral spaces and for subsets. * $\diamond$ Given a subset $A$ in a metric space $X$, we define $\mathop{\rm Ram}\nolimits_{A}X$ as the completion of the universal cover of $X\backslash A$. We assume here that $X\backslash A$ is connected. * $\diamond$ Given a polyhedral space $\mathcal{P}$ (Euclidean, spherical or hyperbolic), the ramification $\mathop{\rm Ram}\nolimits\mathcal{P}$ is defined as $\mathop{\rm Ram}\nolimits_{A}\mathcal{P}$, where $A$ is the singular locus of $\mathcal{P}$. * $\diamond$ Given an isometric and orientation-preserving action $\Gamma\curvearrowright\mathbb{R}^{m}$, the ramification $\mathop{\rm Ram}\nolimits_{\Gamma}\mathcal{P}$ can be defined as $\mathop{\rm Ram}\nolimits(\mathbb{R}^{m}/\Gamma)$; this definition makes sense since $\mathbb{R}^{m}/\Gamma$ is a polyhedral space. Curvature bounds for polyhedral spaces. _A Euclidean polyhedral space_ is a simplicial complex equipped with an intrinsic metric such that each simplex is isometric to a simplex in a Euclidean space. _A spherical polyhedral space_ is a simplicial complex equipped with an intrinsic metric such that each simplex is isometric to a simplex in a unit sphere. The link of any simplex in a polyhedral space (Euclidean or spherical), equipped with the angle metric forms a spherical polyhedral space. The following two propositions give a more combinatorial description of polyhedral spaces with curvature bounded from below or above. 3.1. Proposition. An $m$-dimensional Euclidean (spherical) polyhedral space $\mathcal{P}$ has curvature $\geqslant 0$ (correspondingly $\geqslant 1$) in the sense of Alexandrov if and only if 1. 1. The link of any $(m-1)$-simplex is isometric to the one-point space $\mathfrak{p}$ or $\SS^{0}$; i.e., the two-point space with distance $\pi$ between the distinct points. 2. 2. The link of any $(m-2)$-simplex is isometric to a closed segment of length $\leqslant\pi$ or a circle with length $\leqslant 2{\hskip 0.5pt\cdot\nobreak\hskip 0.5pt}\pi$. 3. 3. The link of any $k$-simplex with $k\leqslant m-2$, is connected. 3.2. Corollary. The simplicial complex of any polyhedral space $\mathcal{P}$ with a lower curvature bound is a pseudomanifold. The following proposition follows from Hadamard–Cartan theorem and its analogue is proved by Bowditch in [6]; see also [1], where both theorems are proved nicely. 3.3. Proposition. A polyhedral space $\mathcal{P}$ is a $\mathrm{CAT}[0]$ space if and only if $\mathcal{P}$ is simply connected and the link of each vertex is a $\mathrm{CAT}[1]$ space. A spherical polyhedral space $\mathcal{P}$ is a $\mathrm{CAT}[1]$ space if and only if the link of each vertex of $\mathcal{P}$ is a $\mathrm{CAT}[1]$ space and any closed curve of length $<2{\hskip 0.5pt\cdot\nobreak\hskip 0.5pt}\pi$ in $\mathcal{P}$ is null-homotopic in the class of curves of length $<2{\hskip 0.5pt\cdot\nobreak\hskip 0.5pt}\pi$. We say that a polyhedral space has _finite shapes_ if the number of isometry types of simplices that compose it is finite. The following proposition is proved in [7, II. 4.17]. 3.4. Proposition. Let $\mathcal{P}$ be a Euclidean or spherical polyhedral space with finite shapes and suppose that $\mathcal{P}$ has curvature $\leqslant 0$ or $\leqslant 1$ correspondingly. If $\mathcal{P}$ is not a $\mathrm{CAT}[0]$ or a $\mathrm{CAT}[1]$ space, then $\mathcal{P}$ contains an isometrically embedded circle; if the space is spherical it contains such a circle shorter than $2{{\hskip 0.5pt\cdot\nobreak\hskip 0.5pt}}\pi$. Spherical polyhedral metrics on $\bm{\SS^{2}}$. The following theorem appears as an intermediate statement in Zalgaller’s proof of rigidity of spherical polygons; see [21]. 3.5. Zalgaller’s theorem. Let $\Sigma$ be a spherical polyhedral space homeomorphic to the $2$-sphere and with curvature $\geqslant 1$ in the sense of Alexandrov. Assume that there is a point $z\in\Sigma$ such that all singular points lie on the distance $>\tfrac{\pi}{2}$ from $z$. Then $\Sigma$ is isometric to the standard sphere. A sketch of Zalgaller’s proof. We apply an induction on the number $n$ of singular points. The base case $n=1$ is trivial. To do the induction step choose two singular points $p,q\in\Sigma$, cut $\Sigma$ along a geodesic $[pq]$ and patch the hole so that the obtained new polyhedron $\Sigma^{\prime}$ has curvature $\geqslant 1$. The patch is obtained by doubling555Given a metric length space $X$ with a closed subset $A\subset X$, the _doubling_ of $X$ across $A$ is obtained by gluing two copies of $X$ along $A$. a convex spherical triangle across two sides. For a unique choice of triangle, the points $p$ and $q$ become regular in $\Sigma^{\prime}$ and exactly one new singular point appears in the patch.666This patch construction was introduced by Alexandrov, the earliest reference we found is [4, VI, §7]. This way, the case with $n$ singular points is reduced to the case with $n-1$ singular points. ∎ A test for homotopy equivalence. The following lemma is a slight modification of Lemma 6.2 in [2]; the proofs of these lemmas are almost identical. 3.6. Allcock’s lemma. Let $\mathcal{S}$ be an $m$-dimensional _pure_ 777i.e., each simplex in $\mathcal{S}$ forms a face in an $m$-dimensional simplex. simplicial complex. Let $K$ be a subcomplex in $\mathcal{S}$ of codimension $\geqslant 1$; set $W=\mathcal{S}\backslash K$. Assume that the link in $\mathcal{S}$ of any simplex in $K$ is contractible. Then the inclusion map $W\hookrightarrow\mathcal{S}$ is a homotopy equivalence. Proof. Denote by $K_{n}$ the $n$-skeleton of $K$; set $W_{n}=\mathcal{S}\backslash K_{n}$ and set $K_{-1}=\varnothing$. For each $n\in\\{0,1,\dots,m-1\\}$ we will construct a homotopy $F_{n}\colon[0,1]\times W_{n-1}\to W_{n-1}$ of the identity map $\mathop{\rm id}\nolimits_{W_{n-1}}$ into a map with the target in $W_{n}$. Note that $W_{m-1}=W$ and $W_{-1}=\mathcal{S}$. Therefore joining all the homotopies $F_{n}$, we construct a homotopy of the identity map on ${\mathcal{S}}$ into a map with the target in $W$. Hence the lemma will follow once we construct $F_{n}$ for all $n$. Existence of $F_{n}$. Note that each open $n$-dimensional simplex $\Delta$ in $\mathcal{S}$ admits a closed neighborhood $N_{\Delta}$ in $W_{n-1}$ which is homeomorphic to $\Delta\times(\mathfrak{p}\star\mathop{\rm Link}\Delta),$ where $\mathop{\rm Link}\Delta$ denotes the link of $\Delta$, $\mathfrak{p}$ denotes a one-point complex, and ${\star}$ denotes the join. Moreover, we can assume that $\Delta$ lies in $N_{\Delta}=\Delta\times(\mathfrak{p}\mathop{\rm Link}\Delta)$ as $\Delta\times\mathfrak{p}$ and $N_{\Delta}\cap N_{\Delta^{\prime}}=\varnothing$ for any two open $(n-1)$-dimensional simplexes $\Delta$ and $\Delta^{\prime}$ in $\mathcal{S}$. Note that if $\mathop{\rm Link}\Delta$ is contractible then $\mathop{\rm Link}\Delta$ is a deformation retract of $\mathfrak{p}\star\mathop{\rm Link}\Delta$. It follows that for any $(n-1)$-dimensional simplex $\Delta$ in $K$, the relative boundary $\partial_{W_{n-1}}N_{\Delta}$ is a deformation retract of $N_{\Delta}$. Clearly $\partial_{W_{n-1}}N_{\Delta}\subset W_{n}$. Hence the existence of $F_{n}$ follows. ∎ An orbi-space version of Hadamard–Cartan theorem. 3.7. Proposition. Let $\mathcal{P}$ be a polyhedral pseudomanifold. Suppose that for any point $x\in\mathcal{P}$ the ramification of the cone at $x$ is $\mathrm{CAT}[0]$. Then 1. (i) $\mathop{\rm Ram}\nolimits\mathcal{P}$ is $\mathrm{CAT}[0]$. 2. (ii) For any $y\in\mathop{\rm Ram}\nolimits\mathcal{P}$ that projects to $x\in\mathcal{P}$ the cone at $y$ is isometric to the ramification of the cone at $x$. The proposition can be proved along the same lines as Hadamard–Cartan theorem; see for example [1]. A closely related statement was rigorously proved by Haefliger in [14]; he shows that if the charts of an orbi-space are $\mathrm{CAT}[0]$ then its universal orbi-cover is $\mathrm{CAT}[0]$. Haefliger’s definition of orbi- space restricts only to finite isotropy groups, but the above proposition requires only minor modifications of Haefliger’s proof. Polyhedral Kähler manifolds. Let us recall some definitions and results from [18] concerning polyhedral Kähler manifolds. We will restrict our consideration to the case of non- negatively curved polyhedra. 3.8. Definition. Let $\mathcal{P}$ be an orientable non-negatively curved Euclidean polyhedral manifold on dimension $2{{\hskip 0.5pt\cdot\nobreak\hskip 0.5pt}}n$. We say that $\mathcal{P}$ is _polyhedral Kähler_ if the holonomy of the metric on $\mathcal{P}^{\circ}$ belongs to $\mathrm{U}(n)\subset\mathrm{SO}(2{\hskip 0.5pt\cdot\nobreak\hskip 0.5pt}n)$. In the case when $\mathcal{P}$ is a metric cone piecewise linearly isomorphic to $\mathbb{R}^{2{\hskip 0.5pt\cdot\nobreak\hskip 0.5pt}n}$ we call it a _polyhedral Kähler cone_. Recall that from a result of Cheeger (see [10] and [18, Proposition 2.3]) it follows that the metric of an orientable simply connected non-negatively curved polyhedral compact $4$-manifold not homeomorphic to $\SS^{4}$ has unitary holonomy. Moreover in the case when the (unitary) holonomy is irreducible, the manifold has to be diffeomorphic to $\mathbb{C}\mathrm{P}^{2}$. Metric singularities form a collection of complex curves on $\mathbb{C}\mathrm{P}^{2}$; see [18] for the details. The following theorem summarizes some results on non-negatively curved $4$-dimensional polyhedral Kähler cones proven in [18, Theorems 1.5, 1.7, 1.8]. 3.9. Theorem. Let $\mathcal{C}^{4}$ be a non-negatively curved polyhedral Kähler cone and let $\Sigma$ be the unit sphere of this cone. 1. (a) There is a canonical isometric $\mathbb{R}$-action on $\mathcal{C}$ such that its orbits on $\Sigma$ are geodesics. This action is generated by the vector field $J(r\frac{\partial}{\partial r})$ in the non-singular part of $\mathcal{C}$, where $J$ is the complex structure on $\mathcal{C}$ and $r\frac{\partial}{\partial r}$ is the radial vector field on $\mathcal{C}$. 2. (b) If the metric singularities of the cone are topologically equivalent to a collection of $n\geqslant 3$ complex lines in $\mathbb{C}^{2}$, then the action of $\mathbb{R}$ on $\Sigma$ factors through $\SS^{1}$ and the map $\Sigma\to\Sigma/\SS^{1}$ is the Hopf fibration. 3. (c) If the metric singularities of the cone are topologically equivalent to a union of $2$ complex lines, then the cone splits as a metric product of two $2$-dimensional cones. Reshetnyak gluing theorem. Let us recall the formulation of Reshetnyak gluing theorem which will be used in the proof of Proposition 4. 3.10. Theorem. Suppose that $\mathcal{U}_{1},\mathcal{U}_{2}$ are $\mathrm{CAT}[\kappa]$ spaces888We always assume that $\mathrm{CAT}[\kappa]$ spaces are complete. with closed convex subsets $A_{i}\subset\mathcal{U}_{i}$ which admit an isometry $\iota\colon A_{1}\to A_{2}$. Let us define a new space $W$ by gluing $\mathcal{U}_{1}$ and $\mathcal{U}_{2}$ along the isometry $\iota$; i.e., consider the new space $\mathcal{W}=\mathcal{U}_{1}\sqcup_{\sim}\mathcal{U}_{2}$ where the equivalence relation $\sim$ is defined by $a\sim\iota(a)$ with the induced length metric. Then the following holds. The space $\mathcal{W}$ is $\mathrm{CAT}[\kappa]$. Moreover, both canonical mappings $\tau_{i}\colon\mathcal{U}_{i}\to\mathcal{W}$ are distance preserving, and the images $\tau_{i}(\mathcal{U}_{i})$ are convex subsets in $\mathcal{W}$. The following corollary is proved by repeated application of Reshetnyak’s theorem. 3.11. Corollary. Let $S$ be a finite tree. Assume a convex Euclidean (or spherical) polyhedron $Q_{\nu}$ corresponds to each node $\nu$ in $S$ and for each edge $[\nu\mu]$ in $S$ there is an isometry $\iota_{\mu\nu}$ from a facet999Facet is a face of codimension 1 $F\subset Q_{\nu}$ to a facet $F^{\prime}\subset Q_{\mu}$. Then the space obtained by gluing all the polyhedra $Q_{\nu}$ along the isometries $\iota_{\mu\nu}$ forms a $\mathrm{CAT}[0]$ space (correspondingly a $\mathrm{CAT}[1]$ space). Flag complexes. Let us recall the definition of a _flag complex_. 3.12. Definition. A simplicial complex $\mathcal{S}$ is _flag_ if whenever $\\{v_{0},\dots\nobreak,v_{k}\\}$ is a set of distinct vertices which are pairwise joined by edges, then $\\{v_{0},\dots,v_{k}\\}$ spans a $k$-simplex in $\mathcal{S}$. Note that every flag complex is determined by its 1-skeleton. Spherical polyhedral $\mathrm{CAT}[1]$ spaces glued from right-angled simplices admit the following combinatorial characterization discovered by Gromov [13, p. 122]. 3.13. Theorem. A piecewise spherical simplicial complex made of right-angled simplices is a $\mathrm{CAT}[1]$ space if and only if it is a flag complex. ## 4 On the reflection groups If the singular locus of a polyhedral space $\mathcal{P}$ coincides with its $(m-2)$-skeleton then $\mathcal{P}^{\circ}$ has the homotopy type of a graph (it vertices correspond to the centers of simplices of $\mathcal{P}$). We will show that in this case the Ramification conjecture can be proven by applying Reshetnyak gluing theorem recursively. We will prove the following stronger statement. 4.1. Proposition. Assume $\mathcal{P}$ is an $m$-dimensional polyhedral space which admits a subdivision into closed sets $\\{Q_{i}\\}$ such that each $Q_{i}$ with the induced metric is isometric to a convex $m$-dimensional polyhedron and each face of dimension $m-2$ of each polyhedron $Q_{i}$ belongs to the singular locus of $\mathcal{P}$. Then $\mathop{\rm Ram}\nolimits\mathcal{P}\in\mathrm{CAT}[0]$. Note that Theorem 1$(\mathrm{R}^{+})$ follows directly from the above proposition. Also the condition in the above proposition holds if $\mathcal{P}$ is isometric to the boundary of a convex polyhedron in Euclidean space and in particular, by Alexandrov’s theorem it holds if $\mathcal{P}$ is homeomorphic to $\SS^{2}$. Proof of Proposition 4. In the subdivision of $\mathcal{P}$ into $Q_{i}$, color all the facetsin different colors. Consider the graph $\Gamma$ with a node for each $Q_{i}$, where two nodes are connected by an edge if the correspondent polyhedra have a common facet. Color each edge of $\Gamma$ in the color of the corresponding facet. Denote by $\tilde{\Gamma}$ the universal cover of $\Gamma$. ($\tilde{\Gamma}$ has to be a tree.) For each node $\nu$ of $\tilde{\Gamma}$, prepare a copy of $Q_{i}$ which corresponds to the projection of $\nu$ in $\Gamma$. Note that the space $\mathop{\rm Ram}\nolimits\mathcal{P}$ can be obtained by gluing the prepared copies. Two copies should be glued along two facets of the same color $z$ if the nodes corresponding to these copies are connected in $\tilde{\Gamma}$ by an edge of color $z$. Given a finite subtree $S$ of $\tilde{\Gamma}$ consider the subset $Q_{S}\subset\mathop{\rm Ram}\nolimits\mathcal{P}$ formed by all the copies of $Q_{i}$ corresponding to the nodes of $S$. Note that $Q_{S}$ is a convex subset of $\mathop{\rm Ram}\nolimits\mathcal{P}$. Indeed, if a path between points of $Q_{S}$ escapes from $Q_{S}$, it has to cross the boundary $\partial Q_{S}$ at the same facet twice, say at the points $x$ and $y$ in a facet $F\subset\partial Q_{S}$. Further note that the natural projection $\mathop{\rm Ram}\nolimits\mathcal{P}\to\mathcal{P}$ is a short map which is distance preserving on $F$. Therefore there is a unique geodesic from $x$ to $y$ and it lies in $F$. In particular, geodesic with ends in $Q_{S}$ can not escape from $Q_{S}$; in other words $Q_{S}$ is convex. Finally, by Corollary 3, the subspace $Q_{S}$ is $\mathrm{CAT}[0]$ for any finite subtree $S$. Clearly, for every triangle $\triangle$ in $\mathop{\rm Ram}\nolimits\mathcal{P}$ there is a finite subtree $S$ such that $Q_{S}\supset\triangle$. Therefore the $\mathrm{CAT}[0]$ comparison holds for any geodesic triangle in $\mathop{\rm Ram}\nolimits\mathcal{P}$. ∎ ## 5 Case $(\mathbb{Z}_{2})$ In this section we reduce the case $(\mathbb{Z}_{2})$ of Theorem 1 to the case $(\mathrm{R}^{+})$. Proof of Theorem 1; case $(\mathbb{Z}_{2})$. Every orientation preserving action of a group $\mathbb{Z}_{2}^{k}$ on $\mathbb{R}^{m}$ arises as the action of a subgroup of the group $\mathbb{Z}_{2}^{m}$ generated by reflections in coordinate hyperplanes. By the definition of ramification, we can assume that the action of $\mathbb{Z}_{2}^{k}$ is generated by reflections in hyperlines. Let us write $i\sim j$ if $i=j$ or the reflection in the hyperline $x_{i}=x_{j}=0$ belongs to $\Gamma$. Note that $\sim$ is an equivalence relation. It follows that $\mathbb{R}^{m}/\Gamma$ splits as a direct product of the subspaces corresponding to the coordinate subspaces of $\mathbb{R}^{m}$ for each equivalence relation. Finally, for each of the factors in this splitting, the statement holds by Theorem 1 $(\mathrm{R}^{+})$. ∎ ## 6 Two dimensional spaces 6.1. Definition. An $m$-dimensional spherical polyhedral space $\Sigma$ is called _$\alpha$ -extendable_ if for any $\varepsilon>0$, every isometric immersion into $\Sigma$ of a ball of radius $\alpha+\epsilon$ from $\SS^{m}$ extends to an isometric immersion of the whole $\SS^{m}$. In other words $\Sigma$ is $\alpha$-extendable if either the distance from any point $x\in\Sigma$ to its singular locus $\Sigma^{{\star}}$ is at most $\alpha$ or $\Sigma$ is a space form. 6.2. Theorem. Let $\Sigma$ be a 2-dimensional spherical polyhedral manifold. Then $\mathop{\rm Ram}\nolimits\Sigma$ is $\mathrm{CAT}[1]$ if and only if $\Sigma$ is $\frac{\pi}{2}$-extendable. Proof. Note that if $\Sigma^{{\star}}=\varnothing$ then $\Sigma$ is a spherical space form. So we assume that $\Sigma^{{\star}}\neq\varnothing$. Let us show that in the case when $\Sigma$ is $\frac{\pi}{2}$-extendable one can decompose $\Sigma$ into a collection of convex spherical polygons with vertices in $\Sigma^{{\star}}$. The proof is almost identical to the proof of [23, Proposition 3.1], so we just recall the construction. In the case if $\Sigma^{{\star}}$ consist of two points, $\Sigma$ can be decomposed into a collection of two-gons. It remains to consider the case when $\Sigma^{{\star}}$ has at least 3 distinct points. Consider the Voronoi decomposition of $\Sigma$ with respect to the points in $\Sigma^{{\star}}$. The vertices of this decomposition consist of points $x$ that have the following property. If $D$ is the maximal open ball in $\Sigma^{\circ}=\Sigma\backslash\Sigma^{{\star}}$ with the center at $x$, then the radius of $D$ is at most $\tfrac{\pi}{2}$ and the the convex hull of points in $\partial D\cap\Sigma^{{\star}}$ contains $x$. Note that such a convex hull is a convex spherical polygon $P(x)$ and $\Sigma$ is decomposed in the union of $P(x)$ for various vertices $x$. Consider finally the Euclidean cone over $\Sigma$ with the induced decomposition into cones over spherical polygons. Applying Proposition 4 to the cone we see that its ramification is $\mathrm{CAT}[0]$. So $\mathop{\rm Ram}\nolimits\Sigma\in\mathrm{CAT}[1]$ by Proposition 3. ∎ The next result follows directly from Theorem 6 and Proposition 3. 6.3. Theorem. Let $\mathcal{Y}$ be a 3-dimensional polyhedral cone. Then $\mathop{\rm Ram}\nolimits\mathcal{Y}$ is $\mathrm{CAT}[0]$ if and only if $\mathcal{Y}$ satisfies one of the following conditions. 1. 1. The singular locus $\mathcal{Y}^{\star}$ is formed by the tip or it is empty. 2. 2. For any direction $v\in\mathcal{Y}$ there is a direction $w\in\mathcal{Y}^{\star}$ such that $\measuredangle(v,w)\leqslant\tfrac{\pi}{2}$. Indeed, the link $\Sigma$ of $\mathcal{Y}$ is a space form if an only if $\mathcal{Y}^{\star}$ is formed by the tip or it is empty. If $\mathcal{Y}^{\star}$ contains more than one point the condition $2$ of this theorem means literally that $\Sigma$ is $\frac{\pi}{2}$-extendable. ## 7 Case $(\mathbb{R}^{3})$ Here we present two proofs of Theorem 1 case $(\mathbb{R}^{3})$, the first one is based on Theorem 6 and the second on Theorem 1 case $(\mathrm{R}^{+})$. Proof 1. Suppose first that $\Gamma$ is finite. In this case, without loss of generality we may assume that $\Gamma$ fixes the origin. By Zalgaller’s theorem 3, the link of the origin in the quotient $\mathbb{R}^{3}/\Gamma$ is $\tfrac{\pi}{2}$-extendable. Applying Theorem 6, we get the result. The case when $\Gamma$ is infinite is treated in the same way as the $\mathbb{C}^{2}$ case, see Section 9. ∎ Proof 2. Suppose first that $\Gamma$ is finite. By Theorem 1 it is sufficient to prove that $\Gamma$ is an index two subgroup in a group $\Gamma_{1}$ generated by reflections in planes. If $\Gamma$ fixes a line in $\mathbb{R}^{3}$ then it is a cyclic group and it is an index two subgroup of a dihedral group. Otherwise $\SS^{2}/\Gamma$ is an orbifold with three orbi-points glued from two copies of a Coxeter spherical triangle. Such an orbifold has an involution $\sigma$ such that $(\SS^{2}/\Gamma)/\sigma$ is a Coxeter triangle $\Delta$. So $\Gamma_{1}$ is the group generated by reflections in the sides of $\Delta$. The case when $\Gamma$ is infinite is treated in the same way as the $\mathbb{C}^{2}$ case, see Section 9. ∎ ## 8 3-spaces with a geodesic actions The following theorem is the main technical result. 8.1. Theorem. Let $\Sigma$ be a 3-dimensional spherical polyhedral manifold. Assume $\Sigma$ admits an isometric action of $\mathbb{R}$ with geodesic orbits. Then $\mathop{\rm Ram}\nolimits\Sigma$ is $\mathrm{CAT}[1]$ if an only if $\Sigma$ is $\frac{\pi}{4}$-extendable or $\mathop{\rm Ram}\nolimits\Sigma$ is the completion of the universal cover of $\SS^{3}\setminus\SS^{1}$. Example. We will further apply this theorem to unit spheres of polyhedral cones that are quotients of $\mathbb{C}^{2}$ by a finite group of unitary isometries. The action of $\mathbb{R}$ in this case comes from the action on $\mathbb{C}^{2}$ by multiplication by complex units. The proof of Theorem 8 relies on several lemmas. The following lemma is spherical analogue of [22, Problem 3]101010The number might change, if so please check an earlier version.. 8.2. Drop lemma. Let $D$ be a disk with a metric of curvature $1$, whose boundary consists of several smooth arcs of curvature at most $\kappa$ that meet at angles larger than $\pi$ at all points except at most one. Then the following hold 1. (a) $D$ contains an isometric copy of a disk whose boundary has curvature $\kappa$. 2. (b) If the length of $\partial D$ is less than the length of the circle with curvature $\kappa$ on the unit sphere then $D$ contains an isometric copy of a unit half sphere. Proof; (a). Recall that the cut locus of $D$ with respect to its boundary $\partial D$ is defined as the closure of the subset of points $x$ in $D$ such that the restriction of the distance function $\mathop{\rm dist}\nolimits_{x}\|_{\partial D}$ attains its global minimum at two or more points of $\partial D$. The cut locus will be denoted as $\mathop{\rm CutLoc}\nolimits D$. After a small perturbation of $\partial D$ we may assume that $\mathop{\rm CutLoc}\nolimits D$ is a graph embedded in $D$ with finite number of edges. $z$$\bar{z}$$y$$\bar{y}$ Note that $\mathop{\rm CutLoc}\nolimits D$ is a deformation retract of $D$. The retraction can be obtained by moving each point $y\in D\setminus\mathop{\rm CutLoc}\nolimits D$ towards $\mathop{\rm CutLoc}\nolimits D$ along the geodesic containing $y$ and the point $\bar{y}\in\partial D$ closest to $y$. In particular $\mathop{\rm CutLoc}\nolimits D$ is a tree. Since $\mathop{\rm CutLoc}\nolimits D$ is a tree, it has at least two vertices of valence one. Among all points of $\partial D$ only the non-smooth point of $\partial D$ with angle less than $\pi$ belongs to $\mathop{\rm CutLoc}\nolimits D$. So there is at least one point $z$ of $\mathop{\rm CutLoc}\nolimits D$ of valence one contained in the interior of $D$. The point $z$ has to be a focal point of $\partial D$; this means that the disk of radius $\mathop{\rm dist}\nolimits_{\partial D}z$ centered at $z$ touches $\partial D$ with multiplicity at least two at some point $\bar{z}$. At $\bar{z}$ the curvature of the boundary of the disk centred at $z$ equals the curvature of $\partial D$ and so it is at most $\kappa$. So this disk contains a disk with boundary of curvature $\kappa$. (b). By (a) we can assume that $\kappa>0$. Consider a locally isometric immersion of $D$ into the unit sphere, $\varphi:D\looparrowright\SS^{2}$. Since the length of $\partial D$ is less than $2{{\hskip 0.5pt\cdot\nobreak\hskip 0.5pt}}\pi$, by Crofton’s formula, $\partial D$ does not intersect one of equators. Therefore the curve $\varphi(\partial D)$ is contained in a half sphere, say $\SS^{2}_{+}$. Note that it is sufficient to show that $\varphi(D)$ contains the complement of $\SS^{2}_{+}$. Suppose the contrary; note that in this case $\varphi(D)\subset\SS^{2}_{+}$. Applying (a), we get that $\varphi(D)$ contains a disc bounded by a circle, say $\sigma_{\kappa}$, of curvature $\kappa$. Note that $\partial[\varphi(D)]$ cuts $\sigma_{\kappa}$ from its antipodal circle; therefore $\mathop{\rm length}\nolimits\partial[\varphi(D)]\geqslant\mathop{\rm length}\nolimits\sigma_{\kappa}.$ Note that $\mathop{\rm length}\nolimits\partial D\geqslant\mathop{\rm length}\nolimits\partial[\varphi(D)].$ On the other hand, by the assumptions $\mathop{\rm length}\nolimits\partial D<\mathop{\rm length}\nolimits\sigma_{\kappa},$ a contradiction. ∎ 8.3. Lemma. Assume that $\Sigma$ is a spherical polyhedral $3$-manifold with an isometric $\mathbb{R}$-action, whose orbits are geodesic. Then the quotient $\Lambda=(\mathop{\rm Ram}\nolimits\Sigma)/\mathbb{R}$ is a spherical polyhedral surface of curvature $4$, and there are two possibilities. 1. (a) If $\Lambda$ is not contractible then it is isometric to the sphere of curvature $4$, further denoted as $\tfrac{1}{2}{\hskip 0.5pt\cdot\nobreak\hskip 0.5pt}\SS^{2}$. In this case, $\mathop{\rm Ram}\nolimits\Sigma$ is isometric to the unit $\SS^{3}$ or to $\mathop{\rm Ram}\nolimits_{\SS^{1}}\SS^{3}$, where $\SS^{1}$ is a closed geodesic in $\SS^{3}$. 2. (b) If $\Lambda$ is contractible then a point $x\in\mathop{\rm Ram}\nolimits\Sigma$ is singular if an only if so is its projection $\bar{x}\in\Lambda$. Moreover, the angle around each singular point $\bar{x}\in\Lambda$ is infinite. Proof. We will consider two cases. Case 1. Assume the action $\mathbb{R}\curvearrowright\Sigma$ is not periodic; i.e., it does not factor through an $\SS^{1}$-action. Then the group of isometries of $\Sigma$ contains $\mathbb{T}^{2}$. From [18, Proposition 3.9] one can deduce that the Euclidean cone over $\mathop{\rm Ram}\nolimits\Sigma$ is isometric to the ramification of $\mathbb{R}^{4}$ in one $2$-plane or in a pair of two orthogonal $2$-planes. It follows that $\mathop{\rm Ram}\nolimits\Sigma$ is either $\mathop{\rm Ram}\nolimits_{\SS^{1}}\SS^{3}\ \ \mathrm{or}\ \ \mathop{\rm Ram}\nolimits_{\SS_{a}^{1}\cup\SS_{b}^{1}}\SS^{3}$ where $\SS_{a}^{1}$ and $\SS_{b}^{1}$ are two opposite Hopf circles. In both cases, the $\mathbb{R}$-action is lifted from the Hopf’s $\SS^{1}$-action on $\SS^{3}$. If $\mathop{\rm Ram}\nolimits\Sigma=\mathop{\rm Ram}\nolimits_{\SS^{1}}\SS^{3}$ then $\Lambda=\tfrac{1}{2}{\hskip 0.5pt\cdot\nobreak\hskip 0.5pt}\SS^{2}$ and therefore (a) holds. If $\mathop{\rm Ram}\nolimits\Sigma=\mathop{\rm Ram}\nolimits_{\SS_{a}^{1}\cup\SS_{b}^{1}}\SS^{3}$ then $\Lambda=\mathop{\rm Ram}\nolimits_{\\{a,b\\}}(\tfrac{1}{2}{\hskip 0.5pt\cdot\nobreak\hskip 0.5pt}\SS^{2})$ where $a$ and $b$ are two poles of the sphere; therefore (b) holds. Case 2. Assume that the $\mathbb{R}$-action is periodic. Let $s$ be the number of orbits in the singular locus $\Sigma^{{\star}}$ and let $m$ be the number of multiple orbits in the regular locus $\Sigma^{\circ}$. Note that the space $\Sigma^{\circ}/\SS^{1}$ is an orbifold with constant curvature $4$; it has $m$ orbi-points. Passing to the completion of $\Sigma^{\circ}/\SS^{1}$, we get $\Sigma/\SS^{1}$. This way we add $s$ points to $\Sigma^{\circ}/\SS^{1}$ which we will call the _punctures_ ; this is a finite set of points formed by the projection of the singular locus $\Sigma^{{\star}}$ in the quotient space $\Sigma/\SS^{1}$. Now we will consider a few subcases. Assume $s=0$; in other words $\Sigma^{{\star}}=\varnothing$. Then $\mathop{\rm Ram}\nolimits\Sigma$ is isometric to $\SS^{3}$ and the $\mathbb{R}$-action factors through the standard Hopf action; i.e., the first part of (a) holds. Assume either $s\geqslant 2$ or $s\geqslant 1$ and $m\geqslant 2$. Then the orbifold fundamental group of $\Sigma^{\circ}/\SS^{1}$ is infinite, the universal orbi-cover is a disk, and it branches infinitely over every puncture of $\Sigma/\SS^{1}$. The completion of the cover is contractible; i.e., (b) holds. It remains to consider the subcase $s=1$ and $m=1$. In this subcase the universal orbi-cover of $\Sigma^{\circ}/\SS^{1}$ is a once punctured $\SS^{2}$ of curvature $4$ and $\mathop{\rm Ram}\nolimits\Sigma=\mathop{\rm Ram}\nolimits_{\SS^{1}}\SS^{3}$; i.e., (a) holds. ∎ Proof of Theorem 8. Suppose first $\Lambda=(\mathop{\rm Ram}\nolimits\Sigma)/\mathbb{R}$ is not contractible. By Lemma 8, $\Lambda$ is isometric to $\SS^{2}$, and the ramification $\mathop{\rm Ram}\nolimits\Sigma$ is either isometric to $\SS^{3}$ or $\mathop{\rm Ram}\nolimits_{\SS^{1}}\SS^{3}$. Both of these spaces are $\mathrm{CAT}[1]$; so the theorem follows. From now on we consider the case when $\Lambda$ is contractible and will prove in this case that $\mathop{\rm Ram}\nolimits\Sigma\in\mathrm{CAT}[1]$ if and only if $\Sigma$ is $\frac{\pi}{4}$-extendable. If part. From Lemma 8 it follows that $\mathop{\rm Ram}\nolimits\Sigma$ branches infinitely over singular circles of $\Sigma$. So $\mathop{\rm Ram}\nolimits\Sigma$ is locally $\mathrm{CAT}[1]$ and we only need to show that any closed geodesic $\gamma$ in $\mathop{\rm Ram}\nolimits\Sigma$ has length at least $2{{\hskip 0.5pt\cdot\nobreak\hskip 0.5pt}}\pi$ (see Proposition 3). Let $\gamma$ be a closed geodesic in $\mathop{\rm Ram}\nolimits\Sigma$; denote by $\bar{\gamma}$ its projection in $\Lambda$. The curve $\bar{\gamma}$ is composed of arks of constant curvature, say $\kappa$, joining singularities of $\Lambda$. Moreover for each singular point $p$ of $\Lambda$ that belongs to $\bar{\gamma}$ the angle between the arcs of $\bar{\gamma}$ at $p$ is at least $\pi$. Both of the above statements are easy to check; the first one is also proved in [19, Lemma 3.1]. The following lemma follows directly from [19, Proposition 3.6 2)]. 8.4. Lemma. Assume $\Sigma$ and $\Lambda$ are as in the formulation of Lemma 8. Then for every geodesic $\gamma$ in $\mathop{\rm Ram}\nolimits\Sigma$ its projection $\bar{\gamma}$ in $\Lambda$ has a point of self-intersections. Summarizing all the above, we can chose two sub loops in $\bar{\gamma}$, say $\bar{\gamma}_{1}$ and $\bar{\gamma}_{2}$, which bound disks on $\mathop{\rm Ram}\nolimits\Sigma/\mathbb{R}$ and both of these disks satisfy the conditions of Lemma 8 for some $\kappa$. Clearly, we can chose $\bar{\gamma}_{1}$ and $\bar{\gamma}_{2}$ so that $\bar{\gamma}_{1}\cap\bar{\gamma}_{2}$ is at most a finite set. By our assumptions the disks bounded by $\bar{\gamma}_{i}$ can not contain points on distance more than $\frac{\pi}{4}$ from their boundary, otherwise $\Sigma$ would not be $\frac{\pi}{4}$-extendable. So we deduce from Lemma 8(b) that $\mathop{\rm length}\nolimits\bar{\gamma}_{i}\geqslant\ell(\kappa),$ $None$ where $\ell(\kappa)=\tfrac{2{\hskip 0.5pt\cdot\nobreak\hskip 0.5pt}\pi}{\sqrt{\kappa^{2}+4}}$ is the length of a circle of curvature $\kappa$ on the sphere of radius $\tfrac{1}{2}$. Let $\alpha$ be an arc of $\gamma$ and $\bar{\alpha}$ be its projection in $\Lambda$. Note that $\mathop{\rm length}\nolimits\alpha=\tfrac{\pi}{\ell(\kappa)}{\hskip 0.5pt\cdot\nobreak\hskip 0.5pt}\mathop{\rm length}\nolimits\bar{\alpha}.$ Together with $({})$, this implies that $\mathop{\rm length}\nolimits\gamma\geqslant 2{{\hskip 0.5pt\cdot\nobreak\hskip 0.5pt}}\pi$. Only if part. Suppose now that $\Sigma$ contains an immersed copy of a ball with radius $\frac{\pi}{4}+\varepsilon$. Consider a lift of this ball to $\mathop{\rm Ram}\nolimits\Sigma$ and denote it by $B$. Set as before $\Lambda=(\mathop{\rm Ram}\nolimits\Sigma)/\mathbb{R}$. The projection of $B$ in $\Lambda$ is a disc, say $D$, of radius $\frac{\pi}{4}+\varepsilon$ and curvature $4$, isometrically immersed in $\Lambda$. Since $\Lambda$ is contractible $D$ has to be embedded in $\Lambda$. Consider a closed geodesic $\bar{\gamma}\subset\Lambda\setminus D$ which is obtained from $\partial D$ by a curve shortening process. Such a geodesic has to contain at least two singular points; let $x$ be one of such points. Choose now a lift of $\bar{\gamma}$ to a horizontal geodesic path $\gamma$ on $\mathop{\rm Ram}\nolimits\Sigma$ with two (possibly distinct) ends at the $\mathbb{R}$-orbit over $x$. Finally consider a deck transformation $\iota$ of $\mathop{\rm Ram}\nolimits\Sigma$ that fixes the $\mathbb{R}$-orbit over $x$ and rotates around it $\mathop{\rm Ram}\nolimits\Sigma$ by an angle larger than $\pi$. The union of $\gamma$ with $\iota\circ\gamma$ forms a closed geodesic in $\mathop{\rm Ram}\nolimits\Sigma$ of length less than $2{{\hskip 0.5pt\cdot\nobreak\hskip 0.5pt}}\pi$. ∎ The following statement is proved by the same methods as in the theorem. 8.5. Corollary from the proof. Let $n\geqslant 2$ be an integer and $X$ be a union of $n$ fibers of the Hopf fibration on the unit $\SS^{3}$. Then $\mathop{\rm Ram}\nolimits_{X}\SS^{3}$ is $\mathrm{CAT}[1]$ if and only if the following condition holds. There is no point on $\SS^{3}$ on distance more than $\frac{\pi}{4}$ from $X$. ## 9 Case $(\mathbb{C}^{2})$ Proof of Theorem 1; case $(\mathbb{C}^{2})$. Let us show that $\mathop{\rm Ram}\nolimits_{\Gamma}\mathbb{C}^{2}$ is $\mathrm{CAT}[0]$. First assume that $\Gamma$ is finite. Without loss of generality, we can assume that the origin is fixed by $\Gamma$. Let $L$ be the union of all the lines in $\mathbb{C}^{2}$ fixed by some non-identity elements of $\Gamma$. Note that $\mathop{\rm Ram}\nolimits_{\Gamma}=\mathop{\rm Ram}\nolimits_{L}\mathbb{C}^{2}$. If $L=\varnothing$ or $L$ is a single line, the statement is clear. Set $\Theta=\SS^{3}\cap L$; this is a union of Hopf circles. If the circles in $\Theta$ satisfy the conditions of Corollary 8 then $\mathop{\rm Ram}\nolimits_{\Theta}\SS^{3}$ is $\mathrm{CAT}[1]$. Therefore $\mathop{\rm Ram}\nolimits_{\Gamma}=\mathop{\rm Ram}\nolimits_{L}\mathbb{C}^{2}=\mathop{\rm Cone}\nolimits(\mathop{\rm Ram}\nolimits_{\Theta}\SS^{3})\in\mathrm{CAT}[0],$ here $\mathop{\rm Ram}\nolimits_{A}X$ denotes the completion of universal cover of $X\backslash A$. Suppose now that the conditions of Corollary 8 are not satisfied. Denote by $\Xi$ the projection of $\Theta$ in $\tfrac{1}{2}{\hskip 0.5pt\cdot\nobreak\hskip 0.5pt}\SS^{2}=\SS^{3}/\SS^{1}$; $\Xi$ is a finite set of points. In this case there is an open half-sphere containing all points $\Xi$. Denote by $P$ be the convex hull of points $\Xi$. Note that $\Xi$ and therefore $P$ are $\Gamma$-invariant sets. Therefore the action on $\SS^{2}$ is cyclic. The latter means that $L$ consists of one line. If $\Gamma$ is infinite, we can apply the above argument to each isotropy group of $\Gamma$. We get that $\mathop{\rm Ram}\nolimits_{\Gamma_{x}}$ is $\mathrm{CAT}[0]$ for the isotropy group $\Gamma_{x}$ at any point $x\in\mathbb{C}^{2}$. Then it remains to apply Proposition 3(i). Now let us show that the inclusion $W_{\Gamma}\hookrightarrow\mathop{\rm Ram}\nolimits_{\Gamma}$ is a homotopy equivalence. Fix a singular point $y$ in $\mathop{\rm Ram}\nolimits_{\Gamma}$ and let $x$ be its projection to $\mathbb{C}^{2}/\Gamma$. By Proposition 3(ii) the link at $y$ is the same as the link of the ramification of the cone at $x$. The latter space is the ramification of $\SS^{3}$ in a non-empty collection of Hopf circles, which is clearly contractible. It remains to apply Allcock’s lemma 3. ∎ ## 10 The counterexample In this section we use the technique introduced above to show that the answer to the Question 2 is negative without additional assumptions on $\mathcal{P}$. 10.1. Theorem. There is a positively curved spherical polyhedral space $\mathcal{P}$ homeomorphic to $\SS^{3}$ with an isometric $\SS^{1}$-action with geodesic orbits, such that $\mathop{\rm Ram}\nolimits\mathcal{P}$ is not $\mathrm{CAT}[1]$. Proof. Consider a triangle $\Delta$ on the sphere of curvature $4$ with one angle $\frac{\pi}{n}$ and the other two $\frac{\pi{\hskip 0.5pt\cdot\nobreak\hskip 0.5pt}(n+1)}{2{\hskip 0.5pt\cdot\nobreak\hskip 0.5pt}n}+\nobreak\varepsilon$; here $n$ is a positive integer and $\varepsilon>0$. Note that two sides of $\Delta$ are longer than $\frac{\pi}{4}$. Denote by $\Lambda$ the doubling of $\Delta$. The space $\Lambda$ is a spherical polyhedral space with curvature $4$; it has three singular points which correspond to the vertices of $\Delta$. Let us call the point with angle $\frac{2{\hskip 0.5pt\cdot\nobreak\hskip 0.5pt}\pi}{n}$ by $x$. According to [18, Theorem 1.8] there is a unique up to isometry polyhedral spherical space $\mathcal{P}$ with an isometric action $\SS^{1}\curvearrowright\mathcal{P}$ such that $\SS^{1}$-orbits are geodesic, $\Lambda$ is isometric to the quotient space $\mathcal{P}/\SS^{1}$ and the point $x$ corresponds to the orbit of multiplicity $n$, while the rest of orbits are simple. Note that the points in $\mathcal{P}$ on the $\SS^{1}$-fiber over $x$ are regular. The distance from this fiber to the singularities of $\SS^{3}$ is more than $\frac{\pi}{4}$; i.e., $\mathcal{P}$ is not $\frac{\pi}{4}$-extendable. By Theorem 8 we conclude that $\mathop{\rm Ram}\nolimits\mathcal{P}$ is not $\mathrm{CAT}[1]$. ∎ ## 11 Line arrangements The following theorem is the main result of this section. 11.1. Theorem. Let $\mathcal{P}$ be a non-negatively curved polyhedral space homeomorphic to $\mathbb{C}\mathrm{P}^{2}$ whose singularities form a complex line arrangement on $\mathbb{C}\mathrm{P}^{2}$. Then $\mathop{\rm Ram}\nolimits\mathcal{P}$ is a $\mathrm{CAT}[0]$ space and the inclusion $(\mathop{\rm Ram}\nolimits\mathcal{P})^{\circ}\hookrightarrow\mathop{\rm Ram}\nolimits\mathcal{P}$ is a homotopy equivalence. It follows that all complex line arrangements in $\mathbb{C}\mathrm{P}^{2}$ appearing as singularities of non-negatively curved polyhedral metrics have aspherical complements. The class of such arrangements is characterized in Theorem 11, this class includes all the arrangements from Theorem 1. Proof of Theorem 11. According to [10] and [18] the metric on $\mathcal{P}$ is polyhedral Kähler. First let us show that $\mathop{\rm Ram}\nolimits\mathcal{P}$ is $\mathrm{CAT}[0]$. By Theorem 3, it is sufficient to show that the ramification of the cone of each singular point $x$ in $\mathcal{P}$ is $\mathrm{CAT}[0]$. If there are exactly two lines meeting at $x$ then the cone of $x$ is a direct product by Theorem 3(c), and the statement is clear. If more than two lines meet at $x$ consider the link $\Sigma$ of the cone at $x$. According to Theorem 3 there is a free $\SS^{1}$-action on $\Sigma$ inducing on it the structure of the Hopf fibration. The quotient $\Sigma/\SS^{1}$ is a $2$-sphere with spherical polyhedral metric of curvature $4$ and the conical angle is at most $2{\hskip 0.5pt\cdot\nobreak\hskip 0.5pt}\pi$ around any point. It follows from Zalgaller’s theorem that $\Sigma$ is $\frac{\pi}{4}$-extendable. So by Theorem 8 $\mathop{\rm Ram}\nolimits\Sigma$ is $\mathrm{CAT}[1]$. It remains to show that $\mathop{\rm Ram}\nolimits\mathcal{P}^{\circ}\hookrightarrow\mathop{\rm Ram}\nolimits\mathcal{P}$ is a homotopy equivalence. The latter follows from Allcock’s lemma 3 the same way as at the end of the proof of Theorem 1; case $\mathbb{C}^{2}$. ∎ Proof of Theorem 1. By Theorem 11 it suffices to know that there is a non- negatively curved polyhedral metric on $\mathbb{C}\mathrm{P}^{2}$ with singularities at the line arrangement. It is shown in [18] that for any arrangement of $3n$ lines that satisfies Hirzebruch’s property and such that no $2n$ lines of the arrangement pass through one point, such a metric exists. We are left with the case when at least $2n$ lines of the arrangement pass through one point, say $p$. Take any other line that does not pass through $p$. This line has at least $2n$ distinct intersections with other lines of the arrangement. So $n+1\geqslant 2n$, and we conclude that the arrangement is composed of three generic lines, hence it complement is aspherical. ∎ General line arrangements. Let $(\ell_{1},\dots,\ell_{n})$ be a line arrangement in $\mathbb{C}\mathrm{P}^{2}$. The number of lines $\ell_{i}$ passing through a given point $x\in\mathbb{C}\mathrm{P}^{2}$ will be called the _multiplicity_ of $x$, briefly $\mathrm{mult}_{x}$. Let us associate to the arrangement a symmetric $n\times n$ matrix $(b_{ij})$. For $i\neq j$ put $b_{ij}=1$ if the point $x_{ij}=\ell_{i}\cap\ell_{j}$ has multiplicity $2$ and $b_{ij}=0$ if its multiplicity is $3$ and higher. The number $b_{jj}+1$ equals the number of points on $\ell_{j}$ with the multiplicity $3$ and higher. Next theorem follows from [18, Theorem 1.12, Lemma 7.9]; it reduces the existence of a non-negatively curved polyhedral Kähler metric on $\mathbb{C}\mathrm{P}^{2}$ with singularities at a given line arrangement to the existence of a solution of certain system of linear equalities and inequalities. 11.2. Theorem. Let $(\ell_{1},\dots,\ell_{n})$ be a line arrangement in $\mathbb{C}\mathrm{P}^{2}$ and $(b_{ij})$ be its matrix. There exists a non- negatively curved polyhedral Kähler metric on $\mathbb{C}\mathrm{P}^{2}$ with the singular locus formed by the lines $\ell_{i}$ if and only there are real numbers $(z_{1},\dots,z_{n})$ such that 1. (i) For each $k$, we have $0<z_{k}<1;$ 2. (ii) For each $j$, we have $\sum_{k}b_{jk}{\hskip 0.5pt\cdot\nobreak\hskip 0.5pt}z_{k}=1$ and $\sum_{k}z_{k}=3;$ 3. (iii) For each point $x\in\mathbb{C}\mathrm{P}^{2}$ with multiplicity at least 3, we have $\alpha_{x}=1-\tfrac{1}{2}{\hskip 0.5pt\cdot\nobreak\hskip 0.5pt}\sum_{\\{j\|x\in\ell_{k}\\}}z_{k}>0.$ Let us explain the geometric meaning of the above conditions. If $(z_{1},\dots,z_{n})$ satisfy the condition then there is a polyhedral Kähler metric on $\mathbb{C}\mathrm{P}^{2}$ with the conical angle around $\ell_{i}$ equal to $2{\hskip 0.5pt\cdot\nobreak\hskip 0.5pt}\pi{\hskip 0.5pt\cdot\nobreak\hskip 0.5pt}(1-z_{i})$. The inequalities (i) say that conical angles are positive and less than $2\pi$. Each of $n$ equalities (ii) is the Gauss–Bonnet formula for the flat metric with conical singularities at a line of the arrangement; the additional equality expresses the fact that the canonical bundle of $\mathbb{C}\mathrm{P}^{2}$ is $O(-3)$. The link $\Sigma_{x}$ at $x$ with the described metric is isometric to a $3$-sphere with an $\SS^{1}$-invariant metric. A straightforward calculation shows that the length of an $\SS^{1}$-fiber in $\Sigma_{x}$ is $2{\hskip 0.5pt\cdot\nobreak\hskip 0.5pt}\pi{\hskip 0.5pt\cdot\nobreak\hskip 0.5pt}\alpha_{x}$, where $\alpha_{x}$ as in (iii). Equivalently, $\pi{\hskip 0.5pt\cdot\nobreak\hskip 0.5pt}\alpha_{x}$ is the area of the quotient space $\Sigma_{x}/\SS^{1}$. The construction of the metric in this theorem relies on a parabolic version of Kobayshi–Hitchin correspondence established by Mochizuki [17]. Surprisingly, the system of $n$ linear equations in (ii) is equivalent to the following quadratic equation. (The equation implies the system by [18, Lemma 7.9] and the converse implication is a direct computation.) $\sum_{\\{x\|\mathrm{mult}_{x}>2\\}}(\alpha_{x}-1)^{2}-\sum_{j=1}^{n}z_{j}^{2}{\hskip 0.5pt\cdot\nobreak\hskip 0.5pt}b_{jj}=\tfrac{3}{2}.$ This equation is the border case of a parabolic Bogomolov–Miayoka inequality. Geometrically it expresses the second Chern class of $\mathbb{C}\mathrm{P}^{2}$ as a sum of contributions of singularities of the metric. The following corollary generalizes Theorem 1. 11.3. Corollary. Any line arrangement $(\ell_{1},\dots,\ell_{n})$ in $\mathbb{C}\mathrm{P}^{2}$ for which one can find positive $z_{j}$ satisfying equalities and inequalities of Theorem 11 has an aspherical complement. The arrangements of lines as in Theorem 1 satisfy the conditions in Theorem 11 with $z_{i}=\tfrac{1}{n}$ at all $3{\hskip 0.5pt\cdot\nobreak\hskip 0.5pt}n$ lines of the arrangement. This is proved by an algebraic computation, see [18, Corollary 7.8]. The restriction that at most $2{\hskip 0.5pt\cdot\nobreak\hskip 0.5pt}n-1$ lines pass through one point follows from (iii). Therefore the corollary above is a generalization of Theorem 11. Proof. 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Topol. 18 (2014), no. 2, 617--631. * [21] Залгаллер, В. А. О деформациях многоугольника на сфере. УМН, 11:5(71) (1956), 177--178 * [22] Petrunin A. _Exercises in orthodox geometry_ , arXiv:0906.0290 [math.HO]. * [23] Thurston, W. Shapes of polyhedra and triangulations of the sphere. The Epstein birthday schrift, Geom. Topol. Monogr., 1. (1998) 511--549.	arxiv-papers	2013-12-24T17:19:44	2024-09-04T02:49:55.913094	{ "license": "Creative Commons Zero - Public Domain - https://creativecommons.org/publicdomain/zero/1.0/", "authors": "Dima Panov and Anton Petrunin", "submitter": "Anton Petrunin", "url": "https://arxiv.org/abs/1312.6856" }
1312.6895	# Optimal shapes and stresses of adherent cells on patterned substrates Shiladitya Banerjee James Franck Institute, The University of Chicago, Chicago, Illinois 60637, USA Rastko Sknepnek Department of Physics and Syracuse Biomaterials Institute, Syracuse University, Syracuse, New York 13244, USA School of Engineering, Physics, and Mathematics, University of Dundee, Dundee DD1 4HN, UK M. Cristina Marchetti Department of Physics and Syracuse Biomaterials Institute, Syracuse University, Syracuse, New York 13244, USA ###### Abstract We investigate a continuum mechanical model for an adherent cell on two dimensional adhesive micropatterned substrates. The cell is modeled as an isotropic and homogeneous elastic material subject to uniform internal contractile stresses. The build-up of tension from cortical actin bundles at the cell periphery is incorporated by introducing an energy cost for bending of the cell boundary, resulting to a resistance to changes in local curvature. Integrin-based adhesions are modeled as harmonic springs, that pin the cell to adhesive patches of a predefined geometry. Using Monte Carlo simulations and analytical techniques we investigate the competing effects of bulk contractility and cortical bending rigidity in regulating cell shapes on non- adherent regions. We show that the crossover from convex to concave cell edges is controlled by the interplay between contractile stresses and boundary bending rigidity. In particular, the cell boundary becomes concave beyond a critical value of the contractile stress that is proportional to the cortical bending rigidity. Furthermore, the intracellular stresses are found largely concentrated at the concave edge of the cell. The model can be used to generate a cell-shape phase diagram for each specific adhesion geometry. ## I Introduction Living cells actively probe physical cues in their environment via receptor- ligand adhesion complexes that link the actomyosin cytoskeleton to the extracellular matrix (ECM) Schwarz and Gardel (2012). The cellular microenvironment, comprising of the ECM and of neighboring cells, imposes specific boundary conditions that can regulate physiological processes such as cell differentiation, division and motility, as well as cell architecture and polarity Discher _et al._ (2005). Myosin motors generate contractile stresses in the actin cytoskeleton that are transmitted to the substrate by focal adhesions. The traction stresses exerted by the cells on the substrate are thus very sensitive to the stiffness of the substrate as well as to the adhesion geometry. Cell morphology in turn is directly affected by traction stresses through the tension that builds up in the actomyosin stress fibers. It has been shown that the substrate stiffness plays a crucial role in regulating the cell spread area, the magnitude of traction forces and the cell morphology Yeung _et al._ (2005); Lo _et al._ (2000); Ghibaudo _et al._ (2008); Chopra _et al._ (2011). Much less explored is the role of adhesion geometry in regulating the spatial distribution of cellular stresses. Micropatterning has emerged as a powerful tool to investigate the interplay of mechanics and cytoskeletal architecture in controlling cell morphology by specific tuning of the geometry of the adhesion sites Théry (2010). When plated on small micropatterns, cells are unable to grow, thus showing high apoptotic rate Chen _et al._ (1997). Large adhesive patches, in contrast, favor cell spreading and promote the assembly of contractile stress fibers along the cell’s perimeter Théry _et al._ (2006). These peripheral stress fibers interconnect focal adhesions and yield concave arcs of constant curvature in the nonadherent portions of the cell boundaries. In addition, traction forces tend to localize in regions of high curvature at the boundary Roca-Cusachs _et al._ (2008); Rape _et al._ (2011). The model proposed here allows to separately study the roles of cell contractility and mechanical properties of peripheral cell fibers in controlling cell shape. Future comparison with experiments where both quantities can be perturbed using pharmacological interventions Bar-Ziv _et al._ (1999); Théry _et al._ (2006) may provide a quantitative understanding of the relative importance of boundary and bulk properties in determining steady state cell shapes. Various successful theoretical models of single and multi-cell mechanics have been proposed over the past decade that address the role of ECM elasticity in regulating cell behavior Schwarz and Safran (2013). Previous work has addressed the interplay between cell mechanics and geometry by either focusing solely on the elasticity of the cell boundary Bischofs _et al._ (2009); Banerjee and Giomi (2013) or by considering only the bulk of the cell, described via continuum mechanics Banerjee and Marchetti (2011); Edwards and Schwarz (2011); Pathak _et al._ (2008); Banerjee and Marchetti (2013), by a cellular Potts model Vianay _et al._ (2010), or as a polymer network Torres _et al._ (2012). These models highlight the competing roles of cell contractility and substrate stiffness in regulating polymorphic cell shapes. Continuum models of cell mechanics have assumed that the material constants describing the cell are spatially homogeneous. Cell material properties are, however, highly heterogeneous. In particular, experiments have shown strong differences in the mechanical properties of the bulk and boundary regions of the cell Heidemann and Wirtz (2004). Increased tension and rigidity of cell boundaries can spontaneously arise during adhesion as a result of the assembly of peripheral stress fibers consisting of thin bundles of semiflexible actin filaments. Due to thermal and active forces these bundles considerably bend generating non-uniform peripheral tensions. Cell boundary can also resist changes in local curvature due to contact forces at the three-phase contact line between the cell, the substrate and the ambient medium. Motivated by these observations, in this paper we couple cell contour elasticity Bischofs _et al._ (2009); Banerjee and Giomi (2013) to a continuum description of bulk cell mechanics Banerjee and Marchetti (2011); Edwards and Schwarz (2011) to investigate the cooperative roles of cortical elasticity, bulk elasticity and active contractility in controlling cell shapes on non-uniform adhesion patterns. Non-uniform tension and elasticity is incorporated in the model by introducing a penalty for bending deformations of the cell periphery Banerjee and Giomi (2013). Using a combination of Monte Carlo simulation and analytical studies, we examine the interplay of bulk contractility and cortical tension in controlling morphological transitions in adherent cells and propose a cell shape phase diagram for specific adhesion geometries. An example of such a phase diagram for a cross-shaped adhesion pattern is shown in Fig. 1. See section 3 for details. Figure 1: Shape phase diagram for a cell adhering to a cross shaped micropattern (grey region of the cell images) as a function of the bending rigidity $\kappa$ of the cell boundary in units of $Y\ell^{3}$ and the bulk contractile stress $\sigma_{a}$ in units of $Y$, where $Y$ is the cellular Young’s modulus and the $\ell$ is the interparticle distance in the triangulation. The blue dashed line is a guide to the eye. The paper is organized as follows. In section II, we describe a continuum mechanical model for a thin adherent cell as an isotropic and homogeneous elastic material, subject to a homogeneous negative pressure, embodying active contractility. Cell-ECM adhesions are modeled as linear springs distributed non-uniformly along the cell-substrate interface. Cortical tension is described via a penalty for bending deformations of the cell periphery. In Section III we discuss the steady shapes of cells adherent to different concave micropatterns obtained via Monte Carlo simulations (numerical details are given in the Appendix). Our simulations suggest a transition between convex and concave morphologies as a function of cell contractility and bending rigidity, that is captured by an analytically solvable model for the cell boundary presented in Section IV. We conclude with a brief discussion. ## II Continuum Mechanical Model We consider the mechanical equilibrium of a stationary cell strongly adherent to a soft elastic substrate. We assume that the cell’s interior can be described as an isotropic homogeneous elastic material and neglect all dissipation. We further neglect all out-of-plane deformations of the cell and assume that its thickness is uniform throughout its entire area and remains unaffected by the substrate-induced deformations. The bulk elastic energy of our model cell is given by $\displaystyle E_{el}$ $\displaystyle=$ $\displaystyle\int_{A_{0}}dA\frac{Eh}{2\left(1+\nu\right)}\left(\frac{\nu}{1-\nu}u_{\gamma\gamma}^{2}+u_{\alpha\beta}^{2}\right)\;$ (1) with $E$ the three-dimensional Young’s modulus, $h$ the cell thickness, $\nu$ the Poisson’s ratio, and $u_{\alpha\beta}=\frac{1}{2}\left(\partial_{\alpha}u_{\beta}+\partial_{\beta}u_{\alpha}+\partial_{\alpha}u_{\gamma}\partial_{\beta}u_{\gamma}\right)$ ($\alpha,\beta,\gamma\in\left\\{x,y\right\\}$) the strain tensor. We retain the nonlinearity in strain tensor to allow for the possibility of large strains that can arise even for small displacements Audoly and Pomeau (2010). The nonlinear terms essentially describe strain stiffening which is indeed expected in crosslinked actin networks Gardel _et al._ (2004). The two- dimensional displacement vector $\vec{u}$ is defined as $\vec{u}\left(\vec{r}_{0}\right)=\vec{r}-\vec{r}_{0}$, where $\vec{r}_{0}$($\vec{r}$) is a material point before (after) the deformation. The integral is calculated over the area $A_{0}$ of the undeformed (reference) state and summation over pairs of repeated indices is assumed. Cell contractility arising from myosin motors is modeled as a homogeneous negative pressure resulting in an additional contribution to the cell’s energy, given by $E_{active}=\sigma_{a}\int_{A_{0}}dA\ u_{\gamma\gamma}\;,$ (2) where $\sigma_{a}>0$ is a parameter controlling active contractility, determine by concentration of myosin motors and rate of ATP consumption. At the continuum scale, it controls active contributions to the cell’s surface tension Mertz _et al._ (2012). Cell adhesion to the substrate is modeled via a harmonic potential with a position-dependent rigidity parameter $\Gamma(\vec{r})$ $E_{adh}=\frac{1}{2}\int dA\ \Gamma\left(\vec{r}\right)\left\|\vec{r}-\vec{r}_{a}\right\|^{2}\;,$ (3) with $\vec{r}_{a}$ the position of focal adhesions on the substrate. The rigidity parameter $\Gamma$ is nonzero over the adhesion region and zero elsewhere. Thus we allow for non-uniformity in the geometry of cell-substrate adhesions as can be realized experimentally using micropatterning techniques Théry _et al._ (2006). The assumption of _local_ adhesive interactions with the underlying substrate strictly holds for elastic substrates that are much thinner than the cell perimeter or on soft microposts Banerjee and Marchetti (2012). The rigidity parameter $\Gamma$ depends on the elastic modulus of the underlying substrate as well as on the stiffness $k_{f}$ of focal adhesions. For an elastic substrate of shear modulus $\mu_{s}$ and thickness $h_{s}$, with focal adhesion density $\rho_{f}$, $\Gamma$ is given by, $\Gamma^{-1}=\left(k_{f}\rho_{f}\right)^{-1}+\left(\mu_{s}/h_{s}\right)^{-1}$. Traction force density is therefore given by, $\vec{T}=\frac{1}{h}\delta E_{\text{adh}}/\delta\vec{u}=\frac{1}{h}\Gamma(\vec{r})(\vec{r}-\vec{r}_{a})$. Finally, we assign a bending penalty to the cell’s perimeter, reflecting the resistance of cortical actin bundles to changes in curvature, $E_{bend}=\kappa\oint ds\ c^{2},$ (4) where $\kappa$ is the bending rigidity, $c=\left\|\gamma^{\prime\prime}\left(s\right)\right\|$ is the curvature of the boundary, with $\gamma\left(s\right)$ a parametric curve describing the cell boundary, and the line integral is calculated along the cell boundary. The optimal shape of the cell is obtained by minimizing the total mechanical energy $E$, that is given as the sum of elastic, active, adhesion, and boundary bending energies, $E=E_{el}+E_{active}+E_{adh}+E_{bend}$. Figure 2: (a) Initial configuration of the ”V-shape” adhesion pattern. The adhesive region where the cell is strongly anchored to the V-shaped micro- pattern on the substrate ($\Gamma\not=0$) is indicated in grey, whereas the non-adherent portion of the cell ($\Gamma=0$) is indicated in yellow. (b) Zoom-in of the upper right corner, showing the triangulation. ## III Numerical Simulations The minimal energy cell shapes have been determined numerically by a Monte Carlo study of a discrete representation of the continuum model introduced in Section II. The discrete representation of the undeformed cell is a triangulated disk. The initial configuration is built by randomly placing $N\approx 10^{4}$ particles on a disk of radius $R_{0}$. Particles are assumed to interact pairwise via a Weeks-Chandler-Andersen potential Weeks _et al._ (1971) and their positions are equilibrated using a standard Monte Carlo simulation with canonical ($NVT$) Metropolis algorithm. A typical equilibrated configuration is stored and the positions of the particles of that configuration are then used as nodes to construct a Delaunay triangulation. The resulting triangulation for a V-pattern is shown in Fig. 2. We note that the initial density of points in the disk is chosen such that even in the equilibrium state there is a substantial overlap between neighbors, thus ensuring a densely packed distribution of points. As a result the equilibrium distribution of the interparticle distances is rather narrow and its mean, denoted as $\ell$, represents a suitable unit of length. In the following, all distances are measured in units of $\ell$ and all energies are measured in units of $Y\ell^{2}$, where $Y=Eh$ is the two-dimensional Young’s modulus. The substrate rigidity $\Gamma$ has units of $Y\ell^{2}$ and the bending rigidity of cortical stress fibers has units of $Y\ell^{3}$. The low energy configurations are obtained using simulated annealing Monte Carlo (see Appendix for details). ### III.1 Optimal shapes Figure 3: Relaxed shapes of (a) $V$-pattern with $\kappa=0.4\ Y\ell^{3}$, (b) $U$-pattern with $\kappa=0.08\ Y\ell^{3}$, and (c) cross-pattern with $\kappa=0.4\ Y\ell^{3}$ and $\sigma_{a}=100Y$. The white dashed lines indicate the boundary of the micropattern: the cell is anchored inside this region and is free to contract outside. The color scale shows the distribution of the displacements with respect to the reference circular configuration. We performed a series of simulations for $V$, $U$, and cross shaped micro- patterns, corresponding to the white dashed outlines shown in Fig. 3. The relaxed shapes for three non-convex patterns ($U$, $V$, and cross) obtained for a fixed value of the rigidity of adhesions $\Gamma_{grey}=10^{6}$, and $\nu=1/3$ are shown in Fig. 3. In all cases, the non-adherent cell edges spanning two pinning regions are clearly concave. The relaxed shapes can be qualitatively compared with experiments on concave micropatterns Théry _et al._ (2006). Our simulations suggest that there is a transition between the concave and convex morphologies as a function of $\sigma_{a}$ and $\kappa$. In Figure 1, we show a sample cell shape phase diagram as a function of the cortical bending rigidity $\kappa$ and the active contractile stress $\sigma_{a}$ for the cross shaped micropattern. The figure indicates that the cell boundary is concave at high values of the contractile stress $\sigma_{a}$, whereas convexity is ensured at high values of bending rigidity. The phase boundary between convex and concave shapes appears to be linear in the $\sigma_{a}-\kappa$ plane. To justify this observation, in the next section we study analytically the shape of the cell boundary in the non- adhesive regions, considering small deformation about a circular configuration. ### III.2 Optimal Stresses Experiments probing the distribution of traction force density $\vec{T}(\vec{r})$ exerted by cells adhering to soft substrates consistently show that such stresses are concentrated at the cell edges, and strongest in region of high cell curvature. Force balance requires $T_{\alpha}=\partial_{\beta}\sigma_{\alpha\beta}$, where $\sigma_{\alpha\beta}$ is the two-dimensional stress tensor of the bulk cellular material, given by, $\displaystyle\sigma_{\alpha\beta}$ $\displaystyle=$ $\displaystyle\frac{\partial\left(E_{el}+E_{active}\right)}{\partial u_{\alpha\beta}}=\sigma_{\alpha\beta}^{el}+\sigma_{a}\delta_{\alpha\beta}$ (5) $\displaystyle=$ $\displaystyle\frac{Eh}{\left(1+\nu\right)}\left(\frac{\nu}{1-\nu}\delta_{\alpha\beta}u_{\gamma\gamma}+u_{\alpha\beta}\right)+\sigma_{a}\delta_{\alpha\beta}\;.$ The distribution of such internal stresses can therefore be inferred experimentally from traction force microscopy measurements Tambe _et al._ (2011). Internal stresses of adhering cells are found to be concentrated at the cell’s interior, with a maximum value proportional to the active cell contractility, here $\sigma_{a}$. To highlight the role of patterned adhesion on the spatial distribution of cellular stress, we display in Fig. 4 the spatial distribution of the so-called Lamé’s stress ellipses Love (1927) for the elastic part $\sigma^{el}_{\alpha\beta}$ of the stress tensor. The constant active contribution $\sigma_{a}$ has been subtracted out to highlight spatial variations. As a result, the displayed stress is largest at the cell edges. The Lamé’s stress ellipses are obtained by computing the elastic part of the two-dimensional stress tensor at a representative subset of triangles in the Delaunay grid. This is achieved by directly evaluating the expression in eqn (5), excluding the active term. We then compute the low and the high eigenvalues, $\sigma_{min}$ and $\sigma_{max}$, respectively, of the elastic stress tensor of a given triangle. Note that since the stress tensor is symmetric, its eigenvalues are always real. The length of the major and minor semi-axes of each ellipse are then given by $\sigma_{max}$ and $\sigma_{min}$, respectively, whereas the orientation of the ellipse axes is given by the directions of the corresponding eigenvectors. As expected, and consistent with experiments Théry (2010), elastic stresses are concentrated at the free boundaries of the adherent cell. Boundary stresses along free edges connecting two adhesion points are directed normal to the edge whereas they are oriented along the edge near the adhesion points. This is most evident in the cross- shaped pattern, Fig. 4(c). The large stresses in the convex regions of the cell spilling outside straight portions of the pinning regions (see, for instance, the V-shape pattern, Fig. 4(a)) are largely an artifact of our model. They arise because we have introduced excluded volume interactions to prevent self-intersections of the triangulation. In other words, we assign a hard core radius of $0.25\ell$ to each vertex of the triangulation, such that no two vertices can came closer than $0.5\ell$ from each other. Once this limit has been reached, the excluded volume prevents further collapse of the cell, thus accounting for the presence of a sizeable portion of the cell that extends outside the pinning region. While steric effects are present in vivo and may describe for instance the role of structural elements capable of carrying compressive loads, such as microtubules, cells on synthetic substrates generally almost completely conform to the micropattern by changing their thickness. This is not possible in our strictly two-dimensional model. As a result, the model captures well the behavior of “free” cell edges spanning two adhesion points, but has limitations for describing the behavior of cell boundaries along straight pinning regions. (a) (b) (c) Figure 4: Stress profile (Lamé’s ellipses, shown in red) at representative sets of points of contracted cells for (a) $V$-pattern with $\kappa=0.16\ Y\ell^{3}$, (b) $U$-pattern with $\kappa=0.24\ Y\ell^{3}$, and (c) cross pattern with $\kappa=0.32\ Y\ell^{3}$; $\sigma_{a}=10Y$ for all patterns. Note that only the elastic part of the stress tensor is shown. The active component, $\sigma_{a}$, has been removed, since it is isotropic and much larger in magnitude than the elastic component of the stress tensor. Length of the ellipse axes is proportional to the two eigenvalues of the stress tensor, $\sigma_{min}$ and $\sigma_{max}$. The stress is not uniform but largest around cell’s perimeter and gradually falls off toward its interior. For clarity, the Lamé’s ellipses are computed only for a subset of all triangles selected from the non-regular triangulation and their sizes are scaled by a factor of $0.5$. ## IV Boundary shapes ### IV.1 Strong pinning at adhesions Figure 5: Curvature, deformation and phase boundary for a pinned contractile string. (a) Radial displacement $u_{r}$ in units of $R_{0}$ and (b) curvature profile $c(\theta)$ in units of $1/R_{0}$ for $\sigma_{a}R_{0}^{3}/\kappa=1$ (solid, black), $\sigma_{a}R_{0}^{3}/\kappa=10$ (dashed, blue) and $\sigma_{a}R_{0}^{3}/\kappa=40$ (dotted, red), where $0<\theta<\pi/2$, corresponding to a cross pattern of width zero. (c) Shape phase diagram for the contractile string pinned to a cross micropattern of width $w$, obtained from the solution to eqn (8). Bending rigidity $\kappa$ and contractile stress $\sigma_{a}$ are given in units of $Y\ell^{3}$ and $Y$ respectively, corresponding to the parameters in Fig. 1, where $R_{0}/\ell=50$, $w/\ell=20$, and $\nu=1/3$. For small deformations about an initially circular configuration of radius $R_{0}$, the cell boundary in the non-adhesive region can be parametrized using polar coordinates as, $r(\theta)=R_{0}+u_{r}(\theta)$, where $u_{r}$ is the radial component of the displacement field at the cell boundary, $\vec{u}=(u_{r},u_{\theta})$. Thus, $\vec{u}$ is solely a function of the angular coordinate $\theta$. In mechanical equilibrium, boundary force balance along the normal and tangential directions requires $\displaystyle 2\kappa\frac{d^{2}c}{ds^{2}}-\sigma_{ij}n_{i}n_{j}=0\;,$ (6a) $\displaystyle\sigma_{ij}t_{i}n_{j}=0\;,$ (6b) where $s$ is the arc-length parameter and $\vec{n}$ and $\vec{t}$ are unit vectors normal and tangent, respectively, to the unperturbed cell boundary. Tangential force-balance in polar coordinates reduces to $\sigma_{r\theta}=0$, which leads to the relation $u_{\theta}=\partial_{\theta}u_{r}$. Thus, the normal component of the elastic stress is given by $\sigma_{rr}^{el}\simeq\lambda\partial_{\theta}u_{\theta}/R_{0}=\lambda\partial_{\theta}^{2}u_{r}/R_{0}$, where $\lambda=Y\nu/(1-\nu^{2})$ is the Lamé elastic constant. Furthermore, for small deformations $u_{r}$, the boundary curvature can be expanded as, $c(\theta)\simeq\frac{1}{R_{0}}-\frac{1}{R_{0}^{2}}(u_{r}+\partial_{\theta}^{2}u_{r})+\mathcal{O}(u_{r}^{2})\;.$ (7) Using eqns (6a)-(7), and letting $ds=R_{0}d\theta$, we obtain an equation for the boundary profile, $\frac{2\kappa}{R_{0}^{3}}\left(\partial_{\theta}^{2}\tilde{u}+\partial_{\theta}^{4}\tilde{u}\right)+\sigma_{a}+\lambda\partial_{\theta}^{2}\tilde{u}=0\;,$ (8) where $\tilde{u}=u_{r}/R_{0}$. Without loss of generality, we can consider solution in the interval $\theta\in[0,\phi]$, where $\phi$ is the angular width of the nonadherent region and depends on the geometry of the adhesion pattern. The boundary conditions for the case of strong pinning at adhesions are given by: $\tilde{u}(0)=\tilde{u}(\phi)=\partial_{\theta}\tilde{u}(0)=\partial_{\theta}\tilde{u}(\phi)=0$. The full solution of eqn (8) is analytically tractable but cumbersome. We instead discuss the solutions in two limiting cases in terms of the dimensionless parameter $K=2\kappa/\lambda R_{0}^{3}$, reflecting the relative contributions of bending and bulk elasticity. This parameter can also be written as $K=(\xi/R_{0})^{3}$ in terms of the ratio of a length scale $\xi=[2\kappa/\lambda]^{1/3}$ to the undeformed cell radius $R_{0}$. The length scale $\xi$ described the interplay between bulk elasticity and boundary tension in controlling the response of the cell. When $\xi\gg R_{0}$ (corresponding to $K\gg 1$) the cel deformation is controlled by the cortical tension at the boundary and the curvature is given by $c(\theta)\simeq\frac{1}{R_{0}}\left[1+\frac{\sigma_{a}R_{0}^{3}}{4\kappa}\left(2+\theta^{2}-\theta\phi-\phi\cot{\frac{\phi}{2}}\right)\right]\;.$ (9) Bulk elasticity drops out and the behavior is controlled by the ratio $\sigma_{a}R_{0}^{3}/\kappa$ of contractility to bending rigidity. The curvature has a minimum at the center of the nonadherent segment. Thus, as one increases contractility $\sigma_{a}$, a region of negative curvature develops near $\theta=\phi/2$, which grows upon increasing $\sigma_{a}$ until convexity is retained within a small neighborhood of the adhesion patch. The onset of concavity is thus given by the condition of reality to the solution of $c(\theta)=0$, which gives the condition, $\sigma_{a}>\frac{2\kappa}{R_{0}^{3}}\left(\frac{1}{\phi^{2}/8+(\phi/2)\cot{\frac{\phi}{2}}-1}\right)\;.$ (10) Since concave shapes are commonly observed in experiments Théry _et al._ (2006), we now turn to estimate the critical value of $\sigma_{a}$ as predicted by our model in order to compare it with experimentally reported values for $\sigma_{a}$. The bending rigidity of cortical stress fibers can be estimated as $\kappa\sim\frac{\pi}{4}E_{act}r_{s}^{4}$, where $E_{act}$ is the Young’s modulus of actin and $r_{s}$ is the typical radius of the stress fibers. Using $E_{act}\simeq 2.6$ GPa Gittes _et al._ (1993) and $r_{s}\sim 0.1\ \mu$m, we get $\kappa\sim 2.0\times 10^{-19}$ Nm2. Using this value in eqn (10) for $\phi\sim 2\pi/3$, corresponding to a thin V-pattern, we get value for the critical $\sigma_{a}\sim$2.7 nN/$\mu$m. This is indeed the order of magnitude value for active stress or surface tension reported in experiments for adherent epithelial cells on continuous elastic substrates or endothelial cells on microposts Bischofs _et al._ (2009); Mertz _et al._ (2012). In the opposite limit of $K\ll 1$ the deformation is controlled by bulk elasticity and the curvature is given by $c(\theta)\simeq\frac{1}{R_{0}}\left[1+\frac{\sigma_{a}}{2\lambda}\left(2+\theta^{2}-\theta\phi\right)\right]$ (11) The condition of concavity is given by $\sigma_{a}>\lambda\left(\frac{1}{\phi^{2}/8-1}\right)$. In the case when $\xi$ is comparable to $R_{0}$, a simpler solution of eqn (8) can be obtained by neglecting the fourth order gradient term and also the derivative boundary conditions. The crossover to concave profiles can be approximated by the following interpolating form between the two limiting cases, $\sigma_{a}>\left(\lambda+\frac{2\kappa}{R_{0}^{3}}\right)\left[\frac{1}{\phi^{2}/8+(\phi/2)\cot{\frac{\phi}{2}}-1}\right]\;.$ (12) In the general case of eqn (8), the solution for curvature and the radial displacement is given in Fig. 5a,b for three different values of $\sigma_{a}R_{0}^{3}/\kappa$ that compares the relative strengths of contractility to bending deformations. Furthermore, to compare numerically with the simulation results for the shape phase diagram of the adherent cell, we show the concave-convex phase boundary in $\sigma_{a}-\kappa$ plane in Fig. 5c, for a cross-shaped micropattern using the same parameters as used in Fig. 1. The resultant phase diagram is in good order-of-magnitude agreement with the simulation results, and the discrepancy in numerical values possibly arise from neglecting non-local bulk elasticity in the theoretical analysis. ### IV.2 Soft pinning We now consider the case of soft pinning, where the free cell boundary is anchored to soft springs at the adhesion sites. Equation (8) is now solved with the boundary conditions $\tilde{u}(0)=\tilde{u}(\phi)=\delta$ and $\partial_{\theta}\tilde{u}(0)=\partial_{\theta}\tilde{u}(\phi)=0$, where we have introduced an unknown displacement $\delta$ of the ends of the segment, which can be self consistently determined by minimizing the total energy of the deformed configuration with respect to $\delta$. For simplicity we ignore bulk elasticity and consider the limit $K\gg 1$. The total energy of the deformed configuration is then given by $U=\kappa R_{0}\int_{0}^{\phi}d\theta c(\theta)^{2}+k_{s}\delta^{2}$, where $k_{s}=\Gamma A_{f}$ and $A_{f}$ is the cross-sectional area of focal adhesions. Note that the contribution due to contractility vanishes in the final energy due to the derivative boundary conditions on $\tilde{u}$. The onset of concavity now depends on the substrate stiffness $k_{s}$ and the condition for convex-concave transition is given by $\sigma_{a}>\frac{4\kappa}{R_{0}^{3}}\left[\frac{k_{s}}{\kappa\phi^{3}/12R_{0}^{3}+k_{s}\left(\phi^{2}/4+\phi\cot{\frac{\phi}{2}}-2\right)}\right]\;.$ (13) Thus, stiffer adhesions with $k_{s}R_{0}^{3}\gg\kappa$, promote concavity transition at a much higher value of contractility. It is favorable for a cell to invaginate at the free edges if the anchoring at adhesions is softer than the effective bending stiffness $\kappa/R_{0}^{3}$. ## V Concluding Remarks Using a simple continuum model, coupling bulk and contour mechanics, we investigate the equilibrium shapes and stresses of adherent cells on substrates with various adhesion patterns. A continuum model without contour elasticity have been studied previously by two of us on convex patterns Banerjee and Marchetti (2011), which was successful in capturing distribution of traction and cellular stresses and their dependence on substrate physical properties Banerjee and Marchetti (2012). Here we focus on the shape and geometry induced stresses of non-adherent cell edges on concave micropatterns. We demonstrate numerically and analytically that the curvature of the non- adherent cell boundary can undergo a shape transition from convex to concave morphology, controlled by the interplay of contractility and bending rigidity. Stiff boundaries with low contractility relax to convex shapes, whereas at higher values of contractility, non-adherent cell edges attain a concave morphology. Previous work has shown that contractile cable network models are capable of reproducing the invaginated circular arc morphology of cell edges connecting strongly adhering sites Torres _et al._ (2012). Here we demonstrate that simple _continuum_ whole-cell models can also predict qualitatively cell shape and the transition between convex and concave cell edges, provided a bending rigidity describing cortical tension is included. For parameters realistic to experiments (see section 4.1) our model suggest that cells prefer to invaginate at their free edges, such that the effective boundary stiffness on non-adhesive zones are softer than myosin induced contractile stresses. Images of actin from experiments on concave micropatterns do indeed show the formation of long and thin stress fibers that are invaginated on non-adherent edges Théry _et al._ (2006), indicating a softer cortical rigidity. In addition, elastic stresses are found to be higher along the free cell boundaries than in the neighborhood of adhesions, since in the absence of mechanotransduction cellular forces along free edges are not shared by the substrate. Previous theoretical study with only contour elasticity indicated that substrate stiffness and contractility can cooperatively control cell morphology and induce hysteresis at the onset of convex-concave transition Banerjee and Giomi (2013). Here we show that even in the presence of rigid adhesions, cell shape can be controlled by regulating the cortical bending rigidity and contractility. Bending rigidity can be experimentally controlled by regulating the amount of actin cross-linking proteins that can impact stress fiber thickness and rigidity, whereas myosin based contractility can be perturbed using the conventional inhibitor Blebbistatin. One limitation of our model is that it is strictly two-dimensional and does not allow for changes in the cell thickness. Due to the presence of steric interactions in the finite element simulations, the cell edges on flat adhesive segments do not fully relax to the flat morphology, but maintain a convex shape. This is in contrast to real cells that contract to adjust to the shape of the micropattern. A fully three-dimensional model can overcome this difficulty, and is a natural extension of our present work. ## VI Appendix : Simulation Details ### VI.1 Discrete Model The discrete version of the elastic and active contraction energies can be expressed as a sum over triangles of the triangulation, $E_{el}=\sum_{T}\left\\{\frac{Eh}{8\left(1+\nu\right)}\left(\frac{\nu}{1-\nu}\left(\mathrm{Tr}\hat{F}_{T}\right)^{2}+\mathrm{Tr}\left(\hat{F}_{T}\right)^{2}\right)\right\\}A_{T}\;,$ (14) $E_{active}=\sigma_{a}\sum_{T}A_{T}\mathrm{Tr}\hat{F}_{T}\;,$ (15) where matrix $\hat{F}=\hat{g}^{-1}\hat{G}-\hat{I}$, with $\hat{g}$ ($\hat{G}$) being discrete metric tensor of the reference (deformed) configuration. $A_{T}=\frac{1}{2}\left\|\vec{a}\times\vec{b}\right\|$ is the area of an undeformed triangle spanned by two vectors $\vec{a}$ and $\vec{b}$ pointing along its sides. The sum is carried over all triangles. Adhesion energy is discretized as $E_{adh}=\frac{1}{2}\sum_{i}\Gamma_{i}\left\|\vec{r}_{i}-\vec{r}_{i}^{\left(o\right)}\right\|^{2}A_{i},$ (16) where $\Gamma_{i}=10^{6}$($0$) for grey (yellow) vertices in Fig. 2, $\vec{r}_{i}$($\vec{r}_{i}^{\left(0\right)})$ is the current (reference) position of the vertex _i_ , and $A_{i}=\frac{1}{3}\sum_{T\in\Omega_{i}}A_{T}$ is the area associated to the vertex (i.e., a third of the sum of areas of all triangles that share the vertex, so-called “vertex star”). Finally, following ref. 31, the boundary bending energy is discretized as $E_{bend}=4\kappa\sum_{i}\frac{1-\cos\left(\vartheta_{i}\right)}{s_{i}+s_{i+1}},$ (17) where $\vartheta_{i}$ is the exterior angle at the boundary vertex _i_ , $s_{i}$ and $s_{i+1}$ are lengths of two boundary edges meeting at _i_ , and the sum is carried over all boundary vertices. ### VI.2 Monte Carlo Sweeps A Monte Carlo sweep consist of an attempted move for each vertex. 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1312.6910	# Quantum Fisher information for density matrices with arbitrary ranks Jing Liu, Xiaoxing Jing, Wei Zhong, Xiaoguang Wang Zhejiang Institute of Modern Physics, Department of Physics, Zhejiang University, Hangzhou 310027, China [email protected] ###### Abstract We provide a new expression of the quantum Fisher information(QFI) for a general system. Utilizing this expression, the QFI for a non-full rank density matrix is only determined by its support. This expression can bring convenience for a infinite dimensional density matrix with a finite support. Besides, a matrix representation of the QFI is also given. ###### pacs: 03.67.-a, 03.65.Ta, 06.20.-f ## I Introduction Quantum metrology is a field that utilizes the character of quantum mechanics to improve the precision of a parameter under detection review . For the past few years, this field has drawn a lot of attention and has been developing rapidly S.loyd ; 3 ; 5 ; 6 ; 7 ; 8 ; 9 ; 10 ; 11 ; 12 ; 13 ; 14 ; 15 ; 16 ; 17 ; 18 . Quantum Fisher information(QFI) is a central concept in quantum metrology because it depicts the lower bound on the variance of the estimator $\hat{\theta}$ for the parameter $\theta$ due to the Cramér-Rao theorem Fisher ; FI ; FI-1 $\mathrm{var}(\hat{\theta})\geq\frac{1}{\nu F},$ (1) where $\mathrm{var(\cdot)}$ is the variance, $\nu$ is the number of repeated experiments and $F$ is the QFI. However, the QFI is not just limited in the field of quantum metrology. It has been widely applied in other aspects of quantum physics Luo2004 ; Luo2006 ; Paris ; Lu ; lu2 ; Ma ; Pezze09 ; wootters ; Braunstein ; Pezze ; Yu ; Toth ; mine , like quantum information and open quantum systems. Thus, it is necessary and meaningful to study the quantum Fisher information as well as its properties and dynamical behaviors under various circumstances. Quantum Fisher information is a local quantity, which can be intuitively interpreted as the “velocity" at which the density matrix moves for a given parameter value. This physical interpretation comes from the fact that the QFI is dependent on the parameterized density matrix $\rho_{\theta}$ and its first derivative $\partial_{\theta}\rho_{\theta}$. Utilizing the spectral decomposition, when the eigenstates of $\rho_{\theta}$ as projectors, act on $\rho_{\theta}$ and its first derivative, the value is only related to the spectral decomposition within the support, which strongly hints that the QFI may be expressed in the representation of the density matrix’s support. To find such an expression is the major motivation of this paper. In this paper, we provide a new expression of the quantum Fisher information in the representation of the density matrix’s support. With this expression, for a non-full rank density matrix, especially for a infinite dimension one, the QFI may be solved in a finite support space without realizing the knowledge out of the support. Recently, it is found Toth ; Yu that the QFI can be written in the form of the convex roof of variance. To obtain the QFI, one should take the minimum value running over all the possible pure-state ensembles. Utilizing the new expression, we give the condition when the ensemble from the spectral decomposition is the optimal ensemble in which the minimum value attains. Besides, we also provide a matrix representation form of the QFI and give two examples of it. ## II Fisher information for a non-full rank density matrix In the following we consider a $\mathrm{N}$-dimensional system ($\mathrm{N}$ can be infinite) with the density operator $\rho_{\theta}$, which is dependent on the parameter $\theta$. Assume that the spectral decomposition of the density operator is given by $\rho_{\theta}=\sum_{i=1}^{\mathrm{s}}p_{i}\|\psi_{i}\rangle\langle\psi_{i}\|,$ (2) where $p_{i}$ is a eigenvalue and $\|\psi_{i}\rangle$ is a eigenstate. $\mathrm{s}$ is the dimension of the support set of $\rho_{\theta}$, denoted as $\mathrm{supp}(\rho_{\theta})$, i.e., $\mathrm{s}=\mathrm{dim}[\mathrm{supp}(\rho_{\theta})]$. For a parameterized quantum state $\rho_{\theta}$, the quantum Fisher information $F$ is defined as below FI ; FI-1 $F:={\rm tr}(\rho_{\theta}L^{2}),$ (3) where $L$ is the so-called symmetric logarithmic derivative operator and determined by $\partial_{\theta}\rho_{\theta}=\frac{1}{2}\left(L\rho_{\theta}+\rho_{\theta}L\right).$ (4) In the eigenbasis of $\rho_{\theta}$, above equation reads $\langle\psi_{i}\|\partial_{\theta}\rho_{\theta}\|\psi_{j}\rangle=\frac{1}{2}(p_{i}+p_{j})L_{ij},$ (5) where $L_{ij}:=\langle\psi_{i}\|L\|\psi_{j}\rangle$. From above equation, one can find that $L_{ij}$ is in principle supported by the full space, but the value of $L_{ij}$ for $i,j>\mathrm{s}$ is arbitrary because above equation is always established for any value of $L_{ij}$ when $i,j>\mathrm{s}$. Nevertheless, the quantum Fisher information is still a determinate quantity because the calculation of it does not use those values of $L_{ij}$ for $i,j>\mathrm{s}$, which we will show below. Thus, one can set $L_{ij}=0$ for $i,j>\mathrm{s}$ as a matter of convenience. By substituting Eq. (2) and the normalization relation $\mathbb{I}=\sum_{j=1}^{\mathrm{N}}\|\psi_{j}\rangle\langle\psi_{j}\|$ into Eq. (3), one can obtain the quantum Fisher information as $F=\sum_{i=1}^{\mathrm{s}}\sum_{j=1}^{\mathrm{N}}p_{i}L_{ij}L_{ji}.$ (6) Here $\mathbb{I}$ is the identity operator. All $p_{i}$ here is greater than zero because the index $i\leq\mathrm{s}$ and satisfies $\sum_{i=1}^{\mathrm{s}}p_{i}=1$. From this equation we see that the randomicity of $L_{ij}$ for $i,j>\mathrm{s}$ does not affect the certainty of the quantum Fisher information. As $p_{i}>0$, Eq. (5) can be rewritten into $L_{ij}=\frac{2(\partial_{\theta}\rho_{\theta})_{ij}}{p_{i}+p_{j}},$ (7) where $(\partial_{\theta}\rho_{\theta})_{ij}:=\langle\psi_{i}\|\partial_{\theta}\rho_{\theta}\|\psi_{j}\rangle$. Utilizing this expression, Eq. (6) can be written in the form $F_{\theta}=\sum_{i=1}^{\mathrm{s}}\sum_{j=1}^{\mathrm{N}}\frac{4p_{i}}{(p_{i}+p_{j})^{2}}\|(\partial_{\theta}\rho_{\theta})_{ij}\|^{2},$ (8) where the Hermiticity of the operator $\partial_{\theta}\rho_{\theta}$ was used. Next, from the spectral decomposition of $\rho_{\theta}$, one can find that $(\partial_{\theta}\rho_{\theta})_{ij}=\partial_{\theta}p_{i}\delta_{ij}+(p_{j}-p_{i})\langle\psi_{i}\|\partial_{\theta}\psi_{j}\rangle,$ (9) where we have used the equation $\langle\psi_{i}\|\partial_{\theta}\psi_{j}\rangle=-\langle\partial_{\theta}\psi_{i}\|\psi_{j}\rangle,$ (10) resulted from the orthogonality $\langle\psi_{i}\|\psi_{j}\rangle=\delta_{ij}$. For $i\in[1,\mathrm{s}]$ and $j\in[\mathrm{s}+1,\mathrm{N}]$, the expression of $(\partial_{\theta}\rho_{\theta})_{ij}$ reduces to $-p_{i}\langle\psi_{i}\|\partial_{\theta}\psi_{j}\rangle$. Substituting Eq. (9) into Eq. (8), we have $F_{\theta}=\sum_{i=1}^{\mathrm{s}}\frac{1}{p_{i}}(\partial_{\theta}p_{i})^{2}+\sum_{i=1}^{\mathrm{s}}\sum_{j=1}^{\mathrm{N}}\frac{4p_{i}(p_{i}-p_{j})^{2}}{(p_{i}+p_{j})^{2}}\|\langle\psi_{i}\|\partial_{\theta}\psi_{j}\rangle\|^{2}.$ (11) Furthermore, with the knowledge that $\sum_{j=1}^{\mathrm{N}}=\sum_{j=1}^{\mathrm{s}}+\sum_{j=\mathrm{s}+1}^{\mathrm{N}}$, the second item of above expression can be separated into two parts $F_{1}$ and $F_{2}$. The first part $F_{1}$ reads $F_{1}=\sum_{i,j=1}^{\mathrm{s}}\frac{4p_{i}(p_{i}-p_{j})^{2}}{(p_{i}+p_{j})^{2}}\|\langle\psi_{i}\|\partial_{\theta}\psi_{j}\rangle\|^{2},$ (12) and the second part $F_{2}$ reads $F_{2}=\sum_{i=1}^{\mathrm{s}}\sum_{j=\mathrm{s}+1}^{\mathrm{N}}4p_{i}\|\langle\psi_{j}\|\partial_{\theta}\psi_{i}\rangle\|^{2}.$ (13) Based on the normalization relation, it is easy to find that $\sum_{j=\mathrm{s}+1}^{\mathrm{N}}\|\psi_{j}\rangle\langle\psi_{j}\|=\mathbb{I}-\sum_{j=1}^{\mathrm{s}}\|\psi_{j}\rangle\langle\psi_{j}\|.$ (14) Substituting this equation into the expression of $F_{2}$, one can obtain $F_{2}=\sum_{i=1}^{\mathrm{s}}4p_{i}\langle\partial_{\theta}\psi_{i}\|\partial_{\theta}\psi_{i}\rangle-\sum_{i,j=1}^{\mathrm{s}}4p_{i}\|\langle\psi_{j}\|\partial_{\theta}\psi_{i}\rangle\|^{2}.$ (15) Then, the quantum Fisher information can be expressed by $\displaystyle F_{\theta}$ $\displaystyle=$ $\displaystyle\sum_{i=1}^{\mathrm{s}}\frac{1}{p_{i}}\left(\partial_{\theta}p_{i}\right)^{2}+\sum_{i=1}^{\mathrm{s}}4p_{i}\langle\partial_{\theta}\psi_{i}\|\partial_{\theta}\psi_{i}\rangle$ (16) $\displaystyle-\sum_{i,j=1}^{\mathrm{s}}\frac{8p_{i}p_{j}}{p_{i}+p_{j}}\|\langle\psi_{i}\|\partial_{\theta}\psi_{j}\rangle\|^{2}.\quad$ From this equation one can find that the quantum Fisher information for a non- full rank density matrix is determined by its support. The information of eigenstates out of the support is not necessary for the calculation of the QFI. This advantage would bring some convenience for the calculation in some cases, especially when $\mathrm{N}$ is infinite and $\mathrm{s}$ is finite. According the theory of the classical Fisher information Fisher ; FI ; FI-1 , it is natural to treat the first item of Eq. (16) as the classical contribution of quantum Fisher information Paris because $\sum_{i=1}^{\mathrm{s}}\frac{1}{p_{i}}(\partial_{\theta}p_{i})^{2}=4\sum_{i=1}^{\mathrm{s}}\left(\partial_{\theta}\sqrt{p_{i}}\right)^{2}$. Then the quantum Fisher information for a quantum system can be separated into two parts, the classical contribution and quantum contribution, namely, $F_{\theta}=F_{\mathrm{ct}}+F_{\mathrm{qt}},$ (17) where the classical contribution reads $F_{\mathrm{ct}}=\sum_{i=1}^{\mathrm{s}}4\left(\partial_{\theta}\sqrt{p_{i}}\right)^{2},$ (18) and the quantum contribution reads $F_{\mathrm{qt}}=\sum_{i=1}^{\mathrm{s}}4p_{i}\langle\partial_{\theta}\psi_{i}\|\partial_{\theta}\psi_{i}\rangle-\sum_{i,j=1}^{\mathrm{s}}\frac{8p_{i}p_{j}}{p_{i}+p_{j}}\|\langle\psi_{i}\|\partial_{\theta}\psi_{j}\rangle\|^{2}.$ (19) The separation of the quantum Fisher information is not just in form. From the equations above, one can find that the classical contribution of the quantum Fisher information is a special case of the classical Fisher information. It can be treated as the classical Fisher information obtained through the measurement $\\{\|\psi_{i}\rangle\\}$ in the eigenspace of $\rho_{\theta}$: $\mathbb{E}^{\mathrm{N}}$. The eigenspace $\mathbb{E}^{\mathrm{N}}$ is spanned by the basis $\left\\{\|\psi_{i}\rangle\right\\}$, and $\left\\{p_{i}\right\\}$ is a classical distribution in this space. From Eq. (18), it is not difficult to find that the classical contribution $F_{\mathrm{ct}}$ is only related to the derivative of the eigenvalues, which indicates that this part of information is coming from the classical distribution in $\mathbb{E}^{\mathrm{N}}$. Moreover, we find that the classical contribution has the following properties: (1) it vanishes for pure states; (2) it vanishes for the unitary parametrization; (3) it is invariant under unitary transformation of density matrix, no matter the transformation is parameter- dependent or not. In the mean time, with some transformation, Eq. (19) can be rewritten as $F_{\mathrm{qt}}=\sum_{i=1}^{\mathrm{s}}p_{i}F_{Q}(\|\psi_{i}\rangle)-\sum_{i\neq j}^{\mathrm{s}}\frac{8p_{i}p_{j}}{p_{i}+p_{j}}\|\langle\psi_{i}\|\partial_{\theta}\psi_{j}\rangle\|^{2},$ (20) where $F_{Q}(\|\psi_{i}\rangle)=4\left(\langle\partial_{\theta}\psi_{i}\|\partial_{\theta}\psi_{i}\rangle-\|\langle\psi_{i}\|\partial_{\theta}\psi_{i}\rangle\|^{2}\right)$ (21) is the quantum Fisher information of the eigenstate $\|\psi_{i}\rangle$. From this equation, it is clear that $F_{\mathrm{qt}}$ is related to the basis of $\mathbb{E}^{\mathrm{N}}$. In $\mathbb{E}^{\mathrm{N}}$, $F_{\mathrm{qt}}$ is determined by the weighted average of all the quantum Fisher information $F_{Q}(\|\psi_{i}\rangle)$ of the basis vector $\|\psi_{i}\rangle$ and the coupling between these vectors. This manifests that this part of information originates from the quantum structure of space $\mathbb{E}^{\mathrm{N}}$. These are the geometric meanings of the classical and quantum contribution as well as the intrinsic reason for the separation. We know the classical contribution of the QFI always vanishes for the unitary parametrization. But for a non-unitary parametrization procedure, including the channel estimation channel_es ; channel_es1 ; channel_es2 ; channel_es3 ; channel_es4 ; channel_es5 and the noise estimation Chaves ; NJP , the classical contribution does have an influence on the precision. However, only improving the classical contribution without enhancing the quantum counterpart, the precision is not available to surpass the shot-noise limit, the lower limit for a total classical scenario. The estimation of the decoherence strength NJP , in which the classical contribution plays the leading role, is an example of this scenario. The quantum Fisher information is a local quantity, which can be intuitively interpreted as the “velocity" at which the matrix moves for a given parameter value. In mathematics, this means that the quantum Fisher information depends on the density matrix $\rho_{\theta}$ and its first derivative $\partial_{\theta}\rho_{\theta}$. Utilizing the spectral decomposition, there exists items such as $\|\psi_{i}\rangle\langle\partial_{\theta}\psi_{j}\|$ and $\|\partial_{\theta}\psi_{i}\rangle\langle\psi_{j}\|$. When these items are traced with the eigenstates out of the support, the values turn out to be zero. This is the intuitive reason that the QFI can be expressed in the representation of the support. If the QFI is related to the higher order derivatives, like the second one $\partial^{2}_{\theta}\rho_{\theta}$, then there would exist the item like $\|\partial_{\theta}\psi_{i}\rangle\langle\partial_{\theta}\psi_{j}\|$. As $\|\partial_{\theta}\psi_{i}\rangle$ is not always orthogonal with $\|\psi_{j}\rangle$, when this item is traced with the projectors out of the support, the value cannot always be zero, then the quantum Fisher information has to be related to the whole Hilbert space, rather than the support only. For the unitary parametrization $\exp(i\theta H)$, the classical contribution vanishes, and the quantum Fisher information reduces to $F_{Q}=\sum_{i=1}^{\mathrm{s}}p_{i}F_{Q}(\|\psi_{i}\rangle)-\sum_{i\neq j}^{\mathrm{s}}\frac{8p_{i}p_{j}}{p_{i}+p_{j}}\|\langle\psi_{i}\|H\|\psi_{j}\rangle\|^{2}.$ (22) In the mean time, $F_{Q}(\|\psi_{i}\rangle)$ reduces to the form that is proportional to the variance of operator $H$ on the eigenstates, i.e., $F_{Q}(\|\psi_{i}\rangle)=4(\Delta H)^{2}_{\|\psi_{i}\rangle},$ (23) where $(\Delta H)^{2}_{\|\psi_{i}\rangle}:=\langle\psi_{i}\|H^{2}\|\psi_{i}\rangle-\|\langle\psi_{i}\|H\|\psi_{i}\rangle\|^{2}$ is the variance. Recently, Tóth and Petz Toth found that for a rank-2 system the quantum Fisher information can be treated as the convex roof of the variance, then Yu Yu proves that this theorem is also established for a general system, namely, $F_{\theta}=\min_{\\{q_{k},\|\Psi_{k}\rangle\\}}4\sum_{k}q_{k}(\Delta H)^{2}_{\|\Psi_{k}\rangle}.$ (24) Here $\\{q_{k},\|\Psi_{k}\rangle\\}$ refers to a set of pure-state ensembles, which satisfies $\rho_{\theta}=\sum_{k}q_{k}\|\Psi_{k}\rangle\langle\Psi_{k}\|.$ (25) One should notice that the ensemble of the eigenvalues and eigenstates $\\{p_{i},\|\psi_{i}\rangle\\}$ is one of these ensembles, but not the only one. Comparing Eq. (22) with Eq. (24), one can find that the condition for the ensemble $\\{p_{i},\|\psi_{i}\rangle\\}$ being the optimal ensemble is that the transition item $\langle\psi_{i}\|H\|\psi_{j}\rangle=0,\mbox{ for any }i\neq j.$ (26) For example, in some Mach-Zehnder interferometer, $H=\frac{1}{2i}(a^{\dagger}b-ab^{\dagger})$, where $a$, $b$ are the annihilation operators of two modes, and $a^{\dagger}$, $b^{\dagger}$ are the creation operators respectively. Choosing an appropriate input state, like an even state mine or a Fock state Pezze in one port, the item $\langle\psi_{i}\|H\|\psi_{j}\rangle$ vanishes for any $i\neq j$, then the ensemble $\\{p_{i},\|\psi_{i}\rangle\\}$ is the optimal ensemble and the QFI reduces to $F_{\theta}=4\sum_{i=1}^{\mathrm{s}}p_{i}(\Delta H)^{2}_{\|\psi_{i}\rangle}$. This condition can be checked through another way. Based on Ref. Yu , we introduce an observable $\mathrm{Y}=\sum_{i,j}\frac{2\sqrt{p_{i}p_{j}}}{p_{i}+p_{j}}H_{ij}\|\psi_{i}\rangle\langle\psi_{j}\|,$ (27) where $H_{ij}=\langle\psi_{i}\|H\|\psi_{j}\rangle$. Denote the spectral decomposition $\mathrm{Y}=\sum_{k}\alpha_{k}\|y_{k}\rangle\langle y_{k}\|$, then the optimal pure state can be constructed as $\|U_{k}\rangle=\frac{1}{\sqrt{u_{k}}}\sum_{i}U_{ki}\sqrt{p_{i}}\|\psi_{i}\rangle,$ (28) with $u_{k}=\sum_{i}\|U_{ki}\|^{2}p_{i}$ and $U_{ki}=\langle\psi_{i}\|y_{k}\rangle$. When $\|U_{k}\rangle=\|\psi_{k}\rangle$, there must be $\|y_{k}\rangle=\|\psi_{k}\rangle$. As $\|y_{k}\rangle$ is the eigenstate of observable Y, then one can see that the condition for $\|y_{k}\rangle=\|\psi_{k}\rangle$ is that all the off-diagonal elements of observable Y have to vanish, i.e., $H_{ij}=0$ for any $i\neq j$, which coincides with our result. ## III Matrix representation In this section we show a matrix representation of the quantum Fisher information. We consider the classical contribution first. Define a $\mathrm{N}$-dimensional diagonal matrix $D$ with elements $D_{ii}=p_{i}$, then the classical contribution can be rewritten in the form $F_{\mathrm{ct}}=4\mathrm{Tr}\left(\partial_{\theta}\sqrt{D}\right)^{2}.$ (29) This equation is equivalent to Eq. (18) as $p_{i}=0$ for $i\in[\mathrm{s}+1,\mathrm{N}]$. Define a $\mathrm{N}$-dimensional matrix $\mathcal{P}$ with the elements $\mathcal{P}_{ij}:=\|\langle\psi_{i}\|\partial_{\theta}\psi_{j}\rangle\|^{2}$. It is easy to see that the matrix $\mathcal{P}$ is real and symmetric. The symmetry can be proved by using Eq. (10) into the definition above. Denote a constant $\mathrm{N}$-dimensional matrix $\mathcal{I}$ whose elements are 1, i.e., $\mathcal{I}_{ij}=1$ for any $i$ and $j$, and define a $\mathrm{N}$-dimensional block diagonal matrix $\mathcal{G}$, which is $\mathcal{G}=\mathrm{diag}[\mathcal{H}_{\mathrm{s}\times\mathrm{s}},0_{(\mathrm{N}-\mathrm{s})\times(\mathrm{N}-\mathrm{s})}]$, where $\mathcal{H}_{\mathrm{s}\times\mathrm{s}}$ is a $\mathrm{s}$-dimensional real symmetric matrix. The elements of $\mathcal{H}$ are the harmonic mean values, $\mathcal{H}_{ij}=2p_{i}p_{j}/(p_{i}+p_{j}).$ With the help of above matrices, as well as the symmetry of $\mathcal{P},$ i.e., $\mathcal{P}_{ij}=\mathcal{P}_{ji}$, the quantum contribution can be written in the form $F_{\mathrm{qt}}=4\mathrm{Tr}\left[\left(D\mathcal{I}-\mathcal{G}\right)\mathcal{P}\right].$ (30) This is the matrix representation of quantum contribution of the QFI. It is easy to see that the coefficient matrix $D\mathcal{I}-\mathcal{G}$ is traceless. The matrix $\mathcal{P}$ can be treated as the “transfer” matrix between the vector of the eigenstates $(\|\psi_{1}\rangle,\cdots,\|\psi_{i}\rangle,\cdots,\|\psi_{\mathrm{N}}\rangle)^{\mathrm{T}}$ and its derivative vector. For a unitary parametrization, the element of $\mathcal{P}$ reads $\mathcal{P}_{ij}=\|\langle\phi_{i}\|H\|\phi_{j}\rangle\|^{2}$. In this case, the diagonal element of $\mathcal{P}$ is the survive probability of the eigenstate $\|\phi_{i}\rangle$ under the evolution $H$ and the non-diagonal element is the transition probability between $\|\phi_{i}\rangle$ and $\|\phi_{j}\rangle$ under $H$. Compared with Eqs. (18) and (19), the matrix representation of the quantum Fisher information is related to the entire $\mathrm{N}$-dimensional space. However, the coefficient matrix $D$, $\mathcal{G}$ and the “transfer” matrix $\mathcal{P}$ are all real and symmetric. For a unitary parametrization, in the matrix representation, one does not need to calculate the average value of $H^{2}$, but the transition item $\langle\psi_{i}\|H\|\psi_{j}\rangle$ has to be calculated through the entire space, not only those in the support. In the mean time, using the expression of Eq. (19), one has to calculate the average value of $H^{2}$ under the eigenstates, but the transition item needn’t to be calculated out of the support. These two representations have their own merits and will bring convenience if being used properly. In the following we give two examples utilizing this matrix representation. First we apply it in the qubit case. In this case, the parameterized density matrix $\rho_{\theta}$ can be decomposed as $\rho_{\theta}=\sum_{i=1}^{2}p_{i}(\theta)\|\psi_{i}(\theta)\rangle\langle\psi_{i}(\theta)\|$. Then the coefficient matrix reads $D\mathcal{I}-\mathcal{G}=\left(\begin{array}[]{cc}0&p_{1}-2\det\rho_{\theta}\\\ p_{2}-2\det\rho_{\theta}&0\end{array}\right),$ (31) where the equation $p_{1}p_{2}=\det\rho_{\theta}$ has been used. Thus, it is easy to obtain the quantum contribution as $F_{\mathrm{qt}}=4\left(1-4\det\rho_{\theta}\right)\mathcal{P}_{12},$ (32) where $\mathcal{P}_{12}=\|\langle\psi_{1}\|\partial_{\theta}\psi_{2}\rangle\|^{2}$. When the state is a pure state, for instance $p_{1}=1$ and $p_{2}=0$, there is $\det\rho_{\theta}=0$, then the quantum contribution reduces to $F_{\mathrm{qt}}=4\mathcal{P}_{12}=4\|\langle\psi_{1}\|\partial_{\theta}\psi_{2}\rangle\|^{2}.$ (33) This form coincides with the traditional quantum Fisher information form for pure state: $F_{Q}=\langle\partial_{\theta}\psi_{1}\|\partial_{\theta}\psi_{1}\rangle-\|\langle\psi_{1}\|\partial_{\theta}\psi_{1}\rangle\|^{2}$, which can be proved by substituting the normalization relation $\mathbb{I}=\|\psi_{1}\rangle\langle\psi_{1}\|+\|\psi_{2}\rangle\langle\psi_{2}\|$ into the item $\langle\partial_{\theta}\psi_{1}\|\partial_{\theta}\psi_{1}\rangle$. The classical contribution can also be obtained in this case, which reads $F_{\mathrm{ct}}=\frac{\left(\partial_{\theta}p_{1,2}\right)^{2}}{\det\rho_{\theta}}=\frac{\det\rho_{\theta}}{1-4\det\rho_{\theta}}\left[\partial_{\theta}\left(\ln\det\rho_{\theta}\right)\right]^{2}$ (34) for mixed states and $F_{\mathrm{ct}}=0$ for pure states. For a unitary parametrization, the quantum contribution reads $F_{\mathrm{qt}}=4\left(1-4\det\rho_{0}\right)\|\langle\phi_{1}\|H\|\phi_{2}\rangle\|^{2},$ (35) with $\|\phi_{i}\rangle$ a eigenstate of $\rho_{0}$. As $D$ is independent of $\theta$, the classical contribution vanishes for both mixed and pure states. Next we give another example. Consider a density matrix with the following form Hyllus $\rho_{\theta}=\sum_{n=0}^{\infty}Q_{n}\rho_{\theta}^{(n)},$ (36) where $Q_{n}$ is a real number and independent of $\theta$. $\rho_{\theta}^{(n)}$ is a state of $n$ particles in the entire Hilbert space. This form is representative for an optical system taking into account the superselection rules Hyllus . For a unitary parametrization, the spectral decomposition of $\rho_{\theta}$ reads $\rho_{\theta}=\sum_{n=0}^{\infty}\sum_{i=0}^{n}Q_{n}q_{i}^{(n)}\|\psi_{i}^{(n)}\rangle\langle\psi_{i}^{(n)}\|,$ (37) where $\|\psi_{i}^{(n)}\rangle=e^{-iH\theta}\|\phi_{i}^{(n)}\rangle$. In this case, the classical contribution vanishes. If the transition between the eigenstates in different particle spaces through the Hamiltonian $H$ is forbidden, which is feasible in some cases Jzrzyna , namely, $\langle\phi_{i}^{(n)}\|H\|\phi_{j}^{(n^{\prime})}\rangle=0$ when $n\neq n^{\prime}$, then the “transfer” matrix $\mathcal{P}$ can be written in a block diagonal form $\mathcal{P}=\sum_{n=0}^{\infty}\mathcal{P}^{(n)}$, where $\mathcal{P}^{(n)}$ is the corresponding “transfer” matrix for fixed $n$ particles. According to the feature of trace operation, only the corresponding block diagonal part of the coefficient matrices $D$, $\mathcal{I}$ and $\mathcal{G}$ matters in the calculation of the quantum contribution. If we define $D^{(n)}$, $\mathcal{I}^{(n)}$ and $\mathcal{G}^{(n)}$ as the coefficient matrices for fixed $n$ particles, then the block diagonal parts of $D$, $\mathcal{I}$ and $\mathcal{G}$ can be expressed as $\sum_{n}Q_{n}D^{(n)}$, $\sum_{n}\mathcal{I}^{(n)}$ and $\sum_{n}Q_{n}\mathcal{G}^{(n)}$. Thus, the quantum Fisher information reads $F_{Q}=4\sum_{n=0}^{\infty}Q_{n}\mathrm{Tr}\left[\left(D^{(n)}\mathcal{I}^{(n)}-\mathcal{G}^{(n)}\right)\mathcal{P}^{(n)}\right].$ (38) Also, one can find that the quantum Fisher information $F^{(n)}$ in the subspace of fixed $n$ particles can be written as $F_{Q}^{(n)}=4\mathrm{Tr}\left[\left(D^{(n)}\mathcal{I}^{(n)}-\mathcal{G}^{(n)}\right)\mathcal{P}^{(n)}\right].$ (39) Thus, one can write the total quantum Fisher information in the form $F_{Q}=\sum_{n=0}^{\infty}Q_{n}F_{Q}^{(n)}.$ (40) This total quantum Fisher information is the weighted average of all the quantum Fisher information for fixed $n$ particles. This form of the QFI has been widely used in the optical interferometry devices when no external global phase reference is present Demkowicz09 . More generally, taking into account the transition between the eigenstates in different particle subspaces, $\mathcal{P}$ can still be separated into blocks according to the particle number. Denote the sub-block in the upper and lower triangular of $\mathcal{P}$ between $n$ and $n^{\prime}$ particle subspaces as $\mathcal{P}^{(nn^{\prime})}$ and $\mathcal{P}^{(n^{\prime}n)}$, respectively. The diagonal block $\mathcal{P}^{(n)}$ is the same as above. Then, $\mathcal{P}$ can be expressed in the form $\mathcal{P}=\sum_{n}\mathcal{P}^{(n)}+\sum_{n\neq n^{\prime}}\mathcal{P}^{(nn^{\prime})}$, so as $\mathcal{I}$ and $\mathcal{G}$. Here all the elements of $\mathcal{P}^{(nn^{\prime})}$ is non- negative based on the property of $\mathcal{P}$. Thus, the total quantum Fisher information can be written as $F_{Q}=\sum_{n}Q_{n}F_{Q}^{(n)}+\sum_{n\neq n^{\prime}}4\mathrm{Tr}\left[C^{(nn^{\prime})}\mathcal{P}^{(n^{\prime}n)}\right],$ (41) where $C^{(nn^{\prime})}=Q_{n}D^{(n)}\mathcal{I}^{(nn^{\prime})}-\mathcal{G}^{(nn^{\prime})}$. From this equation one can find that when all the elements of $C^{(nn^{\prime})}$ is non-negative, the transition between the eigenstates in different particle subspaces, i.e., the second item of Eq. (41), can enhance the total QFI. Apart from this condition, the effect has to be discussed case by case. ## IV Conclusion In this paper, we provide a new analytic expression of the quantum Fisher information. For a non-full rank density matrix, this new expression is only determined by the support of the density matrix. With this new expression, the QFI for some infinite systems can be solved in a finite support space. This would bring significant advantage during the calculation in some scenarios. Besides, we also provide a matrix representation form of the quantum Fisher information and give two examples. ###### Acknowledgements. This work was supported by NFRPC through Grant No. 2012CB921602, the NSFC through Grants No. 11025527 and No. 10935010. ## References * (1) V. Giovannetti, S. 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1312.6977	# A Note on Symmetric properties of the multiple $q$-Euler zeta functions and higher-order $q$-Euler polynomials Dae San Kim and Taekyun Kim ###### Abstract. Recently, the higher-order $q$-Euler polynomials and multiple $q$-Euler zeta functions are introduced by T. Kim ([key-8, key-7]). In this paper, we investigate some symmetric properties of the multiple $q$-Euler zeta function and derive various identities concerning the higher-order $q$-Euler polynomials from the symmetric properties of the multiple $q$-Euler zeta functions. ## 1\. Introduction For $q\in\mathbb{C}$ with $\left\|q\right\|<1$, the $q$-number is defined by $\left[x\right]_{q}=\frac{1-q^{x}}{1-q}$ . Note that ${\displaystyle\lim_{q\rightarrow 1}\left[x\right]_{q}}=x$. As is well known, the Euler polynomials of order $r$$\left(\in\mathbb{N}\right)$ are defined by the generating function to be (1) $\left(\frac{2}{e^{t}+1}\right)^{r}e^{xt}=\underset{r\textrm{-times}}{\underbrace{\left(\frac{2}{e^{t}+1}\right)\times\cdots\times\left(\frac{2}{e^{t}+1}\right)}}e^{xt}=\sum_{n=0}^{\infty}E_{n}^{\left(r\right)}\left(x\right)\frac{t^{n}}{n!}.$ When $x=0$, $E_{n}^{\left(r\right)}=E_{n}^{\left(r\right)}\left(0\right)$ are called the Euler numbers of order $r$ (see [1-13]). In [key-7], T. Kim considered the $q$-extension of higher-order Euler polynomials which are given by the generating function to be (2) $\displaystyle F_{q}^{\left(r\right)}\left(t,x\right)$ $\displaystyle=\left[2\right]_{q}^{r}\sum_{m_{1},\cdots,m_{r}=0}^{\infty}\left(-q\right)^{m_{1}+\cdots+m_{r}}e^{\left[m_{1}+\cdots+m_{r}+x\right]_{q}t}$ $\displaystyle=\sum_{n=0}^{\infty}E_{n,q}^{\left(r\right)}\left(x\right)\frac{t^{n}}{n!}.$ Note that ${\displaystyle\lim_{q\rightarrow 1}}F_{q}^{\left(r\right)}\left(t,x\right)=\left(\frac{2}{e^{t}+1}\right)^{r}e^{xt}={\displaystyle\sum_{n=0}^{\infty}E_{n}^{\left(r\right)}\left(x\right)\frac{t^{n}}{n!}}$. When $x=0$, $E_{n,q}^{\left(r\right)}=E_{n,q}^{\left(r\right)}\left(0\right)$ are called the $q$-Euler numbers of order $r$$\left(\in\mathbb{N}\right)$. In [key-11], Rim et al. have studied the properties of $q$-Euler polynomials due to T. Kim. From (2), we note that (3) $\displaystyle E_{n,q}^{\left(r\right)}\left(x\right)$ $\displaystyle=\sum_{l=0}^{n}\dbinom{n}{l}q^{lx}E_{l,q}^{\left(r\right)}\left[x\right]_{q}^{n-l}$ $\displaystyle=\left(q^{x}E_{q}^{\left(r\right)}+\left[x\right]_{q}\right)^{n},$ with the usual convention about replacing $\left(E_{q}^{\left(r\right)}\right)^{n}$ by $E_{n,q}^{\left(r\right)}$. In [key-7], T. Kim considered the multiple $q$-Euler zeta function which interpolates higher-order $q$-Euler polynomials at negative integers as follows : $\displaystyle\zeta_{q,r}\left(s,x\right)$ $\displaystyle=$ $\displaystyle\left[2\right]_{q}^{r}\sum_{m_{1},\cdots,m_{r}=0}^{\infty}\frac{\left(-q\right)^{m_{1}+\cdots+m_{r}}}{\left[m_{1}+\cdots+m_{r}+x\right]_{q}^{s}}$ $\displaystyle=$ $\displaystyle\left[2\right]_{q}^{r}\sum_{m=0}^{\infty}\dbinom{m+r-1}{m}_{q}\left(-q\right)^{m}\frac{1}{\left[m+x\right]_{q}^{s}},$ where $s\in\mathbb{C}$ and $x\in\mathbb{R}$ with $x\neq 0,\,-1,\,-2,\,-3,\,\cdots$. By using Cauchy residue theorem and Laurent series, we note that (5) $\zeta_{q,r}\left(-n,x\right)=E_{n,q}^{\left(r\right)}\left(x\right),\quad\textrm{where }r\in\mathbb{Z}_{\geq 0}.$ Recently, D.S. Kim et al. ([key-5]) introduced some interesting and important symmetric identities of the $q$-Euler polynomials which are derived from the symmetric properties of $q$-Euler zeta function. Indeed, their identities are a part of an answer to an open question for the symmetric identities of Carlitz’s type $q$-Euler polynomials in [key-6]. In order to find a generalization of identities of D. S. Kim et al. ([key-5]), we consider symmetric properties of the multiple $q$-Euler zeta function. From the symmetric properties of multiple $q$-Euler zeta function, we derive identities of symmetry for the higher-order $q$-Euler polynomials. ## 2\. Some identities of higher-order $q$-Euler polynomials For $a,\,b\in\mathbb{N}$ with $a\equiv 1$ (mod $2$) and $b\equiv 1$ (mod 2), we observe that (6) $\displaystyle\frac{1}{\left[2\right]_{q^{a}}^{r}}\zeta_{q^{a},r}\left(s,\,bx+\frac{b}{a}\left(j_{1}+\cdots+j_{r}\right)\right)$ $\displaystyle=$ $\displaystyle\sum_{n_{1},\cdots,\,n_{r}=0}^{\infty}\frac{\left(-1\right)^{n_{1}+\cdots+n_{r}}q^{a\left(n_{1}+\cdots+n_{r}\right)}}{\left[n_{1}+\cdots+n_{r}+bx+\frac{b}{a}\left(j_{1}+\cdots+j_{r}\right)\right]_{q^{a}}^{s}}$ $\displaystyle=$ $\displaystyle\left[a\right]_{q}^{s}\sum_{n_{1},\cdots,\,n_{r}=0}^{\infty}\frac{\left(-1\right)^{n_{1}+\cdots+n_{r}}q^{a\left(n_{1}+\cdots+n_{r}\right)}}{\left[a\left(n_{1}+\cdots+n_{r}\right)+abx+b\left(j_{1}+\cdots+j_{r}\right)\right]_{q}^{s}}$ $\displaystyle=$ $\displaystyle\left[a\right]_{q}^{s}\sum_{n_{1},\cdots,\,n_{r}=0}^{\infty}\sum_{i_{1},\cdots,i_{r}=0}^{b-1}\frac{\left(-1\right)^{{\displaystyle{\textstyle\sum_{l=1}^{r}\left(i_{l}+bn_{l}\right)}}}q^{a{\displaystyle{\textstyle\sum_{l=1}^{r}\left(i_{l}+bn_{l}\right)}}}}{\left[ab{\displaystyle{\textstyle\sum_{l=1}^{r}\left(x+n_{l}\right)}+b{\textstyle\sum_{l=1}^{r}j_{l}}+a{\textstyle\sum_{l=1}^{r}i_{l}}}\right]_{q}^{s}}.$ From (6), we note that (7) $\displaystyle\frac{\left[b\right]_{q}^{s}}{\left[2\right]_{q^{a}}^{r}}\sum_{j_{1},\cdots,\,j_{r}=0}^{a-1}\left(-1\right)^{{\textstyle{\displaystyle{\textstyle\sum_{l=1}^{r}j_{l}}}}}q^{b{\displaystyle{\textstyle\sum_{l=1}^{r}j_{l}}}}\zeta_{q^{a},r}\left(s,\,bx+\frac{b}{a}\left(j_{1}+\cdots+j_{r}\right)\right)$ $\displaystyle=$ $\displaystyle\left[a\right]_{q}^{s}\left[b\right]_{q}^{s}\sum_{j_{1},\cdots,\,j_{r}=0}^{a-1}\sum_{i_{1},\cdots,\,i_{r}=0}^{b-1}$ $\displaystyle\times\sum_{n_{1},\cdots,\,n_{r}=0}^{\infty}\frac{\left(-1\right)^{{\displaystyle{\textstyle\sum_{l=1}^{r}\left(i_{l}+j_{l}+n_{l}\right)}}}q^{{\displaystyle{\textstyle\sum_{l=1}^{r}\left(bj_{l}+ai_{l}+abn_{l}\right)}}}}{\left[ab{\displaystyle{\textstyle\sum_{l=1}^{r}\left(x+n_{l}\right)+b{\displaystyle{\textstyle\sum_{l=1}^{r}j_{l}}+a{\displaystyle{\textstyle\sum_{l=1}^{r}i_{l}}}}}}\right]_{q}^{s}}.$ By the same method as (7), we get (8) $\displaystyle\frac{\left[a\right]_{q}^{s}}{\left[2\right]_{q^{b}}^{r}}\sum_{j_{1},\cdots,\,j_{r}=0}^{b-1}\left(-1\right)^{{\displaystyle{\textstyle\sum_{l=1}^{r}j_{l}}}}q^{a{\displaystyle{\textstyle\sum_{l=1}^{r}j_{l}}}}\zeta\left(s,\,ax+\frac{a}{b}\left(j_{1}+\cdots+j_{r}\right)\right)$ $\displaystyle=$ $\displaystyle\left[a\right]_{q}^{s}\left[b\right]_{q}^{s}\sum_{j_{1},\cdots,\,j_{r}=0}^{b-1}\sum_{i_{1},\cdots,\,i_{r}=0}^{a-1}$ $\displaystyle\times\sum_{n_{1},\cdots,\,n_{r}=0}^{\infty}\frac{\left(-1\right)^{{\displaystyle{\textstyle\sum_{l=1}^{r}\left(i_{l}+j_{l}+n_{l}\right)}}}q^{{\displaystyle{\textstyle\sum_{l=1}^{r}\left(bi_{l}+aj_{l}+abn_{l}\right)}}}}{\left[ab{\displaystyle{\textstyle\sum_{l=1}^{r}\left(x+n_{l}\right)}+a{\textstyle\sum_{l=1}^{r}j_{l}}+b{\displaystyle{\textstyle\sum_{l=1}^{r}i_{l}}}}\right]_{q}^{s}}.$ Threfore, by (7) and (8), we obtain the following theorem. ###### Theorem 1. For $a,\,b\in\mathbb{N}$ with $a\equiv 1$ $\textnormal{(mod }2\mathnormal{)}$ and $b\equiv 1$ $\textnormal{(mod }2\mathnormal{)}$, we have $\displaystyle\left[2\right]_{q^{b}}^{r}\left[b\right]_{q}^{s}\sum_{j_{1},\cdots,j_{r}=0}^{a-1}\left(-1\right)^{{\displaystyle{\textstyle\sum_{l=1}^{r}j_{l}}}}q^{b{\displaystyle{\textstyle\sum_{l=1}^{r}j_{l}}}}\zeta_{q^{a},r}\left(s,bx+\frac{b}{a}\left(j_{1}+\cdots+j_{r}\right)\right)$ $\displaystyle=$ $\displaystyle\left[2\right]_{q^{a}}^{r}\left[a\right]_{q}^{s}\sum_{j_{1},\cdots,j_{r}=0}^{b-1}\left(-1\right)^{{\displaystyle{\textstyle\sum_{l=1}^{r}j_{l}}}}q^{a{\displaystyle{\textstyle\sum_{l=1}^{r}j_{l}}}}\zeta_{q^{b},r}\left(s,ax+\frac{a}{b}\left(j_{1}+\cdots+j_{r}\right)\right).$ From (5) and Theorem 1, we obtain the following theorem. ###### Theorem 2. For $n\geq 0$ and $a,\,b\in\mathbb{N}$ with $a\equiv 1$ $\textnormal{(mod }2\mathnormal{)}$ and $b\equiv 1$ $\textnormal{(mod }2\mathnormal{)}$, we have $\displaystyle\left[2\right]_{q^{b}}^{r}\left[a\right]_{q}^{n}\sum_{j_{1},\cdots,j_{r}=0}^{a-1}\left(-1\right)^{{\displaystyle{\textstyle\sum_{l=1}^{r}j_{l}}}}q^{b{\textstyle{\displaystyle{\textstyle\sum_{l=1}^{r}j_{l}}}}}E_{n,q^{a}}^{\left(r\right)}\left(bx+\frac{b}{a}\left(j_{1}+\cdots+j_{r}\right)\right)$ $\displaystyle=$ $\displaystyle\left[2\right]_{q^{a}}^{r}\left[b\right]_{q}^{n}\sum_{j_{1},\cdots,j_{r}=0}^{b-1}\left(-1\right)^{{\displaystyle{\textstyle\sum_{l=1}^{r}j_{l}}}}q^{a{\displaystyle{\textstyle\sum_{l=1}^{r}j_{l}}}}E_{n,q^{b}}^{\left(r\right)}\left(ax+\frac{a}{b}\left(j_{1}+\cdots+j_{r}\right)\right).$ By (3), we easily get (9) $E_{n,q}^{\left(r\right)}\left(x+y\right)=\sum_{i=0}^{n}\dbinom{n}{i}q^{xi}E_{i,q}^{\left(r\right)}\left(y\right)\left[x\right]_{q}^{n-i}.$ Thus, from (9), we have (10) $\displaystyle\sum_{j_{1},\cdots,j_{r}=0}^{a-1}\left(-1\right)^{{\displaystyle{\textstyle\sum_{l=1}^{r}j_{l}}}}q^{b{\displaystyle{\textstyle\sum_{l=1}^{r}j_{l}}}}E_{n,q^{a}}^{\left(r\right)}\left(bx+\frac{b}{a}\left(j_{1}+\cdots+j_{r}\right)\right)$ $\displaystyle=$ $\displaystyle\sum_{j_{1},\cdots,j_{r}=0}^{a-1}\left(-1\right)^{{\displaystyle{\textstyle\sum_{l=1}^{r}j_{l}}}}q^{b{\displaystyle{\textstyle\sum_{l=1}^{r}j_{l}}}}\sum_{i=0}^{n}\dbinom{n}{i}q^{ia\left(\frac{b}{a}{\displaystyle{\textstyle\sum_{l=1}^{r}j_{l}}}\right)}E_{i,q^{a}}^{\left(r\right)}\left(bx\right)\left[\frac{b}{a}\sum_{l=1}^{r}j_{l}\right]_{q^{a}}^{n-i}$ $\displaystyle=$ $\displaystyle\sum_{j_{1},\cdots,j_{r}=0}^{a-1}\left(-1\right)^{{\displaystyle{\textstyle\sum_{l=1}^{r}j_{l}}}}q^{b{\displaystyle{\textstyle\sum_{l=1}^{r}j_{l}}}}\sum_{i=0}^{n}\dbinom{n}{i}q^{\left(n-i\right)b{\textstyle{\displaystyle{\textstyle\sum_{l=1}^{r}j_{l}}}}}E_{n-i,q^{a}}^{\left(r\right)}\left(bx\right)\left[\frac{b}{a}\sum_{l=1}^{r}j_{l}\right]_{q^{a}}^{i}$ $\displaystyle=$ $\displaystyle\sum_{i=0}^{n}\dbinom{n}{i}\left(\frac{\left[b\right]_{q}}{\left[a\right]_{q}}\right)^{i}E_{n-i,q^{a}}^{\left(r\right)}\left(bx\right)$ $\displaystyle\times\sum_{j_{1},\cdots,j_{r}=0}^{a-1}\left(-1\right)^{{\displaystyle{\textstyle\sum_{l=1}^{r}j_{l}}}}q^{b{\displaystyle{\textstyle\sum_{l=1}^{r}\left(n-i+1\right)j_{l}}}}\left[j_{1}+\cdots+j_{r}\right]_{q^{b}}^{i}$ $\displaystyle=$ $\displaystyle\sum_{i=0}^{n}\dbinom{n}{i}\left(\frac{\left[b\right]_{q}}{\left[a\right]_{q}}\right)^{i}E_{n-i,q^{a}}^{\left(r\right)}\left(bx\right)S_{n,i,q^{b}}^{\left(r\right)}\left(a\right),$ where (11) $S_{n,i,q^{b}}^{\left(r\right)}\left(a\right)=\sum_{j_{1},\cdots,\,j_{r}=0}^{a-1}\left(-1\right)^{\sum_{l=1}^{r}j_{l}}q^{\sum_{l=1}^{r}\left(n-i+1\right)j_{l}}\left[j_{1}+\cdots+j_{r}\right]_{q}^{i}.$ From (10) and (11), we note that (12) $\displaystyle\left[2\right]_{q^{b}}^{r}\left[a\right]_{q}^{n}\sum_{j_{1},\cdots,j_{r}=0}^{a-1}\left(-1\right)^{\sum_{l=1}^{r}j_{l}}q^{b\sum_{l=1}^{r}j_{l}}E_{n,q^{a}}^{\left(r\right)}\left(bx+\frac{b}{a}\left(j_{1}+\cdots+j_{r}\right)\right)$ $\displaystyle=$ $\displaystyle\left[2\right]_{q^{b}}^{r}\sum_{i=0}^{n}\dbinom{n}{i}\left[a\right]_{q}^{n-i}\left[b\right]_{q}^{i}E_{n-i,q^{a}}^{\left(r\right)}\left(bx\right)S_{n,i,q^{b}}^{\left(r\right)}\left(a\right).$ By the same method as (12), we get (13) $\displaystyle\left[2\right]_{q^{a}}^{r}\left[b\right]_{q}^{n}\sum_{j_{1},\cdots,j_{r}=0}^{b-1}\left(-1\right)^{\sum_{l=1}^{r}j_{l}}q^{a\sum_{l=1}^{r}j_{l}}E_{n,q^{b}}^{\left(r\right)}\left(ax+\frac{a}{b}\left(j_{1}+\cdots+j_{r}\right)\right)$ $\displaystyle=$ $\displaystyle\left[2\right]_{q^{a}}^{r}\sum_{i=0}^{n}\dbinom{n}{i}\left[b\right]_{q}^{n-i}\left[a\right]_{q}^{i}E_{n-i,q^{b}}^{\left(r\right)}\left(ax\right)S_{n,i,q^{a}}^{\left(r\right)}\left(b\right).$ Therefore, by (12) and (13), we obtain the following theorem. ###### Theorem 3. For $n\geq 0$ and $a,\,b\in\mathbb{N}$ with $a\equiv 1$ $\textnormal{(mod }2\mathnormal{)}$ and $b\equiv 1$ $\textnormal{(mod }2\mathnormal{)}$, we have $\displaystyle\left[2\right]_{q^{b}}^{r}\sum_{i=0}^{n}\dbinom{n}{i}\left[a\right]_{q}^{n-i}\left[b\right]_{q}^{i}E_{n-i,q^{a}}^{\left(r\right)}\left(bx\right)S_{n,i,q^{b}}^{\left(r\right)}\left(a\right)$ $\displaystyle=$ $\displaystyle\left[2\right]_{q^{a}}^{r}\sum_{i=0}^{n}\dbinom{n}{i}\left[b\right]_{q}^{n-i}\left[a\right]_{q}^{i}E_{n-i,q^{b}}^{\left(r\right)}\left(ax\right)S_{n,i,q^{a}}^{\left(r\right)}\left(b\right).$ It is not difficult to show that (14) $\displaystyle e^{\left[x\right]_{q}u}\sum_{m_{1},\cdots,m_{r}=0}^{\infty}q^{m_{1}+\cdots+m_{r}}\left(-1\right)^{m_{1}+\cdots+m_{r}}e^{\left[y+m_{1}+\cdots+m_{r}\right]_{q}q^{x}\left(u+v\right)}$ $\displaystyle=$ $\displaystyle e^{-\left[x\right]_{q}v}\sum_{m_{1},\cdots,m_{r}=0}^{\infty}q^{m_{1}+\cdots+m_{r}}\left(-1\right)^{m_{1}+\cdots+m_{r}}e^{\left[x+y+m_{1}+\cdots+m_{r}\right]_{q}\left(u+v\right)}.$ By (2) and (14), we get (15) $\displaystyle\sum_{k=0}^{m}\dbinom{m}{k}q^{\left(k+n\right)x}E_{k+n,q}^{\left(r\right)}\left(y\right)\left[x\right]_{q}^{m-k}$ $\displaystyle=$ $\displaystyle\sum_{k=0}^{n}\dbinom{n}{k}E_{m+k,q}^{\left(r\right)}\left(x+y\right)q^{\left(n-k\right)x}\left[-x\right]_{q}^{n-k},$ where $m,n\geq 0$. Thus, by (15), we see that (16) $\displaystyle\sum_{k=0}^{m}\dbinom{m}{k}q^{kx}E_{k+n,q}^{\left(r\right)}\left(y\right)\left[x\right]_{q}^{m-k}$ $\displaystyle=$ $\displaystyle\sum_{k=0}^{n}\dbinom{n}{k}q^{-kx}E_{m+k,q}^{\left(r\right)}\left(x+y\right)\left[-x\right]_{q}^{n-k},$ where $m,\,n\geq 0$. ACKNOWLEDGEMENTS. This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MOE) (No.2012R1A1A2003786 ). Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea _E-mail_ _address :_ [email protected] Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea _E-mail_ _address :_ [email protected]	arxiv-papers	2013-12-25T14:19:04	2024-09-04T02:49:55.944238	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Dae San Kim and Taekyun Kim", "submitter": "Taekyun Kim", "url": "https://arxiv.org/abs/1312.6977" }
1312.7127	# Prediction of a quantum anomalous Hall state in Co decorated silicene T. P. Kaloni, N. Singh, and U. Schwingenschlögl [email protected],+966(0)544700080 Physical Science & Engineering Division, KAUST, Thuwal 23955-6900, Kingdom of Saudi Arabia ###### Abstract Based on first-principles calculations, we demonstrate that Co decorated silicene can host a quantum anomalous Hall state. The exchange field induced by the Co atoms combined with the strong spin orbit coupling of the silicene opens a nontrivial band gap at the K-point. As compared to other transition metals, Co decorated silicene is unique in this respect, since usually hybridization and spin-polarization induced in the silicene suppress a quantum anomalous Hall state. ###### pacs: 73.22.Gk, 75.50.Pp, 73.43.Cd ## I Introduction Silicene is a single layer of Si atoms arranged in a two-dimensional honeycomb lattice verri and therefore closely related to graphene. It nowadays attracts considerable attention due to its exotic electronic structure and promising applications in Si nanoelectronics. The chemical similarity between silicene and graphene arises from the fact that Si and C belong to the same group in the periodic table. However, Si is subject to $sp^{3}$ hybridization, whereas pure $sp^{2}$ hybridization is energetically favorable in graphene. Silicene has a buckled structure with two sublattices displaced vertically with respect to each other. The synthesis of silicene nanoribbons on the anisotropic Ag(110) surface has been studied early padova ; padova1 , while recently growth on Ag(111) has been reported vogt ; feng . A combined experimental and theoretical study by Fleurence and coworkers antoine on the formation of silicene on ZrB2 thin film has indicated that the buckling is influenced by the interaction with the substrate, leading to a direct electronic band gap at the $\Gamma$-point. Simulations based on density functional theory show that the electronic structure of silicene is governed by a Dirac cone, similar to graphene olle . Around the Fermi energy, the charge carriers behave as massless Dirac fermions in the $\pi$/$\pi^{}$ bands, which approach each other at the K-point. The electronic structure of silicene in a perpendicular electric field has been addressed in Ref. Ni and the results of Drummond and coworkers falko suggest that variation of the electric field with respect to the strength of the spin orbit coupling induces a transition between a topological and a band insulator. Unlike the quantum Hall effect, the quantum anomalous Hall (QAH) effect results from breaking of the time-reversal symmetry by an exchange field combined with strong spin orbit coupling that induces a band inversion. In topological insulators the quantum spin Hall effect has been observed experimentally science1 and the QAH effect has been predicted science2 . A QAH state also has been predicted by Yu and coworkers science for magnetically doped thin films of topological insulators and by Ezawa for silicene nanoribbons izawa . However, the effect has not been demonstrated for silicene so far, although transition metal adsorption has been studied theoretically in Ref. PRB , however, without inclusion of the spin orbit coupling and onsite Coulomb interaction. Therefore, the QAH state could not be detected in this work. In our present study, we analyze the structural, magnetic, and electronic properties of silicene decorated by the transition metals Ti, V, Cr, Mn, Fe, Co, Ni, or Cu in comparison to each other. Co decoration is found to result in a unique nature, because all other elements under investigation turn out to inhibit creation of a QAH state for different reasons. ## II Computational details We perform geometry optimizations within the generalized gradient approximation including the van der Waals interaction grime ; kaloni-jmc as implemented in the Quantum ESPRESSO package paolo . Moreover, we use a plane wave cutoff energy of 544 eV and a Monkhorst-Pack $16\times 16\times 1$ k-mesh for $4\times 4\times 1$ supercells of silicene with a lattice constant of $a=15.44$ Å and a vacuum layer of 20 Å. Our supercells contain 32 Si atoms and 1 transition metal atom, such that the density of the impurities is low enough to neglect their mutual interaction. The atomic positions are optimized until all forces have converged to less than 0.003 eV/Å. We calculate the electronic band structures of transition metal decorated silicene by the full potential augmented plane wave method implemented in the WIEN2k code Wien2k . The application of a finite onsite Coulomb interaction $U$ is necessary for correctly describing the $d$ electrons of the transition metal atoms, as previous studies suggest that the adsorption geometries and electronic configurations are sensitive to correlation effects mousumi ; wehling . A value of $U=4$ eV is expected to give appropriate results for the transition metal atoms under consideration new3 ; new4 and, therefore, is employed in our calculations. We have also tested different values of the onsite interaction from 3 to 5 eV without finding any influence on our conclusions. The convergence threshold is set to 10-4 eV with $R_{mt}K_{max}=7$, where $K_{max}$ is the planewave cutoff and $R_{mt}$ is the smallest muffin-tin radius. A k-mesh with 36 points in the irreducible Brillouin zone is used. Silicene has been demonstrated to exist on various substrates. While we do not take into account a specific substrate in our calculations, our results are valid not only for a suspended sample but also in the case that the interaction with the substrate is small. ## III Structural Considerations \| \| $E_{b}$ top --- (eV) \| $E_{b}$ bottom --- (eV) $h$ (Å) \| $\theta$ $(^{\circ})$ \| \| Total --- ($\mu_{B}$) \| Metal $d$ --- ($\mu_{B}$) \| Si --- ($\mu_{B}$) \| Interstitial --- ($\mu_{B}$) \| spin up --- occup. \| spin down --- occup. Ti \| 3.58 \| 3.60 \| 1.40 \| 112-118 \| 2.05 \| 1.59 \| 0.05 \| 0.40 \| 1.90 \| 0.31 V \| 3.16 \| 3.52 \| 1.36 \| 113-117 \| 2.99 \| 2.55 \| 0.02 \| 0.41 \| 2.80 \| 0.25 Cr \| 2.00 \| 2.33 \| 1.30 \| 115-118 \| 4.76 \| 4.10 \| 0.08 \| 0.59 \| 4.21 \| 0.11 Mn \| 2.39 \| 2.40 \| 1.10 \| 111-115 \| 3.10 \| 3.97 \| $-$0.42 \| $-$0.44 \| 4.40 \| 0.43 Fe \| 2.91 \| 3.45 \| 0.96 \| 111-118 \| 2.10 \| 2.69 \| $-$0.25 \| $-$0.33 \| 4.39 \| 1.70 Co \| 3.42 \| 3.99 \| 0.79 \| 113-118 \| 0.99 \| 1.16 \| $-$0.07 \| $-$0.10 \| 4.48 \| 3.32 Ni \| 3.05 \| 3.57 \| 0.75 \| 113-117 \| 0.00 \| 0.00 \| 0.00 \| 0.00 \| 4.08 \| 4.08 Cu \| 2.10 \| 2.50 \| 0.66 \| 113-117 \| 0.00 \| 0.00 \| 0.00 \| 0.00 \| 4.40 \| 4.40 Table 1: Transition metal decoration at the top site (binding energy) and hollow site (binding energy, structural parameters, total magnetic moment, transition metal $d$ moment, Si moment, interstitial moment, and transition metal occupations for the spin up and down channels). Figure 1: Top and side views of the optimized structure for transition metal decoration of silicene at the hollow site. Because of the hexagonal symmetry of silicene, the possible decoration sites for a single atom can be categorized as top, bridge, and hollow. While all three choices have been studied, decoration at the bridge site will no longer be considered in the following, because it turns out to be an unstable configuration. The structural optimization demonstrates that a transition metal atom added at the top site shifts close to an original Si position and displaces the Si atom. Therefore, one metal$-$Si bond length of 2.43 Å (Ti), 2.41 Å (V), 2.38 Å (Cr), 2.35 Å (Mn), 2.33 Å (Fe), 2.30 Å (Co), 2.28 Å (Ni), or 2.26 Å (Cu) is realized. In addition, the metal atom is bound to three other Si atoms with shorter bond lengths of 2.39 Å (Ti), 2.37 Å (V), 2.35 Å (Cr), 2.31 Å (Mn), 2.29 Å (Fe), 2.28 Å (Co), 2.26 Å (Ni), or 2.24 Å (Cu). The Si$-$Si bond lengths around the impurity have values of 2.27 to 2.33 Å and thus are slightly modified as compared to pristine silicene (bond length 2.27 Å yao ). The buckling in the silicene layer amounts to 0.34 Å to 0.52 Å and the binding energy to $E_{b}=3.58$ eV (Ti), 3.16 eV (V), 2.00 eV (Cr), 2.39 eV (Mn), 2.91 eV (Fe), 3.42 eV (Co), 3.05 eV (Ni), or 2.10 eV (Cu). Defining $h$ as the height of the transition metal atom above the silicene plane we obtain $h=1.90$ Å (Ti), 1.80 Å (V), 1.70 Å (Cr), 1.20 Å (Mn), 1.11 Å (Fe), 1.01 Å (Co), 0.91 Å (Ni), or 0.82 Å (Cu). Moreover, the angle $\theta$ between the Si$-$Si bonds and the normal of the silicene sheet is in pristine silicene $\theta=116^{\circ}$ due to the mentioned mixture of $sp^{2}$ and $sp^{3}$ hybridizations, while around the impurity a wide range of angles, 113∘ to 118∘, is realized. For decoration at the hollow site the transition metal atom does not displace a specific Si atom but stays rather in the center of the Si hexagon, see Fig. 1. It is bound to the six neighboring Si atoms with bond lengths of 2.26 Å to 2.35 Å. The obtained values of $h$ are listed in Table I. The buckling, the Si–Si bond length, and the angle $\theta$ are found to be slightly modified as compared to decoration at the top site. The energy difference between the configurations with the transition metal atom at the top and hollow sites is 0.02 eV (Ti), 0.36 eV (V), 0.33 eV (Cr), 0.01 eV (Mn), 0.54 eV (Fe), 0.57 eV (Co), 0.52 eV (Ni), and 0.40 eV (Cu), indicating that decoration at the hollow site is always energetically favorable. These values are in reasonable agreement with previous results on transition metal decorated silicene PRB , which also applies to the optimized structures. Only the heights $h$ are significantly different, which we attribute to the inclusion of the van der Waal interaction in our structural optimizations and the spin orbit coupling in our electronic structure calculations. It thus is to be expected that our results are more reliable than the previously reported values. ## IV Electronic structure In the analysis of the electronic structure, we consider only the energetically favorable hollow site, for the different transition metal impurities. In the case of Ti, V, and Cr decoration we obtain total magnetic moments of 2.05 $\mu_{B}$, 2.99 $\mu_{B}$, and 4.10 $\mu_{B}$ per unit cell, respectively, see Table I. It should be noted that the main portion to the magnetic moment comes from the transition metal $d$ orbitals with small contributions form Si atoms and the interstitial region. We note that the silicene sheet gets significantly polarized, where the magnetic moments of the Ti, V, and Cr atoms are aligned ferrimagnetically with respect to the induced Si spins. For half filling (Mn) and beyond half filling (Fe and Co) the orientations of the transition metal and Si spins are opposite (antiferromagnetically aligned; negative signs in Table I). In contrast to Mn and Fe decoration, in the case of Co decoration the magnetic moment of 0.99 $\mu_{B}$ is strongly localized and almost completely due to the Co $d$ orbitals. This fact implies that Co affects the silicene sheet much less than the other transition metal atoms with larger local magnetic moments. Finally, silicene decorated with Ni or Cu turns out to be non-magnetic, as to be expected. Figure 2: Electronic band structure with weights of the Co 3$d$ states (size of the dots) for decorated silicene (a) $1\times$1 and (b) $2\times 2$ supercells. We consider different coverages of transition metal atoms by employing $1\times 1\times 1$, $2\times 2\times 1$, $3\times 3\times 1$, and $4\times 4\times 1$ supercells of silicene to which a single transition metal atom is added. We find in none of the supercells besides the $4\times 4\times 1$ supercell signs of a QAH state, indicating that interaction between the transition metal atoms counteracts the creation of this state and that, thus, the impurity density has to be sufficiently low. In the case of a high transition metal coverage, see Fig. 2(a,b) for the example of Co decoration, the Co$-$Co interaction, coming along with the small separation of only 3.86 Å (periodic boundary conditions), modifies the shape of the band structure in the vicinity of the Fermi energy strongly. For different coverages accordingly fundamentally different band structures are obtained. On the other hand, the distance between the transition metal atoms in the $4\times 4\times 1$ supercell is large enough to represent the dilute limit. A close similarity of the $4\times 4\times 1$ and $5\times 5\times 1$ band structures indicates that already in the former case the limit of low impurity density is reached. In the following we address the $4\times 4\times 1$ supercell for this reason. In the band structures of Ti decorated silicene, see Fig. 3(a), and V decorated silicene, see Fig. 3(b), a strong hybridization between the transition metal 3$d$ and Si 3$p$ states is observed. Similar findings previously have been reported for Au and Mo doped graphene new1 ; new2 . Due to the hybridization the QAH effect cannot be realized for Ti and V decoration as the silicene states are perturbed. As a consequence of the relative energetic shift between the impurity and silicene states when another transition metal from the same period of the periodic table is chosen, we can expect that exchange of the impurity can overcome the hybridization problem in the vicinity of the Fermi energy. In the case of Cr decoration, see Fig. 3(c), we observe the remainder of a Dirac cone with only small transition metal weights, which demonstrates that the states near the Fermi energy are mainly due to the Si $p_{z}$ orbitals. However, in this case the local magnetic moment of the impurity is high, see Table I, such that the silicene sheet becomes to some extent spin polarized, which also prevents the creation of a QAH state. The band structure of Mn decorated silicene is addressed in Fig. 3(d). The remainder of a Dirac cone is visible about 0.25 eV below the Fermi energy, reflecting $n$-doping. At the K-point the bands are due to the Si $p_{z}$ states without hybridization with the Mn 3$d$ states. Otherwise the situation is very similar to the Cr case, except for the antiferromagnetic polarization of the silicene as mentioned before. In Fig. 3(e) we deal with Fe decorated silicene. The Dirac cone now is located about 0.12 eV below the Fermi energy at the high symmetry K-point, i.e., it is $n$-doped. The Fe $3d$ states likewise do not hybridize with the Si 4$p_{z}$ states in the vicinity of the Fermi energy. However, similar to Mn decoration, the high Fe magnetic moment induces significant spin polarization in the silicene, such that no QAH state is realized. Figure 3: Electronic band structure with weights of the transition metal 3$d$ states (size of the dots) for (a) Ti, (b) V, (c) Cr, (d) Mn, (e) Fe, (f) Co, (g) Ni, and (h) Cu decorated silicene (at the hollow site). (i) Zoom of the marked region of panel (f). The electronic band structure obtained for Co decorated silicene is shown in Fig. 3(f). We note a tiny energy gap at the K-point and the typical band alignment of a QAH system science , where the Dirac cone is centered at the Fermi energy. The exchange field due to the Co local magnetic moment breaks the time-reversal symmetry and combined with the strong spin orbit coupling this results in a QAH state in Co decorated silicene. From the weighted band structure it is clear that the Dirac cone is essentially purely due to Si $p_{z}$ orbitals. Therefore, the characteristic silicene states are maintained and the way is paved to the QAH state. The magnetic moment of Co is minimal but finite and therefore just right to break the time-reversal symmetry, while maintaining the electronic structure specific to silicene. Our results clearly demonstrate that a QAH state by transition metal decoration is a rare phenomenon, as in all other cases besides Co the $d$-$p_{z}$ hybridization and/or the induced spin polarization of the Si $p_{z}$ electrons perturbs the electronic states. Finally, we mention that for Ni and Cu decorated silicene non-magnetic states are obtained. Interestingly, in both these systems a Dirac cone is observed below the Fermi level, which could by shifted back by means of a gate voltage. Still, a QAH state could not be induced, since no exchange field is left. ## V Conclusion In conclusion, using density functional theory, we find that silicene decoration by the transition metal atoms Ti, V, Cr, Mn, Fe, Co, Ni, and Cu results in occupation of the hollow site of the Si honeycomb. While pristine silicene is subject to spin degeneracy, transition metal decoration can induce substantial spin polarization, that is understood from atomic considerations. We demonstrate that Co decorated silicene is a hybrid material that hosts a QAH state. In the cases of Ti and V decoration a strong hybridization of the transition metal 3$d$ states with the Si 3$p_{z}$ states suppresses this state. Because of the large local magnetic moments induced by Mn and Fe the silicene in these cases becomes spin polarized, which also prohibits the formation of the QAH state. Cr is probably affected by a combination of the two effects, both being weaker but together enough to perturb the silicene electronic structure sufficiently. Ni and Cu decorated silicene are found to be non-magnetic and therefore also not suitable for our purpose. We have demonstrated that realization of a QAH state is possible in Co decorated silicene, as long as the Co atoms do not cluster. On the other hand, both hybridization and induced spin polarization typically destroy the characteristic electronic structure of pristine silicene and therefore exclude a QAH state by transition metal decoration. Only in the case of Co decoration a sufficient but not too large exchange field is achieved which can interact with the strong spin orbit coupling in silicene. ###### Acknowledgements. We thank M. Tahir for fruitful discussions and KAUST IT for providing computational resources. ## References (1) G. G. Guzmán-Verri and L. C. L. Y. Voon, Phys. Rev. B 76, 075131 (2007). * (2) P. De Padova, C. Quaresima, C. Ottaviani, P. M. Sheverdyaeva, P. Moras, C. Carbone, D. Topwal, B. Olivieri, A. Kara, H. Oughaddou, B. Aufray, and G. Le Lay, Appl. Phys. Lett. 96, 261905 (2010). * (3) P. De Padova, C. Quaresima, P. Perfetti, B. Olivieri, B. Leandri, B. Aufray, S. Vizzini, and G. Le Lay, Nano Lett. 8, 271 (2008). * (4) P. Vogt, P. De Padova, C. Quaresima, J. Avila, E. Frantzeskakis, M. C. 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Katsnelson, Phys. Rev. B 84, 235110 (2011). * (21) A. E. Bocquet, T. Mizokawa, T. Saitoh, H. Namatame, and A. Fujimori, Phys. Rev. B 46, 3771 (1992). * (22) P. Wei and Z. Q. Qi, Phys. Rev. B 49, 10864 (1994). * (23) C.-C. Liu, W. Feng, and Y. Yao, Phys. Rev. Lett. 107, 076802 (2011). * (24) C. Kittel, Introduction to Solid State Physics, John Wiley & Son Inc., 6th edition, page 55. * (25) D. Marchenko, A. Varykhalov, M. R. Scholz, G. Bihlmayer, E. I. Rashba, A. Rybkin, A. M. Shikin, and O. Rader, arXiv:1208.4265. * (26) J. Kang, H.-X. Deng, S.-S. Li, and J. Li, J. Phys.: Condens. Matter 23, 346001 (2011).	arxiv-papers	2013-12-26T16:19:20	2024-09-04T02:49:55.965773	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "T. P. Kaloni, N. Singh, and U. Schwingenschl\\\"ogl", "submitter": "Thaneshwor Prashad Kaloni", "url": "https://arxiv.org/abs/1312.7127" }
1312.7133	# Effect of cohesion on shear banding in quasi-static granular material Abhinendra Singh [email protected] Multi Scale Mechanics (MSM), Faculty of Engineering Technology, MESA+, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands. Vanessa Magnanimo Multi Scale Mechanics (MSM), Faculty of Engineering Technology, MESA+, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands. Kuniyasu Saitoh Multi Scale Mechanics (MSM), Faculty of Engineering Technology, MESA+, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands. Stefan Luding Multi Scale Mechanics (MSM), Faculty of Engineering Technology, MESA+, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands. ###### Abstract It is widely recognized in particle technology that adhesive powders show a wide range of different bulk behavior due to the peculiarity of the particle interaction. We use Discrete Element simulations to investigate the effect of contact cohesion on the steady state of dense powders in a slowly sheared split-bottom Couette cell, which imposes a wide stable shear band. The intensity of cohesive forces can be quantified by the granular Bond number ($Bo$), namely the ratio between maximum attractive force and average force due to external compression. We find that the shear banding phenomenon is almost independent of cohesion for Bond numbers $Bo<1$, but for $Bo\geq 1$ cohesive forces start to play an important role, as both width and center position of the band increase for $Bo>1$. Inside the shear band, the mean normal contact force is always independent of cohesion and depends only on the confining stress. In contrast, when the behavior is analyzed focusing on the eigen-directions of the local strain rate tensor, a dependence on cohesion shows up. Forces carried by contacts along the compressive and tensile directions are symmetric about the mean force (larger and smaller respectively), while the force along the third, neutral direction follows the mean force. This anisotropy of the force network increases with cohesion, just like the heterogeneity in all (compressive, tensile and neutral) directions. ## I Introduction Granular materials such as sand and limestone, neither behave like elastic solids nor like normal fluids, which makes their motion difficult to predict. When they yield under slow shear, the relative motion is confined to narrow regions (between large solid-like parts) called shear bands bridgwater80 ; howell99 ; schall2010shear . Shear bands are observed in many complex materials, which range from foams Katgert08 and emulsions Addad05 ; Ovarlez09 to colloids Dhont03 and granular matter Mandl77 ; bridgwater80 ; bardet91 ; oda98 ; Scott96 ; muhlhaus87b ; latzel00 ; latzel03 ; fenistein03 ; mcnamara1996dynamics ; mcnamara1994inelastic ; howell99 . There has been tremendous effort to understand the shear banding in flow of non-cohesive grains Mandl77 ; bridgwater80 ; bardet91 ; oda98 ; Scott96 ; muhlhaus87b ; latzel00 ; latzel03 ; fenistein03 ; mcnamara1996dynamics ; mcnamara1994inelastic ; howell99 ; KamrinKoval2012 ; kamrin2013predictive . However, real granular materials often experience inter-particle attractive forces due to many physical phenomena: van der Waals due to atomic forces for small grains Quintanilla03 ; Valverdejammingpowder04 ; Castellanos05 , capillary forces due to presence of humidity herminghaus05 , solid bridges Micela05 ; brendel-2011 , coagulation of particles hatzes91 , and many more. The question, arises how does the presence of attractive forces affect shear banding? So far, only a few attempts have been made to answer this question, concerning dense metallic glasses Spaepen1977 ; LiSBglasses , adhesive emulsions Becu06 ; Chaudhuri2012 , attractive colloids vermant2001large ; hohler2005rheology ; coussot2010physical , cemented granular media Nicolas10 , wet granular media Roman12 ; Fabian13 and clayey soils Yuan13 . Recently, rheological studies on adhesive emulsions and colloids Becu06 ; Chaudhuri2012 ; vermant2001large ; coussot2010physical reported that the presence of attractive forces at contact affects shear banding by affecting flow heterogeneity and wall slip. Another unique yet not completely understood feature of granular materials is their highly heterogeneous contact force distribution. The heterogeneity in the force distribution has been observed in both experimental and numerical studies liu1995force ; mueth98 ; howell99 ; lovoll0000force ; blair01b ; majmudar2005contact ; liffman1992force ; radjai98b ; silbert2002statistics ; Snoeijer03 . While huge effort has been made to understand the force distribution of non-cohesive particles liu1995force ; mueth98 ; howell99 ; lovoll0000force ; blair01b ; majmudar2005contact ; liffman1992force ; radjai98b ; Silber10 , only limited studies have aimed to understand the same for assemblies with attractive interactions trappe2001jamming ; Valverdejammingpowder04 ; Radjaiwet06 ; Rouxcoh07 ; ABYucohfor08 ; Radjaidry- wet10 . Richefeu et al. Radjaiwet06 studied the stress transmission in wet granular system subjected to isotropic compression. Gilabert et al. Rouxcoh07 focussed on a two-dimensional packing made of particles with short-range interactions (cohesive powders) under weak compaction. Yang et al. ABYucohfor08 studied the effect of cohesion on force structures in a static granular packing by changing the particle size. In a previous study luding11 , the effect of dry cohesion at contact on the critical state yield stress was studied. The critical-state yield stress shows a peculiar non-linear dependence on the confining pressure related to cohesion. But the microscopic origin was not studied. In this paper, we report the effect of varying attractive forces at contact on the steady state flow behavior and the force structure in sheared dry cohesive powders. Discrete Element Method (DEM) simulations are used to investigate the system at micro (partial) and macro level. In order to quantify the intensity of cohesion, a variation of the granular Bond number Nase01 ; Rouxcoh07 ; Rognon08 is introduced. We find that this dimensionless number very well captures the transition from a gravity/shear-dominated regime to the cohesion- dominated regime. To understand this further we look at the effect of cohesion on the mean force and anisotropy, by investigating the forces along the eigen- directions of the local strain rate tensor. Intuitively, one would expect only the tensile direction to be affected by cohesion, but the real behavior is more complex. We also discuss the probability distributions and heterogeneities of the forces in different directions to complete the picture. The paper is organized in four main parts. Section II describes the model system in detail specifying the geometry, details of particle properties, and the interaction laws. In section III, the velocity profiles and shear band from samples with different contact cohesion are presented. In the same section, the force anisotropy and probabilities are studied too. Finally, section IV is dedicated to the discussion of the results, conclusions and an outlook. ## II Discrete element method simulation (DEM) In this section, we explain our DEM simulations. We introduce a model of cohesive grains in Sec. II.1 and show our numerical setup in Sec. II.2. In Sec. II.3, we introduce a control parameter, i.e., the _global Bond number_ , which governs the flow profiles and structure of the system. ### II.1 Model DEM provides numerical solutions of Newton’s equations of motion based on the specification of particle properties viz. stiffness, density, radius and a certain type of interaction laws like Hertzian/Hookean allen87 ; cundall71 . Simulation methodology and material parameters used in this study are the same as in our previous work luding11 ; asingh13 . The adhesive elasto-plastic contact model Luding08gm is used to simulate cohesive bulk flow, as briefly explained below. For fine, dry powders, adhesive properties due to van der Waals forces and plasticity and irreversible deformation in the vicinity of the contact have to be considered at the same time tomas2004fundamentals ; thornton1998theoretical . This complex behavior is modeled using a piece-wise linear hysteretic spring model Luding08gm . Few other contact models in similar spirit are also recently proposed thakur2013experimental ; pasha2014linear . The adhesive, plastic (hysteretic) force is introduced by allowing the normal unloading stiffness to depend on the history of deformation. During initial loading the force increases linearly with overlap $\delta$ along $k_{1}$, until the maximum overlap $\delta_{\rm max}$ is reached, which acts as a history parameter. During unloading the force decreases along $k_{2}$, the value of which depends on the maximum overlap $\delta_{\rm max}$ as given by Eq. (2). The overlap when the unloading force reaches zero, $\delta_{0}=(1-k_{1}/k_{2})\delta_{\rm max}$, resembles the permanent plastic deformation and depends nonlinearly on the previous maximal force $f_{\rm max}=k_{1}\delta_{\rm max}$. The negative forces reached by further unloading are attractive, cohesion forces, which also increase nonlinearly with the previous maximum force experienced. The maximal cohesion force that corresponds to the “pull–off” force, is given by $f_{\rm m}=-k_{c}\delta_{\rm min},$ (1) with $\delta_{\rm min}=\frac{k_{2}-k_{1}}{k_{2}+k_{c}}\delta_{\rm max}$. Figure 1: Schematic graph of the piece-wise linear, hysteretic, and adhesive force-displacement model in normal direction. Three physical phenomena: elasticity, plasticity and cohesion are quantified by three material parameters $k_{p}$, $k_{1}$, and $k_{c}$, respectively. Plasticity disappears for $k_{1}=k_{p}$ and cohesion vanishes for $k_{c}=0$. In the following we focus on the relative importance of cohesion and thus do not provide measurable force magnitudes. Furthermore, the contact model has to be seen as a meso-scale model, where each particle represents an ensemble of primary particles and the contact model represents the respective bulk behavior, see Ref. asingh14 , without a direct match of the magnitude of forces in the model with the forces between the primary particles. Qualitatively, the interpretation of $k_{c}$ is that it describes the increased van der Waals type adhesion due to plastic deformations (both of the particles and the micro-structure) under compression, which increase the contact surface and thus the cohesion. Some considerations on the magnitude and relative importance of the cohesion force can be found in Appendix B. In order to account for realistic load-dependent contact behavior, the $k_{2}$ value is chosen to depend on the maximum overlap $\delta_{\rm max}$, i.e. particles are more stiff for larger previous deformation and so the dissipation is dependent on deformation. The dependence of $k_{2}$ on overlap $\delta_{\rm max}$ is chosen empirically as linear interpolation $k_{2}(\delta_{\rm max})=\left\\{\begin{array}[]{lll}k_{p}\,{\rm~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}if~{}~{}}\delta_{\rm max}/\delta^{p}_{\rm max}\geq 1\\\ k_{1}+(k_{p}-k_{1})\frac{{\delta_{\rm max}}}{{\delta^{p}_{\rm max}}}&\\\ ~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{\rm~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}if~{}~{}}\delta_{\rm max}/\delta^{p}_{\rm max}<1\end{array}\right.$ (2) As discussed in Ref. Luding08gm , very large deformations will lead to a quantitatively different contact behavior, a maximal force overlap $\delta^{p}_{\rm max}=\frac{k_{p}}{k_{p}-k_{1}}\frac{2{a_{1}a_{2}}}{a_{1}+a_{2}}\phi_{f}$ is defined (with $\phi_{f}=0.05$). Above this overlap $k_{2}$ does not increase anymore and is set to the maximal value $k_{2}=k_{p}$. This visco-elastic, reversible branch is referred to as “limit branch”. The contact friction is set to $\mu=0.01$, i.e. artificially small, in order to be able to focus on the effect of contact cohesion only. In order to study the influence of contact cohesion, we analyzed the system for the following set of adhesivity parameters $k_{c}$: $k_{c}\in\left[0,5,10,25,33,50,75,100,200\right]\mathrm{Nm^{-1}}~{},$ (3) which has to be seen in relation to $k_{1}=100$ $\mathrm{Nm^{-1}}$. Other parameters, such as the jump–in force $f_{a}=0$ asingh14 and $\phi_{f}=0.05$ asingh14 are not varied here. We also introduce damping forces proportional to the normal and tangential relative velocities, where the viscous coefficients are given by $\gamma_{n}=0.002$ s-1 and $\gamma_{t}=0.0005s^{-1}$ respectively. ### II.2 Split-bottom ring shear cell Figure 2: (Color online) A sketch of our numerical setup consisting of a fixed inner part (light blue shade) and a rotating outer part (white). The white part of the base and the outer cylinder rotate with the same angular velocity $\Omega$ around the symmetry axis. The inner, split, and outer radii are given by $R_{i}=0.0147$ m, $R_{s}=0.085$ m, and $R_{o}=0.11$ m, respectively, where each radius is measured from the symmetry axis. The gravity $g$ points downwards as shown by arrow. Figure 2 is a sketch of our numerical setup (as introduced in Refs. fenistein03 ; fenistein04 ; luding08p ; luding08crs ; dijksman10 ). In this figure, the inner, split, and outer radii are given by $R_{i}$, $R_{s}$, and $R_{o}$, respectively, where the concentric cylinders rotate relative to each other around the symmetry axis (the dot-dashed line). The ring shaped split at the bottom separates the moving and static parts of the system, where a part of the bottom and the outer cylinder rotate at the same rate. The system is filled with $N\approx 3.7\times 10^{4}$ spherical particles with density $\rho=2000$ kg/m${}^{3}=2$ g/cm3 up to height $H$. The average size of particles is $a_{0}=1.1$ mm, and the width of the homogeneous size- distribution (with $a_{\rm min}/a_{\rm max}=1/2$) is $1-{\cal A}=1-\langle a\rangle^{2}/\langle a^{2}\rangle=0.18922$. The cylindrical walls and the bottom are roughened due to some (about $3\%$ of the total number) attached/glued particles luding08p ; luding08crs . When there is a relative motion at the split, a shear band propagates from the split position $R_{s}$ upwards and inwards, remaining far away from cylinder- walls and bottom in most cases. The qualitative behavior is governed by the ratio $H/R_{s}$ and three different regimes can be identified, as reported in Refs. Unger04shear ; Fenistein06 ; ries07 ; dijksman10 . We keep ${H}/{R_{s}}<0.5$, such that the shear band reaches the free surface and stays away from inner wall Unger04shear ; Fenistein06 . Translational invariance is assumed in the azimuthal $\theta-$direction, and the averaging is performed over toroidal volumes, over many snapshots in time. This leads to fields $Q(r,z)$ as function of the radial and vertical positions. Since we are interested in the quasi-static regime, the rotation rate of the outer cylinder is chosen to be $0.01$ s-1, such that the inertial number $I=\frac{\dot{\gamma}d}{\sqrt{p/\rho}}$ da2005rheophysics is $I\ll 1$, and the simulation runs for more than $50$ s. ### II.3 Bond number Intensity of cohesion can be quantified by the ratio of the maximum attractive force to a typical force scale in the system. For example, Nase et al. Nase01 introduced the granular Bond number under gravity, which compares the maximum attractive force at contact with the weight of a single grain. For plane shear without gravity, other authors Rouxcoh07 ; Rognon08 used a ratio between the maximum attractive force and the average force due to the confining pressure. In our analysis, we introduce a _global Bond number_ as $Bo=\frac{f_{\rm m}}{\langle f\rangle}~{},$ (4) where $f_{\rm m}$ and $\langle f\rangle$ are the maximum allowed attractive force reached at a contact (given by the contact model, Appendix A using $\delta_{\rm max}=\delta^{p}_{\rm max}$ ) and the mean force per contact reached close to the bottom, respectively. For the calculation of the mean force $\langle f\rangle$, a layer of two particle diameters away from the bottom is chosen. Because the shear band initiates from the bottom, we choose the mean force $\langle f\rangle$ close to the bottom to understand the effect of cohesion on these shear bands. It is important to mention that the mean compressive force (at the bottom) corresponds to the weight of the material above, whereas the maximum attractive force corresponds to the pull-off force, which is directly related to the surface energy of the particles. These two material and particle properties are easily accessible experimentally, see Appendix B. The Bond number is a measure of the relative importance of adhesive forces compared to compressive forces. A low Bond number indicates that the system is relatively unaffected by attractive forces; a high number (typically larger than one) indicates that attractive forces dominate. Intermediate numbers indicate a non-trivial competition between the two effects. In parallel with the global Bond number, we also define two local variants of this quantity. A local simulation based Bond number $Bo^{s}_{l}(p)=f^{s}_{\rm m}(p)/\langle f(p)\rangle$ can be defined by comparing the maximum attractive force reached at a given pressure (which can be less than or equal to the maximum allowed attractive force given by the contact model) with the mean force at that pressure (subscript $l$ represents the local quantity, while superscript $s$ denotes that this definition takes input from simulation data). Another variant of this $Bo^{a}_{l}(p)$ is defined in Appendix A, which compares the analytical prediction for the maximum attractive force with mean force at that pressure, and do not use the gravitational Bond number, see Appendix B, since it is only relevant close to the free surface and for single particles in contact with a wall. Figure 3 displays the global Bond number $Bo$ and the mean values of $Bo^{s}_{l}(p)$ and $Bo^{a}_{l}(p)$ (averaged over different pressure) as functions of the adhesivity parameter $k_{c}$, where the figure shows that local and global quantities are comparable with slight divergence for high cohesion $k_{c}$. For the sake of simplicity in the rest of this paper, we use the global Bond number $Bo$ to quantify the intensity of cohesion. Figure 3: (Color online) Variants of granular Bond number plotted against cohesive strength $k_{c}$, where the red circles represent the global Bond number $Bo$, while the blue triangles and black squares represent the average values of $Bo^{s}_{l}(p)$ and $Bo^{a}_{l}(p)$, respectively. ## III Results In this section, we present our results of DEM simulations. In Sec. III.1, we analyze the flow profiles and shear banding in the system. In Sec. III.2, we study distributions and structures of force chain networks in shear bands. In Sec. III.3, we explain anisotropic features of the force chain networks. ### III.1 Effect of cohesion on Flow Profiles Figure 4: (Color online) Snapshots from simulations with different cohesion strengths, but the same number of mobile particles $N=34518$, seen from the top (Top) and from the front (Bottom). The material is (a) without cohesion $Bo=0$, and (b) with strong cohesion $Bo=4.86$. The colors blue, green, and orange denote the particles with displacements in tangential direction per second $r\,{\rm d}\phi\leq 0.5$ mm, $r\,{\rm d}\phi\leq 2$ mm, $r\,{\rm d}\phi\leq 4$ mm, and $r\,{\rm d}\phi>4$ mm, respectively Figure 4 displays both top- and front-view of samples with same filling height, i.e. same number of particles, and different global Bond numbers, $Bo=$ (left) $0$ and (right) $4.86$, respectively, where the color code represents the azimuthal displacement rate of the particles. From the front- view, we observe that the shear band (green colored area) moves inwards and gets wider with increasing height and Bond number. With the goal to extract quantitative data for the shear band area, in Fig. 5 we plot the non-dimensional angular velocity profiles at the top surface against radial coordinate normalized with the mean particle diameter $\langle d\rangle$, where we assume translational invariance in the azimuthal direction and take averages over the toroidal volumes as well as many snapshots in time latzel00 . The angular velocity profile can be well approximated by an error function $\omega=A_{1}+A_{2}{\rm erf}{\left(\frac{r-R_{c}}{W}\right)}$ (5) as in the case of non-cohesive materials luding08p ; luding08crs ; dijksman10 ; fenistein04 ; fenistein03 , where $R_{c}$ and $W$ are the position and width of the shear band, respectively. Here, we use the dimensionless amplitudes, $A_{1}=A_{2}\approx 0.5$, for the whole range of the Bond numbers, while we use $A_{1}=0.6$ and $A_{2}=0.4$ for the strong cohesion with $Bo=4.86$. The dimensionless amplitudes, $A_{1}$ and $A_{2}$ (along with estimated errors), are summarized in Table 1. Then we extract the position of the shear band relative to the split at bottom $R_{s}-R_{c}$ and the width of shear band $W$ (both scaled by mean particle diameter) at the top surface and we plot them in Fig. 6 against the Bond number. Within the error-bars, both the position and width are independent of cohesion if $Bo<1$. However, the shear band moves inside and becomes wider with the Bond number for $Bo>1$. Figure 5: (Color online) Non-dimensional angular velocity profile $\omega$ at the top surface plotted against the radial coordinate $r$ scaled by the mean diameter $\langle d\rangle$. Different symbols represent different values of the global Bond number $Bo$ given in the inset, where the solid lines represent the corresponding fits to Eq. (5). Both $R_{s}-R_{c}$ and $W$ also depend on the height ($z$) in the system. Figure 7 displays the non-dimensional position and width of the shear band for different values of $Bo$ as functions of the height scaled by the filling height, i.e. $z/H$. The shear band moves closer to the inner cylinder and gets wider while approaching the top layer, which is consistent with previous studies luding11 ; luding08p ; luding08crs ; dijksman10 ; ries07 ; fenistein04 ; fenistein03 on cohesive and non-cohesive assemblies. In Fig. 7, the lines are the prediction by Unger et al. Unger04shear : $z=H-R_{c}\left\\{1-\frac{R_{s}}{R_{c}}\left[1-\left(\frac{H}{R_{s}}\right)^{\beta}\right]\right\\}^{1/\beta}~{},$ (6) where the exponent is given by $\beta=2.5$ for non-cohesive particles. If the Bond number is less than one, the data collapse on a unique curve, very well predicted by Eq. (6), with fixed exponent $\beta$. On the other hand, above $Bo=1$, the exponent $\beta$ decreases with the global Bond number (values reported in Table 1). Note that Eq. (6) slightly deviates from the results near the top surface if the cohesion is strong ($Bo=2.22$ and $2.85$). In Fig. 7, the lines are the prediction by Ries. et al. ries07 for non-cohesive system: $W(z)=W_{\rm top}\left[1-\left(1-\frac{z}{H}\right)^{2}\right]^{\gamma}~{},$ (7) where $W_{\rm top}$ is the width at the top surface and the exponent is given by $\gamma=0.5$ for non-cohesive particles. If $Bo<1$, Eq. (7) with $W_{\rm top}=0.012$ and $\gamma=0.5\pm 0.1$ well agrees with our results. However, for $Bo>1$, both the width $W_{\rm top}$ and exponent $\gamma$ increase with the global Bond number as in Table 1. In addition, Eq. (7) deviates from the results near the top layer if the cohesion is strong ($Bo=2.22$ and $2.85$), where $W$ seems to saturates above $z/H\simeq 0.6$. Hence for $Bo>1$, we choose width at that height to be $W_{\rm top}$ and use $\gamma=0.66$ and $0.7$ for $Bo=2.22$ and $2.85$, respectively. Figure 6: (Color online) (a) Position and (b) width (both scaled by mean particle diameter) of shear band at the top surface plotted against the global Bond number $Bo$. Symbols with error-bars are the data, while the lines are only a guide to eye. From the above results, we conclude that the cohesive forces between particles drastically affect the flow profiles. Eqs. (6) and (7) very well predict the position and width of the shear bands for $Bo<1$. For large $Bo$ these equations deviate from observed behavior at large heights since the shear band interferes with the inner cylinder. The shear band, which is the region with large velocity gradient, is caused by _sliding motions_ of particles. However, strong cohesive forces keep particles in contacts (in other words, the cohesive forces promote _collective motions_ of particles) and prevent them from sliding. As a result, the velocity gradient is smoothened and the width of shear-band is broadened. This observation is consistent with previous studies on adhesive dense emulsions Ovarlez08 . Interestingly, such an effect of cohesion is suppressed if the global Bond number is less than one, where our numerical data agrees well with previous theoretical/numerical studies on non-cohesive particles Unger04shear ; ries07 . Hence, the global Bond number, $Bo$, captures the transition between essentially non-cohesive free-flowing granular assemblies $(Bo<1)$ to cohesive ones $(Bo>1)$. Figure 7: (Color online) (a) Position and (b) width (both scaled by mean particle diameter) of shear band in the cell plotted against height $z$ scaled by the filling height $H$. Different symbols correspond to values of the global Bond number $Bo$ given in the inset. The lines in (a) and (b) are the predictions, Eqs. (6) and (7), respectively. $Bo$ \| $A_{1}$ \| $A_{2}$ \| $H$ \| $\beta$ \| $\frac{z}{H}$ range \| $W_{\rm top}$ \| $\gamma$ ---\|---\|---\|---\|---\|---\|---\|--- 0 \| 0.50$\pm$ 0.0005 \| 0.500$\pm$ 0.0005 \| 0.0365 \| 2.52 \| 0.1-1 \| 0.0117 \| 0.507 0.17 \| 0.50$\pm$ 0.0005 \| 0.499$\pm$ 0.0005 \| 0.0365 \| 2.52 \| 0.1-1 \| 0.0118 \| 0.523 0.33 \| 0.49$\pm$ 0.0007 \| 0.500$\pm$ 0.0007 \| 0.0365 \| 2.512 \| 0.1-1 \| 0.0118 \| 0.555 0.81 \| 0.49$\pm$ 0.0008 \| 0.500$\pm$ 0.0008 \| 0.0361 \| 2.494 \| 0.1-1 \| 0.0119 \| 0.583 1.05 \| 0.49$\pm$ 0.001 \| 0.501$\pm$ 0.001 \| 0.0359 \| 2.510 \| 0.1-1 \| 0.0120 \| 0.582 1.50 \| 0.49$\pm$ 0.002 \| 0.501$\pm$ 0.002 \| 0.0364 \| 2.453 \| 0.1-0.8 \| 0.0126 \| 0.613 2.22 \| 0.49$\pm$ 0.003 \| 0.501$\pm$ 0.003 \| 0.0368 \| 2.367 \| 0.1-0.6 \| 0.0138 \| 0.667 2.85 \| 0.49$\pm$ 0.005 \| 0.502$\pm$ 0.005 \| 0.0369 \| 2.259 \| 0.1-0.6 \| 0.0160 \| 0.713 Table 1: Table showing filling height of the system $H$, and fitting range ${z}/{H}$ for Eqs. (6) and (7), together with the fit parameters $A_{1}$, $A_{2}$ in Eq. (5), $\beta$ in Eq. (6), $W_{top}$ and $\gamma$ in Eq. (7) for different values of Bond number $Bo$. ### III.2 Structure and distribution of forces in shear bands Figure 8: (Color online) Force chain networks of positive normal forces for $Bo=$ $0.33$ (a) and $2.85$ (b), and negative normal forces for $Bo=$ $0.33$ (c) and $2.85$ (d) at height $0.02<z<0.025$ m, respectively. In (a) and (b) positive normal force smaller than $0.002$ N is represented by grey, while larger than $0.002$ N is represented by red color. In (c) and (d) negative normal force smaller than $-0.0005$ N is represented by grey, while larger than $-0.0005$ N is represented by blue color. Figure 9: (Color online) Scatter plots of overlaps and forces between all contacts inside (left) and outside (right) of the shear bands for different $Bo=0.33$ and $2.85$. The different symbols represent a zoom into the vertical ranges $z=8$ mm $\pm 1$ mm (green stars), 15 mm $\pm 1$ mm (blue circles), 22 mm $\pm 1$ mm (magenta dots), 29 mm $\pm 1$ mm (cyan squares), with approximate pressure as given in the inset. Note that the points do not collapse on the line $k_{p}(\delta-\delta_{f})$ due to the finite width of the size distribution: pairs of larger than average particles fall out of the indicated triangle. Radial range $0.075$ m $\leq r\leq 0.085$ m (left) signifies data points inside the shear band, while the radial range $0.055$ m $\leq r\leq 0.065$ m (right) signifies the data points outside the shear band. To understand the microscopic origin of the anomalous flow profiles of cohesive aggregates, we study the force network and the statistics of the inter-particle normal forces. Recently Wang et al. ABYu13 reported the shape of probability distribution function (PDF) as an indicator for transition from quasistatic to inertial flows. In this section, we use a similar philosophy and study the change in the shape of PDFs as the cohesive strength is increased. Figure 8 shows force chains of positive ((a) and (b)) and negative ((c) and (d)) normal forces in the systems with low cohesion ((a) and (c)) and strong cohesion ((b) and (d)). Grey color shows the weak forces, while red and blue colors show the strong positive and negative forces, respectively. The strong or weak positive forces are forces larger or smaller than the mean positive force $f_{\rm pos}$. A similar approach is adopted to identify the strong/weak negative forces. In this figure, we observe that both positive and negative forces are fully developed in the cohesive system ((b) and (d)), with the intensity of the force inside the shear band being stronger than outside. In addition, the strong (positive/negative) force chains are percolated through the shear band region. As explained in Sec. III.3, we can also see that the positive and negative force chains are aligned in their preferred directions, i.e. compressive and tensile directions, respectively. Figure 9 displays scatter plots of the inter-particle forces against overlaps between the particles in contacts, where each point corresponds to a contact and different colors represent different height, i.e. pressure level in the system. The higher the pressure $p$, the higher is the average force (or overlap), as it must sustain the weight of the particles. For almost all values of $Bo$, the density of points towards the unloading branch $k_{p}$ is higher inside the shear band compared to outside. We also observe that with increasing $Bo$, most contacts (except for small pressure) drift towards and collapse around the limit branch. This implies that _the cohesive forces are more pronounced in shear bands_ rather than outside. #### III.2.1 Mean force and overlap in shear bands Figure 10: (Color online) The mean normal force $\langle f\rangle$ inside of the shear band plotted against pressure $p$, where different symbols represent the global Bond number (as given in the inset) and the solid line is given by Eq. (8). Figure 10 displays the mean normal forces, $\langle f\rangle$, in the shear band plotted against pressure, for different values of the global Bond number, where the solid line is the prediction by Shaebani et al. Shebani11 for non- cohesive granular systems: $\langle f\rangle=\frac{4\pi\langle a^{2}\rangle}{\phi Cg_{2}}\langle p\rangle$ (8) with the $2^{\rm nd}$ moment of the size distribution $\langle a^{2}\rangle$, coordination number $C$, volume fraction $\phi$, and mean pressure $\langle p\rangle$. Notably, _the mean normal force is almost independent of cohesion_ and linearly increases with pressure as in the cases of static non-cohesive mueth98 ; silbert2002statistics and cohesive systems ABYucohfor08 . We also observe that for low pressure, Eq. (8) slightly over predicts the value of the mean force, while for higher pressure the prediction well captures the data. Figure 11: (Color online) The mean (a) positive force $\langle f_{\rm pos}\rangle$ and (b) negative force $\langle f_{\rm neg}\rangle$ inside the shear band plotted against pressure $p$, where different symbols represent the global Bond number (as given in the inset). While the mean value is insensitive to cohesion, the mean positive and negative normal forces, $\langle f_{\rm pos}\rangle$ and $\langle f_{\rm neg}\rangle$, strongly depend on cohesion, we plot them in Fig. 11 against pressure for different values of $Bo$, where _the intensities increase with cohesion_ in agreement with Fig. 8. Note that the mean positive (negative) force is linear with pressure and independent of cohesion below $Bo=1$, while its dependence on pressure becomes nonlinear above $Bo=1$. Though the origin of this nonlinearity is not clear, it is readily understood that cohesion enhances the collective motion of the particles, i.e. the particles rearrange less and the system is in a mechanically constrained state. Such constrain leads to increase in the magnitude of negative forces. As a consequence, positive forces also increase, in order to balance the negative ones. It is noteworthy that in Fig. 9, the increase of $Bo$ increases the density of points in both positive and negative extremes, inside the shear band, in accordance with the previous considerations. Figure 12: (Color online) The fractions of (a) positive and (b) negative contacts inside the shear band plotted against pressure $p$, where different symbols represent the global Bond number (as given in the inset). Similar to what observed for the mean force, cohesion seems not to affect the average number of contacts, as reported in Ref. asingh13 , where we observed that cohesion had practically no effect on the contact number density (volumetric fabric). Fig. 12 shows the fractions of repulsive and attractive contacts against pressure for different Bond numbers, together with the overall coordination number. An increase of cohesion generates more attractive contacts while the number of repulsive contacts decrease. Interestingly, the overall mean force remains independent of cohesion and contacts simply redistribute between the repulsive and attractive directions. In contrast to the mean force, the mean overlap between particles in contact depends on cohesion non-linearly, as shown in Fig. 13. In our model of cohesive particles Luding08gm , overlaps are always positive for both positive and negative forces. It is worth mentioning that for low $Bo$, $\langle\delta(t)\rangle$ saturates quickly, while for $Bo=1.5,2.22,2.85$ it takes longer to reach the steady state due to the average plastic increase of the overlap luding11 . Figure 13: (Color online) Normalized mean overlap $\frac{<\delta>}{\delta^{p}_{\rm max}}$ inside the shear band plotted against pressure $p$, where different symbols represent the global Bond number (as given in the inset). #### III.2.2 PDFs of forces and structures of strong force chains in shear bands Figure 14: (Color online) Probability distribution of the normalized force $f^{}$ for (a) cohesion-less $Bo=0$ and (b) highly cohesive $Bo=2.85$ systems at different pressures $p$ in the system. Different symbols represent value of local pressure (as given in the inset). Figure 15: (Color online) Probability distribution of normalized force $f^{}$ for (a) low pressure $p=50$ $\mathrm{Nm^{-2}}$ (close to top) and (b) high pressure $p=400$ $\mathrm{Nm^{-2}}$ (close to bottom) in the system for data inside the shear band. Different symbols represent the global Bond number $Bo$ (as given in the inset). The probability distribution function (PDF) of forces are also strongly affected by cohesion. Figure 14 shows the PDFs of normal forces in shear bands for different pressure and cohesion, where the forces are scaled by the mean normal force, i.e. $f^{\ast}\equiv f/\langle f\rangle$. As can be seen, the PDF of cohesion-less particles ($Bo=0$) is almost independent of pressure (Fig. 14), while it _depends on pressure_ if the cohesive forces are very strong (Fig. 14). Figure 15 displays the variations of the PDFs for different intensities of cohesion, where we find that the PDF becomes broad with increasing cohesion and $Bo>1$. Therefore, _strong cohesion, which leads the system to a “mechanically frustrated state” induces larger fluctuations of positive/negative forces_. We note that Yang et al. ABYucohfor08 also found similar trends in static three-dimensional packing for small sized particles, where the PDF becomes broader, as particle size decreases, i.e. cohesion increases. Broadening of the PDFs was also observed by Luding et al. Luding2005 during cooling down of a sintered system. Figure 16: (Color online) Fit parameters (a) $\alpha$ and (b) $f_{0}$ plotted against Bond number $Bo$. Different symbols represent value of local pressure (as given in the inset). The cohesive forces modify not only the shapes of the PDFs, but also their asymptotic behavior, i.e. the structure of strong force chains. The tails can be fitted by a stretched exponential function Hecke_tail05 $P(f^{})\sim e^{-(f^{}/f_{0})^{\alpha}}$ (9) with a characteristic force $f_{0}$ and a fitting exponent $\alpha$. Figure 16 displays the characteristic force and the exponent against the global Bond number $Bo$. If $Bo<1$, we obtain $f_{0}=1.4\pm 0.1$ and $\alpha=1.6\pm 0.1$, which is very close to that predicted by Eerd et al. Hecke_tail05 for three- dimensional non-cohesive ensemble generated by MD simulations. However, for $Bo>1$, both characteristic force and fitting exponent decrease with increasing cohesion. The decreasing fitting exponent hints at stronger fluctuations in the force distribution. A Gaussian tail of the probability distribution would indicate a more homogeneous random spatial distribution of forces. The deviation towards an exponential distribution can be linked to an increase in heterogeneity in the spatial force distribution as mentioned in previous studies radjai1996force ; makse2000packing ; zhang2005jamming . Therefore, we conclude that _the tail of the PDF becomes a wider exponential with increasing cohesion, which implies a more heterogeneous spatial distribution, especially of the strong forces_. Finally we observe that the fitting exponent decreases with increasing pressure, which implies that at high pressure where cohesion is more active due to the contact model the spatial distribution is more heterogeneous compared to low pressure. ### III.3 Anisotropy of force chain networks in shear bands Figure 17: (Color online) A sketch showing the shear band, shear plane, and three eigen-directions of the strain rate tensor. Grey lines show inner and outer cylinders, while solid brown line shows the split, dashed black line shows the shear band which initiates at the split at bottom and moves towards inner cylinder as it moves towards the top. Green arrow represents the eigen- direction for neutral eigenvalue of the strain rate tensor, which is tangential to the shear band, perpendicular to this vector is the shear plane (yellow shaded region), which contains the eigen-directions for compression (red arrow) and tensile (blue arrow) eigenvalues. $R_{i}$, $R_{s}$ and $R_{o}$ show the inner, split, and outer radii respectively. In the case of simple shear as developed in the split-bottom shear cell, there are two non-zero eigenvalues of the strain rate tensor, which are equal in magnitude but opposite in sign, while the third eigenvalue is zero. The plane containing the eigen-vectors associated to non-zero eigenvalues is called the “shear plane”, and the eigen-vector with zero eigenvalue is perpendicular to this plane (tangent to the shear band). In the following we will refer to the eigen-directions associated to positive, negative, and zero eigenvalues as compressive, tensile, and neutral directions, respectively. Note that the shear band here is not vertical, instead its orientation changes with depth as shown by the schematic in Fig. 17. In this figure, the eigen- direction of the neutral (zero) eigenvalue (green arrow) moves with the shear band. This turning of the neutral eigen-direction makes the shear plane tilt as well (which is shown by the yellow shaded regions). To extract the contacts aligned along these directions at a given pressure in the system, we first calculate the local tensor at a given strain-rate and extract the three eigen- directions $\mathbf{n}^{\rm i}_{\mathrm{\gamma}}$ (with $i$ being compressive, neutral and tensile). Next, we look for contacts with unit contact vector $\mathbf{n}_{\mathrm{c}}$, which satisfy the condition ${\|\mathbf{n}_{\mathrm{c}}\cdot{\mathbf{n}^{\rm i}_{\gamma}}\|}$ $\geq 0.9$ . The contacts which satisfy the condition for compressive eigen-direction are termed compressive; tensile and neutral contacts are defined in a similar fashion. The forces carried by compressive, tensile, and neutral contacts are denoted by $f_{\rm com}$, $f_{\rm ten}$, and $f_{\rm neu}$ respectively. Since compressive and tensile directions are associated with loading and unloading of contacts, respectively, it is intuitive that in the absence of any external force other than shear, the mean force would be positive in compressive direction, negative in tensile direction, and almost zero in neutral direction. In our system, both an external load – gravity– coexist with (external) shear. The neutral direction gets a contribution from the additional load only, while the two principal (compressive and tensile) directions get contributions from both shear and gravity. Because the cohesive force is activated by unloading, we expect that it affects the forces along the tensile direction. Figure 18: (Color online) Mean forces in different eigen-directions of the strain rate tensor, relative to the overall mean force plotted against the local pressure $p$ in the system. Different symbols represent the global Bond number $Bo$ (as given in the inset). Figure 18 shows the mean compressive/tensile/neutral forces relative to overall local mean force, $f^{{}^{\prime}}_{\rm{com/ten/neu}}\equiv\langle f_{\rm{com/ten/neu}}\rangle-\langle f\rangle$, plotted against pressure for different values of $Bo$. We find that $f^{{}^{\prime}}_{\rm com}(>0)$ and $f^{{}^{\prime}}_{\rm ten}(<0)$ are symmetric about zero, and $f^{{}^{\prime}}_{\rm neu}\simeq 0$. The mean force along the neutral direction is independent of $Bo$, as the cohesion does not affect $f_{\rm neu}$ due to the absence of shear in this direction. However, $f^{{}^{\prime}}_{\rm ten}$ decreases with pressure and cohesion. At the same time $f^{{}^{\prime}}_{\rm com}$ increases in order to keep the mean overall force independent of cohesion. We point out here that the difference between positive and negative values, i.e., _the anisotropy of forces becomes more pronounced with increasing pressure and cohesion_. This is consistent with the visual observation of force chains of negative and positive forces for different intensity of cohesion, as shown in Fig. 8. Figure 19: (Color online) Probability distributions of normalized forces $f^{\ast}=f/\langle f\rangle$ in compressive, tensile, and neutral directions inside the shear bands for high pressure in (a) non-cohesive $Bo=0$ and (b) highly cohesive $Bo=2.85$ systems. The dashed line curves show the PDFs of the overall normalized forces, while the average force is indicated by the vertical lines. Next, we study the PDFs of forces in the compressive, tensile, and neutral directions. Figure 19 displays the PDFs along each direction for non-cohesive $Bo=0$ and highly cohesive $Bo=2.85$ systems, where the forces along different directions are normalized by the overall mean force. In a non-cohesive system (Fig. 19), we observe that for weak forces, i.e., $f^{}<1$, the PDF along the tensile direction is higher compared to that for the compressive direction. This is intuitive as for the weak forces, the majority of contacts will be aligned along the tensile direction. However, for $f^{}>1$ the PDF along the compressive direction becomes higher compared to that along the tensile direction, as majority of contacts along the compressive direction should carry strong forces Brian05force . For a highly cohesive system (Fig. 19), a similar behavior is observed for strong positive forces $f^{}>1$. While for weak positive and whole range negative forces, the PDF along the tensile direction is higher in comparison to the compressive direction. The PDFs of forces in the neutral direction lie in between those in compressive and tensile directions, suggesting a close to average distribution of forces. It is interesting to note that, both positive and negative forces are present in all directions. However, the positive and negative forces dominate in the compressive and tensile directions, respectively. Figure 20: (Color online) Probability distributions of normalized forces in (a) compressive ($f^{\ast}_{c}=f_{c}/\langle f\rangle$) and (b) tensile ($f^{\ast}_{t}=f_{t}/\langle f\rangle$) directions inside the shear bands for high pressure. Different symbols represent different values of the global Bond number $Bo$ as given in the inset. Figure 20 shows the variations of the PDFs along compressive and tensile directions for different values of $Bo$. If $Bo<1$, the PDFs collapse on top of each other. However, the PDFs get wider with increasing cohesion above $Bo=1$. Such widening is more prominent for positive and negative forces in the compressive and tensile directions, respectively. Again, we confirm that strong cohesion leads to an increases of positive and negative forces in the compressive and tensile directions, respectively. Therefore, as _the force distributions in the principal directions gets more heterogeneous with increasing cohesion for $Bo>1$_, the heterogeneity of the overall force structure increases. Results in this section suggest that for low $Bo$, external load and shear dominate and govern the distribution of forces along compressive and tensile directions. The forces can adapt to external shear, the particles rearrange and can avoid very large forces. In contrast, for high $Bo$, cohesion dominates over external forcing: the contact forces still respond to compression and tension, but their rearrangements are hampered by cohesion. Due to the sticky nature of cohesive forces, rearrangements of the contact network become more difficult, so that very large contact forces as well as strong sticking forces occur together, leading to a more heterogeneous contact network. ## IV Discussion and conclusion In this paper, we have studied the effect of cohesion on shear banding in dry cohesive powders. The global Bond number, $Bo$, can be used to quantify how strong cohesive forces are relative to compressive forces, where $Bo\simeq 1$ very well predicts the transition from a free-flowing, non-cohesive system to a cohesive system. Interestingly, many other features of the system also show a transition at $Bo\approx 1$. Using local $Bo$ has no big advantage in this system, but is recommended in general. ##### Shear band Width and center position of the shear band are fairly un-affected by cohesion for $Bo<1$; only for $Bo\geq 1$ cohesion affects the flow behavior. The width of the shear band increases with $Bo$ increasing above unity; the velocity gradient, as a consequence, decreases, since cohesive forces tend to keep the particles in contact to remain connected for longer. Cohesive forces assist the “collective motion” of particles; implying that attractive forces work against the localization of shear. ##### Forces and their direction dependence The mean force $\langle f\rangle(p)$ (with $p\propto H-z$) is found to be independent of cohesion, just like the number of contacts. With increasing $Bo$, stronger attractive negative forces are possible at contacts (which is intuitive). However, these negative forces must be balanced by some stronger positive forces to maintain the same overall mean force. Due to the planar shear that establishes in steady state, compressive/tensile contact forces are induced in compressive/tensile eigen–directions of the local strain rate tensor, respectively, while along the third, neutral direction neither compression nor tension takes place. The mean force along the neutral direction remains unaffected by cohesion, which implies that cohesive forces in the system are activated by shear; more specific, cohesive forces are activated by the tension in the respective (eigen) direction. In other words, only about one third of all contacts features considerable strain-induced cohesion. The mean force carried by contacts along compressive and tensile directions is symmetric about the mean overall force. For $Bo\leq 1$, this anisotropy of the force network is independent of cohesion, while for $Bo>1$ the anisotropy in the force network increases with cohesion. Macroscopically, this anisotropy in force is directly related to the shear stress; the trend in force anisotropy is very similar to the trends found in the shear stress in previous work luding11 . ##### Force probability distribution Since granular systems are known to be heterogeneous in nature, we also analyzed the effect of cohesion on the force probability distributions. For non-cohesive and weakly cohesive systems, no prominent effect of pressure on force distributions could be seen. For strong cohesion $Bo>1$, pressure affects the distribution of forces, by making the tails wider, and more symmetric, as compared to the cases with $Bo<1$. Splitting up the force distributions along the compressive and tensile directions reveals that, for higher $Bo$, cohesion broadens the force distributions along the tensile direction, which in turn affects the distribution along the compressive direction, which also becomes wider. This suggests, an increase in heterogeneity in forces for $Bo>1$ along all directions, compressive, tensile and neutral as well. For low $Bo$, the kinematics of shear helps the particles to rearrange and avoid very strong forces. In contrast, for high $Bo$, cohesion induces stickiness at the contacts so that rearrangements are suppressed, increasing the heterogeneity of the system, as evidenced by the wider tails of the probability distributions. In conclusion, both the flow profiles (shear banding) and the force structure are unaffected by cohesion for $Bo<1$. In contrast, for $Bo\geq 1$, cohesion strongly affects the flow behavior, the anisotropy, and the internal force structure. Attractive forces thus reduce shear localization for $Bo>1$ and promote heterogeneity of the force-network. These two observations are consistent with previous studies with attractive forces, concerning the rheology Chaudhuri2012 and force structures for static packings ABYucohfor08 . As speculation, for a wider view, our results can be interpreted as follows: In the language of statistical mechanics, the global $Bo$ corresponds to a “control parameter” and $Bo=1$ to a “critical point”. The changes in the characteristic force and the fitting exponents show a weak pressure dependence, which might be better captured using a pressure dependent, local Bond number. In our case, the macroscopic properties (position and width of shear-bands), the anisotropy and the micro-structural signatures (the tails of the PDFs) gradually increase for $Bo\geq 1$. This continuous increase implies a “second-order transition”, however, confirming this would need a further detailed study. Also, experiments performed with controlled, pressure- dependent cohesive strength would be exciting to confirm and validate our results. Finally it would be interesting to reproduce our findings with different contact models, e.g. capillary bridges or simpler cohesive contact models with no pressure dependence. ###### Acknowledgements. We thank M. Wojtkowski, N. Kumar, O. I. Imole, T. Weinhart, and Ruud van Ommen for stimulating discussions. Financial support (project number: 07CJR06) through the “Jamming and Rheology” program of the Stichting voor Fundamenteel Onderzoek der Materie (FOM), which is financially supported by the “Nederlandse Organisatie voor Wetenschappelijk Onderzoek” (NWO), is acknowledged. K. Saitoh is funded by the NWO-STW VICI grant 10828. ## Appendix A Maximum attractive force The extreme loading and unloading branches are reflected by the outer triangle in Fig. 1. Starting from a realized maximum overlap during loading, $\delta_{\rm max}<\delta^{p}_{\rm max}$, the unloading happens within the triangle, as can be characterized by a branch with stiffness $k_{2}=k_{1}+(k_{p}-k_{1}){\delta_{\rm max}}/{\delta^{p}_{\rm max}}$ (10) (as given in asingh13 ). The elastic, reversible force along this branch is given by $k_{2}(\delta-\delta_{0})$ Luding08gm ; asingh14 . The intermediate stiffness $k_{2}$ follows from a linear interpolation between $k_{1}$ and $k_{p}$, as explained in Luding08gm ; asingh14 . The corresponding maximal attractive force is $f_{\rm m}=-k_{c}\delta_{\rm m}=-k_{c}\frac{({k_{2}-k_{1}})}{({k_{2}+k_{c}})}\delta_{\rm max}$. If we assume that the maximal overlap $\delta^{p}_{\rm max}$ is realized under a given external (compressive) pressure $p_{\rm max}$, then we can infer $\frac{p}{p_{\rm max}}=\frac{\delta_{\rm max}}{\delta^{p}_{\rm max}}$, with pressure $p$ being $p=k_{1}\delta_{\rm max}/A$, $A$ being a representative area. This leads to realized maximal attractive force being $f_{\rm m}=-k_{c}\frac{({k_{2}-k_{1}})}{({k_{2}+k_{c}})}\frac{p}{p_{\rm max}}{\delta^{p}_{\rm max}}$ (11) Using Eq. (10) in Eq. (11), we get $f_{\rm m}=-k_{c}\frac{(k_{p}-k_{1})\frac{p_{\rm max}}{k_{1}}\left(\frac{p}{p_{\rm max}}\right)^{2}}{k_{c}+k_{1}+(k_{p}-k_{1})\frac{p}{p_{\rm max}}}.$ (12) This definition can be used to define a local Bond number as $Bo^{a}_{l}(p)=f_{\rm m}(p)/\langle f(p)\rangle$, where mean force at that pressure is discussed in Sec. II.3. This Bond number would be compared with various other definitions in Sec. II.3. ## Appendix B Cohesive force magnitude In order to get a feeling for the magnitude of the adhesion forces in experimental systems, we resort to Ref. tahmasebpoora13 and estimate the attractive force as: $F_{\rm vdW}=\frac{Hd}{24l^{2}}\approx 1.7\times 10^{-10}\,\mathrm{N}\mathrm{~{}~{}or~{}~{}}1.7\times 10^{-9}\,\mathrm{N}\,,$ for SiO2 particles with Hamaker constant $H=6.6\times 10^{-20}$ J, minimal inter-particle distance $l=sd\approx 4\times 10^{-4}d$ (order of surface roughness for 9 $\mu$m particles of high quality is actually a factor ten smaller fuchs2014 , while realistic roughness can be even larger - the value of $s$ is just a rough estimate), and diameter $d=100$ $\mu$m or $10$ $\mu$m. 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Luding, Granular Matter(2014)	arxiv-papers	2013-12-26T17:23:17	2024-09-04T02:49:55.973163	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Abhinendra Singh, Vanessa Magnanimo, Kuniyasu Saitoh, Stefan Luding", "submitter": "Abhinendra Singh", "url": "https://arxiv.org/abs/1312.7133" }
1312.7140	# Computation of multi-leg amplitudes with NJet S Badger1 B Biedermann2 P Uwer3 and V Yundin4,555Speaker; talk given at the Workshop on Advanced Computing and Analysis Techniques in Physics (ACAT), Beijing, China, May 2013. 1 Theory Division, Physics Department, CERN, CH-1211 Geneva 23, Switzerland 2 Universität Würzburg, Institut für Theoretische Physik und Astrophysik, Emil-Hilb-Weg 22, 97074 Würzburg, Germany 3 Humboldt-Universität zu Berlin, Institut für Physik, Newtonstraße 15, 12489 Berlin, Germany 4 Max-Planck-Institut für Physik, Föhringer Ring 6, 80805 Munich, Germany [email protected] ###### Abstract In these proceedings we report our progress in the development of the publicly available C++ library NJet for accurate calculations of high-multiplicity one- loop amplitudes. As a phenomenological application we present the first complete next-to-leading order (NLO) calculation of five jet cross section at hadron colliders. ## 1 Introduction NLO predictions of multi-jet production at hadron colliders have a long history. They are important processes for the LHC both as precision tests of QCD and direct probes of the strong coupling and also as background in many new physics searches. The LHC experiments have been able to measure jet rates for up to 6 hard jets which are now being used in new physics searches [1, 2, 3]. This presents a serious challenge for precise theoretical predictions since high multiplicity computations in QCD are notoriously difficult. Di-jet production has been known at NLO for more than 20 years [4] and has recently seen improvements via NLO plus parton shower (NLO+PS) description [5, 6] and steady progress towards NNLO QCD results [7]. The full three-jet computation was completed and implemented in a public code NLOJET++ 10 years ago [8]. Recently predictions for four-jet production have been presented by two independent groups [9, 10]. The advances in methods of evaluation of multi-leg virtual amplitudes [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25] have inspired many efforts to automate NLO computations [26, 27, 28, 29, 30, 31]. Processes with four final states, previously out of reach, can now be routinely used for phenomenological predictions [32, 33, 34, 35, 36, 37, 38]. We refer the reader to other contributions to these proceedings for further details on the current state-of-the art [39, 40, 41]. Five partons in the final state still constitute a considerable challenge, though steady progress in that direction gives hope for the same level of automation in the near future. Recent state-of-the-art calculations with five QCD partons in the final state include the NLO QCD corrections to $pp\to W+5j$ [34] by the BlackHat collaboration and NLO QCD corrections to $pp\to 5j$ [42]. ## 2 5-Jet production at the LHC at 7 and 8 TeV The different parts of the calculation, which contribute to NLO cross section can be schematically written as $\displaystyle\delta\sigma^{\mbox{\scriptsize NLO}}=\int\limits_{n}\big{(}d\sigma_{n}^{\rm V}+\int\limits_{1}d\sigma_{n+1}^{\rm S}\big{)}+\int\limits_{n}d\sigma_{n}^{\rm Fac}+\int\limits_{n+1}\big{(}d\sigma_{n+1}^{\rm R}-d\sigma_{n+1}^{\rm S}\big{)}.$ (1) We used the Sherpa Monte-Carlo event generator [43] to handle phase-space integration and generation of tree-level amplitudes and Catani-Seymour dipole subtraction terms as implemented in Comix [44, 45]. The one-loop matrix elements for the virtual corrections $d\sigma_{n}^{\rm V}$ are evaluated with the publicly available NJet 111To download NJet visit the project home page at https://bitbucket.org/njet/njet/. package [31] interfaced to Sherpa via the Binoth Les Houches Accord [46, 47]. NJet is based on the NGluon library [26] and uses an on-shell generalized unitarity framework [17, 18, 19, 20, 21, 22] to compute multi-parton one-loop primitive amplitudes from tree-level building blocks [48]. The scalar loop integrals are obtained via the QCDLoop/FF package [49, 50]. NJet implements full-colour expressions for up-to five outgoing QCD partons. The complexity of high-multiplicity virtual corrections motivates us to explore ways to speed up the computation. One of the optimizations implemented in NJet is the usage of de-symmetrized colour sums for multi-gluon final states, which allows us to get full colour result at a small fraction of computational cost by exploiting the Bose symmetry of the phase space [51, 31]. Another possibility is to separate leading and sub-leading contributions, which enables Monte-Carlo integrator to sample the dominant however simpler terms more often and get the same statistical error with fewer evaluations of the expensive sub-leading part. In our leading terms we include all multi-quark processes in the large $N_{c}$ limit and processes with two or more gluons in the final state using the de- symmetrized colour sums. In Figure 1 we compare leading and full virtual contributions to the hadrest jet transverse momentum in $pp\to 5j$. The correction from the sub-leading part is around $10\%$ at low $p_{T}$ and shows a tendency to grow with increasing hardness of the jet. Considering that $d\sigma_{n}^{\rm V}$ contribute $\sim 50\%$ of the total NLO cross section for this process this translates to $5-10$ percent effect depending on the kinematic region. Figure 1: Full colour and leading approximation (as explained in the text) for the virtual corrections to the transverse momentum of the 1st jet in $pp\to 5j$. The calculation is done in QCD with five massless quark flavours including the bottom-quark in the initial state. We neglect contributions from top quark loops. We set the renormalization scale equal to the factorization scale ($\mu_{r}=\mu_{f}=\mu$) and use a dynamical scale based on the total transverse momentum ${\widehat{H}_{T}}$ of the final state partons: ${\widehat{H}_{T}}=\sum_{i=1}^{N_{\mbox{\scriptsize parton}}}p_{T,i}^{\mbox{\scriptsize parton}}.$ (2) For the definition of physical observables we use the anti-kt jet clustering algorithm as implemented in FastJet [52, 53]. We apply asymmetric cuts on the jets ordered in transverse momenta, $p_{T}$, to match the ATLAS multi-jet measurements [1]: $\displaystyle p_{T}^{j_{1}}$ $\displaystyle>80\text{ GeV}$ $\displaystyle p_{T}^{j_{\geq 2}}$ $\displaystyle>60\text{ GeV}$ $\displaystyle R$ $\displaystyle=0.4$ (3) The PDFs are obtained through the LHAPDF interface [54] with all central values using NNPDF2.1 [55] for LO ($\alpha_{s}(M_{Z})=0.119$) and NNPDF2.3 [56] for NLO ($\alpha_{s}(M_{Z})=0.118$) if not mentioned otherwise. Generated events are stored in ROOT Ntuple format [57] which allows for flexible analysis. Renormalization and factorization scales can be changed at the analysis level as well as the PDF set. This technique makes it possible to do extended analysis of PDF uncertainties and scale dependence, which would otherwise be prohibitively expensive for such high multiplicity processes. ### 2.1 Numerical results Using the above setup we obtain for the 5-jet cross section at 7 TeV $\displaystyle\sigma_{5}^{\text{7TeV-LO}}(\mu={\widehat{H}_{T}}/2)$ $\displaystyle=0.699(0.004)^{+0.530}_{-0.280}\>{\rm nb},$ (4) $\displaystyle\sigma_{5}^{\text{7TeV-NLO}}(\mu={\widehat{H}_{T}}/2)$ $\displaystyle=0.544(0.016)^{+0.0}_{-0.177}\>{\rm nb}.$ (5) In parentheses we quote the uncertainty due to the numerical integration. The theoretical uncertainty has been estimated from scale variations over the range $\mu\in[{\widehat{H}_{T}}/4,{\widehat{H}_{T}}]$ and is indicated by the sub- and superscripts. As seen in Fig. 2 the total cross section at the scale $\mu={\widehat{H}_{T}}$ is lower than the central value which is the origin of the zero value of the upper error bound. The total cross section at this scale is $\sigma_{5}^{\text{7TeV-NLO}}(\mu={\widehat{H}_{T}})=0.544(0.016)\>{\rm nb}$. For a centre-of-mass energy of 8 TeV the results read: $\displaystyle\sigma_{5}^{\text{8TeV-LO}}(\mu={\widehat{H}_{T}}/2)$ $\displaystyle=1.044(0.006)^{+0.770}_{-0.413}\>{\rm nb},$ (6) $\displaystyle\sigma_{5}^{\text{8TeV-NLO}}(\mu={\widehat{H}_{T}}/2)$ $\displaystyle=0.790(0.021)^{+0.0}_{-0.313}\>{\rm nb},$ (7) where we have found $\sigma_{5}^{\text{8TeV- NLO}}(\mu={\widehat{H}_{T}})=0.723(0.011)\>{\rm nb}$. As usual for a next-to-leading order correction a significant reduction of the scale uncertainty can be observed. (a) (b) Figure 2: Residual scale dependence of the 5-jet cross section in leading and next-to-leading order using LO (a) and NLO (b) PDFs for LO prediction. In Fig. 2 the scale dependence of the LO and NLO cross section is illustrated. The dashed black line indicates the central scale $\mu={\widehat{H}_{T}}/2$. The horizontal bands show the cross section uncertainty estimated by a scale variation within $\mu\in[{\widehat{H}_{T}}/4,{\widehat{H}_{T}}$]. By comparing Figs. 2a and 2b we observe that a significant part of the NLO corrections comes from using NLO PDFs with the corresponding $\alpha_{s}$. Similar to what has been found in Ref. [10] we conclude that using the NLO PDFs in the LO predictions gives a better approximation to the full result compared to using LO PDFs. In Tab. 1 we show for completeness the cross sections for two, three and four- jet production as calculated with NJet using the same setup as in the five jet case. $n$ \| $\sigma_{n}^{\text{7TeV-LO}}\>[{\rm nb}]$ \| $\sigma_{n}^{\text{7TeV-NLO}}\>[{\rm nb}]$ ---\|---\|--- $2$ \| $768.0(0.9)^{+203.0}_{-151.3}$ \| $1175(3)^{+120}_{-129}$ $3$ \| $71.1(0.1)^{+31.5}_{-20.0}$ \| $52.5(0.3)^{+1.9}_{-19.3}$ $4$ \| $7.23(0.02)^{+4.37}_{-2.50}$ \| $5.65(0.07)^{+0}_{-1.93}$ Table 1: Cross sections for 2, 3 and 4 jets at 7 TeV. The jet rates have been measured recently by ATLAS using the 7 TeV data set [1]. (a) (b) Figure 3: (a) LO and NLO cross sections for jet production calculated with NJet as well as results from ATLAS measurements [1]. (b) NLO NJet predictions with different PDF sets for the jet ratios ${\cal{R}}_{n}$ compared with recent ATLAS measurements [1]. In Fig. 3a we show the data together with the theoretical predictions in leading and next-to-leading order. In case of the six jet rate only LO results are shown. In the lower plot the ratio of theoretical predictions with respect to data is given. With exception of the two-jet cross section the inclusion of the NLO results improves significantly the agreement with data. In addition to inclusive cross sections it is useful to consider their ratios since many theoretical and experimental uncertainties may cancel between numerator and denominator. In particular we consider ${\cal R}_{n}={\sigma_{(n+1)\text{-jet}}\over\sigma_{n\text{-jet}}}.$ (8) This quantity is in leading order proportional to the QCD coupling $\alpha_{s}$ and can be used to determine the value of $\alpha_{s}$ from jet rates. In Fig. 3b we show QCD predictions in NLO using different PDF sets together with the results from ATLAS. The results obtained from NNPDF2.3 are also collected in Tab. 2 where, in addition, the ratios at leading order (using the LO setup with NNPDF2.1) are shown. ${\cal R}_{n}$ \| ATLAS [1] \| LO \| NLO ---\|---\|---\|--- 2 \| $0.070^{+0.007}_{-0.005}$ \| $0.0925(0.0002)$ \| $0.0447(0.0003)$ 3 \| $0.098^{+0.006}_{-0.007}$ \| $0.102(0.000)$ \| $0.108(0.002)$ 4 \| $0.101^{+0.012}_{-0.011}$ \| $0.097(0.001)$ \| $0.096(0.003)$ 5 \| $0.123^{+0.028}_{-0.027}$ \| $0.102(0.001)$ \| $--$ Table 2: Results for the jet ratios ${\cal R}_{n}$ for the central scale of ${\widehat{H}_{T}}/2$ and NNPDF2.3 PDF set. In case of ${\cal R}_{3}$ and ${\cal R}_{4}$ perturbation theory seems to provide stable results. The leading order and next-to-leading order values differ by less than 10%. In addition NNPDF [56], CT10 [58] and MSTW08 [59] give compatible predictions. ABM11 [60] gives slightly smaller results for ${\cal R}_{3}$ and ${\cal R}_{4}$. Within uncertainties the predictions also agree with the ATLAS measurements. The poor description of ${\cal R}_{2}$ can be attributed to the inclusive two-jet cross section which seems to be inadequately described by a fixed order NLO calculation. As a function of the leading jet $p_{T}$, all PDF sets agree well with the 3/2 ratio ATLAS data at large $p_{T}$ as shown in Fig. 4a. (a) (b) Figure 4: (a) The 3/2 jet ratio as a function of the $p_{T}$ of the leading jet compared with ATLAS data [1] ($R=0.6$). (b) The ${\cal R}_{n}$ ratios as functions of the $p_{T}$ of the leading jet ($R=0.4$). In Fig. 4b we compare LO and NLO predictions for ${\cal R}_{n}$ as function of the leading jet $p_{T}$. While for ${\cal R}_{3}$ and ${\cal R}_{4}$ the corrections are moderate for all values of $p_{T}$ we observe large negative corrections independent from $p_{T}$ in case of ${\cal R}_{2}$. (a) (b) Figure 5: The $p_{T}$ and rapidity distributions of the leading jet. Both LO and NLO use the NNPDF2.3 PDF set with $\alpha_{s}(M_{Z})=0.118$ In Fig. 5 we show the transverse momentum and rapidity distributions of the leading jet for five-jet production. Similarly to total cross section we observe significant reduction of the scale uncertainty when going from LO to NLO. Using again the NLO setup to calculate the LO predictions, the NLO calculation gives very small corrections. Over a wide range the LO predictions are modified by less than 10%. A remarkable feature observed already in the 4-jet calculation [9, 10] is the almost constant K-factor. ## 3 Conclusions In this contribution we have presented first results for five-jet production at NLO accuracy in QCD. We find moderate corrections of the order of 10% at NLO with respect to a leading order computation using NLO PDFs. We have compared theoretical predictions for inclusive jet cross sections and jet rates with data from ATLAS. 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1312.7171	# Barnes’ multiple Bernoulli and poly-Bernoulli mixed-type polynomials Dae San Kim Department of Mathematics, Sogang University Seoul 121-741, Republic of Korea [email protected] Taekyun Kim Department of Mathematics, Kwangwoon University Seoul 139-701, Republic of Korea [email protected] Takao Komatsu Graduate School of Science and Technology, Hirosaki University Hirosaki 036-8561, Japan [email protected] This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MOE) (No.2012R1A1A2003786). The third author was supported in part by the Grant- in-Aid for Scientific research (C) (No.22540005), the Japan Society for the Promotion of Science. ( MR Subject Classifications: 05A15, 05A40, 11B68, 11B75, 65Q05) ###### Abstract In this paper, we consider Barnes’ multiple Bernoulli and poly-Bernoulli mixed-type polynomials. From the properties of Sheffer sequences of these polynomials arising from umbral calculus, we derive new and interesting identities. ## 1 Introduction In this paper, we consider the polynomials $S_{n}^{(r,k)}(x\|a_{1},\dots,a_{r})$ whose generating function is given by $\frac{t^{r}}{\prod_{j=1}^{r}(e^{a_{j}t}-1)}\frac{{\rm Li}_{k}(1-e^{-t})}{1-e^{-t}}e^{xt}=\sum_{n=0}^{\infty}S_{n}^{(r,k)}(x\|a_{1},\dots,a_{r})\frac{t^{n}}{n!}\,,$ (1) where $r\in\mathbb{Z}_{>0}$, $k\in\mathbb{Z}$, $a_{1},\dots,a_{r}\neq 0$, and ${\rm Li}_{k}(x)=\sum_{m=1}^{\infty}\frac{x^{m}}{m^{k}}$ is the $k$th polylogarithm function. $S_{n}^{(r,k)}(x\|a_{1},\dots,a_{r})$ will be called Barnes’ multiple Bernoulli and poly-Bernoulli mixed-type polynomials. When $S_{n}^{(r,k)}(a_{1},\dots,a_{r})$ $=S_{n}^{(r,k)}(0\|a_{1},\dots,a_{r})$ will be called Barnes’ multiple Bernoulli and poly-Bernoulli mixed-type numbers. Recall that, for every integer $k$, the poly-Bernoulli polynomials $B_{n}^{(k)}(x)$ are defined by the generating function as $\frac{{\rm Li}_{k}(1-e^{-t})}{1-e^{-t}}e^{xt}=\sum_{n=0}^{\infty}B_{n}^{(k)}(x)\frac{t^{n}}{n!}$ (2) ([1], Cf.[3]). Also, recall that the Barnes’ multiple Bernoulli polynomials $B_{n}(x\|a_{1},\dots,a_{r})$ are defined by the generating function as $\frac{t^{r}}{\prod_{j=1}^{r}(e^{a_{j}t}-1)}e^{xt}=\sum_{n=0}^{\infty}B_{n}(x\|a_{1},\dots,a_{r})\frac{t^{n}}{n!}\,,$ (3) where $a_{1},\dots,a_{r}\neq 0$ ([4]). In this paper, we consider Barnes’ multiple Bernoulli and poly-Bernoulli mixed-type polynomials. From the properties of Sheffer sequences of these polynomials arising from umbral calculus, we derive new and interesting identities. ## 2 Umbral calculus Let $\mathbb{C}$ be the complex number field and let $\mathcal{F}$ be the set of all formal power series in the variable $t$: $\mathcal{F}=\left\\{f(t)=\sum_{k=0}^{\infty}\frac{a_{k}}{k!}t^{k}\Bigg{\|}a_{k}\in\mathbb{C}\right\\}\,.$ (4) Let $\mathbb{P}=\mathbb{C}[x]$ and let $\mathbb{P}^{\ast}$ be the vector space of all linear functionals on $\mathbb{P}$. $\left\langle L\|p(x)\right\rangle$ is the action of the linear functional $L$ on the polynomial $p(x)$, and we recall that the vector space operations on $\mathbb{P}^{\ast}$ are defined by $\left\langle L+M\|p(x)\right\rangle=\left\langle L\|p(x)\right\rangle+\left\langle M\|p(x)\right\rangle$, $\left\langle cL\|p(x)\right\rangle=c\left\langle L\|p(x)\right\rangle$, where $c$ is a complex constant in $\mathbb{C}$. For $f(t)\in\mathcal{F}$, let us define the linear functional on $\mathbb{P}$ by setting $\left\langle f(t)\|x^{n}\right\rangle=a_{n},\quad(n\geq 0).$ (5) In particular, $\left\langle t^{k}\|x^{n}\right\rangle=n!\delta_{n,k}\quad(n,k\geq 0),$ (6) where $\delta_{n,k}$ is the Kronecker’s symbol. For $f_{L}(t)=\sum_{k=0}^{\infty}\frac{\left\langle L\|x^{k}\right\rangle}{k!}t^{k}$, we have $\left\langle f_{L}(t)\|x^{n}\right\rangle=\left\langle L\|x^{n}\right\rangle$. That is, $L=f_{L}(t)$. The map $L\mapsto f_{L}(t)$ is a vector space isomorphism from $\mathbb{P}^{\ast}$ onto $\mathcal{F}$. Henceforth, $\mathcal{F}$ denotes both the algebra of formal power series in $t$ and the vector space of all linear functionals on $\mathbb{P}$, and so an element $f(t)$ of $\mathcal{F}$ will be thought of as both a formal power series and a linear functional. We call $\mathcal{F}$ the umbral algebra and the umbral calculus is the study of umbral algebra. The order $O\bigl{(}f(t)\bigr{)}$ of a power series $f(t)(\neq 0)$ is the smallest integer $k$ for which the coefficient of $t^{k}$ does not vanish. If $O\bigl{(}f(t)\bigr{)}=1$, then $f(t)$ is called a delta series; if $O\bigl{(}f(t)\bigr{)}=0$, then $f(t)$ is called an invertible series. For $f(t),g(t)\in\mathcal{F}$ with $O\bigl{(}f(t)\bigr{)}=1$ and $O\bigl{(}g(t)\bigr{)}=0$, there exists a unique sequence $s_{n}(x)$ ($\deg s_{n}(x)=n$) such that $\left\langle g(t)f(t)^{k}\|s_{n}(x)\right\rangle=n!\delta_{n,k}$, for $n,k\geq 0$. Such a sequence $s_{n}(x)$ is called the Sheffer sequence for $\bigl{(}g(t),f(t)\bigr{)}$ which is denoted by $s_{n}(x)\sim\bigl{(}g(t),f(t)\bigr{)}$. For $f(t),g(t)\in\mathcal{F}$ and $p(x)\in\mathbb{P}$, we have $\left\langle f(t)g(t)\|p(x)\right\rangle=\left\langle f(t)\|g(t)p(x)\right\rangle=\left\langle g(t)\|f(t)p(x)\right\rangle$ (7) and $f(t)=\sum_{k=0}^{\infty}\left\langle f(t)\|x^{k}\right\rangle\frac{t^{k}}{k!},\quad p(x)=\sum_{k=0}^{\infty}\left\langle t^{k}\|p(x)\right\rangle\frac{x^{k}}{k!}$ (8) ([13, Theorem 2.2.5]). Thus, by (8), we get $t^{k}p(x)=p^{(k)}(x)=\frac{d^{k}p(x)}{dx^{k}}\quad\hbox{and}\quad e^{yt}p(x)=p(x+y).$ (9) Sheffer sequences are characterized in the generating function ([13, Theorem 2.3.4]). ###### Lemma 1 The sequence $s_{n}(x)$ is Sheffer for $\big{(}g(t),f(t)\bigr{)}$ if and only if $\frac{1}{g\bigl{(}\bar{f}(t)\bigr{)}}e^{y\bar{f}(t)}=\sum_{k=0}^{\infty}\frac{s_{k}(y)}{k!}t^{k}\quad(y\in\mathbb{C})\,,$ where $\bar{f}(t)$ is the compositional inverse of $f(t)$. For $s_{n}(x)\sim\bigl{(}g(t),f(t)\bigr{)}$, we have the following equations ([13, Theorem 2.3.7, Theorem 2.3.5, Theorem 2.3.9]): $\displaystyle f(t)s_{n}(x)$ $\displaystyle=ns_{n-1}(x)\quad(n\geq 0),$ (10) $\displaystyle s_{n}(x)$ $\displaystyle=\sum_{j=0}^{n}\frac{1}{j!}\left\langle g\bigl{(}\bar{f}(t)\bigr{)}^{-1}\bar{f}(t)^{j}\|x^{n}\right\rangle x^{j},$ (11) $\displaystyle s_{n}(x+y)$ $\displaystyle=\sum_{j=0}^{n}\binom{n}{j}s_{j}(x)p_{n-j}(y)\,,$ (12) where $p_{n}(x)=g(t)s_{n}(x)$. Assume that $p_{n}(x)\sim\bigl{(}1,f(t)\bigr{)}$ and $q_{n}(x)\sim\bigl{(}1,g(t)\bigr{)}$. Then the transfer formula ([13, Corollary 3.8.2]) is given by $q_{n}(x)=x\left(\frac{f(t)}{g(t)}\right)^{n}x^{-1}p_{n}(x)\quad(n\geq 1).$ For $s_{n}(x)\sim\bigl{(}g(t),f(t)\bigr{)}$ and $r_{n}(x)\sim\bigl{(}h(t),l(t)\bigr{)}$, assume that $s_{n}(x)=\sum_{m=0}^{n}C_{n,m}r_{m}(x)\quad(n\geq 0)\,.$ Then we have ([13, p.132]) $C_{n,m}=\frac{1}{m!}\left\langle\frac{h\bigl{(}\bar{f}(t)\bigr{)}}{g\bigl{(}\bar{f}(t)\bigr{)}}l\bigl{(}\bar{f}(t)\bigr{)}^{m}\Bigg{\|}x^{n}\right\rangle\,.$ (13) ## 3 Main results We now note that $B_{n}^{(k)}(x)$, $B_{n}(x\|a_{1},\dots,a_{r})$ and $S_{n}^{(r,k)}(x\|a_{1},\dots,a_{r})$ are the Appell sequences for $g_{k}(t)=\frac{1-e^{-t}}{{\rm Li}_{k}(1-e^{-t})},\quad g_{r}(t)=\frac{\prod_{j=1}^{r}(e^{a_{j}t}-1)}{t^{r}},\quad g_{r,k}(t)=\frac{\prod_{j=1}^{r}(e^{a_{j}t}-1)}{t^{r}}\frac{1-e^{-t}}{{\rm Li}_{k}(1-e^{-t})}\,.$ So, $\displaystyle B_{n}^{(k)}(x)$ $\displaystyle\sim\left(\frac{1-e^{-t}}{{\rm Li}_{k}(1-e^{-t})},t\right)\,,$ (14) $\displaystyle B_{n}(x\|a_{1},\dots,a_{r})$ $\displaystyle\sim\left(\frac{\prod_{j=1}^{r}(e^{a_{j}t}-1)}{t^{r}},t\right)\,,$ (15) $\displaystyle S_{n}^{(r,k)}(x\|a_{1},\dots,a_{r})$ $\displaystyle\sim\left(\frac{\prod_{j=1}^{r}(e^{a_{j}t}-1)}{t^{r}}\frac{1-e^{-t}}{{\rm Li}_{k}(1-e^{-t})},t\right)\,.$ (16) In particular, we have $\displaystyle tB_{n}^{(k)}(x)$ $\displaystyle=\frac{d}{dx}B_{n}^{(k)}(x)=nB_{n-1}^{(k)}(x)\,,$ (17) $\displaystyle tB_{n}(x\|a_{1},\dots,a_{r})$ $\displaystyle=\frac{d}{dx}B_{n}(x\|a_{1},\dots,a_{r})$ $\displaystyle=nB_{n-1}(x\|a_{1},\dots,a_{r})\,,$ (18) $\displaystyle tS_{n}^{(r,k)}(x\|a_{1},\dots,a_{r})$ $\displaystyle=\frac{d}{dx}S_{n}^{(r,k)}(x\|a_{1},\dots,a_{r})$ $\displaystyle=nS_{n-1}^{(r,k)}(x\|a_{1},\dots,a_{r})\,.$ (19) Notice that $\frac{d}{dx}{\rm Li}_{k}(x)=\frac{1}{x}{\rm Li}_{k-1}(x)\,.$ ### 3.1 Explicit expressions Write $B_{n}(a_{1},\dots,a_{r}):=B_{n}(0\|a_{1},\dots,a_{r})$ and $S_{n}^{(r,k)}(a_{1},\dots,a_{r}):=S_{n}^{(r,k)}(0\|a_{1},\dots,a_{r})$. Let $(n)_{j}=n(n-1)\cdots(n-j+1)$ ($j\geq 1$) with $(n)_{0}=1$. ###### Theorem 1 $\displaystyle S_{n}^{(r,k)}(x\|a_{1},\dots,a_{r})$ $\displaystyle=\sum_{l=0}^{n}\binom{n}{l}B_{n-l}(a_{1},\dots,a_{r})B_{l}^{(k)}(x)\,,$ (20) $\displaystyle=\sum_{l=0}^{n}\binom{n}{l}B_{n-l}^{(k)}B_{l}(x\|a_{1},\dots,a_{r})\,,$ (21) $\displaystyle=\sum_{l=0}^{n}\sum_{m=l}^{n}\sum_{j=0}^{m}(-1)^{j}\binom{m}{j}\binom{n}{l}\frac{1}{(m+1)^{k}}B_{n-l}(a_{1},\dots,a_{r})(x-j)^{l}\,,$ (22) $\displaystyle=\sum_{l=0}^{n}\left(\sum_{j=l}^{n}\sum_{m=0}^{n-j}(-1)^{n-m-j}\binom{n}{j}\binom{j}{l}\right.$ $\displaystyle\qquad\qquad\left.\times\frac{m!}{(m+1)^{k}}S_{2}(n-j,m)B_{j-l}(a_{1},\dots,a_{r})\right)x^{l}\,,$ (23) $\displaystyle=\sum_{j=0}^{n}\binom{n}{j}S_{n-j}^{(r,k)}(a_{1},\dots,a_{r})x^{j}\,.$ (24) * Proof. By (1), (2) and (3), we have $\displaystyle S_{n}^{(r,k)}(y\|a_{1},\dots,a_{r})$ $\displaystyle=\left\langle\sum_{i=0}^{\infty}S_{i}^{(r,k)}(y\|a_{1},\dots,a_{r})\frac{t^{i}}{i!}\Big{\|}x^{n}\right\rangle$ $\displaystyle=\left\langle\frac{t^{r}}{\prod_{j=1}^{r}(e^{a_{j}t}-1)}\frac{{\rm Li}_{k}(1-e^{-t})}{1-e^{-t}}e^{yt}\Big{\|}x^{n}\right\rangle$ $\displaystyle=\left\langle\frac{t^{r}}{\prod_{j=1}^{r}(e^{a_{j}t}-1)}\Big{\|}\frac{{\rm Li}_{k}(1-e^{-t})}{1-e^{-t}}e^{yt}x^{n}\right\rangle$ $\displaystyle=\left\langle\frac{t^{r}}{\prod_{j=1}^{r}(e^{a_{j}t}-1)}\Big{\|}\sum_{l=0}^{\infty}B_{l}^{(k)}(y)\frac{t^{l}}{l!}x^{n}\right\rangle$ $\displaystyle=\left\langle\frac{t^{r}}{\prod_{j=1}^{r}(e^{a_{j}t}-1)}\Big{\|}\sum_{l=0}^{n}\binom{n}{l}B_{l}^{(k)}(y)x^{n-l}\right\rangle$ $\displaystyle=\sum_{l=0}^{n}\binom{n}{l}B_{l}^{(k)}(y)\left\langle\frac{t^{r}}{\prod_{j=1}^{r}(e^{a_{j}t}-1)}\Big{\|}x^{n-l}\right\rangle$ $\displaystyle=\sum_{l=0}^{n}\binom{n}{l}B_{l}^{(k)}(y)\left\langle\sum_{i=0}^{\infty}B_{i}(a_{1},\dots,a_{r})\frac{t^{i}}{i!}\Big{\|}x^{n-l}\right\rangle$ $\displaystyle=\sum_{l=0}^{n}\binom{n}{l}B_{l}^{(k)}(y)B_{n-l}(a_{1},\dots,a_{r})\,.$ So, we get (20). We also have $\displaystyle S_{n}^{(r,k)}(y\|a_{1},\dots,a_{r})$ $\displaystyle=\left\langle\sum_{i=0}^{\infty}S_{i}^{(r,k)}(y\|a_{1},\dots,a_{r})\frac{t^{i}}{i!}\Big{\|}x^{n}\right\rangle$ $\displaystyle=\left\langle\frac{{\rm Li}_{k}(1-e^{-t})}{1-e^{-t}}\Big{\|}\frac{t^{r}}{\prod_{j=1}^{r}(e^{a_{j}t}-1)}e^{yt}x^{n}\right\rangle$ $\displaystyle=\left\langle\frac{{\rm Li}_{k}(1-e^{-t})}{1-e^{-t}}\Big{\|}\sum_{l=0}^{\infty}B_{l}(y\|a_{1},\dots,a_{r})\frac{t^{l}}{l!}x^{n}\right\rangle$ $\displaystyle=\left\langle\frac{{\rm Li}_{k}(1-e^{-t})}{1-e^{-t}}\Big{\|}\sum_{l=0}^{n}B_{l}(y\|a_{1},\dots,a_{r})\binom{n}{l}x^{n-l}\right\rangle$ $\displaystyle=\sum_{l=0}^{n}\binom{n}{l}B_{l}(y\|a_{1},\dots,a_{r})\left\langle\frac{{\rm Li}_{k}(1-e^{-t})}{1-e^{-t}}\Big{\|}x^{n-l}\right\rangle$ $\displaystyle=\sum_{l=0}^{n}\binom{n}{l}B_{l}(y\|a_{1},\dots,a_{r})\left\langle\sum_{i=0}^{\infty}B_{i}^{(k)}\frac{t^{i}}{i!}\Big{\|}x^{n-l}\right\rangle$ $\displaystyle=\sum_{l=0}^{n}\binom{n}{l}B_{l}(y\|a_{1},\dots,a_{r})B_{n-l}^{(k)}\,.$ Thus, we get (21). In [8] we obtained that $\frac{{\rm Li}_{k}(1-e^{-t})}{1-e^{-t}}x^{n}=\sum_{m=0}^{n}\frac{1}{(m+1)^{k}}\sum_{j=0}^{m}(-1)^{j}\binom{m}{j}(x-j)^{n}\,.$ So, $\displaystyle S_{n}^{(r,k)}(x\|a_{1},\dots,a_{r})$ $\displaystyle=\frac{t^{r}}{\prod_{j=1}^{r}(e^{a_{j}t}-1)}\frac{{\rm Li}_{k}(1-e^{-t})}{1-e^{-t}}x^{n}$ $\displaystyle=\sum_{m=0}^{n}\frac{1}{(m+1)^{k}}\sum_{j=0}^{m}(-1)^{j}\binom{m}{j}\frac{t^{r}}{\prod_{j=1}^{r}(e^{a_{j}t}-1)}(x-j)^{n}$ $\displaystyle=\sum_{m=0}^{n}\frac{1}{(m+1)^{k}}\sum_{j=0}^{m}(-1)^{j}\binom{m}{j}\sum_{l=0}^{n}\binom{n}{l}B_{n-l}(a_{1},\dots,a_{r})(x-j)^{l}$ $\displaystyle=\sum_{l=0}^{n}\sum_{m=l}^{n}\sum_{j=0}^{m}(-1)^{j}\binom{m}{j}\binom{n}{l}\frac{1}{(m+1)^{k}}B_{n-l}(a_{1},\dots,a_{r})(x-j)^{l}\,,$ which is the identity (22). In [8] we obtained that $\frac{{\rm Li}_{k}(1-e^{-t})}{1-e^{-t}}x^{n}=\sum_{j=0}^{n}\left(\sum_{m=0}^{n-j}\frac{(-1)^{n-m-j}}{(m+1)^{k}}\binom{n}{j}m!S_{2}(n-j,m)\right)x^{j}\,,$ where $S_{2}(l,m)$ are the Stirling numbers of the second kind, defined by $(e^{t}-1)^{m}=m!\sum_{l=m}^{\infty}S_{2}(l,m)\frac{t^{l}}{l!}\,.$ Thus, $\displaystyle S_{n}^{(r,k)}(x\|a_{1},\dots,a_{r})=\sum_{j=0}^{n}\left(\sum_{m=0}^{n-j}\frac{(-1)^{n-m-j}}{(m+1)^{k}}\binom{n}{j}m!S_{2}(n-j,m)\right)\frac{t^{r}}{\prod_{i=1}^{r}(e^{a_{i}t}-1)}x^{j}$ $\displaystyle=\sum_{j=0}^{n}\left(\sum_{m=0}^{n-j}\frac{(-1)^{n-m-j}}{(m+1)^{k}}\binom{n}{j}m!S_{2}(n-j,m)\right)B_{j}(x\|a_{1},\dots,a_{r})$ $\displaystyle=\sum_{j=0}^{n}\left(\sum_{m=0}^{n-j}\frac{(-1)^{n-m-j}}{(m+1)^{k}}\binom{n}{j}m!S_{2}(n-j,m)\right)\sum_{l=0}^{j}\binom{j}{l}B_{j-l}(a_{1},\dots,a_{r})x^{l}$ $\displaystyle=\sum_{l=0}^{n}\left(\sum_{j=l}^{n}\sum_{m=0}^{n-j}(-1)^{n-m-j}\binom{n}{j}\binom{j}{l}\frac{m!}{(m+1)^{k}}S_{2}(n-j,m)B_{j-l}(a_{1},\dots,a_{r})\right)x^{l}\,,$ which is the identity (23). By (11) with (16), we have $\displaystyle\left\langle g\bigl{(}\bar{f}(t)\bigr{)}^{-1}\bar{f}(t)^{j}\|x^{n}\right\rangle$ $\displaystyle=\left\langle\frac{t^{r}}{\prod_{j=1}^{r}(e^{a_{j}t}-1)}\frac{{\rm Li}_{k}(1-e^{-t})}{1-e^{-t}}t^{j}\Big{\|}x^{n}\right\rangle$ $\displaystyle=(n)_{j}\left\langle\frac{t^{r}}{\prod_{j=1}^{r}(e^{a_{j}t}-1)}\frac{{\rm Li}_{k}(1-e^{-t})}{1-e^{-t}}\Big{\|}x^{n-j}\right\rangle$ $\displaystyle=(n)_{j}\left\langle\sum_{i=0}^{\infty}S_{i}^{(r,k)}(a_{1},\dots,a_{r})\frac{t^{i}}{i!}\Big{\|}x^{n-j}\right\rangle$ $\displaystyle=(n)_{j}S_{n-j}^{(r,k)}(a_{1},\dots,a_{r})\,.$ Thus, we get (24). ### 3.2 Sheffer identity ###### Theorem 2 $S_{n}^{(r,k)}(x+y\|a_{1},\dots,a_{r})=\sum_{j=0}^{n}\binom{n}{j}S_{j}^{(r,k)}(x\|a_{1},\dots,a_{r})y^{n-j}\,.$ (25) * Proof. By (16) with $\displaystyle p_{n}(x)$ $\displaystyle=\frac{\prod_{j=1}^{r}(e^{a_{j}t}-1)}{t^{r}}\frac{1-e^{-t}}{{\rm Li}_{k}(1-e^{-t})}S_{n}^{(r,k)}(x\|a_{1},\dots,a_{r})$ $\displaystyle=x^{n}\sim(1,t)\,,$ using (12), we have (25). ### 3.3 Recurrence ###### Theorem 3 $\displaystyle S_{n+1}^{(r,k)}(x\|a_{1},\dots,a_{r})=xS_{n}^{(r,k)}(x\|a_{1},\dots,a_{r})$ $\displaystyle\quad-\frac{1}{n+1}\sum_{j=1}^{r}\sum_{l=0}^{n}\binom{n+1}{l}(-a_{j})^{n+1-l}B_{n+1-l}S_{l}^{(r,k)}(x\|a_{1},\dots,a_{r})$ $\displaystyle\quad-\frac{1}{n+1}\left(S_{n+1}^{(r+1,k)}(x\|a_{1},\dots,a_{r},1)-S_{n+1}^{(r+1,k-1)}(x\|a_{1},\dots,a_{r},1)\right)\,,$ (26) where $B_{n}$ is the $n$th ordinary Bernoulli number. * Proof. By applying $s_{n+1}(x)=\left(x-\frac{g^{\prime}(t)}{g(t)}\right)\frac{1}{f^{\prime}(t)}s_{n}(x)$ ([13, Corollary 3.7.2]) with (16), we get $S_{n+1}^{(r,k)}(x\|a_{1},\dots,a_{r})=\left(x-\frac{g_{r,k}^{\prime}(t)}{g_{r,k}(t)}\right)S_{n}^{(r,k)}(x\|a_{1},\dots,a_{r})\,.$ Now, $\displaystyle\frac{g_{r,k}^{\prime}(t)}{g_{r,k}(t)}=(\ln g_{r,k}(t))^{\prime}$ $\displaystyle=\left(\sum_{j=1}^{r}\ln(e^{a_{j}t}-1)-r\ln t+\ln(1-e^{-t})-\ln{\rm Li}_{k}(1-e^{-t})\right)^{\prime}$ $\displaystyle=\sum_{j=1}^{r}\frac{a_{j}e^{a_{j}t}}{e^{a_{j}t}-1}-\frac{r}{t}+\frac{e^{-t}}{1-e^{-t}}\left(1-\frac{{\rm Li}_{k-1}(1-e^{-t})}{{\rm Li}_{k}(1-e^{-t})}\right)$ $\displaystyle=\frac{\sum_{j=1}^{r}\prod_{i\neq j}(e^{a_{i}t}-1)(a_{j}te^{a_{j}t}-e^{a_{j}t}+1)}{t\prod_{j=1}^{r}(e^{a_{j}t}-1)}+\frac{t}{e^{t}-1}\frac{{\rm Li}_{k}(1-e^{-t})-{\rm Li}_{k-1}(1-e^{-t})}{t{\rm Li}_{k}(1-e^{-t})}\,.$ Since $\displaystyle\frac{\sum_{j=1}^{r}\prod_{i\neq j}(e^{a_{i}t}-1)(a_{j}te^{a_{j}t}-e^{a_{j}t}+1)}{\prod_{j=1}^{r}(e^{a_{j}t}-1)}$ $\displaystyle=\frac{\frac{1}{2}\bigl{(}\sum_{j=1}^{r}a_{1}\cdots a_{j-1}a_{j}^{2}a_{j+1}\cdots a_{r}\bigr{)}t^{r+1}+\cdots}{(a_{1}\cdots a_{r})t^{r}+\cdots}$ $\displaystyle=\frac{1}{2}\left(\sum_{j=1}^{r}a_{j}\right)t+\cdots$ is a series with order$\geq 1$ and $\frac{{\rm Li}_{k}(1-e^{-t})-{\rm Li}_{k-1}(1-e^{-t})}{1-e^{-t}}=\left(\frac{1}{2^{k}}-\frac{1}{2^{k-1}}\right)t+\cdots$ is a delta series, we have $\displaystyle S_{n+1}^{(r,k)}(x\|a_{1},\dots,a_{r})=xS_{n}^{(r,k)}(x\|a_{1},\dots,a_{r})-\frac{g_{r,k}^{\prime}(t)}{g_{r,k}(t)}S_{n}^{(r,k)}(x\|a_{1},\dots,a_{r})$ $\displaystyle=xS_{n}^{(r,k)}(x\|a_{1},\dots,a_{r})-\frac{g_{r,k}^{\prime}(t)}{g_{r,k}(t)}\frac{t^{r}}{\prod_{j=1}^{r}(e^{a_{j}t}-1)}\frac{{\rm Li}_{k}(1-e^{-t})}{1-e^{-t}}x^{n}$ $\displaystyle=xS_{n}^{(r,k)}(x\|a_{1},\dots,a_{r})$ $\displaystyle\quad-\frac{t^{r}}{\prod_{j=1}^{r}(e^{a_{j}t}-1)}\frac{{\rm Li}_{k}(1-e^{-t})}{1-e^{-t}}\frac{\sum_{j=1}^{r}\prod_{i\neq j}(e^{a_{i}t}-1)(a_{j}te^{a_{j}t}-e^{a_{j}t}+1)}{t\prod_{j=1}^{r}(e^{a_{j}t}-1)}x^{n}$ $\displaystyle\quad-\frac{t^{r}}{\prod_{j=1}^{r}(e^{a_{j}t}-1)}\frac{t}{e^{t}-1}\frac{{\rm Li}_{k}(1-e^{-t})-{\rm Li}_{k-1}(1-e^{-t})}{t(1-e^{-t})}x^{n}\,.$ Now, $\displaystyle\frac{\sum_{j=1}^{r}\prod_{i\neq j}(e^{a_{i}t}-1)(a_{j}te^{a_{j}t}-e^{a_{j}t}+1)}{t\prod_{j=1}^{r}(e^{a_{j}t}-1)}x^{n}$ $\displaystyle=\frac{\sum_{j=1}^{r}\prod_{i\neq j}(e^{a_{i}t}-1)(a_{j}te^{a_{j}t}-e^{a_{j}t}+1)}{\prod_{j=1}^{r}(e^{a_{j}t}-1)}\frac{x^{n+1}}{n+1}$ $\displaystyle=\frac{1}{n+1}\sum_{j=1}^{r}\frac{a_{j}te^{a_{j}t}-e^{a_{j}t}+1}{e^{a_{j}t}-1}x^{n+1}$ $\displaystyle=\frac{1}{n+1}\sum_{j=1}^{r}\left(\frac{a_{j}te^{a_{j}t}}{e^{a_{j}t}-1}-1\right)x^{n+1}$ $\displaystyle=\frac{1}{n+1}\sum_{j=1}^{r}\left(\sum_{l=0}^{\infty}\frac{(-1)^{l}B_{l}a_{j}^{l}}{l!}t^{l}-1\right)x^{n+1}$ $\displaystyle=\frac{1}{n+1}\sum_{j=1}^{r}\left(\sum_{l=0}^{n+1}\binom{n+1}{l}(-a_{j})^{l}B_{l}x^{n+1-l}-x^{n+1}\right)$ $\displaystyle=\frac{1}{n+1}\sum_{j=1}^{r}\sum_{l=1}^{n+1}\binom{n+1}{l}(-a_{j})^{l}B_{l}x^{n+1-l}$ $\displaystyle=\frac{1}{n+1}\sum_{j=1}^{r}\sum_{l=0}^{n}\binom{n+1}{l}(-a_{j})^{n+1-l}B_{n+1-l}x^{l}\,.$ Also, $\frac{{\rm Li}_{k}(1-e^{-t})-{\rm Li}_{k-1}(1-e^{-t})}{t(1-e^{-t})}x^{n}=\frac{1}{n+1}\frac{{\rm Li}_{k}(1-e^{-t})-{\rm Li}_{k-1}(1-e^{-t})}{1-e^{-t}}x^{n+1}\,.$ Thus, we get the identity (26). ### 3.4 A more relation ###### Theorem 4 $\displaystyle S_{n}^{(r,k)}(x\|a_{1},\dots,a_{r})=xS_{n-1}^{(r,k)}(x\|a_{1},\dots,a_{r})$ $\displaystyle\quad+\sum_{m=1}^{n}\frac{(-1)^{m-1}\binom{n-1}{m-1}B_{m}}{m}\sum_{j=1}^{r}a_{j}^{m}S_{n-m}^{(r,k)}(x\|a_{1},\dots,a_{r})$ $\displaystyle\quad+\frac{1}{n}S_{n}^{(r+1,k-1)}(x\|a_{1},\dots,a_{r},1)-\frac{1}{n}S_{n}^{(r+1,k)}(x\|a_{1},\dots,a_{r},1)\,.$ (27) * Proof. For $n\geq 1$ we have $\displaystyle S_{n}^{(r,k)}(y\|a_{1},\dots,a_{r})$ $\displaystyle=\left\langle\sum_{l=0}^{\infty}S_{l}^{(r,k)}(y\|a_{1},\dots,a_{r})\frac{t^{l}}{l!}\Big{\|}x^{n}\right\rangle$ $\displaystyle=\left\langle\frac{t^{r}}{\prod_{j=1}^{r}(e^{a_{j}t}-1)}\frac{{\rm Li}_{k}(1-e^{-t})}{1-e^{-t}}e^{yt}\Big{\|}x^{n}\right\rangle$ $\displaystyle=\left\langle\partial_{t}\left(\frac{t^{r}}{\prod_{j=1}^{r}(e^{a_{j}t}-1)}\frac{{\rm Li}_{k}(1-e^{-t})}{1-e^{-t}}e^{yt}\right)\Big{\|}x^{n-1}\right\rangle$ $\displaystyle=\left\langle\left(\partial_{t}\frac{t^{r}}{\prod_{j=1}^{r}(e^{a_{j}t}-1)}\right)\frac{{\rm Li}_{k}(1-e^{-t})}{1-e^{-t}}e^{yt}\Big{\|}x^{n-1}\right\rangle$ $\displaystyle\quad+\left\langle\frac{t^{r}}{\prod_{j=1}^{r}(e^{a_{j}t}-1)}\left(\partial_{t}\frac{{\rm Li}_{k}(1-e^{-t})}{1-e^{-t}}\right)e^{yt}\Big{\|}x^{n-1}\right\rangle$ $\displaystyle\quad+\left\langle\frac{t^{r}}{\prod_{j=1}^{r}(e^{a_{j}t}-1)}\frac{{\rm Li}_{k}(1-e^{-t})}{1-e^{-t}}(\partial_{t}e^{yt})\Big{\|}x^{n-1}\right\rangle$ $\displaystyle=yS_{n-1}^{(r,k)}(y\|a_{1},\dots,a_{r})$ $\displaystyle\quad+\left\langle\left(\partial_{t}\frac{t^{r}}{\prod_{j=1}^{r}(e^{a_{j}t}-1)}\right)\frac{{\rm Li}_{k}(1-e^{-t})}{1-e^{-t}}e^{yt}\Big{\|}x^{n-1}\right\rangle$ $\displaystyle\quad+\left\langle\frac{t^{r}}{\prod_{j=1}^{r}(e^{a_{j}t}-1)}\left(\partial_{t}\frac{{\rm Li}_{k}(1-e^{-t})}{1-e^{-t}}\right)e^{yt}\Big{\|}x^{n-1}\right\rangle\,.$ Observe that $\displaystyle\partial_{t}\left(\frac{t^{r}}{\prod_{j=1}^{r}(e^{a_{j}t}-1)}\right)$ $\displaystyle=\frac{rt^{r-1}-t^{r}\sum_{j=1}^{r}\frac{a_{j}e^{a_{j}t}}{e^{a_{j}t}-1}}{\prod_{j=1}^{r}(e^{a_{j}t}-1)}$ $\displaystyle=\frac{t^{r-1}}{\prod_{j=1}^{r}(e^{a_{j}t}-1)}\left(r-\sum_{j=1}^{r}\frac{a_{j}te^{a_{j}t}}{e^{a_{j}t}-1}\right)$ $\displaystyle=\frac{t^{r-1}}{\prod_{j=1}^{r}(e^{a_{j}t}-1)}\left(r-\sum_{j=1}^{r}\frac{-a_{j}t}{e^{-a_{j}t}-1}\right)$ $\displaystyle=\frac{t^{r-1}}{\prod_{j=1}^{r}(e^{a_{j}t}-1)}\left(r-\sum_{j=1}^{r}\sum_{m=0}^{\infty}\frac{(-a_{j})^{m}B_{m}t^{m}}{m!}\right)$ $\displaystyle=\frac{t^{r-1}}{\prod_{j=1}^{r}(e^{a_{j}t}-1)}\left(r-\sum_{m=0}^{\infty}\left(\sum_{j=1}^{r}(-a_{j})^{m}\right)\frac{B_{m}t^{m}}{m!}\right)$ $\displaystyle=\frac{t^{r}}{\prod_{j=1}^{r}(e^{a_{j}t}-1)}\sum_{m=1}^{\infty}\left(\sum_{j=1}^{r}a_{j}^{m}\right)\frac{(-1)^{m-1}B_{m}}{m!}t^{m-1}\,.$ Thus, $\displaystyle\left\langle\left(\partial_{t}\frac{t^{r}}{\prod_{j=1}^{r}(e^{a_{j}t}-1)}\right)\frac{{\rm Li}_{k}(1-e^{-t})}{1-e^{-t}}e^{yt}\Big{\|}x^{n-1}\right\rangle$ $\displaystyle=\left\langle\frac{t^{r}}{\prod_{j=1}^{r}(e^{a_{j}t}-1)}\frac{{\rm Li}_{k}(1-e^{-t})}{1-e^{-t}}e^{yt}\Big{\|}\sum_{m=1}^{n}\left(\sum_{j=1}^{r}a_{j}^{m}\right)\frac{(-1)^{m-1}B_{m}}{m!}t^{m-1}x^{n-1}\right\rangle$ $\displaystyle=\sum_{m=1}^{n}\frac{(-1)^{m-1}\binom{n-1}{m-1}B_{m}}{m}\sum_{j=1}^{r}a_{j}^{m}\left\langle\frac{t^{r}}{\prod_{j=1}^{r}(e^{a_{j}t}-1)}\frac{{\rm Li}_{k}(1-e^{-t})}{1-e^{-t}}e^{yt}\Big{\|}x^{n-m}\right\rangle$ $\displaystyle=\sum_{m=1}^{n}\frac{(-1)^{m-1}\binom{n-1}{m-1}B_{m}}{m}S_{n-m}^{(r,k)}(y\|a_{1},\dots,a_{r})\sum_{j=1}^{r}a_{j}^{m}\,.$ Since $\frac{{\rm Li}_{k-1}(1-e^{-t})-{\rm Li}_{k}(1-e^{-t})}{1-e^{-t}}=\left(\frac{1}{2^{k-1}}-\frac{1}{2^{k}}\right)t+\cdots$ is a delta series, we have $\displaystyle\left\langle\frac{t^{r}}{\prod_{j=1}^{r}(e^{a_{j}t}-1)}\left(\partial_{t}\frac{{\rm Li}_{k}(1-e^{-t})}{1-e^{-t}}\right)e^{yt}\Big{\|}x^{n-1}\right\rangle$ $\displaystyle=\left\langle\frac{t^{r}}{\prod_{j=1}^{r}(e^{a_{j}t}-1)}\frac{e^{-t}\bigl{(}{\rm Li}_{k-1}(1-e^{-t})-{\rm Li}_{k}(1-e^{-t})\bigr{)}}{(1-e^{-t})^{2}}e^{yt}\Big{\|}x^{n-1}\right\rangle$ $\displaystyle=\left\langle\frac{t^{r}}{\prod_{j=1}^{r}(e^{a_{j}t}-1)}\frac{t}{e^{t}-1}\frac{{\rm Li}_{k-1}(1-e^{-t})-{\rm Li}_{k}(1-e^{-t})}{t(1-e^{-t})}e^{yt}\Big{\|}x^{n-1}\right\rangle$ $\displaystyle=\left\langle\frac{t^{r+1}}{\prod_{j=1}^{r}(e^{a_{j}t}-1)(e^{t}-1)}\frac{{\rm Li}_{k-1}(1-e^{-t})-{\rm Li}_{k}(1-e^{-t})}{1-e^{-t}}e^{yt}\Big{\|}\frac{x^{n}}{n}\right\rangle$ $\displaystyle=\frac{1}{n}\left\langle\frac{t^{r+1}}{\prod_{j=1}^{r}(e^{a_{j}t}-1)(e^{t}-1)}\frac{{\rm Li}_{k-1}(1-e^{-t})}{1-e^{-t}}e^{yt}\Big{\|}x^{n}\right\rangle$ $\displaystyle\quad-\frac{1}{n}\left\langle\frac{t^{r+1}}{\prod_{j=1}^{r}(e^{a_{j}t}-1)(e^{t}-1)}\frac{{\rm Li}_{k}(1-e^{-t})}{1-e^{-t}}e^{yt}\Big{\|}x^{n}\right\rangle$ $\displaystyle=\frac{1}{n}S_{n}^{(r+1,k-1)}(y\|a_{1},\dots,a_{r},1)-\frac{1}{n}S_{n}^{(r+1,k)}(y\|a_{1},\dots,a_{r},1)\,.$ Therefore, we obtain the desired result. Remark. After simple modification, Theorem 4 becomes $\displaystyle S_{n+1}^{(r,k)}(x\|a_{1},\dots,a_{r})=xS_{n}^{(r,k)}(x\|a_{1},\dots,a_{r})$ $\displaystyle\quad+\sum_{l=1}^{n+1}\frac{(-1)^{l-1}\binom{n}{l-1}B_{l}}{l}\sum_{j=1}^{r}a_{j}^{l}S_{n+1-l}^{(r,k)}(x\|a_{1},\dots,a_{r})$ $\displaystyle\quad+\frac{1}{n+1}S_{n+1}^{(r+1,k-1)}(x\|a_{1},\dots,a_{r},1)-\frac{1}{n+1}S_{n+1}^{(r+1,k)}(x\|a_{1},\dots,a_{r},1)\,.$ which is the same as the above recurrence formula (26) upon replacing $n$ by $n-1$. ### 3.5 Relations with poly-Bernoulli numbers and Barnes’ multiple Bernoulli numbers ###### Theorem 5 $\sum_{m=0}^{n}\binom{n+1}{m}(-1)^{n-m}S_{m}^{(r,k)}(a_{1},\dots,a_{r})\\\ =\sum_{l=0}^{n}\sum_{m=0}^{l}(-1)^{l-m}\binom{l}{m}\binom{n+1}{l+1}B_{m}^{(k-1)}B_{n-l}(a_{1},\dots,a_{r})\,.$ (28) * Proof. We shall compute $\left\langle\frac{t^{r}}{\prod_{j=1}^{r}(e^{a_{j}t}-1)}{\rm Li}_{k}(1-e^{-t})\Big{\|}x^{n+1}\right\rangle$ in two different ways. On the one hand, $\displaystyle\left\langle\frac{t^{r}}{\prod_{j=1}^{r}(e^{a_{j}t}-1)}{\rm Li}_{k}(1-e^{-t})\Big{\|}x^{n+1}\right\rangle=\left\langle\frac{t^{r}}{\prod_{j=1}^{r}(e^{a_{j}t}-1)}\frac{{\rm Li}_{k}(1-e^{-t})}{1-e^{-t}}\Big{\|}(1-e^{-t})x^{n+1}\right\rangle$ $\displaystyle=\left\langle\frac{t^{r}}{\prod_{j=1}^{r}(e^{a_{j}t}-1)}\frac{{\rm Li}_{k}(1-e^{-t})}{1-e^{-t}}\Big{\|}x^{n+1}-(x-1)^{n+1}\right\rangle$ $\displaystyle=\sum_{m=0}^{n}\binom{n+1}{m}(-1)^{n-m}\left\langle\frac{t^{r}}{\prod_{j=1}^{r}(e^{a_{j}t}-1)}\frac{{\rm Li}_{k}(1-e^{-t})}{1-e^{-t}}\Big{\|}x^{m}\right\rangle$ $\displaystyle=\sum_{m=0}^{n}\binom{n+1}{m}(-1)^{n-m}S_{m}^{(r,k)}(a_{1},\dots,a_{r})\,.$ On the other hand, $\displaystyle\left\langle\frac{t^{r}}{\prod_{j=1}^{r}(e^{a_{j}t}-1)}{\rm Li}_{k}(1-e^{-t})\Big{\|}x^{n+1}\right\rangle=\left\langle{\rm Li}_{k}(1-e^{-t})\Big{\|}\frac{t^{r}}{\prod_{j=1}^{r}(e^{a_{j}t}-1)}x^{n+1}\right\rangle$ $\displaystyle=\left\langle{\rm Li}_{k}(1-e^{-t})\Big{\|}B_{n+1}(x\|a_{1},\dots,a_{r})\right\rangle$ $\displaystyle=\left\langle\int_{0}^{t}\bigl{(}{\rm Li}_{k}(1-e^{-s})\bigr{)}^{\prime}ds\Big{\|}B_{n+1}(x\|a_{1},\dots,a_{r})\right\rangle$ $\displaystyle=\left\langle\int_{0}^{t}e^{-s}\frac{{\rm Li}_{k-1}(1-e^{-s})}{1-e^{-s}}ds\Big{\|}B_{n+1}(x\|a_{1},\dots,a_{r})\right\rangle$ $\displaystyle=\left\langle\int_{0}^{t}\left(\sum_{j=0}^{\infty}\frac{(-s)^{j}}{j!}\right)\left(\sum_{m=0}^{\infty}\frac{B_{m}^{(k-1)}}{m!}s^{m}\right)ds\Big{\|}B_{n+1}(x\|a_{1},\dots,a_{r})\right\rangle$ $\displaystyle=\left\langle\sum_{l=0}^{\infty}\left(\sum_{m=0}^{l}(-1)^{l-m}\binom{l}{m}B_{m}^{(k-1)}\right)\frac{1}{l!}\int_{0}^{t}s^{l}ds\Big{\|}B_{n+1}(x\|a_{1},\dots,a_{r})\right\rangle$ $\displaystyle=\sum_{l=0}^{n}\sum_{m=0}^{l}(-1)^{l-m}\binom{l}{m}\frac{B_{m}^{(k-1)}}{(l+1)!}\left\langle t^{l+1}\Big{\|}B_{n+1}(x\|a_{1},\dots,a_{r})\right\rangle$ $\displaystyle=\sum_{l=0}^{n}\sum_{m=0}^{l}(-1)^{l-m}\binom{l}{m}\frac{B_{m}^{(k-1)}}{(l+1)!}(n+1)_{l+1}B_{n-l}(a_{1},\dots,a_{r})$ $\displaystyle=\sum_{l=0}^{n}\sum_{m=0}^{l}(-1)^{l-m}\binom{l}{m}\binom{n+1}{l+1}B_{m}^{(k-1)}B_{n-l}(a_{1},\dots,a_{r})\,.$ Here, $B_{n-l}(a_{1},\dots,a_{r})=B_{n-l}(0\|a_{1},\dots,a_{r})$. Thus, we get (28). ### 3.6 Relations with the Stirling numbers of the second kind and the falling factorials ###### Theorem 6 $S_{n}^{(r,k)}(x\|a_{1},\dots,a_{r})=\sum_{m=0}^{n}\left(\sum_{l=m}^{n}S_{2}(l,m)\binom{n}{l}S_{n-l}^{(r,k)}(a_{1},\dots,a_{r})\right)(x)_{m}\,.$ (29) * Proof. For (16) and $(x)_{n}\sim(1,e^{t}-1)$, assume that $S_{n}^{(r,k)}(x\|a_{1},\dots,a_{r})=\sum_{m=0}^{n}C_{n,m}(x)_{m}$. By (13), we have $\displaystyle C_{n,m}$ $\displaystyle=\frac{1}{m!}\left\langle\frac{1}{\frac{\prod_{j=1}^{r}(e^{a_{j}t}-1)}{t^{r}}\frac{1-e^{-t}}{{\rm Li}_{k}(1-e^{-t})}}(e^{t}-1)^{m}\Big{\|}x^{n}\right\rangle$ $\displaystyle=\frac{1}{m!}\left\langle\frac{t^{r}}{\prod_{j=1}^{r}(e^{a_{j}t}-1)}\frac{{\rm Li}_{k}(1-e^{-t})}{1-e^{-t}}\Big{\|}(e^{t}-1)^{m}x^{n}\right\rangle$ $\displaystyle=\frac{1}{m!}\left\langle\frac{t^{r}}{\prod_{j=1}^{r}(e^{a_{j}t}-1)}\frac{{\rm Li}_{k}(1-e^{-t})}{1-e^{-t}}\Big{\|}m!\sum_{l=m}^{n}S_{2}(l,m)\frac{t^{l}}{l!}x^{n}\right\rangle$ $\displaystyle=\sum_{l=m}^{n}S_{2}(l,m)\binom{n}{l}\left\langle\frac{t^{r}}{\prod_{j=1}^{r}(e^{a_{j}t}-1)}\frac{{\rm Li}_{k}(1-e^{-t})}{1-e^{-t}}\Big{\|}x^{n-l}\right\rangle$ $\displaystyle=\sum_{l=m}^{n}S_{2}(l,m)\binom{n}{l}S_{n-l}^{(r,k)}(a_{1},\dots,a_{r})\,.$ Thus, we get the identity (29). ### 3.7 Relations with the Stirling numbers of the second kind and the rising factorials ###### Theorem 7 $S_{n}^{(r,k)}(x\|a_{1},\dots,a_{r})=\sum_{m=0}^{n}\left(\sum_{l=m}^{n}S_{2}(l,m)\binom{n}{l}S_{n-l}^{(r,k)}(-m\|a_{1},\dots,a_{r})\right)(x)^{(m)}\,.$ (30) * Proof. For (16) and $(x)^{(n)}=x(x+1)\cdots(x+n-1)\sim(1,1-e^{-t})$, assume that $S_{n}^{(r,k)}(x\|a_{1},\dots,a_{r})=\sum_{m=0}^{n}C_{n,m}(x)^{(m)}$. By (13), we have $\displaystyle C_{n,m}$ $\displaystyle=\frac{1}{m!}\left\langle\frac{1}{\frac{\prod_{j=1}^{r}(e^{a_{j}t}-1)}{t^{r}}\frac{1-e^{-t}}{{\rm Li}_{k}(1-e^{-t})}}(1-e^{-t})^{m}\Big{\|}x^{n})\right\rangle$ $\displaystyle=\frac{1}{m!}\left\langle\frac{t^{r}}{\prod_{j=1}^{r}(e^{a_{j}t}-1)}\frac{{\rm Li}_{k}(1-e^{-t})}{1-e^{-t}}e^{-mt}\Big{\|}(e^{t}-1)^{m}x^{n})\right\rangle$ $\displaystyle=\sum_{l=m}^{n}S_{2}(l,m)\binom{n}{l}\left\langle e^{-mt}\Big{\|}\frac{t^{r}}{\prod_{j=1}^{r}(e^{a_{j}t}-1)}\frac{{\rm Li}_{k}(1-e^{-t})}{1-e^{-t}}x^{n-l}\right\rangle$ $\displaystyle=\sum_{l=m}^{n}S_{2}(l,m)\binom{n}{l}\left\langle e^{-mt}\Big{\|}S_{n-l}^{(r,k)}(x\|a_{1},\dots,a_{r})\right\rangle$ $\displaystyle=\sum_{l=m}^{n}S_{2}(l,m)\binom{n}{l}S_{n-l}^{(r,k)}(-m\|a_{1},\dots,a_{r})\,.$ Thus, we get the identity (30). ### 3.8 Relations with higher-order Frobenius-Euler polynomials For $\lambda\in\mathbb{C}$ with $\lambda\neq 1$, the Frobenius-Euler polynomials of order $r$, $H_{n}^{(r)}(x\|\lambda)$ are defined by the generating function $\left(\frac{1-\lambda}{e^{t}-\lambda}\right)^{r}e^{xt}=\sum_{n=0}^{\infty}H_{n}^{(r)}(x\|\lambda)\frac{t^{n}}{n!}$ (see e.g. [7]). ###### Theorem 8 $S_{n}^{(r,k)}(x\|a_{1},\dots,a_{r})=\sum_{m=0}^{n}\left(\frac{\binom{n}{m}}{(1-\lambda)^{s}}\sum_{j=0}^{s}\binom{s}{j}(-\lambda)^{s-j}S_{n-m}^{(r,k)}(j\|a_{1},\dots,a_{r})\right)H_{m}^{(s)}(x\|\lambda)\,.$ (31) * Proof. For (16) and $H_{n}^{(s)}(x\|\lambda)\sim\left(\left(\frac{e^{t}-\lambda}{1-\lambda}\right)^{s},t\right)\,,$ assume that $S_{n}^{(r,k)}(x\|a_{1},\dots,a_{r})=\sum_{m=0}^{n}C_{n,m}H_{m}^{(s)}(x\|\lambda)$. By (13), we have $\displaystyle C_{n,m}$ $\displaystyle=\frac{1}{m!}\left\langle\left(\frac{e^{t}-\lambda}{1-\lambda}\right)^{s}\frac{t^{r}}{\prod_{j=1}^{r}(e^{a_{j}t}-1)}\frac{{\rm Li}_{k}(1-e^{-t})}{1-e^{-t}}t^{m}\Big{\|}x^{n}\right\rangle$ $\displaystyle=\frac{1}{m!(1-\lambda)^{s}}\left\langle(e^{t}-\lambda)^{s}\frac{t^{r}}{\prod_{j=1}^{r}(e^{a_{j}t}-1)}\frac{{\rm Li}_{k}(1-e^{-t})}{1-e^{-t}}\Big{\|}t^{m}x^{n}\right\rangle$ $\displaystyle=\frac{\binom{n}{m}}{(1-\lambda)^{s}}\sum_{j=0}^{s}\binom{s}{j}(-\lambda)^{s-j}\left\langle e^{jt}\Big{\|}\frac{t^{r}}{\prod_{j=1}^{r}(e^{a_{j}t}-1)}\frac{{\rm Li}_{k}(1-e^{-t})}{1-e^{-t}}x^{n-m}\right\rangle$ $\displaystyle=\frac{\binom{n}{m}}{(1-\lambda)^{s}}\sum_{j=0}^{s}\binom{s}{j}(-\lambda)^{s-j}S_{n-m}^{(r,k)}(j\|a_{1},\dots,a_{r})\,.$ Thus, we get the identity (31). ### 3.9 Relations with higher-order Bernoulli polynomials Bernoulli polynomials $\mathfrak{B}_{n}^{(r)}(x)$ of order $r$ are defined by $\left(\frac{t}{e^{t}-1}\right)^{r}e^{xt}=\sum_{n=0}^{\infty}\frac{\mathfrak{B}_{n}^{(r)}(x)}{n!}t^{n}$ (see e.g. [13, Section 2.2]). ###### Theorem 9 $S_{n}^{(r,k)}(x\|a_{1},\dots,a_{r})=\sum_{m=0}^{n}\left(\binom{n}{m}\sum_{l=0}^{n-m}\frac{\binom{n-m}{l}}{\binom{l+s}{l}}S_{2}(l+s,s)S_{n-m-l}^{(r,k)}(a_{1},\dots,a_{r})\right)\mathfrak{B}_{m}^{(s)}(x)\,.$ (32) * Proof. For (16) and $\mathfrak{B}_{n}^{(s)}(x)\sim\left(\left(\frac{e^{t}-1}{t}\right)^{s},t\right)\,,$ assume that $S_{n}^{(r,k)}(x\|a_{1},\dots,a_{r})=\sum_{m=0}^{n}C_{n,m}\mathfrak{B}_{m}^{(s)}(x)$. By (13), we have $\displaystyle C_{n,m}$ $\displaystyle=\frac{1}{m!}\left\langle\left(\frac{e^{t}-1}{t}\right)^{s}\frac{t^{r}}{\prod_{j=1}^{r}(e^{a_{j}t}-1)}\frac{{\rm Li}_{k}(1-e^{-t})}{1-e^{-t}}t^{m}\Big{\|}x^{n}\right\rangle$ $\displaystyle=\binom{n}{m}\left\langle\frac{t^{r}}{\prod_{j=1}^{r}(e^{a_{j}t}-1)}\frac{{\rm Li}_{k}(1-e^{-t})}{1-e^{-t}}\Big{\|}\left(\frac{e^{t}-1}{t}\right)^{s}x^{n-m}\right\rangle$ $\displaystyle=\binom{n}{m}\left\langle\frac{t^{r}}{\prod_{j=1}^{r}(e^{a_{j}t}-1)}\frac{{\rm Li}_{k}(1-e^{-t})}{1-e^{-t}}\Big{\|}\sum_{l=0}^{n-m}\frac{s!}{(l+s)!}S_{2}(l+s,s)t^{l}x^{n-m}\right\rangle$ $\displaystyle=\binom{n}{m}\sum_{l=0}^{n-m}\frac{s!}{(l+s)!}S_{2}(l+s,s)(n-m)_{l}\left\langle\frac{t^{r}}{\prod_{j=1}^{r}(e^{a_{j}t}-1)}\frac{{\rm Li}_{k}(1-e^{-t})}{1-e^{-t}}\Big{\|}x^{n-m-l}\right\rangle$ $\displaystyle=\binom{n}{m}\sum_{l=0}^{n-m}\frac{s!}{(l+s)!}S_{2}(l+s,s)(n-m)_{l}S_{n-m-l}^{(r,k)}(a_{1},\dots,a_{r})$ $\displaystyle=\binom{n}{m}\sum_{l=0}^{n-m}\frac{\binom{n-m}{l}}{\binom{l+s}{l}}S_{2}(l+s,s)S_{n-m-l}^{(r,k)}(a_{1},\dots,a_{r})\,.$ Thus, we get the identity (32). ## References * [1] A. Bayad and Y. Hamahata, Polylogarithms and poly-Bernoulli polynomials, Kyushu J. Math. 65 (2011), 15–24. * [2] A. Bayad, T. Kim, W. J. Kim and S. H. Lee, Arithmetic properties of $q$-Barnes polynomials, J. Comput. Anal. Appl. 15 (2013), 111–117. * [3] M. -A. Coppo and B. Candelpergher, The Arakawa-Kaneko zeta functions, Ramanujan J. 22 (2010), 153–162. * [4] L. C. Jang, J. H. Kim, T. Kim, D. H. Lee, D. W. Park and C. S. Ryoo, On Witt’s formula for the Barnes’ multiple Bernoulli polynomials, Far East J. Math. Sci. (FJMS) 13 (2004), 309–317. * [5] L. Jang, T. Kim, Y. -H. Kim, K. -W. Hwang, Note on the $q$ $q$-extension of Barnes’ type multiple Euler polynomials, J. Inequal. Appl. 2009, Art. ID 136532, 8 pp. * [6] L. Jang and T. Kim, $q$-analogue of Euler-Barnes’ numbers and polynomials, Bull. Korean Math. Soc. 42 (2005), 491–499. * [7] D. S. Kim and T. Kim, Some identities of Frobenius-Euler polynomials arising from umbral calculus, Adv. Difference Equ. 2012 (2012), #196. * [8] D. S. Kim, T. Kim and S. -H. Lee, Poly-Bernoulli polynomials arising from umbral calculus, available at http://arxiv.org/pdf/1306.6697.pdf * [9] T. Kim, On Euler-Barnes multiple zeta functions, Russ. J. Math. Phys. 10 (2003), 261–267. * [10] T. Kim, $p$-adic $q$-integrals associated with the Changhee-Barnes’ $q$-Bernoulli polynomials, Integral Transforms Spec. Funct. 15 (2004), 415–420. * [11] T. Kim, Barnes-type multiple $q$-zeta functions and $q$-Euler polynomials, J. Phys. A 43 (2010), 255201, 11pp. * [12] T. Kim and S. -H. Rim, On Changhee-Barnes’ $q$-Euler numbers and polynomials, Adv. Stud. Contemp. Math. (Kyungshang) 9 (2004), 81–86. * [13] S. Roman, The umbral Calculus, Dover, New York, 2005. * [14] Y. Simsek, T. Kim and I. -S. Pyung, Barnes’ type multiple Changhee $q$-zeta functions, Adv. Stud. Contemp. Math. (Kyungshang) 10 (2005), 121–129.	arxiv-papers	2013-12-27T01:56:51	2024-09-04T02:49:55.991650	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "D. S. Kim, T. Kim, T. Komatsu", "submitter": "Taekyun Kim", "url": "https://arxiv.org/abs/1312.7171" }
1312.7253	VBL]Universität Rostock Institut für Informatik 18051 Rostock, Germany FP]University of Colorado at Denver, Department of Mathematics & Statistics, Denver, CO 80202, USA # Complexity Results for Rainbow Matchings Van Bang Le [ [email protected] and Florian Pfender [ [email protected] ###### Abstract. A rainbow matching in an edge-colored graph is a matching whose edges have distinct colors. We address the complexity issue of the following problem, max rainbow matching: Given an edge-colored graph $G$, how large is the largest rainbow matching in $G$? We present several sharp contrasts in the complexity of this problem. We show, among others, that * • max rainbow matching can be approximated by a polynomial algorithm with approximation ratio $2/3-\varepsilon$. * • max rainbow matching is APX-complete, even when restricted to properly edge- colored linear forests without a $5$-vertex path, and is solvable in polynomial time for edge-colored forests without a $4$-vertex path. * • max rainbow matching is APX-complete, even when restricted to properly edge- colored trees without an $8$-vertex path, and is solvable in polynomial time for edge-colored trees without a $7$-vertex path. * • max rainbow matching is APX-complete, even when restricted to properly edge- colored paths. These results provide a dichotomy theorem for the complexity of the problem on forests and trees in terms of forbidding paths. The latter is somewhat surprising, since, to the best of our knowledge, no (unweighted) graph problem prior to our result is known to be NP-hard for simple paths. We also address the parameterized complexity of the problem. ###### Key words and phrases: Rainbow matching; computational complexity; NP-completeness; APX-completeness; parameterized complexity ###### 1991 Mathematics Subject Classification: ## 1\. Introduction and Results Given a graph $G=(E(G),V(G))$, an edge coloring is a function $\phi:E(G)\rightarrow\mathcal{C}$ mapping each edge $e\in E(G)$ to a color $\phi(e)\in\mathcal{C}$; $\phi$ is a _proper_ edge-coloring if, for all distinct edges $e$ and $e^{\prime}$, $\phi(e)\not=\phi(e^{\prime})$ whenever $e$ and $e^{\prime}$ have an endvertex in common. A (properly) edge-colored graph $(G,\phi)$ is a pair of a graph together with a (proper) edge coloring. A rainbow subgraph of an edge-colored graph is a subgraph whose edges have distinct colors. Rainbow subgraphs appear frequently in the literature, for a recent survey we point to [13]. In this paper we are concerned with _rainbow matchings_ , i.e., matchings whose edges have distinct colors. One motivation to look at rainbow matchings is Ryser’s famous conjecture from [20], which states that every Latin square of odd order contains a Latin transversal. Equivalently, the conjecture says that every proper edge coloring of the complete bipartite graph $K_{2n+1,2n+1}$ with $2n+1$ colors contains a rainbow matching with $2n+1$ edges. One often asks for the size of the largest rainbow matching in an edge-colored graph with certain restrictions (see, e.g., [4, 14, 15, 22]). In this paper, we consider the complexity of this problem. In particular, we consider the complexity of the following two problems, and we will restrict them to certain graph classes and edge colorings. rainbow matching Instance: Graph $G$ with an edge-coloring and an integer $k$ Question: Does $G$ have a rainbow matching with at least $k$ edges? rainbow matching is also called multiple choice matching in [8, Problem GT55]. The optimization version of the decision problem rainbow matching is: max rainbow matching Instance: Graph $G$ with an edge-coloring Output: A largest rainbow matching in $G$. Only the following complexity result for rainbow matching is known. Note that the considered graphs in the proof are not properly edge colored. When restricted to properly edge-colored graphs, no complexity result is known prior to this work. ###### Theorem 1 ([12]). rainbow matching is NP-complete, even when restricted to edge-colored bipartite graphs. In this paper, we analyze classes of graphs for which max rainbow matching can be solved and thus rainbow matching can be decided in polynomial time, or is NP-hard. Our results are: * • There is a polynomial-time $(2/3-\varepsilon)$-approximation algorithm for max rainbow matching for every $\varepsilon>0$. * • max rainbow matching is APX-complete, and thus rainbow matching is NP- complete, even for very restricted graphs classes such as * – edge-colored complete graphs, * – properly edge-colored paths, * – properly edge-colored $P_{8}$-free trees in which every color is used at most twice, * – properly edge-colored $P_{5}$-free linear forests in which every color is used at most twice, * – properly edge-colored $P_{4}$-free bipartite graphs in which every color is used at most twice. These results significantly improve Theorem 1. We also provide an inapproximability bound for each of the listed graph classes. * • max rainbow matching is solvable in time $O(m^{3/2})$ for $m$-edge graphs without $P_{4}$ (induced or not); in particular for $P_{4}$-free forests. * • max rainbow matching is polynomially solvable for $P_{7}$-free forests with bounded number of components; in particular for $P_{7}$-free trees. * • max rainbow matching is fixed parameter tractable for $P_{5}$-free forests, when parameterized by the number of the components. The next section contains some relevant notation and definitions. Section 3 deals with approximability and inapproximability results, Section 4 discusses some polynomially solvable cases, and Section 5 addresses the parameterized complexity. We conclude the paper in Section 6 with some open problems. ## 2\. Definitions and Preliminaries We consider only finite, simple, and undirected graphs. For a graph $G$, the vertex set is denoted $V(G)$ and the edge set is denoted $E(G)$. An edge $xy$ of a graph $G$ is a bridge if $G-xy$ has more components than $G$. If $G$ does not contain an induced subgraph isomorphic to another graph $F$, then $G$ is $F$-free. For $\ell\geq 1$, let $P_{\ell}$ denote a chordless path with $\ell$ vertices and $\ell-1$ edges, and for $\ell\geq 3$, let $C_{\ell}$ denote a chordless cycle with $\ell$ vertices and $\ell$ edges. A triangle is a $C_{3}$. For $p,q\geq 1$, $K_{p,q}$ denotes the complete bipartite graph with $p$ vertices of one color class and $q$ vertices of the second color class; a star is a $K_{1,q}$. A complete graph with $p$ vertices is denoted by $K_{p}$; $K_{p}-e$ is the graph obtained from $K_{p}$ by deleting one edge. An $r$-regular graph is one in which each vertex has degree exactly $r$. A forest in which each component is a path is a linear forest. The line graph $L(G)$ of a graph $G$ has vertex set $E(G)$, and two vertices in $L(G)$ are adjacent if the corresponding edges in $G$ are incident. By definition, every matching in $G$ corresponds to an independent set in $L(G)$ of the same size, and vice versa. One of the main tools we use in discussing rainbow matchings is the following concept that generalizes line graphs naturally: ###### Definition 1. The color-line graph $C\\!L(G)$ of an edge-colored graph $G$ has vertex set $E(G)$, and two vertices in $C\\!L(G)$ are adjacent if the corresponding edges in $G$ are incident or have the same color. Notice that, given an edge-colored graph $G$, $C\\!L(G)$ can be constructed in time $O(\|E(G)\|^{2})$ in an obvious way. We will make use of further facts about color-line graphs below that can be verified by definition. ###### Lemma 1. Let $G$ be an edge-colored graph. Then * (i) $C\\!L(G)$ is $K_{1,4}$-free. * (ii) $C\\!L(G)$ is $(K_{7}-e)$-free, provided $G$ is properly edge-colored. * (iii) Every rainbow matching in $G$ corresponds to an independent set in $C\\!L(G)$ of the same size, and vice versa. Lemma 1 allows us to use results about independent sets to obtain results on rainbow matchings. This way, we will relate max rainbow matching to the following two problems, which are very well studied in the literature. MIS (Maximum Independent Set) Instance: A graph $G$. Output: A maximum independent set in $G$. $3$-MIS (Maximum Independent Set in $3$-regular Graphs) Instance: A $3$-regular graph $G$. Output: A maximum independent set in $G$. In the present paper, a polynomial-time algorithm $\mathrm{A}$ with approximation ratio $\alpha$, $0<\alpha<1$, for a (maximization) problem is one that, for all problem instances $I$, $\mathrm{A}(I)\geq\alpha\cdot\mathrm{opt}(I)$, where $\mathrm{A}(I)$ is the objective value of the solution found by $\mathrm{A}$ and $\mathrm{opt}(I)$ is the objective value of an optimal solution. A problem is said to be in APX (for approximable) if it admits an algorithm with a constant approximation ratio. A problem in APX is called APX-complete if all other problems in APX can be $L$-reduced (cf. [18]) to it. It is known that $3$-MIS is APX-complete (see [1]). Thus, if $3$-MIS is $L$-reducible to a problem in APX, then this problem is also APX-complete. All reductions in this paper are $L$-reductions. ## 3\. Approximability and Hardness We first show that max rainbow matching is in APX, by reducing to MIS on $K_{1,4}$-free graphs. The following theorem is due to Hurkens and Schrijver [11]; see also [10]. ###### Theorem 2 ([11]). For every $\varepsilon>0$ and $p\geq 3$, MIS for $K_{1,p+1}$-free graphs can be approximated by a polynomial algorithm with approximation ratio $2/p-\varepsilon$. ###### Theorem 3. For every $\varepsilon>0$, max rainbow matching can be approximated by a polynomial algorithm with approximation ratio $2/3-\varepsilon$. ###### Proof. This follows from Lemma 1 and Theorem 2 with $p=3$. ∎ On the other hand, we show that max rainbow matching is APX-complete, and thus rainbow matching is NP-complete, even when restricted to very simple graph classes, and we give some inapproximability bounds for max rainbow matching. We will reduce max rainbow matching on these graph classes to $3$-MIS, and use the following theorem by Berman and Karpinski [2], where the second part of the statement does not appear in the original statement, but follows directly from their proof. ###### Theorem 4 ([2]). For any $\varepsilon\in(0,1/2)$, it is NP-hard to decide whether an instance of $3$-MIS with $284n$ nodes has the maximum size of an independent set above $(140-\varepsilon)n$ or below $(139+\varepsilon)n$. The statement remains true if we restrict ourselves to the class of bridgeless triangle-free $3$-regular graphs. ###### Theorem 5. max rainbow matching is APX-complete, even when restricted to properly edge- colored $2$-regular graphs in which every color is used exactly twice. Unless P=NP, no polynomial algorithm can guarantee an approximation ratio greater than $\frac{139}{140}$. ###### Proof. Let $G$ be a bridgeless triangle-free $3$-regular graph on $284n$ vertices. By a classical theorem of Petersen [19], $G$ contains a perfect matching $M$. Then, $G-M$ is triangle-free and $2$-regular, and thus the line graph of a triangle-free and $2$-regular graph $H$ on $284n$ vertices. Now it is easy to color the edges of $H$ in such a way that every color is used exactly twice, and $G=C\\!L(H)$. From Theorem 4, it follows that it is NP-hard to decide if the maximal size of a rainbow matching in $H$ is above $(140-\varepsilon)n$ or below $(139+\varepsilon)n$. ∎ ###### Corollary 1. max rainbow matching is APX-complete, even when restricted to complete graphs. Unless P=NP, no polynomial algorithm can guarantee an approximation ratio greater than $\frac{139}{140}$. ###### Proof. Use the same graph $H$ from the previous proof, add two new vertices, and add all missing edges, all colored with the same new color, to get an edge-colored complete graph $H^{\prime}$ on $284n+2$ vertices. Then, it is NP-hard to decide if the maximal size of a rainbow matching in $H^{\prime}$ is above $(140-\varepsilon)n+1$ or below $(139+\varepsilon)n+1$. ∎ ###### Theorem 6. max rainbow matching is APX-complete, even when restricted to properly edge- colored paths. Unless P=NP, no polynomial algorithm can guarantee an approximation ratio greater than $\frac{210}{211}$. ###### Proof. Again, start with the $2$-regular graph $H$ from the proof of Theorem 5. For a cycle $C\subseteq H$, create a path $P_{C}$ as follows. Cut the cycle at a vertex $v$ to get a path of the same length as $C$, with the two end vertices corresponding to the original vertex $v$. Now add an extra edge to each of the two ends, and color this edge with the color $v$—a color not used anywhere else in $H$. The maximum rainbow matching in this new graph $H^{\prime}$ is exactly one greater than the maximum rainbow matching in $H$. To see this, take a maximum rainbow matching in $H$, and notice that it can contain at most one edge incident to $v$. Thus, in $H^{\prime}$ we can add one of the two edges colored $v$ to this matching to get a greater rainbow matching. On the other hand, every rainbow matching in $H$ contains at most two edges incident to the two copies of $v$, and at most one of them is not in $H$. Thus, deleting one edge from a rainbow matching in $H^{\prime}$ yields a rainbow matching in $H$. Now repeat this process for every cycle in $H$ to get a linear forest $L$ with $c$ components, say, and $284n+2c$ edges. Similarly to above, it is NP-hard to decide if the maximal size of a rainbow matching in $L$ is above $(140-\varepsilon)n+c$ or below $(139+\varepsilon)n+c$. We now connect all paths in $L$ with $c-1$ extra edges colored with a new color $1$ to one long path, and add a path on $5$ edges colored $1,2,1,2,1$ (where $2$ is a new color) to one end to get a path $P$ on $284n+3c+4$ edges. The size of a maximum rainbow matching in $P$ is exactly $2$ larger than in $L$, so it is NP-hard to decide if the maximal size of a rainbow matching in $P$ is above $(140-\varepsilon)n+c+2$ or below $(139+\varepsilon)n+c+2$. As $H$ does not contain any triangles, we have $c\leq 71n$, and the theorem follows. ∎ ###### Theorem 7. max rainbow matching is APX-complete, even when restricted to properly edge- colored $P_{5}$-free linear forests in which every color is used at most twice. Unless P=NP, no polynomial algorithm can guarantee an approximation ratio greater than $\frac{423}{424}$. ###### Proof. We again start with the $2$-regular graph $H$ from the proof of Theorem 5. Now construct a linear forest $L$ consisting of $284n=\|E(H)\|$ paths of length $3$, where every edge in $H$ corresponds to one path component in $L$. For an edge $vw\in E(H)$ with color $\phi(vw)$, color the three edges of the corresponding path with the colors $v$, $\phi(vw)$ and $w$ in this order. We claim that a maximum rainbow matching in $L$ is exactly $284n$ greater than a maximum rainbow matching in $H$. Note that this claim implies the theorem. To see the claim, consider first a rainbow matching $M$ in $H$. Note that for every vertex $v\in V(H)$, $M$ can contain at most one edge incident to $v$, so in $L$ we can add one of the two edges labeled $v$ to the matching induced by $M$. This can be done for every vertex in $V(H)$, so the largest rainbow matching in $L$ is at least $284n$ larger than $M$. On the other hand, every rainbow matching $M^{\prime}$ in $L$ contains at most two edges either colored $v$ or incident to an edge colored $v$, and at most one of them is colored $v$. Thus, by deleting at most $284n$ edges from $M^{\prime}$ we can create a rainbow matching in $H$. Therefore, it is NP-hard to decide if the maximal size of a rainbow matching in $L$ is above $(140+284-\varepsilon)n$ or below $(139+284+\varepsilon)n$. ∎ ###### Theorem 8. max rainbow matching is APX-complete, even when restricted to properly edge- colored $P_{4}$-free bipartite graphs in which every color is used at most twice. Unless P=NP, no polynomial algorithm can guarantee an approximation ratio greater than $\frac{423}{424}$. ###### Proof. Take the linear forest $L$ from the proof of Theorem 7, add an edge to every $P_{4}$ to make it a $C_{4}$, and color the new edge with the same color as the middle edge of the $P_{4}$. This graph $G$ is $P_{4}$-free, and every rainbow matching in $G$ corresponds to a rainbow matching in $L$ of the same size. ∎ ###### Theorem 9. max rainbow matching is APX-complete, even when restricted to properly edge- colored $P_{6}$-free linear forests in which every color is used at most twice. Unless P=NP, no polynomial algorithm can guarantee an approximation ratio greater than $\frac{1689}{1694}$. ###### Proof. Similarly to above, we start with the $2$-regular graph $H$ from the proof of Theorem 5, and transform it into a linear forest $L$ similarly to the last two proofs. This time, we try to produce paths of length $4$ by splitting the cycles in $H$ only at every other vertex if possible. As $H$ may contain odd cycles, we have to use one path of length only $3$ for every one of the $o$ odd cycles in $H$. Thus, $L$ has exactly $(284n+o)/2$ component paths, and a maximum rainbow matching in $L$ is exactly $(284n+o)/2$ greater than a maximum rainbow matching in $H$. As $H$ is triangle-free, we know that $o\leq 284n/5$, so it is NP-hard to decide if the maximal size of a rainbow matching in $L$ is above $(140+0.7\times 284-\varepsilon)n$ or below $(139+0.7\times 284+\varepsilon)n$. ∎ ###### Theorem 10. max rainbow matching is APX-complete, even when restricted to properly edge- colored $P_{8}$-free trees in which every color is used at most twice. Unless P=NP, no polynomial algorithm can guarantee an approximation ratio greater than $\frac{1689}{1694}$. ###### Proof. Start with an edge-colored linear forest $L$ as in the proof of Theorem 9. Add an extra vertex $v$, and connect it to a central vertex (i.e., a vertex with maximum distance to the ends) in every path in $L$. Further, add one pending edge to $v$. Color the added edges with colors not appearing on $L$. Then the resulting tree $T$ is $P_{8}$-free, and a maximum rainbow matching in $T$ is exactly one edge larger than a maximum rainbow matching in $L$. ∎ All these theorems imply the following complexity result on rainbow matching. ###### Corollary 2. rainbow matching is NP-complete, even when restricted to one of the following classes of edge-colored graphs. 1. (1) Complete graphs. 2. (2) Properly edge-colored paths. 3. (3) Properly edge-colored $P_{5}$-free linear forests in which every color is used at most twice. 4. (4) Properly edge-colored $P_{4}$-free bipartite graphs in which every color is used at most twice. 5. (5) Properly edge-colored $P_{8}$-free trees in which every color is used at most twice. ## 4\. Polynomial-time Solvable Cases In contrast to Theorem 8, saying that max rainbow matching is hard even for $P_{4}$-free bipartite graphs, we have: ###### Theorem 11. In every graph $G$ which does not contain $P_{4}$ as a not necessarily induced subgraph, max rainbow matching is solvable in time $O(m^{3/2})$, where $m$ is the number of edges in $G$. ###### Proof. As $G$ does not contain a $P_{4}$, every component of $G$ is either a star or a triangle. Now construct a bipartite graph $H$ with partite sets being the components of $G$ and the colors used in $G$. The graph $H$ has an edge between a component and a color if in $G$, the color appears in the component. The graph $H$ has at most as many edges as $G$, and rainbow matchings in $G$ correspond to matchings in $H$ of the same size. As we can find a maximum matching in the bipartite graph in time $O(m^{3/2})$ ([9, 16, 21]), the same is true for $G$. ∎ Since in a forest, every $P_{4}$ is an induced subgraph, we have the following positive result complementing Theorem 7: ###### Corollary 3. In every $P_{4}$-free forest $F$, max rainbow matching is solvable in time $O(n^{3/2})$, where $n$ is the number of vertices in $F$. In contrast to Theorem 10, saying that max rainbow matching is hard even for $P_{8}$-free trees, we have: ###### Theorem 12. In every tree $T$ which does not contain $P_{7}$ as a subgraph, max rainbow matching is solvable in time $O(n^{7/2})$, where $n$ is the number of vertices in $T$. ###### Proof. As $T$ is $P_{7}$-free, we can find an edge $xy$ in $T$ such that $G-\\{x,y\\}$ is a forest consisting of stars; all we have to do is to pick the two most central vertices in a longest path in $T$ and note that every vertex of $T$ must have distance at most $2$ to $\\{x,y\\}$. Every matching $M$ can contain at most $2$ edges incident to $\\{x,y\\}$. Once we have decided on these at most two edges (less than $n^{2}$ choices), we are left with the task of finding a rainbow matching in a $P_{4}$-free forest, which can be done in time $O(n^{3/2})$ by Theorem 11. This gives a total time of $O(n^{7/2})$. ∎ The following theorem describes a more general setting of Theorem 12: ###### Theorem 13. In every forest $F$ which does not contain $P_{7}$ as a subgraph, max rainbow matching is solvable in time $O\left(\frac{1}{2^{k}k^{2k}}n^{(4k+3)/2}\right)$, where $n$ is the number of vertices in $F$ and $k$ is the number of components in $F$. ###### Proof. As in proof of Theorem 12, find a central edge $x_{T}y_{T}$ in every component $T\subseteq F$, such that $\bigcup\big{(}T-\\{x_{T},y_{T}\\}\big{)}$ is a forest consisting of stars. Once we have decided on the at most $2k$ edges in a matching incident to the $x_{T}y_{T}$—a total of at most ${n/k\choose 2}^{k}<\frac{n^{2k}}{2^{k}k^{2k}}$ choices—we are left with the task of finding a rainbow matching in a $P_{4}$-free forest, which can be done in $O(n^{3/2})$ by Theorem 11. This gives a total time of $O\left(\frac{1}{2^{k}k^{2k}}n^{(4k+3)/2}\right)$. ∎ ###### Corollary 4. max rainbow matching is solvable in polynomial time for $P_{7}$-free forests with bounded number of components. ## 5\. Fixed Parameter Aspects An approach to deal with NP-hard problems is to fix a parameter when solving the problems. A problem parameterized by $k$ is fixed parameter tractable, fpt for short, if it can be solved in time $f(k)\cdot n^{O(1)}$, or, equivalently, in time $O\left(n^{O(1)}+f(k)\right)$, where $f(k)$ is a computable function, depending only on the parameter $k$. For an introduction to parameterized complexity theory, see for instance [5, 7, 17]. Observe that max rainbow matching is fpt, when parameterized by the size of the problem solution. In case of properly edge-colored inputs, this is a consequence of Lemma 1, and of the fact that MIS is fpt for $(K_{7}-e)$-free graphs ([3]). In the general case, rainbow matchings can be seen as matching (set packing) in certain $3$-uniform hypergraphs, hence max rainbow matching is fpt by a result of Fellows et al. [6]. Recall that max rainbow matching is already hard for $P_{8}$-free trees. In view of Theorem 13, we now consider max rainbow matching for $P_{7}$-free forests, parametrized by $k$, the number of components in the inputs. Formally, we want to address the following parameterized problem: $k$-forest rainbow matching Instance: A $P_{7}$-free forest $F$ with $k$ components containing a $P_{4}$. Parameter: $k$. Output: A maximum rainbow matching in $F$. Theorem 14 below shows that $k$-forest rainbow matching is fpt for $P_{5}$-free forests. ###### Theorem 14. In every forest $F$ which does not contain $P_{5}$ as a subgraph, max rainbow matching is solvable in time $O(n+2^{k}k^{3})$, where $n$ is the number of vertices in $F$ and $k$ is the number of components in $F$ containing a $P_{4}$. ###### Proof. Let $E^{\prime}\subset E(F)$ be the set of edges between vertices of degree greater than $1$, and observe that $\|E^{\prime}\|\leq k$. Then, $F^{\prime}=F-E^{\prime}$ is a forest consisting of at most $2k$ stars. Delete edges in $F^{\prime}$ until each star is rainbow colored and has at most $2k$ edges. Notice that this does not change the size of a maximum rainbow matching. Call this new graph $F^{\prime\prime}$, and observe that $\|E(F^{\prime\prime})\|\leq 4k^{2}$. Now for every rainbow choice of edges in $E^{\prime}$, we can solve max rainbow matching on a subgraph of $F^{\prime\prime}$. There are at most $2^{k}$ rainbow choices in $E^{\prime}$, and solving max rainbow matching on $F^{\prime\prime}$ takes time $O(k^{3})$ as in Theorem 11. Creating $F^{\prime\prime}$ from $F$ takes time $O(n)$, so the result follows. ∎ We do not know if $k$-forest rainbow matching is fpt for $P_{6}$-free forests or $W[1]$-hard. Note that $k$-forest rainbow matching is in XP by Theorem 13. ###### Remark 1. If $k$-forest rainbow matching for $P_{6}$-free forests is in $W[i]$, then so is $k$-forest rainbow matching for $P_{7}$-free forests. ###### Proof. Let $F$ be a $P_{7}$-free forest with $k$ components containing a $P_{4}$. For every tree $T\subseteq F$ containing a $P_{6}$, find the central edge $x_{T}y_{T}$. Once you choose one of the $2^{k}$ possibilities to include these central edges in a rainbow matching, delete the not-chosen edges, and delete the chosen edges together with their neighborhood, the remaining graph contains at most $2k$ components containing a $P_{4}$, and no components containing a $P_{6}$. ∎ ## 6\. Concluding Remarks We have shown that it is NP-hard to approximate max rainbow matching within certain ratio bounds for very restricted graph classes. Implicit in our results is the following dichotomy theorem for forests and trees in terms of forbidding paths: rainbow matching is NP-complete for $P_{5}$-free forests ($P_{8}$-free trees), and is polynomially solvable for $P_{4}$-free forests ($P_{7}$-free trees). We have also proved that $k$-forest rainbow matching is fixed parameter tractable for $P_{5}$-free forests. What can we find out about the parameterized complexity in the only open case of $P_{6}$-free (and equivalently, $P_{7}$-free) forests? Another open problem of independent interest is the computational complexity of recognizing color-line graphs: Given a graph $G$, does there exist an edge- colored graph $H$ such that $G=C\\!L(H)$? Note that it is well-known that line graphs can be recognized in linear time. ## References * [1] Paola Alimonti and Viggo Kann, Some APX-completeness results for cubic graphs, Theoretical Computer Science 237 (2000) 123–134. * [2] Piotr Berman and Marek Karpinski, On Some Tighter Inapproximability Results, ICALP’99, Lecture Notes in Comput. Sci., 1644 (1999), pp. 200–209. * [3] Konrad Dabrowski, Vadim V. Lozin, Haiko Müller, and Dieter Rautenbach, Parameterized algorithms for the independent set problem in hereditary graph classes, J. of Discrete Algorithms 14 (2012) 207–213. * [4] Jennifer Diemunsch, Michael Ferrara, Allan Lo, Casey Moffatt, Florian Pfender, and Paul S. Wenger, Rainbow matchings of size $\delta(G)$ in properly-colored graphs, Electr. J. Combin. 19(2) (2012), #P52. * [5] Rodney G. Downey and Michael R. Fellows, Parameterized Complexity, Springer-Verlag, New York, 1999. * [6] Michael R. Fellows, Christian Knauer, Naomi Nishimura, Prabhakar Ragde, Frances A. Rosamond, Ulrike Stege, Dimitrios M. Thilikos, Sue Whitesides, Faster Fixed-Parameter Tractable Algorithms for Matching and Packing Problems, Algorithmica 52 (2008) 167–176. * [7] Jörg Flum and Martin Grohe, Parameterized Complexity Theory, Springer-Verlag, Berlin Heidelberg, 2006. * [8] Michael R. Garey and David S. Johnson, _Computers and Intractability: An Introduction to the Theory of NP-completeness_ , Freeman, New York, 1979. * [9] John E. Hopcroft and Richard M. Karp, A $n^{5/2}$ algorithm for maximum matching in bipartite graphs, SIAM J. on Comput. 2 (1973) 225–231. * [10] Magnús M. Halldórsson, Approximations of independent sets in graphs. Approximation algorithms for combinatorial optimization (Aalborg, 1998), 1–13, Lecture Notes in Comput. Sci., 1444, Springer, Berlin, 1998. * [11] Cor A. J. Hurkens and Alexander Schrijver, On the size of systems of sets every $t$ of which have an SDR, with an application to the worst-case ratio of heuristics for packing problems, SIAM Journal on Discrete Mathematics 2 (1989) 68–72. * [12] Alon Itai, Michael Rodeh, and Steven L. Tanimoto, Some Matching Problems for Bipartite Graphs, _Journal of the Association for Computing Machinery_ , 25 (1978) 517–525. * [13] Mikio Kano and Xueliang Li, Monochromatic and Heterochromatic Subgraphs in Edge-Colored Graphs – A Survey, _Graphs and Combinatorics_ 24 (2008) 237–263. * [14] Alexandr V. Kostochka and M. Yancey, Large rainbow matchings in edge-coloured graphs, _Combinatorics, Probability and Computing_ 21 (2012) 255–263. * [15] Allan Lo and Ta Sheng Tan, A note on large rainbow matchings in edge-coloured graphs, to appear in Graphs and Combinatorics. DOI: 10.1007/s00373-012-1271-y * [16] Silvio Micali and Vijay V. Vazirani, An $O(\sqrt{V}E)$ Algorithm for Finding Maximum Matching in General Graphs, in: Proc. 21st Annual IEEE Symposium on Foundations of Computer Science (1980) 17–27. * [17] Rolf Niedermeier, Invitation to Fixed-Parameter Algorithms, Oxford University Press, 2006. * [18] Christos H. Papadimitriou and Mihalis Yannakakis, Optimization, approximation, and complexity classes, J. Comput. Syst. Sci. 43 (1991) 425–440. * [19] Julius Petersen, Die Theorie der regulären Graphen, Acta Math. 15 (1891), 193–220. * [20] Herbert J. Ryser, Neuere Probleme der Kombinatorik, Vorträge über Kombinatorik, Mathematisches Forschungsinstitut Oberwolfach, Oberwolfach, 1967, 24–29. * [21] Vijay V. Vazirani, A Theory of Alternating Paths and Blossoms for Proving Correctness of the $O(\sqrt{V}E)$ Maximum Matching Algorithm, Combinatorica 14 (1994) 71–109. * [22] Guanghui Wang, Rainbow matchings in properly edge colored graphs, Electr. J. Combin. 18(1) (2011), #P162.	arxiv-papers	2013-12-27T12:53:47	2024-09-04T02:49:55.999952	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Van Bang Le and Florian Pfender", "submitter": "Van Bang Le", "url": "https://arxiv.org/abs/1312.7253" }
1312.7387	# Bernstein type theorem for entire weighted minimal graphs in $\mathbb{G}^{n}\times\mathbb{R}.$ Doan The Hieu Departement of Mathematics College of Education, Hue University, Hue, Vietnam [email protected] Tran Le Nam Departement of Mathematics Dong Thap University, Dong Thap, Vietnam [email protected] ###### Abstract Based on a calibration argument, we prove a Bernstein type theorem for entire minimal graphs over Gauss space $\mathbb{G}^{n}$ by a simple proof. AMS Subject Classification (2000): Primary 53C25; Secondary 53A10; 49Q05 Keywords: Gauss space, weighted minimal graphs, Bernstein theorem ## 1 Introduction The classical Bernstein theorem asserts that an entire minimal graph over $\mathbb{R}^{2}$ is a plane (see [4], [8], [13]). The theorem has been generalized to dimensions $n\leq 7$ (see [7], [1], [14]) and has been proved to be false in dimensions $n\geq 8$ (see [2]). Entire minimal graphs over $\mathbb{R}^{n},\ n\geq 8,$ that are not hyperplanes, were found by Bombieri, De Giorgi and Guisti in [2]. For arbitrary dimensions, the theorem is settled under some hypotheses on the growth of the minimal graph. The theorem is also studied in product spaces $M^{n}\times\mathbb{R}$ (see [6]), where $M$ is a Riemannian manifold, as well as for self-similar shrinkers. Ecker and Huisken [5] proved a Bernstein type theorem for self-similar shrinkers that are entire graph and have at most polynomial volume growth. Later the condition on volume growth is removed by Wang [15]. Self-similar shrinkers, a simplest solution to mean curvature flow that satisfies the following equation $H=\frac{1}{2}\langle\vec{x},{\bf n}\rangle,$ are just minimal hypersurfaces in $\mathbb{R}^{n+1}$ under the conformally changed metric $g_{ij}=\exp(-\|\overrightarrow{x}\|/2n)\delta_{ij}$ (see [3]) or a special example of weighted minimal hypersurfaces in $\mathbb{R}^{n+1}$ with density $e^{-\frac{\|x\|^{2}}{4}},$ a modified version of Gauss space. Gauss space $\mathbb{G}^{n+1},$ Euclidean space $\mathbb{R}^{n+1}$ with Gaussian probability density $e^{-f}=(2\pi)^{-\frac{n+1}{2}}e^{-\frac{\|x\|^{2}}{2}},$ is a typical example of a manifold with density and very interesting to probabilists. It should be mentioned that the Bernstein type theorem for self-similar shrinkers can be applied for $f$-minimal graphs in $\mathbb{G}^{n+1}=\mathbb{G}^{n}\times\mathbb{G}^{1}.$ Motivated by these works, we study a Bernstein type theorem for entire weighted minimal graphs in the product space $\mathbb{G}^{n}\times\mathbb{R},$ that is $\mathbb{R}^{n+1}$ with mixed Gaussian-Euclidean density. Based on a calibration argument, we prove a volume growth estimate of a weighted minimal graph in the product space $\mathbb{G}^{n}\times\mathbb{R}$ and the theorem follows easily. Our proof is quite simple, elementary and holds for arbitrary dimensions without any additional conditions. The proof is also adapted for the case in which Gauss space is replaced by $\mathbb{R}^{n}$ with a radial density and finite weighted volume. ## 2 Weighted minimal surfaces in $\mathbb{G}^{n}\times\mathbb{R}$ A manifold with density, also called a weighted manifold, is a Riemannian manifold with a positive function $e^{-f}$ used to weight both volume and perimeter area. The weighted mean curvature of hypersufaces on a such manifold is defined as follows $H_{f}=H+\langle\nabla f,{\bf n}\rangle,$ (1) where $H$ is the Euclidean mean curvature and ${\bf n}$ is the normal vector field of the hypersurface. A hypersurface with $H_{f}=0$ is called a weighted minimal hypersurface or an $f$-minimal hypersurface. For more details about manifolds with density, we refer the reader to [10], [11], [12]. Let $S$ be a regular hypersurface in $\mathbb{G}^{n}\times\mathbb{R},$ $M$ be a point on $S$ and $\rho$ denote the projection onto $x_{n+1}$-axis. The following lemma shows a geometric meaning of $\langle\nabla f,{\bf n}\rangle.$ ###### Lemma 1. $\|\langle\nabla{\varphi},{\bf n}\rangle\|=d(\rho(M),T_{M}S).$ ###### Proof. Let ${\bf n}(a_{1},a_{2},\ldots,a_{n+1})$ be a normal vector of $S$ at $M.$ Then an equation of $T_{M}S$ is $\sum_{i=1}^{n+1}a_{i}x_{i}+d=0.$ Therefore, $d(\rho(M),T_{M}S)=\|a_{n+1}x_{n+1}+d\|=\|\langle(x_{1},x_{2},\ldots,x_{n},0),(a_{1},a_{2},\ldots,a_{n},a_{n+1})\rangle\|=\|\langle\nabla{f},{\bf n}\rangle\|.$ ∎ Below are simple examples of constant weighted mean curvatures and weighted minimal surfaces in $\mathbb{G}^{2}\times\mathbb{R}.$ 1. 1. Planes parallel to the $z$-axis have constant weighted mean curvature and planes containing the $z$-axis are weighted minimal. 2. 2. Planes $z=a$ are weighted minimal. 3. 3. Right circular cylinders about $z$-axis have constant weighted mean curvature and the one of radius 1 is weighted minimal. 4. 4. Consider the deformation determined by a family of parametric minimal surfaces given by $\displaystyle X_{\theta}(u,v)$ $\displaystyle=(x(u,v),y(u,v),z(u,v));$ (2) $\displaystyle x(u,v)$ $\displaystyle=\cos\theta\sinh v\sin u+\sin\theta\cosh v\cos u,$ (3) $\displaystyle y(u,v)$ $\displaystyle=-\cos\theta\sinh v\cos u+\sin\theta\cosh v\sin u,$ (4) $\displaystyle z(u,v)$ $\displaystyle=u\cos\theta+v\sin\theta;$ (5) where $-\pi<u\leq\pi,\ -\infty<v<\infty$ and the defomation parameter $-\pi<\theta\leq\pi.$ A direct computation shows that $X_{\theta}$ is minimal with the normal vector field $N=\left(\frac{\cos u}{\cosh v},\frac{\sin u}{\cosh v},-\frac{\sinh u}{\cosh v}\right).$ Since $\nabla f=(x,y,0),$ $\langle\nabla f,N\rangle=\sin\theta,$ $X_{\pi/2}$ is the catenoid while $X_{0}$ is the helicoid, it follows that 1. (a) In $\mathbb{G}^{2}\times\mathbb{R},$ $X_{\theta}$ has constant weighted curvature. 2. (b) The helicoid $X_{0}$ is a ruled weighted minimal surface in $\mathbb{G}^{2}\times\mathbb{R}.$ 3. (c) The catenoid $X_{\pi/2}$ has constant weighted curvature 1 in $\mathbb{G}^{2}\times\mathbb{R}.$ ## 3 A Bernstein type theorem for weighted minimal graphs in $\mathbb{G}^{n}\times\mathbb{R}$ ### 3.1 Minimality of hyperplanes in $\mathbb{G}^{n}\times(\mathbb{R},e^{-h})$ Consider the product space $\mathbb{G}^{n}\times(\mathbb{R},e^{-h})$ with product density $e^{-(f+h)}.$ A point in $\mathbb{G}^{n}\times(\mathbb{R},e^{-h})$ can be written as $({\bf x},x_{n+1}),$ where ${\bf x}=(x_{1},x_{2},\ldots,x_{n})\in\mathbb{R}^{n}.$ An equation of a non-vertical hyperplane in $\mathbb{G}^{n}\times(\mathbb{R},e^{-h})=\mathbb{R}^{n+1}$ is of the form $\Sigma_{i=1}^{n}a_{i}x_{i}+x_{n+1}+c=0,\ \ \ c,a_{1},a_{2},\ldots,a_{n}\in\mathbb{R}.$ A direct computation shows that $\langle\nabla(f+h),{\bf n}\rangle=0$ if and only if $\Sigma_{i=1}^{n}a_{i}x_{i}+h^{\prime}(x_{n+1})=0.$ Thus, 1. 1. the plane is weighted minimal if and only if $h^{\prime}(-c)=0,$ 2. 2. the plane is weighted minimal and non-horizontal if and only if $h(x_{n+1})=x_{n+1}^{2}/2+cx_{n+1}+b,$ where $b\in\mathbb{R}$ is a constant. If $h$ is monotone, any hyperplane is not weighted minimal and a Bernstein type theorem does not exist for this case. Since weighted minimality of a hypersurface in $\mathbb{G}^{n+1}$ is equivalent to that in $\mathbb{G}^{n}\times(\mathbb{R},e^{-h}),$ where $h(x_{n+1})=x_{n+1}^{2}/2+cx_{n+1}+b,$ the Bernstein type theorem proved by Ecker-Huisken and Wang ([5], [15]) for self-similar shrinkers is adapted for weighted minimal graphs in the later case. The following example shows that in $\mathbb{G}^{n}\times(\mathbb{R}_{+},e^{-h}),$ where $\mathbb{R}_{+}=\\{x\in\mathbb{R}:x\geq 0\\}$ and $h(z)=z^{2}-\ln{\sqrt{1+4z}},$ there exist both hyperplanar and non- hyperplanar entire weighted minimal graphs. Consider the graph of the function $z=u(x,y)=x^{2}$ over $\mathbb{G}^{2}$ in $\mathbb{G}^{2}\times(\mathbb{R},e^{-h}).$ A direct computation yields $H_{(f+g)}=\frac{1}{(1+4z)^{3/2}}-\frac{2z+h^{\prime}(z)}{2\sqrt{1+4z}}=0.$ Moreover, the horizontal planes $z=(1+\sqrt{17})/8$ are also weighted minimal. ### 3.2 Entire weighted minimal graph in $\mathbb{G}^{n}\times\mathbb{R}$ This subsection considers the case where $h=\operatorname{const}.$ and we can assume that $h=1.$ The space $\mathbb{G}^{n}\times\mathbb{R}$ is just $\mathbb{R}^{n+1}=\mathbb{R}^{n}\times\mathbb{R}$ endowed with the Euclidean- Gaussian density $e^{-f}=(2\pi)^{-\frac{n}{2}}e^{-\frac{\|{\bf x}\|^{2}}{2}}.$ Denote by: * • $B^{n+1}(p,R)$ the $(n+1)$-ball in $\mathbb{G}^{n}\times\mathbb{R}$ with center $p$ and radius $R,$ * • $B^{n}(p,R)$ the $n$-ball in $\mathbb{G}^{n}$ with center $p$ and radius $R,$ * • $S^{n+1}(p,R)$ the $(n+1)$-sphere in $\mathbb{G}^{n}\times\mathbb{R}$ with center $p$ and radius $R,$ * • $S^{n}(p,R)$ the $n$-sphere in $\mathbb{G}^{n}$ with center $p$ and radius $R,$ * • $S^{n}_{+}(p,R)$ the upper half of $S^{n}(p,R),$ * • $B^{n+1}_{+}(p,R)$ the upper half of $B^{n+1}(p,R),$ Let $\Sigma$ be the weighted minimal graph of a function $u(x_{1},x_{2},\ldots,x_{n})=x_{n+1}$ over $\mathbb{G}^{n}$ and let $p$ be the intersection point of $\Sigma$ and $x_{n+1}$-axis, then we have the following area estimates. ###### Lemma 2. $\operatorname{Vol}_{f}(\Sigma\cap B^{n+1}(p,R))\leq\operatorname{Vol}_{f}(B^{n}(O,R))+ne^{-R^{2}}C_{n}R^{n-1},$ (6) where $C_{n}=\operatorname{Vol}B^{n}(O,1).$ ###### Proof. Let ${\bf n}$ be a unit normal field of $\Sigma$ and consider the smooth extension of ${\bf n}$ by the translation along $x_{n+1}$-axis, also denoted by ${\bf n}.$ Consider the $n$-differential form defined by $w(X_{1},X_{2},\ldots,X_{n})=\det(X_{1},X_{2},\ldots,X_{n},{\bf n}),$ (7) where $X_{i},\ i=1,2,\ldots,n$ are smooth vector fields. It is not hard to see that: 1. 1. $\|w(X_{1},X_{2},\ldots,X_{n})\|\leq 1,$ for every normal vector fields $X_{i},\ i=1,2,\ldots,n$ and the equality holds if and only if $X_{1},X_{2},\ldots,X_{n}$ are tangent to $\Sigma.$ 2. 2. $d(e^{-f}w)=0,$ because $\Sigma$ is weighted minimal. Such a differential form is called a weighted calibration that calibrates $\Sigma.$ A simple proof by using Stokes’ Theorem proves that $\Sigma$ is weighted area-minimizing, i.e. any compact portion of $\Sigma$ has least area among all surfaces in its homology class (see [9]). Now let $\Sigma\cap B^{n+1}(p,R):=\widetilde{\Sigma_{R}}.$ Since $\widetilde{\Sigma_{R}}$ is weighted area-minimizing and $\partial\widetilde{\Sigma_{R}}\subset S^{n}(p,R),$ we have the following estimate $\operatorname{Vol}_{f}(\widetilde{\Sigma_{R}})\leq\frac{1}{2}\operatorname{Vol}_{f}S^{n}(p,R).$ Note that the density does not dependent on the last coordinate, therefore $\frac{1}{2}\operatorname{Vol}_{f}S^{n}(p,R)=\frac{1}{2}\operatorname{Vol}_{f}S^{n}(O,R)=\operatorname{Vol}_{f}S_{+}^{n}(O,R).$ Let $\eta$ be the volume form of $\overline{S_{+}^{n}(O,R)}$ associated with the outward unit normal defined as in (7). Its extension by the translation along $x_{n+1}$-axis in the cylinder $\overline{S^{n}(O,R)}\times\mathbb{R}$ is also denoted by ${\eta}.$ By applying Stokes’ theorem together by choosing suitable orientations for objects, we have $\displaystyle\operatorname{Vol}_{f}S_{+}^{n}(O,R)$ $\displaystyle=\int_{S_{+}^{n}(O,R)}e^{-f}\eta$ $\displaystyle=\int_{\overline{B^{n}}(O,R)}e^{-f}\eta+\int_{\overline{B^{n+1}_{+}(O,R)}}d(e^{-f}\eta)\hskip 71.13188pt(\text{by Stokes' Theorem})$ $\displaystyle\leq\operatorname{Vol}_{f}(B^{n}(O,R))+\int_{\overline{B^{n}(O,R)}\times[0,R]}d(e^{-f}\eta)\hskip 34.14322pt(\|\eta\|\leq 1\ \text{and}\ \ B^{n+1}_{1/2}(O,R)\subset{\cal C})$ $\displaystyle=\operatorname{Vol}_{f}(B^{n}(O,R))+\int_{S^{n}(O,R)\times[0,R]}e^{-f}\eta\hskip 56.9055pt(\text{by Stokes' Theorem})$ $\displaystyle\leq\operatorname{Vol}_{f}(B^{n}(O,R))+e^{-R^{2}}\operatorname{Vol}(S^{n-1}(O,R)\times[0,R])$ $\displaystyle=\operatorname{Vol}_{f}(B^{n}(O,R))+ne^{-R^{2}}C_{n}R^{n-1}.$ ∎ ###### Corollary 3. $\operatorname{Vol}_{f}(\Sigma)\leq 1.$ ###### Proof. Taking the limit of both side of (6) as $R$ goes to infinity. ∎ ###### Theorem 4. The graph $\Sigma$ of a function $u(x_{1},x_{2},\ldots,x_{n})=x_{n+1}$ over $\mathbb{G}^{n}$ is weighted minimal if and only if it is a hyperplane $\\{x_{n+1}=a\\},$ i.e. $u$ is constant. ###### Proof. Of course, a horizontal hyperplane in $\mathbb{G}^{n}\times\mathbb{R}$ is weighted minimal. Now suppose that $\Sigma$ is weighted minimal. Let $dV=dx_{1}\wedge dx_{2}\wedge\ldots\wedge dx_{n}.$ We have $1\geq\operatorname{Vol}_{f}(\Sigma)=\int_{\mathbb{G}^{n}}e^{-f}\sqrt{1+\|\nabla u\|^{2}}dV\geq\int_{\mathbb{G}^{n}}e^{-f}dV=\operatorname{Vol}_{f}(\mathbb{G}^{n})=1.$ The equality holds if and only if $\|\nabla u\|^{2}=0,$ i.e. $u$ is constant. ∎ ## References * [1] F. J. Almgren, Some interior regularity theorems for minimal surfaces and an extension of Bernstein’s Theorem, Ann. of Math. 84 (1966), 277-292. * [2] E. Bombieri, E. De Giorgi, E. Giusti, Minimal cones and the Bernstein problem, Inv. Math. 7 (1969), 243-268. * [3] T. H. Colding, W. P. Minicozzi, II; Generic mean curvature flow I: generic singularities. Ann. of Math. (2) 175 (2012), no. 2, 755-833. * [4] S. N. Bernstein, Sur un théorème de géométrie et son application aux equation aux dérivées partielles du type elliptique, Soobsc. Har’kov. Math. Obsc. 15 (1915) 38-45. * [5] K. Ecker, G. Huisken, Mean curvature evolution of entire graphs. Ann. of Math. (2) 130 (1989), no. 3, 453-471. * [6] J. M. Espinar, H. Rosenberg, Complete constant mean curvature surfaces in homogeneous spaces, Comment. Math. Helv. 86 (2011), no. 3, 659-674. * [7] E. De Giorgi, Una estensione del teorema di Bernstein, Ann. Suola Normale Sup. di Pisa 19 (1965), 79-80. * [8] E. Heinz, $\ddot{U}$ber die L$\ddot{o}$sungen del Minimalf$\ddot{a}$chengleichung, Nach. Akad. Wiss. G$\ddot{\text{o}}$ttingen Math. Phys. K1 II (1952), 51-56. * [9] D. T. Hieu, Some calibrated surfaces in manifolds with density, J. Geom. Phys. 61 (2011), no. 8, 1625-1629. * [10] F. Morgan, Manifolds with density, Notices Amer. Math. Soc., 52 (2005), 853-858. * [11] F. Morgan, Geometric Measure Theory: a Beginner s Guide, Academic Press, fourth edition, 2008. * [12] F. Morgan, Manifolds with density and Perelman’s proof of the Poincaré Conjecture, Amer. Math. Monthly 116 (Feb., 2009), 134-142. * [13] R. Osserman, A survey of minimal surfaces, Courier Dover Publications, 2002. * [14] J. Simons, Minimal Varieties in Riemannian manifolds, Ann. of Math. 88 (1968) 62-105. * [15] L. Wang, A Bernstein type theorem for self-similar shrinkers, Geom. Dedicata 151 (2011), 297-303.	arxiv-papers	2013-12-28T03:49:30	2024-09-04T02:49:56.013350	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Doan The Hieu, Tran Le Nam", "submitter": "Tran Nam Le", "url": "https://arxiv.org/abs/1312.7387" }
1312.7388	# The classification of constant weighted curvature curves in the plane with a log-linear density Doan The Hieu Departement of Mathematics College of Education, Hue University, Hue, Vietnam [email protected] Tran Le Nam Departement of Mathematics Dong Thap University, Dong Thap, Vietnam [email protected] ###### Abstract In this paper, we classify the class of constant weighted curvature curves in the plane with a log-linear density, or in other words, classify all traveling curved fronts with a constant forcing term in $\mathbb{R}^{2}.$ The classification gives some interesting phenomena and consequences including: the family of curves converge to a round point when the weighted curvature of curves (or equivalently the forcing term of traveling curved fronts) goes to infinity, a simple proof for a main result in [13] as well as some well-known facts concerning to the isoperimetric problem in the plane with density $e^{y}.$ AMS Subject Classification (2000): Primary 53C44, 53A04; Secondary 53C21, 35J60 Keywords: Curve flow with a forcing term, traveling curved front, self similar translator, planes with a log-linear density, weighted curvature. ## 1 Introduction Consider the following equation $\operatorname{div}\left(\frac{\nabla u}{\sqrt{1+\|\nabla u\|^{2}}}\right)=\frac{a}{\sqrt{1+\|\nabla u\|^{2}}}+b,$ (1) where $u:U\subset\mathbb{R}^{n}\longrightarrow\mathbb{R},$ $U$ is an open domain and $b$ is a constant. This is the equation for self-similar translators with constant velocity $a$ of a mean curvature flow (MCF) with constant forcing term ${\cal H}=b$ $\frac{\partial u}{\partial t}=\sqrt{1+\|\nabla u\|^{2}}\operatorname{div}\left(\frac{\nabla u}{\sqrt{1+\nabla u\|^{2}}}-{\cal H}\right).$ (2) MCF (2) can be thought of as a generalization of the classical MCF when ${\cal H}=0$ and $a=1$ or as a modified version of MCF with constant forcing term in Minkowski space, a problem studied by Ecker and Huisken [7], followed by Aarons [1], H. Jian et al. [12], [14] and others. Equation (1) appears in the study of traveling curved fronts, originated by the research of vortex motion of Ginzburg-Landau superconduction equation or by the research of interfacial phenomena, and in several other fields in Physics (see [20], [13]). It also appears in the study of constant weighted mean curvature hypersurfaces in $\mathbb{R}^{n+1}$ with a log-linear density. The study of the geometry of manifolds with density has increased in the last seven years after the appearance of Morgan’s paper “Manifolds with density” published in the Notices Amer. Math. Soc. journal in 2005. A manifold with density is a Riemannian manifold with a positive function, called the density, used to weight both volume and hypersurface area. On a manifold with density $f=e^{\varphi},$ the weighted mean curvature of a hypersurface with unit normal $N$ is defined as follows $H_{f}:=H-\frac{1}{n}\frac{d\varphi}{d{N}},$ (3) where $H$ is the (Riemannian) mean curvature of the hypersurface. For more details about manifolds with density and some relative topics we refer the reader to [2]-[6], [8], [9], [15]-[19], [21]. A typical example of manifolds with density is Gauss space $G^{n},$ the Euclidean space $\mathbb{R}^{n}$ with Gaussian probability density $f=(2\pi)^{-\frac{n}{2}}e^{-\frac{r^{2}}{2}},$ that is very interesting to probabilists. Weighted minimal hypersurfaces, i. e. $H_{f}=0,$ in Gauss spaces are known as self-similar shrinkers. This is one of three types of self- similar solutions of MCF (self-similar shrinkers, self-similar translators and self-similar expanders) that are most studied recently. Self-similar translators are just weighted minimal hypersurfaces in $\mathbb{R}^{n+1}$ with a log-linear density $f,$ that is $f=e^{\varphi},$ where $\varphi(x_{1},x_{2},\ldots,x_{n+1})=\sum_{i=1}^{n+1}a_{i}x_{i}+a_{0}$ (see [8]). Indeed, let $\Gamma_{u}\subset\mathbb{R}^{n+1}$ be the graph of the function $u.$ Because $\operatorname{div}\left({\nabla u}/{\sqrt{1+\|\nabla u\|^{2}}}\right)=nH,$ and $a/{\sqrt{1+\|\nabla u\|^{2}}}=a\langle N,e_{n}\rangle,$ where $H$ is the mean curvature of $\Gamma_{u}$ and $N$ is the unit normal of the graph; Equation (1) can be rewritten as $H_{f}=c,$ (4) where $c=b/n$ and $H_{f}:=H-(a/n)\langle N,e_{n+1}\rangle$ is the weighted mean curvature of $\Gamma_{u}$ with density $e^{ax_{n+1}}.$ Therefore, solutions of Equation (1) are constant weighted mean curvature hypersurfaces in $\mathbb{R}^{n+1}$ with a log-linear density $f=e^{ax_{n+1}}.$ Weighted minimal hypersurfaces in $\mathbb{R}^{n+1}$ with the density $f=e^{x_{n+1}}$ (self-similar translators of MCF) are solutions of (4) for $a=1$ and $c=0.$ The classification of all constant weighted curvature hypersurfaces as well as weighted minimal hypersurfaces in $\mathbb{R}^{n+1}$ with a general log-linear density is equivalent to the one with the density $f=e^{x_{n+1}}$ (see Section 2). In other words, we can assume that the velocity $a=1$ in the classification of all self-similar translators of the MCF (2). Self-similar translators of MCF play a crucial role in the study type II singularities of the flow (see [10], [11]). Beside lines, the first example of self-similar translators in the plane, the Grim Reaper, was discovered by Calabi. For higher dimensional cases, only a few specific examples of self- similar translators are known. In terms of weighted minimal surfaces in $\mathbb{R}^{3}$ with density $e^{z},$ some cylindrical type examples have been showed in [8]. In this paper, we solve Equation (4) for parametric curves in the simplest case $n=1$ and therefore give the classification of constant weighted curvature curves in the plane with a log-linear density, or in other words, the classification of all traveling curved fronts with a constant forcing term in $\mathbb{R}^{2}.$ Although the classification is elementary, it gives some interesting phenomena and consequences including: the family of curves converge to a round point when the weighted curvature of curves goes to infinity (see Section 5); a simple proof for a main result in [13] as well as some well-known facts concerning to the isoperimetric problem in the plane with density $e^{y}.$ Acknowledgement. We would like to thank Professor Frank Morgan for his encouragement and for reading the first draft of the paper. The authors are supported in part by the National Foundation for Science and Technology Development, Vietnam (Grant No. 101.01-2011.26). ## 2 Spaces with log-linear densities Consider $\mathbb{R}^{n}$ with a log-linear density $f=e^{\varphi},$ where $\varphi(x_{1},\ldots,x_{n+1})=\sum_{i=1}^{n+1}a_{i}x_{i}+a_{0}.$ In this space, the weighted mean curvature of a hypersurface with the unit normal $N$ is $H_{f}:=H-\frac{1}{n}\langle N,e_{n}\rangle,$ The quantity $\langle N,e_{n+1}\rangle$ is just the projection of $N$ on the $x_{n+1}$-axis. This quantity and the mean curvature are invariant under rigid motions. Therefore, when study any problem concerning only to weighted mean curvature, without loss of generality we can assume that $\varphi=ax_{n+1}.$ Moreover, we can assume $a=1.$ Now consider the plane with density $e^{ay}.$ This is a plane of zero generalized Gauss curvature, $G_{\varphi}=G-\triangle\varphi,$ where $G$ is the classical Gauss curvature. The notion of generalized Gauss curvature was introduced in Corwin et al. [4] to study the generalization of the Gauss- Bonnet formula for surfaces with densities, which is finally found in the most general form for the case of unrelated densities in [5]. Let $\alpha(s)=(x(s),y(s))$ be a parametric curve with arc length parameter $s,$ and let $\beta(s)=(1/a)(x(as),y(as)).$ Denote by $k_{f}(\alpha)$ the weighted curvature of $\alpha$ corresponding to the density $f=e^{y}$ and $k_{\widetilde{f}}(\beta)$ the weighted curvature of $\beta$ corresponding to the density $\widetilde{f}=e^{ay}.$ By a simple computation, it is showed that the weighted curvature of $\alpha$ in $\mathbb{R}^{2}$ with density $f=e^{y}$ is $k_{f}(\alpha)=x^{\prime}y^{\prime\prime}-x^{\prime\prime}y^{\prime}-x^{\prime},$ (5) while the weighted curvature of $\beta$ in $\mathbb{R}^{2}$ with density $\widetilde{f}=e^{ay}$ is $k_{\widetilde{f}}(\beta)=ax^{\prime}y^{\prime\prime}-ax^{\prime\prime}y^{\prime}-ax^{\prime}.$ (6) Therefore, we get ###### Lemma 1. $ak_{f}(\alpha)=k_{\widetilde{f}}(\beta).$ ## 3 Constant weighted curvature curves in the plane with density $e^{y}$ For solving Equation (4) to classify all constant weighted curvature curves in the plane with density $e^{ay},$ by Lemma 1, we can assume that $a=1.$ Moreover, because the weighted curvature of a curve is invariant under translations, some constant of integration will be omitted for simplicity. Equation (4) for a parametric curve $\alpha(s)=(x(s),y(s))$ with arc length parameter is $x^{\prime}y^{\prime\prime}-x^{\prime\prime}y^{\prime}-x^{\prime}=c.$ (7) Set $\displaystyle\begin{cases}x^{\prime}=-\cos(2\xi)\\\ y^{\prime}=\sin(2\xi)\end{cases}.$ (8) Then, the equation (7) reads as follows: $\displaystyle-2\xi^{\prime}+\cos(2\xi)=c.$ (9) If $\cos(2\xi)\neq c$ for all $s,$ then the equation (9) is rewritten as follows $\displaystyle-2\xi^{\prime}=\dfrac{c-1+(c+1)\tan^{2}\xi}{1+\tan^{2}\xi}.$ (10) ### 3.1 The case of $\boldsymbol{c<-1}$ (see Figure 6.1) Equation (10) can be rewritten as $\displaystyle-\dfrac{2d\tan\xi}{\tan^{2}\xi+a^{2}}=(c+1)ds,\;\mbox{where }\,a=\sqrt{\dfrac{c-1}{c+1}}.$ Thus, $-\dfrac{2}{a}\arctan\dfrac{\tan\xi}{a}=(c+1)s$ or $\tan\xi=a\tan\Big{(}\dfrac{\sqrt{c^{2}-1}}{2}s\Big{)},$ (11) because $c+1<0.$ From (11), we have $\begin{cases}x^{\prime}(s)=\dfrac{a^{2}\tan^{2}\Big{(}\dfrac{\sqrt{c^{2}-1}}{2}s\Big{)}-1}{1+a^{2}\tan^{2}\Big{(}\dfrac{\sqrt{c^{2}-1}}{2}s\Big{)}},\\\ y^{\prime}(s)=\dfrac{2a\tan\Big{(}\dfrac{\sqrt{c^{2}-1}}{2}s\Big{)}}{1+a^{2}\tan^{2}\Big{(}\dfrac{\sqrt{c^{2}-1}}{2}s\Big{)}},\\\ \end{cases}$ A direct computation yields $\displaystyle\begin{cases}x^{\prime}(s)=\dfrac{-2}{\sqrt{c^{2}-1}}\dfrac{\Big{(}1-a^{2}\tan^{2}\dfrac{\sqrt{c^{2}-1}}{2}s\Big{)}d\Big{(}\tan\dfrac{\sqrt{c^{2}-1}}{2}s\Big{)}}{\Big{(}1+a^{2}\tan^{2}\dfrac{\sqrt{c^{2}-1}}{2}s\Big{)}\Big{(}1+\tan^{2}\dfrac{\sqrt{c^{2}-1}}{2}s\Big{)}},\\\\[22.76228pt] y^{\prime}(s)=\dfrac{-4}{(c+1)}\dfrac{\tan\Big{(}\dfrac{\sqrt{c^{2}-1}}{2}s\Big{)}d\Big{(}\tan\dfrac{\sqrt{c^{2}-1}}{2}s\Big{)}}{\Big{(}1+a^{2}\tan^{2}\dfrac{\sqrt{c^{2}-1}}{2}s\Big{)}\Big{(}1+\tan^{2}\dfrac{\sqrt{c^{2}-1}}{2}s\Big{)}}.\end{cases}$ (12) Integrating both sides of (12), we obtain $\displaystyle\begin{cases}x(s)=\dfrac{2(a^{2}+1)}{\sqrt{c^{2}-1}(a^{2}-1)}\dfrac{\sqrt{c^{2}-1}}{2}s-\dfrac{4a}{\sqrt{c^{2}-1}(a^{2}-1)}\arctan\left(\sqrt{\dfrac{c-1}{c+1}}\tan\dfrac{\sqrt{c^{2}-1}}{2}s\right),\\\\[11.38092pt] y(s)=\dfrac{2}{(c+1)(a^{2}-1)}\ln\left(\dfrac{\tan^{2}\left(\dfrac{\sqrt{c^{2}-1}}{2}s\right)+1}{\dfrac{c-1}{c+1}\tan^{2}\left(\dfrac{\sqrt{c^{2}-1}}{2}s\right)+1}\right).\end{cases}$ (13) Substitute $a=\sqrt{\dfrac{c-1}{c+1}}$ into (13), we get $\displaystyle\begin{cases}x(s)=-2\arctan\left(\sqrt{\dfrac{c-1}{c+1}}\tan\dfrac{\sqrt{c^{2}-1}}{2}s\right)-cs,\\\\[11.38092pt] y(s)=-\ln\left(\dfrac{\tan^{2}\left(\dfrac{\sqrt{c^{2}-1}}{2}s\right)+1}{\dfrac{c-1}{c+1}\tan^{2}\left(\dfrac{\sqrt{c^{2}-1}}{2}s\right)+1}\right).\end{cases}$ ### 3.2 The case of $\boldsymbol{c=-1}$ (see Figure 6.2) If there exists $s_{0}$ so that $\cos[2\xi(s_{0})]=-1,$ then $\xi(s)=\xi(s_{0})$ is the unique solution of the equation (9) and the corresponding curves are straight lines parallel to the $x$-axis. If $\cos 2\xi(s)\neq-1$ for all $s,$ then the equation (10) becomes $\xi^{\prime}=\dfrac{1}{1+\tan^{2}\xi}.$ Thus, $d(\tan\xi)=ds,$ and therefore, $\tan\xi=s.$ Then, by Equation (8) $\displaystyle\begin{cases}x^{\prime}(s)=-\dfrac{1-s^{2}}{1+s^{2}},\\\\[5.69046pt] y^{\prime}(s)=\dfrac{2s}{1+s^{2}}.\end{cases}$ (14) Integrating both sides (14), we obtain $\displaystyle\begin{cases}x(s)=-2\arctan s+s,\\\ y(s)=\ln(1+s^{2}).\end{cases}$ (15) ### 3.3 The case of $\boldsymbol{-1<c<1}$ (see Figures 6.3, 6.4, 6.5) If there exists $s_{0}$ so that $\cos[2\xi(s_{0})]=c,$ then $\xi(s)=\xi(s_{0})$ is the unique solution of the equation (9) and the corresponding curves are straight lines with the slope $-\frac{\sqrt{1-c^{2}}}{c}.$ In the case of $c=0,$ the slope of the lines is $\infty,$ i.e. the lines are vertical. If this is not to be the case, Equation (10) is equivalent to $\displaystyle-\dfrac{2d(\tan\xi)}{\tan^{2}\xi-b^{2}}=(c+1)ds,\;\mbox{where }\,b=\sqrt{\dfrac{1-c}{c+1}}\neq 0.$ (16) Solving (16) yields $\dfrac{1}{b}\ln\Big{\|}\dfrac{\tan\xi+b}{\tan\xi-b}\Big{\|}=(c+1)s,$ or $\Big{\|}\dfrac{\tan\xi+b}{\tan\xi-b}\Big{\|}={e}^{\sqrt{1-c^{2}}s}.$ We have two cases to consider: Case 1: $\dfrac{\tan\xi+b}{\tan\xi-b}={e}^{\sqrt{1-c^{2}}s},$ or equivalently, $\tan\xi=b\dfrac{{e}^{\sqrt{1-c^{2}}s}+1}{{e}^{\sqrt{1-c^{2}}s}-1}.$ In this case, we have $\begin{cases}x^{\prime}(s)=-\dfrac{1-\Big{(}b\dfrac{{e}^{\sqrt{1-c^{2}}s}+1}{{e}^{\sqrt{1-c^{2}}s}-1}\Big{)}^{2}}{1+\Big{(}b\dfrac{{e}^{\sqrt{1-c^{2}}s}+1}{{e}^{\sqrt{1-c^{2}}s}-1}\Big{)}^{2}},\\\ y^{\prime}(s)=\dfrac{2\Big{(}b\dfrac{{e}^{\sqrt{1-c^{2}}s}+1}{{e}^{\sqrt{1-c^{2}}s}-1}\Big{)}}{1+\Big{(}b\dfrac{{e}^{\sqrt{1-c^{2}}s}+1}{{e}^{\sqrt{1-c^{2}}s}-1}\Big{)}^{2}};\end{cases}$ or $\begin{cases}x^{\prime}(s)=-\dfrac{(1-b^{2}){e}^{2\sqrt{1-c^{2}}s}-2(b^{2}+1){e}^{\sqrt{1-c^{2}}s}+1-b^{2}}{(1+b^{2}){e}^{2\sqrt{1-c^{2}}s}+2(b^{2}-1){e}^{\sqrt{1-c^{2}}s}+b^{2}+1},\\\\[5.69046pt] y^{\prime}(s)=\dfrac{2b\big{(}{e}^{2\sqrt{1-c^{2}}s}-1\big{)}}{(1+b^{2}){e}^{2\sqrt{1-c^{2}}s}+2(b^{2}-1){e}^{\sqrt{1-c^{2}}s}+b^{2}+1}.\end{cases}$ (17) Substitute $b=\sqrt{\dfrac{1-c}{1+c}}$ into (17), we get $\begin{cases}x^{\prime}(s)=-\dfrac{\frac{2c}{c+1}{e}^{2\sqrt{1-c^{2}}s}-2\frac{2}{c+1}{e}^{\sqrt{1-c^{2}}s}+\frac{2c}{c+1}}{\frac{2}{c+1}{e}^{2\sqrt{1-c^{2}}s}+2\frac{-2c}{c+1}{e}^{\sqrt{1-c^{2}}s}+\frac{2}{c+1}},\\\\[5.69046pt] y^{\prime}(s)=\dfrac{2\sqrt{\frac{1-c}{1+c}}\big{(}{e}^{2\sqrt{1-c^{2}}s}-1\big{)}}{\frac{2}{c+1}{e}^{2\sqrt{1-c^{2}}s}+2\frac{-2c}{c+1}{e}^{\sqrt{1-c^{2}}s}+\frac{2}{c+1}};\end{cases}$ or $\displaystyle\begin{cases}x^{\prime}(s)=-\dfrac{c{e}^{2\sqrt{1-c^{2}}s}-2{e}^{\sqrt{1-c^{2}}s}+c}{{e}^{2\sqrt{1-c^{2}}s}-2c{e}^{\sqrt{1-c^{2}}s}+1},\\\\[11.38092pt] y^{\prime}(s)=\dfrac{\sqrt{1-c^{2}}({e}^{\sqrt{1-c^{2}}s}-{e}^{-\sqrt{1-c^{2}}s})}{{e}^{\sqrt{1-c^{2}}s}-2c+{e}^{-\sqrt{1-c^{2}}s}}.\end{cases}$ (18) Integrating both sides of (18), we obtain $\begin{cases}x(t)=2\,\arctan\left({\dfrac{{{e}^{\sqrt{1-{c}^{2}}s}}-c}{\sqrt{1-{c}^{2}}}}\right)-cs,\\\\[11.38092pt] y(s)=\ln\left({{e}^{\sqrt{1-c^{2}}s}+{e}^{-\sqrt{1-c^{2}}s}}-2c\right).\end{cases}$ (19) Case 2: $\dfrac{\tan\xi+b}{\tan\xi-b}={e}^{\sqrt{1-c^{2}}s},$ or equivalently, $\tan\xi=b\dfrac{{e}^{\sqrt{1-c^{2}}s}-1}{{e}^{\sqrt{1-c^{2}}s}+1}.$ After a quite similar computation as in Case 1, we obtain $\begin{cases}x(s)=-2\,\arctan\left({\dfrac{{{e}^{\sqrt{1-{c}^{2}}s}}+c}{\sqrt{1-{c}^{2}}}}\right)-cs,\\\\[8.5359pt] y(s)=\ln\left({{e}^{\sqrt{1-c^{2}}s}+{e}^{-\sqrt{1-c^{2}}s}}+2c\right).\end{cases}$ (20) ###### Remark 2. 1. 1. Set $f_{1}(c,s)=2\arctan\Big{(}{\dfrac{{{e}^{\sqrt{1-{c}^{2}}s}}-c}{\sqrt{1-{c}^{2}}}}\Big{)}-cs$, $f_{2}(c,s)=-2\,\arctan\Big{(}{\dfrac{{{e}^{\sqrt{1-{c}^{2}}s}}+c}{\sqrt{1-{c}^{2}}}}\Big{)}-cs$, $g_{1}(c,s)=\ln\left({{e}^{\sqrt{1-c^{2}}s}+{e}^{-\sqrt{1-c^{2}}s}}-2c\right)$ and $g_{2}(c,s)=\ln\left({{e}^{\sqrt{1-c^{2}}s}+{e}^{-\sqrt{1-c^{2}}s}}+2c\right),$ we can verify that $\displaystyle\begin{cases}f_{1}(c,s)=-f_{2}(-c,s),\\\ g_{1}(c,s)=g(-c,s).\end{cases}$ Therefore the traces of the curves defined by equation (19) and (20) are symmetric to the $x$-axis. 2. 2. When $c=0,$ Equation (19) and (20) are the equations of the Grim Reapers (see Figure 6.4) $\begin{cases}x(s)=\pm 2\arctan(e^{s}),\\\ y(s)=\ln(e^{s}+e^{-s});\end{cases}\ \ \ \ \ s\in\mathbb{R}.$ (21) 3. 3. From Equation (21) we can deduce some simple facts (see also Figure 6.4). 1. (a) Two zero weighted curvature curves intersect in at most one point. Therefore, the geodesic connecting two points if exists is unique. 2. (b) If the difference of the $x$-coordinates of two given points is not less than $\pi,$ then there exists no geodesic connecting these points. 3. (c) If the difference of the $x$-coordinates of two given points is less than $\pi,$ then there exists a unique zero weighted curvature curve connecting these points and this curve is the shortest path which can be proved directly. ### 3.4 The case of $\boldsymbol{c=1}$ (see Figure 6.2) If there exists $s_{0}$ so that $\cos[2\xi(s_{0})]=1,$ then $\xi(s)=\xi(s_{0})$ is the unique solution of the equation (9) and the corresponding curves are straight lines parallel to the $x$-axis. If $\cos 2\xi(s)\neq 1$ for all $s,$ then the equation (10) becomes $-\xi^{\prime}=\dfrac{\tan^{2}\xi}{1+\tan^{2}\xi}=\dfrac{1}{1+\cot^{2}s}.$ Thus, $d(\cot\xi)=ds,$ and therefore, $\cot\xi=s.$ Then, by Equation (8) $\displaystyle\begin{cases}x^{\prime}(s)=\dfrac{1-s^{2}}{1+s^{2}},\\\\[5.69046pt] y^{\prime}(s)=\dfrac{2s}{1+s^{2}}.\end{cases}$ (22) Integrating both sides (22) yields $\displaystyle\begin{cases}x(s)=2\arctan s-s,\\\ y(s)=\ln(1+s^{2}).\end{cases}$ (23) We can see that the curves determined by (15) and (23) have the same traces but have opposite directions. ### 3.5 The case of $\boldsymbol{c>1}$ (see Figure 6.6) With similar arguments as in the case of $c<-1$, we have $\tan\xi=-a\tan\Big{(}\dfrac{\sqrt{c^{2}-1}}{2}s\Big{)}.$ (24) Therefore, $\displaystyle\begin{cases}x^{\prime}(s)=\dfrac{-2}{\sqrt{c^{2}-1}}\dfrac{\Big{(}1-a^{2}\tan^{2}\dfrac{\sqrt{c^{2}-1}}{2}s\Big{)}d\Big{(}\tan\dfrac{\sqrt{c^{2}-1}}{2}s\Big{)}}{\Big{(}1+a^{2}\tan^{2}\dfrac{\sqrt{c^{2}-1}}{2}s\Big{)}\Big{(}1+\tan^{2}\dfrac{\sqrt{c^{2}-1}}{2}s\Big{)}},\\\\[22.76228pt] y^{\prime}(s)=\dfrac{-4}{(c+1)}\dfrac{\tan\Big{(}\dfrac{\sqrt{c^{2}-1}}{2}s\Big{)}d\Big{(}\tan\dfrac{\sqrt{c^{2}-1}}{2}s\Big{)}}{\Big{(}1+a^{2}\tan^{2}\dfrac{\sqrt{c^{2}-1}}{2}s\Big{)}\Big{(}1+\tan^{2}\dfrac{\sqrt{c^{2}-1}}{2}s\Big{)}}.\end{cases}$ (25) Integrating both sides of (25) and substituting $a=\sqrt{\dfrac{c-1}{c+1}}$ into the result, we get $\displaystyle\begin{cases}x(s)=2\arctan\left(\sqrt{\dfrac{c-1}{c+1}}\tan\dfrac{\sqrt{c^{2}-1}}{2}s\right)-cs,\\\\[11.38092pt] y(s)=-\ln\left(\dfrac{\tan^{2}\left(\dfrac{\sqrt{c^{2}-1}}{2}s\right)+1}{\dfrac{c-1}{c+1}\tan^{2}\left(\dfrac{\sqrt{c^{2}-1}}{2}s\right)+1}\right).\end{cases}$ ## 4 Classification of constant weighted curvature curves Combining the above results, up to translations, the classification of constant weighted curvature curves in the plane with density $e^{y}$ is stated as follows. ###### Theorem 3. 1. 1. A curve with weighted curvature zero is either a straight line (parallel to the $y$-axis) or the Grim Reaper defined by (see Figure 6.4) $\begin{cases}x(s)=2\arctan(e^{s}),\\\ y(s)=\ln(e^{s}+e^{-s}).\end{cases}s\in\mathbb{R}.$ (26) 2. 2. A curve with constant weighted curvature $\|k_{\varphi}\|<1$ is either a straight line or the one defined by (see Figure 6.3, Figure 6.5) $\begin{cases}x(s)=2\,\arctan\left({\dfrac{{{e}^{\sqrt{1-{c}^{2}}s}}-c}{\sqrt{1-{c}^{2}}}}\right)-cs,\\\\[11.38092pt] y(s)=\ln\left({{e}^{\sqrt{1-c^{2}}s}+{e}^{-\sqrt{1-c^{2}}s}}-2c\right).\end{cases}s\in\mathbb{R}.$ 3. 3. A curve with constant weighted curvature $\pm 1$ is either a straight line (parallel to the $x$-axis) or the one defined by (see Figure 6.2) $\begin{cases}x(s)=2\arctan s-s,\\\ y(s)=\ln(1+s^{2}).\end{cases}s\in\mathbb{R}.$ 4. 4. A curve with constant weighted curvature $\|k_{\varphi}\|>1$ is defined by (see Figure 6.1, Figure 6.6) $\displaystyle\begin{cases}x(s)=\pm 2\arctan\left(\sqrt{\dfrac{c-1}{c+1}}\tan\dfrac{\sqrt{c^{2}-1}}{2}s\right)-cs,\\\\[11.38092pt] y(s)=-\ln\left(\dfrac{\tan^{2}\left(\dfrac{\sqrt{c^{2}-1}}{2}s\right)+1}{\dfrac{c-1}{c+1}\tan^{2}\left(\dfrac{\sqrt{c^{2}-1}}{2}s\right)+1}\right).\end{cases}s\in\Big{(}-\dfrac{\pi}{\sqrt{c^{2}-1}},\dfrac{\pi}{\sqrt{c^{2}-1}}\Big{)}.$ ## 5 Some consequenses 1. 1. The figures in the next section seem to show that if the weighted curvature $c$ goes to $\pm\infty$ then the limit of the curves is a point. For looking the behaviour of curves near the limit point we can use the standard technique of rescaling the curves by a scale factor $\sqrt{c^{2}-1}.$ We give a proof for the case of $c\rightarrow\infty.$ The proof for $c\rightarrow-\infty$ is quite similar. Suppose that $\alpha(c)$ is a curve with arc length parameter and of constant weighted curvature $c>1.$ Let $\beta(c)=\sqrt{c^{2}-1}\alpha(c).$ The curvature of $\beta$ is $(1/\sqrt{c^{2}-1})k(\alpha(c)),$ where $k(\alpha(c))$ is the curvature of the curve $\alpha(c).$ By Equation (7), $k(\alpha(c))=x^{\prime}+c.$ Therefore $\frac{1}{\sqrt{c^{2}-1}}k(\alpha(c))=\frac{1}{\sqrt{c^{2}-1}}\left(\dfrac{-c\cos(\sqrt{c^{2}-1}s)-1}{c+\cos(\sqrt{c^{2}-1}s)}+c\right).$ It is not hard to check that $\lim_{c\rightarrow\infty}\Big{\|}\frac{1}{\sqrt{c^{2}-1}}k(\alpha(c))\Big{\|}=1.$ Thus, the family of curves converges to a round point when $c$ goes to infinity. 2. 2. The study of traveling fronts of curve flow with external force field for the simple case $\nabla w=(c_{1},c_{2})$ leads to the following equation (see [13]): $c=\frac{\varphi^{\prime\prime}(x)}{1+\varphi(x)^{2}}+c_{2}-c_{1}\varphi^{\prime}(x).$ (27) One of the main results in [13] stated that, the solution of (27) with an initial conditions is either a line or the Grim Reaper. In terms of weighted curvature, solutions of (27) are just zero weighted curvature curves in the plane with density $f=e^{-c_{1}x+(c_{2}-c)y},$ which are also zero weighted curvature curves in the plane with density $f=e^{y}$ under a suitable change of coordinates. In other words, the solutions are self-similar translators in the plane which are known to be either lines or Grim Reapers. 3. 3. Proposition 4.8 in [2], which states “The plane with density $e^{x}$ contains no isoperimetric region”, can be deduced from the classification. It is clear from the classification (and the figures) that isoperimetric curves, i.e. curves bounding isoperimetric regions (in case of existence), must have infinite weighted length or singularities. ## 6 Figures of constant weighted curvature curves Figure 6.1. Curves of $k_{\varphi}<-1$. \| Figure 6.2. Curves of $k_{\varphi}=\pm 1$. ---\|--- $\begin{cases}x(s)=-2\arctan\left(\sqrt{\dfrac{c-1}{c+1}}\tan\dfrac{\sqrt{c^{2}-1}}{2}s\right)-cs,\\\\[11.38092pt] y(s)=-\ln\left(\dfrac{\tan^{2}\left(\dfrac{\sqrt{c^{2}-1}}{2}s\right)+1}{\dfrac{c-1}{c+1}\tan^{2}\left(\dfrac{\sqrt{c^{2}-1}}{2}s\right)+1}\right).\end{cases}$ \| $\begin{cases}x(s)=s-2\arctan s,\\\ y(s)=\ln(1+s^{2}).\end{cases}$ Figure 6.3. Curves of $k_{\varphi}\in(-1,0)$. \| Figure 6.4. Curve of $k_{\varphi}=0$. ---\|--- $\begin{cases}x(s)=2\,\arctan\left({\dfrac{{{e}^{\sqrt{1-{c}^{2}}s}}-c}{\sqrt{1-{c}^{2}}}}\right)-cs,\\\\[11.38092pt] y(s)=\ln\left({{e}^{\sqrt{1-c^{2}}s}+{e}^{-\sqrt{1-c^{2}}s}}-2c\right).\end{cases}$ \| (the Grim Reaper curve) $\begin{cases}x(s)=2\arctan(e^{s}),\\\ y(s)=\ln(e^{s}+e^{-s}).\end{cases}$ Figure 6.5. Curves of $k_{\varphi}\in(0,1)$. \| Figure 6.6. Curves of $k_{\varphi}>1$. ---\|--- $\begin{cases}x(s)=2\,\arctan\left({\dfrac{{{e}^{\sqrt{1-{c}^{2}}s}}-c}{\sqrt{1-{c}^{2}}}}\right)-cs,\\\\[11.38092pt] y(s)=\ln\left({{e}^{\sqrt{1-c^{2}}s}+{e}^{-\sqrt{1-c^{2}}s}}-2c\right).\end{cases}$ \| $\begin{cases}x(s)=2\arctan\left(\sqrt{\dfrac{c-1}{c+1}}\tan\dfrac{\sqrt{c^{2}-1}}{2}s\right)-cs,\\\\[11.38092pt] y(s)=-\ln\left(\dfrac{\tan^{2}\left(\dfrac{\sqrt{c^{2}-1}}{2}s\right)+1}{\dfrac{c-1}{c+1}\tan^{2}\left(\dfrac{\sqrt{c^{2}-1}}{2}s\right)+1}\right).\end{cases}$ ## References * [1] M. A. S. Aarons, Mean curvature flow with a forcing term in Minkowski space, Calc. Var. Partial Differential Equations 25 (2006), no. 2, 205-246. * [2] C. Carroll, A. Jacob, C. Quinn, R. Walters, The isoperimetric problem on planes with density, Bull. Aust. Math. Soc. 78 (2008), no. 2, 177–197. * [3] A. Cañete, M. Miranda and D. Vittone, Some isoperimetric problems in planes with density, J. Geo. Anal.,Vol. 20, No. 2, 243-290. * [4] I. Corwin, N. Hoffman, S. Hurder, V. Sesum, and Y. Xu, Differential geometry of manifolds with density, Rose-Hulman Und. Math. J., 7 (1) (2006). * [5] I. Corwin, F. Morgan, The Gauss-Bonnet formula on surfaces with densities, Involve 4 (2011), no. 2, 199–202. * [6] J. Dahlberg, A. Dubbs, E. Newkirk, H. Tran, Isoperimetric regions in the plane with density $r^{p}$, New York J. Math. 16 (2010) 31-51. * [7] K. Ecker, G. Huisken, Parabolic methods for the construction of spacelike slices of prescribed mean curvature in cosmological spacetimes, Comm. Math. 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Jian, Translating solutions to mean curvature flow with a forcing term in Minkowski space Commun. Pure Appl. Anal. 9 (2010), no. 4, 963-973. * [15] Q. Maurmann, F. Morgan, Isoperimetric comparison theorems for manifolds with density, Calc. Var. PDE 36 (2009), No. 1, 1-5. * [16] F. Morgan, Manifolds with density, Notices Amer. Math. Soc., 52 (2005), 853-858. * [17] F. Morgan, Myers Theorem with density, Kodai Math. J. 29 (2006), 454-460. * [18] F. Morgan, “Geometric Measure Theory: a Beginner s Guide”, Academic Press, fourth edition, 2009. * [19] F. Morgan, Manifolds with density and Perelman’s proof of the Poincaré Conjecture, Amer. Math. Monthly 116 (Feb., 2009), 134-142. * [20] H. Ninomiya, M. Taniguchi, Traveling curved fronts of a mean curvature flow with constant driving force Free boundary problems: theory and applications, I (Chiba, 1999), 206-221, GAKUTO Internat. Ser. Math. Sci. Appl., 13, Gakkotosho, Tokyo, 2000. * [21] C. Rosales, A. Cañete, V. Bayle and F. Morgan, On the isoperimetric problem in Euclidean space with density, Calc. Var. PDE 31 (2008), no. 1, 27-46.	arxiv-papers	2013-12-28T04:02:29	2024-09-04T02:49:56.019707	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Doan The Hieu, Tran Le Nam", "submitter": "Tran Nam Le", "url": "https://arxiv.org/abs/1312.7388" }
1312.7469	# Collaborative Discriminant Locality Preserving Projections With its Application to Face Recognition Sheng Huang Dan Yang Dong Yang Ahmed Elgammal College of Computer Science at Chongqing University, Chonqing, 400044, China School of Software Engineering at Chongqing University Chonqing, 400044, China Department of Computer Science at Rutgers University, Piscataway, NJ, 08854, USA ###### Abstract We present a novel Discriminant Locality Preserving Projections (DLPP) algorithm named _Collaborative Discriminant Locality Preserving Projection_ (CDLPP). In our algorithm, the discriminating power of DLPP are further exploited from two aspects. On the one hand, the global optimum of class scattering is guaranteed via using the between-class scatter matrix to replace the original denominator of DLPP. On the other hand, motivated by collaborative representation, an $L_{2}$-norm constraint is imposed to the projections to discover the collaborations of dimensions in the sample space. We apply our algorithm to face recognition. Three popular face databases, namely AR, ORL and LFW-A, are employed for evaluating the performance of CDLPP. Extensive experimental results demonstrate that CDLPP significantly improves the discriminating power of DLPP and outperforms the state-of-the- arts. ###### keywords: Discriminant Locality Preserving Projections, Face recognition, Dimensionality reduction, Feature extraction, Collaborative representation [cor1]Corresponding author (Dan Yang):[email protected] ## 1 Introduction Subspace learning is a useful technique in computer vision, pattern recognition and machine learning, particularly for solving the dimensionality reduction, feature selection, feature extraction and face recognition tasks. Subspace learning aims to learn a specific subspace of the original sample space, which has some particular desired properties. This topic has been studied for decades and many impressive algorithms have been proposed. The representative subspace learning algorithms include Principle Component Analysis (PCA) pca ; spca ; nspca , Linear Discriminant Analysis (LDA) lda , Non-negative Matrix factorization (NMF) nmf ; gnmf , Independent Component Analysis (ICA) ica , Locality Preserving Projections (LPP) and so on. In face recognition, subspace learning is also known as appearance-based face recognition. For example, PCA is known as Eigenfaces, LDA is known as Fisherfaces and LPP is known as Laplacianfaces. Some recent studies show that the high-dimensional samples may reside on low- dimensional manifolds lle ; isomap and such manifold structures are essential for data clustering and classification lap ; lpp . The manifold-based subspace learning algorithms may start from Locality Preserving Projections (LPP). LPP constructs an adjacency matrix to weight the distance between each pair of sample points for learning a projection which can preserve the local manifold structures of data. The weight between two nearby points is much greater than that between two distant points. So if two points are close in the original space, then they will be close in the learned subspace as well. However, the conventional LPP only takes the manifold information into consideration. Many researchers make efforts to improve LPP from different perspectives. Discriminant Locality Preserving Projections (DLPP) dlpp ; tdlppfd ; 2dlpp is deemed as one of the most successful extensions of LPP. It improves the discriminating power of LPP via simultaneously maximizing the distance between each two nearby classes and minimizing the original LPP objective. Orthogonal Laplacianfaces (OLPP) olpp imposes an orthogonality constraint to LPP to ensure that the learned projections are mutually orthogonal. Parametric Regularized Locality Preserving Projections (PRLPP) rlpp regulates the LPP space in a parametric manner and extracts useful discriminant information from the whole feature space rather than a reduced projection subspace of PCA. Furthermore, this parametric regularization can be also employed to other LPP based methods, such as Parametric Regularized DLPP (PRDLPP), Parametric Regularized Orthogonal LPP (PROLPP). Inspired by the idea of LPP, Qiao et al spp proposed a novel projection named Sparsity Preserving Projections (SPP) for preserving the sparsity of original sample data and applied it to face recognition. Our work is mainly based on DLPP. In this paper, we intend to further improve the discriminating power of DLPP from two different aspects. Similar to LPP, DLPP constructs a Laplacian matrix of classes and then improves the discriminating power of LPP via maximizing such matrix. Since the distance between two nearby classes has a greater weight, maximizing the Laplacian matrix of classes actually is equal to maximizing the distance between the nearby classes. Clearly, this strategy cannot guarantee the global optimal class scattering, since the distant classes may be projected closer with each other in such DLPP space than before. In order to obtain the global optimal classes scattering, we use the between-class scatter matrix to replace the Laplacian matrix of classes, which is the denominator of DLPP objective. Moreover, inspired by the idea of the collaborative representation crc ; rcr , an $L_{2}$-norm constraint is imposed to the projections, since we believe that not all the dimensions of the samples are equally important and collaboration should exist among the dimensions. For example, if we consider the face images as the samples in face recognition, each dimension of samples is corresponding to a specific pixel in the face images. Clearly, the pixels in the face area of images play a more important role than the pixels in the background area and collaboration exists naturally between the adjacent pixels. We name the proposed improved DLPP algorithm Collaborative Discriminant Locality Preserving Projections (CDLPP) and apply it to face recognition. Three popular face databases, namely ORL, AR, and LFW-A, are chosen to validate the effectiveness of the proposed algorithm. Extensive experimental results demonstrate that CDLPP remarkably improves the discriminating power of DLPP and outperforms the state-of-the-art subspace learning algorithms with a distinct advantage. Moreover, we also compare CDLPP with four of the most popular face recognition approaches in the recent days, namely Linear Regression Classification (LRC) lrc , Sparse Representation Classification (SRC) sparse , Collaborative Representation Classification (CRC) crc and Relaxed Collaborative Representation Classification (RCR) rcr , (These four algorithms are not the subspace learning algorithms). Even so, CDLPP still outperforms them in all experiments and CDLPP improves the recognition accuracy of RCR from 75% to 81% on LFW-A database, which is a very recent challenging face verification and face recognition database. There are three main contributions of our work: 1. 1. The between-class scatter matrix is used to replace the original denominator of DLPP for guaranteeing the global optimum of class scattering. 2. 2. According to the fact that collaboration exist among the dimensions of samples, we improve the quality of projections via imposing a collaboration constraint. To the best of our knowledge, our work is the first paper introducing the collaboration of dimensions to the subspace learning. Moreover, this is generalizable to other subspace learning algorithms. 3. 3. A prominent improvement of recognition accuracy of DLPP is obtained by our approach. For example, the gains of CDLPP over DLPP are 12% and 23% on the subset 1 and subset 2 of LFW-A database respectively. The rest of paper is organized as follows: we introduce related works in section 2; section 3 describes the proposed algorithm; experiments are presented in section 4; the conclusion is finally summarized in section 5. ## 2 Related Works ### 2.1 Discriminant Locality Preserving Projections Discriminant Locality Preserving Projections (DLPP) dlpp is one of the most influential LPP algorithms. It improves the discriminating power of LPP via simultaneously minimizing the original Laplacian matrix of LPP and maximizing the Laplacian matrix of classes. Let $l\times{n}$-dimensional matrix $X=[x_{1},...,x_{n}]$, $x_{i}\subset\mathcal{R}^{l}$ be the samples and the vector $C=[1,2,...,p]$ be class labels where $p$ is the number of classes. Matrix $X_{c},c\in{C}$, denotes the samples belonging to class $c$. The $l\times{p}$-dimensional matrix $U=[u_{1},...,u_{i},...,u_{p}]$ denotes the mean matrix where $u_{i}\subset\mathcal{R}^{l},i\in{C}$ is the mean of the samples belonging to class $i$. A $p$-dimensional row vector $M=w^{T}{U}=[m_{1},...,m_{i},...,m_{p}],i\in{C}$ presents the projected mean matrix where $l$-dimensional column vector $w$ is a learned projection. Similarly, the projected sample matrix is denoted as a $n$-dimensional row vector $Y=w^{T}{X}=[y_{1},...,y_{n}]$. DLPP aims to find a set of projections $W=[w_{1},w_{2},...,w_{d}]$ to map the $l$-dimensional original sample space into a $d$-dimensional subspace which can preserve the local geometric structures and scatter classes simultaneously. The $l\times d$-dimensional matrix $W$ denotes projection matrix where $d\ll{l}$. The original objective of Discriminant Locality Preserving Projections (DLPP) is as follows: $min\frac{{\sum\limits_{c\in C}{\sum\limits_{i,j\in c}{{{({y_{i}}-{y_{j}})}^{2}}}}H_{ij}^{c}}}{{\sum\limits_{i,j\in c}{{{({u_{i}}-{u_{j}})}^{2}}}{B_{ij}}}}$ (1) where $H_{ij}^{c}$ and $B_{ij}$ denote the weights of the distance between each two homogenous points and the distance between each two mean points respectively. They are the entries of the respective adjacency weight matrices $H^{c}$ and $B$. These weights are all determined by the distance between two points (either Cosine distance or Euclidean distance) in the original space. It is not difficult to formulate the numerator and denominator of the objective function in Equation 1 into the forms of Laplacian matrices. $\hat{w}=\arg\underset{w}{\min}{\frac{w^{T}XLX^{T}{w}}{w^{T}UQU^{T}{w}}}\rightarrow\hat{w}=\arg\underset{w}{\min}{\frac{w^{T}K_{l}{w}}{w^{T}K_{c}{w}}}$ (2) where matrix $L$ is exactly the Laplacian matrix of LPP and matrix $Q$ is the Laplacian matrix of classes. ### 2.2 Collaborative Representation Classification In recent decade, the sparse representation is very popular and extensive works have emphasized the importance of sparsity for classification spca ; nspca ; sparse ; dsparse ; gsr . However, some researcher argue that the collaboration actually plays a more important role in the classification rather than the sparsity, since the samples of different classes share similarities and some samples from class $j$ may be very helpful to represent the testing sample with label $i$ crc ; rcr . In order to collaboratively represent the query sample using $X$ with low computational burden, the $L_{2}$-norm is used to replace the $L_{1}$-norm in the objective function of sparse representation. Therefore, the objective function of collaborative representation model is denoted as follows $\hat{p}=\arg\underset{p}{\min}\\{\|\|y-X\cdot p\|\|^{2}_{2}+\lambda\|\|p\|\|^{2}_{2}\\}$ (3) where $\lambda$ is the regularization parameter. The role of the regularization term is two-fold. First, it makes the least square solution stable, and second, it introduces a certain amount of _sparsity_ to the solution $\hat{p}$ , yet this sparsity is much weaker than that by $L_{1}$-norm. The solution of this model is $\hat{p}=(X^{T}X+\lambda\cdot I)^{-1}X^{T}y$ (4) After we get the solution, we can apply it to classify as the way of sparse representation. ## 3 Collaborative Discriminant Locality Preserving Projections In the algorithm, we improve discriminating power from two aspects. The first one is to use the between-class scatter matrix to replace the Laplacian matrix of classes, which is the denominator of the objective function of DLPP. The second one is to impose a dimension collaboration constraint to the model. Figure 1: The illustrations of class scattering abilities of Laplacian matrix of classes and between-class scatter matrix on Yale database yale (15 subjects with 11 samples each). For clarity, we just draw the center of each class. (a)The distribution of classes in a subspace learned by maximizing the Laplacian matrix of classes. (b)The distribution of classes in a subspace learned by maximizing Between-Class Scatter Matrix. The core of LPP algorithms is the construction of affinity matrix and the core of the construction of the affinity matrix is the weighting schemes. Several weighting schemes are available for weighting the distance between two samples. The most common used weighting schemes include the dot-product weighting and the heat-kernel weighting gnmf . The weighting schemes are all nonlinear and the assigned weight will drop sharply while the distance is increasing. So, the closer points own the greater weight and this strategy makes only the distances between close points able to effectively affect the subspace learning. In that way, if we maximize the denominator of DLPP, which is the Laplacian matrix of classes, only the closer classes can be scattered and the distant classes may be projected much closer. In fisher discriminant analysis, we know that maximizing the between-class scatter matrix can obtain the global optimal classes scatter lda . In order to intuitively show the class scattering abilities of between-class scatter matrix and Laplacian matrix of classes, we conduct an experiment via maximizing them respectively on Yale database yale . Figure 1 shows the results and each point is a class center. The experimental result demonstrates that using the between-class scatter matrix obtains a better performance. Consequently, the discriminating power of DLPP can be further improved via using the between-class scatter matrix instead of the Laplacian matrix of classes, and the new objective function of DLPP is denoted as follows $\hat{w}=\arg\underset{w}{\min}{\frac{w^{T}K_{l}{w}}{w^{T}S_{b}{w}}}$ (5) where $\displaystyle\begin{aligned} w^{T}K_{l}{w}=w^{T}XLX^{T}w={\sum\limits_{c\in C}{\sum\limits_{i,j\in c}{{{({y_{i}}-{y_{j}})}^{2}}}}H_{ij}^{c}}\end{aligned}$ and $\displaystyle\begin{aligned} w^{T}S_{b}{w}=w^{T}XC_{p}X^{T}w=\sum\limits_{i\in c}{{n_{i}}{{({u_{i}}-\bar{u})}^{2}}}\end{aligned}$ The $\bar{u}$ is the mean of whole samples, $C_{p}$ is the $p\times p$ dimensional centering matrix and other notations in this section have been already defined in section 2. We also test the effectiveness of this modification on all the face databases in experiment section. In order to distinguish from the conventional DLPP, we name this modified DLPP _Class Scattering Locality Preserving Projections_ (CSLPP). As same as DLPP, the model of CSLPP can be solved by eigenvalue decomposition and the best CSLPP projection $w$ is the eigenvector corresponding to the minimum nonzero eigenvalue of $S_{b}^{-1}K_{l}$. As we known, the low-dimensional representation of the sample is achieved by projecting the sample to the learned projection as $y_{i}=w^{T}x_{i}$. From the perspective of numerical computation, $y_{i}$ is the sum of dot product of the projection $w$ and the original sample $x_{i}$, $y_{i}=x_{i}\cdot{w}=\sum^{l}_{j=1}w^{j}x^{j}_{i}$. So, each element of projection, $w^{j}$, is corresponding to each dimension of the sample, $x_{i}^{j}$. A greater value of a specific element of projection, no matter it is positive or negative, has more impact to the $y$. In other words, the dimension of sample corresponding to a greater value element of $w$ should be valued more. Clearly, the role of each dimension of sample is not equally important. For example, if the sample is the face image, and therefore each pixel of image is corresponding to each dimension of sample, the pixels in the face area should play a more important role than the pixels in the background area. Consequently, a good projection $w$ should satisfy the following conditions: the elements of $w$ corresponding to the more important dimensions of sample should own a greater value; the values of elements of $w$ corresponding to the less important dimensions should tend to zero. Moreover, in the subspace learning, the dimensions of sample often highly exceeds the number of samples. So, a good projection $w$ should tend to sparse. Another fact is that the collaboration exists among dimensions in the subspace learning. It can be easily verified in the case of face recognition, for example, the adjacent pixels always collaboratively represent a specific component in the face. The collaboration constraint is imposed to projection $w$ to address above issues, since it can emphasize the importance of the collaborations of dimensions in subspace learning and it is also a relaxed sparsity constraint. According to this optimization, Equation 5 can be further modified as follows $\hat{w}=\arg\underset{w}{\min}{\frac{w^{T}K_{l}{w}}{w^{T}S_{b}{w}}}+\beta{\Arrowvert{w}\Arrowvert_{2}}$ (6) where $\beta>0$ controls the amount of additional collaborations required and we name this new Discriminant Locality Preserving Projections (DLPP) algorithm _Collaborative Discriminant Locality Preserving Projections_ (CDLPP). Its objective function can be further formatted as a purely matrix format as follows 111 This formulation can be also transformed as a collaborative representation formulation style format as follows: $J(w)=\frac{w^{T}K_{l}{w}}{w^{T}S_{b}{w}}+\beta{\Arrowvert{w}\Arrowvert_{2}}=\frac{{\sum\limits_{c\in C}{\sum\limits_{i,j\in c}{{{\|\|{y_{i}}-{y_{j}}\|\|}_{2}^{2}}}}H_{ij}^{c}}}{\sum\limits_{i\in c}{{n_{i}}{{\|\|{u_{i}}-\bar{u}\|\|)}_{2}^{2}}}}+\beta{\Arrowvert{w}\Arrowvert^{2}_{2}}=\frac{{\sum\limits_{c\in C}{\sum\limits_{i,j\in c}{{{\|\|{x_{i}\cdot w}-{x_{j}\cdot w}\|\|}_{2}^{2}}}}H_{ij}^{c}}}{\sum\limits_{i\in c}{{n_{i}}{{\|\|{\bar{x_{i}}\cdot{w}}-{\bar{x}\cdot{w}}\|\|)}_{2}^{2}}}}+\beta{\Arrowvert{w}\Arrowvert^{2}_{2}}$ where $\bar{x_{i}}$ is the mean of the samples belonging class $i$ and $\bar{x}$ is the mean of all samples. To the dimensionality reduction task, the dimension of sample typically exceeds the number of samples. This fact guarantees the dictionary is over-complete, since the projection $w$ plays the role as the coefficient in the collaborative (or sparse) representation while the dimension of samples is the dictionary. $\displaystyle J(w)=\frac{w^{T}K_{l}{w}}{w^{T}S_{b}{w}}+\beta{\Arrowvert{w}\Arrowvert_{2}}=\frac{w^{T}K_{l}{w}}{w^{T}S_{b}{w}}+\beta w^{T}{w}$ (7) Then, the derivative of $w$ can be calculated and let it be equivalent to zero for obtaining the minimum of $J(w)$. $\displaystyle\frac{\delta{(J(w))}}{\delta w}$ $\displaystyle=$ $\displaystyle\frac{2K_{l}w-2S_{b}w{(w^{T}S_{b}w)^{-1}w^{T}K_{l}w}}{w^{T}S_{b}w}+2\beta{w}=0$ (8) Since items $w^{T}K_{l}{w}$ and $w^{T}S_{b}{w}$ can be treated as two unknown scalars and let them be $\alpha$ and $\gamma$ respectively, Equation 8 can be formulated as follows $\displaystyle\frac{2K_{l}w-2{\gamma^{-1}\alpha}S_{b}w}{\gamma}+2\beta{w}=0$ $\displaystyle\Rightarrow\frac{\gamma K_{l}w-{\alpha}S_{b}w+\gamma^{2}\beta{w}}{\gamma^{2}}=0$ $\displaystyle\Rightarrow\qquad\gamma K_{l}w-\alpha S_{b}w+\beta w=0$ (9) $\displaystyle\Rightarrow\qquad K_{l}w+\beta Iw-\lambda S_{b}w=0$ $\displaystyle\Rightarrow\qquad S_{b}^{-1}(K_{l}+\beta I)w=\lambda w$ where matrix ${I}$ is an identity matrix and $\lambda=\frac{\alpha}{\gamma}=\frac{w^{T}K_{l}{w}}{w^{T}S_{b}{w}}$ is a scalar. According to Equation 3, this problem is also an eigenvalue problem and the best CDLPP projection $w$ is the eigenvector corresponding to the minimum nonzero eigenvalue of $S_{b}^{-1}(K_{l}+\beta I)$. We can yield the first $d$ CDLPP projections $W=[w_{1},...,w_{d}]$ for face recognition. Figure 2: Sample face images from (a) the AR database (b) the ORL database and (c) the LFW-A database ## 4 Experiments ### 4.1 Face Databases Three popular face databases including AR AR , ORL ORL and LFW-A lfwa Databases are used to evaluate the recognition performances of the proposed methods. ORL database contains 400 images from 40 subjects ORL . Each subject has ten images acquired at different times. In this database, the subjects’ facial expressions and facial details are varying. And the images are also taken with a tolerance for some tilting and rotation. The size of face image on ORL database is 32$\times$32 pixels. AR database consists of more than 4,000 color images of 126 subjects AR . The database characterizes divergence from ideal conditions by incorporating various facial expressions, luminance alterations, and occlusion modes. Following paper lrc , a subset contains 1680 images with 120 subjects are constructed in our experiment. The size of face image on AR database is 50$\times$40 pixels. LFW-A database is an automatically aligned version lfwa of LFW (Labeled Faces in the Wild) database which is a very recent database. And it aims at studying the problem of the unconstrained face recognition. This database is considered as one of the most challenging database since it contains 13233 images with great variations in terms of lighting, pose, age, and even image quality. We copped these images to 120$\times$120 pixels around their center and resize these images to 64$\times$64 pixels. On this database, we follow the experimental configuration of rcr that uses Local Binary Pattern (LBP) descriptor lbp as the baseline image representation. ### 4.2 Compared Algorithms Nine state-of-the-art face recognition algorithms are used to compare with Class-Scattering Locality Preserving Projections (CSLPP) and Collaborative Discriminant Locality Preserving Projections (CDLPP). Among them, Principal Component analysis (PCA) pca , Linear Discriminant Analysis (LDA) lda , Locality Preserving Projections (LPP) lpp , Discriminant Locality Preserving Projections (DLPP) dlpp and Neighbour Preserving Embedding (NPE) npe are subspace learning algorithms. Relaxed Collaborative Representation (RCR) rcr , Linear Regression Classification (LRC) lrc , Sparse Representation Classification (SRC) sparse , and Collaborative Representation Classification (CRC) crc are not the subspace learning algorithms, but they are four popular face recognition algorithms in recent days. The experiment results of CRC and SRC are directly referenced from the experiment results reported in rcr while the experiment results of LRC and RCR are obtained via running the codes by ourselves. The codes of LDA, PCA, LPP and NPE is downloaded from Prof. Deng Cai’s web page codes . The code of RCR is provided by Dr. Meng Yang and the code of LRC is provided Mr. Peng Ma. Methods \| Recognition Rates $\pm$ Standard Deviation ---\|--- Leave-one-out \| 7-fold \| 3-fold \| 2-fold PCA pca \| 96.96%$\pm$2.71% \| 93.69%$\pm$7.29% \| 89.17%$\pm$5.20% \| 66.73%$\pm$0.08% LDA lda \| 96.31%$\pm$4.60% \| 96.65%$\pm$3.38% \| 93.04%$\pm$3.03% \| 58.57%$\pm$1.52% NPE npe \| 93.75%$\pm$6.83% \| 92.62%$\pm$5.15% \| 90.83%$\pm$5.46% \| 61.61%$\pm$0.25% LPP lpp \| 93.93%$\pm$6.57% \| 92.56%$\pm$4.84% \| 91.25%$\pm$4.58% \| 61.19%$\pm$0.34% DLPP dlpp \| 95.06%$\pm$5.46% \| 94.29%$\pm$3.81% \| 92.92%$\pm$4.17% \| 65.95%$\pm$2.69% CSLPP \| 97.44%$\pm$3.00% \| 97.02%$\pm$2.61% \| 94.93%$\pm$3.09% \| 63.45%$\pm$3.37% CDLPP \| 99.70%$\pm$0.53% \| 99.52%$\pm$0.85% \| 99.31%$\pm$1.20% \| 69.23%$\pm$3.11% RCR rcr \| 99.40%$\pm$0.69% \| 99.11%$\pm$1.03% \| 98.40%$\pm$1.39% \| 76.96%$\pm$1.43% LRC lrc \| 99.58%$\pm$0.63% \| 99.40%$\pm$0.63% \| 98.47%$\pm$1.77% \| 68.75%$\pm$0.43% Table 1: Recognition performance comparison (in percents) using AR database Methods \| Recognition Rates $\pm$ Standard Deviation ---\|--- Leave-one-out \| 5-fold \| 3-fold \| 2-fold PCA pca \| 94.25%$\pm$3.13% \| 91.25%$\pm$3.19% \| 89.72%$\pm$3.19% \| 85.25%$\pm$0.35% LDA lda \| 96.75%$\pm$3.34% \| 96.25%$\pm$1.98% \| 95.83%$\pm$3.00% \| 93.00%$\pm$0.71% NPE npe \| 97.50%$\pm$3.12% \| 94.50%$\pm$1.90% \| 92.22%$\pm$1.73% \| 90.00%$\pm$4.54% LPP lpp \| 98.00%$\pm$2.58% \| 96.75%$\pm$1.43% \| 94.72%$\pm$3.37% \| 90.75%$\pm$3.89% DLPP dlpp \| 98.25%$\pm$2.06% \| 97.25%$\pm$2.05% \| 97.22%$\pm$2.10% \| 93.75%$\pm$3.18% CSLPP \| 99.00%$\pm$1.75% \| 98.00%$\pm$1.43% \| 97.50%$\pm$1.44% \| 94.50%$\pm$1.41% CDLPP \| 99.00%$\pm$1.75% \| 99.00%$\pm$1.63% \| 98.06%$\pm$2.10% \| 95.75%$\pm$1.77% RCR rcr \| 98.25%$\pm$2.06% \| 97.5%$\pm$1.53% \| 95.83%$\pm$1.67% \| 93.75%$\pm$3.89% LRC lrc \| 97.50%$\pm$2.75% \| 97.75%$\pm$2.05% \| 96.11%$\pm$2.92% \| 88.75%$\pm$3.18% Table 2: Recognition performance comparison (in percents) using ORL database LFW-A database \| Top Recognition Rate (Retained Dimensions) ---\|--- subset1 \| subset2 PCA pca \| 35.34%(529) \| 34.25%(1261) LDA lda \| 58.90%(141) \| 67.17%(125) NPE npe \| 61.37%(145) \| 65.83%(191) LPP lpp \| 58.90%(145) \| 65.12%(169) DLPP dlpp \| 55.34%(289) \| 58.84%(127) CSLPP \| 63.56%(145) \| 71.44%(127) CDLPP \| 67.12%(145) \| 81.41%(127) RCR rcr \| 65.75% \| 75.17% LRC lrc \| 48.49% \| 51.76% SRC sparse 1112 \| 53.00% \| 72.20% CRC crc ††footnotemark: \| 54.50% \| 73.00% Table 3: Recognition performance comparison (in percents) using LFW-A database ### 4.3 Face Recognition Following the conventional subspace learning based face recognition framework, Nearest Neighbour (NN) Classifier is used for classification and the distance metric is the Euclidean distance. With regard to the choice of weighting schemes for the LPP algorithms, we follow the experimental configuration of LPP lpp and apply dot-product weighting to construct Laplacian matrices for other LPP algorithms. We use the cross validation scheme to evaluate different algorithms on both AR and ORL databases, since the sample number of each subject is the same on these two databases. The $n$-fold cross validation is defined as follows: the dataset is averagely divided into $n$ parts, $n$-1 parts are used for training and the remainder is used for testing. With regard to LFW-a database, we cannot directly use the cross validation since the subjects of LFW-a database have different sample numbers. Therefore, we follow the experimental way of paper rcr and divide the LFW-a database into two subsets. The first subset (147 subjects, 1100 samples) is constructed by the subjects whose sample numbers are ranged from 5 to 10 and the second subset (127 subjects, 2891 samples) is constructed by the subjects whose sample numbers are all over 11. In the experiments, the first five samples of each subject in the first subset will be used for training and the rest samples will be used for testing. Similarly, the first ten samples of each subject in the second subset will be used for training and the remainders are used for testing. 11footnotetext: These results of experiments are directly reference from work rcr . However, the experimental configurations of LFW-A database in rcr and our works are different. The baseline feature of rcr is the concatenation of four features, including intensity value, low-frequency Fourier feature, Gabor feature and LBP, with a LDA-based discriminative selection while our baseline feature is LBP. According to the performances of LRC and RCR in our work and rcr , we deduce that the really recognition accuracies of SRC and CRC in the first subset of LFW-A database under our experimental configuration might be a little bit higher than the ones reported in rcr while those accuracies in the second subset of LFW-A database under our experimental configuration might be a little bit lower than the ones reported in rcr . From the observations of Table 1, Table 2 and Table 3, CSLPP outperforms all the compared subspace learning algorithms and CDLPP outperforms all the compared face recognition approaches on all databases. For example, CDLPP obtains absolute improvements around 2% and 6% in comparison with the second best face recognition approach in ORL database and the second subset of LFW-A database respectively. The results of face recognition experiments also show that the CDLPP presents a prominent improvement over DLPP. More specifically, the gains of CDLPP over DLPP are around 2% and around 5% on ORL and AR databases respectively. On LFW-A database, which is a more challenging database, CDLPP performs even better. The gains of CDLPP over DLPP are 15% and 23% in the first subset and the second subset respectively. Moreover, CSLPP also defeats DLPP in all experiments and this verifies that the between-class scatter matrix has a better class scattering ability than the Laplacian matrix of classes. Recently, the linear regression based face recognition approaches are very popular and generally considered as a more advanced face recognition approach than the conventional Nearest Neighbour Classifier based Subspace learning approach. However, another interesting point learned from our experimental results is that the proposed subspace learning algorithm, CDLPP, consistently outperforms LRC, CRC, SRC and RCR, which are four recent representative algorithms of linear regression based face recognition approach. For instance, CDLPP obtains 6% , 8%, 9% and 30% more recognition accuracies than RCR, CRC, SRC and LRC respectively in the second subset of LFW-A database. Therefore we believe such phenomenon demonstrates that the conventional subspace learning algorithms may still have the potential to outperform other categories of face recognition algorithms, such linear regression based face recognition approach. Figure 3: (a) the sparseness of the first base of CDLPP, CSLPP and DLPP from top to bottom, (b) the visualizations of the first five bases of CDLPP, CSLPP and DLPP from top to bottom. In order to show the spareness of the bases of CDLPP and the collaborations between elements of CDLPP base, several experiments are conducted on ORL database. We draw the absolute value of first base of CDLPP, CSLPP and DLPP, $\|w\|$ , from top to bottom in Figure 3. Clearly, among the three bases, the base of CDLPP is the most sparse one, which verifies the imposed constraint is a relaxed sparse constraint. Moreover, we also visualize the first five bases of CDLPP, CSLPP and DLPP from top to bottom. The brighter part of visualized bases are the elements of base owns a greater magnitude. Comparing with CSLPP and DLPP, such brighter elements of CDLPP always group together to present a facial component. For example, we can clearly find the brightest part is the hair of human in the first visualized base of CDLPP. Such phenomenon verifies that the fact of the dimensions exist collaboration. ### 4.4 Dimensionality Reduction In this section, some experiments are conducted to the dimensionality reduction abilities of different subspace learning algorithms. According to the experimental results in Figure 4, we find that CDLPP consistently outperforms all the subspace learning algorithms cross all the dimensions with a distinct advantage on all databases and the proposed algorithm, CSLPP, also gets a second top recognition accuracy among the subspace learning algorithms. Moreover, CDLPP is a more robust subspace learning algorithm. The recognition accuracy of CDLPP almost not drop down along with the dimension increasing after it reached the top. Actually, this is a very desirable property, since it can facilitate the determination of the optimal number of the retained bases. (a) (b) (c) (d) Figure 4: The recognition rates of different methods corresponding to the dimensions on (a)ORL database (five trains), (b) the first subset of LFW-A database (five trains), (c) AR database (ten trains) and (d) the second subset of LFW-A database (ten trains). _Our methods are the red and black curves_. ### 4.5 Training Efficiency We examine the training costs of CSLPP and CDLPP, and compare it with LDA, PCA, NPE, LPP and DLPP. The experimental hardware configuration is CPU: 2.2 GHz, RAM: 2G. Table 4 shows the CPU times spent on the training phases by these linear methods using MATLAB. In this experiment, we select five samples of each subject for training. According to the experimental results of Table 4, CSLPP has a similar training time of the LPP and the training time of CDLPP is the twice of the training time of LPP. Moreover, the training time of all methods on ORL database is almost identical. This is because the size of ORL database is too small and the time of program loading accounts for a great proportion. Methods \| Training Efficiency (Seconds) ---\|--- AR \| ORL \| LFW-A set1 \| LFW-A set2 PCA pca \| 2.7768 \| 0.1560 \| 5.3664 \| 21.7621 LDA lda \| 1.8252 \| 0.1404 \| 3.8064 \| 15.3349 NPE npe \| 3.7752 \| 0.2964 \| 8.6425 \| 35.3498 LPP lpp \| 5.5692 \| 0.3432 \| 10.0777 \| 41.3403 DLPP dlpp \| 8.5489 \| 0.4212 \| 14.6017 \| 63.1336 CSLPP \| 5.5068 \| 0.2340 \| 10.7641 \| 45.9267 CDLPP \| 10.1869 \| 0.3588 \| 19.6717 \| 92.2746 Table 4: Training times comparison (in seconds) using different databases. ### 4.6 Parameter Selection of CDLPP $\beta$ is an important parameter to control the amount of additional collaborations. Figure 5 depicts the effect of $\beta$ to the recognition performance of CDLPP. The curves plot the relationship between recognition rate and $\beta$ on ORL, AR and LFW-A databases. According to the observations of Figure 5, we can know that the recognition accuracies are slowly increasing along with adding more collaborations in the beginning. While, after it reaches the top, the accuracies are decreasing dramatically. This phenomenon verifies that the moderate collaborations of dimension can offer a significant contribution to improve the discriminating power of DLPP while the overmuch collaborations can degrade the model (Equation 6) into the minimizing of the $L_{2}$-norm of the projections, which is meaningless. Another phenomenon is that a large database seems can be better benefitted by the collaborations of dimensions. This is very desirable for a particular application, since the amount of samples in the particular application is always very large. According to the results of these experiments, we let $\beta=0.1$ in the experiments using ORL database and the first subset of LFW-A database and let $\beta=1$ in the experiments using AR database and the second subset of LFW-A database. (a) (b) Figure 5: The recognition rates under different $\beta$ on (a) AR and ORL databases, and (b) LFW-A database. ## 5 Conclusion In this paper, we present a novel DLPP algorithm name Collaborative Discriminant Locality Preserving projections (CDLPP) and apply it to face recognition. In this algorithm, we use the between-class scatter matrix to replace the original denominator of DLPP to guarantee the global optimum of classes scattering. Motivated by the idea of collaborative representation, a $L_{2}$-norm constraint is imposed to the projection $w$ as a collaborations constraint for improving the quality of bases. Three popular face databases, including ORL, AR and LFW-A databases, are employed for testing the proposed algorithms. CDLPP outperforms all the compared state-of-the-art face recognition approaches. Our next work may focus on utilizing the collaborations of dimensions to solve the feature selection and image segmentation tasks. ## Acknowledgement This work has been supported by the Fundamental Research Funds for the Central Universities (No. CDJXS11181162 and CDJZR12098801) and the authors would thank Dr. Lin Zhong and Dr. Amr bakry for their useful suggestions. ## References * [1] Matthew Turk and Alex Pentland. Eigenfaces for recognition. Journal of cognitive neuroscience, 3(1):71–86, 1991. * [2] Hui Zou, Trevor Hastie, and Robert Tibshirani. Sparse principal component analysis. Journal of computational and graphical statistics, 15(2):265–286, 2006. * [3] Ron Zass and Amnon Shashua. Nonnegative sparse pca. Advances in Neural Information Processing Systems (NIPS), 19:1561, 2007. * [4] Peter N. Belhumeur, Joao P. Hespanha, and David J. Kriegman. 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The codes of pca, lda, lpp and npe: http://www.cad.zju.edu.cn/home/dengcai/data/dimensionreduction.html.	arxiv-papers	2013-12-28T20:12:17	2024-09-04T02:49:56.031542	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Sheng Huang and Dan Yang and Dong Yang and Ahmed Elgammal", "submitter": "Sheng Huang", "url": "https://arxiv.org/abs/1312.7469" }
1312.7531	# Goeritz Invariant of Torus Links K. AHARA, and S. WATANABE ###### Key words and phrases: Goeritz invariant, torus links, nulity ###### 2000 Mathematics Subject Classification: 57M25 abstrust. We obtain the full list of Goeritz invariants of all torus knots and links. ## 1\. introduction Goeritz invariant $g(L)$ of a link (or a knot) $L$ is one of classical invariants of knots and it was defined by Goeritz [1] in 1930’s. To obtain Goeritz invariant $g(L)$, we calculate integral elementary divisors of a irreducible Goeritz matrix $G_{1}(D)$ determined by a link projection $D$. (Usually we remove $1$’s from the set of integral elementary divisors.) Recently, Ikeda et al.[2] calculate Goeritz invariants of the torus links $T(3,q)$. They show that there are infinite number of torus links with its Goeritz invariant of length more than one. (In fact, $g(T(3,6k))=(0,0)$ and $g(T(3,6k+3))=(2,2)$.) In this article, we calculate Goeritz invariants of all torus links. Goeritz invariant contains more information than the determinant and the nulity of links do, but it may have less information than Alexander ideals. We compare Goeritz invariant and Alexander polynomial. If $V_{L}$ is a Seifert matrix of a link $L$, Alexander polynomial is given by $\Delta_{L}(t)=\mathrm{det}(tV_{L}-\,{}^{t}V_{L})$. Alexander polynomials $\Delta_{T(p,q)}(t)$ for torus links $T(p,q)$ are computed by Murasugi (see Prop. 5.2). It is known that the symmetrization $V_{L}+\,{}^{t}V_{L}$ has the same integral elementary divisors as those of irreducible Goerits matrix $G_{1}$. So if the Goeritz invariants is of length $1$, it coincides to the determinant $\mathrm{det\,}(L)=\|V_{L}+\,{}^{t}V_{L}\|=\pm\Delta_{L}(-1)$. The nulity $n(L)$ is the number of 0’s in the Goeritz invariant. It is also the order of zero at $t=-1$ of $\Delta_{L}(t)$. The coefficient of the top term at $t=-1$ of $\Delta_{L}(t)$ is the product of non-zero entries of the Goeritz invariant. In this context, we know that if the number of non-zero entries of the Goeritz invariant is more than $1$, it has much information than Alexander polynomial. Here we state our result. ###### Theorem 1.1. Let $p,q$ be a pair of integers more than $1$. Let $g(p,q)$ be the Goeritz invariants of the torus link (or knot) $T(p,q)$. Let $r$ be the GCD (greatest common divisor) of $(p,q)$ and $p=p^{\prime}r,q=q^{\prime}r$. (1) If both of $p,q$ are odd, then $g(p,q)=(2,\cdots,2)$ ($(r-1)$-times $2$.) ( If $p,q$ are co-prime, then $g(p,q)=(1)$.) (2) If $p$ is odd and $q$ is even, then $g(p,q)=(p^{\prime},0,\cdots,0)$ ($(r-1)$-times $0$.) (If $p,q$ are co-prime, then $g(p,q)=(p)$. If $p^{\prime}=1$ then $g(p,q)=(0,\cdots,0)$. ) (3) If both of $p,q$ are even, then $g(p,q)=(2p^{\prime}q^{\prime},0,\cdots,0)$ ($(r-2)$-times $0$.) (If $r=2$, then $g(p,q)=(2p^{\prime}q^{\prime})$.) A Goeritz matrix $G(T(p,q))$ has a big size(, around $(pq/2)\times(pq/2)$ ), so we always consider how to downsize the matrix. In the proof of our theorem, we only pursue how to transform matrices well(, indeed we only use elementary transformations of matrices). The authors would like to indicate an observation on a proof of our theorem. In the ’odd $p$’ case, we use a kind of induction. First we set $(m,n)=(p,q)$ with some parameters, and consider induction for $(m,n)$. See for example, Lemma 3.3. There are two ways of descent for induction, one is $2m<n$ case ($(m,n)\mapsto(m,n-m)$) and the other is $2m>n$ case ($(m,n)\mapsto(2m-n,m)$). There are inductive formula for signature of Torus knots by Murasugi, where there are two kinds of similar recursion formula for $2m<n$ and $2m>n$ cases. Such induction reminds us of a certain relation between Goeritz invariant and the signature of torus knots. In the last section, we mention homology groups of the branched double covers of torus links, which are straightforward determined by Goeritz invariants. This paper is organized as follows. In Section 2 we prepare some notations and basic technical lemmas. In Section 3 we consider the case that $p$ is an odd number. In Section 4 we consider the case that $p,q$ are even numbers. When $p$ is even, the Goeritz matrix is much complicated than that of odd case, so we need very tricky way of calculation for even $p$. In Section 5 we remark some trivial corollaries about our result. The authors would thank Prof. Tohru Ikeda and Prof. Jun Murakami for their kind advises. ## 2\. preliminary ### 2.1. Goeritz invariant In this subsection, we recall the definition of Goeritz matrix and Goeritz invariant of a general link. Let $L$ be a link (or a knot) and let $D$ be the diagram of $L$. We give a checkerboard coloring on $D$ and obtain the signatures of all crossings as follows. Let $R_{0},R_{1},\cdots,R_{n}$ be (black-)colored regions of the link projection. Using these regions, we define the Goeritz matrix as follows. For a pair of different numbers $i,j$, let $g_{ij}=g_{ji}$ be the sum of signatures of all crossings where $R_{i}$ and $R_{j}$ intersect. Let $g_{ii}$ be an integer satisfying an equation $\sum_{k=0}^{n}g_{ik}=0$. Let the Goeritz matrix $G(D)$ be $\begin{pmatrix}g_{ij}\end{pmatrix}$, a matrix with elements $g_{ij}$ for $(i,j)$-entry. Let an irreducible Goeritz matrix $G_{1}(D)$ be a minor matrix of $G(D)$ that results from $G(D)$ by removing an arbitrary row and an arbitrary column. By its definition, an irreducible Goeritz matrix is not well-defined according to choice of a removed row and a removed column. We define the Goeritz invariant $g(K)$ by the sequence of integral elementary divisors (other than $1$s) of $G_{1}(D)$. It is known that $g(K)$ does not depend on a link projection nor a checkerboard coloring nor choice of a removed row nor choice of a removed column. See, for example, section 5.3 of [3]. We describe the definition of Goeritz invariant precisely. ###### Definition 2.1. Two integral matrices $M,N$ are called to be $\mathbb{Z}$\- equivalent ($M\sim N$) if we transform $M$ into $N$ by the followings. (a) Interchange two rows (resp. two columns). (b) Multiply a row (resp. a column) by $-1$. (c) Add a row (resp. column) to another one multiplied by an integer. (d) $M=(1)\oplus N$ or $N=(1)\oplus M$. The relations (a) or (b) or (c) are equivalent to the condition that there exists integral general linear (IGL) matrices $U_{1},U_{2}\in GL(\mathbb{Z})$ such that $M=U_{1}NU_{2}$. In the elementary divisor theory, it is known that any integral square matrix has a normalized form $\mathrm{diag}(1,\cdots,1,d_{1},d_{2},\cdots,d_{k})=(1)^{r\oplus}\oplus(d_{1})\oplus(d_{2})\oplus\cdots\oplus(d_{k}),$ and the next proposition follows. ###### Proposition 2.2 (Integral elementary divisors). Any integral square matrix $X$ is $\mathbb{Z}$-equivalent to $(d_{1})\oplus(d_{2})\oplus\cdots\oplus(d_{k})$. Here $d_{i}$ is a non- negative integer ($d_{i}\neq 1$) and there exists an integer $c_{i}$ such that $d_{i}=c_{i}d_{i-1}$ for each $i$. The sequence $(d_{1},d_{2},\cdots,d_{k})$ is uniquely determined. ###### Definition 2.3 (Goeritz invariant). For a given knot (or a link) $L$, we choose one projection $D$ and one checkerboard coloring on $D$. We define Goeritz invariant $g(L)=(d_{1},d_{2},\cdots,d_{k})$ by the unique sequence of the integral elementary divisors of a irreducible Goeritz matrix $G_{1}(D)$. ### 2.2. Notations of matrices In this subsection, we prepare some notations about matrices. Let $q$ be a positive integer greater than $1$. Let $M_{q}(\mathbb{Z})$ be a set of integral matrices of size $q\times q$. Let $E=E_{q}$ be an identity matrix and $W=W_{q}$ be a cyclic permutation matrix given by $W=W_{q}=\begin{pmatrix}0&1&&\\\ &0&\ddots&\\\ &&\ddots&1\\\ 1&&&0\end{pmatrix},$ where these matrices are of size $q\times q$. Remark that $W^{q}=E$ and $\text{det\,}W=(-1)^{q-1}$. Let $X=X_{q}$ be defined by $X_{q}=W_{q}+W_{q}^{-1}$. That is, $X=X_{q}=\begin{pmatrix}0&1&&1\\\ 1&0&\ddots&\\\ &\ddots&\ddots&1\\\ 1&&1&0\end{pmatrix}$ Let $N=N_{q}$ be a nil matrix given by $N=N_{q}=\begin{pmatrix}0&1&&&\\\ &0&\ddots&&\\\ &&\ddots&\ddots&\\\ &&&0&1\\\ &&&&0\end{pmatrix}.$ Remark that $N^{q}=O$. ### 2.3. technical lemmas In this subsection, we introduce $\mathbb{Z}[w,w^{-1}]$-equivalence and we show some technical lemmas. ###### Definition 2.4. Let $M,N$ be matrices with entries of $\mathbb{Z}[w,w^{-1}]$. $M$ and $N$ are called to be $\mathbb{Z}[w,w^{-1}]$-equivalent ($M\overset{\mathbb{Z}[w,w^{-1}]}{\sim}N$) if we transform $M$ into $N$ by the followings. (a) Interchange two rows (resp. two columns). (b) Multiply a row (resp. a column) by a unit in $\mathbb{Z}[w,w^{-1}]$. (c) Add a row (resp. column) to another one multiplied by a constant in $\mathbb{Z}[w,w^{-1}]$. (d) $M=(1)\oplus N$ or $N=(1)\oplus M$. We prepare two technical lemmas on $\mathbb{Z}[w,w^{-1}]$-equivalence for the proof of our theorem. ###### Lemma 2.5. Let $x$ be defined by $x=w+w^{-1}\in\mathbb{Z}[w,w^{-1}]$, and let $k\times k$ matrix $F_{k}(w)$ be $F_{k}(w)=\begin{pmatrix}-x&1&&&&\\\ 1&-x&1&&&\\\ &1&\ddots&\ddots&&\\\ &&\ddots&\ddots&1&\\\ &&&1&-x&1\\\ &&&&1&-x+1\end{pmatrix}\in M_{k}(\mathbb{Z}[w,w^{-1}]).$ (1) $F_{k}(w)\overset{\mathbb{Z}[w,w^{-1}]}{\sim}\begin{pmatrix}\mathrm{det\,}F_{k}(w)\end{pmatrix}$. (2) $\mathrm{det}F_{k}(w)=\displaystyle\sum_{i=-k}^{k}(-w)^{i}$. ###### Proof. Interchanging rows of $F_{k}(w)$, we move the first row to the lowest row. That is, we have $F_{k}(w)\sim\begin{pmatrix}1&-x&1&&&\\\ &1&\ddots&\ddots&&\\\ &&\ddots&\ddots&1&\\\ &&&1&-x&1\\\ &&&&1&-x+1\\\ -x&1&0&\cdots&\cdots&0\end{pmatrix}.$ Remark that the part (from the first row to $(k-1)$-th row) of this matrix is upper triangle. So, using Gaussian elimination, we have the following matrix for a polynomial $f(w)$. $F_{k}(w)\sim\begin{pmatrix}1&&&\\\ &\ddots&&\\\ &&1&\\\ 0&&&f(w)\end{pmatrix}\quad(f(w)\in\mathbb{Z}[w,w^{-1}])$ Remarking that the Gaussian elimination preserve the determinant of matrices and observing the last matrix, we show that the $(k,k)$-entry $f(w)$ satisfies $f(w)=\pm\mathrm{det}F_{k}(w)$, and the statement (1) follows. (2) We show the formula by induction. When $k=1$, the entry is $-x+1=-w^{-1}+1-w$. When $k=2$, the left hand side is $\text{det}\begin{pmatrix}-w-w^{-1}&1\\\ 1&-w-w^{-1}+1\end{pmatrix}=w^{-2}-w^{-1}+1-w+w^{2}$ and the statement holds. We assume that the formula (2) holds for $k=1,2,\cdots,h$. Using Laplace expansion, we have $\displaystyle\mathrm{det}F_{h+1}(w)$ $\displaystyle=-x\ \mathrm{det}F_{h}(w)-\mathrm{det}F_{h-1}(w)$ $\displaystyle=(w+w^{-1})\left(\displaystyle\sum_{i=-h}^{h}(-w)^{i}\right)-\left(\displaystyle\sum_{i=-(h-1)}^{h-1}(-w)^{i}\right)$ $\displaystyle=\displaystyle\sum_{i=-(h+1)}^{h+1}(-w)^{i},$ and the proof completes. ∎ In the same way, it is easy to show the following lemma. ###### Lemma 2.6. Let a $k\times k$ matrix $\tilde{F}_{k}(w)$ be defined by $\tilde{F}_{k}(w)=\begin{pmatrix}-x+1&1&&&&\\\ 1&-x&1&&&\\\ &1&\ddots&\ddots&&\\\ &&\ddots&\ddots&1&\\\ &&&1&-x&1\\\ &&&&1&-x+1\end{pmatrix}\in M_{k}(\mathbb{Z}[w,w^{-1}]).$ (1) $\tilde{F}_{k}(w)\overset{\mathbb{Z}[w,w^{-1}]}{\sim}\begin{pmatrix}\mathrm{det}\tilde{F}_{k}(w)\end{pmatrix}$ (2) $\mathrm{det}\tilde{F}_{k}(w)=\mathrm{det}F_{k}(w)+\mathrm{det}F_{k-1}(w)$. ## 3\. Proof of Theorem for odd $p$ ### 3.1. Goeritz matrices of torus links In this subsection, we get a Goeritz matrix $G(T(p,q))$ of the torus knot (link) for a odd number $p$ ($3\leq p$) and an integer $q$ ($2\leq q$). ###### Proposition 3.1. Suppose that $p$ is odd and $p=2k+1$. ($k=1,2,\cdots$) A Goeritz matrix $G(T(p,q))$ is given by the following. $G(T(p,q))=\begin{pmatrix}-q&{\boldsymbol{1}}&&&&&\\\ {}^{t}\boldsymbol{1}&-X&E&&&&\\\ &E&-X&E&&&\\\ &&E&\ddots&\ddots&&\\\ &&&\ddots&\ddots&E&\\\ &&&&E&-X&E\\\ &&&&&E&-X+E\end{pmatrix}\in M_{kq+1}(\mathbb{Z})$ Here, $X=X_{q}=W_{q}+W_{q}^{-1}$, $E=E_{q}$, ${\boldsymbol{1}}=\begin{pmatrix}1&1&\cdots&1\end{pmatrix}\in M_{1,q}(\mathbb{Z})$ ###### Proof. First, we make an checkerboard coloring on the diagram of a torus link $T(p,q)$ as in the following figure. We determine names of each regions of the diagram as follows. We obtain a Goeritz matrix immediately from this figure. ∎ Next, let $G_{1}(T(p,q))$ be a minor matrix that results from $G(T(p,q))$ by removing the first row and the first column. We start to transform this matrix into smaller one. ###### Lemma 3.2. @ Suppose that $p=2k+1$, $p\leq q$, and $G_{1}(T(p,q))=\begin{pmatrix}-X&E&&&&\\\ E&-X&E&&&\\\ &E&\ddots&\ddots&&\\\ &&\ddots&\ddots&E&\\\ &&&E&-X&E\\\ &&&&E&-X+E\end{pmatrix}\in M_{kq}(\mathbb{Z}),$ then we obtain $G_{1}(T(p,q))\sim E_{q}+A_{p,q}+W_{q}^{p}.$ Here $A_{p,q}$ is a $q\times q$ matrix given by $A_{p,q}=\left(\begin{array}[]{c\|c}&\\\ O&O\\\ &\\\ \overbrace{\hphantom{-1\ 1\ -1\ \cdots\ -1}}^{p}&\overbrace{\hphantom{0\ 0\ \cdots\ 0}}^{q-p}\\\ -1\ 1\ -1\ \cdots\ -1&0\ 0\ \cdots\ 0\end{array}\right)$. ###### Proof. Here we substitute $w=W$ for Lemma 2.5. In the similar way of the proof of Lemma 2.5 (1), we eliminate entries of $G_{1}(T(p,q))$ and obtain $G_{1}(T(p,q))\sim\displaystyle\sum_{i=-k}^{k}(-W_{q})^{i}.$ Next, we calculate $(E_{q}+N_{q})W_{q}^{k}(\displaystyle\sum_{i=-k}^{k}(-W_{q})^{i})$. Because $(E_{q}+N_{q})W_{q}^{k}$ is unimodular, the resulting matrix is $\mathbb{Z}$-equivalent to $G_{1}(T(p,q))$. $\displaystyle G_{1}(T(p,q))$ $\displaystyle\sim(E_{q}+N_{q})W_{q}^{k}\left(\displaystyle\sum_{i=-k}^{k}(-W_{q})^{i}\right)$ $\displaystyle=E_{q}+N_{q}-(W_{q}+N_{q}W_{q})+(W_{q}^{2}+N_{q}W_{q}^{2})+\cdots$ $\displaystyle\qquad+(-1)^{p-1}(W_{q}^{p-1}+N_{q}W_{q}^{p-1})$ $\displaystyle=E_{q}+(N_{q}-W_{q})(E_{q}-W_{q}+W_{q}^{2}-\cdots+W_{q}^{p-1})+W_{q}^{p}$ Here, remark that $N_{q}-W_{q}=\begin{pmatrix}&\text{\LARGE$O$}\\\ -1&\end{pmatrix}$, and we obtain $=E_{q}+A_{p,q}+W_{q}^{p}.$ ∎ ### 3.2. Proof of 1.1(1) In this subsection, we show Theorem 1.1 for odd $p,q$. We suppose $p\leq q$ without loss of generality. First we define an $n\times n$ matrix $W(m,n,e_{1},e_{2},k,h)$ as follows. $W(m,n,e_{1},e_{2},k,h)=$ $\hskip 31.29802pt\overbrace{\hphantom{{e_{2}}{\ddots}{e_{2}}\hskip 20.0pt}}^{m}\hskip 10.0pt\overbrace{\hphantom{{e_{1}}{\ddots}{\ddots}{e_{1}}\hskip 25.0pt}}^{n-m}\\\ =\left(\begin{array}[]{ccc\|cccc}1&&&e_{1}&&&\\\ &\ddots&&&\ddots&&\\\ &&1&&&\ddots&\\\ &&&\ddots&&&e_{1}\\\ \hline\cr e_{2}&&&&1&&\\\ &\ddots&&&&\ddots&\\\ &&e_{2}&&&&1\end{array}\right)\begin{array}[]{l}\left.\vphantom{\begin{array}[]{l}\\\ \\\ \\\ \\\ \end{array}}\right\\}\,n-m\\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\left.\vphantom{\begin{array}[]{l}\\\ \\\ \\\ \end{array}}\right\\}\,m\end{array}$ $+\left(\begin{array}[]{c\|c}&\\\ O&O\\\ &\\\ \overbrace{\hphantom{-k\ k\ -k\ \cdots\ (-1)^{m}k}}^{m}&\overbrace{\hphantom{-h\ h\ -h\ \cdots\ (-1)^{n-m}h}}^{n-m}\\\ -k\ k\ -k\ \cdots\ (-1)^{m}k&-h\ h\ -h\ \cdots\ (-1)^{n-m}h\end{array}\right)$ Because of the assumption $p\leq q$, $E_{q}+A_{p,q}+W_{q}^{p}$ is $W(p,q,1,1,1,0)$. For a matrix $W(m,n,e_{1},e_{2},k,h)$ of size $n\times n$, we consider the following two operations of elementary transformations. (operation 1) When $2m<n$, we add $j$-th row ($j=1,2,\cdots,m$) multiple by $-e_{1}$ to $(m+j)$-th row. (operation 2) When $2m<n$, we add $j$-th row ($j=1,2,\cdots,n-m$) multiple by $-e_{1}$ to $(m+j)$-th row. After the operation 1, there remain diagonal elements in the $j$-th row ($j=1,2,\cdots,m$). After the operation 2, there remain diagonal elements in the $j$-th row ($j=1,2,\cdots,n-m$). Then Lemma 3.3 follows . ###### Lemma 3.3. (1) When $2m<n$, using the operation 1, we show that $W(m,n,e_{1},e_{2},k,h)$ is $\mathbb{Z}$-equivalent to $W(m,n-m,e_{1},-e_{1}e_{2},h-e_{1}k,(-1)^{m}h)$. (2) When $2m>n$, using the operation 2, we show that $W(m,n,e_{1},e_{2},\\\ k,h)$ is $\mathbb{Z}$-equivalent to $W(2m-n,m,-e_{1}e_{2},e_{2},(-1)^{n-m}k,h-e_{1}k)$. From Lemma 3.3, we show the following key proposition. ###### Proposition 3.4. Suppose that both of $p,q$ are odd and that $r$ is the GCD of $(p,q)$. Then $W(p,q,1,1,1,0)$ is $\mathbb{Z}$-equivalent to $W(r,2r,1,-1,-1,0)$ or $W(r,2r,-1,1,0,1)$. ###### Proof. First, set $m=p$ and $n=q$. When $n\neq 2m$, we can apply one of operations 1 and 2. After the operation, the GCD is preserved and the sum $m+n$ decreases. So we can apply operations until $(m,n)=(r,2r)$ within finite steps. On the other hand, transition of combinations of parities of $m,n$ and values of $e_{1},e_{2},k,h$ are limited to the followings. $\displaystyle W(\text{odd},\text{odd},1,1,1,0)\xLongleftrightarrow{\text{(1)}}W(\text{odd},\text{even},1,-1,-1,0)\xLongleftrightarrow$ $\displaystyle W(\text{even},\text{odd},1,-1,1,1)\xLongleftrightarrow{\text{(1)}}W(\text{even},\text{odd},1,1,0,1)\xLongleftrightarrow{\text{(2)}}$ $\displaystyle W(\text{odd},\text{even},-1,1,0,1)\xLongleftrightarrow{\text{(1)}}W(\text{odd},\text{odd},-1,1,1,-1)\xLongleftrightarrow{\text{(2)}}$ $\displaystyle W(\text{odd},\text{odd},1,1,1,0)$ The GCD $r$ is odd number and the cases corresponding to$(m,n)=(r,2r)$ are $W(\text{odd},\text{even},1,-1,-1,0)$ or $W(\text{odd},\text{even},-1,1,0,1)$. This completes the proof. ∎ Using this proposition, we can show Theorem 1.1 (1) immediately. ###### Proposition 3.5. If $p,q$ are odd and $p\leq q$, then $g(p,q)=(2,\cdots,2)$ ($(r-1)$-times $2$.) ( If $p,q$ are co-prime, then $g(p,q)=(1)$.) ###### Proof. If the irreducible Goeritz matrix $G_{1}(T(p,q))$ is $\mathbb{Z}$-equivalent to $W(r,2r,1,-1,-1,0)$, then we obtain the result by the following calculation. Here $A_{r}=A_{r,r}$. $\displaystyle W(r,2r,1,-1,-1,0)$ $\displaystyle=\begin{pmatrix}E_{r}&E_{r}\\\ -E_{r}-A_{r}&E_{r}\end{pmatrix}$ $\displaystyle\sim\begin{pmatrix}E_{r}&O\\\ -E_{r}-A_{r}&2E_{r}+A_{r}\end{pmatrix}$ $\displaystyle\sim 2E_{r}+A_{r}$ $\displaystyle=\begin{cases}\begin{pmatrix}2&&&&\\\ &2&&&\\\ &&\ddots&&\\\ &&&\ddots&\\\ -1&1&\cdots&1&1\end{pmatrix}=(2)^{(r-1)\oplus}&(r>1)\\\ (1)&(r=1)\end{cases}$ In the similar way, we can show it in the case of $W(r,2r,-1,+1,0,1)$. This completes the proof. ∎ ### 3.3. Proof of Theorem 1.1(2) In this subsection we show our theorem for an odd $p$ and an even $q$. ###### Proposition 3.6. Suppose that $p$ is odd and $q$ is even. Let integers $s,t$ be determined by $p=qs+t$. ($0\leq t<q$, $0\leq s$.) Such integers are uniquely determined. (1) The irreducible Goeritz matrix $G_{1}(T(p,q))$ of $T(p,q)$ is $\mathbb{Z}$-equivalent to $W(t,q,1,1,s+1,-s)$. (2) There exists integers $k,h$ satisfying $\|k-h\|=p^{\prime}=p/r$ such that $W(t,q,1,1,s+1,-s)$ is $\mathbb{Z}$-equivalent to $W(r,2r,1,1,k,h)$. ###### Proof. (1) In the halfway of the proof of Lemma 3.2 we have $\displaystyle G_{1}(T(p,q))\sim$ $\displaystyle E_{q}+(N_{q}-W_{q})(E_{q}-W_{q}+W_{q}^{2}-\cdots+W_{q}^{p-1})+W_{q}^{p}$ $\displaystyle=$ $\displaystyle E_{q}+W_{q}^{p}+\begin{pmatrix}&\text{\LARGE$O$}\\\ -1&\end{pmatrix}(E_{q}-W_{q}+W_{q}^{2}-\cdots+W_{q}^{p-1}).$ Because $q$ is even and $(W_{q})^{q}=E$, (1) follows immediately. (2) We consider in the same way in the former half of the proof of Proposition 3.4. First set $m=t,n=q$ and continue applying operations 1 and 2 until $(m,n)=(r,2r)$. The transition of combinations of parities of $m,n$ and values of $e_{1},e_{2},k,h$ is limited as follows. $\displaystyle W(\text{odd},\text{even},1,1,k,h)\xLongrightarrow{\text{(1)}}W(\text{odd},\text{odd},1,-1,h-k,-h$ (1) $\displaystyle W(\text{odd},\text{even},1,1,k,h)\xLongrightarrow{\text{(2)}}W(\text{even},\text{odd},-1,1,-k,h-k)$ (2) $\displaystyle W(\text{even},\text{odd},-1,1,k,h)\xLongrightarrow{\text{(1)}}W(\text{even},\text{odd},-1,1,h+k,h)$ (3) $\displaystyle W(\text{even},\text{odd},-1,1,k,h)\xLongrightarrow{\text{(2)}}W(\text{odd},\text{even},1,1,-k,h+k)$ (4) $\displaystyle W(\text{odd},\text{odd},1,-1,k,h)\xLongrightarrow{\text{(1)}}W(\text{odd},\text{even},1,1,h-k,-h)$ (5) $\displaystyle W(\text{odd},\text{odd},1,-1,k,h)\xLongrightarrow{\text{(2)}}W(\text{odd},\text{odd},1,-1,k,h-k)$ The case corresponding to $(m,n)=(r,2r)$ is only $W(\text{odd},\text{even},1,1,k,h)$. (Here remark that $r$ is odd.) So there exists integers $k,h$ such that $W(t,q,1,1,s+1,-s)\sim W(r,2r,1,1,k,h)$. We have another observation from the above transition list. When $m$ is odd $e_{1}=1$, and when $m$ is even $e_{1}=-1$. A formula $-e_{1}=(-1)^{m}$ follows. Next we calculate the value of $\|k-h\|$. Consider the alternating sum of the $n$-th column of $W(m,n,e_{1},e_{2},k,h)$. In fact we define an integer $\ell(m,n,e_{1},e_{2},k,h)$ by $\|km+(-1)^{m}h(n-m)\|$. Applying the operation 1, we have a map $W(m,n,e_{1},e_{2},k,h)\mapsto W(m,n-m,e_{1},-e_{1}e_{2},h-e_{1}k,(-1)^{m}h)$. $\displaystyle\ell(m,n-m,e_{1},-e_{1}e_{2},h-e_{1}k,(-1)^{m}h)$ $\displaystyle=\|(h-e_{1}k)m+(-1)^{m}((-1)^{m}h)(n-2m)\|$ $\displaystyle=\|-e_{1}km+h(n-m)\|$ $\displaystyle=\|km+(-1)^{m}h(n-m)\|=\ell(m,n,e_{1},e_{2},k,h).$ Therefore we show that $\ell(m,n,e_{1},e_{2},k,h)=\ell(m,n-m,e_{1},-e_{1}e_{2},h-e_{1}k,(-1)^{m}h)$ holds. Applying the operation 2, we have a map $W(m,n,e_{1},e_{2},k,h)\mapsto W(2m-n,m,-e_{1}e_{2},e_{2},(-1)^{n-m}k,h-e_{1}k)$. $\displaystyle\ell(2m-n,m,-e_{1}e_{2},e_{2},(-1)^{n-m}k,h-e_{1}k)$ $\displaystyle=\|(-1)^{n-m}k(2m-n)+(-1)^{2m-n}(h-e_{1}k)(m-(2m-n))\|$ $\displaystyle=\|(-1)^{n}\\{k((-1)^{m}(2m-n)+e_{1}(n-m))+h(n-m)\\}\|$ $\displaystyle=\|(-1)^{n}\\{(-1)^{m}km+h(n-m)\\}\|$ $\displaystyle=\|(-1)^{n+m}(km+(-1)^{m}h(n-m))\|=\|km+(-1)^{m}h(n-m)\|$ And we have $\ell(m,n,e_{1},e_{2},k,h)=\ell(2m-n,m,-e_{1}e_{2},e_{2},(-1)^{n-m}k,h-e_{1}k).$ From the statement (1), we have $\ell(t,q,1,1,s+1,-s)=\ell(r,2r,1,1,k,h)$. Straightforward $\ell(r,2r,1,1,k,h)=\|rh+(-1)^{r}rk\|$ holds. Since $r$ is odd, we have $\ell(r,2r,1,1,k,h)=r\|k-h\|$. On the other hand, $\ell(t,q,1,1,s+1,s)=(s+1)t+(-1)^{t}(-s)(q-t)=t+sq=p$ is followed by $r\|k-h\|=p$ and we now get the result $\|k-h\|=p^{\prime}$. ∎ Now we show (2) of the main theorem. ###### Proposition 3.7 (Theorem 1.1 (2)). Suppose that $p$ is odd and $q$ is even. Then $g(p,q)=(p^{\prime},0,\cdots,0)$ ($(r-1)$-times $0$.) (If $p,q$ are co-prime then $g(p,q)=(p)$, if $p^{\prime}=1$ then $g(p,q)=(0,\cdots,0)$) ł D ###### Proof. From the above Proposition, the irreducible Goeritz matrix $G_{1}(T(p,q))$ is $\mathbb{Z}$-equivalent to $W(r,2r,1,1,k,h)$ and $\|k-h\|=p^{\prime}$. If $A_{r}=A_{r,r}$ then $\displaystyle W(r,2r,+1,+1,h,k)$ $\displaystyle=\begin{pmatrix}E_{r}&E_{r}\\\ E_{r}+hA_{r}&E_{r}+kA_{r}\end{pmatrix}$ $\displaystyle\sim(h-k)A_{r}\sim p^{\prime}\begin{pmatrix}\begin{matrix}&&&&\\\ &&O&&\\\ &&&&\end{matrix}\\\ \begin{matrix}-1&1&-1&\cdots&1&-1\end{matrix}\end{pmatrix}$ Therefore we have $g(p,q)=(p^{\prime},0,\cdots,0)$. Here there are $(r-1)$ times zeros. If $p,q$ are co-prime, $r=1$ and there are no zero and $g(p,q)=(p)$. If $p^{\prime}=1$(, that is, $p=r$) then $g(p,q)=(0,\cdots,0)$ (, $1$ is removed). ∎ ## 4\. Proof of Theorem for even $p$ ### 4.1. Goeritz matrices of torus links $T(p,q)$ for even $p$ In this subsection, we get a Goeritz matrix $G(T(p,q))$ of the torus link for even numbers $p$ and $q$. ###### Proposition 4.1. Assume that $p$ is an even positive integer. A Goeritz matrix $G(T(p,q))$ is given by the following. $G(T(p,q))=\begin{pmatrix}-X_{q}+E_{q}&E_{q}&&&&\\\ E_{q}&-X_{q}&E_{q}&&&\\\ &E_{q}&\ddots&\ddots&&\\\ &&\ddots&\ddots&E_{q}&\\\ &&&E_{q}&-X_{q}&E_{q}\\\ &&&&E_{q}&-X_{q}+E_{q}\end{pmatrix}$ Here there are $\frac{p}{2}$ blocks in rows and columns. See the following figure. In this case, we first obtain the integral elementary divisors of $G(T(p,q))$. The matrix $G(T(p,q))$ is sigular(, because the sum of elements in every row or in every column is zero), so the set of the integral elementary divisors contains at least one $0$. Using this property we can calculate the set of integral elementary divisors of $G_{1}(T(p,q))$ from that of $G(T(p,q))$. ###### Proposition 4.2. Suppose that the integral elementary divisors of $G(T(p,q))$ are $d_{1},d_{2},\cdots,d_{r},0$. Here $d_{i}$ is a non-negative integer ($d_{i}\neq 1$) and there exists an ingeter $c_{i}$ such that $d_{i}=c_{i}d_{i-1}$. Then the integral elementary divisors of $G_{1}(T(p,q))$are $d_{1},d_{2},\cdots,d_{r}$ and hence the Goeritz invariant $g(p,q)$ is $(d_{1},d_{2},\cdots,d_{r})$. ###### Proof. First let $G$ be $G(T(p,q))$ for simplisity, and let $\ell$ be $\frac{pq}{2}$, the size of $G$. From the assumption of the proposition, there exist IGL (integral general linear) matrices $U,U^{\prime}$ such that () $UGU^{\prime}=\mathrm{diag}(1,\cdots,1,d_{1},d_{2},\cdots,d_{r},0).$ On the other hand, there exist $P,Q,S,P^{\prime},Q^{\prime},S^{\prime}$ such that $U=PQS$, $U^{\prime}=S^{\prime}P^{\prime}Q^{\prime}$, where $P,P^{\prime}$ are lower triangle IGL matrices, $Q,Q^{\prime}$ are upper triangle IGL matrices, and $S,S^{\prime}$ are permutation matrices. (LU decomposition) Let an $\ell\times\ell$ matrix $T$ be as follows. $T=\begin{pmatrix}1&&&\\\ &1&&\\\ &&\ddots&\\\ -1&\cdots&-1&1\end{pmatrix}$ By definition of a Goeritz matrix $G=\begin{pmatrix}g_{ij}\end{pmatrix}$, we have $\sum_{k=0}^{\ell-1}g_{ik}=\sum_{k=0}^{\ell-1}g_{ki}=0$ for any $i$. Hence all entries of $\ell$-th row and those of $\ell$-th column of $T(S\,G\,S^{\prime})\,{}^{t}T$ are zero. So there exists a matrix $G^{\prime}$ such that (*) $T(S\,G\,S^{\prime})\,{}^{t}T=G^{\prime}\oplus(0).$ Here $G^{\prime}$ is a matrix that results from $G$ by changing rows and columns and removing one row and one column. Therefore $G^{\prime}$ is $\mathbb{Z}$-equivalent to a irreducible Goeritz matrix $G_{1}(T(p,q))$. ($G_{1}(T(p,q))\sim G^{\prime}$.) Substituting the formula () to (*), we obtain the following. $(PQT^{-1})(G^{\prime}\oplus(0))(\,^{t}T^{-1}P^{\prime}Q^{\prime})=\mathrm{diag}(1,\cdots,1,d_{1},d_{2},\cdots,d_{r},0).$ Next, we set $PQT^{-1}={\hat{P}}{\hat{Q}}$, and $\,{}^{t}T^{-1}P^{\prime}Q^{\prime}={\hat{P}^{\prime}}{\hat{Q}^{\prime}}$, where ${\hat{P}},{\hat{P}^{\prime}}$ are lower triangle IGL matrices, and ${\hat{Q}},{\hat{Q}^{\prime}}$ are upper triangle IGL matrices. There exists a matrix $G^{\prime\prime}$ such that ${\hat{Q}}(G^{\prime}\oplus(0)){\hat{P}^{\prime}}=G^{\prime\prime}\oplus(0),$ and clearly $G^{\prime}\sim G^{\prime\prime}$ holds(, because the $(\ell,\ell)$-minor of ${\hat{Q}},{\hat{P}^{\prime}}$ are also IGL matrices). And, ${\hat{P}}(G^{\prime\prime}\oplus(0)){\hat{Q}^{\prime}}=\mathrm{diag}(1,\cdots,1,d_{1},d_{2},\cdots,d_{r},0)$ is followed by $G^{\prime\prime}\sim\mathrm{diag}(1,\cdots,1,d_{1},d_{2},\cdots,d_{r})$. (Because the $(\ell,\ell)$-minor of ${\hat{P}},{\hat{Q}^{\prime}}$ are also IGL matrices.) We have a conclusion that the integral elementary divisors of $G_{1}(T(p,q))$ are $d_{1},d_{2},\cdots,d_{r}$ and complete the proof of Proposition 4.2. ∎ In the sequel, we assume that both $p$ and $q$ are even numbers and that $p\leq q$. Let $r$ be the GCD of $(p,q)$ and let $p^{\prime},q^{\prime}$ be integers such that $p=p^{\prime}r,q=q^{\prime}r$. Remark that $r$ is an even number. The size of $G(T(p,q))$ in Proposition 4.1 is $\frac{pq}{2}\times\frac{pq}{2}$. We will downsize this matrix by $\mathbb{Z}$-equivalence with the following three steps. (step 1) Down to $q\times q$ matrix, (step 2) down to $p\times p$ matrix, and (step 3) down to $2r\times 2r$ matrix. ### 4.2. Down to $q\times q$ matrix In this subsection, we downsize $G$ (in Proposition 4.1) into a $q\times q$ matrix by $\mathbb{Z}$-equivalence. Using a similar way of proof of 2.6, we can show that $G(T(p,q))$ is $\mathbb{Z}$-equivalent to ${\tilde{F}}_{k}(W_{q})$. Let $G_{2}$ be ${\tilde{F}}_{k}(W_{q})$. $\displaystyle G(T(p,q))\sim{\tilde{F}}_{k}(W_{q})=F_{k}(W_{q})+F_{k-1}(W_{q})$ $\displaystyle=\displaystyle\sum_{i=-k}^{k}(-W_{q})^{i}+\displaystyle\sum_{i=-(k-1)}^{k-1}(-W_{q})^{i}$ $\displaystyle=(-W_{q})^{-k}+2\displaystyle\sum_{i=-(k-1)}^{k-1}(-W_{q})^{i}+(-W_{q})^{k}$ $\displaystyle=(E_{q}-2W_{q}+W_{q}^{2})(E_{q}+W_{q}^{2}+W_{q}^{4}+\cdots+W_{q}^{p-2})(-W_{q})^{-k}$ $\displaystyle($ $\displaystyle=:G_{2})$ $G_{2}$ is already $q\times q$ matrix, but we furthermore transform $G_{2}$ and regulate the form. Let $G_{3}$ be $(E_{q}+N_{q})^{2}G_{2}(-W_{q})^{k}$. We calculate this matrix and obtain the follows. $\displaystyle G_{3}=(E_{q}+N_{q})^{2}G_{2}(-W_{q})^{p}$ $\displaystyle=\begin{array}[]{@{}l@{}}\hskip 17.07164pt\overbrace{\hphantom{{E_{2}}{-E_{2}}{\ddots}{B_{2}-E_{2}}\hskip 40.0pt}}^{p\ (k\text{ blocks})}\hskip 10.0pt\overbrace{\hphantom{{-E_{2}}{-E_{2}}{-E_{2}}{\ddots}{-E_{2}}\hskip 30.0pt}}^{q-p}\\\ \left(\begin{array}[]{cccc\|ccccc}E_{2}&-E_{2}&&&-E_{2}&E_{2}&&&O\\\ &E_{2}&\ddots&&&-E_{2}&E_{2}&&\\\ &&\ddots&\ddots&&&\ddots&\ddots&\\\ &&&\ddots&\ddots&&&\ddots&E_{2}\\\ E_{2}&&&&\ddots&\ddots&&&-E_{2}\\\ \hline\cr-E_{2}&\ddots&&&&\ddots&\ddots&&\\\ &\ddots&E_{2}&&&&\ddots&\ddots&\\\ &&-E_{2}&E_{2}&&&&\ddots&-E_{2}\\\ B_{2}&\cdots&B_{2}&B_{2}-E_{2}&&&&&E_{2}\\\ \end{array}\right)\begin{array}[]{@{}l@{}}\left.\vphantom{\begin{array}[]{@{}c@{}}\\\ \\\ \\\ \\\ \\\ \\\ \\\ \end{array}}\right\\}\,\text{\scriptsize$q-p$}\\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\left.\vphantom{\begin{array}[]{@{}c@{}}\\\ \\\ \\\ \\\ \\\ \end{array}}\right\\}\,\text{\scriptsize$p$}\end{array}\end{array}$ Here $B_{2}$ is a $2\times 2$ matrix $\begin{pmatrix}-2&2\\\ -2&2\end{pmatrix}$. Since $(E_{q}+N_{q})^{2}$, $(-W_{q})^{p}$ are IGL matrices, we have $G_{2}\sim G_{3}$. ### 4.3. Down to $p\times p$ matrix In the sequel, Let $B_{m,n}$ be defined by $B_{m,n}=\begin{pmatrix}-2&2&-2&2&\cdots\\\ -2&2&-2&2&\cdots\\\ \cdots&&&&\end{pmatrix}\in M_{m,n}(\mathbb{Z}).$ We focus on the right-upper half of the matrix $G_{3}$. A matrix $\displaystyle\begin{array}[]{@{}l@{}}\hskip 9.95845pt\overbrace{\hphantom{{E_{2}}{-E_{2}}{\ddots}{-E_{2}}\hskip 25.0pt}}^{p}\hskip 7.5pt\overbrace{\hphantom{{-E_{2}}{-E_{2}}{\ddots}{-E_{2}}\hskip 25.0pt}}^{q-p}\\\ \left(\begin{array}[]{cccccccc}E_{2}&-E_{2}&&&-E_{2}&E_{2}&&O\\\ &E_{2}&\ddots&&&\ddots&\ddots&\\\ &&\ddots&\ddots&&&\ddots&E_{2}\\\ &&&\ddots&\ddots&&&-E_{2}\\\ &&&&\ddots&\ddots&&\\\ &&&&&\ddots&\ddots&\\\ &O&&&&&\ddots&-E_{2}\\\ &&&&&&&E_{2}\\\ \end{array}\right)\end{array}$ $\displaystyle=(E_{q}-N_{q}^{2})(E_{q}-N_{q}^{p}).$ is an IGL matrix and its inverse matrix is $((E_{q}-N_{q}^{2})(E_{q}-N_{q}^{p}))^{-1}=(E_{q}+N_{q}^{p}+N_{q}^{2p}+\cdots)(E_{q}+N_{q}^{2}+N_{q}^{4}+\cdots).$ We divide $G_{3}$ into three parts, that is, $\displaystyle G_{3}$ $\displaystyle=(E_{q}-N_{q}^{2})(E_{q}-N_{q}^{p})$ $\displaystyle+\left(\begin{array}[]{c\|c}O&O\\\ B_{2,p}&O\end{array}\right)+\left(\begin{array}[]{c\|c}\begin{array}[]{ccc}&&\\\ &O&\\\ E_{2}&&\\\ -E_{2}&\ddots&\\\ &\ddots&E_{2}\\\ &&-E_{2}\end{array}&\begin{array}[]{ccc}&&\\\ &O&\\\ &&\end{array}\end{array}\right)$ We calculate $((E_{q}-N_{q}^{2})(E_{q}-N_{q}^{p}))^{-1}G_{3}$ one by one. $((E_{q}-N_{q}^{2})(E_{q}-N_{q}^{p}))^{-1}\left(\begin{array}[]{c\|c}O&O\\\ B_{2,p}&O\end{array}\right)=\left(\begin{array}[]{c\|c}\vdots&O\\\ 3B_{p,p}&O\\\ 2B_{p,p}&O\\\ B_{p,p}&O\end{array}\right),$ $((E_{q}-N_{q}^{2})(E_{q}-N_{q}^{p}))^{-1}\begin{array}[]{@{}l@{}}\hskip 9.95845pt\overbrace{\hphantom{{-E_{2}}{-E_{2}}{\cdots}\hskip 35.0pt}}^{p}\hskip 7.5pt\overbrace{\hphantom{{O}\hskip 30.0pt}}^{q-p}\\\ \left(\begin{array}[]{c\|c}\begin{array}[]{ccc}&&\\\ &O&\\\ E_{2}&&\\\ -E_{2}&\ddots&\\\ &\ddots&E_{2}\\\ &&-E_{2}\end{array}&\begin{array}[]{ccc}&&\\\ &O&\\\ &&\end{array}\end{array}\right)\end{array}$ $=\left(\begin{array}[]{c\|c}\vdots&O\\\ -E_{p}&O\\\ -E_{p}&O\end{array}\right).$ We sum them up and we know that there exists a matrix $G_{4}\in M_{p}(\mathbb{Z})$ such that $((E_{q}-N_{q}^{2})(E_{q}-N_{q}^{p}))^{-1}G_{3}=\begin{array}[]{@{}l@{}}\hskip 8.53581pt\overbrace{\hphantom{\hskip 11.38109pt}}^{p}\hskip 5.0pt\overbrace{\hphantom{O\hskip 10.0pt}}^{q-p}\\\ \begin{pmatrix}\ G_{4}&O\\\ &E_{q-p}\end{pmatrix}\begin{array}[]{@{}l@{}}\left.\vphantom{3.5mm}\right\\}\,\text{\scriptsize$p$}\\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\left.\vphantom{3.5mm}\right\\}\,\text{\scriptsize$q-p$}\end{array}\end{array}.$ Suppose that integers $m,\alpha$ are given by $q=mp+\alpha$(, $0\leq\alpha<p$), then explicitly we have $G_{4}=E_{p}+\begin{pmatrix}(m+1)B_{\alpha,p}\\\ mB_{p-\alpha,p}\end{pmatrix}-W_{p}^{-\alpha},$ and $G_{3}$ and $G_{4}$ are $\mathbb{Z}$-equivalent to each other. ### 4.4. Down to $2r\times 2r$ matrix and proof of the theorem 1.1(3) We divide into two cases: case 1:$\alpha=0$, and case 2: $\alpha>0$. When $\alpha=0$, $E_{p}-W_{p}^{-\alpha}=O$ and $G_{4}=mB_{p,p}\sim(2m)\oplus(0)^{(p-1)\oplus}$ holds. From Proposition 4.2, we obtain $g(p,mp)=(2m,0,\cdots,0)\quad((p-2)\text{-times }0),$ In this case, $p=r,p^{\prime}=1,q^{\prime}=m$ is followed by Theorem 1.1(3) immediately. Next consider the case $\alpha>0$. Let an integer $\alpha^{\prime}$ be defined by $\alpha^{\prime}=\frac{\alpha}{r}$. Remark that $q^{\prime}=mp^{\prime}+\alpha^{\prime}$. We divide $G_{4}$ into blocks of $r\times r$ matrices. Let $G^{\prime}_{4},G^{\prime\prime}_{4}$ be defined by $G^{\prime}_{4}=\begin{array}[b]{@{}c@{}}\overbrace{\hphantom{{E_{r}}{\ddots}{E_{r}}\hskip 30.0pt}}^{p-\alpha}\hskip 10.0pt\overbrace{\hphantom{{E_{r}}{\ddots}{E_{r}}\hskip 30.0pt}}^{\alpha}\\\ \left(\begin{array}[]{@{\,}ccc\|ccc@{\,}}E_{r}&&&-E_{r}&&\\\ &\ddots&&&\ddots&\\\ &&\ddots&&&-E_{r}\\\ -E_{r}&&&\ddots&&\\\ &\ddots&&&\ddots&\\\ &&-E_{r}&&&E_{r}\end{array}\right)\end{array}\begin{array}[]{@{}l@{}}\left.\vphantom{\begin{array}[]{@{}c@{}}\\\ \\\ \\\ \end{array}}\right\\}\,\text{\scriptsize$\alpha$}\\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\left.\vphantom{\begin{array}[]{@{}c@{}}\\\ \\\ \\\ \end{array}}\right\\}\,\text{\scriptsize$p-\alpha$}\end{array}$ $G^{\prime\prime}_{4}=\begin{pmatrix}(m+1)A_{r}&\cdots&(m+1)A_{r}\\\ \vdots&\\\ (m+1)A_{r}&\cdots&(m+1)A_{r}\\\ mA_{r}&\cdots&mA_{r}\\\ \vdots&\\\ mA_{r}&\cdots&mA_{r}\end{pmatrix}\begin{array}[]{@{}l@{}}\left.\vphantom{\begin{array}[]{@{}c@{}}\\\ \\\ \\\ \end{array}}\right\\}\,\text{\scriptsize$\alpha$}\\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\left.\vphantom{\begin{array}[]{@{}c@{}}\\\ \\\ \\\ \end{array}}\right\\}\,\text{\scriptsize$p-\alpha$}\end{array},$ then we have $G_{4}=G^{\prime}_{4}+G^{\prime\prime}_{4}$. Let $T$ and $T^{\prime}$ be defined by $T=\begin{pmatrix}E_{r}&-E_{r}&\cdots&-E_{r}\\\ &E_{r}&&\\\ &&\ddots&\\\ &&&E_{r}\end{pmatrix},T^{\prime}=\begin{pmatrix}E_{r}&E_{r}&\cdots&E_{r}\\\ &E_{r}&&\\\ &&\ddots&\\\ &&&E_{r}\end{pmatrix}\in M_{p}(\mathbb{Z}),$ then we have $T^{\prime}G^{\prime}_{4}T=\left(\begin{array}[]{@{\,}ccc\|cccc@{\,}}O&&&O&O&\cdots&O\\\ &E_{r}&&&-E_{r}&&\\\ &&\ddots&&&\ddots&\\\ &&&\ddots&&&-E_{r}\\\ -E_{r}&E_{r}&\cdots&E_{r}&2E_{r}&\cdots&E_{r}\\\ &\ddots&&&&\ddots&\\\ &&-E_{r}&&&&E_{r}\end{array}\right).$ Since $p^{\prime}=p/r$ and $\alpha^{\prime}=\alpha/r$ are co-prime, after changing some rows together and changing some columns together, we have $T^{\prime}G^{\prime}_{4}T\sim\left(\begin{array}[]{@{\,}cccccc@{\,}}O&\cdots&\cdots&\cdots&O\\\ -E_{r}&2E_{r}&E_{r}&\cdots&E_{r}&\\\ &-E_{r}&E_{r}&&&\\\ &&\ddots&\ddots&&\\\ &&&-E_{r}&E_{r}\end{array}\right)$ We make a right multiplication by $\begin{pmatrix}E_{r}&O&\cdots&O\\\ O&E_{r}&&O\\\ \vdots&\vdots&\ddots&\\\ O&E_{r}&\cdots&E_{r}\end{pmatrix}$ and we have $\sim\left(\begin{array}[]{@{\,}cccccc@{\,}}O&\cdots&\cdots&\cdots&\cdots&O\\\ -E_{r}&p^{\prime}E_{r}&(p^{\prime}-2)E_{r}&\cdots&2E_{r}&E_{r}\\\ &O&E_{r}&&&O\\\ \vdots&\vdots&&\ddots&&\\\ \vdots&\vdots&&&\ddots&\\\ &O&O&&&E_{r}\end{array}\right).$ We make the same elementary transformations on $G^{\prime\prime}_{4}$ as we did on $G^{\prime}_{4}$, we have $G^{\prime\prime}_{4}\sim\begin{pmatrix}(mp^{\prime}+\alpha^{\prime})B_{r}&O&\cdots&O\\\ mB_{r}&\vdots&&\vdots\\\ &\vdots&&\vdots\\\ \vdots&\vdots&&\vdots\\\ &O&\cdots&O\end{pmatrix}.$ We sum them up and obtain $\displaystyle G_{4}$ $\displaystyle\sim\begin{pmatrix}(mp^{\prime}+\alpha^{\prime})B_{r}&O&\cdots&\cdots&O\\\ mB_{r}-E_{r}&p^{\prime}E_{r}&&\cdots&\\\ &O&E_{r}&&\\\ &\vdots&&E_{r}&\\\ &O&&&E_{r}\end{pmatrix}$ $\displaystyle\sim\begin{pmatrix}(mp^{\prime}+\alpha^{\prime})B_{r}&O\\\ mB_{r}-E_{r}&p^{\prime}E_{r}\end{pmatrix}\sim\begin{pmatrix}(mp^{\prime}+\alpha^{\prime})B_{r}&p^{\prime}(mp^{\prime}+\alpha^{\prime})B_{r}\\\ -E_{r}&O\end{pmatrix}$ $\displaystyle\sim p^{\prime}(mp^{\prime}+\alpha^{\prime})B_{r}\sim(2p^{\prime}q^{\prime})\oplus(0)^{(r-1)\oplus}.$ From Proposition 4.2, $g(p,q)=(2p^{\prime}q^{\prime},0,\cdots,0)\quad((r-2)\text{- times }0)$ and we complete the proof of Theorem 1.1 (3). ## 5\. Relation with Alexander polynomials In this section we mention two topics about relation between Goeritz invariants, double branched cover of $S^{3}$, and Alexander polynomials of torus links. Kawauchi’s book [3] says that Georitz invariant $(d_{1},\cdots,d_{k})$ and integral elementary divisors of the symmetrized matrix $V+\,^{t}V$ of the Seifert matrix $V$ coincide. (See Prop 8.2.2. jThese integral elementary divisors give the first homology of a double branched cover of the link. That is, if $M_{L}$ is a double branched cover of $S^{3}$ along the link $L$, then $H_{1}(M_{L})\equiv\mathbb{Z}_{d_{1}}\oplus\cdots\oplus\mathbb{Z}_{d_{k}}$ holds. The following corollary follows this fact. ###### Corollary 5.1. Let $M=M_{T(p,q)}$ be a double branched cover of torus link $T(p,q)$. (1) If both of $p,q$ are odd, then $H_{1}(M)\cong\mathbb{Z}_{2}^{(r-1)\oplus}$ ( If $p,q$ are co-prime, then $H_{1}(M)=O$.) (2) If $p$ is odd and $q$ is even, then $H_{1}(M)\cong\mathbb{Z}_{p^{\prime}}\oplus\mathbb{Z}^{(r-1)\oplus}$ (If $p,q$ are co-prime, then $H_{1}(M)\cong\mathbb{Z}_{p}$. If $p^{\prime}=1$ then $H_{1}(M)\cong\mathbb{Z}^{(r-1)\oplus}$. ) (3) If both of $p,q$ are even, then $H_{1}(M)\cong\mathbb{Z}_{2p^{\prime}q^{\prime}}\oplus\mathbb{Z}^{(r-2)\oplus}$ (If $r=2$, then $H_{1}(M)\cong\mathbb{Z}_{2p^{\prime}q^{\prime}}$.) The second topic is on Alexander polynomial. It is given by $\Delta_{L}(t)=\det(tV-\,^{t}V)$. Murasugi [4] shows the following formulas on torus links. ###### Proposition 5.2. Alexander polynomial $\Delta(t)$ of the torus link $T(p,q)$ is given by follows. (1) If $p,q$ are co-prime, then $\Delta(t)=t^{-\frac{(p-1)(q-1)}{2}}\dfrac{(1-t)(1-t^{pq})}{(1-t^{p})(1-t^{q})}$ (2) If the GCD $r$ of $(p,q)$ is more than 1, then $\Delta(t)=t^{-\frac{(p-1)(q-1)}{2}}\dfrac{(1-t)(1-t^{pq/r})^{r}}{(1-t^{p})(1-t^{q})}.$ On the above formulas, we define the zero-order $k$ and the top-coefficient $a_{k}$ by $t^{\frac{(p-1)(q-1)}{2}}\Delta(t)=a_{k}(t+1)^{k}+a_{k+1}(t+1)^{k+1}+\cdots.$ It is easy to show that (1) $k$ is nulity and that (2) $\|a_{k}\|$ equals to the product of non-zero elementary divisors of $\Delta(-1)$. This is a small corollary of our theorem. ###### Corollary 5.3. Let $r$ be the GCD of $(p,q)$. (1) If both of $p,q$ are odd, then $k=1$ and $a_{k}=2^{r-1}$. (2) If $p$ is odd and $q$ is even, then $k=r-1$ and $a_{k}=p^{\prime}=p/r$. (3) If both of $p,q$ are even, then $k=r-2$ and $a_{k}=2p^{\prime}q^{\prime}=2pq/r^{2}$. ## References [1] L. Goeritz, Knoten und quadratische Formen, Math. Annalen, 103, (1930), 647-654 * [2] K. Ikeda, T. Ikeda, T. Kawakami, H. Sugimoto, T. Sugiura, J. Yagi, and S. Yamanaka, Goeritz Invariants of Two-bridge Links and Totus Links, Kochi J. Math., 5, (2010), 163-172 * [3] A. Kawauchi, Lecture of Knot Theory (in Japanese), (2007), Kyoritsu Shuppan * [4] K. Murasugi, Knot theory and its application (in Japanese), (1993/2012), Nippon Hyouron-sha Kazushi Ahara, Department of Frontier Media Science, Meiji University, 4-21-1 Nakano, Nakano-ku, Tokyo, 164-8525, Japan [email protected] Shingo Watanabe, Department of Mathematics, Meiji University, 1-1-1 Higashi- Mita, Tama-ku, Kawasaki, Kanagawa, 214-8571, Japan	arxiv-papers	2013-12-29T12:55:59	2024-09-04T02:49:56.041821	{ "license": "Public Domain", "authors": "K. Ahara, and S. Watanabe", "submitter": "Kazushi Ahara", "url": "https://arxiv.org/abs/1312.7531" }
1312.7585	11institutetext: Otakar Svítek 22institutetext: Institute of Theoretical Physics, Faculty of Mathematics and Physics, Charles University in Prague, V Holešovičkách 2, 180 00 Praha 8, Czech Republic , 22email: [email protected] # Connection between horizons and algebraic type Otakar Svítek ###### Abstract We study connections between both event and quasilocal horizons and the algebraic type of the Weyl tensor. The relation regarding spacelike future outer trapping horizon is analysed in four dimensions using double-null foliation. ## 1 Introduction We would like to, at least partially, understand how does the presence of some form of horizon restrict the possible algebraic types on it. Since algebraic type of a tensor is determined locally we need to characterize the horizon without employing global notions. We will concentrate on the Weyl tensor and Petrov types derived from it. Event horizon is a global characteristic and the full spacetime evolution is necessary in order to localize it. In many situations this is not desirable or even attainable and therefore, over the past years different quasi-local characterizations of black hole boundary were developed. The most important ones being apparent horizon hawking-ellis , trapping horizon hayward and isolated or dynamical horizon ashtekar . The basic local condition in the above mentioned horizon definitions is effectively the same: these horizons are sliced by marginally trapped surfaces with vanishing expansion of outgoing (ingoing) null congruence orthogonal to the surface. We adopt the so called spacelike future outer trapping horizon (SFOTH) which merges the properties of trapping and dynamical horizons. Since event horizon in a static spacetime with a well-behaved matter is a Killing horizon one can use the local condition on stationary Killing field in such a situation avoiding the global nature of event horizon. This case was investigated by Pravda and Zaslavskii pravda and we summarize their results in the next section. In the third section the relation of SFOTH to the algebraic type is derived. ## 2 Killing horizons Pravda and Zaslavskii pravda studied curvature scalars in a general static spacetime possessing Killing horizon (generalizing previous results of medved on high degree of symmetry of the Einstein tensor to the non-extremal case). They assumed regularity of all polynomial invariants of the Riemann tensor on a horizon and used two naturally preferred frames for calculations - the static observer and the freely falling observer frames. Note that the static frame is singular on the null horizon. Assuming 1+1+2 decomposition and using the Gauss normal coordinates the metric takes the following form ${\rm d}s^{2}=-M^{2}{\rm d}t^{2}+{\rm d}n^{2}+\gamma_{ab}{\rm d}x^{a}{\rm d}x^{b}$ (1) The stationary Killing field is $\xi^{\mu}=(1,0,0,0)$ with $M^{2}\equiv\xi^{\mu}\xi_{\mu}=0$ on the horizon. The tetrad adapted to the static observer’s four-velocity and the Gaussian normal direction has the form $l^{\mu}=\frac{1}{\sqrt{2}}({\textstyle\frac{1}{M}},1,0,0),\ n^{\mu}=\frac{1}{\sqrt{2}}({\textstyle\frac{1}{M}},-1,0,0),\ m^{\mu}=(0,0,m^{a})$ (2) which immediately implies $\Psi_{4}=\bar{\Psi}_{0}$ and $\Psi_{3}=-\bar{\Psi}_{1}$. Next, one can express the Weyl tensor, the Riemann tensor etc. using the above decomposition in terms of 2-metric $\gamma_{ab}$, extrinsic curvature $K_{ab}$, lapse $M$ and their derivatives. Upon projecting the Weyl tensor onto the tetrad and taking the horizon limit $M\to 0$ one gets the Weyl scalars on the horizon. Petrov type is then determined based on curvature invariants $I,J$ and coefficients $K,L,N$ $I=\Psi_{0}\Psi_{4}-4\Psi_{1}\Psi_{3}+3\Psi_{2}^{2},\ J={\rm det}\left(\begin{array}[]{ccc}\Psi_{4}&\Psi_{3}&\Psi_{2}\\\ \Psi_{3}&\Psi_{2}&\Psi_{1}\\\ \Psi_{2}&\Psi_{1}&\Psi_{0}\end{array}\right)$ (3) $K=\Psi_{1}\Psi_{4}^{2}-3\Psi_{4}\Psi_{3}\Psi_{2}+2\Psi_{3}^{3},\ L=\Psi_{2}\Psi_{4}-\Psi_{3}^{2},\ N=12L^{2}-\Psi_{4}^{2}I$ (4) The resulting Petrov type is either D ($\Psi_{2}\neq 0$) or O ($\Psi_{2}=0$). In the case of the freely falling observer the adapted tetrad is given by simple transformation from (2) $\hat{l}^{\mu}=zl^{\mu},\ \hat{n}^{\mu}=z^{-1}n^{\mu}$ (5) where $z=exp(-\alpha)$, ${\rm cosh}\alpha=\frac{E}{M}$, with $E$ being an energy per unit mass for radially infalling geodesic. In this frame invariants $I,J$ do not change but the coefficients are modified $\hat{K}=z^{-3}K,\ \hat{L}=z^{-2}L,\ \hat{N}=z^{-4}N$ (6) Since $z\to 0$ on the horizon the coefficients $\hat{K},\hat{L},\hat{N}$ can be nonzero in the limit (unlike for static observer). The Petrov type is either II ($\Psi_{2}\neq 0$) or III ($\Psi_{2}=0$) here. Due to singular nature of the static frame on the horizon these results are more physically relevant. ## 3 Quasilocal horizons As mentioned in the Introduction in the general dynamical situation we use the SFOTH - spacelike future outer trapping horizon - which has the following properties: 1. 1. spacelike submanifold foliated by closed 2-surfaces with null normal fields $l,n$ 2. 2. expansion $\Theta_{l}=0$ (marginal) 3. 3. expansion $\Theta_{n}<0$ (future) 4. 4. ${\cal{L}}_{n}\Theta_{l}<0$ (outer) We employ a double-null foliation developed by Hayward hayward (mainly for the characteristic initial value problem and the trapping horizon definition) and adapted by Brady and Chambers brady to study a nonlinear stability of Kerr-type Cauchy horizons. The procedure is based on a local foliation by closed orientable 2-surfaces $S$ with smooth embedding $\phi:S\times[0,U)\times[0,V)\to{\cal{M}}$ and induced spatial metric $h_{ab}$ on $S$ with corresponding covariant derivative $D_{a}$. Null vectors $l^{\mu}$, $n^{\mu}$ are normal to $S$ and there is a spatial two vector $s^{a}$ called shift (encoding freedom in identifying points on subsequent surfaces). Evolution of the induced metric is described using Lie derivatives along $l$ and $n$ $\displaystyle\Sigma_{ab}={\cal L}_{l}h_{ab}\,$ , $\displaystyle\ \tilde{\Sigma}_{ab}={\cal L}_{n}h_{ab}$ (7) $\displaystyle\theta={\textstyle\frac{1}{2}}h^{ab}\Sigma_{ab}\,$ , $\displaystyle\ \tilde{\theta}={\textstyle\frac{1}{2}}h^{ab}\tilde{\Sigma}_{ab}$ (8) $\displaystyle\sigma_{ab}=\Sigma_{ab}-\theta h_{ab}\,$ , $\displaystyle\ \tilde{\sigma}_{ab}=\tilde{\Sigma}_{ab}-\tilde{\theta}h_{ab}$ (9) $\displaystyle\omega_{a}$ $\displaystyle=$ $\displaystyle{\textstyle\frac{1}{2}}h_{ab}{\cal L}_{l}s^{b}$ (10) $\theta,\tilde{\theta}$ being expansions, $\sigma_{ab},\tilde{\sigma}_{ab}$ shears and $\omega_{a}$ anholonomicity (related to normal fundamental form). We assume normalized null vectors thus having zero inaffinities. From vacuum Einstein equations and contracted Bianchi identities one obtains $\displaystyle{\cal{L}}_{l}\theta$ $\displaystyle=$ $\displaystyle-{\textstyle\frac{1}{2}}\theta^{2}-{\textstyle\frac{1}{4}}\sigma_{ab}\sigma^{ab}$ (11) $\displaystyle{\cal{L}}_{l}h$ $\displaystyle=$ $\displaystyle\theta h$ (12) $\displaystyle{\cal{L}}_{l}\omega_{a}$ $\displaystyle=$ $\displaystyle-\theta\omega_{a}+{\textstyle\frac{1}{2}}D^{b}\sigma_{ab}-{\textstyle\frac{1}{2}}D_{a}\theta$ (13) $\displaystyle{\cal{L}}_{l}(h^{-1/2}h_{ab})$ $\displaystyle=$ $\displaystyle h^{-1/2}\sigma_{ab}$ (14) After expressing curvature tensors in the given frame we get the following Weyl scalars in vacuum $\displaystyle 4\Psi_{0}$ $\displaystyle=$ $\displaystyle(2{\cal L}_{l}\Sigma_{ab}-\Sigma_{am}h^{mn}\Sigma_{bn})m^{a}m^{b}$ (15) $\displaystyle 4\Psi_{1}$ $\displaystyle=$ $\displaystyle(2\omega^{m}\Sigma_{am}+4{\cal L}_{l}\omega_{a})m^{a}$ (16) $\displaystyle 4\Psi_{2}$ $\displaystyle=$ $\displaystyle(2{\cal L}_{n}\Sigma_{ab}-4D_{a}\omega_{b}-\Sigma_{am}h^{mn}\tilde{\Sigma}_{bn}-4\omega_{a}\omega_{b})m^{a}\bar{m}^{b}$ (17) $\displaystyle 4\Psi_{3}$ $\displaystyle=$ $\displaystyle-(2\omega^{m}\tilde{\Sigma}_{am}+4{\cal L}_{n}\omega_{a})\bar{m}^{a}$ (18) $\displaystyle 4\Psi_{4}$ $\displaystyle=$ $\displaystyle(2{\cal L}_{n}\tilde{\Sigma}_{ab}-\tilde{\Sigma}_{am}h^{mn}\tilde{\Sigma}_{bn})\bar{m}^{a}\bar{m}^{b}$ (19) Next, we use the vanishing of expansion and further fixing of the spatial part of the frame for simplification. We evaluate the first term on the right-hand side of equation (15) noting that ${\cal L}_{l}\Sigma_{ab}={\cal L}_{l}\sigma_{ab}+\theta\Sigma_{ab}+h_{ab}{\cal L}_{l}\theta$ (20) Using the projection and the horizon condition we obtain $m^{a}m^{b}{\cal L}_{l}\Sigma_{ab}=m^{a}m^{b}{\cal L}_{l}\sigma_{ab}$ (21) Assuming (see brady ) the Lie-propagated spatial part of the frame and equation (14) we arrive at $0={\cal L}_{l}(h^{-1/2}h_{ab}m^{a}m^{b})=h^{-1/2}\sigma_{ab}m^{a}m^{b}$ (22) Next, we may assume that $\omega_{a}=0$ initially on $S$ and would like to have ${\cal L}_{l}\omega_{a}=0$ as well. Indeed, the first and the last terms of equation (13) vanish on the horizon and by further locally fixing the spatial part of the frame we obtain $m^{a}D^{b}\sigma_{ab}=0$. In a similar way, one can show that $\Sigma_{am}h^{mn}\Sigma_{bn}m^{a}m^{b}=0$ on the horizon. Then $\Psi_{0}=0$ and $\Psi_{1}=0$. Assuming regularity of $\Psi_{\\{2,3,4\\}}$ we have $I^{3}=27J^{2}$ and therefore Petrov type II. Clearly the spacetime is generically type I away from the horizon. In the future, we would like to check whether stronger statements are possible (with additional assumptions), generalize the results to well-behaved matter fields and nonzero cosmological constant. Also, we would like to extend the analysis to other important tensors (Ricci etc.). ###### Acknowledgements. This work was supported by grant GACR 202/09/0772 and project UNCE 204020/2012. ## References * (1) Hawking, S.W., Ellis, G.F.R.: The large scale structure of space-time. CUP, Cambridge (1975) * (2) Hayward, S.A.:General laws of black-hole dynamics. Phys. Rev. D 49, 6467-74 (1994) * (3) Ashtekar, A., Beetle, C., Fairhurst, S.: Mechanics of isolated horizons. Class. Quant. Grav. 17, 253-298 (2000) Ashtekar, A., Krishnan, B.: Dynamical horizons and their properties. Phys. Rev. D 68, 104030 (2003) * (4) Pravda, V., Zaslavskii, O.B.: Curvature tensors on distorted Killing horizons and their algebraic classification. Class. Quant. Grav. 22, 5053-5072 (2005) * (5) Medved, A.J.M., Martin, D., Visser, M.: Dirty black holes: spacetime geometry and near-horizon symmetries, Class. Quant. Grav. 21, 3111 (2004) * (6) Brady, P.R., Chambers, C.M.: Non-linear instability of Kerr-type Cauchy horizons, Phys. Rev. D 51, 4177 (1995)	arxiv-papers	2013-12-29T20:46:29	2024-09-04T02:49:56.051053	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Otakar Svitek", "submitter": "Otakar Svitek", "url": "https://arxiv.org/abs/1312.7585" }
1312.7656	# Mass Spectra of $D$, $D_{s}$ Mesons using Dirac formalism with martin-like confinement potential Manan Shah1 [email protected] Bhavin Patel2 [email protected] P C Vinodkumar1 [email protected] 1Department of Physics, Sardar Patel University,Vallabh Vidyanagar-388120, INDIA 2P. D. Patel Institute of Applied Sciences, CHARUSAT, Changa-388421, INDIA ## Introduction Remarkable progress at the experimental side, with various high energy machines such as BaBar, BELLE, B-factories, Tevatron, ARGUS collaborations, CLEO, CDF, D${\O}$ etc., for the study of hadrons has opened up new challenges in the theoretical understanding of light-heavy flavour hadrons. The existing results on excited heavy-light mesons are partially inconclusive, and even contradictory in several cases. The predictions of masses of heavy-light system for low-lying 1S and $1P_{J}$ states of D and $D_{s}$ mesons were known from experiment PDG2012 and few from the theory Godfrey ; Eichen ; Ebert . Here we study the mass spectra of $D$ and $D_{s}$ mesons in a relativistic framework. ## Theoretical Framework The bound constituent quark and antiquark inside the meson are in definite energy states having no definite momenta. However one can find out the momentum distribution amplitude for the constituent quark and antiquark inside the meson immediately before their annihilation to a lepton pair. Though the colour confinement of quarks are understood in terms of multigluon exchanging at the non-perturbative regime of the hadronic size, it is not feasible to compute theoretically from the QCD first principles. Thus one assumes various confinement mechanism to study the hadronic properties. In the present study, we assume that the constituent quarks in a meson core is independently confined by an average Martin-like potential of the form Barik2000 $V(r)=\frac{1}{2}(1+\gamma_{0})(\lambda r^{0.1}+V_{0})$ (1) To a first approximation, the confining part of the interaction is believed to provide the zeroth-order quark dynamics inside the meson core through the quark Lagrangian density Table 1: S-wave $D$ ($c\bar{q}$, q = d,u) spectrum (in MeV). \| \| \| Experiment \| \| \| ---\|---\|---\|---\|---\|---\|--- nL \| State \| Present \| Meson \| Mass PDG2012 \| Badalian2011 \| Ebert2010 1S \| $1{{}^{3}S_{1}}$ \| 2013.3 \| $D^{}$(2010) \| 2010.28$\pm$0.13 \| - \| 2010 \| $1{{}^{1}S_{0}}$ \| 1874.0 \| \| 1869.62$\pm$0.15 \| - \| 1871 2S \| $2{{}^{3}S_{1}}$ \| 2581.0 \| $D^{}$(2600) \| 2608.7 del \| 2639 \| 2632 \| $2{{}^{1}S_{0}}$ \| 2501.7 \| $D$(2550) \| 2539.4 del \| 2567 \| 2581 3S \| $3{{}^{3}S_{1}}$ \| 3088.9 \| \| - \| 3125 \| 3096 \| $3{{}^{1}S_{0}}$ \| 3031.5 \| \| - \| 3065 \| 3062 4S \| $4{{}^{3}S_{1}}$ \| 3567.8 \| \| - \| - \| 3482 \| $4{{}^{1}S_{0}}$ \| 3521.6 \| \| - \| - \| 3452 1P \| $1{{}^{3}P_{2}}$ \| 2455.1 \| $D_{2}^{}$(2460) \| 2462.6 $\pm$ 0.7 \| - \| 2460 \| $1{{}^{3}P_{1}}$ \| 2348.0 \| \| - \| - \| 2469 \| $1{{}^{3}P_{0}}$ \| 2276.6 \| $D_{0}^{}$(2400) \| 2318 $\pm$ 29 \| - \| 2406 \| $1{{}^{1}P_{1}}$ \| 2317.3 \| $D_{1}$(2420) \| 2421.3 $\pm$ 0.6 \| - \| 2426 2P \| $2{{}^{3}P_{2}}$ \| 2907.0 \| \| - \| 2965 \| 3012 \| $2{{}^{3}P_{1}}$ \| 2834.4 \| \| - \| 2960 \| 3021 \| $2{{}^{3}P_{0}}$ \| 2786.0 \| \| - \| 2880 \| 2919 \| $2{{}^{1}P_{1}}$ \| 2812.3 \| \| - \| 2940 \| 2932 ${\cal L}^{0}_{q}(x)=\bar{\psi}_{q}(x)\left[\frac{i}{2}\gamma^{\mu}\overleftrightarrow{\partial_{\mu}}-V(r)-m_{q}\right]\psi_{q}(x).$ (2) The normalized quark wave functions $\psi(\vec{r})$ obtained from Eqn (2) satisfies the Dirac equation given by $[\gamma^{0}E_{q}-\vec{\gamma}.\vec{P}-m_{q}-V(r)]\psi_{q}(\vec{r})=0.$ (3) Table 2: S-wave $D_{s}$ ($c\bar{s}$) spectrum (in MeV). \| \| \| Experiment \| \| \| ---\|---\|---\|---\|---\|---\|--- nL \| State \| Present \| Meson \| Mass PDG2012 \| Badalian2011 \| Ebert2010 1S \| $1{{}^{3}S_{1}}$ \| 2112.3 \| $D_{s}^{}$ \| 2112.3$\pm$0.5 \| - \| 2111 \| $1{{}^{1}S_{0}}$ \| 1970.6 \| $D_{s}$ \| 1968.49$\pm$0.32 \| - \| 1969 2S \| $2{{}^{3}S_{1}}$ \| 2684.4 \| $D_{s1}$(2710) \| $2710_{-7}^{+12}$ \| 2728 \| 2731 \| $2{{}^{1}S_{0}}$ \| 2603.9 \| \| 2638 Evdokimov \| 2656 \| 2688 3S \| $3{{}^{3}S_{1}}$ \| 3195.1 \| \| - \| 3200 \| 3242 \| $3{{}^{1}S_{0}}$ \| 3136.9 \| \| - \| 3140 \| 3219 4S \| $4{{}^{3}S_{1}}$ \| 3676.0 \| \| - \| - \| 3669 \| $4{{}^{1}S_{0}}$ \| 3629.3 \| \| - \| - \| 3652 1P \| $1{{}^{3}P_{2}}$ \| 2572.3 \| $D_{s2}$(2573) \| 2571.9$\pm$0.8 \| - \| 2571 \| $1{{}^{3}P_{1}}$ \| 2433.7 \| $D_{s1}$(2460) \| 2459.6$\pm$0.6 \| - \| 2574 \| $1{{}^{3}P_{0}}$ \| 2341.3 \| $D_{s0}^{}$(2317) \| 2317.8$\pm$0.6 \| - \| 2509 \| $1{{}^{1}P_{1}}$ \| 2420.4 \| $D_{s1}$(2536) \| 2535.12$\pm$0.13 \| - \| 2536 2P \| $2{{}^{3}P_{2}}$ \| 3023.2 \| \| - \| 3045 \| 3142 \| $2{{}^{3}P_{1}}$ \| 2927.7 \| \| - \| 3020 \| 3154 \| $2{{}^{3}P_{0}}$ \| 2864.1 \| \| - \| 2970 \| 3054 \| $2{{}^{1}P_{1}}$ \| 2918.4 \| $D_{sJ}$(3040) \| $3044_{-9}^{+30}$ \| 3040 \| 3067 The two component solution of Dirac equation can be written as $\psi_{nlj}(r)=\left(\begin{array}[]{c}\psi_{A}\\\ \psi_{B}\end{array}\right)$ (4) where the positive and negative energy solutions are written as $\psi_{A}^{(+)}(\vec{r})=N_{nlj}\left(\begin{array}[]{c}\frac{ig(r)}{r}\\\ \frac{(\sigma.\hat{r})f(r)}{r}\end{array}\right){\cal{Y}}_{ljm}(\hat{r})$ (5) $\psi_{B}^{(-)}(\vec{r})=N_{nlj}\left(\begin{array}[]{c}\frac{i(\sigma.\hat{r})f(r)}{r}\\\ \frac{g(r)}{r}\end{array}\right)(-1)^{j+m_{j}-l}{\cal{Y}}_{ljm}(\hat{r})$ (6) and $N_{nlj}$ is the overall normalization constant. The radial solutions f(r) and g(r) is obtained numerically to yield the energy eigen values. The parameters are fixed to get the ground state masses of $D$ and $D_{s}$ mesons. The meson radial wave function for $q\bar{q}$ combination is constructed with the respective quark and anti-quark wave functions given by Eqn. (5) and 6. The quark mass parameters $m_{c}$, $m_{u,d}$ and $m_{s}$ are taken as 1.27 GeV, 0.37 GeV and 0.4 GeV respectively. ## Results and Discussion The predicted S-wave masses of $D$ and $D_{s}$ mesons are in very good agreement with experimental PDG2012 results as given in Table 1 and 2 respectively. The predicted results of P-wave $D$ meson states, $1^{3}P_{2}$ (2455.1 MeV) and $1^{3}P_{0}$ (2276.6 MeV) are in good agreement with experimental results of 2462.6 $\pm$ 0.7 MeV and 2318 $\pm$ 29 MeV PDG2012 respectively. The predicted results of P-wave $D_{s}$ meson states $1^{3}P_{2}$ (2572.3 MeV), $1^{3}P_{1}$ (2433.7 MeV) and $1^{3}P_{0}$ (2341.3 MeV) are also good found in good agreement with experimental results $2571.9\pm 0.8$ MeV, $2459.6\pm 0.6$ MeV and $2317.8\pm 0.6$ MeV PDG2012 respectively. ## Acknowledgments The work is part of Major research project NO. F. 40-457/2011(SR) funded by UGC. One of the author (BP) acknowledges the support through Fast Track project funded by DST (SR/FTP/PS-52/2011). ## References * (1) J. Beringer et al. (Particle Data Group), Phys. Rev. D 86, (2012) 010001. * (2) S Godfrey and R Kokoski, Phys. Rev. D 43, (1991) 1679. * (3) E J Eichen, C T Hill and C Quigg, Phys. Rev. Lett.71, (1993) 4116. * (4) D Ebert, R N Faustov and V O Galkin, Phys. Rev. D 57, (1998) 014027. * (5) N.Barik, B. K. Dash, and M. Das, Phys. Rev. D 31, 1652 (1985). * (6) A. M. Badalian, B. L. G. Bakker, Phys. Rev. D 84, (2011) 034006. * (7) D. Ebert, R. N. Faustov and V. O. Galkin, Eur. Phys. J. C 66 (2010) 197. * (8) P. Del A. Sanchez et al. (BABAR Collaboration), Phys. Rev. D 82, (2010) 111101. * (9) A.V. Evdokimov et al. (SELEX Collaboration), Phys. Rev. Lett. 93 (2004) 242001.	arxiv-papers	2013-12-30T09:14:16	2024-09-04T02:49:56.061205	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Manan Shah, Bhavin Patel and P C Vinodkumar", "submitter": "Manan N. Shah", "url": "https://arxiv.org/abs/1312.7656" }
1312.7664	# Status of $$\backslash$psi $ (3686), $$\backslash$psi $ (4040), $$\backslash$psi $ (4160), Y (4260), $$\backslash$psi $ (4415) and X (4630) charmonia like states 1Department of Physics, Sardar Patel University, Vallabh Vidyanagar-388120, INDIA E-mail Kaushal Thakkar Department of Applied Physics, S V National Institute of Technology, Surat-395007, INDIA E-mail [email protected] Arpit Parmar Department of Physics, Sardar Patel University, Vallabh Vidyanagar-388120, INDIA E-mail [email protected] P C Vinodkumar Department of Physics, Sardar Patel University, Vallabh Vidyanagar-388120, INDIA E-mail [email protected] ###### Abstract: We examine the status of charmonia like states by looking into the behaviour of the energy level differences and regularity in the behaviour of the leptonic decay widths of the excited charmonia states. The spectroscopic states are studied using a phenomenological Martin-like confinement potential and their radial wave functions are employed to compute the di-leptonic decay widths. Their deviations from the expected behaviour provide a clue to consider them as admixtures of the nearby S and D states. The present analysis strongly favour $$\backslash$psi $ (3686) as admixture of $c\bar{c}$ (2S) and $c\bar{c}$g (4.1 GeV) hybrid, $$\backslash$psi $ (4040) and $$\backslash$psi $ (4160) as admixture states of charmonia (3S, 3D) states with mixing angle $$\backslash$theta $ = 11∘ and 45∘ respectively. We identify Y (4260) as a pure $c\bar{c}$ (4S) state whose leptonic decay is predicted as 0.65 keV. While X(4630) is closer to the $c\bar{c}$ (6S) state. The status of $$\backslash$psi $ (4415) is still not clear as it does not fit to be pure or admixture state. ## 1 Introduction Since the discovery of $J/$\backslash$Psi$, large number of charmonium excited states with their masses and decay widths have been recorded experimentally [1]. Though their spectra and decay properties are well studied, there exist disparities related to the decay properties of their excited states. For example, the $$\backslash$rho-$\backslash$pi$ puzzle related to the hadronic decays of $$\backslash$Psi$ (3686) compared to that of $J/$\backslash$Psi $ (1S) [2, 3] is resolved by invoking these higher charmonia states as admixtures of the respective $c\bar{c}$ states with $c\bar{c}g$ hybrids [3]. Further, if we consider the experimental energy level differences and leptonic decay rates of the excited states beyond 2S of the $c\bar{c}$ $(1^{--})$ states, their deviations from the expected behaviour provide a clue to consider them as admixtures of the nearby S and D states [4]. In this context we examine the status of $$\backslash$psi $ (3686), $$\backslash$psi $ (4040), $$\backslash$psi $ (4160), Y (4260), $$\backslash$psi $ (4415) and X (4630) charmonia like states by looking into the behaviour of the energy level differences of charmonia states and their experimental leptonic decay widths. ## 2 Methodology From the experimentally known $J^{PC}=1^{--}$ charmonium states, their energy level differences are shown in Fig (1). One of our recent theoretical predictions of these states [5] are also shown for comparison. It is evident from the plot that $$\backslash$psi $ (3770,4160), X(4630) of charmonia like states are off from the expected trend as seen from the graph. Looking into their leptonic decay widths similar disparities are observed for the states $$\backslash$psi $ (4040,4160). The disparities of the predicted higher S$-$wave masses beyond $nS(n\geq 3)$ states are reported to be due to the admixture of the S$-$states with nearby D$-$states [4]. Thus we adopt a methodology by considering these states as disturbed with suitable mixing with the nearby states having similar parity. Accordingly, the mixed state $R_{nS^{\prime}}$ is represented in terms of the mixing angle $\theta$ as [4] $R_{nS^{\prime}}=\cos\$\backslash theta\$\ R_{nS}-\sin\$\backslash theta\$\ R_{n^{\prime}D}$ (1) where the wave function at zero of the D-wave, $R_{n^{\prime}D}(0)$ is defined in terms of the second derivative of the D$-$wave as $R^{\prime\prime}_{n^{\prime}D}(0)/M^{2}_{n^{\prime}D}$ [4]. For the admixture of $c\bar{c}g$ hybrid case, $R_{n^{{}^{\prime}}D}$ of eqn. (1) is replaced by $R_{c\bar{c}g}$. These disturbed wave functions at the origin are then employed to compute the leptonic decay widths of the mixed states. We have employed the predicted masses and wave functions based on a phenomenological confinement model with Martin-like potential [5] for the present study. When we consider 50 % admixture of $c\bar{c}g$ hybrid state bearing its mass equal to 4.1 GeV given by [6] yield the leptonic decay widths of $$\backslash$psi $ (3686) as 2.376 keV as against the predicted 1.686 keV [5] which is in good agreement with the reported experimental values of 2.35 $\pm$ 0.04 keV. The mixing configuration, the mixing angle and predicted leptonic decay widths of charmonia like states of present interest are listed in Table 1. Figure 1: Behavior of energy level shift of the (n+1)S$-$nS charmonium states Figure 2: Behavior of leptonic decay width of Exp. charmonia states Table 1: Mixing configuration and leptonic widths (in keV) of $c\bar{c}$ states Exp. State \| Mixed \| $$\backslash$theta $ \| $$\backslash$Gamma${}^{e^{+}e^{-}}$ $ \| $$\backslash$Gamma${}^{e^{+}e^{-}}_{[Expt.]}$ $ \| ---\|---\|---\|---\|---\|--- \| config. \| \| \| [1] \| $\psi(3686)$ \| Pure $2{{}^{3}S_{1}}$ \| \| 1.686 \| $2.35^{0.04}_{0.04}$ \| \| ($\psi(2S),c\bar{c}g$) \| 45∘ \| 2.376 \| \| $\psi(4040)$ \| ($3{{}^{3}S_{1}}$,$3{{}^{3}D_{1}}$) \| 11.07∘ \| 0.896 \| $0.86^{+0.07}_{-0.07}$ \| \| ($3{{}^{3}S_{1}}$,$2{{}^{3}D_{1}}$) \| 37.53∘ \| 0.528 \| \| $\psi(4160)$ \| $4{{}^{3}S_{1}}$,$2{{}^{3}D_{1}}$ \| 46.31∘ \| 0.268 \| $0.48^{+0.22}_{-0.22}$ [7] \| \| $3{{}^{3}S_{1}}$,$3{{}^{3}D_{1}}$ \| 44.62∘ \| 0.398 \| \| $Y(4260)$ \| $4{{}^{3}S_{1}}$,$2{{}^{3}D_{1}}$ \| 14.44∘ \| 0.588 \| - \| \| $3{{}^{3}S_{1}}$,$3{{}^{3}D_{1}}$ \| 69.19∘ \| 0.074 \| - \| \| $4{{}^{3}S_{1}}$,$3{{}^{3}D_{1}}$ \| NP \| - \| - \| $\psi(4415)$ \| $5{{}^{3}S_{1}}$,$3{{}^{3}D_{1}}$ \| 32.38∘ \| 0.320 \| $0.58^{+0.07}_{-0.07}$ \| \| $4{{}^{3}S_{1}}$,$4{{}^{3}D_{1}}$ \| 55.87∘ \| 0.158 \| \| \| $5{{}^{3}S_{1}}$,$4{{}^{3}D_{1}}$ \| NP \| - \| \| $X(4630)$ \| $5{{}^{3}S_{1}}$,$5{{}^{3}D_{1}}$ \| 80.23∘ \| 0.005 \| - \| \| $6{{}^{3}S_{1}}$,$5{{}^{3}D_{1}}$ \| 53.52∘ \| 0.112 \| - \| NP= Not Possible ## 3 Results and discussion Our analysis here has also provided a strong support to treat $$\backslash$psi $ (3686) as hybrid admixture states [3]. We find the admixture of hybrid state excludes the radiative correction to the leptonic decay widths. Further we find that $$\backslash$psi $ (4040) is admixture of $3^{3}S_{1}$ and $3^{3}D_{1}$ with mixing angle $$\backslash$theta $ = 11.07∘ correspond to $96.31\%$ of $3S$ state and 3.69 $\%$ of 3D state with its leptonic decay width 0.896 keV which is in close agreement with the experimental value of $0.86\pm 0.07$ keV. The leptonic decay widths of $$\backslash$psi $ (4160) obtained here with the mixing configuration of ($3^{3}S_{1}$, $3^{3}D_{1}$) and ($4^{3}S_{1}$, $2^{3}D_{1}$) are in agreement with the experimental value $0.48\pm 0.22$ and lie within the error bar reported by Belle Collaboration and BES [8] but completely in disagreement with the value of $0.83\pm 0.07$ reported by [1]. Though Y (4260) can be interpreted as 4S$-$2D admixture state with mixing angle $$\backslash$theta $ = 14.44∘ that predicts its leptonic decay width 0.588 keV, the mixing may not be possible as the 4S and 2D masses differ by more than 200 MeV. So, we consider Y(4260) close to the $c\bar{c}$ (4S) state with predicted leptonic decay width of 0.65 keV. However experimental determination of this width is awaited. While the state $$\backslash$psi $ $(4421\pm 4)$ does not qualify to be the pure 5S state or S$-$D admixture. Experimental measurements of the leptonic decay widths of Y (4260) and X (4630) will justify their status discussed in this paper. ## 4 Acknowledgments The work is part of a Major research project NO. F. 40-457/2011(SR) funded by UGC. ## References * [1] K Nakamura et al. (Particle Data Group), J.Phys. G 37, 075021 (2010). * [2] Y-Q Chen and E. Braaten, Phys. Rev. Lett. 80 (1998) 5060. * [3] Leonard S. Kisslinger, Phys. Rev. D 79 (2009) 114026. * [4] A.M. Badalian, B.L.G. Bakker and I.V. Danilkin. Physics of Atomic Nuclei,73 (2010) 138. * [5] Manan Shah, Arpit Parmar, P C Vinodkumar, Phys. Rev. D 86, 034015 (2012). arXiv:1203.6184. * [6] F.Iddir and L.Semlala, arXiv:1101.2431v2 [hep-ph] (2011). * [7] M. Ablikim et al. (BES Collaboration), Phys. Lett. B 660 (2008) 315. arXiv:0705.4500 [hep-ex] * [8] S.K. Choi et al. (Belle Collaboration), Phys. Rev. Lett. 100 (2008) 142001. arXiv:0708.1790 [hep-ex]	arxiv-papers	2013-12-30T09:32:48	2024-09-04T02:49:56.066592	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Manan Shah, Kaushal Thakkar, Arpit Parmar and P C Vinodkumar", "submitter": "Manan N. Shah", "url": "https://arxiv.org/abs/1312.7664" }
1312.7665	# Ground state search, hysteretic behaviour, and reversal mechanism of skyrmionic textures in confined helimagnetic nanostructures Marijan Beg [email protected] Rebecca Carey Weiwei Wang David Cortés- Ortuño Mark Vousden Marc-Antonio Bisotti Maximilian Albert Dmitri Chernyshenko Ondrej Hovorka Faculty of Engineering and the Environment, University of Southampton, Southampton SO17 1BJ, United Kingdom Robert L. Stamps SUPA School of Physics and Astronomy, University of Glasgow, Glasgow G12 8QQ, United Kingdom Hans Fangohr [email protected] Faculty of Engineering and the Environment, University of Southampton, Southampton SO17 1BJ, United Kingdom ###### Abstract Magnetic skyrmions have the potential to provide solutions for low-power, high-density data storage and processing. One of the major challenges in developing skyrmion-based devices is the skyrmions’ magnetic stability in confined helimagnetic nanostructures. Through a systematic study of equilibrium states, using a full three-dimensional micromagnetic model including demagnetisation effects, we demonstrate that skyrmionic textures are the lowest energy states in helimagnetic thin film nanostructures at zero external magnetic field and in absence of magnetocrystalline anisotropy. We also report the regions of metastability for non-ground state equilibrium configurations. We show that bistable skyrmionic textures undergo hysteretic behaviour between two energetically equivalent skyrmionic states with different core orientation, even in absence of both magnetocrystalline and demagnetisation-based shape anisotropies, suggesting the existence of Dzyaloshinskii-Moriya-based shape anisotropy. Finally, we show that the skyrmionic texture core reversal dynamics is facilitated by the Bloch point occurrence and propagation. An ever increasing need for data storage creates great challenges for the development of high-capacity storage devices that are cheap, fast, reliable, and robust. Nowadays, hard disk drive technology uses magnetic grains pointing up or down to encode binary data (0 or 1) in so-called perpendicular recording media. Practical limitations are well understood and dubbed the “magnetic recording trilemma” Richter2007 . It defines a trade-off between three conflicting requirements: signal-to-noise ratio, thermal stability of the stored data, and the ability to imprint information. Because of these fundamental constraints, further progress requires radically different approaches. Recent research demonstrated that topologically stable magnetic skyrmions have the potential for the development of future data storage and information processing devices. For instance, a skyrmion lattice formed in a monoatomic Fe layer grown on a Ir(111) surface Heinze2011 revealed skyrmions with diameters as small as a few atom spacings. In addition, it has been demonstrated that skyrmions can be easily manipulated using spin-polarised currents of the $10^{6}\,\text{A}\,\text{m}^{-2}$ order Jonietz2010 ; Yu2012 which is a factor $10^{5}$ to $10^{6}$ smaller than the current densities required in conventional magneto-electronics. These unique skyrmion properties point to an opportunity for the realisation of ambitious novel high-density, power- efficient storage Kiselev2011 ; Fert2013 and logic Zhang2015 devices. Skyrmionic textures emerge as a consequence of chiral interactions, also called the Dzyaloshinskii-Moriya Interactions (DMI), that appear when there is no inversion symmetry in the magnetic system structure. The lack of inversion symmetry can be either due to a non-centrosymmetric crystal lattice structure Dzyaloshinsky1958 ; Moriya1960 in so-called helimagnetic materials, or at interfaces between different materials that inherently lack inversion symmetry Fert1980 ; Crepieux1998 . According to this, the Dzyaloshinskii-Moriya interaction can be classified either as bulk or interfacial, respectively. Skyrmions, after being predicted Bogdanov1989 ; Bogdanov1999 ; Rossler2006 , were later experimentally observed in magnetic systems with both bulk Muhlbauer2009 ; Yu2011 ; Yu2010 ; Seki2012 ; Kanazawa2012 and interfacial Heinze2011 ; Romming2013 types of DMI. So far, a major challenge obstructing the development of skyrmion-based devices has been their thermal and magnetic stability SkyrmionicsEditorial2013 . Only recently, skyrmions were observed at the room temperature in magnetic systems with bulk Tokunaga2015 and interfacial Woo2015 ; Moreauluchaire2015 ; Jiang2015 DMI. However, the magnetic stability of skyrmions in absence of external magnetic field was reported only for magnetic systems with interfacial DMI in one-atom layer thin films Heinze2011 ; Sampaio2013 , where the skyrmion state is stabilised in the presence of magnetocrystalline anisotropy. The focus of this work is on the zero-field stability of skyrmionic textures in confined geometries of bulk DMI materials. Zero-field stability is a crucial requirement for the development of skyrmion-based devices: devices that require external magnetic fields to be stabilised are volatile, harder to engineer and consume more energy. We address the following questions that are relevant for the skyrmion-based data storage and processing nanotechnology. Can skyrmionic textures be the ground state (i.e. have the lowest energy) in helimagnetic materials at zero external magnetic field, and if they can, what is the mechanism responsible for this stability? Do the demagnetisation energy and magnetisation variation along the out-of-film direction Rybakov2013 have important contribution to the stability of skyrmionic textures? Is the magnetocrystalline anisotropy an essential stabilisation mechanism? Are there any other equilibrium states that emerge in confined helimagnetic nanostructures? How robust are skyrmionic textures against varying geometry? Do skyrmionic textures undergo hysteretic behaviour in the presence of an external magnetic field (crucial for data imprint), and if they do, what is the skyrmionic texture reversal mechanism? To resolve these unknowns, we use a full three-dimensional simulation model that makes no assumption about translational invariance of magnetisation in the out-of-film direction and takes full account of the demagnetisation energy. We demonstrate, using this full model, that DMI-induced skyrmionic textures in confined thin film helimagnetic nanostructures are the lowest energy states in the absence of both the stabilising external magnetic field and the magnetocrystalline anisotropy and are able to adapt their size to hosting nanostructures, providing the robustness for their practical use. We demonstrate that both the demagnetisation energy and the magnetisation variation in the out-of-film direction play an important role for the stability of skyrmionic textures. In addition, we report the parameter space regions where other magnetisation configurations are in equilibrium. Moreover, we demonstrate that these zero-field stable skyrmionic textures undergo hysteretic behaviour when their core orientation is changed using an external magnetic field, which is crucial for data imprint. The hysteretic behaviour remains present even in the absence of all relevant magnetic anisotropies (magnetocrystalline and demagnetisation-based shape anisotropies), suggesting the existence of a novel Dzyaloshinskii-Moriya-based shape anisotropy. We conclude the study by showing that the skyrmionic texture core orientation reversal is facilitated by the Bloch point occurrence and propagation, where the Bloch point may propagate in either of the two possible directions. This work is based on the specific cubic helimagnetic material, FeGe with $70\,\text{nm}$ helical period, in order to encourage the experimental verification of our predictions. Other materials could allow either to reduce the helical period Muhlbauer2009 ; Kanazawa2012 and therefore the hosting nanostructure size or increase the operating temperature Tokunaga2015 . Some stability properties of DMI-induced isolated skyrmions in two-dimensional confined systems have been studied analytically Rohart2013 ; Du2013a ; Leonov2013 and using simulations Sampaio2013 ; Du2013 . However, in all these studies, either magnetocrystalline anisotropy or an external magnetic field (or both) are crucial for the stabilisation of skyrmionic textures. In addition, an alternative approach to the similar problem, in absence of chiral interactions, where skyrmionic textures can be stabilised at zero external magnetic field and at room temperature using a strong perpendicular anisotropy, has been studied analytically Guslienko2015 , experimentally Buda2001 ; Moutafis2007 , as well as using simulations Moutafis2006 . Our new results, and in particular the zero-field skyrmionic ground state in isotropic helimagnetic materials, can only be obtained by allowing the chiral modulation of magnetisation direction along the film normal, which has recently been shown to radically change the skyrmion energetics Rybakov2013 . ## I Results Equilibrium states. In order to identify the lowest energy magnetisation state in confined helimagnetic nanostructures, firstly, all equilibrium magnetisation states (local energy minima) must be identified, and secondly, their energies compared. In this section, we focus on the first step – identifying the equilibrium magnetisation states. We compute them by solving a full three-dimensional model using a finite element based micromagnetic simulator. In particular, we simulate a thin film helimagnetic FeGe disk nanostructure with thickness $t=10\,\text{nm}$ and diameter $d$, as shown in Fig. 1 inset. The finite element mesh discretisation is such that the maximum spacing between two neighbouring mesh nodes is below $3\,\text{nm}$. The material parameters are $M_{\text{s}}=384\,\text{kA}\,\text{m}^{-1}$, $A=8.78\,\text{pJ}\,\text{m}^{-1}$, and $D=1.58\,\text{mJ}\,\text{m}^{-2}$. We apply a uniform external magnetic field perpendicular to the thin film sample, i.e. in the positive $z$-direction. The Methods section contains the details about the model, FeGe material parameters estimation, as well as the simulator software. In this section, we determine what magnetisation configurations emerge as the equilibrium states at different $d$–$H$ parameter space points. In order to do that, we systematically explore the parameter space by varying the disk sample diameter $d$ from $40\,\text{nm}$ to $180\,\text{nm}$ and the external magnetic field $\mu_{0}H$ from $0\,\text{T}$ to $1.2\,\text{T}$ in steps of $\Delta d=4\,\text{nm}$ and $\mu_{0}\Delta H=20\,\text{mT}$, respectively. At every point in the parameter space, we minimise the energy for a set of different initial magnetisation configurations: (i) five different skyrmionic configurations, (ii) three helical-like configurations with different helical period, (iii) the uniform out-of-plane configuration, and (iv) three random magnetisation configurations. We use the random magnetisation configurations in order to capture other equilibrium states not obtained by relaxing the well-defined initial magnetisation configurations. The details on how we define and generate initial magnetisation configurations are provided in the Supplementary Section S1. The equilibrium states to which different initial magnetisation configurations relax in the energy minimisation process (at every $d$–$H$ parameter space point) we present in the Supplementary Section S2 as a set of “relaxation diagrams”. We summarise these relaxation diagrams and determine the phase space regions where different magnetisation states are in equilibrium, and show them in Fig. 1. Among the eight computed equilibrium states, three are radially symmetric and we label them as iSk, Sk, and T, whereas the other states, marked as H2, H3, H4, 2Sk, and 3Sk, are not. Subsequently, we discuss the meaning of the chosen labels. Figure 1: The metastability phase diagram and magnetisation configurations of all identified equilibrium states. The phase diagram with regions where different states are in equilibrium together with magnetisation configurations and out-of-plane magnetisation component $m_{z}(x)$ along the horizontal symmetry line corresponding to different regions in the phase diagram. Now, we focus on the analysis of radially symmetric skyrmionic equilibrium states, supported by computing the skyrmion number $S$ and scalar value $S_{\text{a}}$ as defined in the Methods section. In the first configuration, marked in Fig. 1 as iSk, the out-of-plane magnetisation component $m_{z}(x)$ profile along the horizontal symmetry line does not cover the entire $[-1,1]$ range, as would be the case for a skyrmion configuration (where the magnetisation vector field $\mathbf{m}$ needs to cover the whole sphere). Accordingly, the scalar value $S_{\text{a}}$ (Eq. (6) in the Methods section, and plotted in Supplementary Fig. 2 (b) for a range of configurations), is smaller than 1. For these reasons we refer to this skyrmionic equilibrium state as the incomplete Skyrmion (iSk) state. A similar magnetisation configuration has been predicted and observed in other works for the case of two-dimensional systems in the presence of magnetocrystalline anisotropy where it is called either the quasi-ferromagnetic Rohart2013 ; Sampaio2013 or edged vortex state Du2013 ; Du2013a . Because the iSk equilibrium state clearly differs from the ferromagnetic configuration and using the word vortex implies the topological charge of $1/2$, we prefer calling this state the incomplete skyrmion state. The incomplete Skyrmion (iSk) state emerges as an equilibrium state in the entire simulated $d$–$H$ parameter space range. In the second equilibrium state, marked as Sk in Fig. 1, $m_{z}(x)$ covers the entire $[-1,1]$ range, the magnetisation covers the sphere at least once and, consequently, the skyrmion configuration is present in the simulated sample. Although the skyrmion number value (Eq. (5) in the Methods section) for this solution is $\|S\|<1$ due to the additional magnetisation tilting at the disk boundary Rohart2013 , which makes it indistinguishable from the previously described iSk equilibrium state, the scalar value is $1<S_{\text{a}}<2$. This state is referred to as the isolated Skyrmion or just Skyrmion (Sk), in two- dimensional systems Rohart2013 ; Sampaio2013 , and we use the same name subsequently in this work. We find that the Sk state is not in equilibrium for sample diameters smaller than $56\,\text{nm}$ and external magnetic field values larger than approximately $1.14\,\text{T}$. Finally, the equilibrium magnetisation state marked as T in Fig. 1 covers the sphere at least twice. In other works, this state together with all other predicted higher-order solutions (not observed in this work) are called the “target states” Leonov2013 , and we use the same Target (T) state name. The analytic model, used for generating initial states, also predicts the existence of higher- order target states (Supplementary Fig. 2 (c)). The T magnetisation configuration emerges as an equilibrium state for samples with diameter $d\geq 144\,\text{nm}$ and field values $\mu_{0}H\leq 0.24\,\text{T}$. The equilibrium states lacking radial symmetry can be classified into two groups: helical-like (marked as H2, H3, and H4) and multiple skyrmion (marked as 2Sk and 3Sk) states. The difference between the three helical-like states is in their helical period. More precisely, in the studied range of disk sample diameter values, either 2, 3, or 4 helical half-periods, including the additional magnetisation tilting at the disk sample edge due to the specific boundary conditions Rohart2013 , fit in the sample diameter. Consequently, we refer to these states, that occur as an equilibrium state for samples larger than $88\,\text{nm}$ and field values lower than $0.2\,\text{T}$, as H2, H3, and H4 . The other two radially non-symmetric equilibrium states are the multiple skyrmion configurations with 2 or 3 skyrmions present in the sample and we call these equilibrium states 2Sk and 3Sk, respectively. These configurations emerge as equilibrium states for samples with $d\geq 132\,\text{nm}$ and external magnetic field values between $0.28\,\text{T}\leq\mu_{0}H\leq 1.06\,\text{T}$. Ground state. After we identified all observed equilibrium states in confined helimagnetic nanostructures, in this section we focus on finding the equilibrium state with the lowest energy at all $d$–$H$ parameter space points. For every parameter space point ($d$, $H$), after we compute and compare the energies of all found equilibrium states, we determine the lowest energy state, and refer to it, in this context, as the ground state. For the identified ground state, we compute the scalar value $S_{\text{a}}$ and use it for plotting a $d$–$H$ phase diagram shown in Fig. 2 (a). Discontinuous changes in the scalar value $S_{\text{a}}$ define the boundaries between regions where different magnetisation configurations are the ground state. In the studied phase space, two different ground states emerge in the confined helimagnetic FeGe thin film disk samples: one with $S_{\text{a}}<1$ and the other with $1<S_{\text{a}}<2$. The previous discussion of the $S_{\text{a}}$ value suggests that these two regions correspond to the incomplete Skyrmion (iSk) and the isolated Skyrmion (Sk) states. We confirm this by visually inspecting two identified ground states, taken from the two phase space points (marked with circle and triangle symbols) in different regions, and show them in Fig. 2 (b) together with their out-of-plane magnetisation component $m_{z}(x)$ along the horizontal symmetry line. Figure 2: Thin film disk ground state phase diagram and corresponding magnetisation states. (a) The scalar value $S_{\text{a}}$ for the thin film disk sample with thickness $t=10\,\text{nm}$ as a function of disk diameter $d$ and external out-of-plane magnetic field $\mathbf{H}$ (as shown in an inset). (b) Two identified ground states: incomplete Skyrmion (iSk) and isolated Skyrmion (Sk) magnetisation configurations at single phase diagram points together with their out-of-plane magnetisation component $m_{z}(x)$ profiles along the horizontal symmetry line. A key result of this study is that both incomplete Skyrmion (iSk) and isolated Skyrmion (Sk) are the ground states at zero external magnetic field for different disk sample diameters. More precisely, iSk is the ground state for samples with diameter $d<140\,\text{nm}$ and Sk is the ground state for $d\geq 140\,\text{nm}$. The Sk changes to the iSk ground state for large values of external magnetic field. The phase diagram in Fig. 2 shows the phase space regions where iSk and Sk are the ground states, which means that all other previously identified equilibrium states are metastable. Now, we focus on computing the energies of metastable states relative to the identified ground state. Firstly, we compute the energy density $E/V$ for all equilibrium states, where $E$ is the total energy of the system and $V$ is the disk sample volume, and then subtract the ground state energy density corresponding to that phase space point. We show the computed energy density differences $\Delta E/V$ when the disk sample diameter is changed in steps of $\Delta d=2\,\text{nm}$ at zero external magnetic field in Fig. 3 (a). Similarly, the case when the disk sample diameter is $d=160\,\text{nm}$ and the external magnetic field is changed in steps of $\mu_{0}\Delta H=20\,\text{mT}$ is shown in Fig. 3 (b). The magnetisation configurations are the equilibrium states in the $d$ or $H$ values range where the line is shown and collapse otherwise. Figure 3: The energy density difference between identified equilibrium states and the corresponding ground state. Energy density differences $\Delta E/V$ at (a) zero field for different sample diameters $d$ and for (b) sample diameter $d=160\,\text{nm}$ and different external magnetic field values. Configurations are in equilibrium where the line is shown and collapse for other diameter or external magnetic field values. Figure 4: The $m_{z}(x)$ profiles and skyrmionic texture sizes $s$ for different sizes of hosting nanostructures at zero external magnetic field. (a) Profiles of the out-of- plane magnetisation component $m_{z}(x)$ along the horizontal symmetry line for different thin film disk sample diameters with thickness $t=10\,\text{nm}$ at zero external magnetic field $\mu_{0}H=0\,\text{T}$. The curves for $d\leq 120\,\text{nm}$ represent incomplete skyrmion ($\circ$) states, and for $d\geq 140\,\text{nm}$ represent isolated skyrmion ($\times$) states. (b) The skyrmionic texture size $s=2\pi/k$ (that can be interpreted as the length along which the full magnetisation rotation occurs) as a function of the hosting nanostructure size, obtained by fitting $m_{z}(x)=\pm\cos(kx)$ to the simulated profile. (c) The ratio of skyrmionic texture size to disk sample diameter ($s/d$) as a function of hosting nanostructure size $d$. For the practical use of ground state skyrmionic textures in helimagnetic nanostructures, their robustness is of great significance due to the unavoidable variations in the patterning process. Because of that, in Fig. 4 (a) we plot the out-of-plane magnetisation component $m_{z}(x)$ along the horizontal symmetry line for the iSk and the Sk ground state at zero external magnetic field for six different diameters $d$ of the hosting disk nanostructure: three iSk profiles for $d\leq 120\,\text{nm}$, and three Sk profiles for $d\geq 140\,\text{nm}$. The profiles show that the two skyrmionic ground states have the opposite core orientations. In the case of the Sk states, the magnetisation at the core is antiparallel and at the outskirt parallel to the external magnetic field. This reduces the Zeeman energy $E_{\text{z}}=-\mu_{0}\int\mathbf{H}\cdot\mathbf{M}\,\text{d}^{3}\mathbf{r}$ because the majority of the magnetisation in the isolated skyrmion outskirts points in the same direction as the external magnetic field $\mathbf{H}$. Once the disk diameter is sufficiently small that less than a complete spin rotation fits into the sample, this orientation is not energetically favourable anymore and the iSk state emerges. In this iSk state, the core magnetisation points in the same direction as the external magnetic field in order to minimise the Zeeman energy. We compute and plot the skyrmionic texture size $s=2\pi/k$ as a function of the disk sample diameter $d$ in Fig. 4 (b). We obtain the size $s$, that can be interpreted as the length along which the full magnetisation rotation occurs, by fitting $k$ in the $f(x)=\pm\cos(kx)$ function to the simulated iSk and Sk $m_{z}(x)$ profiles. In Fig. 4 (c), we show how the ratio of skyrmionic texture size to disk sample diameter ($s/d$) depends on the hosting nanostructure size. Although this ratio is constant ($s/d\approx 0.6$) for the Sk state, in the iSk case, it is larger for smaller samples and decreases to $s/d\approx 1.5$ in larger nanostructures. In agreement with related findings for two-dimensional disk samples Du2013a we find that both iSk and Sk are able to change their size $s$ in order to accommodate the size of hosting nanostructure, which provides robustness for the technological use. Figure 5: The ground state phase diagram in absence of demagnetisation energy contribution. The scalar value $S_{\text{a}}$ as a function of disk sample diameter $d$ and external magnetic field $H$ computed for the ground state at every phase space point in absence of demagnetisation energy contribution for (a) a 3d mesh and (c) for a 2d mesh. In order to better resolve the boundaries of the Helical (H) state region, the skyrmion number $S$ is shown in (b) and (d). (e) The magnetisation configurations of three identified ground states as well as the out-of-plane magnetisation component $m_{z}(x)$ along the horizontal symmetry line. The emergence of skyrmionic texture ground state in helimagnetic nanostructures at zero external magnetic field and in absence of magnetocrystalline anisotropy is unexpected SkyrmionicsEditorial2013 . Now, we discuss the possible mechanisms, apart from the geometrical confinement, responsible for this stability, in particular (i) the demagnetisation energy contribution, and (ii) the magnetisation variation along the out-of-film direction which can radically change the skyrmion energetics in infinitely large helimagnetic thin films Rybakov2013 . We repeat the simulations using the same method and model as above but ignoring the demagnetisation energy contribution (i.e. setting the demagnetisation energy density $w_{\text{d}}$ in Eq. (1) artificially to zero). We then carry out the calculations (i) on a three-dimensional (3d) mesh (i.e. with spatial resolution in $z$-direction) and (ii) on a two-dimensional (2d) mesh (i.e. with no spatial resolution in $z$-direction, and thus not allowing a variation of the magnetisation along the $z$-direction). The disk sample diameter $d$ is changed between $40\,\text{nm}$ and $180\,\text{nm}$ in steps of $\Delta d=5\,\text{nm}$ and the external magnetic field $\mu_{0}H$ is changed systematically between $0\,\text{T}$ and $0.5\,\text{T}$ in steps of $\mu_{0}\Delta H=25\,\text{mT}$. The two resulting phase diagrams are shown in Fig. 5, where subplots (a) and (c) show $S_{\text{a}}$ as a function of $d$ and $H$. Because the scalar value $S_{\text{a}}$ does not provide enough contrast to determine the boundaries of the new Helical (H) ground state region, the skyrmion number $S$ is plotted for the relevant phase diagram areas and shown in Fig. 5 (b) and Fig. 5 (d). We demonstrate the importance of including demagnetisation effects into the model by comparing Fig. 5 (a) (without demagnetisation energy) and Fig. 2 (a) (with demagnetisation energy). In the absence of the demagnetisation energy, the isolated Skyrmion (Sk) configuration is not found as the ground state at zero applied field; instead, Helical (H) configurations have lower energies. At the same time, the external magnetic field at which the skyrmion configuration ground state disappears is reduced from about $0.7\,\text{T}$ to about $0.44\,\text{T}$. By comparing Fig. 5 (a) computed on a 3d mesh and Fig. 5 (c) computed on a 2d mesh, we can see the importance of spatial resolution in the out-of-plane direction of the thin film, and how it contributes to the stabilisation of isolated Skyrmion (Sk) state. In the 2d model, the field range over which skyrmions can be observed as the ground state is further reduced to approximately [$0.05\,\text{T}$, $0.28\,\text{T}$]. In the 3d mesh model the Sk configuration can reduce its energy by twisting the magnetisation at the top of the disk relative to the bottom of the disk so that along the $z$-direction the magnetisation starts to exhibit (a part of) the helix that arises from the competition between symmetric exchange and DMI energy terms, similar to Ref. Rybakov2013 . A similar twist provides no energetic advantage to the helix configuration, thus the Sk state region in Fig. 5 (a) is significantly larger than the Sk state region in Fig. 5 (c) where the 2d mesh does not allow any variation of the magnetisation along the $z$-direction and thus the partial helix cannot form. While the isolated Skyrmion (Sk) configuration at zero field is a metastable state in the absence of demagnetisation energy, or in 2d models, it is not the ground state anymore as there are Helical (H) equilibrium configurations that have lower total energy. The demagnetisation energy appears to suppress these helical configurations which have a lower energy than the skyrmion. The variation of the magnetisation along the $z$-direction stabilises the skyrmion configuration substantially. These findings demonstrate the subtle nature of competition between symmetric exchange, DM and demagnetisation interactions, and show that ignoring the demagnetisation energy or approximating the thin film helimagnetic samples using two-dimensional models is not generally justified. Hysteretic behaviour. The phase diagram in Fig. 2 (a) shows the regions in which incomplete Skyrmion (iSk) and isolated Skyrmion (Sk) configurations are the ground states. Intuitively, one can assume that for every sample diameter $d$ at zero external magnetic field, there are two possible skyrmionic magnetisation configurations of equivalent energy: core pointing up or core pointing down, suggesting that these textures can be used for an information bit (0 or 1) encoding. We now investigate this hypothesis and study whether an external magnetic field can be used to switch the skyrmionic state orientation (crucial for data imprint) by simulating the hysteretic behaviour of ground state skyrmionic textures. We obtain the hysteresis loops in the usual way by evolving the system to an equilibrium state after changing the external magnetic field, and then using the resulting state as the starting point for a new evolution. In this way, a magnetisation loop takes into account the history of the magnetisation configuration. The external magnetic field $\mu_{0}\mathbf{H}$ is applied in the positive $z$-direction and changed between $-0.5\,\text{T}$ and $0.5\,\text{T}$ in steps of $\mu_{0}\Delta H=5\,\text{mT}$. The hysteresis loops are represented as the dependence of the average out-of-plane magnetisation component $\langle m_{z}\rangle$ on the external magnetic field $H$. The hysteresis loop for a $10\,\text{nm}$ thin film disk sample with $d=80\,\text{nm}$ diameter in which the incomplete Skyrmion (iSk) is the ground state is shown in Fig. 6 (a) as a solid line. Similarly, a solid line in Fig. 6 (b) shows the corresponding hysteresis loop for a larger disk sample with $d=150\,\text{nm}$ diameter in which the isolated Skyrmion (Sk) is the ground state. The hysteresis between two energetically equivalent skyrmionic magnetisation states with the opposite core orientation at zero external magnetic field, shown in Fig. 6 (c), is evident. Moreover, the system does not relax to any other equilibrium state at any point in the hysteresis loop, which demonstrates the bistability of skyrmionic textures in studied system. The area of the open loop in the hysteresis curve is a measure of the work needed to reverse the core orientation by overcoming the energy barrier separating the two skyrmionic states with opposite core orientation. Figure 6: Hysteresis loops and obtained zero-field skyrmionic states with different orientations. The average out-of-plane magnetisation component $\langle m_{z}\rangle$ hysteretic dependence on the external out-of-plane magnetic field $H$ for $10\,\text{nm}$ thin film disk samples for (a) incomplete Skyrmion (iSk) magnetisation configuration in $d=80\,\text{nm}$ diameter sample and (b) isolated Skyrmion (Sk) magnetisation configuration in $d=150\,\text{nm}$ diameter sample. (c) The magnetisation states and $m_{z}(x)$ profiles along the horizontal symmetry lines for positive and negative iSk and Sk core orientations from $H=0$ in the hysteresis loop, both in presence and in absence of demagnetisation energy (demagnetisation-based shape anisotropy). As throughout this work, it is assumed that the simulated helimagnetic material is isotropic, and thus, the magnetocrystalline anisotropy energy contribution is neglected. Due to that, one might expect that the obtained hysteresis loops are the consequence of demagnetisation-based shape anisotropy. To address this, we simulate hysteresis using the same method, but this time in absence of the demagnetisation energy contribution. More precisely, the minimalistic energy model contains only the symmetric exchange and Dzyaloshinskii-Moriya interactions together with Zeeman coupling to an external magnetic field. We show the obtained hysteresis loops in Fig. 6 (a) and (b) as dashed lines. The hysteretic behaviour remains, although all energy terms that usually give rise to the hysteretic behaviour (magnetocrystalline anisotropy and demagnetisation energies) were neglected. This suggests the existence of a new magnetic anisotropy that we refer to as the Dzyaloshinskii- Moriya-based shape anisotropy. Reversal mechanism. The hysteresis loops in Fig. 6 show that skyrmionic textures in confined thin film helimagnetic nanostructures undergo hysteretic behaviour and that an external magnetic field can be used to change their orientation from core pointing up to core pointing down and vice versa. In this section, we discuss the mechanism by which the skyrmionic texture core orientation reversal occurs. We simulate a $150\,\text{nm}$ diameter thin film FeGe disk sample with $t=10\,\text{nm}$ thickness. The maximum spacing between two neighbouring finite element mesh nodes is reduced to $1.5\,\text{nm}$ in order to better resolve the magnetisation field. According to the hysteresis loop in Fig. 6 (b), the switching field $H_{\text{s}}$ of the isolated skyrmion state in this geometry from core orientation down to core orientation up is $\mu_{0}H_{\text{s}}\approx-235\,\text{mT}$. Therefore, we first relax the system at $-210\,\text{mT}$ external magnetic field and then decrease it abruptly to $-250\,\text{mT}$. We simulate the magnetisation dynamics for $1\,\text{ns}$, governed by a dissipative LLG equation Gilbert2004 with Gilbert damping $\alpha=0.3$ Sampaio2013 , and record it every $\Delta t=0.5\,\text{ps}$. Figure 7: The isolated skyrmion orientation reversal in confined three- dimensional helimagnetic nanostructure. (a) The spatially averaged magnetisation components $\langle m_{x}\rangle$, $\langle m_{y}\rangle$, and $\langle m_{z}\rangle$ and (b) skyrmion number $S$, scalar value $S_{\text{a}}$, and total energy $E$ time evolutions in the reversal process over $1\,\text{ns}$. The simulated sample is a $10\,\text{nm}$ thin film disk with $150\,\text{nm}$ diameter. (c) The magnetisation states at different instances of time (points A to F) together with $m_{z}$ colourmap in the $xz$ cross section and $m_{z}(x)$ profiles along the horizontal symmetry line. (d) The $m_{z}$ colourmap and magnetisation field in the central part of $xz$ cross section as shown in an inset together with the position of Bloch point (BP). (e) The BP structure along with colourmaps of magnetisation components which shows that the magnetisation covers the closed surface (sphere surrounding the BP) exactly once. We now look at how certain magnetisation configuration parameters evolve during the reversal process. We show the time-dependent average magnetisation components $\langle m_{x}\rangle$, $\langle m_{y}\rangle$, and $\langle m_{z}\rangle$ in Fig. 7 (a), and on the same time axis, the skyrmion number $S$, scalar value $S_{\text{a}}$ and total energy $E$ in Fig. 7 (b). The initial magnetisation configuration at $t=0\,\text{ns}$ is denoted as A and the final relaxed magnetisation at $t=1\,\text{ns}$ as F. We show in Fig. 7 (c) the out-of-plane magnetisation field component $m_{z}$ in the whole sample, in the $xz$ cross section, as well as along the horizontal symmetry line. At approximately $662\,\text{ps}$ the skyrmionic core reversal occurs and Fig. 7 (b) shows an abrupt change both in skyrmion number $S$ and total energy $E$. We summarise the reversal process with the help of six snapshots shown in Fig. 7 (c). Firstly, in (A-B), the isolated skyrmion core shrinks. At some point the maximum $m_{z}$ value lowers from $1$ to approximately $0.1$ (C). After that, the core reverses its direction (D) and an isolated skyrmion of different orientation is formed (E). From that time onwards, the core expands in order to accommodate the size of hosting nanostructure, until the final state (F) is reached. The whole reversal process is also provided in Supplementary Video 1. In order to better understand the actual reversal of the skyrmionic texture core between $t_{1}\approx 661\,\text{ps}$ and $t_{2}\approx 663\,\text{ps}$, we show additional snapshots of the magnetisation vector field and $m_{z}$ colourmap in the $xz$ cross section in Fig. 7 (d). The location marked by a circle in subplots L, M, and N identifies a Bloch Point (BP): a noncontinuous singularity in the magnetisation pattern where the magnetisation magnitude vanishes to zero Feldtkeller1965 ; Doring1968 . Because micromagnetic models assume constant magnetisation magnitude, the precise magnetisation configuration at the BP cannot be obtained using micromagnetic simulations Andreas2014 . However, it is known how to identify the signature of the BP in such situations: the magnetisation direction covers any sufficiently small closed surface surrounding the BP exactly once Slonczewski1975 ; Thiaville2003 . We illustrate this property in Fig. 7 (e) using a vector plot together with $m_{x}$, $m_{y}$, and $m_{z}$ colour plots that show the structure of a Bloch point. We conclude that the isolated skyrmion core reversal occurs via Bloch Point (BP) occurrence and propagation. Firstly, at $t\approx 661.5\,\text{ps}$ the BP enters the sample at the bottom boundary and propagates upwards until $t\approx 663\,\text{ps}$ when it leaves the sample at the top boundary. In the Supplementary Video 2 the isolated skyrmion core reversal dynamics is shown. We note that the Bloch point moves upwards in Fig. 7 (d) but one may ask whether an opposite propagation direction can occur and how the Bloch point structure is going to change. We demonstrate that which of these two propagation directions will occur in the reversal process depends on the simulation parameters. The reversal mechanism simulation was repeated with increased Gilbert damping ($\alpha=0.35$ instead of $\alpha=0.3$) and the results showing the downwards propagation are shown in the Supplementary Section S3. We hypothesise that both reversal paths (Bloch point moving upwards or downwards) exhibit the same energy barriers and that the choice of path is a stochastic process. By analysing the results from Fig. 7 (d) and (e) and Supplementary Fig. 6, we also observe that the change in the BP propagation direction implies the change of the BP structure since the out-of- plane magnetisation component $m_{z}$ field reverses in the vicinity of BP. ## II Discussion Through systematic micromagnetic study of equilibrium states in helimagnetic confined nanostructures, we identified the ground states and reported the (meta)stability regions of other equilibrium states. We demonstrated in Fig. 2 that skyrmionic textures in the form of incomplete Skyrmion (iSk) and isolated Skyrmion (Sk) configurations are the ground states in disk nanostructures, and that this occurs in a wide $d$–$H$ parameter space range. We have carried out similar studies for a square geometry and obtain qualitatively similar results. Of particular importance is that iSk and Sk states are the ground states at zero external magnetic field which is in contrast to infinite thin film and bulk helimagnetic samples. We note that neither an external magnetic field is necessary nor magnetocrystalline anisotropy is required for this stability. We also note in Fig. 4 (c) that there is significant flexibility in the skyrmionic texture size which provides robustness for technology built on skyrmions, where fabrication of nanostructures and devices introduces unavoidable variation in geometries. We have established that including the demagnetisation interaction is crucial for the system investigated here, i.e. in the absence of demagnetisation effects, there are other magnetisation configurations with energies lower than that of the incomplete and isolated skyrmion. We also note that the translational variance of the magnetisation from the lower side of the thin film (at $z=0\,\text{nm}$) to the top (at $z=10\,\text{nm}$) is essential for the physics reported here: if we use a two-dimensional micromagnetic simulation (i.e. assuming translational invariance of the magnetisation $\mathbf{m}$ in the out-of-plane direction), the isolated skyrmion configuration does not arise as the ground state. Our interpretation is that for skyrmion-like configurations the twist of $\mathbf{m}$ between top and bottom layer allows the system’s energy to reduce significantly while such a reduction is less beneficial for other configurations such as helices; inline with recent predictions in the case of infinite thin films Rybakov2013 . Accordingly, we conclude that three-dimensional helimagnetic nanostructure models, where demagnetisation energy contribution is neglected, or the geometry approximated using a two-dimensional mesh, are not generally justified. Because of the specific boundary conditions Rohart2013 and the importance of including the demagnetisation energy contribution, our predictions cannot be directly applied to other helimagnetic materials without repeating the stability study. For instance, although the size of skyrmionic textures in this study was based on cubic FeGe helimagnetic material with helical period $L_{\text{D}}=70\,\text{nm}$, in order to encourage the experimental verification of our predictions, this study could be repeated for materials with smaller $L_{\text{D}}$. In such materials the skyrmionic core size is considerably reduced, which allows the reduction of hosting nanostructure size and is an essential requirement for advancing future information storage technologies. Similarly, the ordering temperature of simulated FeGe helimagnetic material, $T_{\text{C}}=278.7\,\text{K}$ Lebech1989 , is lower than the room temperature, which means that a device operating at the room temperature cannot be constructed using this material. Because of that, in Supplementary Section S4, we demonstrate that our predictions are still valid if the ordering temperature of simulated B20 helimagnetic material is artificially increased to $350\,\text{K}$. We demonstrate in Fig. 6 that skyrmionic textures in confined helimagnetic nanostructures exhibit hysteretic behaviour as a consequence of energy barriers between energetically equivalent stable configurations (skyrmionic texture core pointing up or down). In the absence of magnetocrystalline anisotropy and if the demagnetisation energy (demagnetisation-based shape anisotropy) is removed from the system’s Hamiltonian, the hysteretic behaviour is still present, demonstrating the existence of a novel Dzyaloshinskii- Moriya-based shape anisotropy. Finally, we show how the reversal of the isolated skyrmion core orientation is facilitated by the Bloch point occurrence and propagation, and demonstrate that the Bloch point can propagate in both directions along the out-of-plane $z$-direction. All data obtained by micromagnetic simulations in this study and used to create figures both in the main text and in the Supplementary Information are included in Supplementary Data. ## III Methods Model. We use an energy model consistent with a non-centrosymmetric cubic B20 (P213 space group) crystal structure. This is appropriate for a range of isostructural compounds and pseudo-binary alloys in which skyrmionic textures have been experimentally observed Muhlbauer2009 ; Jonietz2010 ; Yu2010 ; Yu2011 ; Yu2012 ; Wilhelm2012 ; Huang2012 ; Seki2012 . The magnetic free energy of the system $E$ contains several contributions and can be written in the form: $E=\int\left[w_{\text{ex}}+w_{\text{dmi}}+w_{\text{z}}+w_{\text{d}}+w_{\text{a}}\right]\,\text{d}^{3}r.$ (1) The first term is the symmetric exchange energy density $w_{\text{ex}}=A\left[(\nabla m_{x})^{2}+(\nabla m_{y})^{2}+(\nabla m_{z})^{2}\right]$ with exchange stiffness material parameter $A$, where $m_{x}$, $m_{y}$, and $m_{z}$ are the Cartesian components of the vector $\mathbf{m}=\mathbf{M}/M_{\text{s}}$ that describes the magnetisation $\mathbf{M}$, with $M_{\text{s}}=\|\mathbf{M}\|$ being the saturation magnetisation. The second term is the Dzyaloshinskii-Moriya Interaction (DMI) energy density $w_{\text{dmi}}=D\mathbf{m}\cdot\left(\nabla\times\mathbf{m}\right)$, obtained by constructing the allowed Lifshitz invariants for the crystallographic class T Bak1980 ; Bogdanov1989 , where $D$ is the material parameter. The third term is the Zeeman energy density term $w_{\text{z}}=-\mu_{0}\mathbf{H}\cdot\mathbf{M}$ which defines the coupling of magnetisation to an external magnetic field $\mathbf{H}$. The $w_{\text{d}}$ term represents the demagnetisation (magnetostatic) energy density. The last term $w_{\text{a}}$ is the magnetocrystalline anisotropy energy density, and because the simulated material is assumed to be isotropic, we neglect it throughout this work. Neglecting this term also allows us to determine whether the magnetocrystalline anisotropy is a crucial mechanism allowing the stability of skyrmionic textures in confined helimagnetic nanostructures. The Landau-Lifshitz-Gilbert (LLG) equation Gilbert2004 : $\frac{\partial\mathbf{m}}{\partial t}=\gamma^{}\mathbf{m}\times\mathbf{H}_{\text{eff}}+\alpha\mathbf{m}\times\frac{\partial\mathbf{m}}{\partial t},$ (2) governs the magnetisation dynamics, where $\gamma^{}=\gamma(1+\alpha^{2})$, with $\gamma<0$ and $\alpha$ being the gyromagnetic ratio and Gilbert damping, respectively. We compute the effective magnetic field using $\mathbf{H}_{\text{eff}}=-(\delta w/\delta\mathbf{m})/(\mu_{0}M_{\text{s}})$, where $w$ is the total energy density functional. With this model, we solve for magnetic configurations $\mathbf{m}$ using the condition of minimum torque arrived by integrating a set of dissipative, time-dependent equations. We validated the boundary conditions by a series of simulations reproducing the results in Ref. Rohart2013 ; Sampaio2013 . Simulator. We developed a micromagnetic simulation software, inspired by the Nmag simulation tool Fischbacher2007 ; Nmag . Unlike Nmag, we use the FEniCS project Logg2012 instead of the Nsim multi-physics library Fischbacher2007 for the finite element low-level operations. In addition, we use IPython Perez2007 ; IPython and Matplotlib Hunter2007 ; Matplotlib extensively in this work. Material parameters. We estimate the material parameters in our simulations to represent the cubic B20 FeGe helimagnet with four Fe and four Ge atoms per unit cell Pauling1948 and crystal lattice constant $a=4.7\,\text{\AA}$ Richardson1967 . The local magnetic moments of iron and germanium atoms are $1.16\mu_{\text{B}}$ and $-0.086\mu_{\text{B}}$ Yamada2003 , respectively, where $\mu_{\text{B}}$ is the Bohr magneton constant. Accordingly, we estimate the saturation magnetisation as $M_{\text{s}}=4N(1.16-0.086)\mu_{\text{B}}=384\,\text{kA}\,\text{m}^{-1}$, with $N=a^{-3}$ being the number of lattice unit cells in a cubic metre. The spin-wave stiffness is $D_{\text{sw}}=a^{2}T_{\text{C}}$ Grigoriev2005 , where the FeGe ordering temperature is $T_{\text{C}}=278.7\,\text{K}$ Lebech1989 . Consequently, the exchange stiffness parameter value is $A=D_{\text{sw}}M_{\text{s}}/(2g\mu_{\text{B}})=8.78\,\text{pJ}\,\text{m}^{-1}$ Hamrle2009 , where $g\approx 2$ is the Landé $g$-factor. The estimated DMI material parameter $D$ from the long-range FeGe helical period $L_{\text{D}}=70\,\text{nm}$ Lebech1989 , using $L_{\text{D}}=4\pi A/\|D\|$ Wilhelm2012 , is $\|D\|=1.58\,\text{mJ}\,\text{m}^{-2}$. Skyrmion number $S$ and injective scalar value $S_{\mathrm{a}}$. In order to support the discussion of skyrmionic textures, the topological skyrmion number Heinze2011 $S^{\text{2D}}=\frac{1}{4\pi}\int\mathbf{m}\cdot\left(\frac{\partial\mathbf{m}}{\partial x}\times\frac{\partial\mathbf{m}}{\partial y}\right)\,\text{d}^{2}r,$ (3) can be computed for two-dimensional samples hosting the magnetisation configuration. However, for confined systems, the skyrmion number $S^{\text{2D}}$ is not quantised into integers Sampaio2013 ; Du2013 , and therefore, a more suitable name for $S^{\text{2D}}$ may be the “scalar spin chirality” (and consequently the expression under an integral would be called the “spin chirality density”), but we will follow the existing literature Sampaio2013 ; Du2013 and refer to $S^{\text{2D}}$ as the skyrmion number. We show its dependence on different skyrmionic textures that can be observed in confined helimagnetic nanostructures in Supplementary Fig. 2 (b), demonstrating that the skyrmion number in confined geometries is not an injective function since it does not preserve distinctness (one-to-one mapping between skyrmionic textures and skyrmion number value $S^{\text{2D}}$). Therefore, for two-dimensional samples, we define a different scalar value $S^{\text{2D}}_{\text{a}}=\frac{1}{4\pi}\int\left\|\mathbf{m}\cdot\left(\frac{\partial\mathbf{m}}{\partial x}\times\frac{\partial\mathbf{m}}{\partial y}\right)\right\|\,\text{d}^{2}r,$ (4) and show its dependence on different skyrmionic textures in Supplementary Fig. 2 (b). This scalar value is injective and provides necessary distinctness between $S_{\text{a}}^{2D}$ values for different skyrmionic states. In terms of the terminology discussion above regarding $S^{\text{2D}}$, the entity $S_{\text{a}}^{2D}$ describes the “scalar absolute spin chirality”. We also emphasise that although the skyrmion number $S^{\text{2D}}$ has a clear mathematical Braun2012 and physical Schulz2012 interpretation, we define the artificial injective scalar value $S_{\text{a}}$ only to support the classification and discussion of different skyrmionic textures observed in this work. Skyrmion number $S^{\text{2D}}$ and artificially defined scalar value $S_{\text{a}}^{\text{2D}}$, given by Eq. (3) and Eq. (4), respectively, are valid only for the two-dimensional samples hosting the magnetisation configuration. However, in this work, we also study three-dimensional samples and, because of that, we now define a new set of expressions taking into account the third dimension. The skyrmion number in three-dimensional samples $S^{\text{3D}}$ we compute using $S^{\text{3D}}=\frac{1}{8\pi}\int\mathbf{m}\cdot\left(\frac{\partial\mathbf{m}}{\partial x}\times\frac{\partial\mathbf{m}}{\partial y}\right)\,\text{d}^{3}r,$ (5) as suggested by Lee et al. Lee2009 , which results in a value proportional to the anomalous Hall conductivity. Similar to the two-dimensional case, we also define the artificial injective scalar value $S_{\text{a}}^{\text{3D}}$ for three-dimensional samples as $S^{\text{3D}}_{\text{a}}=\frac{1}{8\pi}\int\left\|\mathbf{m}\cdot\left(\frac{\partial\mathbf{m}}{\partial x}\times\frac{\partial\mathbf{m}}{\partial y}\right)\right\|\,\text{d}^{3}r.$ (6) In order to allow the $S^{\text{3D}}_{\text{a}}$ value to fall within the two- dimensional skyrmionic textures classification scheme, we normalise the computed $S^{\text{3D}}_{\text{a}}$ value by a constant ($t/2$, where $t$ is the sample thickness). For simplicity, in this work, we refer to both two-dimensional and three- dimensional skyrmion number and scalar value expressions as $S$ and $S_{\text{a}}$ because it is always clear what expression has been used according to the dimensionality of the sample. ## IV References ## References * (1) Richter, H. J. The transition from longitudinal to perpendicular recording. _J. Phys. D. Appl. 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Phys._ 61, 1–116 (2012). * (59) Schulz, T. _et al._ Emergent electrodynamics of skyrmions in a chiral magnet. _Nat. Phys._ 8, 301–304 (2012). * (60) Lee, M., Kang, W., Onose, Y., Tokura, Y. & Ong, N. Unusual Hall effect anomaly in MnSi under pressure. _Phys. Rev. Lett._ 102, 186601 (2009). ## V Acknowledgements This work was financially supported by the EPSRC’s Doctoral Training Centre (DTC) grant EP/ G03690X/1. R.L.S. acknowledges the EPSRC’s EP/M024423/1 grant support. D.C.-O. acknowledges the financial support from CONICYT Chilean scholarship programme Becas Chile (72140061). We acknowledge the use of the IRIDIS High Performance Computing Facility, and associated support services at the University of Southampton, in the completion of this work. We also thank Karin Everschor-Sitte for helpful discussions. ## VI Author Contributions M.B. and H.F conceived the study, and M.B. performed micromagnetic simulations. R.L.S. devised the analytic model and discussed its implications. R.C. contributed to the simulations and analysis of equilibrium states. D.C., M.-A.B., M.A., W.W., M.B., R.C., M.V., D.C.-O. and H.F. developed the micromagnetic finite element based simulator. M.V. and M.A. enabled running simulations on IRIDIS High Performance Computing Facility. M.B., H.F., R.L.S., and O.H. interpreted the data and prepared the manuscript. ## VII Competing financial interests The authors declare no competing financial interests. See pages 1 of supplementary_information.pdfSee pages 2 of supplementary_information.pdfSee pages 3 of supplementary_information.pdfSee pages 4 of supplementary_information.pdfSee pages 5 of supplementary_information.pdfSee pages 6 of supplementary_information.pdfSee pages 7 of supplementary_information.pdfSee pages 8 of supplementary_information.pdf	arxiv-papers	2013-12-30T09:33:00	2024-09-04T02:49:56.072865	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "authors": "Marijan Beg, Rebecca Carey, Weiwei Wang, David Cort\\'es-Ortu\\~no, Mark\n Vousden, Marc-Antonio Bisotti, Maximilian Albert, Dmitri Chernyshenko, Ondrej\n Hovorka, Robert L. Stamps, and Hans Fangohr", "submitter": "Marijan Beg", "url": "https://arxiv.org/abs/1312.7665" }
1312.7669	# Spectroscopy of di-meson bound states in charm and beauty sector Smruti Patel [email protected] Manan Shah Arpit Parmar P.C.Vinodkumar p.c. [email protected] Deptartment of Physics, Sardar Patel University,Vallabh Vidyanagar-388120, Gujarat, INDIA ## Introduction Very recently there exists increasing attention towards the study of four quark states as di-hadronic molecular states followed by the recent discovery of $Z_{c}$(3900) state by two seperate experimental groups BES III Ablikim and BELLE Collabaration Liu . The interpretation of the new state has triggered a considerable amount of theoretical work, especially due to the controversies related their internal structure. Moreover, very recently BELLE Collaboration has made the tantalizing observation of two new charged bottom resonances, namely Zb(10610) and Zb(10650). Since all the standard bottomonia are neutrally charged, these two resonances have a flavour only compatible with $b\bar{b}u\bar{d}$ tetraquarks Adachi ; Bondar . Motivated by striking observation of tetraquark states, here we wish to predict the interpretation of these states as di-mesonic molecules composed of a pair of heavy mesons such as $D\bar{D}$, $D\bar{D}^{}$, $D^{}\bar{D}^{}$, $D^{+}\bar{D}^{}$ in the charm sector and $B\bar{B}$, $B^{}\bar{B}^{}$, $B\bar{B}^{}$ in the bottom sector. ## Phenomenology Different attempts have been made for the interpretation of exotic hadronic states of four quark system such as di-mesonic states, hadroquarkonium states and tetraquark states. Investigations into the existence of multiquark states have begun in the early days of QCD Jaffe ; Strottman . However, little success has been achieved in understanding tetraquark states due to the non- perturbative nature of QCD at the hadronic scale. The hadron molecular considerations does simplify this difficulty by replacing interquark color interacion with a residual strong interactions between two color singlet hadrons. Thus, for the present study of di-mesonic molecules, we employ Woods Saxon plus coulomb type of potential between two color singlet hadrons of the form $V(r)=\frac{V_{0}}{1+\exp^{(\frac{r-R}{a})}}-\frac{B}{r}$ (1) Table 1: Mass spectra of di-mesonic systems (in MeV). Molecule \| $J^{pc}$ \| BE \| $E_{(j_{1},j_{2};J)}$ \| $M_{SA}$ \| $M_{J}$ \| EXP. \| Others ---\|---\|---\|---\|---\|---\|---\|--- \| \| \| \| \| \| \| 3738rai $D\bar{D}$ \| $0^{++}$ \| 28.87 \| 0.0 \| 3758.87 \| 3758.87 \| - \| 3760$\pm 100$Jian \| \| \| \| \| \| \| $3715^{+24}_{-27}$Hidalgo \| \| \| \| \| \| \| 3878[9] $D\bar{D}^{}$ \| $1^{++}$ \| 28.79 \| 0.0 \| 3900.8 \| 3900.8 \| X(3871.68$\pm 0.17$)Beringer \| 3880$\pm 110$Jian $D^{+}\bar{D}^{}$ \| $1^{+-}$ \| 28.78 \| 0.0 \| 3904.79 \| 3904.79 \| $Z_{c}^{+}(3898\pm 5)$ \| - $D^{}\bar{D}^{}$ \| $2^{++}$ \| 28.69 \| 0.358 \| 4042.69 \| 4043.05 \| - \| 4062[9], $4012^{+4}_{-9}$Hidalgo \| $1^{+-}$ \| \| -0.358 \| \| 4042.33 \| - \| 3974rai , $3958^{+24}_{-27}$Hidalgo \| $0^{++}$ \| \| -0.72 \| \| 4041.97 \| - \| 3930rai $B\bar{B}$ \| $0^{++}$ \| 19.06 \| 0.0 \| 10577.1 \| 10577.1 \| - \| - $B\bar{B}^{}$ \| $1^{+-}$ \| 18.99 \| 0.0 \| 10623 \| 10623 \| $Z_{b}$(10607$\pm 2.0$)Bondar \| - $B^{}\bar{B}^{}$ \| $2^{++}$ \| 18.92 \| 1.367 \| 10668.9 \| 10670.28 \| - \| - \| $1^{+-}$ \| \| -1.367 \| \| 10667.53 \| $Z_{b}$(10652.2$\pm 1.5$)Bondar \| - \| $0^{++}$ \| \| -2.73 \| \| 10666.16 \| - \| - The potential parameters employed here are as follows: a$=$-0.0387 fm; $V_{0}$=0.03 GeV; B=0.04; R=0.8875 fm. Binding energy is obtained by numerically solving Schrödinger equation using mathematica notebook of Range- Kutta method. The non-relativistic Schrodinger bound-state mass (spin average mass) of the di-mesonic system is obtained as $M_{SA}=m{{}_{1}}+m{{}_{2}}+BE$ (2) We introduce j-j coupling term to obtain the hyperfine splitting of the different di-meson states. Accordingly, the di-mesonic molecular mass is obtained as $M_{J}=M_{SA}+E_{(j_{1},j_{2};J)}$ (3) Where m1 and m2 are the masses of the constituent mesons, BE represents the binding energy of the di-mesonic system and $E_{(j_{1},j_{2};J)}$ represents the spin-dependent term. The hyperfine interaction is computed using the expression similar to the hyperfine interactions for quarkonia but without considering color factor and is taken as $E_{(j_{1},j_{2};J)}=\frac{2<j_{1}.j_{2}>_{J}\|R(0)\|^{2}}{3m_{1}m_{2}}$ (4) ## Results and conclusion Table 1 summarizes the binding energies and low lying masses of the di-mesonic states. The recent experimental exotic states and other theoretical results are also presented for comparision. In the present work, the mass of $D\bar{D}^{}$ is 18 MeV above the experimental value. Overall agreement of the present results gives us a clue to believe that X(3872) state is nothing but the loosely bound $D\bar{D}^{}$ meson molecule and its companion $D\bar{D}$$(0^{++})$ is predicted to be at 3759 MeV. In the same spirit two bottomonium-like twin resonances Zb(10610) and Zb(10650) are found to be $B\bar{B}^{}$ and $B^{}\bar{B}^{}$ molecules respectively. And recent $Z_{c}^{+}$(3900) state is found to be the $D^{+}\bar{D}^{}$ molecular state. Other positive parity molecular states ($D^{}\bar{D}^{})_{J=0,1,2}$ close to $\psi$(4040) are predicted around 4042 MeV. Other di-mesonic molecular states in the charm and beauty sector are also presented in table 1. Many of these states require further experimental support. ## Acknowledgments The work is part of a Major research project No. F. 40-457/2011(SR) funded by UGC. ## References * (1) M. Ablikim et al. (BESIII Collaboration), Phys. Rev. Lett. 110, 252001 (2013). * (2) Z. Q. Liu et al. (Belle Collaboration),Phys. Rev. Lett. 110, 252002 (2013). * (3) I. Adachi et al. (Belle Collaboration), arXiv:hep-ex/1105.4583. * (4) A. Bondar et al. (Belle Collaboration), Phys. Rev. Lett. 108, 122001 (2012). * (5) R L Jaffe, Phys. Rev. D 15, 281 (1977). * (6) D Strottman,Phys. Rev. D 20, 748 (1979). * (7) J. Beringer et al. (Particle Data Group), Phys. Rev. D 86, 010001 (2012). * (8) Jian-rong et al. Phys. Rev. D 80, 056004 (2009). * (9) Ajay Kumar rai et al. Nuclear Physics A 782, 406 (2007) * (10) C. Hidalgo- Duque et al.Phys. Rev. D 87,076006 (2013)	arxiv-papers	2013-12-30T09:52:38	2024-09-04T02:49:56.083387	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Smruti Patel, Manan Shah, Arpit Parmar and P.C.Vinodkumar", "submitter": "Smruti Patel", "url": "https://arxiv.org/abs/1312.7669" }
1312.7806	∎ e1e-mail: [email protected] e2e-mail: [email protected] 11institutetext: Department of Physics, Eastern Mediterranean University, G. Magusa, north Cyprus, Mersin 10, Turkey 22institutetext: Institute of Theoretical Physics, Faculty of Mathematics and Physics, Charles University in Prague, V Holešovičkách 2, 180 00 Prague 8, Czech Republic # Resolution of curvature singularities from quantum mechanical and loop perspective T. Tahamtane1,addr1 O. Svíteke2,addr2 (Received: date / Accepted: date) ###### Abstract We analyze the persistence of curvature singularities when analyzed using quantum theory. First, quantum test particles obeying the Klein-Gordon and Chandrasekhar-Dirac equation are used to probe the classical timelike naked singularity. We show that the classical singularity is felt even by our quantum probes. Next, we use loop quantization to resolve singularity hidden beneath the horizon. The singularity is resolved in this case. ###### Keywords: Singularity resolution global monopole loop quantization ## 1 Introduction One of the important predictions of the Einstein’s theory of general relativity is the formation of spacetime singularities. In classical general relativity, singularities are defined as points in which the evolution of timelike or null geodesics is not defined after a finite proper time. According to the classification of the classical singularities devised by Ellis and Schmidt 01 , scalar curvature singularities are the most strongest one in the sense that the spacetime posses incomplete geodesics ending in them and all the physical quantities such as the gravitational field (scalars formed from curvature tensor), energy density and tidal forces diverge at the singular point. But such divergence of physical quantities signify the breakdown of predictive power of classical general relativity. If these singularities are covered by horizon (as supposed by Cosmic Censorship Conjecture) then at least the physically most relevant region of spacetime is under control. Naked singularities (those not covered by horizon), on the other hand, provide an observer with causal access to the region of diverging quantities and should be avoided. However, even singularities covered by the horizon can be accessed by an infalling observer and, more importantly, we would like to have a theory that lacks divergences, at least effectively. The natural direction for resolving the problem of singularities in classical theory is investigating their persistence in quantum picture. Although we do not have a final quantum theory of gravity we still have several tools for analyzing quantum singularities. The first approach relies on examining properties of quantum particle wave functions on the background represented by the studied geometry. This is a frequently used technique based on well understood properties of operators on a Hilbert space. To move further, one might proceed to using quantum fields and possibly even the backreaction of background geometry using semiclassical Einstein equations with suitably regularized stress energy tensor. Finally, one can apply quantization of the geometry itself. The last approach is in principle the most precise but relies on the selected quantization method and we have no generally accepted one in case of gravity. Quantum singularities were studied for different specific situations (and using also generalizations), mainly using the first approach 6 ; 7 ; 8 ; 9 ; 10 ; 11 ; 12 ; 13 ; 140 ; 14 ; 15 ; 16 ; 17 . Recently, singularities in f(R) gravity were investigated in the presence of linear electromagnetic field 170 . We will apply two of the above mentioned approaches for analysis of singularity in case of the general metric of global monopole 1 , which is determined by two parameters - one characterizing the "Schwarzschild-type mass" and the other one the deficit of solid angle. The singularity is generally covered by single horizon but the class of metrics also contains, as a special case, a naked singularity which is analyzed from quantum mechanical point of view using the technique of Horowitz and Marolf 4 (who continued the pioneering work of Wald 3 ). This method for analyzing timelike singularities is based on investigation of self-adjoint extensions of the evolution operator associated with the given wave equation. If it is unique the spacetime is deemed quantum mechanically non-singular. The analysis is carried out for relativistic quantum particle wave equations on a fixed background. Specifically, we review the previous results for Klein-Gordon equation and show the calculation using Newman-Penrose formalism for the Dirac equation, both in the case of pure global monopole with naked singularity for which the method was developed. But as already mentioned, the most reliable method when trying to investigate the possible removal of the singularities from geometry is quantum gravity. Here we have selected loop quantization method inspired by 22 ; 23 ; 24 , where the spacetime beneath the horizon (in the non-naked subclass) is isometric to the Kantowski-Sachs cosmology. Then one can apply methods from Loop Quantum Cosmology (LQC), that are based on loop quantization on the restricted configuration space. In this way, the results for resolution of initial cosmological singularity are translated to statements about the singularity at the origin $r=0$. ## 2 The General Metric for Global monopole It is well known that different types of non-standard topological objects may have been formed during initial Universe evolution, such as domain walls, cosmic strings and monopoles 1 ; 2 . The basic idea is that these topological defects have formed as a result of a breakdown of local or global gauge symmetries. The simplest model that gives rise to global monopole is described by the Lagrangian $L=\frac{1}{2}\partial_{\mu}\phi^{a}\partial^{\mu}\phi^{a},$ (1) where $\phi^{a}$ is a triplet of scalar fields, $a=1,2,3.$ The model has a global $O\left(3\right)$ symmetry, which is spontaneously broken to $U\left(1\right)$. The field configuration describing the monopole is $\phi^{a}=\eta\frac{x^{a}}{r}$ where $x^{a}x^{a}=r^{2}$. We assume that underlying geometry is general static spherically symmetric described by the line element $ds^{2}=-B\left(r\right)dt^{2}+\frac{dr^{2}}{A\left(r\right)}+r^{2}\left(d\theta^{2}+\sin^{2}\theta d\phi^{2}\right),$ (2) with the usual relation between the spherical coordinates, $r,\theta,\phi$ and the Cartesian coordinates $x^{a}$. The Lagrangian for the above given field configuration simplifies in the following way $L=\frac{1}{2}\left(\partial_{\theta}\phi^{a}\partial^{\theta}\phi^{a}+\partial_{\phi}\phi^{a}\partial^{\phi}\phi^{a}\right)=\frac{\eta^{2}}{r^{2}},$ (3) and the diagonal energy momentum tensor is given by these components $T_{t}^{t}=T_{r}^{r}=-\frac{\eta^{2}}{r^{2}},\text{ \ \ }T_{\theta}^{\theta}=T_{\phi}^{\phi}=0.\text{\ }$ (4) The general solution of the Einstein equations with this $T_{\mu}^{\nu}$ is $B=A^{-1}=1-8\pi G\eta^{2}-\frac{2GM}{r}$ (5) where $M$ is a constant of integration. The metric describes a black hole of mass $M$, carrying a global monopole charge characterized by $\eta$. Such a black hole can be formed if a global monopole is swallowed by an ordinary black hole 1 . The Kretschmann scalar which indicates the formation of curvature singularity is given by $\mathcal{K}=\frac{48M^{2}G^{2}}{r^{6}}+\frac{128M\pi G^{2}\eta^{2}}{r^{5}}+\frac{256\pi^{2}G^{2}\eta^{4}}{r^{4}}.$ (6) It is obvious that $r=0$ is a typical central curvature singularity (scalar curvature singularity according to above mentioned classification) and the dominant contribution comes from term corresponding to black hole mass $M$. If $M>0$ the singularity is evidently spacelike and covered by a single horizon. ## 3 Global monopole and its singularity If we assume that the mass term is negligible on the astrophysical scale or vanishing, we will have $ds^{2}=-\left(1-8\pi G\eta^{2}\right)dt^{2}+\frac{dr^{2}}{\left(1-8\pi G\eta^{2}\right)}+r^{2}d\Omega^{2},$ (7) For simplicity we choose $\alpha^{2}=1-8\pi G\eta^{2}$ and by rescaling $r$ and $t$ variables, we can rewrite the monopole metric as $ds^{2}=-dt^{2}+dr^{2}+\alpha^{2}r^{2}\left(d\theta^{2}+\sin^{2}\theta d\varphi^{2}\right),$ (8) If we calculate the Kretschmann scalar, $\mathcal{K}=4\frac{\left(\alpha^{2}-1\right)^{2}}{r^{4}\alpha^{4}}.$ still there is a weaker singularity at $r=0$. From the metric (7) one can immediately see that the singularity is timelike. This time, because our simplified metric does not have the horizon the singularity is naked. ## 4 Naked Singularity As mentioned in the Introduction naked singularity poses a serious problems and its resolution would be desirable. In this section, the occurrence of naked singularities in global monopole will be analyzed from quantum mechanical point of view. In probing the singularity, quantum test particles obeying the Klein-Gordon and Dirac equations are used. The reason for using two different types of fields is to clarify whether the classical singularity is sensitive to spin of the fields. According to Horowitz and Marolf (HM) 4 , the singular character of the spacetime is defined as the ambiguity in the evolution of the wave functions. That is to say, the singular character is determined based on the number of self-adjoint extensions of the evolution operator to the entire Hilbert space. If the extension is unique, it is said that the spacetime is quantum mechanically regular. The brief review of the method follows: Consider a static spacetime $\left(\mathcal{M},g_{\mu\nu}\right)$ with a timelike Killing vector field $\xi^{\mu}$. Let $t$ denote the Killing parameter and $\Sigma$ denote a static slice. The Klein-Gordon equation in this space is $\left(\nabla^{\mu}\nabla_{\mu}-M^{2}\right)\psi=0.$ (9) This equation can be written in the form $\frac{\partial^{2}\psi}{\partial t^{2}}=\sqrt{f}D^{i}\left(\sqrt{f}D_{i}\psi\right)-fM^{2}\psi=-A\psi,$ (10) in which $f=-\xi^{\mu}\xi_{\mu}$ and $D_{i}$ is the spatial covariant derivative on $\Sigma$. We assume that the Hilbert space $\mathcal{H}=L^{2}\left(\Sigma,\mu\right)$ is the space of square integrable functions on $\Sigma$ with appropriate measure $\mu$. Initially the operator $A$ is defined on smooth functions with compact support $C_{0}^{\infty}(\Sigma)$. Since the operator $A$ is real, positive and symmetric its self-adjoint extensions always exist. If it has a unique extension $A_{E},$ then $A$ is called essentially self-adjoint 18 ; 19 ; 20 . Accordingly, the Klein-Gordon equation for a free particle satisfies $i\frac{d\psi}{dt}=\sqrt{A_{E}}\psi,$ (11) with the solution $\psi\left(t\right)=\exp\left[-it\sqrt{A_{E}}\right]\psi\left(0\right).$ (12) If $A$ is not essentially self-adjoint, the future time evolution of the wave function (12) is ambiguous. Then, HM criterion defines the spacetime as quantum mechanically singular. However, if there is only a single self-adjoint extension, the operator $A$ is said to be essentially self-adjoint and the quantum evolution described by equation (12) is uniquely determined by the initial conditions. According to the HM criterion, this spacetime is said to be quantum mechanically non-singular. In order to determine the number of self-adjoint extensions, the concept of deficiency indices is used. The deficiency subspaces $N_{\pm}$ are defined by ( see Ref.5 for a detailed mathematical background), $\displaystyle N_{+}$ $\displaystyle=\\{\psi\in D(A^{\ast}),\text{\ \ }A^{\ast}\psi=Z_{+}\psi,\text{\ \ }ImZ_{+}>0\\}$ $\displaystyle\text{\ \ with dimension }n_{+}$ (13) $\displaystyle N_{-}$ $\displaystyle=\\{\psi\in D(A^{\ast}),\text{ \ \ }A^{\ast}\psi=Z_{-}\psi,\text{\ \ }ImZ_{-}<0\\}$ $\displaystyle\text{\ \ with dimension }n_{-}$ The dimensions $\left(\text{ }n_{+},n_{-}\right)$ are the deficiency indices of the operator $A$. The indices $n_{+}(n_{-})$ are completely independent of the choice of $Z_{+}(Z_{-})$ depending only on whether $Z$ lies in the upper (lower) half complex plane. Generally one takes $Z_{+}=i\lambda$ and $Z_{-}=-i\lambda$ , where $\lambda$ is an arbitrary positive constant necessary for dimensional reasons. The determination of deficiency indices then reduces to counting the number of solutions of $A^{\ast}\psi=Z\psi$ ; (for $\lambda=1$), $A^{\ast}\psi\pm i\psi=0$ (14) that belong to the Hilbert space $\mathcal{H}$. If there are no square integrable solutions ( i.e. $n_{+}=n_{-}=0)$, the operator $A$ possesses a unique self-adjoint extension and it is essentially self-adjoint. Consequently, a sufficient condition for the operator $A$ to be essentially self-adjoint is to find only solutions satisfying Eq. (14) that do not belong to the Hilbert space. ### 4.1 Klein - Gordon Fields The Klein-Gordon equation for a massless scalar particle is given by, $\square\psi=g^{-1/2}\partial_{\mu}\left[g^{1/2}g^{\mu\nu}\partial_{\nu}\right]\psi=M^{2}\psi.$ (15) For the metric (8), the Klein-Gordon equation becomes, $\frac{\partial^{2}\psi}{\partial t^{2}}=-\left\\{\frac{\partial^{2}\psi}{\partial r^{2}}+\frac{1}{r^{2}\alpha^{2}}\frac{\partial^{2}\psi}{\partial\theta^{2}}+\frac{1}{r^{2}\alpha^{2}\sin^{2}\theta}\frac{\partial^{2}\psi}{\partial\varphi^{2}}+\right.$ (16) $\left.\qquad\qquad+\frac{\cos\theta}{r^{2}\alpha^{2}\sin\theta}\frac{\partial\psi}{\partial\theta}+\frac{2}{r}\frac{\partial\psi}{\partial r}\right\\}.$ In analogy with the equation (10), the spatial operator $A$ is $\emph{A}=\left\\{\frac{\partial^{2}}{\partial r^{2}}+\frac{1}{r^{2}\alpha^{2}}\frac{\partial^{2}}{\partial\theta^{2}}+\frac{1}{r^{2}\alpha^{2}\sin^{2}\theta}\frac{\partial^{2}}{\partial\varphi^{2}}\right.+$ (17) $\left.\qquad\qquad+\frac{\cos\theta}{r^{2}\alpha^{2}\sin\theta}\frac{\partial}{\partial\theta}+\frac{2}{r}\frac{\partial}{\partial r}\right\\}.$ and the equation to be solved is $\left(\emph{A}^{\ast}\pm i\right)\psi=0$. Using separation of variables, $\psi=R\left(r\right)Y_{l}^{m}\left(\theta,\varphi\right)$, we get the radial portion of equation (14) as, $\frac{d^{2}R\left(r\right)}{dr^{2}}+\frac{2}{r}\frac{dR\left(r\right)}{dr}+\left(\frac{-l\left(l+1\right)}{r^{2}\alpha^{2}}\pm i\right)R\left(r\right)=0.$ (18) The square integrability of the above solution is checked by calculating the squared norm of the above solution in which the function space on each $t=$ constant hypersurface $\Sigma$ is defined as $\mathcal{H=}L^{2}\left(\Sigma,\mu\right)$ where $\mu$ is the measure given by the spatial metric volume element. We easily recover the results showed in 12 : The spacetime of global monopole remains singular in the view of relativistic quantum mechanics: the future of a given initial wave packet obeying the Klein-Gordon equation is not generally well determined, similarly to the future of a classical particle which reaches the classical singularity at $r=0$. ### 4.2 Dirac Fields The Newman-Penrose formalism will be used here to analyze massless Dirac particle propagating in the space of global monopole. The signature of the metric (8) is changed to $-2$ in order to use the Dirac equation in Newman- Penrose formalism. Thus, the metric is given by, $ds^{2}=dt^{2}-dr^{2}-r^{2}\alpha^{2}\left(d\theta^{2}+\sin^{2}\theta d\varphi^{2}\right).$ (19) The Chandrasekhar-Dirac (CD) 140 equations in Newman-Penrose formalism are given by $\displaystyle\left(D+\epsilon-\rho\right)F_{1}+\left(\bar{\delta}+\pi-\alpha\right)F_{2}$ $\displaystyle=$ $\displaystyle 0,$ (20) $\displaystyle\left(\nabla+\mu-\gamma\right)F_{2}+\left(\delta+\beta-\tau\right)F_{1}$ $\displaystyle=$ $\displaystyle 0,$ $\displaystyle\left(D+\bar{\epsilon}-\bar{\rho}\right)G_{2}-\left(\delta+\bar{\pi}-\bar{\alpha}\right)G_{1}$ $\displaystyle=$ $\displaystyle 0,$ $\displaystyle\left(\nabla+\bar{\mu}-\bar{\gamma}\right)G_{1}-\left(\bar{\delta}+\bar{\beta}-\bar{\tau}\right)G_{2}$ $\displaystyle=$ $\displaystyle 0,$ where $F_{1},F_{2},G_{1}$ and $G_{2}$ are the components of the wave function, $\epsilon,\rho,\pi,\alpha,\mu,\gamma,\beta$ and $\tau$ are the spin coefficients to be found and the "bar" denotes complex conjugation. The null tetrad vectors for the metric (19) are defined by $\displaystyle l^{a}$ $\displaystyle=$ $\displaystyle\left(1,1,0,0\right),$ (21) $\displaystyle n^{a}$ $\displaystyle=$ $\displaystyle\left(\frac{1}{2},-\frac{1}{2},0,0\right),$ $\displaystyle m^{a}$ $\displaystyle=$ $\displaystyle\frac{1}{\sqrt{2}}\left(0,0,\frac{1}{\alpha r},\frac{i}{r\alpha\sin\theta}\right).$ The directional derivatives in the Dirac equation are defined by $D=l^{a}\partial_{a},\nabla=n^{a}\partial_{a}$ and $\delta=m^{a}\partial_{a}.$ We define operators in the following way $\displaystyle\mathbf{D}_{0}$ $\displaystyle=$ $\displaystyle D$ $\displaystyle\mathbf{D}_{0}^{\dagger}$ $\displaystyle=$ $\displaystyle-2\nabla$ (22) $\displaystyle\mathbf{L}_{0}^{\dagger}$ $\displaystyle=$ $\displaystyle\sqrt{2}r\text{ }\alpha\delta\text{ and }\mathbf{L}_{1}^{\dagger}=\mathbf{L}_{0}^{\dagger}+\frac{\cot\theta}{2}$ $\displaystyle\mathbf{L}_{0}$ $\displaystyle=$ $\displaystyle\sqrt{2}r\alpha\text{ }\bar{\delta}\text{ and }\mathbf{L}_{1}=\mathbf{L}_{0}+\frac{\cot\theta}{2}$ The nonzero spin coefficients are, $\mu=-\frac{1}{2r},\rho=-\frac{1}{r},\beta=-\mathbf{\alpha}=\frac{1}{2\sqrt{2}}\frac{\cot\theta}{r\alpha}.$ (23) Substituting nonzero spin coefficients and the definitions of the operators given above into the CD equations leads to $\displaystyle\left(\mathbf{D}_{0}+\frac{1}{r}\right)F_{1}+\frac{1}{r\alpha\sqrt{2}}\mathbf{L}_{1}F_{2}=0,$ $\displaystyle-\frac{1}{2}\left(\mathbf{D}_{0}^{\dagger}+\frac{1}{r}\right)F_{2}+\frac{1}{r\alpha\sqrt{2}}\mathbf{L}_{1}^{\dagger}F_{1}=0,$ $\displaystyle\left(\mathbf{D}_{0}+\frac{1}{r}\right)G_{2}-\frac{1}{r\alpha\sqrt{2}}\mathbf{L}_{1}^{\dagger}G_{1}=0,$ $\displaystyle\frac{1}{2}\left(\mathbf{D}_{0}^{\dagger}+\frac{1}{r}\right)G_{1}+\frac{1}{r\alpha\sqrt{2}}\mathbf{L}_{1}G_{2}=0.$ (24) For the solution of the CD equations, we assume separable solution in the form of $\displaystyle F_{1}$ $\displaystyle=$ $\displaystyle f_{1}(r)Y_{1}(\theta)e^{i\left(kt+m\varphi\right)},$ $\displaystyle F_{2}$ $\displaystyle=$ $\displaystyle f_{2}(r)Y_{2}(\theta)e^{i\left(kt+m\varphi\right)},$ $\displaystyle G_{1}$ $\displaystyle=$ $\displaystyle g_{1}(r)Y_{3}(\theta)e^{i\left(kt+m\varphi\right)},$ $\displaystyle G_{2}$ $\displaystyle=$ $\displaystyle g_{2}(r)Y_{4}(\theta)e^{i\left(kt+m\varphi\right)}.$ (25) Here $\left\\{f_{1},f_{2},g_{1},g_{2}\right\\}$ and $\left\\{Y_{1},Y_{2},Y_{3},Y_{4}\right\\}$ are functions of $r$ and $\theta$ respectively, $m$ is the azimuthal quantum number and $k$ is the frequency of the Dirac spinor, which is assumed to be positive and real. By substituting (25) in (24) we will see that with these assumptions $\displaystyle\text{\ }f_{1}(r)$ $\displaystyle=$ $\displaystyle g_{2}(r)\text{ \ \ \ \ and \ \ \ }f_{2}(r)=g_{1}(r)\text{\ \ },$ (26) $\displaystyle Y_{1}(\theta)$ $\displaystyle=$ $\displaystyle Y_{3}(\theta)\text{ \ \ \ \ and \ \ \ }Y_{2}(\theta)=Y_{4}(\theta)$ (27) Dirac equation reduces to two equations. The radial part of the Dirac equations become $\displaystyle\left(\mathbf{D}_{0}+\frac{1}{r}\right)f_{1}\left(r\right)=\frac{\lambda}{r\alpha\sqrt{2}}f_{2}\left(r\right),$ (28) $\displaystyle\frac{1}{2}\left(\mathbf{D}_{0}^{\dagger}+\frac{1}{r}\right)f_{2}\left(r\right)=\frac{\lambda}{r\alpha\sqrt{2}}f_{1}\left(r\right).$ where $\lambda$ comes from separation of variables. We further assume that $\displaystyle f_{1}\left(r\right)$ $\displaystyle=$ $\displaystyle\frac{\Psi_{1}\left(r\right)}{r},$ $\displaystyle f_{2}\left(r\right)$ $\displaystyle=$ $\displaystyle\frac{\sqrt{2}\Psi_{2}\left(r\right)}{r},$ then equation (28) transforms into, $\displaystyle\mathbf{D}_{0}\Psi_{1}=\frac{\lambda^{{}^{\prime}}}{r}\Psi_{2},$ (29) $\displaystyle\mathbf{D}_{0}^{\dagger}\Psi_{2}=\frac{\lambda^{{}^{\prime}}}{r}\Psi_{1}.$ where $\lambda^{{}^{\prime}}=\frac{\lambda}{\alpha},$ so we will have $\displaystyle\left(\frac{d}{dr}+ik\right)\Psi_{1}$ $\displaystyle=$ $\displaystyle\frac{\lambda^{{}^{\prime}}}{r}\Psi_{2},$ (30) $\displaystyle\left(\frac{d}{dr}-ik\right)\Psi_{2}$ $\displaystyle=$ $\displaystyle\frac{\lambda^{{}^{\prime}}}{r}\Psi_{1},$ In order to write the above equation in a more compact form we combine the solutions in the following way, $\displaystyle Z_{+}$ $\displaystyle=$ $\displaystyle\Psi_{1}+\Psi_{2},$ $\displaystyle Z_{-}$ $\displaystyle=$ $\displaystyle\Psi_{2}-\Psi_{1}.$ After doing some calculations we end up with a pair of one-dimensional Schrödinger-like wave equations with effective potentials, $\displaystyle\left(\frac{d^{2}}{dr^{2}}+k^{2}\right)Z_{\pm}=V_{\pm}Z_{\pm},$ (31) $\displaystyle V_{\pm}=\frac{\lambda^{{}^{\prime}2}}{r^{2}}\mp\frac{\lambda^{{}^{\prime}}}{r^{2}}.$ (32) In analogy with the equation (10), the spatial operator $A$ for the massless case is $A=-\frac{d^{2}}{dr^{2}}+V_{\pm},$ so we have $\left(\frac{d^{2}}{dr^{2}}-\left[\frac{\lambda^{{}^{\prime}2}}{r^{2}}\mp\frac{\lambda^{{}^{\prime}}}{r^{2}}\right]\mp i\right)Z_{\pm}=0.$ (33) The solutions of the above equations are expressible using Bessel functions of the first and second kind in the following way $\displaystyle Z_{+}$ $\displaystyle=$ $\displaystyle C_{1}\sqrt{r}J\left(\lambda^{{}^{\prime}}-\frac{1}{2},\frac{r}{\sqrt{2}}\left(1-i\right)\right)+$ $\displaystyle C_{2}\sqrt{r}Y\left(\lambda^{{}^{\prime}}-\frac{1}{2},\frac{r}{\sqrt{2}}\left(1-i\right)\right),$ $\displaystyle Z_{-}$ $\displaystyle=$ $\displaystyle C_{1}^{{}^{\prime}}\sqrt{r}J\left(\lambda^{{}^{\prime}}+\frac{1}{2},\frac{r}{\sqrt{2}}\left(1+i\right)\right)+$ (34) $\displaystyle C_{2}^{{}^{\prime}}\sqrt{r}Y\left(\lambda^{{}^{\prime}}+\frac{1}{2},\frac{r}{\sqrt{2}}\left(1+i\right)\right).$ Using the asymptotic formulas for Bessel functions when $r\rightarrow\infty$ ($Y(\kappa,z)\approx z^{-1/2}\sin(z-\kappa\pi/2-\pi/4)$ and $J(\kappa,z)\approx z^{-1/2}\cos(z-\kappa\pi/2-\pi/4)$) and noting the complex argument in both solutions one can find a combination of constants $C_{1},C_{2}$ or $C_{1}^{\prime},C_{2}^{\prime}$ which is square integrable near infinity. (But, it is also possible to choose the constants differently so that both solutions are not square integrable!). When $r\rightarrow 0$ the approximate expressions for Bessel functions ($Y(\kappa,z)\approx z^{-\kappa}$ for $\kappa\neq 0$, $Y(0,z)\approx\ln(z/2)$ and $J(\kappa,z)\approx z^{\kappa}$) imply that for $C_{2}=0$ and $C_{2}^{\prime}=0$ we have square integrable solution near zero. (Here again if we suppose $C_{1}=0$ and $C_{1}^{\prime}=0$, for $\kappa\geq 3/2$, the solutions are not square integrable!. One could restrict an analysis to only certain wave modes and purposely choose the modes to be quantum regular). But since we have a solution of equations valid on the whole domain (not just asymptotic forms of equations) we can match the behaviour at zero and infinity. Based on the results we can have solution square integrable over the whole domain and therefore our deficiency indices are nonzero. The operator is not essentially self-adjoint and the spacetime is quantum mechanically singular. ## 5 Quantum Gravity Now we are going to investigate the singularity of general global monopole using techniques from loop quantization in the manner of 23 . Consider equation (2), for $r<\frac{2GM}{1-8\pi G\eta^{2}}$. This metric describes spacetime inside the horizon of a black hole. The coordinate $r$ is timelike and the coordinate $t$ is spatial there; for convenience we rename them as $r\equiv T$ and $t\equiv r$ with $T\in[0,\frac{2GM}{1-8\pi G\eta^{2}}]$ and $r\in[-\infty,+\infty]$ and the metric becomes $ds^{2}=-\left(\alpha^{2}-\frac{2GM}{T}\right)dr^{2}+\frac{dT^{2}}{\left(\alpha^{2}-\frac{2GM}{T}\right)}+T^{2}d\Omega^{2},$ (35) we eliminate the coefficient of $dT^{2}$ by defining a new temporal variable $\tau$ via $d\tau=\frac{dT}{\sqrt{\frac{2GM}{T}-\alpha^{2}}}.$ (36) Accordingly, the metric becomes $ds^{2}=-d\tau^{2}+\left(\frac{2GM}{T}-\alpha^{2}\right)dr^{2}+T^{2}\left(d\theta^{2}+\sin^{2}\theta d\varphi^{2}\right).$ (37) We introduce two functions $a^{2}\left(\tau\right)\equiv\frac{2Gm}{T}-\alpha^{2}$ and $b^{2}\left(\tau\right)\equiv T^{2}\left(\tau\right)$ and redefine $\tau\equiv t$. The metric becomes $ds^{2}=-dt^{2}+a^{2}\left(t\right)dr^{2}+b^{2}\left(t\right)\left(d\theta^{2}+\sin^{2}\theta d\varphi^{2}\right),$ (38) this metric describes a homogeneous, anisotropic Kanto-wski-Sachs cosmological mmodel with spatial section having topology $\mathbf{R\times S}^{2}$. From this observation comes the motivation to use LQC approach. In our case $a\left(t\right)$ is a function of $b\left(t\right)$. ### 5.1 Classical observables The corresponding action for gravity minimally coupled with scalar field can be written in the form $S=\frac{1}{16\pi G}\int dtd^{3}xNh^{1/2}\left[K_{ij}K^{ij}-K^{2}+\right.$ $\left.\qquad\qquad\qquad+^{\left(3\right)}R-\frac{16\pi G\eta^{2}}{b^{2}}\right],$ (39) by considering the metric (38), the action becomes $\displaystyle S$ $\displaystyle=$ $\displaystyle\frac{-1}{8\pi G}\int dt\int_{0}^{R}dr\int_{0}^{2\pi}d\phi\int_{0}^{\pi}d\theta\sin\theta ab^{2}\times$ (40) $\displaystyle\times\left[\frac{\dot{b}^{2}}{b^{2}}+\frac{2\dot{a}\dot{b}}{ab}-\frac{\alpha^{2}}{b^{2}}\right]$ by using the relation between $a$ and $b$, we will be able to write the action in terms of a single function $\displaystyle S$ $\displaystyle=$ $\displaystyle\frac{R\alpha^{2}}{2G}\int dt\sqrt{\frac{b}{2GM}}\left(1-\frac{\alpha^{2}b}{2GM}\right)^{-1/2}\times$ (41) $\displaystyle\times\left[\dot{b}^{2}+\frac{2GM}{b}\left(1-\frac{\alpha^{2}b}{2GM}\right)\right].$ Now, we will compute the Hamiltonian (Hamiltonian constraint). The momentum associated to the chosen configuration variable is $p_{b}=\frac{R\alpha^{2}\dot{b}}{G}\sqrt{\frac{b}{2GM}}\left(1-\frac{\alpha^{2}b}{2GM}\right)^{-1/2},$ (42) and therefore we obtain $\displaystyle H$ $\displaystyle=$ $\displaystyle p_{b}\dot{b}-L$ $\displaystyle=$ $\displaystyle\sqrt{\frac{2GM}{b}}\sqrt{1-\frac{\alpha^{2}b}{2GM}}\left[\frac{Gp_{b}^{2}}{2R\alpha^{2}}-\frac{R\alpha^{2}}{2G}\right].$ Now, we calculate the Hamiltonian constraint in terms of $\dot{b}$ $\displaystyle H$ $\displaystyle=$ $\displaystyle\frac{R\alpha^{2}}{2G}\sqrt{\frac{b}{2GM}}\left(1-\frac{\alpha^{2}b}{2GM}\right)^{-1/2}\times$ (44) $\displaystyle\times\left[\dot{b}^{2}-\frac{2GM}{b}\left(1-\frac{\alpha^{2}b}{2GM}\right)\right]=0,$ and immediately get the following solution $\dot{b}^{2}=\frac{2GM}{b}-\alpha^{2},$ (45) which is exactly the equation (36). When the horizon radius, $r_{h}=\frac{2GM}{\alpha^{2}}$, is much larger than the scale on which we are probing the singularity, we can write $1-\frac{\alpha^{2}b}{2GM}\sim 1$ so the Hamiltonian would be $H=\sqrt{\frac{2GM}{b}}\left[\frac{GP_{b}^{2}}{2R\alpha^{2}}-\frac{R\alpha^{2}}{2G}\right].$ (46) The volume $\displaystyle V$ $\displaystyle=$ $\displaystyle\int drd\theta d\phi\sqrt{h}=4\pi Rab^{2}$ (47) $\displaystyle=$ $\displaystyle 4\pi Rb^{3/2}\sqrt{2GM}\sqrt{1-\frac{\alpha^{2}b}{2GM}}$ simplifies when using the above approximation and we obtain $\displaystyle V$ $\displaystyle=$ $\displaystyle l_{0}b^{3/2}$ $\displaystyle l_{0}$ $\displaystyle=$ $\displaystyle 4\pi R\sqrt{2GM}$ (48) The canonical pair is given by $b\equiv x$ and $p_{b}$, with Poisson bracket $\left\\{x,p_{b}\right\\}=1$. For isotropic models, only holonomies evaluated in isotropic connections $A_{a}^{i}=\tilde{c}\delta_{a}^{i}$ appear. Along straight lines in the direction of translation symmetries $X_{I}^{a}=\left(\partial/\partial X^{I}\right)^{a}$, holonomies $\exp\left(\int X_{I}^{a}A_{a}^{i}\tau_{i}\right)$ in the fundamental representation of $SU\left(2\right)$ have matrix elements of the form $\exp\left(i\mu c\right)$, where $\mu$ depends on the length of the curve used. Here, it turns out to be useful to introduce $c:=V_{0}^{1/3}\tilde{c}$ defined in terms of the coordinate size $V_{0}$ of the region used to define the isotropic phase space 21 . Using this motivation we introduce the following function which will be used instead of the momentum (from now on we leave out the subscript $b$ for momentum associated with this observable) 23 $U_{\gamma}\left(p\right)\equiv\exp\left(8\pi G\frac{i\gamma}{L}p\right)$ (49) where $\gamma$ is a real parameter and $L$ fixes the length scale. The parameter $\gamma$ determines the separation of momentum points in the phase space. The pair $\left(x,U_{\gamma}\left(p\right)\right)$ has the following Poisson bracket algebra $\left\\{x,U_{\gamma}\left(p\right)\right\\}=8\pi G\frac{i\gamma}{L}U_{\gamma}\left(p\right)$ (50) A straightforward calculation gives $\displaystyle U_{\gamma}^{-1}\left\\{V^{n},U_{\gamma}\right\\}$ $\displaystyle=$ $\displaystyle l_{0}^{n}U_{\gamma}^{-1}\left\\{\left\|x\right\|^{3n/2},U_{\gamma}\right\\}$ $\displaystyle=$ $\displaystyle i8\pi Gl_{0}^{n}\frac{\gamma}{L}\frac{3n}{2}sgn\left(x\right)\left\|x\right\|^{3n/2-1}$ We are concerned with the quantity $\frac{1}{\left\|x\right\|}$ which can serve as an indicator for singularity presence because classically it diverges for $\left\|x\right\|\rightarrow 0$ thus producing singularity. From this moment we choose $n=1/3$ $\frac{sgn\left(x\right)}{\sqrt{\left\|x\right\|}}=-\frac{2Li}{8\pi Gl_{0}^{1/3}\gamma}U_{\gamma}^{-1}\left\\{V^{1/3},U_{\gamma}\right\\}.$ (52) ### 5.2 Quantization We will use the basis of Hillbert space introduced in 22 ; 23 , which is formed by eigenstates of $\hat{x}$. This implies the existence of a self- adjoint operator $\hat{x}$, acting on the basis states according to $\hat{x}\left\|\mathbf{\mu}\right\rangle=L\mu\left\|\mathbf{\mu}\right\rangle$ (53) Next, we want to promote the classical momentum function $U_{\gamma}=e^{\left(8\pi G\frac{i\gamma}{L}p\right)}$ to operator. We can do so by defining the action of $\hat{U}_{\gamma}$ on the basis states with the help of the definition equation (53) and using commutation relation based on the Poisson bracket between $x$ and $U_{\gamma}$ we obtain $\hat{U}_{\gamma}\left\|\mathbf{\mu}\right\rangle=\left\|\mathbf{\mu-\gamma}\right\rangle,\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left[\hat{x},\hat{U}_{\gamma}\right]=-\gamma L\hat{U}_{\gamma}.$ (54) Using canonical quantization of Poison bracket $\left[,\right]\rightarrow i\hbar\left\\{,\right\\}$, and using equation (51) we get a relation for the length scale $L=\sqrt{8\pi}l_{p}$ (55) ### 5.3 Volume operator and disappearance of the singularity In the vicinity of the singularity we assume the approximate equation (48). Then the volume operator acts in the following way on the basis states $\hat{V}\left\|\mathbf{\mu}\right\rangle=l_{0}\left\|x\right\|^{3/2}\left\|\mathbf{\mu}\right\rangle=l_{0}\left\|L\mu\right\|^{3/2}\left\|\mathbf{\mu}\right\rangle$ (56) Using the equation (52) and promoting the Poisson brackets to commutators, while setting $\gamma=1$, we find $\frac{\widehat{1}}{\left\|x\right\|}=\frac{1}{2\pi l_{p}^{2}l_{0}^{2/3}}\left(\hat{U}_{\gamma}^{-1}\left[\hat{V}^{1/3},\hat{U}_{\gamma}\right]\right)^{2}.$ (57) On the basis states this operator acts in the following way $\displaystyle\hat{U}_{\gamma}^{-1}\left[\hat{V}^{1/3},\hat{U}_{\gamma}\right]\left\|\mathbf{\mu}\right\rangle$ $\displaystyle=$ $\displaystyle\left(\hat{U}_{\gamma}^{-1}\hat{V}^{1/3}\hat{U}_{\gamma}-\hat{U}_{\gamma}^{-1}\hat{U}_{\gamma}\hat{V}^{1/3}\right)\left\|\mathbf{\mu}\right\rangle$ (58) $\displaystyle=$ $\displaystyle l_{0}^{1/3}l_{p}^{1/2}\left(\sqrt{\mu-1}-\sqrt{\mu}\right)\left\|\mathbf{\mu}\right\rangle$ so finally we get $\frac{\widehat{1}}{\left\|x\right\|}\left\|\mathbf{\mu}\right\rangle=\sqrt{\frac{2}{\pi l_{p}^{2}}}\left(\sqrt{\mu-1}-\sqrt{\mu}\right)^{2}\left\|\mathbf{\mu}\right\rangle.$ (59) We can see that the spectrum is bounded from above and so the singularity is resolved in the quantum theory (the theory gives finite predictions for observables related to singularity). In fact, the eigenvalue of operator $\frac{\widehat{1}}{\left\|x\right\|}$ corresponding to the state $\left\|\mathbf{0}\right\rangle$ which probes the classical singularity is equal to $\sqrt{\frac{2}{\pi l_{p}^{2}}}$, which is the highest eigenvalue of the spectrum. Specifically, the operator corresponding to the curvature invariant $\displaystyle\mathcal{~{}R}_{\mu\nu\rho\sigma}\mathcal{R}^{\mu\nu\rho\sigma}$ $\displaystyle=$ $\displaystyle\frac{48M^{2}G^{2}}{r^{6}}+\frac{128M\pi G^{2}\eta^{2}}{r^{5}}+\frac{256G^{2}\pi^{2}\eta^{4}}{r^{4}}$ $\displaystyle\equiv\frac{48M^{2}G^{2}}{b\left(t\right)^{6}}$ $\displaystyle+$ $\displaystyle\frac{128M\pi G^{2}\eta^{2}}{b\left(t\right)^{5}}+\frac{256G^{2}\pi^{2}\eta^{4}}{b\left(t\right)^{4}}$ (60) is then automatically finite in quantum mechanics. Promoting it to operator and evaluating on $\left\|\mathbf{0}\right\rangle$ we get $\displaystyle\mathcal{~{}}\widehat{\mathcal{R}_{\mu\nu\rho\sigma}\mathcal{R}^{\mu\nu\rho\sigma}}\left\|\mathbf{0}\right\rangle$ $\displaystyle=\left(\widehat{\frac{48M^{2}G^{2}}{\left\|x\right\|^{6}}}+\widehat{\frac{128M\pi G^{2}\eta^{2}}{\left\|x\right\|^{5}}}+\widehat{\frac{256\pi^{2}G^{2}\eta^{4}}{\left\|x\right\|^{4}}}\right)\left\|\mathbf{0}\right\rangle$ $\displaystyle=\left(\textstyle{\frac{384M^{2}G^{2}}{\pi^{3}l_{p}^{6}}+\sqrt{\frac{2}{\pi^{5}}}\frac{512M\pi G^{2}\eta^{2}}{l_{p}^{5}}+\frac{1024\pi^{2}G^{2}\eta^{4}}{\pi^{2}l_{p}^{4}}}\right)\left\|\mathbf{0}\right\rangle$ (61) On the other hand, when $\left\|\mu\right\|\rightarrow\infty$ the eigenvalue of $\frac{\widehat{1}}{\left\|x\right\|}$ goes to zero which is natural behaviour for large $\left\|x\right\|$. Also, it is possible to show that the quantum Hamiltonian constraint gives a discrete difference equation for the coefficients of the physical states. ## 6 Conclusion We have seen that we have not been successful in removing the naked singularity by using relativistic quantum mechanics (for both Klein-Gordon and Dirac equations). On the other hand we have shown that the curvature singularity of general global monopole is resolved when the geometry is quantized using loop techniques. Unfortunately, one cannot directly compare the results because the loop quantization relied on radial coordinate being timelike beneath the horizon which is not the case for naked singularity of pure monopole. But still, this might be an indication that the first method is not reliable for determining the fate of singularities in quantum theory and one should rather focus on quantization of the geometry itself. But even the approach using loop quantization that relied on restricted class of geometries should not be trusted completely. One should allow, e.g., for deviations from spherical symmetry to be completely sure about the fate of singularities. ###### Acknowledgements. T.T. gratefully acknowledges the hospitality of Institute of Theoretical Physics (Charles University in Prague) during her stay. O. S. was supported by grant GAČR 202/09/0772. ## Appendix A Geometric quantities The spatial metric is $h_{ij}=\left(a^{2}\left(t\right),b^{2}\left(t\right),b^{2}\left(t\right)\sin^{2}\theta\right),$ (62) The extrinsic curvature is $K_{ij}=-\frac{1}{2}\frac{\partial h_{ij}}{\partial t}$, and so $\displaystyle K_{ij}=-\left(a\dot{a},b\dot{b},b\dot{b}\sin^{2}\theta\right),$ $\displaystyle K=K_{ij}h^{ij}=-\left(\frac{\dot{a}}{a}+2\frac{\dot{b}}{b}\right)$ (63) $\displaystyle K_{ij}K^{ij}=\left(\frac{\dot{a}^{2}}{a^{2}}+2\frac{\dot{b}^{2}}{b^{2}}\right)\text{\ \ \ \ \ \ \ }$ $\displaystyle K_{ij}K^{ij}-K^{2}=-2\left(\frac{\dot{b}^{2}}{b^{2}}+2\frac{\dot{a}\dot{b}}{ab}\right)$ (64) The Ricci curvature for the space section is ${}^{\left(3\right)}R=\frac{2}{b^{2}}.$ (65) ## References * (1) G. F. R. Ellis and B. G. Schmidt, Gen. Rel. Grav. 8, 915 (1977). * (2) D. A. Konkowski and T. M. Helliwell, Gen. Rel. and Grav. 33, 1131, (2001). * (3) T. M. Helliwell, D. A. Konkowski and V. Arndt, Gen. 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1401.0051	# Wall Adhesion and Constitutive Modelling of Strong Colloidal Gels D. R. Lester CSIRO Mathematics, Informatics and Statistics, PO Box 56, Highett, Vic. 3190, Australia [email protected] R. Buscall MSACT Research and Consulting, Exeter, United Kingdom Particulate Fluids Processing Centre, Dept. of Chemical and Biomolecular Engineering, University of Melbourne, Victoria 3010, Australia A. D. Stickland Particulate Fluids Processing Centre, Dept. of Chemical and Biomolecular Engineering, University of Melbourne, Victoria 3010, Australia P. J. Scales Particulate Fluids Processing Centre, Dept. of Chemical and Biomolecular Engineering, University of Melbourne, Victoria 3010, Australia ###### Abstract Wall adhesion effects during batch sedimentation of strongly flocculated colloidal gels are commonly assumed to be negligible. In this study _in-situ_ measurements of gel rheology and solids volume fraction distribution suggest the contrary, where significant wall adhesion effects are observed in a 110mm diameter settling column. We develop and validate a mathematical model for the equilibrium stress state in the presence of wall adhesion under both viscoplastic and viscoelastic constitutive models. These formulations highlight fundamental issues regarding the constitutive modeling of colloidal gels, specifically the relative utility and validity of viscoplastic and viscoelastic rheological models under arbitrary tensorial loadings. The developed model is validated against experimental data, which points toward a novel method to estimate the shear and compressive yield strength of strongly flocculated colloidal gels from a series of equilibrium solids volume fraction profiles over various column widths. colloidal gel, wall adhesion, constitutive modelling ††preprint: APS/123-QED ## I Introduction The batch settling test is widely utilized as a means to characterize both the sedimentation and consolidation properties of colloidal suspensions (Kynch, 1952; Michaels and Bolger, 1962; Tiller and Shirato, 1964; Howells et al., 1990; Landman and White, 1994; Bürger and Tory, 2000; Lester et al., 2005; Diehl, 2007; Grassia et al., 2011), where the relevant material properties act as inputs for the modeling of a wide range of solid-liquid separation processes, ranging from tailings disposal and gravity settling, through to continuous thickening and pressure filtration. In the minerals industry, as is exemplified herein, these suspensions are typified by a broad range of particle size distributions, ranging from 0.1 to 200 micron. To improve the rate of sedimentation, flocculation of the suspensions is augmented through the addition of very high molecular weight ($>$10 million) polymeric flocculants, whereas in other applications, electrolyte coagulants are used to aid aggregation. In either case, the suspensions under consideration are strongly flocculated (often with effective well-depths $>$ 20 kT) and so are non-Brownian (athermal) and stable over long time scales. Hence the behaviour of strongly flocculated suspensions can be very different to that of weak- or partially-aggregated systems; time-dependent phenomena such as spontaneous creep, ripening and collapse cannot occur. As strong systems are simpler, these materials form a benchmark against which more complex, labile weak systems can be compared. At a critical solids concentration, sometimes as low as a few volume %, particulate aggregates in strongly flocculated colloidal suspensions can form a continuous space-filling particulate network (or colloidal gel) which can both withstand and transmit stress. This particulate network can consolidate significantly under differential stress (e.g. pressure filtration, gravitation or centrifugation), however as the network is strongly volume-strain hardening, the system reaches an equilibrium concentration (typically significantly less than the close-packing or frictional limit) for a given load. Conversely, the particulate network is strongly strain-softening in shear, and by the standards of polymer rheology, these systems are very brittle, being able to only withstand shear strains of less than 1% and quite often less than 0.001% prior to yield and flow. Despite the brittle nature of these strongly flocculated suspensions in shear, imposition of shear is not common in traditional batch sedimentation tests. To characterize the compressive strength of the particulate network, the batch settling test has significant advantages in that it is simple, cheap and highly portable, and the range of compressive stress (typically $\lesssim 1$ kPa) involved is commensurate with many gravity settling applications. The most commonly measured data is the height of the transient sediment/supernatant interface over one or more experiments, and in more sophisticated experiments the equilibrium and/or transient local average solids volume fraction profile $\phi$ is also determined via e.g. gamma ray (Labbett et al., 2006) or ultrasonic attenuation (Auzerais et al., 1990). Although deconvolution of the measured data set into accurate estimates of the relevant material properties is not trivial, significant advances (Grassia et al., 2011; Lester et al., 2005) have been made in recent years regarding this problem, facilitating accurate and complete suspension characterization from a small number of batch settling tests. An important assumption underpinning these deconvolution techniques (and sedimentation theory in general) is that effects arising from adhesion between the settling suspension and the container wall are negligible, effectively allowing the sedimentation and consolidation processes to be quantified via a one-dimensional vertical force balance. However, as strongly flocculated gels are both strongly cohesive and adhesive, these materials readily adhere to container walls with a wall adhesive shear strength $\tau_{w}(\phi)$, as is well-known from studies of wall-slip in colloidal suspension rheometry. The assumption of negligible wall adhesion effect is motivated by estimates that $\tau_{w}(\phi)$ is small in comparison to the suspension compressive yield strength $P_{y}(\phi)$, both of which serve as inputs for an equilibrium momentum balance (Michaels and Bolger, 1962) over the particulate phase in the vertical direction $\frac{dP_{y}}{d\phi}\frac{\partial\phi}{\partial z}-\Delta\rho g\phi+\frac{2\tau_{w}(\phi)}{R}=0,$ (1) where $z$ is the vertical bed depth (downwards from the suspension/supernatant interface), $\Delta\rho$ the interphase density difference, $g$ gravitational acceleration constant, and $R$ is the radius of the settling container. As the apparent wall adhesion strength $\tau_{w}(\phi)$ typically appears (Seth et al., 2008; Buscall et al., 1993; Barnes, 1995) to be somewhat smaller but of the same order to the bulk suspension shear yield strength $\tau_{y}(\phi)$, and the ratio of shear to compressive yield strength $S(\phi)=\tau_{y}(\phi)/P_{y}(\phi)$ appears to vary over the range 0.001-0.2 (Buscall et al., 1987, 1988; de Kretser et al., 2002; Zhou et al., 2001; Channell and Zukoski, 1997), these effects may be neglected for all but narrow settling columns. However, if wall adhesion effects are significant - i.e. if $\tau_{w}$ is large and/or $R$ is small - then the assumption of a one-dimensional force balance governing the suspension mechanics breaks down. Now the particulate network experiences a combination of both shear and compressive stress, and this arbitrary stress state varies both vertically and radially. From (1), it is clear that the wall adhesion strength acts to counteract the gravitational force, and in some cases, the entire suspension weight can be supported by shear stress alone. Under the approximation $\tau_{w}\approx\tau_{y}(\phi)$, this state is given in terms of a critical solids concentration $\phi_{c}$ which only depends upon the container radius and shear yield strength as $\tau_{y}(\phi_{c})\approx\frac{1}{2}\Delta\rho g\phi_{c}R.$ (2) In principle, once the critical volume fraction $\phi=\phi_{c}$ is reached (at the critical bed depth $z_{c}$), the network pressure is constant for bed depths beyond $z_{c}$. Hence, a clear signature of significant wall adhesion effects is given by a constant equilibrium solids volume fraction profile. Such behaviour is clearly illustrated in Fig. 1, which depicts the vertical solids volume fraction profiles of a polymer flocculated calcium carbonate suspension in 22 mm and 110 mm diameter columns - the vertical profile for the 22 mm column can only be reasonably explained by wall adhesion effects. As the shear yield strength is a nonlinear monotonic increasing function of $\phi$, this critical state $\phi=\phi_{c}$ is reached by strongly adhesive colloidal suspensions in narrow containers, given sufficient bed depth. As this critical state is approached ($\phi\rightarrow\phi_{c}$), estimates of the compressive yield strength $P_{y}(\phi)$ which neglect wall adhesion effects diverge to $+\infty$, hence wall adhesion can introduce unbounded errors in estimates of suspension material parameters. Such errors can also contaminate the estimate of other suspension properties such as the hindered settling function $R(\phi)$ (Lester et al., 2005) which quantifies the hydrodynamic drag between particulate and fluid phases. The data presented herein on the gravity batch settling of mineral particles flocculated with high molecular weight polymers have suggested that wall adhesion effects are by no means always secondary or insignificant. Here the suspensions are seldom far from the gel-point and the ratio of shear to compressive strength is expected to be at a maximum of order unity at the gel- point, decreasing rapidly away from it (Buscall, 2009). Note that for strongly-flocculated colloidal suspensions the wall adhesion force arises directly from the “sticky” nature of the flocculated particles, and for the range of stresses typical of batch settling applications, frictional forces do not contribute to wall adhesion. Wall adhesion (typically weaker than particle cohesion) also arises as wall slip in shear rheometry of colloidal suspensions; ironically the lack of total adhesion is problematic in shear rheology, whilst the presence of adhesion is problematic in compressive rheology (i.e. batch settling). In-situ measurements (i.e. within the settling container itself) of the shear yield strength $\tau_{y}$ of flocculated colloidal suspensions appear to be significantly higher than those for a decanted suspension. Colloidal gels flocculated with high molecular weight polymer flocculants exhibit rapid irreversible breakdown under shear, and so significant degradation can occur during the decanting process. Conversely, the compressive yield stress is typically measured _in-situ_ , hence such inconsistency can significantly underestimate the magnitude of $S(\phi)$. The results herein demonstrate wall adhesion effects are significant in a 110 mm diameter settling column, typically considered to be wide enough to render such effects negligible. These observations suggest that wall adhesion effects for colloidal gels in batch settling tests are more prevalent than previously appreciated, and have motivated us to investigate the problem of wall adhesion in batch sedimentation in greater detail. In particular, we aim to develop and validate a mathematical model of the suspension equilibrium stress state in the presence of wall adhesion, and develop error estimates for the one-dimensional approximation (1) under such conditions. Analysis of the governing multidimensional force balance and suspension behaviour under arbitrary tensorial loadings also raises fundamental questions regarding the constitutive modeling of strongly flocculated colloidal suspensions, particularly the validity and utility of viscoplastic rheological models as opposed to more general but less mathematically tractable viscoelastic formulations. Strongly flocculated colloidal gels exhibit a wide array of complex rheological behaviour (Grenard et al., 2014; Sprakel et al., 2011; Lindstrom et al., 2012; Gibaud et al., 2010; Santos et al., 2013; Koumakis and Petekidis, 2011; Gibaud et al., 2008; Ovarlez et al., 2013; Ramos and Cipelletti, 2001; Ovarlez and Coussot, 2007; Cloitre et al., 2000; Tindley, 2007; Kumar et al., 2012; Uhlherr et al., 2005), including nonlinear creep and time-dependent yield under small shear strains, followed by rapid strain- softening which is described as shear yield prior to viscous flow. Whilst constitutive modelling is still being developed to resolve such complex flow phenomena, the different constitutive approaches (broadly categorized as viscoplastic and viscoelastic models) constitute different levels of resolution of the rheology of colloidal gels. In this study we find such issues are also of direct relevance with respect to resolution of the wall adhesion problem, hence we consider the properties of each constitutive framework, and utilize an appropriate combination to resolve the wall adhesion problem. Specifically, we seek a macroscopic phenomenological description of the suspension rheology which quantifies the wall adhesion problem in the simplest manner possible, and make no claim as to the network behaviour at the particle level. As such, we seek a minimum extension of traditional 1D compressive rheology of strongly flocculated colloidal suspensions which is capable of addressing the wall adhesion problem. The nature of traditional critical state compressive rheology and the tensorial nature of the wall adhesion problem means that a large-strain visco- elastic framework is required to resolve this problem. This solution is then compared with independent experimental measurements, validating the constitutive approach used herein, and pointing to a novel method to extract accurate estimates of both the compressive $P_{y}(\phi)$ and shear $\tau_{y}(\phi)$ yield strength from a series of batch sedimentation tests in columns of varying diameter. Whilst the static equilibrium problem of wall adhesion in batch settling appears to be somewhat divorced from dynamic suspension rheology, these fundamental issues with regard to constitutive modelling apply more broadly to colloidal suspension rheology in general. In the following Section we develop governing equations for the wall adhesion problem and review constitutive modeling approaches for strongly flocculated colloidal gels. In Sections III and IV the hyper-elastic and viscoplastic constitutive models respectively are examined in greater detail, and in Section V we present a closure approximation and solution of the viscoplastic model for the equilibrium stress state. A small strain solution of the hyperelastic model is presented by relaxation of the viscoplastic closure in Section VI, and in Section VII we validate the viscoplastic solution against experimental data, before conclusions are made in Section VIII. Figure 1: Equilibrium solids volume fraction profiles $\phi_{\infty}$ of a strongly flocculated colloidal gel in narrow ($R_{s}$=0.011 m, black) and wide ($R_{l}$=0.055 m, grey) settling columns. Note that a constant solids volume fraction profile can only be explained by wall adhesion supporting the particulate phase. ## II Constitutive Modeling of Colloidal Gels To develop a quantitative model of batch settling in the presence of wall adhesion effects, we consider the transient dynamics of an attractive colloidal gel within a batch settling experiment starting at the initial condition $\phi=\phi_{0}<\phi_{g}$. Over the past few decades, several phenomenological theories of the behaviour of strongly flocculated colloidal gels have been developed (Richardson and Zaki, 1954; Kim et al., 2007; Philip and Smiles, 1982; Toorman, 1996; Howells et al., 1990; Bürger and Concha, 1998; Auzerais et al., 1990; Buscall and White, 1987) across a variety of diverse fields, with a significant degree of duplication and fragmentation. The majority of these formulations have focussed upon a one-dimensional force balance between the solid and fluid phases, and whilst these have been very successful in capturing the gross features of 1D processes such as pressure filtration and continuous thickening, their ability to resolve multi- dimensional phenomena such as the wall adhesion problem is limited. Despite these limitations, these constitutive models utilize several simplifying assumptions which provide significant insights into the nature of colloidal suspensions under arbitrary tensorial loads. The most significant of these assumptions is that the impact of anisotropy at the particle scale due to consolidation history is negligible with respect to macroscopic rheology. Whilst consolidation processes such as 1D pressure filtration in confined domains involve uniaxial rather than isotropic consolidation, microstructural re-arrangement via collapse and buckling of particle chains acts to maintain isotropy of the particulate network (Seto et al., 2013). This assumption, central to compressive rheology, is further supported by the fact that strongly flocculated colloidal gels can only support small deviatoric strains prior to yield, and so the macroscopic rheology is essentially identical under confined uniaxial compression and isotropic volumetric strain, quantified in terms of the solids volume fraction $\phi$. A multi-dimensional theory of the flow and separation of flocculated colloidal suspensions has been developed (Lester et al., 2010) which quantifies the evolution of $\phi$ as $\frac{\partial\phi}{\partial t}+\mathbf{q}\cdot\nabla\phi=\nabla\cdot\frac{(1-\phi)^{2}}{R(\phi)}\left(\Delta\rho\phi\frac{D_{\mathbf{q}}\mathbf{q}}{Dt}-\nabla\cdot\Sigma^{N}-\Delta\rho\mathbf{g}\phi\right),$ (3) where $D_{\mathbf{q}}/Dt$ is the material derivative with respect to the volume-averaged suspension velocity $\mathbf{q}$, $R(\phi)$ is the hindered settling function or (or inverse Darcy permeability) (Howells et al., 1990; Bürger and Concha, 1998; Buscall and White, 1987) which quantifies interphase drag, and $\Sigma^{N}$ is the network stress tensor, defined (Batchelor, 1977) as the difference between the total suspension stress $\Sigma$ and the fluid stress $\Sigma^{f}$: $\Sigma^{N}\equiv\Sigma-\Sigma^{f},$ (4) which may be decomposed in terms of the network pressure $p_{N}$ and deviatoric stress $\bm{\sigma}_{N}$ $\Sigma^{N}=-p_{N}\mathbf{I}+\bm{\sigma}_{N}.$ (5) Under the assumption that during sedimentation the bulk suspension velocity $\mathbf{q}$ is zero, the network force balance simplifies to $\frac{\partial\phi}{\partial t}+\nabla\cdot\frac{(1-\phi)^{2}}{R(\phi)}\left(\nabla\cdot\Sigma^{N}+\Delta\rho\mathbf{g}\phi\right)=0,$ (6) which at equilibrium yields a balance between the network stress gradient and gravitational force $\nabla\cdot\Sigma^{N}+\Delta\rho\mathbf{g}\phi=0.$ (7) Central to this model is the specification of a constitutive equation for the colloidal suspension network stress tensor $\Sigma^{N}$ to close the transient (3) and equilibrium (7) momentum balances. Henceforth we explore several constitutive modeling approaches for strongly flocculated colloidal gels. As mentioned in the Introduction, particulate aggregates in colloidal suspensions form a particulate network (or colloidal gel) with finite strength above a critical solids concentration termed the gel point $\phi_{g}$. For strongly flocculated suspensions such as coagulated or polymer flocculated suspensions, the network stress tensor $\Sigma^{N}$ is identically zero for $\phi<\phi_{g}$, whereas for $\phi>\phi_{g}$, attractive inter-particle forces result in an apparent network strength (in shear and/or differential compression) which strongly increases with solids volume fraction. Hence suspension sedimentation (which involves the settling of hydrodynamically interacting particulate aggregates) occurs for $\phi<\phi_{g}$, whereas suspension consolidation (which involves simultaneous compression of the particulate network and hydrodynamic drainage) occurs in the range $\phi\geqslant\phi_{g}$. The athermal nature of the continuous particulate network imparts solid-like properties to strongly flocculated colloidal gels, which leads to a rich array of complex rheological behaviour (de Kretser et al., 2002; Zhou et al., 2001; Channell and Zukoski, 1997; Tindley, 2007; Kumar et al., 2012; Uhlherr et al., 2005; Grenard et al., 2014) under both shear and compressive loads. Of the array of constitutive models for the network stress $\Sigma^{N}$ of colloidal gels, there exist two distinct modelling approaches which are most clearly delineated via description of network compression. Flocculated colloidal gels are strongly strain-hardening in compression due to the increase in local solids volume fraction, and so may be described as poro- elastic materials with a volumetric strain-hardening compressional bulk modulus $K(\phi)$. As the inter-particle potential of a strongly flocculated colloidal gel typically contains a deep energy well, the compression of such gels is essentially irreversible. Some workers (Buscall, 2009; Liétor-Santos et al., 2009; Manley et al., 2005; Kim et al., 2007) describe such gels as “ratchet poro-elastic”, which quantifies the evolution of the network pressure $p_{N}$ as $p_{N}=\int_{\phi_{0}}^{\phi}K(\varphi)d\ln\varphi,\,\,\frac{D_{s}\phi}{Dt}\geqslant 0,$ (8) where $\phi_{0}$ is the initial concentration, and $D_{s}/Dt$ denotes the material derivative with respect to the particulate phase. Alternately, compressional behaviour of colloidal gels is described by several workers (Philip and Smiles, 1982; Toorman, 1996; Howells et al., 1990; Bürger and Concha, 1998; Auzerais et al., 1990; Buscall and White, 1987) as a viscoplastic process in terms of the so-called compressive yield strength $P_{y}(\phi)$, which implicitly encodes the irreversible nature of compression. The terminology “compressive yield” is somewhat misleading in that it implies an elastic strain limit, whereas in reality the particulate network strain-hardens without limit, and so $P_{y}(\phi)$ represents the volumetric strain (given by $\phi$) at which an applied network pressure is in equilibrium with the strength of the particulate network: $p_{N}=P_{y}(\phi),\,\,\frac{D_{s}\phi}{Dt}\geqslant 0.$ (9) Hence, in terms of compressive strength, the poro-elastic and viscoplastic formulations are equivalent under the approximation $P_{y}(\phi)\approx P(\phi,\phi_{0})\equiv\int_{\phi_{0}}^{\phi}K(\varphi)d\ln\varphi,$ (10) which is exact for $\phi_{0}<\phi_{g}$. For $\phi_{0}>\phi_{g}$, the compressive yield strength is not a true material property, but rather an experimental artefact as it is not dependent upon the initial volume fraction $\phi_{0}$, i.e. $P_{y}\neq P$. However, for $\phi_{0}-\phi_{g}\ll 1$, the compressive yield strength represents an accurate approximation to the true network pressure $P$. This approximation typifies the relationship between the poro-elastic and viscoplastic formulations; whilst the former more accurately reflects the colloidal gel rheology, the latter leads to more tractable formulation as the strain history need not be evaluated. In contrast to compression, the shear response of particulate gels is strain- softening, typically, and hence not self-limiting. Indeed, many colloidal gels strain-soften so rapidly that they can be considered to yield. Where this is the case, the notion of a critical yield stress $\tau_{y}$, or, sometimes, a yield strain $\gamma_{c}$, is adequate for many purposes. While detailed experiments (Uhlherr et al., 2005; Tindley, 2007; Kumar et al., 2012) indicate particulate gels actually undergo this transition over a range of stresses and strains, suggesting the true yield criterion is more complicated than a critical stress or strain condition, one can still identify a representative critical strain $\gamma_{c}$ associated with the rapid transition to viscous flow. Colloidal gels are typically brittle in shear, and the representative critical strain is of the order $10^{-4}-10^{-2}$ (Channell and Zukoski, 1997; Buscall et al., 1987; Uhlherr et al., 2005; Tindley, 2007). In many applications (including experimental studies and numerical simulations) it is often neither feasible nor desirable to resolve such strains and the detailed sub-yield dynamics of colloidal gels. In this case, a viscoplastic constitutive model (such as a Herschel-Bulkley or Bingham model) serves as a useful engineering approximation for the deviatoric network stress tensor $\bm{\sigma}_{N}$, which for simple shear may be quantified as $\tau_{N}=\left(\frac{\tau_{y}(\phi)}{\dot{\gamma}}+\eta(\phi,\dot{\gamma})\right)\dot{\gamma}\,\,\,\text{for}\,\,\tau_{N}\geqslant\tau_{y}(\phi),$ (11) where $\tau_{N}$ is the 2nd invariant of $\bm{\sigma}_{N}$, $\eta(\phi,\dot{\gamma})$ is the apparent suspension viscosity (which is typically non-Newtonian), and $\dot{\gamma}$ is the rate of shear strain. One disadvantage of the viscoplastic constitutive model is that in general the deviatoric stress $\bm{\sigma}_{N}$ is unresolved below the yield stress $\tau_{N}<\tau_{y}(\phi)$, as $\dot{\gamma}\rightarrow 0$ and the effective viscosity diverges. Although specialized regularization methods have been developed for numerical calculations (Balmforth et al., 2014), the conceptual problem of an undefined sub-yield stress state persists. Conversely, the poro-elastic constitutive model resolves the detailed elastic strain in terms of the shear modulus $G(\phi,\gamma)$ and memory function $m(t)$ via the quasi-linear viscoelastic (Fung, 1993) constitutive model $\tau_{N}=\int_{-\infty}^{t}m(t-s)G(\phi,\gamma)\frac{\partial\gamma}{\partial s}ds+\eta(\phi,\gamma)\dot{\gamma}\,$ (12) in which time-strain separability has been invoked as a first approximation for the sake of clarity; most real colloidal gets are not expected to be so obliging, necessarily, even though there are examples, remarkably (Yin and Solomon, 2008). Strain softening is encoded via the shear modulus $G(\phi,\gamma)$, and the strain rate $\dot{\gamma}$ is small prior to strain softening, which is interpreted as yield in the viscoplastic model. Hence for rapid shear strain (i.e. significantly faster than the relaxation timescale of $m(t)$), the shear yield stress and shear modulus are related via the critical strain as $\tau_{y}(\phi)=\int_{0}^{\gamma_{c}}G(\phi,\gamma)d\gamma.$ (13) As such, the viscoplastic and poro-elastic constitutive models can be reconciled as different levels of approximation for the rheology of a colloidal gel, with distinct advantages and disadvantages. Whilst the tensorial form of the poro-elastic model (detailed in Section 3) is a more accurate representation of the dynamics of a strongly flocculated colloidal gel, the viscoplastic model represents a lower-order approximation which has utility in a wide range of applications. In this study we are primarily interested in solution of the equilibrium stress state, however under the poro-elastic formulation the transient problem (6) must be evolved from the initial condition ($\phi=\phi_{0}$) toward the limit $t\rightarrow\infty$ to determine the distribution of stress and strain at the equilibrium state. Furthermore, as the strains associated with consolidation are large, finite strain measures are required to track material displacements in the Lagrangian frame, which adds further computational complexity. Conversely, the viscoplastic formulation allows one to analyse the equilibrium state (7) directly without need for temporal evolution. However, in multiple dimensions the viscoplastic model can lead to an under-determined stress state, analogous to statically indeterminate problems in structural mechanics. To circumvent this problem, we use a combination of both formulations to address the wall adhesion problem, which greatly simplifies the solution methodology. ## III Hyperelastic Constitutive Model The poro-elastic constitutive model under 1D compression (8) or simple shear (12) may be extended to arbitrary tensorial loadings via a hyperelastic constitutive model which is general enough to capture most observed phenomena of colloidal gels (Grenard et al., 2014; Sprakel et al., 2011; Uhlherr et al., 2005; Kumar et al., 2012). As particulate gels can undergo large volumetric strains, a finite strain measure is required as a basis for the hyperelastic model, as is provided by the Hencky strain tensor $\mathbf{H}=\ln\mathbf{U}$, where $\mathbf{U}$ is the right stretch tensor, i.e. $\mathbf{F}=\mathbf{R}\mathbf{U}$ where $\mathbf{R}$ is a proper orthogonal tensor, and $\mathbf{F}=\frac{\partial\mathbf{x}}{\partial\mathbf{X}}$ is the deformation gradient tensor arising from the Eulerian $\mathbf{x}$ and Lagrangian $\mathbf{X}$ coordinate frames. The Hencky strain tensor provides a convenient basis for constitutive modelling as $\mathbf{H}$ is work-conjugate with the Cauchy stress tensor $\Sigma^{N}$. Furthermore, the set of modified invariants $K_{i}$, $i=1:3$ of $\mathbf{H}$ introduced by Criscione et al. (2000) give rise to response terms which are mutually orthogonal, providing a clear elucidation between the invariants and various modes of deformation and their underlying symmetries. The first such invariant $K_{1}$ is associated with volumetric strain $K_{1}=\text{tr}(\mathbf{H})=\ln\frac{\phi_{0}}{\phi},$ (14) whilst the second invariant $K_{2}$ quantifies the magnitude of shear strain $K_{2}=\sqrt{\text{dev}(\mathbf{H}):\text{dev}(\mathbf{H})},$ (15) where $\mathbf{H}=\frac{1}{3}K_{1}\mathbf{I}+K_{2}\bm{\Phi}$, and the normalized deviatoric strain $\bm{\Phi}=\text{dev}(\mathbf{H})/K_{2}$, with $\bm{\Phi}:\bm{\Phi}=1$. The third invariant $K_{3}$ is associated with the mode of distortion $K_{3}=3\sqrt{6}\text{det}(\bm{\Phi}),$ (16) where $K_{3}\in[-1,1]$ such that $K_{3}=-1$ corresponds to uniaxial extension, $K_{3}=1$ uniaxial compression, and $K_{3}=0$ to pure shear. The original hyperelastic model is based upon an elastic potential $\psi=\psi(K_{1},K_{2},K_{3})$ which for perfectly elastic materials stores all work done by material deformations as internal strain energy. For such materials, the isotropic Cauchy stress $\mathbf{t}$ is $J\mathbf{t}=\frac{\partial\psi}{\partial\mathbf{H}},$ (17) where $J=\text{det}(\mathbf{F})$ is the total volumetric strain. The hyperelastic framework can also be extended to dissipative materials (as per the K-BKZ or Rivlin-Saywers type viscoelastic models), in which case the potential $\psi$ loses its strict thermodynamic interpretation (via decomposition into conservative and dissipative components $\psi=\psi_{c}+\psi_{d}$) as strain energy is no longer fully conserved, leading to irreversible deformations characteristic of strongly flocculated colloidal suspensions. In general, the potential $\psi$ may be dependent upon both strain-rate and strain history, as per the invariants $K_{i}$ $\psi=\psi(K_{i},\dot{K}_{i},t-s),$ (18) where $s\in(-\infty,t)$ is the strain history. As for a general nonlinear viscoelastic material (Wineman, 2009), the network stress $\Sigma^{N}$ is given by the generalisation of the Cauchy stress in (17) for a dissipative materials as $\Sigma^{N}=\mathcal{H}[\psi(K_{i},\dot{K}_{i},t-s)]_{-\infty}^{t}.$ (19) where the functional $\mathcal{H}[\,\,]^{t}_{-\infty}$ acts over the entire strain history. As such, the hyperelastic potential $\psi$ encodes the full viscoelastic rheology of the particulate network, including compressive and shear deformations and tensorial combinations thereof. For this dissipative potential $\psi$, from (19) the network stress tensor is quantified via the integro-differential equation $\begin{split}\Sigma^{N}=&\int_{-\infty}^{t}\frac{1}{\exp{K_{1}}}\sum_{i=1}^{3}\frac{\partial\psi}{\partial\mathbf{H}}ds\\\ =&\int_{-\infty}^{t}\frac{1}{\exp{K_{1}}}\left(\frac{\partial\psi}{\partial K_{1}}\mathbf{I}+\frac{\partial\psi}{\partial K_{2}}\bm{\Phi}-\frac{1}{K_{2}}\frac{\partial\psi}{\partial K_{3}}\mathbf{Y}\right)ds,\end{split}$ (20) where $\mathbf{Y}=3\sqrt{6}\bm{\Phi}^{2}=\sqrt{6}\mathbf{I}-3K_{3}\bm{\Phi}$. Hence $\partial\psi/\partial K_{i}$ encode the rheological properties of the suspension, namely the shear and compressive moduli. As strongly flocculated colloidal gels can only support small deviatoric strains prior to yield, then the compressive behaviour is essentially identical under differential uniaxial compression ($K_{3}=-1$) or spherical volumetric strain ($K_{2}=0$, $K_{3}$ indeterminate), and so the dependence of these materials upon the $K_{3}$ invariant is negligible. This simplification follows directly from the arguments in Section II that the macroscopic rheology can be quantified in terms of the total volumetric strain (encoded as $K_{1}$ or $\phi$) alone. Hence the rheology of strong colloidal gels only depends upon the magnitude of the deviatoric ($K_{2}$) and isotropic ($K_{1}$) strains. Whilst this simplification does not necessarily preclude interaction between combined shear and compression loadings, and experimental evidence Channell and Zukoski (1997) suggests such interactions can be significant, for simplicity we assume herein that these deformation modes act independently. Under these assumptions, the isotropic $p_{N}$ and deviatoric $\bm{\sigma}_{N}$ components of the network stress tensor $\Sigma^{N}$ may be generalized from (8) (12) as $\displaystyle p_{N}=\int_{-\infty}^{t}\frac{1}{\exp K_{1}}\frac{\partial\psi}{\partial K_{1}}ds=\int_{-\infty}^{t}\frac{K(\phi)}{\phi}\frac{\partial\phi}{\partial s}ds=\int_{\phi_{0}}^{\phi(t)}K(\phi)d\ln\phi,$ (21) $\displaystyle\bm{\sigma}_{N}=\int_{-\infty}^{t}\frac{1}{\exp K_{1}}\frac{\partial\psi}{\partial K_{2}}\bm{\Phi}ds=\int_{-\infty}^{t}\frac{\partial G(\phi,\gamma,t-s)}{\partial s}\bm{\Phi}(s)ds,$ (22) where $K(\phi)$, $G(\phi,\gamma,t)$ are the bulk and shear moduli respectively, which are related to $\psi$ as $\displaystyle\frac{\partial\psi}{\partial K_{1}}=-\phi_{0}\frac{\partial}{\partial s}\left(\frac{K(\phi)}{\phi}\right),$ (23) $\displaystyle\frac{\partial\psi}{\partial K_{2}}=\frac{\phi_{0}}{\phi}\frac{\partial G(\phi,\gamma,t-s)}{\partial s}.$ (24) Equations (21), (22) represent tensorial forms of (8), (12) under an appropriate finite strain measure ($\mathbf{H}$) for colloidal gels. This hyperelastic constitutive model describes the solid mechanics of the particulate network as a viscoelastic material which via (3) describe the deformation and flow and separation of colloidal gels. ## IV Viscoplastic Constitutive Model Given appropriate assumptions regarding the evolution of a batch settling experiment under the influence of wall adhesion effects, the viscoplastic constitutive model allows the equilibrium state $\phi=\phi_{\infty}$ to be approximated directly via the force balance (7) without need to solve the full material evolution equation (3). The primary assumption underpinning the equilibrium state is that the suspension is in a _critical_ state, whereby the network pressure $p_{N}$ is balanced by the compressive yield strength $P_{y}(\phi_{\infty})$ throughout $p_{N}=P_{y}(\phi_{\infty}),$ (25) This critical state arises for all colloidal gels regardless of the reversibility of consolidation, if _(i)_ the suspension is initially unnetworked, i.e. $\phi_{0}\leqslant\phi_{g}$, and _(ii)_ the network pressure $p_{N}$ for each material element monotonically increases with time over the course of the experiment. Under these conditions, the network pressure $p_{N}$ in (25) is identical to that of the hyperelastic formulation (22). This assumption is supported by the fact that the hydrodynamic drag between phases in a batch settling experiment decreases monotonically with time, from the initial condition in an asymptotic fashion toward the equilibrium state when the gravitational stress is supported solely by the inherent strength of the particulate network. During the sedimentation and consolidation process, fluid upflow at the walls (via a lubrication film) can be observed which prevents the network from adhering. It is proposed that as the equilibrium state is approached, the shear stress associated with this film generates a microscopic stick-slip mechanism between the particulate network and the container wall which allows the suspension interface to subside so long as this shear stress exceeds the wall adhesion strength. This mechanism is supported by experimental observations that the equilibrium suspension/supernantant interface is flat, whereas adhesion without slip would generate a concave interface due to subsidence of material in the interior. As the compressive stress due to gravitational acceleration increases with depth, this flat interface means that the particulate network experiences a compressive load throughout, whereas a convex interface could impart tensile load near the walls. The stick/slip mechanism suggests that the suspension is in a critical state of shear stress at the container walls, i.e. the suspension shear stress is equivalent to the wall adhesion strength $\tau_{N}\|_{r=R}=\tau_{w}(\phi\|_{r=R}).$ (26) This wall adhesion boundary condition for strongly flocculated colloidal suspensions is well-established in shear rheometry for the problem of wall- slip. Whilst it is conceivable that at large network pressures $p_{N}$, Coulombic friction may augment the wall boundary condition, the compressive stresses typical of batch settling experiments are such that frictional effects are dominated by the strong adhesive force between flocculated particles and the container wall. As this adhesive force (which gives rise to $\tau_{w}$) is weaker than the cohesive force between particles (which gives rise to $\tau_{y}$), this wall adhesion boundary condition ensures that throughout the internal shear stress $\tau_{N}\leqslant\tau_{y}(\phi)$, and by symmetry the shear stress is zero at the central axis: $\tau_{N}\|_{r=0}=0.$ (27) As the wall adhesion boundary condition ensures that the critical shear strain is never exceeded in the batch settling experiment, the suspension can only undergo very small strains in shear. Conversely, due to consolidation, the suspension can undergo large volumetric strains, as reflected in the change in local solids volume fraction from a few % to order 20% at equilibrium. This behaviour is typical of all strongly flocculated colloidal suspensions, these materials are brittle in shear and so can only support small deviatoric strains prior to flow, but are poroelastic in compression and so can support large volumetric strains which are largely irreversible (hence “ratchet poroelastic”). Further boundary conditions at the top of the bed are given by $\displaystyle\phi\|_{z=0}=\phi_{g},$ (28) $\displaystyle p_{N}\|_{z=0}=0,$ (29) $\displaystyle\bm{\sigma}_{N}\|_{z=0}=\mathbf{0}.$ (30) For the axis-symmetric batch settling problem, the equilibrium network force balance (7) may be directly expanded in cylindrical coordinates $(r,\theta,z)$ as $\displaystyle\frac{\partial\Sigma^{N}_{rr}}{\partial r}+\frac{\Sigma^{N}_{rr}-\Sigma^{N}_{\theta\theta}}{r}+\frac{\partial\Sigma^{N}_{rz}}{\partial z}=0,$ (31) $\displaystyle\frac{\partial\Sigma_{\theta\theta}}{\partial\theta}=0,$ (32) $\displaystyle\frac{\partial\Sigma^{N}_{rz}}{\partial r}+\frac{\Sigma^{N}_{rz}}{r}+\frac{\partial\Sigma^{N}_{zz}}{\partial z}+\Delta\rho g\phi=0,$ (33) where $z$ is the vertical coordinate down from the suspension/supernatant interface, hence $\mathbf{g}=g\,\hat{\mathbf{e}}_{z}$. Due to symmetry the transverse angular stresses $\Sigma^{N}_{r\theta}$, $\Sigma^{N}_{z\theta}$ are zero, and all gradients with respect to $\theta$ are zero. We define the network pressure as $p_{N}=-\frac{1}{3}\text{tr}(\Sigma^{N})$, the network shear stress magnitude $\tau_{N}=\|\text{dev}(\Sigma^{N})\|$, and the first and second normal stress differences respectively are $N_{1}=\Sigma^{N}_{zz}-\Sigma^{N}_{rr}$, $N_{2}=\Sigma^{N}_{rr}-\Sigma^{N}_{\theta\theta}$, following the usual convention where $(z,r,\theta)$ denote the (“flow”, “gradient”, “vorticity”) directions for the batch settling problem. As the number of field variables ($\phi$, $\Sigma^{N}_{rr}$, $\Sigma^{N}_{\theta\theta}$, $\Sigma^{N}_{zz}$, $\Sigma^{N}_{rz}$) exceeds the number of field equations (25), (31), (32), (33), the system is under-determined. Such statically indeterminate problems are common in plasticity theory; typically these arise from the application of constitutive models which do not possess well-defined stress-strain relationships below the critical yield stress. This deficiency is not a physical problem _per se_ , but rather stems from the over-simplified constitutive model; there exists a large class of problems for which plastic models generate under-determined systems (Hill, 1950; Balmforth et al., 2014). A common approach to resolve statically indeterminate systems is to invoke small-strain elasticity to solve deformations away from the equilibrium state and thus determine the equilibrium stress distribution. We utilise a similar approach here in that a closure approximation is invoked to generate a viscoplastic approximation to the equilibrium stress state, which may then be converted into the hyperelastic frame and closure approximation relaxed at the expense of disrupting the force balance away from equilibrium. As such, the viscoplastic estimate serves as a psuedo-initial condition for the hyperelastic formulation, from which the equilibrium state can be approached via the temporal evolution equation (3). If the closure approximation used to generate the viscoplastic estimate is accurate, only small deviatoric strains are required to evolve this estimate toward the true hyperelastic equilibrium condition, greatly simplifying the solution process. Furthermore, propagation toward the true equilibrium state generates quantitative estimates of the accuracy of the viscoplastic approximation. ## V Closure and Solution of Viscoplastic Formulation A closure approximation for the viscoplastic formulation may be generated by consideration of the equilibrium state below the critical bed depth $z_{c}$, where the stress equilibrium conditions render all derivatives with respect to $z$ to be zero. Under these conditions the viscoplastic solution simplifies to $\frac{\partial}{\partial r}\Sigma^{N}_{rr}+(\Sigma^{N}_{rr}-\Sigma^{N}_{\theta\theta})/r=0$, and $\frac{\partial}{\partial r}\Sigma^{N}_{rz}=\Delta\rho g\phi$, which under the axisymmetric boundary condition (27) gives $-p_{N}-\frac{1}{3}N_{1}+\frac{1}{3}N_{2}=\int_{0}^{r}\frac{N_{2}}{r^{\prime}}dr^{\prime}-P_{y}(\phi\|_{r=0}),\quad\text{for}\,\,z>z_{c}.$ (34) As $N_{1},N_{2}\leqslant\tau_{N}\leqslant\tau_{y}(\phi)\ll P_{y}(\phi)$ and the compressive stress $P_{y}(\phi)$ is a strongly increasing function of $\phi$, the radial solids volume fraction distribution varies weakly as $\phi(r)=P_{y}^{-1}\left[P_{y}(\phi\|_{r=0})-\int_{0}^{r}\frac{N_{2}}{r^{\prime}}dr^{\prime}-\frac{1}{3}N_{1}+\frac{1}{3}N_{2}\right]=\phi\|_{r=0}+\delta\phi(r),$ (35) where $\|\delta\phi(r)\|\ll\|\phi_{r=0}\|$ for $z>z_{c}$. Hence the $rz$ stress component $\Sigma^{N}_{rz}$ also deviates weakly from the linear distribution $\Sigma^{N}_{rz}=-\Delta\rho g\phi\|_{r=0}\frac{r^{2}}{2}+\Delta\rho g\int_{0}^{r}\delta\phi(r^{\prime})r^{\prime}dr^{\prime}.$ (36) These scalings motivate us to consider the approximation $\delta\phi=0$, which corresponds to the assumption that the first and second normal stress differences $N_{1}$, $N_{2}$ are negligible throughout the entire suspension, both above and below the critical bed depth $z_{c}$. Although the validity of this assumption for $z<z_{c}$ is unknown, this assumption is invoked temporarily as an intermediate step prior to relaxation of this closure under the hyperelastic formulation. Invoking the closure approximation $N_{1}=N_{2}=0$ simplifies the total shear stress to $\tau_{N}=\|\Sigma^{N}_{rz}\|$, and closes the set of governing equations which can be diagonalized in terms of the coupled hyperbolic system $\displaystyle\frac{\partial\omega_{1}}{\partial z}+\frac{\partial\omega_{1}}{\partial r}=-\frac{\omega_{1}-\omega_{2}}{2r}+\Delta\rho gf\left(\frac{\omega_{1}+\omega_{2}}{2}\right),$ (37) $\displaystyle\frac{\partial\omega_{2}}{\partial z}-\frac{\partial\omega_{2}}{\partial r}=-\frac{\omega_{1}-\omega_{2}}{2r}+\Delta\rho gf\left(\frac{\omega_{1}+\omega_{2}}{2}\right),$ (38) where $\omega_{1}=p_{N}+\tau_{N}$, $\omega_{2}=p_{N}-\tau_{N}$, and $f(p)=P_{y}^{-1}(p)$. From (27)-(30), the initial and boundary conditions are $\displaystyle\omega_{1}\|_{z=0}=\omega_{2}\|_{z=0}=0,$ (39) $\displaystyle\omega_{1}\|_{r=0}=\omega_{2}\|_{r=0},$ (40) $\displaystyle\omega_{1}\|_{r=R}=\mathcal{F}\left(\omega_{2}\|_{r=R}\right),$ (41) where $\mathcal{F}$ describes the relationship between the shear and compressive stress at the container wall, such that $\tau_{N}=\tau_{y}(\phi\|_{r=R})$, $p_{N}=P_{y}(\phi\|_{r=R})$. Based upon physical arguments and experimental data, Buscall (2009) proposes a relationship for the ratio $S(\phi)=\tau_{y}(\phi)/P_{y}(\phi)$ between the shear and compressive yield strength of a particulate gel, which rapidly decreases from around 1 at the gel point $\phi_{g}$ to the asymptotic value $S_{\infty}$ with increasing $\phi$. For a compressive yield strength of the form $P_{y}(\phi)=k\left(\left(\frac{\phi}{\phi_{g}}\right)^{n}-1\right),$ (42) the asymptotic value is $S_{\infty}=\kappa n\gamma_{c}.$ (43) where $\gamma_{c}$ is the critical shear strain and $\kappa$ the ratio of shear to compressive moduli, which is related to the the Poisson ratio $\nu$ as $\kappa=\frac{2}{3}\left(\frac{1-\nu}{1-2\nu}\right),$ (44) where $\nu=3/8$, $\kappa=5/3$ for systems bound by central forces as per Cauchy’s relationships. For a compressive yield strength with the functional form (42), $S(\phi)$ is then $S(\phi)=\left(\left(\frac{1}{S_{\infty}}-1\right)\left(1-\left(\frac{\phi}{\phi_{g}}\right)^{-n}\right)+1\right)^{-1}.$ (45) For these relationships, the operator $\mathcal{F}$ is explicitly $\mathcal{F}(\omega_{1})=\frac{-S_{\infty}(2k+\omega_{1})+\sqrt{4kS_{\infty}^{2}(k+\omega_{1})+\omega_{1}^{2}}}{1+S_{\infty}},$ (46) however other forms arise for different functional forms of $P_{y}(\phi)$. \| ---\|--- (a) \| (b) Figure 2: Schematic (a) of $\omega_{1}$ (red), $\omega_{2}$ (blue) characteristics and boundary conditions for the viscoplastic formulation, and typical distribution (b) of the network shear stress $\tau_{N}$. Solutions to the coupled hyperbolic system (37), (38) are organized by characteristics which propagate from the top of the bed ($z=0$) at 45 degrees to the container wall and normal to each other as per Fig 2(a), and manifest as $C^{1}$ shocks (i.e. non Lifshitz-continuous) contours in the shear stress distribution as shown in Fig 2(b). Whilst the mathematical details differ, such 45 degree characteristics are typical of viscoplastic flows, such as slip-line fields in plasticity theory (Balmforth et al., 2014; Hill, 1950). The reflection conditions (40), (41) correspond to the boundary conditions $\tau_{N}=0$ and $\tau_{N}=\tau_{y}(\phi)$ respectively, and $\omega_{1}$, $\omega_{2}$ increase from zero at the top of the bed (39) due to the gravitational source in (37) (38). Predictions of the network shear stress, network pressure and solids volume fraction distributions for the colloidal suspension (a) in Table 1 in 22 mm and 110 mm diameter settling columns are shown in Fig.s 3 and 4 respectively. The signature of the $\omega_{1}$, $\omega_{2}$ characteristics are clearly shown in the network shear stress distribution, and the number of times the characteristics are reflected from the $r=0$ and $r=R$ boundaries plays a significant role. The curved contours are due to the radial source terms $(\omega_{1}-\omega_{2})/2r$ in (37), (38), whereas straight contours arise in Cartesian geometries. For both columns represented in Fig.s 3 and 4, the network pressure and shear stress increase asymptotically with bed depth $z$ toward the equilibrium condition where all of the gravitational stress is borne by the shear stress. As such, the critical bed depth $z_{c}$ is never reached in practice, however it is possible to identify a finite representative depth $z_{c}^{\prime}$ at which deviations from the equilibrium state are negligible. For the larger diameter column (Fig. 4), the wider spacing between boundary reflections means that this equilibrium state is approached more slowly than in narrower columns. In general, these solutions of the equilibrium stress state appear to be physically plausible apart from the $C^{1}$ shocks in the network shear stress distribution, which arise from idealizations of the viscoplastic constitutive model and the closure approximation of zero normal stress differences. To resolve the accuracy of this viscoplastic approximation, the small strain hyperelastic formulation can be used to determine the true equilibrium state. \| \| ---\|---\|--- (a) \| (b) \| (c) Figure 3: Equilibrium (a) network shear stress $\tau_{N}$, (b) network pressure $p_{N}$, and (c) solids volume fraction $\phi$ distributions for suspension (a) in Table 1 as predicted by viscoplastic formulation in a 22 mm diameter container. \| \| ---\|---\|--- (a) \| (b) \| (c) Figure 4: Equilibrium (a) network shear stress $\tau_{N}$, (b) network pressure $p_{N}$, and (c) solids volume fraction $\phi$ distributions for suspension (a) in Table 1 as predicted by viscoplastic formulation in a 110 mm diameter container. ## VI Small Strain Solution of Hyperelastic Formulation Central to solution of the equilibrium state $\zeta_{\infty}$ under the hyperelastic formulation are the assumptions stated above that (i) volumetric and isochoric strains for the suspension increase monotonically with time in an asymptotic fashion toward the equilibrium state, (ii) the equilibrium state is described by a critical state where the compressive yield stress balances the network pressure everywhere, and the network shear stress is equal to the shear yield strength at the container wall. These assumptions mean that the compressive irreversibility constraint need not be explicitly invoked, and similarly the shear yield criterion is never exceeded. As such, a unique equilibrium state is reached for all reasonable initial conditions where $\phi_{0}<\phi_{g}$. This behaviour provides a significant simplification of the hyperelastic formulation, as it is no longer necessary to determine the path integral (20) from the initial condition $\zeta_{0}$, as any reasonable state which satisfies the conditions above shall ultimately converge to the unique equilibrium state $\zeta_{\infty}$. Hence, whilst the viscoplastic solution $\zeta_{v}$ may not represent a point on a true solution path from $\zeta_{0}$, under the hyperelastic formulation this state will still converge to the true equilibrium solution $\zeta_{\infty}$. The relationship between these states may be represented as $\hat{\zeta_{0}}\xrightarrow{\mathbf{d}_{v}}\hat{\zeta_{v}}\Rightarrow\underset{\bm{\kappa}_{v}}{\zeta_{v}}\xrightarrow{\mathbf{d}_{h}}\zeta_{\infty},$ where the hat refers to the viscoplastic formulation, $\Rightarrow$ denotes conversion from the viscoplastic to the hyperelastic frame, and $\bm{\kappa}_{v}$ represents a reference state given by the hyperelastic frame. The total deformation $\mathbf{d}$ from the initial condition $\zeta_{0}$ in the frame $\bm{\kappa}_{v}$ is the sum of deformations under the viscoplastic and hyperelastic formulations $\mathbf{d}=\mathbf{d}_{v}+\mathbf{d}_{h},$ (47) and total Hencky strain is given by the sum of the strains $\mathbf{H}=\mathbf{H}_{v}+\mathbf{H}_{h}.$ (48) Due to the condition $\tau_{N}\leqslant\tau_{y}(\phi)$ and the brittleness of colloidal suspensions under shear, all deviatoric strains associated with relaxation from the viscoplastic frame to the hyperelastic frame are small $\gamma\leqslant\gamma_{c}$, and so in general the deviatoric Hencky strain is well-approximated by the infinitesimal strain tensor $\text{dev}(\mathbf{H})\approxeq\text{dev}(\bm{\epsilon})=\text{dev}\left(\frac{1}{2}(\nabla_{v}\mathbf{d}+(\nabla_{v}\mathbf{d})^{T})\right),$ (49) with $\gamma\approxeq\sqrt{\text{dev}(\bm{\epsilon}):\text{dev}(\bm{\epsilon})}$. Conversely, whilst the deviatoric strains are small, the particulate network can undergo large-scale consolidation, and hence support large isotropic strains, the volumetric component of the finite strain measure (14) in the material frame $\bm{\kappa}_{v}$ must be preserved as $\phi=\phi_{0}\exp(-\nabla\cdot\mathbf{d}),$ (50) and so the total Hencky strain for strongly flocculated colloidal gels may be well approximated as $\mathbf{H}=\frac{1}{3}\ln\left(\frac{\phi_{0}}{\phi}\right)\mathbf{I}+\text{dev}(\bm{\epsilon}).$ (51) Under these strain measures, the suspension network pressure (21) and deviatoric stress (22) are now $\displaystyle p_{N}=\int_{\phi_{0}}^{\phi(t)}K(\phi)d\ln\phi=P_{y}(\phi_{v})+\int_{\phi_{v}}^{\phi(t)}K(\phi)d\ln\phi,$ (52) $\displaystyle\bm{\sigma}_{N}=\int_{-\infty}^{t}\frac{\partial G(\phi,\gamma,t-s)}{\partial s}\text{dev}(\bm{\epsilon}(s))ds.$ (53) As the timescale of material deformations of the particulate network is slow (due to the magnitude of the interphase drag coefficient $R(\phi)$), the shear modulus $G(\phi,\gamma,t)$ is well-approximated by the infinite-time modulus $G_{\infty}(\phi,\gamma)=\lim_{t\rightarrow\infty}G(\phi,\gamma,t)$. Furthermore, as the critical shear strain $\gamma_{c}$ is small, it is unnecessary to resolve the nonlinear shear strain prior to yield, and so the infinite-time modulus may be linearized from (13) as $G_{\infty}(\phi)=\frac{1}{\gamma_{c}}\tau_{y}(\phi),$ (54) and integration by parts of (53) yields $\begin{split}\bm{\sigma}_{N}=\int_{-\infty}^{t}G_{\infty}(\phi)\frac{\partial}{\partial s}\text{dev}(\bm{\epsilon}(s))ds.\end{split}$ (55) In converting the viscoplastic solution $\hat{\zeta}_{v}$ to the hyperelastic formulation $\zeta_{v}$, this solution is now over-determined in the hyperelastic frame due to the closure $N_{1}=N_{2}=0$ developed in Section V. Relaxation of this constraint perturbs the force balance (7) away from equilibrium, and so the relaxed viscoplastic solution forms an initial condition for the evolution equation (3) which can then evolve to the true equilibrium state. This process can be formally represented as $\nabla\cdot\hat{\Sigma}^{N,v}+\Delta\rho\mathbf{g}\phi_{v}=0\Rightarrow\nabla\cdot\Sigma^{N,v}+\Delta\rho\mathbf{g}\phi_{v}=\mathbf{S},$ (56) where $\mathbf{S}$ is the force imbalance due to conversion of the network stress $\hat{\Sigma}^{N,v}$ in the viscoplastic frame to the network stress $\Sigma^{N,v}$ in the hyperelastic frame. This conversion requires calculation of the deformation vector $\mathbf{d}_{v}$ from the solids volume fraction distribution $\phi_{v}$ as $\nabla\cdot\mathbf{d}_{v}=\log\left(\frac{\phi_{0}}{\phi_{v}}\right),$ (57) subject to the boundary condition $\mathbf{d}_{v}\|_{r=0,z=0}=0$. Although it is not possible to determine the temporal evolution of $\mathbf{d}$ up to the viscoplastic solution, the arguments above justify that convergence to the unique equilibrium state are independent of the solution path. Hence, for simplicity we assume $\mathbf{d}$ evolves linearly under the viscoplastic solution as $\mathbf{d}(t)=h(t)\mathbf{d}_{v},\quad\text{for}\,\,\,\,t\leqslant t_{v},$ (58) where $h(t)$ is the ramp function $h(t)=\begin{cases}&0\quad t\leqslant 0,\\\ &t/t_{v}\quad 0<t\leqslant t_{v},\\\ &1\quad t_{v}<t,\end{cases}$ (59) and $t_{v}$ is a nominal time at which the viscoplastic solution occurs. As such, the deviatoric network stress at $t_{v}$ is $\bm{\sigma}_{N}^{v}=\int_{0}^{t_{v}}G_{\infty}(\phi(s))h^{\prime}(s)\text{dev}(\bm{\epsilon}_{v})ds=H(\phi_{v})\frac{\text{dev}(\bm{\epsilon}_{v})}{\nabla\cdot\mathbf{d}_{v}},$ (60) where $H(\phi_{v})=\int_{\phi_{0}}^{\phi_{v}}\frac{1}{\varphi}G_{\infty}(\varphi)d\varphi$. Hence the stress imbalance arising from relaxation of the closure constraint manifests as $\mathbf{S}=\nabla\cdot\left(\Sigma^{N,v}-\hat{\Sigma}^{N,v}\right)=\nabla\cdot\left(H(\phi_{v})\frac{\text{dev}(\bm{\epsilon}_{v})}{\nabla\cdot\mathbf{d}_{v}}-\hat{\tau}_{N}^{v}\left(\begin{array}[]{cc}0&1\\\ 1&0\end{array}\right)\right),$ (61) where $\hat{\tau}_{N}^{v}$ is the network shear stress calculated for the viscoplastic solution. As the total network stress is $\Sigma^{N}=\Sigma^{N,v}+\Sigma^{N,h}$, where $\Sigma^{N,h}$ is the hyperelastic component of the total stress, then from (56) the evolution equation (3) from the viscoplastic solution $\zeta_{v}$ is $\frac{\partial\phi}{\partial t}+\mathbf{q}\cdot\nabla\phi+\nabla\cdot\left[\frac{(1-\phi)^{2}}{R(\phi)}\left(\nabla\cdot\Sigma^{N,h}+\Delta\rho\mathbf{g}\delta\phi+\mathbf{S}\right)\right]=0,$ (62) where $\delta\phi=\phi-\phi_{v}$, $\phi\|_{t=t_{v}}=\phi_{v}$, and $\Sigma^{N,h}\|_{t=t_{v}}=\mathbf{0}$. The solids phase velocity is defined as $\mathbf{v}_{s}\equiv\frac{\partial}{\partial t}\mathbf{d}$, and the evolution equation (3) is of the form (Lester et al., 2010) $\frac{\partial\phi}{\partial t}+\mathbf{q}\cdot\nabla\phi+\nabla\cdot((1-\phi)\mathbf{v}_{r})=0,$ (63) where $\mathbf{v}_{r}=\frac{\mathbf{v}_{s}-\mathbf{q}}{1-\phi}$ is the inter- phase velocity. Under the assumption $\mathbf{q}=0$, (62), (63) may be re-cast as an evolution equation for the hyperelastic deformation $\mathbf{d}_{h}$ $\frac{\partial}{\partial t}\mathbf{d}_{h}=\frac{(1-\phi)^{2}}{R(\phi)}\left(\nabla\cdot\Sigma^{N,h}+\Delta\rho\mathbf{g}\delta\phi+\mathbf{S}\right),$ (64) subject to the initial condition $\mathbf{d}_{h}\|_{t=t_{v}}=\mathbf{0}$ and the same boundary conditions (26)-(30) as for the viscoplastic problem. The network tensor $\Sigma^{N,h}$ is explicitly $\Sigma^{N,h}=\int_{\phi_{v}}^{\phi}K(\varphi)d\ln\varphi\mathbf{I}+\int_{t_{v}}^{t}G_{\infty}(\phi)\frac{\partial}{\partial s}\text{dev}(\bm{\epsilon}_{h})ds,$ (65) where the bulk and shear moduli are given from $P_{y}(\phi)$, $\tau_{y}(\phi)$, $\gamma_{c}$ in (45), (42) as $\displaystyle K(\phi)=\frac{1}{\phi}\frac{dP_{y}(\phi)}{d\phi},$ (66) $\displaystyle G_{\infty}(\phi)=\frac{1}{\gamma_{c}}\tau_{y}(\phi).$ (67) \| \| ---\|---\|--- (a) \| (b) \| (c) Figure 5: Equilibrium (a) network shear stress $\tau_{N}$, (b) network pressure $p_{N}$, and (c) solids volume fraction $\phi$ distributions for suspension (a) in Table 1 as predicted by the hyperelastic formulation in a 22 mm diameter container. \| ---\|--- (a) \| (b) Figure 6: distributions of (a) radial $d_{h,r}$ and (b) vertical $d_{h,z}$ components of the hyperelastic deformation $\mathbf{d}_{h}$ as predicted by (47) in a 22 mm diameter container. The evolution equation (64) describes relaxation of the particulate network from the viscoplastic solution to the hyperelastic equilibrium condition as a nonlinear elliptic equation. A finite difference routine is used to numerically solve (64) on a series of increasingly fine spatial grids subject to successive over-relaxation to aid both robustness and convergence. Under this routine, (64) converges relatively quickly, and the equilibrium distributions of $\tau_{N}$, $p_{N}$ and $\phi_{N}$ for suspension (a) in the 22 mm column are shown in Fig. 5 (a)-(c), with the associated hyperelastic deformation $\mathbf{d}_{h}$ shown in Fig. 6. The most striking impact of the hyperelastic equilibrium compared to Fig. 3 is the smoothing of the $C^{1}$ shocks from the network shear stress distribution, resulting in a smooth radial profile of $\tau_{N}$. This removal is related to the change in governing equations from hyperbolic to elliptic, and demonstrates clearly that the $C^{1}$ shocks are spurious artifacts of the viscoplastic formulation under the closure approximation $N_{1}=N_{2}=0$. Whilst the smoothing of $C^{1}$ shocks appears markedly in Fig. 5(a), the $L_{2}$ norm of $\tau_{N}$ between the viscoplastic and hyperelastic solutions is of the order 2%, and so the gross distribution of shear stress is preserved. Likewise the network pressure and solid volume fraction distributions are only slightly altered ($L_{2}$ norm 0.5%, 0.3% respectively). The magnitude of the hyperelastic strain is also small, $\|\mathbf{d}_{h}\|\leqslant 3.2\times 10^{-6}$ m. From Fig. 6, these strains are primarily located near the $C^{1}$ shocks, and so most of the deformation relaxation of the hyperelastic body occurs as isochoric strains due to the stress imbalance generated by these shocks. As such, whilst the viscoplastic formulation under the closure assumption $N_{1}=N_{2}=0$ does introduce spurious artifacts in the form of the reflected $C^{1}$ shocks in the shear stress field, this model still yields accurate estimates of the solids volume fraction $\phi$ and network pressure $p_{N}$ distributions, and captures the gross features of the shear stress $\tau_{N}$ distribution. Despite this deficiency, this model serves as a useful tool for the modeling and characterization of sedimentation in the presence of significant wall adhesion effects. Most importantly, the hyperelastic solution above directly quantifies the nature and extent of the errors associated with the viscoplastic solution. ## VII Application of Viscoplastic Solution to Experimental Data Solution of the small-strain hyperelastic formulation above shows that the viscoplastic formulation under the closure assumption ($N_{1}=N_{2}=0$) for wall adhesion in batch sedimentation yields an accurate approximation of the solids volume fraction $\phi$ and network pressure $p_{N}$ distributions, and captures the gross features of the network shear stress $\tau_{N}$. This represents a significant simplification, as the hyperelastic formulation (20) (3) is significantly more computationally intensive, even if the small-strain hyperelastic formulation (64) is invoked. To test the viscoplastic model and develop methods to extract the relevant rheological functions, we apply the viscoplastic model to equilibrium solids volume fraction profile data (measured using a gamma-ray attenuation device (Labbett et al., 2006)) across batch settling columns of various widths to estimate both the compressive and shear yield strength functions. These estimates are then compared to _in-situ_ measurements of the shear yield stress using a vane rheometer, which serves as an indirect test of the viscoplastic model. In this study we consider three suspensions which consist of mean size 4 $\mu$m calcium carbonate primary particles (Omyacarb 2, Omya Australia Pty Ltd.) of density difference $\Delta\rho=1720$kg m-3 with respect to an aqueous solution. The primary particles are flocculated in a 22 mm ID continuous flow pipe reactor (Owen et al., 2008) using two different commercial high molecular weight polymer flocculants (Magnafloc 336 and Rheomax DR 1050, BASF). Both flocculants were made up at 0.1 w/w% and diluted immediately prior to use to 0.01 w/w% prior to application at a solids concentration of 90 g/L for a residence time of 9.9 s in the continuous flow pipe reactor operating at a flowrate of 20 L/min. Further details of the flocculation conditions are described by Owen et al. (2008), and the different dosages and remaining flocculation conditions are summarized in Table 1. The settling behaviour of these three suspensions was measured in two different diameter (22 mm and 110 mm) cylindrical settling columns of initial height $h_{0}\approx 2000$ mm and initial solids concentration $\phi_{0}\approx 0.033$. These suspensions exhibit markedly different transient and equilibrium sedimentation behaviour, as reflected in equilibrium solids volume fraction profile data under batch settling (shown in Fig. 7), as measured by gamma ray attenuation in both 22 mm and 110 mm diameter cylindrical settling columns. Evidence of wall adhesion is clearly shown in the 22 mm columns, where the solids volume fraction profile evolves to a constant value (within scatter) with increasing bed depth, suggesting that beyond a critical depth the suspension weight is entirely supported by the container walls. Suspension \| Flocculant \| Dosage [g/t] \| $\phi_{g}$ [-] \| $k$ [Pa] \| $n$ [-] \| $S_{\infty}$ [-] ---\|---\|---\|---\|---\|---\|--- (a) \| Magnafloc 336 \| 46 \| 0.0918 \| 3.21 \| 5.48 \| 0.157 (b) \| Rheomax DR 1050 \| 30 \| 0.1042 \| 0.63 \| 7.03 \| 0.112 (c) \| Rheomax DR 1050 \| 46 \| 0.0890 \| 0.16 \| 7.01 \| 0.113 Table 1: Suspension flocculant type, dosage and fitted rheological parameters. Estimation of the compressive and shear yield stress functions is performed by fitting of the relevant rheological parameters via minimization of the $L_{2}$ error between the solids volume fraction profiles as predicted by the viscoplastic model and the measured data over the 22 mm and 110 mm columns. The functional form (42) is used for the compressive yield strength function $P_{y}(\phi)$, and the critical strain relationship proposed by Buscall (2009) is used for the shear yield strength function $\tau_{y}(\phi)$, where $\tau_{y}(\phi)=S(\phi)P_{y}(\phi)$, and $S(\phi)$ is given by (45). Hence fitting of the equilibrium solids volume fraction profiles comprises of four rheological constants: the suspension gel point $\phi_{g}$, the consistency $k$ in (42), the index $n$ in (42), and the asymptotic shear/compressive yield strength ratio $S_{\infty}$ in (45). Note that the suspension gel point can be accurately estimated directly from the equilibrium solids volume fraction data a priori, and so the numerical fitting method only involves 3 variable parameters. Minimization of the $L_{2}$ error between model predictions and experimental data is performed via a simplex optimisation routine, where the numerical resolution of the finite difference routine used to solve the viscoplastic model (37) (38) is increased as the rheological parameters converge. Although the viscoplastic model predicts variation in the radial network pressure distribution (e.g. Fig. 5 (c)), the resultant variation in solids volume fraction profile is weak, and so such variations are neglected in comparison between experimental data and model predictions. The vertical solids volume fraction profiles from the fitted viscoplastic model are shown in Fig. 7 along with the experimental gamma ray attenuation, and the fitted compressive and shear yield strength curves are shown in Fig. 8. The rheological parameters associated with these fits are also summarized in Table 1, of note is the large asymptotic yield strength ratio $S_{\infty}\approx 0.1-0.15$ required to fit the measured solids volume fraction profiles. We found no means of fitting the experimental data in Fig. 7 without incorporating such values of $S_{\infty}$, regardless of the functional form of $\tau_{y}(\phi)$, $P_{y}(\phi)$, hence we conclude that these values are an accurate representation of the asymptotic yield strength ratio, independent of fitting methodology. The range $S(\phi)\approx$ 0.1-1 found here for the materials summarised in Table 1 is overall significantly larger than the range $S_{\infty}\approx$ 0.001-0.2 reported previously (Buscall et al., 1987, 1988; de Kretser et al., 2002; Zhou et al., 2001; Channell and Zukoski, 1997) for a range of strongly flocculated suspensions (and summarised in Buscall (2009)), leading to much stronger wall effects. Overall, $S(\phi)$ is expected to decay from unity at the gel-point towards the asymptotic value $S_{\infty}$ dependent _inter alia_ upon the elasticity of the interparticle bonds and the particle-size (Buscall, 2009). Much of the earlier data refers to electrolyte-coagulated systems away from the gel-point and, with the benefit of hindsight, it is perhaps not too surprising to find that high-polymer flocculated systems are different. Furthermore, more than a dozen types or mechanisms of flocculation are known and this alone means that wall effects are likely to be more important for some systems than others. \| \| ---\|---\|--- (a) \| (b) \| (c) Figure 7: Measured data (points) and model predictions (lines) of the equilibrium solids volume fraction profile $\phi_{\infty}$ for $R_{s}$=0.011[m] (black) and $R_{l}$=0.055[m] (gray) column widths for calcium carbonate suspensions under flocculant types and dosages (a)-(c) summarized in Table 1. Figure 8: Fitted compressive yield strength $P_{y}(\phi)$ (solid) and shear yield strength $\tau_{y}(\phi)$ (dashed) curves from viscoplastic model for suspensions (a) (red), (b) (blue), (c) (green) summarized in Table 1. Figure 9: Schematic of shear yield stress measurement protocol in the 110 mm ID column using an _in-situ_ vane rheometer. The top schematic shows the vane placement with respect to the column cross-section, and the bottom schematic shows a side profile of the vane locations at different heights of the column. All dimensions are in mm, and the vane measurement order is given by $A$-$D$. Measured data is shown in Table 2 Suspension \| Flocculant \| Dosage [g/t] \| A [Pa] \| B [Pa] \| C [Pa] \| D [Pa] ---\|---\|---\|---\|---\|---\|--- (a) \| Magnafloc 336 \| 46 \| 14.58 \| 80.68 \| 61.24 \| 95.01 (b) \| Rheomax DR 1050 \| 30 \| 4.37 \| 41.55 \| 26.97 \| 56.38 (c) \| Rheomax DR 1050 \| 46 \| 5.35 \| 49.33 \| 27.46 \| 57.83 Table 2: Table of _in-situ_ vane shear yield strength measurements for vane protocol A-D shown in Fig. 9 To validate the fits predicted from the viscoplastic model, _in-situ_ shear yield stress measurements were performed in the 110 mm column as per the protocol shown in Fig. 9, using a Haake VT500 rheometer fitted with a cruciform vane (diameter 22 mm, height 31 mm) following the procedure of Nguyen and Boger (1983). As the shear vane spans a significant bed height (31 mm), the measured shear yield strength is an average of the shear yield strength distribution (arising from the solids volume fraction distribution) over the height of the vane. These average shear yield strengths are shown in Table 2, and the corresponding average solids volume fraction for each measurement is determined from the fitted solids volume fraction profile for the 110 mm column shown in Fig. 7. These averaged values are shown in Fig. 10 as data points, and the shear yield stress functions predicted from fitting of the profile data in Fig. 7 are shown as continuous curves. The fitted shear yield stress functions agree within experimental error (estimated to be order 20%) of the _in-situ_ vane rheometer measurements. Also note that the accuracy of the fitted shear yield stress function is contingent upon both the validity of the assumption $\tau_{y}(\phi)\approx\tau_{w}(\phi)$, and linearisation of $\tau_{y}(\phi)$ over the vane height during the averaging process. \| \| ---\|---\|--- (a) \| (b) \| (c) Figure 10: _In-situ_ measurements (points) and model predictions (lines) of the shear yield stress $\tau_{y}(\phi)$ for calcium carbonate suspensions under flocculant types and dosages (a)-(c) summarized in Table 1. \| ---\|--- (a) \| (b) Figure 11: (a) Errors for uncorrected (i.e. not accounting for wall adhesion) estimation of the compressive yield stress $P_{y}(\phi)$ for suspension (c) in Table 1 in the 22 mm and 110 mm columns, and (b) prediction of equilibrium solids volume fraction profiles for suspension (c) in Table 1 across various column diameters. Agreement between the viscoplastic model and _in-situ_ shear yield strength and volume fraction profile measurements suggests that the ratio $S(\phi)$ of shear to compressive yield strength for these colloidal gels is significantly higher (0̃.1-1) than previously reported, and as such wall adhesion effects are also much more prevalent. Neglect of wall adhesion effects can lead to very serious errors in the estimation of the compressive yield strength, as illustrated in Fig. 11(a), which shows significant deviations from the corrected compressive yield strength in both the 22 mm and 110 mm columns, where the divergence of $P_{y}(\phi)$ for the uncorrected estimate in the 22 mm column around $\phi\approx 0.23$ arises from the vertical solids volume fraction profile in Fig. 1. Fig. 11(a) also suggests significant wall effects arise in the 110 mm column, where the uncorrected compressive yield strength estimate deviates by up to 100% at large $\phi$. Traditionally, such wide columns would have been considered free from wall adhesion effects, and even very wide columns still exhibit significant errors as shown in Fig. 11(b). For such batch sedimentation problems involving significant wall adhesion effects, the viscoplastic formulation (37) (38) under the closure approximation $N_{1}=N_{2}=0$, and wall adhesion assumption $\tau_{y}(\phi)\approx\tau_{w}(\phi)$ serves as a useful analysis tool, and leads to a useful methodology to generate accurate estimates of the shear $\tau_{y}(\phi)$ and compressive $P_{y}(\phi)$ yield strengths from the solids volume fraction profile data across several columns of various widths. A useful experimental parameter is the minimum column diameter $D_{\min}$ required to render wall adhesion effects negligible. To derive a relationship between $D_{\min}$ and the relevant experimental and suspension parameters, we define the relative error $\epsilon$ from the vertical force balance (1) as the relative contribution of wall adhesion stress $\epsilon\equiv\frac{\frac{4}{D_{\min}}\tau_{w}(\phi)}{\Delta\rho g\phi},$ (68) where in the limit of vanishing $\epsilon$, gravitational stress is balanced by the compressive yield strength of the suspension, and a 1D stress analysis is valid. Under the assumption that wall shear strength is well approximated by the bulk shear strength, $\tau_{w}(\phi)\approx\tau_{y}(\phi)=S(\phi)P_{y}(\phi)$, then for particulate gels whose shear yield strength is well characterized by (42) and (45), (68) may be expressed as $D_{\min}\epsilon\frac{\phi_{g}}{S_{\infty}}\frac{\Delta\rho g}{k}=\frac{4}{S_{\infty}}\frac{\frac{p_{\infty}}{k}\left(1+\frac{p_{\infty}}{k}\right)^{1-\frac{1}{n}}}{1+\frac{p_{\infty}}{k}\frac{1}{S_{\infty}}},$ (69) where $p_{\infty}=\Delta\rho g\phi_{0}h_{0}$ is the equilibrium network pressure at the base of a batch settling experiment with initial height $h_{0}$ and volume fraction $\phi_{0}$. From (45), $S(\phi)$ decays rapidly from $S\sim 1$ toward the asymptotic value $S\rightarrow S_{\infty}$ as $\phi$ increases from $\phi_{g}$ for $n>2$, and so (69) only varies very weakly with changes in $S_{\infty}$ for $S_{\infty}<0.1$. Under these physically reasonable conditions, the right hand side of (69) only varies significantly with $p_{\infty}/k$ and the index $n$, and this relationship is plotted in Fig. 12 for various indices $n$ and relative stress $p_{\infty}/k$. Figure 12: Scaled minimum column diameter $D_{\text{min}}$ and relative error $\epsilon$ as a function of scaled equilibrium network pressure $p_{\infty}/k$ for various indices $n$ for functional forms (42), (45). Suspension \| $p_{\infty}$ [Pa] \| $S_{\infty}$ [-] \| $D_{\text{min}}$ [mm] \| $z_{\text{max},D=22\text{mm}}$ [mm] \| $z_{\text{max},D=110\text{mm}}$ [mm] \| $\phi_{\text{max},D=22\text{mm}}$ [-] \| $\phi_{\text{max},D=110\text{mm}}$ [-] ---\|---\|---\|---\|---\|---\|---\|--- (a) \| 1142.9 \| 0.157 \| 3197 \| 0.649 \| 8.520 \| 0.0966 \| 0.1274 (b) \| 1125.1 \| 0.112 \| 2016 \| 2.488 \| 13.962 \| 0.1436 \| 0.1872 (c) \| 1137.6 \| 0.113 \| 1960 \| 2.714 \| 14.033 \| 0.1496 \| 0.1955 Table 3: Estimation of minimum column diameter $D_{\text{min}}$, maximum bed depth $z_{\text{max}}$, and maximum volume fraction $\phi_{\text{max}}$ for suspensions (a)-(c) in 22 mm and 110 mm diameter columns for an experimental error $\epsilon$=5%. Application of (69) to the three suspensions considered experimentally clearly illustrates the prevalence of wall adhesion effects for high molecular weight polymer flocculated colloidal gels. For a modest experimental error $\epsilon=5$%, minimum column diameters $D_{\min}$ of at least 2 m are required to render wall adhesion effects negligible. Conversely, one may calculate the maximum bed depth $z_{\max}$ and solids volume fraction $\phi_{\max}$ below which the error exceeds $\epsilon$. All of these measures indicate that either very shallow bed depths or impractically wide columns are required to avoid significant errors, hence correction of wall adhesion effects is necessary for many experiments involving strongly flocculated colloidal suspensions. Development of this model for batch sedimentation with wall adhesion effects raises fundamental issues regarding the constitutive modeling of strongly flocculated colloidal gels, specifically the validity and utility of the viscoplastic and hyperelastic formulations. Whilst selection of an appropriate modelling framework is contingent upon the application at hand and the requisite level of fidelity and tractability, an understanding of the nature and range of validity of each approach is of critical importance. It is anticipated that such issues shall play a central role in the continuing development and application of constitutive models for strongly flocculated colloidal gels. ## VIII Conclusions The assumption that wall adhesion effects are negligible for the batch settling of strongly flocculated colloidal gels is commonly invoked via the justification that the shear yield strength is small compared to the compressive yield strength. In this study, _in-situ_ measurements of both colloidal gel rheology and solids volume fraction distribution in equilibrium batch settling experiments suggest the contrary for polymer flocculated colloidal gels, where wall adhesion effects are found to be significant in a 110 mm diameter column, normally considered to be sufficient to render wall effects negligible. Neglect of such effects can lead to serious errors in estimation of e.g. the compressive yield stress, where errors of order 100% are observed at higher concentrations in the 110 mm diameter column, and divergence of the compressive yield stress for a 22 mm diameter column. Consideration of a mathematical model for the batch settling equilibrium stress state in the presence of wall adhesion raises fundamental issues regarding the constitutive modeling of strongly flocculated colloidal gels, namely the relative utility of viscoplastic and viscoelastic rheological models under arbitrary tensorial loadings. More commonly used viscoplastic models (e.g. generalisation of Herschel-Bulkely or Bingham models to the compressible case (Lester et al., 2010; Stickland and Buscall, 2009; Michaels and Bolger, 1962)) quantify the shear and compressive rheology of colloidal gels solely in terms of critical yield strength, ignoring the detailed mechanisms of shear strain softening and compressive strain hardening, whereas hyperelastic models treat the particulate network as a history-dependent viscoelastic material which facilitates detailed resolution of the complex rheological behaviour (Lindstrom et al., 2012; Sprakel et al., 2011; Gibaud et al., 2010; Santos et al., 2013; Koumakis and Petekidis, 2011; Gibaud et al., 2008; Ovarlez et al., 2013; Ramos and Cipelletti, 2001; Ovarlez and Coussot, 2007; Cloitre et al., 2000; Tindley, 2007; Kumar et al., 2012; Uhlherr et al., 2005; Grenard et al., 2014) inherent to colloidal gels. In the context of batch settling with wall adhesion effects, the viscoplastic formulation leads to a statically indeterminate formulation due to the strain being undefined for stresses below the yield value. This formulation is closed by assuming the first normal stress difference is negligible, and this closure is found to be a reasonable approximation by conversion of the viscoplastic solution to the hyperelastic frame and evolution to the hyperelastic equilibrium state. Whilst the viscoplastic model is appropriate for this problem, the hyperelastic formulation is required to resolve a wide class of colloidal gel flow phenomena, and it is of critical importance to determine the limitations of the viscoplastic model in a given application. Application of the viscoplastic model to the _in situ_ measurements serves as an indirect validation of this model, and points to a methodology for estimation of the shear and compressive yield strengths from a series of batch settling experiments in various width columns. These estimates are shown to fall within experimental error of the _in-situ_ shear yield stress measurements, and furthermore suggest the strength ratio and hence wall adhesion effects in strongly flocculated colloidal gels can be much greater than was appreciated hitherto. They should always be expected near the gel- point where the ratio of shear to compressive strength is largest and it has been found that they can be much stronger overall for particles flocculated with high molecular weight polymers than has been found for coagulated systems. ## IX Acknowledgements This work was conducted as part of AMIRA P266F “Improving Thickener Technology” project, supported by the following companies: Alcoa World Alumina, Alunorte, Anglo American, BASF, Bateman Engineering, BHP Billiton, Cytec Australia Holdings, Exxaro, FL Smidth Minerals, Freeport-McMoran, Hatch, Metso, MMG, Nalco, Outotec, Rio Tinto, Rusal, Shell Energy Canada, Teck Resources, Total E&P Canada and WesTech. The authors are indebted to Jon Halewood for undertaking the flocculation, rheological and solid volume fraction profile measurements, Andrew Chryss for design of rheological measurements, and Michel Tanguay for valuable discussions. ## References * Auzerais et al. (1990) François M. 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1401.0121	# THE GROUND STATE STRUCTURE OF ELECTRON’S ENSEMBLE ON ONE-DIMENSION DISORDERED LATTICE L.A.Pastur, V.V.Slavin, A.A.Krivchikov B.I Verkin Institute for Low-Temperature Physics and Engineering of National Academy of Sciences of Ukraine. 47 Lenin Ave., Kharkov, 61103, Ukraine. e-mail:[email protected] ###### Abstract The ground state of interacting particles on a disordered one-dimensional host-lattice is studied by a direct numerical method. It is shown that if the concentration of particles is small, then even a weak disorder of the host- lattice breaks the long-range order of Generalized Wigner Crystal, replacing it by the sequence of blocks (domains) of particles with random lengths. The mean domains length as a function of the host-lattice disorder parameter is also found. It is shown that the domain structure can be detected by a weak random field, whose form is similar to that of the ground state but has fluctuating domain walls positions. This is because the generalized magnetization corresponding to the field has a sufficiently sharp peak as a function of the amplitude of fluctuations for small amplitudes. ## 1 Introduction Low-dimensional and layered conductors possess a number of interesting and unusual properties and have been of considerable interest in the last decades. In particular, it is possible to separate electron subsystem and that of dopant ions in a number of these conductors. As a result, the potential, produced by the dopants in the conducting electron layers is almost constant and conducting characteristics are essentially determined by the inter- electron (or inter-holes) repulsion. Another interesting subclass of these conductors consists of lattice electrons systems, where the charge carriers tunneling between host-lattice sites is suppressed by their mutual Coulomb repulsion. As a result, the charge carriers are _self-localized_ [1, 2].These are the MOSFETs (metal-oxide-semiconductor field-effect transistors) and a variety of other semiconducting heterostructures. The quasi-one-dimensional organic conductors [3] and artificial systems like arrays of quantum dots [4], networks and chains of metallic nano-grains with tunnel junctions (provided by various organic molecules) [5] also belong to this class. The ground state (GS) and the low-temperature properties of these systems have been extensively studied. The first theoretical result is due to Hubbard [3], who considered the GS of one-dimensional (1D) repulsing particles on a periodic host-lattice in the limit of strong interaction, where the dynamic effects are negligible and particles are localized with high accuracy on the lost-lattices sites. It was shown in [3] that the corresponding GS has, in general, an incommensurable structure determined only by the particle density $c_{e}=N_{e}/L,$ (1) where $N_{e}\gg 1$ and $L\gg 1$ are the total numbers of particles and host lattice sites, but does not depend on pair potential $V$, provided that $V$ is non-negative, convex and decays at infinity faster then $\|x\|^{-1}$. Hubbard also suggested an explicit form of the GS in this case. The form was justified and clarified in [6, 7] and it is known now as the Generalized Wigner crystals” (GWL). According to the GWL theory, the position $x_{k}$ of $k$-th particle is given by the simple formula: $x_{k}=a_{0}[k/c_{e}+\phi],.k=1,...,N_{e},$ (2) where $[\ldots]$ denotes the integral part, $\phi$ is an arbitrary constant (initial) phase and $a_{0}$ is the distance between the host lattice sites. It follows from the formula that the only two inter-particle distances $x_{k+1}-x_{k}=[1/c_{e}],\;[1/c_{e}+1]$ can appear in the GS, depending on $\phi$ and $c_{e}$ [3]. Thus, the inter- particle distances do not coincides in general with the minima of potential energy of the system. This leads, in particular, to rather unusual zero- temperature dependence of $c_{e}$ on the chemical potential $\mu$, which proved to be fractal (devil staircase) [7]. The low-temperature thermodynamics of 1D GWL was considered in [8]. In [9] it was shown that 2D systems are characterized by an effective reduction of the dimension that allows one to find their GS, which proved to be a natural 2D generalization of 1D Hubbard description (see (2)). Note, however, that the majority of 1D self-localized conductors are disordered. For example, in semiconductors on the base of MOSFET the disorder is due random impurities, in nanostructures [4, 5] the disorder is due to the fluctuations of tunneling junctions, and in the 1D charge transfer salts [3] the disorder results from imperfections of their chemical structure. That is why the influence of host-lattice disorder on the GS properties is of great interest, especially taking into account that in the 1D case even a weak disorder of the host-lattice can affects essentially the GS structure and low temperature thermodynamic properties. It suffices to recall the Larkin-Imry-Ma result [10, 11], according to which an arbitrary weak random field destroys the long-range order in 1D and 2D systems. As was mentioned above, the systems under consideration have been extensively studied both theoretically and experimentally. However, the majority of theoretical results are obtained in the frameworks of rather simple models, providing often only qualitative estimates. In this situation is it natural to use numerical methods of analysis of thermodynamic properties of 1D systems, since in the 1D case rather large systems (up to $N_{e}\sim 10^{4}-10^{5}$ particles) can by studied by the transfer-matrix method. However, the use of the method to find the the low temperature asymptotic regime is rather limited by the exponential growth (or decay) of the entries of transfer matrix. This implies the considerable decrease of calculation accuracy or even an overflow in numerical co-processor registers. In this paper we propose a new numeric method for the study of GS of the system in question, i.e., classical 1D repulsing particle on a disordered host-lattice in the limit of low particle density (1). This condition is the case for many 1D lattice systems, including quasi-one-dimensional organic conductors, chains of nano-objects, etc. ## 2 Model We will describe the one-dimensional system of repulsing particles [3, 12, 13] on disordered host-lattice by the extended Hubbard model determined by the Hamiltonian [14] ${\hat{H}}=-t\sum\limits_{<\alpha,\beta>,\sigma}{\hat{c}}_{R_{\alpha},\sigma}^{+}{\hat{c}}_{R_{\beta},\sigma}+U\sum\limits_{\alpha}{\hat{n}}_{R_{\alpha},\uparrow}{\hat{n}}_{R_{\beta},\downarrow}+\frac{1}{2}\sum\limits_{\alpha\neq\beta}V(\|R_{\alpha}-R_{\beta}\|){\hat{n}}_{R_{\alpha}}{\hat{n}}_{R_{\beta}}.$ (3) where $<\alpha,\beta>$ means summation over the nearest neighbors, $\\{R_{\alpha}\\}$ are the sites of host-lattice ($\alpha=1,2,\ldots,L$), ${\hat{c}}_{R_{\alpha},\sigma}^{+}$ and ${\hat{c}}_{R_{\alpha},\sigma}$ are the creation and annihilation operators on the site $R_{\alpha}$ with spin $\sigma$ ($\sigma=\uparrow,\downarrow$), $U$ is the interaction energy of two particles on the same site of the host-lattice, ${\hat{n}}_{R_{\alpha},\sigma}={\hat{c}}_{R_{\alpha},\sigma}^{+}{\hat{c}}_{R_{\alpha},\sigma}$ is the number operator with a fixed spin, ${\hat{n}}_{R_{\alpha}}={\hat{n}}_{R_{\alpha},\uparrow}+{\hat{n}}_{R_{\alpha},\downarrow}$ is the total number operator at $R_{\alpha}$. We will consider systems for which the potential has the power-low decay at infinity, i.e., $V(\|x\|)\sim\|x\|^{-\gamma},\;\;\gamma>0,\quad\|x\|>>a_{0}.$ (4) where $a_{0}$ is the length fixing the order of magnitude the distance between the adjacent host-lattice sites, i.e., is the period of translation invariant lattice, the mean distance between the sites of a disordered lattice, etc. (cf. (2)). The thermodynamic stability requires that $\gamma>1$ [7]. Thus, we write $\gamma=1+\delta,\;\delta>0$ (5) and assume that $V$ of (4) is positive (repulsive) and convex. The last condition is important and its violation may change significantly the results (see e.g. [15]). Note that the systems for which $-1<\delta\leq 0$ has also been studied recently (see e.g. [16, 17, 18]). One can call these systems the “strong” long ranged. In this case one has to either take into account certain truncations procedures (screening, confining the system to a finite box, etc.) or to be prepared to obtain rather unusual properties, especially if $V$ is attractive or anisotropic [16, 17]. On the other hand, the systems where with $\delta>0$) can be called the “weak” long ranged. They admit the traditional statistical mechanics description, possessing, however, certain special properties if $\delta$ is small (say, $0<\delta\leq 2$). We will consider in this paper the limit of low particle density $c_{e}$ of (1) $c_{e}=N_{e}/L<<1.$ (6) Besides, we will neglect both the dynamic effects ($t=0$) and the effects of Fermi statistics ($U\rightarrow\infty$) [3]. The last limit leads to so called spinless fermions model. As a result, the only potential energy of particle repulsion has to be taken into account and the Hamiltonian (3) can be replaced by $H=\frac{1}{2}\sum\limits_{\alpha\neq\beta}V(\|R_{\alpha}-R_{\beta}\|)n_{R_{\alpha}}n_{R_{\beta}},$ (7) where $\\{n_{R_{a}}\\}$ are the classical occupation numbers: $n_{R_{\alpha}}=0,1$. It is convenient to pass from the occupation numbers $\\{n_{R_{a}}\\}$ of the host-lattice sites $\\{R_{\alpha}\\}$ to the coordinates $\\{x_{k}\\}$ of particles ($k=1,...,N_{e}$), i.e., the coordinates of points of the host- lattice where the occupation numbers equal 1. We obtain instead of (7) $H=\sum\limits_{j<k}V(\|x_{j}-x_{k}\|).$ (8) It was shown in [19] that in the low-temperature limit the major contribution into partition function of (7) is due to the particle configurations which are close to equidistant ones, i.e., to the ground state configuration of the Wigner crystal with the same $c_{e}$. In other words, if $l_{0}=a_{0}/c_{e}$ is the period of the Wigner crystal, hence the coordinate $r_{k}$ of its $k$-th site is $r_{k}=kl_{0}$, and if $x_{k}$ is the coordinate of $k$-th site of Generalized Wigner Crystal, then the typical shifts between $r_{k}$ and $x_{k}$ are $\overline{s}=<\|x_{k}-r_{k}\|>\sim a_{0}<<l_{0},$ (9) where the symbol $<\ldots>$ denotes the averaging with respect to the host- lattice disorder which we assume as usually to be translation invariant in the mean [20] We denote the shifts $s_{k}=x_{k}-r_{k}$ (10) and expand the Hamiltonian (7) with respect to the small parameter $\overline{s}/l_{0}\simeq a_{0}/l_{0}<<1$: $H\approx H_{WC}(c_{e})+\frac{1}{2}\sum\limits_{\overset{j,k=1}{j\neq k}}^{N_{e}}b_{j-k}(s_{j}-s_{k})^{2}.$ (11) Here $H_{WC}$ is a constant (the ground state energy of Wigner crystal with a given density) and $b_{j-k}=b(\|r_{j}-r_{k}\|)=b(\|j-k\|l_{0})=\frac{1}{2}\frac{\partial^{2}V(x)}{\partial x^{2}}\|_{x=\|j-k\|l_{0}}.$ Figure 1: 1D disordered host-lattice (empty circles) and clusters. The crosses are the sites of Wigner lattice with the same particle density $c_{e}$. The bottom shows the same clusters in larger scale. Omitting $H_{WC}$ and using the nearest neighbor approximation, we obtain from (11) $H=b_{1}\sum\limits_{\alpha=1}^{N_{e}-1}(s_{\alpha}-s_{\alpha+1})^{2}.$ (12) Note that in the above nearest neighbor approximation we take into account the interaction between the nearest particles but not the nearest host-lattice sites. Since the typical distance between particles is $l_{0}=a_{0}/c_{e}$, hence $l_{0}>>a_{0}$ (see (9)) in the low concentration limit (6), the approximation seems fairly reasonable. It is convenient to measure the energy in units of $b_{1}a_{0}^{2}$. Then (12) became $H=\sum\limits_{k=1}^{N_{e}-1}(s_{k}-s_{k+1})^{2},$ (13) where now $H$ and the dynamic variables $\\{s_{k}\\}$ are dimensionless. It was shown in [19] that if the temperature is low enough, then it suffices to consider the case where $k$-th particle occupies only one of two host-lattice sites adjacent to the $k$-th site $r_{k}$ of the corresponding Wigner crystal (see Fig. 1). We will call these sites clusters, each cluster consists of two sites of the host lattice and contains only one particle which can occupy one of these two host-lattice sites. In this case the shifts (10) can be written as: $s_{k}=\sigma_{k}\lambda_{k}^{\sigma_{k}},\;\lambda_{k}^{\sigma}\geq 0,$ (14) where $\\{\sigma_{k}=\pm 1\\}$ are the standard Ising spins and $\\{\lambda_{k}^{\sigma}\\}$ are the random distances of the cluster sites from the site $r_{k}$ of the Wigner crystal of the same density. We will assume that the distances $\\{\lambda_{k}^{\sigma}\\}$ are independent and identically distributed for all $k=1,2,\ldots,N_{e}$, hence the left hand and the right hand distances $\lambda_{k}^{+}$ and $\lambda_{k}^{-}$ are typically different. It follows from (14) that the shifts (10) can be viewed as Ising-type spins with random lengths $\lambda_{k}^{\sigma_{k}},\;k=1,...,N_{e}$. To make more explicit the dependence of Hamiltonian (13) on the dynamic variables $\\{\sigma_{k}\\}$ and the frozen disorder described by the random spin lengths $\\{\lambda_{k}^{\sigma_{k}}\\}$, we write $\lambda_{k}^{\sigma_{k}}=\alpha_{k}\sigma_{k}+\beta_{k},\;\alpha_{k}=(\lambda_{k}^{+}-\lambda_{k}^{-})/2,\;\beta_{k}=(\lambda_{k}^{+}+\lambda_{k}^{-})/2\geq 0,$ (15) i.e., $\\{\alpha_{k}\\}$ and $\\{\beta_{k}\\}$ do not depend on $\\{\sigma_{k}\\}$ and are independent for different $k$. By using this parametrization, we can rewrite (13) as $H(\\{\sigma_{k}\\})=C-2\sum_{k}\beta_{k}\beta_{k+1}\sigma_{k}\sigma_{k+1}-\sum_{k}h_{k}\sigma_{k}$ (16) where $C$ does not depend on $\\{\sigma_{k}\\}$ and can be omitted, $\\{\beta_{k}\\}$ are given by (15) and $h_{k}=2\beta_{k}(\alpha_{k+1}+\alpha_{k-1}-2\alpha_{k}).$ (17) Thus, the Hamiltonian (13) is thermodynamically equivalent to that (16) of the one-dimensional Ising model with random interaction and random short correlated external field (recall that $\\{\alpha_{k}\\}$ and $\\{\beta_{k}\\}$ are independent for different $k$). Since $\\{\lambda_{k}^{\sigma}\\}$ are non-negative, independent and identically distributed for all $k=1,2,\ldots,N_{e}$ and $\sigma=\pm 1$, it follows from (15) that $\\{\beta_{k}\\}$ are non-negative, independent and identically distributed and $\\{h_{k}\\}$ are symmetrically distributed, in particular $<h_{k}>=0.$ (18) We obtain the periodic host-lattice setting putting $\lambda_{k}^{\sigma_{k}}=1$ for all $k$. In this case (16) corresponds to the one-dimensional ferromagnetic Ising model with the nearest-neighbor interaction. In general, each realizations of $2N_{e}$ random lengths $\\{\lambda_{k}^{\sigma_{k}}\\}$, hence random variables $\\{\beta_{k}\\}$ and $\\{h_{k}\\}$ provides a realization of disorder, thus fixing the Hamiltonian to be used to find the partition function $Z(L,N_{e})=\sum_{\\{\sigma_{k}=\pm\\}}e^{H/T}$ (19) of our disordered system or its ground state $\\{\sigma_{k}^{GS}\\}$: $E_{GS}=\min_{\\{\sigma_{k}=\pm\\}}H(\\{\sigma_{k}\\})=H(\\{\sigma_{k}^{GS}\\}).$ (20) ## 3 The Ground State. The low temperature thermodynamic properties of the model (13) have been studied in [19] using the transfer-matrix formalism. In particular, a weak local external field $h=\\{\varepsilon h_{k}\\},\;h_{k}=\delta_{k,k_{0}}$ (21) was introduced into the Hamiltonian (13) as a tool of analysis of the ground state. Here $\varepsilon$ is a small constant and $\delta_{k,k_{0}}$ is Kronecker symbol. In other words, the field affects only the $k_{0}$-th spin and the corresponding Hamiltonian is $H=\sum\limits_{k=1}^{N}(s_{k}-s_{k+1})^{2}-\varepsilon s_{k_{0}}.$ Using the transfer-matrix techniques, one can calculate the free energy $F(T,h)$ of the system and the corresponding magnetization per spin $M_{loc}(T,k_{0})=-\frac{1}{N_{e}}\left.\frac{\partial F(T,h)}{\partial\varepsilon}\right\|_{\varepsilon=0}\approx\frac{F(T,0)-F(T,h)}{N_{e}\varepsilon}.$ (22) It is reasonably to believe that if $M_{loc}(T,k_{0})>0$ in the low temperature limit, then the $k_{0}$-th spin of the ground state is parallel to the field, while if $M_{loc}(T,k_{0})<0$, then the $k_{0}$-th spin is antiparallel to the field. Thus, calculating $M_{loc}(T,k_{0})$ for $k_{0}=1,2,\ldots,N_{e}$ one can obtain the orientations of all the spins of ground state. However, the method seems to have certain disadvantages. First, since $h$ affects only one spin, the local magnetization (22) is of the order $1/N_{e}<<1$ for sufficiently large systems. Hence, the method can only be applied to the cases, where the length of the system is not too large, thus the boundary conditions may seriously effect the ground state. Second, the amplitude $\varepsilon$ of the local field should be small: $\varepsilon<<T,e_{sf},$ (23) where $e_{sf}$ is typical “spin flip” energy. However, in view of possible local energy degeneration $e_{sf}$ can be zero for certain spins, thus the sign of $M(T,k_{0})$ is rather sensitive to the value of $\varepsilon$ and the applicability of the method to rather large systems is again questionable. In this paper we study the ground state by a new method, which is free from the above disadvantage. The main idea of the method is as follows. Let us divide the system into $n$ parts (subchains) $C_{m},\;m=1,...,n$ with the endpoints $p_{0},p_{2},\ldots,p_{n}$, where $p_{0}=1$, $p_{n}=N_{e}$ and $p_{0}<p_{1}<p_{2}<\ldots<p_{n-1}<p_{n}.$ The lengths of the $m$-th subchain is $l_{m}=p_{m+1}-p_{m}+1$,$\;m=0,1,\ldots,n-1$). Thus, the $m$-th subchain contans $l_{m}$ spins $\sigma_{p_{m}},\;\sigma_{p_{m}+1},\ldots,\sigma_{p_{m+1}-1},\;\sigma_{p_{m+1}},$ and the spin $\sigma_{p_{m}}$ ($m=1,2,...,n-1$) belongs to the two neighboring subchains, i.e., $\sigma_{p_{m}}$ is the last “spin” of $m$-th subchain and the first spin of $(m+1)$-th subchain. It is convenient to index the spins in each subchain as $\sigma_{p_{m}+j-1}=\sigma_{m,j},$ i.e., the first index indicates that spin belongs to the $m$-th subchain and the second one is the number of the spin in the subchain. In this notation the $m$-th subchains consists of the spins $\sigma_{m,1},\sigma_{m,2},\ldots,\sigma_{m,l_{m}-1},\sigma_{m,l_{m}}.$ Now we carry out the direct enumeration of the states (the direct search for the configurations with minimum energy) in each of $n$ subchains. According to (13), the energy of $m$-th subchain is: $\displaystyle H_{m}$ $\displaystyle=$ $\displaystyle H_{m}(\sigma_{m,1},\sigma_{m,2},\ldots,\sigma_{m,l_{m}})$ $\displaystyle=$ $\displaystyle-2\sum_{j=1}^{l_{m}-1}\beta_{m,j}\beta_{m,j+1}\sigma_{m,j}\sigma_{m,j+1}-\sum_{j=1}^{l_{m}-1}h_{m,j}\sigma_{m,j}$ Let $H_{m}^{\min}$ is the minimum of $H_{m}$ over the spin configurations of the subchain and let $\sigma_{m,1}^{\min},\sigma_{m,2}^{\min},\ldots,\sigma_{m,l_{m}-1}^{\min},\sigma_{m,l_{m}}^{\min}.$ (24) be a corresponding spin configuration: $H_{m}^{\min}=H_{m}(\sigma_{m,1}^{\min},\ldots,\sigma_{m,l_{m}}^{\min})$. Now we note that if the last spin of each subchain is equal to the first spin of the next subchain, i.e., if $\sigma_{m,l_{m}}^{\min}=\sigma_{m+1,1}^{\min},\;m=1,2\ldots,n-1,$ (25) then the union of all the subchain minimizing configurations (24) is a ground state configuration $\sigma^{GS}=\\{\sigma_{k}^{GS}\\}$ of the whole Hamiltonian (16): $\displaystyle\sigma^{GS}$ $\displaystyle=$ $\displaystyle(\sigma_{1}^{GS},\sigma_{2}^{GS},\ldots,\sigma_{n-1}^{GS})$ $\displaystyle=$ $\displaystyle(\sigma_{0,1}^{min},\sigma_{0,2}^{min},\ldots,\sigma_{0,l_{0}}^{min},\sigma_{1,2}^{min},\ldots,\sigma_{1,l_{1}}^{min},\ldots,\sigma_{n-1,2}^{min},\ldots,\sigma_{n-1,l_{n-1}}^{min}),$ and the corresponding total minimum energy (20) is $E_{GS}=\sum\limits_{m=0}^{n-1}H_{m}^{\min}.$ (26) A similar idea has been recently used to study ground states of certain rather complex (in particular frustrated) translation invariant spin models [21, 22]. The above suggests a direct numerical algorithm to search ground states: split the system into subchains, minimize the energy of every subchain and check the matching conditions (25). If the conditions are not satisfied, repeat the procedure. Note, however, that the procedure does not guarantee that the ground state is unique. The choice of an optimal number $n$ of subchains depends on the computer efficiency. Since in this scheme we perform the direct enumeration of the states (direct energy minimization) for each “subchain”, the typical calculation time is $t_{0}\sim n2^{N_{e}/n}$. It is thus reasonable to choose $n$ so that $t_{0}$ is several seconds, i.e., $n\sim N_{e}/10-N_{e}/20$, where $N_{e}=10^{4}-10^{5}$ and even more. Increase in $n$ leads to decrease in enumeration time, but at the same time, to increase in the number of attempts (number of generations of division points $\\{p_{m}\\}$). It should be noted that the proposed method is rather universal and can be applied for a wide class of 1D disordered systems. An important advantage of the methods is that the direct minimization of energy is carried out independently in each subchain, thus the corresponding operations can be easily adapted to the parallel calculations. Besides, the method can be modified to deal with systems with larger number of interaction neighbors. In this case the conditions (25) is modified. For example, if we take into account the near- and next-neighbors interaction, then the matching conditions (25) are $\left\\{\begin{array}[]{l}\sigma_{m,l_{m}-1}^{\min}=\sigma_{m+1,1}^{\min}\\\ \sigma_{m,l_{m}}^{\min}=\sigma_{m+1,2}^{\min}.\end{array}\right.$ By using the method, we studied the ground state of the Hamiltonian (16). It is convenient to quantify the amount of disorder by writing the random spin length (14) as $\lambda_{k}^{\sigma}=1-\xi_{k}^{\sigma},\;\xi_{k}^{\sigma}=2A\eta_{k}^{\sigma},$ (27) where $\\{\eta_{k}^{\sigma}\\}$ are independent random variables uniformly distributed over the interval $[0,1)$ ($<\eta_{k}^{\sigma}>=1/2$) and $A$ is the “disorder” strength, $0\leq A\leq 1/2$). The limiting case $A=0$ corresponds to the ordered system (the translation invariant Ising model) and $A=1/2$ corresponds to complete disorder, where the spin lengths $\\{\lambda_{k}^{\sigma}\\}$ are uniformly distributed over the interval $[0,1]$. The examples of the ground state spin configurations for $N_{e}=10^{4}$ are presented in Fig. 2. We see that the ground state consists of “domains” [19] of blocks of spins of the same sign, the domains concentration increasing rapidly with the increase of disorder parameter $A$. It follows from (15) and (16) that interaction in our Ising model is ferromagnetic although random. Thus, the ground state of (16 ) without the second term is ferromagnetic (collinear), i.e., with all the spins either “up” or “down”. Since, however, in our case the second term (random field) is of the same order of magnitude as the first one (interaction), an argument similar to that of the well known Larkin-Imry-Ma criterion [10, 11, 23] implies that the ferromagnetic ground state is unlikely for any $0\leq A\leq 1/2$. Thus, it is not completely unexpected that the ground state is not collinear. However, the Larkin-Imry-Ma argument does not suggest a detailed form of the genuine ground state, except that it has to be of a “spin glass” non-collinear type. On the other hand, our numerical method allows us to detect an explicit form of a ground state, having the domain structure. Figure 2: The orientation of spins in the ground state of the systems for $N_{e}=10^{4}$ and two values of the disorder parameter $A$ of (27). The top corresponds to $A=0.4$, the bottom to $A=0.2$. The ground states have the domain structures and the domain concentration increases with $A$. We can also estimate the concentration $c_{dom}$ of the domain walls if $A$ is small enough. Indeed, the typical fluctuations of the pair interaction of (13) (or of (16)) are $\delta\varepsilon\sim A^{2},$ while energy of creation of a domain wall is $\varepsilon_{dom}\sim(\lambda_{k}^{+}+\lambda_{k}^{-})^{2}\sim 1,$ if $A$ is not too close to 1. Thus, the domain is stable if its length $l_{dom}$ satisfies the inequality $\delta\varepsilon\sqrt{l_{dom}}\lesssim\varepsilon_{dom}.$ We obtain then [19, 24]: $l_{dom}\sim A^{-4},$ (28) hence $c_{dom}=l_{dom}^{-1}\sim A^{4}.$ (29) In particular, it follows from (28) and (29) that the ground state consists of a single “ferromagnetic” domain for $A=0$. We have calculated by the same method the domain concentration $c_{dom}$ for a series of disorder parameter value $A$. The results are presented in Fig. 3 (solid boxes). The approximation by the function $c_{dom}=(A/A_{0})^{d}$ is given by the solid line. The best fitting is for $A_{0}=0.88885$ and $d=4.06031$ (quite close to those, obtained in [19] and (29)). Thus, we have a sufficiently good agreement between the numerical data and the fitting curve for $A\leq 1/2$. ## 4 Probing field. In this section we confirm our results of previous section on the form of the ground states of model (16) by probing its ground state by an external field of special form. Denote $H_{I}$ the r.h.s. of (16) without $C$. Given a collection $b=\\{b_{k}\\}$, consider the perturbed Hamiltonian $H_{I}-\varepsilon\sum\limits_{k=1}^{N}b_{k}\sigma_{k}$ (30) and the corresponding generalized magnetization $M(T,b)=-\frac{1}{N_{e}}\left.\frac{\partial F(T,b,\varepsilon)}{\partial\varepsilon}\right\|_{\varepsilon=0}=\frac{1}{N_{e}}\sum\limits_{k=1}^{N_{e}}b_{k}<\sigma_{k}>_{G},$ (31) where $F(T,b,\varepsilon)$ is the free energy of (30) and $<...>_{G}$ denotes the corresponding Gibbs mean. By using the terminology of spin glass theory (see e.g. [25]), we can view (31) as the overlap between external field $\\{b_{k}\\}$ and the local magnetization $\\{<\sigma_{k}>_{G}\\}$. It follows from the r.h.s. of (31) that the inequality $\|M(T,b)\|\leq\left(\frac{1}{N_{e}}\sum\limits_{k=1}^{N_{e}}b_{k}^{2}\frac{1}{N_{e}}\sum\limits_{k=1}^{N_{e}}<\sigma_{k}>_{G}^{2}\right)^{1/2}$ holds for a generic $\\{b_{k}\\}$ and that it becomes the equality only for an external field proportional to $\\{<\sigma_{k}>_{G}\\}$: $\overline{b}_{k}=a<\sigma_{k}>_{G},\;k=1,...,N_{e},$ (32) where $a$ is a constant, i.e., $M(T,\overline{b})=a\sum\limits_{k=1}^{N_{e}}<\sigma_{k}>_{G}^{2}.$ (33) The constant $a$ can be chosen to be $1$ if we normalize $\\{b_{k}\\}$ by the condition $\sum\limits_{k=1}^{N_{e}}b_{k}^{2}=1.$ (34) Since we are interested in the ground state $\\{\sigma_{k}^{GS}=\pm 1\\}$ of the Hamiltonian (16), which we identify with the domain walls configuration of the previous section, we put $T=0$ in the above formulas and arrive to the following algorithm to detect $\\{\sigma_{k}^{GS}\\}$. Pick a class of external fields containing $\\{\sigma_{k}^{GS}\\}$ and satisfying (34) and vary $\\{b_{k}\\}$ over the class. The configuration $\\{\sigma_{k}^{GS}\\}$ will be obtained as a maximizer of the generalized magnetization (31): $M(0,\\{\sigma_{k}^{GS}\\})=\frac{1}{N_{e}}\sum\limits_{k=1}^{N_{e}}(\sigma_{k}^{GS})^{2}=1.$ (35) In general, this will only prove that the corresponding maximizer is a local minimum of the Hamiltonian (13) of the model, it is reasonable to believe but the larger is the class the closer is the minimizer to the genuine ground state. Figure 3: The domain concentration $c_{dom}$ as the function of disorder parameter $A$. Solid boxes are the results of our numeric calculation, solid line is the fitting by function $c_{dom}=(A/A_{0})^{d}$. Let now $d_{1},d_{2},\ldots$ be the coordinates of domain walls of the spin configuration found in the previous section ($<d_{m}-d_{m-1}>=l_{dom}$). Consider the following class of random external (probing) fields: $b_{k}=\sigma_{k}^{GS}+f_{k},$ (36) where $f_{k}=\left\\{\begin{array}[]{lr}-2\sigma_{k}^{GS}&k\in(d_{m},d_{m}+\Delta_{m})\quad m=1,2,...,n,\\\ 0,&\mathrm{otherwise};\end{array}\right.,$ (37) $\Delta_{m}=(-1)^{\eta_{m}}[B\rho_{m}],$ (38) $B\geq 0$ is a non-negative constant, $[\ldots]$ denotes the integer part, $\\{\eta_{m}\\}$ are the independent distributed random variables, assuming ($0,1$) with probability $1/2$ and $\rho_{m}$ are independent random variables with exponential distribution $P(x)=B^{-1}\exp(-x/B)$. Thus, the random variables $\\{f_{k}\\}$ provide generic fluctuations of the positions of domain walls of the ground state configuration $\\{\sigma_{k}^{GS}\\}$. The direction of the shift of the $m$-th domain wall is determined by $\eta_{m}$ and the amplitude of the shift is determined by $B$. In particular, the probe field (36) coincides with $\\{\sigma_{k}^{GS}\\}$ if $B=0$. To find the free energy $F(T,b,\varepsilon)$ corresponding to (16), we use the transfer-matrix method. We set for $s_{k}^{\pm}=\pm\lambda_{k}^{\pm},\;k=1,2,...,N_{e}$ ${\hat{P}_{k}}(T,b)=\left(\begin{array}[]{cc}\exp\left(-\frac{(s_{k}^{+}-s_{k+1}^{+})^{2}-\varepsilon b_{k}s_{k}^{+}}{T}\right)&\exp\left(-\frac{(s_{k}^{+}-s_{k+1}^{-})^{2}}{T}\right)\\\ \exp\left(-\frac{(s_{k}^{-}-s_{k+1}^{+})^{2}}{T}\right)&\exp\left(-\frac{(s_{k}^{-}-s_{k+1}^{-})^{2}-\varepsilon b_{k}s_{k}^{-}}{T}\right)\end{array}\right),$ (39) assume the periodic boundary conditions ($\sigma_{N_{e}+1}=\sigma_{1}$) and write $F(T,b,\varepsilon)=-T\log\left(\mathrm{Tr}\left[\prod_{k=1}^{N_{e}}{\hat{P}}_{k}(T,b)\right]\right),$ (40) where $\mathrm{Tr}$ denotes the trace of a $2\times 2$ matrix. Using (39) - (40), one can calculate numerically the generalized magnetization (31) for the class (36) – (38) of probing fields as the function of amplitude $B$ of fluctuations of the domains walls: $M(T,b)\approx\frac{F(T,b,0)-F(T,b,\varepsilon)}{N_{e}\varepsilon}.$ (41) Fig. 4 (curve 1) presents the generalized magnetization (31) as a functions $B$ for field (36) and $T\rightarrow 0$. Figure 4: Curve 1. The generalized magnetization $M(0,b)$ as the function of $B$ for the fields (36) – (38 ). Curve 2. The analogous curve for the probe fields with $s_{k}^{GS}=\sigma_{k}^{GS}\lambda_{k}^{\sigma_{k}^{GS}}$ instead of $\sigma_{k}^{GS}$ in the r.h.s. of (36). We see that the magnetization is maximal for $B=0$, where the fluctuations (37) are absent, hence the probe field (36) coincides with $\\{\sigma_{k}^{GS}\\}$. However, for $B>0$ the generalized magnetization decays rather fast with the growth of $B$, i.e., it is a rather sensitive characteristic of the proximity of the external field to the maximizing one. It is also worth noting that the if we replace the maximizing field $\sigma_{k}^{GS}$ in (36) by $s_{k}^{GS}=\sigma_{k}^{GS}\lambda_{k}^{\sigma_{k}^{GS}}$ (see (14)) and compute again numerically the corresponding generalized magnetization, we obtain a curve (see curve 2 of Fig. 4), which is quite similar to that of curve 1 of Fig.4, except the magnitude of maximum, which is now $1-A$. The magnitude can be explained as follows. In view of (14) and (15) the maximum is $\frac{1}{N_{e}}\sum\limits_{k=1}^{N_{e}}s_{k}^{GS}\sigma_{k}^{GS}=\frac{1}{N_{e}}\sum\limits_{k=1}^{N_{e}}\alpha_{k}\sigma_{k}^{GS}+\frac{1}{N_{e}}\sum\limits_{k=1}^{N_{e}}\beta_{k}.$ The second term on the right is $<\beta_{k}>=1-A$, if $N_{e}$ is large enough, thus the first term has to vanish for $N_{e}>>1$ according our numerical results. This can be viewed as a manifestation of a certain robustness of our numerical algorithm to detect the ground state provided that the probe field takes into account the sign structure of the ground state. ## 5 Results and Discussion. The ground state of interacting particles on a disordered one-dimensional host-lattice is studied using a new numerical method. It is shown that if the concentration of particles is small enough, then even a weak disorder in host- lattice site positions leads to the formation of the “domains” of particles and to the breaking the long-range order pertinent to the Generalized Wigner crystal in the absence of disorder. The nature of the domains can be explained by using the Hubbard formula (2) for particle positions in the Generalized Wigner crystal. Indeed, in the case of the translation invariant host-lattice the phase $\phi$ in the formula is an arbitrary constant, just fixing the origin of the host-lattice. In the disordered case each domains has its own phase, i.e., $\phi$ is a step function with random steps and the mean domain length $l_{dom}$ is just the mean length of steps. It is also shown that the formula $l_{dom}\sim A^{-4}$ (28) is valid in a sufficiently wide range of disorder parameter $A$: $0\leq A\leq 0.5$ (see Fig. 3). The above results are then confirmed by studying the weak perturbations of the Hamiltonian (16) of the model by random external fields, which “probes” the domain structure of the ground state via random variations (fluctuations) of the domain walls. It is shown that the generalized magnetization per particle (41) corresponding to the field is maximal if the amplitude of fluctuations is small and decays sufficiently fast with the growth of the amplitude, i.e., that the magnetization is rather sensitive to the proximity of the form of the field to that of the ground state. ## References * [1] N. Mott, Metal–Insulator Transitions (Taylor and Francis, London, 1974). * [2] A. A. Slutskin and L. Yu. Gorelik, Low Temp. Phys. 19, 852 (1993) [Fiz .Niz. Temp. 19, 1199 (1993)]. * [3] J. Hubbard, Phys. Rev. B 17, 494 (1978). * [4] E. Y. Andrei, 2D Electron Systems on Helium and Other Substrates (Kluwer, New York, 1997). * [5] H. Nejoh, M. Aono, Appl. Phys. Lett. 64, 2803 (1995). * [6] P. Bak, R. Bruinsma, Phys. Rev. Lett. 49, 249 (1982). * [7] S. E. Burkov, Ya. G. Sinai, Russ. Math. Surv. 38, 235 (1983). * [8] V. V. Slavin, A. A. Slutskin, Phys. Rev. B 54, 8095 (1996). * [9] A. A. Slutskin, V. V. Slavin, and H. A. Kovtun, Phys. Rev. B 61, 14184 (2000). * [10] A. I. Larkin, Sov. Phys. - JETP 31, 784 (1970). * [11] Y. Imry, S. K. Ma, Phys.Rev.Let. 35, 1399 (1975). * [12] S. Fratini, B. Valenzuela and D. Baeriswyl, arXiv:0209518v1. * [13] S. Fratini, B. Valenzuela and D. Baeriswyl, arXiv:0302020v2. * [14] E. H. Lieb, F. Y. Wu, Physica A 321, 1 (2003). * [15] J. Jedrzejewski, J. Miekisz, arXiv:9903163. * [16] A. Campa, T. Dauxois, S. Ruffo, Physics Reports 480, 57 (2009). * [17] T. Dauxois, S. Ruffo, L. Cugliandolo (Editors), Long-Range Interacting Systems, Lecture Notes of the Les Houches Summer School 2008 (Oxford University Press, Oxford 2009). * [18] F. Bouchet, S. Gupta, D. Mukamel, Physica A 389, 389 (2010). * [19] V. V. Slavin, Phys. Stat. Sol. (b) 241, 2928 (2004). * [20] I. M. Lifshitz, S. A. Gredeskul, L. A. Pastur, Introduction to the Theory of Disordered Systems (Wiley, NY, 1988). * [21] A. Grechnev, arXiv:1212.2320. * [22] Y. I. Dublenych, Phys. Rev. Lett. 109, 167202 (2012). * [23] J. Cardy, Scaling and Renormalization in Statistical Physics (Cambridge University Press, 1996). * [24] A. A. Slutskin and H. A. Kovtun, Low Temp.Phys. 31 594 (2005) [Fiz. Niz. Temp. 31, 784 (2005)]. * [25] M. Mezard, G. Parisi, M. A. Virasoro, Spin Glass and Beyond (World Scientific, Singapore, 1987).	arxiv-papers	2013-12-31T10:06:58	2024-09-04T02:49:56.121447	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "L.A.Pastur, V.V.Slavin, A.A.Krivchikov", "submitter": "Victor Slavin V", "url": "https://arxiv.org/abs/1401.0121" }
1401.0142	# Diverse forms of $\sigma$ bonding in two-dimensional Si allotropes: Nematic orbitals in the MoS2 structure Florian Gimbert Chi-Cheng Lee Rainer Friedlein Antoine Fleurence Yukiko Yamada-Takamura School of Materials Science, Japan Advanced Institute of Science and Technology (JAIST), 1-1 Asahidai, Nomi, Ishikawa 923-1292, Japan Taisuke Ozaki School of Materials Science, Japan Advanced Institute of Science and Technology (JAIST), 1-1 Asahidai, Nomi, Ishikawa 923-1292, Japan Research Center for Simulation Science, Japan Advanced Institute of Science and Technology (JAIST), 1-1 Asahidai, Nomi, Ishikawa 923-1292, Japan ###### Abstract The interplay of $sp^{2}$\- and $sp^{3}$-type bonding defines silicon allotropes in two- and three-dimensional forms. A novel two-dimensional phase bearing structural resembleance to a single MoS2 layer is found to possess a lower total energy than low-buckled silicene and to be stable in terms of its phonon dispersion relations. A new set of cigar-shaped, nematic orbitals originating from the Si $sp^{2}$ orbitals realizes bonding with a 6-fold coordination of the inner Si atoms of the layer. The identification of these nematic orbitals advocates diverse Si bonding configurations different from those of C atoms. ###### pacs: 73.22.-f, 71.20.Mq, 61.48.-c With its forefront runner graphene and its unique and exotic properties, at present, two-dimensional materials experience an explosion of interest in scientific and technological aspects Geim07 . While the excellent electronic properties of graphene are derived from its structural robustness, the same property makes it a challenging task to engineer the optical and transport properties. This challenge is stimulating the search for alternative two- dimensional layered materials that are more flexible in terms of its structural and electronic properties Cahangirov07 ; Fleurence12 . In this context, in particular, two new promising two-dimensional materials with a honeycomb structure made of silicon or germanium atoms have been studied as theoretical objects since 1994 Takeda94 . Most importantly, in yet hypothetical, slightly buckled, free-standing forms, silicene and germanene, as they were later called Guzman07 , exhibit a band structure similar to that of graphene, merging linear dispersion of $\pi$ and $\pi^{}$ bands at the Fermi level to form Dirac cones at the $K$ points Takeda94 ; Guzman07 ; Cahangirov07 . Experimentally, it has been shown that two-dimensional Si honeycomb lattices can be formed epitaxially on Er layers Wetzel94 as well as on the Ag(111)Voigt12 ; Lin12 ; Jamgotchian12 , ZrB2(0001)Fleurence12 and Ir(111)Meng13 surfaces. It is established that the interactions with the substrates have a distinct influence on the structural and electronic properties of the layers Chen12 ; Chen13 ; Lee13 . No experimental evidence for the existence of germanene has been reported yet. As density functional theory (DFT) calculations predict consistently that slighly or low-buckled (LB) silicene is the most stable form of freestanding Si allotropes, very recently, it has been shown that the addition of Si adatoms on pristine silicene results in the formation of a dumbbell structure with a lower total energy Kaltsas13 ; Ongun13 . Interestingly, a higher cohesive energy can be achieved towards the complete coverage of the adatoms, which possesses even a higher cohesive energy than the low-buckled silicene. In fact, the periodic dumbbells can be recognized as the structure of a well- known single-layer of MoS2. By considering the 4-fold coordination realized in the $sp^{3}$ bonding of Si atoms and also for the Si atom connecting the low- buckled silicene and a dumbbell, it is diffcult to understand why bonding with a 6-fold coordination could be formed by Si atoms in the MoS2 structure. Therefore, it is timely and interesting to investigate the properties of this new Si phase beyond the view related to the introduction of defects or adatoms to low-buckled silicene. In this Letter, by first-principles calculations Ozaki03 ; DFT , we investigate the stability of this new Si phase (MoS2-Si) together with a possible similar Ge allotrope (MoS2-Ge), whose structures are that of a single layer of molybdenum disulfide, or MoS2 Radisavljevic11 , in a wide range of lattice constants and compare it with that of other two-dimensional silicon structures. The phonon dispersion further demonstrates that MoS2-Si is stable on the Born-Oppenheimer surface. To understand the 6-fold coordinated bonding in MoS2-Si, we construct symmetry-respecting Wannier functions Vanderbilt ; Weng . A new form of $\sigma$ bonding expressed by three cigar-shaped orbitals co-exists with $\pi$ bands that are formed by three additional reconstructed $p_{z}$ orbitals. The direction of these orbitals has changed from the typical in-plane direction of the $sp^{2}$ to the out-of-plane direction to form cigar-shaped orbitals. In analogy to the nematic electronic structure Chuang , the aligned and cigar-shaped orbitals may be called ”nematic”. While as a common feature in the two-dimensional Si allotropes, the three $\sigma$ bands show dispersions similar to those of bands in low-buckled silicene, values of bond lengths and buckling heights vary significantly. Our finding suggests that the $\sigma$ bonds of Si atoms are more flexible than one could expect such that diverse forms of $\sigma$ bonding can allow the existence of a number of Si allotropes. Figure 1: Top and side view of the MoS2-type single layer for silicon atoms with the lattice constant (a), the bonding distance (d) and the thickness (t) indicated. As shown in Figure 1, a single layer of MoS2-Si crystallizes in an A-B-A stacking structure. In preserving the honeycomb structure, the A-B layer taken by itself is exactly freestanding low-buckled silicene. In the MoS2-Si structure, the primitive unit cell contains three atoms in comparison with two atoms in silicene. The middle atom is bound to six atoms while the top and bottom atoms have three neighbors. Clearly, the coordination is very different from that in graphene-like structures where each atom is bound to three neighboring atoms. \| $a$ \| $t$ \| $d$ \| $E_{r}$ ---\|---\|---\|---\|--- silicene \| 3.90 \| 0.49 \| 2.30 \| 0.032 MoS2-Si \| 3.64 \| 2.63 \| 2.47 \| 0 germanene \| 4.07 \| 0.74 \| 2.47 \| 0.140 MoS2-Ge \| 3.90 \| 2.88 \| 2.67 \| 0 Table 1: Values of the lattice constant ($a$ in Å), the thickness ($t$ in Å), the bonding distance ($d$ in Å) and the relative energy ($E_{r}$ in eV/atom) for silicene and MoS2-Si (upper part), as well as germanene and MoS2-Ge (lower part). Next, we compare the total energy per atom and structural parameters of the phases under consideration as a function of $a$. Since the high-buckled forms of silicene and germanene are unstable Cahangirov07 , we restrict the investigations to lattice parameters around to the region of the LB phase. In Fig. 2(a) is shown the relationship between the total energy per atom $E_{r}$ and the lattice constant $a$, for both LB silicene (empty squares) and MoS2-Si (filled circles). The evolution of the buckling is plotted in Fig. 2(b). In order to facillitate comparison between the two phases, for the MoS2 structure, half of the thickness is taken as the value of the buckling. With the energy minimum occuring at a lattice constant of 3.90 Å, the LB silicene prefers a buckling of 0.49 Å and the energy minimum occurs at $a=3.90$ Å. These parameters compare well with values reported previously Cahangirov07 ; Houssa10 ; Wang13 . Interestingly, in equilibrium, the total energy of MoS2-Si is lower than that of LB silicene, stabilized at a shorter lattice constant of $a=3.64$ Å. Due to the stacking of three atoms, with 2.63 Å, the thickness $t$ of the MoS2-Si layer is larger than that of silicene. Similar to the Si phases, MoS2-Ge possess a lower total energy than LB germanene as well. The total energy and lattice parameters of both MoS2-Si and MoS2-Ge are given in Tab. 1. Figure 2: (a): Relative energy (in eV/atom) of silicene (empty squares) and MoS2-Si (filled circles). (b): Buckling of silicene (empty squares) and MoS2-Si(filled circles). For MoS2-Si, the buckling is defined as the distance beween two planes, corresponding to half of the thickness. (c): Phonon dispersion relations of MoS2-Si as obtained by the force-constant method. With MoS2-Si being more stable than LB silicene, it is relevant to understand its stability by investigating the Born-Oppenheimer surface. This can be done by calculating the phonon frequencies in the harmonic approximation. In order to do so, a dynamical matrix has been constructed by calculating real-space force constants. Using a $12\times 12$ supercell, the atoms have been chosen to be displaced by 0.02 Å out of the equilibrium positions. The phonon dispersion relations of MoS2-Si are shown in Fig. 2(c). The frequencies are overall lower than those of LB silicene which can be understood from the elongated bond lengths that represent a weaker bonding. No branch with imaginary frequencies is found. This suggests that the freestanding MoS2-Si is a stable phase that is preferentially formed instead of the commonly studied low-bucked silicene. For MoS2-Ge, on the other hand, the situation is not as clear since a small amount of computed imaginary phonon frequencies maybe due to either a possible instability of the structure or artifact derived from numerical noise in the calculated forces. The following discussion will therefore focus on MoS2-Si. Figure 3: (a): Band structure of low-buckled silicene with a lattice parameter of $a=3.90$ Å, (b): Band structure of MoS2-Si with $a=3.64$ Å. (c): Symmetry- respecting Wannier functions of MoS2-Si related to the bands labeled 1-6 in (b). The top, bottom and middle Si atoms are shown with different colors. The electronic band structure of LB silicene and MoS2-Si are presented in Figs. 3(a) and (b), respectively. Given that the bond length and the degree of buckling are larger and that the lattice constants are shorter for MoS2-Si as compared to silicene, it is surprising that the band structure is not far away from that of silicene. Major differences observed around the Fermi energy relate to the disapearance of the Dirac cone at the K point typical for the LB silicene and to the appearance of two new bands, labeled 5 and 6 in Fig. 3(b). To further understand similarities and differences between these two phases, we construct the symmetry-respecting Wannier functions of MoS2-Si. Technically, in order to do so, the commonly adopted procedure for maximizing the localization of Wannier functions was not be performed Weng . The energy window is chosen to allow for a reproduction of the six occupied bands, labeled 1-6 in Fig. 3(b). As shown in Fig. 3(c), the respective orbitals represented by Wannier functions have contributions in different bands and adopt particular shapes. The orbitals dominating the bands 1-3 originate from the $sp^{2}$ orbitals of the middle Si atom. Interestingly, in order to accomodate bonding between the top and bottom Si atoms, these orbitals have a nematic shape. For clarity, only one of the three symmetric nematic orbitals is shown in Fig. 3(c). The band dispersions related to the nematic orbitals resemble those of the $\sigma$ bands of the LB silicene in Fig. 3(a). Without these nematic orbitals, it is difficult to provide fully occupied $\sigma$ band dispersions that are similar to the ones in LB silicene. Note that for the silicene structure of the A-B stacking obtained directly from the equilibrium lattice parameters of MoS2-Si, the $\sigma$ bands of silicene can only be partially occupied. Another interesting finding relates to orbitals with $p_{z}$ contributions: in particular, the 4th orbital is derived from the $p_{z}$ orbitals of the top and bottom Si atoms and the $sp^{2}$ orbitals of the middle Si atoms. At the K point, the corresponding band crosses the 7th band which, however, does not have any $p_{z}$ character. The crossing can therefore not be considered to be derived from an original Dirac point of silicene. The 5th and 6th orbitals form orbitals with $p_{z}$ symmetry having a node at the height of the middle Si atom. As it can be recognized in Fig. 3(c), while the 5th orbital is mainly derived from the $p_{z}$ and $s$ orbitals of the top and bottom Si atoms, the 6th orbital stems from the $sp^{2}$ orbitals of the top and bottom Si atoms such displaying $p_{z}$ symmetry. Although the band dispersions of the $p_{z}$ orbitals bear some resemblance to the $\pi$ and $\pi^{}$ bands of LB silicene, no Dirac cones are formed since the orbital nature of the two $p_{z}$ orbitals is essentially different. The modification of the electronic properties is also evident from the plot of the charge density shown in Fig. 4(a). The top and bottom atoms are bound to the central atom via three of these nematic orbitals which allow for the coordination of the central atoms with its six neighbors. For comparison, in Figs. 4(b) and (c) are displayed the charge densities of silicene and of the corresponding Si layer in the diamond structure, in which atoms are 3- or 4-fold coordinated, respectively. Note that as single layers, these two structures have a higher total energy per atom than MoS2-Si. This suggests that for two-dimensional silicon, $\sigma$ bonding of the planar $sp^{2}$ type is preferred. This symmetry is well respected by the nematic orbital. In order to allow for an even higher flexibility in the bonding, one can imagine to twist the nematic orbitals to form a new A-B-C stacking structure. Such an asymmetric bonding with respect to the planar $sp^{2}$ orbitals is by 180 meV per Si atom energetically unfavourable. In addition, the middle Si atom of A-B-C stacking structure shares a bonding similar to that of the 6-fold coordinated Si atom in the $\beta$-tin Si phase that can only be stabilized under a high pressure Betatin . The charge density of the A-B-C phase is shown in Fig. 4(d). Figure 4: Valence charge density for different structures composed of Si atoms. (a): silicon MoS2-type layer, corresponding to A-B-A stacking. (b): low-buckled silicene. (c): silicon layer with a diamond structure. (d): A-B-C stacking structure for silicon. To summarize, a new two-dimensional Si phase structurally equal to a single MoS2 layer is identified. Within DFT, this phase is stable on the Born- Oppenheimer surface in terms of the total energy and the phonon frequencies. Instead of the commonly accepted bonding configurations in silicene or the diamond structure of silicon that are related to a mixture of $sp^{2}$ and $sp^{3}$ or pure $sp^{3}$ hybridization, respectively, this phase exhibits nematic orbitals that allow $\sigma$ bonding with a 6-fold coordination for the middle atoms in the MoS2 structure. With bond lengths longer than for silicene, the three nematic orbitals exhibit $\sigma$ band dispersions similar to those of LB silicene which represents a common feature of $\sigma$ bonding in two-dimensional Si phases. On the other hand, the reconstructed $p_{z}$ orbitals, or super $p_{z}$ orbitals, are prominently different from those of low-buckled silicene. Per Si atom, the MoS2-Si phase is lower in total energy than the low-buckled silicene making it the the most stable two-dimensional Si allotrope predicted so far. Our study demonstrates that Si atoms are capable of forming diverse types of $\sigma$ bonds even under ambient pressure conditions that are by themselves quite different from those formed by its smaller and larger cousins carbon and germanium. Even more, the presence of an extended $\pi$ electronic system with properties different to those in silicene and graphene will lead to properties that still must be explored. In a wider context, this finding does not only open new opportunities in the engineering of novel nanostructures to be employed in future applications but it also leads to intriguing questions for instance related to the Si-based chemistry. This work has been supported by the Strategic Programs for Innovative Research (SPIRE), MEXT, the Computational Materials Science Initiative (CMSI), by Materials Design through Computics: Complex Correlation and Non-Equilibrium Dynamics, A Grant in Aid for Scientific Research on Innovative Areas, MEXT, Japan, and by the Funding Program for Next Generation World-Leading Researchers (GR046). The calculations have been performed using the Cray XC30 machine at Japan Advanced Institute of Science and Technology (JAIST). ## References * (1) A. K. Geim and K. S. Novoselov, Nature Materials 6, 183 (2007). * (2) K. Takeda and K.Shiraishi, Phys. Rev. B 50, 14917 (1994). * (3) G. G. Guzman-Verri and L. C. Lew Yan Voon, Phys. Rev. B 76, 075131 (2007). * (4) S. Cahangirov, M. Topsakal, E. Akturk, H. Sahin, and S. Ciraci, Phys. Rev. Lett. 102, 236804 (2007). * (5) M. Houssa, G. Pourtois, V. V. Afanas’ev, and A. Stesmans, Appl. Phys. Lett. 96, 082111 (2010). * (6) Y. Wang and Y. 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For Si and Ge atoms, $s2p2d1$ configurations have been adopted. The spin-orbit coupling has not been considered in our calculation. In order to avoid interactions, the distance between supercells is larger than 10 Å in the z-direction (direction perpendicular to the plane of layers). The k-mesh has been set to 12x12x1 for the primitive cell. The structures have been fully optimized until the maximum force became less than $10^{-4}$ Hartree/Bohr. All the geometric structures are plotted using the XCrySDen software. The energy curves for the phases shown in Fig. 2 have also been confirmed by calculations with the first principles code WIEN2k. [P. Blaha, K. Schwarz, G. K. H. Madsen, D. Kvasnicka, and J. Luits, Wien2k, An Augmented Plane Wave+Local Orbitals Program for Calculating Crystal Properties (K. Schwarz, Vienna, Austria, 2001)] * (21) B. Radisavljevic, A. Radenovic, J. Brivio, V. Giacometti, and A. Kis, Nature Nano. 6, 147 (2011). * (22) N. Marzari and D. Vanderbilt, Phys. Rev. B 56, 12847 (1997). * (23) H. Weng, T. Ozaki, and K. Terakura, Phys. Rev. B 79, 235118 (2009). * (24) T.-M. Chuang, M. P. Allan, J. Lee, Y. Xie, N. Ni, S. L. Bud’ko, G. S. Boebinger, P. C. Canfield, J. C. Davis, Science 327, 181 (2010). * (25) A. Mujica, A. Rubio, A. Munoz, and R. J. Needs, Rev. Mod. Phys. 75, 863 (2003).	arxiv-papers	2013-12-31T13:21:56	2024-09-04T02:49:56.130371	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Florian Gimbert, Chi-Cheng Lee, Rainer Friedlein, Antoine Fleurence,\n Yukiko Yamada-Takamura and Taisuke Ozaki", "submitter": "Florian Gimbert", "url": "https://arxiv.org/abs/1401.0142" }
1401.0251	# The General Stationary Gaussian Markov Process Larry Brown,1 Philip Ernst,1 Larry Shepp1 and Robert Wolpert2 1Department of Statistics, The Wharton School of the University of Pennsylvania 2Department of Statistical Science, Duke University ###### Abstract We find the class, ${\cal{C}}_{k},k\geq 0$, of all zero mean stationary Gaussian processes, $Y(t),~{}t\in\mathbb{R}$ with $k$ derivatives, for which $\displaystyle Z(t)\equiv(Y^{(0)}(t),Y^{(1)}(t),\ldots,Y^{(k)}(t)),~{}t\geq 0$ is a $(k+1)$-vector Markov process. (here, $Y^{(0)}(t)=Y(t)$). ## 1 Introduction We show that the process, $Y$, can be described in three equivalent ways: (i). Each member of ${\cal{C}}_{k}$ is given uniquely by a certain polynomial $P(z)$ via the covariance of any such $Y$: $\displaystyle\mathbb{E}\left[Y(s)Y(t)\right]=R(s,t)=r(t-s)=\int_{-\infty}^{\infty}\frac{e^{i(t-s)zdz}}{\|P(z)\|^{2}}$ where $P(z)$ is a polynomial of degree $k+1$ with positive leading coefficiant and all complex roots $\zeta_{j}=\rho_{j}+i\sigma_{j},j=0,\ldots,k$ with $\sigma_{j}>0$, and $\rho_{j}$ real. If some $\rho_{j}\neq 0$, then there is another $\zeta_{k}=-\zeta_{j}^{}$ which is the negative conjugate of $\zeta_{j}$. Note $r(t)$ automatically has $2k$ derivatives at each $t$, but not $2k+1$. (ii) Equivalently, it is necessary and sufficient that $Y\in{\cal{C}}$ has the representions via Wiener integrals with a standard Brownian motions, $W$ $\displaystyle Y(t)=\int_{-\infty}^{\infty}g(t-\theta)dW(\theta)$ where $g$ has $L^{2}$ Fourier transform, $\displaystyle{\hat{g}}(z)=\frac{1}{\|P(z)\|}$ or, equivalently, $Y$ has the spectral representation, with a pair of independent standard Brownian motions, $W_{1},W_{2}$, with $f(z)\equiv{\hat{g}}$, $\displaystyle Y(t)=\int_{-\infty}^{\infty}\cos{tz}f(z)dW_{1}(z)+\int_{-\infty}^{\infty}\sin{tz}f(z)dW_{2}(z)$ (iii) Equivalently, it is necessary and sufficient for $Y\in{\cal{C}}$ that $Z$ has a representation as an Ito vector diffusion process: $\displaystyle dY^{(i)}(t)$ $\displaystyle=$ $\displaystyle Y^{(i+1)}(t)dt,~{}0\leq i<k,t\in\mathbb{R}$ $\displaystyle dY^{(k)}(t)$ $\displaystyle=$ $\displaystyle\sum_{j=0}^{k}a_{j}Y^{(j)}(t)dt+bdW(t),~{}t\in\mathbb{R}$ where the coefficients, $a_{j}$, are the unique solution of the equations: $\displaystyle r^{(k+i+1)}(0^{+})=\sum_{j=0}^{k}a_{j}r^{(i+j)}(0),~{}i=0,1,\ldots,k$ Note that the left and right derivatives of $r^{(j)}$ are equal except for $j=2k$. The diffusion coefficient, $b$, is given by $\displaystyle b^{2}=\sum_{j=0}^{k}a_{j}r^{(j+k)}(0)(-1)^{j+1}+(-1)^{k}r^{(2k+1)}(0^{-})$ Given the polynomial, $P(z)$, in (i), which uniquely determines each process $Y\in{\cal{C}}$, the derivatives $r^{(j)}(0)$ are easily obtained from the representation in (i) above as $\displaystyle r^{(j)}(0)=\int_{-\infty}^{\infty}\frac{(iz)^{j}dz}{\|P(z)\|^{2}},~{}0\leq j\leq 2k+1$ where for $j=2k+1$ the integral is not $L^{1}$ convergent, but is understood as a principal value. Then the coefficients $a_{j},b$ of the Ito equation is determined as indicated above. This allows one to determine exactly which Ito vector equations have stationary solutions, and what the stationary distribution is. Of course it is the Gaussian vector, $Y^{(j)}(0)$, with covariance $\displaystyle r^{(i+j)}(0),~{}i,j=0,\ldots,k$ It seems that these results are new despite the fact that the problem has been around for nearly fifty years to describe the class ${\cal{C}}$. ## 2 The case $k=0$ . For $k=0$, we ask what is the set of all stationary Gauss-Markov processes and the answer is well-known as the set of processes with covariance $r(t)=Ae^{-\alpha\|t\|},~{}A\geq 0,~{}\alpha\geq 0$, with representation $\displaystyle Y(t)=\sqrt{A}e^{-\alpha t}W\left(e^{2\alpha t}\right)$ with $W$ a standard Wiener process, which satisfies an Ito equation of the form $\displaystyle dY(t)=aY(t)dt+bdW(t),~{}~{}\text{where}~{}~{}a<0~{}~{}\text{and}~{}~{}b>0$ Let us give the simple proof for $k=0$ that the covariance $r$ must be as stated, because we will use the same method for the general case, $k$, and this will make things clearer. The idea is to find or define the covariance, $R(s,t)=r(\|t-s\|)$, under the stated assumptions. We must have $Y(s)$ and $Y(u)$ given $Y(t)$ to be uncorrelated for Markovianness to hold, whenever $s<t<u$. Since $\displaystyle\mathbb{E}\left[Y(u)~{}\|~{}Y(t)\right]=Y(t)\frac{r(t-s)}{r(0)}$ this means that $r$ satisfies for any positive $u,v$, $\displaystyle\mathbb{E}\left[Y(u)-\frac{r(u)}{r(0)}Y(0)\right]Y(-v)=0,~{}~{}\text{or}~{}~{}r(u+v)=\frac{r(u)r(v)}{r(0)}$ Since $r$ is continuous and nonnegative definite, it follows that $r(h)=Ae^{-h\alpha},$ $h\geq 0$. To see this note that $f(t)=\log{\frac{r(t)}{r(0)}}$ satisfies $f(u+v)=f(u)+f(v)$ and so $f(\frac{m}{n})=f(1)\frac{m}{n}$. Since $f$ is continuous we have $f(u)=uf(1)$ and the claim follows. Since $r$ is to be nonnegative definite we must have $A\geq 0$ and $\alpha\geq 0$. We have found a necessary condition for $r$ to be the covariance. In fact we see that $r$ is infinitely differentiable, except at $u=0$, where $r$ has finite left and right derivatives. An analogous property will hold for every $k$. The only thing missing is to prove sufficiency, i.e., the existence of a process with this covariance. This is easy if we use the representation: $\displaystyle Y(t)=\sqrt{A}e^{-t\alpha}W(e^{2\alpha t})$ which has covariance $r(t-s)=Ae^{-\|t-s\|\alpha}$. We know $Y$ is Markovian, so we can obtain the coefficients of the Ito equation by the formula, $\displaystyle aY(0)=\lim_{h\downarrow 0}\frac{\mathbb{E}\left[Y(h)-Y(0)~{}\|~{}Y(0)\right]}{h}=\lim_{h\downarrow 0}\frac{Y(0)(r(h)-r(0))}{hr(0)}=-\alpha Y(0)$ and the other Ito coefficient is $\displaystyle b=\lim_{h\downarrow 0}\frac{\left(\mathbb{E}\left[Y(h\right]-\mathbb{E}\left[Y(h)~{}\|~{}Y(0)\right]\right)^{2}}{h}=\lim_{h\downarrow 0}\frac{r(0)-\frac{r^{2}(h)}{r(0)}}{h}=2\alpha A$ so the Ito equation is $\displaystyle dY(t)=-\alpha Y(t)dt+\sqrt{2A\alpha}dW(t)$ Finally, the stationary measure has $\sigma^{2}=r(0)=A$. ## 3 The case $k=1$ . We will give the approach for general $k$ but let’s do $k=1$. Since $Y(u)$ and $Y(s)$ are conditionally uncorrelated given $Y(t)$ for $s<t<u$, we need $\displaystyle(\mathbb{E}\left[Y(u)\right]-\mathbb{E}\left[Y(u)~{}\|~{}Y(0),Y^{\prime}(0)\right])Y(-v)=0,$ for $u>0>-v$, and since we must have $r^{\prime}(0)=0$ since $r$ is even and is differentiable at zero because $Y$ is differentiable, we have $\displaystyle\mathbb{E}\left[Y(u)~{}\|~{}Y(0),Y^{\prime}(0)\right]=\frac{r(u))}{r(0)}Y(0)+\frac{r^{\prime}(u)}{r^{\prime\prime}(0)}Y^{\prime}(0)$ Since $(\mathbb{E}\left[Y(u)\right]-\mathbb{E}\left[Y(u)~{}\|~{}Y(0),Y^{\prime}(0)\right])Y(-v)=0$, for $u>0,v>0$, this gives $\displaystyle()\hskip 72.26999ptr(u+v)-\frac{r(u)r(v)}{r(0)}-\frac{r^{\prime}(u)r^{\prime}(v)}{r^{\prime\prime}(0)}=0,u>0,v>0$ This shows that $r(u),u>0$ is infinitely differentiable because $r$ is differentiable and $()$ exhibits $r^{\prime}$ in terms of $r$, so that $r^{\prime}$ is differentiable, and by induction $r$ has all derivatives at $u\neq 0$. Moreover, since $r$ is even we must have $r^{\prime}(0)=0$. We next show that $r(u)$ satisfies a second order ODE with constant coefficients, namely: $\displaystyle r^{(2)}(u)=r^{(0)}(u)\frac{r^{(2)}(0)}{r^{(0)}(0)}+r^{(1)}(u)\frac{r^{(3)}(0+)}{r^{(2)}(0)}$ We claim first that $r$ is twice differentiable at $u=0$. This follows from the fact that $Y^{\prime}(t)$ is a Gaussian variable and so $\displaystyle r^{\prime\prime}(0)=-\mathbb{E}\left[Y^{\prime}(0)Y^{\prime}(0)\right]$ exists. If we expand each side of $()$ into power series in $v$, then we get that up to a term, $o(v^{2})$, $\displaystyle\sum_{j=0}^{2}\frac{r^{(j)}(u)v^{j}}{j!}=\frac{r(u)}{r(0)}\sum_{j=0}^{2}\frac{r^{(j)}(0)v^{j}}{j!}+\frac{r^{(1)}(u)}{r^{(2)}(0)}\sum_{j=0}^{2}\frac{r^{(j+1)}(0)v^{j}}{j!}$ and we see that the coefficients of $v^{0},v^{1}$ vanish automatically, but the coefficient of $v^{2}$ shows that $r^{(2)}(0+)$ exists and then the coefficient $v^{2}$ being equal on both sides gives a second degree differential equation for $r$, for $u>0$. Thinking of $r(0),r^{(2)}(0),r^{(3)}(0+)$ as constants, we see that for $u>0$, the differential equation for $r$ has constant coefficients. If the indicial equation has distict roots, then this means that $r(u)=\sum_{j=1}^{2}A_{j}e^{-ua_{j}}.$ (1) If the two roots are not distinct, then one gets a limiting covariance e.g., for the case when $a_{1}=a,a_{2}=a+\epsilon$, and $A_{1}=-A_{2}=\frac{1}{\epsilon}$, $r$ becomes the derivative, $\displaystyle r(u)=(1+au)e^{-ua},u\geq 0$ Next, we have to check that any such $r$ satisfies equation (). This is easy to check in this case, so that satisfying () imposes no additional restrictions than satisfying a second order ode with constant coefficients. It is not true that every such $r$ is realizable because $r$ must be a covariance and conditions to ensure this must be placed on $a_{j}$ and $A_{j}$. For example we must have $a_{j}>0$. For $Y$ to be differentiable we need that $r(h)$ be twice differentiable at $h=0$. In turn, this means that $-r^{\prime}(0)=A_{1}a_{1}+A_{2}a_{2}=0$. We also need that $r(0)=A_{1}+A_{2}>0$. Let us use the process represented below to show that with these restrictions the condition is also sufficient to realize the covariance in (1): $\displaystyle Y(t)=\int_{-\infty}^{\infty}f(t-\theta)dW(\theta)$ where $f$ is any $L^{2}$ function to get a class of processes with covariance $r$ of the form above. Set: $\displaystyle f(x)=A^{-}e^{xa^{-}},~{}x<0,~{}f(x)=A^{+}e^{-xa^{+}},~{}x>0$ Now, $Y$ will only be well defined when $f\in L^{2}$, so we need $a^{\pm}>0$. The covariance of the representation is easily seen to be $\displaystyle r(u)=\left(\frac{(A^{-})^{2}}{2a^{-}}-\frac{A^{-}A^{+}}{a^{-}-a^{+}}\right)e^{-ha^{-}}+\left(\frac{(A^{+})^{2}}{2a^{+}}+\frac{A^{-}A^{+}}{a^{-}-a^{+}}\right)e^{-ha^{+}}$ Also $Y$ will only be differentiable when $f$ is continuous. so this means $f(0-)=f(0+)$ or $A^{-}=A^{+}$. We get a certain class of covariances of our form. Wolog we can choose $a_{1}=a^{-},a_{2}=a^{+}$. If we set $A^{+}=A^{-}=A$, we need to choose $A$ so that $\displaystyle A_{1}=A^{2}\left(\frac{1}{2a_{-}}-\frac{1}{a_{-}-a_{+}}\right),~{}A_{2}=A^{2}\left(\frac{1}{2a_{+}}+\frac{1}{a_{-}-a_{+}}\right)$ It is easy to check that $a_{1}A_{1}+a_{2}A_{2}=0$ holds. ### 3.1 Discrete Considerations Using definition (iii) to classify the elements ${\cal{C}}_{k}$, we ask for the form of an AR(2) process that will give rise to the continuous AR(2) process. ## 4 General $k\geq 2$ . We must have $Y(s)$ and $Y(u)$ conditionally independent given $Y^{(j)}(t),j=0,\ldots,k$, and since the process is Gaussian this means that $Y(s)$ and $Y(t)$ are conditionally uncorrelated. This means that $\displaystyle\mathbb{E}\left[Y(u)-\sum_{j=0}^{k}\alpha_{j}(u)Y^{j)}(0)\right]Y(-v)=0$ for $u>0,v>0$ where $\displaystyle\mathbb{E}\left[Y(u)~{}\|~{}Y^{(j)}(0),j=0,\ldots,k\right]=\sum_{j=0}^{k}\alpha_{j}(u)Y^{(j)}(0)$ because conditional expectations are linear for Gaussian processes. Note the $\alpha_{j}$’s are defined uniquely by the equations $\displaystyle r^{(i)}(u)=\sum_{j=0}^{k}\alpha_{j}(u)r^{(i+j)}(0),~{}i=0,\ldots,k.$ We would like to show that, without any further assumptions than the fact that $r$ satisfies an ode of degree $k+1$, that the first equation holds, because then we can conclude that the first equation poses no additional restrictions. The first equation is the same as $\displaystyle r(u+v)=\sum_{j=0}^{k}\alpha_{j}(u)r^{(j)}(v),~{}u>0,~{}v>0$ i.e., $r(u+v)$ is of rank $k+1$, i.e., (4) holds if the $\alpha_{j}$’s are defined by (4). Here is where using the first approach pays off to avoid a lot of algebra. Imagine solving (4) for the $\alpha_{j}(u)$’s and then placing these $\alpha_{j}(u)$ into (4). We now let $v$ be small and positive and use power series. We get that $r(u)$ satisfies a differential equation of degree $k+1$ with constant coefficients as in the cases $k=0,1$. But to avoid checking that no further condition is required to prove that $r$ also satisfies (4) we can argue as follows. Let for $v$ fixed, $\displaystyle f(u)=r(u+v)-\sum_{j=0}^{k}\alpha_{j}(u)r^{(j)}(v),~{}u>0$ Note that $r(u+v)$ and $r^{(j)}(u)$, as functions of $u$, for all $j$ and any fixed $v$ satisfy the same differential equation because the differential equation has constant coefficients. Also, there are $k+1$ zero boundary conditions $f^{(j)}(0+)=0$, so that $f\equiv 0$. This is quite subtle, and we need that $u>0$ and $v>0$ here to get the required differentiability. It follows that (4) holds. So we have proved that if $r$ is a covariance for which $Z$ is a $(k+1)$-vector Markov process, then $r(u),u>0$ satisfies a differential equation of degree $k+1$ with constant coefficients. The general solution for $r(u)$ must also be of the form (since $r$ is nonengative definite): $\displaystyle r(u)=\int_{\mathbb{R}}e^{iux}\mu(dx)$ for some nonnegative (spectral) measure, $\mu$. Since $r(u)=r(-u)$, $\mu$ is even, and since $r$ satisfies a differential equation of order $k+1$ with constant coefficients, we must have for $u\neq 0$, $\displaystyle\sum_{j=0}^{k}b_{j}r^{(j)}(u)=0=\int_{\mathbb{R}}\sum_{j=0}^{k}b_{j}(-ix)^{j}e^{iux}\mu(dx)=\int_{\mathbb{R}}e^{ixu}P(x)\mu(dx)$ We next prove that we must have, with $c>0$, $\displaystyle r(t)=\int_{\mathbb{R}}\frac{e^{izt}}{\|P(z)\|^{2}}dz$ where $P(z)=c\prod_{j=0}^{k}(1-\frac{z}{\zeta_{j}})$, with $c>0$, and with the $k+1$ complex numbers, $\zeta_{j},j=0,\ldots,k$ having strictly positive imaginary part. We require that $f(z)=f(-z)$ so we must have for each $\zeta_{j}$ another $\zeta_{j^{\prime}}$ for which $\zeta_{j}=-\zeta_{j^{\prime}}^{}$ is the negative complex conjugate. It may be that $j^{\prime}=j$ in which case $\zeta_{j}$ is on the positive imaginary axis. For such a polynomial $P$, there is an ode with constant coefficients satisfied by $r(t),t>0$ because, by Cauchy’s theorem, the differential operator $P(-iD)r(t)$, $D=\frac{d}{dt}$, is just $\displaystyle P(-iD)r(t)=\int_{\mathbb{R}}\frac{P(z)}{P(z)P^{}(z)}e^{izt}dz=\int_{\mathbb{R}}\frac{e^{itz}}{P^{}(z)}dz=0$ because we can complete the integral by adding a semicircle above the real $z$-axis along which, for $u>0$, $e^{iuz}$ is bounded, and since the last integrand is analytic in the upper half plane the integral is zero, and as the radius of the semicircle goes to infinity the contribution from the arc is negligible because $P^{}(z)$ is large. The representation of $r$ as a covariance is now immediate, since we can just set $\displaystyle Y(t,\omega)=\int_{\mathbb{R}}\cos{tz}\frac{dW_{1}(z,\omega)}{\|P(z)\|}+\int_{\mathbb{R}}\sin{tz}\frac{dW_{2}(z,\omega)}{\|P(z)\|}$ where $W_{i}$ are iid standard Brownian motions, and check that $Y$ has covariance $r$. Note that we cannot have any polynomial factor in the numerator of the above equation for $r$ because $r^{(2k)}(u)$ must exist. The Ito equation for $\displaystyle Z(t)=(Y^{(0)}(t),Y^{(1)}(t),\ldots,Y^{(k)}(t))$ is degenerate for the coefficients of $dY^{(j)}(t),j<k$, since $\displaystyle dY^{(i)}(t)\equiv Y^{(i+1)}(t)dt,i<k$ but for $i=k$, we need to compute $a_{j}$ and $b$ in $\displaystyle dY^{(k)}(t)=\sum_{j=0}^{k}a_{j}Y^{(j)}(t)dt+bdW(t)$ The $a_{j}$’s are found as follows, we may as well take $t=0$, so $\displaystyle\sum_{j=0}^{k}a_{j}Y^{(j)}(0)$ $\displaystyle=$ $\displaystyle\lim_{h\downarrow 0}\frac{\mathbb{E}\left[Y^{(k)}(h)-Y^{(k)}(0)~{}\|~{}Y^{(0)}(0),\ldots,Y^{(k)}(0)\right]}{h}$ $\displaystyle=$ $\displaystyle\lim_{h\downarrow 0}\frac{\sum_{j=0}^{k}\alpha_{j}^{(k)}(h)Y^{(j)}(0)-Y^{(k)}(0)}{h}=\sum_{j=0}^{k}\alpha_{j}^{(k+1)}(0)Y^{(j)}(0)$ where we have used the fact that $\alpha_{j}^{(i)}(0)=\delta_{i,j}$ because $\alpha_{j}^{(i)}(h)$ satisfies, for any $i\geq 0$, $\displaystyle r^{(i)}(h)=\sum_{j=0}^{k}\alpha_{j}^{(i)}(h)r^{(j)}(0)$ and we may set $h=0$. Comparing coefficeints, we can read off the result, $a_{j}=\alpha_{j}^{(k+1)}(0)$. Moreover, the values of $a_{j}$ are given as the solutions of the equations satisfied by the $\alpha_{j}^{(i)}$’s, i.e., $\displaystyle r^{(i+k+1)}(0)=\sum_{j=0}^{k}a_{j}r^{(i+j)}(0),i=0,1,\ldots,k$ so the values of $a_{j}$ are now determined. To determine $b$, we use $\displaystyle b^{2}$ $\displaystyle=$ $\displaystyle\lim_{h\downarrow 0}\frac{\left(\mathbb{E}\left[Y^{(k)}(h)\right]-\mathbb{E}\left[Y^{(k)}(h)~{}\|~{}Y^{(0)},\ldots,Y^{(k)}(0)\right]\right)^{2}}{h}$ $\displaystyle=$ $\displaystyle\lim_{h\downarrow 0}\frac{r^{(2k)}(0)(-1)^{k}-\sum_{j=0}^{k}\alpha_{j}^{(k)}(h)r^{(k+j)}(h)(-1)^{j}}{h}$ $\displaystyle=$ $\displaystyle\sum_{j=0}^{k-1}\alpha_{j}^{(k+1)}(0)r^{(k+j)}(0)(-1)^{j+1}+(-1)^{k+1}\lim_{h\downarrow 0}\frac{r^{(2k)}(h)\alpha_{k}^{(k)}(h)-r^{(2k)}(0)\alpha_{k}^{(k)}(0)}{h}.$ but the last limit is just the value of the derivative of the product of $r^{(2k)}(h)\alpha_{k}^{(k)}(h)$ at $h=0$, so we finally arrive at $\displaystyle b^{2}=\sum_{j=0}^{k}\alpha_{j}^{(k+1)}(0)r^{(k+j)}(0)(-1)^{j+1}+(-1)^{k+1}r^{(2k+1)}(0^{+})$ or in terms of the already calculated $a_{j}$’s, switching to $0^{-}$ $\displaystyle b^{2}=\sum_{j=0}^{k}a_{j}r^{(k+j)}(0)(-1)^{j+1}+(-1)^{k}r^{(2k+1)}(0^{-})$ . All the coefficients can be computed in terms of the unique polynomial $P$ which corresponds to any process $Y$ in ${\cal{C}}$ because the only thing we need to know are the derivatives of $r(t)$ at $t=0$, which are given by $\displaystyle r^{(j)}(0^{\pm})=\mp\int_{\mathbb{R}}\frac{(iz)^{j}dz}{\|P(z)\|^{2}}$ where the integral is absolutely convergent for $j\leq 2k$ and is understood as a principle value integral for $j=2k+1$. Generalizations The problem also makes sense for $k=\infty$: the paths of $Y$ are then entire analytic functions. In the case $k=\infty$, the Markov process degenerates because the values of $Y^{(j)}(0)$ for all $j$ completely determines the past and the future of $Y$ because of the power series representation, $\displaystyle Y(t)=\sum_{j=0}^{\infty}\frac{Y^{(j)}(0)t^{j}}{j!}$ . Another generalization is to allow $Y$ itself to be a vector process, $\displaystyle{\bf{Y}}(t)=(Y_{1}(t),\ldots,Y_{n}(t))$ and ask the same question. When is ${\bf{Y}}$ together with its first $k$ derivatives a mean zero stationary Gaussian Markov process. It seems there is no real trouble making this generalization, although noncommuting matrices enter. Indeed, the equation for the covariance, $\displaystyle{\bf{R}}(s,t)=\mathbb{E}\left[{\bf{Y}}(s){\bf{Y}}(t)\right]={\bf{r}}(t-s)$ even when $k=0$ is $\displaystyle r_{ij}(u+v)=\sum_{k,l}r_{ik}(u)A_{kl}r_{lj}(v)$ for some matrix $A$ which may not commute with the matrix $r_{ij}(u)$.	arxiv-papers	2014-01-01T04:59:19	2024-09-04T02:49:56.141986	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Larry Brown, Philip Ernst, Larry Shepp, Bob Wolpert", "submitter": "Philip Ernst", "url": "https://arxiv.org/abs/1401.0251" }
1401.0262	# Persistent charge and spin currents in a quantum ring using Green’s function technique: Interplay between magnetic flux and spin-orbit interactions Santanu K. Maiti [email protected] Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 Barrackpore Trunk Road, Kolkata-700 108, India Moumita Dey Condensed Matter Physics Division, Saha Institute of Nuclear Physics, Sector-I, Block-AF, Bidhannagar, Kolkata-700 064, India S. N. Karmakar Condensed Matter Physics Division, Saha Institute of Nuclear Physics, Sector-I, Block-AF, Bidhannagar, Kolkata-700 064, India ###### Abstract We put forward a new approach based on Green’s function formalism to evaluate precisely persistent charge and spin currents in an Aharonov-Bohm ring subjected to Rashba and Dresselhaus spin-orbit interactions. Unlike conventional methods our present scheme circumvents direct evaluation of eigenvalues and eigenstates of the system Hamiltonian to determine persistent currents which essentially reduces possible numerical errors, especially for larger rings. The interplay of Aharonov-Bohm flux and spin-orbit interactions in persistent charge and spin currents of quantum rings is analyzed in detail and our results lead to a possibility of estimating the strength of any one of the spin-orbit fields provided the other one is known. All these features are exactly invariant even in presence of impurities, and therefore, can be substantiated experimentally. ###### pacs: 73.23.Ra, 73.23.-b, 73.21.Hb, 71.70.Ej ## I Introduction The promise of new technological breakthroughs has been a major driving force for studying transport in meso- and nano-structures whose dimensions are comparable to and even smaller than the mean free paths or wavelengths of electrons datta1 ; imry1 ; datta2 . Simultaneously, inspection of electronic transport in low-dimensional systems comprising simple and complex structures has brought up several new underlying questions. The progress in experimental techniques has allowed for systematic investigations of artificially made nanostructures whose transport properties are affected or even governed by quantum effects and this makes it possible to perform experiments that directly probe quantum properties of phase coherent many-body systems. The appearance of circular currents, induced by external magnetic fields in isolated (no source and drain electrodes) quantum rings, commonly known as persistent currents, is an astonishing quantum effect which reveals the significance of phase coherence of electronic wave functions in low- dimensional quantum systems. The phenomenon of persistent current in normal metal rings in presence of Aharonov-Bohm (AB) flux $\phi$ has been first exposed yang in the early $60$’s, and then, in $1983$ Büttiker et al. butt1 have successfully revived it and they have established that an isolated normal metal mesoscopic ring threaded by an AB flux $\phi$ carries an equilibrium current which does not decay over time and circulates within the sample. Following this pioneering work, interest in this subject has rapidly picked up with substantial theoretical gefen ; ambe ; schm1 ; schm2 ; bary ; maiti4 ; maiti5 ; maiti6 ; maiti7 ; maiti9 ; skm1 ; skm2 ; skm3 ; skm4 and experimental levy ; jari ; bir ; chand works. Still, many open questions persist in this particular issue. For instance, persistent current examined in disordered rings is considerably larger than the corresponding theoretical predictions chand ; mailly1 ; mailly . In $2009$, Bluhm et al. blu have made in situ measurements using scanning SQUID microscope for studying magnetic properties of $33$ discrete mesoscopic gold rings, taking one ring at a time. Their experimental results fit reasonably well with the theoretically predicted value gefen only in an ensemble of $16$ nearly ballistic rings mailly1 and in a single ballistic ring mailly . But, the current amplitudes in single isolated diffusive gold rings chand are still order of magnitude larger than the theoretical predictions. In presence of electron-electron (e-e) interaction and disorder an explanation has been proposed ambe ; smt ; mont ; ph , based on a perturbative calculation, which reveals persistent currents with greater amplitude compared to the non-interacting case, but still off by an order of magnitude. Moreover, the origin of the e-e interaction parameters taken into account in the theory is not suitably unraveled. Thus, it demands further studies to resolve these controversial issues. Another challenging topic is the possible existence of a spin current qf1 in mesoscopic rings with spin-orbit (SO) interaction, even in the absence of a magnetic field. This phenomenon may be observed through the recently developed Doppler and spin modulation relaxation techniques vla ; and1 ; and2 . The SO interaction is a rudimentary mechanism that is manifested in several fascinating properties that pertain to the anticipations of semiconducting structures as potential quantum devices. In conventional semiconducting materials two typical SO interactions are encountered. One is called as Rashba spin-orbit interaction (RSOI) rash and the other is named as Dresselhaus spin-orbit interaction (DSOI) dress . The previous one is associated with electric field that is generated from the structural inversion asymmetry at interfaces, while the later results from the bulk inversion asymmetry gini . An additional contribution can also arise from surface anisotropies prem together with simple Rashba SO interaction, which is associated with the interfacial electric field normal to the surface that results from the band offset at the interface of two different semiconductors. Quantum rings - ring- shaped quantum wells fabricated at such heterojunctions - comprise such anisotropies are exemplary candidates for examining SO coupling effects in persistent currents. Note that if one of the components that make the interface is characterized by bulk asymmetry, a corresponding contribution of the DSOI will also exist at the interface wnk . In such quantum rings electronic transport will exhibit the interplay between these different contributions to the SO coupling at the interface. A sizable amount of related theoretical work has already revealed the distinctive features of persistent charge and spin currents in mesoscopic rings subjected to Rashba and Dresselhaus SO fields qf1 ; sng ; spl ; loss , however a well defined methodology for the prediction of persistent current in large samples is still missing and the magneto-transport properties of such structures are fascinating and remain controversial. The accurate determination of persistent current in such systems, in the presence of an AB flux, is a route for analyzing its magnetic properties. A clear understanding of the role played by SO interactions in the phenomena of persistent charge and spin currents necessiates proper estimation of the strength of these interactions. The Rashba SO interaction which is controlled by an external gate voltage placed in the vicinity of the sample gini ; prem can be determined by the structure of the interface. This yields, in principle, a wide range of possible values of RSOI and its determination in any given material is crucial nit . The feasible routes of measuring the strength of DSOI are mainly based on the photo-galvanic methods yi , measurement of electrical conductance of nano-wires sc , and an optical monitoring of the spin precession of the electrons stu . A unitary transformation has been explored eg ; zh which brings out a hidden symmetry, when applied to the SO Hamiltonian, that has been used to establish that by making the strengths of the two SO interactions equal one achieves a zero spin current in the material maiti1 , and this vanishing spin current is a robust effect which is observed even in presence of disorder maiti1 , and thus, can be established experimentally. Observing the persistent charge current maiti2 one can estimate the strength of DSOI, and, by monitoring the vanishing of persistent spin current one can determine both the RSOI and DSOI in a single mesoscopic ring maiti1 ; maiti3 . The established approach to the determination of persistent charge gefen ; spl ; maiti2 ; bouz ; giam ; yu ; maiti8 ; maiti10 ; skm5 and spin qf1 ; spl ; qf2 ; maiti3 currents in isolated conducting rings is based on the evaluation of eigenvalues and eigenvectors of the system Hamiltonian. For large size rings such an approach becomes highly numerically unreliable, and most importantly - hard to speculate in presence of interaction with external baths. Here we propose a new approach, based on Green’s function formalism, that circumvents the need to evaluate system eigenvalues and eigenfunctions. In particular, this Green’s function methodology for determining persistent currents should give us access to evaluation of the magnetic properties of large conducting rings as well as molecular rings encountered in biopolymers. We firmly believe that the Green’s function technique will yield persistent charge and spin currents a very high degree of accuracy, and this will definitely make it possible to consider the interplay between molecular structure and geometry and the resulting persistent currents obtained in the presence of an AB flux $\phi$ and SO interactions. The rest of the paper is organized as follows. In Section II, the model quantum system and the calculation method are described. In Section III, the numerical results are presented which describe the (i) behavior of persistent charge current, (ii) characteristic features of persistent spin current, and (iii) possible route of estimating the strength of RSOI and DSOI in a single mesoscopic ring. Finally, in Section IV, we summarize our essential results. ## II Theoretical framework ### II.1 Model and Hamiltonian We consider a mesoscopic ring which is subjected to both Rashba and Dresselhaus SO fields. The ring is threaded by an AB flux $\phi$ which is measured in unit of $\phi_{0}=ch/e$, the elementary flux-quantum. A schematic view of this ring is illustrated in Fig. 1. Figure 1: (Color online). Schematic representation of the model quantum system where a mesoscopic ring is threaded by an AB flux $\phi$. The filled circles correspond to the positions of the atomic sites. The TB Hamiltonian of such a $N$-site ring in the site representation reads as maiti1 ; maiti2 ; maiti3 , $\displaystyle\mbox{\boldmath$H$}_{\mbox{\tiny R}}$ $\displaystyle=$ $\displaystyle\sum_{n}\mbox{\boldmath$c_{n}^{\dagger}\epsilon_{n}c_{n}$}+\sum_{n}\left(\mbox{\boldmath$c_{n+1}^{\dagger}t$}\,e^{i\theta}\mbox{\boldmath$c_{n}$}+h.c.\right)$ (1) $\displaystyle-$ $\displaystyle\sum_{n}\left(\mbox{\boldmath$c_{n+1}^{\dagger}($}i\mbox{\boldmath$\sigma_{x})\alpha$}\cos\varphi_{n,n+1}\,e^{i\theta}\mbox{\boldmath$c_{n}$}+h.c.\right)$ $\displaystyle-$ $\displaystyle\sum_{n}\left(\mbox{\boldmath$c_{n+1}^{\dagger}($}i\mbox{\boldmath$\sigma_{y})\alpha$}\sin\varphi_{n,n+1}\,e^{i\theta}\mbox{\boldmath$c_{n}$}+h.c.\right)$ $\displaystyle+$ $\displaystyle\sum_{n}\left(\mbox{\boldmath$c_{n+1}^{\dagger}($}i\mbox{\boldmath$\sigma_{y})\beta$}\cos\varphi_{n,n+1}\,e^{i\theta}\mbox{\boldmath$c_{n}$}+h.c.\right)$ $\displaystyle+$ $\displaystyle\sum_{n}\left(\mbox{\boldmath$c_{n+1}^{\dagger}($}i\mbox{\boldmath$\sigma_{x})\beta$}\sin\varphi_{n,n+1}\,e^{i\theta}\mbox{\boldmath$c_{n}$}+h.c.\right),$ where, $\theta=2\pi\phi/N$ is the phase factor associated with the hopping of an electron between nearest-neighbor sites in presence of the AB flux $\phi$ and $\varphi_{n,n+1}=(\varphi_{n}+\varphi_{n+1})/2$, where $\varphi_{n}=2\pi(n-1)/N$. The other factors are defined as follows. $c_{n}^{\dagger}$=$\left(\begin{array}[]{cc}c_{n\uparrow}^{\dagger}&c_{n\downarrow}^{\dagger}\end{array}\right),$ $c_{n}$=$\left(\begin{array}[]{c}c_{n\uparrow}\\\ c_{n\downarrow}\end{array}\right),$ $\epsilon_{n}$=$\left(\begin{array}[]{cc}\epsilon_{n\uparrow}&0\\\ 0&\epsilon_{n\downarrow}\end{array}\right),$ $t$=$t\left(\begin{array}[]{cc}1&0\\\ 0&1\end{array}\right),$ $\alpha$=$\left(\begin{array}[]{cc}\alpha&0\\\ 0&\alpha\end{array}\right),$ $\beta$=$\left(\begin{array}[]{cc}\beta&0\\\ 0&\beta\end{array}\right),$ $\sigma_{x}$=$\left(\begin{array}[]{cc}0&1\\\ 1&0\end{array}\right),$ $\sigma_{y}$=$\left(\begin{array}[]{cc}0&-i\\\ i&0\end{array}\right),$ $\sigma_{z}$=$\left(\begin{array}[]{cc}1&0\\\ 0&-1\end{array}\right)$, where, $c_{n\sigma}^{\dagger}$ and $c_{n\sigma}$ are the creation and annihilation operators, respectively, for an electron with spin $\sigma$ ($\uparrow,\downarrow$) at $n$-th site. The nearest-neighbor hopping integral is described by the parameter $t$ and $\epsilon_{n\sigma}$ denotes the on-site energy of an electron at the site $n$ of the ring with spin $\sigma$. The factors $\alpha$ and $\beta$ corresponds to the strengths of Rashba and Dresselhaus SO fields, respectively, and $\sigma_{x}$, $\sigma_{y}$ and $\sigma_{z}$ are the conventional Pauli spin matrices. ### II.2 Persistent charge current At absolute zero temperature ($T=0\,$K), net persistent charge current carried by a ring for a particular electron filling is obtained from the expression, $I_{c}(\phi)=\int\limits_{-\infty}^{\mu}J_{c}(E)\,dE,$ (2) where, $\mu$ describes the chemical potential of the ring and $J_{c}(E)$ represents the charge current density. In terms of Green’s functions (see Appendix A and Appendix B for comprehensive derivations) it ($J_{c}(E)$) gets the form: $\displaystyle J_{c}(E)$ $\displaystyle=$ $\displaystyle-\frac{e}{N}\sum_{n}\left\\{t\left(\mbox{\boldmath$G$}_{n+1\uparrow,n\uparrow}^{r}-\mbox{\boldmath$G$}_{n+1\uparrow,n\uparrow}^{a}\right)e^{-i\theta}\right.$ (3) $\displaystyle\left.-t\left(\mbox{\boldmath$G$}_{n\uparrow,n+1\uparrow}^{r}-\mbox{\boldmath$G$}_{n\uparrow,n+1\uparrow}^{a}\right)e^{i\theta}\right\\}$ $\displaystyle-\frac{e}{N}\sum_{n}\left\\{t\left(\mbox{\boldmath$G$}_{n+1\downarrow,n\downarrow}^{r}-\mbox{\boldmath$G$}_{n+1\downarrow,n\downarrow}^{a}\right)e^{-i\theta}\right.$ $\displaystyle\left.-t\left(\mbox{\boldmath$G$}_{n\downarrow,n+1\downarrow}^{r}-\mbox{\boldmath$G$}_{n\downarrow,n+1\downarrow}^{a}\right)e^{i\theta}\right\\}$ $\displaystyle-\frac{e}{N}\sum_{n}\left(i\alpha e^{-i\varphi_{n,n+1}}-\beta e^{i\varphi_{n,n+1}}\right)$ $\displaystyle\times\left(\mbox{\boldmath$G$}_{n+1\downarrow,n\uparrow}^{r}-\mbox{\boldmath$G$}_{n+1\downarrow,n\uparrow}^{a}\right)e^{-i\theta}$ $\displaystyle-\frac{e}{N}\sum_{n}\left(i\alpha e^{i\varphi_{n,n+1}}+\beta e^{-i\varphi_{n,n+1}}\right)$ $\displaystyle\times\left(\mbox{\boldmath$G$}_{n\uparrow,n+1\downarrow}^{r}-\mbox{\boldmath$G$}_{n\uparrow,n+1\downarrow}^{a}\right)e^{i\theta}$ $\displaystyle-\frac{e}{N}\sum_{n}\left(i\alpha e^{i\varphi_{n,n+1}}+\beta e^{-i\varphi_{n,n+1}}\right)$ $\displaystyle\times\left(\mbox{\boldmath$G$}_{n+1\uparrow,n\downarrow}^{r}-\mbox{\boldmath$G$}_{n+1\uparrow,n\downarrow}^{a}\right)e^{-i\theta}$ $\displaystyle-\frac{e}{N}\sum_{n}\left(i\alpha e^{-i\varphi_{n,n+1}}-\beta e^{i\varphi_{n,n+1}}\right)$ $\displaystyle\times\left(\mbox{\boldmath$G$}_{n\downarrow,n+1\uparrow}^{r}-\mbox{\boldmath$G$}_{n\downarrow,n+1\uparrow}^{a}\right)e^{i\theta},$ where, $\mbox{\boldmath$G$}^{r}$ is the retarded Green’s function defined as $\mbox{\boldmath$G$}^{r}=\left(\mbox{\boldmath$E$}-\mbox{\boldmath$H$}_{\mbox{\tiny R}}+i\eta\mbox{\boldmath$I$}\right)^{-1}$ with $\eta\rightarrow 0^{+}$, and, $\mbox{\boldmath$G$}^{a}=\left(\mbox{\boldmath$G$}^{r}\right)^{\dagger}$. $I$ being the identity matrix. ### II.3 Persistent spin current Similar to Eq. 2, we determine polarized spin current along the quantized direction ($+Z$) at absolute zero temperature ($T=0\,$K) from the relation, $I_{s}(\phi)=\int\limits_{-\infty}^{\mu}J_{s}(E)\,dE,$ (4) where, $J_{s}(E)$ corresponds to the spin current density and it becomes (see Appendix C for complete derivations), $\displaystyle J_{s}(E)$ $\displaystyle=$ $\displaystyle-\frac{1}{N}\sum_{n}\left\\{t\left(\mbox{\boldmath$G$}_{n+1\uparrow,n\uparrow}^{r}-\mbox{\boldmath$G$}_{n+1\uparrow,n\uparrow}^{a}\right)e^{-i\theta}\right.$ (5) $\displaystyle\left.-t\left(\mbox{\boldmath$G$}_{n\uparrow,n+1\uparrow}^{r}-\mbox{\boldmath$G$}_{n\uparrow,n+1\uparrow}^{a}\right)e^{i\theta}\right\\}$ $\displaystyle+\frac{1}{N}\sum_{n}\left\\{t\left(\mbox{\boldmath$G$}_{n+1\downarrow,n\downarrow}^{r}-\mbox{\boldmath$G$}_{n+1\downarrow,n\downarrow}^{a}\right)e^{-i\theta}\right.$ $\displaystyle\left.-t\left(\mbox{\boldmath$G$}_{n\downarrow,n+1\downarrow}^{r}-\mbox{\boldmath$G$}_{n\downarrow,n+1\downarrow}^{a}\right)e^{i\theta}\right\\}.$ Thus, introducing the notion of persistent charge and spin current densities $J_{c}(E)$ and $J_{s}(E)$ in terms of the retared and advanced Green’s functions, expressed in Eqs. 3 and 5 respectively, we eventually determine persistent charge and spin currents by integrating the current densities over a suitable energy window (see Eqs. 2 and 4) associated with the electron filling. The detailed and long calculations of these charge and spin current densities as a function of retared and advanced Green’s functions are presented in the Appendices A-C, to have a complete idea for calculating the desired quantities. Our new approach, the so-called Green’s function approach, clearly suggests how to circumvent the need to evaluate system eigenvalues and eigenfunctions for evaluating persistent currents, as used in conventional methods. Here, it should be noted that the present scheme is well applicable both for the ordered and disordered systems since all the mathematical expressions are exactly invariant in both these two cases. Only the magnitudes of different elements of the Green’s functions get changed depending on impurity strength of the sample. This behavior essentially demands the robustness of the present new technique. ## III Numerical results and discussion In what follows, we will present numerical results computed for circulating charge and spin currents in mesoscopic rings based on Green’s function formalism. In all calculations we measure the energy scale in unit of the hopping integral $t$ which is set equal to $1$. The Rashba and Dresselhaus SO coupling strengths are also scaled in unit of this hopping parameter $t$. Throughout the numerical analysis we restrict ourselves to absolute zero temperature and fix $c=h=e=1$. First, we focus on the impurity-free mesoscopic rings, and, for such a ring we put $\epsilon_{n\uparrow}=\epsilon_{n\downarrow}=0$ for all $n$ in the TB Hamiltonian Eq. 1. In Fig. 2 we present the variation of persistent charge current density $J_{c}$ as a function of energy $E$ for some typical Figure 2: (Color online). Persistent charge current density as a function of energy for a $20$-site ordered ring setting $\phi=\phi_{0}/4$, where (a) $\alpha=\beta=0$, (b) $\alpha=0.2$ and $\beta=0$, and (c) $\alpha=0$ and $\beta=0.2$. values of Rashba and Dresselhaus spin-orbit fields. The results are computed for a $20$-site impurity-free mesoscopic ring when the AB flux $\phi$ is set at $\phi_{0}/4$. For a better viewing of distinct peaks in the density spectrum here we display the results for such a small size ring. Several interesting patterns are obtained those can be analyzed as follows. From the spectra it is observed that the charge current density exhibits sharp peaks and dips for some particular energy values, while it drops to zero for other energies. All these peaks and dips are associated with the energy eigenvalues of the ring. For the particular case when the ring is free from any kind of SO interaction and subjected to a non-zero magnetic flux, apart from integer or half-integer multiples of the elementary flux-quantum, the energy levels are two-fold degenerate which results in total $20$ peaks and dips in the current density spectrum (see Fig. 2(a)). It is quite interesting to note that the peaks and dips appear alternately throughout the band spectrum which essentially leads to an important conclusion that successive energy levels carry currents in opposite directions. This behavior suggests the vanishing net current for the complete band filling. Figure 3: (Color online). Persistent charge current as a function of flux $\phi$ for a $60$-site ordered ring, where (a) $\alpha=0.3$ and $\beta=0$, (b) $\alpha=0$ and $\beta=0.3$, and (c) $\alpha=\beta=0.3$. The chemical potential $\mu$ is fixed at $0$. Furthermore, one can also utilize this charge current density spectrum to predict the nature of extendedness of different energy levels by superimposing the average density of states (ADOS) on it. A non-zero contribution, viz, a peak or a dip, to the charge current will be obtained from the conducting states, while it becomes zero for the localized ones. This is another way of estimating the localization phenomenon in addition to the conventional methodologies maiti11 ; maiti12 ; maiti13 . The charge current density spectrum gets significantly modified when we include SO coupling in this AB ring. The results are shown in Figs. 2(b) and (c), where in (b) we set a finite value of RSOI strength keeping the strength of DSOI as zero, while in (c) these parameter values get interchanged. It shows that the total number of peaks in the current density profiles, Figs. 2(b) and (c), associated with energy levels of the ring becomes exactly twice compared to the interacting free mesoscopic ring, viz., $\alpha=\beta=0$ (see Fig. 2(a)). This is because of the complete removal of degenerate energy eigenstates of the AB ring subjected to SO interaction. One more important property is also observed from these spectra (Figs. 2(b) and (c)) that the magnitude and sign of persistent charge current density $J_{c}$ for any energy window when the AB ring is subjected to RSOI only (Fig. 2(b)) are exactly identical to the ring described with only DSOI (Fig. 2(c)). Thus, it should be emphasized that the phase reversal in charge current density does not take place by interchanging the role played by $\alpha$ and $\beta$ into the Hamiltonian Eq. 1. This phenomenon can be implemented from the following analytical prescription. The Rashba and Dresselhaus SO interaction terms, called as, $H_{\mbox{\tiny RSOI}}$ and $H_{\mbox{\tiny DSOI}}$ in the TB Hamiltonian Eq. 1 can be transformed into each other through a simple relation: $U^{{\dagger}}H_{\mbox{\tiny RSOI}}U$ = $H_{\mbox{\tiny DSOI}}$. Here, $U$ = $\left(\mbox{\boldmath$\sigma$}_{x}+\mbox{\boldmath$\sigma$}_{y}\right)/\sqrt{2}$ is the unitary matrix. Thus, any energy eigenstate $\|\mathcal{M}^{\prime}\rangle$ of the transformed Hamiltonian $H_{\mbox{\tiny DSOI}}$ can be demonstrated in terms of the eigenstate $\|\mathcal{M}\rangle$ of the Hamiltonian $H_{\mbox{\tiny RSOI}}$ through the relation $\|\mathcal{M}^{\prime}\rangle$=$\mbox{\boldmath$U$}\|\mathcal{M}\rangle$. This transformation leads to the charge current for the ring with only Dresselhaus SO coupling as, $\displaystyle J_{c}^{m}\|_{\mbox{\tiny DSOI}}$ $\displaystyle=$ $\displaystyle\langle\mathcal{M}^{\prime}\|\mbox{\boldmath$J$}_{c}\|\mathcal{M}^{\prime}\rangle$ (6) $\displaystyle=$ $\displaystyle\langle\mathcal{M}\|\mbox{\boldmath$U$}^{\dagger}\mbox{\boldmath$J$}_{c}\mbox{\boldmath$U$}\|\mathcal{M}\rangle$ $\displaystyle=$ $\displaystyle\langle\mathcal{M}\|\mbox{\boldmath$U$}^{\dagger}\frac{e}{N}\mbox{\boldmath$\dot{x}$}\mbox{\boldmath$U$}\|\mathcal{M}\rangle$ $\displaystyle=$ $\displaystyle\langle\mathcal{M}\|\frac{e}{N}\mbox{\boldmath$\dot{x}$}\|\mathcal{M}\rangle$ $\displaystyle=$ $\displaystyle J_{c}^{m}\|_{\mbox{\tiny RSOI}}.$ The above expression clearly establishes the reason for not affecting the sign and magnitude of charge current density upon the exchange of the role played by RSOI and DSOI. Once the charge current density, $J_{c}(E)$, is evaluated using the relation presented in Eq. 3, the net persistent charge current $I_{c}$ in the ring can be easily determined by integrating the density spectrum (see Eq. 2) up to a certain energy range depending on the electron filling. As illustrative example, in Fig. 3 we present the variation of persistent charge current as a function of magnetic flux $\phi$ for an ordered ring considering different values of the SO coupling strengths. Here we set $N=60$ and $\mu=0$. The effect of the Rashba SO coupling is examined in the spectrum Fig. 3(a), setting the Dresselhaus SO interaction to zero. It is observed that the persistent charge current exhibits kink-like structures together with phase reversals at several values of flux $\phi$, which are however in general not unusual even in the absence of any SO coupling. These are essentially due to the band crossing in energy spectra and are immensely sensitive to the filling factor $\mu$. Though we have computed charge currents for different band fillings through ample numerical calculations, here we present results for a particular electron filling, as a test example, to establish our Green’s function approach for the estimation of persistent current in a Figure 4: (Color online). Persistent spin current density as a function of energy for a $16$-site ordered ring, where (a) $\alpha=0.3$ and $\beta=0$, (b) $\alpha=0$ and $\beta=0.3$, and (c) $\alpha=\beta=0.3$. Here, we set $\phi=\phi_{0}/4$. mesoscopic ring. In addition to the above issues it is also very important to point out that, for a particular value of the AB flux $\phi$ the current amplitude strongly depends on the chemical potential of the sample. An exhaustive analysis has already been given by Splettstoesser et al. spl in this line. Figure 3(b) illustrates the situation in which the ring is described with DSOI only i.e., the other SO coupling term (RSOI) is set equal to zero. Since, the reversal of the roles governed by the variables $\alpha$ and $\beta$ into the TB Hamiltonian Eq. 1 does not anyway alter the physical picture of charge current density spectrum, the current-flux characteristics for the rings described with RSOI only (see Fig. 3(a)) become exactly identical to those with the rings in presence of DSOI only (see Fig. 3(b)). This phenomenon can also be justified from our analytical arguments presented in Eq. 6. The combined outcome of both these two SO fields on persistent charge current is scrutinized in Fig. 3(c), where we specify $\alpha=\beta=0.3$. The other parameters are kept unchanged as taken in Figs. 3(a) and (b). In presence of both these two Figure 5: (Color online). Persistent spin current as a function of flux $\phi$ for a $100$-site ordered ring when the chemical potential $\mu$ is fixed at $0$, where (a) $\alpha=0.4$ and $\beta=0$, (b) $\alpha=0$ and $\beta=0.4$, and (c) $\alpha=\beta=0.4$. interactions, the nature of persistent current for different values of $\phi$ changes appreciably compared to the case when only one SO interaction is present. This is due to the fact that the inclusion of both these two SO fields reforms the electronic band structure of the ring and thus strongly affects the pattern of the circulating current. In short, it can be emphasized that, charge current is distinctly sensitive to the SO coupling strength and magnetic flux threaded by the ring. All these currents vary repeatedly bearing $\phi_{0}$ ($=1$, in our choice of units where $c=e=h=1$) flux-quantum periodicity. Now, we extend our discussion on persistent spin current in SO interaction induced impurity-free mesoscopic rings, where currents are computed from the Green’s function formalism. Before addressing these results, we first analyze the behavior of persistent spin current density, determined from the relation given in Eq. 5, to make this communication a self contained study. In Fig. 4 the energy dependent spin current density spectra are presented for a $16$-site ordered ring considering $\phi=\phi_{0}/4$, where the upper, middle and lower panels correspond to the results for three different set of parameter values of $\alpha$ and $\beta$. Interestingly, we see that the spin current density in the ring described with only Dresselhaus SO coupling (see Fig. 4(b)) changes its sign keeping the magnitude unaltered compared to the ring with RSOI only (see Fig. 4(a)). This sign reversal behavior can be viewed as follows. Using the similar prescription presented in Eq. 6, the spin current for the ring described with only DSOI gets the form, $\displaystyle J_{s}^{m}\|_{\mbox{\tiny DSOI}}$ $\displaystyle=$ $\displaystyle\langle\mathcal{M}^{\prime}\|\mbox{\boldmath$J$}_{s}\|\mathcal{M}^{\prime}\rangle$ (7) $\displaystyle=$ $\displaystyle\langle\mathcal{M}\|\mbox{\boldmath$U$}^{\dagger}\mbox{\boldmath$J$}_{s}\mbox{\boldmath$U$}\|\mathcal{M}\rangle$ $\displaystyle=$ $\displaystyle\langle\mathcal{M}\|\mbox{\boldmath$U$}^{\dagger}\frac{1}{2N}\left(\mbox{\boldmath$\sigma_{z}\dot{x}$}+\mbox{\boldmath$\dot{x}\sigma_{z}$}\right)\mbox{\boldmath$U$}\|\mathcal{M}\rangle$ $\displaystyle=$ $\displaystyle\langle\mathcal{M}\|\frac{1}{2N}\left(-\mbox{\boldmath$\sigma_{z}\dot{x}$}-\mbox{\boldmath$\dot{x}\sigma_{z}$}\right)\|\mathcal{M}\rangle$ $\displaystyle=$ $\displaystyle-J_{s}^{m}\|_{\mbox{\tiny RSOI}}.$ This relation clearly describes the sign reversal of spin current density upon the interchange of the parameters $\alpha$ and $\beta$ into the Hamiltonian (Eq. 1) of the ring. From this analysis we can also justify the vanishing nature of persistent spin current density for the entire density spectrum (see Fig. 4(c)) when the Dresselhaus SO interaction strength becomes precisely identical to the strength of the Rashba term. This is an interesting observation and may lead to a possible route for estimating the strength of anyone of the SO fields provided the other one is known. A detailed analysis of it can be obtained from the following current-flux characteristics. Similar to persistent charge current, we also compute net persistent spin current $I_{s}$ in the ring by integrating the spin current density $J_{s}(E)$ (see Eq. 4) over a finite energy range associated with the filling factor $\mu$. As representative example, in Fig. 5 we show the variation of persistent spin current as a function of AB flux $\phi$ for a $100$-site ordered ring when the chemical potential $\mu$ is set at zero. Figure 5(a) illustrates the situation in which the ring is described with RSOI only, while in Figure 5(b) the effect of Dresselhaus SO coupling on spin current is presented. From these spectra (Figs. 5(a) and (b)) we observe that, spin current in the ring characterized with DSOI only alters its sign keeping the magnitude unchanged compared to the ring with Rashba term only, which is however exactly what we expect from the spin current density spectra shown in Fig. 4, since current is computed by integrating the density function $J_{s}$. The usual phase reversals at several values of AB flux $\phi$ associated with the band crossing in energy spectra together with $\phi_{0}$ flux-quantum periodicity are also noticed from these current-flux characteristics. Certainly, whenever the strength of Dresselhaus SO coupling becomes exactly identical to that of Rashba SO interaction, spin current becomes zero for the entire flux window. It is shown in Fig. 5(c), where we set $\alpha=\beta=0.4$. This vanishing nature of persistent spin current is Figure 6: (Color online). Persistent charge current as a function of flux $\phi$ for a $60$-site ring in presence of disorder ($W=1$) for the same parameter values used in Fig. 3. detected for any non-zero value of Rashba SO interaction provided it becomes equal to the Dresselhaus term, and also this behavior is independent of the band filling which we establish through our vast numerical calculations. This phenomenon, in principle, gives a possibility of estimating anyone of the SO fields if the other one is known. By means of an outside gate voltage one can control the Rashba SO coupling, and thus its strength can be determined. This suggests that, monitoring the RSOI in a mesoscopic ring we will get an absolute vanishing spin current when the strength of the Dresselhaus SO coupling becomes identical to that of the RSOI. Thus, from a realizable experimental measurement of persistent spin current one can evaluate the strength of Dresselhaus SO coupling. Additionally, it is also important to state that one may determine the strength of Rashba term employing this same mechanism provided the other term is known. Up to now we have demonstrated the results for perfect rings. For more practical implications we now focus our attention on mesoscopic rings in presence of impurities. Impurities are introduced randomly in site potentials ($\epsilon_{n\uparrow}$ and $\epsilon_{n\downarrow}$) i.e., Figure 7: (Color online). Persistent spin current as a function of flux $\phi$ for a $100$-site ring in presence of disorder ($W=1$). The other model parameters are kept unchanged as used in Fig. 5. diagonal disorder, through a ‘Box’ distribution function of width $W$, and the results averaged over $1000$ disorder configurations are presented. In Fig. 6 we present the results of persistent charge current for a disordered ring considering $W=1$ for different values of SO coupling strengths. Here we set $N=60$ and the results are computed for $\mu=0$, as a typical example. Several interesting features are obtained. Firstly, the current shows a continuous variation with flux $\phi$. This is essentially due to the fact that disorder makes a smooth variation of energy levels and eliminate band crossings those are mostly observed in impurity-free rings. The other important observation is that, in presence of impurities the current amplitude gets suppressed compared to the ordered case which can be clearly visible from the spectra plotted in Figs. 3 and 6. In presence of impurities the energy eigenstates are localized which results a reduction of persistent current, though this reduction is quite small compared to the ring without any SO interaction. In the absence of any SO coupling, disorder suppresses current amplitude almost to zero which is not unfamiliar in conventional disordered rings. But, with the inclusion of SO interaction current increases significantly and becomes quite proportionate to that of a perfect ring. A detailed analysis behind this mechanism has already been reported in our recent work maiti2 . Finally, in Fig. 7 we present the results of persistent spin current in a mesoscopic ring to give a complete exposure of our Green’s function formalism for the evaluation of persistent current even in presence of impurities. Here we set $W=1$ and all the other parameters are kept unchanged as taken in Fig. 5. As usual the current varies continuously with flux $\phi$ exhibiting $\phi_{0}$ flux-quantum periodicity and it gets a reduced amplitude compared to the perfectly ordered ring. All the other properties i.e., the sign reversal upon the interchange of the role played by the parameters $\alpha$ and $\beta$ into the Hamiltonian Eq. 1 and the vanishing nature of spin current when the RSOI becomes equal to DSOI, remain exactly valid in the presence of impurities. ## IV Summary and outlook In the present work, we have proposed a new approach based on Green’s function formalism within a tight-binding framework to evaluate precisely the persistent charge and spin currents in spin-orbit interaction induced AB rings. The essential results are summarized as follows. As already pointed out that, the standard methodology to the determination of circulating charge and spin currents in isolated conducting loops is based on the evaluation of eigenvalues and eigenvectors of the system Hamiltonian, which is highly numerically unstable especially for large size rings. Not only that it is really very hard to generalize in presence of interaction with external baths, if any. The present approach, the so-called Green’s function technique, circumvents the need to determine eigenvalues and eigenvectors of the system, and in particular, this methodology should give us access to predict the magnetic properties of large conducting rings as well as molecular rings encountered in biopolymers. We strongly believe that the present analysis yields persistent currents a very high degree of accuracy and leads to consider interplay of AB flux and geometry in magneto-transport of conducting loops subjected to Rashba and Dresselhaus SO fields. It is worth pointing out that, in the present work we mainly concentrate on the new technique for the determination of persistent charge and spin currents. With this technique, we have also provided one possible route of estimating the strength of anyone of the SO fields provided the other one is known. This can be done by measuring persistent spin current which vanishes completely when the DSOI becomes equal to the RSOI, and this vanishing effect is also observed even in presence of impurities. It essentially supports us to propose an experiment towards this direction. Before we end, we would like to mention that though we have computed persistent charge and spin currents for different band fillings considering different size rings through extensive numerical calculations, but here we have presented our results for some typical parameter values to explain the physical phenomena computed from our theoretical framework. All these physical pictures will be absolutely invariant for other parameter values also and thus demands the robustness of this new technique. In a forthcoming paper, we will provide the way of determining persistent charge and spin currents using this Green’s function technique considering the effect of electron-electron interaction. ###### Acknowledgements. First author is thankful to Prof. A. Nitzan for useful conversations, and S. Saha and P. Dutta for helpful discussions. ## Appendix A Green’s function approach for persistent charge current density when the ring is free from SO interactions First, we consider the model quantum ring, presented in Fig. 1, and establish persistent charge current density in terms of Green’s function setting $\alpha=\beta=0$. Under this condition, TB Hamiltonian of the ring with site energy $\epsilon_{n}$ and nearest-neighbor hopping interaction $t$ becomes, $H$ $\displaystyle=$ $\displaystyle\sum_{n}\epsilon_{n}\,c_{n}^{\dagger}c_{n}+\sum_{n}\left(c_{n+1}^{\dagger}\,t\,c_{n}e^{i\theta}+c_{n}^{\dagger}\,t\,c_{n+1}e^{-i\theta}\right).$ In terms of the velocity operator $\dot{x}$ the charge current operator $J_{c}$ can be written as, $\mbox{\boldmath$J_{c}$}=\frac{1}{N}e\mbox{\boldmath${\dot{x}}$}=\frac{2\pi ie}{Nh}\left[\mbox{\boldmath${H}$},\mbox{\boldmath${x}$}\right],$ (9) where, ${x}$=$\sum\limits_{n}c_{n}^{\dagger}\,n\,c_{n}$ is the position operator. Substituting $H$ and $x$ in Eq. 9 and doing a quite long but straightforward calculation we eventually reach to the expression, $\mbox{\boldmath$J_{c}$}=\frac{2\pi ie}{Nh}\sum_{n}\left(c_{n}^{\dagger}\,t\,c_{n+1}e^{-i\theta}-c_{n+1}^{\dagger}\,t\,c_{n}e^{i\theta}\right),$ (10) and, for a particular energy eigenstate $\|\mathcal{M}\rangle=\sum\limits_{p}a_{p}^{m}\|p\rangle$, it leads to the persistent charge current: $\displaystyle J_{c}^{m}$ $\displaystyle=$ $\displaystyle\langle\mathcal{M}\|\mbox{\boldmath$J_{c}$}\|\mathcal{M}\rangle$ (11) $\displaystyle=$ $\displaystyle\frac{2\pi iet}{Nh}\sum_{n}\left(a_{n}^{m\,}a_{n+1}^{m}e^{-i\theta}-a_{n+1}^{m\,}a_{n}^{m}e^{i\theta}\right),$ where, $\|p\rangle$’s are the Wannier states and $a_{p}^{m}$’s (the superscript $m$ is used for the eigenstate $\|\mathcal{M}\rangle$) are the corresponding coefficients. Utilizing this relation one can determine persistent charge currents for discrete energy levels, and therefore, at absolute zero temperature net current carried by the ring will be the sum of individual contributions of some specific energy levels associated with the electron filling. This approach requires direct evaluation of energy eigenvalues and eigenstates, like other conventional methods available in literature gefen ; spl ; maiti2 ; bouz ; giam ; yu ; maiti8 ; maiti10 . To avoid it, we reframe the above current expression (Eq. 11) in terms of Green’s functions introducing the concept of current density, instead of defining conventional currents for individual energy levels. The prescription is as follows. We start with the Green’s function of the ring $\mbox{\boldmath$G$}^{r}=\left(\mbox{\boldmath$E$}-\mbox{\boldmath$H$}+i\eta\mbox{\boldmath$I$}\right)^{-1}$. It leads to $\displaystyle\mbox{\boldmath$G$}_{ij}^{r}$ $\displaystyle=$ $\displaystyle\langle i\|\mbox{\boldmath$G$}^{r}\|j\rangle$ (12) $\displaystyle=$ $\displaystyle\sum_{m}\langle i\|\mbox{\boldmath$G$}^{r}\|\mathcal{M}\rangle\langle\mathcal{M}\|j\rangle$ $\displaystyle=$ $\displaystyle\sum_{m}\langle i\|\left(\mbox{\boldmath$E$}-\mbox{\boldmath$H$}+i\eta\mbox{\boldmath$I$}\right)^{-1}\|\mathcal{M}\rangle\langle\mathcal{M}\|j\rangle$ $\displaystyle=$ $\displaystyle\sum_{m}\frac{\langle i\|\mathcal{M}\rangle\langle\mathcal{M}\|j\rangle}{E-\mathcal{E}_{m}+i\eta}$ $\displaystyle=$ $\displaystyle\sum_{m}\frac{a_{i}^{m}\,a_{j}^{m\,}}{E-\mathcal{E}_{m}+i\eta},$ where, $\|\mathcal{M}\rangle$’s are the eigenstates of $H$ satisfying the relation $\sum\limits_{m}\|\mathcal{M}\rangle\langle\mathcal{M}\|=\mbox{\boldmath$I$}$ and $\mathcal{E}_{m}$ is the eigenvalue for the state $\|\mathcal{M}\rangle$. $a_{i}^{m}$’s are the coefficients as described earlier. In a similar way we find, $\mbox{\boldmath$G$}_{ij}^{a}=\sum_{m}\frac{a_{i}^{m}\,a_{j}^{m\,}}{E-\mathcal{E}_{m}-i\eta}.$ (13) Equations 12 and 13 yield, $\displaystyle\mbox{\boldmath$G$}_{ij}^{r}-\mbox{\boldmath$G$}_{ij}^{a}$ $\displaystyle=$ $\displaystyle\sum_{m}a_{i}^{m}a_{j}^{m\,}\left(\frac{1}{E-\mathcal{E}_{m}+i\eta}-\frac{1}{E-\mathcal{E}_{m}-i\eta}\right)$ (14) $\displaystyle=$ $\displaystyle\sum_{m}a_{i}^{m}a_{j}^{m\,}\left(\frac{-2i\eta}{(E-\mathcal{E}_{m})^{2}+\eta^{2}}\right)$ $\displaystyle=$ $\displaystyle\sum_{m}a_{i}^{m}a_{j}^{m\,}(-2i\eta)\,\frac{\pi}{\eta}\,\delta(E-\mathcal{E}_{m})$ $\displaystyle\hskip 56.9055pt(\mbox{in the limit}~{}\eta\rightarrow 0^{+})$ $\displaystyle=$ $\displaystyle-2i\pi\sum_{m}a_{i}^{m}a_{j}^{m\,}\,\delta(E-\mathcal{E}_{m}).$ Interchange of the indices $i$ and $j$ in Eq. 14 generates, $\mbox{\boldmath$G$}_{ji}^{r}-\mbox{\boldmath$G$}_{ji}^{a}=-2i\pi\sum_{m}a_{i}^{m\,}a_{j}^{m}\,\delta(E-\mathcal{E}_{m}).$ (15) It is clearly seen from Eqs. 14 and 15 that the non-zero contributions will only appear when the energy $E$ becomes equal to the discrete energy eigenvalues $\mathcal{E}_{m}$. This immediately allows us to express charge current density combining Eqs. 11, 14 and 15 as, $\displaystyle J_{c}(E)$ $\displaystyle=$ $\displaystyle-\frac{et}{Nh}\sum_{n}\left\\{\left(\mbox{\boldmath$G$}_{n+1,n}^{r}-\mbox{\boldmath$G$}_{n+1,n}^{a}\right)e^{-i\theta}\right.$ (16) $\displaystyle\left.-\left(\mbox{\boldmath$G$}_{n,n+1}^{r}-\mbox{\boldmath$G$}_{n,n+1}^{a}\right)e^{i\theta}\right\\}.$ This is the desired expression of charge current density in terms of Green’s functions when the ring is free from SO interactions. ## Appendix B Green’s function approach for persistent charge current density when the ring is subjected to Rashba and Dresselhaus SO interactions Next, we consider the ring with both Rashba and Dresselhaus SO interactions. The TB Hamiltonian of the ring presented in Eq. 1 can be expressed in a similar look of Eq. LABEL:ap1 like, $\displaystyle\mbox{\boldmath$H$}_{R}$ $\displaystyle=$ $\displaystyle\sum_{n}\mbox{\boldmath$c_{n}^{\dagger}\epsilon_{n}c_{n}$}+\sum_{n}\left(\mbox{\boldmath$c_{n+1}^{\dagger}t_{\varphi}^{n,n+1}c_{n}$}e^{i\theta}\right.$ (17) $\displaystyle\left.+\,\mbox{\boldmath$c_{n}^{\dagger}t_{\varphi}^{\dagger n,n+1}c_{n+1}$}e^{-i\theta}\right),$ where, different elements of the matrix $t_{\varphi}^{n,n+1}$ are: $\displaystyle\mbox{\boldmath$t_{\varphi}^{n,n+1}$}_{1,1}$ $\displaystyle=$ $\displaystyle t$ $\displaystyle\mbox{\boldmath$t_{\varphi}^{n,n+1}$}_{1,2}$ $\displaystyle=$ $\displaystyle-i\,\alpha\,e^{-i\varphi_{n,n+1}}+\beta\,e^{i\varphi_{n,n+1}}$ $\displaystyle\mbox{\boldmath$t_{\varphi}^{n,n+1}$}_{2,1}$ $\displaystyle=$ $\displaystyle-i\,\alpha\,e^{i\varphi_{n,n+1}}-\beta\,e^{-i\varphi_{n,n+1}}$ $\displaystyle\mbox{\boldmath$t_{\varphi}^{n,n+1}$}_{2,2}$ $\displaystyle=$ $\displaystyle t.$ This TB Hamiltonian leads to the charge current operator following the prescription given in Eq. 9 and considering $\mbox{\boldmath$x$}=\sum\limits_{n}\mbox{\boldmath$c_{n}^{\dagger}nc_{n}$}$ in the form: $J_{c}$ $\displaystyle=$ $\displaystyle\frac{2\pi ie}{N}\sum_{n}\left(\mbox{\boldmath$c_{n}^{\dagger}t_{\varphi}^{\dagger\,n,n+1}$}\mbox{\boldmath$c_{n+1}$}\,e^{-i\theta}\right.$ (18) $\displaystyle\left.-\,\mbox{\boldmath$c_{n+1}^{\dagger}t_{\varphi}^{n,n+1}$}\mbox{\boldmath$c_{n}$}\,e^{i\theta}\right).$ Hence, for a particular eigenstate $\|\mathcal{M}\rangle$ ($=\sum\limits_{p}a_{p,\uparrow}^{m}\|p\uparrow\rangle+a_{p,\downarrow}^{m}\|p\downarrow\rangle$) the charge current is written as, $\displaystyle J_{c}^{m}$ $\displaystyle=$ $\displaystyle\frac{2\pi ie}{N}\sum_{n}\left(\mbox{\boldmath$a_{n}^{m\,}\,t_{\varphi}^{\dagger\,n,n+1}$}\,\mbox{\boldmath$a_{n+1}^{m}$}\,e^{-i\theta}\right.$ (19) $\displaystyle\left.-\,\mbox{\boldmath$a_{n+1}^{m\,}\,t_{\varphi}^{n,n+1}$}\,\mbox{\boldmath$a_{n}^{m}$}\,e^{i\theta}\right),$ where, $a_{n}^{m}$=$\left(\begin{array}[]{c}a_{n\uparrow}^{m}\vspace{2mm}\\\ a_{n\downarrow}^{m}\end{array}\right)$ and $a_{n}^{m\,}$=$\left(\begin{array}[]{cc}a_{n\uparrow}^{m\,}&a_{n\downarrow}^{m\,}\end{array}\right)$. After simplification Eq. 19 yields, $\displaystyle J_{c}^{m}$ $\displaystyle=$ $\displaystyle\frac{2\pi ie}{N}\sum_{n}\left\\{t\,a_{n,\uparrow}^{m\,}a_{n+1,\uparrow}^{m}\,e^{-i\theta}-t\,a_{n+1,\uparrow}^{m\,}a_{n,\uparrow}^{m}\,e^{i\theta}\right\\}$ (20) $\displaystyle+$ $\displaystyle\frac{2\pi ie}{N}\sum_{n}\left\\{t\,a_{n,\downarrow}^{m\,}a_{n+1,\downarrow}^{m}\,e^{-i\theta}-t\,a_{n+1,\downarrow}^{m\,}a_{n,\downarrow}^{m}\,e^{i\theta}\right\\}$ $\displaystyle+$ $\displaystyle\frac{2\pi ie}{N}\sum_{n}\left\\{\left(i\alpha e^{-i\varphi_{n,n+1}}-\beta e^{i\varphi_{n,n+1}}\right)\right.$ $\displaystyle\left.\hskip 56.9055pt\times\,a_{n,\uparrow}^{m\,}a_{n+1,\downarrow}^{m}e^{-i\theta}\right.$ $\displaystyle+$ $\displaystyle\left.\left(i\alpha e^{i\varphi_{n,n+1}}+\beta e^{-i\varphi_{n,n+1}}\right)a_{n+1,\downarrow}^{m\,}a_{n,\uparrow}^{m}e^{i\theta}\right\\}$ $\displaystyle+$ $\displaystyle\frac{2\pi ie}{N}\sum_{n}\left\\{\left(i\alpha e^{i\varphi_{n,n+1}}+\beta e^{-i\varphi_{n,n+1}}\right)\right.$ $\displaystyle\left.\hskip 56.9055pt\times\,a_{n,\downarrow}^{m\,}a_{n+1,\uparrow}^{m}e^{-i\theta}\right.$ $\displaystyle+$ $\displaystyle\left.\left(i\alpha e^{-i\varphi_{n,n+1}}-\beta e^{i\varphi_{n,n+1}}\right)a_{n+1,\uparrow}^{m\,}a_{n,\downarrow}^{m}e^{i\theta}\right\\}.$ With this explicit expression (Eq. 20) and following the above prescription described in Appendix A, we eventually get the final result Eq. 3 for persistent charge current density in presence of SO fields. ## Appendix C Green’s function approach for polarized spin current density when the ring is subjected to Rashba and Dresselhaus SO interactions Finally, we derive the expression of persistent spin current density in presence of SO fields. To do this we begin with the spin current operator, $\mbox{\boldmath$J_{s}$}=\frac{1}{2N}\left(\mbox{\boldmath${\sigma}$}\mbox{\boldmath${\dot{x}}$}+\mbox{\boldmath${\dot{x}}$}\mbox{\boldmath${\sigma}$}\right),$ (21) where, $\mbox{\boldmath${\sigma}$}=\\{\mbox{\boldmath${\sigma_{x}}$},\mbox{\boldmath${\sigma_{y}}$},\mbox{\boldmath${\sigma_{z}}$}\\}$. Along the spin quantized direction ($+Z$) this equation (Eq. 21) reduces to, $\mbox{\boldmath$J_{s}^{z}$}=\frac{1}{2N}\left(\mbox{\boldmath${\sigma_{z}}$}\mbox{\boldmath${\dot{x}}$}+\mbox{\boldmath${\dot{x}}$}\mbox{\boldmath${\sigma_{z}}$}\right).$ (22) Now, using the TB form given in Eq. 17 and following the same prescription of Eq. LABEL:ap1 we can express Eq. 22 doing a straightforward and somewhat lengthy algebra as, $J_{s}^{z}$ $\displaystyle=$ $\displaystyle\frac{i\pi}{N}\sum_{n}\left(\mbox{\boldmath$c_{n}^{\dagger}\sigma_{z}t_{\varphi}^{\dagger\,n,n+1}$}\mbox{\boldmath$c_{n+1}$}\,e^{-i\theta}\right.$ (23) $\displaystyle-$ $\displaystyle\left.\mbox{\boldmath$c_{n+1}^{\dagger}\sigma_{z}t_{\varphi}^{n,n+1}$}\mbox{\boldmath$c_{n}$}\,e^{i\theta}\right)$ $\displaystyle+$ $\displaystyle\frac{i\pi}{N}\sum_{n}\left(\mbox{\boldmath$c_{n}^{\dagger}t_{\varphi}^{\dagger\,n,n+1}\sigma_{z}$}\mbox{\boldmath$c_{n+1}$}\,e^{-i\theta}\right.$ $\displaystyle-$ $\displaystyle\left.\mbox{\boldmath$c_{n+1}^{\dagger}t_{\varphi}^{n,n+1}\sigma_{z}$}\mbox{\boldmath$c_{n}$}\,e^{i\theta}\right).$ This spin current operator under the operation $\langle\mathcal{M}\|\mbox{\boldmath$J_{s}^{z}$}\|\mathcal{M}\rangle$ yields persistent spin current for a particular eigenstate $\|\mathcal{M}\rangle$, $\displaystyle J_{s}^{z,m}$ $\displaystyle=$ $\displaystyle\frac{2\pi it}{N}\sum_{n}\left\\{a_{n,\uparrow}^{m\,}a_{n+1,\uparrow}^{m}\,e^{-i\theta}-a_{n+1,\uparrow}^{m\,}a_{n,\uparrow}^{m}\,e^{i\theta}\right\\}$ $\displaystyle-$ $\displaystyle\frac{2\pi it}{N}\sum_{n}\left\\{a_{n,\downarrow}^{m\,}a_{n+1,\downarrow}^{m}\,e^{-i\theta}-a_{n+1,\downarrow}^{m\,}a_{n,\downarrow}^{m}\,e^{i\theta}\right\\}.$ This relation leads to the final result Eq. 5 following the approach given in Appendix A. ## References * (1) S. 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1401.0387	###### Abstract We review a selection of methods for performing enhanced sampling in molecular dynamics simulations. We consider methods based on collective variable biasing and on tempering, and offer both historical and contemporary perspectives. In collective-variable biasing, we first discuss methods stemming from thermodynamic integration that use mean force biasing, including the adaptive biasing force algorithm and temperature acceleration. We then turn to methods that use bias potentials, including umbrella sampling and metadynamics. We next consider parallel tempering and replica-exchange methods. We conclude with a brief presentation of some combination methods. ###### keywords: collective variables, free energy, blue-moon sampling, adaptive-biasing force algorithm, temperature-acceleration, umbrella sampling, metadynamics 10.3390/—— xx Received: xx / Accepted: xx / Published: xx Enhanced sampling in molecular dynamics using metadynamics, replica-exchange, and temperature- acceleration Cameron Abrams 1,⋆ and Giovanni Bussi 2 [email protected]; tel +1-215-895-2231 ## 1 Introduction The purpose of molecular dynamics (MD) is to compute the positions and velocities of a set of interacting atoms at the present time instant given these quantities one time increment in the past. Uniform sampling from the discrete trajectories one can generate using MD has long been seen as synonymous with sampling from a statistical-mechanical ensemble; this just expresses our collective wish that the ergodic hypothesis holds at finite times. Unfortunately, most MD trajectories are not ergodic and leave many relevant regions of configuration space unexplored. This stems from the separation of high-probability “metastable” regions by low-probability “transition” regions and the inherent difficulty of sampling a 3$N$-dimensional space by embedding into it a one-dimensional dynamical trajectory. This review concerns a selection of methods to use MD simulation to enhance the sampling of configuration space. A central concern with any enhanced sampling method is guaranteeing that the statistical weights of the samples generated are known and correct (or at least correctable) while simultaneously ensuring that as much of the relevant regions of configuration space are sampled. Because of the tight relationship between probability and free energy, many of these methods are known as “free-energy” methods. To be sure, there are a large number of excellent reviews of free-energy methods in the literature (e.g., Kollman (1993); Trzesniak et al. (2007); Vanden-Eijnden (2009); Dellago and Bolhuis (2009); Christ et al. (2010)). The present review is in no way intended to be as comprehensive as these; as the title indicates, we will mostly focus on enhanced sampling methods of three flavors: tempering, metadynamics, and temperature-acceleration. Along the way, we will point out important related methods, but in the interest of brevity we will not spend much time explaining these. The methods we have chosen to focus on reflect our own preferences to some extent, but they also represent popular and growing classes of methods that find ever more use in biomolecular simulations and beyond. We divide our review into three main sections. In the first, we discuss enhanced sampling approaches that rely on collective variable biasing. These include the historically important methods of thermodynamic integration and umbrella sampling, and we pay particular attention to the more recent approaches of the adaptive-biasing force algorithm, temperature-acceleration, and metadynamics. In the second section, we discuss approaches based on tempering, which is dominated by a discussion of the parallel tempering/replica exchange approaches. In the third section, we briefly present some relatively new methods derived from either collective-variable- based or tempering-based approaches, or their combinations. ## 2 Approaches Based on Collective-Variable Biasing ### 2.1 Background: Collective Variables and Free Energy For our purposes, the term “collective variable” or CV refers to any multidimensional function ${\boldsymbol{\theta}}$ of 3$N$-dimensional atomic configuration ${\boldsymbol{x}}\equiv\left(x_{i}\|i=1\dots 3N\right)$. The functions $\theta_{1}({\boldsymbol{x}})$, $\theta_{2}({\boldsymbol{x}})$,$\dots$,$\theta_{M}({\boldsymbol{x}})$ map configuration ${\boldsymbol{x}}$ onto an $M$-dimensional CV space ${\boldsymbol{z}}\equiv\left(z_{j}\|j=1\dots M\right)$, where usually $M\ll 3N$. At equilibrium, the probability of observing the system at CV-point ${\boldsymbol{z}}$ is the weight of all configurations ${\boldsymbol{x}}$ which map to ${\boldsymbol{z}}$: $P({\boldsymbol{z}})=\left<\delta[{\boldsymbol{\theta}}({\boldsymbol{x}})-{\boldsymbol{z}}]\right>,$ (1) The Dirac delta function picks out only those configurations for which the CV ${\boldsymbol{\theta}}({\boldsymbol{x}})$ is ${\boldsymbol{z}}$, and $\left<\cdot\right>$ denotes averaging its argument over the equilibrium probability distribution of ${\boldsymbol{x}}$. The probability can be expressed as a free energy: $F({\boldsymbol{z}})=-k_{B}T\ln\left<\delta[{\boldsymbol{\theta}}({\boldsymbol{x}})-{\boldsymbol{z}}]\right>.$ (2) Here, $k_{B}$ is Boltzmann’s constant and $T$ is temperature. Local minima in $F$ are metastable equilibrium states. $F$ also measures the energetic cost of a maximally efficient (i.e., reversible) transition from one region of CV space to another. If, for example, we choose a CV space such that two well-separated regions define two important allosteric states of a given protein, we could perform a free-energy calculation to estimate the change in free energy required to realize the conformational transition. Indeed, the promise of being able to observe with atomic detail the transition states along some pathway connecting two distinct states of a biomacromolecule is strong motivation for exploring these transitions with CV’s. Given the limitations of standard MD, how does one “discover” such states in a proposed CV space? A perfectly ergodic (infinitely long) MD trajectory would visit these minima much more frequently than it would the intervening spaces, allowing one to tally how often each point in CV space is visited; normalizing this histogram into a probability $P({\boldsymbol{z}})$ would be the most straightforward way to compute $F$ via Eq. 2. In all too many actual cases, MD trajectories remain close to only one minimum (the one closest to the initial state of the simulation) and only very rarely, if ever, visit others. In the CV sense, we therefore speak of standard MD simulations failing to overcome barriers in free energy. “Enhanced sampling” in this context refers then to methods by which free-energy barriers in a chosen CV space are surmounted to allow as broad as possible an extent of CV space to be explored and statistically characterized with limited computational resources. In this section, we focus on methods of enhanced sampling of CV’s based on MD simulations that are directly biased on those CV’s; that is, we focus on methods in which an investigator must identify the CV’s of interest as an input to the calculation. We have chosen to limit discussion to two broad classes of biasing: those whose objective is direct computation of the gradient of the free energy $(\partial F/\partial{\boldsymbol{z}})$ at local points throughout CV space, and those in which non-Boltzmann sampling with bias potentials is used to force exploration of otherwise hard-to-visit regions of CV space. The canonical methods in these two classes are thermodynamic integration and umbrella sampling, respectively, and a discussion of these two methods sets the stage for discussion of three relatively modern variants: the Adaptive-Biasing Force Algorithm Darve et al. (2008), Temperature-Accelerated MD Maragliano and Vanden-Eijnden (2006) and Metadynamics Laio and Parrinello (2002). ### 2.2 Gradient Methods: Blue-Moon Sampling, Adaptive-Biasing Force Algorithm, and Temperature-Accelerated MD #### 2.2.1 Overview: Thermodynamic Integration Naively, one way to have an MD system visit a hard-to-reach point ${\boldsymbol{z}}$ in CV space is simply to create a realization of the configuration ${\boldsymbol{x}}$ at that point (i.e., such that ${\boldsymbol{\theta}}({\boldsymbol{x}})={\boldsymbol{z}}$). This is an inverse problem, since the number of degrees of freedom in ${\boldsymbol{x}}$ is usually much larger than in ${\boldsymbol{z}}$. One way to perform this inversion is by introducing external forces that guide the configuration to the desired point from some easy-to-create initial state; both targeted MD Schlitter et al. (1993) and steered MD Grubmüller et al. (1996) are ways to do this. Of course, one would like MD to explore CV space in the vicinity of ${\boldsymbol{z}}$, so after creating the configuration ${\boldsymbol{x}}$, one would just let it run. Unfortunately, this would likely result in the system drifting away from ${\boldsymbol{z}}$ rather quickly, and there would be no way from such calculations to estimate the likelihood of observing an unbiased long MD simulation visit ${\boldsymbol{z}}$. But there is information in the fact that the system drifts away; if one knows on average which direction and how strongly the system would like to move if initialized at ${\boldsymbol{z}}$, this would be a measure of negative gradient of the free energy, $-(\partial F/\partial{\boldsymbol{z}})$, or the “mean force”. We have then a glimpse of a three-step method to compute $F$ (i.e., the statistics of CV’s) over a meaningfully broad extent of CV space: 1. 1. visit a select number of local points in that space, and at each one, 2. 2. compute the mean force, then 3. 3. use numerical integration to reconstruct $F$ from these local mean forces; formally expressed as $F({\boldsymbol{z}})-F({\boldsymbol{z}}_{0})=\int_{{\boldsymbol{z}}_{0}}^{{\boldsymbol{z}}}\left(\frac{\partial F}{\partial{\boldsymbol{z}}}\right)d{\boldsymbol{z}}$ (3) Inspired by Kirkwood’s original suggestion involving switching parameters Kirkwood (1935), such an approach is generally referred to as “thermodynamic integration” or TI. TI allows us to reconstruct the statistical weights of any point in CV space by accumulating information on the gradients of free energy at selected points. #### 2.2.2 Blue-Moon Sampling The discussion so far leaves open the correct way to compute the local free- energy gradients. A gradient is a local quantity, so a natural choice is to compute it from an MD simulation localized at a point in CV space by a constraint. Consider a long MD simulation with a holonomic constraint fixing the system at the point ${\boldsymbol{z}}$. Uniform samples from this constrained trajectory ${\boldsymbol{x}}(t)$ then represent an ensemble at fixed ${\boldsymbol{z}}$ over which the averaging needed to convert gradients in potential energy to gradients in free energy could be done. However, this constrained ensemble has the undesired property that the velocities $\dot{\boldsymbol{\theta}}({\boldsymbol{x}})$ are zero. This is a bit problematic because virtually none of the samples plucked from a long unconstrained MD simulation (as is implied by Eq. 1), would have $\dot{\boldsymbol{\theta}}=0$, and $\dot{\boldsymbol{\theta}}=0$ acts as a set of $M$ unphysical constraints on the system velocities $\dot{\boldsymbol{x}}$, since $\dot{\theta}_{j}=\sum_{i}(\partial\theta_{j}/\partial x_{i})\dot{x}_{i}$. Probably the best-known example of a method to correct for this bias is the so-called “blue-moon” sampling method Carter et al. (1989); Sprik and Ciccotti (1998); Ciccotti and Ferrario (2004); Ciccotti et al. (2005) or the constrained ensemble method den Otter and Briels (1998); Schlitter and Klähn (2003). The essence of the method is a decomposition of free energy gradients into components along the CV gradients and thermal components orthogonal to them: $\frac{\partial F}{\partial z_{j}}=\left<{\boldsymbol{b}}_{j}({\boldsymbol{x}})\cdot\nabla V({\boldsymbol{x}})-k_{B}T\nabla\cdot{\boldsymbol{b}}_{j}({\boldsymbol{x}})\right>_{{\boldsymbol{\theta}}({\boldsymbol{x}})={\boldsymbol{z}}}$ (4) where $\left<\cdot\right>_{{\boldsymbol{\theta}}({\boldsymbol{x}})={\boldsymbol{z}}}$ denotes averaging across samples drawn uniformly from the MD simulation constrained at ${{\boldsymbol{\theta}}({\boldsymbol{x}})={\boldsymbol{z}}}$, and the ${\boldsymbol{b}}_{j}({\boldsymbol{x}})$ is the vector field orthogonal to the gradients of every component $k$ of ${\boldsymbol{\theta}}$ for $k\neq j$: ${\boldsymbol{b}}_{j}({\boldsymbol{x}})\cdot\nabla\theta_{k}({\boldsymbol{x}})=\delta_{jk}$ (5) where $\delta_{jk}$ is the Kroenecker delta. (For brevity, we have omitted the consideration of holonomic constraints other than that on the CV; the reader is referred to the paper by Ciccotti et al. for details Ciccotti et al. (2005).) The vector fields ${\boldsymbol{b}}_{j}$ for each $\theta_{j}$ can be constructed by orthogonalization. The first term in the angle brackets in Eq. 4 implements the chain rule one needs to account for how energy $V$ changes with ${\boldsymbol{z}}$ through all the ways ${\boldsymbol{z}}$ can change with ${\boldsymbol{x}}$. The second term corrects for the thermal bias imposed by the constraint. Although nowhere near exhaustive, below is a listing of common types of problems to which blue-moon sampling has been applied with some representative examples: 1. 1. sampling conformations of small flexible molecules and peptides Depaepe et al. (1993); Zhao et al. (2008); Kim et al. (2009) 2. 2. environmental effects on covalent bond formation/breaking (usually in combination with ab initio MD) Hytha et al. (2001); Fois et al. (2004); Ivanov and Klein (2005); Stubbs and Marx (2005); Trinh et al. (2009); Liu et al. (2010); Bucko and Hafner (2010) 3. 3. solvation and non-covalent binding of small molecules in solvent Paci and Marchi (1994); Sa et al. (2006); Mugnai et al. (2007); Chunsrivirot and Trout (2011); Sato et al. (2012) 4. 4. protein dimerization Sergi et al. (2002); Maragliano et al. (2004) #### 2.2.3 The Adaptive Biasing Force Algorithm The blue-moon approach requires multiple independent constrained MD simulations to cover the region of CV space in which one wants internal statistics. The care taken in choosing these quadrature points can often dictate the accuracy of the resulting free energy reconstruction. It is therefore sometimes advantageous to consider ways to avoid having to choose such points ahead of time, and adaptive methods attempt to address this problem. One example is the adaptive-biasing force (ABF) algorithm of Darve et al. Darve and Pohorille (2001); Darve et al. (2008) The essence of ABF is two- fold: (1) recognition that external bias forces of the form $\nabla_{\boldsymbol{x}}\theta_{j}\left(\partial F/\partial z_{j}\right)$ for $j=1\dots M$ exactly oppose mean forces and should lead to more uniform sampling of CV space, and (2) that these bias forces can be converged upon adaptively during a single unconstrained MD simulation. The first of those two ideas is motivated by the fact that “forces” that keep normal MD simulations effectively confined to free energy minima are mean forces on the collective variables projected onto the atomic coordinates, and balancing those forces against their exact opposite should allow for thermal motion to take the system out of those minima. The second idea is a bit more subtle; after all, in a running MD simulation with no CV constraints, the constrained ensemble expression for the mean force (Eq. 4) does not directly apply, because a constrained ensemble is not what is being sampled. However, Darve et al. showed how to relate these ensembles so that the samples generated in the MD simulation could be used to build mean forces Darve and Pohorille (2001). Further, they showed using a clever choice of the fields of Eq. 4 an equivalence between ($i$) the spatial gradients needed to computed forces, and ($ii$) time-derivatives of the CV’s Darve et al. (2008): $\frac{\partial F}{\partial z_{i}}=-k_{B}T\left<\frac{d}{dt}\left(M_{\theta}\frac{d\theta_{i}}{dt}\right)\right>_{{\boldsymbol{\theta}}={\boldsymbol{z}}}$ (6) where $M_{\theta}$ is the transformed mass matrix given by $M_{\theta}^{-1}=J_{\theta}M^{-1}J_{\theta}$ (7) where $J_{\theta}$ is the $M\times 3N$ matrix with elements $\partial\theta_{i}/\partial x_{j}$ ($i=1\dots M$, $j=1\dots 3N$), and $M$ is the diagonal matrix of atomic masses. Eq. 7 is the result of a particular choice for the fields ${\boldsymbol{b}}_{j}({\boldsymbol{x}})$. This reformulation of the instantaneous mean forces computed on-the-fly makes ABF exceptionally easy to implement in most modern MD packages. Darve et al. present a clear demonstration of the ABF algorithm in a pseudocode Darve et al. (2008) that attests to this fact. ABF has found rather wide application in CV-based free energy calculations in recent years. Below is a representative sample of some types of problems subjected to ABF calculations in the recent literature: 1. 1. Peptide backbone angle sampling Fogolari et al. (2011); Faller et al. (2013); 2. 2. Nucleoside Wei and Pohorile (2011), protein Vivcharuk and Kaznessis (2011) and fullerene Kraszewski et al. (2011, 2012) insertion into a lipid bilayer; 3. 3. Interactions of small molecules with polymers in water Liu et al. (2010); Caballero et al. (2013); 4. 4. Molecule/ion transport through protein complexes Wilson et al. (2011); Cheng and Coalson (2012); Wang et al. (2012); Tillman et al. (2013) and DNA superstructures Akhshi et al. (2012); 5. 5. Calculation of octanol-water partition coefficients Kamath et al. (2012); Bhatnagar et al. (2012); 6. 6. Large-scale protein conformational changes Wereszczynski and McCammon (2012); 7. 7. Protein-nanotube Jana and Sengupta (2012) and nanotube-nanotube Uddin et al. (2010) association. #### 2.2.4 Temperature-Accelerated Molecular Dynamics Both blue-moon sampling and ABF are based on statistics in the constrained ensemble. However, estimation of mean forces need not only use this ensemble. One can instead relax the constraint and work with a “mollified” version of the free energy: $F_{\kappa}({\boldsymbol{z}})=-k_{B}T\ln\left<\delta_{\kappa}\left[{\boldsymbol{\theta}}({\boldsymbol{x}})-{\boldsymbol{z}}\right]\right>$ (8) where $\delta_{\kappa}$ refers to the Gaussian (or “mollified delta function”): $\delta_{\kappa}=\sqrt{\frac{\beta\kappa}{2\pi}}\exp\left[-\frac{1}{2}\beta\kappa\left\|{\boldsymbol{\theta}}({\boldsymbol{x}})-{\boldsymbol{z}}\right\|^{2}\right],$ (9) where $\beta$ is just shorthand for $1/k_{B}T$. Since $\lim_{\beta\kappa\rightarrow\infty}\delta_{\kappa}=\delta$, we know that $\lim_{\beta\kappa\rightarrow\infty}F_{\kappa}=F$. One way to view this Gaussian is that it “smoothes out” the true free energy to a tunable degree; the factor $1/\sqrt{\beta\kappa}$ is a length-scale in CV space below which details are smeared. Because the Gaussian has continuous gradients, it can be used directly in an MD simulation. Suppose we have a CV space ${\boldsymbol{\theta}}({\boldsymbol{x}})$, and we extend our MD system to include variables ${\boldsymbol{z}}$ such that the combined set $({\boldsymbol{x}},{\boldsymbol{z}})$ obeys the following extended potential: $U({\boldsymbol{x}},{\boldsymbol{z}})=V({\boldsymbol{x}})+\sum_{j=1}^{M}\frac{1}{2}\kappa\left\|\theta_{j}({\boldsymbol{x}})-z_{j}\right\|^{2}$ (10) where $V({\boldsymbol{x}})$ is the interatomic potential, and $\kappa$ is a constant. Clearly, if we fix ${\boldsymbol{z}}$, then the resulting free energy is to within an additive constant the mollified free energy of Eq. 8. (The additive constant is related to the prefactor of the mollified delta function and has nothing to do with the number of CV’s.) Further, we can directly express the gradient of this mollified free energy with respect to ${\boldsymbol{z}}$: Kaestner (2009) $\nabla_{\boldsymbol{z}}F_{\kappa}=-\left<\kappa\left[{\boldsymbol{\theta}}({\boldsymbol{x}})-{\boldsymbol{z}}\right]\right>$ (11) This suggests that, instead of using constrained ensemble MD to accumulate mean forces, we could work in the restrained ensemble and get very good approximations to the mean force. By “restrained”, we refer to the fact that the term giving rise to the mollified delta function in the configurational integral is essentially a harmonic restraining potential with a “spring constant” $\kappa$. In this restrained-ensemble approach, no velocities are held fixed, and the larger we choose $\kappa$ the more closely we can approximate the true free energy. Notice however that large values of $\kappa$ could lead to numerical instabilities in integrating equations of motion, and a balance should be found. (In practice, we have found that for CV’s with dimensions of length, values of $\kappa$ less than about 1,000 kcal/mol/Å2 can be stably handled, and values of around 100 kcal/mol/Å2 are typically adequate.) Temperature-accelerated MD (TAMD) Maragliano and Vanden-Eijnden (2006) takes advantage of the restrained-ensemble approach to directly evolve the variables ${\boldsymbol{z}}$ in such a way to accelerate the sampling of CV space. First, consider how the atomic variables ${\boldsymbol{x}}$ evolve under the extended potential (assuming Langevin dynamics): $m_{i}\ddot{x}_{i}=-\frac{\partial V({\boldsymbol{x}})}{\partial x_{i}}-\kappa\sum_{j=1}^{m}\left[\theta_{j}({\boldsymbol{x}})-z_{j}\right]\frac{\partial\theta_{j}({\boldsymbol{x}})}{\partial x_{i}}-\gamma m_{i}\dot{x_{i}}+\eta_{i}(t;\beta)$ (12) Here, $m_{i}$ is the mass of $x_{i}$, $\gamma$ is the friction coefficient for the Langevin thermostat, and ${\boldsymbol{\eta}}$ is the thermostat white noise satisfying the fluctuation-dissipation theorem at physical temperature $\beta^{-1}$: $\left<\eta_{i}(t;\beta)\eta_{j}(t^{\prime};\beta)\right>=\beta^{-1}\gamma m_{i}\delta_{ij}\delta(t-t^{\prime})$ (13) Key to TAMD is that the ${\boldsymbol{z}}$ are treated as slow variables that evolve according to their own equations of motion, which here we take as diffusive (though other choices are possible Maragliano and Vanden-Eijnden (2006)): $\bar{\gamma}\bar{m}_{j}\dot{z}_{j}=\kappa\left[\theta_{j}({\boldsymbol{x}})-z_{j}\right]+\xi_{j}(t;\bar{\beta}).$ (14) Here, $\bar{\gamma}$ is a fictitious friction, $\bar{m}_{j}$ is a mass, and the first term on the right-hand side represents the instantaneous force on variable $z_{j}$, and the second term represents thermal noise at the fictitious thermal energy $\bar{\beta}^{-1}\not=\beta^{-1}$. The advantage of TAMD is that if (1) $\bar{\gamma}$ is chosen sufficiently large so as to guarantee that the slow variables indeed evolve slowly relative to the fundamental variables, and (2) $\kappa$ is sufficiently large such that ${\boldsymbol{\theta}}({\boldsymbol{x}}(t))\approx{\boldsymbol{z}}(t)$ at any given time, then the force acting on ${\boldsymbol{z}}$ is approximately equal to minus the gradient of the free energy (Eq. 11) Maragliano and Vanden- Eijnden (2006). This is because the MD integration repeatedly samples $\kappa\left[{\boldsymbol{\theta}}({\boldsymbol{x}})-{\boldsymbol{z}}\right]$ for an essentially fixed (but actually very slowly moving) ${\boldsymbol{z}}$, so ${\boldsymbol{z}}$ evolution effectively feels these samples as a mean force. In other words, the dynamics of ${\boldsymbol{z}}(t)$ is effectively $\bar{\gamma}\bar{m}_{j}\dot{z}_{j}=-\frac{\partial F({\boldsymbol{z}})}{\partial z_{j}}+\xi_{j}(t;\bar{\beta}).$ (15) This shows that the ${\boldsymbol{z}}$-dynamics describes an equilibrium constant-temperature ensemble at fictitious temperature $\bar{\beta}^{-1}$ acted on by the “potential” $F({\boldsymbol{z}})$, which is the free energy evaluated at the physical temperature $\beta^{-1}$. That is, under TAMD, ${\boldsymbol{z}}$ conforms to a probability distribution of the form $\exp\left[-\bar{\beta}F({\boldsymbol{z}};\beta)\right]$, whereas under normal MD it would conform to $\exp\left[-\beta F({\boldsymbol{z}};\beta)\right]$. The all-atom MD simulation (at $\beta$) simply serves to approximate the local gradients of $F({\boldsymbol{z}})$. Sampling is enhanced by taking $\bar{\beta}^{-1}>\beta^{-1}$, which has the effect of attenuating the ruggedness of $F$. TAMD therefore can accelerate a trajectory ${\boldsymbol{z}}(t)$ through CV space by increasing the likelihood of visiting points with relatively low physical Boltzmann factors. This borrows directly from the main idea of adiabatic free-energy dynamics Rosso and Tuckerman (2002) (AFED), in that one deliberately makes some variables hot (to overcome barriers) but slow (to keep them adiabatically separated from all other variables). In TAMD, however, the use of the mollified free energy means no cumbersome variable transformations are required. (The authors of AFED refer to TAMD as “driven”-AFED, or d-AFED Abrams and Tuckerman (2008).) It is also worth mentioning in this review that TAMD borrows heavily from an early version of metadynamics Iannuzzi et al. (2003), which was formulated as a way to evolve the auxiliary variables ${\boldsymbol{z}}$ on a mollified free energy. However, unlike metadynamics (which we discuss below in Sec. 2.3.3), there is no history-dependent bias in TAMD. Unlike TI, ABF, and the methods of umbrella sampling and metadynamics discussed in the next section, TAMD is not a method for direct calculation of the free energy. Rather, it is a way to overcome free energy barriers in a chosen CV space quickly without visiting irrelevant regions of CV space. (However, we discuss briefly a method in Sec. 4.2.2 in which TAMD gradients are used in a spirit similar to ABF to reconstruct a free energy.) That is, we consider TAMD a way to efficiently explore relevant regions CV space that are practically inaccessible to standard MD simulation. It is also worth pointing out that, unlike ABF, TAMD does not operate by opposing the natural gradients in free energy, but rather by using them to guide accelerated sampling. ABF can only use forces in locations in CV space the trajectory has visited, which means nothing opposes the trajectory going to regions of very high free energy. However, under TAMD, an acceleration of $\bar{\beta}^{-1}$= 6 kcal/mol on the CV’s will greatly accelerate transitions over barriers of 6-12 kcal/mol, but will still not (in theory) accelerate excursions to regions requiring climbs of hundreds of kcal/mol. TAMD and ABF have in common the ability to handle rather high-dimensional CV’s. Although it was presented theoretically in 2006 Maragliano and Vanden-Eijnden (2006), TAMD was not applied directly to large-scale MD until much later Abrams and Vanden-Eijnden (2010). Since then, there has been growing interest in using TAMD in a variety of applications requiring enhanced sampling: 1. 1. TAMD-enhanced flexible fitting of all-atom protein and RNA models into low- resolution electron microscopy density maps Vashisth et al. (2012, 2013); 2. 2. Large-scale (interdomain) protein conformational sampling Abrams and Vanden- Eijnden (2010); Vashisth and Brooks (2012); Hu et al. (2012); 3. 3. Loop conformational sampling in proteins Vashisth and Abrams (2012); 4. 4. Mapping of diffusion pathways for small molecules in globular proteins Maragliano et al. (2010); Lapelosa and Abrams (2013); 5. 5. Vacancy diffusion Geslin et al. (2013); 6. 6. Conformational sampling and packing in dense polymer systems Lucid et al. (2013). Finally, we mention briefly that TAMD can be used as a quick way to generate trajectories from which samples can be drawn for subsequent mean-force estimation for later reconstruction of a multidimensional free energy; this is the essence of the single-sweep method Maragliano and Vanden-Eijnden (2008), which is an efficient means of computing multidimensional free energies. Rather than using straight numerical TI, single sweep posits the free energy as a basis function expansion and uses standard optimization methods to find the expansion coefficients that best reproduce the measured mean forces. Single-sweep has been used to map diffusion pathways of CO and H2O in myoglobin Maragliano et al. (2010); Lapelosa and Abrams (2013). ### 2.3 Bias Potential Methods: Umbrella Sampling and Metadynamics #### 2.3.1 Overview: Non-Boltzmann Sampling In the previous section, we considered methods that achieve enhanced sampling by using mean forces: in TI, these are integrated to reconstruct a free energy; in ABF, these are built on-the-fly to drive uniform CV sampling; and in TAMD, these are used on-the-fly to guide accelerated evolution of CV’s. In this section, we consider methods that achieve enhanced sampling by means of controlled bias potentials. As a class, we refer to these as non-Boltzmann sampling methods. Non-Boltzmann sampling is generally a way to derive statistics on a system whose energetics differ from the energetics used to perform the sampling. Imagine we have an MD system with bare interatomic potential $V({\boldsymbol{x}})$, and we add a bias $\Delta V({\boldsymbol{x}})$ to arrive at a biased total potential: $V_{b}({\boldsymbol{x}})=V({\boldsymbol{x}})+\Delta V({\boldsymbol{x}})$ (16) The statistics on the CV’s on this biased potential are then given as $\displaystyle P_{b}({\boldsymbol{z}})$ $\displaystyle=\frac{\displaystyle\int\\!d{\boldsymbol{x}}\ e^{-\beta V_{0}({\boldsymbol{x}})}e^{-\beta\Delta V({\boldsymbol{x}})}\delta\left[{\boldsymbol{\theta}}({\boldsymbol{x}})-{\boldsymbol{z}}\right]}{\displaystyle\int\\!d{\boldsymbol{x}}\ e^{-\beta V_{0}({\boldsymbol{x}})}e^{-\beta\Delta V({\boldsymbol{x}})}}$ $\displaystyle=\frac{\displaystyle\displaystyle\int\\!d{\boldsymbol{x}}\ e^{-\beta V_{0}({\boldsymbol{x}})}e^{-\beta\Delta V}\delta\left[{\boldsymbol{\theta}}({\boldsymbol{x}})-{\boldsymbol{z}}\right]}{\displaystyle\int\\!d{\boldsymbol{x}}\ e^{-\beta V_{0}({\boldsymbol{x}})}}\frac{\int\\!d{\boldsymbol{x}}\ e^{-\beta V_{0}({\boldsymbol{x}})}}{\displaystyle\int\\!d{\boldsymbol{x}}\ e^{-\beta V_{0}({\boldsymbol{x}})}e^{-\beta\Delta V({\boldsymbol{x}})}}$ $\displaystyle=\frac{\left<e^{-\beta\Delta V({\boldsymbol{x}})}\delta\left[{\boldsymbol{\theta}}({\boldsymbol{x}})-{\boldsymbol{z}}\right]\right>}{\left<e^{-\beta\Delta V({\boldsymbol{x}})}\right>}$ (17) where $\left<\cdot\right>$ denotes ensemble averaging on the unbiased potential $V({\boldsymbol{x}})$. Further, if we take the bias potential $\Delta V$ to be explicitly a function only of the CV’s ${\boldsymbol{\theta}}$, then it becomes invariant in the averaging of the numerator thanks to the delta function, and we have $P_{b}({\boldsymbol{x}})=\frac{e^{-\beta\Delta V({\boldsymbol{z}})}\left<\delta\left[{\boldsymbol{\theta}}({\boldsymbol{x}})-{\boldsymbol{z}}\right]\right>}{\left<e^{-\beta\Delta V\left[\theta({\boldsymbol{x}})\right]}\right>}$ (18) Finally, since the unbiased statistics are $P({\boldsymbol{z}})=\left<\delta\left[{\boldsymbol{\theta}}({\boldsymbol{x}})-{\boldsymbol{z}}\right]\right>$, we arrive at $P({\boldsymbol{z}})=P_{b}({\boldsymbol{z}})e^{\beta\Delta V({\boldsymbol{z}})}\left<e^{-\beta\Delta V\left[\theta({\boldsymbol{x}})\right]}\right>$ (19) Taking samples from an ergodic MD simulation on the biased potential $V_{b}$, Eq. 19 provides the recipe for reconstructing the statistics the CV’s would present were they generated using the unbiased potential $V$. However, the probability $P({\boldsymbol{z}})$ is implicit in this equation, because $\left<e^{-\beta\Delta V}\right>=\int d{\boldsymbol{z}}P({\boldsymbol{z}})e^{-\beta\Delta V\left[{\boldsymbol{\theta}}({\boldsymbol{x}})\right]}$ (20) This is not really a problem, since we can treat $\left<e^{-\beta\Delta V}\right>$ as a constant we can get from normalizing $P_{b}({\boldsymbol{z}})e^{\beta\Delta V({\boldsymbol{z}})}$. How does one choose $\Delta V$ so as to enhance the sampling of CV space? Evidently, from the standpoint of non-Boltzmann sampling, the closer the bias potential is to the negative free energy $-F({\boldsymbol{z}})$, the more uniform the sampling of CV space will be. To wit: if $\Delta V\left[{\boldsymbol{\theta}}({\boldsymbol{x}})\right]=-F\left[{\boldsymbol{\theta}}({\boldsymbol{x}})\right]$, then $e^{\beta\Delta V({\boldsymbol{z}})}=e^{-\beta F({\boldsymbol{z}})}=P({\boldsymbol{z}})$, and Eq. 19 can be inverted for $P_{b}$ to yield $P_{b}({\boldsymbol{z}})=\frac{1}{\left<e^{\beta F({\boldsymbol{z}})}\right>}=\frac{1}{\displaystyle\int d{\boldsymbol{z}}P({\boldsymbol{z}})e^{\beta F({\boldsymbol{z}})}}=\frac{1}{\displaystyle\int d{\boldsymbol{z}}e^{-\beta F}e^{\beta F}}=\frac{1}{\displaystyle\int d{\boldsymbol{z}}}$ (21) So we see that taking the bias potential to be the negative free energy makes all states ${\boldsymbol{z}}$ in CV space equiprobable. This is indeed the limit to which ABF strives by applying negative mean forces, for example Darve et al. (2008). We usually do not know the free energy ahead of time; if we did, we would already know the statistics of CV space and no enhanced sampling would be necessary. Moreover, perfectly uniform sampling of the entire CV space is usually far from necessary, since most CV spaces have many irrelevant regions that should be ignored. And in reference to the mean-force methods of the last section, uniform sampling is likely not necessary to achieve accurate mean force values; how good an estimate of $\nabla F$ is at some point ${\boldsymbol{z}}_{0}$ should not depend on how well we sampled at some other point ${\boldsymbol{z}}_{1}$. Yet achieving uniform sampling is an idealization since, if we do, this means we know the free energy. We now consider two other biasing methods that aim for this ideal, either in relatively small regions of CV space using fixed biases, or over broader extents using adaptive biases. #### 2.3.2 Umbrella Sampling Umbrella sampling is the standard way of using non-Boltzmann sampling to overcome free energy barriers. In its debut Torrie and Valleau (1977), umbrella sampling used a function $w({\boldsymbol{x}})$ that weights hard-to- sample configurations, equivalent to adding a bias potential of the form $\Delta V({\boldsymbol{x}})=-k_{B}T\ln w({\boldsymbol{x}}).$ (22) $w$ is found by trial-and-error such that configurations that are easy to sample on the unbiased potential are still easy to sample; that is, $w$ acts like an “umbrella” covering both the easy- and hard-to-sample regions of configuration space. Nearly always, $w$ is an explicit function of the CV’s, $w({\boldsymbol{x}})=W[{\boldsymbol{\theta}}({\boldsymbol{x}})]$. Coming up with the umbrella potential that would enable exploration of CV space with a single umbrella sampling simulation that takes the system far from its initial point is not straightforward. Akin to TI, it is therefore advantageous to combine results from several independent trajectories, each with its own umbrella potential that localizes it to a small volume of CV space that overlaps with nearby volumes. The most popular way to combine the statistics of such a set of independent umbrella sampling runs is the weighted-histogram analysis method (WHAM) Kumar et al. (1992). To compute statistics of CV space using WHAM, one first chooses the points in CV space that define the little local neighborhoods, or “windows” to be sampled and chooses the bias potential used to localize the sampling. Not knowing how the free energy changes in CV space makes the first task somewhat challenging, since more densely packed windows are preferred in regions where the free energy changes rapidly; however, since the calculations are independent, more can be added later if needed. A convenient choice for the bias potential is a simple harmonic spring that tethers the trajectory to a reference point ${\boldsymbol{z}}_{i}$ in CV space: $\Delta V_{i}({\boldsymbol{x}})=\frac{1}{2}\kappa\left\|{\boldsymbol{\theta}}({\boldsymbol{x}})-{\boldsymbol{z}}_{i}\right\|^{2}$ (23) which means the dynamics of the atomic variables ${\boldsymbol{x}}$ are identical to Eq. 12 at fixed ${\boldsymbol{z}}={\boldsymbol{z}}_{i}$. The points $\left\\{{\boldsymbol{z}}_{i}\right\\}$ and the value of $\kappa$ (which may be point-dependent) must be chosen such that ${\boldsymbol{\theta}}\left[{\boldsymbol{x}}(t)\right]$ from any one window’s trajectory makes excursions into the window of each of its nearest neighbors in CV space. Each window-restrained trajectory is directly histogrammed to yield apparent (i.e., biased) statistics on ${\boldsymbol{\theta}}$; let us call the biased probability in the $i$th window $P_{b,i}({\boldsymbol{z}})$. Eq. 19 again gives the recipe to reconstruct the unbiased statistics $P_{i}({\boldsymbol{z}})$ for ${\boldsymbol{z}}$ in the window of ${\boldsymbol{z}}_{i}$: $P_{i}({\boldsymbol{z}})=P_{b,i}({\boldsymbol{z}})e^{\frac{1}{2}\beta\kappa\left\|{\boldsymbol{z}}-{\boldsymbol{z}}_{i}\right\|^{2}}\left<e^{-\beta\frac{1}{2}\kappa\left\|{\boldsymbol{\theta}}({\boldsymbol{x}})-{\boldsymbol{z}}_{i}\right\|^{2}}\right>$ (24) We could use Eq. 24 directly assuming the biased MD trajectory is ergodic, but we know that regions far from the reference point will be explored very rarely and thus their free energy would be estimated with large uncertainty. This means that, although we can use sampling to compute $P_{b,i}$ knowing it effectively vanishes outside the neighborhood of ${\boldsymbol{z}}_{i}$, we cannot use sampling to compute $\left<e^{-\beta\frac{1}{2}\kappa\left\|{\boldsymbol{\theta}}({\boldsymbol{x}})-{\boldsymbol{z}}_{i}\right\|^{2}}\right>$. WHAM solves this problem by renormalizing the probabilities in each window into a single composite probability. Where there is overlap among windows, WHAM renormalizes such that the statistical variance of the probability is minimal. That is, it treats the factor $\left<e^{-\beta\frac{1}{2}\kappa\left\|{\boldsymbol{\theta}}({\boldsymbol{x}})-{\boldsymbol{z}}_{i}\right\|^{2}}\right>$ as an undetermined constant $C_{i}$ for each window, and solves for specific values such that the composite unbiased probability $P({\boldsymbol{z}})$ is continuous across all overlap regions with minimal statistical error. An alternative to WHAM, termed “umbrella integration”, solves the problem of renormalization across windows by constructing the composite mean force Kaestner and Thiel (2005); Kaestner (2011). The literature on umbrella sampling is vast (by simulation standards), so we present here a very condensed listing of some of its more recent application areas with representative citations: 1. 1. Small molecule conformational sampling Schaefer et al. (1998); Banavali and MacKerell (2002); Cruz et al. (2011); Islam et al. (2011); 2. 2. Protein-folding Young and Brooks (1996); Sheinerman and Brooks (1998); Bursulaya and Brooks (1999) and large-scale protein conformational sampling Rick et al. (1998); Allen et al. (2003); Shams et al. (2012); Yildirim et al. (2013); 3. 3. Protein-protein/peptide-peptide interactions Masunov and Lazaridis (2003); Tarus et al. (2005); Makowski et al. (2011); Casalini et al. (2011); Wanasundara et al. (2011); Zhang et al. (2011); Periole et al. (2012); Vijayaraj et al. (2012); Mahdavi and Kuyucak (2013); 4. 4. DNA conformational changes Banavali and Roux (2005) and DNA-DNA interactions Giudice et al. (2003); Matek et al. (2012); Bagai et al. (2013); 5. 5. Binding and association free-energies Czaplewski et al. (2000); Lee and Olson (2006); Peri et al. (2011); St-Pierre et al. (2011); Rashid and Kuyucak (2012); Chen and Chung (2012); Wilhelm et al. (2012); Louet et al. (2012); Zhang et al. (2012); Kessler et al. (2012); Mascarenhas and Kaestner (2013); 6. 6. Adsorption on and permeation through lipid bilayers MacCallum and Tieleman (2006); Tieleman and Marrink (2006); Kyrychenko et al. (2011); Lemkul and Bevan (2011); Paloncyova et al. (2012); Samanta et al. (2012); Grafmueller et al. (2013); Cerezo et al. (2013); Tian et al. (2013); Karlsson et al. (2013); 7. 7. Adsorption onto inorganic surfaces/interfaces Euston et al. (2011); Doudou et al. (2012); 8. 8. Water ionization Pomes and Roux (1998); Jagoda-Cwiklik et al. (2011); 9. 9. Phase transitions Calvo and Mottet (2011); Sharma and Debenedetti (2012); 10. 10. Enzymatic mechanisms Ridder et al. (2002); Kaestner et al. (2006); Wang et al. (2007); Ke et al. (2011); Yan et al. (2011); Mujika et al. (2012); Lonsdale et al. (2012); Rooklin et al. (2012); Lior-Hoffmann et al. (2012); 11. 11. Molecule/ion transport through protein complexes Crouzy et al. (2001); Allen et al. (2003); Hub and De Groot (2008); Xin et al. (2011); Furini and Domene (2011); Kim and Allen (2011); Domene and Furini (2012); Zhu and Hummer (2012) and other macromolecules Zhongjin and Jian (2011); Nalaparaju and Jiang (2012). #### 2.3.3 Metadynamics As already mentioned, one of the difficulties of the umbrella sampling method is the choice and construction of the bias potential. As we already saw with the relationship among TI, ABF, and TAMD, an adaptive method for building a bias potential in a running MD simulation may be advantageous. Metadynamics Laio and Parrinello (2002); Barducci et al. (2011) represents just such a method. Metadynamics is rooted in the original idea of “local elevation” Huber et al. (1994), in which a supplemental bias potential is progressively grown in the dihedral space of a molecule to prevent it from remaining in one region of configuration space. However, at variance with metadynamics, local elevation does not provide any means to reconstruct the unbiased free-energy landscape and as such it is mostly aimed at fast generation of plausible conformers. In metadynamics, configurational variables ${\boldsymbol{x}}$ evolve in response to a biased total potential: $V({\boldsymbol{x}})=V_{0}({\boldsymbol{x}})+\Delta V({\boldsymbol{x}},t)$ (25) where $V_{0}$ is the bare interatomic potential and $\Delta V({\boldsymbol{x}},t)$ is a time-dependent bias potential. The key element of metadynamics is that the bias is built as a sum of Gaussian functions centered on the points in CV space already visited: $\Delta V\left[{\boldsymbol{\theta}}({\boldsymbol{x}}),t\right]=w\sum_{\begin{array}[]{c}t^{\prime}=\tau_{G},2\tau_{G},\dots\\\ t^{\prime}<t\end{array}}\exp\left(-\frac{\left\|{\boldsymbol{\theta}}\left[{\boldsymbol{x}}(t)\right]-{\boldsymbol{\theta}}\left[{\boldsymbol{x}}(t^{\prime})\right]\right\|^{2}}{2\delta{\boldsymbol{\theta}}^{2}}\right)$ (26) Here, $w$ is the height of each Gaussian, $\tau_{G}$ is the size of the time interval between successive Gaussian depositions, and $\delta{\boldsymbol{\theta}}$ is the Gaussian width. It has been first empirically Laio et al. (2005) then analytically Bussi et al. (2006) demonstrated that in the limit in which the CV evolve according to a Langevin dynamics, the bias indeed converges to the negative of the free energy, thus providing an optimal bias to enhance transition events. Multiple simulations can also be used to allow for a quicker filling of the free-energy landscape Raiteri et al. (2006). The difference between the metadynamics estimate of the free energy and the true free energy can be shown to be related to the diffusion coefficient of the collective variables and to the rate at which the bias is grown. A possible way to decrease this error as a simulation progresses is to decrease the growth rate of the bias. Well-tempered metadynamics Barducci et al. (2008) used an optimized schedule to decrease the deposition rate of bias by modulating the Gaussian height: $w=\omega_{0}\tau_{G}e^{-\frac{\Delta V({\boldsymbol{\theta}},t)}{k_{B}\Delta T}}$ (27) Here, $\omega_{0}$ is the initial “deposition rate”, measured Gaussian height per unit time, and $\Delta T$ is a parameter that controls the degree to which the biased trajectory makes excursions away from free-energy minima. It is possible to show that using well-tempered metadynamics the bias does not converge to the negative of the free-energy but to a fraction of it, thus resulting in sampling the CVs at an effectively higher temperature $T+\Delta T$, where normal metadynamics is recovered for $\Delta T\rightarrow\infty$. We notice that other deposition schedules can be used aimed, e.g., at maximizing the number of round-trips in the CV space Singh et al. (2011). Importantly, it is possible to recover equilibrium Boltzmann statistics of unbiased collective variables from samples drawn throughout a well-tempered metadynamics trajectory Bonomi et al. (2009); it does not seem clear that one can do this from an ABF trajectory. Finally, it is possible to tune the shape of the Gaussians on the fly using schemes based on the geometric compression of the phase space or on the variance of the CVs Branduardi et al. (2012). In the well-tempered ensemble, the parameter $\Delta T$ can be used to tune the size of the explored region, in a fashion similar to the fictitious temperature in TAMD. So both TAMD and well-tempered metadynamics can be used to explore relevant regions of CV space while surmounting relevant free energy barriers. However, there are important distictions between the two methods. First, the main source of error in TAMD rests with how well mean-forces are approximated, and adiabatic separation, realizable only when the auxiliary variables ${\boldsymbol{z}}$ never move, is the only way to guarantee they are perfectly accurate. In practical application, TAMD never achieves perfect adiabatic separation. In contrast, because the deposition rate of decreases as a well-tempered trajectory progresses, errors related to poor adiabatic separation are progressively damped. Second, as already mentioned, TAMD alone cannot report the free energy, but it also is therefore not practically limited by the dimensionality of CV space; multicomponent gradients are just as accurately calculated in TAMD as are single-component gradients. Metadynamics, as a histogram-filling method, must exhaustively sample a finite region around any point to know the free energy and its gradients are correct, which can sometimes limit its utility. Metadynamics is a powerful method whose popularity continues to grow. In either its original formulation or in more recent variants, metadynamics has been employed successfully in several fields, some of which we point out below with some representative examples: 1. 1. Chemical reactions Iannuzzi et al. (2003); McGrath et al. (2013); 2. 2. Peptide backbone angle sampling Mantz et al. (2009); Leone et al. (2009); Melis et al. (2009); 3. 3. Protein folding Bussi et al. (2006); Gangupomu and Abrams (2010); Berteotti et al. (2011); Granata et al. (2013); 4. 4. Protein aggregation Baftizadeh et al. (2012); 5. 5. Molecular docking Gervasio et al. (2005); Soederhjelm et al. (2012); Limongelli et al. (2013); 6. 6. Conformational rearrangement of proteins Sutto and Gervasio (2013 (published ahead of print); 7. 7. Crystal structure prediction Martonak et al. (2006); 8. 8. Nucleation and crystal growth Trudu et al. (2006); Stack et al. (2011); 9. 9. and proton diffusion Zhang et al. (2012). ### 2.4 Some Comments on Collective Variables #### 2.4.1 The Physical Fidelity of CV-Spaces Given a potential $V({\boldsymbol{x}})$, any multidimensional CV ${\boldsymbol{\theta}}({\boldsymbol{x}})$ has a mathematically determined free energy $F({\boldsymbol{z}})$, and in principle the free-energy methods we describe here (and others) can use and/or compute it. However, this does not guarantee that $F$ is meaningful, and a poor choice for ${\boldsymbol{\theta}}({\boldsymbol{x}})$ can render the results of even the most sophisticated free-energy methods useless for understanding the nature of actual metastable states and the transitions among them. This puts two major requirements on any CV space: 1. 1. Metastable states and transition states must be unambiguously identified as energetically separate regions in CV space. 2. 2. The CV space must not contain hidden barriers. The first of these may seem obvious: CV’s are chosen to provide a low- dimensional description of some important process, say a conformational change or a chemical reaction or a binding event, and one can’t describe a process without being able to discriminate states. However, it is not always easy to find CV’s that do this. Even given representative configurations of two distinct metastable states, standard MD from these two different initial configurations may sample partially overlapping regions of CV space, making ambiguous the assignation of an arbitrary configuration to a state. It may be in this case that the two representative configurations actually belong to the same state, or that if there are two states, that no matter what CV space is overlaid, the barrier separating them is so small that, on MD timescales, they can be considered rapidly exchanging substates of some larger state. But a third possibility exists: the two MD simulations mentioned above may in fact represent very different states. The overlap might just be an artifact of neglecting to include one or more CV’s that are truly necessary to distinguish those states. If there is a significant free energy barrier along this neglected variable, an MD simulation will not cross it, yet may still sample regions in CV space also sampled by an MD simulation launched from the other side of this hidden barrier. And it is even worse: if TI or umbrella sampling is used along a pathway in CV space that neglects an important variable, the free-energy barriers along that pathway might be totally meaningless. Hidden barriers can be a significant problem in CV-based free-energy calculations. Generally speaking, one only learns of a hidden barrier after postulating its existence and testing it with a new calculation. Detecting them is not straightforward and often involves a good deal of CV space exploration. Methods such as TAMD and well-tempered metadynamics offer this capability, but much more work could be done in the automated detection of hidden barriers and the “right” CV’s (e.g., Das et al. (2006); Perilla and Woolf (2012); Ceriotti et al. (2011)). An obvious way of reducing the likelihood of hidden barriers is to use increase the dimensionality of CV space. TAMD is well-suited to this because it is a gradient method, but standard metadynamics, because it is a histogram- filling method, is not. A recent variant of metadynamics termed “reconnaissance metadynamics” Tribello et al. (2010) does have the capability of handling high-dimensional CV spaces. In reconnaissance metadynamics, bias potential kernels are deposited at the CV space points identified as centers of clusters detected and measured by an on-the-fly clusterization scheme. These kernels are hyperspherically symmetric but grow as cluster sizes grow and are able to push a system out of a CV space basin to discover other basins. As such, reconnaissance metadynamics is an automated way of identifying free-energy minima in high-dimensional CV spaces. It has been applied the identification of configurations of small clusters of molecules Tribello et al. (2011) and identification of protein-ligand binding poses Soederhjelm et al. (2012). #### 2.4.2 Some Common and Emerging Types of CV’s There are very few “best practices” codified for choosing CV’s for any given system. Most CV’s are developed ad hoc based on the processes that investigators would like to study, for instance, center-of-mass distance between two molecules for studying binding/unbinding, or torsion angles for studying conformational changes, or number of contacts for studying order- disorder transitions. Cartesian coordinates of centers of mass of groups of atoms are also often used as CV’s, as are functions of these coordinates. The potential energy $V({\boldsymbol{x}})$ is also an example of a 1-D CV, and there have been several examples of using it in CV-based enhanced sampling methods, such as umbrella sampling Bartels and Karplus (1998), metadynamics Micheletti et al. (2004) well-tempered metadynamics Bonomi and Parrinello (2010). In a recent work based on steered MD, it has been shown that also relevant reductions of the potential energy (e.g. the electrostatic interaction free-energy) can be used as effective CV’s Do et al. (2013). The basic rationale for enhanced sampling of $V$ is that states with higher potential energy often correspond to transition states, and one need make no assumptions about precise physical mechanisms. Key to its successful use as a CV, as it is for any CV, is a proper accounting for its entropy; i.e., the classical density-of-states. Coarse-graining of particle positions onto Eulerian fields was used early on in enhanced sampling Roitberg and Elber (1991); here, the value of the field at any Cartesian point is a CV, and the entire field represents a very high- dimensional CV. This idea has been put to use recently in the “indirect umbrella sampling” method of Patel et al. Patel et al. (2011) for computing free energies of solvation, and string method (Sec. 4.2.1) calculations of lipid bilayer fusion Mueller et al. (2012). In a similar vein, there have been recent attempts at variables designed to count the recurrency of groups of atoms positioned according to given templates, such as $\alpha$-helices paired $\beta$-strands in proteins Pietrucci and Laio (2009). We finally mention the possibility of building collective variables based on set of frames which might be available from experimental data or generated by means of previous MD simulations. Some of these variables are based on the idea of computing the distances between the present configuration and a set of precomputed snapshots. These distances, here indicated with $d_{i}$, where $i$ is the index of the snapshot, are then combined to obtain a coarse representation of the present configuration, which is then used as a CV. As an example, one might combine the distances as $s=\frac{\sum_{i}e^{-\lambda d_{i}}i}{\sum_{i}e^{-\lambda d_{i}}}$ (28) If the parameter $\lambda$ is properly chosen, this function returns a continuous interpolation between the indexes of the snapshots which are closer to the present conformation. If the snapshots are disposed along a putative path connecting two experimental structures, this CV can be used as a path CV to monitor and bias the progression along the path Branduardi et al. (2007). A nice feature of path CVs is that it is straighforward to also monitor the distance from the putative path. The standard way to do it is by looking at the distance from the closest reference snapshot, which can be approximately computed with the following continuous function: $z=-\lambda^{-1}\log\sum_{i}e^{-\lambda d_{i}}$ (29) This approach, modified to use internal coordinates, was used recently by Zinovjev et al. to study the aqueous phase reaction of pyruvate to salycilate, and in the CO bond-breaking/proton transfer in PchB Zinovjev et al. (2012). A generalization to multidimensional paths (i.e. sheets) can be obtained by assigning a generic vector $v_{i}$ to each of the precomputed snapshot and computing its average Spiwok and Králová (2011): $s=\frac{\sum_{i}e^{-\lambda d_{i}}v_{i}}{\sum_{i}e^{-\lambda d_{i}}}$ (30) ## 3 Tempering Approaches “Tempering” refers to a class of methods based on increasing the temperature of an MD system to overcome barriers. Tempering relies on the fact that according to the Arrhenius law the rate at which activated (barrier-crossing) events happen is strongly dependent on the temperature. Thus, an annealing procedure where the system is first heated and then cooled allows one to produce quickly samples which are largely uncorrelated. The root of all these ideas indeed lies in the simulated annealing procedure Kirkpatrick et al. (1983), a well-known method successfully used in many optimization problems. ### 3.1 Simulated tempering Simulated annealing is a form of Markov-chain Monte Carlo sampling where the temperature is artificially modified during the simulation. In particular, sampling is initially done at a temperature high enough that the simulation can easily overcome high free-energy barriers. Then, the temperature is decreased as the simulation proceeds, thus smoothly bringing the simulation to a local energy minimum. In simulated annealing, a critical parameter is the cooling speed. Indeed, the probability to reach the global minimum grows as this speed is decreased. The search for the global minimum can be interpreted in the same way as sampling an energy landscape at zero temperature. One could thus imagine to use simulated annealing to generate conformations at, e.g., room temperature by slowly cooling conformations starting at high temperature. However, the resulting ensemble will strongly depend on the cooling speed, thus possibly providing a biased result. A better approach consists of the the so-called simulated tempering methods Marinari and Parisi (1992). Here, a discrete list of temperatures $T_{i}$, with $i\in 1\dots N$ are chosen _a priori_ , typically spanning a range going from the physical temperature of interest to a temperature which is high enough to overcome all relevant free energy barriers. (Note that we do not have to stipulate a CV-space in which those barriers live.) Then, the index $i$, which indicates at which temperature the system should be simulated, is evolved with time. Two kind of moves are possible: (a) normal evolution of the system at fixed temperature, which can be done with a usual Markov Chain Monte Carlo or molecular dynamics and (b) change of the index $i$ at fixed atomic coordinates. It is easy to show that the latter can be performed as a Monte Carlo step with acceptance equal to $\alpha=\min\left(1,\frac{Z_{j}}{Z_{i}}e^{-\frac{U(x)}{k_{B}T_{j}}+\frac{U(x)}{k_{B}T_{i}}}\right)$ (31) where $i$ and $j$ are the indexes corresponding to the present temperature and the new one. The weights $Z_{i}$ should be choosen so as to sample equivalently all the value of $i$. It must be noticed that also within molecular dynamics simulations only the potential energy usually appears in the acceptance. This is due to the fact that the velocities are typically scaled by a factor $\sqrt{\frac{T_{j}}{T_{i}}}$ upon acceptance. This scaling leads to a cancellation of the contribution to the acceptance coming from the kinetic energy. Ultimately, this is related to the fact that the ensemble of velocities is analytically known _a priori_ , such that it is possible to adapt the velocities to the new temperature instantaneously. Estimating these weights $Z_{i}$ is nontrivial and typically requires a preliminary step. Moreover, if this estimate is poor the system could spend no time at the physical temperature, thus spoiling the result. Iterative algorithms for adjusting these weights have been proposed (see e.g. Park and Pande (2007)). We also observe that since the temperature sets the typical value of the potential energy, an effect much similar to that of simulated tempering with adaptive weights can be obtained by performing a metadynamics simulation using the potential energy as a CV (Sec. 2.4.2). ### 3.2 Parallel tempering A smart way to alleviate the issue of finding the correct weights is that of simulating several replicas at the same time Hansmann (1997); Sugita and Okamoto (1999). Rather that changing the temperature of a single system, the defining move proposal in parallel tempering consists of a coordinate swap between two $T$-replicas with acceptance probability $\alpha=\min\left(1,e^{\left(\frac{1}{k_{B}T_{j}}-\frac{1}{k_{B}T_{i}}\right)\left[U({\boldsymbol{x}}_{i})-U({\boldsymbol{x}}_{j})\right]}\right)$ (32) This method is the root of a class of techniques collectively known as “replica exchange” methods, and the latter name is often used as a synonimous of parallel tempering. Notably, within this framework it is not necessary to precompute a set of weights. Indeed, the equal time spent by each replica at each temperature is enforced by the constraint that only pairwise swaps are allowed. Moreover, parallel tempering has an additional advantage: since the replicas are weakly coupled and only interact when exchanges are attempted, they can be simulated on different computers without the need of a very fast interconnection (provided, of course, that a single replica is small enough to run on a single node). The calculation of the acceptance is very cheap as it is based on the potential energy which is often computed alongside force evaluation. Thus, one could in theory exploit also a large number of virtual, rejected exchanges so as to enhance statistical sampling Frenkel (2004); Coluzza and Frenkel (2005). Since efficiency of parallel tempering simulation can deteriorate if the stride between subsequent exchanges is too large Sindhikara et al. (2008); Bussi (2009), a typical recipe is to choose this stride as small as possible, with the only limitation of avoiding extra costs due to replica synchronization. One can push this idea further and implement asynchronous versions of parallel tempering, where overhead related to exchanges is minimized Gallicchio et al. (2008); Bussi (2009). One should be however aware that, especially at high exchange rate, artifacts coming from e.g. the use of wrong thermostating schemes could spoil the results Rosta et al. (2009); Sindhikara et al. (2010). Parallel tempering is popular in simulations of protein conformational sampling Vreede et al. (2005); Zhang and Mu (2012), protein folding Sugita and Okamoto (1999); Zhou (2003); Garcia and Onuchic (2003); Im et al. (2003); Mei et al. (2012); Berhanu et al. (2013) and aggregation Kokubo and Okamoto (2004); Oshaben et al. (2012), due at least in part to the fact that one need not choose CV’s to use it, and CV’s for describing these processes are not always straightforward to determine. ### 3.3 Generalized replica exchange The difference between the replicas is not restricted to be a change in temperature. Any control parameter can be changed, and even the expression of the Hamiltonian can be modified Sugita and Okamoto (2000). In the most general case every replica is simulated at a different temperature (and or pressure) and a different Hamiltonian, and the acceptance reads $\alpha=\min\left(1,\frac{e^{-\left(\frac{U_{i}(x_{j})}{k_{B}T_{i}}+\frac{U_{j}(x_{i})}{k_{B}T_{j}}\right)}}{e^{-\left(\frac{U_{i}(x_{i})}{k_{B}T_{i}}+\frac{U_{j}(x_{j})}{k_{B}T_{j}}\right)}}\right)$ (33) Several recipes for choosing the modified Hamiltonian have been proposed in the literature Fukunishi et al. (2002); Liu et al. (2005); Affentranger et al. (2006); Fajer et al. (2008); Xu et al. (2008); Zacharias (2008); Vreede et al. (2009); Itoh et al. (2010); Meng and Roitberg (2010); Terakawa et al. (2011); Wang et al. (2011); Zhang and Ma (2012); Bussi (2013). Among these, a notable idea is that of solute tempering Liu et al. (2005); Wang et al. (2011) which is used for the simulation of solvated biomolecules. Here, only the Hamiltonian of the solute is modified. More precisely, one could notice that a scaling of the Hamiltonian by a factor $\lambda$ is completely equivalent to a scaling of the temperature by a factor $\lambda^{-1}$. Hamiltonian scaling however can take advantage of the fact that the total energy of the system is an extensive property. Thus, one can limit the scaling to the portion of the system which is considered to be interesting and which has the relevant bottlenecks. With solute tempering, the solute energy is scaled whereas the solvent energy is left unchanged. This is equivalent to keeping the solute at a high effective temperature and the solvent at the physical temperature. Since in the simulation of solvated molecules most of the atoms belong to the solvent, this turns in a much smaller modification to the explored ensemble when compared with parallel tempering. In spite of this, the effect on the solute resemble much that of increasing the physical temperature. A sometimes-overlooked subtlety in solute tempering is the choice for the treatment of solvent-solute interactions. Indeed, whereas solute-solute interactions are scaled with a factor $\lambda<1$ and solvent-solvent interactions are not scaled, any intermediate choice (scaling factor between $\lambda$ and 1) could intuitively make sense for solvent-solute coupling. In the original formulation, the authors used a factor $(1+\lambda)/2$ for the solute-solvent interaction. This choice however was later shown to be suboptimal Huang et al. (2007); Wang et al. (2011), and refined to be $\sqrt{\lambda}$. This latter choice appears to be more physically sound, since it allows one to just simulate the biased replicas with a modified force-field. Indeed, if one scales the charges of the solute by a factor $\sqrt{\lambda}$, electrostatic interactions are changed by a factor $\lambda$ for solute-solute coupling and $\sqrt{\lambda}$ for solute-solvent coupling. The same is true for Lennard-Jones terms, albeit in this case it depends on the specific combination rules used. Notably, the same rules for scaling were used in a previous work Affentranger et al. (2006). As a final remark, we point out that solute tempering can be also used in a serial manner _a là_ simulated tempering, in a simulated solute tempering scheme Denschlag et al. (2009). ### 3.4 General comments In general, the advantage of these tempering methods over straighforward sampling can be rationalized as follows. A simulation is evolved so as to sample a modified ensemble by e.g. raising temperature or artificially modifying the Hamiltonian. The change in the ensemble could be drastic, so that trying to extract canonical averages by reweighting from such a simulation would be pointless. For this reason, a ladder of intermediate ensembles is built, interpolating between the physical one (i.e. room temperature, physical Hamiltonian) and the modified one. Then, transitions between consecutive steps in this ladder (or, in parallel schemes, coordinate swaps) are performed using a Monte Carlo scheme. Assuming that the dynamics of the most modified ensemble is ergodic, independent samples will be generated every time a new simulation reaches the highest step of the ladder. Thus, efficiency of these methods is often based on the evaluation of the round trip time required for a replica to traverse the entire ladder. Tempering methods are thus relying on the ergodicity of the most modified ensemble. This assumption is not always correct. A very simple example is parallel tempering used to accelerate the sampling over an entropic barrier. Since the height of an entropic barrier grows with the temperature, in this conditions the barrier in the most modified ensembles are unaffected Zuckerman and Lyman (2006). Moreover, since a lot of time is spent in sampling states in non-physical situations (e.g. high temperature), the overall computational efficiency could even be lower than that of straightforward sampling. Real applications are often in an intermediate situation, and usefulness of parallel tempering should be evaluated case by case. The number of intermediate steps in the ladder can be shown to grow with the square root of the specific heat of the system in the case of parallel tempering simulations. No general relationship can be drawn in the case of Hamiltonian replica exchange, but one can expect approximately that the number of replicas should be proportional to the square root of the number of degrees of freedom affected by the modification of the Hamiltonian. Thus, Hamiltonian replica exchange methods could be much more effective than simple parallel tempering as they allow the effort to be focused and the number of replicas to be minimized. Parallel tempering has the advantage that all the replicas can be analyzed to obtain meaningful results, e.g., to predict the melting curve of a molecule. This procedure should be used with caution, especially with empirically parametrized potentials, which are often tuned to be realistic only at room temperature. On the other hand, Hamiltonian replica exchange often relies on unphysically modified ensembles which have no interest but for the fact that they increase ergodicity. As a final note, we observe that data obtained at different temperature (or with modified Hamiltonians) could be combined to enhance statistics at the physical temperature Chodera et al. (2007). However, the effectiveness of this data recycling is limited by the fact that high temperature replicas visit very rarely low energy conformations, thus decreasing the amount of additional information that can be extracted. ## 4 Combinations and Advanced Approaches ### 4.1 Combination of tempering methods and biased sampling The algorithms presented in Section 3 and based on tempering are typically considered to be simpler to apply when compared with those discussed in Section 2 and based on biasing the sampling of selected collective variables. Indeed, by avoiding the problem of choosing collective variables which properly describe the reaction path, most of the burden of setting up a simulation is removed. However, this comes at a price: considering the computational cost, tempering methods are extremely expensive. This cost is related to the fact that they are able to accelerate all degrees of freedom to the same extent, without an _a priori_ knowledge of the sampling bottlenecks. In this sense, Hamiltonian replica exchange methods are in an intermediate situation, since they are typically less expensive than parallel tempering but allow to embed part of the knowledge of the system in the simulation set up. Because of the conceptual difference between tempering methods and CV-based methods, these approaches can be easily and efficiently combined. As an example, the combination of metadynamics and parallel tempering can be used to take advantage of the known bottlenecks with biased collective variables at the same time accelerating the overall sampling with parallel tempering Bussi et al. (2006). In that work, the free energy landscape for the folding of a small hairpin was computed by biasing a small number of selected CVs (gyration radius and the number of hydrogen bonds). These CVs alone are not enough to describe folding, as can be easily shown by performing a metadynamics simulation using these CVs. However, the combination with parallel tempering allowed acceleration of all the degrees of freedom blindly and reversible folding of the hairpin. This combined approach also improves the results when compared with parallel tempering alone, since it accelerates exploration of phase-space. Moreover, since parallel tempering samples the unbiased canonical distribution, it is very difficult to use it to compute free-energy differences which are larger than a few $k_{B}T$. The metadynamics bias can be used to disfavor, e.g., the folded state so as to better estimate the free- energy difference between the folded and unfolded states. It is also possible to combine metadynamics with the solute tempering method so as to decrease the number of required replicas and the computational cost Camilloni et al. (2008). As an alternative to solute tempering, metadynamics in the well-tempered ensemble can be effectively used to enhance the acceptance in parallel tempering simulations and to decrease the number of necessary replicas Bonomi and Parrinello (2010). This combination of parallel tempering with well-tempered ensemble can be pushed further and combined with metadynamics on a few selected degrees of freedom Deighan et al. (2012). As a final note, bias exchange molecular dynamics Piana and Laio (2007) combines metadynamics and replica echange in a completely different spirit: every replica is run using a different CV, thus allowing many CVs to be tried at the same time. This technique has been succesfully applied to several problems. For a recent review, we refer the reader to Ref. Baftizadeh et al. (2012). ### 4.2 Some methods based on TAMD #### 4.2.1 String method in collective variables The string method is generally an approach to find pathways of minimal energy connecting two points in phase space E et al. (2002). When working in CV’s, the string method is used to find minimal free-energy paths (MFEP’s) Maragliano et al. (2006). String method calculations involve multiple replicas, each representing a point ${\boldsymbol{z}}_{s}$ in CV space at position $s$ along a discretized string connecting two points of interest (reactant and product states, say). The forces on each replica’s ${\boldsymbol{z}}_{s}$ are computed and their ${\boldsymbol{z}}_{s}$’s updated, as in TAMD, with the addition of forces that act to keep the ${\boldsymbol{z}}$’s equidistant along the string (so-called reparameterization forces): $\bar{\gamma}\dot{z}_{j}(s,t)=\displaystyle\sum_{k}\biggl{[}\tilde{M}_{jk}(\mathbf{x}(s,t))\kappa[\theta_{k}(\mathbf{x}(s,t))-z_{k}(s,t)]\biggr{]}+\eta_{z}(t)+\lambda(s,t)\displaystyle\frac{\partial{}z_{j}}{\partial{}s}$ (34) Here, $\tilde{M}_{jk}$ is the metric tensor mapping distances on the manifold of atomic coordinates to the manifold of CV space, $\eta$ is thermal noise and $\lambda(s,t)\displaystyle\frac{\partial{}z_{j}}{\partial{}s}$ represents the reparameterization force tangent to the string that is sufficient to maintain equidistant images along the string. String method has been used to study activation of the insulin-receptor kinase Vashisth and Abrams (2012), docking of insulin to its receptor Vashisth and Abrams (2013), myosin Ovchinnikov et al. (2011), In these examples, the update of the string coordinates is done at a lower frequency than the atomic variables in each image. In contrast, in the on-the-fly variant of string method in CV’s, the friction on the ${\boldsymbol{z}}_{s}$’s is set high enough to make the effective averaging of the forces approach the true mean forces, and the ${\boldsymbol{z}}$ updates occur in lockstep with the ${\boldsymbol{x}}$ updates of the MD system Maragliano and Vanden-Eijnden (2007). Just as in TAMD, the atomic variables obey an equation of motion like Eq. 12 tethering them to the ${\boldsymbol{z}}_{s}$. Stober and Abrams recently demonstrated an implementation of on-the-fly string method to study the thermodynamics of the normal-to-amyloidogenic transition of $\beta$2-microglobulin Stober and Abrams (2012). Unique in this approach was the construction of a single composite MD system containing 27 individual $\beta$2 molecules restrained to points on 3 $\times$ 3 $\times$ 3 grid inside a single large solvent box. Zinovjev et al. used a combination of the on-the-fly string method and of path-collective variables (see Equations 28 and 29) in a quantum-mechanics/molecular-mechanics approach to study a methyltransferase reaction Zinovjev et al. (2013). #### 4.2.2 On-the-fly free energy parameterization Because TAMD provides mean-force estimates as it is exploring CV space, it stands to reason that those mean forces could be used to compute a free energy. In contrast, in the single-sweep method Maragliano and Vanden-Eijnden (2008), the TAMD forces are only used in the CV space exploration phase, not the free-energy calculation itself. Recently, Abrams and Vanden-Eijnden proposed a method for using TAMD directly to parameterize a free energy; that is, to determine the best set of some parameters ${\boldsymbol{\lambda}}$ on which a free energy of known functional form depends Abrams and Vanden-Eijnden (2012): $F({\boldsymbol{z}})=F({\boldsymbol{z}};{\boldsymbol{\lambda}}^{})$ (35) The approach, termed “on-the-fly free energy parameterization”, uses forces from a running TAMD simulation to progressively optimize ${\boldsymbol{\lambda}}$ using a time-averaged gradient error: $E({\boldsymbol{\lambda}})=\frac{1}{2t}\int_{0}^{t}\left\|\nabla_{z}F\left[{\boldsymbol{z}}(s),{\boldsymbol{\lambda}}(t)\right]+\kappa\left[\theta({\boldsymbol{x}}(s))-{\boldsymbol{z}}(s)\right]\right\|^{2}ds,$ (36) If constructed so that $F$ is linear in ${\boldsymbol{\lambda}}=(\lambda_{1},\lambda_{2},\dots,\lambda_{M})$, minimization of $E$ can be expressed as a simple linear algebra problem $\sum_{j}A_{ij}\lambda_{j}=b_{i},\ \ \ i=1,\dots,M$ (37) and the running TAMD simulation provides progressively better estimates of $A$ and $b$ until the ${\boldsymbol{\lambda}}$ converge. In the cited work, it was shown that this method is an efficient way to derive potentials of mean force between particles in coarse-grained molecular simulations as basis-function expansions. It is currently being investigated as a means to parameterize free energies associated with conformational changes of proteins. Chen, Cuendet, and Tuckermann developed a very similar approach that in addition to parameterizing a free energy using d-AFED-computed gradients uses a metadynamics-like bias on the potential Chen et al. (2012). These authors demonstrated efficient reconstruction of the four-dimensional free-energy of vacuum alanine dipeptide with this approach. ## 5 Concluding Remarks In this review, we have summarized some of the current and emerging enhanced sampling methods that sit atop MD simulation. These have been broadly classified as methods that use collective variable biasing and methods that use tempering. CV biasing is a much more prevalent approach than tempering, due partially to the fact that it is perceived to be cheaper, since tempering simulations are really only useful for enhanced sampling of configuration space when run in parallel. CV-biasing also reflects the desire to rein in the complexity of all-atom simulations by projecting configurations into a much lower dimensional space. (Parallel tempering can be thought of as increasing the dimensionality of the system by a factor equal to the number of simulated replicas.) But the drawback of all CV-biasing approaches is the risk that the chosen CV space does not provide the most faithful representation of the true spectrum of metastable subensembles and the barriers that separate them. Guaranteeing that sampling of CV space is not stymied by hidden barriers must be of paramount concern in the continued evolution of such methods. For this reason, methods that specifically allow broad exploration of CV space, like TAMD (which can handle large numbers of CV’s) and well-tempered metadynamics will continue to be valuable. So too will parallel tempering because its broad sampling of configuration space can be used to inform the choice of better CV’s. Accelerating development of combined CV-tempering methods bodes well for enhanced sampling generally. Although some of these methods involve time-varying forces (ABF, TAMD, and metadynamics), all methods we’ve discussed have the underlying rationale of the equilibrium ensemble. TI uses the constrained ensemble, ABF and metadynamics ideally converge to an ensemble in which a bias erases free- energy variations, and TAMD samples an attenuated/mollified equilibrium ensemble. There is an entirely separate class of methods that inherently rely on non-equilibrium thermodynamics. We have not discussed at all the several free-energy methods based on non-equilibrium MD simulations; we refer interested readers to the article by Christoph Dellago in this issue. Finally, we have also not really touched on any of the practical issues of implementing and using these methods in conjunction with modern MD packages (e.g., NAMD Phillips et al. (2005), LAMMPS Plimpton (1995), Gromacs Hess et al. (2008), Amber Case et al. (2005), and CHARMM Brooks et al. (1983), to name a few). At least two packages (NAMD and CHARMM) have native support for collective variable biasing, and NAMD in particular offers both native ABF and a TcL-based interface which has been used to implement TAMD Abrams and Vanden- Eijnden (2010). The native collective variable module for NAMD has been recently ported to LAMMPS Fiorin et al. (2013). Gromacs offers native support for parallel tempering. Generally speaking, however, modifying MD codes to handle CV-biasing and multiple replicas is not straightforward, since one would like access to the data structures that store coordinates and forces. A major help in this regard is the PLUMED package Bonomi et al. (2009); Tribello et al. (2013), which patches a variety of MD codes to enable users to use many of the techniques discussed here. ## 6 Abbreviations • ABF: adaptive-biasing force * • AFED: adiabatic free-energy dynamics * • CV: collective-variable * • MD: molecular dynamics * • MFEP: minimum free-energy path * • TAMD: temperature-accelerated molecular dynamics * • TI: thermodynamic integration * • WHAM: weighted-histogram analysis method ## Acknowledgments CFA would like to acknowledge support of NSF (DMR-1207389) and NIH (1R01GM100472). GB would like to acknowledge the European Research Council (Starting Grant S-RNA-S, no. 306662) for financial support. 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DOI:10.1016/j.cpc.2013.09.018.	arxiv-papers	2014-01-02T08:30:03	2024-09-04T02:49:56.164127	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Cameron Abrams and Giovanni Bussi", "submitter": "Giovanni Bussi", "url": "https://arxiv.org/abs/1401.0387" }
1401.0391	# Semileptonic decays $B_{c}^{+}\to D^{()}_{(s)}(l^{+}\nu_{l},l^{+}l^{-},\nu\bar{\nu})$ in the perturbative QCD approach Wen-Fei [email protected], Xin Yu1, Cai-Dian Lü1, and Zhen-Jun Xiao2 1 Center for Future High Energy Physics, Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, People’s Republic of China, 2 Department of Physics and Institute of Theoretical Physics, Nanjing Normal University, Nanjing, Jiangsu 210023, People’s Republic of China ###### Abstract In this paper we study the semileptonic decays of $B_{c}^{+}\to D^{()}_{(s)}(l^{+}\nu_{l},l^{+}l^{-},\nu\bar{\nu})$ (here $l$ stands for $e$, $\mu$, or $\tau$). After evaluating the $B_{c}^{+}\to(D_{(s)},D^{}_{(s)})$ transition form factors $F_{0,+,T}(q^{2})$ and $V(q^{2}),A_{0,1,2}(q^{2}),T_{1,2,3}(q^{2})$ by employing the perturbative QCD factorization approach, we calculate the branching ratios for all these semileptonic decays. Our predictions for the values of the $B_{c}^{+}\to D_{(s)}$ and $B_{c}^{+}\to D^{}_{(s)}$ transition form factors are consistent with those obtained by using other methods. The branching ratios of the decay modes with $\bar{\nu}\nu$ are almost an order of magnitude larger than the corresponding decays with $l^{+}l^{-}$ after the summation over the three neutrino generations. The branching ratios for the decays with $b\to d$ transitions are much smaller than those decays with the $b\to s$ transitions, due to the Cabibbo-Kobayashi-Maskawa suppression. We define ratios $R_{D}$ and $R_{D^{}}$ for the branching ratios with the $\tau$ lepton versus $\mu$, $e$ lepton final states to cancel the uncertainties of the form factors, which could possibly be tested in the near future. ###### pacs: 13.20.He, 12.38.Bx, 14.40.Nd ## I Introduction The $B_{c}$ meson is a pseudoscalar ground state of $b$ and $c$ quarks, and thus the electromagnetic interaction cannot transform the $B_{c}$ meson into other hadrons containing $b$ and $c$ quarks. The two difference of quark flavors forbid its annihilation into gluons and being below the $B-D$ threshold makes the $B_{c}$ meson stable for strong interaction. The $B_{c}$ meson can only decay through weak interactions, so it is an ideal system to study weak decays of heavy quarks. Either the heavy quark ($b$ or $c$) can decay individually, which makes it different from the $B_{u,d}$ or $B_{s}$ meson. The phase space in the $c\to s$ transition is smaller than that in the $b\to c$ transition, but the Cabibbo-Kobayashi-Maskawa (CKM) matrix element $\|V_{cs}\|\sim 1$ is much larger than the CKM matrix element $\|V_{cb}\|\sim 0.04$. Thus the $c$-quark decays provide the dominant contribution (about $70\%$) to the decay width of the $B_{c}$ meson PAN67-1559 . Because the mass of a $B_{c}\bar{B}_{c}$ pair exceeds the threshold of $\Upsilon(4S)$, the $B_{c}$ meson cannot be produced at the $B$ factories. So comparing with $B_{u,d}$ or $B_{s}$ meson, the $B_{c}$ meson decays received much less experimental attention in the past decades. However, at LHC experiments, around $5\times 10^{10}$ $B_{c}$ events per year are expected pap-LHC events ; PAN67-1559 due to the relatively large production cross section, which provides a very good platform to study various $B_{c}$ meson decay modes. Because there is only one hadronic final product, the $B_{c}$ meson semileptonic decays among the abundant decay modes are relatively clean in the theoretical treatment. These semileptonic decays provide good opportunities to measure not only the CKM matrix elements, such as $\|V_{cb}\|,\|V_{ub}\|$, and $\|V_{cd}\|$, but also the form factors of the $B_{c}$ to bottom and charmed mesons transitions. The rare semileptonic decays governed by the flavor- changing neutral currents are forbidden at tree level in the standard model (SM). Those decays, which are very sensitive to the contributions of new intermediate particles or interactions are especially interesting. There are various approaches working on the semileptonic $B_{c}$ decays. In Ref. arxiv0903.2234 , for example, Dhir and Verma presented a detailed analysis of the exclusive semileptonic $B_{c}$ decays in the Bauer-Stech-Wirbel framework. The authors of the Refs. epjcd4-18 ; prd65-094037 ; arxiv1006-4231 studied the semileptonic $B_{c}$ decays in the relativistic and/or constituent quark model. In Refs. M-Jamil-Aslam ; M-J-Aslam , $B_{c}\to D^{}_{s}l^{+}l^{-}$ decays were studied in the SM with the fourth-generation and supersymmetric models. The three point QCD sum rules approach was adopted to investigate the $B_{c}^{+}\to D^{+}_{(s)}l^{+}l^{-}$ in prd77-114024 and $B_{c}^{+}\to D_{(s)}^{+}(l^{+}l^{-},\bar{\nu}\nu)$ in prd78-036005 . In this paper, we will study the semileptonic decays of $B_{c}^{+}\to D^{()}_{(s)}(l^{+}\nu_{l},l^{+}l^{-},\nu\bar{\nu})$ (here $l$ stands for leptons $e,\mu,$ or $\tau$) the perturbative QCD (pQCD) approach pap-pQCD . These semileptonic decays are governed by the form factors. At the maximum recoil region, the final state meson is collinear with a large momentum. he spectator $c$ quark in $B_{c}$ meson thus needs a hard gluon to kick it off from almost zero momentum to a collinear state. However, when doing integrations of momentum fractions of valence quarks, endpoint singularity occurs. A natural way to kill this singularity is to pick up the neglected transverse momentum in the collinear factorization. With the additional transverse momentum cale $k_{T}$ pqcd , double logarithms appear in the calculation. We have to use the renormalization group equation to perform the resummation resulting in the so-called Sudakov form factors Suda-factor and make the perturbative calculation of the hard amplitudes (form factors) infrared safe. The pQCD approach is widely adopted to calculate the transition form factors of $B_{u,d}$ and $B_{s}$ mesonprd86-114025 ; prd87-097501 ; pQCD- FF . Furthermore, various $B_{c}$ decay modes have also been studied in Refs. cpc37-093102 ; PRD81-014022-EPJC45-711-EPJC60-107 in the pQCD approach. The structure of this paper is as follows. After this Introduction, we collect the distribution amplitudes of the $B_{c}$, $D^{()}$ and $D_{s}^{()}$ mesons in Sec. II. Based on the $k_{\rm T}$ factorization formalism, we calculate and present the expressions for the $B_{c}\to(D^{()},D_{s}^{()})$ transition form factors in the large recoil regions in Section III. The numerical results and relevant discussions are given in Sec. IV. And Sec. V contains a short summary. ## II Kinematics and the wave functions Figure 1: The leading-order Feynman diagrams for the transition of $B_{c}^{+}\to(D^{()},D_{s}^{()})$, where $M$ stands for a $D^{()}$ or $D_{s}^{()}$ meson, and $\otimes$ is the weak vertex. The lowest-order diagrams for $B_{c}^{+}\to(D^{()},D_{s}^{()})$ transitions are displayed in Fig. 1, where $M$ stands for $D^{()}$ or $D_{s}^{()}$ meson, the $\otimes$ is the weak vertex for the leptonic pairs to come out. In the rest frame of $B_{c}$ meson, with the $m_{B_{c}}$ standing for the mass of the $B_{c}$ meson, and $m$ for the $D_{(s)}$ or $D^{}_{(s)}$ mesons, the momenta of $B_{c}$ and $D^{()}_{(s)}$ mesons are defined in the light-cone coordinates as prd67-054028 ; cpc37-093102 $\displaystyle p_{1}=\frac{m_{B_{c}}}{\sqrt{2}}(1,1,0_{\bot}),\quad p_{2}=\frac{m_{B_{c}}}{\sqrt{2}}(r\eta^{+},r\eta^{-},0_{\bot}),$ (1) with $r=m/m_{B_{c}}$ and $\eta^{\pm}=\eta\pm\sqrt{\eta^{2}-1}$. As for the $\eta$ in $\eta^{\pm}$, the expression $\displaystyle\eta=\frac{1}{2r}\left[1+r^{2}-\frac{q^{2}}{m^{2}_{B_{c}}}\right]$ (2) can be evaluated from $q^{2}=(p_{1}-p_{2})^{2}$ which is the invariant mass of the lepton pairs. The momenta of the spectator quarks in the $B_{c}$ and $D^{()}_{(s)}$ mesons are parameterized as $\displaystyle k_{1}=(x_{1}\frac{m_{B_{c}}}{\sqrt{2}},x_{1}\frac{m_{B_{c}}}{\sqrt{2}},k_{1\bot}),\quad k_{2}=(x_{2}\frac{m_{B_{c}}}{\sqrt{2}}r\eta^{+},x_{2}\frac{m_{B_{c}}}{\sqrt{2}}r\eta^{-},k_{2\bot}).$ (3) For the $D^{}_{(s)}$ mesons, we define their polarization vector $\epsilon$ as $\displaystyle\epsilon_{L}=\frac{1}{\sqrt{2}}(\eta^{+},-\eta^{-},0_{\bot}),\quad\epsilon_{T}=(0,0,1),$ (4) where $\epsilon_{L}$ and $\epsilon_{T}$ denote the longitudinal and transverse polarization of the $D^{}_{(s)}$ mesons, respectively. In this work, we use the same distribution amplitude for the $B_{c}$ meson as that used in Refs. cpc37-093102 ; epjc45-711 ; prd81-014002 ; prd87-074027 , $\displaystyle\Phi_{B_{c}}(x)=\frac{i}{\sqrt{2N_{c}}}[(p/+m_{B_{c}})\gamma_{5}\phi_{B_{c}}(x)]_{\alpha\beta}\;,$ (5) with $\displaystyle\phi_{B_{c}}(x)=\frac{f_{B_{c}}}{2\sqrt{2N_{c}}}\delta(x-m_{c}/m_{B_{c}})\exp[-\omega^{2}_{B_{c}}b^{2}/2]\;,$ (6) where $m_{c}$ is the mass of $c$-quark. Because the $B_{c}$ meson consists of two heavy quarks $b$ and $c$, just like a heavy quarkonium, the non- relativistic QCD framework can be applied, which means the leading-order wave function should be just the zero-point wave function shown in Eq. (6). For the $D^{()}_{(s)}$ mesons, up to twist-3 accuracy, the two-parton light- cone distribution amplitudes are defined as prd67-054028 ; prd78-014018 $\displaystyle\langle D_{(s)}(p)\|q_{\alpha}(z)\bar{c}_{\beta}(0)\|0\rangle$ $\displaystyle=$ $\displaystyle\frac{i}{\sqrt{2N_{C}}}\int_{0}^{1}dxe^{ixp\cdot z}\left[\gamma_{5}\left(p/+m\right)\phi_{D_{(s)}}(x,b)\right]_{\alpha\beta}\;,$ $\displaystyle\langle D_{(s)}^{}(p)\|q_{\alpha}(z)\bar{c}_{\beta}(0)\|0\rangle$ $\displaystyle=$ $\displaystyle-\frac{1}{\sqrt{2N_{C}}}\int_{0}^{1}dxe^{ixp\cdot z}\big{[}\epsilon/_{L}(p/+m)\phi_{D^{}_{(s)}}(x,b)$ (7) $\displaystyle+\;\epsilon/_{T}(p/+m)\phi_{D^{}_{(s)}}(x,b)\big{]}_{\alpha\beta}\;,$ with $\displaystyle\int_{0}^{1}dx\phi_{D_{(s)}}(x,0)$ $\displaystyle=$ $\displaystyle\frac{f_{D_{(s)}}}{2\sqrt{2N_{c}}}\;,$ $\displaystyle\int_{0}^{1}dx\phi_{D^{}_{(s)}}(x,0)$ $\displaystyle=$ $\displaystyle\frac{f_{D^{}_{(s)}}}{2\sqrt{2N_{c}}}\;,$ (8) as the normalization conditions. We adopt $f_{D}=206.7\pm 8.9$ MeV and $f_{D_{s}}=260.0\pm 5.6$ MeV in PDG pdg2012 by experimental average for $D$ and $D_{s}$ mesons, respectively. For the $D^{}$ or $D^{}_{s}$ meson, we adopt the same decay constant and distribution amplitude for the longitudinal and transverse components. Since there is no experimental data, we use $f_{D^{}}=270$ MeV and $f_{D^{}_{s}}=310$ MeV for $D^{}$ and $D^{}_{s}$ meson considering of the results in Refs. decay-cons and assume a $10\%$ uncertainty. The distribution amplitude for the $D_{(s)}$ meson is $\displaystyle\phi_{D_{(s)}}=\frac{1}{2\sqrt{2N_{C}}}f_{D_{(s)}}6x(1-x)\left[1+C_{D_{(s)}}(1-2x)\right]\exp\left[-\frac{\omega_{D_{(s)}}^{2}b^{2}}{2}\right],$ (9) which is a $k_{T}$-dependent form with $C_{D}=0.5,\omega_{D}=0.1$ and $C_{D_{s}}=0.4,\omega_{D_{s}}=0.2$ for $D$ and $D_{s}$ mesons, respectively prd78-014018 . In this work, we also adopt the same distribution amplitude for both the vector meson $D^{}_{(s)}$ and pseudoscalar meson $D_{(s)}$ because of their small mass difference prd78-014018 . ## III Form factors of semileptonic decays The form factors $F_{+}(q^{2}),F_{0}(q^{2})$ for the $B_{c}$ to pseudoscalar meson $D_{(s)}$ transition induced by the vector current can be defined as zpc29-637 ; npb592-3 $\displaystyle\langle D_{(s)}(p_{2})\|\bar{q}(0)\gamma_{\mu}b(0)\|B_{c}(p_{1})\rangle$ $\displaystyle=$ $\displaystyle\left[(p_{1}+p_{2})_{\mu}-\frac{m_{B_{c}}^{2}-m^{2}}{q^{2}}q_{\mu}\right]F_{+}(q^{2})$ (10) $\displaystyle+$ $\displaystyle\frac{m_{B_{c}}^{2}-m^{2}}{q^{2}}q_{\mu}F_{0}(q^{2}),$ where $q=p_{1}-p_{2}$ is the momentum of the lepton pairs. In order to cancel the poles at $q^{2}=0$, $F_{+}(0)$ should be equal to $F_{0}(0)$. For the sake of convenience, we define the auxiliary form factors $f_{1}(q^{2})$ and $f_{2}(q^{2})$, $\displaystyle\langle D_{(s)}(p_{2})\|\bar{q}(0)\gamma_{\mu}b(0)\|B_{c}(p_{1})\rangle=f_{1}(q^{2})p_{1\mu}+f_{2}(q^{2})p_{2\mu}.$ (11) In terms of $f_{1}(q^{2})$ and $f_{2}(q^{2})$ the form factors $F_{+}(q^{2})$ and $F_{0}(q^{2})$ are $\displaystyle F_{+}(q^{2})$ $\displaystyle=$ $\displaystyle\frac{1}{2}\left[f_{1}(q^{2})+f_{2}(q^{2})\right],$ $\displaystyle F_{0}(q^{2})$ $\displaystyle=$ $\displaystyle\frac{1}{2}f_{1}(q^{2})\left[1+\frac{q^{2}}{m_{B_{c}}^{2}-m^{2}}\right]+\frac{1}{2}f_{2}(q^{2})\left[1-\frac{q^{2}}{m_{B_{c}}^{2}-m^{2}}\right].$ (12) The form factor $F_{T}(q^{2})$ for the $B_{c}\to D_{(s)}$ transition induced by the tensor current can be defined as npb592-3 $\displaystyle\langle D_{(s)}(p_{2})\|\bar{q}(0)\sigma_{\mu\nu}b(0)\|B_{c}(p_{1})\rangle=i\left[p_{2\mu}q_{\nu}-q_{\mu}p_{2\nu}\right]\frac{2F_{T}(q^{2})}{m_{B_{c}}+m}\;.$ (13) There are seven form factors $V(q^{2}),A_{0,1,2}(q^{2})$ and $T_{1,2,3}(q^{2})$ needed for the transition of $B_{c}\to D^{}_{(s)}$ in this work. The form factors $V(q^{2})$ and $A_{0,1,2}(q^{2})$ are defined by npb592-3 ; epjc28-515 ; prd65-014007 $\displaystyle\langle D^{}_{(s)}(p_{2})\|\bar{q}(0)\gamma_{\mu}b(0)\|B_{c}(p_{1})\rangle$ $\displaystyle=$ $\displaystyle\epsilon_{\mu\nu\alpha\beta}\epsilon^{\nu}p_{1}^{\alpha}p_{2}^{\beta}\frac{2V(q^{2})}{m_{B_{c}}+m}\;,$ (14) $\displaystyle\langle D^{}_{(s)}(p_{2})\|\bar{q}(0)\gamma_{\mu}\gamma_{5}b(0)\|B_{c}(p_{1})\rangle$ $\displaystyle=$ $\displaystyle i\left[\epsilon_{\mu}^{}-\frac{\epsilon^{}\cdot q}{q^{2}}q_{\mu}\right](m_{B_{c}}+m)A_{1}(q^{2})$ (15) $\displaystyle-$ $\displaystyle i\left[(p_{1}+p_{2})_{\mu}-\frac{m_{B_{c}}^{2}-m^{2}}{q^{2}}q_{\mu}\right](\epsilon^{}\cdot q)\frac{A_{2}(q^{2})}{m_{B_{c}}+m}$ $\displaystyle+$ $\displaystyle i\frac{2m(\epsilon^{}\cdot q)}{q^{2}}q_{\mu}A_{0}(q^{2}),$ where $\epsilon^{}$ is the polarization vector of the $D^{}_{(s)}$ meson. The form factors $T_{1,2,3}$ are defined by npb592-3 ; prd53-3672 $\displaystyle\langle D^{}_{(s)}(p_{2})\|\bar{q}(0)\sigma_{\mu\nu}q^{\nu}(1$ $\displaystyle+$ $\displaystyle\gamma_{5})b(0)\|B_{c}(p_{1})\rangle=i\epsilon_{\mu\nu\alpha\beta}\epsilon^{\nu}p_{1}^{\alpha}p_{2}^{\beta}2T_{1}(q^{2})$ (16) $\displaystyle+$ $\displaystyle\left[\epsilon^{}_{\mu}(m^{2}_{B_{c}}-m^{2})-(\epsilon^{}\cdot q)(p_{1}+p_{2})_{\mu}\right]T_{2}(q^{2})$ $\displaystyle+$ $\displaystyle(\epsilon^{}\cdot q)\left[q_{\mu}-\frac{q^{2}}{m^{2}_{B_{c}}-m^{2}}(p_{1}+p_{2})_{\mu}\right]T_{3}(q^{2})\;,$ with $T_{1}(0)=T_{2}(0)$ implied by the identity $\displaystyle\sigma_{\mu\nu}\gamma_{5}=-\frac{i}{2}\epsilon_{\mu\nu\alpha\beta}\sigma^{\alpha\beta}\;.$ (17) In the transverse configuration $b$-space and by including the Sudakov form factors and the threshold resummation effects, we obtain the $B_{c}\to D_{(s)}$ form factors $f_{1}(q^{2}),f_{2}(q^{2})$ and $F_{T}(q^{2})$ as follows $\displaystyle f_{1}(q^{2})$ $\displaystyle=$ $\displaystyle 16\pi m_{B_{c}}^{2}rC_{F}\int dx_{1}dx_{2}\int b_{1}db_{1}b_{2}db_{2}\;\phi_{B_{c}}(x_{1})\phi_{D_{(s)}}(x_{2},b_{2})$ (18) $\displaystyle\times$ $\displaystyle\Bigl{\\{}\left[1-rx_{2}\right]\cdot h_{1}(x_{1},x_{2},b_{1},b_{2})\cdot\alpha_{s}(t_{1})\exp\left[-S_{ab}(t_{1})\right]$ $\displaystyle-$ $\displaystyle\left[r+2x_{1}(1-\eta)\right]\cdot h_{2}(x_{1},x_{2},b_{1},b_{2})\cdot\alpha_{s}(t_{2})\exp\left[-S_{ab}(t_{2})\right]\Bigr{\\}}\;,$ $\displaystyle f_{2}(q^{2})$ $\displaystyle=$ $\displaystyle 16\pi m_{B_{c}}^{2}C_{F}\int dx_{1}dx_{2}\int b_{1}db_{1}b_{2}db_{2}\;\phi_{B_{c}}(x_{1})\phi_{D_{(s)}}(x_{2},b_{2})$ (19) $\displaystyle\times$ $\displaystyle\Bigl{\\{}\left[1-2rx_{2}(1-\eta)\right]\cdot h_{1}(x_{1},x_{2},b_{1},b_{2})\cdot\alpha_{s}(t_{1})\exp\left[-S_{ab}(t_{1})\right]$ $\displaystyle+$ $\displaystyle\left[2r-x_{1}\right]\cdot h_{2}(x_{1},x_{2},b_{1},b_{2})\cdot\alpha_{s}(t_{2})\exp\left[-S_{ab}(t_{2})\right]\Bigr{\\}}\;,$ $\displaystyle F_{T}(q^{2})$ $\displaystyle=$ $\displaystyle 8\pi m_{B_{c}}^{2}C_{F}(1+r)\int dx_{1}dx_{2}\int b_{1}db_{1}b_{2}db_{2}\;\phi_{B_{c}}(x_{1})\phi_{D_{(s)}}(x_{2},b_{2})$ (20) $\displaystyle\times$ $\displaystyle\Bigl{\\{}\left[1-rx_{2}\right]\cdot h_{1}(x_{1},x_{2},b_{1},b_{2})\cdot\alpha_{s}(t_{1})\exp\left[-S_{ab}(t_{1})\right]$ $\displaystyle+$ $\displaystyle\left[2r-x_{1}\right]\cdot h_{2}(x_{1},x_{2},b_{1},b_{2})\cdot\alpha_{s}(t_{2})\exp\left[-S_{ab}(t_{2})\right]\Bigr{\\}}\;,$ where $C_{F}=4/3$ is the color factor. The functions $h_{1}$ and $h_{2}$, the scales $t_{1}$, $t_{2}$ and the Sudakov factors $S_{ab}$ are the same as those given in Refs. prd67-054028 ; cpc37-093102 . The expressions of form factors $V(q^{2}),A_{0,1,2}(q^{2})$ and $T_{1,2,3}(q^{2})$ for the $B_{c}\to D^{}_{(s)}$ transition in the pQCD approach are: $\displaystyle V(q^{2})$ $\displaystyle=$ $\displaystyle 8\pi m_{B_{c}}^{2}C_{F}(1+r)\int dx_{1}dx_{2}\int b_{1}db_{1}b_{2}db_{2}\;\phi_{B_{c}}(x_{1})\phi^{T}_{D^{}_{(s)}}(x_{2},b_{2})$ (21) $\displaystyle\times$ $\displaystyle\Bigl{\\{}\left[1-rx_{2}\right]\cdot h_{1}(x_{1},x_{2},b_{1},b_{2})\cdot\alpha_{s}(t_{1})\exp\left[-S_{ab}(t_{1})\right]$ $\displaystyle+$ $\displaystyle r\cdot h_{2}(x_{1},x_{2},b_{1},b_{2})\cdot\alpha_{s}(t_{2})\exp\left[-S_{ab}(t_{2})\right]\Bigr{\\}}\;,$ $\displaystyle A_{0}(q^{2})$ $\displaystyle=$ $\displaystyle 8\pi m_{B_{c}}^{2}C_{F}\int dx_{1}dx_{2}\int b_{1}db_{1}b_{2}db_{2}\;\phi_{B_{c}}(x_{1})\phi^{L}_{D^{}_{(s)}}(x_{2},b_{2})$ (22) $\displaystyle\times$ $\displaystyle\Bigl{\\{}\left[1-rx_{2}(r-2\eta)+r\left(1-2x_{2}\right)\right]$ $\displaystyle\times$ $\displaystyle h_{1}(x_{1},x_{2},b_{1},b_{2})\cdot\alpha_{s}(t_{1})\exp\left[-S_{ab}(t_{1})\right]$ $\displaystyle+$ $\displaystyle\left[r^{2}+x_{1}(1-2r\eta)\right]\cdot h_{2}(x_{1},x_{2},b_{1},b_{2})\cdot\alpha_{s}(t_{2})\exp\left[-S_{ab}(t_{2})\right]\Bigr{\\}}\;,$ $\displaystyle A_{1}(q^{2})$ $\displaystyle=$ $\displaystyle 16\pi m_{B_{c}}^{2}C_{F}\frac{r}{1+r}\int dx_{1}dx_{2}\int b_{1}db_{1}b_{2}db_{2}\;\phi_{B_{c}}(x_{1})\phi^{T}_{D^{}_{(s)}}(x_{2},b_{2})$ (23) $\displaystyle\times$ $\displaystyle\Bigl{\\{}\left[1+rx_{2}\eta-2rx_{2}+\eta\right]\cdot h_{1}(x_{1},x_{2},b_{1},b_{2})\cdot\alpha_{s}(t_{1})\exp[-S_{ab}(t_{1})]$ $\displaystyle+$ $\displaystyle\left[r\eta-x_{1}\right]\cdot h_{2}(x_{1},x_{2},b_{1},b_{2})\cdot\alpha_{s}(t_{2})\exp[-S_{ab}(t_{2})]\Bigr{\\}}\;,$ $\displaystyle A_{2}(q^{2})$ $\displaystyle=$ $\displaystyle\frac{(1+r)^{2}(\eta-r)}{2r(\eta^{2}-1)}\cdot A_{1}(q^{2})-8\pi m_{B_{c}}^{2}C_{F}\frac{1+r}{\eta^{2}-1}\int dx_{1}dx_{2}\int b_{1}db_{1}b_{2}db_{2}\;\phi_{B_{c}}(x_{1})$ (24) $\displaystyle\times$ $\displaystyle\phi^{L}_{D^{}_{(s)}}(x_{2},b_{2})\cdot\Bigl{\\{}\left[\eta(1-r^{2}x_{2})-rx_{2}(1-2\eta^{2}-2r)+(1-r)-r\eta(1+2x_{2})\right]$ $\displaystyle\times$ $\displaystyle h_{1}(x_{1},x_{2},b_{1},b_{2})\cdot\alpha_{s}(t_{1})\exp\left[-S_{ab}(t_{1})\right]$ $\displaystyle+$ $\displaystyle\left[r(1-x_{1}+2x_{1}\eta^{2})-\eta(r^{2}+x_{1})\right]$ $\displaystyle\times$ $\displaystyle h_{2}(x_{1},x_{2},b_{1},b_{2})\cdot\alpha_{s}(t_{2})\exp\left[-S_{ab}(t_{2})\right]\Bigr{\\}}\;,$ $\displaystyle T_{1}(q^{2})$ $\displaystyle=$ $\displaystyle 8\pi m_{B_{c}}^{2}C_{F}\int dx_{1}dx_{2}\int b_{1}db_{1}b_{2}db_{2}\;\phi_{B_{c}}(x_{1})\phi^{T}_{D^{}_{(s)}}(x_{2},b_{2})$ (25) $\displaystyle\times$ $\displaystyle\Bigl{\\{}\left[1+r(1-x_{2}(2+r-2\eta))\right]\cdot h_{1}(x_{1},x_{2},b_{1},b_{2})\cdot\alpha_{s}(t_{1})\exp\left[-S_{ab}(t_{1})\right]$ $\displaystyle+$ $\displaystyle r\left[1-x_{1}\right]\cdot h_{2}(x_{1},x_{2},b_{1},b_{2})\cdot\alpha_{s}(t_{2})\exp\left[-S_{ab}(t_{2})\right]\Bigr{\\}}\;,$ $\displaystyle T_{2}(q^{2})$ $\displaystyle=$ $\displaystyle 16\pi m_{B_{c}}^{2}C_{F}\frac{r}{1-r^{2}}\int dx_{1}dx_{2}\int b_{1}db_{1}b_{2}db_{2}\;\phi_{B_{c}}(x_{1})\phi^{T}_{D^{}_{(s)}}(x_{2},b_{2})$ (26) $\displaystyle\times$ $\displaystyle\Bigl{\\{}\left[(1-r)(1+\eta)+2rx_{2}(r-\eta)+rx_{2}(2\eta^{2}-r\eta-1)\right]$ $\displaystyle\times$ $\displaystyle h_{1}(x_{1},x_{2},b_{1},b_{2})\cdot\alpha_{s}(t_{1})\exp\left[-S_{ab}(t_{1})\right]$ $\displaystyle+$ $\displaystyle\left[r(1+x_{1})\eta-r^{2}-x_{1}\right]\cdot h_{2}(x_{1},x_{2},b_{1},b_{2})\cdot\alpha_{s}(t_{2})\exp\left[-S_{ab}(t_{2})\right]\Bigr{\\}}\;,$ $\displaystyle T_{3}(q^{2})$ $\displaystyle=$ $\displaystyle\frac{r+\eta}{r}\cdot\frac{1-r^{2}}{2(\eta^{2}-1)}\cdot T_{2}(q^{2})-\frac{1-r^{2}}{(\eta^{2}-1)}$ (27) $\displaystyle\times$ $\displaystyle 8\pi m_{B_{c}}^{2}C_{F}\int dx_{1}dx_{2}\int b_{1}db_{1}b_{2}db_{2}\;\phi_{B_{c}}(x_{1})\phi^{L}_{D^{}_{(s)}}(x_{2},b_{2})$ $\displaystyle\times$ $\displaystyle\Bigl{\\{}\left[1+rx_{2}(\eta-2)+\eta\right]\cdot h_{1}(x_{1},x_{2},b_{1},b_{2})\cdot\alpha_{s}(t_{1})\exp\left[-S_{ab}(t_{1})\right]$ $\displaystyle+$ $\displaystyle\left[x_{1}\eta-r\right]\cdot h_{2}(x_{1},x_{2},b_{1},b_{2})\cdot\alpha_{s}(t_{2})\exp\left[-S_{ab}(t_{2})\right]\Bigr{\\}}\;.$ One should note that the expressions for the form factors $f_{1,2}(q^{2})$, $F_{T}(q^{2})$, $V(q^{2})$, $A_{0,1,2}(q^{2})$ and $T_{1,2,3}(q^{2})$ given in Eqs. (18)-(27) are the results at leading order of the pQCD approach. The next-to-leading-order contributions to the form factors of $B\to(\pi,K,\eta^{(\prime)})$ transitions given in Refs. prd85-074004 ; prd86-114025 ; prd87-097501 are not available here because of the large mass of $c$-quark and $(D_{(s)},D^{}_{(s)})$ mesons. One should note that the pQCD predictions for the considered form factors are reliable only for the small values of $q^{2}$. For the form factors in the large-$q^{2}$ region, one has to make an extrapolation for them from the low-$q^{2}$ region to large-$q^{2}$ region. In this work we make the extrapolation by using the formula in Refs. cpc37-093102 ; prd79-054012 $\displaystyle F(q^{2})=F(0)\cdot\exp{\left[a\cdot q^{2}+b\cdot(q^{2})^{2}\right]},$ (28) where $F$ stands for the form factors $F_{0,+,T},V,A_{0,1,2}$ and $T_{1,2,3}$, and $a,b$ are the constants to be determined by the fitting procedure. The $B_{c}^{-}\to\bar{D}^{0}l^{-}\bar{\nu}_{l}$ and $B_{c}^{-}\to\bar{D}^{0}l^{-}\bar{\nu}_{l}$ decays are from the quark level $b\to ul^{-}\bar{\nu}$ charged current transition. The effective Hamiltonian for such transition is eff-ham $\displaystyle{\cal H}_{eff}(b\to ul^{-}\bar{\nu}_{l})=\frac{G_{F}}{\sqrt{2}}V_{ub}\;\bar{u}\gamma_{\mu}(1-\gamma_{5})b\cdot\bar{l}\gamma^{\mu}(1-\gamma_{5})\nu_{l},$ (29) where $G_{F}=1.16637\times 10^{-5}GeV^{-2}$ is the Fermi-coupling constant and $V_{ub}$ is one of the CKM matrix elements. With the form factors calculated in Eqs. (18,19,21-24), one can easily get the differential decay width expression for $B_{c}^{-}\to\bar{D}^{0}l^{-}\bar{\nu}_{l}$ and $B_{c}^{-}\to\bar{D}^{0}l^{-}\bar{\nu}_{l}$. For those flavor-changing neutral-current one-loop decay modes, such as $B_{c}\to D^{()}l^{+}l^{-}$ and $B_{c}\to D_{s}^{()}l^{+}l^{-}$, are transitions of $b\to dl^{+}l^{-}$ and $b\to sl^{+}l^{-}$ at quark level, respectively. The effective Hamiltonians and the corresponding differential decay widths are more complicated, we refer the readers to Refs. prd86-114025 ; prd53-3672 ; npb612-25 ; epjc40-565 ; prd61-074024 . For the decay modes of $B_{c}\to D^{()}_{s}\nu\bar{\nu}$, the effective Hamiltonian is eff-ham $\displaystyle{\cal H}_{eff}(b\to s\nu\bar{\nu})$ $\displaystyle=$ $\displaystyle\frac{G_{F}}{\sqrt{2}}\frac{\alpha_{em}}{2\pi\sin^{2}(\theta_{W})}V_{tb}V_{ts}^{}\eta_{X}X(x_{t})\;\left[{\bar{s}}\gamma^{\mu}(1-\gamma_{5})b\right]\left[{\bar{\nu}}\gamma_{\mu}(1-\gamma_{5})\nu\right]$ (30) where $\theta_{W}$ is the Weinberg angle with $\sin^{2}(\theta_{W})=0.231$ pdg2012 , $V_{tb}$ and $V_{ts}$ are CKM matrix elements and $\alpha_{em}\approx 1/137$ is the fine structure constant. The function $X(x_{t})$ can be found in Ref. eff-ham , while $\eta_{X}\approx 1$ is the QCD radiative correction factor eff-ham . As for the decay modes of $B_{c}\to D^{()}\nu\bar{\nu}$, their effective Hamiltonian can be obtained by a simple replacement of $s\to d$ in Eq. (30). The corresponding differential decay widths for $B_{c}\to D_{(s)}\nu\bar{\nu}$ is the same as $B\to\pi(K)\nu\bar{\nu}$ in Ref. prd86-114025 except the replacements $m_{B}\to m_{B_{c}}$ and $m_{P}\to m$. While for the decay modes of $B_{c}\to D^{}_{s}\nu\bar{\nu}$, the differential decay width is jhep0707-072 $\displaystyle\frac{d\Gamma(B_{c}\to D^{}_{s}\nu\bar{\nu})}{dq^{2}}$ $\displaystyle=$ $\displaystyle\frac{G^{2}_{F}\alpha^{2}_{em}}{2^{10}\pi^{5}m^{3}_{B_{c}}}\cdot\left\|\frac{X(x_{t})}{sin^{2}(\theta_{w})}\right\|^{2}\cdot\eta^{2}_{X}\cdot\left\|V_{tb}V^{}_{ts}\right\|^{2}\lambda^{\frac{1}{2}}\bigg{\\{}8\lambda q^{2}\frac{V^{2}}{(m_{B_{c}}+m)^{2}}$ (31) $\displaystyle+\frac{1}{m^{2}}\bigg{[}\lambda^{2}\frac{A^{2}_{2}}{(m_{B_{c}}+m)^{2}}+(m_{B_{c}}+m)^{2}(\lambda+12m^{2}q^{2})\cdot A^{2}_{1}$ $\displaystyle-2\lambda(m^{2}_{B_{c}}-m^{2}-q^{2})\cdot Re[A^{}_{1}A_{2}]\bigg{]}\bigg{\\}}\;,$ where $V,A_{1}$ and $A_{2}$ are the form factors of $B_{c}\to D^{}_{s}$ transition, and the phase-space factor $\displaystyle\lambda=(m^{2}_{B_{c}}+m^{2}-q^{2})^{2}-4m^{2}_{B_{c}}m^{2}\;.$ (32) ## IV Numerical results and discussions In the numerical calculations we adopt the following input parameters pdg2012 $\displaystyle m_{B^{-}_{c}}$ $\displaystyle=$ $\displaystyle 6.277~{}{\rm GeV},\quad m_{\bar{D}^{0}}=1.865~{}{\rm GeV},\quad m_{D^{-}}=1.870~{}{\rm GeV},$ $\displaystyle m_{\bar{D}^{0}}$ $\displaystyle=$ $\displaystyle 2.007~{}{\rm GeV},\quad m_{D^{-}}=2.010~{}{\rm GeV},\quad m_{D_{s}^{-}}=1.969~{}{\rm GeV},$ $\displaystyle m_{\bar{D}_{s}^{-}}$ $\displaystyle=$ $\displaystyle 2.112~{}{\rm GeV},\quad m_{\tau}=1.777~{}{\rm GeV},\quad m_{c}=1.275\pm 0.025~{}{\rm GeV},$ $\displaystyle\tau_{B_{c}}$ $\displaystyle=$ $\displaystyle(0.45\pm 0.04)~{}{\rm ps},$ (33) For the CKM matrix element $V_{ub}$, we adopt the value in Refs. prd86-114025 ; babar-Vub . And we use $\|V_{tb}\|=0.999,\|V_{ts}\|=0.0404$ and $\|V_{td}/V_{ts}\|=0.211$ pdg2012 in this work. As for the decay constant of the $B_{c}$ meson, we adopt $0.489~{}{\rm GeV}$ plb651-171 as its central value, and give it an uncertainty of $0.050~{}{\rm GeV}$. The numerical values of the $B_{c}\to D$ and $B_{c}\to D_{s}$ transition form factors $F_{0,+,T}$ at $q^{2}=0$ and their fitted parameters $a,b$ are listed in Table 1. The numerical values of the form factors $V,A_{0,1,2}$ and $T_{1,2,3}$ at $q^{2}=0$ for the $B_{c}\to D^{}$ and $B_{c}\to D^{}_{s}$ transitions are listed in Table 2. The first error of the pQCD predictions for the form factors in Table 1 and Table 2 is induced by the $B_{c}$ meson wave function parameter $\omega_{B_{c}}=1.0\pm 0.1$; the second error comes from the uncertainty of decay constant $f_{B_{c}}$; the third error comes from the uncertainty of decay constants of the $D^{()}_{(s)}$ mesons; the fourth error in Tables 1 and 2 comes from the uncertainty of $D_{(s)}^{()}$ wave function $C_{D^{()}}=0.5\pm 0.1$ or $C_{D^{()}_{s}}=0.4\pm 0.1$; the fifth error comes from $m_{c}=1.275\pm 0.025$ GeV. The errors from the uncertainty of $\omega_{D^{()}}=0.10\pm 0.02$ or $\omega_{D^{()}_{s}}=0.20\pm 0.04$ are very small that have been neglected. Unlike the form factors at maximum recoil, the extrapolation parameters $a$, $b$ of the form factors are less sensitive to the decay constant and wave function of $D_{(s)}^{()}$ meson. In Tables 1 and 2, we only show uncertainties for the parameter $a$ and $b$ from $B_{c}$ meson wave function parameter $\omega_{B_{c}}$, and from quark mass uncertainty $m_{c}=1.275\pm 0.025$ GeV. As a comparison, we also present some results obtained by other authors based on different methods in Table 3. It is easy to see that our results are consistent with the results in literature. Table 1: The pQCD predictions for form factors $F_{0},F_{+}$ and $F_{T}$ at $q^{2}=0$ and the parametrization constants $a$ and $b$ for $B_{c}\to D$ and $B_{c}\to D_{s}$ transitions. \| $\ \ F(0)$ \| $\ \ a$ \| $\ \ b$ ---\|---\|---\|--- $F_{0}^{B_{c}\to D}$ \| $0.19\pm 0.02\pm 0.02\pm 0.01\pm 0.01\pm 0.01$ \| $0.038\pm 0.001\pm 0.000$ \| $0.0013\pm 0.0001$ $F_{+}^{B_{c}\to D}$ \| $0.19\pm 0.02\pm 0.02\pm 0.01\pm 0.01\pm 0.01$ \| $0.059\pm 0.001\pm 0.001$ \| $0.0020\pm 0.0001$ $F_{T}^{B_{c}\to D}$ \| $0.20\pm 0.02\pm 0.02\pm 0.01\pm 0.01\pm 0.01$ \| $0.070\pm 0.001\pm 0.001$ \| $0.0021^{+0.0000}_{-0.0001}$ $F_{0}^{B_{c}\to D_{s}}$ \| $0.27\pm 0.03\pm 0.03\pm 0.02\pm 0.01\pm 0.01$ \| $0.039\pm 0.002\pm 0.001$ \| $0.0015^{+0.0001}_{-0.0000}$ $F_{+}^{B_{c}\to D_{s}}$ \| $0.27\pm 0.03\pm 0.03\pm 0.02\pm 0.01\pm 0.01$ \| $0.061\pm 0.002\pm 0.001$ \| $0.0023^{+0.0001}_{-0.0000}$ $F_{T}^{B_{c}\to D_{s}}$ \| $0.28\pm 0.03\pm 0.03\pm 0.02\pm 0.01\pm 0.01$ \| $0.073\pm 0.002\pm 0.001$ \| $0.0025^{+0.0000}_{-0.0001}$ Table 2: The pQCD predictions for form factors $A_{0,1,2},V$ and $T_{1,2,3}$ at $q^{2}=0$ and the parametrization constants $a$ and $b$ for $B_{c}\to D^{}$ and $B_{c}\to D^{}_{s}$ transitions. \| $\ F(0)$ \| $\ a$ \| $\ b$ ---\|---\|---\|--- $A_{0}^{B_{c}\to D^{}}$ \| $0.17\pm 0.02\pm 0.02\pm 0.02\pm 0.01\pm 0.00$ \| $0.063\pm 0.001\pm 0.001$ \| $0.0024\pm 0.0000\pm 0.0000$ $A_{1}^{B_{c}\to D^{}}$ \| $0.18\pm 0.02\pm 0.02\pm 0.02\pm 0.01\pm 0.01$ \| $0.043\pm 0.001\pm 0.001$ \| $0.0018\pm 0.0001\pm 0.0001$ $A_{2}^{B_{c}\to D^{}}$ \| $0.20\pm 0.02\pm 0.02\pm 0.02\pm 0.01\pm 0.01$ \| $0.067\pm 0.001\pm 0.001$ \| $0.0026\pm 0.0001\pm 0.0001$ $V^{B_{c}\to D^{}}$ \| $0.25\pm 0.03\pm 0.03\pm 0.03\pm 0.01\pm 0.01$ \| $0.073\pm 0.002\pm 0.001$ \| $0.0029\pm 0.0001\pm 0.0001$ $T_{1}^{B_{c}\to D^{}}$ \| $0.22\pm 0.02\pm 0.02\pm 0.02\pm 0.01\pm 0.01$ \| $0.063\pm 0.001\pm 0.001$ \| $0.0027\pm 0.0001\pm 0.0001$ $T_{2}^{B_{c}\to D^{}}$ \| $0.22\pm 0.02\pm 0.02\pm 0.02\pm 0.01\pm 0.01$ \| $0.038\pm 0.001\pm 0.001$ \| $0.0017\pm 0.0001\pm 0.0001$ $T_{3}^{B_{c}\to D^{}}$ \| $0.20\pm 0.02\pm 0.02\pm 0.02\pm 0.01\pm 0.01$ \| $0.077\pm 0.002\pm 0.001$ \| $0.0049\pm 0.0001\pm 0.0001$ $A_{0}^{B_{c}\to D^{}_{s}}$ \| $0.21\pm 0.02\pm 0.02\pm 0.02\pm 0.01\pm 0.01$ \| $0.064\pm 0.001\pm 0.001$ \| $0.0031\pm 0.0002\pm 0.0001$ $A_{1}^{B_{c}\to D^{}_{s}}$ \| $0.23\pm 0.02\pm 0.02\pm 0.02\pm 0.01\pm 0.01$ \| $0.044\pm 0.002\pm 0.001$ \| $0.0022\pm 0.0002\pm 0.0001$ $A_{2}^{B_{c}\to D^{}_{s}}$ \| $0.25\pm 0.03\pm 0.03\pm 0.03\pm 0.01\pm 0.01$ \| $0.069\pm 0.002\pm 0.001$ \| $0.0035\pm 0.0002^{+0.0002}_{-0.0001}$ $V^{B_{c}\to D^{}_{s}}$ \| $0.33\pm 0.03\pm 0.03\pm 0.03\pm 0.02\pm 0.01$ \| $0.075\pm 0.002\pm 0.001$ \| $0.0039\pm 0.0002\pm 0.0001$ $T_{1}^{B_{c}\to D^{}_{s}}$ \| $0.28\pm 0.03\pm 0.03\pm 0.03\pm 0.02\pm 0.01$ \| $0.064\pm 0.001\pm 0.001$ \| $0.0035\pm 0.0002\pm 0.0001$ $T_{2}^{B_{c}\to D^{}_{s}}$ \| $0.28\pm 0.03\pm 0.03\pm 0.03\pm 0.02\pm 0.01$ \| $0.039\pm 0.001\pm 0.001$ \| $0.0023\pm 0.0002\pm 0.0001$ $T_{3}^{B_{c}\to D^{}_{s}}$ \| $0.27\pm 0.03\pm 0.03\pm 0.03\pm 0.02\pm 0.01$ \| $0.082\pm 0.002\pm 0.001$ \| $0.0068\pm 0.0002\pm 0.0002$ Table 3: Comparison of $B_{c}\to D^{()}_{(s)}$ transition form factors at $q^{2}=0$ evaluated in this paper with other methods. $B_{c}\to D^{()}$ \| $F_{+}(0)=F_{0}(0)$ \| $F_{T}(0)$ \| $A_{0}(0)$ \| $A_{1}(0)$ \| $A_{2}(0)$ \| $V(0)$ \| $T_{1}(0)=T_{2}(0)$ \| $T_{3}(0)$ ---\|---\|---\|---\|---\|---\|---\|---\|--- pQCD \| $0.19$ \| $0.20$ \| $0.17$ \| $0.18$ \| $0.20$ \| $0.25$ \| $0.22$ \| $0.20$ Ref.prd79-054012 \| $0.16$ \| $-$ \| $0.09$ \| $0.08$ \| $0.07$ \| $0.13$ \| $-$ \| $-$ Ref.prd68-094020 \| $0.14$ \| $-$ \| $0.14$ \| $0.17$ \| $0.19$ \| $0.18$ \| $-$ \| $-$ Ref.prd63-074010 \| $0.189$ \| $-$ \| $0.284$ \| $0.146$ \| $0.158$ \| $0.296$ \| $-$ \| $-$ Ref.epjc51-833 \| $0.35$ \| $-$ \| $0.05$ \| $0.32$ \| $0.57$ \| $0.57$ \| $-$ \| $-$ Ref.hepph-0211021 \| $0.32$ \| $-$ \| $0.35$ \| $0.43$ \| $0.51$ \| $1.66$ \| $-$ \| $-$ Ref.prd79-304004 \| $0.075$ \| $-$ \| $0.081$ \| $0.095$ \| $0.11$ \| $0.16$ \| $-$ \| $-$ $B_{c}\to D_{s}^{()}$ \| $F_{+}(0)=F_{0}(0)$ \| $F_{T}(0)$ \| $A_{0}(0)$ \| $A_{1}(0)$ \| $A_{2}(0)$ \| $V(0)$ \| $T_{1}(0)=T_{2}(0)$ \| $T_{3}(0)$ pQCD \| $0.27$ \| $0.28$ \| $0.21$ \| $0.23$ \| $0.25$ \| $0.33$ \| $0.28$ \| $0.27$ Ref.prd79-054012 \| $0.28$ \| $-$ \| $0.17$ \| $0.14$ \| $0.12$ \| $0.23$ \| $-$ \| $-$ Ref.hepph-0211021 \| $0.45$ \| $-$ \| $0.47$ \| $0.56$ \| $0.65$ \| $2.02$ \| $-$ \| $-$ Ref.prd79-304004 \| $0.15$ \| $-$ \| $0.16$ \| $0.18$ \| $0.20$ \| $0.29$ \| $-$ \| $-$ With the form factors given, it is straightforward to calculate the branching ratios for all the considered semileptonic decays by performing the numerical integration over the whole range of $q^{2}$. For the $b\to u$ charged current process, with $l=(e,\mu)$, the decay rates are the following: $\displaystyle Br(B^{-}_{c}\to\bar{D}^{0}l^{-}\bar{\nu}_{l})$ $\displaystyle=$ $\displaystyle(3.15^{+0.97}_{-0.72}(\omega_{B_{c}})^{+0.68}_{-0.61}(f_{B_{c}})^{+0.29}_{-0.27}(m_{c})^{+0.31}_{-0.29}(C_{D})^{+0.28}_{-0.27}(f_{D})\pm 0.28(\tau_{B_{c}}))\cdot 10^{-5},$ $\displaystyle Br(B^{-}_{c}\to\bar{D}^{0}\tau^{-}\bar{\nu}_{\tau})$ $\displaystyle=$ $\displaystyle(2.16^{+0.72}_{-0.52}(\omega_{B_{c}})^{+0.46}_{-0.42}(f_{B_{c}})^{+0.22}_{-0.19}(m_{c})^{+0.20}_{-0.19}(C_{D})^{+0.19}_{-0.18}(f_{D})\pm 0.19(\tau_{B_{c}}))\cdot 10^{-5},$ $\displaystyle Br(B^{-}_{c}\to\bar{D}^{0}l^{-}\bar{\nu}_{l})$ $\displaystyle=$ $\displaystyle(1.09^{+0.34}_{-0.26}(\omega_{B_{c}})^{+0.23}_{-0.21}(f_{B_{c}})^{+0.13}_{-0.11}(m_{c})\pm 0.10(C_{D^{}})^{+0.23}_{-0.21}(f_{D^{}})\pm 0.10(\tau_{B_{c}}))\cdot 10^{-4},$ $\displaystyle Br(B^{-}_{c}\to\bar{D}^{0}\tau^{-}\bar{\nu}_{\tau})$ $\displaystyle=$ $\displaystyle(0.64^{+0.20}_{-0.16}(\omega_{B_{c}})^{+0.14}_{-0.12}(f_{B_{c}})^{+0.08}_{-0.07}(m_{c})^{+0.06}_{-0.05}(C_{D^{}})^{+0.13}_{-0.12}(f_{D^{}})\pm 0.06(\tau_{B_{c}}))\cdot 10^{-4},$ (34) where the errors come from the uncertainties of $\omega_{B_{c}}=1.0\pm 0.1$, $f_{B_{c}}=0.489\pm 0.050$ GeV, $m_{c}=(1.275\pm 0.025)$ GeV, $C_{D^{()}}=0.5\pm 0.1$, $f_{D}=(206.7\pm 8.9)$ MeV or $f_{D^{}}=(270\pm 27)$ MeV and $\tau_{B_{c}}=(0.45\pm 0.04)$ ps, respectively. For the flavor-changing neutral-current processes, after making the numerical integration over the whole range of $4m_{l}^{2}\leq q^{2}\leq(m_{B_{c}}-m)^{2}$, we get the pQCD predictions for the branching ratios of considered decay modes which are listed in Table 4. The errors of the pQCD predictions in Table 4 come from the uncertainties of $\omega_{B_{c}}$, $m_{c}$, $C_{D^{()}}$ or $C_{D^{()}_{s}}$, $f_{D^{()}}$ or $f_{D_{s}^{()}}$ and $\tau_{B_{c}}$, respectively. Table 4: The pQCD predictions for the branching ratios of the considered decays ($l=e,\mu$). Decay modes \| pQCD predictions ---\|--- $Br(B^{-}_{c}\to D^{-}l^{+}l^{-})$ \| $(3.79^{+1.16}_{-0.86}(\omega_{B_{c}})^{+0.81}_{-0.74}(f_{B_{c}})^{+0.35}_{-0.32}(m_{c})^{+0.37}_{-0.35}(C_{D})^{+0.33}_{-0.32}(f_{D})\pm 0.34(\tau_{B_{c}}))\cdot 10^{-9}$ $Br(B^{-}_{c}\to D^{-}\tau^{+}\tau^{-})$ \| $(1.03^{+0.38}_{-0.27}(\omega_{B_{c}})^{+0.22}_{-0.20}(f_{B_{c}})^{+0.12}_{-0.10}(m_{c})^{+0.09}_{-0.08}(C_{D})\pm 0.09(f_{D})\pm 0.09(\tau_{B_{c}}))\cdot 10^{-9}$ $Br(B^{-}_{c}\to D^{-}\bar{\nu}\nu)$ \| $(3.13^{+0.96}_{-0.71}(\omega_{B_{c}})^{+0.67}_{-0.61}(f_{B_{c}})^{+0.30}_{-0.26}(m_{c})^{+0.31}_{-0.29}(C_{D})^{+0.28}_{-0.26}(f_{D})\pm 0.28(\tau_{B_{c}}))\cdot 10^{-8}$ $Br(B^{-}_{c}\to D_{s}^{-}l^{+}l^{-})$ \| $(1.56^{+0.46}_{-0.36}(\omega_{B_{c}})^{+0.33}_{-0.30}(f_{B_{c}})^{+0.17}_{-0.15}(m_{c})^{+0.13}_{-0.12}(C_{D_{s}})\pm 0.07(f_{D_{s}})\pm 0.14(\tau_{B_{c}}))\cdot 10^{-7}$ $Br(B^{-}_{c}\to D_{s}^{-}\tau^{+}\tau^{-})$ \| $(0.38^{+0.13}_{-0.10}(\omega_{B_{c}})^{+0.08}_{-0.07}(f_{B_{c}})^{+0.05}_{-0.04}(m_{c})\pm 0.03(C_{D_{s}})\pm 0.02(f_{D_{s}})\pm 0.03(\tau_{B_{c}}))\cdot 10^{-7}$ $Br(B^{-}_{c}\to D_{s}^{-}\bar{\nu}\nu)$ \| $(1.29^{+0.39}_{-0.30}(\omega_{B_{c}})^{+0.28}_{-0.25}(f_{B_{c}})^{+0.14}_{-0.12}(m_{c})^{+0.11}_{-0.10}(C_{D_{s}})\pm 0.06(f_{D_{s}})\pm 0.11(\tau_{B_{c}}))\cdot 10^{-6}$ $Br(B^{-}_{c}\to D^{-}l^{+}l^{-})$ \| $(1.21^{+0.36}_{-0.28}(\omega_{B_{c}})^{+0.26}_{-0.23}(f_{B_{c}})^{+0.14}_{-0.12}(m_{c})\pm 0.11(C_{D^{}})^{+0.25}_{-0.23}(f_{D^{}})\pm 0.11(\tau_{B_{c}}))\cdot 10^{-8}$ $Br(B^{-}_{c}\to D^{-}\tau^{+}\tau^{-})$ \| $(0.16^{+0.05}_{-0.04}(\omega_{B_{c}})\pm 0.03(f_{B_{c}})\pm 0.02(m_{c})\pm 0.01(C_{D^{}})^{\pm}0.03(f_{D^{}})\pm 0.01(\tau_{B_{c}}))\cdot 10^{-8}$ $Br(B^{-}_{c}\to D^{-}\bar{\nu}\nu)$ \| $(1.10^{+0.34}_{-0.26}(\omega_{B_{c}})^{+0.24}_{-0.21}(f_{B_{c}})^{+0.13}_{-0.11}(m_{c})\pm 0.10(C_{D^{}})^{+0.23}_{-0.21}(f_{D^{}})\pm 0.10(\tau_{B_{c}}))\cdot 10^{-7}$ $Br(B^{-}_{c}\to D_{s}^{-}l^{+}l^{-})$ \| $(4.40^{+1.40}_{-1.05}(\omega_{B_{c}})^{+0.95}_{-0.85}(f_{B_{c}})^{+0.72}_{-0.57}(m_{c})^{+0.32}_{-0.31}(C_{D^{}_{s}})^{+0.92}_{-0.84}(f_{D^{}_{s}})\pm 0.39(\tau_{B_{c}}))\cdot 10^{-7}$ $Br(B^{-}_{c}\to D_{s}^{-}\tau^{+}\tau^{-})$ \| $(0.52^{+0.18}_{-0.13}(\omega_{B_{c}})^{+0.11}_{-0.10}(f_{B_{c}})^{+0.10}_{-0.08}(m_{c})\pm 0.03(C_{D^{}_{s}})^{+0.11}_{-0.10}(f_{D^{}_{s}})\pm 0.05(\tau_{B_{c}}))\cdot 10^{-7}$ $Br(B^{-}_{c}\to D_{s}^{-}\bar{\nu}\nu)$ \| $(4.04^{+1.30}_{-0.97}(\omega_{B_{c}})^{+0.87}_{-0.78}(f_{B_{c}})^{+0.68}_{-0.53}(m_{c})^{+0.29}_{-0.28}(C_{D^{}_{s}})^{+0.85}_{-0.77}(f_{D_{s}^{}})\pm 0.36(\tau_{B_{c}}))\cdot 10^{-6}$ From the pQCD predictions for the form factors $F_{0,+,T}$ in Table 1, the form factors $V,A_{0,1,2}$ and $T_{1,2,3}$ in Table 2 and the pQCD predictions for the branching ratios as listed in Eq. (IV) and in Table 4, we have the following points: 1. (i) All the form factors for the transitions $B_{c}\to D^{()}_{s}$ are larger than the corresponding values for the transitions $B_{c}\to D^{()}$ at $q^{2}=0$, which characterizes the SU(3) breaking effect. 2. (ii) $F_{0}(0)$ equals to $F_{+}(0)$ by definition for the $B_{c}\to D$ or $B_{c}\to D_{s}$ transition, but they have different $q^{2}$ dependence by the different parameters $(a,b)$. $T_{1}(0)$ equals to $T_{2}(0)$ for the $B_{c}\to D^{}$ or $B_{c}\to D^{}_{s}$ transition claimed by the Eq. (17) as they are given in Table 2 although their expressions are different in Eqs.(25, 26). 3. (iii) Because of the phase space suppression, the branching ratios of the decay modes with a $\tau$ in the final product are smaller than those decay modes with electron or muon in the final product for the the charged current process. And for the flavor changing neutral current processes, with two $\tau$’s in the final product, the branching ratios are much smaller than the corresponding decays with electron or muon pairs in the final product. 4. (iv) The branching ratios of the decay modes with $\bar{\nu}\nu$ are almost an order magnitude larger than the corresponding decays with $l^{+}l^{-}$ after the summation over the three neutrino generations. Because of the strong suppression of the CKM factor $\|V_{td}/V_{ts}\|^{2}=\|0.211\|^{2}$ pdg2012 , the branching ratios for the decay modes with $b\to d$ transitions are much smaller than those decay modes with the $b\to s$ transitions. In order to reduce the theoretical uncertainty of the form factor calculations, we define two ratios $R_{D}$ and $R_{D^{}}$ among the branching ratios for the the charged-current processes $\displaystyle R_{D}$ $\displaystyle=$ $\displaystyle\frac{Br(B^{-}_{c}\to\bar{D}^{0}\tau^{-}\bar{\nu}_{\tau})}{Br(B^{-}_{c}\to\bar{D}^{0}l^{-}\nu_{l})}=0.69\pm 0.01(\omega_{B_{c}})^{+0.01}_{-0.00}(m_{c})\;,$ (35) $\displaystyle R_{D^{}}$ $\displaystyle=$ $\displaystyle\frac{Br(B^{-}_{c}\to\bar{D}^{0}\tau^{-}\bar{\nu}_{\tau})}{Br(B^{-}_{c}\to\bar{D}^{0}l^{-}\nu_{l})}=0.59^{+0.00}_{-0.01}(\omega_{B_{c}})^{+0.00}_{-0.01}(m_{c})\;,$ (36) with $l=(e,\mu)$. These two relations will be tested by experiments. ## V Summary In this paper we studed the $B_{c}\to(D_{(s)},D^{}_{(s)})$ transition form factors $F_{0,+,T}(q^{2})$ and $V(q^{2}),A_{0,1,2}(q^{2}),T_{1,2,3}(q^{2})$ in the pQCD factorization approach based on $k_{T}$ factorization. The pQCD predictions for the values of the $B_{c}\to D_{(s)}$ and $B_{c}\to D^{}_{(s)}$ transition form factors agree with those obtained using other methods. Utilizing these form factors, we calculated the branching ratios for all the semileptonic decays of $B_{c}^{+}\to D^{()}_{(s)}(l^{+}\nu_{l},l^{+}l^{-},\nu\bar{\nu})$. Because of phase space suppression, the production ratios of the decay modes with lepton $\tau$ in the final product are smaller than the corresponding decays with electron or muon in the final product. The branching ratios of the decay modes with $\bar{\nu}\nu$ are almost an order magnitude larger than the corresponding decays with $l^{+}l^{-}$ after the summation over the three neutrino generations. The branching ratios for the decays with the $b\to d$ transition are much smaller than those with the $b\to s$ transitions. In order to reduce the theoretical uncertainty of the pQCD predictions, we defined two ratios $R_{D}$ and $R_{D^{}}$ among the branching ratios for the the charged-current processes. The pQCD predictions are $\displaystyle R_{D}$ $\displaystyle=$ $\displaystyle\frac{Br(B^{-}_{c}\to\bar{D}^{0}\tau^{-}\bar{\nu}_{\tau})}{Br(B^{-}_{c}\to\bar{D}^{0}l^{-}\nu_{l})}\approx 0.7\;,$ (37) $\displaystyle R_{D^{}}$ $\displaystyle=$ $\displaystyle\frac{Br(B^{-}_{c}\to\bar{D}^{0}\tau^{-}\bar{\nu}_{\tau})}{Br(B^{-}_{c}\to\bar{D}^{0}l^{-}\nu_{l})}\approx 0.6\;,$ (38) with $l=(e,\mu)$. It would possible to test these predictions by LHCb and the forthcoming Super-B experiments. ###### Acknowledgements. This work is supported in part by the National Natural Science Foundation of China under Grant No. 11375208, 11228512, and 11235005. ## References * (1) I.P. Gouz, V.V. Kiselev, A.K. Likhoded, V.I. Romanovsky, and O.P. Yushchenko, Phys. At. Nucl. 67, 1559 (2004). * (2) N. 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1401.0479	Rational curves on hyperkähler manifolds Ekaterina Amerik111Partially supported by AG Laboratory NRU-HSE, RF government grant, ag. 11.G34.31.0023, and the NRU HSE Academic Fund Program for 2013-2014, research grant No. 12-01-0107., Misha Verbitsky222Partially supported by RFBR grants 12-01-00944- , NRU-HSE Academic Fund Program in 2013-2014, research grant 12-01-0179, and AG Laboratory NRU-HSE, RF government grant, ag. 11.G34.31.0023. Keywords: hyperkähler manifold, moduli space, period map, Torelli theorem 2010 Mathematics Subject Classification: 53C26, 32G13 Abstract Let $M$ be an irreducible holomorphically symplectic manifold. We show that all faces of the Kähler cone of $M$ are hyperplanes $H_{i}$ orthogonal to certain homology classes, called monodromy birationally minimal (MBM) classes. Moreover, the Kähler cone is a connected component of a complement of the positive cone to the union of all $H_{i}$. We provide several characterizations of the MBM-classes. We show the invariance of MBM property by deformations, as long as the class in question stays of type $(1,1)$. For hyperkähler manifolds with Picard group generated by a negative class $z$, we prove that $\pm z$ is ${\mathbb{Q}}$-effective if and only if it is an MBM class. We also prove some results towards the Morrison-Kawamata cone conjecture for hyperkähler manifolds. ###### Contents 1. 1 Introduction 1. 1.1 Kähler cone and MBM classes: an introduction 2. 1.2 Hyperkähler manifolds 3. 1.3 Main results 2. 2 Global Torelli theorem, hyperkähler structures and monodromy group 1. 2.1 Hyperkähler structures and twistor spaces 2. 2.2 Global Torelli theorem and monodromy 3. 3 Morrison-Kawamata cone conjecture for hyperkähler manifolds 1. 3.1 Morrison-Kawamata cone conjecture 2. 3.2 Morrison-Kawamata cone conjecture for K3 surfaces 3. 3.3 Finiteness of polyhedral tesselations 4. 3.4 Morrison-Kawamata cone conjecture: birational version 4. 4 Deformation spaces of rational curves 5. 5 Twistor lines in the Teichmüller space 1. 5.1 Twistor lines and 3-dimensional planes in $H^{2}(M,{\mathbb{R}})$. 2. 5.2 Twistor lines in $\operatorname{\sf Teich}_{z}$. 3. 5.3 Twistor lines and MBM classes. 6. 6 Monodromy group and the Kähler cone 1. 6.1 Geometry of Kähler-Weyl chambers 2. 6.2 Morrison-Kawamata cone conjecture and minimal curves ## 1 Introduction ### 1.1 Kähler cone and MBM classes: an introduction Let $M$ be a hyperkähler manifold, that is, a compact, holomorphically symplectic Kähler manifold. We assume that $\pi_{1}(M)=0$ and $H^{2,0}(M)={\mathbb{C}}$ (the general case reduces to this by 1.2). In this paper we give a description of the Kähler cone of $M$ in terms of a set of cohomology classes $S\subset H^{2}(M,{\mathbb{Z}})$ called MBM classes (1.3). This set is of topological nature, that is, it depends only on the deformation type of $M$. It is known since [Mori] that the Kähler cone of a Fano manifold is polyhedral. Each of its finitely many faces is formed by classes vanishing on a certain rational curve: one says that the numerical class of such a curve generates an extremal ray of the Mori cone. The notion of an extremal ray also makes sense for projective manifolds which are not Fano: the number of extremal rays then does not have to be finite. However, they are discrete in the half-space where the canonical class restricts negatively. Huybrechts [H3] and Boucksom [Bou1] have studied the Kähler cone of hyperkähler manifolds (not necessarily algebraic). They have proved that the Kähler classes are exactly those positive classes (i.e. classes with positive Beauville-Bogomolov-Fujiki square; see 1.2) which restrict positively to all rational curves (1.2). Our work puts these results in a deformation-invariant setting. Let $\operatorname{Pos}\subset H^{1,1}(M)$ be the positive cone, and $S(I)$ the set of all MBM classes which are of type (1,1) on $M$ with its given complex structure $I$. Then the Kähler cone is a connected component of $\operatorname{Pos}\backslash S(I)^{\bot}$, where $S(I)^{\bot}$ is the union of all orthogonal complements to all $z\in S(I)$. We describe the MBM classes in terms of the minimal curves on deformations of $M$, birational maps and the monodromy group action (Sections 4, 5), and formulate a finiteness conjecture (6.2) claiming that primitive integral MBM classes have bounded square. We deduce the Morrison-Kawamata cone conjecture from this conjecture. For deformations of the Hilbert scheme of points on a K3 surface our finiteness conjecture is known ([BHT, Proposition 2]). This gives a proof of Morrison-Kawamata cone conjecture for deformations of the Hilbert scheme of K3 (1.3). Its proof is independently obtained by Markman and Yoshioka using different methods (forthcoming). ### 1.2 Hyperkähler manifolds Definition 1.1: A hyperkähler manifold is a compact, Kähler, holomorphically symplectic manifold. Definition 1.2: A hyperkähler manifold $M$ is called simple if $\pi_{1}(M)=0$, $H^{2,0}(M)={\mathbb{C}}$. This definition is motivated by the following theorem of Bogomolov. Theorem 1.3: ([Bo1]) Any hyperkähler manifold admits a finite covering which is a product of a torus and several simple hyperkähler manifolds. Remark 1.4: Further on, we shall assume (sometimes, implicitly) that all hyperkähler manifolds we consider are simple. Remark 1.5: A hyperkähler manifold naturally possesses a whole 2-sphere of complex structures (see Section 2). We shall use the notation $(M,I)$ or $M_{I}$ to stress that a particular complex structure is chosen, and $M$ to discuss the topological properties (or simply when there is no risque of confusion). The Bogomolov-Beauville-Fujiki form was defined in [Bo2] and [Bea], but it is easiest to describe it using the Fujiki theorem, proved in [F1]. Theorem 1.6: (Fujiki) Let $M$ be a simple hyperkähler manifold, $\eta\in H^{2}(M)$, and $n=\frac{1}{2}\dim M$. Then $\int_{M}\eta^{2n}=cq(\eta,\eta)^{n}$, where $q$ is a primitive integer quadratic form on $H^{2}(M,{\mathbb{Z}})$, and $c>0$ an integer. Remark 1.7: Fujiki formula (1.2) determines the form $q$ uniquely up to a sign. For odd $n$, the sign is unambiguously determined as well. For even $n$, one needs the following explicit formula, which is due to Bogomolov and Beauville. $\displaystyle\lambda q(\eta,\eta)$ $\displaystyle=\int_{X}\eta\wedge\eta\wedge\Omega^{n-1}\wedge\overline{\Omega}^{n-1}-$ (1.1) $\displaystyle-\frac{n-1}{n}\left(\int_{X}\eta\wedge\Omega^{n-1}\wedge\overline{\Omega}^{n}\right)\left(\int_{X}\eta\wedge\Omega^{n}\wedge\overline{\Omega}^{n-1}\right)$ where $\Omega$ is the holomorphic symplectic form, and $\lambda>0$. Definition 1.8: A cohomology class $\eta\in H^{1,1}_{\mathbb{R}}(M,I)$ is called negative if $q(\eta,\eta)<0$, and positive if $q(\eta,\eta)>0$. Since the signature of $q$ on $H^{1,1}(M,I)$ is $(1,b_{2}-3)$, the set of positive vectors is disconnected. The positive cone $\operatorname{Pos}(M,I)$ is the connected component of the set $\\{\eta\in H^{1,1}_{\mathbb{R}}(M,I)\ \ \|\ \ q(\eta,\eta)>0\\}$ which contains the classes of the Kähler forms. It is easy to check that the positive cone is convex. The following theorem is crucial for our work. Theorem 1.9: (Huybrechts, Boucksom; see [Bou1]) Let $(M,I)$ be a simple hyperkähler manifold. The Kähler cone $\operatorname{Kah}(M,I)$ can be described as follows: $\operatorname{Kah}(M,I)=\\{\alpha\in\operatorname{Pos}(M,I)\|\alpha\cdot C>0\ \forall C\in RC(M,I)\\}$ where $RC(M,I)$ denotes the set of classes of rational curves on $(M,I)$. ### 1.3 Main results Remark 1.10: Let $(M,I)$ be a hyperkähler manifold, and $\varphi:\;(M,I)\dashrightarrow(M,I^{\prime})$ a bimeromorphic (also called “birational”) map to another hyperkähler manifold (note that by a result of Huybrechts [H1], birational hyperkähler manifolds are deformation equivalent; that is why there is the same letter $M$ used for the source and the target of $\varphi$). Since the canonical bundle of $(M,I)$ and $(M,I^{\prime})$ is trivial, $\varphi$ is an isomorphism in codimension 1 (see for example [H1], Lemma 2.6). This allows one to identify $H^{2}(M,I)$ and $H^{2}(M,I^{\prime})$. Clearly, this identification is compatible with the Hodge structure. Further on, we call $(M,I^{\prime})$ “a birational model” for $(M,I)$, and identify $H^{2}(M)$ for all birational models. Definition 1.11: Let $M$ be a hyperkähler manifold. The monodromy group of $M$ is a subgroup of $GL(H^{2}(M,{\mathbb{Z}}))$ generated by monodromy transforms for all Gauss-Manin local systems. This group can also be characterized in terms of the mapping class group action (2.2). Definition 1.12: Let $(M,I)$ be a hyperkähler manifold. A rational homology class $z\in H_{1,1}(M,I)$ is called ${\mathbb{Q}}$-effective if $Nz$ can be represented as a homology class of a curve, for some $N\in{\mathbb{Z}}^{>0}$, and extremal if for any ${\mathbb{Q}}$-effective homology classes $z_{1},z_{2}\in H_{1,1}(M,I)$ satisfying $z_{1}+z_{2}=z$, the classes $z_{1},z_{2}$ are proportional. In the projective case, a negative extremal class is ${\mathbb{Q}}$-effective and some multiple of it is represented by a rational curve. Moreover the negative part of the cone of ${\mathbb{Q}}$-effective classes is locally rational polyhedral. This is shown by a version of Mori theory adapted to the case of hyperkähler manifolds (see [HT]). The Beauville-Bogomolov-Fujiki form allows one to identify $H^{2}(M,{\mathbb{Q}})$ and $H_{2}(M,{\mathbb{Q}})$. More precisely, it provides an embedding $H^{2}(M,{\mathbb{Z}})\to H_{2}(M,{\mathbb{Z}})$ which is not an isomorphism (indeed $q$ is not necessarily unimodular) but becomes an isomorphism after tensoring with ${\mathbb{Q}}$. We thus can talk of extremal classes in $H^{1,1}(M,{\mathbb{Q}})$, meaning that the corresponding classes in $H_{1,1}(M,{\mathbb{Q}})$ are extremal. In general, we shall switch from the second homology to the second cohomology as it suits us and transfer all definitions from one situation to the other one by means of the BBF form, with one important exception, namely, the notion of effectiveness. Indeed, an effective class in $H_{1,1}(M,{\mathbb{Q}})$ is a class of a curve, whereas an effective class in $H^{1,1}(M,{\mathbb{Q}})$ is a class of a hypersurface. We also shall extend the BBF form to $H_{2}(M,{\mathbb{Z}})$ as a rational- valued quadratic form, and to $H_{2}(M,{\mathbb{Q}})$, without further notice. The following property, with which we shall work in this paper, looks stronger than extremality modulo monodromy and birational equivalence, but is equivalent to it whenever the negative part of the Mori cone is locally rational polyhedral. Recall that a face of a convex cone in a vector space $V$ is an intersection of its boundary and a hyperplane which has non-empty interior (3.1). Definition 1.13: A non-zero negative rational homology class $z\in H^{1,1}(M,I)$ is called monodromy birationally minimal (MBM) if for some isometry $\gamma\in O(H^{2}(M,{\mathbb{Z}}))$ belonging to the monodromy group, $\gamma(z)^{\bot}\subset H^{1,1}(M,I)$ contains a face of the Kähler cone of one of birational models $(M,I^{\prime})$ of $(M,I)$. The definition has an obvious counterpart for homology classes of type $(1,1)$, where one replaces the orthogonality with respect to the BBF form by the usual duality between homology and cohomology, given by integration of forms over cycles. Remark 1.14: A face of $\operatorname{Kah}(M,I^{\prime})$ is, by definition, of maximal dimension $h^{1,1}(M,I^{\prime})-1$. So the definition of $z$ being MBM means that $\gamma(z)^{\bot}\cap\partial\operatorname{Kah}(M,I^{\prime})$ contains an open subset of $\gamma(z)^{\bot}$. The point of 1.3 is, roughly, as follows. As we shall see, rational curves have nice local deformation properties when they are minimal (extremal). However, globally a deformation of an extremal rational curve on a variety $(M,I)$ does not have to remain extremal on a deformation $(M,I^{\prime})$. One can only hope to show the deformation-invariance of the property of $z$ being extremal (as long, of course, as it remains of type $(1,1)$) modulo monodromy and birational equivalence. In what follows we shall solve this problem for MBM-classes and apply it to the study of the Kähler cone. The deformation equivalence is especially useful because when $\operatorname{Pic}(M)$ has rank one, the somewhat obscure notion of monodromy birationally minimal classes becomes much more streamlined. Theorem 1.15: Let $(M,I)$ be a hyperkähler manifold, $\operatorname{rk}\operatorname{Pic}(M,I)=1$, and $z\in H_{1,1}(M,I)$ a non- zero negative rational class. Then $z$ is monodromy birationally minimal if and only if $\pm z$ is ${\mathbb{Q}}$-effective. Proof: See 5.3. Definition 1.16: Let $(M,I)$ be a hyperkähler manifold. A negative rational class $z\in H^{1,1}_{\mathbb{Q}}(M,I)$ is called divisorial if $z=\lambda[D]$ for some effective divisor $D$ and $\lambda\in{\mathbb{Q}}$. Our first main results concern the deformational invariance of these notions. Theorem 1.17: Let $(M,I)$ be a hyperkähler manifold, $z\in H_{1,1}(M,I)$ an integer homology class, $q(z,z)<0$, and $I^{\prime}$ a deformation of $I$ such that $z$ is of type (1,1) with respect to $I^{\prime}$. Assume that $z$ is monodromy birationally minimal on $(M,I)$. Then $z$ is monodromy birationally minimal on $(M,I^{\prime})$. The property of $z\in H^{1,1}(M,I)$ being divisorial is likewise deformation-invariant, provided that one restricts oneself to the complex structures with Picard number one (i.e. such that the Picard group is generated by $z$ over ${\mathbb{Q}}$). Proof: See 4, 4 and 5.3. Remark 1.18: We expect that the property of $z$ being divisorial is deformation-invariant as long as $z$ stays of type $(1,1)$, and plan to return to this question in a forthcoming paper. The MBM classes can be used to determine the Kähler cone of $(M,I)$ explicitly. Theorem 1.19: Let $(M,I)$ be a hyperkähler manifold, and $S\subset H_{1,1}(M,I)$ the set of all MBM classes. Consider the corresponding set of hyperplanes $S^{\bot}:=\\{W=z^{\bot}\ \ \|\ \ z\in S\\}$ in $H^{1,1}(M,I)$. Then the Kähler cone of $(M,I)$ is a connected component of $\operatorname{Pos}(M,I)\backslash\cup S^{\bot}$, where $\operatorname{Pos}(M,I)$ is the positive cone of $(M,I)$. Moreover, for any connected component $K$ of $\operatorname{Pos}(M,I)\backslash\cup S^{\bot}$, there exists $\gamma\in O(H^{2}(M,{\mathbb{Z}}))$ in the monodromy group of $M$, and a hyperkähler manifold $(M,I^{\prime})$ birationally equivalent to $(M,I)$, such that $\gamma(K)$ is the Kähler cone of $(M,I^{\prime})$. Proof: See 6.1. Remark 1.20: In particular, $z^{\bot}\cap\operatorname{Pos}(M,I)$ either has dense intersection with the interior of the Kähler chambers (if $z$ is not MBM), or is a union of walls of those (if $z$ is MBM); that is, there are no “barycentric partitions” in the decomposition of the positive cone into the Kähler chambers. Allowed partition Prohibited partition Finally, we apply this to the Morrison-Kawamata cone conjecture: Theorem 1.21: Let $M$ be a simple hyperkähler manifold, and $q$ the Bogomolov-Beauville-Fujiki form. Suppose that there exists $C>0$ such that $\|q(\eta,\eta)\|<C$ for all primitive integral MBM classes (or, alternatively, for all extremal rational curves on all deformations of $M$). Then the Morrison-Kawamata cone conjecture holds for $M$: the group $\operatorname{Aut}(M)$ acts on the set of faces of the Kähler cone with finitely many orbits. Proof: See 6.2. The condition of the theorem is satisfied for manifolds which are deformation equivalent to the Hilbert scheme of points on a K3 surface, see [BHT]; for such manifolds one therefore obtains a proof of the Morrison-Kawamata cone conjecture. ## 2 Global Torelli theorem, hyperkähler structures and monodromy group In this Section, we recall a number of results about hyperkähler manifolds, used further on in this paper. For more details and references, please see [Bes] and [V4]. ### 2.1 Hyperkähler structures and twistor spaces Definition 2.1: Let $(M,g)$ be a Riemannian manifold, and $I,J,K$ endomorphisms of the tangent bundle $TM$ satisfying the quaternionic relations $I^{2}=J^{2}=K^{2}=IJK=-\operatorname{Id}_{TM}.$ The triple $(I,J,K)$ together with the metric $g$ is called a hyperkähler structure if $I,J$ and $K$ are integrable and Kähler with respect to $g$. Consider the Kähler forms $\omega_{I},\omega_{J},\omega_{K}$ on $M$: $\omega_{I}(\cdot,\cdot):=g(\cdot,I\cdot),\ \ \omega_{J}(\cdot,\cdot):=g(\cdot,J\cdot),\ \ \omega_{K}(\cdot,\cdot):=g(\cdot,K\cdot).$ An elementary linear-algebraic calculation implies that the 2-form $\Omega:=\omega_{J}+\sqrt{-1}\>\omega_{K}$ is of Hodge type $(2,0)$ on $(M,I)$. This form is clearly closed and non-degenerate, hence it is a holomorphic symplectic form. In algebraic geometry, the word “hyperkähler” is essentially synonymous with “holomorphically symplectic”, due to the following theorem, which is implied by Yau’s solution of Calabi conjecture ([Bes], [Bea]). Theorem 2.2: Let $M$ be a compact, Kähler, holomorphically symplectic manifold, $\omega$ its Kähler form, $\dim_{\mathbb{C}}M=2n$. Denote by $\Omega$ the holomorphic symplectic form on $M$. Suppose that $\int_{M}\omega^{2n}=\int_{M}(\operatorname{Re}\Omega)^{2n}$. Then there exists a unique hyperkähler metric $g$ with the same Kähler class as $\omega$, and a unique hyperkähler structure $(I,J,K,g)$, with $\omega_{J}=\operatorname{Re}\Omega$, $\omega_{K}=\operatorname{Im}\Omega$. Further on, we shall speak of “hyperkähler manifolds” meaning “holomorphic symplectic manifolds of Kähler type”, and “hyperkähler structures” meaning the quaternionic triples together with a metric. Every hyperkähler structure induces a whole 2-dimensional sphere of complex structures on $M$, as follows. Consider a triple $a,b,c\in R$, $a^{2}+b^{2}+c^{2}=1$, and let $L:=aI+bJ+cK$ be the corresponging quaternion. Quaternionic relations imply immediately that $L^{2}=-1$, hence $L$ is an almost complex structure. Since $I,J,K$ are Kähler, they are parallel with respect to the Levi-Civita connection. Therefore, $L$ is also parallel. Any parallel complex structure is integrable, and Kähler. We call such a complex structure $L=aI+bJ+cK$ a complex structure induced by a hyperkähler structure. There is a 2-dimensional holomorphic family of induced complex structures, and the total space of this family is called the twistor space of a hyperkähler manifold. ### 2.2 Global Torelli theorem and monodromy Definition 2.3: Let $M$ be a compact complex manifold, and $\operatorname{\sf Diff}_{0}(M)$ a connected component of its diffeomorphism group (the group of isotopies). Denote by $\operatorname{\sf Comp}$ the space of complex structures on $M$, equipped with its structure of a Fréchet manifold, and let $\operatorname{\sf Teich}:=\operatorname{\sf Comp}/\operatorname{\sf Diff}_{0}(M)$. We call it the Teichmüller space. Remark 2.4: In many important cases, such as for manifolds with trivial canonical class ([Cat]), $\operatorname{\sf Teich}$ is a finite-dimensional complex space; usually it is non-Hausdorff. Definition 2.5: The universal family of complex manifolds over $\operatorname{\sf Teich}$ is defined as $\operatorname{\sf Univ}/\operatorname{\sf Diff}_{0}(M)$, where $\operatorname{\sf Univ}$ is the natural universal family over $\operatorname{\sf Comp}$ with its Fréchet manifold structure. Locally, it is isomorphic to the universal family over the Kuranishi space.111We are grateful to Claire Voisin for this observation. Definition 2.6: The mapping class group is $\Gamma=\operatorname{\sf Diff}(M)/\operatorname{\sf Diff}_{0}(M)$. It naturally acts on $\operatorname{\sf Teich}$ (the quotient of $\operatorname{\sf Teich}$ by $\Gamma$ may be viewed as the “moduli space” for $M$, but in general it has very bad properties; see below). Remark 2.7: Let $M$ be a hyperkähler manifold (as usually, we assume $M$ to be simple). For any $J\in\operatorname{\sf Teich}$, $(M,J)$ is also a simple hyperkähler manifold, because the Hodge numbers are constant in families. Therefore, $H^{2,0}(M,J)$ is one-dimensional. Definition 2.8: Let $\operatorname{\sf Per}:\;\operatorname{\sf Teich}{\>\longrightarrow\>}{\mathbb{P}}H^{2}(M,{\mathbb{C}})$ map $J$ to the line $H^{2,0}(M,J)\in{\mathbb{P}}H^{2}(M,{\mathbb{C}})$. The map $\operatorname{\sf Per}$ is called the period map. Remark 2.9: The period map $\operatorname{\sf Per}$ maps $\operatorname{\sf Teich}$ into an open subset of a quadric, defined by $\operatorname{{\mathbb{P}}\sf er}:=\\{l\in{\mathbb{P}}H^{2}(M,{\mathbb{C}})\ \ \|\ \ q(l,l)=0,q(l,\overline{l})>0\\}.$ It is called the period domain of $M$. Indeed, any holomorphic symplectic form $l$ satisfies the relations $q(l,l)=0,q(l,\overline{l})>0$, as follows from (1.1). Proposition 2.10: The period domain $\operatorname{{\mathbb{P}}\sf er}$ is identified with the quotient $SO(b_{2}-3,3)/SO(2)\times SO(b_{2}-3,1)$, which is the Grassmannian of positive oriented 2-planes in $H^{2}(M,{\mathbb{R}})$. Proof: This statement is well known, but we shall sketch its proof for the reader’s convenience. Step 1: Given $l\in{\mathbb{P}}H^{2}(M,{\mathbb{C}})$, the space generated by $\operatorname{Im}l,\operatorname{Re}l$ is 2-dimensional, because $q(l,l)=0,q(l,\overline{l})>0$ implies that $l\cap H^{2}(M,{\mathbb{R}})=0$. Step 2: This 2-dimensional plane is positive, because $q(\operatorname{Re}l,\operatorname{Re}l)=q(l+\overline{l},l+\overline{l})=2q(l,\overline{l})>0$. Step 3: Conversely, for any 2-dimensional positive plane $V\in H^{2}(M,{\mathbb{R}})$, the quadric $\\{l\in V\otimes_{\mathbb{R}}{\mathbb{C}}\ \ \|\ \ q(l,l)=0\\}$ consists of two lines; a choice of a line is determined by the orientation. Definition 2.11: Let $M$ be a topological space. We say that $x,y\in M$ are non-separable (denoted by $x\sim y$) if for any open sets $V\ni x,U\ni y$, $U\cap V\neq\emptyset$. Theorem 2.12: (Huybrechts; [H1]). If two points $I,I^{\prime}\in\operatorname{\sf Teich}$ are non-separable, then there exists a bimeromorphism $(M,I)\dasharrow(M,I^{\prime})$. Remark 2.13: The converse is not true since many points in the Teichmüller space correspond to isomorphic manifolds, and they are not always pairwise non-separable (consider orbits of the mapping class group action). But it is true that if two hyperkähler manifolds are bimeromorphic, then one can find non-separable points in the Teichmüller space representing them. Definition 2.14: The space $\operatorname{\sf Teich}_{b}:=\operatorname{\sf Teich}\\!/\\!\\!\sim$ is called the birational Teichmüller space of $M$. Remark 2.15: This terminology is slightly misleading since there are non- separable points of the Teichmüller space which correspond to biregular, not just birational, complex structures. Even for K3 surfaces, the Teichmüller space is non-Hausdorff. Theorem 2.16: (Global Torelli theorem; [V4]) The period map $\operatorname{\sf Teich}_{b}\stackrel{{\scriptstyle\operatorname{\sf Per}}}{{{\>\longrightarrow\>}}}\operatorname{{\mathbb{P}}\sf er}$ is an isomorphism on each connected component of $\operatorname{\sf Teich}_{b}$. Remark 2.17: By a result of Huybrechts ([H4]), $\operatorname{\sf Teich}$ has only finitely many connected components. We shall denote by $\operatorname{\sf Teich}_{I}$ the component containing the parameter point for the complex structure $I$, and by $\Gamma_{I}$ the subgroup of the mapping class group $\Gamma$ fixing this component. Obviously $\Gamma_{I}$ is of finite index in $\Gamma$. Definition 2.18: Let $M$ be a hyperkaehler manifold, $\operatorname{\sf Teich}_{b}$ its birational Teichmüller space, and $\Gamma$ the mapping class group $\operatorname{\sf Diff}(M)/\operatorname{\sf Diff}_{0}(M)$. The quotient $\operatorname{\sf Teich}_{b}/\Gamma$ is called the birational moduli space of $M$. Its points are in bijective correspondence with the complex structures of hyperkähler type on $M$ up to a bimeromorphic equivalence. Remark 2.19: The word “space” in this context is misleading. In fact, the quotient topology on $\operatorname{\sf Teich}_{b}/\Gamma$ is extremely non- Hausdorff, e.g. every two open sets would intersect ([V5]). The Global Torelli theorem can be stated as a result about the birational moduli space. Theorem 2.20: ([V4, Theorem 7.2, Remark 7.4, Theorem 3.5]) Let $(M,I)$ be a hyperkähler manifold, and $W$ a connected component of its birational moduli space. Then $W$ is isomorphic to ${\operatorname{{\mathbb{P}}\sf er}}/\operatorname{\sf Mon}_{I}$, where ${\operatorname{{\mathbb{P}}\sf er}}=SO(b_{2}-3,3)/SO(2)\times SO(b_{2}-3,1)$ and $\operatorname{\sf Mon}_{I}$ is an arithmetic group in $O(H^{2}(M,{\mathbb{R}}),q)$, called the monodromy group of $(M,I)$. In fact $\operatorname{\sf Mon}_{I}$ is the image of $\Gamma_{I}$ in $O(H^{2}(M,{\mathbb{R}}),q)$. Remark 2.21: The monodromy group of $(M,I)$ can be also described as a subgroup of the group $O(H^{2}(M,{\mathbb{Z}}),q)$ generated by monodromy transform maps for Gauss-Manin local systems obtained from all deformations of $(M,I)$ over a complex base ([V4, Definition 7.1]). This is how this group was originally defined by Markman ([M2], [M3]). The fact that it is of finite index in $O(H^{2}(M,{\mathbb{Z}}),q)$ is crucial for the Morrison-Kawamata conjecture, see next section. Remark 2.22: A caution: usually “the global Torelli theorem” is understood as a theorem about Hodge structures. For K3 surfaces, the Hodge structure on $H^{2}(M,{\mathbb{Z}})$ determines the complex structure. For $\dim_{\mathbb{C}}M>2$, it is false. ## 3 Morrison-Kawamata cone conjecture for hyperkähler manifolds In this section, we introduce the Morrison-Kawamata cone conjecture, and give a brief survey of what is known, following [M3] and [T], with some easy generalizations. Originally, this conjecture was stated in [Mo] and proven by Kawamata in [Ka] for Calabi-Yau threefolds admitting a holomorphic fibtation over a positive- dimensional base. ### 3.1 Morrison-Kawamata cone conjecture Definition 3.1: Let $M$ be a compact, Kähler manifold, $\operatorname{Kah}\subset H^{1,1}(M,{\mathbb{R}})$ the Kähler cone, and $\overline{\operatorname{Kah}}$ its closure in $H^{1,1}(M,{\mathbb{R}})$, called the nef cone. A face of the Kähler cone is the intersection of the boundary of $\overline{\operatorname{Kah}}$ and a hyperplane $V\subset H^{1,1}(M,{\mathbb{R}})$, which has non-empty interior. Conjecture 3.2: (Morrison-Kawamata cone conjecture) Let $M$ be a Calabi-Yau manifold. Then the group $\operatorname{Aut}(M)$ of biholomorphic automorphisms of $M$ acts on the set of faces of $\operatorname{Kah}$ with finite number of orbits. This conjecture has a birational version, which has many important implications. For projective hyperkähler manifolds, the birational version of Morrison-Kawamata cone conjecture has been proved by E. Markman in [M3]. Definition 3.3: Let $M,M^{\prime}$ be compact complex manifolds. Define a pseudo-isomorphism $M\dasharrow M^{\prime}$ as a birational map which is an isomorphism outside of codimension $\geqslant 2$ subsets of $M,M^{\prime}$. Remark 3.4: For any pseudo-isomorphic manifolds $M,M^{\prime}$, the second cohomologies $H^{2}(M)$ and $H^{2}(M^{\prime})$ are naturally identified. As we have already remarked, any birational map of hyperkähler varieties is a pseudo-isomorphism; more generally, this is true for all varieties with nef canonical class. Definition 3.5: The movable cone, also known as birational nef cone is the closure of the union of $\operatorname{Kah}(M^{\prime})$ for all $M^{\prime}$ pseudo-isomorphic to $M$. The union of $\operatorname{Kah}(M^{\prime})$ for all $M^{\prime}$ pseudo-isomorphic to $M$ is called birational Kähler cone. Conjecture 3.6: (Morrison-Kawamata birational cone conjecture) Let $M$ be a Calabi-Yau manifold. Then the group $\operatorname{Bir}(M)$ of birational automorphisms of $M$ acts on the set of faces of its movable cone with finite number of orbits. ### 3.2 Morrison-Kawamata cone conjecture for K3 surfaces In this subsection, we prove the Morrison-Kawamata cone conjecture for K3 surfaces. Originally it was proven by Sterk (see [St]). The proof we are giving has two advantages: it works in the non-algebraic situation, and to some extent generalizes to arbitrary dimension. Notice that the pseudo- isomorphisms of smooth surfaces are isomorphisms, hence for K3 surfaces both flavours of Morrison-Kawamata cone conjecture are equivalent. Definition 3.7: A cohomology class $\eta\in H^{2}(M,{\mathbb{Z}})$ on a K3 surface is called a $(-2)$-class if $\int_{M}\eta\wedge\eta=-2$. Remark 3.8: Let $M$ be a K3 surface, and $\eta\in H^{1,1}(M,{\mathbb{Z}})$ a $(-2)$-class. Then either $\eta$ or $-\eta$ is effective. Indeed, $\chi(\eta)=2+\frac{\eta^{2}}{2}=1$ by Riemann-Roch. The following theorem is well-known. Theorem 3.9: Let $M$ be a K3 surface, and $S$ the set of all effective $(-2)$-classes. Then $\operatorname{Kah}(M)$ is the set of all $v\in\operatorname{Pos}(M)$ such that $q(v,s)>0$ for all $s\in S$. Proof: By adjunction formula, a curve $C$ on a K3 surface satisfies $C^{2}=2g-2$, where $g$ is its genus, hence the cone of effective curves is generated by positive classes and effective $(-2)$-classes. By [DP], the Kähler cone is the intersection of the positive cone and the cone dual to the cone of effective curves, hence it is $v\in\operatorname{Pos}(M)$ such that $q(v,s)>0$ for all effective $s$. However, for all $s,v\in\operatorname{Pos}(M)$ the inequality $q(v,s)>0$ automatically follows from the Hodge index formula. Definition 3.10: A Weyl chamber, or Kähler chamber, on a K3 surface is a connected component of $\operatorname{Pos}(M)\backslash S^{\bot}$, where $S^{\bot}$ is the union of all planes $s^{\bot}$ for all $(-2)$-classes $s\in S$. The reflection group of a K3 surface is the group $W$ generated by reflections with respect to all $s\in S$. Remark 3.11: Clearly, a Weyl chamber is a fundamental domain of $W$, and $W$ acts transitively on the set of all Weyl chambers. Moreover, the Kähler cone of $M$ is one of its Weyl chambers. To a certain extent, this is a pattern which is repeated in all dimensions, as we shall see. The following theorem is well-known, too, but we want to sketch a proof in some detail since it is also important for what follows. Theorem 3.12: Let $M$ be a K3 surface, and $\operatorname{Aut}(M)$ the group of all automorphisms of $M$. Then $\operatorname{Aut}(M)$ is the group of all isometries of $H^{2}(M,{\mathbb{Z}})$ preserving the Kähler chamber $\operatorname{Kah}$ and the Hodge decomposition. Proof: First, let us show that the natural map $\operatorname{Aut}(M)\stackrel{{\scriptstyle\varphi}}{{\to}}O(H^{2}(M,{\mathbb{Z}}))$ is injective. Our argument is similar to Buchdahl’s in [Buc]. Clearly, $\operatorname{Aut}(M)$ acts on $H^{2}(M,{\mathbb{Z}})$; its kernel is formed by automorphisms preserving all Kähler classes, hence acting as isometries on all Calabi-Yau metrics on all deformations of $M$. Deforming $M$ to a Kummer surface, we find that this isometry must also induce an isometry on the underlying 2-torus, acting trivially on cohomology. Therefore $\ker\varphi$ acts as identity on all Kummer surfaces, and hence on all K3 surfaces as well. To describe the image of $\varphi$, we use the Torelli theorem. Let $\gamma\in O(H^{2}(M,{\mathbb{Z}}))$ preserve the Hodge decomposition on $(M,I)$ and the Kähler cone. Global Torelli theorem affirms that the period map $\operatorname{\sf Per}:\;\operatorname{\sf Teich}_{b}{\>\longrightarrow\>}\operatorname{{\mathbb{P}}\sf er}$ for K3 surfaces is an isomorphism. The mapping class group $\Gamma=\operatorname{\sf Diff}(M)/\operatorname{\sf Diff}_{0}(M)$ acting on $\operatorname{\sf Teich}$ embeds as the orientation-preserving subgroup in $O(H^{2}(M,{\mathbb{Z}}))$ ([Bor]). We deduce that $\gamma$ comes from an element $\widetilde{\gamma}$ of the mapping class group. Since $\gamma$ preserves the Hodge decomposition, the corresponding point in the period space $\operatorname{\sf Per}(I)\in\operatorname{{\mathbb{P}}\sf er}$ is a fixed point of $\widetilde{\gamma}$. Therefore, $\widetilde{\gamma}$ exchanges the non- separable points in $\operatorname{\sf Per}^{-1}(\operatorname{\sf Per}(I))$. However, as shown by Huybrechts, two manifolds which correspond to distinct non-separable points in $\operatorname{\sf Teich}$ have different Kähler cones, so that the fibers of the period map parametrize the Kähler (Weyl) chambers ([H1]). Therefore, $\widetilde{\gamma}$ maps $(M,I)\in\operatorname{\sf Per}^{-1}(\operatorname{\sf Per}(I))$ to itself whenever it fixes the Kähler cone. Hence $\widetilde{\gamma}$, considered as a diffeomorphism of $M$, is an automorphism of the complex manifold $(M,I)$, and $\gamma$ is in the image of $\varphi$ . Now we can prove the Morrison-Kawamata cone conjecture for K3 surfaces. Our argument is based on the following general result about lattices. Theorem 3.13: Let $q$ be an integer-valued quadratic form on $\Lambda={\mathbb{Z}}^{n},\ n\geqslant 2$ (not necessarily unimodular) such that its kernel is at most one-dimensional, and $O(\Lambda)$ the corresponding group of isometries. Fix $0\neq r\in{\mathbb{Z}}$, and let $S_{r}$ be the set $\\{v\in\Lambda\ \ \|\ \ q(v,v)=r\\}$. Then $O(\Lambda)$ acts on $S_{r}$ with finite number of orbits. Proof: The non-degenerate case is [Kn, Satz 30.2]. In the case of one- dimensional kernel $L=\ker q$, we write $\Lambda=L\oplus\Lambda_{0}$ and the elements of $\Lambda$ as $a_{0}+kl$ where $a_{0}\in\Lambda_{0}$, $k\in{\mathbb{Z}}$ and $l$ is a fixed generator of $L$. There are finitely many representatives of $S_{r}\cap\Lambda_{0}$ under the action of $O(\Lambda_{0})$, say $a_{0}^{1},\dots,a_{0}^{m}$. For each $j$, take a system of representatives $k_{1},\dots,k_{t_{j}}$ of ${\mathbb{Z}}$ modulo the ideal $\operatorname{Hom}(\Lambda_{0},L)\cdot a_{0}^{j}$. Then representatives of the orbits in $S_{r}$ are elements of the form $a_{0}^{j}+k_{i}l$. We shall apply this to study the Picard lattice $H^{2}(M,{\mathbb{Z}})\cap H^{1,1}(M)$. Notice that it is non-degenerate when $M$ is projective, and can have at most one-dimensional kernel when $M$ is arbitrary. Theorem 3.14: Morrison-Kawamata cone conjecture holds for any K3 surface. Proof, step 1: setting a goal. Let $\operatorname{\sf Mon}^{\operatorname{\sf Hdg}}(M,I)$ be the group of all oriented isometries of $H^{2}(M,{\mathbb{Z}})$ preserving the Hodge decomposition (such isometries are known as Hodge isometries). We have already remarked that when $M$ is a K3 surface, all these are monodromy operators, hence the notation. Since $\operatorname{\sf Mon}^{\operatorname{\sf Hdg}}(M,I)$ acts on the Picard lattice $\Lambda$ as a subgroup of index at most two in $O(\Lambda)=O(H^{1,1}(M,{\mathbb{Z}}))$, 3.2 implies that $\operatorname{\sf Mon}^{\operatorname{\sf Hdg}}(M,I)$ acts with finitely many orbits on the set of $(-2)$-vectors in $\Lambda$. Our goal in the next steps is to prove that the group $\operatorname{\sf Mon}^{\operatorname{\sf Hdg}}(M,I)$ acts on the set of faces of the Weyl chambers with finitely many orbits. Step 2: describing a face by a flag. Recall that an orientation of a hyperplane in an oriented real vector space is the choice of a “positive” normal direction. Then a face $F$ of a Weyl chamber is determined by the following data: a hyperplane $P_{s-1}$ it sits on, with the orientation pointing to the interior of the chamber; a hyperplane $P_{s-2}$ in $P_{s-1}$ supporting one of the faces $F_{1}$ of $F$, together with the orientation pointing to the interior of the face $F$; an oriented hyperplane $P_{s-3}$ in $P_{s-2}$ supporting some face $F_{2}$ of $F_{1}$; and so forth. Here $s$ is the dimension of the ambient space $H^{1,1}_{{\mathbb{R}}}(M)$. In other words, a face is determined by a full flag of linear subspaces, oriented step- by-step as above. To prove that $\operatorname{\sf Mon}^{\operatorname{\sf Hdg}}(M,I)$ acts on the set of faces with finitely many orbits, it suffices to prove that it acts with finitely many orbits on flags $P_{s-1}\supset P_{s-2}\supset\dots\supset P_{1}$ of the above form, orientation forgotten. Step 3: bounding squares. Notice that the hyperplane $P_{s-1}$ is $x^{\bot}$, where $x$ is a $(-2)$-class in $H^{1,1}_{{\mathbb{Z}}}(M,I)$, and $P_{s-2}=x^{\bot}\cap y^{\bot}$ where $y$ is another $(-2)$-class in $H^{1,1}_{{\mathbb{Z}}}(M,I)$. We claim that $P_{s-2}$ as a hyperplane in $P_{s-1}$ is given by $(y^{\prime})^{\bot}$, where $y^{\prime}\in x^{\bot}$ is an integral negative class of square at least $-8$. Indeed, write $y=\frac{\langle x,y\rangle}{\langle x,x\rangle}x+\widetilde{y}$ (orthogonal projection to $x^{\bot}$). Obviously, $P_{s-2}$ as a hyperplane in $P_{s-1}$ is just the orthogonal to $\widetilde{y}\in x^{\bot}$. The first summand on the right has non-positive square since it is proportional to a $(-2)$-class $x$. We claim that the second summand has strictly negative square (this square has then to be at least $-2$). Indeed, since the hyperplanes $x^{\bot}$ and $y^{\bot}$ define a Kähler chamber, the intersection of $x^{\bot}$ and $y^{\bot}$ is within the positive cone. But then the orthogonal to $\widetilde{y}$ in $x^{\bot}$ intersects the positive cone in $x^{\bot}$, so that, since the signature of the intersection form restricted to $x^{\bot}$ is $(+,-,\dots,-)$, $\widetilde{y}$ is of strictly negative square. To make $\widetilde{y}$ integral, it suffices to replace it by $y^{\prime}=\langle x,x\rangle\widetilde{y}=-2\widetilde{y}$; one has $0>\langle y^{\prime},y^{\prime}\rangle\geqslant-8$. Step 4: monodromy action. We have just seen that the intersections of $x^{\bot}$ with the other hyperplanes defining the Kähler chambers are given, in $x^{\bot}$, as orthogonals to integer vectors of strictly negative bounded square. By 3.2, the orthogonal group of $H^{1,1}_{{\mathbb{Z}}}(M,I)\cap x^{\bot}$ acts with finitely many orbits on such vectors. We deduce that the orthogonal group of $\Lambda=H^{1,1}_{{\mathbb{Z}}}(M,I)$ acts with finitely many orbits on partial flags $P_{s-1}\supset P_{s-2}$ arising from the faces of Kähler chambers. Iterating the argument of Step 3, we see that $O(\Lambda)$, and thus also $\operatorname{\sf Mon}^{\operatorname{\sf Hdg}}(M,I)$, acts with finitely many orbits on full flags and therefore on the set of faces of Kähler chambers. Step 5: conclusion. For each pair of faces $F,F^{\prime}$ of a Kähler cone and $w\in\operatorname{\sf Mon}^{\operatorname{\sf Hdg}}(M,I)$ mapping $F$ to $F^{\prime}$, $w$ maps $\operatorname{Kah}$ to itself or to an adjoint Weyl chamber $K^{\prime}$. Then $K^{\prime}=r(K)$, where $r$ is the orthogonal reflection in $H^{2}(M,{\mathbb{Z}})$ fixing $F^{\prime}$. In the first case, $w\in\operatorname{Aut}(M)$. In the second case, $rw$ maps $F$ to $F^{\prime}$ and maps $\operatorname{Kah}$ to itself, hence $rw\in\operatorname{Aut}(M)$. ### 3.3 Finiteness of polyhedral tesselations The argument used to prove 3.2 is valid in an abstract setting, which is worth describing here. Let $V_{\mathbb{Z}}$ is a torsion-free ${\mathbb{Z}}$-module of rank $n+1$, equipped with an integer-valued (but not necessarily unimodular) quadratic form of signature $(1,n)$, $\Gamma\subset O(V_{\mathbb{Z}})$ a finite index subgroup, and $V=V_{\mathbb{Z}}\otimes_{\mathbb{Z}}{\mathbb{R}}$. Let $S_{0}\subset V_{\mathbb{Z}}$ be a finite set of negative vectors, $S:=\Gamma\cdot S_{0}$ its orbit, and $Z:=\bigcup_{s\in S}s^{\bot}$ the union of all orthogonal complements to $s$ in $V$. We associate to $V$ a hyperbolic space ${\mathbb{H}}={\mathbb{P}}V^{+}$ obtained as a projectivization of the set of positive vectors. Let $\\{P_{i}\\}$ be the set of connected components of ${\mathbb{P}}V^{+}\backslash{\mathbb{P}}Z$. This is a polyhedral tesselation of the hyperbolic space. Definition 3.15: We call such a tesselation a tesselation cut out by the set of hyperplanes orthogonal to $S$. Definition 3.16: Let $S_{d}$ be an intersection of $n-d$ transversal hyperplanes $s^{\bot}$, with $s\in S$. A $d$-dimensional face of a tesselation is a connected component of $S_{d}\backslash S_{d-1}$. Theorem 3.17: Let $\\{P_{i}\\}$ be a tesselation obtained in 3.3, and $F_{d}$ the set of all $d$-dimensional faces of the tesselation. Then $\Gamma$ acts on $F_{d}$ with finitely many orbits. Proof: We encode a $d$-dimensional face $S_{d}^{0}$ (up to a finite choice of orientations) by a sequence of hyperplanes $s_{1}^{\bot}$, $s_{2}^{\bot}$, … $s_{n-d}^{\bot}$ such that their intersection supports $S_{d}^{0}$, plus an oriented flag as in 3.2. Using induction, we may assume that up to the action of $\Gamma$ there are only finitely many $(d+1)$-dimensional faces. The same inductive argument as in 3.2, Step 3 is used to show that the projection of $s_{n-d}$ to $S_{d+1}:=\bigcap_{i=1}^{n-d-1}s_{i}^{\bot}$ has bounded square after multiplying by the denominator. Therefore, there are finitely many orbits of the group $\Gamma_{S_{d+1}}:=\\{\gamma\in\Gamma\ \ \|\ \ \gamma(S_{d+1})=S_{d+1}\\}$ on the set $s_{n-d}^{\bot}\cap S_{d+1}$ for all $s_{n-d}\in S$, and the reasoning is the same for the oriented flag. ### 3.4 Morrison-Kawamata cone conjecture: birational version Definition 3.18: The birational Kähler cone of a hyperkähler manifold $M$ is the union of pullbacks of the Kähler cones under all birational maps $M\dasharrow M^{\prime}$ where $M^{\prime}$ is also a hyperkähler manifold. Remark 3.19: These birational maps are actually pseudo-isomorphisms ([H3], Proposition 4.7, [Bou2], Proposition 4.4), so that the second cohomologies of $M$ and $M^{\prime}$ are naturally identified. We shall sometimes say that the birational Kähler cone is the union of the Kähler cones of birational models, omitting to mention pullbacks. Remark 3.20: All this is a slight abuse of language: the birational Kähler cone is not what one would normally call a cone (but its closure, the moving cone, or birational nef cone, is). Definition 3.21: An exceptional prime divisor on a hyperkähler manifold is a prime divisor with negative square (with respect to the BBF form). The birational Kähler (or, rather, nef) cone is characterized in terms of prime exceptional divisors in the same way as the Kähler cone is characterized in terms of rational curves. Theorem 3.22: (Huybrechts, [H3], Prop. 4.2) Let $\eta\in\operatorname{Pos}(M)$ be an element of a positive cone on a hyperkähler manifold. Then $\eta$ is birationally nef if and only if $q(\eta,E)\geqslant 0$ for any exceptional divisor $E$. Remark 3.23: In other words, the faces of birational Kähler cone are dual to the classes of exceptional divisors. Theorem 3.24: (Markman) For each prime exceptional divisor $E$ on a hyperkähler manifold, there exists a reflection $r_{E}\in O(H^{2}(M,{\mathbb{Z}}))$ in the monodromy group, fixing $E^{\bot}$. Proof: In the projective case, this is [M4, Theorem 1.1]. The non-projective case reduces to this by deformation theory. Indeed, as we shall show in the next section (4), a prime exceptional divisor deforms locally as long as its cohomology class stays of type $(1,1)$ (and its small deformations are obviously prime exceptional again): in fact this also has already been remarked by Markman in [M4]), but he restricted himself to the projective case. Any hyperkähler manifold $(M,I)$ has a small deformation $(M,I^{\prime})$ which is projective. The divisor $E$ deforms to $E^{\prime}$ on $(M,I^{\prime})$, so one obtains a reflexion $r_{E}^{\prime}=r_{E}\in O(H^{2}(M,{\mathbb{Z}}))$ in the monodromy group (the monodromy being of topological nature, it is deformation-invariant in a tautological way). Definition 3.25: Such a reflection is called a divisorial reflection. Definition 3.26: An exceptional chamber, or a Weyl chamber on a hyperkähler manifold is a connected component of $\operatorname{Pos}(M)\backslash E^{\bot}$, where $E^{\bot}$ is a union of all planes $e^{\bot}$ orthogonal to all prime exceptional divisors $e$. Remark 3.27: An exceptional chamber is a fundamental domain of a group generated by divisorial reflections. The birational Kähler cone is a dense open subset of one of the exceptional chambers, which we shall call birational Kähler chamber. Theorem 3.28: (Markman) Let $(M,I)$ be a hyperkähler manifold, and $\operatorname{\sf Mon}^{\operatorname{\sf Hdg}}(M,I)$ a subgroup of the monodromy group fixing the Hodge decomposition on $(M,I)$. Then the image of $\operatorname{Bir}(M,I)$ in the orthogonal group of the Picard lattice is the group of all $\gamma\in\operatorname{\sf Mon}^{\operatorname{\sf Hdg}}(M,I)$ preserving the birational Kähler chamber $\operatorname{Kah}_{B}$. Proof: Please see [M3, Lemma 5.11 (6)]. The proof of 3.4 is similar to the proof of 3.2. Now we can generalize the birational Morrison-Kawamata cone conjecture proved by Markman for projective hyperkähler manifolds ([M3, Theorem 6.25]). Theorem 3.29: Let $(M,I)$ be a hyperkähler manifold, and $\operatorname{Bir}(M,I)$ the group of birational automorphisms of $(M,I)$. Then $\operatorname{Bir}(M,I)$ acts on the set of faces of the birational nef cone with finite number of orbits. Proof. Step 1: Let $\delta$ be the discriminant of a lattice $H^{2}(M,{\mathbb{Z}})$, and $E$ an exceptional divisor. Then $\|E^{2}\|\leqslant 2\delta$. Indeed, otherwise the reflection $x{\>\longrightarrow\>}x-2\frac{q(x,E)}{q(E,E)}E$ would not be integral. Step 2: The group of isometries of a lattice $\Lambda$ acts with finitely many orbits on the set $\\{l\in\Lambda\ \ \|\ \ l^{2}=x\\}$ for any given $x$ (3.2), and the monodromy group is of finite index in the isometry group of the lattice $H^{2}(M,{\mathbb{Z}})$. Therefore, $\operatorname{\sf Mon}^{\operatorname{\sf Hdg}}(M,I)$ is of finite index in the isometry group of the Picard lattice and so all classes of exceptional divisors belong to finitely many orbits of $\operatorname{\sf Mon}^{\operatorname{\sf Hdg}}(M,I)$. Step 3: We repeat the argument of 3.2 to show that $\operatorname{\sf Mon}^{\operatorname{\sf Hdg}}(M,I)$ acts with finite number of orbits on the set of faces of all exceptional chambers. It suffices to show that it acts with finitely many orbits on the set of full flags $P_{s-1}\supset P_{s-2}\supset\dots\supset P_{1}$ (notations as in the proof of 3.2) formed by intersections of orthogonal hyperplanes to the classes of exceptional divisors. Using the fact that the squares of those classes are bounded in absolute value by $C=2\delta$, we show that $P_{s-2}$ is described inside $P_{s-1}$ as $y_{1}^{\bot}$ where $y_{1}$ is integral and $\|y_{1}^{2}\|\leqslant C^{3}$, $P_{s-3}$ is defined in $P_{s-2}$ as $y_{2}^{\bot}$ where $y_{2}$ is integral and $\|y_{1}^{2}\|\leqslant C^{9}$, and so on. We deduce by 3.2 that $O(\Lambda)$ acts with finitely many orbits on the set of such full flags. Step 4: Thus $\operatorname{\sf Mon}^{\operatorname{\sf Hdg}}(M,I)$ acts with finite number of orbits on the set of faces of all exceptional chambers. For each pair of faces $F,F^{\prime}$ of a birational Kähler cone and $w\in O(\Lambda)$ mapping $F$ to $F^{\prime}$, $w$ maps $\operatorname{Kah}_{B}$ to itself or to an adjoint Weyl chamber $K^{\prime}$. Then $K^{\prime}=r(K)$, where $r$ is the reflection fixing $F^{\prime}$. In the first case, $w\in\operatorname{Aut}(M)$. In the second case, $rw$ maps $F$ to $F^{\prime}$ and maps $\operatorname{Kah}_{B}$ to itself, hence $rw\in\operatorname{Aut}(M)$. Therefore, there are as many orbits of $\operatorname{\sf Mon}^{\operatorname{\sf Hdg}}(M,I)$ on the faces of exceptional chambers as there are orbits of $\operatorname{Bir}(M,I)$. ## 4 Deformation spaces of rational curves By a rational curve on a manifold $X$, we mean a curve $C\subset X$ such that its normalization is ${\mathbb{P}}^{1}$; in other words, the image of a generically injective map $f:{\mathbb{P}}^{1}\to X$. Let $\operatorname{Mor}({\mathbb{P}}^{1},X)$ denote the parameter space for such maps. Then by deformation theory (see [Ko]) one has $\dim_{[f]}(\operatorname{Mor}({\mathbb{P}}^{1},X))\geqslant\chi(f^{}(TX))=-K_{X}C+\dim(X),$ so that if $H$ denotes the space of deformations of $C$ in $X$, then $\dim(H)\geqslant-K_{X}C+\dim(X)-3.$ A rational curve on an $m$-fold with trivial canonical class must therefore move in a family of dimension at least $m-3$. The following observation due to Z. Ran states that on holomorphic symplectic manifolds, this estimate can be slightly improved. Theorem 4.1: Let $M$ be a hyperkähler manifold of dimension $2n$. Then any rational curve $C\subset M$ deforms in a family of dimension at least $2n-2$. Proof: See [R], Corollary 5.2. Alternatively (we thank Eyal Markman for this argument), one may notice that an extra parameter is due to the existence of the twistor space $\operatorname{Tw}(M)$. This is a complex manifold of dimension $n+1$, fibered over ${\mathbb{P}}^{1}$ in such a way that $M$ is one of the fibers and the other fibers correspond to the other complex structures coming from the hyperkähler data on $M$. The map $f$, seen as a map from ${\mathbb{P}}^{1}$ to $\operatorname{Tw}(M)$, deforms in a family of dimension at least $n+1$. But all deformations have image contained in $M$ since the neighbouring fibers contain no curves at all by [V0] and the rational curves with dominant projection to ${\mathbb{P}}^{1}$ belong to a different cohomology class. Before making the following observation, let us recall a few definitions. Definition 4.2: A complex analytic subvariety $Z$ of a holomorphically symplectic manifold $(M,\Omega)$ is called holomorphic Lagrangian if $\Omega{\left\|{}_{{\phantom{\|}\\!\\!}_{Z}}\right.}=0$ and $\dim_{\mathbb{C}}Z=\frac{1}{2}\dim_{\mathbb{C}}M$, and isotropic if $\Omega{\left\|{}_{{\phantom{\|}\\!\\!}_{Z}}\right.}=0$ (since $\Omega$ is non- degenerate, this implies $\dim_{\mathbb{C}}Z\leqslant\frac{1}{2}\dim_{\mathbb{C}}M$). It is called coisotropic if $\Omega$ has rank $\frac{1}{2}\dim_{\mathbb{C}}M-\operatorname{codim}_{\mathbb{C}}Z$ on $TZ$ in all smooth points of $Z$, which is the minimal possible rank for a $2n-p$-dimensional subspace in a $2n$-dimensional symplectic space. Let now $Z$ be a compact Kähler manifold covered by rational curves. By [Cam], there is an almost holomorphic111An almost holomorphic fibration is a rational map $\pi:\;X{\>\longrightarrow\>}Y$, holomorphic on $X_{0}\subset X$, and with $\pi^{-1}(y)\cap X_{0}$ non-empty for all $y\in Y$. fibration $R:Z\dasharrow Q$, called the rational quotient of $Z$, such that its fiber through a sufficiently general point $x$ consists of all $y$ which can be joined from $x$ by a chain of rational curves. That is, the general fiber of $R$ is rationally connected. It is well-known that rationally connected manifolds do not carry any holomorphic forms, and it follows from [Cam] that they are projective. Remark 4.3: By a theorem due to Graber, Harris and Starr in the projective setting, the base $Q$ of the rational quotient is not uniruled; this is also true in the compact Kähler case (the reason being that the total space of a family of rationally connected varieties over a curve is automatically algebraic, so that the arguments of Graber-Harris-Starr apply), but we shall not need this. Theorem 4.4: Let $M$ be a hyperkähler manifold, $C\subset M$ a rational curve, and $Z\subset M$ be an irreducible component of the locus covered by the deformations of $C$ in $M$. Then $Z$ is a coisotropic subvariety of $M$. The fibers of the rational quotient of the desingularization of $Z$ have dimension equal to the codimension of $Z$ in $M$. Proof: We want to prove that at a general point of $Z$, the restriction of the symplectic form $\Omega$ to $TZ$ has kernel of dimension equal to $k=\operatorname{codim}(Z)$. Recall that this kernel cannot be of dimension greater than $k$ by nondegeneracy of $\Omega$, and the equality means that $Z$ is coisotropic. Let $h:Z^{\prime}\to Z$ be the desingularization. The manifold $Z^{\prime}$ is covered by rational curves which are lifts of deformations of $C$. By dimension count, through a general point $p$ of $Z^{\prime}$ there is a family of such curves of dimension at least $k-1$. In fact any rational curve through $p$ deforms in a family of dimension at least $k-1$, as seen by projecting it to $M$ and applying 4. Take a minimal rational curve $C^{\prime}$ through $p$, then its deformations through $p$ cover a subvariety of dimension at least $k$. This is because by bend-and-break, there is only a finite number of minimal rational curves through two general points (notice that bend-and-break applies here since all rational curves through $p$ are in the fiber of the rational quotient, and this fiber is projective). This means that the fibers of the rational quotient fibration $R:Z^{\prime}\dasharrow Q^{\prime}$ are at least $k$-dimensional. But any holomorphic form on $Z^{\prime}$ is a pull-back of a holomorphic form on $Q^{\prime}$ (since the rationally connected varieties do not carry any holomorphic $m$-forms for $m>0$). So the tangent space to the fiber of $R$ through $p$ is in the kernel of $h^{}(\Omega)$, and thus the kernel of $\Omega\|_{TZ}$ at $h(p)$ is of dimension $k$, and the same is true for the fiber of $R$ through $p$. In the above argument, we have called a curve $C\subset Z$ minimal if it is a rational curve of minimal degree (say, with respect to the Kähler form from the hyperkähler structure) through a general point of $Z$. Corollary 4.5: If deformations of a rational curve $C$ in $M$ cover a divisor $Z$, then $C$ is a minimal rational curve in this divisor. Proof: Indeed by 4, there is no other rational curve through a general point of $Z$. Corollary 4.6: Minimal rational curves on holomorphic symplectic manifolds deform in a $2n-2$-parameter family. Proof: This is obvious from the proof of 4. Indeed, if the deformations of a minimal $C$ cover a subvariety $Z$ of codimension $k$, then these deformations form a family of dimension $k-1+dim(Z)-1=k-1+2n-k-1=2n-2$, since there is a $k-1$-parametric family of them through the general point of $Z$. Remark 4.7: These results have well-known analogues (which are consequences of the work by Wierzba and Kaledin) in the case when $Z$ is contractible, that is, one has a birational morphism $\pi:M\to Y$ whose exceptional set is $Z$. The image of $Z$ by $\pi$ replaces the base of the rational quotient. In the case when moreover $Z$ is a divisor, the fibers of $\pi$ are one-dimensional and therefore are trees of smooth rational curves by Grauert-Riemenschneider theorem. In general, it is not obvious whether the minimal rational curves are smooth. In one important case, though, they are smooth: namely, when $Z$ is a negative-square divisor on $M$. The reason is that one can deform the pair $(M,Z)$ so that $M$ becomes projective (see below), and then use results by Druel from [D] to reduce to the contractible case. In this case, the inverse image of the tangent bundle $T_{M}$ on the normalization of a minimal rational curve $C$ splits as follows: $f^{}T_{M}={\cal O}(-2)\oplus{\cal O}(2)\oplus{\cal O}\oplus\dots\oplus{\cal O}$ This is because it should be isomorphic to its dual (by the pull-back of the symplectic form), have at most one negative summand, and contain $T_{{\mathbb{P}}^{1}}$ as a subbundle. This also remains true when the normalization map of $C$ is an immersion. In general, there are problems related to the singularities (for instance, as soon as there are cusps, $T_{{\mathbb{P}}^{1}}$ is only a subsheaf and not a subbundle of $f^{}T_{M}$) and it is not obvious how to avoid them in order to get a similar splitting, which one would like to be $f^{}T_{M}={\cal O}(-2)\oplus{\cal O}(2)\oplus{\cal O}^{\oplus k}\oplus({\cal O}(-1)\oplus{\cal O}(1))^{\oplus l}$. Corollary 4.8: If $C$ is minimal, any small deformation $M_{t}$ of $M=M_{0}$ such that the dual class $z$ of $C$ stays of type $(1,1)$ on $M_{t}$, contains a deformation of $C$. Proof: Consider the universal family ${\cal M}\to\operatorname{Def}(M)$ of small deformations of $M=M_{0}$. The $t\in\operatorname{Def}(M)$ such that $z$ is of type $(1,1)$ on $M_{t}$ form a smooth hypersurface in $\operatorname{Def}(M)$ (if one identifies the tangent space to $\operatorname{Def}(M)$ at $0$ with $H^{1}(M,T_{M})\cong H^{1}(M,\Omega^{1}_{M})$, then the tangent space to this hypersurface is the hyperplane orthogonal to $z$). Now let ${\cal C}\to\operatorname{Def}(M)$ be the family of deformations of $C$ in ${\cal M}$: the image of ${\cal C}$ in $\operatorname{Def}(M)$ is a subvariety by Grauert’s proper mapping theorem, and it suffices to prove that it is a hypersurface. This is a simple dimension count. Indeed one obtains from Riemann-Roch theorem, as in the proof of 4, that $C$ deforms in a family of dimension at least $2n-3+\dim(\operatorname{Def}(M))$. Since the deformations of $C$ inside any $M_{t}$ form a family of dimension $2n-2$ (when nonempty), the conclusion follows. Corollary 4.9: If $C$ is minimal, any deformation of $M=M_{0}$ such that the corresponding homology class remains of type $(1,1)$, has a birational model containing a rational curve in that homology class. Proof: This is the same argument as in the previous corollary, but we have to consider the universal family over $\operatorname{\sf Teich}_{z}(M)^{0}$, where $\operatorname{\sf Teich}(M)^{0}$ is the connected component of $\operatorname{\sf Teich}(M)$ containing the parameter point for our complex manifold $M_{0}$, and $\operatorname{\sf Teich}_{z}(M)^{0}$ is the part of it where $z$ remains of type $(1,1)$. Birational models appear since $\operatorname{\sf Teich}_{z}(M)$ is not Hausdorff, so that a subvariety of $\operatorname{\sf Teich}_{z}(M)^{0}$ of maximal dimension is not necessarily equal to $\operatorname{\sf Teich}_{z}(M)^{0}$; on the other hand it is known by the work of Huybrechts that unseparable points of $\operatorname{\sf Teich}_{z}(M)$ correspond to birational complex manifolds. The deformations of a minimal $C$ as above are obviously minimal on the neighbouring fibers (indeed, new effective classes only appear on closed subsets of the parameter space). However, globally a limit of minimal curves does not have to be minimal, at least apriori; it can also become reducible (and in fact does so in many examples). The deformation theory of non-minimal curves is not as nice as described in 4. For instance, in an example from [BHT], credited by the authors to Claire Voisin, there is a holomorphic symplectic fourfold $X$ containing a lagrangian quadric $Q$, and rational curves of type $(1,1)$ on $Q$ deform only together with $Q$, that is over a codimension-two subspace of the base space, as follows from [Voi]. However this problem disappears, at least locally, in the case when the dual class $z$ is negative divisorial, that is, when $q(z,z)<0$ and $z$ is ${\mathbb{Q}}$-effective. The following theorem is due to Markman in the projective case. Theorem 4.10: Let $z$ be a negative $(1,1)$-class on $M$ which is represented by an irreducible divisor $D$. Then $z$ is effective on any deformation $M_{t}$ where it stays of type $(1,1)$, and represented by an irreducible divisor on an open part of this parameter space. Proof: By [Bou2], $D$ is uniruled. By 4, there is only one dominating family of rational curves on $D$. Take a general member $C$ in this family. We have seen that it deforms on all neighbouring $M_{t}$, yielding rational curves $C_{t}$. In particular its dual class is proportional (with some positive coefficient) to that of $D$, since both stay of type $(1,1)$ on the same small deformations of $M$; therefore $CD<0$ and all deformations of $C$ in $M$ stay inside $D$. We claim that the deformations of $C_{t}$ on $M_{t}$ also cover divisors on $M_{t}$, which have then to be deformations of $D$. Indeed, let $Z_{t}$ be the subvariety of $M_{t}$ covered by the curves $C_{t}$. Suppose it is not a divisor for all $t$. Since the dimension jumps over closed subsets of the parameter space, one has an open subset $U$ of such $t$. By 4, through a general point of $Z_{t},t\in U,$ there is a positive-dimensional family of curves $C_{t}$. Now consider a parameter space for the following triples over $T$: $\\{(C_{t},x,C^{\prime}_{t}):x\in C_{t}\cap C^{\prime}_{t}\\}$. The dimension of its fibers over $t\in U$ is strictly greater than the dimension of the central fiber, and this is a contradiction. Therefore $D$ deforms everywhere locally. Now consider the universal family ${\cal M}$ over $\operatorname{\sf Teich}_{z}$, and the “universal divisor” ${\cal D}$ in this family. The image of ${\cal D}$ is a subvariety in $\operatorname{\sf Teich}_{z}$, of the same dimension as $\operatorname{\sf Teich}_{z}$. This means that any deformation of $M$ preserving the type $(1,1)$ of $z$ has a birational model such that $z$ is effective. As birational hyperkähler manifolds differ only in codimension two, the theorem follows. Corollary 4.11: A negative class $z\in H^{2}(M,{\mathbb{Z}})$ is effective or not simultaneously in all complex structures where it is of type $(1,1)$ and generates the Picard group over ${\mathbb{Q}}$. Proof: If the only integral $(1,1)$-classes are rational multiples of $z$, one cannot have more than one effective divisor on $M$: indeed if $D$ and $D^{\prime}$ are prime effective divisors and $q(D,D^{\prime})<0$, then $D=D^{\prime}$ (if not, remark that by definition $q(D,D^{\prime})$ is obtained by integration of a positive form over $D\cap D^{\prime}$). In particular, every effective divisor is irreducible. Now apply the previous theorem. When one wants to deform curves rather than divisors, the notion of a minimal curve that we have used above is not quite natural since it depends, apriori, on the subvariety $Z\subset M$ covered by deformations of the curve, and not only on the complex manifold $M$ itself. In the projective setting, one typically considers rational curves generating an extremal ray of the Mori cone: the class of such a curve is extremal in the sense of the Introduction. In the non-projective setting, the following notion of extremality looks better-behaved. Definition 4.12: Let $X$ be a Kähler manifold and $z$ an integral homology class of type $(1,1)$. We say that $z$ is minimal if the intersection of $z^{\bot}$ with the boundary of the Kähler cone contains an open subset of $z^{\bot}$. An example due to Markman, [M4], shows that a limit of minimal (or extremal) curves does not have to be minimal (extremal). Example 4.13: ([M4], Example 5.3) Let $\overline{X}_{0}$ be an intersection of a quadric and a cubic in ${\mathbb{P}}^{4}$ with one double point. The resolution $p:X_{0}\to\overline{X}_{0}$ is a $K3$ surface, and $p^{}H+2E$, where $H$ is a hyperplane section and $E$ is the exceptional curve, is not minimal (or extremal). Now deform $X_{0}$ to a smooth non-projective K3 surface $X_{t}$ in such a way that only the multiples of $p^{}H+2E$ survive in $H^{1,1}$. Since $p^{}H+2E$ is a $(-2)$-class, it is easy to show that those are effective. They are obviously extremal (and minimal, too). In what follows, we shall define the MBM-classes as classes which are minimal modulo monodromy and birational equivalence. We shall show that this notion is deformation-invariant, and that if a class is MBM, then the intersection of its orthogonal hyperplane and the positive cone is a union of faces of Kähler- Weyl chambers (6.1, 5.3). It is an interesting question whether the notions of minimality and extremality are equivalent for hyperkähler manifolds. Minimal classes are extremal, and it follows from our proofs that minimal classes are effective up to a rational multiple and represented by rational curves, as it is the case of extremal rays in the Mori theory. But extremal classes do not, apriori, have to be minimal, since extremal rays of the cone of curves could accumulate. One can conjecture that in real life it never happens (as this is true in the projective case by [HT]). ## 5 Twistor lines in the Teichmüller space ### 5.1 Twistor lines and 3-dimensional planes in $H^{2}(M,{\mathbb{R}})$. Recall that any hyperkähler structure $(M,I,J,K,g)$ defines a triple of Kähler forms $\omega_{I},\omega_{J},\omega_{K}\in\Lambda^{2}(M)$ (Subsection 2.1). A hyperkähler structure on a simple hyperkähler manifold is determined by a complex structure and a Kähler class (2.1). Definition 5.1: Each hyperkähler structure induces a family $S\subset\operatorname{\sf Teich}$ of deformations of complex structures parametrized by ${\mathbb{C}}P^{1}$ (Subsection 2.1). The curve $S$ is called the twistor line associated with the hyperkähler structure $(M,I,J,K,g)$. We identify the period space $\operatorname{{\mathbb{P}}\sf er}$ with the Grassmannian of positive oriented 2-planes in $H^{2}(M,{\mathbb{R}})$ (2.2). For any point $l\in S$ on the twistor line, the corresponding 2-dimensional space $\operatorname{\sf Per}(l)\in\operatorname{{\mathbb{P}}\sf er}$ is a 2-plane in the 3-dimensional space $\langle\omega_{I},\omega_{J},\omega_{K}\rangle$. Therefore, $\langle\omega_{I},\omega_{J},\omega_{K}\rangle$ is determined by the twistor line $S$ uniquely, as the linear span of the planes in $\operatorname{\sf Per}(S)$. We call two hyperkähler structures equivalent if one can be obtained from the other by a homothety and a quaternionic reparametrization: $(M,I,J,K,g)\sim(M,hIh^{-1},hJh^{-1},hKh^{-1},\lambda g),$ for $h\in{\mathbb{H}}^{}$, $\lambda\in{\mathbb{R}}^{>0}$. Clearly, equivalent hyperkähler structures produce the same twistor lines in $\operatorname{\sf Teich}$. However, a hyperkähler structure is determined by a complex structure, which yields a 2-dimensional subspace $\operatorname{\sf Per}(I)=\langle\omega_{J},\omega_{K}\rangle$ in $H^{2}(M,{\mathbb{R}})$, and a Kähler structure $\omega_{I}$, as in 2.1. The form $\omega_{I}$ can be reconstructed up to a constant from the 3-dimensional space $\langle\omega_{I},\omega_{J},\omega_{K}\rangle$ and the plane $\operatorname{\sf Per}(I)$. This proves the following result, which is essentially a form of Calabi-Yau theorem. Claim 5.2: Let $(M,I,J,K,g)$ be a hyperkähler structure on a compact manifold, and $S\subset\operatorname{\sf Teich}$ the corresponding twistor line. Then $S$ is sufficient to recover the equivalence class of $(M,I,J,K,g)$. Proposition 5.3: Let $S\subset\operatorname{\sf Teich}$ be a twistor line, $W\subset H^{2}(M,{\mathbb{R}})$ be the corresponding 3-dimensional plane $W:=\langle\omega_{I},\omega_{J},\omega_{K}\rangle$, and $z\in H^{2}(M,{\mathbb{R}})$ a non-zero real cohomology class. Then $S$ lies in $\operatorname{\sf Teich}_{z}$ if and only if $W\bot z$. Proof: By definition, $\operatorname{\sf Teich}_{z}$ is the set of all $I\in\operatorname{\sf Teich}$ such that the 2-plane $\operatorname{\sf Per}(I)$ is orthogonal to $z$. However, any point of $\operatorname{\sf Per}(S)$ lies in $W$, hence all points in $S$ belong to $\operatorname{\sf Teich}_{z}$ whenever $W\bot z$. Conversely, the planes corresponding to points in $\operatorname{\sf Per}(S)$ generate $W$, so that $W$ is orthogonal to $z$ if all those planes are. ### 5.2 Twistor lines in $\operatorname{\sf Teich}_{z}$. Remark 5.4: Let $z\in H^{1,1}(M,I)$ be a non-zero cohomology class on a hyperkähler manifold $(M,I)$, $\operatorname{\sf Teich}^{I}$ the connected component of the Teichmüller space containing $I$, and $\operatorname{\sf Teich}_{z}$ the set of all $J\in\operatorname{\sf Teich}^{I}$ such that $z$ is of type $(1,1)$ on $(M,J)$. Given a Kähler form $\omega$ on $(M,I)$ such that $q(\omega,z)=0$, consider the corresponding hyperkähler structure $(M,I,J,K)$, and let $S\subset\operatorname{\sf Teich}$ be the corresponding twistor line. By 5.1, $S$ lies in $\operatorname{\sf Teich}_{z}$. In other words, there is a twistor line on $\operatorname{\sf Teich}_{z}$ through the point $I$ if and only if $z$ is orthogonal to a Kähler form on $(M,I)$. Definition 5.5: In the assumptions of 5.2, let $W_{S}:=\langle\omega_{I},\omega_{J},\omega_{K}\rangle$ be a 3-dimensional plane associated with the hyperkähler structure $(M,I,J,K)$, and $S\subset\operatorname{\sf Teich}_{z}$ the corresponding twistor line. The twistor line $S$ is called $z$-GHK ($z$-general hyperkähler) if $W^{\bot}\cap H^{2}(M,{\mathbb{Q}})=\langle z\rangle$. The utility of $z$-GHK lines is that they can be lifted from the period space to $\operatorname{\sf Teich}_{z}$ uniquely, if $\operatorname{\sf Teich}_{z}$ contains a twistor line (5.3). For other twistor lines, such a lift, even if it exists, is not necessarily unique because of non-separable points, but a $z$-GHK line has to pass through a Hausdorff point of $\operatorname{\sf Teich}_{z}$. Claim 5.6: Let $z\in H^{1,1}(M,I)$ be a non-zero cohomology class on a hyperkähler manifold $(M,I)$. Assume that $(M,I)$ admits a Kähler form $\omega$ such that $q(\omega,z)=0$. Then $(M,I)$ admits a $z$-GHK line. Proof: Let $z^{\bot}\subset H^{1,1}_{I}(M,{\mathbb{R}})$ be the orthogonal complement of $z$. The set of Kähler classes is non-empty and open in $z^{\bot}$. For each integer vector $z_{1}\in H^{2}(M,{\mathbb{Z}})$, non- collinear with $z$, the orthogonal complement $z_{1}^{\bot}$ intersects with $z^{\bot}$ transversally. Removing all the hyperplanes $z^{\bot}\cap z_{1}^{\bot}$ from $z^{\bot}$, we obtain a dense set which containes a Kähler form $\omega_{1}$, such that $\omega_{1}^{\bot}$ contains none of the integer vectors $z_{1}$. The hyperkähler structure associated with $I$ and $\omega_{1}$ satisfies $W^{\bot}\cap H^{2}(M,{\mathbb{Q}})=\langle z\rangle$, because $W\ni\omega_{1}$, and $\omega_{1}^{\bot}\cap H^{2}(M,{\mathbb{Q}})=\langle z\rangle$. The following result is easy to prove, however, we give a reference to [V4], where the proof is spelled out in a situation which is almost the same as ours. Claim 5.7: Let $S\subset\operatorname{\sf Teich}_{z}$ be a twistor line, associated with a 3-dimensional subspace $W\subset H^{2}(M,{\mathbb{R}})$. Then the following assumptions are equivalent. (i) $S$ is a $z$-GHK line. (ii) For all $l\in S$, except a countable number, the space $H^{1,1}(M_{l},{\mathbb{Q}})$ is generated by $z$, where $M_{l}$ is the manifold corresponding to $l$. (iii) For some $w\in W$, the space of rational classes in the orthogonal complement $w^{\bot}\subset H^{2}(M,{\mathbb{R}})$ is generated by $z$. Proof: [V4, Claim 5.4]. Given a $z$-GHK line $S$, we may chose a point $l\in S$ where $H^{1,1}(M_{l},{\mathbb{Q}})=\langle z\rangle$. For this point, the Kähler cone coincides with the positive cone by 1.2. Indeed, $M_{l}$ contains no curves, as the class $z$, being orthogonal to a Kähler form, cannot be effective. Chosing a different Kähler form in $\operatorname{Pos}(M_{l})\cap z^{\bot}$, we obtain a different twistor line in $\operatorname{\sf Teich}_{z}$, intersecting $S$. It was shown that such a procedure can be used to connect any two points of $\operatorname{\sf Teich}_{z}$, up to non- separatedness issues (in [V4] it was proven for the whole $\operatorname{\sf Teich}$, but for $\operatorname{\sf Teich}_{z}$ the argument is literally the same). Proposition 5.8: Let $x,y\in\operatorname{\sf Teich}_{z}$. Suppose that $\operatorname{\sf Teich}_{z}$ contains a twistor line. Then a point $\widetilde{x}$ non-separable from $x$ can be connected to a point $\widetilde{y}$ non-separable from $y$ by a sequence of at most 5 sequentially intersecting $z$-GHK lines. Proof: From [V4, Proposition 5.8] it follows that any two points $a=\operatorname{\sf Per}(x),b=\operatorname{\sf Per}(y)$ in the period domain $\operatorname{\sf Per}(\operatorname{\sf Teich}_{z})$ can be connected by at most 5 sequentially intersecting $z$-GHK lines. Moreover, the intersection points can be chosen generic, in particular, separable. Lifting these $z$-GHK lines to $\operatorname{\sf Teich}_{z}$, we connect a point $\widetilde{x}$ in $\operatorname{\sf Per}^{-1}(x)$ to a point $\widetilde{y}$ in $\operatorname{\sf Per}^{-1}(y)$ by a sequence of at most 5 sequentially intersecting $z$-GHK lines. However, these preimages are non-separable from $x$, $y$ by the global Torelli theorem (2.2). Remark 5.9: Clearly, if our twistor line already passes through $x$, we can connect $x$ itself to a $\widetilde{y}$ non-separable from $y$. ### 5.3 Twistor lines and MBM classes. As we have already mentioned in the Introduction, the Bogomolov-Beauville- Fujiki form identifies the rational cohomology and homology of a hyperkähler manifold $M$, inducing an injective map $q:H^{2}(M,{\mathbb{Z}})\to H_{2}(M,{\mathbb{Z}})$. Since $q$ is not necessarily unimodular, this map is not an isomorphism over ${\mathbb{Z}}$. However, it is an isomorphism over ${\mathbb{Q}}$, and we shall identify $H^{2}(M,{\mathbb{Q}})$ with $H_{2}(M,{\mathbb{Q}})$ without further comment (in particular, we shall often view the classes in $H_{2}(M,{\mathbb{Z}})$ as rational classes in $H^{2}$). The only exception to this rule is the notion of effectiveness: an effective homology class is a class of a curve, whereas an effective cohomology class is a class of an effective divisor. Recall that a negative integer homology class $\eta\in H_{2}(M,{\mathbb{Z}})$ is called an MBM class if some image of $\eta$ under monodromy contains in its orthogonal a face of the Kähler cone of some birational hyperkähler model $M^{\prime}$ (which is necessarily pseudo-isomorphic to $M$). To study the MBM classes it is convenient to work with hyperkähler manifolds which satisfy $\operatorname{rk}\operatorname{Pic}(M)=1$, with $\operatorname{Pic}(M)\subset H^{2}(M,{\mathbb{Z}})$ generated over ${\mathbb{Q}}$ by $\eta$. Here $\eta$ is considered as an element of $H^{2}(M,{\mathbb{Q}})$, identified with $H_{2}(M,{\mathbb{Q}})$ as above. In this case $M$ is clearly non-algebraic since $\eta$ is negative. Theorem 5.10: Let $(M,I)$ be a hyperkähler manifold, such that $\operatorname{Pic}(M,I)=\langle z\rangle$, where $z\in H_{1,1}(M,I)$ is a non-zero negative class. Then the following statements are equivalent. (i) The class $\pm z$ is ${\mathbb{Q}}$-effective. (ii) The class $\pm z$ is extremal. (iii) The Kähler cone of $(M,I)$ is not equal to its positive cone. (iv) The class $z$ is minimal, in the sense of 4. (v) The class $z$ is MBM. Proof: First of all, the equivalence of (i) and (ii) is clear since the cone of curves on $M$ consists of multiples of $\pm z$ if it is ${\mathbb{Q}}$-effective and is empty otherwise. The equivalence of (i) and (iii) is proven as follows: first of all, for signature reasons one can find a positive class $\alpha$ with $q(\alpha,z)=0$, and if $\pm z$ is ${\mathbb{Q}}$-effective then $\alpha$ cannot be Kähler. The converse follows by Huybrechts-Boucksom’s description of the Kähler cone, see 1.2 (if $\pm z$ is not ${\mathbb{Q}}$-effective then there are no curves at all, in particular no rational curves, and thus any positive class is Kähler). Finally, let us show the equivalence of (iii), (iv) and (v). If $\operatorname{Kah}(M,I)\neq\operatorname{Pos}(M,I)$, then by Huybrechts- Boucksom the Kähler cone has faces supported on hyperplanes of the form $x^{\bot}$ where $x$ is a class of a rational curve. However, the only integral $(1,1)$-classes on $(M,I)$ are multiples of $z$, so such a face can only be $z^{\bot}$. In this case $z$ must be minimal; this proves (iii) $\Rightarrow$ (iv). The implication (iv) $\Rightarrow$ (v) is a tautology. Finally, if $\operatorname{Kah}(M,I)=\operatorname{Pos}(M,I)$, there are no minimal classes and no non-trivial birational models of $M$, so there cannot be any MBM classes either, which proves (v) $\Rightarrow$ (iii). In general, the notion of an MBM class is much more complicated, but one can hope to study them by deforming to the case we have just described. This is indeed what we are going to do, using 5.2 as a key tool. The following useful proposition, which allows one to identify generic positive 3-dimensional subspaces of $H^{2}(M,{\mathbb{R}})$ and twistor lines, is an illustration of our method. Proposition 5.11: Let $z\in H^{2}(M,{\mathbb{Q}})$ be a non-zero vector, such that $\operatorname{\sf Teich}_{z}$ contains a twistor line, and $W\subset z^{\bot}$ a 3-dimensional positive subspace of $H^{2}(M,{\mathbb{R}})$ which satisfies $W^{\bot}\cap H^{2}(M,{\mathbb{Q}})\subset\langle z\rangle$. Then there exists a $z$-GHK twistor line $S$ such that the corresponding 3-dimensional space $W_{S}\subset H^{2}(M,{\mathbb{R}})$ is equal to $W$. Proof: Let $V\subset W$ be a 2-dimensional plane which satisfies $V^{\bot}\cap H^{2}(M,{\mathbb{Q}})\subset\langle z\rangle$ (by 5.2 (ii) this is true for all planes except at most a countable set). Then $V=\operatorname{\sf Per}(I)$, where $I\in\operatorname{\sf Teich}_{z}$, because the period map is surjective. The space $H^{1,1}_{\mathbb{Q}}(M,I)=V^{\bot}\cap H^{2}(M,{\mathbb{Q}})$ is generated by $z$. Since $\operatorname{\sf Teich}_{z}$ contains a twistor line, by 5.2 there must be a twistor line through the point $I^{\prime}\in\operatorname{\sf Teich}_{z}$ which is inseparable from $I$. Now on $I^{\prime}$, $z$ cannot be effective since it is orthogonal to a Kähler form. But this means that there are no curves on $(M,I^{\prime})$, so the Kähler cone of $(M,I^{\prime})$ is equal to the positive cone, $\operatorname{\sf Teich}_{z}$ is separated at $I^{\prime}$, and $I=I^{\prime}$. Therefore the Kähler cone of $(M,I)$ coincides with its positive cone, hence there exists a Kähler form $\omega$ on $(M,I)$ such that $\langle V,\omega\rangle=W$. The corresponding hyperkähler structure gives a 3-plane $W_{S}=\langle V,\omega\rangle=W$. Our strategy in proving the deformation invariance of MBM property is as follows: we first show that the property of being orthogonal to a Kähler form, modulo monodromy and birational transformations, is deformation-invariant, and then show that the MBM classes are exactly those which do not have this property. Theorem 5.12: Let $M$ be a simple hyperkähler manifold, and $z\in H^{2}(M,{\mathbb{Q}})$ a cohomology class. Suppose that $z$ is orthogonal to a Kähler form on $(M,I_{0})$, where $I_{0}\in\operatorname{\sf Teich}_{z}$. Then: (i) For any $I\in\operatorname{\sf Teich}_{z}$, there is some $I^{\prime}$ non- separable from $I$ such that $z$ is orthogonal to a Kähler form on $(M,I^{\prime})$. (ii) On the manifold $(M,I)$, the class $z$ is orthogonal to an element of a Kähler-Weyl chamber (that is, $z$ is orthogonal to $\gamma(\alpha)$ for some $\gamma$ in the monodromy group and $\alpha$ lying in the birational Kähler cone, i.e. Kähler on a birational model, see 6.1). (iii) The elements of Kähler-Weyl chambers are dense in the intersection of $z^{\bot}$ and the positive cone. Proof: (i) By 5.2, there is a $z$-GHK line through $I_{0}$. By 5.2 , $I_{0}$ is connected to $I^{\prime}$ by a sequence of $z$-GHK lines. Therefore on $(M,I^{\prime})$, $z$ is orthogonal to a Kähler form $\omega^{\prime}$. (ii) The group $\operatorname{\sf Mon}^{\operatorname{\sf Hdg}}(M,I)$ acts transitively on the set of the Weyl chambers (see [M3]; in fact this easily follows from 3.4). Let $W(I)$ denote the fundamental Weyl chamber, that is, the interior of the birational nef cone. Then there is an element $\gamma\in\operatorname{\sf Mon}^{\operatorname{\sf Hdg}}(M,I)$ such that $W(I)=\gamma W(I^{\prime})$. Consider $\widetilde{\gamma}$ which is a lift of $\gamma$ in the mapping class group $\Gamma_{I}$, and the complex structure $\widetilde{\gamma}(I^{\prime})$. This is again non-separable from $I$ and $I^{\prime}$ and it carries a Kähler form $\widetilde{\gamma}(\omega^{\prime})$, orthogonal to $\gamma(z)$. Its cohomology class is an element of the birational Kähler cone of $I$. The inverse image of this class by $\gamma$ is orthogonal to $z$, q.e.d.. (iii) We claim that any positive $(1,1)$-class $\omega$ which is orthogonal to $z$ but not to any other rational cohomology class is Kähler modulo monodromy and birational transforms. Indeed, consider the twistor line $L\subset\operatorname{{\mathbb{P}}\sf er}$ corresponding to the 3-subspace generated by the period of $I_{0}$ and $\omega$. By 5.3, it lifts to $\operatorname{\sf Teich}_{z}$ as a $z$-GHK line. This line does not have to pass through $I_{0}$, but it passes through some $I^{\prime}_{0}$ nonseparable from $I_{0}$. On this $I^{\prime}_{0}$, $\omega$ is Kähler, and we conclude in the same way as in (ii) that on $I_{0}$ itself it is Kähler modulo monodromy and birational equivalence. Notice that by definition, the property obtained in (iii) expresses exactly the fact that $z$ is not MBM. We thus obtain the deformation-invariance of MBM classes. Corollary 5.13: A negative class $z$ is MBM or not simultaneously in all complex structures where it is of type $(1,1)$. Corollary 5.14: An MBM class is $\pm{\mathbb{Q}}$-effective and represented by a rational curve (up to a scalar) on a birational model of $(M,I)$. Proof: We have seen that such is the case (even on $(M,I)$ itself) when $z$ generates the Picard group over ${\mathbb{Q}}$. The rest follows by deformation invariance of MBM property and the results on deformations of minimal rational curves in section 4. Putting all we have done in this section together, we arrive at the following. Theorem 5.15: Let $M$ be a simple hyperkähler manifold, and $z\in H_{2}(M,{\mathbb{Q}})$ a homology class. Then the following statements are equivalent. (i) The space $\operatorname{\sf Teich}_{z}$ (the subset of all $I\in\operatorname{\sf Teich}$ such that $z$ lies in $H_{1,1}(M,I)$) contains a twistor line. (ii) For each $I\in\operatorname{\sf Teich}_{z}$, there exists $I^{\prime}\in\operatorname{\sf Teich}_{z}$ non-separable from $I$ which is contained in a twistor line. (iii) For each $I\in\operatorname{\sf Teich}_{z}$ with Picard number one, $\pm z$ is not effective; (iv) For some $I\in\operatorname{\sf Teich}_{z}$ with Picard number one, $\pm z$ is not effective; (v) $z$ is not MBM. Moreover in the items (iii) and (iv) one can replace “$\pm z$ is not effective” by “$(M,I)$ contains no rational curves”. Proof: The implication (i) $\Rightarrow$ (ii) follows from 5.2. If (ii) holds, then there is $I^{\prime}$ non-separable from $I$ such that $z$ is orthogonal to a Kähler form on $I^{\prime}$, but, as we have already seen, one then has $\operatorname{Kah}(I^{\prime})=\operatorname{Pos}(I^{\prime})$ and $I=I^{\prime}$, so $z$ cannot be effective on $I^{\prime}$, hence (iii). (iii) $\Rightarrow$ (iv) is obvious. To get (i) from (iv), notice that $\operatorname{Kah}(I)=\operatorname{Pos}(I)$ so there are Kähler classes orthogonal to $z$ on $(M,I)$. This implies existence of twistor lines in $\operatorname{\sf Teich}_{z}$. The property (v) is equivalent to (iv) since these properties are deformation-invariant, and the equivalence for complex structures with Picard number one has already been verified. Finally, if $z$ is effective on $(M,I)$ with Picard number one, it is automatically represented by a rational curve. Indeed otherwise there are no rational curves at all, so the Kähler cone should be equal to the positive cone, but one easily finds a positive $(1,1)$-form orthogonal to $z$. ## 6 Monodromy group and the Kähler cone In this section we prove the results on the Kähler cone stated in the Introduction. ### 6.1 Geometry of Kähler-Weyl chambers Definition 6.1: Let $(M,I)$ be a hyperkähler manifold, and $\operatorname{\sf Mon}^{\operatorname{\sf Hdg}}(M,I)$ the group of all monodromy elements preserving the Hodge decomposition on $(M,I)$. A Kähler-Weyl chamber of a hyperkähler manifold is the image of the Kähler cone of $M^{\prime}$ under some $\gamma\in\operatorname{\sf Mon}^{\operatorname{\sf Hdg}}(M,I)$, where $M^{\prime}$ runs through the set of all birational models of $M$. Theorem 6.2: Let $(M,I)$ be a hyperkähler manifold, and $S\subset H_{1,1}(M,I)$ the set of all MBM classes in $H_{1,1}(M,I)$. Consider the corresponding set of hyperplanes $S^{\bot}:=\\{W=z^{\bot}\ \ \|\ \ z\in S\\}$ in $H^{1,1}(M,I)$. Then the Kähler cone of $(M,I)$ is a connected component of $\operatorname{Pos}(M,I)\backslash S^{\bot}$, where $\operatorname{Pos}(M,I)$ is a positive cone of $(M,I)$. Moreover, the connected components of $\operatorname{Pos}(M,I)\backslash S^{\bot}$ are Kähler-Weyl chambers of $(M,I)$. Proof: First of all, none of the classes $v\in W,\ W\in S^{\bot}$ belong to interior of any of the Kähler-Weyl chambers: indeed it follows from 5.3 that if $v\in W$ is in the interior of a Kähler-Weyl chamber, then the classes belonging to the interior of Kähler-Weyl chambers are dense in $W$ so $W\not\in S^{\bot}$. It remains to show that any connected component of $\operatorname{Pos}\backslash S^{\bot}$ is a Kähler-Weyl chamber. Consider now a cohomology class $v\notin S^{\bot}$, and let $W=\langle\operatorname{Re}\Omega,\operatorname{Im}\Omega,v\rangle$ be the corresponding 3-dimensional plane in $H^{2}(M,{\mathbb{R}})$. We say that $v$ is KW-generic if $W^{\bot}\cap H^{1,1}(M,{\mathbb{Q}})$ is at most 1-dimensional. Denote by ${\mathfrak{W}}$ the set of non-generic $(1,1)$-classes. Clearly, ${\mathfrak{W}}$ is a union of codimension 2 hyperplanes, hence removing ${\mathfrak{W}}$ from $\operatorname{Pos}\backslash S^{\bot}$ does not affect the set of connected components. We are going to show that any $v\in\operatorname{Pos}(M,I)\backslash({\mathfrak{W}}\cap S^{\bot})$ belongs to the interior of a Kähler-Weyl chamber. This would imply that the connected components of $\operatorname{Pos}(M,I)\backslash S^{\bot}$ are Kähler-Weyl chambers, finishing the proof of 6.1. Since $v$ is KW-generic, and not in $S^{\bot}$, the space $W^{\bot}\cap H^{1,1}(M,{\mathbb{Q}})$ contains no MBM-classes. This implies that $\operatorname{\sf Teich}_{z}$ is covered by twistor lines, where $z$ is a generator of $W^{\bot}\cap H^{1,1}(M,{\mathbb{Q}})$. By 5.3, there exists a hyperkähler structure $(I^{\prime},J,K)$ such that $W$ is the corresponding 3-dimensional plane $\langle\omega_{I^{\prime}},\omega_{J},\omega_{K}\rangle$ and $\operatorname{\sf Per}(I^{\prime})=\operatorname{\sf Per}(I)$. Therefore, $v$ is Kähler on $(M,I^{\prime})$. As we have already seen in the proof of 5.3, this implies that $v$ belongs to the interior of a Kähler-Weyl chamber on $(M,I)$. ### 6.2 Morrison-Kawamata cone conjecture and minimal curves Recall the following theorem which follows from the global Torelli theorem. Theorem 6.3: Let $(M,I)$ be a hyperkähler manifold, and $\operatorname{\sf Mon}(M,I)\subset O(H^{2}(M,{\mathbb{Z}}))$ its monodromy group. Let $G$ the image of $\operatorname{Aut}(M)$ in $O(H^{2}(M,{\mathbb{Z}}))$. Then $G$ is the set of all $\gamma\in\operatorname{\sf Mon}^{\operatorname{\sf Hdg}}(M,I)$ fixing the Kähler chamber. Proof: Similar to the proof of 3.2 (second part); see [M3]. Recall also that the image of the mapping class group is a finite index subgroup in $O(H^{2}(M,{\mathbb{Z}}))$, and, accordingly, $\operatorname{\sf Mon}^{\operatorname{\sf Hdg}}(M,I)$ is of finite index in the group of isometries of the Picard lattice. Let $M$ be a hyperkähler manifold, and $s\in H_{2}(M,{\mathbb{Z}})$ a homology class. The BBF form defines an injection $H_{2}(M,{\mathbb{Z}})\stackrel{{\scriptstyle j}}{{\hookrightarrow}}H^{2}(M,{\mathbb{Q}})$, hence $q(s,s)$ can be rational. However, the denominators of $\operatorname{im}j$ are divisors of the discriminant $\delta$ of $q$, hence $\delta j(s)$ is always integer, and $\delta^{2}q(s,s)\in{\mathbb{Z}}$. Conjecture 6.4: Let $M$ be a hyperkähler manifold. Then there exists a constant $C>0$, depending only on the deformation type of $M$, such that for any primitive MBM class $s$ in $H^{2}(M,{\mathbb{Z}})$ one has $\|q(s,s)\|<C$. Remark 6.5: This conjecture is slightly weaker than its following, more algebraic-geometric version, implicitely appearing in [BHT]: let $M$ be a hyperkähler manifold. Then there exists a constant $C>0$ such that for any extremal rational curve (of minimal degree) $R$ on any deformation $(M,I),\ I\in\operatorname{\sf Teich}$, one has $\|q(R,R)\|<C$. Indeed, for any primitive integral MBM class $s$, some integral multiple $Ns$ is represented by an extremal rational curve $R$ on such a deformation $(M,I)$ that $s$ generates its Picard group, by 5.3. So the boundedness of $\|q(R,R)\|$ means the boundedness of $N$ plus the boundedness of $\|q(s,s)\|$. Theorem 6.6: Let $M$ be a hyperkähler manifold. Then 6.2 implies the Morrison-Kawamata cone conjecture for all deformations of $M$. Proof: Fix a complex structure $I$ on $M$, and let $S(I)$ be the set of MBM classes which are of type $(1,1)$ on $(M,I)$. The faces of the Kähler-Weyl chambers are pieces of $s^{\bot}$, where $s^{\bot}$ runs through $S(I)$. If 6.2 is true, the monodromy group $\operatorname{\sf Mon}^{\operatorname{\sf Hdg}}(M,I)$ acts on $S(I)$ with finitely many orbits (3.2). Then the argument as in 3.2 proves that $\operatorname{\sf Mon}^{\operatorname{\sf Hdg}}(M,I)$ acts on the set of faces of Kähler-Weyl chambers with finitely many orbits. Let ${\mathfrak{F}}$ be the set of all pairs $(F,\nu)$, where $F$ is a face of a Kähler-Weyl chamber, and $\nu$ is orientation on a normal bundle $NF$. Then the monodromy acts on ${\mathfrak{F}}$ with finitely many orbits, as we have already indicated. Each face of a Kähler cone $\operatorname{Kah}$ gives an element of ${\mathfrak{F}}$: we pick orientation determined by the side of a face adjoint to the cone. Denote by ${\mathfrak{F}}_{0}\subset{\mathfrak{F}}$ the set of all faces of the Kähler cone with their orientations. There are finitely many orbits of $\operatorname{\sf Mon}^{\operatorname{\sf Hdg}}(M,I)$ acting on ${\mathfrak{F}}_{0}$. However, each $\gamma\in\operatorname{\sf Mon}^{\operatorname{\sf Hdg}}(M,I)$ which maps an element $f\in{\mathfrak{F}}_{0}$ to an element $\gamma(f)\in{\mathfrak{F}}_{0}$ maps the Kähler chamber to itself. Indeed, there are two Kähler-Weyl chambers adjoint to each face, and $\gamma(\operatorname{Kah})$ is one of the two chambers adjoint to $\gamma(f)$. But since the orientation is preserved, $\gamma(\operatorname{Kah})=\operatorname{Kah}$ and not the other one. Therefore $\gamma$ is induced by an automorphism of $(M,I)$. For hyperkähler manifolds which are deformation equivalent to the Hilbert scheme of length $n$ subschemes on a K3 surface, 6.2 is easily deduced from [BHT], Proposition 2. Indeed it is shown there that for $M$ as above and projective, any extremal ray of the Mori cone contains an effective curve class $R$ with $q(R,R)\geqslant-\frac{n+3}{2}$. But if we want to bound the “length” (that is, the square of a minimal representative curve) of an MBM class on $(M,I)$, we can look at the extremal rays on projective deformations, because MBM classes are deformation equivalent and the monodromy acts by isometries. We thus obtain the following Corollary 6.7: The Morrison-Kawamata cone conjecture holds for deformations of the Hilbert scheme of length $n$ subschemes on a K3 surface. Acknowledgements: We are grateful to Eyal Markman for many interesting discussions and an inspiration. Many thanks to Brendan Hassett and to Justin Sawon who disabused us of some wrong assumptions. We are grateful to Claire Voisin for patiently answering many questions, and to Frédéric Campana for his expert advice on cycle spaces of non-algebraic manifolds. 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Anal. 5 (1995), no. 1, 92-104. * [V1] Verbitsky, M., Cohomology of compact hyperkähler manifolds, alg-geom electronic preprint 9501001, 89 pages, LaTeX. * [V2] Verbitsky, M., Cohomology of compact hyperkähler manifolds and its applications, GAFA vol. 6 (4) pp. 601-612 (1996). * [V3] Verbitsky, M., Parabolic nef currents on hyperkähler manifolds, 19 pages, arXiv:0907.4217. * [V4] Verbitsky, M., A global Torelli theorem for hyperkähler manifolds, Duke Math. J. Volume 162, Number 15 (2013), 2929-2986. * [V5] Verbitsky, M., Ergodic complex structures on hyperkahler manifolds, arXiv:1306.1498, 22 pages. * [Voi] Voisin, C., Sur la stabilité des sous-variétés lagrangiennes des variétés symplectiques holomorphes, Complex projective geometry (Trieste, 1989/Bergen, 1989), 294–303, London Math. Soc. Lecture Note Ser., 179, Cambridge Univ. Press, Cambridge, 1992. Ekaterina Amerik Laboratory of Algebraic Geometry, National Research University HSE, Department of Mathematics, 7 Vavilova Str. Moscow, Russia, [email protected], also: Université Paris-11, Laboratoire de Mathématiques, Campus d’Orsay, Bâtiment 425, 91405 Orsay, France Misha Verbitsky Laboratory of Algebraic Geometry, National Research University HSE, Department of Mathematics, 7 Vavilova Str. Moscow, Russia, [email protected], also: Kavli IPMU (WPI), the University of Tokyo	arxiv-papers	2014-01-02T17:35:20	2024-09-04T02:49:56.192131	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Ekaterina Amerik, Misha Verbitsky", "submitter": "Misha Verbitsky", "url": "https://arxiv.org/abs/1401.0479" }
1401.0483	11institutetext: Universita’ di Firenze, Sez. di Astronomia, Largo E. Fermi 2, I-50125 Florence, Italy 22institutetext: I.N.A.F.-Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, I-50125 Florence, Italy 33institutetext: Thüringer Landessternwarte Tautenburg, Sternwarte 5, D-07778 Tautenburg, Germany 44institutetext: School of Cosmic Physics, Dublin Institute for Advanced Studies, 31 Fitzwilliam Place, Dublin 2, Ireland 55institutetext: Max-Planck- Institut für Astronomie, Königstuhl 17, D-69117 Heidelberg, Germany 66institutetext: Dipartimento di Matematica, Università degli Studi di Roma “Tor Vergata”, Via della Ricerca Scientifica 1, 00133 Roma, Italy, 77institutetext: School of Physics, University College Dublin, Belfield, Dublin 4, Ireland. # Physical properties of the jet from DG Tauri on sub-arcsecond scales with HST/STIS ††thanks: Based on observations made with the NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. L. Maurri 11 F. Bacciotti 22 L. Podio 22 J. Eislöffel 33 T. P. Ray 44 R. Mundt 55 U. Locatelli 66 D. Coffey 77 (Received / Accepted) ###### Abstract Context. Stellar jets are believed to play a key role in star formation, but the question of how they originate is still being debated. Aims. We derive the physical properties at the base of the jet from DG Tau both along and across the flow and as a function of velocity. Methods. We analysed seven optical spectra of the DG Tau jet, taken with the Hubble Space Telescope Imaging Spectrograph. The spectra were obtained by placing a long-slit parallel to the jet axis and stepping it across the jet width. The resulting position-velocity diagrams in optical forbidden emission lines allowed access to plasma conditions via calculation of emission line ratios. In this way, we produced a 3-D map (2-D in space and 1-D in velocity) of the jet’s physical parameters i.e. electron density ne, hydrogen ionisation fraction xe, and total hydrogen density nH. The method used is a new version of the BE-technique. Results. A fundamental improvement is that the new diagnostic method allows us to overcome the upper density limit of the standard [S ii] diagnostics. As a result, we find at the base of the jet high electron density, $n_{e}\sim$ 105, and very low ionisation, $x_{e}\sim 0.02-0.05$, which combine to give a total density up to $n_{H}\sim$ 3 106. This analysis confirms previous reports of variations in plasma parameters along the jet, (i.e. decrease in density by several orders of magnitude, increase of $x_{e}$ from 0.05 to a plateau at 0.7 downstream at 2′′ from the star). Furthermore, a spatial coincidence is revealed between sharp gradients in the total density and supersonic velocity jumps. This strongly suggests that the emission is caused by shock excitation. No evidence was found of variations in the parameters across the jet, within a given velocity interval. The position-velocity diagrams indicate the presence of both fast accelerating gas and slower, less collimated material. We derive the mass outflow rate, $\dot{M}_{j}$, in the blue-shifted lobe in different velocity channels, that contribute to a total of $\dot{M}_{j}\sim$ 8 $\pm$ 4 10-9 M⊙ yr-1. We estimate that a symmetric bipolar jet would transport at the low and intermediate velocities probed by rotation measurements, an angular momentum flux of $\dot{L}_{j}\sim$ 2.9 $\pm$ 1.5 10-6 M⊙ yr-1 AU km s-1. We discuss implications of these findings for jet launch theories. Conclusions. The derived properties of the DG Tau jet are demonstrated to be consistent with magneto-centrifugal theory. However, non-stationary modelling is required in order to explain all of the features revealed at high resolution. ###### Key Words.: ISM: Herbig-Haro objects - ISM: jets and outflows - Stars: formation - Stars: pre-main sequence ## 1 Introduction Herbig-Haro (HH) jets emanating from young stars have been widely studied in the recent past (Bally et al. 2007, Ray et al. 2007). Jets are believed to regulate important processes, such as the extraction of the angular momentum in excess from the star-disk system and the dispersion of the parent cloud. The phenomenon, however, is far from completely understood, and open questions still remain. For example, the elegant magneto-centrifugal theory behind the proposed models for the jet launch (e.g., Shu et al. 2000, Ferreira et al. 2006, Pudritz et al. 2007, Edwards 2009) still lacks a strong observational confirmation. In this scenario, particles that have been lifted from the disk are then accelerated and collimated by the combined action of centrifugal and magnetic forces along hour-glass shaped magnetic surfaces anchored to the disk. This process takes place within the first few AU from the central star, a region that is not directly observable even for the closest jet-disk systems. Another important open issue is the nature of the gas excitation. The observed jet emission is generally attributed to the presence of shocks that heat the gas locally (Hartmann & Raymond 1989, Hartigan et al. 1994, Hartigan et al. 1995, and Bacciotti et al. 1999). However, other heating mechanisms may also be in operation, such as ambipolar diffusion (Safier 1993; Garcia et al. 2001) or turbulent dissipation in a viscous mixing layer (Raymond et al., 1994). To clarify these issues it is important to rely on high resolution facilities, such as the Hubble Space Telescope (HST), to examine young stars for which the immediate stellar environment is not opaque, i.e. classical T Tauri stars (CTTS). In 1999, we used the Hubble Space Telescope Imaging Spectrograph (HST/STIS) to observe the jet from the CTTS DG Tau at optical wavelengths. The angular resolution of 0.′′1 corresponds to a spatial scale of 14 AU at the distance of the Taurus cloud (140 pc). Although this scale is much greater than that of the magneto-centrifugal engine (a few AU), we can expect to see an imprint of the jet launching conditions. The DG Tau jet (HH 158) was one of the first HH jets discovered (Mundt & Fried, 1983) and, because of its brightness, proximity, and structure, it is still one of the most well-studied stellar jets. The blue-shifted lobe of the bipolar jet is inclined by about 38∘ to the line of sight (Eislöffel & Mundt, 1998). The flow presents a diverging geometry and appears to blow a sequence of ‘bubbles’ that terminate in luminous bow-like features. In particular, a large bright bow-like structure was imaged at 2$\aas@@fstack{\prime\prime}$7 arcseconds from the source in 1997 by Lavalley et al. (1997) (labelled B1 in their nomenclature), but other knots are also seen farther away (Eislöffel & Mundt, 1998; McGroarty et al., 2007). Subsequently, the DG Tau jet has been studied at near infrared wavelengths (e.g., Takami et al. 2002, Pyo et al. 2003, and Agra-Amboage et al. 2011), and most recently in the X-ray domain (Güdel et al., 2008). A high resolution study of the DG Tau jet morphology and kinematics was conducted based on HST/STIS observations. In 1999, our team obtained a valuable HST/STIS dataset of the DG Tau jet, consisting of seven long-slit spectra. The slit was placed along the jet and stepped by 0$\aas@@fstack{\prime\prime}$07 across the jet width, to build a three- dimensional datacube (i.e. two spatial dimensions and one spectral dimension). Firstly, Bacciotti et al. (2000) presented examples of high spatial resolution velocity-channel maps of the jet, within the first 2$\arcsec$ from the source. These maps outline well-defined features in the flow, as well as an onion-like kinematic structure in which the low velocity gas is less collimated than that at higher velocities. Subsequently, Bacciotti et al. (2002) described how these data provide possible indications for rotation of the jet about its symmetry axis close to the base of the flow. Further signatures of jet rotation from complementary HST/STIS observations were presented in (Coffey et al., 2004, 2007). These rotation results supported the magneto-centrifugal jet launch scenario (e.g., Ferreira et al. 2006, Pudritz et al. 2007) and, for the first time, tested the idea that jets can extract angular momentum from the disk, in order to permit accretion onto the star at the observed rate. In the present study, we continue to exploit the 1999 HST/STIS dataset, in order to achieve a detailed parameterisation of the jet plasma physics. Previous studies of the gas conditions include: Lavalley-Fouquet et al. (2000); Bacciotti (2002) (preliminary analysis of HST/STIS 1999 data); and Coffey et al. (2008). These studies relied on the so-called BE-technique (Bacciotti et al., 1999), and yet give contradictory reports regarding the correlation between gas excitation and gas velocity. The present study aims to clarify the issue, while providing high resolution maps of the gas physics in three dimensions (two spatial and one in velocity). No jet plasma study to- date (Bacciotti (2002), Melnikov et al. (2008, 2009), Coffey et al. (2008) Hartigan & Morse (2007)) has been in a position to present the combination of high resolution with all three dimensions. The most important outcome of this study is the determination of the total hydrogen density, resolved in space and velocity. This is a fundamental parameter for the characterisation of the jet dynamics, allowing estimates of the mass outflow rate ($\dot{M}_{j}$) and of the angular momentum flux ($\dot{L}_{j}$). The paper is organised as follows. The observations and the diagnostic techniques are presented in section 2. The position-velocity (PV) diagrams of the emission lines and their ratios are illustrated in section 3. The results of the spectral diagnostic analysis are given in section 4, and their implications for the dynamics of the system are discussed in section 5. Finally, section 6 summarises our conclusions. ## 2 Observations and method of analysis ### 2.1 Observations and data reduction As described in Bacciotti et al. (2000), seven optical spectra of DG Tau and its jet were taken with HST/STIS in January 1999 (Proposal ID. GO 7311). The slit was placed parallel to the jet axis (P.A. $\sim$226∘), and stepped by 0$\aas@@fstack{\prime\prime}$07 in the transverse direction thus covering a total jet width of $\sim$ 0$\aas@@fstack{\prime\prime}$5\. The spectra (labelled S1, S2, … S7, from south-east to north-west) constitute a 3-D data- cube, comprising two spatial dimensions and one spectral dimension. To observe strong jet tracers such as [O i] $\lambda\lambda$6300,6363, [N ii] $\lambda\lambda$6548,6583, [S ii] $\lambda\lambda$6716,6731, and H$\alpha$, the G750M grating was used, covering a wavelength band of 652 Å centred at 6581 Å. The stronger component of the OI doublet, [O i]$\lambda$6300, is partially blue-shifted off the detector. However, since this doublet is emitted in a fixed ratio of 3:1, we use the [O i]$\lambda$6363 line in its place. At the chosen wavelengths, the angular resolution of HST is 0$\aas@@fstack{\prime\prime}$1, with two-pixel sampling. The slit aperture is 52$\times$0.1 arcsec2. and the spectral sampling 0.554 Å pixel-1. The effective velocity resolution is $\sim$ 50 km s-1 for extended sources. Standard data reduction was carried out by the HST/STIS pipeline. IRAF tasks were used to remove the effects of bad pixels and cosmic rays, to conduct continuum subtraction, and to convert to a velocity scale. Velocities have been corrected for the heliocentric velocity of the star, v${}_{\star,hel}\sim$ +17.0 km s-1, as derived from a Gaussian fit to the LiI $\lambda$6707 photospheric absorption line in the central slit position. A velocity resampling was applied to achieve the same dispersion in all lines (24.67 km s-1 per pixel), to ensure the utmost accuracy in the line ratios. ### 2.2 Application of the BE diagnostic technique The BE-technique is a method of obtaining information on the gas physics by comparing ratios of the gas emission lines (Bacciotti & Eislöffel 1999, Podio et al. 2006). The technique relies on the fact that, in low excitation conditions and far from strong sources of ionising radiation, sulphur is ionised only once, and the ionisation state of oxygen and nitrogen is dominated by charge-exchange with hydrogen. Under these conditions, the ratios between the forbidden emission lines emitted by S+, O, and N+ are a known function of electron density, $n_{e}$, ionisation fraction, $x_{e}$ (where $x_{e}$ = $n_{e}$ / $n_{H}$), and electron temperature, $T_{e}$. With the lines in our dataset, $n_{e}$ can be calculated from the ratio [S ii]$\lambda 6731$ /[S ii]$\lambda 6716$ (hereafter [S ii]31/16). Then, using $n_{e}$, a dedicated numerical code (see Melnikov et al. 2008) evaluates the ratios [N ii]($\lambda$6583+$\lambda$6548)/[O i]($\lambda$6300+$\lambda$6363) and [O i]($\lambda$6300+$\lambda$6363)/[S ii]($\lambda$6716+$\lambda$6731) (hereafter [N ii]/[O i] and [O i]/[S ii], respectively) against a grid of $x_{e}$ and $T_{e}$ values. The best fit gives the ionisation (and temperature) of the emitting gas, leading ultimately to an estimate of the total hydrogran density, $n_{H}$, a fundamental parameter in jet dynamics. The procedure is independent of the assumed heating mechanism and its simplicity of application allows speedy investigation of large datasets. Excitation conditions are assumed to remain constant along the line of sight, which results in smoothing the gradients of the quantities (DeColle et al., 2010). However, in the present case the problem is mitigated by the velocity resolution, which naturally sorts the different jet layers. A further limitation is that in spatially unresolved shock waves, $T_{e}$ varies rapidly over the line emission region, while the evolution of $n_{e}$ and $x_{e}$ is slow. Therefore, as discussed in Bacciotti & Eislöffel (1999), the observed line ratios, averaged over the resolution element and along the line of sight, can only give a rough indication of the local excitation temperature. In order to illustrate the uncertainties of the BE-technique in the case of unresolved shocks, a determination of $x_{e}$ has been attempted from the grid of shock models of Hartigan et al. (1994), assuming that the ionisation in the gas is produced locally in each resolution element by a shock. To this aim, we used the grid of shock models by Hartigan et al. (1994), finding values of $x_{e}$ lower than those derived with the BE-technique by about 30%. Finally, extinction is not taken into account, as this should be determined locally around the jet base using emission lines across a broader wavelength range. However, the lines used in the BE technique are close in wavelength, and typically in the case of CTTS, the uncertainty introduced by not accounting for extinction is found to be lower than the error due to noise (Podio et al., 2006). We also note that to increase the number of positions where an indication of the plasma conditions can be given, in regions where one of the emission line intensities falls below 3$\sigma$, the 3$\sigma$ value is used in order to obtain an upper/lower limit for the ratio. In this case the results of the diagnostics are given in terms of upper or lower limits. In this work we use an updated version of the technique with respect to recent papers (Coffey et al., 2008; Melnikov et al., 2009). As in Podio et al. (2011), we use values for the collision strengths derived from the results of Keenan et al. (1996) for S+, and of Hudson & Bell (2005) for N+. For neutral oxygen, we use the values of the collisional coefficients reported in Berrington & Burke (1981), which integrate the compilation by Mendoza (1983). The interpolation of these coefficients gives better values of the collision strengths over a wider temperature range than in previous studies. Elemental abundances are taken from Asplund et al. (2005). In addition, where the plasma density is higher than the high density limit for the [S ii] ratio, thus preventing derivation of electron density, we use an extension of the diagnostic code which relies instead on the [N ii]/[O i] and[O i]/[S ii] line ratios to find $n_{e}$ and $x_{e}$. This extension assumes a value of Te derived for neighbouring points where the standard BE technique can be safely applied (see sect.4). In this case the uncertainty in the determination of $n_{e}$ and $x_{e}$ is estimated to be of about 25% and 15%, respectively, for variations of the assumed temperature of 30%, with both quantities decreasing for increasing $T_{e}$. When [N ii] is below the 3$\sigma$ threshold, the inferred values of $x_{e}$ and $n_{e}$ are upper limits. ## 3 Results: 3-D kinematic structure and line ratios We present our data and results as PV maps of the emission lines, of the line ratios, and of the derived plasma parameters. In each figure the seven PV maps obtained from the stepped slit positions cover a combined field of view of $\sim$ 5$\arcsec\times$ 0$\aas@@fstack{\prime\prime}$5\. The dashed lines indicate the position of emission peaks which were identified in previous studies: A2 at 0$\aas@@fstack{\prime\prime}$75 and A1 at 1$\aas@@fstack{\prime\prime}$45 in this dataset, Bacciotti et al. (2000); B1 at 2$\aas@@fstack{\prime\prime}$7 Lavalley et al. (1997) in 1998, seen in this dataset at 3$\aas@@fstack{\prime\prime}$8 (B1 is the same feature as the X-ray feature at 6′′ identified in 2010 Güdel et al. (2011), based on proper motion of 0$\aas@@fstack{\prime\prime}$275 yr-1 (Pyo et al., 2003)); and a secondary peak, B0, at 3$\aas@@fstack{\prime\prime}$3, identified in the channel maps at high velocity of Lavalley-Fouquet et al. (2000). ### 3.1 Position-velocity diagrams of the surface brightness Figs. 1, 2, and 3 show the [S ii]$\lambda$6731, [O i]$\lambda$6363 and [N ii]$\lambda$6583 emission lines respectively, in three dimensions: along the jet, across the jet, and in velocity space. Figure 1: Continuum-subtracted HST/STIS position-velocity (PV) plots of the jet from DG Tau, in [S ii]$\lambda$6731 emission in slit positions S1 to S7 (south-east to north-west). Contours are from 1.1 10-15 erg s-1 arcsec-2 cm-2 Å-1 (3 $\sigma$), with a ratio of 22/5. The solid lines mark the position of the star and zero velocity, while dashed lines mark the positions of identified features in images of this flow (see text). Figure 2: Same as Fig. 1, but for [O ii]$\lambda$6363\. The stronger doublet component, [O i]$\lambda$6300, is blue-shifted off the detector. However, since this doublet is emitted in a fixed ratio of 3:1, we use the [O i]$\lambda$6363 line in its place. Figure 3: Same as Fig. 1, but for [N ii]$\lambda$6583. At least three velocity components can be identified in the jet up to location A1: a low velocity component, between about -150 and +20, more evident in [S ii] and [O i], and in the lateral positions; a medium velocity component between -300 and -150 km s-1, more evident in [N ii], characterised by an emission peak at about 0$\aas@@fstack{\prime\prime}$45 and a slow acceleration; and a high velocity component between about -300 and -400 km s-1, with an emission spot at 1$\aas@@fstack{\prime\prime}$4, hereafter A1HV, brighter in slits 1, 2, 3 and 4, already identified as a knot in Lavalley- Fouquet et al. (2000) at 0$\aas@@fstack{\prime\prime}$93 from the source. A bifurcation of the emission is evident between the low and the medium velocity components (e.g. slit 1 to 6 in [S ii] at A2, and slit 5 in [N ii] at 0$\aas@@fstack{\prime\prime}$45). A steep gradient towards lower velocities, of $\sim$ 30 km s-1, is apparent about 0$\aas@@fstack{\prime\prime}$2 downstream from A1HV in [N ii], and less evidently in [S ii]. Futher along the jet, between A1 and $\sim 3^{\prime\prime}$, emission is very low in [O i] and [S ii], while the [N ii] line is stronger, mainly at medium velocities. This region corresponds to a faint stripe connecting A1 and B1 in the images of Dougados et al. (2000). The PV plots are very difficult to read here, but there are indications of material flowing at two different velocities (about -150 and -320 km s-1), and marginal indications of a localised gradient in velocity (of about 30 km s-1) at 2$\aas@@fstack{\prime\prime}$8. Beyond 3″, the system of slits intercepts the large ($\sim$ 2′′) bow-like feature of which B0 and B1 form a part. Here all lines are detected, with [N ii] being the strongest, but only at velocities between -150 and -350 km s-1, and brighter in slits S1 to S4. Another gradient towards lower velocities, of $\sim$ 70 km s-1, is seen at 3$\aas@@fstack{\prime\prime}$45 between the emission peaks at B0 and B1, and a less evident one at 4$\aas@@fstack{\prime\prime}$1, of $\sim$ 30 km s-1, downstream of knot B1. Interestingly, these apparent abrupt reductions in velocity occur immediately downsteam of a peak in emission. This is expected in radiative shocks, in which the discontinuity in velocity is at the front, while the optical emission arises behind the shock front on scales resolvable by HST (Hartigan et al., 1994). ### 3.2 Position-velocity diagrams of the line ratios Figure 4: PV plots of the [S ii]$\lambda$6731/[S ii]$\lambda$6716 line ratio in linear greyscale. Cyan and orange contours indicate the [S ii]$\lambda$6716 and [S ii]$\lambda$6731 emission at $3\sigma$, respectively. Where only one of the two lines is above the $3\sigma$ threshold the upper/lower limit of the line ratio is reported. Figure 5: Same as Fig. 4 for the logarithm of the [N ii]($\lambda$6583+$\lambda$6548)/ [O i]($\lambda$6300+$\lambda$6363) line ratio. The superposed cyan and red contours indicate the [O i]$\lambda$6363 and the [N ii]$\lambda$6583 emission at $3\sigma$ ([O i]$\lambda$6300 = 3 [OII]$\lambda$6363 and [N ii]$\lambda$6548 = 1/3 [N ii]$\lambda$6583 is assumed everywhere, see text). Figure 6: Same as Fig. 4 for [O i]($\lambda$6300+$\lambda$6363)/[S ii]($\lambda$6716+$\lambda$6731) line ratio. Blue and green contours indicate [O i]$\lambda$6363 and the [S ii]6731+6716 emission at $3\sigma$ respectively. Figures 4, 5, and 6 show emission line ratio PV plots for [S ii]31/16, [N ii]/[O i] and [O i]/[S ii], respectively. Plots are produced after two-pixel binning in both spatial and spectral dimensions, to reflect resolution (i.e. 0$\aas@@fstack{\prime\prime}$1 and 50 km s-1). A 3$\sigma$ contour ($\sim$2.2 10 -15 erg s-1 arcsec-2 cm-2 Å-1 after binning) for each emission line is overlaid. The [S ii] ratio is in many regions at the high density limit of 2.35 (HDL), beyond which the ratio is no longer a valid diagnostic tool. In these regions, encircled by green contours, the electron density is $\geq$ 2 104 cm-3, i.e. high when compared to published values at large distances along many jets of typically 103 cm-3. The [N ii]/[O i] ratio is a good indicator of the hydrogen ionisation fraction since it increases monotonically with it, and is nearly independent of $n_{e}$ and $T_{e}$, as long as the electron density is below the [N ii] critical electron density of $n_{e}<10^{5}$ cm-3. The value of this ratio is low near the star, but smoothly increases with distance and velocity. It reaches higher values at high speeds approaching A2, and at medium velocities between A2 and A1. At A1HV, both the [O i] and [N ii] lines are intense giving a moderate ratio. In the A1-B0 ridge the reported value of $\sim$0.4 is a lower limit, as [O i] has been set to 3$\sigma$. Meanwhile, high ratio values return in the B0 - B1 region. Finally, the [O i]/[S ii] line ratio depends on both $n_{e}$ and $T_{e}$, and weakly on $x_{e}$ through [O i]. The ratio shows moderate values almost everywhere except for the shoulder between the star and A2 at progressively higher speeds, and at A1HV. This ’shoulder’ appears not to correspond to any of the kinematic components identified above. Figure 7: PV plots of the logarithm of the electron density $n_{e}$. Contours indicate regions inside which: cyan \- [S ii]$\lambda 6716\geq\leavevmode\nobreak\ 3\sigma$; orange \- [S ii]$\lambda 6731\geq\leavevmode\nobreak\ 3\sigma$; green \- [S ii]31/16 ratio at the high density limit. Where only [S ii]$\lambda$6731 ([S ii]$\lambda$6731) $\geq\leavevmode\nobreak\ 3\sigma$, the derived $n_{e}$ is a lower (upper) limit. Inside the green contours $n_{e}$ is derived with the modified HDL-BE procedure (cfr. Sect. 2.2). Figure 8: Position-velocity diagrams of the ionisation fraction xe. Contours: blue \- [O i]$\lambda$6363 at $3\sigma$; red \- [N ii]$\lambda$6583 at $3\sigma$; green \- high density limit for the [S ii]31/16 ratio. Where only [O i] ([N ii]) is above $3\sigma$, $x_{e}$ is an upper (lower) limit. Inside the HDL regions $x_{e}$ is determined with the HDL-BE procedure (cfr. Sect. 2.2). Figure 9: Position-velocity diagrams of the logarithm of the hydrogen density derived as n${}_{H}=$ ne/xe. Contour colour coding is as in Fig.8. Figure 10: 2-D velocity channel maps of the logarithm of the electron density (left), of the ionisation fraction (centre) and of the logarithm of the total density (right). Greyscales are linear. Low velocity interval (LVI) is defined as -120 to +25 km s-1, medium velocity interval (MVI) is defined as -270 to -120 km s-1 and high velocity interval (HVI) is defined as -420 to -270 km s-1. Contours: cyan \- [S ii]$\lambda$6716 at $3\sigma$; orange \- [S ii]$\lambda$6731 at $3\sigma$; green \- region above [S ii] critical density; blue \- [O i]$\lambda$6363 at $3\sigma$; red \- [N ii]$\lambda$6583 at $3\sigma$. ## 4 Results: physical quantities and jet widths The diagnostic analysis was performed in three ways, in order to optimise the output and to facilitate comparisons with the literature. The first preserves all high resolution information in 3-D (i.e. 2-D space and 1-D velocity), and the results are presented as PV plots of $n_{e}$, $x_{e}$, and $n_{H}$ (Fig. 7, 8, 9) giving a global picture of the jet excitation conditions. The results highlight regions of low signal-to-noise where the diagnostics would benefit from a further binning of the input data in space and/or velocity. Therefore, the second way of carrying out the analysis bins the velocity information into three velocity channels (Fig. 10), defined as low velocity interval (LVI), from -120 to +25 km s-1, medium velocity interval (MVI), from -270 to -120 km s-1 and high velocity interval (HVI) from -420 to -270 km s-1. The third way bins in velocity and jet width giving plasma parameters in 1-D along the jet as it propagates (Figs. 11, 12, 13). 1-D profiles of the line ratios are provided as on-line material, Figs. 16 \- 18. In all cases, in positions in which the [SII] ratio is at the high density limit, the modified HDL technique has been applied (cfr. Sect. 2.2) adopting in the affected regions T${}_{e}=10^{4}K$ for the LVI, T${}_{e}=$ 1 and 2 104 K for the MVI, upstream and downstream A2, respectively, and T${}_{e}=$ 3 104 K for the HVI. Lastly, the jet width of the various velocity components is estimated, Fig. 14. ### 4.1 Electron density The results for $n_{e}$ are illustrated in Fig. 7, 10 (left panels) and 11. The electron density is higher than 103 cm-3 almost everywhere in the jet (Fig. 7). The densest portion is near the jet base as far as A1, and for high velocities. While this trend was previously reported by Bacciotti et al. (2000) and Bacciotti (2002), these studies were limited by the critical density of [S ii]. Closer inspection via the modified BE-technique reveals that $n_{e}$ is higher in the MVI and HVI than in the LVI (Fig. 10), with values reaching close to 105 cm-3 near the star. Between the star and A2, there is marginal evidence of a separation between the LVI and the other velocity components (Fig. 7). Meanwhile, there is no variation of $n_{e}$ across the jet, from S1 to S7 in each velocity interval (Fig. 10) . Further along the jet, a slight increase in $n_{e}$ is noted at A1HV (Fig. 7), while downstream of A1, where the flow is seen only in the MVI and HVI, $n_{e}$ continues to decrease in the MVI, reaching 103 cm-3 at 2$\aas@@fstack{\prime\prime}$8, in contrast to the HVI which reaches a maximum at this position. Continuing to the B0-B1 region, $n_{e}$ increases again in the MVI. Finally, the HVI contribution fades after B0, while in the MVI $n_{e}$ is detectable well beyond B1. ### 4.2 Ionisation fraction The results for $x_{e}$ are illustrated in Fig. 8, 10 (middle panels) and 12. It is worth noting that the ’missing’ $x_{e}$ data points result from a low signal in both [S ii] lines, or lack of an unique solution from the code where [O i]/[S ii] $>>$1 (i.e. between the star and A2). However, the trends in those regions can be gleaned from the [N ii]/[O i] ratio (Fig.5), which is almost directly proportional to $x_{e}$. At the base of the jet, $x_{e}$ is not larger than 0.07 (Fig. 8), but it increases with distance and velocity up to a high value of over 0.6 in the region between A2 and A1, peaking at different distances in each velocity interval. Before A1, $x_{e}$ drops in MVI and HVI, while in LVI it increases toward A1 (Fig. 8). There is no strong enhancement of $x_{e}$ at A1HV, contrary to what would be expected given the strong emission. Moving along the jet, we note that between A1 and B0 the determination of $x_{e}$ is poor, and given often in terms of lower limits. Binning over space and velocity increases the [O i] signal-to-noise, making more evident the existence of a high ionisation region between 2′′ and 4$\aas@@fstack{\prime\prime}$1 in both the MVI and the HVI. In the 1-D profiles $x_{e}$ shows a plateau in $x_{e}$ of $\sim$ 0.7 in this region. At 3$\aas@@fstack{\prime\prime}$9 a sharp drop occurs for the MVI, possibly coincident with the 30 km s-1 velocity gradient (Section 3.1). After this point one sees a more gradual decrease in $x_{e}$, for the MVI, represented mainly by lower limits (Fig. 12). Finally, as for $n_{e}$, no large transverse variations are evident in the velocity channel maps of $x_{e}$ (Fig. 10). ### 4.3 Total hydrogen density The results for $n_{H}$ are illustrated in Fig. 9, 10 (right panels) and 13. We stress that our mapping of this quantity is more extended than in previous works, thanks to the application of the modified BE-technique. Similar to $n_{e}$, the total hydrogen density is at its maximum of 3 106 cm-3 close to the star, and then decreases by several orders of magnitude along the flow. The range of variation is larger than in ne, because of the effect of the increase in ionisation. Again, no spatial variation is evident in the transverse direction. In contrast to MVI and LVI trends, moving from A2 to A1 we see a plateau in $n_{H}$ for HVI, giving rise to the high emission of the AHV1 spot. Then $n_{H}$ drops back again downstream of A1HV (Fig. 13). Other $n_{H}$ increases in HVI followed by a drop are seen at 2$\aas@@fstack{\prime\prime}$8 and at B0 and B1. We note that all these locations are also sites of velocity gradients (Section 3.1) and this coincidence points toward a shock nature of the exciting mechanism, as it is dicussed in Sect. 5.2. At B0, the HVI jet slows down and merges with the MVI component, that remains dense to beyond B1 (Fig. 10). ### 4.4 Jet width in the initial channel Measurements of the jet width close to its base are useful in constraining models for launching jets. We estimate the jet width, for the first 0$\aas@@fstack{\prime\prime}$7 of the flow, as the full-width-half-maximum (FWHM) of the intensity profile across the jet. The intensity profiles are obtained by using the seven slit positions to construct an image of the jet in each of four emission lines ([O i]$\lambda$6563, [N ii]$\lambda$6583, [S ii]$\lambda$6716, [S ii]$\lambda$6731), and in three velocity intervals (similar to Fig. 10). The measured FWHM is deconvolved by subtracting in quadrature the FWHM of a reconstructed image of the stellar continuum in a wavelength interval of 3.2 Å, which turns out to be 13 AU (due to the overlap of the slit width). Figure 11: 1-D profiles of log(ne) along the flow in discrete velocity intervals (see Fig. 10). Error bars (smaller than the symbol size where the signal is strong) are determined a posteriori by conducting the analysis with input values of the line ratios and their uncertainties (evaluated from the 3$\sigma$ error on the fluxes). Empty symbols indicate positions where ne is above the high density limit ofr the [S ii] diagnostics, and the derivation has been made through the modified BE-technique described in Sect. 2.2. Figure 12: Same as Fig. 11 for xe. Figure 13: Same as Fig. 11 for log(nH). Figure 14: DG Tau jet width in the first 0$\aas@@fstack{\prime\prime}$7\. Each point is the average obtained from four different lines, with the error bars indicating the dispersion. Dashed line: FWHM of the stellar continuum. The jet width estimates are shown in Fig. 14, where each point is the average of the values obtained in the different lines in each of the velocity intervals (with the error bar indicating the dispersion). The magnitude of the opening angle of the jet depends on the velocity interval, with the LVI being wider and the HVI narrower. However, within each velocity interval, the slope remains almost constant. Our results for the jet width are in agreement with the values estimated in Dougados et al. (2000) and Woitas et al. (2002), where these velocity- integrated results are close to our MVI values. In a recent infra-red study of the DG Tau jet Agra-Amboage et al. (2011), the [Fe ii] emission at velocities of -300 to -160 km s-1 appears to have a smaller opening angle than our corresponding MVI emission, but similar to our HVI emission. This difference may be due to variability of the flow over a six year interval. For example, the data in Agra-Amboage et al. (2011) may have been taken between two episodes of intermittent inflation of hot plasma close to the source. Indeed, in contrast to our spectra, the [Fe ii] emission did not show any emission at velocities above -300 km s-1. Monitoring of [Fe ii] emission over time intervals of 1-2 years would help clarifying this issue. ## 5 Discussion ### 5.1 Flow structure The emerging picture is that of a flow initially collimated and characterised by smoothly varying properties, which soon undergoes strong perturbations represented by bright spots in the PV plots corresponding to the tips of the bow-like features described by Bacciotti et al. (2000); Lavalley et al. (1997). The initial jet channel, up to at least position A2, has characteristics consistent with an overall onion-like kinematic structure, like the one predicted by classical models of jet launching, such as the so-called Disk- wind (e.g. Ferreira et al. 2006; Pudritz et al. 2007) or X-wind (Shu et al., 2000; Shang et al., 2002). In this context an interesting aspect is the apparent acceleration of the gas between the star and A2. Magnetocentrifugal models of disk winds predict an acceleration to asymptotic velocities on a scale proportional to the footpoint radius, $r_{0}$ (i.e. the radius from the star in the disk plane from which the jet is launched). The terminal speed is reached at about 1000$r_{0}$ for the solutions in Garcia et al. 2001 (cfr. their Fig. 1) and at about 100$r_{0}$ for the solutions in Pesenti et al. 2004. Our observed acceleration would then imply a launch radius of 0.3 or 3 AU, respectively. Alternatively, the ejection velocity has decreased over time, or the apparent acceleration is actually the effect of the superposition of separate components. Pyo03 examine the jet structure at lower spatial resolution in [Fe II] infra- red emission. Two well-separated velocity components are detected at epoch 10/2001. Their low radial velocity component of -80 km s-1, located at 0$\aas@@fstack{\prime\prime}$4, may correspond to the bright spot seen at the stellar position in our LVI, while their “high radial velocity component” at -220 km s-1, located at 0$\aas@@fstack{\prime\prime}$6 - 0$\aas@@fstack{\prime\prime}$8, may correspond to the bright elongated feature seen in our MVI in the [N ii] PV plots at 0 to 0$\aas@@fstack{\prime\prime}$4 from the star. No HVI material is reported in Pyo et al. (2003), but the A1HV feature would have moved out of their 1$\aas@@fstack{\prime\prime}$6-long diagrams in two years. ### 5.2 Gas heating While our results are in agreement with previous studies, such as Lavalley- Fouquet et al. (2000), our analysis improves on these studies by quantifying the higher jet densities in the region close to the star. In addition, the HST resolution reveals that the ionisation peaks upstream A1 have different positions in the different velocity bins (0$\aas@@fstack{\prime\prime}$8 for the HVI, 1$\aas@@fstack{\prime\prime}$1 for the MVI, and 1$\aas@@fstack{\prime\prime}$4 for LVI (with a smaller peak at 0$\aas@@fstack{\prime\prime}$5)), followed by a decrease. This behaviour is reminiscent of the excitation produced by a Disk-wind heated by ambipolar diffusion (Garcia et al. (2001), Figs. 1, 2). In this model the slower outer material reaches its maximum $x_{e}$ at a greater distance than the inner higher velocity gas. The $x_{e}$ value, however, is much smaller than our value at the observed location. Alternatively, the HVI and MVI $x_{e}$ peaks may be due to marginally resolved shocks which are not evident in the forbidden emission lines. Indeed, in Bacciotti et al. (2000), small condensations are visible at 0$\aas@@fstack{\prime\prime}$5 and 1$\aas@@fstack{\prime\prime}$1 but only in the strong H$\alpha$ line, in reconstructed images at various velocities. This would be in line with the conclusions of Lavalley-Fouquet et al. (2000), who find that the observed [N ii]/[O i] ratio are compatible only with shock heating, and of the numerical study of Massaglia et al. (2005), where the variations of $x_{e}$ and $n_{e}$ in distance were reproduced by a continuous series of identical shocks travelling along a jet of decreasing density. We stress that we see an increase in total density $n_{H}$ toward each knot, at the same location of a sharp velocity gradient. This is a key result as these sudden compressions indicate clearly that the luminous knots in the DG Tau jet are generated by propagating shock fronts. For example, at A1HV, the high velocity tip of the A1 structure, there is a velocity gradient associated to a high $n_{H}$ value in the HVI. Then at B0, the velocity jump and the local increase in density suggests that this knot is shocking the slower gas at B1. Again feature B1 has the properties expected for a shock front generated by higher velocity gas catching up with slower material emitted at an earlier time. At 4$\aas@@fstack{\prime\prime}$1, just downstream of the B1 emission peak, we see a supersonic velocity jump associated with a gradient in density and ionisation. The increase in total density in proximity to velocity jumps, however, does not always correspond to an increase in ionisation. At A1HV, for example, no peak in $x_{e}$ is found. Again at B0 one would expect an increase in ionisation, but $x_{e}$ was already high upstream of B0. Spatial offsets between the position of peaks in $x_{e}$ and the position of shock fronts have been found in other works conducted on similar spatial scales, see e.g. Hartigan & Morse 2007 for the HH 30 jet. The lack of variation in $x_{e}$ does not necessarily exclude a shock. In a number of cases, in fact, line ratio modelling has shown evidence of substantial pre-shock ionisation (see, e.g., Hartigan et al. (2004) in HN Tau jet and Teşileanu et al. (2012) for the jet from RW Aur). If a strong pre-shock ionisation was created, for example by the passage of a previous front, or by the x-ray field associated with this jet (Güdel et al., 2011), this ionisation could endure in a low density gas because of the slow recombination time (Bacciotti et al., 1999). The half-life of free electrons is given by $t_{\rm rec}=(n_{e}\alpha_{H}(T_{e}))^{-1}$, where $\alpha$ is the hydrogen recombination coefficient (which is weakly dependent on $T_{e}$). Taking $\alpha_{H}=2.5$ 10 -13 cm3 s-1 (Osterbrock, 1989) and for the HVI upstream of B0 $n_{e}=7\leavevmode\nobreak\ 10^{3}$ cm-3, we obtain a recombination time of $t_{\rm rec}=6\leavevmode\nobreak\ 10^{8}$ s. Combining this with a radial velocity of -300 km s-1 and the jet inclination of 38∘ to the line of sight, we find that the free electrons can travel a distance of about 3′′ in the plane of the sky before recombination. Finally, all PV plots show a slight asymmetry with respect to the axis. This jet wiggle was first detected by Dougados et al. (2000), and is clear in Fig. 10 in the faint region between A1 and B0. Following Lavalley-Fouquet et al. (2000), the wiggling may explain the high excitation in this region, as a bending of the jet by only 10∘ at the observed velocities can produce oblique fronts with shock speeds of up to 70 km s-1. ### 5.3 Mass outflow rate An important parameter in any jet launching model is the mass outflow rate. To ensure accuracy in estimates, it is vital to make measurements as close as possible to the base of the jet. Here it is hoped that the plasma parameters are still dominated by the physics of the launching mechanism, rather than interactions with the environment through which it propagates. Our HST/STIS dataset allows us to estimate the mass outflow rate of the jet, $\dot{M}_{j}$, for the first 0$\aas@@fstack{\prime\prime}$7 from the star. Inspired by the magneto-centrifugal models (Cabrit et al. 1999, Pudritz et al. 2007, Shang et al. 2002), we assume that the jet flows along nested magnetic surfaces, with a poloidal velocity which decreases with distance from the jet axis. Based on Fig. 14, the flow is structured in three nested cones around a hollow core, with boundary surfaces labelled $k=$1,2,3 and 4 according to increasing opening angle. Therefore, k=2, 3 and 4 mark the outer surfaces of the HVI, MVI and LVI jet cones, respectively. Once the jet reaches a distance, $d$, of 50 AU above the disk plane, the k=1 surface opens out to a diameter of either 6 AU for the disk-wind (Cabrit et al., 1999) or 2 AU, for the X-wind model (Shang et al., 2002). At each distance, $d$, the jet material of each cone crosses an annulus of area $\pi\leavevmode\nobreak\ (r_{k}^{2}-r_{k-1}^{2})$, where $r_{k}(d)=FWHM(d)/2$ is the radius of a given cone surface. We define $\alpha_{k}$ as half of the opening angle of a given cone. From Fig. 14, we find that $\alpha_{k}=8^{\circ},11,^{\circ},19^{\circ}$ for $k=2,3,4$, respectively, and $\alpha_{k}<3^{\circ}$ for k=1 if we consider the Disk-wind model. The mass flux $\dot{M}_{j}$ in each velocity interval, labelled by the outer cone surface $k$ of that interval, is then given by $\dot{M}_{j,k}=\mu m_{p}f_{c}n_{H,k}\leavevmode\nobreak\ \|\bar{v}_{pk}\|\cos{\beta_{k}}\leavevmode\nobreak\ \pi\leavevmode\nobreak\ (r_{k}^{2}-r_{k-1}^{2})\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ {\rm for}\leavevmode\nobreak\ k=2,3,4,$ (1) where $n_{H,k}$ is the hydrogen density in the velocity interval region (with values taken from Fig. 13), $\mu=1.41$ is the average atomic weight per hydrogen atom (Allen & Cox, 2001), $m_{p}$ the proton mass, $i=38^{\circ}$ is the jet inclination angle with respect to the line of sight, and $\beta_{k}=(\alpha_{k}+\alpha_{k-1})/2$. In the above expression, $\|\bar{v}_{pk}\|$ is the module of the average poloidal velocity in the considered interval, given as $\|\bar{v}_{pk}\|={\cos{i}\cos{\beta_{k}}\leavevmode\nobreak\ \|v_{k}\|-0.5\sin{i}\sin{\beta_{k}}\leavevmode\nobreak\ \|\delta v_{k}\|\over\cos^{2}{i}-\sin^{2}{\beta_{k}}},$ $v_{k}$ is the radial velocity at the middle of the interval and $\delta v_{k}$ the width of the velocity interval. Finally, $f_{c}$ is a correction factor which accounts for compression of the flow in unresolved shocks, estimated as the square root of the inverse of the post-shock compression (Hartigan et al., 1994). The latter is found to be 8-10 after 10 years evolution for an average shock velocity around 50 km s-1 and pre-shock magnetic field around 100 $\mu G$, (Massaglia et al., 2005), leading to $f_{c}=1/3$. The results for $\alpha_{1}=3^{\circ}$ (i.e. the Disk-wind model) are presented in Fig. 15. Figure 15: Mass outflow rate of the jet, $\dot{M}_{j}$, using two methods: (i) jet density traversing annuli of nested cones, represented by data points broken down by velocity interval and also totalled; (ii) jet density obtained via emission line luminosities from a uniform slab, as in Hartigan et al. (1995), represented by curves. Uncertainties are about 50% in every velocity interval. Neglecting the errors on distance and inclination, the uncertainty arises in the density and FWHM estimates. On average the uncertainty is of 20%, 30%, and 40% for the HVI, MVI, and LVI mass loss rates, respectively, as well as an additional 20 to 30% error introduced by the modified BE-technique for the MVI and HVI densities. The net effect is an uncertainty of 50% in every velocity interval. Fig. 15 shows that the contribution of the LVI and HVI is dominant initially, but it decreases with distance by 75%, due to the fall of density. By contrast, the MVI contribution increases beyond the LVI and HVI contributions from 0$\aas@@fstack{\prime\prime}$5\. The average total mass outflow rate of the jet is $\dot{M}_{j}$ = 8 $\pm$ 4 $10^{-9}$ M⊙ yr-1, slightly decreasing with distance which is probably due to flux falling outside the 0$\aas@@fstack{\prime\prime}$5 slit coverage as the jet diverges. Assuming a smaller central hollow cone, as in the X-wind model, only the HVI contribution increases, to a maximum of 25% for $\alpha_{1}=0$. Since the data are flux- calibrated, $\dot{M}_{j}$ could also be estimated using the luminosity-based method of Hartigan et al. (1995), finding good agreement (Fig. 15). The derived $\dot{M_{j}}$ values of $10^{-9}$ to $10^{-8}$ match typical values for jets from CTTSs (e.g. for RW AUR $\dot{M}_{j}\sim 4.6\leavevmode\nobreak\ 10^{-9}$ M⊙ yr-1 (Melnikov et al., 2009) and in CW Tau $\dot{M}_{j}\sim 7\leavevmode\nobreak\ 10^{-9}$ M⊙ yr-1 (Coffey et al., 2008)), and compare well with previous estimates for DG Tau. At 0$\aas@@fstack{\prime\prime}$3 from the star, Coffey et al. (2008) find $\dot{M}_{j}$= 2.6 and 4.1 10-8 M⊙ yr-1 at -100 and -225 km s-1. These values are reconciled with ours by: applying FWHM deconvolution (i.e. yielding jet widths of 24 and 15 AU, respectively); replacing the unreliable value for $x_{e}$ at high velocity; and introducing $f_{c}=$ 1/3. Meanwhile, Agra- Amboage et al. (2011) obtain a value of $\dot{M}_{j}\sim$ 1.6 and 1.7 10-8 M⊙ yr-1 at -300 and -135 km s-1. Their ‘jet density and cross-section’ method at 0$\aas@@fstack{\prime\prime}$4 – 0$\aas@@fstack{\prime\prime}$7 (their Fig. 9) gives results in agreement with ours, if $f_{c}=$ 1/3 is applied. Finally, Lavalley-Fouquet et al. (2000) reported $\dot{M}_{j}$ $\sim$ 0.2, 0.8, and 0.4 10-8 M⊙ yr-1 in their LVI, MVI, and HVI components beyond 1$\aas@@fstack{\prime\prime}$2 from the star, again in line with our estimates. Given that the range of accretion rates prevailing in DG Tau is found to be $\dot{M}_{acc}=3\pm 2\,10^{-7}\,M_{\odot}\,yr^{-1}$ (Agra-Amboage et al., 2011), for the two-sided jet the ejection to accretion ratio $\dot{M}_{j}\,/\,\dot{M}_{acc}$ turns out to vary between 0.03$\pm$0.01 and 0.16$\pm$0.08, which is compatible with the range predicted by Disk-wind models (e.g. 0.01 $<\dot{M}_{j}/\dot{M}_{acc}<$ 0.2, Ferreira et al. 2006). ### 5.4 Angular momentum flux In these spectra Bacciotti et al. (2002) found systematic differences in the Doppler shift of the LVI emission either side of the jet axis. These measurements were tentatively interpreted as the jet rotating about its axis as it propagates. Toroidal velocities were found to be $\sim$ 5-20 km s-1 at a radius of 20-30 AU from the jet axis and a distance of 40-80 AU from the star- disk plane. Subsequently, similar differences in Doppler shift were detected in this and other jets Coffey et al. 2004, Woitas et al. 2005, Coffey et al. 2007). These results seem to support magneto-centrifugal jet launch and, in particular, the fact that jets extract the excess angular momentum from the star-disk system. Our estimate of $\dot{M}_{j}$ now enables us to test the extraction efficiency. We approximate the angular momentum of the jet in a given velocity interval (where k labels the outer cone surface of that interval) as follows: $\dot{L}_{j,k}\,=\,\int_{r_{k-1}}^{r_{k}}\,\mu m_{p}n_{H,k}v_{\phi}v_{k}\,2\pi r^{2}\,dr\approx\bar{v}_{\phi,k}\dot{M}_{j,k}\bar{r}_{k}$ (2) where $v_{k}$ and $v_{\phi}$ are the poloidal and toroidal components of the jet velocity. In DG Tau, $v_{\phi}$ could only be measured for LVI and MVI (Bacciotti, 2002; Coffey et al., 2007), so Equation 2 simplifies to $\dot{L}_{j}\,=\,v_{\phi}\,(\dot{M}_{j,MVI}\,{\bar{r}_{MVI}}\,+\,\dot{M}_{j,LVI}\,{\bar{r}_{LVI}}).$ (3) where $\overline{r}_{MVI}$ and $\overline{r}_{LVI}$ =($r_{k}$ \+ rk-1)/2, for $k=3,4$ respectively. Given $v_{\phi}\sim 15\pm 5$ km s-1 for DG Tau at 0$\aas@@fstack{\prime\prime}$3 from the star (Coffey et al., 2007), and taking the values of mass outflow rates and FWHMs (with their error) derived at this distance, we obtain $\dot{L}_{j}\sim 6.1\pm 3.7$ 10-7 and $1.2\pm 0.8$ 10-7 M⊙ yr-1 AU km s-1 for the LVI and MVI components, respectively . The difference with higher values given by Bacciotti et al. (2002) and Coffey et al. (2008), $\dot{L}_{j}\sim=$ 3.8 10-5 and 1.3 10-5 M⊙ yr-1 AU km s-1, respectively, arises because we calculate lower mass outflow rates and smaller radii. Ferreira et al. (2006), however, show that for DG Tau at 50 AU from the star, the outer streamlines of the wind have not yet reached the asymptotic regime and contain only half of their final angular momentum, contrary to the inner streamlines. Accounting for this effect and for the presence of a symmetric red-shifted jet we obtain globally $\dot{L}_{j}\sim 2.9\pm 1.5$ 10-6 M⊙ yr-1 AU km s-1. In order to allow accretion to proceed, the disk must loose angular momentum. The excess angular momentum to be removed from the disk, $\dot{L}_{D}$, can be calculated as in Woitas et al. (2005), Sect. 4.2. We consider a Disk-wind launched at a broad range of disk radii, from the innermost region where the disk is truncated by the stellar magnetosphere at $r_{in}\sim$ 0.03 AU, to the outermost region at $r_{out}\sim$ 3 AU (the latter value is dictated by limiting poloidal velocities of 50 km s-1 (Bacciotti et al., 2002; Anderson et al., 2003; Pesenti et al., 2004)). The disk material falling below the inner radius is accreted in full onto the star, such that the disk mass flux at the inner radius $\dot{M}_{D,in}$ is equal to $\dot{M}_{acc}$, the mass accretion flux. Conservation of mass in the disk dictates that the mass flux in the disk at the outer radius, $\dot{M}_{D,out}$, satisfies $\dot{M}_{D,out}=\dot{M}_{acc}+2\dot{M}_{j}$. Considering the range of $\dot{M}_{acc}$ found in DG Tau and the estimate of $\dot{M}_{j}$ reported in the previous section, we obtain $\dot{M}_{D,out}=$ 3.2 $\pm$ 2.1 10-7 M⊙ yr-1. The angular momentum which the disk looses between the outer and inner radii is given by $\dot{L}_{D}=\dot{M}_{D,out}v_{K,out}r_{out}-\dot{M}_{D,in}v_{K,in}r_{in}$ (4) where $v_{K}$ is the Keplerian velocity of the disk. With M⋆ = 0.5 M⊙, $v_{K}$ = 122 and 11 km s-1 at the inner and outer radii, respectively, and therefore $\dot{L}_{D}\sim$ 9.5 $\pm$ 7.7 10-6 M⊙ yr-1 AU km s-1. $\dot{L}_{D}$ and $\dot{L}_{j}$ are of the same order of magnitude, and both present a wide range of variation. It cannot be excluded that the atomic jet is carrying away all the excess angular momentum from the disk out to 3 AU, if the accretion rate was at the low end of the range of $\dot{M}_{acc}$ measured for DG Tau over the period 1988 - 2003 (Agra-Amboage et al., 2011). Unfortunately, however, the large uncertainties still affecting the mass outflow and accretion rates prevent a firm conclusion on this point. ## 6 Summary and conclusions We analysed a set of seven high angular resolution HST/STIS spectra of the first 5 arcseconds of the outflow from DG Tau, taken in January 1999 with spectral and spatial resolution of $\sim 50$ km s-1 and 0$\aas@@fstack{\prime\prime}$1\. Previously published results based on this extraordinarily rich dataset include: the basic morphology of the jet in the first 2″ from the star (Bacciotti et al., 2000); a preliminary set of spectral diagnostics over the same region (Bacciotti, 2002); indications of jet rotation (Bacciotti et al., 2002). Here, we continue to exploit the 1999 HST/STIS dataset, to achieve a high resolution parameterisation of the jet plasma physics, investigating it in three dimensions: along the first 5″of the jet; across the jet width; and in velocity space. We provide the PV plots of the forbidden emission lines and their ratios. From these we derive PV plots of the electron density $n_{e}$, hydrogen ionisation fraction $x_{e}$, and total hydrogen density $n_{H}$, by applying an updated version of the BE- technique (first published in Bacciotti et al. (1999)). The presentation of the results as PV plots has the advantage of retaining all the spatio- kinematic information available. To assist with the interpretation, we also create 2-D images, and 1-D profiles along the jet, of the plasma parameters in each of three velocity intervals, defined as LVI from -120 to +25 km s-1, MVI from -270 to -120 km s-1 and HVI from -420 to -270 km s-1, applying the updated technique to binned data. Our main conclusions are listed below. Within the first arcsecond, the flow presents smoothly varying kinematic properties, with an apparent continuous acceleration in [S ii] and [O i] PV plots, reminiscent of a magneto-centrifugal jet launch mechanism. The [N ii] emission is concentrated in medium to high velocities, and the identified features in the flow (A2, A1, B0, B1) are associated with supersonic velocity jumps as expected of shocked gas. Previous reports of an onion-like kinematic structure in the initial (first 1″) channel are confirmed, but the new estimates of the jet excitation properties indicate that the efficiency of their ionising mechanism is different in the moderate-high velocity components of the flow with respect to the surrounding slower, wider flow. To find the plasma parameters above the high density limit of [S ii], the modified BE-technique was applied. At the beginning of the jet, values up to $n_{e}\sim$ 105 cm-3 were found for MVI and HVI, while lower values of $n_{e}\sim$ 104 cm-3 were found in the LVI. In the same region, $x_{e}$ increases markedly in the MVI and HVI, up to values of 0.7 and 0.6, respectively, close to A2. By contrast, the LVI value remains low i.e. $x_{e}\leq$ 0.3 within the location of A1. This suggests a fundamental difference between the dominant ionisation process in the MVI and HVI components with respect to the LVI component. Proceeding along the jet beyond feature A1, $n_{e}$ decreases in all velocity intervals. Meanwhile, $x_{e}$ is found to remain high (0.7-0.8) in the MVI and HVI (the LVI is not visible here), both in the A2-B0 stream, and between features B0 and B1. After B1, $n_{e}$ and $x_{e}$ can only be determined for the MVI, and both decrease to low values, along the jet to 5 ″. The total hydrogen density $n_{H}$ for the three velocity intervals is similar within a factor 3-4 all along the jet. Beyond A1 we have no $n_{H}$ estimates for the LVI, but values similar to our HVI are found here by Lavalley-Fouquet et al. (2000). Overall, the total density drops in magnitude over 4 orders along the jet (from almost 107 to 103 cm-3). Our results are in agreement with previous determinations of the excitation parameters, in the regions where the comparison was feasible. Remarkably, our analysis shows absence of significant variations in the plasma parameters across the jet width in each velocity interval. Slightly downstream the emission peaks at AHV1, B0 and B1, and in the HVI, local pronounced gradients of the total density are found to be coincident with velocity jumps. This is direct evidence that the gas is compressed locally at the position of sharp velocity gradients, supporting a shock origin for the observed knots, as proposed by Lavalley-Fouquet et al. (2000). This conclusion holds true even if not in all cases an increase in ionisation is detected at the velocity jumps, as it is in the MVI at B1. In fact the absence of variation in the ionisation level does not exclude a shock, as the ionisation created by the passage of a previous shock or by a radiation field could endure because of the slow recombination time in a rarefied medium (Bacciotti et al., 1999). We confirm reports of wiggling between A1 and B0, supporting the suggestion by Lavalley- Fouquet et al. (2000), that the high excitation values in this region may be maintained via the occurence of lateral shocks. The mass outflow rate of the jet, $\dot{M_{j}}$, is found as a function of velocity and distance from the star, in the initial portion of the jet (0$\aas@@fstack{\prime\prime}$1 - 0$\aas@@fstack{\prime\prime}$7), using the jet density and cross-section measurements. $\dot{M_{j}}$ is similar in the HVI and LVI, and decreases along the flow. Meanwhile, the MVI value is initially lower, but then increases with distance until it dominates after 0$\aas@@fstack{\prime\prime}$5\. The total mass flux, of all velocity intervals, is on average $\dot{M_{j}}\sim$ 8$\pm$4 10-9 M⊙ yr-1, and it is found to decrease slightly with distance, but this could be a bias introduced by the line flux loss at the slit borders. Results were cross-checked with mass outflow rates obtained via emission line luminosity (cfr. Hartigan et al. (1995)), finding good agreement. Taking into account differences in the derivation procedure, our results are also in agreement with previous estimates for this jet, and in the typical range for CTTS jets. Given the range of mass accretion rates found in DG Tau of $\dot{M}_{acc}\,\sim\,(3\pm 2)\,10^{-7}$ M⊙ yr-1 (Agra-Amboage et al., 2011), the ratio of mass ejection to mass accretion, $\dot{M}_{j}\,/\,\dot{M}_{acc}$, for the supposedly simmetric bipolar jet can vary between 0.03$\pm$0.01 and 0.16$\pm$0.08, which is compatible with the range predicted by Disk-wind models. Combining the derived mass outflow rates with previously published toroidal velocities for the LVI and MVI material at 0$\aas@@fstack{\prime\prime}$3 form the star, we estimate the angular momentum transported by these components. Considering two symmetric jet lobes and allowing a correction for the fraction of the angular momentum still in the disk-wind magnetic field before the asymptotic regime is reached, $\dot{L}_{j}$ turns out to be $\sim$ (2.9 $\pm$ 1.5) 10-6 M⊙ yr-1 AU km s-1. We tentatively compare this estimate to the amount of angular momentum lost by the disk to allow accretion, $\dot{L}_{D}$. Proceeding as in Woitas et al. (2005), we find $\dot{L}_{D}\sim$ 9.5 $\pm$ 7.7 10-6 M⊙ yr-1 AU km s-1, which indicates that the two quantities are of the same order of magnitude, and comparable if the accretion rate was at its lower values when the material probed by our data was ejected. The large uncertainties affecting both estimates, however, prevent further conclusions. In summary, the physical structure of the DG Tau jet reveals patterns of variation in parameters which are expected of magneto-centrifugal jet launch models. However, the situation is complicated by the simultaneous presence of other features, like hints of a multi-component flow, and shock fronts formed on different temporal scales, which seem to reach beyond this simple scenario. The presented plasma maps constitute a powerful benchmark for testing new alternatives. ###### Acknowledgements. The authors wish to thank the referee, Sylve Cabrit, for her detailed and thorough reports, that led to a significant improvement in the derivation and presentation of the results. L.M. thanks S. Cabrit and C. Dougados for the hospitality at IAP during the preliminary analysis of the data. T.P.R. acknowledges support from Science Foundation Ireland under grant 07/RFP/PHYF790. ## References * Agra-Amboage et al. 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Upper and lower limits arise when one of the lines is undetected, and so its flux has been set to 3$\sigma$. Figure 17: Same as Fig. 16 for the logarithm of the [N ii]/[O i] line ratio. Figure 18: Same as Fig. 16 for the [O i]/[S ii] line ratio. Due to the prominence of [O i] emission over the [S ii] lines close to the star, a few points are off the scale. Their values are: 6.58 at 0$\aas@@fstack{\prime\prime}$175, for the MVI, and 12.10, 14.49, 9.54 and 4.99 at 0$\aas@@fstack{\prime\prime}$175, 0$\aas@@fstack{\prime\prime}$275, 0$\aas@@fstack{\prime\prime}$375 and 0$\aas@@fstack{\prime\prime}$475, respectively, for the HVI.	arxiv-papers	2014-01-02T17:41:34	2024-09-04T02:49:56.206420	{ "license": "Public Domain", "authors": "L. Maurri, F. Bacciotti, L. Podio, J. Eisl\\\"offel, T. P. Ray, R.\n Mundt, U. Locatelli, D. Coffey", "submitter": "Lorenzo Maurri", "url": "https://arxiv.org/abs/1401.0483" }
1401.0489	# Element order versus minimal degree in permutation groups: an old lemma with new applications László Babai and Ákos Seress University of Chicago. http://people.cs.uchicago.edu/l̃aci/people.cs.uchicago.edu/$\sim$laciOhio State University. Deceased February 13, 2013. ###### Abstract In this note we present a simplified and slightly generalized version of a lemma the authors published in 1987. The lemma as stated here asserts that if the order of a permutation of $n$ elements is greater than $n^{\alpha}$ then some nonidentity power of the permutation has support size less than $n/\alpha$. The original version made an unnecessary additional assumption on the cycle structure of the permutation; the proof of the present cleaner version follows the original proof verbatim. Application areas include parallel and sequential algorithms for permutation groups, the diameter of Cayley graphs of permutation groups, and the automorphisms of combinatorial structures with regularity constraints such as Latin squares, Steiner 2-designs, and strongly regular graphs. This note also serves as a modest tribute to the junior author whose untimely passing is deeply mourned. ## 1 The lemma For a permutation $\pi$ on the set $K$, the _support_ of $\pi$ is defined as the set $\operatorname{supp}(\pi)=\\{x\in K\mid x^{\pi}\neq x\\}$. $S_{n}$ denotes the symmetric group of degree $n$ (and order $n!$). The _degree_ of a permutation is the size of its support, and the _minimal degree_ of a permutation group is the smallest degree of its non-identity elements. The following lemma shows that if there are elements of large order in a permutation group then its minimal degree is small. Lemma. Let $\pi\in S_{n}$ have order $\geq n^{\alpha}$ for some $\alpha>0$. Then $\|\operatorname{supp}(\pi^{m})\|\leq n/\alpha$ for some power $\pi^{m}\neq 1$. ###### Proof. Let $\pi$ act on the set $K$ where $\|K\|=n$. Let the order of $\pi$ be $N=n^{\alpha}=\prod_{i=1}^{r}q_{i}$ where $q_{i}=p_{i}^{\beta_{i}}>1$ are powers of distinct primes $p_{i}$. For each $x\in K$, let us consider the set $P(x)$ of those $i$ for which $q_{i}$ divides the length of the $\pi$-cycle through $x$. Clearly, for each $x\in K$, $\prod_{i\in P(x)}q_{i}\leq n.$ (1) Let $n(i)$ denote the number of points $x\in K$ such that $i\in P(x)$. Let us estimate the weighted average $W$ of the $n(i)$ with weights $\log q_{i}$. Recall that the sum of weights is $\sum\log q_{i}=\log N\geq\alpha\log n$, therefore (using Eq. (1)) $W\leq\sum_{x\in K}\sum_{i\in P(x)}\log q_{i}/(\alpha\log n)\leq(n\log n)/(\alpha\log n)=n/\alpha.$ (2) Thus we infer that $n(i)\leq n/\alpha$ for some $i\leq r$. Now let $m=N/p_{i}$ be the corresponding maximal divisor of $N$. Clearly $\pi^{m}$ is not the identity and it fixes all but $n(i)$ points. ∎ ## 2 History and applications The original version of this lemma, published as [BS1, Lemma 3], assumed that the permutation $\pi$ involved cycles of prime lengths $p_{i}$ such that $\prod p_{i}\geq n^{\alpha}$. As we have seen, this assumption is unnecessary. The proof of the above cleaner form requires no new ideas, however; it is an essentially verbatim copy of the original proof. Applications of this lemma are manifold, both old and new. ### 2.1 Degree of transitivity, diameter, parallel and sequential complexity of permutation groups In [BS1], the authors used this lemma to obtain a short proof of a theorem of Jordan [Jo1, Jo2] on the degree of transitivity of permutation groups. In another simultaneous paper [BS2], the authors used this lemma to prove an $\exp(\sqrt{n\ln n}(1+o(1)))$ upper bound on the diameter of all Cayley graphs of $S_{n}$; this bound was not improved until very recently [HS]. The breakthrough 2013 paper [HS] by Helfgott and Seress that reduced the diameter bound to quasipolynomial (exponential in a constant power of $\log n$) makes substantial use of a slight generalization of the lemma. In [BS3] the present authors, again using the Lemma, extended the $\exp(\sqrt{n\ln n}(1+o(1)))$ bound to all permutation groups of degree $n$ (subgroups of $S_{n}$), which in this form is tight since $S_{n}$ has cyclic subgroups of order $\exp(\sqrt{n\ln n}(1+o(1)))$ (product of small prime length cycles). In [BLS1], the Lemma played a key role in settling the parallel complexity of the permutation group membership problem; the subsequent papers [BLS2, BLS3] used the Lemma to design and analyze improved sequential algorithms for the same problem. ### 2.2 Latin squares The present note was prompted by a conversation between Babai and Ian Wanless in July 2013 at a conference celebrating Peter Cameron at Queen Mary, University of London. Wanless mentioned the following recent result of his with Brendan McKay and Xiande Zhang. Recall that a _quasigroup_ is a set with a binary operation such that all equations of the form $ax=b$ and $ya=b$ are uniquely solvable. In other words, a quasigroup is a set with a binary operation of which the multiplication table is a Latin square. ###### Theorem 2.1 (McKay, Wanless, Zhang [MWZ]). No automorphism of a quasigroup of order $n$ has order greater than $n^{2}/4.$ We show that a slightly weaker bound, $n^{2}$, is immediate from the Lemma. Indeed, let $\pi$ be a non-identity automorphism of some quasigroup $G$ of order $n$. Assume the order of $\pi$ is greater than $n^{2}$. Then some power $\pi^{m}\neq 1$ fixes more than half the elements of $G$. But the set of fixed points of a non-identity automorphism is a proper sub-quasigroup and therefore has order $\leq n/2$, a contradiction. ∎ The McKay–Wanless–Zhang argument is not dissimilar to ours. They in fact prove the following stronger result. ###### Theorem 2.2 (McKay, Wanless, Zhang [MWZ]). No autotopism of a quasigroup of order $n$ has order greater than $n^{2}/4.$ An _autotopism_ of a quasigroup $G$ is a triple $(\alpha,\beta,\gamma)$ of permutations of the set $G$ such that for all $g,h\in G$ we have $\alpha(g)\beta(h)=\gamma(gh)$. Wanless pointed out that a quadratic bound, $9n^{2}$, on the order of autotopisms of quasigroups of order $n$ also follows quickly from the Lemma. Slightly modifying his argument, we infer a bound of $4n^{2}$ on the order of any autotopism. (Of course this is still a factor of 16 worse than their result.) Indeed, assume $\theta=(\alpha,\beta,\gamma)$ is an autotopism of order greater than $(2n)^{2}$. We can view the autotopisms as acting on the union $G_{1}\,\dot{\cup}\,G_{2}\,\dot{\cup}\,G_{3}$ of three disjoint copies of $G$. In fact, the action on $U=G_{1}\,\dot{\cup}\,G_{2}$ is faithful; let us consider this action. By the Lemma, some power $\theta^{m}\neq 1$ has more than $n$ fixed points on $U$. Therefore it has at least one fixed point in each of $G_{1}$ and $G_{2}$. A result by McKay, Meynert, and Myrvold [MMM] asserts that if a non-identity autotopism has a fixed point in $G_{1}$ and a fixed point in $G_{2}$ then the number of fixed points in each part is the same and that number cannot exceed $n/2$, a contradiction. ∎ We do not expect the quadratic rate of growth in these results to be optimal. Let $f(n)$ denote the maximum of the orders of automorphisms of quasigroups of order $\leq n$. By the above, we have $f(n)=O(n^{2})$. ###### Conjecture 2.1. $f(n)=o(n^{2}).$ On the other hand, McKay et al. [MWZ] conjecture that 2 is the best possible exponent. ###### Conjecture 2.2 ([MWZ]). For every $\epsilon>0$ and for infinitely many values of $n$, $f(n)>n^{2-\epsilon}.$ The construction of quasigroups with automorphisms of nearly quadratic order seems to face significant obstacles; currently, no superlinear examples are known (cf. [MWZ]). It is noted in [MWZ] that the quadratic upper bound on the order of automorphisms holds in particular for Steiner Triple Systems (STSs) because such systems can be viewed as quasigroups: if $x,y$ are distinct points of an STS then define $xy$ as the third point in the unique triple containing $x,y$; and set $xx=x$. We observe that the quadratic upper bound for STSs extends to all Steiner 2-designs. A Steiner 2-design (also called a “regular linear space”) is an incidence geometry with lines of uniform length and exactly one line through every pair of points. If it has $n$ points and each line has $k$ points then this is a $2-(n,k,1)$-design. ###### Proposition 2.3. Let $X$ be a Steiner 2-design with $n$ points and with lines of length $k\geq 3$. Let $m$ be the maximum order of automorphisms of $X$. Then * (a) $m<n^{2}$ * (b) $m=O\left(n^{1+\frac{1}{k-2}}\right)$ where the implied constant is absolute. The proof follows by combining the Lemma with a bound on the number of fixed points of automorphisms of a Steiner 2-design by Davies [Da]. The details will appear in [Ba2]. ### 2.3 Strongly regular graphs In another paper [Ba1], the Lemma is used to establish strong structural constraints on the automorphism groups of strongly regular (SR) graphs. Recall that a SR graph with parameters $(n,k,\lambda,\mu)$ has $n$ vertices, is regular of degree $k$, each pair of adjacent vertices has $\lambda$ common neighbors, and each pair of distinct, non-adjacent vertices has $\mu$ common neighbors. Disjoint unions of cliques of equal size and their complements are _trivial_ examples of SR graphs. The line graphs of complete graphs and of complete bipartite graphs with equal parts are also SR; we refer to them and to their complements as _graphic_ SR graphs. The senior author has long suspected that the automorphism groups of non- trivial, non-graphic SR graphs are “small.” The following result gives a specific interpretation to this statement. We say that a group $H$ is _involved_ in a group $G$ if $G$ has subgroups$L\triangleleft K\leq G$ such that $H\cong K/L$ ($H$ is isomorphic to a quotient of a subgroup of $G$). We note that the automorphism groups of the trivial and the graphic SR graphs involve alternating groups of degree $\geq\sqrt{n}$. It turns out that in all other cases, the alternating groups involved are tiny. This has significant implications to attempts at subexponential and possibly even quasi-polynomial-time isomorphism tests for SR graphs, and it limits the possible primitive group actions on SR graphs. We state the result. ###### Theorem 2.4 ([Ba1]). If the alternating group $A_{t}$ is involved in the automorphism group of a non-trivial, non-graphic SR graph with $n$ vertices then$t=O((\ln n)^{2}/\ln\ln n)$. Theorem 2.4 is derived from a lemma that limits the number of fixed points of any non-identity automorphism of a regular graph in terms of its combinatorial and spectral parameters; from this, a bound on the orders of automorphisms follows via the Lemma above, and a bound on $t$ is then immediate. To apply this general result to SR graphs, bounds on the second largest eigenvalue and on the parameters $\lambda,\mu$ are derived from known results. ## 3 Ákos Seress (November 24, 1958 – February 13, 2013) Apart from minor updates, this note was written in August 2013, about six months after the untimely passing of Ákos Seress [BO13]. I (Babai) had known Ákos, 8 years my junior, from his undergraduate years in the late 70s at Eötvös University, Budapest, where I was teaching at the time and he was a star student, already active in research. But our real meeting of minds occurred at a conference in Szeged, Hungary, in summer 1986, when by serendipity both of us missed the boat for a scenic afternoon ride. It was there, on the banks of the Tisza river, that I began to introduce Ákos, then a fresh Ph. D. in combinatorics, to asymptotic group theory and algorithmic group theory. That conversation evolved into a lifelong collaboration that produced 15 joint papers spanning a quarter century. I count several of our joint papers among the best of my career; this includes our last paper [BBS]. Ákos became a leader in algorithmic and computational group theory, an author of the definitive monograph on the subject [Ser1], a major contributor to the GAP symbolic algebra package, and a speaker at ICM 2006 in the algebra section. Ákos was a most generous friend and colleague. He died at the age of 54 of renal cell carcinoma, a particularly aggressive form of cancer, diagnosed only six months earlier. The disease struck at the height of his creative powers, shortly after he had finished two breakthrough papers, one mathematical and one computational: the above-mentioned paper [HS] with Helfgott, a _tour de force_ in the combinatorial theory of permutation groups, and a computational work on the Monster group [Ser2] that received the “Distinguished paper award” at the ISSAC 2012 conference and was hailed as “a groundbreaking work” that “marks a turning point in Majorana Theory.” The Lemma discussed in this note was the fruit of the first hours of our collaboration, conceived even before the return of the boat, so I find it most appropriate to list Ákos as a coauthor. Acknowledgment. I wish to thank Ian Wanless for the inspiring conversation at the CameronFest last July and for his helpful comments on earlier versions of this note. ## References * [Ba1] László Babai: On the automorphisms of strongly regular graphs I. _Proc. 5th Innovations in Theoretical Computer Science conf. (ITCS 2014)_ , ACM Press, to appear. * [Ba2] László Babai: On the automorphisms of strongly regular graphs II. In preparation. * [BBS] László Babai, Robert Beals, Ákos Seress: Polynomial-time theory of matrix groups. In _Proc. 41st STOC_ , pp. 55–64. ACM Press, 2009. DOI: 10.1145/1536414.1536425. * [BLS1] László Babai, Eugene M. Luks, Ákos Seress: Permutation groups in NC. In: _Proc. 19th STOC_ , pp. 409–420. ACM Press, 1987 * [BLS2] László Babai, Eugene M. Luks, Ákos Seress: Fast management of permutation groups. In _Proc. 29th FOCS_ , pp. 272–282. IEEE Comp. Soc. 1988 * [BLS3] László Babai, Eugene M. Luks, Ákos Seress: Fast management of permutation groups I. _SIAM J. Comput._ 26 (1997), 1310–1342 * [BO13] László Babai, Eamonn O’Brien: Ákos Seress (1958–2013). Obituary on the GAP Forum, Feb. 15, 2013. * [BS1] László Babai, Ákos Seress: On the degree of transitivity of permutation groups: a short proof. _J. Combinatorial Theory-A_ 45 (1987), 310–315 * [BS2] László Babai, Ákos Seress: On the diameter of Cayley graphs of the symmetric group. _J. Combinatorial Theory-A_ 49 (1988), 175–179 * [BS3] László Babai, Ákos Seress: On the diameter of permutation groups. _Europ. J. Comb._ 13 (1992), 231–243 * [CNP] Alan R. Camina, Peter M. Neumann, Cheryl E. Praeger: Alternating groups acting on finite linear spaces. Proc. London Math. Soc. (3) 87 (2003) 29–53. * [Da] D. Huw Davies: _Automorphisms of Designs._ Ph. D. Thesis, University of East Anglia, 1987. Cited by [CNP]. * [HS] Harald Helfgott, Ákos Seress: On the diameter of permutation groups. _Annals of Mathematics._ To appear * [Jo1] Camille Jordan: Sur la limite de transitivité des groupes non alternés. _Bull. Soc. Math. France_ 1 (1873), 40–71 * [Jo2] Camille Jordan: Nouvelles recherches sur la limite de transitivité des groupes qui ne continennent pas le groupe alterné. _J. Math._ 1 (1895), 35–60 * [MMM] Brendan D. McKay, Alison Meynert, Wendy Myrvold: Small Latin squares, quasigroups, and loops. _Journal of Combinatorial Designs._ 15 (2007), 98–119. * [MWZ] Brendan D. McKay, Ian M. Wanless, Xiande Zhang: The order of automorphisms of quasigroups. _Journal of Combinatorial Designs._ To appear. * [Ser1] Ákos Seress: _Permutation Group Algorithms._ Cambridge Univ. Press, 2003 * [Ser2] Ákos Seress: Construction of 2-closed M-representations. In: _Proc. Internat. Symp. on Symbolic and Algebraic Computation (ISSAC’12)_ , pp. 311–318. ACM Press, 2012	arxiv-papers	2014-01-02T18:07:54	2024-09-04T02:49:56.218555	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "L\\'aszl\\'o Babai and \\'Akos Seress", "submitter": "Laszlo Babai", "url": "https://arxiv.org/abs/1401.0489" }
1401.0500	# Kinser inequalities and related matroids Amanda Cameron ###### Abstract Kinser developed a hierarchy of inequalities dealing with the dimensions of certain spaces constructed from a given quantity of subspaces. These inequalities can be applied to the rank function of a matroid, a geometric object concerned with dependencies of subsets of a ground set. A matroid which is representable by a matrix with entries from some finite field must satisfy each of the Kinser inequalities. We provide results on the matroids which satisfy each inequality and the structure of the hierarchy of such matroids. ###### Contents 1. 1 Introduction 2. 2 Fundamentals 1. 2.1 Dependencies 2. 2.2 Representability 3. 2.3 Minors 4. 2.4 Duality 5. 2.5 Transversals 3. 3 Kinser Inequalities 1. 3.1 Inequalities 2. 3.2 Kinser matroids 3. 3.3 Kinser classes 4. 3.4 Sums 4. 4 Kinser Hierarchy 5. 5 $\mathcal{K}_{5}\neq\mathcal{K}_{5}^{}$ 6. 6 A Complexity Theorem 7. 7 Excluded minors 8. 8 Conjectures ###### List of Figures 1. 3.1 $\mathrm{Kin}(4)$ 2. 3.2 $\mathrm{Kin}(5)$ 3. 4.1 Kinser classes 4. 4.2 Kinser classes (2) 5. 4.3 Kinser classes (3a) 6. 4.4 Kinser classes (3b) 7. 8.1 Kinser classes (4) ## Chapter 1 Introduction A fundamental question in matroid theory is whether it is possible to find a characterisation of the class of representable matroids. In particular, we wish to know whether this can be achieved with a finite number of axioms, by adding additional rank axioms to the exisiting three. This was first alluded to by Whitney [9], in the paper which initiated the area of matroid theory, and the problem remains open today. Ingleton [2] introduced one new axiom which a matroid must satisfy in order to be representable. ###### Definition 1.1. Let $M=(E,r)$ be a matroid. For subsets ${X}_{1},\dots,{X}_{4}$ of $E$, the _Ingleton inequality_ is: $\displaystyle r(X_{3})+r(X_{4})+r(X_{1}\cup X_{2})+r(X_{1}\cup X_{3}\cup X_{4})+r(X_{2}\cup X_{3}\cup X_{4})$ $\displaystyle\leq r(X_{1}\cup X_{3})+r(X_{1}\cup X_{4})+r(X_{2}\cup X_{3})+r(X_{2}\cup X_{4})+r(X_{3}\cup X_{4})$ This new condition, while necessary, is not sufficient to characterise representability. For instance, the direct sum of the Fano and the non-Fano matroids satisfies the Ingleton condition but is not representable, as later proven in Lemma 4.1. () Fano matroid, $F_{7}$ () Non-Fano matroid, $F_{7}^{-}$ Recently, Kinser [3] introduced an infinite family of new representability conditions, the first of which is equivalent to the Ingleton condition. ###### Definition 1.2. Let $M$ be a matroid, and let ${X}_{1},\dots,{X}_{n}$ be any collection of subsets of $E(M)$. The _$n$ -th Kinser inequality_, where $n\geq 4$, is $\displaystyle\sum_{i=3}^{n}r(X_{i})+r(X_{1}\cup X_{2})+r(X_{1}\cup X_{3}\cup X_{n})+\sum_{i=4}^{n}r(X_{2}\cup X_{i-1}\cup X_{i})$ $\displaystyle\leq r(X_{1}\cup X_{3})+r(X_{1}\cup X_{n})+\sum_{i=3}^{n}r(X_{2}\cup X_{i})+\sum_{i=4}^{n}r(X_{i-1}\cup X_{i})$ This hierarchy of inequalities is also not sufficient to guarantee representability of a matroid – the direct sum of the Fano and the non-Fano is again a counter-example to this. Briefly putting aside the use of an infinite list of axioms, we have the following conjecture, which is due to Mayhew, Newman, and Whittle [4]. ###### Conjecture 1.3. It is impossible to characterise the class of representable matroids with a finite number of rank axioms. Note that [4] is a response to an paper of Vámos’ [8] dealing with the same question. In this paper, Vámos introduced the following geometric construction, which we call a $V$-matroid: ###### Definition 1.4. A _V-matroid_ consists of a (possibly infinite) set $E$ and a collection of finite subsets $\mathcal{I}\subseteq E$ such that: I1. $\varnothing\in\mathcal{I}$ * I2. If $I\in\mathcal{I}$ and $J\subseteq I$, then $J\in\mathcal{I}$ * I3. If $I,J\in\mathcal{I}$ and $\|I\|=\|J\|+1$, there exists $x\in I-J$ such that $J\cup x\in\mathcal{I}$ Instead of using rank axioms, Vámos describes $V$-matroids with an infinite list of first-order axioms. In [8], Vámos proved that it is not possible to characterise representable $V$-matroids by adding a further first-order axiom to this infinite list. Note that a first-order axiom is not equivalent to a rank axiom. However, as every finite $V$-matroid is a matroid, Conjecture 1.3 could be regarded as a strengthening of this result. This thesis is dedicated to investigating the classes of matroids which satisfy each of the Kinser inequalities. We will cover invariant properties of the classes, such as being minor closed and direct sum closed, and, more importantly, we will provide results on how the classes interact with each other to form an infinite hierarchy. We will touch on the complexity of verifying a matroid satisfies a given Kinser inequality, which will show that gaining certain information on the Kinser classes, such as which classes are closed under duality, could involve a great amount of computational work. This thesis will conclude by considering a question which arises naturally in conjunction with representability, that of excluded minors. The following theorem was proven by Mayhew, Newman, and Whittle in 2008 [5], settling a conjecture by J. Geelen. ###### Theorem 1.5. For any infinite field $\mathbb{K}$ and any matroid $N$ representable over $\mathbb{K}$, there is an excluded minor for $\mathbb{K}$-representability that has $N$ as a minor. We will provide a strengthening of this result, showing that there is in fact an infinite number of such excluded minors. Specifically, we will show that for each layer of the Kinser class hierarchy, we can find an excluded minor which is contained inside that layer. ## Chapter 2 Fundamentals To begin with, we will cover the basic concepts in matroid theory which will be used throughout this thesis. All of the following concepts and results can be found in [6]. ###### Definition 2.1. A _matroid_ $M=(E,\mathcal{I})$ consists of a finite ground set $E$ and a collection of subsets $\mathcal{I}\subseteq E$ such that: * I1. $\varnothing\in\mathcal{I}$ * I2. If $I\in\mathcal{I}$ and $J\subseteq I$, then $J\in\mathcal{I}$ * I3. If $I,J\in\mathcal{I}$ and $\|I\|<\|J\|$, there exists $x\in J-I$ such that $I\cup x\in\mathcal{I}$ Any subset of $E$ contained in $\mathcal{I}$ is referred to as an _independent set_ , while any subset of $E$ which is not contained in $\mathcal{I}$ is called _dependent_. A dependent set of cardinality one is called a _loop_. We may use $E(M)$ in the place of $E$ at times, in order to make it clear which matroid is being referred to. ###### Definition 2.2. Take a matroid $M$ with ground set $E$. The _rank_ of a subset $X$ of $E$, denoted by $r(X)$, is the cardinality of the largest independent subset of $X$. ###### Lemma 2.3. A matroid $M$ can be described by the ground set $E$ and a _rank function_ $r:\mathcal{P}(E)\rightarrow\mathbb{Z}\cup\\{0\\}$ such that, for $X,Y\in\mathcal{P}(E)$, the following conditions hold: * R1. $r(X)\leq\|X\|$ * R2. If $Y\subseteq X$, $r(Y)\leq r(X)$ * R3. $r(X\cup Y)+r(X\cap Y)\leq r(X)+r(Y)$ A set $X$ is independent if and only if $r(X)=\|X\|$. If $r(X)=r(M)$ we call $X$ a _basis_ of $M$. If a set contains a basis, it is called _spanning_. ### 2.1 Dependencies ###### Definition 2.4. The _closure_ of a set $X$ is denoted by $cl(X)$, where $cl(X)=X\cup\\{e\in E-X\ \|\ r(X\cup e)=r(X)\\}$ ###### Lemma 2.5. The closure function of a matroid satisfies the following conditions: * CL1. If $X\subseteq E$, then $X\subseteq cl(X)$. * CL2. If $X\subseteq Y$, then $cl(X)\subseteq cl(Y)$. * CL3. If $X\subseteq E$, then $cl(cl(X))=cl(X)$. * CL4. If $X\subseteq E$ and $x\in E$, and $y\in cl(X\cup x)-cl(X)$, then $x\in cl(X\cup y)$. The closure function corresponds to the notion of span of a vector space, and is sometimes referred to as such. A _flat_ is a set whose closure is equal to the set itself, i.e. $cl(X)=X$. If a flat has rank $r(M)-1$, it is called a _hyperplane_. A minimally dependent set, i.e. a dependent set whose every proper subset is independent, is called a _circuit_. A matroid can be described entirely by its set of circuits $\mathcal{C}$. ###### Lemma 2.6. $(E,\mathcal{C})$ describes a matroid when the following conditions hold. * C1. $\varnothing\notin\mathcal{C}$ * C2. If $C,D\in\mathcal{C}$ and $C\subseteq D$, then $C=D$ * C3. If $C,D$ are distinct elements of $\mathcal{C}$ amd $e\in C\cup D$, then $(C\cup D)-e$ contains a circuit A _circuit-hyperplane_ is a set which is both a circuit and a hyperplane. ###### Definition 2.7. Let $M$ be a matroid and let $H$ be a circuit-hyperplane of $M$. $H$ has rank equal to $r(M)-1$. We say that we _relax_ $H$ when we make it independent, i.e. $r(H)=r(M)$. When we reverse this operation, we say that we _tighten_ $H$. ### 2.2 Representability ###### Definition 2.8. If $V$ is a set of vectors in a vector space, and for every subset $X$ of $V$, we define $r(X)$ to be the linear rank of $X$, then $(V,r)$ is a matroid, which we say is _representable_. If these vectors come from a finite field $\mathbb{K}$, we say that $M$ is _$\mathbb{K}$ -representable._ ### 2.3 Minors ###### Definition 2.9. We can remove an element $e$ of a matroid $M=(E,r)$ by _deleting_ it. This yields a matroid $M\backslash e=(E-e,r_{M\backslash e})$, where $r_{M\backslash e}(X)=r_{M}(X)$ for all $X\subseteq E-\\{e\\}$. ###### Definition 2.10. We can also remove an element $e$ of a matroid $M=(E,r)$ by _contracting_ it. This gives a matroid $M/e=(E-e,r_{M/e})$ where $r_{M/e}(X)=r_{M}(X\cup\\{e\\})-r(\\{e\\})$ for all $X\subseteq E-\\{e\\}$. Any matroid producted by a sequence of deletions and contractions is called a _minor_ of $M$. We say that a class of matroids $\mathcal{M}$ is _minor closed_ if, for every matroid $M$ in $\mathcal{M}$, each of its minors is also in $\mathcal{M}$. A matroid $M$ is an _excluded minor_ for a minor closed class of matroids $\mathcal{M}$ if $M\notin\mathcal{M}$ but deleting or contracting any element from $M$ produces a matroid in $\mathcal{M}$. A matroid $M$ is contained in $\mathcal{M}$ if and only if $M$ does not contain an excluded minor for $\mathcal{M}$. ### 2.4 Duality ###### Definition 2.11. From $M$ we can construct the _dual matroid_ $M^{}$. This has ground set equal to the ground set $E$ of $M$, and the rank of any subset is found using the function $r^{}(X)=\|X\|+r(E^{}-X)-r(M)$. A basis of $M^{}$ is is called a _cobasis_ of $M$. Note that if $B$ is a basis of $M$, then $E-B$ is a cobasis of $M$. Similarly, the rank function, circuits and independent sets of $M^{}$ are called the _corank_ function, _cocircuits_ and _coindependent_ sets of $M$. ###### Lemma 2.12. _([6, Proposition 2.1.7])_ Let $M$ be a matroid. Relax a circuit-hyperplane $H$ of $M$ to yield the matroid $M^{\prime}$. Then $(M^{\prime})^{}$ is identical to the matroid yielded from $M^{}$ by relaxing the circuit- hyperplane $E-H$ of $M^{}$. ###### Lemma 2.13. _([6, Proposition 3.3.5])_ Let $H$ be a cicuit-hyperplane of a matroid $M$, and let $M^{\prime}$ be the matriod obtained from $M$ by relaxing $H$. * i. When $e\in E(M)-H$, $M/e=M^{\prime}/e$, and, unless $e$ is a coloop of $M$, $M^{\prime}\backslash e$ is obtained from $M\backslash e$ by relaxing the circuit-hyperplane $H$ of $M\backslash e$. * ii. Dually, when $f\in H$, $M\backslash f=M^{\prime}\backslash f$ and, unless $f$ is a loop of $M$, $M^{\prime}/f$ is obtained from $M/f$ by relaxing the circuit-hyperplane $X-f$ of $M/f$. ### 2.5 Transversals ###### Definition 2.14. Let $S$ be any set. Take a family of subsets $\mathcal{A}=(A_{1},\ldots,A_{k})$ of $S$. A _transversal_ or _system of distinct representatives_ of $\mathcal{A}$ is a subset $\\{s_{1},\ldots,s_{m}\\}$ of $S$ such that $s_{i}\in A_{i}$ for all $i\in\\{1,\ldots,m\\}$ and $s_{1},\ldots,s_{m}$ are distinct. ###### Definition 2.15. Let $S$ be any set. $X\subseteq S$ is a _partial transversal_ of a family of subsets $\mathcal{A}=(A_{1},\ldots,A_{j})$ if $X$ is a transversal of $(A_{1},\ldots,A_{k})$ for some $A_{1},\ldots,A_{k}\subseteq S$. ###### Lemma 2.16. Let $\mathcal{A}=(A_{1},\ldots,A_{m})$ be a family of subsets of a set $S$. When $\mathcal{A}$ is a partition of $S$, the collection of partial transversals of $\mathcal{A}$ is the collection of independent sets of a matroid on $S$. This matroid is denoted by $M[\mathcal{A}]$. If a matroid $M$ is isomorphic to $M[\mathcal{A}]$ for some family of subsets $\mathcal{A}$, we say that M is a _transversal matroid_ and that $\mathcal{A}$ is a _presentation_ of $M$. Every transversal matroid is representable over all sufficiently large fields, as proven in [6, Proposition 11.2.16]. A transversal matroid can be represented by a bipartite graph. Let $\mathcal{A}=(A_{1},\ldots,A_{m})$ be a family of subsets of $S$, and let $J=\\{1,\ldots,m\\}$. Construct the graph $G[\mathcal{A}]$ which has vertex set $S\cup J$ and edge set $\\{xj\ \|\ x\in S,j\in J,x\in A_{j}\\}$. Recall that a matching of a graph is a collection of edges such that no two share a common endpoint. A subset $X$ is a partial transversal of $\mathcal{A}$ if and only if there is a matching in $G[\mathcal{A}]$ in which every edge has an endpoint in $X$, i.e. $X$ is matched into $J$. ###### Definition 2.17. Take a matroid $M=(E,r)$ with independent sets $\mathcal{I}$. Let $J=\\{I\in\mathcal{I}\ \|\ \|I\|=r(M)\\}$. The _truncation_ of $M$ is a matroid $T(M)=(E,r)$ with independent sets $\mathcal{I}-J$. ## Chapter 3 Kinser Inequalities We will now introduce the Kinser inequalities, developed by Kinser in 2009 in [3]. An example of a matroid which exemplifies inequality $n$ for all $n\geq 4$ will be described, as the Vámos matroid exemplifies the Ingleton inequality. If a single circuit-hyperplane of this matroid is relaxed, it no longer satisfies the inequality. These matroids will be used in further results in this thesis. We will show that the class of matroids which satisfy Kinser inequality $n$ for all $n\geq 4$ is minor-closed. These classes are also closed under direct sums. ### 3.1 Inequalities ###### Definition 3.1. Let $M$ be a matroid, and let ${X}_{1},\dots,{X}_{n}$ be any collection of subsets of $E(M)$. The _$n$ -th Kinser inequality_, where $n\geq 4$, is $\displaystyle\sum_{i=3}^{n}r(X_{i})+r(X_{1}\cup X_{2})+r(X_{1}\cup X_{3}\cup X_{n})+\sum_{i=4}^{n}r(X_{2}\cup X_{i-1}\cup X_{i})$ $\displaystyle\leq r(X_{1}\cup X_{3})+r(X_{1}\cup X_{n})+\sum_{i=3}^{n}r(X_{2}\cup X_{i})+\sum_{i=4}^{n}r(X_{i-1}\cup X_{i})$ Note that inequality $n$ has $2n-3$ terms on each side. The above diagram gives a representation of Kinser inequality $n$. The ovals represent the $n$ subsets of $E(M)$, and each edge aside from the dotted one between $X_{1}$ and $X_{2}$ represents a term on the right-hand side of the inequality. On the left-hand side, we have the singleton sets starting from $X_{3}$, the triple $X_{1}\cup X_{3}\cup X_{n}$ at the very top, the dashed $X_{1}\cup X_{2}$ edge, and every triangle of edges involving $X_{2}$, excluding those using $X_{1}$. When $n=4$, this yields the _Ingleton inequality_ [1971], which holds for any four subspaces $X_{1},\ldots,X_{4}$ of a vector space: $\displaystyle\text{dim}(V_{3})+\text{dim}(V_{4})+\text{dim}(V_{1}+V_{2})+\text{dim}(V_{1}+V_{3}+V_{4})+\text{dim}(V_{2}+V_{3}+V_{4})$ $\displaystyle\leq\text{dim}(V_{1}+V_{3})+\text{dim}(V_{1}+V_{4})+\text{dim}(V_{2}+V_{3})+\text{dim}(V_{2}+V_{4})+\text{dim}(V_{3}+V_{4})$ As a representable matroid can be embedded inside a vector space, this inequality clearly holds for such matroids. In fact, in order for a matroid to be representable, it must satisfy each Kinser inequality for all choices of families ${X}_{1},\dots,{X}_{n}$. Recall that if $X$ and $Y$ are subspaces of some vector space $\mathcal{V}$, then $X+Y=\\{{\bf x}+{\bf y}\ \|\ {\bf x}\in X,{\bf y}\in Y\\}$ is a subspace of $\mathcal{V}$ as well. The following proof is adapted from that of [3, Theorem 1], which was stated in terms of an arrangement of $n$ subspaces. ###### Lemma 3.2. A representable matroid $M$ satisfies each Kinser inequality. ###### Proof. Let $M$ be a representable matroid, and let ${V}_{1},\dots,{V}_{n}$ be subsets of $E(M)$. Embed $M$ in the projective geometry $PG(r-1,\mathcal{K})$ and replace each $V_{i}$ with its closure, $\langle V_{i}\rangle$, in the projective geometry. Let $W=\langle V_{3}\rangle\cap\ldots\cap\langle V_{n}\rangle$. Let $\|\langle V_{i}\rangle\|$ denote the dimension of $\langle V_{i}\rangle$. Using submodularity, we have that $\displaystyle\|\langle W\rangle+\langle V_{1}\rangle\|+\|\langle W\rangle+\langle V_{2}\rangle\|$ $\displaystyle\geq\|(\langle W\rangle+\langle V_{1}\rangle)\cap(\langle W\rangle+\langle V_{2}\rangle)\|$ $\displaystyle\qquad+\|\langle W\rangle+\langle V_{1}\rangle+\langle V_{2}\rangle\|$ $\displaystyle\geq\|\langle W\rangle+(\langle V_{1}\rangle\cap\langle V_{2}\rangle)\|+\|\langle W\rangle+\langle V_{1}\rangle+\langle V_{2}\rangle\|$ $\displaystyle\geq\|\langle W\rangle\|+\|\langle W\rangle+\langle V_{1}\rangle+\langle V_{2}\rangle\|$ Rearranging this, we get that $\|\langle W\rangle+\langle V_{1}\rangle+\langle V_{2}\rangle\|-\|\langle W\rangle+\langle V_{1}\rangle\|\leq\|\langle W\rangle+\langle V_{2}\rangle\|-\|\langle W\rangle\|$ (3.2.1) We will give a bound on each side of this inequality. Note that $\|\langle W\rangle+\langle V_{1}\rangle+\langle V_{2}\rangle\|\geq\|\langle V_{1}\rangle+\langle V_{2}\rangle\|$. As $\langle W\rangle+\langle V_{1}\rangle\subseteq(\langle V_{1}\rangle+\langle V_{3}\rangle)\cap(\langle V_{1}\rangle+\langle V_{n}\rangle)$, we have by submodularity that $\|\langle W\rangle+\langle V_{1}\rangle\|\leq\|\langle V_{1}\rangle+\langle V_{3}\rangle\|+\|\langle V_{1}\rangle+\langle V_{n}\rangle\|-\|\langle V_{1}\rangle+\langle V_{3}\rangle+\langle V_{n}\rangle\|$ This gives us a lower bound for the left-hand side of (3.2.1): $\displaystyle\|\langle V_{1}\rangle+\langle V_{2}\rangle\|-\|\langle V_{1}\rangle+\langle V_{3}\rangle\|-\|\langle V_{1}\rangle+\langle V_{n}\rangle\|+\|\langle V_{1}\rangle+\langle V_{3}\rangle+\langle V_{n}\rangle\|$ $\displaystyle\leq\|\langle W\rangle+\langle V_{1}\rangle+\langle V_{2}\rangle\|-\|\langle W\rangle+\langle V_{1}\rangle\|$ Now take the right-hand side. We have that $\|\langle W\rangle+\langle V_{2}\rangle\|-\|\langle W\rangle\|=\|\langle V_{2}\rangle\|-\|\langle V_{2}\rangle\cap\langle W\rangle\|$ Note that $V_{2}\supseteq V_{2}\cap V_{3}\supseteq\ldots\supseteq V_{2}\cap\ldots\cap V_{n}=V_{2}\cap W$. This gives that $\|\langle V_{2}\rangle\|-\|\langle V_{2}\rangle\cap\langle W\rangle\|=\sum_{i=3}^{n}(\|\langle V_{2}\rangle\cap\ldots\cap\langle V_{i-1}\rangle\|-\|\langle V_{2}\rangle\cap\ldots\cap\langle V_{i}\rangle\|)$ (3.2.2) For each summand above, we give an upper bound: for $3\leq i\leq n$, submodularity gives that $\|\langle V_{2}\rangle\cap\ldots\cap\langle V_{i-1}\rangle\|-\|\langle V_{2}\rangle\cap\ldots\cap\langle V_{i}\rangle\|=\|\langle V_{i}\rangle+(\langle V_{2}\rangle\cap\ldots\cap\langle V_{i-1})\rangle\|-\|\langle V_{i}\rangle\|$ As $\langle V_{i}\rangle+(\langle V_{2}\rangle\cap\ldots\cap\langle V_{i-1}\rangle)\subseteq(\langle V_{i}\rangle+\langle V_{2}\rangle)\cap(\langle V_{i}\rangle+\langle V_{i-1}\rangle)$, we have $\displaystyle\|\langle V_{i}\rangle+(\langle V_{2}\rangle\cap\ldots\cap\langle V_{i-1}\rangle)\|-\|\langle V_{i}\rangle\|$ $\displaystyle\leq\|(\langle V_{i}\rangle+\langle V_{2}\rangle)\cap(\langle V_{i}\rangle+\langle V_{i-1})\rangle\|-\|\langle V_{i}\rangle\|$ $\displaystyle=\|\langle V_{i}\rangle+\langle V_{2}\rangle\|+\|\langle V_{i}\rangle+\langle V_{i-1}\rangle\|$ $\displaystyle\qquad-\|\langle V_{2}\rangle+\langle V_{i-1}\rangle+\langle V_{i}\rangle\|-\|\langle V_{i}\rangle\|$ Note that when $i=3$ this simplifies to $\displaystyle\|\langle V_{3}\rangle+\langle V_{2}\rangle\|-\|\langle V_{3}\rangle\|$ $\displaystyle\leq\|\langle V_{3}\rangle+\langle V_{2}\rangle\|+\|\langle V_{3}\rangle+\langle V_{2}\rangle\|-\|\langle V_{2}\rangle+\langle V_{3}\rangle\|-\|\langle V_{3}\rangle\|$ $\displaystyle=\|\langle V_{2}\rangle+\langle V_{3}\rangle\|-\|\langle V_{3}\rangle\|$ Plugging this into (3.2.2) then (3.2.1) gives, after rearranging, $\displaystyle\sum_{i=3}^{n}\|\langle V_{i}\rangle\|+\|\langle V_{1}\rangle+\langle V_{2}\rangle\|+\|\langle V_{1}\rangle+\langle V_{3}\rangle+\langle V_{n}\rangle\|+\sum_{i=4}^{n}\|\langle V_{2}\rangle+\langle V_{i-1}\rangle+\langle X_{i}\rangle\|$ $\displaystyle\leq\|\langle V_{1}\rangle+\langle V_{3}\rangle\|+\|\langle V_{1}\rangle+\langle V_{n}\rangle\|+\sum_{i=3}^{n}\|\langle V_{2}\rangle+\langle V_{i}\rangle\|+\sum_{i=4}^{n}\|\langle V_{i-1}\rangle+\langle V_{i}\rangle\|$ Note that $\|\langle V_{i}\rangle\|=r(V_{i})$. In order to show that inequality $n$ holds, we must show that $\|\langle V_{i}\rangle+\langle V_{j}\rangle\|=r(V_{i}\cup V_{j})$. We have that $r(V_{i}\cup V_{j})=\|\langle V_{i}\cup V_{j}\rangle\|$ We will show that this is equal to $\|\langle V_{i}\rangle+\langle V_{j}\rangle\|$. Let $x\in\langle V_{i}\rangle+\langle V_{j}\rangle$. This means that $x=x_{1}+x_{2}$ where $x_{1}\in\langle V_{i}\rangle$ and $x_{2}\in\langle V_{j}\rangle$. We have that $x_{1}\in\langle V_{i}\cup V_{j}\rangle$ and $x_{2}\in\langle V_{i}\cup V_{j}\rangle$, so $x\in\langle V_{i}\cup V_{j}\rangle$. Now take $x\in\langle V_{i}\cup V_{j}\rangle$. We can write $x$ as a linear combination of elements $S_{i}$ from $V_{i}$ and elements $S_{j}$ from $V_{j}$. We have that $S_{i}\subseteq\langle V_{i}\rangle$ and that $S_{j}\subseteq\langle V_{j}\rangle$, so $x\in\langle V_{i}\rangle+\langle V_{j}\rangle$. Thus $\|\langle V_{i}\cup V_{j}\rangle\|=\|\langle V_{i}\rangle+\langle V_{j}\rangle\|$. We can therefore replace every term $\|\langle V_{i}\rangle+\langle V_{j}\rangle\|$ with $r(V_{i}\cup V_{j})$. Similarly, $\|\langle V_{i}\cup V_{j}\cup V_{k}\rangle\|=\|\langle V_{i}\rangle+\langle V_{j}\rangle+\langle V_{k}\rangle\|$. Making all such replacements yields inequality $n$. ∎ We say that a _bad family_ for a matroid $M$, relative to $n$, is a family of subsets ${X}_{1},\dots,{X}_{n}$ which does not satisfy Kinser inequality $n$. We can also represent an inequality as applied to a specific matroid with a graph. Let ${X}_{1},\dots,{X}_{n}$ be a family of subsets of a matroid $M$. Take a graph $G$ on vertices $V=\\{X_{1},\ldots,X_{n}\\}$ with adjacency relation $a$ such that $a(X_{i})=\\{X_{j}\ \|\ X_{i}\cup X_{j}\ \mathrm{is}\ \mathrm{a}\ \mathrm{term}\ \mathrm{on}\ \mathrm{the}\ \text{right-hand side}\ \mathrm{of}\ \mathrm{inequality}\ n\\}$ In other words, two vertices are joined by an edge if the union of the two vertices is a term in inequality $n$. Recall that when $G[V,E]$ is any graph with vertex set $V$ and edge set $E$, an induced subgraph $G[E^{\prime}]$, has edge set $E^{\prime}$ and vertex set equal to the vertices incident with edges in $E^{\prime}$. We will use this construction to show that certain subgraph structures cannot exist, when attempting to find a bad family in a matroid. ###### Definition 3.3. _Kinser class_ $n$, denoted by $\mathcal{K}_{n}$, is the set of matroids which satisfy Kinser inequality $n$ for all families of subsets ${X}_{1},\dots,{X}_{n}$ of the ground set. We define $\mathcal{K}_{\infty}=\bigcap_{i\geq 4}\mathcal{K}_{i}$. A matroid $M$ has a bad family relative to $n$ if and only if $M\notin\mathcal{K}_{n}$. ###### Definition 3.4. The _dual Kinser class_ $n$ is $\mathcal{K}_{n}^{}=\\{M^{}\ \|\ M\in\mathcal{K}_{n}\\}$ ### 3.2 Kinser matroids Next we will construct a class of matroids called _Kinser matroids_ , relating to the Kinser inequalities as the Vámos matroid relates to the Ingleton inequality. In fact, the Vámos matroid is the fourth Kinser matroid after a circuit-hyperplane having been relaxed. The rank $r$ Kinser matroid, for $r\geq 4$, is denoted by $\mathrm{Kin}(r)$, and has a ground set of size $r^{2}-3r+4$. First, we will define a rank $r+1$ tranversal matroid, $M_{r+1}$. Let $\mathcal{A}=(A_{1},\ldots,A_{r-1},A,A^{\prime})$. Also let $V_{1},\ldots,V_{r}$ be pairwise disjoint sets such that $\|V_{1}\|=\|V_{3}\|=\cdots=\|V_{r}\|=r-2$ and $V_{2}=\\{e,f\\}$. The ground set of $M_{r+1}$ is $V_{1}\cup\cdots\cup V_{r}$. Let $A=E(M_{r+1})$ and let $A^{\prime}=V_{2}$. Let $\displaystyle A_{1}$ $\displaystyle=(V_{1}\cup V_{3}\cup\cdots\cup V_{r})-(V_{1}\cup V_{r})$ $\displaystyle A_{3}$ $\displaystyle=(V_{1}\cup V_{3}\cup\cdots\cup V_{r})-(V_{1}\cup V_{3})$ For $i\in\\{4,\ldots,r\\}$, let $A_{i}=(V_{1}\cup V_{3}\cup\cdots\cup V_{r})-(V_{i-1}\cup V_{i})$ Then $M_{r+1}$ is the tranversal matroid $M[\mathcal{A}]$. Note $\\{e,f\\}$ is a series pair in $M_{r+1}$. Define $\mathrm{Kin}(r)$ to be the truncation of $M_{r+1}$. $V_{1}$$V_{2}$$V_{3}$$V_{4}$$e$$f$ Figure 3.1: $\mathrm{Kin}(4)$ $V_{2}$$e$$f$$V_{1}$$V_{3}$$V_{4}$$V_{5}$ Figure 3.2: $\mathrm{Kin}(5)$ The following result is Proposition 4.3 of [4] ###### Lemma 3.5. Let $\mathbb{K}$ be an infinite field. Then $\mathrm{Kin}(r)$ is $\mathbb{K}$-representable for any $r\geq 4$. As $M_{r+1}$ is a tranversal matroid, is it representable over every infinite field by [6, Proposition 11.2.16]. We obtain $\mathrm{Kin}(r)$ by truncating $M_{r+1}$. This is equivalent to freely adding an element to the ground set of $M_{r+1}$ and then contracting it. As the class of representable matroids is closed under free extensions, $\mathrm{Kin}(r)$ is also representable. The following result is proven in [4, Proposition 4.4]. ###### Lemma 3.6. Let $r\geq 4$ be an integer. Then $V_{2}\cup V_{i}$ is a circuit-hyperplane of $\mathrm{Kin}(r)$ for any $i\in\\{1,3,\ldots,r\\}$. Define $\mathrm{Kin}(r)^{-}$ to be the matroid obtained from $\mathrm{Kin}(r)$ by relaxing the circuit-hyperplane $V_{1}\cup V_{2}$. Also define $\mathrm{Kin}(r)_{i}^{=}$ to be the matroid obtained from $\mathrm{Kin}(r)$ by relaxing the circuit-hyperplanes $V_{1}\cup V_{2}$ and $V_{2}\cup V_{i}$, for some $i\in\\{3,\ldots,r\\}$. The next two results are Proposition 4.5 and Lemma 4.6 of [4]. ###### Lemma 3.7. Let $r\geq 4$. The matroid $\mathrm{Kin}(r)^{-}$ is not in $\mathcal{K}_{r}$, and is therefore not representable over any field. The family of subsets ${V}_{1},\dots,{V}_{n}$ in $\mathrm{Kin}(r)^{-}$ is a bad family relative to $r$, as will be proven in Lemma 4.3. ###### Lemma 3.8. Let $r\geq 4$ and let $\mathbb{K}$ be an infinite field. The matroid $\mathrm{Kin}(r)_{i}^{=}$ is $\mathbb{K}$-representable. ### 3.3 Kinser classes ###### Lemma 3.9. $\mathcal{K}_{n}$ is minor-closed for all $n\geq 4$. ###### Proof. Take some $M\in\mathcal{K}_{n}$ and $e\in E(M)$ such that $M/e\notin\mathcal{K}_{n}$. Assume $e$ is not a loop. Assume ${X}_{1},\dots,{X}_{n}$ is a bad family in $M/e$ and let ${X_{i}}^{\prime}=X_{i}\cup e$ for all $i$. Recall that $r_{M/x}(X)=r_{M}(X\cup x)-r_{M}(x)$. Thus $r_{M/e}(X_{i})=r_{M}(X_{i}\cup e)-r_{M}(e)$. When $e$ is not a loop, we have that $r_{M}({X_{i}}^{\prime})=r_{M/e}(X_{i})+1$ for all $i$, and $r_{M}({X_{i}}^{\prime}\cup{X_{j}}^{\prime}\cup{X_{k}}^{\prime})=r_{M/e}(X_{i}\cup X_{j}\cup X_{k})+1$. Now evaluate inequality $n$ for $X_{1}^{\prime},\ldots,X_{n}^{\prime}$ in $M$. $\sum_{i=3}^{n}r_{M}(X_{i}^{\prime})+r_{M}(X_{1}^{\prime}\cup X_{2}^{\prime})+r_{M}(X_{1}^{\prime}\cup X_{3}^{\prime}\cup X_{n}^{\prime})+\sum_{i=4}^{n}r_{M}(X_{2}^{\prime}\cup X_{i-1}^{\prime}\cup X_{i}^{\prime})\\\ \leq r_{M}(X_{1}^{\prime}\cup X_{3}^{\prime})+r_{M}(X_{1}^{\prime}\cup X_{n}^{\prime})+\sum_{i=3}^{n}r_{M}(X_{2}^{\prime}\cup X_{i}^{\prime})+\sum_{i=4}^{n}r_{M}(X_{i-1}^{\prime}\cup X_{i}^{\prime})$ Using the rank equalities calculated above, this is equivalent to $\sum_{i=3}^{n}(r_{M/x}(X_{i})+1)+r_{M/x}(X_{1}\cup X_{2})+1\\\ +r_{M/x}(X_{1}\cup X_{3}\cup X_{n})+1+\sum_{i=4}^{n}(r_{M/x}(X_{2}\cup X_{i-1}\cup X_{i})+1)\\\ \leq r_{M/x}(X_{1}\cup X_{3})+1+r_{M/x}(X_{1}\cup X_{n})+1\\\ +\sum_{i=3}^{n}(r_{M/x}(X_{2}\cup X_{i})+1)+\sum_{i=4}^{n}(r_{M/x}(X_{i-1}\cup X_{i})+1)$ All the constant terms cancel out, leaving inequality $n$ as applied to ${X}_{1},\dots,{X}_{n}$ in $M/e$, contradicting ${X}_{1},\dots,{X}_{n}$ being a bad family in $M/e$. Thus there is no $e$ such that $M/e\notin\mathcal{K}_{n}$. Now consider $M\backslash e$. Assume that $M\backslash e$ has a bad family ${X}_{1},\dots,{X}_{n}$. These subsets are also subsets of $M$ and their rank is unchanged in $M$, so they must form a bad family in $M$ as well, contradicting $M\in\mathcal{K}_{n}$. ∎ ###### Lemma 3.10. Suppose ${X}_{1},\dots,{X}_{n}$ is a bad family for Kinser inequality $n$. We can assume that each set $X_{i}$ is independent. ###### Proof. For all $i\in\\{1,\ldots,n\\}$, let $I_{k}\subseteq X_{j}$ be a basis of $X_{j}$. We will show that we can replace each set $X_{j}$ with its basis $I_{j}$. We have that $r(X_{j})=r(I_{j})$. Now consider $r(X_{j}\cup X_{k})$. Clearly $r(I_{j}\cup I_{k})\leq r(X_{j}\cup X_{k})$. If $x\in X_{j}\cup X_{k}$, then $x$ is either in $X_{j}$ or it is in $X_{j}$, so $x\in cl_{M}(I_{j})$ or $x\in cl_{M}(I_{k})$. In either case, $x\in cl_{M}(I_{j}\cup I_{k})$, so $X_{j}\cup X_{k}\subseteq cl_{M}(I_{j}\cup I_{k})$. Thus $\displaystyle r(X_{j}\cup X_{k})$ $\displaystyle\leq r(cl_{M}(I_{j}\cup I_{k}))$ $\displaystyle=r(I_{j}\cup I_{k})$ Thus $r(I_{j}\cup I_{k})\leq r(X_{j}\cup X_{k})\leq r(I_{j}\cup I_{k})$, so $r(X_{j}\cup X_{k})=r(I_{j}\cup I_{k})$. Similarly, $r(X_{j}\cup X_{k}\cup X_{l})=r(I_{j}\cup I_{k}\cup I_{l})$. Thus ${I}_{1},\dots,{I}_{n}$ is a bad family for Kinser inequality $n$. ∎ ###### Lemma 3.11. Suppose ${X}_{1},\dots,{X}_{n}$ is a bad family for Kinser inequality $n$. We can assume that each set $X_{i}$ is a flat. ###### Proof. Recall that a flat is a set $X$ such that $cl(X)=X$. We simply replace each $X_{i}$ with $cl(X_{i})$. First note that $r(cl(X_{i}))=r(X_{i})$. Now consider $r(X_{i}\cup X_{j})=r(cl(X_{i}\cup X_{j}))$. We need to show that this is equal to $r(cl(X_{i})\cup cl(X_{j}))$. As $X_{i}\subseteq cl(X_{i})$, we must have that $r(X_{i}\cup X_{j})\leq r(cl(X_{i})\cup cl(X_{j}))$ Now note that if $e\in cl(X_{i})$, then $e\in cl(X_{i}\cup X_{j})$. This implies that $cl(X_{i})\subseteq cl(X_{i}\cup X_{j})$ Likewise, every element in the closure of $X_{j}$ is also in the closure of $X_{i}\cup X_{j}$, so $cl(X_{j})\subseteq cl(X_{i}\cup X_{j})$. We thus have that $cl(X_{i})\cup cl(X_{j})\subseteq cl(X_{i}\cup X_{j})$, so $\displaystyle r(cl(X_{i})\cup cl(X_{j}))$ $\displaystyle\leq r(cl(X_{i}\cup X_{j}))$ $\displaystyle=r(X_{i}\cup X_{j})$ We thus have that $r(cl(X_{i})\cup cl(X_{j}))\leq r(X_{i}\cup X_{j})\leq r(cl(X_{i})\cup cl(X_{j}))$ so $r(X_{i}\cup X_{j})=r(cl(X_{i})\cup cl(X_{j}))$, and so $cl(X_{i}),\ldots,cl(X_{n})$ is a bad family for inequality $n$ as well. ∎ ### 3.4 Sums ###### Definition 3.12. Let $M=(E,r)$ and $M^{\prime}=(E^{\prime},r^{\prime})$ where $E\cap E^{\prime}=\varnothing$. The _direct sum_ of these matroids is denoted by $M\oplus M^{\prime}$, and has ground set $E\cup E^{\prime}$ and rank of $X\subseteq E\cup E^{\prime}$ equal to $r(X\cap E)+r^{\prime}(X\cap E^{\prime})$. ###### Lemma 3.13. $\mathcal{K}_{n}$ is closed under direct sum for all $n$. ###### Proof. Take two matroid $M=(E,r)$ and $M^{\prime}=(E^{\prime},r^{\prime})$ which are contained in $\mathcal{K}_{n}$. Take the direct sum $M\oplus M^{\prime}$. We wish to show that for any family ${X}_{1},\dots,{X}_{n}$ of $E\cup E^{\prime}$ the following inequality holds: $\displaystyle\sum_{i=3}^{n}r_{M\oplus M^{\prime}}(X_{i})+r_{M\oplus M^{\prime}}(X_{1}\cup X_{2})+r_{M\oplus M^{\prime}}(X_{1}\cup X_{3}\cup X_{n})+\sum_{i=4}^{n}r_{M\oplus M^{\prime}}(X_{2}\cup X_{i-1}\cup X_{i})$ $\displaystyle\leq r_{M\oplus M^{\prime}}(X_{1}\cup X_{3})+r_{M\oplus M^{\prime}}(X_{1}\cup X_{n})+\sum_{i=3}^{n}r_{M\oplus M^{\prime}}(X_{2}\cup X_{i})+\sum_{i=4}^{n}r_{M\oplus M^{\prime}}(X_{i-1}\cup X_{i})$ This is equivalent to $\begin{split}&\sum_{i=3}^{n}r(X_{i}\cap E)+r((X_{1}\cup X_{2})\cap E)+r((X_{1}\cup X_{3}\cup X_{n})\cap E)\\\ &\qquad+\sum_{i=4}^{n}r((X_{2}\cup X_{i-1}\cup X_{i})\cap E)+\sum_{i=3}^{n}r^{\prime}(X_{i}\cap E^{\prime})+r^{\prime}((X_{1}\cup X_{2})\cap E^{\prime})\\\ &\qquad+r^{\prime}((X_{1}\cup X_{3}\cup X_{n})\cap E^{\prime})+\sum_{i=4}^{n}r^{\prime}((X_{2}\cup X_{i-1}\cup X_{i})\cap E^{\prime})\\\ \leq&\ r((X_{1}\cup X_{3})\cap E)+r((X_{1}\cup X_{n})\cap E)+\sum_{i=3}^{n}r((X_{2}\cup X_{i})\cap E)\\\ &\qquad+\sum_{i=4}^{n}r((X_{i-1}\cup X_{i})\cap E)+r^{\prime}((X_{1}\cup X_{3})\cap E^{\prime})+r^{\prime}((X_{1}\cup X_{n})\cap E^{\prime})\\\ &\qquad+\sum_{i=3}^{n}r^{\prime}((X_{2}\cup X_{i})\cap E^{\prime})+\sum_{i=4}^{n}r^{\prime}((X_{i-1}\cup X_{i})\cap E^{\prime})\end{split}$ (3.13.1) As $M\in\mathcal{K}_{n}$, we have that $\begin{split}&\sum_{i=3}^{n}r(X_{i}\cap E)+r((X_{1}\cup X_{2})\cap E)\\\ &\qquad+r((X_{1}\cup X_{3}\cup X_{n})\cap E)+\sum_{i=4}^{n}r((X_{2}\cup X_{i-1}\cup X_{i})\cap E)\\\ &\leq r((X_{1}\cup X_{3})\cap E)+r((X_{1}\cup X_{n})\cap E)\\\ &\qquad+\sum_{i=3}^{n}r((X_{2}\cup X_{i})\cap E)+\sum_{i=4}^{n}r((X_{i-1}\cup X_{i})\cap E)\end{split}$ As $M^{\prime}\in\mathcal{K}_{n}$, we have that $\begin{split}&r^{\prime}(X_{i}\cap E^{\prime})+r^{\prime}((X_{1}\cup X_{2})\cap E^{\prime})\\\ &\qquad+r^{\prime}((X_{1}\cup X_{3}\cup X_{n})\cap E^{\prime})+\sum_{i=4}^{n}r^{\prime}((X_{2}\cup X_{i-1}\cup X_{i})\cap E^{\prime})\\\ &\leq r^{\prime}((X_{1}\cup X_{3})\cap E^{\prime})+r^{\prime}((X_{1}\cup X_{n})\cap E^{\prime})\\\ &\qquad+\sum_{i=3}^{n}r^{\prime}((X_{2}\cup X_{i})\cap E^{\prime})+\sum_{i=4}^{n}r^{\prime}((X_{i-1}\cup X_{i})\cap E^{\prime})\end{split}$ The values of the terms on the left of inequality (3.0.21.1) are thus bounded by the terms on the right-hand side, and so the inequality holds. ∎ ## Chapter 4 Kinser Hierarchy In this chapter we will investigate how the Kinser classes interact with each other. We will first show that representable matroids are properly contained inside every Kinser class. Next we will show that the classes form a descending chain, and show that the relaxed Kinser matroid of rank $n$ is contained inside $\mathcal{K}_{n-1}$ but not $\mathcal{K}_{n}$. Next we will consider the issue of duality, and prove that the class of matroids which satisfy Kinser inequality $4$ is dual closed. The class of matroids which satisfy Kinser inequality $5$ is, in contrast, not dual closed. The proof of this is given in the next chapter. ###### Lemma 4.1. The class of representable matroids is properly contained in $\mathcal{K}_{\infty}$ ###### Proof. Note that as a consequence of Lemma 3.2, the class of representable matroids are contained inside every Kinser class, and so is a subset of $\mathcal{K}_{\infty}$. We will now show the class of representable matroids forms a proper subset of $\mathcal{K}_{\infty}$. Recall that $\mathcal{K}_{n}$ is closed under direct sum for all $n$. Take two matroids in $\mathcal{K}_{\infty}$. As these matroid is in the intersection of every Kinser class, their direct sum is also contained in every Kinser class, and thus inside $\mathcal{K}_{\infty}$. Thus $\mathcal{K}_{\infty}$ is also closed under direct sums. Define $F_{7}$ to be the matroid represented over GF(2) by $\bordermatrix{\text{}\cr&1&0&0&1&1&0&1\cr&0&1&0&1&0&1&1\cr&0&0&1&0&1&1&1}$ Define $F_{7}^{-}$ to be the matroid represented by the same matrix, but over GF(3). By [6, Proposition 6.4.8], $F_{7}$ can be represented over a field only if it has characteristic $2$, while $F_{7}^{-}$ can be represented over a field only if it has characteristic different from $2$. Therefore $F_{7}\oplus F_{7}^{-}$ is not representable. However, it is contained in $K_{\infty}$, since $F_{7}$ and $F_{7}^{-}$ are both representable, and hence in $\mathcal{K}_{\infty}$. ∎ ###### Lemma 4.2. $\mathcal{K}_{n}\supseteq\mathcal{K}_{n+1}$. ###### Proof. Assume inequality $n+1$ holds for the matroid $M$. Let $X_{1},\ldots,X_{n}$ be arbitrary subsets of $E(M)$. We show inequality $n$ holds for ${X}_{1},\dots,{X}_{n}$. Let $X_{n+1}=X_{n}$. We have that $\displaystyle\sum_{i=3}^{n+1}r(X_{i})+r(X_{1}\cup X_{2})+r(X_{1}\cup X_{3}\cup X_{n+1})+\sum_{i=4}^{n+1}r(X_{2}\cup X_{i-1}\cup X_{i})$ $\displaystyle\leq r(X_{1}\cup X_{3})+r(X_{1}\cup X_{n+1})+\sum_{i=3}^{n+1}r(X_{2}\cup X_{i})+\sum_{i=4}^{n+1}r(X_{i-1}\cup X_{i})$ Bringing out the last term of each sum, $\begin{split}&\sum_{i=3}^{n}r(X_{i})+r(X_{n+1})+r(X_{1}\cup X_{2})+r(X_{1}\cup X_{3}\cup X_{n})\\\ &\qquad+\sum_{i=4}^{n}r(X_{2}\cup X_{i-1}\cup X_{i})+r(X_{2}\cup X_{n+1})\\\ &\leq r(X_{1}\cup X_{3})+r(X_{1}\cup X_{n})+\sum_{i=3}^{n}r(X_{2}\cup X_{i})\\\ &\qquad+r(X_{2}\cup X_{n+1})+\sum_{i=4}^{n}r(X_{i-1}\cup X_{i})+r(X_{n}\cup X_{n+1})\end{split}$ Now bringing these terms to the start of each side of the inequality and using $X_{n+1}=X_{n}$, we have that this is the same as $\begin{split}&r(X_{n})+r(X_{2}\cup X_{n})+\sum_{i=3}^{n}r(X_{i})+r(X_{1}\cup X_{2})\\\ &\qquad+r(X_{1}\cup X_{3}\cup X_{n})+\sum_{i=4}^{n}r(X_{2}\cup X_{i-1}\cup X_{i})\\\ &\leq r(X_{2}\cup X_{n})+r(X_{n})+r(X_{1}\cup X_{3})+r(X_{1}\cup X_{n})\\\ &\qquad+\sum_{i=3}^{n}r(X_{2}\cup X_{i})+\sum_{i=4}^{n}r(X_{i-1}\cup X_{i})\end{split}$ The two terms at the start of each side of the inequality cancel out, leaving inequality $n$: $\displaystyle\sum_{i=3}^{n}r(X_{i})+r(X_{1}\cup X_{2})+r(X_{1}\cup X_{3}\cup X_{n})+\sum_{i=4}^{n}r(X_{2}\cup X_{i-1}\cup X_{i})$ $\displaystyle\leq r(X_{1}\cup X_{3})+r(X_{1}\cup X_{n})+\sum_{i=3}^{n}r(X_{2}\cup X_{i})+\sum_{i=4}^{n}r(X_{i-1}\cup X_{i}).$ ∎ We now have the following diagram of the Kinser hierarchy. Figure 4.1: Kinser classes Next, we will give an example of a matroid which lies in the gap between two of these classes. ###### Lemma 4.3. For $n\geq 5$, $\mathrm{Kin}(n)$${}^{-}\in\mathcal{K}_{n-1}-\mathcal{K}_{n}$. ###### Proof. Take $\mathrm{Kin}(n)^{-}$. We will first show that ${V}_{1},\dots,{V}_{n}$ as in the definition of $\mathrm{Kin}(n)$ forms a bad family for inequality $n$ – that is, $\displaystyle\sum_{i=3}^{n}r(V_{i})+r(V_{1}\cup V_{2})+r(V_{1}\cup V_{3}\cup V_{n})+\sum_{i=4}^{n}r(V_{2}\cup V_{i-1}\cup V_{i})$ $\displaystyle\nleq r(V_{1}\cup V_{3})+r(V_{1}\cup V_{n})+\sum_{i=3}^{n}r(V_{2}\cup V_{i})+\sum_{i=4}^{n}r(V_{i-1}\cup V_{i})$ We sketch the proof of [4, Proposition 4.5]. Recall that $V_{1}\cup V_{2}$ is a relaxed circuit-hyperplane, while $V_{2}\cup V_{i}$ is a circuit-hyperplane for all $i\in\\{3,\ldots,n\\}$. Also, $V_{i}\cup V_{i+1}$ is a hyperplane for all $i\geq 4$, as is $V_{1}\cup V_{3}$, while $V_{i}\cup V_{k}$ is spanning for inconsecutive $i,k$. Each $V_{i}$ is indepedent, while the union of any three $V_{i}$ is spanning. Substituting these results into inequality $n$ gives $\begin{array}[]{rrcl}&\sum_{i=3}^{n}(n-2)+n+n+\sum_{i=4}^{n}n&\not\leq&(n-1)+n(n-1)\\\ &&&\quad+\sum_{i=3}^{n}(n-1)+\sum_{i=4}^{n}(n-1)\\\ \Leftrightarrow&(n-2)(n-2)+2n+(n-3)n&\not\leq&(2n-3)(n-1)\\\ \Leftrightarrow&2n^{2}-5n+4&\not\leq&2n^{2}-5n+3\end{array}$ Thus $\mathrm{Kin}(n)$${}^{-}\notin\mathcal{K}_{n}$. Assume $\mathrm{Kin}(n)$- has a family $X_{1},\ldots,X_{n-1}$ which violates inequality $n-1$. Recall that if we take a second hyperplane of the form $V_{2}\cup V_{j}$, for an arbitrary $j\in\\{3,\ldots,n\\}$ and relax it, this yields a representable matroid, $\mathrm{Kin}(n)_{j}^{=}$, by Lemma 3.8. As relaxing a circuit-hyperplane causes that subset to become independent, the rank of $V_{2}\cup V_{j}$ increases by one. The rank of all other subsets are unchanged. Suppose $V_{2}\cup V_{j}$ was not a term in inequality $n-1$ as applied to ${X}_{1},\dots,{X}_{n-1}$. Relaxing $V_{2}\cup V_{j}$ would then have no effect on the value of the inequality, giving that ${X}_{1},\dots,{X}_{n-1}$ is a bad family in $\mathrm{Kin}(n)$=. This contradicts the matroid being representable. Suppose that $V_{2}\cup V_{j}$ is a term on the left-hand side of inequality $n-1$. Then, when we relax $V_{2}\cup V_{j}$, the left-hand side of the inquality increases by one while the right-hand side remains the same. As the inequality previously did not hold, i.e. the left-hand side was in fact greater than the right-hand side, it still cannot hold. Thus $V_{2}\cup V_{j}$ must be a term on the right-hand side of inequality $n$, for all $j$. There are $n-2$ choices of $j$, hence $n-2$ terms on the right-hand side of inequality $n-1$ must be these circuit- hyperplanes, $V_{2}\cup V_{j}$. Next, note that if we tighten $V_{1}\cup V_{2}$, the resulting matroid is representable, by [4, Lemma 4.6]. This decreases the rank of $V_{1}\cup V_{2}$ by one and leaves the ranks of all other subsets unchanged. As with $V_{2}\cup V_{j}$, $V_{1}\cup V_{2}$ must be a term of inequality $n-1$ in order for the inequality to be able to reflect the change in representability. If $V_{1}\cup V_{2}$ was a term on the right-hand side, after tightening $V_{1}\cup V_{2}$ the right-hand side would decrease by one and the left-hand side would remain the same. As prior to tightening the left-hand side was greater than the right-hand side, this fact is still true, meaning the inequality does not hold, contradicting the matroid being representable. A term on the left-hand side of inequality $n-1$ must thus be equal to $V_{1}\cup V_{2}$. Now consider a graph $G$ on vertices $V=\\{X_{1},...,X_{n-1}\\}$ with adjacency relation $a$ such that $a(X_{i})=\\{X_{j}\ \|\ X_{i}\cup X_{j}\ \mathrm{is}\ \mathrm{a}\ \mathrm{term}\ \mathrm{on}\ \mathrm{the}\ \text{right-hand side}\ \mathrm{of}\ \mathrm{inequality}\ n-1\\}$. Take the subgraph $G^{\prime}$ induced by the edges of $G$ corresponding to the circuit-hyperplanes $V_{2}\cup V_{3},\ldots,V_{2}\cup V_{n}$. Suppose $G^{\prime}$ has a path of length three. Call this path $X_{a},e,X_{b},f,X_{c},g,X_{d}$ and let the edges $e,f,g$ refer to the circuit-hyperplanes $V_{2}\cup V_{i}$, $V_{2}\cup V_{j}$, $V_{2}\cup V_{k}$ respectively. $X_{a}$$X_{b}$$X_{c}$$X_{d}$$V_{2}\cup V_{i}$$V_{2}\cup V_{j}$$V_{2}\cup V_{k}$ Consider what elements must lie where; $X_{b}\cup X_{c}$ must be equal to $V_{2}\cup V_{j}$. $X_{b}$ cannot contain any elements of $V_{j}$ as $X_{a}\cup X_{b}=V_{2}\cup V_{i}$, and $V_{i}$ and $V_{j}$ are disjoint. However, $X_{c}$ cannot contain any of the elements of $V_{j}$ either, since $X_{c}\cup X_{d}=V_{2}\cup V_{k}$ does not. This structure can thus not exist, i.e. $G^{\prime}$ can have no paths of length three. The same reasoning shows $G^{\prime}$ cannot have a cycle of length three. We will now show that $G^{\prime}$ does in fact contain a path or cycle of length three. As $V_{1}\cup V_{2}$ must be a term on the left-hand side of inequality $n-1$, one of $X_{1},...,X_{n-1}$ must contain elements from $V_{1}$. This subset cannot be incident with any edge representing a circuit-hyperplane $V_{2}\cup V_{i}$, so $G^{\prime}$ has at most $n-2$ vertices. $G^{\prime}$ must have $n-2$ edges. As trees must have an edge set of size one less than the number of vertices, this shows $G^{\prime}$ is not a tree, and thus contains a path or cycle of length three. Hence $\mathrm{Kin}(n)$- cannot have a bad family for inequality $n-1$. ∎ Figure 4.2: Kinser classes (2) Next we show that the first Kinser class, $\mathcal{K}_{4}$, is dual-closed. ###### Lemma 4.4. Let $M$ be an excluded minor for the class $\mathcal{K}_{4}$. Let ${X}_{1},\dots,{X}_{4}$ be a bad family in $M$ for inequality $4$. Then ${X}_{1},\dots,{X}_{4}$ is a partition of $E(M)$. ###### Proof. Suppose there is an element of $E(M)$ which is not contained in some $X_{i}$. We could then delete this element and ${X}_{1},\dots,{X}_{n}$ would still form a bad family, contradicting $M$ being minor-minimal with respect to not being in $\mathcal{K}_{4}$. Thus ${X}_{1},\dots,{X}_{n}$ must cover the entire ground set. Now assume the sets in the bad family are not disjoint, so there exists an $x$ which is in $X_{i}\cap X_{j}$ for some $i,j$. Contract $x$, and for each set $X_{k}$, let $X^{\prime}_{k}=X_{k}-x$. Let $L$ be some union of the $X_{i}$, so that $r(L)$ appears somewhere in the inequality. Say that $L$ is stable if $r_{M/x}(L-x)=r_{M}(L)$. If none of the five terms on the right-hand side of the inequality are stable, then the sum on the right-hand side of the inequality decreases by exactly five when we contract $x$ and remove $x$ from each term. Since the left-hand side can decrease by at most five, this means that $X_{1}/x,\ldots,X_{n}/x$ is a bad family in $M/x$, which is impossible as $M$ is an excluded minor for $\mathcal{K}_{4}$, and so $M/x\in\mathcal{K}_{4}$. Therefore there is a stable set on the right-hand side. As $x\in X_{i}\cap X_{j}$, the rank of any term that includes $X_{i}$ or $X_{j}$ decreases when we contract $x$ and remove it from the term, and so these terms cannot be stable. There is at most one term on the right-hand side of the inequality that does not involve $X_{i}$ or $X_{j}$, so the right-hand side has exactly one stable term. Now this stable term is equal to $X_{m}\cup X_{n}$, where either $m$ or $n$ is equal to $3$ or $4$. Note $x\notin cl(X_{m})$ and $x\notin cl(X_{n})$, or else $X_{m}\cup X_{n}$ would not be stable. Therefore either $X_{3}$ or $X_{4}$ is stable, so there is a stable term on the left-hand side of the inequality also. This means that $X_{1}/x,\ldots,X_{n}/x$ is a bad family in $M/x$, and we get another contradiction. ∎ ###### Lemma 4.5. $\mathcal{K}_{4}=\mathcal{K}_{4}^{}$ ###### Proof. Assume for a contradiction that $M\in\mathcal{K}_{4}$, $M^{}\notin\mathcal{K}_{4}$. Note that $M^{}$ contains a minor-minimal matroid not in $\mathcal{K}_{4}$. Let $N$ be a minor of $M$ such that $N^{}$ is an excluded minor for $\mathcal{K}_{4}$. Let ${X}_{1},\dots,{X}_{4}$ be a bad family of $N^{}$. By the previous lemma, we have that ${X}_{1},\dots,{X}_{4}$ partitions $E(N^{})$. By assumption we have that $\begin{split}&\qquad r^{}(X_{3})+r^{}(X_{4})+r^{}(X_{1}\cup X_{2})+r^{}(X_{1}\cup X_{3}\cup X_{4})+r^{}(X_{2}\cup X_{3}\cup X_{4})\\\ &\nleq r^{}(X_{1}\cup X_{3})+r^{}(X_{1}\cup X_{4})+r^{}(X_{2}\cup X_{3})+r^{}(X_{2}\cup X_{4})+r^{}(X_{3}\cup X_{4})\end{split}$ (4.5.1) Recall $r^{}(X)=\|X\|+r(E-X)-r(N)$. Use this identity on every term in the inequality. Now we see that (4.4.1) is true if and only if (4.4.2) is true. $\begin{split}&\|X_{3}\|+\|X_{4}\|+r(\overline{X_{3}})+r(\overline{X_{4}})+\|X_{1}\cup X_{2}\|+r(\overline{X_{1}\cup X_{2}})\\\ &\qquad+\|X_{1}\cup X_{3}\cup X_{4}\|+r(\overline{X_{1}\cup X_{3}\cup X_{4}})\\\ &\qquad+\|X_{2}\cup X_{3}\cup X_{4}\|+r(\overline{X_{2}\cup X_{3}\cup X_{4}})\\\ &\nleq\|X_{1}\cup X_{3}\|+r(\overline{X_{1}\cup X_{3}})+\|X_{1}\cup X_{4}\|+r(\overline{X_{1}\cup X_{4}})\\\ &\qquad+\|X_{2}\cup X_{3}\|+r(\overline{X_{2}\cup X_{3}})+\|X_{2}\cup X_{4}\|\\\ &\qquad+r(\overline{X_{2}\cup X_{4}})+\|X_{3}\cup X_{4}\|+r(\overline{X_{2}\cup X_{4}})\end{split}$ (4.5.2) where $\overline{X_{i}}=E-X$. Note that every $-r(N)$ cancelled out as there is an equal number of terms on each side of the inequality. Using the identities $\|X_{i}\cup X_{j}\|=\|X_{i}\|+\|X_{j}\|-\|X_{1}\cap X_{j}\|$ and $\|X_{i}\cup X_{j}\cup X_{k}\|=\|X_{i}\|+\|X_{j}\|+\|X_{k}\|-\|X_{i}\cap X_{j}\|-\|X_{i}\cap X_{k}\|-\|X_{j}\cap X_{k}\|+\|X_{i}\cap X_{j}\cap X_{k}\|$, we can simplify this. Fully apply these identities to each cardinality term. We now have that (4.4.2) is true if and only if (4.4.3) is true. $\begin{split}&\|X_{3}\|+\|X_{4}\|+r(\overline{X_{3}})+r(\overline{X_{4}})\\\ &\qquad+\|X_{1}\|+\|X_{2}\|-\|X_{1}\cap X_{2}\|+r(\overline{X_{1}\cup X_{2}})\\\ &\qquad+\|X_{1}\|+\|X_{3}\|+\|X_{4}\|-\|X_{1}\cap X_{3}\|-\|X_{1}\cap X_{4}\|\\\ &\qquad\qquad-\|X_{3}\cap X_{4}\|+\|X_{1}\cap X_{3}\cap X_{4}\|+r(\overline{X_{1}\cup X_{3}\cup X_{4}})\\\ &\qquad+\|X_{2}\|+\|X_{3}\|+\|X_{4}\|-\|X_{2}\cap X_{3}\|-\|X_{2}\cap X_{4}\|-\|X_{3}\cap X_{4}\|\\\ &\qquad\qquad+\|X_{2}\cap X_{3}\cap X_{4}\|+r(\overline{X_{2}\cup X_{3}\cup X_{4}})\\\ &\nleq\|X_{1}\|+\|X_{3}\|-\|X_{1}\cap X_{3}\|+r(\overline{X_{1}\cup X_{3}})\\\ &\qquad+\|X_{1}\|+\|X_{4}\|-\|X_{1}\cap X_{4}\|+r(\overline{X_{1}\cup X_{4}})\\\ &\qquad+\|X_{2}\|+\|X_{3}\|-\|X_{2}\cap X_{3}\|+r(\overline{X_{2}\cup X_{3}})\\\ &\qquad+\|X_{2}\|+\|X_{4}\|-\|X_{2}\cap X_{4}\|+r(\overline{X_{2}\cup X_{4}})\\\ &\qquad+\|X_{3}\|+\|X_{4}\|-\|X_{3}\cap X_{4}\|+r(\overline{X_{2}\cup X_{4}})\end{split}$ (4.5.3) As ${X}_{1},\dots,{X}_{4}$ is a partition of $E(M)$, any term $X_{i}\cap X_{j}$ is empty. Cancelling out these terms, and all common terms, this yields $\begin{split}&\qquad r(\overline{X_{3}})+r(\overline{X_{4}})+r(\overline{X_{1}\cup X_{2}})+r(\overline{X_{1}\cup X_{3}\cup X_{4}})+r(\overline{X_{2}\cup X_{3}\cup X_{4}})\\\ &\nleq r(\overline{X_{1}\cup X_{3}})+r(\overline{X_{1}\cup X_{4}})+r(\overline{X_{2}\cup X_{3}})+r(\overline{X_{2}\cup X_{4}})+r(\overline{X_{3}\cup X_{4}})\end{split}$ (4.5.4) Now let $Y_{1}=\overline{X_{1}\cup X_{2}\cup X_{3}}$, $Y_{2}=\overline{X_{1}\cup X_{2}\cup X_{4}}$, $Y_{3}=\overline{X_{1}\cup X_{3}\cup X_{4}}$, and $Y_{4}=\overline{X_{2}\cup X_{3}\cup X_{4}}$. We have the following equalities $\displaystyle Y_{1}\cup Y_{3}\cup Y_{4}$ $\displaystyle=\overline{X_{1}\cup X_{2}\cup X_{3}}\cup\overline{X_{1}\cup X_{3}\cup X_{4}}\cup\overline{X_{2}\cup X_{3}\cup X_{4}}$ $\displaystyle=X_{4}\cup X_{2}\cup X_{1}$ $\displaystyle=\overline{X_{3}}$ $\displaystyle Y_{2}\cup Y_{3}\cup Y_{4}$ $\displaystyle=\overline{X_{1}\cup X_{2}\cup X_{4}}\cup\overline{X_{1}\cup X_{3}\cup X_{4}}\cup\overline{X_{2}\cup X_{3}\cup X_{4}}$ $\displaystyle=X_{3}\cup X_{2}\cup X_{1}$ $\displaystyle=\overline{X_{4}}$ $\displaystyle Y_{1}\cup Y_{2}$ $\displaystyle=\overline{X_{1}\cup X_{2}\cup X_{3}}\cup\overline{X_{1}\cup X_{2}\cup X_{4}}$ $\displaystyle=X_{4}\cup X_{3}$ $\displaystyle=\overline{X_{1}\cup X_{2}}$ $\displaystyle Y_{1}\cup Y_{3}$ $\displaystyle=\overline{X_{1}\cup X_{2}\cup X_{3}}\cup\overline{X_{1}\cup X_{3}\cup X_{4}}$ $\displaystyle=X_{4}\cup X_{2}$ $\displaystyle=\overline{X_{1}\cup X_{3}}$ $\displaystyle Y_{2}\cup Y_{3}$ $\displaystyle=\overline{X_{1}\cup X_{2}\cup X_{4}}\cup\overline{X_{1}\cup X_{3}\cup X_{4}}$ $\displaystyle=X_{3}\cup X_{2}$ $\displaystyle=\overline{X_{1}\cup X_{4}}$ $\displaystyle Y_{1}\cup Y_{4}$ $\displaystyle=\overline{X_{1}\cup X_{2}\cup X_{3}}\cup\overline{X_{2}\cup X_{3}\cup X_{4}}$ $\displaystyle=X_{4}\cup X_{1}$ $\displaystyle=\overline{X_{2}\cup X_{3}}$ $\displaystyle Y_{2}\cup Y_{4}$ $\displaystyle=\overline{X_{1}\cup X_{2}\cup X_{4}}\cup\overline{X_{2}\cup X_{3}\cup X_{4}}$ $\displaystyle=X_{3}\cup X_{1}$ $\displaystyle=\overline{X_{2}\cup X_{4}}$ $\displaystyle Y_{3}\cup Y_{4}$ $\displaystyle=\overline{X_{1}\cup X_{3}\cup X_{4}}\cup\overline{X_{2}\cup X_{3}\cup X_{4}}$ $\displaystyle=X_{2}\cup X_{1}$ $\displaystyle=\overline{X_{3}\cup X_{4}}$ Inequality (4.5.4) thus becomes $\displaystyle r(Y_{1}\cup Y_{3}\cup Y_{4})+r(Y_{2}\cup Y_{3}\cup Y_{4})+r(Y_{1}\cup Y_{2})+r(Y_{3})+r(Y_{4})$ $\displaystyle\nleq r(Y_{1}\cup Y_{3})+r(Y_{2}\cup Y_{3})+r(Y_{1}\cup Y_{4})+r(Y_{2}\cup Y_{4})+r(Y_{3}\cup Y_{4})$ These rank terms are exactly those of inequality $4$, which holds for ${Y}_{1},\dots,{Y}_{4}$ as $N\in\mathcal{K}_{4}$. Thus $N^{}$ also satisfies inequality $4$, for all choices of ${X}_{1},\dots,{X}_{4}$. ∎ We conjecture that this is the only Kinser class which is dual-closed. We provide a proof of this for only the second class. There appears to be no simple way to verify that a matroid satisfies a particular Kinser inequality, which leads to the difficulty involved in the next chapter. In a later chapter, we will provide a result on the complexity of verifying that a matroid satisfies Kinser inequality $n$. ###### Theorem 4.6. $\mathcal{K}_{5}\neq\mathcal{K}_{5}^{}$ The proof of this theorem forms the next chapter. There are now two possibilities of how the Kinser classes sit in the hierarchy, shown in the following diagrams. Figure 4.3: Kinser classes (3a) Figure 4.4: Kinser classes (3b) ## Chapter 5 $\mathcal{K}_{5}\neq\mathcal{K}_{5}^{}$ ###### Theorem 5.1. $\mathrm{Kin}(5)^{-}\in\mathcal{K}_{5}^{}$ ###### Proof. Recall that $\mathrm{Kin}(5)^{-}$ is the matroid obtained from the rank $5$ Kinser matroid by relaxing the circuit-hyperplane $V_{1}\cup V_{2}$. Assume ($\mathrm{Kin}(5)^{-})^{}$ has at least one bad family $X_{1},\ldots,X_{5}$ violating Kinser inequality $5$, where each set $X_{i}$ is a flat of $(\mathrm{Kin}(5)^{-})^{}$. Note the complement of $V_{1}\cup V_{2}$ is a relaxed circuit-hyperplane in $(\mathrm{Kin}(5)^{-})^{}$, and tightening it produces Kin($5$)∗, which is representable. This operation affects the rank function by decreasing the rank of exactly one set: $V_{3}\cup V_{4}\cup V_{5}$. Suppose $V_{3}\cup V_{4}\cup V_{5}$ is not a term in inequality $5$ as applied to ${X}_{1},\dots,{X}_{5}$. Then evaluating the inequality after relaxing the circuit-hyperplane would give the same value as before relaxing, contradicting the change in representability. As the rank of $V_{3}\cup V_{4}\cup V_{5}$ decreases by one when we tighten it, the only way for the inequality to hold only after the tighening is for the left-hand side of the inequality to decrease and the right-hand side to remain the same. So $V_{3}\cup V_{4}\cup V_{5}$ must be a term on the left-hand side of the inequality. Next note that $V_{2}\cup V_{3}$, $V_{2}\cup V_{4}$, and $V_{2}\cup V_{5}$ are circuit-hyperplanes on $\mathrm{Kin}(5)^{-}$, so $V_{1}\cup V_{4}\cup V_{5}$, $V_{1}\cup V_{3}\cup V_{5}$, $V_{1}\cup V_{3}\cup V_{4}$ are circuit- hyperplanes of $(\mathrm{Kin}(5)^{-})^{}$. Relaxing any one of these will make $(\mathrm{Kin}(5)^{-})^{}$ representable again. These subsets must thus be terms in the inequality. The rank of each of these subsets increases by one when relaxed, so for the inequality to hold the right-hand side must increase, meaning these three sets must be terms on the right-hand side of inequality $5$. Let $e\in V_{2}$. If $e\notin{X}_{1},\dots,{X}_{5}$, then $(\mathrm{Kin}(5)^{-})^{}\backslash e=(\mathrm{Kin}(5)^{-}/e)^{}$ has a bad family. This means that $\mathrm{Kin}(5)^{-}/e$ is not representable. We will show that the elements of $V_{1}$ are freely placed in $\mathrm{Kin}(5)^{-}/e$ . Pick $z\in V_{1}$. Let $Z$ be a non-spanning subset in $\mathrm{Kin}(5)^{-}/e$ such that $z\in cl_{\mathrm{Kin}(5)^{-}/e}(Z)-Z$. This implies by [6, Proposition lookitup] $z\in cl_{\mathrm{Kin}(5)^{-}}(Z\cup\\{e\\})$. Note that the closure of a set in $\mathrm{Kin}(5)$ only differs from that in $\mathrm{Kin}(5)^{-}$ in that it may contain additional elements in $\mathrm{Kin}(5)$. Thus $z\in cl_{\mathrm{Kin}(5)}(Z\cup\\{e\\})$. Recall that $M_{6}$ is the transversal matroid whose truncation is defined to be $\mathrm{Kin}(5)$. As $Z\cup\\{e\\}$ does not span $\mathrm{Kin}(5)$, truncation does not affect the rank of this subset, and so $z\in cl_{M_{6}}(Z\cup\\{e\\})$. Now consider the neighbours of $Z\cup\\{e\\}$ in the transversal system $\mathcal{A}$. The neighbours of $z$ must be contained in this set, as otherwise the rank of $Z\cup\\{e,f\\}$ would be greater than the rank of $Z\cup\\{e\\}$, contradicting $z\in cl(Z\cup\\{e\\})$. Recall that in $M_{r+1}$, $V_{i}$ is incident with $A_{0},\ldots,A_{i-1},A_{i+2},\ldots,A_{r}$. $Z\cup\\{e\\}$ is thus neighbours with $A_{0},A_{1},A_{3}$, and $A_{4}$. Recall $Z$ is non-spanning in $\mathrm{Kin}(5)^{-}/e$. $\mathrm{Kin}(5)^{-}/e$ has rank $4$, so $r_{\mathrm{Kin}(5)^{-}/e}(Z)\leq 3$, and $r_{\mathrm{Kin}(5)^{-}}(Z\cup\\{e\\})\leq 4$. This rank cannot change after tightening $V_{1}\cup V_{2}$, so $r_{\mathrm{Kin}(5)}(Z\cup\\{e\\})\leq 4$. As $Z\cup\\{e\\}$ is hence non-spanning in $\mathrm{Kin}(5)$, $r_{M_{6}}(Z\cup\\{e\\}\leq 4$. This means that $Z\cup\\{e\\}$ has exactly $A_{0},A_{1},A_{3},$ and $A_{4}$ as neighbours. Thus $Z\cup\\{e\\}\subseteq V_{1}\cup\\{e,f\\}$. This implies that $Z\cup\\{e,z\\}$ is independent in $\mathrm{Kin}(5)^{-}$, and so $Z\cup\\{z\\}$ is independent in $\mathrm{Kin}(5)^{-}/e$. This is a contradiction, as $z\in cl_{\mathrm{Kin}(5)^{-}/e}(Z)-Z$. Thus $z$ is free in $\mathrm{Kin}(5)^{-}/e$ and $\mathrm{Kin}(5)^{-}/e$ is a free-extension of $\mathrm{Kin}(5)^{-}/e\backslash z=\mathrm{Kin}(5)/e\backslash z$ which is $\mathbb{K}$-representable. This contradicts $(\mathrm{Kin}(5)^{-})^{}\backslash e=(\mathrm{Kin}(5)^{-}/e)^{}$ having a bad family, and so all elements of $V_{2}$ must be contained in some set $X_{i}$. Suppose the elements of $V_{2}$ are contained in different sets in the bad family. That is, suppose $e\in X_{i}$ and $f\in X_{j}$ for some $i,j$.The three circuit-hyperplanes $V_{1}\cup V_{3}\cup V_{4}$, $V_{1}\cup V_{4}\cup V_{5}$, $V_{1}\cup V_{3}\cup V_{5}$ and the relaxed circuit-hyperplane $V_{3}\cup V_{4}\cup V_{5}$ do not include elements of $V_{2}$, and thus cannot be equal to terms in the inequality which use $X_{i}$ or $X_{j}$. Removing these terms from the inequality leaves at most three possible terms for these circuit-hyperplanes, regardless of the values of $i$ and $j$. Thus both elements of $V_{2}$, $e$ and $f$, must be contained in the same set $X_{i}$ of the bad family. Considering each possible location for the elements of $V_{2}$ gives us five cases – $V_{2}\cap X_{1}\neq\varnothing,V_{2}\cap X_{2}\neq\varnothing,V_{2}\cap X_{3}\neq\varnothing,V_{2}\cap X_{4}\neq\varnothing,V_{2}\cap X_{5}\neq\varnothing$. Take the case $X_{5}\cap V_{2}\neq\varnothing$. Swapping $X_{3}$ and $X_{5}$ gives the inequality, and this case is thus identical to the case where $V_{2}\cap X_{3}\neq\varnothing.$ We thus reduce to the four cases examined below. ###### Case 1. $X_{3}\cap V_{2}\neq\varnothing$ Consider which term on the left-hand side of the inequality must be $V_{3}\cup V_{4}\cup V_{5}$. As this term does not contain any elements of $V_{2}$ and these elements are contained in $X_{3}$, the term cannot involve $X_{3}$. In the following diagram, the ovals represent the sets ${X}_{1},\dots,{X}_{5}$. An edge between two sets represents the union of those two sets. The edges shown below are on the right-hand side of the inequality, and thus possible locations for the necessary circuit-hyperplanes $V_{1}\cup V_{3}\cup V_{5}$, $V_{1}\cup V_{4}\cup V_{5}$, $V_{1}\cup V_{3}\cup V_{4}$. Note that $X_{1}\cup X_{2}$, indicated by the dashed line, appears on the left-hand side of the inequality. The edges coming from $X_{3}$ have been left off as, since the elements of $V_{2}$ are contained in $X_{3}$, none of these edges could represent the three circuit-hyperplanes. $X_{1}$$X_{2}$$X_{3}$$X_{4}$$X_{5}$ If $V_{3}\cup V_{4}\cup V_{5}$ is equal to $X_{4}$ or $X_{5}$ then any term involving these sets cannot be one of the circuit-hyperplanes $V_{1}\cup V_{3}\cup V_{5}$, $V_{1}\cup V_{4}\cup V_{5}$, $V_{1}\cup V_{3}\cup V_{4}$ which must appear on the right-hand side of the inequality. This will not leave us with enough terms on the right-hand side which could be these three circuit-hyperplanes – if $V_{3}\cup V_{4}\cup V_{5}=X_{4}$, then the terms available for the circuit-hyperplanes are $X_{1}\cup X_{5}$ and $X_{2}\cup X_{5}$, while if $V_{3}\cup V_{4}\cup V_{5}=X_{5}$, there are again only two terms available, $X_{1}\cup X_{2}$ and $X_{2}\cup X_{4}$. If $V_{3}\cup V_{4}\cup V_{5}=X_{2}\cup X_{4}\cup X_{5}$, then elements of $V_{1}$ can only be in $X_{1}$. Note that all of the three circuit-hyperplanes which must appear on the right-hand side include $V_{1}$. However, as $X_{3}\cap V_{2}\neq\varnothing$, the only free term using $X_{1}$ is $X_{1}\cup X_{5}$, so we are unable to have the three needed circuit-hyperplanes on the right- hand side. The only remaining possibility is that $V_{3}\cup V_{4}\cup V_{5}=X_{1}\cup X_{2}$. We now have four possibilities of where the circuit-hyperplanes on the right- hand side of the inequality lie, and we will consider various subcases. Let $\\{i,j,k\\}=\\{3,4,5\\}$. The following diagrams show these subcases, as indicated by the bold lines. $X_{1}$$X_{2}$$X_{3}$$X_{4}$$X_{5}$ () Subcase 1 $X_{1}$$X_{2}$$X_{3}$$X_{4}$$X_{5}$ () Subcase 2 $X_{1}$$X_{2}$$X_{3}$$X_{4}$$X_{5}$ () Subcase 3 $X_{1}$$X_{2}$$X_{3}$$X_{4}$$X_{5}$ () Subcase 4 In each subcase examined below, we are considering the circuit-hyperplanes to lie as indicated by the diagram at the start of each subcase. In all of these, there are multiple options for which elements are in what sets, as described below. Subcase 1.1. $X_{1}$$X_{2}$$X_{3}$$X_{4}$$X_{5}$$V_{1}\cup V_{j}\cup V_{k}$$V_{1}\cup V_{i}\cup V_{j}$$V_{1}\cup V_{i}\cup V_{k}$ Here, we have assumed that $V_{2}\subseteq X_{3}$. $X_{3}$ may contain other elements of the ground set of $M$, thus any terms in the inequality involving $X_{3}$ will not be immediately evaluated. Recall that $X_{1}\cup X_{2}=V_{3}\cup V_{4}\cup V_{5}$. This means that $V_{1}$ can only be a subset of $X_{4}$ or $X_{5}$. In order for the three circuit-hyperplanes to lie as indicated, both $X_{4}$ and $X_{5}$ must contain $V_{1}$. $X_{2}$ must be equal to $V_{i}$, as this forms the common values of the two circuit- hyperplanes involving $X_{2}$, excluding $V_{1}$. Finally, $X_{5}$ must be equal to $V_{1}\cup V_{k}$, while $X_{1}$ must contain $V_{j}\cup V_{k}$. As we assumed the sets $X_{i}$ to be flats, $X_{1}$ could also include one or two elements of $V_{i}$. It cannot contain all three elements, as $V_{i}\cup V_{j}\cup V_{k}$ is a relaxed circuit-hyperplane, and thus has closure equal to the entire ground set. This gives the following three subcases, which differ only in elements of $X_{1}$. In each subcase, we will begin by evaluating terms of the left-hand and right-hand sides of inequality $5$, then show that the left-hand side must be lower in value, meaning that the inequality holds. * Subcase 1.1a: $X_{1}=V_{j}\cup V_{k}$ $X_{2}=V_{i}$ $X_{3}=V_{2}\cup Z$ where $Z\subseteq E(M)$ $X_{4}=V_{1}\cup V_{j}$ $X_{5}=V_{1}\cup V_{k}$ Note that $\overline{X_{1}\cup X_{2}}=V_{1}\cup V_{2}$ and that $\overline{X_{2}\cup X_{4}\cup X_{5}}=\overline{V_{1}\cup V_{i}\cup V_{j}\cup V_{k}}=V_{2}$. We will use $M$ to refer to $\mathrm{Kin}(5)^{-}$. $\begin{split}LHS&=r^{}(X_{3})+r^{}(X_{4})+r^{}(X_{5})+r^{}(X_{1}\cup X_{2})+r^{}(X_{1}\cup X_{3}\cup X_{5})\\\ &\qquad+r^{}(X_{2}\cup X_{3}\cup X_{4})+r^{}(X_{2}\cup X_{4}\cup X_{5})\\\ &=r^{}(X_{3})+r^{}(X_{1}\cup X_{3}\cup X_{5})+r^{}(X_{2}\cup X_{3}\cup X_{4})\\\ &\qquad+\|X_{4}\|+\|X_{5}\|+\|X_{1}\cup X_{2}\|+\|X_{2}\cup X_{4}\cup X_{5}\|\\\ &\qquad+r(\overline{X_{4}})+r(\overline{X_{5}})+r(\overline{X_{1}\cup X_{2}})+r(\overline{X_{2}\cup X_{4}\cup X_{5}})-4r(M)\\\ &=r^{}(X_{3})+r^{}(X_{1}\cup X_{3}\cup X_{5})+r^{}(X_{2}\cup X_{3}\cup X_{4})-4r(M)\\\ &\qquad+6+6+9+12+5+5+5+2\\\ &=r^{}(X_{3})+r^{}(X_{1}\cup X_{3}\cup X_{5})+r^{}(X_{2}\cup X_{3}\cup X_{4})-4r(M)+50\\\ \end{split}$ $\begin{split}RHS&=r^{}(X_{1}\cup X_{3})+r^{}(X_{1}\cup X_{5})+r^{}(X_{2}\cup X_{3})+r^{}(X_{2}\cup X_{4})\\\ &\qquad+r^{}(X_{2}\cup X_{5})+r^{}(X_{3}\cup X_{4})+r^{}(X_{4}\cup X_{5})\\\ &=r^{}(X_{1}\cup X_{3})+r^{}(X_{2}\cup X_{3})+r^{}(X_{3}\cup X_{4})+\|X_{1}\cup X_{5}\|\\\ &\qquad+\|X_{2}\cup X_{4}\|+\|X_{2}\cup X_{5}\|+\|X_{4}\cup X_{5}\|+r(\overline{X_{1}\cup X_{5}})\\\ &\qquad+r(\overline{X_{2}\cup X_{4}})+r(\overline{X_{2}\cup X_{5}})+r(\overline{X_{4}\cup X_{5}})-4r(M)\\\ &\geq r^{}(X_{1}\cup X_{3})+r^{}(X_{2}\cup X_{3})+r^{}(X_{3}\cup X_{4})-4r(M)\\\ &\qquad+9+9+9+9+4+4+4+4\\\ &=r^{}(X_{1}\cup X_{3})+r^{}(X_{2}\cup X_{3})+r^{}(X_{3}\cup X_{4})-4r(M)+52\\\ \end{split}$ We thus must have that $\begin{split}&\qquad r^{}(X_{3})+r^{}(X_{1}\cup X_{3}\cup X_{4})+r^{}(X_{2}\cup X_{3}\cup X_{5})\\\ &>r^{}(X_{1}\cup X_{3})+r^{}(X_{2}\cup X_{3})+r^{}(X_{3}\cup X_{4})+2\end{split}$ in order for ${X}_{1},\dots,{X}_{5}$ to be a bad family. Suppose $Z=\varnothing$, i.e. $X_{3}=V_{2}$. We then have $2+9+9>8+5+8+2$ which is untrue. Suppose we increase the size of $Z$. If the cardinality of one of these unions does not change, then neither does the rank of the complement, and thus the corank is unchanged. Note that $X_{1}\cup X_{3}\cup X_{5}=V_{1}\cup V_{2}\cup V_{j}\cup V_{k}\cup Z$. If the cardinality of this set is greater when $Z$ is non-empty, then $Z\subseteq V_{i}$. As $\overline{X_{1}\cup X_{3}\cup X_{5}}=V_{i}-Z$, this will cause the rank of $\overline{X_{1}\cup X_{3}\cup X_{5}}$ to decrease. The corank is thus unchanged from when $Z=\varnothing$. A similar argument shows that the corank of $X_{2}\cup X_{3}\cup X_{4}=V_{1}\cup V_{2}\cup V_{i}\cup V_{j}\cup Z$ must be unchanged as its complement $V_{k}-Z$ is independent. Also, $\displaystyle r^{}(X_{3}\cup X_{4})+2$ $\displaystyle\geq r^{}(X_{3})+r^{}(X_{4})-r^{}(X_{3}\cap X_{4})+2$ $\displaystyle\geq r^{}(X_{3})+r^{}(X_{4})-r^{}(Z\cap X_{4})+2$ $\displaystyle\geq r^{}(X_{3})+2$ $\displaystyle\geq r^{}(X_{3})$ Thus increasing the size of $Z$ can only cause the left-hand side to decrease by more than the right-hand side, and so we cannot have a bad family for any choice of $Z$. Subcase 1.1b: $X_{1}=V_{j}\cup V_{k}\cup\\{a\\}$ where $a\in V_{i}$ $X_{2}=V_{i}$ $X_{3}=V_{2}\cup Z$ where $Z\subseteq E(M)$ $X_{4}=V_{1}\cup V_{j}$ $X_{5}=V_{1}\cup V_{k}$ The only terms whose value changes from before is $\|X_{1}\cup X_{5}\|$ which increases by one. So we must have that $\begin{split}&\qquad r^{}(X_{3})+r^{}(X_{1}\cup X_{3}\cup X_{4})+r^{}(X_{2}\cup X_{3}\cup X_{4})\\\ &>r^{}(X_{1}\cup X_{3})+r^{}(X_{2}\cup X_{3})+r^{}(X_{3}\cup X_{4})+3\end{split}$ The same argument as before shows that we do not have a bad family, for any choice of $Z$. * Subcase 1c: $X_{1}=V_{j}\cup V_{k}\cup\\{a,b\\}$ where $a,b\in V_{i}$ $X_{2}=V_{i}$ $X_{3}=V_{2}\cup Z$ where $Z\subseteq E(M)$ $X_{4}=V_{1}\cup V_{j}$ $X_{5}=V_{1}\cup V_{k}$ Here, $\|X_{1}\cup X_{5}\|$ is one higher than in the previous subcase, while $r(\overline{X_{1}\cup X_{5}})$ is one lower, and the inequality is thus identical to subcase 1b. Subcase 1.2. $X_{1}$$X_{2}$$X_{3}$$X_{4}$$X_{5}$$V_{1}\cup V_{i}\cup V_{j}$$V_{1}\cup V_{j}\cup V_{k}$$V_{1}\cup V_{i}\cup V_{k}$ Again, we have by assumption that $V_{2}\subseteq X_{3}$. $X_{3}$ may contain other elements of the ground set of $M$, thus any terms in the inequality involving $X_{3}$ will not be immediately evaluated. $X_{5}$ must be equal to $V_{1}$, as these are the only common elements of the three circuit- hyperplanes. This forces $X_{1}$ to be equal to $V_{i}\cup V_{j}$, as $X_{1}\cup X_{2}=V_{3}\cup V_{4}\cup V_{5}$ and thus $X_{1}$ cannot contain elements of $V_{1}$. Likewise, $X_{2}$ must be equal to $V_{i}\cup V_{k}$. $X_{4}$ must contain $V_{j}\cup V_{k}$. As we assumed the sets $X_{i}$ to be flats, $X_{4}$ could also include one or three elements of $V_{1}$. It cannot contain exactly two elements of $V_{1}$, as $V_{1}\cup V_{j}\cup V_{k}$ is a circuit-hyperplane. This gives the following three subcases, which differ only in elements of $X_{4}$. As before, in each subcase we will begin by evaluating terms of the left-hand and right-hand sides of inequality $5$, then show that the left-hand side must be lower in value, meaning that the inequality holds. * Subcase 1.2a: $X_{1}=V_{i}\cup V_{j}$ $X_{2}=V_{i}\cup V_{k}$ $X_{3}=V_{2}\cup Z$ where $Z\subseteq E(M)$ $X_{4}=V_{j}\cup V_{k}$ $X_{5}=V_{1}$ Note that $\overline{X_{1}\cup X_{2}}=V_{1}\cup V_{2}$ and $\overline{X_{2}\cup X_{4}\cup X_{5}}=\overline{V_{1}\cup V_{i}\cup V_{j}\cup V_{k}}=V_{2}$. $\begin{split}LHS&=r^{}(X_{3})+r^{}(X_{4})+r^{}(X_{5})+r^{}(X_{1}\cup X_{2})+r^{}(X_{1}\cup X_{3}\cup X_{5})\\\ &\qquad+r^{}(X_{2}\cup X_{3}\cup X_{4})+r^{}(X_{2}\cup X_{4}\cup X_{5})\\\ &=r^{}(X_{3})+r^{}(X_{1}\cup X_{3}\cup X_{5})+r^{}(X_{2}\cup X_{3}\cup X_{4})\\\ &\qquad+\|X_{4}\|+\|X_{5}\|+\|X_{1}\cup X_{2}\|+\|X_{2}\cup X_{4}\cup X_{5}\|\\\ &\qquad+r(\overline{X_{4}})+r(\overline{X_{5}})+r(\overline{X_{1}\cup X_{2}})+r(\overline{X_{2}\cup X_{4}\cup X_{5}})-4r(M)\\\ &=r^{}(X_{3})+r^{}(X_{1}\cup X_{3}\cup X_{5})+r^{}(X_{2}\cup X_{3}\cup X_{4})-4r(M)\\\ &\qquad+6+3+9+12+5+5+5+2\\\ &=r^{}(X_{3})+r^{}(X_{1}\cup X_{3}\cup X_{5})+r^{}(X_{2}\cup X_{3}\cup X_{4})-4r(M)+47\\\ \end{split}$ $\begin{split}RHS&=r^{}(X_{1}\cup X_{3})+r^{}(X_{1}\cup X_{5})+r^{}(X_{2}\cup X_{3})+r^{}(X_{2}\cup X_{4})\\\ &\qquad+r^{}(X_{2}\cup X_{5})+r^{}(X_{3}\cup X_{4})+r^{}(X_{4}\cup X_{5})\\\ &=r^{}(X_{1}\cup X_{3})+r^{}(X_{2}\cup X_{3})+r^{}(X_{3}\cup X_{4})+\|X_{1}\cup X_{5}\|\\\ &\qquad+\|X_{2}\cup X_{4}\|+\|X_{2}\cup X_{5}\|+\|X_{4}\cup X_{5}\|+r(\overline{X_{1}\cup X_{5}})\\\ &\qquad+r(\overline{X_{2}\cup X_{4}})+r(\overline{X_{2}\cup X_{5}})+r(\overline{X_{4}\cup X_{5}})-4r(M)\\\ &\geq r^{}(X_{1}\cup X_{3})+r^{}(X_{2}\cup X_{3})+r^{}(X_{3}\cup X_{4})-4r(M)\\\ &\qquad+9+9+9+9+4+5+4+4\\\ &=r^{}(X_{1}\cup X_{3})+r^{}(X_{2}\cup X_{3})+r^{}(X_{3}\cup X_{4})-4r(M)+53\\\ \end{split}$ We thus must have that $\begin{split}&\qquad r^{}(X_{3})+r^{}(X_{1}\cup X_{3}\cup X_{5})+r^{}(X_{2}\cup X_{3}\cup X_{4})\\\ &>r^{}(X_{1}\cup X_{3})+r^{}(X_{2}\cup X_{3})+r^{}(X_{3}\cup X_{4})+6\end{split}$ Suppose $Z=\varnothing$. Then $2+9+9>8+8+8+6$ which is untrue. The same argument as in subcase 1a shows that if $Z$ is non-empty, we still cannot have a bad family. * Subcase 1.2b: $X_{1}=V_{i}\cup V_{j}$ $X_{2}=V_{i}\cup V_{k}$ $X_{3}=V_{2}\cup Z$ where $Z\subseteq E(M)$ $X_{4}=V_{j}\cup V_{k}\cup\\{a\\}$ where $a\in V_{1}$ $X_{5}=V_{1}$ Compared to subcase 2a, $\|X_{4}\|$ increases by one on the left-hand side. On the right-hand side, $\|X_{2}\cup X_{4}\|$ increases by one. The overall inequality is unchanged, so the same argument as in subcase 2a holds. * Subcase 1.2c: $X_{1}=V_{i}\cup V_{j}$ $X_{2}=V_{i}\cup V_{k}$ $X_{3}=V_{2}\cup Z$ where $Z\subseteq E(M)$ $X_{4}=V_{j}\cup V_{k}\cup V_{i}$ $X_{5}=V_{1}$ Compared to subcase 2a, both $\|X_{4}\|$ and $\|X_{2}\cup X_{4}\|$ increase by three, leaving the inequality unchanged. Subcase 1.3. $X_{1}$$X_{2}$$X_{3}$$X_{4}$$X_{5}$$V_{1}\cup V_{i}\cup V_{k}$$V_{1}\cup V_{j}\cup V_{k}$$V_{1}\cup V_{i}\cup V_{j}$ Again, we have by assumption that $V_{2}\subseteq X_{3}$. $X_{3}$ may contain other elements of the ground set of $M$, thus any terms in the inequality involving $X_{3}$ will not be immediately evaluated. Recall that $X_{1}\cup X_{2}=V_{3}\cup V_{4}\cup V_{5}$. This means that $V_{1}$ can only be a subset of $X_{4}$ or $X_{5}$. In order for the three circuit-hyperplanes to lie as indicated, both $X_{4}$ and $X_{5}$ must contain $V_{1}$. $X_{2}$ must be equal to $V_{j}$, as this forms the common values of the two circuit- hyperplanes involving $X_{2}$, excluding $V_{1}$. Finally, $X_{5}$ must be equal to $V_{1}\cup V_{i}$, while $X_{4}$ must be equal to $V_{i}\cup V_{k}$. As we assumed the sets $X_{i}$ to be flats, $X_{1}$ could also include one or two elements of $V_{j}$. It cannot contain three elements of $V_{j}$, as $V_{i}\cup V_{j}\cup V_{k}$ is a relaxed circuit-hyperplane in $(\mathrm{Kin}(5)^{-})^{}$ and thus has closure equal to the entire ground set. This gives the following three subcases, which differ only in elements of $X_{1}$. As before, in each subcase we will begin by evaluating terms of the left-hand and right-hand sides of inequality $5$, then show that the left-hand side must be lower in value, meaning that the inequality holds. Subcase 1.3a: $X_{1}=V_{i}\cup V_{k}$ $X_{2}=V_{j}$ $X_{3}=V_{2}\cup Z$ where $Z\subseteq E(M)$ $X_{4}=V_{1}\cup V_{k}$ $X_{5}=V_{1}\cup V_{i}$ Note that $\overline{X_{1}\cup X_{2}}=V_{1}\cup V_{2}$ and $\overline{X_{2}\cup X_{4}\cup X_{5}}=\overline{V_{1}\cup V_{i}\cup V_{j}\cup V_{k}}=V_{2}$. $\begin{split}LHS&=r^{}(X_{3})+r^{}(X_{4})+r^{}(X_{5})+r^{}(X_{1}\cup X_{2})+r^{}(X_{1}\cup X_{3}\cup X_{5})\\\ &\qquad+r^{}(X_{2}\cup X_{3}\cup X_{4})+r^{}(X_{2}\cup X_{4}\cup X_{5})\\\ &=r^{}(X_{3})+r^{}(X_{1}\cup X_{3}\cup X_{5})+r^{}(X_{2}\cup X_{3}\cup X_{4})\\\ &\qquad+\|X_{4}\|+\|X_{5}\|+\|X_{1}\cup X_{2}\|+\|X_{2}\cup X_{4}\cup X_{5}\|\\\ &\qquad+r(\overline{X_{4}})+r(\overline{X_{5}})+r(\overline{X_{1}\cup X_{2}})+r(\overline{X_{2}\cup X_{4}\cup X_{5}})-4r(M)\\\ &=r^{}(X_{3})+r^{}(X_{1}\cup X_{3}\cup X_{5})+r^{}(X_{2}\cup X_{3}\cup X_{4})-4r(M)\\\ &\qquad+6+6+9+12+5+5+5+2\\\ &=r^{}(X_{3})+r^{}(X_{1}\cup X_{3}\cup X_{5})+r^{}(X_{2}\cup X_{3}\cup X_{4})-4r(M)+50\\\ \end{split}$ $\begin{split}RHS&=r^{}(X_{1}\cup X_{3})+r^{}(X_{1}\cup X_{5})+r^{}(X_{2}\cup X_{3})+r^{}(X_{2}\cup X_{4})\\\ &\qquad+r^{}(X_{2}\cup X_{5})+r^{}(X_{3}\cup X_{4})+r^{}(X_{4}\cup X_{5})\\\ &=r^{}(X_{1}\cup X_{3})+r^{}(X_{2}\cup X_{3})+r^{}(X_{3}\cup X_{4})+\|X_{1}\cup X_{5}\|\\\ &\qquad+\|X_{2}\cup X_{4}\|+\|X_{2}\cup X_{5}\|+\|X_{4}\cup X_{5}\|+r(\overline{X_{1}\cup X_{5}})\\\ &\qquad+r(\overline{X_{2}\cup X_{4}})+r(\overline{X_{2}\cup X_{5}})+r(\overline{X_{4}\cup X_{5}})-4r(M)\\\ &\geq r^{}(X_{1}\cup X_{3})+r^{}(X_{2}\cup X_{3})+r^{}(X_{3}\cup X_{4})-4r(M)\\\ &\qquad+9+9+9+9+4+4+4+4\\\ &=r^{}(X_{1}\cup X_{3})+r^{}(X_{2}\cup X_{3})+r^{}(X_{3}\cup X_{4})-4r(M)+52\\\ \end{split}$ We thus must have that $\begin{split}&\qquad r^{}(X_{3})+r^{}(X_{1}\cup X_{3}\cup X_{5})+r^{}(X_{2}\cup X_{3}\cup X_{4})\\\ &>r^{}(X_{1}\cup X_{3})+r^{}(X_{2}\cup X_{3})+r^{}(X_{3}\cup X_{4})+2\end{split}$ Suppose $Z=\varnothing$. Then we have $2+9+9>8+5+8+2$ which is untrue. If we increase the size of $Z$, the same argument as in subcase 1a shows that we still cannot have a bad family. * Subcase 1.3b: $X_{1}=V_{i}\cup V_{k}\cup\\{a\\}$ where $a\in V_{j}$ $X_{2}=V_{j}$ $X_{3}=V_{2}\cup Z$ where $Z\subseteq E(M)$ $X_{4}=V_{1}\cup V_{k}$ $X_{5}=V_{1}\cup V_{i}$ Compared to subcase 3a, $\|X_{1}\cup X_{5}\|$ increases by one, and the same argument still holds. * Subcase 1.3c: $X_{1}=V_{i}\cup V_{k}\cup\\{a,b\\}$ where $a,b\in V_{j}$ $X_{2}=V_{j}$ $X_{3}=V_{2}\cup Z$ where $Z\subseteq E(M)$ $X_{4}=V_{1}\cup V_{k}$ $X_{5}=V_{1}\cup V_{i}$ Compared to subcase 3a, $\|X_{1}\cup X_{5}\|$ increases by two, while $r(\overline{X_{1}\cup X_{5}})$ falls by one. The right-hand side of the inequality increases by one, and the same argument as before holds. Subcase 1.4. $X_{1}$$X_{2}$$X_{3}$$X_{4}$$X_{5}$$V_{1}\cup V_{i}\cup V_{k}$$V_{1}\cup V_{i}\cup V_{j}$$V_{1}\cup V_{j}\cup V_{k}$ Again, we have by assumption that $V_{2}\subseteq X_{3}$. $X_{3}$ may contain other elements of the ground set of $M$, thus any terms in the inequality involving $X_{3}$ will not be immediately evaluated. Recall that $X_{1}\cup X_{2}=V_{3}\cup V_{4}\cup V_{5}$. This means that $V_{1}$ can only be a subset of $X_{4}$ or $X_{5}$. In order for the three circuit-hyperplanes to lie as indicated, both $X_{4}$ and $X_{5}$ must contain $V_{1}$. As $X_{2}\cup X_{4}$ cannot contain $V_{i}$, $X_{5}$ must be equal to $V_{1}\cup V_{i}$. Likewise, $X_{2}$ must contain $V_{k}$. This forces $X_{4}$ to contain $V_{j}$, and $X_{1}$ to contain $V_{i}$. Finally, in order for $X_{1}\cup X_{2}=V_{3}\cup V_{4}\cup V_{5}$, it must be that $X_{2}$ is equal to $V_{j}\cup V_{k}$. $X_{2}$ must be equal to $V_{j}\cup V_{k}$, as this forms the common values of the two circuit-hyperplanes involving $X_{2}$, excluding $V_{1}$. Finally, $X_{5}$ must be equal to $V_{1}\cup V_{i}$, while $X_{4}$ must contain $V_{k}$. As we assumed the sets $X_{i}$ to be flats, $X_{4}$ could also include one or two elements of $V_{j}$. It cannot contain all three elements, as $V_{i}\cup V_{j}\cup V_{k}$ is a relaxed circuit-hyperplane, and thus has closure equal to the entire ground set. This gives the following three subcases, which differ only in elements of $X_{4}$. As before, in each subcase we will begin by evaluating terms of the left-hand and right-hand sides of inequality $5$, then show that the left-hand side must be lower in value, meaning that the inequality holds. * $X_{1}=V_{i}\cup V_{k}$ $X_{2}=V_{j}\cup V_{k}$ $X_{3}=V_{2}\cup Z$ where $Z\subseteq E(M)$ $X_{4}=V_{1}\cup V_{j}$ $X_{5}=V_{1}\cup Vi$ Note that $\overline{X_{1}\cup X_{2}}=V_{1}\cup V_{2}$ and $\overline{X_{2}\cup X_{4}\cup X_{5}}=\overline{V_{1}\cup V_{i}\cup V_{j}\cup V_{k}}=V_{2}$. $\begin{split}LHS&=r^{}(X_{3})+r^{}(X_{4})+r^{}(X_{5})+r^{}(X_{1}\cup X_{2})+r^{}(X_{1}\cup X_{3}\cup X_{5})\\\ &\qquad+r^{}(X_{2}\cup X_{3}\cup X_{4})+r^{}(X_{2}\cup X_{4}\cup X_{5})\\\ &=r^{}(X_{3})+r^{}(X_{1}\cup X_{3}\cup X_{5})+r^{}(X_{2}\cup X_{3}\cup X_{4})\\\ &\qquad+\|X_{4}\|+\|X_{5}\|+\|X_{1}\cup X_{2}\|+\|X_{2}\cup X_{4}\cup X_{5}\|\\\ &\qquad+r(\overline{X_{4}})+r(\overline{X_{5}})+r(\overline{X_{1}\cup X_{2}})+r(\overline{X_{2}\cup X_{4}\cup X_{5}})-4r(M)\\\ &=r^{}(X_{3})+r^{}(X_{1}\cup X_{3}\cup X_{5})+r^{}(X_{2}\cup X_{3}\cup X_{4})-4r(M)\\\ &\qquad+6+6+9+12+5+5+5+2\\\ &=r^{}(X_{3})+r^{}(X_{1}\cup X_{3}\cup X_{5})+r^{}(X_{2}\cup X_{3}\cup X_{4})-4r(M)+50\\\ \end{split}$ $\begin{split}RHS&=r^{}(X_{1}\cup X_{3})+r^{}(X_{1}\cup X_{5})+r^{}(X_{2}\cup X_{3})+r^{}(X_{2}\cup X_{4})\\\ &\qquad+r^{}(X_{2}\cup X_{5})+r^{}(X_{3}\cup X_{4})+r^{}(X_{4}\cup X_{5})\\\ &=r^{}(X_{1}\cup X_{3})+r^{}(X_{2}\cup X_{3})+r^{}(X_{3}\cup X_{4})+\|X_{1}\cup X_{5}\|\\\ &\qquad+\|X_{2}\cup X_{4}\|+\|X_{2}\cup X_{5}\|+\|X_{4}\cup X_{5}\|+r(\overline{X_{1}\cup X_{5}})\\\ &\qquad+r(\overline{X_{2}\cup X_{4}})+r(\overline{X_{2}\cup X_{5}})+r(\overline{X_{4}\cup X_{5}})-4r(M)\\\ &\geq r^{}(X_{1}\cup X_{3})+r^{}(X_{2}\cup X_{3})+r^{}(X_{3}\cup X_{4})-4r(M)\\\ &\qquad+9+9+9+9+4+4+2+4\\\ &=r^{}(X_{1}\cup X_{3})+r^{}(X_{2}\cup X_{3})+r^{}(X_{3}\cup X_{4})-4r(M)+50\\\ \end{split}$ We thus must have that $\begin{split}&\qquad r^{}(X_{3})+r^{}(X_{1}\cup X_{3}\cup X_{5})+r^{}(X_{2}\cup X_{3}\cup X_{4})\\\ &>r^{}(X_{1}\cup X_{3})+r^{}(X_{2}\cup X_{3})+r^{}(X_{3}\cup X_{4})\end{split}$ Suppose $Z=\varnothing$. Then we have $2+9+9>8+8+8$ which is untrue. If we increase the size of $Z$, the same argument as in subcase 1a shows that we still cannot have a bad family. ###### Case 2. $X_{4}\cap V_{2}\neq\varnothing$ As in Case 1, we will consider which term on the left-hand side dual inequality must be $V_{3}\cup V_{4}\cup V_{5}$. As this term does not contain any elements of $V_{2}$ and these elements are contained in $X_{4}$, the term cannot involve $X_{4}$. We now have the following representation of the locations remaining as possibilities for all necessary circuit-hyperplanes, where the dashed line again indicates a term which falls on the left-hand side of the inequality. $X_{1}$$X_{2}$$X_{3}$$X_{4}$$X_{5}$ If it $V_{3}\cup V_{4}\cup V_{5}$ is equal to $X_{3}$ or $X_{5}$ then any term involving these sets cannot be one of the circuit-hyperplanes $V_{1}\cup X_{4}\cup V_{5}$, $V_{1}\cup V_{3}\cup V_{5}$. When $V_{3}\cup V_{4}\cup V_{5}=X_{3}$, the possible terms left are $X_{1}\cup X_{5}$ and $X_{2}\cup X_{5}$. When $V_{3}\cup V_{4}\cup V_{5}=X_{5}$, the possible terms left are $X_{1}\cup X_{3}$ and $X_{2}\cup X_{3}$. In both cases there are not enough terms on the right-hand side for the three circuit-hyperplanes. Thus $V_{3}\cup V_{4}\cup V_{5}=X_{1}\cup X_{2}$. As $X_{3}$ and $X_{5}$ can be switched with no effect on the inequality, we now have, up to symmetry, two possibilities of where the circuit-hyperplanes lie, indicated by the bold lines in the following diagrams. $X_{1}$$X_{2}$$X_{3}$$X_{4}$$X_{5}$ () Subcase 1 $X_{1}$$X_{2}$$X_{3}$$X_{4}$$X_{5}$ () Subcase 2 We will again consider both of these subcases, with circuit-hyperplanes lying as shown in the diagrams, which give multiple options within each subcase for the choice of sets $X_{1},\ldots,X_{5}$. Subcase 2.1. $X_{1}$$X_{2}$$X_{3}$$X_{4}$$X_{5}$$V_{1}\cup V_{j}\cup V_{k}$$V_{1}\cup V_{i}\cup V_{j}$$V_{1}\cup V_{i}\cup V_{k}$ Here, we have assumed that $V_{2}\subseteq X_{4}$. $X_{4}$ may contain other elements of the ground set of $M$, thus any terms in the inequality involving $X_{4}$ will not be immediately evaluated. Recall that $X_{1}\cup X_{2}=V_{3}\cup V_{4}\cup V_{5}$. This means that $V_{1}$ can only be a subset of $X_{3}$ or $X_{5}$. In order for the three circuit-hyperplanes to lie as indicated, both $X_{3}$ and $X_{5}$ must contain $V_{1}$. $X_{2}$ must be equal to $V_{i}$, as this forms the common values of the two circuit- hyperplanes involving $X_{2}$, excluding $V_{1}$. This forces $X_{1}$ to be equal to $V_{j}\cup V_{k}$. Note that $X_{1}$ cannot contain any elements of $V_{i}$ as $X_{1}\cup X_{3}$ does not. Also, $X_{3}$ must contain $V_{j}$ and $X_{5}$ must contain $V_{k}$. $X_{3}$ must be equal to $V_{1}\cup V_{j}$. As we assumed the sets $X_{i}$ to be flats, $X_{5}$ could also include one or three elements of $V_{i}$. $X_{5}$ cannot contain exactly two elements of $V_{i}$ as $V_{1}\cup V_{k}\cup V_{i}$ is a circuit-hyperplane in the dual.This gives the following three subcases, which differ only in elements of $X_{5}$. * Subcase 2.1a: $X_{1}=V_{j}\cup V_{k}$ $X_{2}=V_{i}$ $X_{3}=V_{1}\cup V_{j}$ $X_{4}=V_{2}\cup Z$ where $Z\subseteq E(M)$ $X_{5}=V_{1}\cup V_{k}$ Note that $\overline{X_{1}\cup X_{2}}=V_{1}\cup V_{2}$ and $\overline{X_{1}\cup X_{3}\cup X_{5}}=\overline{V_{1}\cup V_{j}\cup V_{k}}=V_{2}\cup V_{i}$. $\begin{split}LHS&=r^{}(X_{3})+r^{}(X_{4})+r^{}(X_{5})+r^{}(X_{1}\cup X_{2})+r^{}(X_{1}\cup X_{3}\cup X_{5})\\\ &\qquad+r^{}(X_{2}\cup X_{3}\cup X_{4})+r^{}(X_{2}\cup X_{4}\cup X_{5})\\\ &=r^{}(X_{4})+r^{}(X_{2}\cup X_{3}\cup X_{4})+r^{}(X_{2}\cup X_{4}\cup X_{5})\\\ &\qquad+\|X_{3}\|+\|X_{5}\|+\|X_{1}\cup X_{2}\|+\|X_{1}\cup X_{3}\cup X_{5}\|\\\ &\qquad+r(\overline{X_{3}})+r(\overline{X_{5}})+r(\overline{X_{1}\cup X_{2}})+r(\overline{X_{1}\cup X_{3}\cup X_{5}})-4r(M)\\\ &=r^{}(X_{4})+r^{}(X_{2}\cup X_{3}\cup X_{4})+r^{}(X_{2}\cup X_{4}\cup X_{5})-4r(M)\\\ &\qquad+6+6+9+9+5+5+5+4\\\ &=r^{}(X_{4})+r^{}(X_{2}\cup X_{3}\cup X_{4})+r^{}(X_{2}\cup X_{4}\cup X_{5})-4r(M)+49\\\ \end{split}$ $\begin{split}RHS&=r^{}(X_{1}\cup X_{3})+r^{}(X_{1}\cup X_{5})+r^{}(X_{2}\cup X_{3})+r^{}(X_{2}\cup X_{4})\\\ &\qquad+r^{}(X_{2}\cup X_{5})+r^{}(X_{3}\cup X_{4})+r^{}(X_{4}\cup X_{5})\\\ &=r^{}(X_{2}\cup X_{4})+r^{}(X_{3}\cup X_{4})+r^{}(X_{4}\cup X_{5})+\|X_{1}\cup X_{3}\|\\\ &\qquad+\|X_{1}\cup X_{5}\|+\|X_{2}\cup X_{3}\|+\|X_{2}\cup X_{5}\|+r(\overline{X_{1}\cup X_{3}})\\\ &\qquad+r(\overline{X_{1}\cup X_{5}})+r(\overline{X_{2}\cup X_{3}})+r(\overline{X_{2}\cup X_{5}})-4r(M)\\\ &=r^{}(X_{2}\cup X_{4})+r^{}(X_{3}\cup X_{4})+r^{}(X_{4}\cup X_{5})-4r(M)\\\ &\qquad+9+9+9+9+4+4+4+4\\\ &=r^{}(X_{2}\cup X_{4})+r^{}(X_{3}\cup X_{4})+r^{}(X_{4}\cup X_{5})-4r(M)+52\\\ \end{split}$ We thus must have that $\begin{split}&\qquad r^{}(X_{4})+r^{}(X_{2}\cup X_{3}\cup X_{4})+r^{}(X_{2}\cup X_{4}\cup X_{5})\\\ &>r^{}(X_{2}\cup X_{4})+r^{}(X_{3}\cup X_{4})+r^{}(X_{4}\cup X_{5})+3\end{split}$ Suppose $Z=\varnothing$. Then we have that $2+8+8>5+8+8+3$ which is untrue. Now suppose $Z$ is non-empty. $X_{2}\cup X_{3}\cup X_{4}=V_{1}\cup V_{2}\cup V_{i}\cup V_{j}\cup Z$ and $\overline{X_{2}\cup X_{3}\cup X_{4}}=V_{k}-Z$, so if $\|X_{2}\cup X_{3}\cup X_{4}\|$ increases, then $Z\subseteq V_{k}$, and $r(\overline{X_{2}\cup X_{3}\cup X_{4}})$ must decrease by the same amount. Thus $r^{}(X_{2}\cup X_{3}\cup X_{4})$ remains unchanged. Likewise, $r^{}(X_{2}\cup X_{4}\cup X_{5})$ cannot change, as $X_{2}\cup X_{4}\cup X_{5}=V_{1}\cup V_{2}\cup V_{i}\cup V_{j}\cup Z$ and has complement $V_{k}-Z$. Finally, $r^{}(X_{4})$ must be no greater than $r^{}(X_{4}\cup X_{5})$. We thus do not have a bad family. * Subcase 2.1b: $X_{1}=V_{j}\cup V_{k}$ $X_{2}=V_{i}$ $X_{3}=V_{1}\cup V_{j}$ $X_{4}=V_{2}\cup Z$ where $Z\subseteq E(M)$ $X_{5}=V_{1}\cup V_{k}\cup\\{a\\}$ where $a\in V_{i}$ On the left-hand side, $\|X_{5}\|$ and $\|X_{1}\cup X_{3}\cup X_{5}\|$ both increase in size by two. On the right-hand side, $\|X_{1}\cup X_{5}\|$ increases in size by one. The same argument as in Subcase 1a thus shows there can be no bad family regardless of $Z$. * Subcase 2.1c: $X_{1}=V_{j}\cup V_{k}$ $X_{2}=V_{i}$ $X_{3}=V_{1}\cup V_{j}$ $X_{4}=V_{2}\cup Z$ where $Z\subseteq E(M)$ $X_{5}=V_{1}\cup V_{k}\cup\\{a,b\\}$ where $a,b\in V_{i}$ On the left-hand side, $\|X_{5}\|$ and $\|X_{1}\cup X_{3}\cup X_{5}\|$ both increase in size by two, while $r(X_{1}\cup X_{3}\cup X_{5})$ falls by one. On the right-hand side, $\|X_{1}\cup X_{5}\|$ increases in size by two while $r(X_{1}\cup X_{5})$ falls by one. We still have no bad family. Subcase 2.2. $X_{1}$$X_{2}$$X_{3}$$X_{4}$$X_{5}$$V_{1}\cup V_{i}\cup V_{j}$$V_{1}\cup V_{j}\cup V_{k}$$V_{1}\cup V_{i}\cup V_{k}$ Again, we have assumed that $V_{2}\subseteq X_{4}$. $X_{4}$ may contain other elements of the ground set of $M$, thus any terms in the inequality involving $X_{4}$ will not be immediately evaluated. Recall that $X_{1}\cup X_{2}=V_{3}\cup V_{4}\cup V_{5}$. This means that $V_{1}$ can only be a subset of $X_{3}$ or $X_{5}$. In order for the three circuit-hyperplanes to lie as indicated, both $X_{3}$ and $X_{5}$ must contain $V_{1}$. $X_{1}$ must be equal to $V_{i}$, as this forms the common values of the two circuit- hyperplanes involving $X_{2}$, excluding $V_{1}$. This forces $X_{2}$ to be equal to $V_{j}\cup V_{k}$, forces $X_{3}$ to be equal to $V_{1}\cup V_{j}$, and forces $X_{5}$ to contain $V_{1}\cup V_{k}$. As we assumed the sets $X_{i}$ to be flats, $X_{5}$ could also include one or three elements of $V_{i}$. $X_{5}$ cannot contain exactly two elements of $V_{i}$ as $V_{1}\cup V_{k}\cup V_{i}$ is a circuit-hyperplane in the dual.This gives the following three subcases, which differ only in elements of $X_{5}$. * Subcase 2.2a: $X_{1}=V_{i}$ $X_{2}=V_{j}\cup V_{k}$ $X_{3}=V_{1}\cup V_{j}$ $X_{4}=V_{2}\cup Z$ where $Z\subseteq E(M)$ $X_{5}=V_{1}\cup V_{k}$ Note that $\overline{X_{1}\cup X_{2}}=V_{1}\cup V_{2}$ and $\overline{X_{1}\cup X_{3}\cup X_{5}}=\overline{V_{1}\cup V_{i}\cup V_{j}\cup V_{k}}=V_{2}$. $\begin{split}LHS&=r^{}(X_{3})+r^{}(X_{4})+r^{}(X_{5})+r^{}(X_{1}\cup X_{2})+r^{}(X_{1}\cup X_{3}\cup X_{5})\\\ &\qquad+r^{}(X_{2}\cup X_{3}\cup X_{4})+r^{}(X_{2}\cup X_{4}\cup X_{5})\\\ &=r^{}(X_{4})+r^{}(X_{2}\cup X_{3}\cup X_{4})+r^{}(X_{2}\cup X_{4}\cup X_{5})\\\ &\qquad+\|X_{3}\|+\|X_{5}\|+\|X_{1}\cup X_{2}\|+\|X_{1}\cup X_{3}\cup X_{5}\|\\\ &\qquad+r(\overline{X_{3}})+r(\overline{X_{5}})+r(\overline{X_{1}\cup X_{2}})+r(\overline{X_{1}\cup X_{3}\cup X_{5}})-4r(M)\\\ &=r^{}(X_{4})+r^{}(X_{2}\cup X_{3}\cup X_{4})+r^{}(X_{2}\cup X_{4}\cup X_{5})-4r(M)\\\ &\qquad+6+6+9+12+5+5+5+2\\\ &=r^{}(X_{4})+r^{}(X_{2}\cup X_{3}\cup X_{4})+r^{}(X_{2}\cup X_{4}\cup X_{5})-4r(M)+50\\\ \end{split}$ $\begin{split}RHS&=r^{}(X_{1}\cup X_{3})+r^{}(X_{1}\cup X_{5})+r^{}(X_{2}\cup X_{3})+r^{}(X_{2}\cup X_{4})\\\ &\qquad+r^{}(X_{2}\cup X_{5})+r^{}(X_{3}\cup X_{4})+r^{}(X_{4}\cup X_{5})\\\ &=r^{}(X_{2}\cup X_{4})+r^{}(X_{3}\cup X_{4})+r^{}(X_{4}\cup X_{5})+\|X_{1}\cup X_{3}\|\\\ &\qquad+\|X_{1}\cup X_{5}\|+\|X_{2}\cup X_{3}\|+\|X_{2}\cup X_{5}\|+r(\overline{X_{1}\cup X_{3}})\\\ &\qquad+r(\overline{X_{1}\cup X_{5}})+r(\overline{X_{2}\cup X_{3}})+r(\overline{X_{2}\cup X_{5}})-4r(M)\\\ &=r^{}(X_{2}\cup X_{4})+r^{}(X_{3}\cup X_{4})+r^{}(X_{4}\cup X_{5})-4r(M)\\\ &\qquad+9+9+9+9+4+4+4+4\\\ &=r^{}(X_{2}\cup X_{4})+r^{}(X_{3}\cup X_{4})+r^{}(X_{4}\cup X_{5})-4r(M)+52\\\ \end{split}$ We thus must have that $\begin{split}&\qquad r^{}(X_{4})+r^{}(X_{2}\cup X_{3}\cup X_{4})+r^{}(X_{2}\cup X_{4}\cup X_{5})\\\ &>r^{}(X_{2}\cup X_{4})+r^{}(X_{3}\cup X_{4})+r^{}(X_{4}\cup X_{5})+2\end{split}$ Suppose $Z=\varnothing$. Then we have that $2+8+8>8+8+8+2$ which is untrue. Now suppose $Z$ is non-empty. The same argument from Subcase 1a shows we still cannot have a bad family. * Subcase 2.2b: $X_{1}=V_{i}$ $X_{2}=V_{j}\cup V_{k}$ $X_{3}=V_{1}\cup V_{j}$ $X_{4}=V_{2}\cup Z$ where $Z\subseteq E(M)$ $X_{5}=V_{1}\cup V_{k}\cup\\{a\\}$ where $a\in V_{i}$ On the left-hand side, $\|X_{5}\|$ increases by one. On the right-hand side, $\|X_{2}\cup X_{5}\|$ increases by one. We still have no bad family. * Subcase 2.2c: $X_{1}=V_{i}$ $X_{2}=V_{j}\cup V_{k}$ $X_{3}=V_{1}\cup V_{j}$ $X_{4}=V_{2}\cup Z$ where $Z\subseteq E(M)$ $X_{5}=V_{1}\cup V_{k}\cup\\{a,b\\}$ where $a,b\in V_{i}$ Both $\|X_{5}\|$ and $\|X_{2}\cup X_{5}\|$ increase by two, while $r(\overline{X_{2}\cup X_{5}})$ falls by one. We still have no bad family. ###### Case 3. $X_{2}\cap V_{2}\neq\varnothing$ Again consider which term on the left-hand side dual inequality must be $V_{3}\cup V_{4}\cup V_{5}$. As this term does not contain any elements of $V_{2}$ and these elements are contained in $X_{2}$, the term cannot involve $X_{2}$. We have the following representation of remaining possible locations for the necessary circuit-hyperplanes. $X_{1}$$X_{2}$$X_{3}$$X_{4}$$X_{5}$ If $V_{3}\cup V_{4}\cup V_{5}$ is equal to $X_{3}$, $X_{4}$, or $X_{5}$ then any term involving these sets cannot be one of the circuit-hyperplanes $V_{1}\cup V_{3}\cup V_{4}$, $V_{1}\cup X_{4}\cup V_{5}$, $V_{1}\cup V_{3}\cup V_{5}$. This will not leave us with two terms on the right-hand side which could be the circuit-hyperplanes, but three are needed. If it is equal to $X_{1}\cup X_{3}\cup X_{5}$, then only $X_{4}$ can contain elements of $V_{1}$. As $V_{1}$ appears in all of the three circuit-hyperplanes needed on the right-hand side of the inequality, and there are only two terms available using $X_{4}$, we do not have enough terms left for the circuit-hyperplanes. This covers all terms on the left-hand side, and thus $X_{2}\cup V_{2}=\varnothing.$ ###### Case 4. $X_{1}\cap V_{2}\neq\varnothing$ Consider which term on the left-hand side dual inquality must be $V_{3}\cup V_{4}\cup V_{5}$. It cannot be any of the terms involving $X_{1}$ . We have the following representation of the remaining possible locations for the necessary circuit-hyperplanes. $X_{1}$$X_{2}$$X_{3}$$X_{4}$$X_{5}$ If $V_{3}\cup V_{4}\cup V_{5}$ is $X_{1}\cup X_{3}\cup X_{5}$, $X_{2}\cup X_{3}\cup X_{4}$ or $X_{2}\cup X_{4}\cup X_{5}$, only two sets could include elements of $V_{1}$. We will not have enough terms left on the right-hand side to be the circuit-hyperplanes $V_{1}\cup V_{3}\cup V_{4}$, $V_{1}\cup X_{4}\cup V_{5}$, $V_{1}\cup V_{3}\cup V_{5}$. If $V_{3}\cup V_{4}\cup V_{5}=X_{4}$, the circuit-hyerplanes cannot be terms using $X_{4}$, so the only possible edges remaining are $X_{2}\cup X_{3}$ and $X_{2}\cup X_{5}$ – one edge less than is necessary. Thus $V_{3}\cup V_{4}\cup V_{5}$ must be either $X_{3}$ or $X_{5}$. These two possibilities are symmetric, so assume $V_{3}\cup V_{4}\cup V_{5}=X_{3}$. The necessary circuit-hyperplanes now cannot use $X_{3}$ or $X_{1}$. This gives only one possible assignment to the terms in the inequality, as shown in the following diagram. $X_{1}$$X_{2}$$X_{3}$$X_{4}$$X_{5}$$V_{1}\cup V_{j}\cup V_{k}$$V_{1}\cup V_{i}\cup V_{j}$$V_{1}\cup V_{i}\cup V_{k}$ We have that $X_{1}=V_{2}\cup Z$ where $Z$ is a possibly empty subset of the ground set and that $X_{3}=V_{3}\cup V_{4}\cup V_{5}$. $X_{2}$ must contain $V_{i}$ but cannot contain $V_{j}$ or $V_{k}$, $X_{4}$ must contain $V_{k}$ but cannot contain $V_{i}$ or $V_{j}$, and $X_{5}$ must contain $V_{j}$ but cannot contain $V_{i}$ or $V_{k}$. All three of these sets could also contain one of three elements from $V_{1}$, such that whenever we take the union of two of the sets, the union contains $V_{1}$. A set cannot contain two elements of $V_{1}$ as this would mean it was not a flat. Note that if one of the sets $X_{2}$, $X_{4}$, or $X_{5}$ contains no elements of $V_{1}$, both the other two sets must contain all elements of $V_{1}$. If one of the sets contains one element of $V_{1}$, the other two sets must again contain all elements of $V_{1}$. The third possibility is that all three sets contain $V_{1}$. * Subcase 4.1a: $X_{1}=V_{2}\cup Z$ where $Z\subseteq E(M)$ $X_{2}=V_{i}\cup V_{1}$ $X_{3}=V_{3}\cup V_{4}\cup V_{5}$ $X_{4}=V_{k}$ $X_{5}=V_{j}\cup V_{1}$ Note that $\overline{X_{2}\cup X_{3}\cup X_{4}}=\overline{X_{2}\cup X_{4}\cup X_{5}}=\overline{V_{1}\cup V_{i}\cup V_{j}\cup V_{k}}=V_{2}$. $\begin{split}LHS&=r^{}(X_{3})+r^{}(X_{4})+r^{}(X_{5})+r^{}(X_{1}\cup X_{2})+r^{}(X_{1}\cup X_{3}\cup X_{5})\\\ &\qquad+r^{}(X_{2}\cup X_{3}\cup X_{4})+r^{}(X_{2}\cup X_{4}\cup X_{5})\\\ &=r^{}(X_{1}\cup X_{2})+r^{}(X_{1}\cup X_{3}\cup X_{5})+\|X_{3}\|+\|X_{4}\|+\|X_{5}\|\\\ &\qquad+\|X_{2}\cup X_{3}\cup X_{4}\|+\|X_{2}\cup X_{4}\cup X_{5}\|+r(\overline{X_{3}})+r(\overline{X_{4}})\\\ &\qquad+r(\overline{X_{5}})+r(\overline{X_{2}\cup X_{3}\cup X_{4}})+r(\overline{X_{2}\cup X_{4}\cup X_{5}})-5r(M)\\\ &=r^{}(X_{1}\cup X_{2})+r^{}(X_{1}\cup X_{3}\cup X_{5})-5r(M)+9+3+6\\\ &\qquad+12+12+5+5+5+2+2\\\ &=r^{}(X_{1}\cup X_{2})+r^{}(X_{1}\cup X_{3}\cup X_{5})-5r(M)+61\\\ \end{split}$ $\begin{split}RHS&=r^{}(X_{1}\cup X_{3})+r^{}(X_{1}\cup X_{5})+r^{}(X_{2}\cup X_{3})+r^{}(X_{2}\cup X_{4})\\\ &\qquad+r^{}(X_{2}\cup X_{5})+r^{}(X_{3}\cup X_{4})+r^{}(X_{4}\cup X_{5})\\\ &=r^{}(X_{1}\cup X_{3})+r^{}(X_{1}\cup X_{5})+\|X_{2}\cup X_{3}\|+\|X_{2}\cup X_{4}\|+\|X_{2}\cup X_{5}\|\\\ &\qquad+\|X_{3}\cup X_{4}\|+\|X_{4}\cup X_{5}\|+r(\overline{X_{2}\cup X_{3}})+r(\overline{X_{2}\cup X_{4}})\\\ &\qquad+r(\overline{X_{2}\cup X_{5}})+r(\overline{X_{3}\cup X_{4}})+r(\overline{X_{4}\cup X_{5}})-5r(M)\\\ &=r^{}(X_{1}\cup X_{3})+r^{}(X_{1}\cup X_{5})-5r(M)+12+9+9\\\ &\qquad+9+9+2+4+4+5+4\\\ &=r^{}(X_{1}\cup X_{3})+r^{}(X_{1}\cup X_{5})-5r(M)+67\\\ \end{split}$ We thus must have that $\begin{split}&\qquad r^{}(X_{1}\cup X_{2})+r^{}(X_{1}\cup X_{3}\cup X_{5})\\\ &>r^{}(X_{1}\cup X_{3})+r^{}(X_{1}\cup X_{5})+6\end{split}$ in order for this to be a bad family. Suppose $Z=\varnothing$. Then we have that $7+9>9+8+6$ which is untrue. Now suppose $Z$ is non-empty. $X_{1}\cup X_{3}\cup X_{5}$ is equal to the entire ground set, so changing $Z$ has no effect on this term and it is thus still spanning. Now take $r^{}(X_{1}\cup X_{3})=r^{}(V_{2}\cup V_{3}\cup V_{4}\cup V_{5}\cup Z)$. As $\overline{X_{1}\cup X_{3}}=V_{1}-Z$ is coindependent for any choice of $Z$, $X_{1}\cup X_{3}$ is spanning for any choice of $Z$. Now note that $X_{1}\cup X_{2}=V_{1}\cup V_{2}\cup V_{i}\cup Z$ and $\overline{X_{1}\cup X_{2}}=(V_{j}\cup V_{k})-Z$. Note that $r(V_{j}\cup V_{k})=4$. If $Z=\varnothing$, $r(\overline{X_{1}\cup X_{2}})=4$. If $Z$ is equal to one element in $V_{j}\cup V_{k}$, the cardinality of $X_{1}\cup X_{2}$ will increase by one but the rank of $\overline{X_{1}\cup X_{2}}$ will be unchanged. This increases $r^{}(X_{1}\cup X_{2})$ by one. Likewise, if $Z$ is equal to two elements of $V_{j}\cup V_{k}$, $r^{}(X_{1}\cup X_{2})$ increases by two. If $Z$ has cardinality greater than or equal to two, $(V_{j}\cup V_{k})-Z$ will be coindependent, making $X_{1}\cup X_{2}$ spanning for all such $Z$. This means the left-hand side of the inequality can increase by at most two. Finally, $\|X_{1}\cup X_{5}\|=\|V_{1}\cup V_{2}\cup V_{j}\cup Z\|$ and $\overline{X_{1}\cup X_{5}}=(V_{i}\cup V_{k})-Z$. Note that $r(V_{i}\cup V_{k})=5$. If $Z$ is equal to one element in $V_{i}\cup V_{k}$, the cardinality of $X_{1}\cup X_{5}$ will increase by one but the rank of $\overline{X_{1}\cup X_{5}}$ will be unchanged. This increases $r^{}(X_{1}\cup X_{5})$ by one. When $Z$ is equal to one or more elements in $V_{i}\cup V_{k}$, $(V_{i}\cup V_{k})-Z$ is coindependent. This means that $X_{1}\cup X_{5}$ is spanning for all such $Z$, and increasing $Z$ further can have no effect. This means the right-hand side of the inequality can only increase in value by at most one with a non-empty $Z$. The inequality will thus still not be satisfied for any choice of $Z$. Subcase 4.1b: $X_{1}=V_{2}\cup Z$ where $Z\subseteq E(M)$ $X_{2}=V_{i}\cup V_{1}$ $X_{3}=V_{3}\cup V_{4}\cup V_{5}$ $X_{4}=V_{k}\cup V_{1}$ $X_{5}=V_{j}$ Compare this to Subcase 4.1a. $X_{4}$ now contains $V_{1}$ and $X_{5}$ does not, while in 4.1a this was the opposite way around. First consider the left- hand side of the inequality. $r^{}(X_{4})+r^{}(X_{5})$ is unchanged, as is $r^{}(X_{2}\cup X_{4}\cup X_{5})$. As $V_{1}$ is contained in $X_{2}$, $r^{}(X_{2}\cup X_{3}\cup X_{4})$ and $r^{}(X_{2}\cup X_{4}\cup X_{5})$ are also unchanged. The only possible change from 4.1a is thus in $r^{}(X_{1}\cup X_{3}\cup X_{5})$. Note that $X_{1}\cup X_{3}\cup X_{5}=V_{2}\cup V_{i}\cup V_{j}\cup V_{k}\cup Z$ and $\overline{X_{1}\cup X_{3}\cup X_{5}}=V_{1}-Z$. As $V_{1}-Z$ is coindependent for any value of $Z$, $X_{1}\cup X_{3}\cup X_{5}$ is spanning, as in Subcase 4.1a. The left-hand side of the inequality is thus unchanged from Subcase 4.1a. Now consider the right-hand side of the inequality. $r^{}(X_{2}\cup X_{4})+r^{}(X_{2}\cup X_{5})$ remains the same, as does $r^{}(X_{4}\cup X_{5})$. This leaves $r^{}(X_{1}\cup X_{5})$ and $r^{}(X_{3}\cup X_{4})$. First note that $X_{3}\cup X_{4}=V_{1}\cup V_{3}\cup V_{4}\cup V_{5}$ and $\overline{X_{3}\cup X_{4}}=V_{2}$, which is coindependent, and so $X_{3}\cup X_{4}$ is still spanning. Now take $X_{1}\cup X_{5}=V_{2}\cup V_{j}\cup Z$, where $\overline{X_{1}\cup X_{5}}=(V_{1}\cup V_{i}\cup V_{k})-Z$. Suppose $Z=\varnothing$. As $V_{i}\cup V_{k}$ is spanning, $r^{}(X_{1}\cup X_{5})$ will fall by three in comparision to Subcase 4.1a. As $V_{1}\cup V_{i}\cup V_{k}$ is a dependent set of rank $5$, we can remove at most three elements from it without affecting the rank. Thus we can increase $\|X_{1}\cup X_{5}\|$ by three without affecting $r^{}(X_{1}\cup X_{5})$, but, after that, any change in $\|X_{1}\cup X_{5}\|$ is matched by a decrease in $r(\overline{X_{1}\cup X_{5}})$, causing $r^{}(X_{1}\cup X_{5})$ to remain the same. Thus, in comparision to Subcase 4.1a, the inequality is at worst three lower on the right-hand side. As we showed in that subcase that the left-hand side can increase by at most two with a non-empty choice of $Z$, the inequality still cannot be satisfied. * Subcase 4.1c: $X_{1}=V_{2}\cup Z$ where $Z\subseteq E(M)$ $X_{2}=V_{i}$ $X_{3}=V_{3}\cup V_{4}\cup V_{5}$ $X_{4}=V_{k}\cup V_{1}$ $X_{5}=V_{j}\cup V_{1}$ Compared to subcase 4.1a, $X_{5}$ now contains $V_{1}$ and $X_{2}$ does not. Consider the left-hand side of the inequality. Any terms which do not involve $X_{2}$ or $X_{5}$ will be unchanged from Subcase 4.1a. $\|X_{5}\|$ increases by three, while $r(\overline{X_{5}})$ remains the same, causing $r^{}(X_{5})$ to increase by three. $r^{}(X_{2}\cup X_{3}\cup X_{4})$ and $r^{}(X_{2}\cup X_{4}\cup X_{5})$ remain the same, as $V_{1}\subset X_{4}$. Finally, take $X_{1}\cup X_{2}=V_{2}\cup V_{i}\cup Z$. We have that $r^{}(V_{2}\cup V_{i})=5$, which is two less than in Subcase 4.1a. As $\overline{X_{1}\cup X_{2}}=(V_{1}\cup V_{j}\cup V_{k})-Z$ is a dependent set of rank $5$, we can increase $\|X_{1}\cup X_{2}\|$ by three without changing $r(\overline{X_{1}\cup X_{2}})$. If the cardinality of $Z\subseteq V_{1}\cup V_{j}\cup V_{k}$ is any greater, $\overline{X_{1}\cup X_{2}}$ is coindependent, and so $X_{1}\cup X_{2}$ is spanning for all such $Z$. Thus, in total, the left-hand side increases in value by at most one. Now take the right-hand side of the inequality. Note that $X_{2}\cup X_{3}$ is still spanning, as $\overline{X_{2}\cup X_{3}}=V_{2}$ is coindependent, and note that $r^{}(X_{2}\cup X_{4})$ is unchanged. Also, $r^{}(X_{2}\cup X_{5})$ and $r^{}(X_{4}\cup X_{5})$ remain the same, as $V_{1}\subset X_{5}$. The right-hand side thus does not change in value. The inequality still does not hold, and we thus have no bad family for any choice of $Z$. Subcase 4.2a: $X_{1}=V_{2}\cup Z$ where $Z\subseteq E(M)$ $X_{2}=V_{i}\cup V_{1}$ $X_{3}=V_{3}\cup V_{4}\cup V_{5}$ $X_{4}=V_{k}\cup V_{1}$ $X_{5}=V_{j}\cup\\{a\\}$ where $a\in V_{1}$ Note that $\overline{X_{2}\cup X_{3}\cup X_{4}}=\overline{X_{2}\cup X_{4}\cup X_{5}}=\overline{V_{1}\cup V_{i}\cup V_{j}\cup V_{k}}=V_{2}$. $\begin{split}LHS&=r^{}(X_{3})+r^{}(X_{4})+r^{}(X_{5})+r^{}(X_{1}\cup X_{2})+r^{}(X_{1}\cup X_{3}\cup X_{5})\\\ &\qquad+r^{}(X_{2}\cup X_{3}\cup X_{4})+r^{}(X_{2}\cup X_{4}\cup X_{5})\\\ &=r^{}(X_{1}\cup X_{2})+r^{}(X_{1}\cup X_{3}\cup X_{5})+\|X_{3}\|+\|X_{4}\|+\|X_{5}\|\\\ &\qquad+\|X_{2}\cup X_{3}\cup X_{4}\|+\|X_{2}\cup X_{4}\cup X_{5}\|+r(\overline{X_{3}})+r(\overline{X_{4}})\\\ &\qquad+r(\overline{X_{5}})+r(\overline{X_{2}\cup X_{3}\cup X_{4}})+r(\overline{X_{2}\cup X_{4}\cup X_{5}})-5r(M)\\\ &=r^{}(X_{1}\cup X_{2})+r^{}(X_{1}\cup X_{3}\cup X_{5})-5r(M)+9+6+4\\\ &\qquad+12+12+5+5+5+2+2\\\ &=r^{}(X_{1}\cup X_{2})+r^{}(X_{1}\cup X_{3}\cup X_{5})-5r(M)+62\\\ \end{split}$ $\begin{split}RHS&=r^{}(X_{1}\cup X_{3})+r^{}(X_{1}\cup X_{5})+r^{}(X_{2}\cup X_{3})+r^{}(X_{2}\cup X_{4})\\\ &\qquad+r^{}(X_{2}\cup X_{5})+r^{}(X_{3}\cup X_{4})+r^{}(X_{4}\cup X_{5})\\\ &=r^{}(X_{1}\cup X_{3})+r^{}(X_{1}\cup X_{5})+\|X_{2}\cup X_{3}\|+\|X_{2}\cup X_{4}\|+\|X_{2}\cup X_{5}\|\\\ &\qquad+\|X_{3}\cup X_{4}\|+\|X_{4}\cup X_{5}\|+r(\overline{X_{2}\cup X_{3}})+r(\overline{X_{2}\cup X_{4}})\\\ &\qquad+r(\overline{X_{2}\cup X_{5}})+r(\overline{X_{3}\cup X_{4}})+r(\overline{X_{4}\cup X_{5}})-5r(M)\\\ &=r^{}(X_{1}\cup X_{3})+r^{}(X_{1}\cup X_{5})-5r(M)+12+9+9\\\ &\qquad+12+9+2+4+4+2+4\\\ &=r^{}(X_{1}\cup X_{3})+r^{}(X_{1}\cup X_{5})-5r(M)+67\end{split}$ We thus must have that $\begin{split}&\qquad r^{}(X_{1}\cup X_{2})+r^{}(X_{1}\cup X_{3}\cup X_{5})\\\ &>r^{}(X_{1}\cup X_{3})+r^{}(X_{1}\cup X_{5})+5\end{split}$ in order for this to be a bad family. Suppose $Z=\varnothing$. We have that $7+9>9+6+5$ which is untrue. Now suppose $Z\neq\varnothing$. We have shown in Subcase 4.1a that $r^{}(X_{1}\cup X_{3})$ cannot change, and that $r^{}(X_{1}\cup X_{2})$ can increase by at most two. The sets ${X}_{1},\dots,{X}_{3}$ are the same in the current subcase and hence the same facts apply. Note that $X_{1}\cup X_{3}\cup X_{5}$ is equal to the entire ground set, and thus changing $Z$ will have no effect on this term, as in Subcase 4.1a. Finally, note that $X_{1}\cup X_{5}=\\{a\\}\cup V_{2}\cup V_{j}\cup Z$ and $\overline{X_{1}\cup X_{5}}=(\\{b,c\\}\cup V_{i}\cup V_{k})-Z$. As $\\{b,c\\}\cup V_{i}\cup V_{k}$ is a dependent set of rank $5$, we can remove at most three elements from it without affecting the rank. Thus we can increase $\|X_{1}\cup X_{5}\|$ by three without affecting $r^{}(X_{1}\cup X_{5})$, but, after that, $\overline{X_{1}\cup X_{5}}$ is coindependent, causing $r^{}(X_{1}\cup X_{5})$ to be spanning for all such $Z$. Thus the right-hand side of the inequality can increase by at most three when we make $Z$ non-empty. The left-hand side can increase by at most two, and so the inequality still cannot be satisfied. * Subcase 4.2b: $X_{1}=V_{2}\cup Z$ where $Z\subseteq E(M)$ $X_{2}=V_{i}\cup\\{a\\}$ where $a\in V_{1}$ $X_{3}=V_{3}\cup V_{4}\cup V_{5}$ $X_{4}=V_{k}\cup V_{1}$ $X_{5}=V_{j}\cup V_{1}$ Compared to Subcase 4.2a, $X_{2}$ contains two less elements of $V_{1}$ while $X_{5}$ contains two more. Consider the left-hand side of the inequality. Any terms not involving $X_{2}$ or $X_{5}$ are unchanged from Subcase 4.2a. $\|X_{5}\|$ increases by two in comparision to Subcase 4.2a, while $r(\overline{X_{5}})$ remains the same, causing $r^{}(X_{5})$ to increase by two. As $V_{1}\in X_{4}$, $r^{}(X_{2}\cup X_{3}\cup X_{4})$ and $r^{}(X_{2}\cup X_{4}\cup X_{5})$ remain the same. Note that $X_{1}\cup X_{3}\cup X_{5}$ is equal to the entire ground set and thus must be spanning, as in Subcase 4.2a. Now note that $X_{1}\cup X_{2}=\\{a\\}\cup V_{2}\cup V_{i}\cup Z$ and $\overline{X_{1}\cup X_{2}}=(\\{b,c\\}\cup V_{i}\cup V_{k})-Z$. As $\\{b,c\\}\cup V_{i}\cup V_{k}$ is a dependent set of rank $5$, we can remove at most three elements from it without affecting the rank. Thus we can increase $\|X_{1}\cup X_{2}\|$ by three without affecting $r^{}(X_{1}\cup X_{2})$, but, after that, $\overline{X_{1}\cup X_{2}}$ is coindependent, causing $r^{}(X_{1}\cup X_{2})$ to be spanning for all such $Z$.. The left-hand side can thus increase by at most five in comparision to Subcase 4.2a. Take the right-hand side of the inequality. $r^{}(X_{2}\cup X_{5})$ is unchanged. As $V_{1}\subset X_{4}$, $r^{}(X_{2}\cup X_{4})$ and $r^{}(X_{4}\cup X_{5})$ are also unchanged. $\|X_{2}\cup X_{3}\|$ decreases by two, but $r(\overline{X_{2}\cup X_{3}})$ increases by two, meaning that $r^{}(X_{2}\cup X_{3})$ is unchanged. Finally, take $X_{1}\cup X_{5}=V_{1}\cup V_{2}\cup V_{j}\cup Z$, where $\overline{X_{1}\cup X_{5}}=(V_{i}\cup V_{k})-Z$. Suppose that, to begin with, $Z=\varnothing$. In Subcase 4.2a, $r^{}(X_{1}\cup X_{5})=6$. As $\overline{X_{1}\cup X_{5}}$ is still spanning in the current subcase, $r^{}(X_{1}\cup X_{5})$ increases by two due to the increase in $\|X_{1}\cup X_{2}\|$. Now suppose $Z$ is non-empty. As $V_{i}\cup V_{k}$ has rank $5$, we can increase $\|X_{1}\cup X_{5}\|$ by one without decreasing $r(\overline{X_{1}\cup X_{5}})$. For any $Z\subseteq V_{i}\cup V_{k}$ with a cardinality greater than or equal to one, $\overline{X_{1}\cup X_{5}}$ is coindependent. This means $X_{1}\cup X_{5}$ is spanning for all such $Z$, and so $r^{}(X_{1}\cup X_{5})$ can increase by at most one with a non-empty $Z$. Thus, in sum, the right-hand side of the inequality can increase by at most three. We cannot have a bad family, for any choice of $Z$. * Subcase 4.2c: $X_{1}=V_{2}\cup Z$ where $Z\subseteq E(M)$ $X_{2}=V_{i}\cup V_{1}$ $X_{3}=V_{3}\cup V_{4}\cup V_{5}$ $X_{4}=V_{k}\cup\\{a\\}$ where $a\in V_{1}$ $X_{5}=V_{j}\cup V_{1}$ Compared to Subcase 4.2a, $X_{4}$ contains two less elements of $V_{1}$ while $X_{5}$ contains two more. On the left-hand side of the inequality, there is no change in value compared to Subcase 4.2a. $r^{}(X_{4})+r^{}(X_{5})$ remains the same, as does $r^{}(X_{2}\cup X_{4}\cup X_{5})$. In Subcase 4.2a, $X_{1}\cup X_{3}\cup X_{5}$ was spanning and the same is true after adding additional elements to it. As $V_{1}\subset X_{2}$, $r^{}(X_{2}\cup X_{4}\cup X_{4})$ is also unchanged. Now take the right-hand side of the inequality. $r^{}(X_{4}\cup X_{5})$ is unchanged, as are $r^{}(X_{2}\cup X_{4})$ and $r^{}(X_{2}\cup X_{5})$ since $V_{1}\subset X_{2}$. As in Subcase 4.2b, $r^{}(X_{1}\cup X_{5})$ can increase by at most three. Finally, take $X_{3}\cup X_{4}=\\{a\\}\cup V_{3}\cup V_{4}\cup V_{5}$. This set is spanning, as in Subcase 4.2a. We have that, in comparision to Subcase 4.2a, the left-hand side remains the same while the right-hand side increases by at most three. We still have no bad family, for any choice of $Z$. * Subcase 4.3: $X_{1}=V_{2}\cup Z$ where $Z\subseteq E(M)$ $X_{2}=V_{i}\cup V_{1}$ $X_{3}=V_{3}\cup V_{4}\cup V_{5}$ $X_{4}=V_{k}\cup V_{1}$ $X_{5}=V_{j}\cup V_{1}$ Note that $\overline{X_{2}\cup X_{3}\cup X_{4}}=\overline{X_{2}\cup X_{4}\cup X_{5}}=\overline{V_{1}\cup V_{i}\cup V_{j}\cup V_{k}}=V_{2}$. $\begin{split}LHS&=r^{}(X_{3})+r^{}(X_{4})+r^{}(X_{5})+r^{}(X_{1}\cup X_{2})+r^{}(X_{1}\cup X_{3}\cup X_{5})\\\ &\qquad+r^{}(X_{2}\cup X_{3}\cup X_{4})+r^{}(X_{2}\cup X_{4}\cup X_{5})\\\ &=r^{}(X_{1}\cup X_{2})+r^{}(X_{1}\cup X_{3}\cup X_{5})+\|X_{3}\|+\|X_{4}\|+\|X_{5}\|\\\ &\qquad+\|X_{2}\cup X_{3}\cup X_{4}\|+\|X_{2}\cup X_{4}\cup X_{5}\|+r(\overline{X_{3}})+r(\overline{X_{4}})\\\ &\qquad+r(\overline{X_{5}})+r(\overline{X_{2}\cup X_{3}\cup X_{4}})+r(\overline{X_{2}\cup X_{4}\cup X_{5}})-5r(M)\\\ &=r^{}(X_{1}\cup X_{2})+r^{}(X_{1}\cup X_{3}\cup X_{5})-5r(M)+9+6+6\\\ &\qquad+12+12+5+5+5+2+2\\\ &=r^{}(X_{1}\cup X_{2})+r^{}(X_{1}\cup X_{3}\cup X_{5})-5r(M)+64\\\ \end{split}$ $\begin{split}RHS&=r^{}(X_{1}\cup X_{3})+r^{}(X_{1}\cup X_{5})+r^{}(X_{2}\cup X_{3})+r^{}(X_{2}\cup X_{4})\\\ &\qquad+r^{}(X_{2}\cup X_{5})+r^{}(X_{3}\cup X_{4})+r^{}(X_{4}\cup X_{5})\\\ &=r^{}(X_{1}\cup X_{3})+r^{}(X_{1}\cup X_{5})+\|X_{2}\cup X_{3}\|+\|X_{2}\cup X_{4}\|+\|X_{2}\cup X_{5}\|\\\ &\qquad+\|X_{3}\cup X_{4}\|+\|X_{4}\cup X_{5}\|+r(\overline{X_{2}\cup X_{3}})+r(\overline{X_{2}\cup X_{4}})\\\ &\qquad+r(\overline{X_{2}\cup X_{5}})+r(\overline{X_{3}\cup X_{4}})+r(\overline{X_{4}\cup X_{5}})-5r(M)\\\ &=r^{}(X_{1}\cup X_{3})+r^{}(X_{1}\cup X_{5})-5r(M)+12+12+9\\\ &\qquad+12+9+2+4+4+2+4\\\ &=r^{}(X_{1}\cup X_{3})+r^{}(X_{1}\cup X_{5})-5r(M)+70\end{split}$ We thus must have that $\begin{split}&\qquad r^{}(X_{1}\cup X_{2})+r^{}(X_{1}\cup X_{3}\cup X_{5})\\\ &>r^{}(X_{1}\cup X_{3})+r^{}(X_{1}\cup X_{5})+6\end{split}$ in order for this to be a bad family. Suppose $Z=\varnothing$. Then we have that $7+9>9+8+6$ which is untrue. We have shown in Subcase 4.1a that $r^{}(X_{1}\cup X_{3})$ cannot change when $Z$ is non-empty, and that $r^{}(X_{1}\cup X_{2})$ can increase by at most two. The sets ${X}_{1},\dots,{X}_{3}$ are the same in the current subcase and hence the same facts apply. Note that $X_{1}\cup X_{3}\cup X_{5}$ is equal to the entire ground set, and thus changing $Z$ will have no effect on this term. Finally, $X_{1}\cup X_{5}=V_{1}\cup V_{2}\cup V_{j}\cup Z$ and $\overline{X_{1}\cup X_{5}}=(V_{i}\cup V_{k})-Z$. As $r(V_{i}\cup V_{k})=5$, we can remove one element from it without decreasing the rank. This increases $\|X_{1}\cup X_{5}\|$ by one and therefore $r^{}(X_{1}\cup X_{5})$. For any $Z\subseteq V_{i}\cup V_{k}$ with cardinality one or higher, $\overline{X_{1}\cup X_{5}}$ is coindependent, so $X_{1}\cup X_{5}$ is spanning for all such $Z$. We thus have that the left-hand side can increase by at most two, while the right-hand side can increase by at most one, giving us no possible bad family. ∎ ## Chapter 6 A Complexity Theorem As yet no method of testing whether a matroid satisfies a Kinser equality has presented itself other than brute force. This leads to the question of whether it is possible to do this in polynomial time. Given the increasing number of terms in each inequality and the lack of bounds on a matroid’s possible ground set, this is an important question in terms of the results it is feasible to get – in particular, whether it would be feasible to construct a matroid similar to that used in Theorem 4.4 and test whether it satisfies inequality $n$ for $n\geq 5$, in order to show that the higher Kinser classes are not dual closed. We give a proof that it would in fact be impossible to test these in polynomial time. An _oracle machine_ consists of a Turing machine with a black box attached, which is referred to as the oracle. Given some question about a particular matroid, inputs are fed to the oracle, which then gives an output answering the question. The time the machine takes to produce an output is given as a function of the number on inputs necessary to answer the question. We wish to know the time an oracle machine would take to answer whether a matroid satisfies Kinser inequality $n$. ###### Definition 6.1. Let $r\geq 4$ and take two distinct $r$-element sets $A=\\{a_{1},\ldots,a_{r}\\},B=\\{b_{1},\ldots,b_{r}\\}$. We will define the circuit-hyperplanes of the _rank- $r$ binary spike_, denoted by $Z_{r}$, on ground set $E=A\cup B$ by its set of circuits. First, define the set of circuit-hyperplanes to be the subsets $\\{z_{1},\ldots,z_{r}\\}$, where $z_{i}\in\\{a_{i},b_{i}\\}$, such that $\|\\{z_{1},\ldots,z_{r}\\}\cap\\{b_{1},\ldots,b_{r}\\}\|$ is even. The non- spanning circuits of $Z_{r}$ consist of the circuit-hyperplanes as defined above and subsets of $E$ of the form $\\{a_{i},b_{i},a_{k},b_{k}\\}$. $Z_{r}$ can be represented by the following matrix: ###### Lemma 6.2. Take an arbitrary rank $r$ binary spike where $r$ is even. If we relax any circuit-hyperplane other than $A$, the resulting matroid violates the Ingleton condition. ###### Proof. Take a binary spike $Z_{r}$. Take one of the circuit-hyperplanes of $Z_{r}$ and call it $Z$, where $Z$ is chosen so that $Z\cap A$ and $Z\cap B$ are non- empty. Define $I\subseteq\\{1,\ldots,r\\}$ such that $i\in I$ if and only if $a_{i}\in Z$, and define $J\subseteq\\{1,\ldots,r\\}$ such that $j\in J$ if and only if $b_{i}\in Z$. Now let $X_{1}=\\{a_{i}\ \|\ i\in I\\}$, $X_{2}=\\{b_{j}\ \|\ j\in J\\}$, $X_{3}=\\{b_{i}\ \|\ i\in I\\}$, and $X_{4}=\\{a_{j}\ \|\ j\in J\\}$. In other words, $X_{1}$ and $X_{2}$ consist of the elements in the circuit-hyperplane $Z$ contained in $A$ and $B$ respectively, while $X_{3}$ and $X_{4}$ consist of all the remaining elements in $B$ and $A$. Note that $X_{2}$ contains an even number of elements from $B$ and that $\|X_{1}\cup X_{2}\|=r$, making it a circuit-hyperplane. Relax $X_{1}\cup X_{2}$ to get the matroid $Z^{-}_{r}$ and evaluate the Ingleton condition: $\displaystyle r(X_{3})+r(X_{4})+r(X_{1}\cup X_{2})+r(X_{1}\cup X_{3}\cup X_{4})+r(X_{2}\cup X_{3}\cup X_{4})$ $\displaystyle\leq r(X_{1}\cup X_{3})+r(X_{1}\cup X_{4})+r(X_{2}\cup X_{3})+r(X_{2}\cup X_{4})+r(X_{3}\cup X_{4})$ The set of non-spanning circuits of $Z_{r}$ consists of the circuit- hyperplanes as defined above and subsets of $E$ of the form $\\{a_{i},b_{i},a_{k},b_{k}\\}$. $X_{3}$ and $X_{4}$ do not fit into this category and are thus independent. The ground set of $Z_{r}^{-}$ has size $2r$ and $Z$ has size $r$, so $X_{3}$ and $X_{4}$ have ranks which sum to $r$. Recall that a leg is a subset $\\{a_{k},b_{k}\\}$ of the ground set for some $k$. A proper subset of the legs has rank one greater than the number of legs. $X_{1}\cup X_{3}=\\{a_{i}\cup b_{i}\ \|\ i\in I\\}$ and $X_{2}\cup X_{4}=\\{a_{j}\cup b_{j}\ \|\ j\in J\\}$ are both collections of legs, the former having $\|X_{1}\|=\|X_{3}\|$ legs and the latter having $\|X_{2}\|=\|X_{4}\|$ legs. Thus $r(X_{1}\cup X_{3})=\|X_{1}\|+1$ and $r(X_{2}\cup X_{4})=\|X_{2}\|+1$. As $Z$ is a circuit-hyperplane, $\|X_{2}\|$ is even by definition. This means $X_{2}\cup X_{3}$ is a circuit-hyperplane. Recall that $X_{2}$ and $X_{3}$ partition $B$. As $r$ is even, $\|X_{3}\|=r-\|X_{2}\|$ must be even as well. Thus $X_{3}\cup X_{4}$ is a circuit-hyperplane, as is $X_{1}\cup X_{4}=A$. tNow consider $X_{1}\cup X_{3}\cup X_{4}$. This set properly contains the circuit- hyperplane $X_{3}\cup X_{4}$, and all the sets $X_{i}$ are non-empty. Thus $X_{1}\cup X_{3}\cup X_{4}$ is spanning. $X_{2}\cup X_{3}\cup X_{4}$ also properly contains a circuit-hyperplane, and so is also spanning. Using these calculations we can now evaluate the inequality. $4r\leq(\|X_{1}\|+1)+(r-1)+(r-1)+(\|X_{2}\|+1)+(r-1)$ Since $\|X_{1}\|+\|X_{2}\|=r$, this simplifies to $4r\leq 4r-1$ which is untrue. ${X}_{1},\dots,{X}_{4}$ therefore form a bad family. ∎ ###### Theorem 6.3. Let $n\geq 4$. There does not exist a polynomial time oracle machine testing Kinser inequality $n$ or its dual. ###### Proof. As proven above, each binary spike $Z_{r}$ of even rank is representable, therefore satisfies the inequality, while its relaxation $Z^{-}_{r}$ does not. This means that in order to test whether a matroid satisfies Kinser inequality $n$ or its dual, the oracle machine must distinguish between each $Z_{r}$ and $Z^{-}_{r}$. Recall $Z^{-}_{r}$ can be constructed by relaxing any circuit- hyperplane, which consists of an $r$ element subset $\\{z_{1},...,z_{r}\\}$ of the ground set $A\cup B$ where $z_{i}\in\\{a_{i},b_{i}\\}$ and $\|\\{z_{1},...,z_{r}\\}\cap\\{b_{1},...,b_{r}\\}\|$ is even. Suppose the oracle did not check the rank of the relaxed circuit-hyperplane. This would mean it yields the same result as before the circuit-hyperplane was relaxed, as that is the only subset which changes in rank. Thus the oracle must check the rank of each possible circuit-hyperplane. There are $2^{r}$ $r$-element sets using one element from each leg, and half of these contain an even number of elements from $\\{b_{1},...,b_{r}\\}$. The algorithm hence takes at least $2^{r-1}=2^{\frac{E}{2}-1}$ checks, and therefore is exponential in the size of the ground set. As the class of spikes is dual-closed, testing whether the dual of a matroid satisfies Kinser inequality $n$ is also exponential in the size of the ground set. ∎ ## Chapter 7 Excluded minors The following theorem was proved by Mayhew, Newman, and Whittle in 2008 [5], settling a conjecture by J. Geelen. ###### Theorem 7.1. For any infinite field $\mathbb{K}$ and any matroid $N$ representable over $\mathbb{K}$, there is an excluded minor for $\mathbb{K}$-representability that has $N$ as a minor. The proof of Theorem 7.1 constructed an excluded minor which contained $N$ and which was not contained in $\mathcal{K}_{4}$, and thus was not contained inside any Kinser class. In this chapter we will give a strengthening of this result, which states that the excluded minors can actually be contained inside any layer of the hierarchy. ###### Lemma 7.2. Let $r\geq 3$ be an integer. Let $P$ be the projective geometry PG$(r-1,\mathcal{K})$, where $\mathcal{K}$ is an infinite field, and let ${S}_{1},\dots,{S}_{t}$ be a finite collection of proper subspaces of $P$. If $S$ is a subspace of $P$ that is not contained in any of ${S}_{1},\dots,{S}_{t}$, then $S$ is not contained in $S_{1}\cup\ldots\cup S_{t}$. This is Proposition 4.2 of [4] and we will make frequent reference to it throughout this chapter. Whenever we add points freely to a subspace, it is justified by this result. ###### Theorem 7.3. Let $n\geq 5$ be an integer. Let $\mathbb{K}$ be a infinite field and let $M$ be a $\mathbb{K}$-representable matroid. Then $M$ is contained in an excluded minor for $\mathcal{K}_{n+1}$ which is in $\mathcal{K}_{n}$. ###### Proof. As we can add coloops as desired, we can asssume $M$ has rank $r$ where $r\geq n$. By [5, Lemma 2.2], we can assume that $M$ is partitioned into two independent hyperplanes. Call these $H_{0}$ and $H_{n-1}$. Let $\mathbb{K}$ be an infinite field. Imbed $M$ in the projective geometry $P=PG(r,\mathbb{K})$, so that the elements in the ground set of $M$ are identified with points in $P$. Note that this geometry has rank $r+1$, so $M$ spans a hyperplane of $P$. If $X$ is any set of points in $P$, let $\langle X\rangle$ denote the closure of $X$ in $P$. We will now extend $M$ to get $N$, an excluded minor for $\mathcal{K}_{n+1}$. First we will choose points which will not be added to the ground set of $M$, but will enable us to freely place points within $M$. Begin by arbitrarily choosing $x_{0}$ in $P-\langle E(M)\rangle$. Next freely place $x_{n-1}$ with respect to $\langle H_{0}\rangle$ – i.e., choose $x_{n-1}$ in $\langle H_{0}\rangle$ so that $x_{n-1}$ is not spanned by any subset of $E(M)\cup\\{x_{0}\\}$ that doesn’t span $H_{0}$. We are able to do this using [4, Proposition 4.2]. Choose $x_{1}$ in $\langle H_{n-1}\rangle$ so that it is not spanned by any subset of $E(M)\cup\\{x_{0},x_{n-1}\\}$ that doesn’t span $H_{n-1}$. Now choose $x_{2}$ in $\langle H_{0}\rangle\cap\langle H_{n-1}\rangle$ so that it is not spanned by any supset of $E(M)\cup\\{x_{0},x_{1},x_{n-1}\\}$ unless that subset spans $\langle H_{0}\rangle\cap\langle H_{n-1}\rangle$. Choose $x_{3}$ in $\langle H_{0}\rangle\cap\langle H_{n-1}\rangle$ so that it is not spanned by any subset of $E(M)\cup\\{x_{0},x_{1},x_{2},x_{n-1}\\}$ unless that subset spans $\langle H_{0}\rangle\cap\langle H_{n-1}\rangle$. Continue in this way until $x_{0},x_{1},\ldots,x_{n-1}$ have been chosen. Now choose $r-n+1$ points in the same space, $\langle H_{0}\rangle\cap\langle H_{n-1}\rangle$ using the same technique. Call this set of points $X$, and note that $X\cup\\{x_{2},\ldots,x_{n-2}\\}$ is an independent set that spans $\langle H_{0}\rangle\cap\langle H_{n-1}\rangle$. The points chosen so far, $X\cup\\{x_{0},\ldots,x_{n-1}\\}$, will act as guides for adding points to the ground set of $N$. Add a point $e_{1}$ to $\langle(X\cup\\{x_{0},\ldots,x_{n-1}\\})-\\{x_{1},x_{2}\\}\rangle$ so that it is not spanned by any subset of $E(M)\cup X\cup\\{x_{0},\ldots,x_{n-1}\\}$ unless that subset spans $(X\cup\\{x_{0},\ldots,x_{n-1}\\}-\\{x_{1},x_{2}\\}$. Now add another point to the same space so that it is not spanned by any subset of $E(M)\cup X\cup\\{x_{0},\ldots,x_{n-1},e_{1}\\}$ unless that subset spans $X\cup\\{x_{0},\ldots,x_{n-1}\\}-\\{x_{1},x_{2}\\}$. Contine in this way until $r-1$ points have been added to the space. Call this set of $r-1$ points $H_{1}$. Follow this same method to create $r-1$ points to form the set $H_{2}$, this time adding the points to the space $\langle(X\cup\\{x_{0},\ldots,x_{n-1}\\})-\\{x_{2},x_{3}\\}\rangle$. In this way we create $H_{1},\ldots,H_{n-2}$ – i.e. for $i\in\\{1,\ldots,n-2\\}$, create $H_{i}$ by freely placing $r-1$ points in the space $\langle(X\cup\\{x_{0},\ldots,x_{n-1}\\})-\\{x_{i},x_{i+1}\\}\rangle$. Note that the points of $H_{0}$ are in $\langle(X\cup\\{x_{0},\ldots,x_{n-1}\\})-\\{x_{0},x_{1}\\}\rangle$ and the points of $H_{n-1}$ are in $\langle(X\cup\\{x_{0},\ldots,x_{n-1}\\})-\\{x_{n-1},x_{0}\\}\rangle$. Finally, add a point $p$ freely to $\langle X\rangle$, then add another point $p^{\prime}$ freely to $P$. Freely place a point $e$ on the line spanned by $p$ and $p^{\prime}$, $\langle\\{p,p^{\prime}\\}\rangle$, then do the same with another point $f$. Let $N$ be the matroid consisting of the points $H_{0}\cup\ldots H_{n-1}\cup\\{e,f\\}$. ###### Lemma 7.4. $N$ is $\mathbb{K}$-representable. This lemma is true by construction. ###### Lemma 7.5. $H_{i}\cup\\{e,f\\}$ is a circuit-hyperplane of $N$ for every $i\in\\{0,\ldots,n-1\\}$. ###### Proof. $H_{i}\cup\\{e,f\\}$ has $r+1$ points, and by construction is contained in $\langle(X\cup\\{p^{\prime},x_{0},\ldots,x_{n-1}\\})-\\{x_{i},x_{i+1}\\}\rangle$. This is a rank $r$ space and so $H_{i}\cup\\{e,f\\}$ must be dependent. Suppose $H_{i}$ is dependent for some $i$. Then at some point in constructing $N$, we would have added a point $g$ to already chosen elements of $H_{i}$ so that the point was contained in $cl(H_{i}-g)$, i.e. contained in $\langle(X\cup\\{x_{0},\ldots,x_{n-1}\\})-\\{x_{i},x_{i+1}\\}\rangle$. This contradicts every point of $H_{i}$ being freely placed in the space $\langle(X\cup\\{x_{o},\ldots,x_{n-1}\\})-\\{x_{i},x_{i+1}\\}\rangle$. Thus $H_{i}$ is independent. Now suppose $H_{i}\cup\\{e\\}$ is dependent. Then $e\in cl(H_{i})$. That is, $e\in\langle(X\cup\\{x_{0},\ldots,x_{n-1}\\}-\\{x_{i},x_{i+1}\\}\rangle$. This contradicts $e$ being a point on the line spanned by $p$ and $p^{\prime}$. Likewise, $H_{i}\cup\\{f\\}$ is also independent. We have shown that every subset of $H_{i}\cup\\{e,f\\}$ is independent, meaning that $H_{i}\cup\\{e,f\\}$ must be a circuit. Now suppose $H_{i}\cup\\{e,f\\}$ is not a flat. There must be some element $g\in E(N)-(H_{i}\cup\\{e,f\\})$ such that $r(H_{i}\cup\\{e,f,g\\})=r(H_{i}\cup\\{e,f\\})$ – that is, $g\in cl(H_{i}\cup\\{e,f\\})$. This implies $g\in cl(H_{i}\cup\\{e\\})$. Let $g\in H_{j}$ for some $j$. Assume $g\in cl(H_{i})$. Then we have that $g\in\langle(X\cup\\{x_{0},\ldots,x_{n-1}\\})-\\{x_{i},_{i+1}\\}\rangle$. This is a contradiction, as $(X\cup\\{x_{0},\ldots,x_{n-1}\\})-\\{x_{i},x_{i+1}\\}$ does not span $(X\cup\\{x_{0},\ldots,x_{n-1}\\})-\\{x_{j},x_{j+1}\\}$. Now suppose $g\notin cl(H_{i})$. By the fourth closure axiom, $e\in cl(H_{i}\cup\\{g\\})$. Recall that $p\in cl(H_{i})$. As $p$ and $e$ form a line, we must have that $\langle\\{p,e\\}\rangle\subseteq cl(H_{i}\cup\\{g\\})$. Thus $p^{\prime}\in cl(H_{i}\cup g)$. As $p^{\prime}$ was added freely to the projective geometry, the only way this is possible is if $H_{i}\cup g$ is spanning, which is a contradiction. ∎ ###### Lemma 7.6. Relaxing $H_{0}\cup\\{e,f\\}$ produces a matroid not in $\mathbb{K}_{n+1}$. ###### Proof. We will show that $(X_{1},X_{2},\ldots,X_{n+1})=(H_{0},\\{e,f\\},H_{1},\ldots,H_{n-1})$ violates inequality $n+1$, i.e. $\displaystyle\sum_{i=3}^{n+1}r(X_{i})+r(X_{1}\cup X_{2})+r(X_{1}\cup X_{3}\cup X_{n+1})+\sum_{i=4}^{n+1}r(X_{2}\cup X_{i-1}\cup X_{i})$ $\displaystyle>r(X_{1}\cup X_{3})+r(X_{1}\cup X_{n+1})+\sum_{i=3}^{n+1}r(X_{2}\cup X_{i})+\sum_{i=4}^{n+1}r(X_{i-1}\cup X_{i})$ $X_{i}$ is independent by construction, as proven in the previous result, with rank $r-1$. Recall $X_{i}\cup X_{2}$ is a circuit-hyperplane for all $i$ as proven in the previous lemma. Note that $X_{i}\subseteq\langle X\cup\\{x_{0},\ldots,x_{n-1}\\}-\\{x_{i},x_{i+1}\\}\rangle$ for all $i\neq 2$, and that the points were chosen so as to make it an independent set of rank $r-1$. Take two consecutive sets $X_{i},X_{j}$, where $i,j\neq 2$. $\begin{split}r(&X_{i}\cup X_{j})\\\ &\leq\quad r(\langle X\cup\\{x_{0},\ldots,x_{n-1}\\}-\\{x_{i},x_{i+1}\\}\rangle\cup\langle X\cup\\{x_{0},\ldots,x_{n-1}\\}-\\{x_{i+1},x_{j}\\}\rangle)\\\ &=\quad r(\langle X\cup\\{x_{0},\ldots,x_{n-1}\\}-\\{x_{i},x_{i+1}\\}\rangle)+r(\langle X\cup\\{x_{0},\ldots,x_{n-1}\\}-\\{x_{i+1},x_{j}\\}\rangle)\\\ &\hskip 14.22636pt-r(\langle X\cup\\{x_{0},\ldots,x_{n-1}\\}-\\{x_{i},x_{i+1}\\}\rangle\cap\langle X\cup\\{x_{0},\ldots,x_{n-1}\\}-\\{x_{i+1},x_{j}\\}\rangle\\\ &=\quad(r-1)+(r-1)-(r-2)\\\ &=\quad r\end{split}$ Now suppose $X_{i},X_{j}$ are inconsecutive. The intersection term will now have rank $r-3$, one less than before, so $r(X_{i}\cup X_{j})=r+1$. Note that these two calculations imply the rank of the union of any three $X_{i}$ must be $r+1$. We can now show that the inequality above holds: $\sum_{i=3}^{n+1}(r-1)+(r+1)+(r+1)+\sum_{i=4}^{n+1}(r+1)>r+r+\sum_{i=3}^{n+1}r+\sum_{i=4}^{n+1}r$ Therefore ${X}_{1},\dots,{X}_{n+1}$ is a bad family if and only if $\begin{split}&(n+1-2)(r-1)+2(r+1)+(n+1-3)(r+1)\\\ &\hskip 14.22636pt>2r+(n+1-2)r+(n+1-3)r\end{split}$ which is true if and only if $\begin{array}[]{ccrcl}&&(n-1)(r-1)+n(r+1)&>&(2n-1)r\\\ \Leftrightarrow&&(2n-1)r+1&>&(2n-1)r\\\ \end{array}$ and this completes the proof. ∎ Call this relaxation $N^{\prime}$. ###### Lemma 7.7. Relaxing $H_{i}\cup\\{e,f\\}$ in $N$ creates a $\mathbb{K}$-representable matroid. ###### Proof. Construct $L$ in exactly the same way as $N$, up until the point where $p$ and $p^{\prime}$ are added. Instead of adding $p$ to $\langle X\rangle$, add it freely to $\langle X\cup\\{x_{1},\ldots,x_{i}\\}\rangle$. Now add $p^{\prime}$ freely to $\langle X\cup\\{x_{i+1},\ldots,x_{n-1},x_{0}\\}\rangle$. Then add $e$ and $f$ freely to the line $\langle\\{p,p^{\prime}\\}\rangle$ as before. This matroid $L$ is $\mathbb{K}$-representable by construction. We will show that it is the same as the matroid obtained from $N^{\prime}$ by relaxing $H_{i}\cup\\{e,f\\}$, referred to as $N^{\prime\prime}$. Note that by [6, Proposition 3.3.5], we have that $N\backslash e\backslash f=N^{\prime}\backslash e\backslash f=N^{\prime\prime}\backslash e\backslash f$, and also that $N\backslash e\backslash f=L\backslash e\backslash f$ by construction. If $Z\subseteq E(N\backslash e)$ spans $f$, then, as we chose $f$ to be freely placed on the line spanned by $p$ and $p^{\prime}$, $Z$ must span $\langle\\{p,p^{\prime}\\}\rangle$. This implies that $p^{\prime}\in\langle Z\rangle$. As $p^{\prime}$ was freely placed in $E(N\backslash e)$, this implies $Z$ is spanning. Thus $N\backslash e$ is a free extension of $N\backslash e\backslash f$ by the element $f$. Now suppose $Z\subseteq E(L\backslash e)$ spans $f$. Then again we have that $\langle\\{p,p^{\prime}\\}\rangle\subseteq\langle Z\rangle$. As this gives that $p\in\langle Z\rangle$, we have that $X\cup\\{x_{1},\ldots,x_{i}\\}\subseteq\langle Z\rangle$ by the way $p$ was chosen in the construction of $L$. As $p^{\prime}\in\langle Z\rangle$, we have that $X\cup\\{x_{i+1},\ldots,x_{n-1},x_{0}\\}\subseteq\langle Z\rangle$. Putting these together gives $X\cup\\{x_{0},\ldots,x_{n-1}\\}\subseteq\langle Z\rangle$. As $X\cup\\{x_{0},\ldots,x_{n-1}\\}$ was chosen so as to be a basis of $L$, $Z$ must be spanning. This tells us that $f$ is freely placed in $L\backslash e$, so $L\backslash e$ is a free extension of $L\backslash e\backslash f$ by the element $f$. As $L\backslash e\backslash f=N\backslash e\backslash f$, we have that $L\backslash e=N\backslash e$. Note also that $N\backslash e=N^{\prime}\backslash e=N^{\prime\prime}\backslash e$, so $L\backslash e=N^{\prime\prime}\backslash e$. The same argument shows that $L\backslash f=N^{\prime\prime}\backslash f$. Suppose $L\neq N^{\prime\prime}$. There must exist a set $A$ which is a non- spanning circuit in $N^{\prime\prime}$ and independent in $L$ or vice versa. The previous results tell us that $e,f\in A$, as otherwise $A$ would have the same rank in both matroids. Suppose $A$ is a non-spanning circuit in $L$. Say that the points in $E(L)-\\{e,f\\}$ were added in the order $e_{1},....,e_{t}$. Let $e_{j}$ be the largest element of $A$ according to this ordering, and let $e_{j}\in H_{k}$. As $e_{j}$ was freely placed, $A$ must span $\langle(X\cup\\{x_{0},\ldots,x_{n-1}\\})-\\{x_{k},x_{k+1}\\}\rangle$. This means that $(A-H_{k})\cup(X\cup\\{x_{0},...,x_{n-1}\\})-\\{x_{k},x_{k+1}\\})$ spans the same set as $A$. Suppose the last element added to $A$ before those in $H_{k}$ is $e_{l}\in H_{j}$ where $j<k$. Then $(A-H_{k})\cup(X\cup\\{x_{0},...,x_{n-1}\\})-\\{x_{k},x_{k+1}\\}$ spans an element from $H_{j}$, and by construction, as every element in the set above was added before $H_{j}$, we see that this set spans $(X\cup\\{x_{0},...,x_{n-1}\\})-\\{x_{j},x_{j+1}\\}$. Thus $A$ spans both $\langle(X\cup\\{x_{0},\ldots,x_{n-1}\\})-\\{x_{k},x_{k+1}\\}\rangle$ and $\langle(X\cup\\{x_{0},\ldots,x_{n-1}\\})-\\{x_{j},x_{j+1}\\}\rangle$. As shown in the previous lemma, if $H_{j}$ and $H_{k}$ are inconsecutive, $A$ will have rank $r+1$ and be spanning. $H_{j}$ and $H_{k}$ thus must be consecutive in order for $A$ to be non-spanning. Take a dependent subset of $H_{j}\cup H_{k}$ in $L$. As this subset does not include $e$ nor $f$, it has the same rank in $L\backslash e$. Likewise, the rank of the subset in $N^{\prime\prime}$ has the same rank in $N^{\prime\prime}\backslash e$. As we have already shown $L\backslash e=N^{\prime\prime}\backslash e$, we have that any dependent subset of $H_{j}\cup H_{k}$ in $L$ is also dependent in $N^{\prime\prime}$. This contradicts the assumption that $A$ is independent in $N^{\prime\prime}$. If there is no point contained in a set $H_{j}$ where $j<k$, in order for $A$ to be a circuit, $A$ must be equal to $H_{k}\cup\\{e,f\\}$, where $k\notin\\{0,i\\}$, as any subset of this is independent in $L$, as proved in the next lemma. ###### Lemma 7.8. $H_{k}\cup\\{e,f\\}$ is a circuit of $L$ for all $k\in\\{1,\ldots,i-1,i+1,\ldots,n\\}$. ###### Proof. Consider $H_{k}\cup\\{e,f\\}$. Recall that $p$ was added freely to $\langle X\cup\\{x_{1},\ldots,x_{i}\\}\rangle$ while $p^{\prime}$ was added freely to $\langle X\cup\\{x_{i+1},\ldots,x_{n-1},x_{0}\\}\rangle$. $H_{k}$ is contained in $\langle(X\cup\\{x_{0},\ldots,x_{n-1}\\}-\\{x_{k},x_{k+1}\\}\rangle$. When $i\leq k$, this subspace spans $\langle X\cup\\{x_{1},\ldots,x_{i}\\}\rangle$ and so spans $p$. When $i\geq k$, this subspace spans $\langle X\cup\\{x_{i+1},\ldots,x_{n-1},x_{0}\\}\rangle$ and so spans $p^{\prime}$. As $e$ and $f$ were freely placed on the line spanned by $p$ and $p^{\prime}$, in either case we have that $H_{k}\cup\\{e,f\\}\in cl(H_{k}\cup\\{e\\})$ and so $H_{k}\cup\\{e,f\\}$ is dependent. Suppose $H_{k}$ is dependent for some $k$. Then at some point in constructing $L$, we would have added a point $g$ to already chosen elements of $H_{k}$ so that the point was contained in $cl(H_{k}-g)$, i.e. contained in $\langle(X\cup\\{x_{0},\ldots,x_{n-1}\\})-\\{x_{k},x_{k+1}\\}\rangle$. This contradicts each of the $r-1$ points of $H_{k}$ being freely placed in the rank $r-1$ space $\langle(X\cup\\{x_{o},\ldots,x_{n-1}\\})-\\{x_{k},x_{k+1}\\}\rangle$. Thus $H_{k}$ is independent. Now suppose $H_{k}\cup\\{e\\}$ is dependent. Then $e\in cl(H_{k})$ – that is, $e\in\langle(X\cup\\{x_{0},\ldots,x_{n-1}\\}-\\{x_{k},x_{k+1}\\}\rangle$. However, $e$ was freely placed on the line spanned by $p$ and $p^{\prime}$. Thus if $H_{k}$ spans $e$, it must span this line. As the line itself is free in the matroid, for this to happen, $H_{k}$ must be spanning – contradiction. Likewise, $H_{k}\cup\\{f\\}$ is also independent. We have shown that every subset of $H_{k}\cup\\{e,f\\}$ is independent, meaning that $H_{k}\cup\\{e,f\\}$ must be a circuit. ∎ Thus $H_{k}\cup\\{e,f\\}$ is dependent in $L$. We have shown that it is also dependent in $N^{\prime\prime}$, so again have a contradiction to $A$ being independent in $L$. The same argument shows that if $A$ is dependent in $N^{\prime\prime}$, $A$ is also dependent in $L$. Thus $L=N^{\prime\prime}$. ∎ We constructed $N$ to be representable, so $N$ must satisfy every Kinser inequality. In particular, it must be contained inside $\mathcal{K}_{n+1}$. Next we have shown that if we relax a single circuit-hyperplane of $N$, the resulting matroid $N^{\prime}$ has a bad family for $\mathcal{K}_{n+1}$. We will now show that $N^{\prime}$ is in fact an excluded minor for $\mathcal{K}_{n+1}$ – that is, we will show that each proper minor of $N^{\prime}$ is representable and thus is contained in $\mathcal{K}_{n+1}$. First suppose that $x\in H_{j}$ where $j\neq 0$. Let $N^{\prime\prime}=N^{\prime}$ with the circuit-hyperplane $H_{j}\cup\\{e,f\\}$ relaxed. By [6, Proposition 3.3.5], $N^{\prime\prime}\backslash x=N^{\prime}\backslash x$. As $N^{\prime\prime}$ is $\mathbb{K}$-representable by Theorem 7.7, and representability is preserved under minors, $N^{\prime}\backslash x$ is $\mathbb{K}$-representable. Say $l\in\\{0,\ldots,n-1\\}-\\{0,j\\}$. Now let $N^{\prime\prime}=N^{\prime}$ with $H_{l}\cup\\{e,f\\}$ relaxed. Also by [6, Proposition 3.3.5], we have that $N^{\prime\prime}/x=N^{\prime}/x$, and so $N^{\prime}/x$ is $\mathbb{K}$-representable. Next, suppose $x\in H_{0}$. As $N^{\prime}=N$ with the circuit-hyperplane $H_{0}\cup\\{e,f\\}$ relaxed, we have that $N^{\prime}\backslash x=N\backslash x$, so $N^{\prime}\backslash x$ is $\mathbb{K}$-representable. Let $N^{\prime\prime}=N^{\prime}$ with $H_{i}\cup\\{e,f\\}$ relaxed. We have that $N^{\prime\prime}/x=N^{\prime}/x$, so $N^{\prime}/x$ is $\mathbb{K}$-representable. Now suppose $x$ is equal to $e$. As $e$ and $f$ were freely placed on the line spanned by $p$ and $p^{\prime}$, the same argument as follows works for $x=f$. We have that $N^{\prime}\backslash e=N\backslash e$, so $N^{\prime}\backslash e$ is $\mathbb{K}$-representable. Finally, consider $N^{\prime}/e$. Take some $z\in H_{0}$. Recall that $N^{\prime}=N$ with the circuit-hyperplane $H_{0}\cup\\{e,f\\}$ relaxed. Note that $N^{\prime}/e$ is obtained from $N/e$ by relaxing $H_{0}\cup\\{f\\}$. This gives us that $N^{\prime}/e\backslash z=N/e\backslash z$, as deleting $z$ effectively undoes the relaxation. As $N$ is $\mathbb{K}$-representable, and thus $N/e\backslash z$ is $\mathbb{K}$-representable, $N^{\prime}/e\backslash z$ is also $\mathbb{K}$-representable. Let $Z\subseteq E(N^{\prime}/e)$ be such that $z\notin Z$ and $z\in cl_{N^{\prime}/e}(Z)$. $N^{\prime}/e$ is a relaxation of $N/e$ which can only affect closures in so far as that some may contain additional elements in $N/e$, so $z\in cl_{N/e}(Z)$. This implies that $z\in cl_{N}(Z\cup\\{e\\})$ by [6, Proposition 3.1.11]. Due to the way $H_{0}$ was constructed, we thus have that $\langle Z\cup\\{e\\}\rangle\supseteq(X\cup\\{x_{0},\ldots,x_{n-1}\\})-\\{x_{0},x_{1}\\}$. As we have that $z\in cl_{N}(Z\cup\\{e\\})$ and all elements of $H_{0}$ are freely placed in the relevant subspace, $Z\cup\\{e,f\\}$ must thus also span every other element of $H_{0}$. As $e$ and $f$ were freely placed on the line spanned by $p$ and $p^{\prime}$, we also have that $f\in cl_{N}(Z\cup\\{e\\})$. Thus, in $N$, $H_{0}\cup\\{e,f\\}$ is contained in $\langle Z\cup\\{e\\}\rangle$. As $H_{0}\cup\\{e,f\\}$ is a circuit- hyperplane, this implies that either $Z\cup\\{e\\}$ is spanning in $N$ or that $Z\cup\\{e\\}=H_{0}\cup\\{e,f\\}$. If $Z\cup\\{e\\}=H_{0}\cup\\{e,f\\}$, we have a contradiction to the assumption that $z\notin Z$. We thus have that $Z\cup\\{e\\}$ is spanning in $N$. This means that $Z\cup\\{e\\}$ is also spanning in $N^{\prime}$, and, as $r(N^{\prime}/e)=r(N^{\prime})-1$, that $Z$ is spanning in $N^{\prime}/e$. We have that $z$ is only in the closure of a subset of $N^{\prime}/e$ when that subset spans $N^{\prime}/e$ – that is, we have shown that $z$ is freely placed in $N^{\prime}/e$. Thus $N^{\prime}/e$ is a free extension of $N^{\prime}/e\backslash z$ by $z$. As $N^{\prime}/e\backslash z$ is $\mathbb{K}$-representable and this fact is preserved under free extentions, we have that $N^{\prime}/e$ is $\mathbb{K}$-representable. We have now shown that every minor of $N^{\prime}$ is $\mathbb{K}$-representable and so contained in $\mathcal{K}_{n+1}$, making $N^{\prime}$ an excluded minor for $\mathcal{K}_{n+1}$. This completes the proof of Theorem 7.1. ∎ ## Chapter 8 Conjectures Finally, we give some conjectures on the hierachy of the Kinser classes. ###### Conjecture 8.1. Let $n>5$. $\mathcal{K}_{n}\neq\mathcal{K}_{n}^{}$. As shown in the previous chapter, verifying that a matroid satisfies a Kinser inequality is very difficult. Given the amount of difficulty involved in proving that the fifth Kinser class is not dual closed, proving this result in general would involve an even greater amount of work. Based on that case, however, we give a strengthening of the above conjecture. ###### Conjecture 8.2. $\mathrm{Kin}(n)^{-}\in\mathcal{K}_{n}^{}-\mathcal{K}_{n}$ One further question about the structure of the hierarchy is how each dual class sits within the previous Kinser class. There are two possibilities here, and we conjecture that the following is true. ###### Conjecture 8.3. Let $n>4$. $\mathcal{K}_{n+1}^{}\subseteq\mathcal{K}_{n}$. The following conjecture is true if Conjecture 8.3 is as well. ###### Conjecture 8.4. $\mathcal{K}_{\infty}^{}=\mathcal{K}_{\infty}$ To see that this follows from Conjecture 8.3, assume $M\in\mathcal{K}_{\infty}$, but $M\not\in\mathcal{K}_{\infty}^{}$. Then there exists an integer $n$ such that $M\not\in\mathcal{K}_{n}^{}$. However, this contradicts Conjecture 8.3, which gives that $M\in\mathcal{K}_{n+1}\subseteq\mathcal{K}_{n}^{}$. Assuming these conjectures to hold true, we have a final diagram of the hierarchy. Figure 8.1: Kinser classes (4) Now we will consider two classes of matroids which we conjecture satisfy every Kinser inequality. ###### Definition 8.5. Let $G$ be an abelian group. Take a complete graph and add a loop to very vertex, replace every edge with a parallel class of $\|G\|$ edges. Call this graph $H$. Orient every edge which is not a loop so that parallel edges have the same direction. Bijectively label each parallel class with the elements of $G$, and label loops with non-identities. Let $C$ be a cycle of $H$. Consider the product of group labels in $C$, taken in cyclic order, where if an edge is oriented against the cyclic order we take the inverse of its label instead. If the result is the identity, call $C$ _positive_. Otherwise, call $C$ _negative_. Take a Dowling geometry $H$. There exists a matroid which has $E(H)$ as its ground set, and set of circuits equal to the positive cycles of $H$ and minimal connected subgraphs that contain two negative cycles. Call this matroid a _Dowling geometry_. Take a field $\mathbb{F}$. Recall that $\mathbb{F}^{\times}$ is the multiplicative group consisting of the non-zero elements of $\mathbb{F}$. ###### Lemma 8.6. [6, Theorem 6.10.10] Take a Dowling geometry of rank $r$ over a finite group $G$. This matroid is representable over a field $\mathbb{F}$ if and only if $G$ is isomorphic to a subgroup of $\mathbb{F}^{\times}$. In this case, the Dowling matroid satisfies every Kinser inequality. We also have that if $G$ is a finite subgroup of the multiplicative group of a field, then $G$ is cyclic by [1]. The following conjecture is thus open when $G$ is both finite and non-cyclic. ###### Conjecture 8.7. A Dowling geometry satisfies every Kinser inequality. Now we will consider matroids which are representable over skew partial fields. All of the following definitions and results can be found in [7]. ###### Definition 8.8. A _skew partial field_ is a pair $(R,G)$ where $R$ is a ring, and $G$ is a subgroup of the group of units of $R$, such that $-1\in G$. ###### Definition 8.9. Let $R$ be a ring, and let $E$ be a finite set. An _R-chain group_ on $E$ is a subset $C\subseteq R^{E}$ such that, for all $f,g\in C$ and $r\in R$, * i. $0\in C$, * ii. $f+g\in C$, * iii. $rf\in C$ The elements of $C$ are called _chains_ , and the _support_ of a chain $c=\\{c_{1},\ldots,c_{e}\\}\in C$ is $\|\|c\|\|=\\{i\in E\ \|\ c_{i}\neq 0\\}$ ###### Definition 8.10. A chain $c\in C$ is _elementary_ if $c\neq 0$ and there is no $c^{\prime}\in C-\\{0\\}$ with $\|\|c^{\prime}\|\|\subset\|\|c\|\|$. ###### Definition 8.11. Let $G$ be a subgroup of the group of units of $R$. A chain $c\in C$ is _G- primitive_ if $c\in(G\cup\\{0\\})^{E}$. ###### Definition 8.12. Let $\mathbb{P}=(R,G)$ be a skew partial field, and $E$ a finite set. A _$\mathbb{P}$ -chain group_ on $E$ is an $R$-chain group $C$ on $E$ such that every elementary chain $c\in C$ can be written as $c=rc^{\prime}$ for some $G$-primitive chain $c^{\prime}\in C$ and some $r\in R$. ###### Lemma 8.13. Let $\mathbb{P}=(R,G)$ be a skew partial field, and let $C$ be a $\mathbb{P}$-chain group on $E$. Then $\mathcal{C}^{}=\\{\|\|c\|\|\ \|\ c\in C,\ c\ \text{is elementary}\\}$ is the set of cocircuits of a matroid on $E$. A matroid $M$ is said to be _$\mathbb{P}$ -representable_ if there exists a $\mathbb{P}$-chain group $C$ such that $M=M(C)$. If a matroid is representable and thus satisfies every Kinser inequality, it is representable over a skew partial field. ###### Conjecture 8.14. Take a matroid $M$ which is $\mathbb{P}$-representable. $M$ satisfies every Kinser inequality. ## Bibliography [1] http://www.math.uwo.ca/~srankin/courses/413/2011/finite_subgroup_field_cyclic.pdf. * [2] Ingleton, A. Conditions for representability and transversality of matroids. Théorie des Matroïdes (1971), 62–66. * [3] Kinser, R. New inequalities for subspace arrangements. J. Combin. Theory Ser. A 118, 1 (2011), 152–161. * [4] Mayhew, D., Newman, M., and Whittle, G. Is the missing axiom of matroid theory lost forever? Submitted. * [5] Mayhew, D., Newman, M., and Whittle, G. On excluded minors for real-representability. J. Combin. Theory Ser. B 99, 4 (2009), 685–689. * [6] Oxley, J. Matroid theory, second ed., vol. 21 of Oxford Graduate Texts in Mathematics. Oxford University Press, Oxford, 2011. * [7] Pendavingh, R., and van Zwam, S. Skew partial fields, multilinear representations of matroids, and a matrix tree theorem. Adv. in Appl. Math. 50, 1 (2013), 201–227. * [8] Vámos, P. The missing axiom of matroid theory is lost forever. J. London Math. Soc. (2) 18, 3 (1978), 403–408. * [9] Whitney, H. On the Abstract Properties of Linear Dependence. Amer. J. Math. 57, 3 (1935), 509–533.	arxiv-papers	2014-01-02T18:39:04	2024-09-04T02:49:56.228171	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Amanda Cameron", "submitter": "Amanda Cameron", "url": "https://arxiv.org/abs/1401.0500" }
1401.0509	# Zero-Shot Learning for Semantic Utterance Classification Yann N. Dauphin1 Gokhan Tur2 Dilek Hakkani-Tür2 Larry Heck2 1University of Montreal, Montreal, Canada 2Microsoft Research, Mountain View, CA, USA ###### Abstract We propose a novel zero-shot learning method for semantic utterance classification (SUC). It learns a classifier $f:X\to Y$ for problems where none of the semantic categories $Y$ are present in the training set. The framework uncovers the link between categories and utterances through a semantic space. We show that this semantic space can be learned by deep neural networks trained on large amounts of search engine query log data. What’s more, we propose a novel method that can learn discriminative semantic features without supervision. It uses the zero-shot learning framework to guide the learning of the semantic features. We demonstrate the effectiveness of the zero-shot semantic learning algorithm on the SUC dataset collected by (Tur et al., 2012). Furthermore, we achieve state-of-the-art results by combining the semantic features with a supervised method. ## 1 Introduction Conversational understanding systems aim to automatically classify user requests into predefined semantic categories and extract related parameters (Tur and Mori, 2011). For instance, such a system might classify the natural language query “I want to fly from San Francisco to New York next Sunday” into the semantic domain flights. This is known as semantic utterance classification (SUC). Typically, these systems use supervised classification methods such as Boosting (Schapire and Singer, 2000), support vector machines (SVMs) (Haffner et al., 2003), or maximum entropy models (Yaman et al., 2008). These methods can produce state-of-the-art results but they require significant amounts of labelled data. This data is mostly obtained through manual labor and becomes costly as the number of semantic domains increases. This limits the applicability of these methods to problems with relatively few semantic categories. We consider two problems here. First, we examine the problem of predicting the semantic domain of utterances without having seen examples of any of the domains. Formally, the goal is to learn a classifier $f:X\to Y$ without any values of $Y$ in the training set. In constrast to traditional SUC systems, adding a domain is as easy as including it in the set of domains. This is a form of zero-shot learning (Palatucci et al., 2009) and is possible through the use of a knowledge base of semantic properties of the classes to extrapolate to unseen classes. Typically this requires seeing examples of at least some of the semantic categories. Second, we consider the problem of easing the task of supervised classifiers when there are only few examples per domain. This is done by augmenting the input with a feature vector $H$ for a classifier $f:(X,H)\to Y$. The difficulty is that $H$ must be learned without any knowledge of the semantic domains $Y$. In this paper, we introduce a zero-shot learning framework for SUC where none of the classes have been seen. We propose to use a knowledge base which can output the semantic properties of both the input and the classes. The classifier matches the input to the class with the best matching semantic features. We show that a knowledge-base of semantic properties can be learned automatically for SUC by deep neural networks using large amounts of data. The recent advances in deep learning have shown that deep networks trained at large scale can reach state-of-the-art results. We use the Bing search query click logs, which consists of user queries and associated clicked URLs. We hypothesize that the clicked URLs reflect high level meaning or intent of the queries. Surprinsingly, we show that is is possible to learn semantic properties which are discriminative of our unseen classes without any labels. We call this method zero-shot discriminative embedding (ZDE). It uses the zero-shot learning framework to provide weak supervision during learning. Our experiments show that the zero-shot learning framework for SUC yields competitive results on the tasks considered. We demonstrate that zero-shot discriminative embedding produces more discriminative semantic properties. Notably, we reach state-of-the-art results by feeding these features to an SVM. In the next section, we formally define the task of semantic utterance classification. We provide a quick overview of zero-shot learning in Section 3. Sections 4 and 5 present the zero-shot learning framework and a method for learning semantic features using deep networks. Section 6 introduces the zero- shot discriminative embedding method. We review the related work on this task in Section 7 In Section 8 we provide experimental results. ## 2 Semantic Utterance Classification The semantic utterance classification (SUC) task aims at classifying a given speech utterance $X_{r}$ into one of $M$ semantic classes, $\hat{C}_{r}\in\mathcal{C}=\\{C_{1},\ldots,C_{M}\\}$ (where $r$ is the utterance index). Upon the observation of $X_{r}$, $\hat{C}_{r}$ is chosen so that the class-posterior probability given $X_{r}$, $P(C_{r}\|X_{r})$, is maximized. More formally, $\hat{C}_{r}=\hbox{arg}\max_{C_{r}}P(C_{r}\|X_{r})$. Semantic classifiers need to allow significant utterance variations. A user may say “I want to fly from San Francisco to New York next Sunday” and another user may express the same information by saying “Show me weekend flights between JFK and SFO”. Not only is there no a priori constraint on what the user can say, these systems also need to generalize well from a tractably small amount of training data. On the other hand, the command “Show me the weekend snow forecast” should be interpreted as an instance of another semantic class, say, “Weather.” In order to do this, the selection of the feature functions $f_{i}(C,W)$ aims at capturing the relation between the class $C$ and word sequence $W$. Typically, binary or weighted $n$-gram features, with $n=1,2,3$, to capture the likelihood of the $n$-grams, are generated to express the user intent for the semantic class $C$ (Tur and Deng, 2011). Once the features are extracted from the text, the task becomes a text classification problem. Traditional text categorization techniques devise learning methods to maximize the probability of $C_{r}$, given the text $W_{r}$; i.e., the class-posterior probability $P(C_{r}\|W_{r})$. ## 3 Zero-shot learning In general, zero-shot learning (Palatucci et al., 2009) is concerned with learning a classifier $f:X\to Y$ that can predict novel values of $Y$ not present in the training set. It is an important problem setting for tasks where the set of classes is large and in cases where the cost of labelled examples is high. It has found application in vision where the number of classes can be very large (Frome et al., 2013). A zero-shot learner uses semantic knowledge to extrapolate to novel classes. Instead of predicting the classes directly, the learner predicts semantic properties or features of the input. Thanks to a knowledge-base of semantic features for the classes it can match the inputs to the classes. The semantic feature space is a euclidean space of $d$ dimensions. Each dimension encodes a semantic property. In vision for instance, one dimension might encode the size of the object, another the color. The knowledge base $\mathcal{K}$ stores a semantic feature vector $H$ for each of the classes. The zero-shot classifier $f=m\circ n$ is the composition of two classifiers. The first classifier $m:X\to H$ predicts the semantic properties of the input. The training set is found by replacing the class values in the training set by their semantic features. The second classifier $n:H\to Y$ matches the semantic code to the class. This can be done by a $k$-NN classifier. In applying zero-shot learning to semantic utterance classification there are several challenges. The framework described by (Palatucci et al., 2009) requires some of the classes to be present in the training data in order to train the $m$ classifier. We are interested in the setting where none of classes have training data. Furthermore, an adequate knowledge-base must be found for SUC. ## 4 Zero-Shot Learning for Semantic Utterance Classification In this section, we introduce a zero-shot learning framework for SUC where none of the classes are seen during training. It is based on the observation that in SUC both the semantic categories and the inputs reside in the same semantic space. In this framework, classification can be done by finding the best matching semantic category for a given input. Semantic utterance classification is concerned with finding the semantic category for a natural language utterance. Traditionally, conversational systems learn this task using labelled data. This overlooks the fact that classification would be much easier in a space that reveals the semantic meaning of utterances. Interestingly, the semantics of language can be discovered without labelled data. What’s more, the name of semantic classes are not chosen randomly. They are in the same language as the sentences and are often chosen because they describe the essence of the class. These two facts can easily be used by humans to classify without task-specific labels. For instance, it is easy to see that the utterance _the accelerator has exploded_ belongs more to the class _physics_ than _outdoors_. This is the very human ability that we wish to replicate here. Figure 1: Visualization of the 2d semantic space learned by a deep neural net. We see that the two axis differentiate between phrases relating to hotels and movies. More details in Section 8. We propose a framework called zero-shot semantic learning (ZSL) that leverages these observations. In this framework, the knowledge-base $\mathcal{K}$ is a function which can output the semantic properties of any sentence. The classification procedure can be done in one step because both the input and the categories reside in the same space. The zero-shot classifier finds the category which best matches the input. More formally, the zero-shot classifier is given by $\displaystyle P(C_{r}\|X_{r})=\frac{1}{Z}e^{-\|\mathcal{K}(X_{r})-\mathcal{K}(C_{r})\|}$ (1) where $Z=\sum_{C}e^{-\|\mathcal{K}(X_{r})-\mathcal{K}(C)\|}$ and $\|x-y\|$ is a distance measure like the euclidean distance. The knowledge-base maps the input $\mathcal{K}(X_{r})$ and the category $\mathcal{K}(X_{r})$ in a space that reveals their meaning. An example 2d semantic space is given in Figure 1 which maps sentences relating to movies close to each other and those relating to hotels further away. In this space, given the categories _hotel_ and _movies_ , the sentence _motels in aurora colorado_ will be classified to _hotel_ because $\mathcal{K}(\textit{motels in aurora colorado})$ is closer to $\mathcal{K}(\textit{hotel})$. This framework will classify properly if * • The semantics of the language are properly captured by $\mathcal{K}$. In other words, utterances are clustered according to their meaning. * • The class name $C_{r}$ describes the semantic core of the class well. Meaning that $\mathcal{K}(C_{r})$ resides close to the semantic representation of sentences of that class. The success of this framework rests on the quality of the knowledge-base $\mathcal{K}$. Following the success of learning methods with language, we are interested in learning this knowledge-base from data. Unsupervised learning methods like LSA, and LDA have had some success but it is hard to ensure that the semantic properties will be useful for SUC. ## 5 Learning Semantic Features for SUC using Deep Nets In this section, we describe a method for learning a semantic features for SUC using deep networks trained on Bing search query click logs. We use the query click logs to define a task that makes the networks learn the meaning or intent behind the queries. The semantic features are found at the last hidden layer of the deep neural network. Query Click Logs (QCL) are logs of unstructured text including both the users queries sent to a search engine and the links that the users clicked on from the list of sites returned by that search engine. Some of the challenges in extracting useful information from QCL is that the feature space is very high dimensional (there are thousands of url clicks linked to many queries), and there are millions of queries logged daily. We make the mild hypothesis that the website clicked following a query reveals the meaning or intent behind a query. The queries which have similar meaning or intent will map to the same website. For example, it is easy to see that queries associated with the website _imdb.com_ share a semantic connection to movies. Figure 2: Depiction of the deep network from queries to URLs. We train the network with the query as input and the website as the output (see Figure 2). This learning scheme is inspired by the neural language models (Bengio, 2008) who learn word embeddings by learning to predict the next word in a sentence. The idea is that the last hidden layer of the network has to learn an embedding space which is helpful to classification. To do this, it will map similar inputs in terms of the classification task close in the embedding space. The key difference with word embeddings methods like (Bengio, 2008) is that we are learning sentence-level embeddings. We train deep neural networks with softmax output units and rectified linear hidden units. The inputs $X_{r}$ are queries represented in bag-of-words format. The labels $Y_{r}$ are the index of the website that was clicked. We train the network to minimize the negative log-likelihood of the data $\mathcal{L}(X,Y)=-\log P(Y_{r}\|X_{r})$. The network has the form $P(Y=i\|X_{r})=\frac{e^{W^{n+1}_{i}H^{n}(X_{r})+b^{n+1}_{i}}}{\sum_{j}e^{W^{n+1}_{j}H^{n}(X_{r})+b^{n+1}_{j}}}$ The latent representation function $H^{n}$ is composed on $n$ hidden layers $\displaystyle H^{n}(X_{r})$ $\displaystyle=$ $\displaystyle\max(0,W^{n}H^{n-1}(X_{r})+b^{n})$ $\displaystyle H^{1}(X_{r})$ $\displaystyle=$ $\displaystyle\max(0,W^{1}X_{r}+b^{1})$ We have a set of weight matrices $W$ and biases $b$ for each layer giving us the parameters $\theta=\\{W^{1},b^{1},\dots,W^{n+1},b^{n+1}\\}$ for the full network. We train the network using stochastic gradient descent with minibatches. The knowledge-base function is given by the last hidden layer $\mathcal{K}=H^{n}(X_{r})$. In this scheme, the embeddings are used as the semantic properties of the knowledge-base. However, it is not clear that the semantic space will be discriminative of the semantic categories we care about for SUC. ## 6 Learning Discriminative Semantic Features without Supervision We introduce a novel regularization that encourages deep networks to learn discriminative semantic features for the SUC task without labelled data. More precisely, we define a clustering measure for the semantic classes using the zero-shot learning framework of Section 4. We hypothesize the classes are well clustered hence we minimize this measure. In the past section, we have described a method for learning semantic features using query click logs. The features are given by finding the best semantic space for the query click logs task. In general, there might be a mismatch between what qualifies as a good semantic space for the QCL and SUC tasks. For example, the network might learn an embedding that clusters sentences of the category _movies_ and _events_ close together because they both relate to activities. In this case the features would have been more discriminative if the sentences were far from each other. However, there is no pressure for the network to do that because it doesn’t know about the SUC task. Figure 3: Visualization of an actual 2d embedding space learned by a DNN (left) and DNN trained with ZDE (right). The points are sentences with different colors for each class and the arrows point to the location of the class name in the embedding space. ZDE significantly improves the clustering of the classes. More details in Section 8. This problem could have been addressed by multi-task or semi-supervised learning methods if we had access to labelled data. Research has shown adding even a little bit of supervision is often helpful (Larochelle et al., 2009). The simplest solution would be to train the network on the QCL and SUC task simultaneously. In other words, we would train the network to minimize the sum of the QCL objective $-\log P(Y\|X)$ and the SUC objective $-\log P(C\|X)$. This would allow the model to leverage the large amount of QCL data while learning a better representation for SUC. We cannot miminize $-\log P(C\|X)$ but we can minimize a similar measure which does not require labels. We can measure the overlap of the semantic categories using the conditional entropy $\displaystyle H(P(C_{r}\|X_{r}))$ $\displaystyle=$ $\displaystyle E[I(P(C_{r}\|X_{r}))]$ $\displaystyle=$ $\displaystyle E[-\sum_{i}P(C_{r}=i\|X_{r})\log P(C_{r}=i\|X_{r})].$ The measure is lowest when the overlap is small. Interestingly, calculating the entropy does not require labelled data. We can recover a zero-shot classifier $P(C\|X)$ from the semantic space using Equation 1. The entropy $H(P(C_{r}\|X_{r}))$ of this classifier measures the clustering of the categories in the semantic space. Spaces with the lowest entropy are those where the examples $\mathcal{K}(X_{r})$ cluster around category names $\mathcal{K}(C_{r})$ and where the categories have low-overlap in the semantic space. Figure 3 illustrates a semantic space with high conditional entropy on the left, and one with a low entropy on the right side. Zero-shot Discriminative Embedding (ZDE) combines the embedding method of Section 5 with the minimization of the entropy of a zero-shot classifier on that embedding. The objective has the form $\displaystyle\mathcal{L}(X,Y)=-\log P(Y\|X)+\lambda H(P(C\|X)).$ (3) The variable $X$ is the input, $Y$ is the website that was clicked, $C$ is a semantic class. The hyper-parameter $\lambda$ controls the strength of entropy objective in the overall objective. We find this value by cross-validation. ## 7 Related work Early work on spoken utterance classification has been done mostly for call routing or intent determination system, such as the AT&T How May I Help You? (HMIHY) system (Gorin et al., 1997), relying on salience phrases, or the Lucent Bell Labs vector space model (Chu-Carroll and Carpenter, 1999). Typically word $n$-grams are used as features after preprocessing with generic entities, such as dates, locations, or phone numbers. Because of the very large dimensions of the input space, large margin classifiers such as SVMs (Haffner et al., 2003) or Boosting (Schapire and Singer, 2000) were found to be very good candidates. Deep learning methods have first been used for semantic utterance classification by Sarikaya et al. (Sarikaya et al., 2011). Deep Convex Networks (DCNs) (Tur et al., 2012) and Kernel DCNs (K-DCNs) (Deng et al., 2012) have also been applied to SUC. K-DCNs allow the use of kernel functions during training, combining the power of kernel based methods and deep learning. While both approaches resulted in performances better than a Boosting-based baseline, K-DCNs have shown significantly bigger performance gains due to the use of query click features. Entropy minimization (Grandvalet and Bengio, 2005) is a semi-supervised learning framework which also uses the conditional entropy. In this framework, both labelled and unlabelled data are available, which is an important difference with ZDE. In (Grandvalet and Bengio, 2005), a classifier is trained to minimize its conditional likelihood and its conditional entropy. ZDE avoids the need for labels by minimizing the entropy of a zero-shot classifier. (Grandvalet and Bengio, 2005) shows that this approach produces good results especially when generative models are mispecified. ## 8 Experiments In this section, we evaluate the zero-shot semantic learning framework and the zero-shot discriminative embedding method proposed in the previous sections. ### 8.1 Setup We have gathered a month of query click log data from Bing to learn the embeddings. We restricted the websites to the the 1000 most popular websites in this log. The words in the bag-of-words vocabulary are the 9521 found in the supervised SUC task we will use. All queries containing only unknown words were filtered out. We found that using a list of stop-words improved the results. After these restrictions, the dataset comprises 620,474 different queries. We evaluate the performance of the methods for SUC on the dataset gathered by (Tur et al., 2012). It was compiled from utterances by users of a spoken dialog system. There are 16,000 training utterances, 2000 utterances for validation and 2000 utterances for testing. Each utterance is labelled with one of 25 domains. The hyper-parameters of the models are tuned on the validation set. The learning rate parameter of gradient descent is found by grid search with $\\{0.1,0.01,0.001\\}$. The number of layers is between 1 and 3. The number of hidden units is kept constant through layers and is found by sampling a random number from 300 to 800 units. We found that it was helpful to regularize the networks using dropout (Hinton et al., 2012). We sample the dropout rate randomly between 0% dropout and 20%. The $\lambda$ of the zero-shot embedding method is found through grid-search with $\\{0.1,0.01,0.001\\}$. The models are trained on a cluster of computers with double quad-core Intel(R) Xeon(R) CPUs with 2.33GHz and 8Gb of RAM. Training either the ZDE method on the QCL data requires 4 hours of computation time. ### 8.2 Results First, we want to see what is learned by the embedding method described in Section 5. A first step is to look at the nearest neighbor of words in the embedding space. Table 1 shows the nearest neighbours of specific words in the embedding space. We observe that the neighbors of the words al share the semantic domain of the word. This confirms that the network learns some semantics of the language. Restaurant Hotel Flight Events Transportation steakhouse suites airline festivals distributing diner hyatt airfaire upcoming dfw seafood resorts plane fireworks petroleum tavern ramada baggage happening hospitality Table 1: Nearest neighbours in the embedding space. Each column displays the 5 nearest neighbours of the word at the top. We can see that the embedding captures the semantics of the words. We can better visualize the embedding space using a network with a special architecture. Following (Hinton and Salakhutdinov, 2006), we train deep networks where the last hidden layer contains only 2 dimensions. The depth allows the network to progressively reduce the dimensionality of the data. This approach enables us to visualize exactly what the network has learned. Figure 1 shows the embedding a deep network with 3 layers (with size 200-10-2) trained on the QCL task. We observe that the embedding distinguishes between sentences related to movies and hotels. In Figure 3, we compare the embedding spaces of a DNN trained on the QCL (left) and a DNN trained using ZDE (right) both with hidden layers of sizes 200-10-2. The comparison suggests that minimizing the conditional entropy of the zero-shot classifier successfully improves the clustering. Method Restaurant Hotel Flight Events Transportation ZSL with Bag-of-words 0.616 0.641 0.683 0.559 0.5 ZSL with $p(Y\|X)$ (LR) 0.779 0.821 0.457 0.677 0.472 ZSL with $p(Y\|X)$ (DNN) 0.838 0.862 0.46 0.631 0.503 ZSL with DNN Embedding 0.858 0.935 0.870 0.727 0.667 ZSL with ZDE Embedding 0.863 0.940 0.906 0.841 0.826 Representative URL heuristic (DNN) 0.798 0.892 0.769 0.707 0.577 Table 2: Comparison of several zero-shot semantic learning methods for 5 semantic classes. Our proposed zero-shot learning system with DNN embeddings outperforms other approaches. Second, we want to confirm that good classification results can be achieved using zero-shot semantic learning. To do this, we evaluate the classification results of our method on the SUC task. Our results are given in Table 2. The performance is measured using the AUC (Area under the curve of the precision- recall curve) for which higher is better. We compare our ZDE method against various means of obtaining the semantic features $H$. We compare with using the bag-of-words representation (denoted _ZSL with Bag-of-words_) as semantic features. _ZSL with $p(Y\|X)$ (LR)_ and _ZSL with $p(Y\|X)$ (DNN)_ are models trained from the QCL to predict the website associated with queries. The semantic features are the vector of probability that each website is associated with the query. _ZSL with $p(Y\|X)$ (LR)_ is a logistic regression model, _ZSL with $p(Y\|X)$ (DNN)_ is a DNN model. We also compare with a sensible heuristic method denoted _Representative URL heuristic_. For this heuristic, we associate each semantic category with a representative website (i.e. _flights_ with _expedia.com_ , _movies_ with imdb.com). We train a DNN using the QCL to predict which of these websites is clicked given an utterance. The semantic category distribution $P(C\|X)$ is the probability that each associated website was clicked. Table 2 shows that the proposed zero-shot learning method with ZDE achieves the best results. In particular, ZDE improves performance by a wide margin for hard categories like _transportation_. These results confirm the hypothesis behind both ZSL and the ZDE method. Figure 4: Comparison between the proposed zero-shot learning method and an SVM trained with increasing amount of examples. The curve shows that ZSL compares favorably with SVMs when there are few labels. We also compare the zero-shot learning system with a supervised SUC system. We compare ZSL with a linear SVM. The task is identify utterances of the _restaurant_ semantic class. Figure 4 shows the performance of the linear SVM as the number of labelled training examples increases. The performance of ZSL is shown as a straight line because it does not use labelled data. Predictably, the SVM achieves better results when the labelled training set is large. However, ZSL achieves better performance in the low-data regime. This confirms that ZSL can be useful in cases where labelled data is costly, or the number of classes is large. Features Kernel DCN SVM Bag-of-words 9.52% 10.09% QCL features (Hakkani-Tür et al., 2011) 5.94% 6.36% DNN urls 6.88% DNN embeddings 6.2% ZDE embeddings 5.73% Table 3: Test error rate of various methods on the SUC task. The best results are achieved with by augmenting the input with ZDE embeddings. Finally, we consider the problem of using semantic features $H$ to increase the performance of a classifier $f:(X,H)\to Y$. The input X is a bag-of-words representation of the utterances. We compare with state-of-the-art approaches in Table 3. The state-of-the-art method is the Kernel DCN on QCL features with 5.94% test error. However, we train using the more scalable linear SVM which leads to 6.36% with the same input features. The linear SVM is better to compare features because it cannot non-linearly transform the input by itself. Using the embeddings learned from the QCL data as described in Section 4 yields 6.2% errors. Using zero-shot discriminative embedding further reduces the error t 5.73%. ## 9 Conclusion We have introduced a zero-shot learning framework for SUC. The proposed method learns a knowledge-base using deep networks trained on large amounts of search engine query log data. We have proposed a novel way to learn embeddings that are discriminative without access to labelled data. Finally, we have shown experimentally that these methods are effective. ## References * Bengio (2008) Bengio, Y. (2008). Neural net language models. Scholarpedia, 3(1), 3881. * Chu-Carroll and Carpenter (1999) Chu-Carroll, J. and Carpenter, B. (1999). Vector-based natural language call routing. Computational Linguistics, 25(3), 361–388. * Deng et al. (2012) Deng, L., Tur, G., He, X., and Hakkani-Tür, D. (2012). Use of kernel deep convex networks and end-to-end learning for spoken language understanding. 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An integrative and discriminative technique for spoken utterance classification. IEEE Transactions on Audio, Speech, and Language Processing, 16(6), 1207–1214.	arxiv-papers	2013-12-20T17:08:26	2024-09-04T02:49:56.245568	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Yann N. Dauphin, Gokhan Tur, Dilek Hakkani-Tur, Larry Heck", "submitter": "Yann Dauphin", "url": "https://arxiv.org/abs/1401.0509" }
1401.0672	# Spin Dephasing as a Probe of Mode Temperature, Motional State Distributions, and Heating Rates in a 2D Ion Crystal Brian C. Sawyer [email protected] Joseph W. Britton John J. Bollinger Time and Frequency Division, National Institute of Standards and Technology, Boulder, CO 80305 ###### Abstract We employ spin-dependent optical dipole forces to characterize the transverse center-of-mass (COM) motional mode of a two-dimensional Wigner crystal of hundreds of 9Be+. By comparing the measured spin dephasing produced by the spin-dependent force with the predictions of a semiclassical dephasing model, we obtain absolute mode temperatures in excellent agreement with both the Doppler laser cooling limit and measurements obtained from a previously published technique (B. C. Sawyer et al. Phys. Rev. Lett. 108, 213003 (2012)). Furthermore, the structure of the dephasing histograms allows for discrimination between initial thermal and coherent states of motion. We also apply the techniques discussed here to measure, for the first time, the ambient heating rate of the COM mode of a 2D Coulomb crystal in a Penning trap. This measurement places an upper limit on the anomalous single-ion heating rate due to electric field noise from the trap electrode surfaces of $\frac{d\bar{n}}{dt}\sim 5$ s-1 for our trap at a frequency of 795 kHz, where $\bar{n}$ is the mean occupation of quantized COM motion in the axial harmonic well. ###### pacs: 52.27.Jt, 52.27.Aj, 03.65.Ud, 03.67.Bg ## I Introduction Laser-cooled ions stored in radiofrequency (RF) or Penning traps readily form crystalline arrays. Sensitive measurements of the motion of ions in these arrays are important for a variety of studies in atomic physics, quantum information science, and plasma physics. In atomic physics the small residual motion of trapped ions produces systematic errors that can limit the performance of atomic clocks and precision measurements Rosenband et al. (2008); Shiga et al. (2011). In quantum information science, where trapped-ion crystals provide a promising platform for quantum computation and simulation Blatt and Wineland (2008); Home (2013); Blatt and Roos (2012); Islam et al. (2013); Britton et al. (2012), the residual motion of the trapped ions produces infidelities that require careful evaluation. In plasma physics, trapped-ion crystals provide a convenient laboratory platform for studies of strongly coupled plasmas, which model dense astrophysical matter Horn (1991); Ichimaru et al. (1987); Dubin and Dewitt (1994); Baiko (2009). Careful measurements of ion motion are used to determine the ion energy and the plasma coupling. Energy transport studies require detailed measurements of the ion motion, typically as a function of time and resolved spatially or between different modes Anderegg et al. (2009). Here we discuss a new technique for measuring the temperature and, more generally, the energy state distribution of a trapped-ion crystal. The technique is mode specific in that it can resolve the energy distribution of different modes of the crystal. We demonstrate the technique by presenting measurements of the energy distribution of the axial center-of-mass (COM) mode of a single-plane array of several hundred Be+ ions stored in a Penning trap. The technique requires isolating and controlling a two-level system – an effective spin-1/2 in each ion – and employs a weak, global spin-dependent force that couples the spin and motional degrees of freedom of each ion. This general technique should be applicable to other systems such as neutral atoms in optical lattices Bloch (2005) or opto-mechanical systems Brahms et al. (2012) where spin degrees of freedom can be controlled and coupled to motional degrees of freedom. Spin-dependent forces are a key tool in trapped-ion quantum simulation and quantum computing work. Application of a spin-dependent force to a superposition of different spin states can generate entanglement between the spins, while the concomitant coupling of the spin and motional degrees of freedom typically produces infidelities that must be mitigated Garcia-Ripoll et al. (2005); Kim et al. (2009); Sørensen and Mølmer (1999, 2000); Leibfried et al. (2003); Sawyer et al. (2012); Wang and Freericks (2012). Here we focus on the coupling and potential entanglement between the spin and motional degrees of freedom produced by a spin-dependent force, and work in a regime where the induced spin-spin entanglement is negligible. However, the discussion and measurements presented here provide insight into the size and nature of trapped-ion quantum gate errors produced by coupling of the spins to thermal fluctuations of the motional modes Kirchmair et al. (2009). Spin-echo as well as other dynamical decoupling techniques can remove the coupling of the spin and motional degrees of freedom Garcia-Ripoll et al. (2005); Hayes et al. (2011, 2012), but their efficacy depends on the size of the error and the coherence of the motional state throughout an experiment, which can be evaluated with the techniques discussed here. This study extends the results of Ref. Sawyer et al. (2012), where we measured the decrease in the composite Bloch vector length produced by the application of a homogeneous spin-dependent optical dipole force. We showed that this decrease (or decoherence) of the Bloch vector depended on the average energy or temperature of the initial motional state. Here we show that the dephasing responsible for this decoherence may be directly measured, revealing more detailed information about the motional state. In addition to the average energy or temperature of a mode, information on the energy distribution can also be obtained. Spin-dephasing produced through the application of a spin- dependent force provides an alternative to the well known Raman sideband technique for determining the energy distribution of motional states of trapped-ion crystals Leibfried et al. (1996). The spin-dephasing technique is particularly well-suited for many-ion crystals, and for some setups – in particular for higher frequency two-level systems such as the 124 GHz spin- flip transition discussed here (see Sec. II) – can be simpler to implement. To illustrate the basic idea of spin dephasing produced through the application of a spin-dependent force, we consider the simple case of a single trapped ion whose motional degree of freedom along the z-axis (trap frequency $\omega_{z}$) is coupled to two internal spin states through a sinusoidally time-varying spin-dependent force. The interaction Hamiltonian for this system is $\hat{H}=F_{0}\cos\left(\mu t\right)\hat{z}\hat{\sigma}^{z},$ (1) where $\hat{z}$ is the position operator of the ion in the z-direction, $\hat{\sigma}^{z}$ is the Pauli spin matrix associated with the two internal energy levels, and $\mu$ is the frequency of the applied spin-dependent force. We assume the ion spin state is initialized in an equal superposition $\left\\{\left\|\uparrow\right\rangle+\left\|\downarrow\right\rangle\right\\}/\sqrt{2}$ of the $\left\|\uparrow\right\rangle,\left\|\downarrow\right\rangle$ internal levels. This spin state can be represented as pointing along the $x$-axis in the rotating frame of the Bloch sphere. Suppose the ion temperature is large compared to $\hbar\omega_{z}/k_{B}$, where $\hbar$ and $k_{B}$ are the Planck and Boltzmann constants, and we may treat the ion motion as classical. The initial motional state of the ion can then be written as $z(t)=Z_{A}\cos(\omega_{z}t+\phi)$, where $Z_{A}$ and $\phi$ fluctuate from one shot (or realization) of the experiment to the next, consistent with a thermal distribution. Application of Eq. 1 produces an additional spin- dependent motion, but we assume this driven spin-dependent motion is small compared with the initial thermal fluctuation ($Z_{A})$ (valid for our work with hundreds of trapped Be+ ions), and we can approximate the Hamiltonian, $\hat{H}$, as $\begin{array}[]{ccc}\hat{H}&\approx&F_{0}\cos\left(\mu t\right)Z_{A}\cos\left(\omega_{z}t+\phi\right)\hat{\sigma}^{z}\\\ &=&\frac{F_{0}Z_{A}}{2}\left\\{\cos\left[\left(\mu-\omega_{z}\right)t-\phi\right]+\cos\left[\left(\mu+\omega_{z}\right)t+\phi\right]\right\\}\hat{\sigma}^{z}\>.\end{array}$ (2) For $\mu=\omega_{z}$ this Hamiltonian is simply a constant shift $F_{0}Z_{A}\cos\left(\phi\right)/\hbar$ in the frequency difference between the $\left\|\uparrow\right\rangle,\left\|\downarrow\right\rangle$ levels, plus a rapidly varying term that averages to zero for time intervals long compared to $\pi/\omega_{z}$. If the spin-dependent force is applied for a time interval $\tau$, then the Bloch vector undergoes a precession by an angle $\Phi_{p}=\left(F_{0}Z_{A}\cos\left(\phi\right)/\hbar\right)\tau$. Fluctuations in $Z_{A}$ and $\phi$ from one shot (or realization) of the experiment to the next produce spin dephasing when averaged over many experimental realizations. By measuring this dephasing directly we show that it is possible to acquire information on the initial motional state (for example, the energy distribution) of the trapped-ion harmonic oscillator. The sensitivity to motion of this technique is very high. For the modest parameters used in the measurements of Sec. III and IV, $F_{0}=10^{-23}$ N and $\tau=1$ ms give $\Phi_{P}=30^{\circ}$ for $Z_{A}\simeq 6$ nm. With $N\gtrsim 100$ trapped ions, a $30^{\circ}$ precession is much larger than the quantum projection noise Itano et al. (1993), and can be measured with good signal-to- noise in one experimental shot. The rest of the manuscript is structured as follows. In Section II we describe the Penning trap setup where we implement the spin dephasing technique to characterize the energy distribution of the axial COM mode of a 2D trapped-ion crystal of hundreds of 9Be+ ions. In Section III we discuss a more detailed dephasing model assuming a thermal distribution of coherent states. In Section IV we discuss dephasing measurements of the COM mode energy distribution for both thermal states and coherently-excited thermal states. We also present measurements of the heating rate of the axial COM mode. In Section V we summarize and conclude. ## II Experimental Setup As described in previous publications, we employ a Penning trap to confine crystals of hundreds of 9Be+ ions Britton et al. (2012); Sawyer et al. (2012). Depicted in Fig. 1a, the trap consists of a static electric ($E$) quadrupole produced from a stack of cylindrical electrodes (inner radius of 2.0 cm) placed within the room-temperature bore of a $\sim$4.46 T superconducting magnet. The orientation of this uniform magnetic ($B$) field defines the $z$-axis in our system. Harmonic axial ($z$-axis) ion confinement with a frequency of $\omega_{z}=2\pi\times 795$ kHz is obtained by applying -1 kV to the central ring electrodes relative to grounded upper and lower endcap electrodes. The cylindrical axis of the trap electrodes is aligned with the uniform magnetic field. Radial ion confinement results from $\vec{E}\times\vec{B}$ induced rotation through the magnetic field. We apply a weak quadrupolar “rotating wall” potential to precisely control the rotation frequency ($\omega_{r}$) and hence, radial confining force, of the ion cloud Hasegawa et al. (2005). Neglecting the weak azimuthal dependence of the rotating wall potential, the following trap potential describes ion confinement in the Penning trap as seen in a frame rotating at $\omega_{r}$ Dubin and O’Neil (1999): $\displaystyle q\Phi_{\text{trap}}(r,z)$ $\displaystyle=$ $\displaystyle\frac{1}{2}M\omega_{z}^{2}\left(z^{2}+\beta_{r}r^{2}\right),$ (3) $\displaystyle\beta_{r}$ $\displaystyle\equiv$ $\displaystyle\frac{\omega_{r}(\Omega_{c}-\omega_{r})}{\omega_{z}^{2}}-\frac{1}{2},$ (4) where $M$ ($q$) is the mass (charge) of a single 9Be+, $\Omega_{c}=2\pi\times 7.597$ MHz is the cyclotron frequency, and $z$ ($r$) is the axial (radial) distance from the trap center. For rotation frequencies where the radial confinement is weak relative to transverse confinement ($\beta_{r}\ll 1$), we obtain a single ion plane. The rotation frequency at which the ion configuration transitions from two planes to one plane depends sensitively on the ion number Mitchell et al. (1998). For most experiments, we operate with 100 to 300 ions, which necessitates $\omega_{r}\lesssim 2\pi\times 48\text{ kHz}$ for single-plane conditions with $\omega_{z}=2\pi\times 795$ kHz. Figure 1: (color online) (a) Simplified illustration of the Penning trap electrode structure in cross section (not to scale). Static voltages are applied to the electrodes in the foreground (orange), while three of the six rotating wall electrodes (red) are shown in the background. A simulated single-plane ion configuration is shown at the trap center and magnified by $\sim$100 for visibility. The 313-nm intersecting optical dipole force beams are also illustrated. (b) Low-lying electronic levels of 9Be+ in the 4.46 T $B$-field of the Penning trap. Projection of total electronic angular momentum $(m_{J})$ is given to the right of each level. Nuclear spin projections have been excluded for clarity. Relevant transitions near 313 nm are labeled. We use Doppler laser cooling along both the axial ($z$-axis, out-of-plane) and radial (in-plane) trap dimensions to produce the Coulomb crystal. As shown in Fig. 1b, the hyperfine structure of 9Be+ exhibits a strong Zeeman shift in the large uniform $B$-field of the Penning trap. We laser cool along the $\sim$313 nm $\|J=1/2,m_{J}=+1/2\rangle\rightarrow\|3/2,+3/2\rangle$ cycling transition between the 2S1/2 and 2P3/2 manifolds, where $J$ and $m_{J}$ are the total electronic angular momentum and its projection along the $B$-field axis, respectively. The linewidth of this cooling transition is $\Gamma\sim 2\pi\times 17.97$ MHz Kramida et al. (2013), yielding a Doppler cooling limit of $\hbar\Gamma/2k_{B}\sim 0.43$ mK. For all experiments described here, the 9Be+ are optically pumped to the $\|I=3/2,m_{I}=+3/2\rangle$ nuclear spin state Itano and Wineland (1981). The two qubit states for our experiments are the ${\|\\!\\!\uparrow\rangle}\equiv\|1/2,+1/2\rangle$ and ${\|\\!\\!\downarrow\rangle}\equiv\|1/2,-1/2\rangle$ valence electron spin projections of the 2S1/2 electronic ground state. The cooling and repump transitions illustrated in Fig. 1b allow for efficient preparation of all $N$ trapped ions to the state ${\|\\!\\!\uparrow\rangle}_{N}\equiv{\|\\!\\!\uparrow\uparrow...\uparrow\rangle}$. The splitting between qubit levels is $\sim$124 GHz, and we perform global qubit rotations via direct application of resonant millimeter-wave radiation to the ions. We typically achieve $\pi$-pulse times ($t_{\pi}$) of $\sim$70 $\mu$s. As discussed below, for the experiments described here we perform global readouts of the qubit state through state-dependent resonance fluorescence on the Doppler cooling transition. Figure 2a illustrates a typical pulse sequence for qubit manipulation. We first prepare ${\|\\!\\!\uparrow\rangle}_{N}$ using the Doppler cooling and repump lasers. The spin echo sequence shown includes both $\pi/2$\- and $\pi$-pulses about the given Bloch sphere axes. The phase of the final pulse ($\Delta\phi$) is defined relative to that of the first $\pi/2$-pulse, which we define to be a rotation about the $y$-axis, and is varied depending on the intended final spin state. We use a spin echo sequence with free evolution periods of $\tau\sim 0.1$ to 1 ms to mitigate the deleterious effects of radial $B$-field inhomogeneity over the $\sim$400 $\mu$m ion plane diameter, and to cancel $B$-field fluctuations at frequencies below $\tau^{-1}$ Hahn (1950); Biercuk et al. (2009a). After the pulse sequence, we measure the population of spins in state ${\|\\!\\!\uparrow\rangle}$ ($P_{\uparrow}$) by switching on the Doppler cooling beams and counting scattered photons collected by an ultraviolet-sensitive photomultiplier. An $f/5$ objective imaging the side of the ion plane collects the scattered cooling photons, and a typical photon count rate per ion is $10^{3}$ s-1. Figure 2: (color online) (a) An example spin echo pulse sequence used for qubit state manipulation. Spins are initialized to ${\|\\!\\!\uparrow\rangle}_{N}$, and the 124 GHz is pulsed for the given duration at a specific phase, where $\Delta\phi$ is defined relative to the $y$-axis and is varied depending on the intended final spin state. (b) Detected histograms for three collective spin orientations of the qubit ensemble consisting of $174(10)$ spins. The collective state at the end of the spin-echo sequence is represented on a Bloch sphere above the corresponding histogram. The horizontal axis is scaled such that the mean photon counts detected when in state ${\|\\!\\!\uparrow\rangle}_{N}$ corresponds to unity. Each histogram is the result of 1000 state preparation and detection sequences, and bin widths have been adjusted between the three histograms to place them on the same vertical scale. The histograms of Fig. 2b give experimental results for three different values of $\Delta\phi$ as measured in a system of $N=174(10)$ spins. Each color-coded histogram is the result of 1000 pulse sequences (see Fig. 2a) and subsequent qubit state readouts. Bin widths for the three histograms are adjusted for clear presentation on a single vertical scale, and the horizontal axis is scaled to the photon counts collected for the state ${\|\\!\\!\uparrow\rangle}_{N}$. The standard deviation of the $\Delta\phi=\pi$ histogram suggests technical noise that is comparable to shot noise with 3510 photons collected. The increase in the standard deviation of the $\Delta\phi=\frac{\pi}{2}$ data is due to quantum spin projection noise ($\propto N^{-1/2}$) Itano et al. (1993). We generate a spin-dependent optical dipole force (ODF) by interfering two $\sim$313 nm laser beams at the ion plane to create an optical lattice (see Fig. 1a) Sawyer et al. (2012); Britton et al. (2012). The resulting lattice wavelength ($\lambda_{l}$) is determined by the crossing angle of the ODF beams as $\lambda_{l}=\lambda_{ODF}\left[2\sin(\frac{\theta_{R}}{2})\right]^{-1}$, where $\theta_{R}$ is the full beam crossing angle and $\lambda_{ODF}$ is the ODF laser wavelength. For this work, $\theta_{R}=4.2(2)^{\circ}$, which results in $\lambda_{l}\sim 3.7$ $\mu$m. The ODF laser frequency is detuned by $\sim$20 GHz from the nearest resonances for the ${\|\\!\\!\uparrow\rangle}$ and ${\|\\!\\!\downarrow\rangle}$ states (see Fig. 1b) and the linear polarization of each beam is chosen so as to produce a polarization gradient at the ion plane that imparts equal-magnitude, opposite-sign forces to the two qubit states with a magnitude of $\sim 10^{-23}$ N per spin at 1 W cm-2 per beam. The optical lattice wavevector ($\overrightarrow{\Delta k}$) is oriented along the $z$-axis of the trap to preferentially excite motion transverse to the crystal plane. The two ODF laser beams are produced from a single beam using a 50/50 beamsplitter, and their relative frequency is adjusted from zero to $\sim$10 MHz using acousto-optic modulators, enabling production of a standing- or running-wave spin-dependent optical lattice. ## III Thermal Dephasing Model In Ref. Sawyer et al. (2012) we excited arbitrary drumhead modes of a 2D trapped-ion crystal through application of a homogeneous spin-dependent force. The spin-dependent force coupled the 9Be+ ground-state valence electron spin and transverse motional degrees of freedom. We measured the decrease in the composite Bloch vector of the spins due to this coupling, and showed that this decrease (or decoherence) depended on the average energy or temperature of the motional state. We sketch the calculation of Ref. Sawyer et al. (2012) in Appendix A. In the Fock state basis thermal motional states are described by a diagonal density matrix. The calculation proceeds by assuming an initial Fock state $\left\|n\right\rangle$ for a mode. Application of a spin-dependent force produces spin-dependent displacements of the Fock state and decoherence of the spins is naturally described in terms of spin-motion entanglement and the increasing displacement sensitivity of Fock states with $n$. Here we use a model motivated by the dephasing picture of the Introduction that does not require quantum entanglement of the spin and motional degrees of freedom for thermal excitations large compared with the ground state size. For simplicity, we describe the dephasing model for the axial COM mode, although a generalization to other drumhead modes is straightforward. Center- of-mass motional modes play an important role in quantum information experiments with trapped ions. If all trapped ions possess the same charge-to- mass ratio, the COM mode frequency is independent of ion number and, in the case of transverse modes, constitutes the highest-frequency and longest- wavelength oscillation. In traps whose electrode dimensions are much larger than those of the ion crystal (i.e. Penning traps), the COM mode is the transverse mode most susceptible to noise from fluctuating potentials on trap electrodes. The interaction Hamiltonian for the spins and the COM degree of freedom is $\hat{H}_{ODF}=F_{0}\cos\left(\mu t+\varphi\right)\frac{z_{0}}{\sqrt{N}}\left(\hat{a}\,e^{-i\omega_{z}t}+\hat{a}^{\dagger}\,e^{i\omega_{z}t}\right)\sum_{i=1}^{N}\hat{\sigma}_{i}^{z},$ (5) where the sum is over the $N$ spins, $z_{0}=\sqrt{\hbar/\left(2M\omega_{z}\right)}$ is the ground state wavefunction size of a single trapped ion, $\hat{a}$ ($\hat{a}^{\dagger}$) are the lowering (raising) operators for the COM mode, and $\varphi$ is the ODF phase. In general the time evolution operator for the above Hamiltonian can be written as the product of a spin-dependent displacement operator, $\exp\left(\left[\alpha\hat{a}^{\dagger}-\alpha^{}\hat{a}\right]\sum_{i=1}^{N}\hat{\sigma}_{i}^{z}\right)$, and an evolution operator for a general Ising interaction that involves only pairwise spin interactions Kim et al. (2009); Sawyer et al. (2012). For resonant drives ($\mu\approx\omega_{z}$), the effect of the spin-dependent displacement typically dominates, and we neglect the induced Ising interaction throughout this manuscript. In this case the evolution operator for $\hat{H}_{ODF}$ separates into a product of $N$ individual spin-dependent displacement operators, $\hat{D}_{\text{SD}}\left(\alpha\right)=\prod_{i=1}^{N}\exp\left(\left[\alpha\hat{a}^{\dagger}-\alpha^{}\hat{a}\right]\hat{\sigma}^{z}_{i}\right).$ (6) The displacement amplitude, $\alpha$, for resonant ($\mu=\omega_{z}$) spin- dependent excitation of the COM mode for a time, $\tau$, and phase, $\varphi$, is $\alpha(\tau,\varphi)=-i\frac{F_{0}z_{0}\tau}{2\hbar\sqrt{N}}e^{i\varphi}.$ (7) For pulse sequences involving separated periods of ODF excitation (e.g. spin echo), the final motional displacement is simply a sum of individual displacements of the form of Eq. 7 with appropriate phases ($\varphi$) and times ($\tau$) for each of the ODF excitations within the sequence. We define spin-independent displacements, $\hat{D}(\alpha_{0})$, in the usual way: $\hat{D}\left(\alpha_{0}\right)=\exp\left(\alpha_{0}\hat{a}^{\dagger}-\alpha_{0}^{}\hat{a}\right).$ (8) In contrast to Ref. Sawyer et al. (2012), we consider the initial state of the COM mode for each experiment to be a coherent state $\left\|\alpha_{0}\right\rangle$. We denote the expectation value of a quantum operator $\hat{O}$ at the end of an experiment as $\left\langle\hat{O}\right\rangle$. We then perform an average over a thermal distribution of expectation values which we denote as $\left\langle\left\langle\hat{O}\right\rangle\right\rangle_{th}$. More precisely, we calculate thermal averages of a function, $A(\xi)$, of the continuous variable $\xi\equiv\|\alpha_{0}\|^{2}$ as $\langle A(\xi)\rangle_{th}\equiv\beta\int_{0}^{\infty}A(\xi)e^{-\beta\xi}d\xi,$ (9) where $\beta=\hbar\omega_{z}(k_{B}T)^{-1}$ for COM mode temperature $T$. For completeness, the Fock state calculations in the Appendix involve the corresponding thermal average over discrete Fock state expectation values, $A_{n}$, as $\langle A_{n}\rangle_{th}\equiv\left(1-e^{-\beta}\right)\sum_{n=0}^{\infty}A_{n}e^{-\beta n}.$ (10) ### III.1 Bloch Vector Length ($\Delta\phi=0$) Here we are interested in calculating the expectation value of a component of the composite Bloch vector, $\left(\hat{S}_{x},\hat{S}_{y},\hat{S}_{z}\right)=\left(\sum_{i=1}^{N}\frac{\hat{\sigma}^{x}_{i}}{2},\sum_{i=1}^{N}\frac{\hat{\sigma}^{y}_{i}}{2},\sum_{i=1}^{N}\frac{\hat{\sigma}^{z}_{i}}{2}\right)$, for initial spin states which are product states. This reduces to calculating the expectation value of a component of an individual spin $\vec{\sigma}_{i}$. For the evolution operator of Eq. 6, which is a product of commuting displacement operators involving individual spins, only the displacement operator involving $\hat{\sigma}^{z}_{i}$ non-trivially enters into the calculation. We assume that each experiment begins with the state, ${\|\\!\\!\uparrow\rangle}\|\alpha_{0}\rangle={\|\\!\\!\uparrow\rangle}\hat{D}(\alpha_{0})\|0\rangle$, where $\|\alpha_{0}\rangle$ is a coherent state of COM motion. In Sec. III.1 and III.2, we consider a Ramsey pulse sequence consisting of two $\pi/2$ pulses separated by a time $\tau$ as shown in Fig. 3. For this Ramsey sequence consisting of a single ODF excitation period, Eq. 7 is used to calculate spin- dependent displacements, $\alpha$. The measurements of Section IV involve spin echo sequences, but all of the theory results of Sections III.1 and III.2 apply with small modifications for calculating the spin-dependent displacement, $\alpha$. The first $\pi/2$ pulse of the sequence of Fig. 3 yields the qubit rotation $\displaystyle\|\psi_{1}\rangle$ $\displaystyle=$ $\displaystyle\hat{R}(\frac{\pi}{2},0){\|\\!\\!\uparrow\rangle}\|\alpha_{0}\rangle$ (11) $\displaystyle=$ $\displaystyle\frac{1}{\sqrt{2}}\left({\|\\!\\!\uparrow\rangle}+{\|\\!\\!\downarrow\rangle}\right)\hat{D}(\alpha_{0})\|0\rangle,$ where we define the following qubit rotation matrix $\hat{R}(\theta,\phi)=\left(\begin{array}[]{cc}\cos{(\frac{\theta}{2})}&-e^{-i\phi}\sin{(\frac{\theta}{2})}\\\ e^{i\phi}\sin{(\frac{\theta}{2})}&\cos{(\frac{\theta}{2})}\\\ \end{array}\right).$ (12) We now consider the effect of the spin-dependent ODF acting for the free evolution time, $\tau$. This yields the state $\displaystyle\|\psi_{2}\rangle$ $\displaystyle=$ $\displaystyle\frac{1}{\sqrt{2}}{\|\\!\\!\uparrow\rangle}\hat{D}(\alpha)\hat{D}(\alpha_{0})\|0\rangle$ (14) $\displaystyle+\frac{1}{\sqrt{2}}{\|\\!\\!\downarrow\rangle}\hat{D}(-\alpha)\hat{D}(\alpha_{0})\|0\rangle$ $\displaystyle=$ $\displaystyle\frac{1}{\sqrt{2}}e^{-i\theta_{0}}{\|\\!\\!\uparrow\rangle}\|\alpha+\alpha_{0}\rangle$ $\displaystyle+\frac{1}{\sqrt{2}}e^{i\theta_{0}}{\|\\!\\!\downarrow\rangle}\|-\alpha+\alpha_{0}\rangle,$ where $\theta_{0}=\text{Im}\\{\alpha\alpha_{0}^{\ast}\\}$. It is useful to pause at Eq. 14 before applying the final microwave pulse and evaluate $\langle\hat{S}_{x}\rangle$ and $\langle\hat{S}_{y}\rangle$. We find $\displaystyle\langle\hat{S}_{x}\rangle$ $\displaystyle=$ $\displaystyle\frac{N}{2}\langle\hat{\sigma}^{x}\rangle=\frac{N}{2}\cos{\left(4\,\text{Im}\\{\alpha^{\ast}\alpha_{0}\\}\right)e^{-2\|\alpha\|^{2}}}$ (15) $\displaystyle\langle\hat{S}_{y}\rangle$ $\displaystyle=$ $\displaystyle\frac{N}{2}\langle\hat{\sigma}^{y}\rangle=\frac{N}{2}\sin{\left(4\,\text{Im}\\{\alpha^{\ast}\alpha_{0}\\}\right)e^{-2\|\alpha\|^{2}}}.$ (16) From Eqs. 15 and 16, we see that the ODF has caused a coherent rotation of the composite Bloch vector about the $z$-axis by $\theta_{coh}=\arctan{\left(\langle\hat{S}_{y}\rangle/\langle\hat{S}_{x}\rangle\right)}=4\,\text{Im}\\{\alpha^{\ast}\alpha_{0}\\}.$ (17) The effect of spin-motion entanglement is reflected in the term $e^{-2\|\alpha\|^{2}}$ of Eqs. 15 and 16, which deviates negligibly from unity for the dephasing measurements discussed in Sec. IV. Thermal averages may be performed over the continuous variable $\xi\equiv\|\alpha_{0}\|^{2}$, where the magnitude $\xi$ is now weighted according to Boltzmann statistics and the phase of $\alpha_{0}\in\mathbb{C}$ is evenly distributed over $2\pi$ radians. Defining $\alpha=\|\alpha\|e^{i\phi^{\prime}}$ and $\alpha_{0}=\|\alpha_{0}\|e^{i\phi_{0}}$, we calculate: $\displaystyle\langle\theta_{coh}^{2}\rangle_{th}$ $\displaystyle=$ $\displaystyle\frac{\beta}{2\pi}\int_{0}^{2\pi}\int_{0}^{\infty}16\|\alpha\|^{2}\xi\sin^{2}{(\phi^{\prime}-\phi_{0})}e^{-\beta\xi}d\xi d\phi_{0}$ (18) $\displaystyle=$ $\displaystyle 8\|\alpha\|^{2}\beta^{-1}.$ In Sec. IV, a typical $\|\alpha\|$ for a 30-yN ODF driving at $\omega_{z}$ for 100 $\mu$s is $\sim\\!0.05$, while $\beta^{-1}\sim 12$ at the Be+ Doppler cooling limit, producing a non-negligible rotation angle standard deviation of $\sim\\!30^{\circ}$. Figure 3: (color online) Ramsey pulse sequence for global 124-GHz qubit rotations (upper) and ODF excitation (lower) as used for the derivations of Section III. The quantum states $\|\psi_{1}\rangle$, $\|\psi_{2}\rangle$, and $\|\psi_{f}\rangle$ are labeled at appropriate points in the sequence as described in the text. For completeness, we present $\|\psi_{f}\rangle$, $P_{\uparrow}$ and $\langle\hat{S}_{z}\rangle$ results following the final $\pi/2$-pulse, $\hat{R}(\frac{\pi}{2},0)$, $\displaystyle\|\psi_{f}\rangle$ $\displaystyle=$ $\displaystyle\frac{1}{2}{\|\\!\\!\uparrow\rangle}\left(e^{-i\theta_{0}}\|\alpha+\alpha_{0}\rangle-e^{i\theta_{0}}\|-\alpha+\alpha_{0}\rangle\right)$ (19) $\displaystyle+\frac{1}{2}{\|\\!\\!\downarrow\rangle}\left(e^{-i\theta_{0}}\|\alpha+\alpha_{0}\rangle+e^{i\theta_{0}}\|-\alpha+\alpha_{0}\rangle\right)$ $\displaystyle P_{\uparrow}^{(\alpha_{0})}$ $\displaystyle\equiv$ $\displaystyle\frac{1}{2}\left(1-\cos{\left(4\,\text{Im}\\{\alpha^{\ast}\alpha_{0}\\}\right)}e^{-2\|\alpha\|^{2}}\right)$ (20) $\displaystyle P_{\uparrow}$ $\displaystyle=$ $\displaystyle\langle P_{\uparrow}^{(\alpha_{0})}\rangle_{th}$ (21) $\displaystyle=$ $\displaystyle\frac{1}{2}\left(1-e^{-2\|\alpha\|^{2}(2\beta^{-1}+1)}\right)$ $\displaystyle\langle\hat{S}_{z}\rangle$ $\displaystyle=$ $\displaystyle-\frac{N}{2}\cos{\left(4\,\text{Im}\\{\alpha^{\ast}\alpha_{0}\\}\right)}e^{-2\|\alpha\|^{2}}$ (22) $\displaystyle\langle\hat{S}_{z}\rangle_{th}$ $\displaystyle=$ $\displaystyle-\frac{N}{2}e^{-2\|\alpha\|^{2}(2\beta^{-1}+1)}.$ (23) We note that $\beta^{-1}=\langle\|\alpha_{0}\|^{2}\rangle_{th}$. Also, for $\bar{n}\gg 1$, $\beta^{-1}\sim\bar{n}$ and the $P_{\uparrow}$ of Eq. 21 agrees with the treatment that assumes a thermal distribution of Fock states (Eq. 37) given in Appendix A, but offers a more classical description. That is, for each possible coherent state amplitude and phase, the result of the experimental sequence is that the composite Bloch vector undergoes coherent rotation by some angle $\theta_{coh}$. Over many such experimental sequences, we measure a dephasing of the composite Bloch vector associated with the angular variance, $\langle\theta_{coh}^{2}\rangle_{th}$, of Eq. 18. ### III.2 Dephasing ($\Delta\phi=\frac{\pi}{2}$) A final qubit rotation of $\hat{R}(\frac{\pi}{2},\frac{\pi}{2})$ ($\Delta\phi=\pi/2$) transforms rotations and dephasing in the $xy$-plane of the Bloch sphere to the detection ($z$) basis. Below we calculate the thermal average of the expectation value $\langle\hat{S}_{z}^{2}\rangle$ from which a temperature determination can be obtained. In addition, we discuss the implementation of a Monte Carlo analysis (see Sec. IV) that is in excellent agreement with measurements of thermal as well as non-thermal motional state distributions with $\bar{n}\gg 1$. For the $\Delta\phi=\frac{\pi}{2}$ pulse sequence, we obtain the following expression for $\langle\hat{S}_{z}\rangle$ after the final $\pi/2$ pulse: $\langle\hat{S}_{z}\rangle=\frac{N}{2}\sin{(4\text{Im}\\{\alpha^{\ast}\alpha_{0}\\})}e^{-2\|\alpha\|^{2}}.$ (24) Note that the thermal average of the expression in Eq. 24 vanishes, so we instead calculate dephasing through the second moment of $\hat{S}_{z}$: $\displaystyle\langle\hat{S}_{z}^{2}\rangle$ $\displaystyle=$ $\displaystyle\frac{1}{4}\sum_{i=1}^{N}\langle\hat{\sigma}^{z}_{i}\hat{\sigma}^{z}_{i}\rangle+\frac{1}{4}\sum_{i\neq j}\langle\hat{\sigma}^{z}_{i}\hat{\sigma}^{z}_{j}\rangle$ (25) $\displaystyle=$ $\displaystyle\frac{N}{4}+\frac{N(N-1)}{4}\langle\hat{\sigma}^{z}_{1}\hat{\sigma}^{z}_{2}\rangle$ (26) where $\langle\hat{\sigma}^{z}_{1}\hat{\sigma}^{z}_{2}\rangle$ is an expectation value involving any two non-identical spins within the ensemble. We simplify to Eq. 26 since, for COM excitation, all spins feel a force of equal magnitude and there is no differentiation between spin pairs. In evaluating the two-spin expectation, $\langle\hat{\sigma}^{z}_{1}\hat{\sigma}^{z}_{2}\rangle$, only the displacement operators in Eq. 6 involving $\hat{\sigma}^{z}_{1}$ and $\hat{\sigma}^{z}_{2}$ non-trivially enter into the calculation. We then compute the thermal average of this expectation value and obtain: $\displaystyle\langle\langle\hat{S}_{z}^{2}\rangle\rangle_{th}$ $\displaystyle=$ $\displaystyle\frac{N}{4}+\frac{N(N-1)}{4}\sigma^{2}$ (27) $\displaystyle\sigma^{2}$ $\displaystyle\equiv$ $\displaystyle\left\langle\langle\hat{\sigma}^{z}_{1}\hat{\sigma}^{z}_{2}\rangle\right\rangle_{th}$ (28) $\displaystyle=$ $\displaystyle\frac{1}{2}\left(1-\langle\cos{\left(8\,\text{Im}\\{\alpha^{\ast}\alpha_{0}\\}\right)}\rangle_{th}e^{-8\|\alpha\|^{2}}\right)$ (29) $\displaystyle=$ $\displaystyle\frac{1}{2}\left(1-e^{-8\|\alpha\|^{2}\left(2\beta^{-1}+1\right)}\right).$ (30) The first term of Eq. 27 is the contribution of spin projection noise while the second is due to dephasing. The quadratic scaling of the dephasing with $N$ relative to the linear scaling of projection noise indicates that thermal dephasing can be more accurately measured with larger ion numbers. We may recast the normalized thermal dephasing portion ($\sigma^{2}$) in terms of the COM mean phonon number ($\bar{n}\sim\beta^{-1}$): $\bar{n}\sim\frac{1}{16\|\alpha\|^{2}}\ln{\left[\frac{1}{1-2\sigma^{2}}\right]-\frac{1}{2}}.$ (31) Using Eq. 31, we can now extract the mean phonon occupation of the COM mode from measurements of spin dephasing. To determine $\sigma^{2}$ we subtract the calculable spin variance ($\frac{N}{4}$) from the measured $\langle\langle\hat{S}_{z}^{2}\rangle\rangle_{th}$ and normalize by the squared length of the composite Bloch vector ($\|\vec{S}\|^{2}=\frac{N^{2}}{4}$). For $N>100$, the difference between $\frac{N^{2}}{4}$ and $\frac{N(N-1)}{4}$ is below 1 %. In Appendix B we derive an analogous expression to Eq. 30 using Fock states that agrees in the limit $\beta^{-1}\sim\bar{n}$. As detailed in Sec. IV, this approach to calculating spin dephasing using coherent states motivates a straightforward Monte Carlo analysis in which, for each experiment, we choose a random initial coherent state of motion and subsequently apply the experimental pulse sequences and state readout. The initial coherent state magnitudes are weighted according to a thermal distribution while the phase is random and unweighted in the range $[0,2\pi)$. Each simulation includes a Bloch vector rotation of $\theta_{coh}$ determined by the randomized initial coherent state and fixed spin-dependent displacement $\alpha$. After many such Monte Carlo runs, we bin the outcomes into simulated histograms for direct comparison with experimental histograms. ## IV Center-of-Mass Measurements In this Section, we describe measurements of spin dephasing for different initial states of COM motion. We show experimentally that such measurements reveal not only the effective temperature of the COM mode ($\bar{n}$) Sawyer et al. (2012), but allow for a more detailed characterization of the initial motional state (e.g. thermal, coherent, or a mixture of the two). Figure 4: (color online) Experimental pulse sequences for global 124-GHz qubit rotations (upper) and ODF excitation (lower) along with corresponding data. Relative phases and rotation angles are given for each qubit rotation. The ODF drive frequencies $(\mu)$ and phases $(\phi_{\text{ODF}})$ are given as $(\mu,\phi_{\text{ODF}})$. (a) Measurement of COM temperature using collective Bloch vector length. The microwave spin echo sequence produces the state ${\|\\!\\!\downarrow\rangle}_{N}$ ($P_{\uparrow}=0$) in the absence of the ODF, and the ODF frequency is swept over the COM resonance ($\omega_{z}=2\pi\times 795$ kHz) with $\tau_{1}=500$ $\mu$s. Frequency-dependent deviation from $P_{\uparrow}=0$ is fit to a theoretical expression derived in Ref. Sawyer et al. (2012). The resulting experimental data (black points with error bars) is fit (solid red line) to extract a mode temperature of 0.4(1) mK given the applied ODF. (b) Extraction of COM mode temperature through direct measurement of spin dephasing. The COM mode is resonantly driven in each arm for $\tau_{2}=100$ $\mu$s and $\phi_{\text{ODF}}$ for the second arm is chosen to either undo ($\phi_{\text{ODF}}=0$) or enhance ($\phi_{\text{ODF}}=\pi$) the Bloch vector rotation produced by the spin-dependent force. Temperature is extracted from the excess width of the $\phi_{\text{ODF}}=\pi$ histogram (red bars) using Eq. 31, and the solid black-line histogram is the result of a Monte Carlo simulation of 50,000 experimental runs with no adjustable parameters. We determine a COM temperature of 0.42(7) mK using this dephasing measurement. Figure 4a shows the experimental pulse sequences for ODF laser beams and qubit rotations used to measure the Bloch vector length ($\Delta\phi=0$, Sec. IIIA) as in Ref. Sawyer et al. (2012), along with a plot of experimental data and corresponding theory fit. The collective spin of the trapped ions is first prepared in the state, ${\|\\!\\!\uparrow\rangle}_{N}$, and the pulses at $\sim$124 GHz constitute a Hahn spin echo that, in the absence of the ODF beams, leaves the spins in state ${\|\\!\\!\downarrow\rangle}_{N}$ with $>$99 % fidelity Biercuk et al. (2009b). We apply the ODF laser beams during each free evolution period of the spin echo for a duration $\tau_{1}=500$ $\mu$s. The relative frequency of the two beams is $\mu$, and the relative phase of the ODF beat between the first and second arms is given by $\phi_{\text{ODF}}$. Note that, for $\phi_{\text{ODF}}=\pi$ and $\mu=\omega_{z}$, the net result of the experimental sequence is identical to that of the Ramsey sequence of Fig. 3 with no intermediate $\pi$-pulse and a single free-precession period of $2\tau_{1}$. This is due to the condition that the ODF on state ${\|\\!\\!\uparrow\rangle}$ ($F_{\uparrow}$) is opposite in sign but equal in magnitude to that on ${\|\\!\\!\downarrow\rangle}$ ($F_{\downarrow}$), where $(F_{\uparrow}-F_{\downarrow})=2F_{0}$ (see Eq. 5). In this case the ODF of the second arm reinforces that of the first. In other words, the phase advance of the ODF beat by $\pi$ radians reverses the effect of the intermediate qubit $\pi$-pulse but retains the suppression of spin decoherence inherent in the spin echo. However, for the case $\mu\neq\omega_{z}$, the finite duration of the qubit $\pi$-pulse leads to a phase offset between the ion crystal oscillation and ODF drive at the start of the second arm given by $\delta(\tau_{1}+t_{\pi})$, where $\delta\equiv(\mu-\omega_{z})$, that must be included in the theoretical analysis. Assuming that we interact exclusively with the COM mode, the final position ($\alpha_{\text{SE}}$) of the ion crystal in phase space after application of the two ODF pulses of Fig. 4a is given by: $\displaystyle\alpha_{\text{SE}}(\tau)$ $\displaystyle=$ $\displaystyle\frac{F_{0}z_{0}}{2\hbar\sqrt{N}}\frac{\left(1-e^{i\delta\tau}\right)}{\delta}$ (32) $\displaystyle\times$ $\displaystyle\left(e^{i\varphi_{0}}-e^{i[\varphi_{0}+\delta(\tau+t_{\pi})+\phi_{\text{ODF}}]}\right)$ where $\tau$ is the duration of each evolution period, $\varphi_{0}$ is the ODF phase at the start of the first free evolution period, and we have included the additional $\phi_{\text{ODF}}$ to denote the added phase advance in the second free evolution period of the spin echo sequence. The common phase, $\varphi_{0}$, does not contribute to any experimental observables and may be disregarded. The relative minus sign between phase factors in the final term of Eq. 32 is due to the intermediate $\pi$-pulse of the spin echo sequence, which removes displacements common to both free evolution periods since $F_{\uparrow}=-F_{\downarrow}$. Previous experiments with ions confined within RF Paul traps have shown that Doppler laser cooling produces thermal states of ion motion Meekhoff et al. (1996). In Fig. 4a, we show that measurements of the Bloch vector length under application of a spin-dependent force with $\mu\sim\omega_{z}$ are consistent with that of a thermal state of motion whose temperature is 0.4(1) mK, the Doppler cooling limit. The frequency width of the spectral feature is approximately given by the Fourier width of the ODF pulse duration of $2\tau_{1}=1$ ms, and the degree of decoherence measured near $\delta=0$ is determined by the ODF magnitude and $\bar{n}$ according to Eqs. 21 and 32 Sawyer et al. (2012). The detuning-independent background decoherence level of $P_{\uparrow}\sim 0.07$ in Fig. 4a is due to spontaneous emission from the off-resonant ODF laser beams, and is fully characterized for this system Uys et al. (2010); Sawyer et al. (2012). ### IV.1 Thermal Distributions The measurement of Fig. 4a is one of composite Bloch vector length, and is only second-order sensitive to spin dephasing. To measure dephasing more directly and gain more complete knowledge of the spin statistics, we implement the experimental pulse sequence of Fig. 4b. For this experiment, we resonantly drive the COM mode ($\delta=0$) with the spin-dependent ODF during free- evolution periods of $\tau_{2}=100$ $\mu$s. We set the phase of the final $\pi/2$-pulse of the sequence to be the same as the intermediate $\pi$-pulse ($\Delta\phi=\frac{\pi}{2}$), thereby rotating any dephasing within the $xy$-plane to lie along the qubit axis. After each such sequence, we perform a projective measurement of $P_{\uparrow}$ and repeat for a total of 1000 experiments. The collection of 1000 $P_{\uparrow}$ values are binned and displayed as histograms in Fig. 4b. As an additional check, we choose $\phi_{\text{ODF}}$ to be either 0 or $\pi$. In the case of $\phi_{\text{ODF}}=0$ (light gray bars), the spin echo effectively cancels the spin-dependent excitation and concomitant Bloch vector rotation, allowing for characterization of other sources of dephasing such as spin projection noise, photon shot noise, AC Stark shift fluctuations from the ODF lasers between the two spin echo arms, and excess magnetic field fluctuations not fully canceled by the spin echo. For $\phi_{\text{ODF}}=\pi$ (red bars), the precession induced by the spin-dependent force in the first arm is enhanced in the second and we observe that the detected spin variance is greatly increased relative to the $\phi_{\text{ODF}}=0$ case. We extract a COM temperature of 0.42(7) mK from this measurement by applying Eq. 31 to the excess variance of the $\phi_{\text{ODF}}=\pi$ experiments. This temperature is in excellent agreement with the theoretical Doppler cooling limit as well as the Bloch vector length measurement of Fig. 4a. To further compare the measurements of Fig. 4b with the model of Sec. III.2, we use a Monte Carlo algorithm to produce simulated histograms consisting of 50,000 ‘detection’ events. This is a factor of 50 more detections than for the experimental measurements, and is so chosen to reduce noise in the Monte Carlo histograms for clearer distinction between experiment and simulation. Importantly, the simulations have no adjustable parameters – all inputs to the Monte Carlo algorithm are experimental parameters (e.g. $\alpha_{\text{SE}}$, $\tau$, $t_{\pi}$) or obtained from measurement (e.g. $\bar{n}$ from the measured $\sigma$). The simulated histograms are scaled vertically to match the experimental data given bin widths and total detection events. The simulation procedure is the following: for each Monte Carlo run, $k\in\\{1,...,5\times 10^{4}\\}$, we choose a random initial coherent state of COM motion given by $\|\alpha_{k}\|e^{i\phi_{k}}$. The probability of choosing a given magnitude, $\|\alpha_{k}\|$, is given by Boltzmann statistics for a thermal state. As such, the chosen magnitudes $\|\alpha_{k}\|$ follow a probability distribution proportional to $\exp(-\beta\|\alpha_{k}\|^{2})$. The random value of $\phi_{k}$ is unweighted and assigned from the set $[0,2\pi)$, which assumes the phase of the initial state of COM motion is uncorrelated with that of the ODF and varies for each experimental sequence. Following the pulse sequence of Fig. 4b with $\phi_{ODF}=\pi$, we apply relevant qubit rotations in sequence with coherent $z$-axis rotations due to the spin- dependent ODF given by Eqs. 15 and 16, where $\alpha_{k}$ constitutes the initial coherent state for each run and $\alpha_{\text{SE}}$ is the spin- dependent displacement calibrated as in Refs. Britton et al. (2012); Sawyer et al. (2012). For completeness, we also include the smaller measured variance in the $\phi_{ODF}=0$ case phenomenologically by adding another, uncorrelated random rotation to the composite Bloch vector. The angle of this additional rotation follows a Gaussian probability distribution with the measured variance (light gray bars in Fig. 4b)111This background dephasing contributes negligibly to the final dephasing of interest for the given experimental conditions, but is described here for completeness.. The simulated histogram (black line) of Fig. 4b shows good agreement with experimental measurements, and further supports the absolute temperature measurement of Fig. 4a. Note that the higher sensitivity of the direct spin-dephasing measurements enables the use of a shorter free-evolution period. As a result, direct measurements of spin dephasing are less sensitive to COM frequency and ODF phase drift within a single experiment. The effects of spontaneous emission decoherence are also negligible in this parameter regime. ### IV.2 Thermal Distributions with Large Coherent Displacements We now describe the spin dephasing signature of relatively large coherent motional displacements, $\alpha_{d}$, acting in addition to thermal fluctuations characterized by $\bar{n}$. Such analysis may be relevant when narrow-bandwidth electric field fluctuations exist on trap electrodes at frequencies near that of the COM. As in the previous section, we assume that the phases of thermal, coherent, and ODF displacements are all uncorrelated. It may seem that little can be inferred from dephasing in the presence of coherent excitations with a random phase, but we demonstrate that a constant magnitude is all that is required to distinguish such noise sources. We use resonant RF excitation of the COM mode to produce coherent states of motion whose square magnitude, $\|\alpha_{d}\|^{2}$, is larger than the thermal magnitude given by $\bar{n}$. To this end, we apply an oscillating voltage with a frequency of $\omega_{z}$ to the upper endcap electrode of the Penning trap for a period of 20 $\mu$s as in Ref. Biercuk et al. (2010). This homogeneous COM mode excitation is applied following the initial Doppler cooling and state preparation pulses but before the experimental pulse sequence of Fig. 4b. The amplitude of the applied voltage is 110 $\mu$V (1.2 mV m-1 at the ion position), corresponding to a coherent excitation of $\|\alpha_{d}\|\sim 8$ ($\sim 30$ nm amplitude). Histograms compiled from 1000 experimental runs without and with the resonant RF excitation are shown in Figs. 5a and 5b, respectively. The absolute COM temperature is determined to be 0.7(1) mK for the sequence without coherent excitation. This elevated COM temperature is due to the connection of the direct digital synthesizer used to apply resonant RF to the upper endcap. Nevertheless, we achieve $\|\alpha_{d}\|^{2}/\bar{n}\sim 3.5$ for these experiments. Figure 5b shows the effect of applying the coherent motional excitation, namely that the spin distribution is split into two peaks whose separation is determined by the combined coherent and ODF displacements. If the relative phase of $\alpha_{d}$ and $\alpha_{\text{SE}}$ were fixed for every experimental sequence, then the mean of the histogram of Fig. 5a would simply be shifted to a new position corresponding to a coherent Bloch vector rotation, with the standard deviation still reflecting the COM temperature of 0.7(1) mK. However, the randomness of the relative phases of $\alpha_{d}$ and $\alpha_{\text{SE}}$ in combination with their constant amplitudes leads to a characteristic splitting of the spin distribution about the mean. Figure 5: (color online) (a) Histogram of 1000 experimental runs (red bars) under similar conditions as that shown in Fig. 4b with a measured $\bar{n}=18(3)$. These data were taken without an initial coherent drive of the COM motion. Solid black histograms are obtained from Monte Carlo simulations of 50,000 experiment sequences with no adjustable parameters. (b) Measured and simulated histograms resulting from insertion of a spin- independent coherent excitation ($\|\alpha_{d}\|=8$) before the spin dephasing measurement. The effect of coherent excitation is visible despite the fact that the relative phases of $\alpha_{d}$ and the ODF $\alpha_{\text{SE}}$ are uncontrolled from one experiment to another. (c, d) Example phase space trajectories of the COM mode in a frame rotating at $\omega_{z}$ for each Monte Carlo simulation step, $k$, both without and with the coherent RF drive. The initial displacement, $\alpha_{k}$, for each simulation run is chosen randomly according to Boltzmann statistics to reflect thermal fluctuations of COM motion. As in Sec. IV.1, we use a Monte Carlo algorithm to simulate the experimental histograms of Fig. 5a and 5b. We simulate 50,000 runs to minimize noise relative to the experimental data. The simulation results given in Fig. 5a (black-line histogram) use the same procedure as described in Sec. IV.1. To include the coherent excitation for Fig. 5b, we simply redefine the initial coherent state amplitude to be $\left(\|\alpha_{k}\|e^{i\phi_{k}}+\|\alpha_{d}\|e^{i\phi^{\prime}_{k}}\right)$. Only the variables with the $k$ subscript change with each simulation run, as $\|\alpha_{d}\|=8$ is fixed for all pulse sequences. The amplitudes $\|\alpha_{k}\|^{2}$ once again follow a Boltzmann distribution, but here we use $T=0.7$ mK to reflect the elevated COM temperature for this set of measurements. The phases $\phi_{k}$ and $\phi^{\prime}_{k}$ are uncorrelated and chosen randomly from the set $[0,2\pi)$ for each simulation run. Example paths of COM motion through phase space for each Monte Carlo simulation run are shown schematically in Figs. 5c and 5d both without and with the initial RF pulse, respectively. The effect of the spin-dependent ODF is depicted as two oppositely-oriented vectors leading to a separation of the spin states by $2\alpha_{\text{SE}}\sim 0.1$, while all other excitations are common to both spins. ### IV.3 Heating Rates Motional heating in RF ion traps has gained increased prominence in recent years due primarily to studies of so-called “anomalous heating” from ion- surface proximity Deslauriers et al. (2006); Labaziewicz et al. (2008a, b); Wineland et al. (1998); Turchette et al. (2000); Wang et al. (2010). Recent work suggests surface contamination as the culprit Allcock et al. (2011); Hite et al. (2012). Additionally, micromotion in RF traps limits the useable size of Coulomb crystals for quantum information and quantum simulation experiments. Penning traps use static potentials for ion confinement and therefore do not induce micromotion, enabling the formation of large ion crystals Tan et al. (1995); Itano et al. (1998). Furthermore, the large physical size of typical Penning traps means a likely insensitivity to anomalous heating processes. Despite these encouraging features, no measurements have yet been reported for ambient heating of a resolved motional mode of an ion crystal in a Penning trap. A previous study of global (not mode-resolved) ambient heating of large 3D crystals in our Penning trap estimates that background gas collisions are a primary contributor Jensen et al. (2004). Heating rates of $\sim$65 mK/s were measured for background pressures of $\sim 4\times 10^{-9}$ Pa ($3\times 10^{-11}$ Torr), which translates to $\frac{d\bar{n}}{dt}\sim 1.7\times 10^{3}$ s-1 at our trap frequency of 795 kHz. We apply the thermometry techniques summarized in Fig. 4 to obtain an initial measurement of the ambient heating rate of the axial COM mode of our 2D crystals. We include a variable delay between initial Doppler cooling/state preparation and application of the experimental pulse sequences. Any increase in $\bar{n}$ over this initial delay period is measured in the subsequent decoherence or dephasing measurement. Figure 6 shows measured absolute COM temperatures as a function of initial delay (points with error bars) along with linear fits to each data set (solid lines). The data of panels 6a and 6b are identical, but plotted on linear and logarithmic vertical axes, respectively, for clarity. The fitted slopes reflecting $\frac{d\bar{n}}{dt}$ for each curve are displayed in Fig 6b and color-coded to match the corresponding data set. We measure the largest heating rate of $1.4(2)\times 10^{4}$ s-1 (black points) when the trap endcap electrodes are held at 0 V using the high voltage power supplies responsible for initial ion loading and transport. Small voltage fluctuations from these power supplies as well as electromagnetic interference along the connecting cables are the likely cause of this COM heating. Upon grounding the endcaps directly to the vacuum chamber at the high-voltage feedthrough, we observe an order-of-magnitude drop in the heating rate to between $4.7(8)\times 10^{2}$ s-1 (blue points) and $1.2(3)\times 10^{3}$ s-1 (red points). This approximate factor-of-two variation in heating rates is representative of our day-to-day observations with different ion samples, and we see no evidence of a correlation between total ion number and heating rate with the endcap electrodes grounded at the feedthrough. The mean of the two lowest heating rates corresponds to $\sim$30 mK/s, which is near previous collisional heating estimates Jensen et al. (2004). Additional filtering of the central ring and rotating-wall electrodes outside the vacuum envelope yielded no measurable improvement in COM heating 222Heating rate measurements of single ions in small RF traps are generally thought to be insensitive to background gas collisions. However, we believe our ambient heating measurements on hundreds of ions in a deep Penning trap should be sensitive to background gas collisions.. Because the distance from the trapped-ion arrays to the trap electrode surfaces ($\geq$2 cm) is large compared to the diameter of the planar array ($<$0.5 cm), electric field noise from trap electrode surfaces will be uniform across the array and preferentially heat the COM mode. For an array with $N$ ions this results in a linear dependence of the COM heating rate on ion number due to uniform electric field noise Home (2013). The lack of an observed $N$-dependence in the measured ambient heating rate indicates the source of the heating is likely not electric field noise. However, any potential anomalous heating must be less than the measured $\sim$103 s-1 ambient heating rate. Dividing this limit by the number of trapped ions ($N\sim 200$) gives a limit on the anomalous heating rate for a single trapped ion of $\sim$5 s-1 at a trap frequency of 795 kHz. Figure 6: (color online) Measured COM heating rates (points with error bars) and corresponding linear fits (solid lines) plotted with both (a) linear and (b) logarithmic vertical scales. The highest heating rate of $\frac{d\bar{n}}{dt}=1.4(2)\times 10^{4}$ s-1 is obtained with the trap endcap electrodes grounded through the high voltage power supplies used for initial ion loading. The two lower heating rates are measured with the endcaps shorted directly to the trap vacuum ground at the high voltage vacuum feedthrough. The variation between the two lowest rates is representative of the range of heating rates measured thus far under the given conditions. ## V Conclusion In summary, we have demonstrated a new technique for analyzing the motional state of a resolved ion crystal mode. The methods presented here do not rely on stimulated Raman transitions or Doppler linewidth analysis, and are in principle applicable to any resolved motional mode at any temperature provided the Lamb-Dicke confinement criterion is satisfied for the given mode. The sensitivity of our spin dephasing measurements allows for a regime of operation with negligible spin-motion entanglement and spontaneous emission decoherence. Monte Carlo simulations based on the semiclassical description of Sec. III are in excellent agreement with spin dephasing measurements. Furthermore, we observe a clear distinction between coherent and thermal states of motion, despite the randomness of the RF drive phase relative to our optical dipole force. The methods and analysis presented here enable very sensitive detection of coherently driven motion of a trapped-ion crystal, and may be used to phase- sensitively detect weak forces Biercuk et al. (2010); Schreppler et al. (2013). Section III can be used to estimate and optimize the force detection sensitivity for a given $N$ and temperature of the trapped-ion crystal. We estimate that the spin-motion coupling technique discussed here could improve on the force sensitivity obtained in Ref. Biercuk et al. (2010) by more than an order-of-magnitude. We also present the first measurements of ambient heating of a resolved mode of motion in a Penning trap. Future crystal heating measurements will include other resolved transverse motional modes with the goal of more clearly distinguishing between electric field fluctuations (mode-specific, $N$-dependent) and background gas collisions (mode- and $N$-independent). Nevertheless, we demonstrate that low heating rates are indeed achievable in Penning ion traps. ###### Acknowledgements. This work was supported by the DARPA-OLE program and NIST. The authors thank J. P. Home, K. R. A. Hazzard, M. Foss-Feig, and A. M. Rey for useful discussions as well as S. Kotler and J. P. Gaebler for comments on the manuscript. This manuscript is a contribution of NIST and not subject to U.S. copyright. ## Appendix A Calculating Bloch Vector Length using Fock States A thermal motional state is described by a density matrix that is a statistical mixture of Fock states. This motivates a calculation that assumes each experiment begins with the system in the state, $\|\\!\\!\uparrow\rangle\|n\rangle$, where $\|n\rangle$ is the harmonic oscillator Fock state of COM motion. The first $\pi/2$ pulse of the Ramsey sequence yields the qubit rotation $\|\psi_{1}\rangle=\hat{R}(\frac{\pi}{2},0)\|\\!\\!\uparrow\rangle\|n\rangle=\frac{1}{\sqrt{2}}\left({\|\\!\\!\uparrow\rangle}+{\|\\!\\!\downarrow\rangle}\right)\|n\rangle.$ (33) The spin-dependent ODF then produces displaced Fock states, $\|\alpha,n\rangle$, as Wünsche (1991) $\|\psi_{2}\rangle=\hat{D}_{\text{SD}}(\alpha)\|\psi_{1}\rangle=\frac{1}{\sqrt{2}}\left({\|\\!\\!\uparrow\rangle}\|\alpha,n\rangle+{\|\\!\\!\downarrow\rangle}\|-\alpha,n\rangle\right).$ (34) Note that $\|\psi_{2}\rangle$ involves entanglement of spin and motional degrees of freedom for nonzero $\alpha$. We now apply the final $\pi/2$ pulse whose phase is identical to the first ($\Delta\phi=0$) to obtain $\displaystyle\|\psi_{f}\rangle$ $\displaystyle=$ $\displaystyle\frac{1}{2}{\|\\!\\!\uparrow\rangle}\left(\|\alpha,n\rangle-\|-\alpha,n\rangle\right)$ (35) $\displaystyle+\frac{1}{2}{\|\\!\\!\downarrow\rangle}\left(\|\alpha,n\rangle+\|-\alpha,n\rangle\right).$ The probability of measuring ${\|\\!\\!\uparrow\rangle}$ for state $\|\psi_{f}\rangle$ depends on the overlap of $\|\alpha,n\rangle$ and $\|-\alpha,n\rangle$, and is given by $P_{\uparrow}^{(n)}=\frac{1}{2}\left(1-L_{n}\left(4\|\alpha\|^{2}\right)e^{-2\|\alpha\|^{2}}\right)$ (36) where $L_{n}$ is the Laguerre polynomial of order $n$. Our fluorescence detection is insensitive to the ion motional state, so we perform a Boltzmann- weighted thermal average over all Fock states to obtain $\displaystyle P_{\uparrow}$ $\displaystyle\equiv$ $\displaystyle\langle P_{\uparrow}^{(n)}\rangle_{th}$ (37) $\displaystyle=$ $\displaystyle\frac{1}{2}\left(1-e^{-2\|\alpha\|^{2}}\langle L_{n}\left(4\|\alpha\|^{2}\right)\rangle_{th}\right)$ $\displaystyle=$ $\displaystyle\frac{1}{2}\left(1-e^{-2\|\alpha\|^{2}(2\bar{n}+1)}\right).$ In the above equation, $\bar{n}=\left(e^{\beta}-1\right)^{-1}$ is the average COM mode occupation number for a thermal state at temperature, $T$, and $\beta=\hbar\omega_{z}/k_{B}T$. In the absence of the spin-dependent displacement, $P_{\uparrow}=0$ for this pulse sequence. However, as the displacement amplitude increases, $P_{\uparrow}$ takes on positive values that increase with mode occupation – eventually saturating at $P_{\uparrow}=0.5$ corresponding to complete loss of spin coherence. In Ref. Sawyer et al. (2012), we describe using this decoherence signature to perform mode spectroscopy and thermometry on a planar array of ions. We may also calculate the expectation value for the $N$-spin composite Bloch vector, $\langle\hat{S}_{z}\rangle=\langle\sum_{i=1}^{N}\frac{\hat{\sigma}^{z}_{i}}{2}\rangle=\frac{N}{2}\langle\hat{\sigma}^{z}\rangle$ using the final state of Eq. 35. We obtain $\displaystyle\langle\hat{S}_{z}\rangle$ $\displaystyle=$ $\displaystyle-\frac{N}{2}L_{n}\left(4\|\alpha\|^{2}\right)e^{-2\|\alpha\|^{2}}$ (38) $\displaystyle\langle\langle\hat{S}_{z}\rangle\rangle_{th}$ $\displaystyle=$ $\displaystyle-\frac{N}{2}e^{-2\|\alpha\|^{2}(2\bar{n}+1)}.$ (39) We note that the calculation in this Appendix, which uses the Fock state basis, motivates a picture of spin decoherence produced by entanglement of the spin and motional degrees of freedom. However, in the manuscript we show that for coherent input states of motion, spin decoherence can be explained by dephasing without resorting to quantum entanglement of spin and motion. ## Appendix B Calculating Dephasing using Fock States If the phase of the final microwave $\pi/2$-pulse is shifted by $\pi/2$ (e.g. $\hat{R}(\frac{\pi}{2},\frac{\pi}{2})$, $\Delta\phi=\pi/2$), then the composite Bloch vector will remain in the equatorial plane of the Bloch sphere and $\langle\hat{S}_{z}\rangle=\langle\hat{S}_{z}\rangle_{th}=0$. Fock states therefore produce no coherent spin rotation due to the spin-dependent ODF. To calculate dephasing, we must compute pairwise spin correlations of the form $\langle\hat{\sigma}^{z}_{1}\hat{\sigma}^{z}_{2}\rangle$ as shown in Eq. 26. As in Appendix A, we will consider the initial state of COM motion to be a Fock state, $\|n\rangle$. In contrast with the Bloch vector length calculation that requires only a single spin, we here consider a two-spin system whose full initial state is $\|\\!\\!\uparrow\uparrow\rangle\|n\rangle$. We construct the necessary two-spin rotation matrices, $\hat{R}^{(2)}(\frac{\pi}{2},0)$ and $\hat{R}^{(2)}(\frac{\pi}{2},\frac{\pi}{2})$, using Kronecker products as follows: $\displaystyle\hat{R}^{(2)}(\frac{\pi}{2},0)$ $\displaystyle\equiv$ $\displaystyle\hat{R}(\frac{\pi}{2},0)\otimes\hat{R}(\frac{\pi}{2},0)$ (44) $\displaystyle=$ $\displaystyle\frac{1}{2}\left(\begin{array}[]{rrrr}1&-1&-1&1\\\ 1&1&-1&-1\\\ 1&-1&1&-1\\\ 1&1&1&1\\\ \end{array}\right)$ $\displaystyle\hat{R}^{(2)}(\frac{\pi}{2},\frac{\pi}{2})$ $\displaystyle\equiv$ $\displaystyle\hat{R}(\frac{\pi}{2},\frac{\pi}{2})\otimes\hat{R}(\frac{\pi}{2},\frac{\pi}{2})$ (49) $\displaystyle=$ $\displaystyle\frac{1}{2}\left(\begin{array}[]{rrrr}1&i&i&-1\\\ i&1&-1&i\\\ i&-1&1&i\\\ -1&i&i&1\end{array}\right).$ We can now define the final state, $\|\psi_{f}\rangle$, after the pulse sequence of Fig. 3 ($\Delta\phi=\frac{\pi}{2}$) as $\|\psi_{f}\rangle=\hat{R}^{(2)}(\frac{\pi}{2},\frac{\pi}{2})\hat{D}_{\text{SD}}(\alpha)\hat{R}^{(2)}(\frac{\pi}{2},0)\|\\!\\!\uparrow\uparrow\rangle\|n\rangle,$ (50) where $\hat{D}_{\text{SD}}=\exp\left(\left[\alpha\hat{a}^{\dagger}-\alpha^{}\hat{a}\right]\sum_{i=1}^{2}\hat{\sigma}_{i}^{z}\right)$ and $\alpha$ is the spin-dependent displacement amplitude. 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Stamper-Kurn (2013), eprint arXiv:1312.4896. * Wünsche (1991) A. Wünsche, Quantum Opt. 3, 359 (1991).	arxiv-papers	2014-01-03T16:51:16	2024-09-04T02:49:56.259804	{ "license": "Public Domain", "authors": "Brian C. Sawyer, Joseph W. Britton, and John J. Bollinger", "submitter": "Brian Sawyer", "url": "https://arxiv.org/abs/1401.0672" }
1401.0718	Generalization of the Poisson distribution for the case of changing probabilities of consecutive events E.A.Kushnirenko† ###### Abstract In this paper the generalization of the Poisson distribution is derived for the case when each consecutive event changes event rate. A simple formula for the probability of observing of a given number of events for the selected period of time is derived for a given set of rates. Application of this distribution in high-energy physics calculations is discussed. Preface: This distribution was used to perform calculations of the backgrounds in the electron-photon beam interactions [1],[2]. Mathematical properties of the distribution are discussed here in more detail. This distribution can be used when interaction causes change of the detector properties or of the incident beam. Also it can be of certain interest as an example of yet another generalization of the Poisson distribution. ## 1 Introduction The Poisson distribution is used to describe a variety of different processes. It is used very often in the analysis of data from the experimental sets in accelerator physics, in cosmic rays physics, in radioactive decay studies, in fluctuations of energy losses of a particle moving in the matter and in many other cases. The only parameter that describes the distribution is the rate of events $\mu$ or the average number of events per unit time. The assumption that this parameter is constant gives the Poisson distribution unique features and a certain beauty, however limits its applications. Consider a several hundred $GeV$ electron beam going through a monochromatic photon beam going in the opposite direction. An electron going through the photon beam, can have several interactions with photons, and the energy of the electron will drop down by an order of magnitude after 2 interactions. The probability of the interaction of the electron with the monochromatic photon beam of a particular energy is defined by the energy of the electron. In this situation the rate $\mu$ of the Poisson process, which describes the probability of the electron-photon interaction changes dramatically after each interaction, for this reason even for an estimate of the number of the electron-photon interactions the Poisson distribution can not be used. Next we consider a process of shooting down a military aircraft with shells. The probability of hitting the aircraft grows after each consecutive hit, as its flying characteristics as well as defense systems go down and it becomes an easier target to hit. There could be different scenarios which describe variations of $\mu_{n}$. Clearly the classical Poison distribution will not describe the process correctly and this problem requires generalization of the Poisson distribution. Also it is quite natural, that after the interaction the properties of the detector change, which leads to the change of the rate of the detection of events. For example the detection efficiency can degrade after the interaction of the particle with the detector. In these examples, the rate of observed events during the observation period varies considerably, while during the period of constant rate $\mu$ the observed process is the Poisson process. These examples show that it can be useful to find distribution of events in processes where average rate changes. In this paper we consider a process where the rate $\mu$ changes after each consecutive event is happening, and the process for any length of time between events is Poisson, with different rates $\mu_{i}$. ## 2 The generalization of the Poisson process In general the problem can be formulated as follows: find a probability that for a given period of time ${(0,t)}$ there would be $n$ consecutive events, when the average rate of events $\mu$ is not constant, but changes instantaneously after each consequent event. Before the first event the rate is $\mu_{1}$, immediately after the first event and before the second event it is $\mu_{2}$, immediately after the second event and before third event $\mu_{3}$, immediately after event ${n-1}$ and before event $n$ the rate is $\mu_{n}$. In each interval between consecutive events the rate is constant. Generally speaking $\mu_{1},\mu_{2},\mu_{3},...,\mu_{n},....$ are independent finite positive numbers. We will consider that all of these numbers are different unless specifically stated otherwise. The solution of the problem is the generalization of the Poisson distribution which we look for. Obviously when ${\mu_{1}=\mu_{2}=...=\mu_{n}=...=\mu}$ the distribution should transform into the Poisson distribution. We will make the following assumptions: 1. 1. The probability ${P_{n}(t)}$ that during the period of time ${(0,t)}$ there would be ${n\geq 1}$ events is defined by the rates ${\mu_{1},\mu_{2},...,\mu_{n+1}}$ which characterize the rates before $1,2,\ldots,n$-th events and rate before event $n+1$. 2. 2. The probability of an event during the small period of time $\Delta t$ is proportional to the duration $\Delta t$ and the expected rate $\mu_{i}$ of events. 3. 3. The probability of 2 or more events during the small period of time $\Delta t$ is vanishingly small. 4. 4. For any $n\geq 1$ the event with number $n+1$ takes place later than the event with number $n$. Using these assumptions the probability $P_{0}(t+\Delta t)$ that during the period of time $(0,t+\Delta t)$ there are no events means that there are no events neither during the period $(0,t)$, nor during the period $(t,t+\Delta t)$. For short periods of time $\Delta t$ this probability in accordance with assumptions (2) and (3) is equal to: $\displaystyle P_{0}(t+\Delta t)=P_{0}(t)\cdot(1-\mu_{1}\cdot\Delta t)$ (1) where $P_{0}(t)$ and $(1-\mu_{1}\cdot\Delta t)$ are probabilities that there are no events neither during the period of time $(0,t)$, nor during the period of time $(t+\Delta t)$. With $\Delta t\rightarrow 0$ we have a differential equation: $\displaystyle\frac{dP_{0}(t)}{dt}=-\mu_{1}P_{0}(t)$ (2) with the initial condition: $\displaystyle P_{0}(0)=1$ (3) which has the solution $\displaystyle P_{0}(t)=e^{-\mu_{1}\cdot t}$ (4) Let us now find out the probability $P_{1}(t+\Delta t)$ of single event happening during the period of time $(t+\Delta t)$. In accordance with assumptions (1),(2),(3) there could be 2 possibilities: either the first event happens during the period $(0,t)$ and no events during the period $(t,t+\Delta t)$, or there are no events during the period of time $(0,t)$ and single event during the period $(t+\Delta t)$. So in accordance with assumptions (2),(3): $\displaystyle P_{1}(t+\Delta t)=P_{1}(t)\cdot(1-\mu_{2}\cdot\Delta t)+P_{0}(t)\cdot\mu_{1}\cdot\Delta t$ (5) Considering the difference $P_{1}(t+\Delta t)-P_{1}(t)$ and taking $\Delta t\rightarrow 0$ we obtain differential equation: $\displaystyle\frac{dP_{1}(t)}{dt}=-\mu_{2}P_{1}(t)+\mu_{1}P_{0}(t)$ (6) The solution of this equation using (Eq. 4) and initial condition $\displaystyle P_{1}(0)=0$ (7) is given by the formula: $\displaystyle P_{1}(t)=\mu_{1}\left[\frac{e^{-\mu_{1}t}}{\mu_{2}-\mu_{1}}+\frac{e^{-\mu_{2}t}}{\mu_{1}-\mu_{2}}\right]$ (8) Analogous to the above one can solve the equation for 2 events with the initial conditions $P_{2}(0)=P_{3}(0)=0$ (9) we find that: $\displaystyle P_{2}(t)=\mu_{1}\mu_{2}\left[\frac{e^{-\mu_{1}t}}{(\mu_{3}-\mu_{1})(\mu_{2}-\mu_{1})}+\right.$ $\displaystyle\left.+\frac{e^{-\mu_{2}t}}{(\mu_{3}-\mu_{2})(\mu_{1}-\mu_{2})}+\right.$ $\displaystyle\left.+\frac{e^{-\mu_{3}t}}{(\mu_{2}-\mu_{3})(\mu_{1}-\mu_{3})}\right]$ (10) $\displaystyle P_{3}(t)=\mu_{1}\mu_{2}\mu_{3}\left[\frac{e^{-\mu_{1}t}}{(\mu_{4}-\mu_{1})(\mu_{3}-\mu_{1})(\mu_{2}-\mu_{1})}+\right.$ $\displaystyle\left.+\frac{e^{-\mu_{2}t}}{(\mu_{4}-\mu_{2})(\mu_{3}-\mu_{2})(\mu_{1}-\mu_{2})}+\right.$ $\displaystyle\left.+\frac{e^{-\mu_{3}t}}{(\mu_{4}-\mu_{3})(\mu_{2}-\mu_{3})(\mu_{1}-\mu_{3})}+\right.$ $\displaystyle\left.+\frac{e^{-\mu_{4}t}}{(\mu_{3}-\mu_{4})(\mu_{2}-\mu_{4})(\mu_{1}-\mu_{4})}\right]$ (11) In general case the differential equation is $\displaystyle\frac{dP_{n}(t)}{dt}$ $\displaystyle=$ $\displaystyle-\mu_{n+1}P_{n}(t)+\mu_{n}P_{n-1}(t),{\rm when\,\,}n>0$ $\displaystyle\frac{dP_{n}(t)}{dt}$ $\displaystyle=$ $\displaystyle-\mu_{n+1}P_{n}(t){\hskip 55.97205pt\rm when\,\,}n=0$ (12) with initial condition $\displaystyle P_{n}(0)$ $\displaystyle=$ $\displaystyle 0,{\rm when\,\,}n>0$ $\displaystyle P_{n}(0)$ $\displaystyle=$ $\displaystyle 1,{\rm when\,\,}n=0$ (13) Comparing expressions for $P_{0}(t),P_{1}(t),P_{2}(t),P_{3}(t)$ and using method of mathematical induction it is straightforward to show that the solution for the probability $P_{n}(t)$ of $n$ events during the period $(0,t)$ $\displaystyle P_{n}(t)=\mu_{1}\mu_{2}...\mu_{n}\left[\frac{e^{-\mu_{1}t}}{(\mu_{n+1}-\mu_{1})(\mu_{n}-\mu_{1})...(\mu_{2}-\mu_{1})}+\right.$ $\displaystyle\left.+\frac{e^{-\mu_{2}t}}{(\mu_{n+1}-\mu_{2})(\mu_{n}-\mu_{2})...(\mu_{1}-\mu_{2})}+\right.\cdot\cdot\cdot$ $\displaystyle\left.\cdot\cdot\cdot+\frac{e^{-\mu_{n+1}t}}{(\mu_{n}-\mu_{n+1})(\mu_{n-1}-\mu_{n+1})...(\mu_{1}-\mu_{n+1})}\right]$ (14) The structure of the obtained formula is fairly straightforward: $P_{n}(t)$ contains $n+1$ terms, where every $j$-th term $(1\leq j\leq n+1)$ contains a fraction with $e^{-\mu_{j}t}$ in the numerator and the product $(\mu_{n+1}-\mu_{j})(\mu_{n}-\mu_{j})\ldots(\mu_{1}-\mu_{j})$ with $n$ factors but without $(\mu_{j}-\mu_{j})$ factor in the denominator. Figure 1: The distributions $P_{n}(t)$ for $\mu_{1}=1$, $\mu_{2}=2$, $\mu_{3}=3$, $\mu_{4}=0.5$, $\mu_{5}=1.50$, and $\mu_{6}=0$. Expression $P_{n}(t)$ can be written in more compact form: $\displaystyle P_{n}(t)=\left(\mathop{\displaystyle\prod}\limits_{i=1}^{n}\mu_{i}\right)\cdot\sum\limits_{i=1}^{n+1}\frac{e^{-\mu_{i}t}}{\mathop{\displaystyle\prod}\limits_{j=1;j\neq i}^{n+1}(\mu_{j}-\mu_{i})}$ (15) Remark: Clearly $P_{n}(t)$ is a function of not only time $t$, but also of parameters $\mu_{1},\mu_{2},\ldots,\mu_{n},\mu_{n+1}$, so it would be more appropriate to use notation $P_{n}(t,\mu_{1},\mu_{2},\ldots,\mu_{n+1})$. Nevertheless we will use the notation $P_{n}(t)$ for the sake of shortness, and will use the notation $P_{n}(t,\mu_{1},\mu_{2},\ldots,\mu_{n+1})$ in case when it is absolutely necessary. ## 3 Analysis of the solution ### 3.1 Expression for the $m$-th derivative of $P_{n}(t)$ and some other useful formulas Writing down differential equations (Eq. 12) for $n=1,2,3$ and taking into account (Eq. 2) we obtain quite helpful for the further calculations system of differential equations: $\displaystyle\frac{dP_{0}(t)}{dt}$ $\displaystyle=$ $\displaystyle-\mu_{1}P_{0}(t)$ $\displaystyle\frac{dP_{1}(t)}{dt}$ $\displaystyle=$ $\displaystyle-\mu_{2}P_{1}(t)+\mu_{1}P_{0}(t)$ $\displaystyle\frac{dP_{2}(t)}{dt}$ $\displaystyle=$ $\displaystyle-\mu_{3}P_{2}(t)+\mu_{2}P_{1}(t)$ $\displaystyle\ldots$ $\displaystyle\frac{dP_{n}(t)}{dt}$ $\displaystyle=$ $\displaystyle-\mu_{n+1}P_{n}(t)+\mu_{n}P_{n-1}(t)$ (16) Then $\displaystyle\frac{dP_{n}(t)}{dt}$ $\displaystyle=$ $\displaystyle(-1)^{1}\cdot\mu_{n+1}\cdot P_{n}(t)$ $\displaystyle+$ $\displaystyle(-1)^{0}\cdot\mu_{n}\cdot P_{n-1}(t)$ $\displaystyle\frac{d^{2}P_{n}(t)}{dt^{2}}$ $\displaystyle=$ $\displaystyle(-1)^{2}\cdot\mu_{n+1}^{2}\cdot P_{n}(t)$ $\displaystyle+$ $\displaystyle(-1)^{1}\cdot\mu_{n}\left(\mu_{n+1}+\mu_{n}\right)\cdot P_{n-1}(t)$ $\displaystyle+$ $\displaystyle(-1)^{0}\cdot\mu_{n}\mu_{n-1}\cdot P_{n-2}(t)$ $\displaystyle\frac{d^{3}P_{n}(t)}{dt^{3}}$ $\displaystyle=$ $\displaystyle(-1)^{3}\cdot\mu_{n+1}^{3}\cdot P_{n}(t)$ (17) $\displaystyle+$ $\displaystyle(-1)^{2}\cdot\mu_{n}\left(\mu_{n+1}^{2}+\mu_{n+1}\mu_{n}+\mu_{n}^{2}\right)\cdot P_{n-1}(t)$ $\displaystyle+$ $\displaystyle(-1)^{1}\cdot\mu_{n}\mu_{n-1}\left(\mu_{n+1}+\mu_{n}+\mu_{n-1}\right)\cdot P_{n-2}(t)$ $\displaystyle+$ $\displaystyle(-1)^{0}\cdot\mu_{n}\mu_{n-1}\mu_{n-2}\cdot P_{n-3}(t)$ General formula for the $m$-th derivative of $P_{n}(t)$, where $(n\geq 0)$, which includes system of equations (Eq. 16) is given by: $\displaystyle\frac{d^{m}P_{n}(t)}{dt^{m}}$ $\displaystyle=$ $\displaystyle(-1)^{m}\mu_{n+1}^{m}P_{n}(t)+$ (18) $\displaystyle+$ $\displaystyle\sum\limits_{s=1}^{m}(-1)^{m-s}\cdot\prod\limits_{j=n-s+1}^{n}\mu_{j}\cdot\left(\sum\limits_{i=n-s+1}^{n+1}\mu_{i}\right)^{\otimes(m-s)}\cdot P_{n-s}(t)$ where the following notation is used: $\displaystyle\left(\sum\limits_{i=1}^{r}\mu_{i}\right)^{\otimes n}=\sum\limits_{k_{1}+k_{2}+\ldots+k_{r}=n}\mu_{1}^{k_{1}}\cdot\mu_{2}^{k_{2}}\cdot\ldots\cdot\mu_{r}^{k_{r}},$ (19) For example: $\displaystyle\left(\mu_{1}+\mu_{2}\right)^{\otimes 3}$ $\displaystyle=$ $\displaystyle\mu_{1}^{3}+\mu_{1}^{2}\cdot\mu_{2}+\mu_{1}\cdot\mu_{2}^{2}+\mu_{2}^{3}$ $\displaystyle\left(\mu_{1}+\mu_{2}+\mu_{3}\right)^{\otimes 2}$ $\displaystyle=$ $\displaystyle\mu_{1}^{2}+\mu_{1}\mu_{2}+\mu_{2}^{2}+\mu_{2}\mu_{3}+\mu_{1}\mu_{3}+\mu_{3}^{2}$ Also in usual notation we have: $\displaystyle\left(\sum\limits_{i=1}^{3}\mu_{i}\right)^{\otimes n}=\sum\limits_{s=0}^{n}\sum\limits_{m=0}^{s}\mu_{1}^{s-m}\mu_{2}^{m}\mu_{3}^{n-s}=\sum\limits_{s=0}^{n}\sum\limits_{m=0}^{n-s}\mu_{1}^{n-s-m}\mu_{2}^{m}\mu_{3}^{s}$ (20) Expression for the derivative (Eq.18) was derived for $n\geq 0$ and $m<n$, but it can be expanded for any $n$ or $m$ if we assume that: $\displaystyle\left(\sum\limits_{i=1}^{r}\mu_{i}\right)^{\otimes n}$ $\displaystyle=$ $\displaystyle 1,\mbox{~{}when~{}}n=0,$ $\displaystyle\left(\sum\limits_{i=1}^{r}\mu_{i}\right)^{\otimes n}$ $\displaystyle=$ $\displaystyle 0,\mbox{~{}when~{}}n<0.$ (21) $\displaystyle P_{n}(t)$ $\displaystyle=$ $\displaystyle 0,\mbox{~{}when~{}}n<0$ It is interesting to note the behaviour of (Eq. 18) at $t=0$. First of all at $t=0$ $\displaystyle P_{n}(0)=0\mbox{ for }n\geq 1$ (22) $\displaystyle P_{n}(0)=1\mbox{ for }n=0$ (23) Then the first term in (Eq. 18) is equal to 0 for $n\geq 1$. In the second term all the terms of the sum vanish except for the term with $s=n$. According to (Eq. 21) we have non-zero terms only when $m\geq n$. So, given that $P_{0}(0)=1$, the formula (Eq. 18) at $t=0$ transforms into: $\displaystyle\left.\frac{d^{m}P_{n}(t)}{dt^{m}}\right\|_{t=0}=(-1)^{(m-n)}\cdot\mathop{\displaystyle\coprod}\limits_{j=1}^{n}\mu_{j}\cdot\left(\sum\limits_{i=1}^{n+1}\mu_{i}\right)^{\otimes(m-n)}\mbox{~{}when~{}}m\geq n.$ (24) Let us use $m=n+r$ for convenience, then the above formula transforms into $\displaystyle\left.\frac{d^{n+r}P_{n}(t)}{dt^{n+r}}\right\|_{t=0}=(-1)^{r}\cdot\mathop{\displaystyle\coprod}\limits_{j=1}^{n}\mu_{j}\cdot\left(\sum\limits_{i=1}^{n+1}\mu_{i}\right)^{\otimes(r)}\mbox{~{}when~{}}r\geq 0.$ (25) From expression (Eq. 18) given the remark (Eq. 21) we conclude that at $t=0$: the derivatives : $\displaystyle\left.\frac{d^{m}P_{n}(t)}{dt^{m}}\right\|_{t=0}=0,\mbox{~{}when~{}}m<n$ (26) According to (Eq. 25) the first derivative which does not vanish at $t=0$ of the function $P_{n}(t)$ has the rank $m=n$ and it is equal to $\displaystyle\left.\frac{d^{n}P_{n}(t)}{dt^{n}}\right\|_{t=0}=\mathop{\displaystyle\prod}\limits_{i=1}^{n}\mu_{i}$ (27) So we can combine the results in the following form: $\displaystyle\left.\frac{d^{m}P_{n}(t)}{dt^{m}}\right\|_{t=0}$ $\displaystyle=$ $\displaystyle(-1)^{(m-n)}\cdot\mathop{\displaystyle\coprod}\limits_{j=1}^{n}\mu_{j}\cdot\left(\sum\limits_{i=1}^{n+1}\mu_{i}\right)^{\otimes(m-n)}\mbox{~{}when~{}}m>n.$ (28) $\displaystyle\left.\frac{d^{m}P_{n}(t)}{dt^{m}}\right\|_{t=0}$ $\displaystyle=$ $\displaystyle\mathop{\displaystyle\coprod}\limits_{j=1}^{n}\mu_{j},\mbox{~{}when~{}}m=n$ (29) $\displaystyle\left.\frac{d^{m}P_{n}(t)}{dt^{m}}\right\|_{t=0}$ $\displaystyle=$ $\displaystyle 0,\mbox{~{}when~{}}m<n$ (30) taking derivative $d^{m}/dt^{m}$ in (Eq. 15) we obtain that $\displaystyle P_{n}(t)=\left(\mathop{\displaystyle\prod}\limits_{i=1}^{n}\mu_{i}\right)\cdot\sum\limits_{i=1}^{n+1}\frac{e^{-\mu_{i}t}}{\mathop{\displaystyle\prod}\limits_{j=1;j\neq i}^{n+1}(\mu_{j}-\mu_{i})}$ $\displaystyle\frac{d^{m}P_{n}(t)}{dt^{m}}=\left(\mathop{\displaystyle\prod}\limits_{i=1}^{n}\mu_{i}\right)\cdot\sum\limits_{i=1}^{n+1}\frac{(-1)^{m}\cdot\mu_{i}^{m}\cdot e^{-\mu_{i}t}}{\mathop{\displaystyle\prod}\limits_{j=1;j\neq i}^{n+1}(\mu_{j}-\mu_{i})}$ (31) Then for any given $m<n$ the sum of the set vanish: $\displaystyle\sum\limits_{i=1}^{n+1}\frac{(-1)^{m}\cdot\mu_{i}^{m}}{\mathop{\displaystyle\prod}\limits_{j=1;j\neq i}^{n+1}(\mu_{j}-\mu_{i})}=0$ (32) Given the condition $P_{n}(0)=0$ for $n>0$ the following set vanishes: $\displaystyle\sum\limits_{i=1}^{n+1}\frac{1}{\mathop{\displaystyle\prod}\limits_{j=1;j\neq i}^{n+1}(\mu_{j}-\mu_{i})}=0,\mbox{~{}when~{}}n>0,$ (33) which is a special case of (Eq. 32) when $m=0$ Taking the derivative $d^{n}/dt^{n}$ of the (Eq. 15) we conclude that the sum of the following set is equal to 1: $\displaystyle\sum\limits_{i=1}^{n+1}\frac{(-1)^{n}\cdot\mu_{i}^{n}}{\mathop{\displaystyle\prod}\limits_{j=1;j\neq i}^{n+1}(\mu_{j}-\mu_{i})}=1$ (34) In the general case, which applies to any $m$ and $n$ we have: $\displaystyle\sum\limits_{i=1}^{n+1}\frac{(-1)^{m}\cdot\mu_{i}^{m}}{\mathop{\displaystyle\prod}\limits_{j=1;j\neq i}^{n+1}(\mu_{j}-\mu_{i})}=(-1)^{m-n}\cdot\left(\sum\limits_{i=1}^{n+1}\mu_{i}\right)^{\otimes(m-n)},$ (35) assuming (Eq. 21) for ($m\leq n$) Taking into account (Eq. 24) and (Eq. 27) the Taylor series expansion in the vicinity of $t=0$ can be written as: $\displaystyle P_{n}(t)$ $\displaystyle=$ $\displaystyle P_{n}(0)+\sum\limits_{m=1}^{\infty}\frac{t^{m}}{m!}\cdot\left.\frac{d^{m}P_{n}(t)}{dt^{m}}\right\|_{t=0}=P_{n}(0)+\sum\limits_{m=n}^{\infty}\frac{t^{m}}{m!}\cdot\left.\frac{d^{m}P_{n}(t)}{dt^{m}}\right\|_{t=0}$ $\displaystyle=$ $\displaystyle P_{n}(0)+\sum\limits_{r=0}^{\infty}\frac{t^{n+r}}{(n+r)!}\cdot\left.\frac{d^{n+r}P_{n}(t)}{dt^{n+r}}\right\|_{t=0}$ $\displaystyle=$ $\displaystyle P_{n}(0)+\left(\mathop{\displaystyle\coprod}\limits_{j=1}^{n}\mu_{j}\right)\cdot\sum\limits_{r=0}^{\infty}(-1)^{r}\cdot\frac{t^{n+r}}{(n+r)!}\cdot\left(\sum\limits_{i=1}^{n+1}\mu_{i}\right)^{\otimes(r)}$ $\displaystyle=$ $\displaystyle P_{n}(0)+\left(\mathop{\displaystyle\coprod}\limits_{j=1}^{n}\mu_{j}\right)\cdot\frac{t^{n}}{n!}\sum\limits_{r=0}^{\infty}(-1)^{r}\cdot\left(\sum\limits_{i=1}^{n+1}\mu_{i}\right)^{\otimes(r)}\cdot\frac{t^{r}\cdot n!}{(n+r)!}$ ### 3.2 The limiting transition of $P_{n}(\mu_{1},\mu_{2},\ldots,\mu_{n+1},t)$ for the case when $(\mu_{1},\mu_{2},\ldots,\mu_{n+1})\rightarrow\mu$ For $n>0$ let us consider $P_{n}(t)$ in the form $\displaystyle P_{n}(t)=\left(\mathop{\displaystyle\prod}\limits_{j=1}^{n}\mu_{j}\right)\cdot\frac{t^{n}}{n!}\cdot\left[\sum\limits_{r=0}^{\infty}(-1)^{r}\frac{n!}{(n+r)!}\cdot t^{r}\cdot\left(\sum\limits_{i=1}^{n+1}\mu_{i}\right)^{\otimes r}\right]$ (37) Let all $\mu_{i}=\mu$. According to the notation (Eq. 19): $\displaystyle\left(\sum\limits_{i=1}^{n+1}\mu_{i}\right)^{\otimes r}=\sum\limits_{k_{1}+k_{2}+\ldots+k_{n+1}=r}\mu_{1}^{k_{1}}\cdot\mu_{2}^{k_{2}}\cdot\ldots\cdot\mu_{n+1}^{k_{n+1}}$ (38) The number $N$ of terms in the expression (Eq. 38) is given by the formula: $\displaystyle N=C_{n+r}^{r}=\frac{(n+r)!}{n!\cdot r!}.$ (39) Hence using formula (Eq. 37) $\displaystyle\lim_{\mu_{i}\rightarrow\mu}P_{n}(t)$ $\displaystyle=$ $\displaystyle\frac{\mu^{n}t^{n}}{n!}\sum\limits_{r=0}^{\infty}(-1)^{r}\cdot\frac{t^{r}}{r!}\cdot\mu^{r}$ (40) $\displaystyle=$ $\displaystyle\frac{\left(\mu\cdot t\right)^{n}}{n!}\cdot e^{-\mu\cdot t},\mbox{~{}when~{}}n>1$ $\displaystyle\lim_{\mu_{1}\rightarrow\mu}P_{0}(t)$ $\displaystyle=$ $\displaystyle e^{-\mu t},\mbox{~{}when~{}}n=0$ (41) So for any $n\geq 0$ generalized Poisson distribution transforms into Poisson distribution when $(\mu_{1},\mu_{2},\ldots,\mu_{n+1})\rightarrow\mu$ ### 3.3 Case of $\mu_{n+1}=0$ It is quite clear that if $\mu_{n+1}=0$, which basically means that there are only $n$ events, and event $n+1$ would never happen, then $P_{n}(t)$ is assimptotically growing to 1, and $P_{n+1}(t)=P_{n+2}(t)=\ldots=0$. Let us consider specific cases: $\mu_{1}>0,\mu_{2}=0$, then $P_{1}(t)=\mu_{1}\left[\frac{e^{-\mu_{1}t}}{\mu_{2}-\mu_{1}}+\frac{e^{-\mu_{2}t}}{\mu_{1}-\mu_{2}}\right]=1-e^{-\mu_{1}t}=1-P_{0}(t)$ (42) $\mu_{1,2}>0,\mu_{3}=0$, then $\displaystyle P_{2}(t)$ $\displaystyle=$ $\displaystyle\mu_{1}\mu_{2}\left[\frac{e^{-\mu_{1}t}}{(-\mu_{1})(\mu_{2}-\mu_{1})}+\frac{e^{-\mu_{2}t}}{(-\mu_{2})(\mu_{1}-\mu_{2})}+\frac{1}{\mu_{2}\cdot\mu_{1}}\right]$ (43) $\displaystyle=$ $\displaystyle 1-\mu_{1}\frac{e^{-\mu_{2}t}}{\mu_{1}-\mu_{2}}-\mu_{2}\frac{e^{-\mu_{1}t}}{\mu_{2}-\mu_{1}}$ $\displaystyle=$ $\displaystyle 1-P_{1}(t)+\mu_{1}\frac{e^{-\mu_{1}t}}{\mu_{2}-\mu_{1}}-\mu_{2}\frac{e^{-\mu_{1}t}}{\mu_{2}-\mu_{1}}$ $\displaystyle=$ $\displaystyle 1-P_{1}(t)-P_{0}(t)$ It is straightforward to show than when $\mu_{1,\ldots,n-1}>0,\mu_{n}=0$ then $\displaystyle P_{n-1}(t)$ $\displaystyle=$ $\displaystyle\mathop{\displaystyle\prod}\limits_{i=1}^{n-1}\mu_{i}\cdot\sum\limits_{i=1}^{n}\frac{e^{-\mu_{i}t}}{\mathop{\displaystyle\prod}\limits_{j=1;j\neq i}^{n}(\mu_{j}-\mu_{i})}$ (44) $\displaystyle=$ $\displaystyle 1+\mathop{\displaystyle\prod}\limits_{i=1}^{n-1}\mu_{i}\cdot\sum\limits_{i=1}^{n-1}\frac{e^{-\mu_{i}t}}{(-\mu_{i})\cdot\mathop{\displaystyle\prod}\limits_{j=1;j\neq i}^{n-1}(\mu_{j}-\mu_{i})}$ $\displaystyle=$ $\displaystyle 1-P_{n-2}(t)-\ldots-P_{1}(t)-P_{0}(t)$ Indeed assuming (Eq. 44) being true, we can derive that, when $\mu_{1,\ldots,n}>0$, and $\mu_{n+1}=0$ then $\displaystyle P_{n}(t)$ $\displaystyle=$ $\displaystyle\mathop{\displaystyle\prod}\limits_{i=1}^{n}\mu_{i}\cdot\sum\limits_{i=1}^{n+1}\frac{e^{-\mu_{i}t}}{\mathop{\displaystyle\prod}\limits_{j=1;j\neq i}^{n+1}(\mu_{j}-\mu_{i})}$ (45) $\displaystyle=$ $\displaystyle 1+\mathop{\displaystyle\prod}\limits_{i=1}^{n}\mu_{i}\cdot\sum\limits_{i=1}^{n}\frac{e^{-\mu_{i}t}}{(-\mu_{i})\cdot\mathop{\displaystyle\prod}\limits_{j=1;j\neq i}^{n}(\mu_{j}-\mu_{i})}$ $\displaystyle=$ $\displaystyle 1-\mathop{\displaystyle\prod}\limits_{i=1}^{n-1}\mu_{i}\cdot\frac{e^{-\mu_{n}t}}{\mathop{\displaystyle\prod}\limits_{j=1;j\neq i}^{n}(\mu_{j}-\mu_{i})}+\mathop{\displaystyle\prod}\limits_{i=1}^{n}\mu_{i}\cdot\sum\limits_{i=1}^{n-1}\frac{e^{-\mu_{i}t}}{(-\mu_{i})\cdot\mathop{\displaystyle\prod}\limits_{j=1;j\neq i}^{n}(\mu_{j}-\mu_{i})}$ $\displaystyle=$ $\displaystyle 1-P_{n-1}(t)+\mathop{\displaystyle\prod}\limits_{i=1}^{n-1}\mu_{i}\cdot\sum\limits_{i=1}^{n-1}\frac{e^{-\mu_{i}t}}{\mathop{\displaystyle\prod}\limits_{j=1;j\neq i}^{n}(\mu_{j}-\mu_{i})}+\mathop{\displaystyle\prod}\limits_{i=1}^{n-1}\mu_{i}\cdot\sum\limits_{i=1}^{n-1}\frac{\mu_{n}\cdot e^{-\mu_{i}t}}{(-\mu_{i})\cdot\mathop{\displaystyle\prod}\limits_{j=1;j\neq i}^{n}(\mu_{j}-\mu_{i})}$ $\displaystyle=$ $\displaystyle 1-P_{n-1}(t)+\mathop{\displaystyle\prod}\limits_{i=1}^{n-1}\mu_{i}\cdot\sum\limits_{i=1}^{n-1}\frac{e^{-\mu_{i}t}}{\mathop{\displaystyle\prod}\limits_{j=1;j\neq i}^{n}(\mu_{j}-\mu_{i})}\left[1+\frac{\mu_{n}}{-\mu_{i}}\right]$ $\displaystyle=$ $\displaystyle 1-P_{n-1}(t)+\mathop{\displaystyle\prod}\limits_{i=1}^{n-1}\mu_{i}\cdot\sum\limits_{i=1}^{n-1}\frac{e^{-\mu_{i}t}}{(-\mu_{i})\mathop{\displaystyle\prod}\limits_{j=1;j\neq i}^{n-1}(\mu_{j}-\mu_{i})}{\mathrm{\;\;hence\,\,using\,(Eq.\ref{eq:p_n-1_when0})}}$ $\displaystyle=$ $\displaystyle 1-P_{n-1}(t)-P_{n-2}(t)-\ldots- P_{1}(t)-P_{0}(t)$ ### 3.4 The Integral of $P_{n}(t)$ There are several ways to prove that: $\displaystyle\mu_{n+1}\cdot\int\limits_{0}^{\infty}P_{n}(t)\cdot dt=1$ (46) Indeed using direct integration and (Eq. 45) $\displaystyle\int\limits_{0}^{\infty}P_{n}(t)\cdot dt$ $\displaystyle=$ $\displaystyle\mathop{\displaystyle\prod}\limits_{i=1}^{n}\mu_{i}\cdot\sum\limits_{i=1}^{n+1}\frac{1}{\mu_{i}\mathop{\displaystyle\prod}\limits_{j=1;j\neq i}^{n+1}(\mu_{j}-\mu_{i})}=\frac{1}{\mu_{n+1}}\mathop{\displaystyle\prod}\limits_{i=1}^{n+1}\mu_{i}\cdot\sum\limits_{i=1}^{n+1}\frac{1}{\mu_{i}\mathop{\displaystyle\prod}\limits_{j=1;j\neq i}^{n+1}(\mu_{j}-\mu_{i})}$ (47) $\displaystyle=$ $\displaystyle\frac{1}{\mu_{n+1}}\left[P_{n}(0)+P_{n-1}(0)+\ldots+P_{1}(0)+P_{0}(0)\right]=\frac{1}{\mu_{n+1}}$ Similarly as $\mu_{1}\int\limits_{0}^{\infty}P_{0}(t)\cdot dt=1$ then using mathematical induction applied to (Eq. 12) one can prove that $\displaystyle\frac{dP_{n}(t)}{dt}$ $\displaystyle=$ $\displaystyle-\mu_{n+1}P_{n}(t)+\mu_{n}P_{n-1}(t)$ $\displaystyle\mu_{n+1}\int\limits_{0}^{\infty}P_{n}(t)\cdot dt$ $\displaystyle=$ $\displaystyle P_{n}(0)-P_{n}(\infty)+\mu_{n}\cdot\int\limits_{0}^{\infty}P_{n-1}(t)\cdot dt=1$ (48) ### 3.5 Normalisation of $P_{n}(t)$ Let us prove that the limit of the sum $S(t)=\sum\limits_{n=0}^{\infty}P_{n}(t)$ is equal to 1, at any given time $t$. In the special case when one of the $\mu_{i}=0$ the formula (Eq. 45) can be applied: $\displaystyle\sum\limits_{i=0}^{n-1}P_{i}(t)$ $\displaystyle=$ $\displaystyle 1-P_{n}(t)$ $\displaystyle P_{n+1}$ $\displaystyle=$ $\displaystyle P_{n+2}=\ldots=0,$ then clearly $\sum\limits_{n=0}^{\infty}P_{n}(t)=1$ For the case when every $\mu_{i}>0$ let us consider the sum: $\displaystyle S_{n}(t)=\sum\limits_{i=0}^{n}P_{i}(t)$ (49) Let us find the limit, to which the first derivative $S_{n}(t)$ is approximating when $n$ is growing. $\displaystyle\lim_{n\rightarrow\infty}\frac{dS_{n}(t)}{dt}=\sum\limits_{i=0}^{\infty}\frac{dP_{i}(t)}{dt}$ (50) Considering the system of equations (Eq. 16) and (Eq. 46) $\displaystyle\sum\limits_{i=0}^{n}\frac{dP_{i}(t)}{dt}=-\mu_{n+1}\cdot P_{n}(t)=-\frac{\int\limits_{0}^{t}P_{n}(t)\cdot dt}{\int\limits_{0}^{\infty}P_{n}(t)\cdot dt}$ (51) Clearly if we fix $t$ and keep increasing $n$ the fraction in (Eq. 51) vanishes. So for any given $t\geq 0$ and for any given set $\mu_{i}>0$: $\displaystyle\frac{dS(t)}{dt}=\lim_{n\rightarrow\infty}\frac{dS_{n}(t)}{dt}=0$ (52) and hence $S(t)=S(0)=\mbox{const}$ So, taking into account (Eq. 13) we derive that $S_{n}(0)=1$, for $n>0$. That means that for any $t\geq 0$ and for any given set $\mu_{i}$ the normalization of $P_{n}(t)$ is: $\displaystyle\sum\limits_{i=0}^{\infty}P_{i}(t)=S(t)=S(0)=1$ (53) ### 3.6 The symmetry of $P_{n}(\mu_{1},\mu_{2},\ldots,\mu_{n+1},t)$ under the permutation of parameters $\mu_{i},\mu_{j}$ One of very obvious properties of functions $P_{n}(t)$ is its independence from mutual interchange of parameters $\mu_{1},\mu_{2},\ldots,\mu_{n}$, because of terms simply interchange in (Eq. 15). $\displaystyle P_{n}(t)=\left(\mathop{\displaystyle\prod}\limits_{i=1}^{n}\mu_{i}\right)\cdot\sum\limits_{i=1}^{n+1}\frac{e^{-\mu_{i}t}}{\mathop{\displaystyle\prod}\limits_{k=1;k\neq i}^{n+1}(\mu_{k}-\mu_{i})}$ This gives us the possibility to put parameters $\mu_{i}$ in any order, for example in the order of growing $\mu_{i}$. Let us remark, that this does not apply to parameter $\mu_{n+1}$. In this case when we interchange $\mu_{i}$ with $\mu_{n+1}$, the distribution changes according to: $P_{n}(\mu_{1},\ldots,\mu_{i},\ldots,\mu_{n},\mu_{n+1},t)\cdot\mu_{n+1}=P_{n}(\mu_{1},\ldots,\mu_{n+1},\ldots,\mu_{n},\mu_{i},t)\cdot\mu_{i}$ So the distribution changes under the permutation of parameters $\mu_{i},\mu_{j}$ according to: $\displaystyle P_{n}(\mu_{1},\ldots,\mu_{i},\mu_{j},\ldots,\mu_{n},\mu_{n+1},t)$ $\displaystyle=$ $\displaystyle P_{n}(\mu_{1},\ldots,\mu_{j},\mu_{i},\ldots,\mu_{n},\mu_{n+1},t)$ (54) $\displaystyle P_{n}(\mu_{1},\ldots,\mu_{i},\ldots,\mu_{n},\mu_{n+1},t)$ $\displaystyle=$ $\displaystyle P_{n}(\mu_{1},\ldots,\mu_{n+1},\ldots,\mu_{n},\mu_{i},t)\frac{\mu_{i}}{\mu_{n+1}}$ (55) Given this property when studying properties of function $P_{n}(t)$ it is usually enough to study dependency $P_{n}(\mu_{1},\mu_{2},\ldots,\mu_{n},\mu_{n+1},t)$ with respect to $\mu_{n+1}$ and with respect to any of the parameters $\mu_{1},\ldots,\mu_{n}$, and we can use $\mu_{1}$ for example. ### 3.7 The limiting transition in $P_{n}(\mu_{1},\mu_{2},\ldots,\mu_{n+1},t)$ when several parameters $\mu_{i}$ are equal to each other . Let us assume that $k$ different parameters $\mu_{i}$ are equal to each other. Due to the symmetry of $P_{n}$ under the permutation of parameters $\mu_{i}$ we can calculate the case of $P_{n}(\mu_{1}=\mu_{2}=\ldots=\mu_{k}=\mu,\mu_{k+1},\ldots,\mu_{n+1})$ Let us consider $k=2$, then $\mu_{1}=\mu$ and we calculate the limit when $\mu_{2}\rightarrow\mu$: $\displaystyle\frac{P_{n}}{\mathop{\displaystyle\prod}\limits_{i=1}^{n}\mu_{i}}$ $\displaystyle=$ $\displaystyle\frac{e^{-\mu\cdot t}}{(\mu_{2}-\mu)\ldots(\mu_{n+1}-\mu)}+\frac{e^{-\mu_{2}\cdot t}}{(\mu-\mu_{2})\ldots(\mu_{n+1}-\mu_{2})}+\sum\limits_{i=3}^{n+1}\frac{e^{-\mu_{i}\cdot t}}{\mathop{\displaystyle\prod}\limits_{j=1;j\neq i}^{n+1}(\mu_{j}-\mu_{i})}$ Let us define the third term as $R(3)$ and define $\Delta\mu=\mu_{2}-\mu$, then expanding in terms power of $\Delta\mu$ $\displaystyle\frac{P_{n}}{\mathop{\displaystyle\prod}\limits_{i=1}^{n}\mu_{i}}$ $\displaystyle=$ $\displaystyle\frac{e^{-\mu\cdot t}}{\Delta\mu(\mu_{3}-\mu)\ldots(\mu_{n+1}-\mu)}\left[1+\frac{e^{-\Delta\mu\cdot t}}{(-1)(1-\frac{\Delta\mu}{\mu_{3}-\mu})\ldots(1-\frac{\Delta\mu}{\mu_{n+1}-\mu})}\right]+R(3)$ (57) $\displaystyle=$ $\displaystyle\frac{e^{-\mu\cdot t}}{\Delta\mu(\mu_{3}-\mu)\ldots(\mu_{n+1}-\mu)}\left[1-\left(1+(-\Delta\mu\cdot t)+\frac{(-\Delta\mu\cdot t)^{2}}{2!}+\ldots\right)\times\right.$ $\displaystyle\times$ $\displaystyle\left.\mathop{\displaystyle\prod}\limits_{i=3}^{n+1}\left(1+\frac{\Delta\mu}{\mu_{i}-\mu}+\left(\frac{\Delta\mu}{\mu_{i}-\mu}\right)^{2}+\ldots\right)\right]+R(3)$ $\displaystyle=$ $\displaystyle\frac{e^{-\mu\cdot t}}{\Delta\mu(\mu_{3}-\mu)\ldots(\mu_{n+1}-\mu)}\left[1-\left(1+\left[\sum_{i=3}^{n+1}\frac{\Delta\mu}{\mu_{i}-\mu}-\Delta\mu\cdot t\right]+\right.\right.$ $\displaystyle+$ $\displaystyle\left.\left.\left[\frac{(-\Delta\mu\cdot t)^{2}}{2!}+(-\Delta\mu\cdot t)\left(\sum_{i=3}^{n+1}\frac{\Delta\mu}{\mu_{i}-\mu}\right)^{\otimes 1}+\left(\sum_{i=3}^{n+1}\frac{\Delta\mu}{\mu_{i}-\mu}\right)^{\otimes 2}\right]+\ldots\right.\right.$ $\displaystyle+$ $\displaystyle\left.\left.\sum_{k=0}^{n}\left[\frac{(-\Delta\mu\cdot t)^{k}}{k!}\left(\sum_{i=3}^{n+1}\frac{\Delta\mu}{\mu_{i}-\mu}\right)^{\otimes{n-k}}\right]+\ldots\right)\right]+R(3)$ where notation (Eq. 19) was used. In the limit $\Delta\mu\rightarrow 0$ $\displaystyle\frac{P_{n}(\mu_{1},\mu_{2}=\mu)}{\mathop{\displaystyle\prod}\limits_{i=1}^{n}\mu_{i}}$ $\displaystyle=$ $\displaystyle\frac{e^{-\mu\cdot t}}{\mathop{\displaystyle\prod}\limits_{i=3}^{n+1}(\mu_{i}-\mu)}\left[t+\sum_{i=3}^{n+1}\frac{-1}{\mu_{i}-\mu}\right]+R(3)$ $\displaystyle\frac{P_{n}(\mu_{1},\mu_{2},\mu_{3}=\mu)}{\mathop{\displaystyle\prod}\limits_{i=1}^{n}\mu_{i}}$ $\displaystyle=$ $\displaystyle\frac{e^{-\mu\cdot t}}{\mathop{\displaystyle\prod}\limits_{i=4}^{n+1}(\mu_{i}-\mu)}\left[\frac{t^{2}}{2!}+t\cdot\sum_{i=4}^{n+1}\frac{-1}{\mu_{i}-\mu}+\left(\cdot\sum_{i=4}^{n+1}\frac{-1}{\mu_{i}-\mu}\right)^{\otimes 2}\right]+R(4)$ $\displaystyle\frac{P_{n}(\mu_{1},\mu_{2},\ldots,\mu_{k}=\mu)}{\mathop{\displaystyle\prod}\limits_{i=1}^{n}\mu_{i}}$ $\displaystyle=$ $\displaystyle\frac{e^{-\mu\cdot t}}{\mathop{\displaystyle\prod}\limits_{i=k+1}^{n+1}(\mu_{i}-\mu)}\left[\sum_{j=0}^{k-1}\frac{t^{k-j}}{(k-j)!}\times\left(\sum_{i=k+1}^{n+1}\frac{-1}{\mu_{i}-\mu}\right)^{\otimes j}\right]+R(k+1)$ where we use the notation $R(k)=\sum\limits_{i=k}^{n+1}\frac{e^{-\mu_{i}\cdot t}}{\mathop{\displaystyle\prod}\limits_{j=1;j\neq i}^{n+1}(\mu_{j}-\mu_{i})}$ (59) In a particular case when only $\mu_{n+1}$ differs from the rest of $\mu_{i}=\mu$ we obtain: $\displaystyle P_{2}(\mu_{1},\mu_{2}=\mu,\mu_{3})$ $\displaystyle=$ $\displaystyle\mu^{2}\left(\frac{e^{-\mu_{3}\cdot t}}{(\mu-\mu_{3})^{2}}-e^{-\mu\cdot t}\left[\frac{1}{(\mu-\mu_{3})^{2}}+\frac{t}{(\mu-\mu_{3})}\right]\right)$ $\displaystyle P_{3}(\mu_{1},\mu_{2},\mu_{3}=\mu,\mu_{4})$ $\displaystyle=$ $\displaystyle\mu^{3}\left(\frac{e^{-\mu_{4}\cdot t}}{(\mu-\mu_{4})^{3}}-e^{-\mu\cdot t}\left[\frac{1}{(\mu-\mu_{4})^{3}}+\frac{t}{(\mu-\mu_{4})^{2}}+\frac{t^{2}}{2!(\mu-\mu_{4})}\right]\right)$ $\displaystyle P_{n}(\mu_{1},\ldots,\mu_{n}=\mu,\mu_{n+1})$ $\displaystyle=$ $\displaystyle\mu^{n}\left(\frac{e^{-\mu_{n+1}\cdot t}}{(\mu-\mu_{n+1})^{n}}-e^{-\mu\cdot t}\left[\sum\limits_{j=0}^{n-1}\frac{t^{j}}{j!\left(\mu-\mu_{n+1}\right)^{(n-j)}}\right]\right)$ (60) ### 3.8 The limiting transition in $P_{n}(\mu_{1},\mu_{2},\ldots,\mu_{n+1},t)$ for the case when $\mu_{i}\rightarrow\infty$ It is easy to see from compact form of the distribution (Eq. 15): $\displaystyle P_{n}(t)=\left(\mathop{\displaystyle\prod}\limits_{i=1}^{n}\mu_{i}\right)\cdot\sum\limits_{i=1}^{n+1}\frac{e^{-\mu_{i}t}}{\mathop{\displaystyle\prod}\limits_{j=1;j\neq i}^{n+1}(\mu_{j}-\mu_{i})}$ that $\displaystyle\lim_{\mu_{1}\rightarrow\infty}P_{n}(\mu_{1},\mu_{2},\ldots,\mu_{n+1},t)$ $\displaystyle=$ $\displaystyle\left(\mathop{\displaystyle\prod}\limits_{i=2}^{n}\mu_{i}\right)\cdot\sum\limits_{i=2}^{n+1}\frac{e^{-\mu_{i}\cdot t}}{\mathop{\displaystyle\prod}\limits_{j=2,j\neq i}^{n+1}(\mu_{j}-\mu_{i})}\,\rm{,or}$ $\displaystyle\lim_{\mu_{1}\rightarrow\infty}P_{n}(\mu_{1},\mu_{2},\ldots,\mu_{n+1},t)$ $\displaystyle=$ $\displaystyle P_{n-1}(\mu_{2},\ldots,\mu_{n+1},t)$ which certainly makes sense, as in the limit $\mu_{1}\rightarrow\infty$ 1-st event happens right away and following events obey $P_{n-1}(t)$ distribution. Due to the symmetry under the permutations (Eq. 54) same formula applies to any $\mu_{i}$ where $i\leq n$. When the last parameter $\mu_{n+1}\rightarrow\infty$, then using (Eq. 55) $\displaystyle\lim_{\mu_{n+1}\rightarrow\infty}P_{n}(\mu_{1},\ldots,\mu_{n},\mu_{n+1},t)$ $\displaystyle=$ $\displaystyle\lim_{\mu_{n+1}\rightarrow\infty}P_{n}(\mu_{1},\ldots,\mu_{n+1},\mu_{n},t)\frac{\mu_{n}}{\mu_{n+1}}$ $\displaystyle=$ $\displaystyle P_{n-1}(\mu_{1},\ldots,\mu_{n},t)\frac{\mu_{n}}{\mu_{n+1}}$ we can unify the result in the following way: $\displaystyle\lim_{\mu_{i}\rightarrow\infty}P_{n}(t)$ $\displaystyle=$ $\displaystyle P_{n-1}(\mu_{1},\ldots,\mu_{i-1},\mu_{i+1},\ldots,\mu_{n+1},t)\,\,{\rm when}\,i\leq n$ (61) $\displaystyle\lim_{\mu_{i}\rightarrow\infty}P_{n}(t)$ $\displaystyle=$ $\displaystyle P_{n-1}(\mu_{1},\ldots,\mu_{n},t)\frac{\mu_{n}}{\mu_{n+1}}\,\,{\rm when}\,i=n+1$ (62) ### 3.9 Partial derivative $\partial P_{n}/\partial\mu_{i}$ Let us first calculate $\partial P_{n}/\partial\mu_{n+1}$ and for any other $\mu_{i}$ the partial derivative can be calculated using the symmetry (Eq. 54) and (Eq. 55) under the permutation of the parameters $\mu_{i}$. $\displaystyle\frac{\partial P_{1}(\mu_{1},\mu_{2},t)}{\partial\mu_{2}}$ $\displaystyle=$ $\displaystyle-\mu_{1}\exp(-\mu_{2}t)\cdot\left[\frac{e^{-(\mu_{1}-\mu_{2})t}+(\mu_{1}-\mu_{2})t-1}{(\mu_{2}-\mu_{1})^{2}}\right]$ $\displaystyle\frac{\partial P_{2}(\mu_{1},\mu_{2},\mu_{3},t)}{\partial\mu_{3}}$ $\displaystyle=$ $\displaystyle-\mu_{1}\mu_{2}e^{-\mu_{3}t}\cdot\left[\frac{e^{-(\mu_{1}-\mu_{3})t}+(\mu_{1}-\mu_{3})t-1}{(\mu_{3}-\mu_{1})^{2}(\mu_{2}-\mu_{1})}\right.$ $\displaystyle+$ $\displaystyle\left.\frac{e^{-(\mu_{2}-\mu_{3})t}+(\mu_{2}-\mu_{3})t-1}{(\mu_{3}-\mu_{2})^{2}(\mu_{1}-\mu_{2})}\right]$ $\displaystyle\frac{\partial P_{3}(\mu_{1},...,\mu_{4},t)}{\partial\mu_{4}}$ $\displaystyle=$ $\displaystyle-\left(\mathop{\displaystyle\prod}\limits_{i=1}^{3}\mu_{i}\right)e^{-\mu_{4}t}\left[\frac{e^{-(\mu_{1}-\mu_{4})t}+(\mu_{1}-\mu_{4})t-1}{(\mu_{4}-\mu_{1})^{2}(\mu_{3}-\mu_{1})(\mu_{2}-\mu_{1})}\right.$ $\displaystyle+$ $\displaystyle\left.\frac{e^{-(\mu_{2}-\mu_{4})t}+(\mu_{2}-\mu_{4})t-1}{(\mu_{4}-\mu_{2})^{2}(\mu_{3}-\mu_{2})(\mu_{1}-\mu_{2})}+\right.$ $\displaystyle+$ $\displaystyle\left.\frac{e^{-(\mu_{3}-\mu_{4})t}+(\mu_{3}-\mu_{4})t-1}{(\mu_{4}-\mu_{3})^{2}(\mu_{2}-\mu_{3})(\mu_{1}-\mu_{3})}\right]$ $\displaystyle\frac{\partial P_{n}(\mu_{1},\ldots,\mu_{n+1},t)}{\partial\mu_{n+1}}$ $\displaystyle=$ $\displaystyle-\left(\mathop{\displaystyle\prod}\limits_{i=1}^{n}\mu_{i}\right)\cdot e^{-\mu_{n+1}t}\sum\limits_{i=1}^{n}\frac{e^{-(\mu_{i}-\mu_{n+1})t}+(\mu_{i}-\mu_{n+1})t-1}{(\mu_{n+1}-\mu_{i})^{2}\cdot\mathop{\displaystyle\prod}\limits_{j=1,j\neq i}^{n}(\mu_{j}-\mu_{i})}$ (63) It is straightforward to show that $\partial P_{n}/\partial\mu_{n+1}<0$. Indeed expanding $e^{-(\mu_{i}-\mu_{n+1})t}$ we obtain: $\displaystyle\frac{\partial P_{n}(\mu_{1},\ldots,\mu_{n+1},t)}{\partial\mu_{n+1}}$ $\displaystyle=$ $\displaystyle-\mathop{\displaystyle\prod}\limits_{i=1}^{n}\mu_{i}\cdot e^{-\mu_{n+1}t}\sum\limits_{i=1}^{n}\frac{\sum\limits_{k=2}^{\infty}(-1)^{k}(\mu_{i}-\mu_{n+1})^{k}\frac{\cdot t^{k}}{k!}}{(\mu_{n+1}-\mu_{i})^{2}\cdot\mathop{\displaystyle\prod}\limits_{j=1,j\neq i}^{n}(\mu_{j}-\mu_{i})}$ $\displaystyle=$ $\displaystyle-\mathop{\displaystyle\prod}\limits_{i=1}^{n}\mu_{i}\cdot e^{-\mu_{n+1}t}\sum\limits_{k=0}^{\infty}\frac{t^{k+2}}{(k+2)!}\sum\limits_{i=1}^{n}\frac{(-1)^{k}(\mu_{i}-\mu_{n+1})^{k}}{\mathop{\displaystyle\prod}\limits_{j=1,j\neq i}^{n}(\mu_{j}-\mu_{i})}$ Then using (Eq. 35) $\displaystyle\frac{\partial P_{n}(\mu_{1},\ldots,\mu_{n+1},t)}{\partial\mu_{n+1}}$ $\displaystyle=$ $\displaystyle-\mathop{\displaystyle\prod}\limits_{i=1}^{n}\mu_{i}\cdot e^{-\mu_{n+1}t}\sum\limits_{k=0}^{\infty}\frac{t^{k+2}}{(k+2)!}(-1)^{k-n+1}\left(\sum\limits_{i=1}^{n}\mu_{i}-\mu_{n+1}\right)^{\otimes{k-n+1}}$ $\displaystyle=$ $\displaystyle-\mathop{\displaystyle\prod}\limits_{i=1}^{n}\mu_{i}\cdot e^{-\mu_{n+1}t}\sum\limits_{r=0}^{\infty}\frac{t^{n+r+1}}{(n+r+1)!}(-1)^{r}\left(\sum\limits_{i=1}^{n}\mu_{i}-\mu_{n+1}\right)^{\otimes{r}}$ The last expression is similar to the Taylor series expansion of $P_{n+1}(\mu_{i}-\mu_{n+1},t)$ (Eq. LABEL:eq:Pn_2) which is always positive. Hence $\partial P_{n}/\partial\mu_{n+1}<0$ ### 3.10 Analysis of functions $P_{n}(t)$ Expression for $P_{n}(t)$ where $(n\geq 1)$ (Eq. 15) is the sum of monotonically decreasing functions or monotonically increasing functions, depending on the sign of coefficeint before exponent. It is clear that $P_{n}(t)$ is unique, smooth, differentiable function with no discontinuities, which is equal to 0 when $t=0$ and when $t\rightarrow\infty$. From the Taylor series expansion (Eq. LABEL:eq:Pn_2) the function $P_{n}(t)$ obeys the following inequality: $\displaystyle\left(\mathop{\displaystyle\prod}\limits_{i=1}^{n}\mu_{i}\right)\cdot\frac{t^{n}}{n!}\cdot\exp{\left[-\sum\limits_{i=1}^{n+1}\mu_{i}\cdot t\right]}<P_{n}(t)<\left(\mathop{\displaystyle\prod}\limits_{i=1}^{n}\mu_{i}\right)\cdot\frac{t^{n}}{n!}\cdot e^{-\mu_{\min}\cdot t}$ (66) where $\mu_{\min}=\min(\mu_{1},\mu_{2},\ldots,\mu_{n+1}$). And it quite clear that $P_{n}(t)$ is positive, bounded function. ## 4 Conclusion So in this paper the generalization of the Poisson distribution is found for the case of changing rates caused by consecutive events. In case of constant event rate the distribution naturally transforms in the classical Poisson distribution. The derived generalization can have different applications, especially in simulation of cascade processes. It is possible that this problem was already solved and published, but the author did not manage to find such a publication. The author is grateful to S.M.Sergeev for the usefull discussion. ## References * [1] E.Kushnirenko, ”Generalization of Poisson Distribution for the Case of Changing Probabilities of Consequential Events”, IX Workshop on High Energy Physics and Quantum Field Theory, NPI MSU, Zvenigorod, September 16-22, 1994. Workshop materials were published in Moscow State University, Moscow, 1995, pp.362-368. http://inspirehep.net/record/387538 * [2] E.Kushnirenko, ”Estimates of the Electron-Gamma Conversion Background using Generalized Poisson Distribution”, VII International Workshop on Linear Colliders, Sept.29-Oct.3, 1997, Zvenigorod, Russia, Volume III, pp.1473-1476. * [3] Arsenin, V. Ya., Methods of Mathematical Physics, and Special Functions, ”Nauka”, Moscow, 1974. (Russian) * [4] G.Korn, T.Korn., Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review, 1970 (Russian) * [5] I.N. Bronshtein and K.A. Semendyayev. A Guide-Book to Mathematics for scientist and engineers, ”Nauka”, Moscow 1981. (Russian) * [6] Prudnikov, A.P. and Brychkov, I.U.A. and Marichev, O.I., Integrals and Series: Special functions, ”Nauka”, Moscow 1981. (Russian)	arxiv-papers	2014-01-03T20:59:14	2024-09-04T02:49:56.275276	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "E.A.Kushnirenko", "submitter": "Alexander Y. Kushnirenko", "url": "https://arxiv.org/abs/1401.0718" }
1401.0807	??–?? # A two-phase flow model of sediment transport: transition from bedload to suspended load FILIPPOCHIODI PHILIPPE CLAUDIN BRUNOANDREOTTI Laboratoire de Physique et Mécanique des Milieux Hétérogènes, (PMMH UMR 7636 ESPCI - CNRS - Univ. Paris Diderot - Univ. P. M. Curie) 10 rue Vauquelin, 75005 Paris, France. (???) ###### Abstract The transport of dense particles by a turbulent flow depends on two dimensionless numbers. Depending on the ratio of the shear velocity of the flow to the settling velocity of the particles (or the Rouse number), sediment transport takes place in a thin layer localized at the surface of the sediment bed (bedload) or over the whole water depth (suspended load). Moreover, depending on the sedimentation Reynolds number, the bedload layer is embedded in the viscous sublayer or is larger. We propose here a two-phase flow model able to describe both viscous and turbulent shear flows. Particle migration is described as resulting from normal stresses, but is limited by turbulent mixing and shear-induced diffusion of particles. Using this framework, we theoretically investigate the transition between bedload and suspended load. ††volume: ??? ## 1 Introduction When a sedimentary bed is sheared by a water flow of sufficient strength, the particles are entrained into motion. In a homogeneous and steady situation, the fluid flow can be characterized by a unique quantity: the shear velocity $u_{}$. The flux of sediments transported by the flow is an increasing function of $u_{}$, for which numerous transport laws have been proposed, for both turbulent flows (Meyer-Peter & Müller, 1948; Einstein, 1950; Bagnold, 1956; Yalin, 1963; van Rijn, 1984; Ribberink, 1998; Camemen & Larson, 2005; Wong & Parker, 2006; Lajeunesse et al., 2010) and laminar flows (Charru & Mouilleron-Arnould, 2002; Cheng, 2004; Charru et al. 2004, Charru & Hinch 2006). Despite a wide literature, some fundamental aspects of sediment transport are still only partially understood. For instance, the dynamical mechanisms limiting transport, in particular the role of the bed disorder (Charru, 2006) and turbulent fluctuations (Marchioli et al., 2006; Le Louvetel-Poilly et al., 2009), remain a matter of discussion. Also, derivations of transport laws have a strong empirical or semi-empirical basis, thus lacking more physics-related inputs. Here we investigate the properties of sediment transport using a two-phase continuum description. In particular, we examine the transition from bedload to suspension when the shear velocity is increased. A classical reference work for two-phase continuum models describing particle- laden flows, is the formulation of Anderson & Jackson (1967), later revisited by Jackson (1997). With this type of description, Ouriemi et al. (2009) and Aussillous et al. (2013) have addressed the case of bedload transport in laminar sheared flows. Similarly, Revil-Baudard & Chauchat (2012) have modelled granular sheet flows in the turbulent regime. In their description, these authors decompose the domain into two layers. An upper ‘fluid layer’ where only the fluid phase momentum equation is solved, and a lower ‘sediment bed layer’ where they apply a two-phase description. A layered structure of collisional sheet flows with turbulent suspension has also been proposed by Berzi (2011, 2013), based on kinetic theory of granular gases. Inspired by these works, we propose in this paper a general model which is able to describe sediment transport both in the laminar viscous regime and in the turbulent regime, and which does not require the domain to be split into several layers. In the literature, considerable emphasis has been placed on the process of averaging the equations of motion and defining stresses (Jackson 2000). However, an important issue beyond the averaging problem is the choice of closures consistent with observations. Here, the fluid phase is described by a Reynolds-dependent mixing length that has been recently proposed in direct numerical simulations of sediment transport (Durán et al., 2012). In the same spirit as Ouriemi et al. (2009), Revil-Baudard & Chauchat (2013) and Aussillous et al. (2013), the granular phase is described by a constitutive relation based on recent developments made in dense suspension and granular flows (GDR MIDI, 2004; Cassar et al., 2005; Jop et al., 2006; Forterre & Pouliquen, 2008; Boyer et al., 2011; Andreotti et al. 2012; Trulsson et al. 2012; Lerner et al. 2012). Our model then takes into account both fluid and particle velocity fluctuations. This paper is constructed as follows. In section 2, we describe our two-phase flow model, putting emphasis on the novelties we propose with respect to the previous works, namely the introduction of a Reynolds stress and particle diffusion induced both by turbulent fluctuations and by the motion of the particles themselves. Section 3 is devoted to the study of homogeneous and steady transport, simplifying the equations under the assumption of a quasi- parallel flow. The results of the model integration are presented in section 4. We discuss the evolution of the velocity, stress and concentration profiles when the shear velocity is increased, as well as the transport law. We end the paper with a summary and draw a few perspectives. ## 2 The two-phase model In this paper, we adopt a continuum description of a fluid-particle system. This approximation is valid whenever the various quantities of interest vary slowly at the scale of grains. On the contrary, such a model cannot accurately describe phenomena that involve particle-scale processes, like the sediment transport threshold (see below). ### 2.1 Continuity equation We define the particle volume fraction $\phi$ so that $1-\phi$ is the fraction of space occupied by the fluid. We respectively denote by $u_{i}^{p}$ and $u_{i}^{f}$, the particle and fluid Eulerian velocities. We assume incompressibility for both the granular and the fluid phases: the densities $\rho_{p}$ and $\rho_{f}$ are considered as constant. The continuity equation then reads: $\displaystyle\frac{\partial\phi}{\partial t}+\frac{\partial u_{i}^{p}\phi}{\partial x_{i}}$ $\displaystyle=$ $\displaystyle-\frac{\partial j_{i}}{\partial x_{i}}\;,$ (1) $\displaystyle\frac{\partial(1-\phi)}{\partial t}+\frac{\partial u_{i}^{f}(1-\phi)}{\partial x_{i}}$ $\displaystyle=$ $\displaystyle\frac{\partial j_{i}}{\partial x_{i}}\;,$ (2) where $j_{i}$ is a particle flux resulting from Reynolds averaging (see Appendix) and equal to the mean product of velocity and volume fraction fluctuations. We assume here that these correlated fluctuations lead to an effective diffusion of particles, quantified by a coefficient $D$: $j_{i}=-D\frac{\partial\phi}{\partial x_{i}}$ (3) We will propose later on a closure for the diffusion coefficient $D$. Here, it is important to underline that two types of velocity fluctuations will be taken into account: (i) those induced by the shearing of the grains, which are associated with the large non-affine displacements of particles in the dense limit (see details below) and with hydrodynamic interactions in the dilute limit (Eckstein et al., 1977; Leighton & Acrivos,1987, Nott & Brady, 1994; Foss & Brady, 2000) and (ii) those induced by turbulent velocity fluctuations. The introduction of this term corrects a major flaw of previously proposed two-phase models. Whenever particle-borne stresses tend to create a migration of particles, diffusion counteracts and tends to homogenize $\phi$. The formulation chosen here assumes that the system composed of the particles and the fluid is globally incompressible and does not diffuse: adding (1) and (2), one simply gets a relation between the Euler velocities for the two phases: $\frac{\partial[u_{i}^{p}\phi+u_{i}^{f}(1-\phi)]}{\partial x_{i}}=0$ (4) ### 2.2 Equations of motion Following Jackson (2000), the equations of motion are simply written as two Eulerian equations expressing the conservation of momentum for each phase: $\displaystyle\rho_{p}\left(\frac{\partial\phi u_{i}^{p}}{\partial t}+\frac{\partial\phi u_{j}^{p}u_{i}^{p}}{\partial x_{j}}\right)$ $\displaystyle=$ $\displaystyle\rho_{p}\phi{g_{i}}+\frac{\partial\sigma_{ij}^{p}}{\partial x_{j}}+f_{i}\;,$ (5) $\displaystyle\rho_{f}\left(\frac{\partial(1-\phi)u_{i}^{f}}{\partial t}+\frac{\partial(1-\phi)u_{j}^{f}u_{i}^{f}}{\partial x_{j}}\right)$ $\displaystyle=$ $\displaystyle\rho_{f}(1-\phi){g_{i}}+\frac{\partial\sigma_{ij}^{f}}{\partial x_{j}}-f_{i}\;.$ (6) The force density $f_{i}$ couples the two equations and represents the average resultant force exerted by the fluid on the particles. Here, $g_{i}$ is the gravity acceleration and $\sigma_{ij}^{p}$ and $\sigma_{ij}^{f}$ are respectively the stresses exerted on the particles and on the fluid. The stresses are additive: the total stress exerted on the effective medium (grains and fluid) is given by $\sigma_{ij}^{p}+\sigma_{ij}^{f}$ and $f_{i}$ becomes an internal force for this system. ### 2.3 Force exerted by one phase on the other The standard hypothesis, introduced by Jackson (2000), is to split the force density $\mathbf{f}$ exerted by the liquid on the solid into a component due to the Archimedes effect – the resultant force exerted on the contour of the solid, replacing the latter by liquid – and a drag force due to the relative velocity of the two phases. We write this force in the form $f_{i}=\phi\frac{\partial\sigma_{ij}^{f}}{\partial x_{j}}+\frac{3}{4}\phi C_{d}(\mathcal{R})\rho_{f}\,\frac{\left\|\mathbf{u}^{r}\right\|\,{u}_{i}^{r}}{d}\;,$ (7) where $\mathbf{u}^{r}=\mathbf{u}^{f}-\mathbf{u}^{p}$ is the relative velocity and $d$ is the grain size. The drag coefficient $C_{d}$ is a function of the grain-based Reynolds number $\mathcal{R}$ defined by $\mathcal{R}=\frac{\left\|\mathbf{u}^{r}\right\|\;d}{\nu}$ (8) using the fluid kinematic viscosity $\nu$. We write the drag coefficient in the convenient phenomenological form $C_{d}=\left(C_{\infty}^{1/2}+s\mathcal{R}^{-1/2}\right)^{2}$ (9) to capture the transition from viscous to inertial drag (Ferguson & Church, 2004). Both $C_{\infty}$ and $s$ _a priori_ depend on $\phi$. However, for the sake of simplicity, we will neglect this dependence and consider that $s$ is on the order of $\simeq\sqrt{24}\simeq 5$, as in the dilute limit. Similarly, we will take a constant asymptotic drag coefficient $C_{\infty}$ equal to $1$. After inserting the expression of $f_{i}$, the equations of motion are modifed to $\displaystyle\rho_{p}\left(\frac{\partial\phi u_{i}^{p}}{\partial t}+\frac{\partial\phi u_{j}^{p}u_{i}^{p}}{\partial x_{j}}\right)$ $\displaystyle=$ $\displaystyle\rho_{p}\phi{g_{i}}+\frac{\partial\sigma_{ij}^{p}}{\partial x_{j}}+\phi\frac{\partial\sigma_{ij}^{f}}{\partial x_{j}}+\frac{3}{4}\phi\,C_{d}(\mathcal{R})\rho_{f}\frac{\left\|\mathbf{u}^{r}\right\|\,{u}_{i}^{r}}{d},$ $\displaystyle\rho_{f}\left(\frac{\partial(1-\phi)u_{i}^{f}}{\partial t}+\frac{\partial(1-\phi)u_{j}^{f}u_{i}^{f}}{\partial x_{j}}\right)$ $\displaystyle=$ $\displaystyle\rho_{f}(1-\phi){g_{i}}+(1-\phi)\frac{\partial\sigma_{ij}^{f}}{\partial x_{j}}-\frac{3}{4}\phi\,C_{d}(\mathcal{R})\rho_{f}\frac{\left\|\mathbf{u}^{r}\right\|\,{u}_{i}^{r}}{d}.$ ### 2.4 Fluid constitutive relation In order to close the equations, one needs to express the stress tensors. For the fluid, we choose a local isotropic constitutive relation which allows us to recover the well-known regimes: $\sigma^{f}_{ij}=-P^{f}\delta_{ij}+\eta_{\rm eff}\;{\dot{\gamma}}^{f}_{\rm ij}\;+\rho_{f}(u_{i}^{f}j_{j}+u_{j}^{f}j_{i}),\quad{\rm with}\quad{\dot{\gamma}}^{f}_{\rm ij}=\frac{\partial u_{i}^{f}}{\partial x_{j}}+\frac{\partial u_{j}^{f}}{\partial x_{i}}$ (11) The last term is the flux of momentum associated with particle diffusion (see Appendix). The effective viscosity $\eta_{\rm eff}$ takes into account both the molecular viscosity and the mixing of momentum induced by turbulent fluctuations. The Reynolds stress, which results from correlated velocity fluctuations (see Appendix), is modelled using a Prandtl mixing length closure, which works well for turbulent shear flows: $\eta_{\rm eff}=\rho_{f}\left(\nu+\ell^{2}\|\dot{\gamma}^{f}\|\right)$ (12) where $\ell$ is the mixing length and $\|\dot{\gamma}^{f}\|$ is the typical mixing rate. This formulation neglects the influence of particles on the fluid effective rheology. This equation can be easily generalized to include a multiplicative factor $(1+5\phi/2)$ in front of the viscosity $\nu$. The Einstein viscosity is then recovered in the limit of dilute suspensions. Even close to the jamming transition, the effective fluid viscosity remains finite; the presence of particles only adds new solid boundary conditions and therefore reduces the fraction of volume where shear is possible. The dominant effect actually results from particle interactions, as discussed below. We have checked that the Einstein correction can then be safely ignored for the description of sediment transport, i.e. that this corrective factor has a negligible effect on the results. In the fully developed turbulent regime, far from the sand bed, the mixing length $\ell$ is proportional to the distance $z$ to the bed. Conversely, in the viscous regime, below some Reynolds number $R_{t}$, there is no velocity fluctuation so that $\ell$ must vanish. A common phenomenological approach (van Driest’s model for instance, see van Driest (1956) or Pope (2000)) is to express the turbulent mixing length as a function of the Reynolds number and $z$. However, this imposes the definition of an interface between the static and mobile zones, below which $\ell$ must vanish. To avoid the need for such an arbitrary definition, we propose instead a differential equation $\frac{d\ell}{dz}=\kappa\left[1-\exp\left(-\sqrt{\frac{\ell u^{f}}{R_{t}\nu}}\right)\right]$ (13) where $\kappa\simeq 0.4$ is the von Kármán constant and the dimensionless parameter $R_{t}\simeq 7$ is determined to recover the experimental ‘law of the wall’. The ratio $u^{f}\ell/\nu$ is the local Reynolds number based on the mixing length. It should be noted that a function other than the exponential can in principle be used, provided that it has the same behaviour in $0$ and $-\infty$, although the present choice provides a quantitative agreement with standard data (Durán et al., 2012). This formulation allows us to define $\ell$ both inside and above the static granular bed. The mixing length also depends on convective effects. In a first-order closure, this can be encoded as a dependence of $\kappa$ on the Richardson number ${\rm Ri}=\frac{N^{2}}{\|\dot{\gamma}\|^{2}},\quad{\rm with}\quad N^{2}=\frac{(\rho_{f}-\rho_{p})g}{\phi\rho_{p}+(1-\phi)\rho_{f}}\,\frac{d\phi}{dz}\,.$ (14) Whenever $N^{2}>0$, the flow is stably stratified. Here, $N$ is called the Brunt-Väisälä frequency. For the sake of simplicity, we have omitted this dependence on ${\rm Ri}$ but we acknowledge that it could lead to major effects in the context of particle and droplet transport in the convective boundary layer and in the stably stratified upper layer. Finally, a term must be, in principle, added to Eq. 11 to take into account the third-order correlation between velocity and volume fraction fluctuations on the right hand side of Eq. 45. Such a term would correspond to a transport of momentum resulting from a gradient of $\phi$ (independently of the convection effect), but, once again, we wish here to study this two-phase model in its simplest version. ### 2.5 Granular constitutive relation For the granular phase, we first decompose the stress tensor into pressure $P^{p}$ and deviatoric stress $\mathbf{\tilde{\sigma}^{p}}$: $\sigma^{p}_{ij}=-P^{p}\,\delta_{ij}\;+\tilde{\sigma}^{p}_{ij}\;-\rho_{p}(u_{i}^{p}j_{j}+u_{j}^{p}j_{i})$ (15) The last term is the flux of momentum associated with particle diffusion (see Appendix). We focus here on a first-order closure where the strain rate tensor and the volume fraction are the only state variables. We therefore assume that the granular temperature, defined as the variance of velocity fluctuations, is not an independent field but relaxes over a very short time scale to a value determined by the strain rate and the volume fraction. This approximation is excellent in the dense regime $\phi\to\phi_{c}$, where numerical simulations show no dependence of the rheology on the grain restitution coefficient. One expects it to be less and less accurate in the dilute limit $\phi\to 0$ and when the restitution coefficient is close enough to $1$. As in Ouriemi et al. (2009), we write the constitutive relation as a simple friction law: $P^{p}=\frac{\|\mathbf{\tilde{\sigma}^{p}}\|}{\mu(\phi)}$ (16) It should be noted that in Ouriemi et al. (2009) the friction law was expressed in terms of the inertial number $I=\dot{\gamma}d/\sqrt{P^{p}/\rho_{p}}$ (GDR MiDi, 2004). This was made possible thanks to the assumption that the volume fraction in the transport layer was almost a constant, equal to $\phi_{c}$. In general, the pressure is not a state variable when the variations of $\phi$ are taken into account self-consistently. We therefore express here the stress tensor (including pressure) as a function of the true state variables: $\phi$ and the strain rate tensor. As $I$ is no longer a state variable, a friction law $\mu(\phi)$ becomes the only possible choice. The equality (16) becomes an inequality, i.e. a Coulomb failure criterion, for a static bed, for which $\|\dot{\gamma}^{p}\|=0$. The friction coefficient $\mu$ decreases with the volume fraction $\phi$. Here we use a linear expansion around jamming ($\phi=\phi_{c}$) $\mu(\phi)\simeq\mu_{c}+\mu^{\prime}_{c}\left(\phi_{c}-\phi\right),$ (17) with $\mu_{c}=0.5$ and $\mu^{\prime}_{c}=0.1$. Following recent experimental and numerical results, the granular viscosity is written as: $\tilde{\sigma}^{p}_{ij}=\psi(\phi)\;\left(\rho_{f}\nu+\alpha\rho_{p}d^{2}\|{\dot{\gamma}}^{p}\|\right){\dot{\gamma}}^{p}_{\rm ij},\quad{\rm with}\quad{\dot{\gamma}}^{p}_{\rm ij}=\frac{\partial u_{i}^{p}}{\partial x_{j}}+\frac{\partial u_{j}^{p}}{\partial x_{i}}.$ (18) Here, $\alpha$ is the inverse of the Stokes number at which the transition from viscous to inertial suspension takes place. Trulsson et al. (2012) have found a value of $\alpha\simeq 0.6$ in a 2D simulation. $\psi(\phi)$ is a function that diverges at the critical volume fraction $\phi_{c}$ as $\psi(\phi)=\upsilon\frac{\phi^{2}}{\left(\phi_{c}-\phi\right)^{2}},$ (19) where $\upsilon$ is a numerical constant. The experimental values are approximately $\phi_{c}=0.615$ and $\upsilon=2.1$ (Ovarlez et al., 2006; Bonnoit et al., 2010). It should be noted, however, that the results presented here have been computed for $\alpha=1$, $\phi_{c}=0.64$ and $\upsilon=1$. In the dilute limit, the dynamics of suspension is determined by long-range hydrodynamic interactions (Eckstein et al., 1977; Leighton & Acrivos,1987, Nott & Brady, 1994; Foss & Brady, 2000). However, close to jamming, the mechanical properties of dense suspensions are related to the contact network geometry (Wyart et al. 2005; van Hecke, 2010; Lerner et al., 2012). Due to steric effects, the grains in a linear shear flow do not follow the average flow, but move cooperatively following ”floppy” modes. This deviation of the actual motion with respect to the mean motion is a type of fluctuation called non-affine motion of particles. To move by a distance as the crow flies equal to its diameter $d$, a grain makes a random-like motion whose average length diverges as $\sim d(\phi_{c}-\phi)^{-1}$. The statistical properties of grain trajectories and in particular their cooperative non-affine motions are mostly controlled by the volume fraction $\phi$, whatever the nature of the dissipative mechanisms (Andreotti et al., 2012). The enhanced viscosity close to jamming is directly related to the amplitude of non-affine motion, leading to a stress tensor diverging as $\psi(\phi)\sim(\phi_{c}-\phi)^{-2}$ (Boyer et al., 2011; Andreotti et al., 2012; Trulsson et al., 2012; Lerner et al., 2012). Numerical simulations have furthermore suggested that the function $\mu(\phi)$ does not depend on the microscopic interparticle friction coefficient nor on the flow regime (overdamped or inertial), see Fig. 1. However, the critical volume fraction $\phi_{c}$ does depend on this microscopic interparticle friction coefficient. This constitutes the most important consequence of having lubricated quasi-contacts rather than true frictional ones. Figure 1: Calibration of the functions $\mu(\phi)$ and $\psi(\phi)$ in a 2D numerical simulation taking into account viscous drag, inertia, contact force, solid friction and lubrication forces (Trulsson et al., 2012). The different colors correspond to different values of the Stokes number $\rho_{p}d^{2}\|{\dot{\gamma}}^{p}\|/\rho_{f}\nu$, which characterizes the relative amplitude of the two terms in Eq. 18. Here again, we argue due to the simplicity of the asymptotic expression of $\psi(\phi)$ one should use it, even in the dilute limit. The model could be easily generalized to a more complex choice for $\psi(\phi)$. We have checked that the results presented here do not depend much on such refinements. ### 2.6 Diffusion Apart from convective transport associated with the mean flow, velocity fluctuations lead to a mixing of particles and tend to homogenize the volume fraction $\phi$. We model these effects by two diffusive terms. The first is associated to turbulent fluctuations. Turbulent diffusivity is usually found to scale with the turbulent viscosity and is thus proportional to $\ell^{2}\|\dot{\gamma}^{f}\|$. The second source of diffusion, dominant in the dense regime, originates from the non-affine motion of particles. The particle-induced diffusivity therefore scales as $d^{2}\|\dot{\gamma}^{p}\|\psi(\phi)$. A similar scaling law has also been proposed to describe shear-induced diffusion in relatively dilute suspensions (Eckstein et al., 1977; Leighton & Acrivos,1987). In that case, the equivalent of the function $\psi(\phi)$, typically increasing like $\phi^{2}$ up to $\phi\simeq 30\%$, represents the interactions between particles mediated by hydrodynamics. This diffusivity has been measured from simulations based on Stokesian dynamics (e.g., Nott & Brady, 1994; Foss & Brady, 2000). Adding the two contributions, the diffusion coefficient reads $D=\frac{\ell^{2}\|\dot{\gamma}^{f}\|}{{\rm Sc}}+\frac{d^{2}\|\dot{\gamma}^{p}\|\psi(\phi)}{S_{\phi}}$ (20) The constant ${\rm Sc}$ is called the turbulent Schmidt number and lies in the range $0.5$–$1$ (Coleman 1970; Celik & Rodi 1988; Nielsen 1992). For the results presented here, we have chosen ${\rm Sc}=1$. We have introduced here a second phenomenological constant $S_{\phi}$, which we take equal to $1$ as well. ## 3 Homogeneous steady transport ### 3.1 Dimensional analysis We now apply this two-phase flow model to the description of sediment transport. In this article, we limit ourselves to the analysis of saturated transport over a flat bed, for which both phases are homogeneous along the flow ($x$) direction (but not along $z$) and in a steady state. It is worth emphasizing that vertical velocities do not vanish. The total vertical mass flux can still vanish, due to particle diffusion along the vertical direction. As discussed below, the condition of stationarity is ensured when the erosion rate of the bed is exactly balanced by a deposition rate induced by the settling of the particles. In the following, we will keep the equations dimensional. It is still interesting to perform the dimensional analysis of the problem. We will consider that the characteristic length is set by the grain diameter $d$, that the characteristic density is given by $\rho_{f}$ and that $(\rho_{p}-\rho_{f})gd$ provides the characteristic stress. Apart from the constants of the model, there are two control parameters and therefore two dimensionless numbers. The first, called the Shields number, is the rescaled shear stress: $\Theta=\frac{\rho_{f}u_{}^{2}}{(\rho_{p}-\rho_{f})gd}\,.$ (21) It compares the fluid-borne shear stress with the buoyancy-free gravity. The second parameter is based on the kinematic viscosity $\nu$ of the fluid. Combining this viscosity with gravity, one builds the viscous diameter: $d_{\nu}=\left(\frac{\rho_{p}}{\rho_{f}}-1\right)^{-1/3}~{}\nu^{2/3}~{}g^{-1/3},$ (22) which corresponds to the grain size for which inertial, gravity and viscous effects are of the same order of magnitude. The dimensionless number is then the ratio $d/d_{\nu}$. Equivalently, the Galileo number is defined as $(d/d_{\nu})^{3}$; the Reynolds number in Stokes sedimentation is $\frac{1}{18}(d/d_{\nu})^{3}$. Other dimensionless numbers, like the particle Reynolds number in the flow are obtained by combining the Shields number $\Theta$ and the dimensionless diameter $d/d_{\nu}$. ### 3.2 Quasi-parallel flow assumption We consider a flat sand bed homogeneous along the $x$-axis. The continuity equation then reads $\phi u^{p}_{z}=(\phi-1)u^{f}_{z}=-j_{z}=D\frac{d\phi}{dz}\,.$ (23) It expresses the balance between the convective flux associated with the downward particle motion and the turbulent diffusive flux. As they are related by the continuity equation, the vertical velocities can be expressed as functions of their relative velocity $u^{r}_{z}$: $u^{f}_{z}=\phi u^{r}_{z}$ and $u^{p}_{z}=(\phi-1)u^{r}_{z}$. The diffusive flux then reads $j_{z}=\phi(1-\phi)u^{r}_{z}$. We consider the asymptotic limit where horizontal velocities are much larger than vertical ones. Figure 2: Vertical profiles $\phi(z)$ of the volume fraction in lin-lin (left) and log-log (right) representation, for five values of the shear velocity: $u_{}/\sqrt{\left(\frac{\rho_{p}}{\rho_{f}}-1\right)gd}=0.3,0.4,0.5,0.6,0.7$. The dotted line correspond to the power-law obtained in the dilute asymptotic limit (Eq. 39). Making use of the homogeneity along $x$, the vertical equations of motion simplify to $\displaystyle\rho_{p}\frac{d\phi\left(u_{z}^{p}\right)^{2}}{dz}$ $\displaystyle=$ $\displaystyle-(\rho_{p}-\rho_{f})\phi g+\frac{d\sigma_{zz}^{p}}{dz}+\phi\frac{d\sigma_{zz}^{f}}{dz}+\frac{3}{4}\phi\,C_{d}(\mathcal{R})\rho_{f}\frac{\left\|\mathbf{u}^{r}\right\|\,{u}_{z}^{r}}{d},$ $\displaystyle\rho_{f}\frac{d(1-\phi)\left(u_{z}^{f}\right)^{2}}{dz}$ $\displaystyle=$ $\displaystyle(1-\phi)\frac{d\sigma_{zz}^{f}}{dz}-\frac{3}{4}\phi\,C_{d}(\mathcal{R})\rho_{f}\frac{\left\|\mathbf{u}^{r}\right\|\,{u}_{z}^{r}}{d}.$ One can eliminate the fluid normal stress $\sigma_{zz}^{f}$ between these two equations and, under the quasi-parallel flow assumption, neglect the left-hand side inertial terms, leading to $\frac{d\sigma_{zz}^{p}}{dz}=(\rho_{p}-\rho_{f})g\phi-\frac{3\phi C_{d}(\mathcal{R})}{4(1-\phi)}\;\rho_{f}\frac{\left\|\mathbf{u}^{r}\right\|\,{u}_{z}^{r}}{d}$ (24) In the dense regime, this equation reduces to the hydrostatic equation. In the dilute regime, the particle normal stress vanishes and the vertical velocity tends to the settling velocity (see Eq. 37 below). The horizontal equations of motion read: $\displaystyle\rho_{p}\frac{d\phi u_{z}^{p}u_{x}^{p}}{dz}$ $\displaystyle=$ $\displaystyle\frac{d\sigma_{xz}^{p}}{dz}+\phi\frac{d\sigma_{xz}^{f}}{dz}+\frac{3}{4}\phi C_{d}(\mathcal{R})\rho_{f}\frac{\left\|\mathbf{u}^{r}\right\|\,{u}_{x}^{r}}{d},$ $\displaystyle\rho_{f}\frac{d(1-\phi)u_{z}^{f}u_{x}^{f}}{dz}$ $\displaystyle=$ $\displaystyle(1-\phi)\frac{d\sigma_{xz}^{f}}{dz}-\frac{3}{4}\phi C_{d}(\mathcal{R})\rho_{f}\frac{\left\|\mathbf{u}^{r}\right\|\,{u}_{x}^{r}}{d}.$ Under the quasi-parallel flow assumption, inertial terms on the left-hand side can again be neglected and these equations lead to: $\frac{d\sigma_{xz}^{f}}{dz}=-\frac{d\sigma_{xz}^{p}}{dz}=\frac{3\phi}{4(1-\phi)}C_{d}(\mathcal{R})\rho_{f}\frac{\left\|\mathbf{u}^{r}\right\|\,{u}_{x}^{r}}{d}.$ (25) Due to an overall conservation of the momentum flux, the total stress $\sigma_{xz}^{p}+\sigma_{xz}^{f}$ is a constant, denoted $\tau$. It can be identified with the fluid shear stress far from the bed, where the particle- borne shear stress vanishes. We also introduce the shear velocity $u_{}$, related to the total stress by $\tau\equiv\rho_{f}u_{}^{2}=\sigma_{xz}^{p}+\sigma_{xz}^{f}$ (26) The fraction of this momentum flux transported by the diffusive motion of particles is (see Appendix) $(\rho_{f}u_{x}^{f}-\rho_{p}u_{x}^{p})j_{z}=\phi(1-\phi)(\rho_{f}u_{x}^{f}-\rho_{p}u_{x}^{p})u_{z}^{r}$. In the limit where the vertical velocity is much smaller than horizontal ones, this effect can be neglected. Under the quasi-parallel flow assumption, the strain rate modulus is simply the shear rate: $\|\dot{\gamma}^{f}\|={\dot{\gamma}}^{f}_{\rm xz}=\frac{du_{x}^{f}}{dz}\quad{\rm and}\quad\|\dot{\gamma}^{p}\|={\dot{\gamma}}^{p}_{\rm xz}=\frac{du_{x}^{p}}{dz}\,.$ (27) The fluid and particle constitutive relations are then simply written as $\displaystyle\sigma^{f}_{xz}$ $\displaystyle=$ $\displaystyle\rho_{f}\left(\nu+\ell^{2}\left\|\frac{du_{x}^{f}}{dz}\right\|\right)\frac{du_{x}^{f}}{dz}$ (28) $\displaystyle\sigma^{p}_{xz}$ $\displaystyle=$ $\displaystyle\psi(\phi)\left(\rho_{f}\nu+\alpha\rho_{p}d^{2}\left\|\frac{du_{x}^{p}}{dz}\right\|\right)\frac{du_{x}^{p}}{dz}$ (29) which must be complemented, when the granular phase is unjammed, by the frictional relation $\|\sigma^{p}_{xz}\|=\mu(\phi)\sigma^{p}_{zz}$. Combining the vertical and horizontal equations of motion, one obtains an equation relating the derivative of the volume fraction to the vertical velocity: $-(\rho_{p}-\rho_{f})g+\frac{3\;C_{d}(\mathcal{R})}{4(1-\phi)}\;\rho_{f}\frac{\left\|\mathbf{u}^{r}\right\|}{d}\,\left(\frac{{u}_{x}^{r}}{\mu(\phi)}+{u}_{z}^{r}\right)+\frac{\mu^{\prime}(\phi)\sigma^{p}_{xz}}{\phi\mu(\phi)^{2}}\;\frac{d\phi}{dz}=0.$ (30) Combining it with Eq. (23), which involves the same quantities, one can deduce ${u}_{z}^{r}$ and $d\phi/dz$ Figure 3: Vertical profiles of the particle-borne shear stress $\sigma_{xz}^{p}$ (solid lines) and of the fluid-borne shear stress $\sigma_{xz}^{f}$ (dotted lines) for the same shear velocity values as in Figure 2. The sum of $\sigma_{xz}^{p}$ and $\sigma_{xz}^{f}$ is constant, equal to $\rho_{f}u_{}^{2}$. Note the evolution of the altitude at which $\sigma_{xz}^{f}=\sigma_{xz}^{p}$ when $u_{}$ is varied. ### 3.3 Static bed The description of the static bed poses a specific problem associated with interparticle friction: it can be prepared at different values of the volume fraction $\phi$, and with different microstructures. This property is at the origin of the hysteresis between the static and dynamic friction coefficients. The problem comes from the formulation of the two-phase flow model, which does not describe the static granular phase. However, if the static bed is prepared in the critical state, i.e. by shearing the grains at a rate $\dot{\gamma}$ going to $0$, the volume fraction is equal to the critical value $\phi_{c}$. We will only consider this situation here. A simple extension would be to consider the bed volume fraction as a parameter, and to impose the continuity of $\phi$ at the static-mobile interface. We define $z=0$ as the position at which the Coulomb criterion is reached, so that, for $z<0$, the bed is strictly static i.e. the grain velocity is exactly null ($u_{x}^{p}=0$ and $u_{z}^{p}=0$). Right at $z=0$, the volume fraction reaches the critical volume fraction $\phi_{c}$. We therefore obtain for $z<0$: $\phi=\phi_{c}\quad{\rm and}\quad P^{p}=P_{0}-(\rho_{p}-\rho_{f})\phi_{c}{g}z.$ (31) The equations describing the fluid in the bed then simplify to $\displaystyle\frac{d\sigma_{xz}^{f}}{dz}$ $\displaystyle=$ $\displaystyle\frac{3\phi_{c}}{4(1-\phi_{c})}\;C_{d}(\mathcal{R})\rho_{f}\frac{\left\|{u}_{x}^{f}\right\|\,{u}_{x}^{f}}{d}\,,$ (32) $\displaystyle\sigma^{f}_{xz}$ $\displaystyle=$ $\displaystyle\rho_{f}\left(\nu+\ell^{2}\left\|\frac{du_{x}^{f}}{dz}\right\|\right)\;\frac{du_{x}^{f}}{dz}.$ (33) Deep inside the static bed, the fluid velocity $u^{f}$ tends to $0$, and one obtains the asymptotic solution in the limit $z\rightarrow-\infty$: $u_{x}^{f}\approx U\,\exp(z/\zeta)\quad{\rm with}\quad\zeta=\sqrt{\frac{4(1-\phi_{c})}{3\phi_{c}}}\;\frac{d}{s}\;.$ (34) where the coefficient $s$ is defined by Eq. 9. The flow velocity thus decays over a distance that is a fraction of the grain diameter. $U$ is a shooting parameter that is determined by the matching with the velocity far above the bed. In the same limit, the mixing length equation can be approximated by: $\frac{d\sqrt{\ell}}{dz}\approx\frac{\kappa}{2}\,\sqrt{\frac{u^{f}}{R_{t}\nu}}\approx\frac{\kappa}{2}\,\sqrt{\frac{U}{R_{t}\nu}}\,\exp\left(\frac{z}{2\zeta}\right)$ (35) As $\ell$ must vanish when $z\rightarrow-\infty$, we obtain the asymptotic expression: $\ell\approx\frac{\kappa^{2}\zeta^{2}U}{R_{t}\nu}\,\,\exp\left(\frac{z}{\zeta}\right)\approx\frac{\kappa^{2}\zeta^{2}}{R_{t}\nu}\;u_{x}^{f}$ (36) Figure 4: Vertical profiles of the two components of the velocity difference between the two phases. Top: horizontal component $u_{x}^{r}$. Bottom: vertical component $u_{z}^{r}$. The different lines correspond to the same shear velocity values as in Figure 2. ### 3.4 Dilute zone When $z$ tends to $+\infty$, the volume fraction $\phi$, the horizontal component of the relative velocity ${u}_{x}^{r}$ and the grain-borne shear stress $\sigma^{p}_{xz}$ all tend to zero. As a consequence, the vertical velocity $u^{r}_{z}=-u^{p}_{z}=V_{\rm fall}$ is the settling velocity defined by $\frac{3}{4}\;C_{d}(\mathcal{R})\;\rho_{f}\frac{V_{\rm fall}^{2}}{d}=(\rho_{p}-\rho_{f})g$ (37) The vertical fluid velocity vanishes like $u^{f}_{z}=\phi u^{r}_{z}$. The mixing length tends to $\ell=\kappa(z+z_{0})$, where $z_{0}$, called the hydrodynamic roughness, is a result of the integration. The horizontal velocities tend to: $u_{x}^{f}\simeq u_{x}^{p}\sim\frac{u_{}}{\kappa}\ln(1+z/z_{0})$ (38) Using the asymptotic expression of the diffusivity $D\sim\kappa u_{}(z+z_{0})/{\rm Sc}$ and the velocity $u^{r}_{z}\sim V_{\rm fall}$, the continuity equation integrates into the well-known volume fraction profile: $\phi\sim\frac{\phi_{\rm 0}}{\left(1+z/z_{0}\right)^{\beta}}\quad{\rm with}\quad\beta=\frac{{\rm Sc}\,V_{\rm fall}}{\kappa\,u_{}},$ (39) where the exponent $\beta$ is known as the Rouse number. This asymptotic behaviour selects the only physical solution, and therefore the value of $U$. By contrast, unphysical solutions present a singularity ($\phi\to\phi_{c}$ or $\phi\to 0$) at finite height. Just like $z_{0}$, the multiplicative constant $\phi_{\rm 0}$ is selected by the asymptotic expressions derived inside the static bed. Figure 5: Vertical profiles of the horizontal fluid velocity $u_{x}^{f}$, for the same shear velocities as in Figure 2. ## 4 Results The equations governing the evolution of ${u}_{x}^{f}$, $\sigma_{xz}^{f}$, $\ell$, $u_{x}^{p}$ and $\phi$ have been integrated numerically, using a shooting method to satisfy the asymptotic expansions on both sides of the integration domain. In practice, the equations are integrated upward using the Runge-Kutta algorithm. The integration is started from a point deep enough in the static bed for the asymptotic expressions to be valid within numerical errors. The unique solution whose volume fraction decays algebraically at infinity is obtained by bracketing the value of $U$. Figure 6: Vertical profiles of the volume fraction (left) and of the horizontal particle velocity $u_{x}^{p}$ (right), for the same shear velocities as in Figure 2. The position $z$ is measured with respect to the altitude $z_{s}$ at which $\phi$ crosses $\phi_{c}/2$. We detail and comment on in this section the results obtained for a rescaled viscosity $\nu/\sqrt{\left(\frac{\rho_{p}}{\rho_{f}}-1\right)gd^{3}}=10^{-1}$, or $d/d_{\nu}\simeq 4.64$. For quartz grains in water, this value corresponds to a mean diameter of $d\simeq 180{\rm\mu m}$. For the shear velocities we have considered (see caption of Fig. 2), the viscous length $\nu/u_{}$ is a fraction of the grain diameter, in the range $0.14$–$0.33d$. As shown below, this is small in comparison to the size of the transport layer (typically $5$–$20d$), so that the curves we present here correspond to transport in the turbulent regime. By systematically varying the shear velocity or, equivalently, the Shields number, we find that the asymptotic conditions on both sides cannot be matched (i.e. there is no solution) below a threshold value $u_{\rm th}\simeq 0.2\,\sqrt{\left(\frac{\rho_{p}}{\rho_{f}}-1\right)gd}$. Experimentally, a similar dimensionless threshold velocity is found, although slightly larger, equal to $0.26$ (see Andreotti et al. 2013 and references therein). Figure 2 shows the volume fraction vertical profiles $\phi(z)$ in log-log and lin-lin scales. Close to the static sand bed, grains are transported in a rather dense layer whose thickness increases with $u_{}$. This corresponds to a form of bed load where several granular layers are entrained (sheet flow). It is worth noting that the scale separation between the transport layer thickness and the grain diameter is never good enough to expect a quantitive description by a continuum model. Far from the static bed, the volume fraction decreases as a power law of the height, as expected (Eq. 39). Figure 3 shows the vertical profile of the grain-borne and the particle-borne shear stresses for five values of $u_{}$. The sum of $\sigma_{xz}^{p}$ and $\sigma_{xz}^{f}$ is constant, equal to $\rho_{f}u_{}^{2}$. Far above the static bed the whole shear stress is carried by the fluid. Furthermore, close to the static bed, it is carried by the grains and transmitted through contact forces. The transfer of momentum flux from the fluid to the particles occurs at a position that can be interpreted as the top of the bedload layer. One observes in Figure 4a) a corresponding peak in the profile of the horizontal velocity difference $u_{x}^{r}$, whose position, on the order of $10d$, increases with $u_{}$. This relative velocity between the two phases is associated with a horizontal drag force: the momentum flux serves to balance friction and entrain the particles from the bed into motion. Figure 4b) shows the vertical velocity profile. As expected, it tends to the settling velocity $V_{\rm fall}$ far above the bedload layer. Inside the bedload layer, one observes that $u_{z}^{r}$ is still large, which means that there is a balance between pressure-induced migration and shear-induced diffusion. We emphasize again that the equations solved are valid throughout the system. The two-layer structure of the transport layer comes out of the solution but was not imposed as in Revil-Baudard & Chauchat (2012). Also, it is not clear whether the specific sublayers characterized by Berzi (2011, 2013) can be identified here. In particular, $\phi$ is not constant in the bedload layer. Figure 7: Vertical profiles of the flux density $\phi u_{x}^{p}$, for the same shear velocities as in Figure 2. The fluid velocity profile $u_{x}^{f}(z)$ (see Figure 5) is, as expected, logarithmic far above the static bed, but is strongly reduced in the dense transport layer. When particles are accelerated by the flow, the flow, in turn, is decelerated. This negative feedback mechanism takes place over a thicker and thicker region, as $u_{}$ increases, which coincides with the bedload layer (Fig. 6). Because of this strong feedback, the fluid contribution to particle diffusion is negligible close to the static bed. As a consequence, the value of $u_{z}^{r}$ when $z\to 0$ is found to be independent of the shear velocity (Fig. 4 bottom). The structure predicted for the bed load layer resembles qualitatively that observed experimentally by Capart & Fraccarollo (2011): the volume fraction $\phi$ is not constant but decreases with the height (roughly linearly) while the particle velocity $u_{x}^{p}$ increases. The average sediment transport at the altitude $z$ is characterized by the product of the volume fraction $\phi$ and the particle velocity $u_{x}^{p}$. More precisely, $\rho_{p}\phi u_{x}^{p}$ is the density of the mass flux; $\phi u_{x}^{p}/\phi_{c}$ is the density of the volume flux. Figure 7 shows that $\phi u_{x}^{p}$ presents a peak at the top of the bed load layer, followed by a tail associated with the dilute turbulent suspension. The total sediment flux is obtained by integration over the vertical direction: $q=\int_{0}^{\infty}\phi\,u_{x}^{p}\,dz$ (40) By definition it quantifies the volume of grain crossing a unit length transverse to the flow per unit time. The mass flux is $\rho_{p}q$ and the volumetric flux at the bed volume fraction is $q/\phi_{c}$. Figure 8: Dependence of the sediment flux $q$ on the Shields number $\Theta$. (a) $q$ as a function of the rescaled shear velocity $\sqrt{\Theta}$. (b) $q$ as a function of $\Theta-\Theta_{\rm th}$. Solid line: total flux. Dashed line: contribution of the suspended load, determined from the asymptotic expansions obtained in the dilute limit (Eq. 41). Dotted line: contribution of bed load, determined from calculations performed in the limit $Sc\to\infty$. The bed load flux $q$ follows the Meyer Peter-Müller scaling in $(\Theta-\Theta_{\rm th})^{3/2}$. Figure 8a) shows the dependence of the sediment flux $q$ on the Shields number $\Theta$. It vanishes below the threshold Shields number $\Theta_{\rm th}$ and beyond, it increases with $\Theta$. It diverges at the Shields number $\Theta_{m}$ for which the Rouse number $\beta$ reaches $1$. In real conditions, the total sediment flux does not diverge at $\Theta_{m}$ because it is limited by the finite flow thickness. We have kept here an unbounded flow in order to highlight the transition to a turbulent suspension. Solutions at higher Shields numbers can be deduced by asymptotic matching, when there is a separation of length scale between the flow thickness and the bedload layer thickness. By integrating the asymptotic expressions obtained far from the sediment bed, we can deduce the contribution of turbulent suspension to the sediment flux: $q_{\rm sus}=\frac{\phi_{\rm 0}u_{}}{\kappa}\int_{0}^{\infty}\left(1+z/z_{0}\right)^{-\beta}\ln(1+z/z_{0})\,dz=\frac{\phi_{\rm 0}u_{}z_{0}}{\kappa(1-\beta)^{2}}\,.$ (41) With this definition, $q_{\rm sus}$ corresponds to the sediment flux for which the water flow and the concentration profile would behave everywhere as in the asymptotic limit $z\to\infty$. One observes in Figure 8a) that $q_{\rm sus}$ (dashed line) indeed becomes dominant around the inflection point of the curve $q(\Theta)$. Conversely, in order to determine precisely the contribution $q_{\rm bl}$ of the bedload to the total flux, we have integrated the equations in the limit of an infinite turbulent Schmidt number ${\rm Sc}$, i.e. without any diffusion of particles resulting from turbulent fluctuations. Importantly, and although this was not given, we have checked that $q_{\rm bl}+q_{\rm sus}$ accurately gives $q$. The bed load contribution is dominant at low shear velocity. It is extremely well fitted by a Meyer & Peter-Müller- like relation: $q_{\rm bl}\propto(\Theta-\Theta_{\rm th})^{3/2}$ (42) For the parameters chosen in this paper, the best fit gives a threshold shear velocity of $u_{\rm th}\simeq 0.19\,\sqrt{\left(\frac{\rho_{p}}{\rho_{f}}-1\right)gd}$ and equivalently a threshold Shields number of $\Theta_{\rm th}\simeq 0.037$. We emphasize again that such a continuum model eventually becomes inaccurate close to the transport threshold, as particle-scale processes become dominant. Figure 9: (a) Dependence of the apparent basal volume fraction $\phi_{0}$ on the Shields number $\Theta$. Starting from the expansion of Eq. (39) in the region of suspension, $\phi_{0}$ appears as an extrapolated value of the volume fraction at the bed. From the point of view of the turbulent suspension, $V_{\rm fall}\,\phi_{0}$ is the effective erosion rate. (b) Dependence of the hydrodynamic roughness on $\Theta$, with (solid line) and without transport (dashed line). Dotted line: roughness obtained with pure bed load, in the limit $Sc\to\infty$. Amongst the novel aspects of our approach, we are able to predict the effective erosion rate seen by the turbulent suspension or, more precisely, the rate at which the bed load particles are injected in this suspension. Figure 9a) shows the parameters extracted from the asymptotic expansions in the upper zone. The apparent basal volume fraction $\phi_{0}$ is defined from Eq. (39). It increases rapidly and saturates to a value of order $1$ at the suspension threshold. Figure 9b) shows the hydrodynamic roughness. In the absence of transport, $z_{0}$ decreases as $\nu/u_{}$. At the transport threshold, it suddenly increases by one order of magnitude before dropping again when suspension becomes dominant. As this quantity can be accessed experimentally at a distance from the bed, it could be used to show the existence of a negative feedback of sediment transport on the flow, in the turbulent regime. Both curves can be prolonged at higher Shields numbers, for a finite flow thickness. ## 5 Concluding remarks The two-phase flow model derived here is an extension of models previously proposed to describe dense viscous flows and turbulent suspensions. We have added two novelties with respect to these works: the description of dense flows at large Stokes numbers and the shear-induced diffusion of particles. In the dilute limit, the diffusion is due to hydrodynamic interactions (Leighton & Acrivos,1987). It had only been qualitatively discussed in the literature (Fall et al., 2010) in the very dense case (for $\phi\simeq\phi_{c}$). We have shown here that it results from non-affine particle motion and identified its scaling law. From the point of view of sediment transport, the two-phase flow model derived here is able to describe all regimes: viscous and turbulent, bedload and suspension. Regarding bedload, we have relaxed a hypothesis used in previous models: the volume fraction is not assumed to be homogeneous, equal to $\phi_{c}$, in the bedload layer. We predict in contrast that the sediment transport layer is rather dilute, with a volume fraction profile continuously decreasing to $0$ away from the static bed. At large shear velocity, when the sediment transport mostly takes place in suspension, our model is able to predict the erosion/deposition rate into the bulk from the equations of mechanics, by matching to the bed load zone. One does not have to introduce a phenomenological erosion rate, balanced by sedimentation, as in previous models of turbulent suspension. An important future work would be to compare these theoretical predictions with experimental or numerical data. At present, several ingredients of the model involve parameters that have never been measured. For a comprehensive comparison, a first calibration step of these parameters is needed before fitting concentration and velocity profiles (Nnadi & Wilson, 1992; Sumer et al., 1996; Cowen et al., 2010). For example, the factor $S_{\phi}$, related to the diffusion of the particles of particles induced by non-affine motion, could be assessed using heterogeneous shear flows (Bonnoit et al., 2010). Also, the predicted erosion/deposition rate, which sensitively depends on the particle shear-induced diffusion, could be tuned using numerical simulations in the spirit of those of Durán et al., (2012). With this two-phase description, we recover the scaling law phenomenologically proposed by Meyer-Peter & Müller (1948) for bedload transport. However, one expects the model to become inaccurate close to the transport threshold, where the sparse mobile grains hardly constitute an eulerian phase. The analysis of forces and torques on a single grain (Andreotti et al., 2013) seems to provide a better understanding of the transport threshold than the continuum modelling, where the grain size appears indirectly in the decay of the flow velocity inside the static bed. If a fine tuning of parameters can improve the agreement of the model with observations in this range of velocities, it may well be that the dynamics is actually dominated by fluctuations (of the bed surface structure in particular), and can therefore not be described in a mean-field manner. On the other hand, we expect the two-phase approach to provide an accurate description of the transition from bed load to suspension. It can therefore be applied to different problems of river morphodynamics like the formation of ripples, alternate bars and meanders by linear instability (for recent reviews, see Charru et al (2013) and Andreotti & Claudin (2013), as well as references therein). Apart from the expression of the saturated flux, as a function of the shear velocity, morphodynamical models must incorporate two important dynamical mechanisms: the dependence of sediment transport on the bed slope and the relaxation of the sediment flux towards saturation. We expect such a two-phase modelling to give some clues to help in the resolution of ongoing controversies on the emergence of bedforms. ## Appendix A Two-phase Reynolds-averaged equations We derive here the terms in the two-phase flow description originating from fluctuations. Here, $\phi$, $u_{i}^{p}$ and $u_{i}^{f}$ are defined as Eulerian averages. The fluctuations (non-affine velocity field and turbulent fluctuations) around these average quantities are denoted $\phi^{\prime}$, ${u_{i}^{p}}^{\prime}$ and ${u_{i}^{f}}^{\prime}$. The averaging operation is denoted $<.>$. The Reynolds-averaged mass conservation equation gives a diffusive flux: $j_{i}=<\phi^{\prime}{u_{i}^{p}}^{\prime}>=<\phi^{\prime}{u_{i}^{f}}^{\prime}>$ (43) The particle momentum equation involves the total stress $\sigma_{ij}^{p}$ as the sum of the contact stress $s_{ij}^{p}$ and of the flux of momentum originating from fluctuations, $\sigma_{ij}^{p}=s_{ij}^{p}-\rho_{p}\left(\phi<{u_{j}^{p}}^{\prime}{u_{i}^{p}}^{\prime}>+u_{i}^{p}j_{j}+u_{j}^{p}j_{i}+<\phi^{\prime}{u_{i}^{p}}^{\prime}{u_{j}^{p}}^{\prime}>\right).$ (44) The term $-\rho_{p}\phi<{u_{j}^{p}}^{\prime}{u_{i}^{p}}^{\prime}>$ is the kinetic stress associated with non-affine motion. The two following terms $-\rho_{p}\phi\left(u_{i}^{p}j_{j}+u_{j}^{p}j_{i}\right)$ quantify the transport of momentum associated with particle diffusion. If the fluctuations are small enough, the last term can be neglected. We work here under this assumption. Similarly, for the fluid, we introduce the viscous stress $s_{ij}^{f}$ and obtain: $\sigma_{ij}^{f}=s_{ij}^{f}-\rho_{f}\left((1-\phi)<{u_{j}^{f}}^{\prime}{u_{i}^{f}}^{\prime}>-u_{i}^{f}j_{j}-u_{j}^{f}j_{i}-<\phi^{\prime}{u_{i}^{f}}^{\prime}{u_{j}^{f}}^{\prime}>\right).$ (45) The term $-\rho_{f}(1-\phi)<{u_{j}^{f}}^{\prime}{u_{i}^{f}}^{\prime}>$ is the Reynolds stress. The two following terms quantify the transport of momentum associated with particle diffusion. Again the last term can be neglected arguing that the fluctuations remain small. This work has benefited from the financial support of the Agence Nationale de la Recherche, grant ‘Zephyr’ ($\\#$ERCS07 18). ## References * [] Anderson, T. 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1401.0827	# Interaction entre mathématique et informatique par le logiciel mathématique Libre/Open Source K.I.A.Derouiche Résumé.— Cet article porte sur l’application du modèle de développement et l’ouverture du code source, disponible et mis en oeuvre par les Logiciels Libres et Open Sources (LL/OS) à la didactique et l’enseignement a la fois des mathématiques et l’informatique par la lecture-écriture (L/E) de logiciel mathématique, dont les cas les plus connus sont le calcul numérique et formel. L’article analyse le modèle de développement du logiciel mathématique Libre /Open Source (L/OS) dont l’importance est avérée dans le secteur de la recherche en mathématique et informatique. En revanche, leur utilisation bien que réelle, est peu lisible dans les formations d’enseignement supérieur. Nous discutons de la faisabilité de ce modèle, qui concerne les caractéristiques du domaine, des acteurs, des interactions qu’ils entretiennent et des communautés qu’ils forment pendant le développement de ces logiciels. Finalement, on propose un exemple de logiciel mathématique Libre/Open Source (LML/OS) comme dispositif d’analyse. Mots-clefs : Enseignement LL/OSS, Open Source, Coopération, logiciel mathématique, didactique de l’informatique. (Séminaire National sur la didactique des Mathématiques 25-26 Novembre Tebessa, Algérie (SNDM’13)) ## 1 Introduction Après avoir été un outil réservé aux centres de recherches. LL/OS s’est implanté dans l’industrie, et depuis quelque années, il est omniprésent dans pratiquement tous les secteurs d’activités de la vie quotidienne, notamment dans les domaines de la gestion, de l’industrie, des sciences et techniques et l’éducation dont il sont souvent issue. Leur principale innovation concerne leur modèle de développement coopératif et décentralisé qui s’est avérée, dans plusieurs des cas[4], plus efficace que le modèle propriétaire. Notre objectif vise à montrer l’apport des LML/OS dans l’enseignement et l’apprentissage des mathématiques, qui nécessitent. la qualification des enseignants, le savoir acquis de la part des étudiants, une dynamique de coopération, partage de connaissances, la participation et la discussion, l’adaptation des contenus pédagogiques mathématiques à l’apprentissages. D’où, notre intérêt envers le modèle de développement des LML/OS dans l’enseignement et l’apprentissage des mathématiques, que nous allons aborder à travers les questions suivantes : 1. 1. Quels logiciels mathématiques L/OS utiliser ? 2. 2. Quels impacts des modifications/contributions des LML/OS sur l’enseignement et l’apprentissage des mathématiques ? ## 2 Définitions de Logiciels Libres/Open source Il existe plusieurs définitions du ”logiciel libre”. Voir [6, 9] de comparaison entre open source, free software et des divers sens qu’on peut donner au LL/OS. Ce qui importe dans notre étude, c’est à la fois un outil et modèle de développement. Nous nous plaçons dans la définition donnée par la Free Software Fondation[3] qui fait référence à la liberté pour les utilisateurs d’exécuter, de copier, de distribuer, d’étudier, de modifier et d’améliorer le logiciel. Plus précisément, elle fait référence à quatre types de libertés: 1. 1. La liberté d’exécuter le programme, pour tous les usages (liberté 0) 2. 2. La liberté d’étudier le fonctionnement du programme, et de l’adapter à vos besoins (liberté 1). Pour ceci l’accés au code source est une condition requise 3. 3. La liberté de redistribuer des copies, donc d’aider votre voisin, (liberté 2) 4. 4. La liberté d’améliorer le programme et de publier vos améliorations, pour en faire profiter toute la communauté (liberté 3). Pour ceci l’accès au code source est une condition requise Un programme est un logiciel libre si les utilisateurs. ont toutes les libertés 0, 1, 2, 3. ## 3 Le modèle de développement Le modèle de développement de LL/OS est apparu comme une alternative au modèle propriétaire de développement du logiciel. Il propose une approche nouvelle de collaboration entre acteurs (développeurs, testeurs, mainteneurs, utilisateurs avancées newbies)[5], de l’utilisation des droits de propriété intellectuelle et de la dynamique interne du processus (outils de communications, outils de développements, code source)[1] Cette approche alternative serait à la source du succès de plusieurs LL/OS, comme les logiciels mathématiques. ### 3.1 Le modèle du logiciel propriétaire Le modèle de logiciel propriétaire, ou ”modèle d’investissement privé”[2], constitue la manière la plus répandue de création de technologie, associée aux grandes découvertes technologiques qui support la croissance économique depuis un siècle et demi. Son idée de base et d’encourager l’innovation par la création de monopoles temporaires protégés par les mécanismes de protection de la propriété intellectuelle, qui excluent les non créateurs d’une nouvelle technologie de son utilisation. Ce modèle de développement ne permet pas un contrôle sur le logiciel, seulement dans le cadre d’une utilisation limitée par la licence Figure 1: Modèle d’interaction des logiciels mathématiques propriétaires. Un cas pratique est le besoin de faire des corrections après avoir détecté une erreur (figure 1). Pèrez et Varona[7] ont été confronté à une erreur de calcul de matrices de nombres entiers dans le logiciel Mathematica, qui donne des résultats différents s’il évalue le même déterminant à deux reprises, leur démarche été d’isoler l’erreur par un ScriptConnect, par manque de code source qui leur aurait permis d’apporter une correction par le biais du code métier MathematicalCodeComponent(MCC) ### 3.2 Le modèle LL/OS: le logiciel mathématique Une caractéristique distinctive de ces logiciels dans l’enseignement des mathématiques, est l’encouragement à la motivation individuelle et collective. Il vise à développer Figure 2: Modèle d’interaction des LML/OS chez les étudiants, conjointement et progressivement les capacités d’expérimentation et de raisonnement (exemple: imagination, analyse critique à travers une démarche de résolution de problèmes, modélisation des situations et d’apprentissage progressif de la démonstration). Ils peuvent prendre conscience de la pertinence des activités mathématiques, identifier un bug (informatique, dysfonctionnement d’un algorithme mathématique), le corriger et le redistribuer, l’améliorer et l’expérimenter sur des exemples et changé avec l’environnement de travail sans aucune restriction voir (figure 2) ## 4 Quelques logiciels mathématiques L/OS On peut classer les logiciels mathématiques L/OS dédiés à l’enseignementselon six catégories: 1. 1. Logiciel bibliothèque: gsl, blas, lapack, cgal, sympy, networkx 2. 2. Logiciel interactif: IPython 3. 3. Logiciel interface graphique: Cantor 4. 4. Logiciel Environnement de développement intégré: Aesthete, Spyder, Reinteract 5. 5. Logiciel on-web: wims, , sagemath, SymPyLive ## 5 Exemple de SymPy SymPy[8] est une bibliothèque pour les mathématiques symboliques et l’algèbre informatique, écrite en Python pure sans dépendance externe, Le projet contient trois outils un interpréteur intégré (isympy) est deux interface web SymPy Live et SymPy Gamma qui permettent aux utilisateurs d’utilisés du code scientifique en ligne. La bibliothèque SymPy offre d’autres fonctionnalités telles que LaTeX, la cryptographie et la mécanique et l’informatique quantique, géométrie différentielle. ⬇ IPython console for SymPy 0.7.4 (Python 2.7.5-64-bit) (ground types: python) These commands were executed: >>> from __future__ import division >>> from sympy import * >>> x, y, z, t = symbols(’x␣y␣z␣t’) >>> k, m, n = symbols(’k␣m␣n’, integer=True) >>> f, g, h = symbols(’f␣g␣h’, cls=Function) Documentation can be found at http://www.sympy.org ### 5.1 Expérience de modification/contribution Cette expérience constitue un exemple d’interaction entre mathématique et informatique dans un environnement de travail coopératif basé sur le modèle du LL/OS. L’expérience consiste calculer la dérivée de la fonction $\cos(3x)$ avec la notation df en utilisation la bibliothèque de calcul symbolique SymPy. Pour la mise en oeuvre pratique, nous avons divisé la l’expérience en deux tâche mathématique (m_tâche) et une tâche informatique (i_tâche) * • Tâche mathématique m_tâche: statbilisation du milieu Les étudiants doivent: 1. 1. Faire la différence entre $\mathrm{df}$ et $\mathrm{diff}$ 2. 2. Calculer la dérivée de la fonction $f(x)=\cos(3x)$, avec $x\in R$ en utilisant la notation de Leibniz * • Tâche informatique 1. 1. i_tâche0: écrire lexpression symbolique de $\cos(3x)$ dans isympy en utilisant la notation de Leibniz $\mathrm{df}$ 2. 2. i_tâche1: comprendre le message d’erreur, localisé le fichier source et la fonction qui l’implémente 3. 3. i_tâche2: changer le nom de la fonction $\mathrm{diff}$ en $\mathrm{df}$, puis exécuter 1. 1. Dimension didactique: Pour l’enseignant, il s’agit d’apprendre à construire et à conduire des groupes d’étudiants dans un environnement de travail coopératif issu du LL/OS. Pour les étudiants, il s’agit d’apprendre à construire des connaissances significatives de calcul de dérivée $\cos(3x)$ à travers d’une part, les interactions entre le registre symbolique de l’écriture mathématique et le registre historique d’autre part, à travers des tâches à réaliser alternativement dans un environnement de travail coopératif LL/OS. 2. 2. m_tâche: stabilité du milieu Durant toute la tâche, les étudiants travaillent en papier/crayon. Deux objectifs sont particulièrement travaillés: 1. (a) Comprendre l’origine de la notation df (recours à l’histoire des mathématiques et a l’enseignent) 2. (b) Calculer la dérivée de $\cos(3x)$ Solution : Explication et discussion de l’enseignent Application immédiat du théorème de la dérivée des fonctions trigonométriques $cos(3x)=-3sin(3x)$ 3. 3. i_tâche0 : Dans cette tâche, les étudiants utilisent l’invite de commande isympy 1. (a) Calculer la dérivée de la fonction sous forme symbolique dans isympy en utilisant $\mathrm{diff}$ Première tentative: certains étudiants ont appliqué littéralement l’expression mathématique $\cos(3x)$, et ont été surpris du message d’erreur affiché dans isympy. ⬇ In [1]:diff(cos(3x), x) File "<ipython-input-4-762bbb5de069>", line 1 diff (cos(3x), x) ^ SyntaxError: invalid syntax Solution attendu: les étudiants lance une recherche documentaire, dans isympy, en tapent help(cos) ⬇ In [2]:diff(cos(3x), x) In [2]:-3.sin(3.x) 1. (a) Calculé la dérivée de la fonction cos(3x) en utilisant la notation de Leibniz $df$ L’enseignant pousse l’étudiant à exploré d’autre possibilité pour écrire la dérivée de $\cos(3x)$ ⬇ In [6]:df(cos(3x), x) ———————————————- NameError Traceback (most recent call last) <ipython-input-6-9a8e0302dd96> in <module>() —-> 1 df (cos(3x), x) NameError: name ’df’ is not defined i_tâche1: Analyser le message d’erreur, localiser le fichier source de la fonction qui implémente $\mathrm{diff}$ ⬇ In [8]: cos(2x).df(x) ————————————————– AttributeError Traceback (most recent call last) <ipython-input-8-b449e4a43e07> in <module>() —-> 1 cos(2x).df(x) AttributeError: ’cos’ object has no attribute ’df’ La réaction des étudiants face a cette erreur est divisée entre deux groupes, ceux qui sont surpris du résultat pour eux les LML/OS c’est avant tout quelque chose de très flexible cela implique que il doit être dynamique et immédiat, l’autre groupe essai de comprendre la cause des erreurs affichées sur l’écran AttributeError:’sin object has no attribute ’df’ et NameError: name ’diff’ is not defined. L’enseignant explique aux étudiants la différence qui existe entre les deux messages d’erreurs. Localisation: L’enseignant demande aux étudiants de lancer une commande, pour trouver les sources de la bibliothèque SymPy dans le système d’exploitation: fournissent à la commande le nom du répertoire et un mot clef sur fonction obtenue par le biais de la commande d’aide help(diff) dans l’invite isympy, sympy/mpmath/calculus/differentiation.py i_tâche2: Changer le nom de la fonction dans le fichier source de $\mathrm{diff}$ en $\mathrm{df}$, puis tester avec isympy Le rôle de l’enseignant est d’impliquer les étudiants dans la L/E approfondie du code source et de découvrir tout les enjeux. Dans cette tâche l’enseignant demande aux étudiant de travailler en collaboration sur la L/E du fichier source. Code source Avant les modifications : ⬇ $./sympy/mpmath/calculus/differentiation.py$ …. def df(ctx, f, x, n=1, options): """ … """ partial = False try: orders = list(n) x = list(x) partial = True except TypeError: pass … Code source aprés les modifications ⬇ … def df(ctx, f, x, n=1, options): … Exécution du calcul ⬇ In [30]:df(cos(3x), x) In [30]:-3.sin(3.x) ## 6 Quelles leçons peut-on tirer de cette expérience L’utilisation des LML/OS s’avèrent un moyen intéressant pour mettre en oeuvre une véritable activité mathématique: Avantages * • Les étudiants perçoivent les problèmes mathématiques autrement, tout en respectant un niveau de rigueur. Leur attitude face aux mathématiques évolue positivement. * • Aux étudiants de reprendre plusieurs fois un algorithme mathématique, ou de revenir plus tard sur certains fragment. * • La progressivité des difficultés dans les codes sources qui traites le problème mathématique, ce qui constitue un réel atout de motivation pour l’étudiant en difficulté. * • L’utilisation de ces logiciels est exposée à des effets réseaux positifs relier différents cadres (algébrique, géométrique, programmation, correction de bug …) d’un même concept ou d’une même situation. * • Les étudiants: peuvent juger, avec relative facilité, la viabilité des programmes et algorithmes mathématiques de leur camarade * • De procéder rapidement à la vérification de certains résultats obtenus. Inconvénients * • L’étudiant se préoccupe le plus souvent par l’amélioration du code source que par le problème mathématique lui-même. * • Ces logiciels nécessitent parfois la maitrise de divers langage de programmation, ayant des syntaxes différentes. * • Il y a un manque de formation chez les enseignants. ## 7 Conclusion L’interaction entre informatique et mathématique à travers l’utilisation des LML/OS dans l’enseignement et l’apprentissage des mathématiques s’inscrit dans le champ (discussion, évaluation par l’étudiant, travail coopératif, partage de contenus, recherche sur des documents et sur le web…). Les LML/OS offrent des opportunités intéressantes d’exploration dans des situations variées pour l’enseignant et pour l’étudiant en l’amenant à réfléchir sur ce qu’il fait et comment il doit le faire et avec quel moyen. Cependant, nous avons signalé un certain nombre d’obstacles qui peuvent entraver l’utilisation efficace des LML/OS comme. la maitrise de la lecture du code source, la maintenance de logiciel, l’exécution dans de nouvelle architecture matériel, le facteur liés à la gestion du temps, le manque de documentation en français ou en arabe pour certain logiciels. Les sources de ces difficultés sont le manque de formation des enseignants dans ce domaine, voire pour les enseignants, le manque de temps(emploi du temps chargé, le manque d’intérêts et de volonté de la part des étudiants ainsi que l’absence d’une véritable politique encourageant l’utilisation des logiciels libres dans d’apprentissage et d’enseignement des mathématiques. ## References * [1] Fitzgerald Brian. The transformation of open source software. Quarterly, 30(2), 2006. * [2] Von Hippel Eric. Democratizing Innovation. Mit Press, February 2005. * [3] FSF. Qu’est-ce que le logiciel libre? http://www.gnu.org/philosophy/free-sw.html, 2013. * [4] Ouattara Hadja, Ouoba Jonathan, and Bissyandé Tegawendé. Open source in africa: An opportunity wasted? -why and how floss should make sense for africa. In Fourth International IEEE EAI Conference on e-infrastructure and e-Services for Developing Countries, Yaoundé, Cameroon, 2012. * [5] Howison James and Herbsleb James D. Scientific software production:incentives and collaboration. In Computer Supported Cooperative Work. Computer Supported Cooperative Work, March 19–23 2011. * [6] Yi-Hsuan Lin, Tung-Mei ko, Tyng-Ruey Chunag, and Kwei-Jay Lin. Open source licenses and the creative commons framework: License selection and comparison. Journal of information science and engineering, 1(22):1–17, 2006\. * [7] Pérez Antonio J. Durán Mario and Varona Juan L. Misfortunes of a mathematicians’ trio using computer algebra systems: Can we trust? prépublication., 2013. * [8] SymPy Team. Sympy documentation library for symbolic mathematics. http://sympy.org/fr/index.html, 2013. Manuscript non publié. * [9] Masashi Ueda, Takahiro Uzuki, and Chihiro Suematsu. A cluster analysis of open source licenses. In Proceedings of the First International Conference on Open Source Systems, pages 50–53. Proceedings of the First International Conference on Open Source Systems, 2005. Kamel Ibn Aziz Derouiche Algerian IT Security Group Cyber Park Sidi Abdellah Incubateur Techno-bridge Route Nationale n°63 Rahmania, Zeralada [email protected]	arxiv-papers	2014-01-04T16:42:05	2024-09-04T02:49:56.296695	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "K.I.A.Derouiche", "submitter": "Kamel Ibn Aziz Derouiche kiaderouiche", "url": "https://arxiv.org/abs/1401.0827" }
1401.0836	# Sequential edge-coloring on the subset of vertices of almost regular graphs Petros A. Petrosyan Department of Informatics and Applied Mathematics, Yerevan State University, 0025, Armenia Institute for Informatics and Automation Problems, National Academy of Sciences, 0014, Armenia E-mail: [email protected] ###### Abstract Let $G$ be a graph and $R\subseteq V(G)$. A proper edge-coloring of a graph $G$ with colors $1,\ldots,t$ is called an $R$-sequential $t$-coloring if the edges incident to each vertex $v\in R$ are colored by the colors $1,\ldots,d_{G}(v)$, where $d_{G}(v)$ is the degree of the vertex $v$ in $G$. In this note, we show that if $G$ is a graph with $\Delta(G)-\delta(G)\leq 1$ and $\chi^{\prime}(G)=\Delta(G)=r$ ($r\geq 3$), then $G$ has an $R$-sequential $r$-coloring with $\|R\|\geq\left\lceil\frac{(r-1)n_{r}+n}{r}\right\rceil$, where $n=\|V(G)\|$ and $n_{r}=\|\\{v\in V(G):d_{G}(v)=r\\}\|$. As a corollary, we obtain the following result: if $G$ is a graph with $\Delta(G)-\delta(G)\leq 1$ and $\chi^{\prime}(G)=\Delta(G)=r$ ($r\geq 3$), then $\Sigma^{\prime}(G)\leq\left\lfloor\frac{2n_{r}(2r-1)+n(r-1)(r^{2}+2r-2)}{4r}\right\rfloor$, where $\Sigma^{\prime}(G)$ is the edge-chromatic sum of $G$. ## 1 Introduction In this note we consider graphs which are finite, undirected, and have no loops or multiple edges. Let $V(G)$ and $E(G)$ denote the sets of vertices and edges of a graph $G$, respectively. The degree of a vertex $v\in V(G)$ is denoted by $d_{G}(v)$ and the chromatic index of $G$ by $\chi^{\prime}\left(G\right)$. For a graph $G$, let $\Delta(G)$ and $\delta(G)$ denote the maximum and minimum degrees of vertices in $G$, respectively. An $(a,b)$-biregular bipartite graph $G$ is a bipartite graph $G$ with the vertices in one part all having degree $a$ and the vertices in the other part all having degree $b$. The terms and concepts that we do not define can be found in [5]. A proper edge-coloring of a graph $G$ is a mapping $\alpha:E(G)\rightarrow\mathbf{N}$ such that $\alpha(e)\neq\alpha(e^{\prime})$ for every pair of adjacent edges $e,e^{\prime}\in E(G)$. If $\alpha$ is a proper edge-coloring of a graph $G$, then $\Sigma^{\prime}(G,\alpha)$ denotes the sum of the colors of the edges of $G$. For a graph $G$, define the edge- chromatic sum $\Sigma^{\prime}(G)$ as follows: $\Sigma^{\prime}(G)=\min_{\alpha}\Sigma^{\prime}(G,\alpha)$, where minimum is taken among all possible proper edge-colorings of $G$. A proper $t$-coloring is a proper edge-coloring which makes use of $t$ different colors. If $\alpha$ is a proper $t$-coloring of $G$ and $v\in V(G)$, then $S\left(v,\alpha\right)$ denotes set of colors appearing on edges incident to $v$. Let $G$ be a graph and $R\subseteq V(G)$. A proper edge-coloring of a graph $G$ with colors $1,\ldots,t$ is called an $R$-sequential $t$-coloring if the edges incident to each vertex $v\in R$ are colored by the colors $1,\ldots,d_{G}(v)$. The concept of sequential edge-coloring of graphs was introduced by Asratian [1]. In [1, 2], he proved the following result. Theorem 1. If $G=(X\cup Y,E)$ is a bipartite graph with $d_{G}(x)\geq d_{G}(y)$ for every $xy\in E(G)$, where $x\in X$ and $y\in Y$, then $G$ has an $X$-sequential $\Delta(G)$-coloring. On the other hand, in [2] Asratian and Kamalian showed that the problem of deciding whether a bipartite graph $G=(X\cup Y,E)$ with $\Delta(G)=3$ has an $X$-sequential $3$-coloring is $NP$-complete. Some other results on sequential edge-colorings of graphs were obtained in [3, 4]. In particular, in [4] Kamalian proved the following result. Theorem 2. If $G$ is an $(r-1,r)$-biregular ($r\geq 3$) bipartite graph with $n$ vertices, then $G$ has an $R$-sequential $r$-coloring with $\|R\|\geq\left\lceil\frac{rn}{2r-1}\right\rceil$. In this note we generalize last theorem. As a corollary, we also obtain the following result: if $G$ is a graph with $\Delta(G)-\delta(G)\leq 1$ and $\chi^{\prime}(G)=\Delta(G)=r$ ($r\geq 3$), then $\Sigma^{\prime}(G)\leq\left\lfloor\frac{2n_{r}(2r-1)+n(r-1)(r^{2}+2r-2)}{4r}\right\rfloor$. ## 2 The Result Theorem 3. If $G$ a graph with $\Delta(G)-\delta(G)\leq 1$ and $\chi^{\prime}(G)=\Delta(G)=r$ ($r\geq 3$), then $G$ has an $R$-sequential $r$-coloring with $\|R\|\geq\left\lceil\frac{(r-1)n_{r}+n}{r}\right\rceil$, where $n=\|V(G)\|$ and $n_{r}=\|\\{v\in V(G):d_{G}(v)=r\\}\|$. Proof. Since $\chi^{\prime}(G)=\Delta(G)=r$, there exists a proper $r$-coloring $\alpha$ of the graph $G$ with colors $1,2,\ldots,r$. For $i=1,2,\ldots,r$, define the set $V_{\alpha}(i)$ as follows: $V_{\alpha}(i)=\left\\{v\in V(G):i\notin S(v,\alpha)\right\\}$. Clearly, for any $i^{\prime},i^{\prime\prime},1\leq i^{\prime}<i^{\prime\prime}\leq r$, we have $V_{\alpha}(i^{\prime})\cap V_{\alpha}(i^{\prime\prime})=\emptyset$ and $\underset{i=1}{\overset{r}{\bigcup}}V_{\alpha}(i)=V(G)\setminus V_{r}$, where $V_{r}=\\{v\in V(G):d_{G}(v)=r\\}$. Hence, $n-n_{r}=\|V(G)\|-\|V_{r}\|=\left\|\underset{i=1}{\overset{r}{\bigcup}}V_{\alpha}(i)\right\|=\underset{i=1}{\overset{r}{\sum}}\|V_{\alpha}(i)\|$. This implies that there exists $i_{0}$, $1\leq i_{0}\leq r$, for which $\|V_{\alpha}(i_{0})\|\geq\left\lceil\frac{n-n_{r}}{r}\right\rceil$. Let $R=V_{r}\cup V_{\alpha}(i_{0})$. Clearly, $\|R\|\geq n_{r}+\left\lceil\frac{n-n_{r}}{r}\right\rceil$. If $i_{0}=r$, then $\alpha$ is an $R$-sequential $r$-coloring of $G$; otherwise define an edge-coloring $\beta$ as follows: for any $e\in E(G)$, let $\beta(e)=\left\\{\begin{tabular}[]{ll}$\alpha(e)$,&if $\alpha(e)\neq i_{0},r$,\\\ $i_{0}$,&if $\alpha(e)=r$,\\\ $r$,&if $\alpha(e)=i_{0}$.\\\ \end{tabular}\right.$ It is easy to see that $\beta$ is an $R$-sequential $r$-coloring of $G$ with $\|R\|\geq\left\lceil\frac{(r-1)n_{r}+n}{r}\right\rceil$. $\square$ Corollary 1. If $G$ is an $(r-1,r)$-biregular ($r\geq 3$) bipartite graph with $n$ vertices, then $G$ has an $R$-sequential $r$-coloring with $\|R\|\geq\left\lceil\frac{rn}{2r-1}\right\rceil$. Corollary 2. If $G$ is a graph with $\Delta(G)-\delta(G)\leq 1$ and $\chi^{\prime}(G)=\Delta(G)=r$ ($r\geq 3$), then $\Sigma^{\prime}(G)\leq\left\lfloor\frac{2n_{r}(2r-1)+n(r-1)(r^{2}+2r-2)}{4r}\right\rfloor$ Proof. Let $\alpha$ be an $R$-sequential $r$-coloring of $G$ with $\|R\|\geq\left\lceil\frac{(r-1)n_{r}+n}{r}\right\rceil$ described in the proof of Theorem 3. Now, we have $\displaystyle\Sigma^{\prime}(G)\leq\Sigma^{\prime}\left(G,\alpha\right)$ $\displaystyle\leq$ $\displaystyle\frac{\frac{n_{r}\cdot r(r+1)}{2}+\left\lceil\frac{n-n_{r}}{r}\right\rceil\frac{r(r-1)}{2}+\left(n-n_{r}-\left\lceil\frac{n-n_{r}}{r}\right\rceil\right)\frac{(r+2)(r-1)}{2}}{2}$ $\displaystyle\leq$ $\displaystyle\frac{\frac{n_{r}\cdot r(r+1)}{2}+\frac{(n-n_{r})r(r-1)}{2r}+\left(n-n_{r}-\frac{n-n_{r}}{r}\right)\frac{(r+2)(r-1)}{2}}{2}$ $\displaystyle=$ $\displaystyle\frac{\frac{n_{r}\cdot r(r+1)}{2}+\frac{(n-n_{r})r(r-1)}{2r}+\frac{(n-n_{r})(r+2)(r-1)^{2}}{2r}}{2}$ $\displaystyle=$ $\displaystyle\frac{n_{r}\cdot r(r+1)}{4}+\frac{(n-n_{r})(r-1)(r^{2}+2r-2)}{4r}$ $\displaystyle=$ $\displaystyle\frac{2n_{r}(2r-1)+n(r-1)(r^{2}+2r-2)}{4r}.$ $\square$ ## References * [1] A.S. Asratian, Investigation of some mathematical model of scheduling theory, Doctoral Thesis, Moscow, 1980. * [2] A.S. Asratian, R.R. Kamalian, Interval colorings of edges of a multigraph, Appl. Math. 5 (1987) 25-34 (in Russian). * [3] R.R. Kamalian, Interval edge-colorings of graphs, Doctoral Thesis, Novosibirsk, 1990. * [4] R.R. Kamalian, On a number of vertices with an interval spectrum in proper edge colorings of some graphs, LiTH-MAT-R-2011/03-SE, Linkoping University, 2011. * [5] D.B. West, Introduction to Graph Theory, Prentice-Hall, New Jersey, 2001.	arxiv-papers	2014-01-04T18:48:01	2024-09-04T02:49:56.303604	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Petros A. Petrosyan", "submitter": "Petros Petrosyan", "url": "https://arxiv.org/abs/1401.0836" }
1401.0962	# Investigating the significance of zero-point motion in small molecular clusters of sulphuric acid and water Jake L. Stinson Email: [email protected] Department of Physics and Astronomy and London Centre for Nanotechnology, University College London, Gower Street, London, WC1E 6BT, United Kingdom Shawn M. Kathmann Physical Sciences Division, Pacific Northwest National Laboratory, Richland, Washington 99352, United States Ian J. Ford Department of Physics and Astronomy and London Centre for Nanotechnology, University College London, Gower Street, London, WC1E 6BT, United Kingdom ###### Abstract The nucleation of particles from trace gases in the atmosphere is an important source of cloud condensation nuclei (CCN), and these are vital for the formation of clouds in view of the high supersaturations required for homogeneous water droplet nucleation. The methods of quantum chemistry have increasingly been employed to model nucleation due to their high accuracy and efficiency in calculating configurational energies; and nucleation rates can be obtained from the associated free energies of particle formation. However, even in such advanced approaches, it is typically assumed that the nuclei have a classical nature, which is questionable for some systems. The importance of zero-point motion (also known as quantum nuclear dynamics) in modelling small clusters of sulphuric acid and water is tested here using the path integral molecular dynamics (PIMD) method at the density functional theory (DFT) level of theory. The general effect of zero-point motion is to distort the mean structure slightly, and to promote the extent of proton transfer with respect to classical behaviour. In a particular configuration of one sulphuric acid molecule with three waters, the range of positions explored by a proton between a sulphuric acid and a water molecule at 300 K (a broad range in contrast to the confinement suggested by geometry optimisation at 0 K) is clearly affected by the inclusion of zero point motion, and similar effects are observed for other configurations. ## I Introduction The role of sulphuric acid in the formation of cloud condensation nuclei (CCN) is believed to be significant(Roy, 2009; Zhang _et al._ , 2012), on account of its low vapour pressure, relatively high atmospheric concentration and its affinity to water. However, simple attempts to understand the binary nucleation of sulphuric acid and water in detail have proved problematic. It is clear that classical nucleation theory (CNT) is insufficient for describing this process, since the critical cluster size suggested from experimental data appears to be small, and consequently several extensions and alternatives have been studied (Ford, 2004; Vehkamäki, 2006; Napari _et al._ , 2010). One approach, the use of atomistic models that explicitly treat individual molecules or atoms within numerical simulations, has proliferated as a consequence of increasing computational power; especially based on quantum chemistry methods which treat the electronic interactions explicitly. Popular quantum chemistry methods include electronic density functional theory (DFT) (Re _et al._ , 1999; Bandy and Ianni, 1998; Ianni and Bandy, 1999; Arstila _et al._ , 1998; Larson _et al._ , 2000; Ding _et al._ , 2003; A. Natsheh _et al._ , 2004; Arrouvel _et al._ , 2005; Ding and Laasonen, 2004; Nadykto _et al._ , 2008; Kurtén _et al._ , 2009) and Møller-Plesset perturbation theory (MPn where n refers to the order of the perturbation) (Larson _et al._ , 2000; Ding and Laasonen, 2004; Kurtén _et al._ , 2009; Temelso _et al._ , 2012). The usual strategy is to identify the lowest energy molecular configuration and then to use the rigid-rotor-harmonic-approximation (RRHO) to compute free energies, and thereby investigate cluster stability and nucleation through specific growth and decay routes. The Born-Oppenheimer approximation (Martin, 2004) is employed by both DFT and MPn. It involves the separation of the wavefunctions of electrons and nuclei followed by a classical treatment of the dynamics of the nuclei. The DFT approach has been used to describe sulphuric acid and water clusters (Choe _et al._ , 2007; Anderson _et al._ , 2008; Hammerich _et al._ , 2008). In simulations based on such approaches, the sulphuric acid and water system has been observed to exhibit proton transfers. Such events are of particular importance in this system and a challenge to the modelling. A question that arises is whether we can account for such processes correctly while representing the nuclei as classical particles. Might a quantum treatment of the proton dynamics be more accurate? Perhaps the additional uncertainty in proton position can alter the delicate balance between neutral and ionised structures? In this paper we employ Path Integral Molecular Dynamics (PIMD) to study the quantum nuclear degrees of freedom (also known as zero-point motion) of sulphuric acid/water molecular clusters to address this question. A particular issue for consideration is the level of hydration of a single sulphuric acid molecule that is required for proton transfer to occur, a matter that can be addressed either by zero temperature calculations or dynamics performed at finite temperature. It has been suggested that the threshold is around three or more water molecules(Arrouvel _et al._ , 2005). Transfer of the second proton was studied by Ding and Laasonen (Ding and Laasonen, 2004) who concluded that it is likely for a level of hydration of around eight or nine water molecules. PIMD emulates the quantum behaviour of a particle by using a classical quasiparticle or bead description, a detailed derivation of which is given by Tuckerman (2010). The PIMD method has been shown to have a significant effect on the properties in some hydrogen bonded systems (Li _et al._ , 2011; Walker and Michaelides, 2010). PIMD has been employed previously together with a parametrised version of the PM6 model (Stewart, 2007) (a semi-empirical model of electronic structure) to study sulphuric acid and water clusters (Kakizaki _et al._ , 2009; Sugawara _et al._ , 2011). Kakizaki _et al._ (2009) concluded that the PIMD technique (using the normal mode transformation (Tuckerman, 2010)) increased thermal fluctuations and produced more liquid- like behaviour in systems at a temperature of ${\rm 250\>K}$ (Kakizaki _et al._ , 2009). Sugawara _et al._ (2011) studied the degree of hydration required for the first and second ionisation events for the sulphuric acid molecule, and concluded that the first ionisation takes place when four water molecules are present in the cluster in agreement with earlier work (Arrouvel _et al._ , 2005). The second ionisation event occurred in the presence of ${\rm 10-12}$ water molecules in contrast with the study by Ding and Laasonen (Ding and Laasonen, 2004) though the latter was based on geometry optimisation techniques rather than on molecular dynamics. As the purpose of this paper was to gauge the importance of zero-point motion in the sulphuric acid and water system as accurately as possible, it was decided to use DFT rather than the semi-empirical PM6 model developed by Kakizaki _et al._ (2009) . We study the importance of zero-point motion in a small cluster of sulphuric acid and water using PIMD (Feynman and Hibbs, 2010; Tuckerman, 2010) as implemented in the CASTEP code(Clark _et al._ , 2005). Section II describes the theory used, section III details our results, and section IV concludes our study where we comment on the significance of zero-point motion in the sulphuric acid and water system. Figure 1: A 16 bead representation of a system containing one sulphuric acid and four water molecules: the distribution of bead positions conveys the quantum uncertainty. ## II Methods According to the PIMD technique each particle (nucleus) is represented by a set of quasiparticles (known as beads) connected by harmonic springs. The following Hamiltonian describing the bead dynamics can be derived using the Trotter approximation (Tuckerman, 2010): $\mathcal{H}(x_{k},p_{k})=\sum_{k=1}^{P}\left[\frac{p_{k}^{2}}{2m^{\prime}}+\frac{1}{2}m\omega_{P}^{2}(x_{k+1}-x_{k})^{2}+\frac{1}{P}U(x_{k})\right]$ under the condition $x_{P+1}=x_{1}$, where $P$ is the number of beads and $x_{k}$ and $p_{k}$ are respectively the position and momentum of bead $k$. $\omega_{P}$ is the harmonic frequency of the inter-bead springs and is given by $\sqrt{P}/\beta\hbar$ where $\beta=(k_{B}T)^{-1}$ and $T$ and $k_{B}$ are the system temperature and the Boltzmann constant respectively. While $m$ denotes the mass of the particle, the mass of the beads is represented by $m^{\prime}$, and $U(x_{k})$ is the classical potential in which the particle moves. The quantum nuclear behaviour is reflected in both the position and momentum of the beads under the influence of this Hamiltonian, which is controlled by the stiffness of the inter-bead springs. Since the latter is proportional to the mass of the particle, the hydrogen nucleus is expected to be the most susceptible to zero-point effects. \| Time step [${\rm fs}$] \| Simulation time [${\rm ps}$] ---\|---\|--- SATH ${\rm 1}$ bead \| ${\rm 1.00}$ \| ${\rm 11.000}$ SATH ${\rm 4}$ bead \| ${\rm 0.50}$ \| $\,\,\,{\rm 1.500}$ SATH ${\rm 8}$ bead \| ${\rm 0.25}$ \| $\,\,\,{\rm 0.875}$ SATH ${\rm 16}$ bead \| ${\rm 0.50}$ \| ${\rm 10.673}$ SATH ${\rm 32}$ bead \| ${\rm 0.50}$ \| $\,\,\,{\rm 0.512}$ SAQH ${\rm 1}$ bead \| ${\rm 1.00}$ \| $\,\,\,{\rm 1.000}$ SAQH ${\rm 4}$ bead \| ${\rm 0.50}$ \| $\,\,\,{\rm 1.500}$ SAQH ${\rm 8}$ bead \| ${\rm 0.50}$ \| $\,\,\,{\rm 1.500}$ SAQH ${\rm 16}$ bead \| ${\rm 0.50}$ \| $\,\,\,{\rm 1.500}$ config H ${\rm 1}$ bead \| ${\rm 1.00}$ \| ${\rm 10.900}$ config H ${\rm 16}$ bead \| ${\rm 1.00}$ \| ${\rm 10.647}$ Table 1: Compilation of the simulation length and time step for the MD runs performed. SATH refers to sulphuric acid trihydrate and SAQH refers to sulphuric acid tetrahydrate, structures that assume typical configurations shown in Figure 2a and 2b respectively. Config H refers to the trihydrate configuration shown in Figure 4a. Note that the longest simulations were performed for SATH and config H. Figure 1 is a snapshot from a 16 bead simulation representing the behaviour of a cluster of one sulphuric acid and four water molecules. The spatial separation of the beads clearly illustrates the greater positional uncertainty of the hydrogen nuclei compared to that of the oxygen and the sulphur nuclei. (a) (b) (c) Figure 2: Geometry optimised configurations for tri- and tetrahydrated (SATH and SAQH) clusters are shown in (a) and (b) respectively. The labelling of various hydrogen bonds is referred to in Section III.2. (c) shows the binding energies of configurations (a) and (b) as a function of the system box size, converging to values obtained by Temelso _et al._ (2012) at the MP2 level. Molecular dynamics simulations at $\mathrm{300\>K}$ incorporating both classical nuclear dynamics and PIMD were performed using the CASTEP (Clark _et al._ , 2005) (version ${\rm 5.5}$) code. The standard on-the-fly ultrasoft pseudopotential provided internally by the CASTEP code was employed for all calculations. The Perdew-Burke-Ernzerhof (Perdew _et al._ , 1996) (PBE) functional was used with a plane wave basis set. The PBE functional has been found to perform well for hydrogen bonded systems(Thanthiriwatte _et al._ , 2011; Ireta _et al._ , 2004). A cut off energy of ${\rm 550\>eV}$ was found to converge the plane wave basis set sufficiently for all systems studied. A time step of $\mathrm{1\>fs}$ was used for classical (single bead) simulations and a time step of $\mathrm{0.5\>fs}$ or shorter was used for the PIMD simulations due to the stiffness of the inter-bead springs. CASTEP utilizes the Born-Oppenheimer version of ab initio MD and the Langevin thermostat with a friction constant of ${\rm 0.01\>fs^{-1}}$ was used in all simulations. The equilibration period was judged by observing when the running mean energy of the system had relaxed (usually requiring less than $\mathrm{0.5\>ps}$) and also by monitoring the distribution of cluster ’temperature’ (or kinetic energy in the centre of mass frame), which ought to be approximately Gaussian (Haile, 1997) with a standard deviation ($\sigma$) obeying $\sigma/\left\langle T\right\rangle\sim N^{-1/2}$. A typical temperature histogram satisfying this requirement is shown in Figure 3. The inset in Figure 3 shows that the typical relaxation time was in the order of $\mathrm{0.5\>ps}$ relaxation time. Initial configurations of sulphuric acid and water identified from the literature were constructed under a classical potential (MMFF94s) using the Avogadro (Hanwell _et al._ , 2012) (version ${\rm 1.0.3}$) package. The choices of time step and simulation time for various cases are given in Table 1. Figure 3: Histogram of cluster kinetic energy (represented as a temperature) from the equilibrated simulation referred to as SATH ${\rm 1}$ bead in Table 1. The inset shows the system’s energy as a function of time for the first $\mathrm{1\>ps}$ of the same simulation. Configurations of ${\rm[H_{2}SO_{4}][H_{2}O]_{n=3-4}}$ were studied at a target temperature of ${\rm 300\>K}$. PIMD uses a certain number of beads to approximate the zero-point motion, and cases with $P=1$, ${\rm 4}$, ${\rm 8}$, ${\rm 16}$ and ${\rm 32}$ beads were tested in this study. The staging transformation(Tuckerman, 2010) was used for all PIMD simulations. The $P=1$ case represents the classical limit of the PIMD technique and corresponds to the complete neglect of zero-point motion. The box size of the system was optimised against MP${\rm 2}$ level data (Temelso _et al._ , 2012) as shown in Figure 2c(c). The binding energies, at zero temperature, of the two configurations in Figures 2a and 2b are compared against MP${\rm 2}$ level data. A box size of ${\rm 15\>\hbox{\AA}}$ was chosen as a compromise between accuracy and computational demand. ## III Results ### III.1 DFT without zero-point motion Molecular configurations likely to feature a dissociated sulphuric acid molecule were identified from the literature and investigated. One such configuration was labelled III-i-1 by Re _et al._ (1999) and is illustrated here in Figure 4a and denoted config H. Our single bead simulations at $\mathrm{300\>K}$ show that the proton labelled H1 moves with considerable freedom between oxygens O1 and O5. Furthermore, Figure 4b demonstrates an anticorrelation between the length $R_{{\rm c}}$ of the dissociating bond O1-H1 and the sum of the lengths of the neighbouring hydrogen bonds, labelled O3-H7 and O4-H6 in Figure 4a, and denoted $R_{{\rm hy}}$. The formation of the ‘ionised’ state due to the switch to the O5-H1 bond (such that the value of $R_{{\rm c}}$ is large) is seen to depend upon the prior existence of both the neighbouring hydrogen bonds (namely a low value of $R_{{\rm hy}}$). If either neighbouring hydrogen bond is broken the system remains ‘neutral’ (with a low value of $R_{{\rm c}}$), which is not surprising since the configuration is then similar to the SATH structure shown in Figure 2. This is an important corollary to conclusions acquired from consideration of geometry optimisation at $\mathrm{0\>K}$, where config H has been shown to ionise (Re _et al._ , 1999). At ${\rm 300\>K}$ the behaviour can most certainly not be represented by harmonic fluctuations about an ionised mean structure and a free energy based on the rigid-rotor-harmonic-approximation for this configuration would fail due to significant anharmonic contributions. We shall return to this system in the next section. (a) (b) Figure 4: The configuration denoted config H is shown in (a) with labels that identify certain O-H pairs. Plot (b) illustrates the probability density (given in arbitrary units) as a function of two structural features labelled $R_{{\rm c}}$ (the length of the covalent bond O1-H1) and $R_{{\rm hy}}$ (the sum of the lengths of prospective hydrogen bonds O4-H6 and O3-H7), obtained at DFT level, equivalent to using a single bead in PIMD. The associated potential of mean force takes the form of a broad, shallow well where the ionisation of the configuration is correlated with the status of the adjacent hydrogen bonds, as denoted by $R_{{\rm hy}}$. ### III.2 PIMD A PIMD study was performed first for two low energy configurations (denoted SATH and SAQH) identified in the literature (Re _et al._ , 1999; Bandy and Ianni, 1998) and shown in Figures 2a and 2b. It is envisaged that hydrogen bonds, in particular those associated with the sulphuric acid, would be the most susceptible to zero-point effects due to the inherent tendency of sulphuric acid to dissociate. Figure 5 shows the average oxygen-oxygen distance (${\rm d_{OO}}$) of specific hydrogen bonds as a function of the number of beads representing atoms in the system. The bonds labelled hb1 and hb2 in the SATH structure contract in length by around ${\rm 2-5\%}$ with respect to the outcome of classical dynamics while the situation for hb3 is less clear. Note that the longest simulations were performed for the single bead and 16 bead representations of the SATH structure, as indicated in Table 1. For other cases shorter studies were performed to illustrate the trends, though the accuracy of the results is lower. Figure 5: The average oxygen-oxygen separation ${\rm d_{OO}}$ of specific hydrogen bonds as a function of the number of beads used in the simulation. Labels hb1 and hb2 refer to Figure 2a and hb3 is shown in Figure 2b. The error bars were determined by the standard blocking procedure (Flyvbjerg and Petersen, 1989; Frenkel and Smit, 2001) and a blocking length of $\mathrm{0.256\>ps}$ was found to give independent sampling. The calculations correspond to the cases listed in Table 1. Next we examine in detail how the behaviour of the hydrogen atom in hydrogen bond hb2 is affected by PIMD. This is explored by constructing a potential of mean force (PMF) for the hydrogen, defined by: $W(R,\beta)=-k_{B}T\ln g(R,\beta)$ where $R$ and $\beta$ are geometric parameters illustrated in Figure 6a and $g(R,\beta)$ is the proportion of simulation snapshots with the hydrogen located within the region defined by $R\rightarrow R+dR$ and $\beta\rightarrow\beta+d\beta$ divided by the equivalent proportion for noninteracting particles. For the PIMD simulations the centroid of the beads representing the hydrogen atom was used to produce the PMF. The method is described extensively by Kumar _et al._ (2007). Figure 6b and 6c show the PMFs acquired using classical MD and PIMD, respectively, for hydrogen bond hb2. The PMF plots in Figure 6 visualise the differences between the dynamics of the hb2 bond in Figure 2a under classical MD and the PIMD schemes. Such a comparison is limited by the computationally expensive techniques employed. However it does offer an insight into the importance of zero-point effects in small clusters of sulphuric acid and water. The main effect is a shift in the minimum of the PMF of hydrogen bond length $R$ by about $\mathrm{0.2\>}\textrm{\AA}$ going from the DFT to the PIMD result indicating that the zero-point motion has a mean configurational influence on this bond. Figure 6d is a one dimensional version of Figures 6b and 6b obtained by integrating over the $\beta$ parameter. (a) (b) (c) (d) Figure 6: Contour plots of the potential of mean force $W(R,\beta)$ in units of $k_{B}T$ for the hydrogen in the bond labelled hb2 in Figure 2a. The green dashed lines indicate contour levels of $\mathrm{0}$, $\mathrm{-2}$, $\mathrm{-4}$ and $\mathrm{-6}$. The coordinates for the PMF are defined by sketch (a) and the method follows the approach described by Kumar _et al._ (2007). Plot (b) shows results from standard DFT molecular dynamics and plot (c) arises from PIMD using 16 beads. The simulation times are given in Table 1. Plot (d) shows a 1D version of plots (b) and (c) obtained by integrating over the $\beta$ parameter. Figure 7: Plot of the O1-H1 bond length (denoted as $\mathrm{R_{O1H1}}$) against time in config H in Figure 4a from the $\mathrm{1}$ bead simulation as detailed in Table 1. The plot clearly illustrates the motion between the neutral and ionised states. The horizontal line drawn at ${\rm 1.22}\>{\rm\hbox{\AA}}$ provides a simple threshold between covalent and hydrogen bond-like behaviour of the O1-H1 bond. Under quantum nuclear dynamics the fraction of time spent above this threshold increases in the order of 10%. The effects of zero-point motion are clearly rather subtle. To explore this further, we return to the delicate switching behaviour of the O1-H1-O5 bonds discussed in section III.1 and contrast the classical and quantum nuclear dynamics. Figure 7 illustrates the motion of the proton between the neutral and ionised positions, discussed earlier, in terms of the O1-H1 bond length. Which of the nuclei O1 or O5 was the nearest neighbour to the H1 nucleus (see Figure 4a) was monitored to quantify this hopping behaviour. It was found that in the classical case H1 was closer to O1 for $\mathrm{21.5\%}$ percent of the simulation with standard error $\sigma_{SE}=3.2\%$ whereas in the 16 bead PIMD simulation this figure dropped to $\mathrm{14.8\%}$ with $\sigma_{SE}=2.7\%$. . This property was further investigated by defining a threshold for the O1-H1 bond length below which the system is considered neutral, and beyond which it is better described as ionised. We define a ${\rm 1.22\>}\hbox{\AA}$ distance to separate the two regimes, and this is shown as a horizontal line in Figure 7. For the classical dynamics, the percentage of time the system remains neutral according to this criterion is ${\rm 20.1\%}$ with ${\rm\sigma_{SE}=2.9\%}$. An analysis of the PIMD simulation with 16 beads yields a corresponding percentage of neutral residence time of ${\rm 12.5\%}$ with $\sigma_{SE}={\rm 2.4\%}$. These results are consistent with those determined from the nearest neighbour criterion. The proportion of time spent in the ionised configuration rises from $\mathrm{79.1\%}$ to $\mathrm{87.5\%}$. This suggests that the inclusion of zero-point motion promotes the formation of the ionised state; quantum uncertainty favours proton transfer. ## IV Conclusions As a consequence of the computational expense of the PIMD technique, especially when using many beads, the simulations presented are limited in duration to around $\mathrm{10\>ps}$ for some configurations, and rather less for others. The statistics on the structural and dynamical behaviour are therefore preliminary. However, it is possible to extract some important features from these simulations that correspond to intuitive expectation, and which can be explored further with more extensive calculations. Our study of small clusters of water and sulphuric acid molecules leads us to two main conclusions. Firstly, we have demonstrated that molecular dynamics can reveal features that are not available from knowledge of the geometry optimised structure at zero temperature. The prime example of this is the complex behaviour of cluster configuration III-i-1 identified by Re _et al._ (Re _et al._ , 1999) and here denoted config H. This configuration has been regarded as the most stable ionised configuration for the trihydrated sulphuric acid molecule (Bandy and Ianni, 1998; Re _et al._ , 1999; Temelso _et al._ , 2012), but our results indicate that the structure exhibits both neutral and ionised characteristics at ${\rm 300\>K}$. This conclusion highlights the limitations of the RRHO approximation(Kathmann _et al._ , 2007a, b) for free energy estimation using a single optimised structure. Secondly, the inclusion of zero-point motion through PIMD simulations of the sulphuric acid-water system has been shown to produce small but clear structural distortion at ${\rm 300\>K}$ in a selected number of configurations when compared with classical dynamics. The mean oxygen-oxygen separation of hydrogen bonds hb1 and hb2 in the structure shown in Figure 2a is reduced by ${\rm 2-5\%}$. We observe a mild shortening of the hb2 hydrogen bond length, shown by constructing potentials of mean force for the classical and PIMD schemes, as illustrated in Figure 6. Furthermore, our results indicate that zero-point motion brings about a greater propensity for proton transfer in the O1-H1-O5 substructure of the configuration shown in Figure 4a at ${\rm 300\>K}$. This conclusion is consistent with the paper by Li _et al._ (2011) where quantum nuclear effects on the hydrogen bond are studied, specifically Figure 3 in reference (23) where the OO length is compared with the length of the projection of the covalent OH bond on the OO vector. The implication is that the projected covalent bond length is increased by quantum effects when the hydrogen bond is considered to be strong, as judged by a shift in vibrational frequency of the covalent OH bond due to the presence of the hydrogen bond. Our research supports the view that the zero-point effect is most significant in configurations where proton transfer is intrinsically likely. Classical and PIMD simulations of the cluster shown in Figure 4a have demonstrated frequent proton transfer. Using an O-H separation of ${\rm 1.22}\>{\rm\hbox{\AA}}$ as a threshold for distinguishing the ionised from the neutral state, the cluster is found to remain neutral ${\rm 20.1\%}$ of the time (${\rm\sigma_{SE}=2.9\%}$) with classical MD and ${\rm 12.5\%}$ ($\sigma_{SE}={\rm 2.4\%}$) according to PIMD. It is possible to infer that quantum effects have increased the degree of proton transfer. It is expected that simulations at lower temperatures would increase the significance of the zero-point effects, making this an avenue for future research. In addition, since substances such as ammonia and amines are increasingly thought to be relevant to atmospheric nucleation (Kurtén _et al._ , 2008; Kirkby _et al._ , 2011), assessing the importance of zero-point motion in these systems would also be of interest. In summary, zero-point motion does affect the structure of small clusters of sulphuric acid and water, particularly the lengths of hydrogen bonds. At $\mathrm{300\;K}$, the contribution appears to be most significant for cases that are intrinsically susceptible to proton transfer. ## V Acknowledgements We thank Prof. Angelos Michaelides and his group at UCL for practical advice and helpful discussions and this work benefited from interactions within the Thomas Young Centre. SMK was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Chemical Sciences, Geosciences, and Biosciences. JLS was supported by the IMPACT scheme at UCL and by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Chemical Sciences, Geosciences, and Biosciences. 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Kulmala, Nature 476, 429 (2011).	arxiv-papers	2014-01-06T00:58:46	2024-09-04T02:49:56.311991	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Jake Stinson, Shawn Kathmann and Ian Ford", "submitter": "Jake Stinson", "url": "https://arxiv.org/abs/1401.0962" }
1401.1051	# Action Minimizing Solutions of The One-Dimensional $N$-Body Problem With Equal Masses ††thanks: Supported partially by NSF of China Xiang Yu111Email:[email protected] and Shiqing Zhang222Email:[email protected] Department of Mathematics, Sichuan University, Chengdu 610064, China Abstract. When we use variational methods to study the Newtonian $N$-body problem, the main problem is how to avoid collisions. C.Marchal got a remarkable result, that is, a path minimizing the Lagrangian action functional between two given configurations is always a true (collision-free) solution, so long as the dimension $d$ of physical space $\mathbb{R}^{d}$ satisfies $d\geq 2$. But Marchal’s idea can’t apply to the case of the one-dimensional physical space. In this paper, we will study the fixed-ends problem for the one-dimensional Newtonian $N$-body problem with equal masses to supplement Marchal’s result. More precisely, we first get the isolated property of collision moments for a path minimizing the action functional between two given configurations, then, if the particles at two endpoints have the same order, the path minimizing the action functional is always a true (collision- free) solution; otherwise, although there must be collisions for any path, we can prove that there are at most $N!-1$ collisions for any action minimizing path. Key Words: N-body problem; Collisions; Variational methods; Central configurations; The fixed-ends problem. 2010 Mathematics Subject Classifications: 34B15; 70F10; 70F16; 70G75. ## 1 Introduction and Main Results In Euclidean space ${\mathbb{R}}^{d}$, we consider $N\geq 2$ particles with positive masses , affected by their gravitational interactions. The equation of motion of the $N$-body problem is written as $m_{k}\ddot{q}_{k}=\sum_{1\leq j\leq N,j\neq k}\frac{m_{j}m_{k}(q_{j}-q_{k})}{\|q_{j}-q_{k}\|^{3}}.$ (1.1) where $m_{k}$ is the mass and $q_{k}$ the position of the $k$-th body. Since these equations are invariant by translation, we can assume that the center of masses is at the origin. Firstly, we set some notations and describe preliminary results that will be needed later. Let $\mathcal{X}_{d}$ denote the space of configurations for $N$ point particles in Euclidean space $\mathbb{R}^{d}$ with dimension $d$, whose center of masses is at the origin, that is, $\mathcal{X}_{d}=\\{q=(q_{1},\cdots,q_{N})\in(\mathbb{R}^{d})^{N}:\sum_{k=1}^{N}{m_{k}q_{k}}=0\\}$. For each pair of indices $j,k\in\\{1,\ldots,N\\}$, let $\Delta_{(j,k)}$ denote the collision set of the j-th and k-th particles $\Delta_{(j,k)}=\\{q\in\mathcal{X}_{d}:q_{j}=q_{k}\\}$. Let $\Delta_{d}=\bigcup_{j,k}\Delta_{(j,k)}$ be the collision set in $\mathcal{X}_{d}$. The space of collision-free configurations $\mathcal{X}_{d}\backslash\Delta_{d}$ is denoted by $\hat{\mathcal{X}}_{d}$. Let $\mathbb{T}$ denote the time interval $[T_{1},T_{2}]$.By the path space $\Lambda$, we mean the Sobolev space $\Lambda=H^{1}(\mathbb{T},\mathcal{X}_{d})$; we denote by $\Lambda(q_{i},q_{f})$ the space of paths $q(t)\in\Lambda$ beginning in the configuration ${q_{i}}$ at the moment $T_{1}$ and ending in the configuration ${q_{f}}$ at the moment $T_{2}$. For a motion $q(t)$ of the $N$-body problem, we say there is a collision at time $t_{0}$ if, for at least two indices, say $j$ and $k$, $q_{k}(t)\rightarrow c_{k}$, $q_{l}(t)\rightarrow c_{l}$ as $t\rightarrow t_{0}$, and $c_{j}=c_{k}$. We now ‘cluster’ the particles according to their limit points, that is, according to which particles are colliding each other. So, let the different limit points be $c_{1},\cdots,c_{n}$, and let $S_{k}=\\{j\in\\{1,\cdots,N\\}:q_{j}(t)\rightarrow c_{k}~{}{as}~{}t\rightarrow t_{0}\\},~{}k=1,\cdots,n$. We consider the opposite of the potential energy (force function) defined by $U(q)=\sum_{k<j}{\frac{m_{k}m_{j}}{\|q_{k}-q_{j}\|}}.$ (1.2) The kinetic energy is defined (on the tangent bundle of $\mathcal{X}_{d}$) by $K=\sum_{j=1}^{N}{\frac{1}{2}{m_{j}\|\dot{q}_{j}\|^{2}}}$, the total energy is $E=K-U$ and the Lagrangian is $L(q,\dot{q})=L=K+U=\sum_{j}\frac{1}{2}m_{j}\|\dot{q}\|^{2}+\sum_{k<j}{\frac{m_{k}m_{j}}{\|q_{k}-q_{j}\|}}$. Given the Lagrangian L, the positive definite functional $\mathcal{{A}}:\Lambda\rightarrow\mathbb{R}\cup\\{+\infty\\}$ defined by $\mathcal{{A}}(q)=\int_{\mathbb{T}}{L(q(t),\dot{q}(t))dt}$ (1.3) is termed as action functional (or the Lagrangian action). The action functional $\mathcal{{A}}$ is of class $C^{1}$ on the collision- free space $\hat{\Lambda}(q_{i},q_{f})\subset\Lambda(q_{i},q_{f})$. Hence the critical point of $\mathcal{{A}}$ in $\hat{\Lambda}(q_{i},q_{f})$ is a classical solution (of class $C^{2}$) of Newtonian equations $m_{j}\ddot{q}_{j}=\frac{\partial U}{\partial q_{j}}.$ (1.4) From the viewpoint of the Least Action Principle, action minimizing solutions of the N-body problem are the most important and the simplest, so it is natural to search for minimizers of the Lagrangian action joining two given configurations in a fixed time. It’s worth noticing that a lot of results have been founded by the action minimization methods just in recent years, please see [1, 2, 3, 4, 6, 7, 8, 9, 10, 12, 14, 15, 16, 24, 25, 26] and the references therein. Recently, the interest in this problem has grown considerably due to the discovery of the figure eight solution [9]. Since the potential of the $N$-body problem is singular at collision configurations, the main problem involved in variational minimizations is that collision could occur for an action minimizer, even if the set of collision times has necessarily zero measure, the system undergoes a collision of two or more bodies, which prevents it form being a true solution. Some techniques are created to overcome the difficulty, ultimately, one got a major advance (essentially due to Christian Marchal) in this subject. More specifically, the advance is the following remarkable theorem [16, 8, 12]. ###### Theorem 1.1 _(Marchal)_ Given the initial moment $T_{1}$,the final moment $T_{2}$ $(T_{2}>T_{1})$ and two corresponding N-body configurations $q_{i}=(q_{i1},\cdots,q_{iN})$, $q_{f}=(q_{f1},\cdots,q_{fN})$ in $\mathbb{R}^{d}$ $(d>1)$, an action minimizing path joining $q_{i}$ to $q_{f}$ in time $T_{2}-T_{1}$ is collision-free for $t\in(T_{1},T_{2})$. This theorem, together with the lower semicontinuity of the action, implies in particular that there always exists a collision-free minimizing solution joining two given collision-free N-body configurations in a given time. The idea of Christian Marchal is to compare the average of the Lagrangian action for local deformations in all possible directions for a local isolated collision with the original Lagrangian action. Roughly speaking, Marchal’s idea is as following : let $a={2}$, by $\frac{1}{2\pi}\int^{2\pi}_{0}{(\frac{1}{\|a+e^{\sqrt{-1}\theta}\|}-\frac{1}{\|a\|})}<0$(i.e., the average of the Lagrangian action on local deformations is smaller than the original Lagrangian action), then there must be some $\theta$ satisfying ${\frac{1}{\|a+e^{\sqrt{-1}\theta}\|}<\frac{1}{\|a\|}}$; however, in the case of $d=1$, we have $\frac{\frac{1}{\|a+1\|}+\frac{1}{\|a-1\|}}{2}-\frac{1}{\|a\|}>0$(i.e., the average of Lagrangian action on local deformations is bigger than the original Lagrangian action), so Marchal’s idea can’t apply to the case of the one- dimensional physical space. In fact, Marchal’s method is local, but the fixed- ends problem for the one-dimensional Newtonian $N$-body problem is a more global problem, since given two collinear configurations, if the particles at two configurations have different order, then any path joining two given configurations suffers collisions for topological reasons, hence Marchal’s theorem does not hold for the one-dimensional physical space. Fortunately, the one-dimensional Newtonian $N$-body problem has its particular characteristics, in particular, the fact that all collinear central configurations are non- degenerate gives us the other facility. Thus, in this paper, by using a different approach, we will study the fixed-ends (Bolza) problem for the one- dimensional Newtonian $N$-body problem. More precisely, we will prove that the path minimizing the Lagrangian action functional between two given configurations is always a true (collision-free) solution of the one- dimensional $N$-body problem, if the particles at two endpoints have the same order, where, we say that the particles at configurations $q_{i}=(q_{i1},\cdots,q_{iN})$ and $q_{f}=(q_{f1},\cdots,q_{fN})$ have the same order if $q_{ij}-q_{ik}\geq 0\Leftrightarrow q_{fj}-q_{fk}\geq 0$ for any $j\neq k$, in other words, the relations $q_{ij}>q_{ik}$ and$q_{fj}<q_{fk}$ can’t hold for any $j\neq k$ at the same time. In particular, if $q_{j_{1}}<q_{j_{2}}<\cdots<q_{j_{N}}$, we call $({j_{1}},{j_{2}},\cdots,{j_{N}})$ is the order of the configuration $(q_{1},q_{2},\cdots,q_{N})$. This requirement is necessary, since it is obvious that there must be collisions for any path if the particles at two endpoints have different order. In this paper, we will study the fixed-ends problem for the one-dimensional Newtonian $N$-body problem with equal masses. Our main results are the following Propositions. ###### Theorem 1.2 Suppose the critical path $q(t)$ of the Lagrangian action for the one- dimensional Newtonian $N$-body problem has a collision at some moment $t_{0}$, every corresponding colliding cluster $S_{k}$ has $n_{k}$ elements. If the collision is isolated at time $t_{0}$ for some right neighborhood or left neighborhood of $t_{0}$, then we have the following results for some right neighborhood or left neighborhood of $t_{0}$: if $n_{k}=1$, that is,the cluster $S_{k}$ is singleton, the body in the cluster is not in a collision, let $j\in S_{k}$, then $q_{j}(t)=q_{j}(t_{0})+\dot{q}_{j}(t_{0})(t-t_{0})+o(t-t_{0})$; if $n_{k}\geq 2$, let $j\in S_{k}$, then $q_{j}(t)=q_{j}(t_{0})+s_{j}(t-t_{0})^{\frac{2}{3}}+o((t-t_{0})^{\frac{2}{3}})$, where $s_{j},j\in S_{k}$ is a central configuration for the particles corresponding to the colliding cluster $S_{k}$. ###### Remark 1.1 Our results depend strongly on the fact that all collinear central configurations are non-degenerate. ###### Theorem 1.3 Suppose the action minimizer $q(t)$ of the Lagrangian action for the one- dimensional Newtonian $N$-body problem with equal masses has a collision at moment $t_{0}$, then the collision moment $t_{0}$ is isolated, that is, there exists some $\varepsilon>0$, $q(t)$ is collision-free in $(t_{0}-\varepsilon,t_{0}+\varepsilon)$ except at time $t_{0}$. Hence there are at most finitely many collision moments for the fixed-ends (Bolza) problem. ###### Remark 1.2 There are some studies about the isolated collision for the general $N$-body problem(see [8, 12, 22]). However, all the results of them only said that: there exists an isolated collision for the general $N$-body problem. Our results show that we can say more about the one-dimensional Newtonian $N$-body problem with equal masses: all the collisions are isolated and finite. ###### Theorem 1.4 For the one-dimensional $N$-body problem with equal masses, given the initial moment $T_{1}$,the final moment $T_{2}$ $(T_{2}>T_{1})$ and two corresponding N-body configurations $q_{i}=(q_{i1},\cdots,q_{iN})$, $q_{f}=(q_{f1},\cdots,q_{fN})$ in $\mathbb{R}^{1}$, if $q_{i}$, $q_{f}$ have the same order in $\mathbb{R}^{1}$, then the action minimizing path of the fixed-ends problem joining $q_{i}$ to $q_{f}$ in time $T_{2}-T_{1}$ is collision-free for $t\in(T_{1},T_{2})$. ###### Theorem 1.5 If the given two configurations $q_{i}$, $q_{f}$ have the different order in $\mathbb{R}^{1}$, then the action minimizing path of the fixed-ends problem with equal masses joining $q_{i}$ to $q_{f}$ in time $T_{2}-T_{1}$ has some collisions for some $t\in(T_{1},T_{2})$, but there are at most $N!-1$ collision moments in $(T_{1},T_{2})$. ###### Remark 1.3 Our results and methods remain valid for more general force function defined by $U(q)=\sum_{k<j}{\frac{m_{k}m_{j}}{\|q_{k}-q_{j}\|^{\alpha}}}$, where $\alpha$ is any positive real number such that $0<\alpha<2$ . It is natural to ask the following questions. Question. 1\. Do the Theorem 1.3,1.4 and 1.5 hold for the one-dimensional $N$-body problem with any masses? 2.Given two configurations which have the different order in $\mathbb{R}^{1}$ and a time $T=T_{2}-T_{1}>0$, what is the largest number of collision times in $(T_{1},T_{2})$? Is the largest number of collision times in $(T_{1},T_{2})$ one? The similar questions can be asked for the fixed-ends problem with any masses. We hope that the answers of these questions are all positive. The paper is structured as follows. Section 2 introduces some definitions and some lemmas, Section 3 gives the proofs of the main results by using the concepts and results introduced in Section 1 and Section 2. ## 2 Some Definitions and Some Lemmas In this section, we give some definitions and recall some classical results. The first one is the important concept of the central configuration [23], ###### Definition 2.1 A configuration $q=(q_{1},\cdots,q_{N})\in{\mathcal{X}}_{d}\setminus\Delta_{d}$ is called a central configuration if there exists a constant $\lambda\in{\mathbb{R}}$ such that $\sum_{j=1,j\neq k}^{N}\frac{m_{j}m_{k}}{\|q_{j}-q_{k}\|^{3}}(q_{j}-q_{k})=-\lambda m_{k}q_{k},1\leq k\leq N,$ (2.1) the value of $\lambda$ in (2.1) is uniquely determined by $\lambda=\frac{U(q)}{I(q)},$ (2.2) where $I(q)=\sum_{1\leq j\leq N}m_{j}\|q_{j}\|^{2}.$ (2.3) Let us recall that, for a motion $q(t)$ of $N$-body problem, we say there is a collision at time $t_{0}$ if as $t\rightarrow t_{0}$, $q_{j}(t)\rightarrow c_{j},~{}j\in\\{1,\cdots,N\\}$ and for at least two different indices, say $j$ and $k$ such that $c_{j}=c_{k}$. Without loss of generality, we can assume that the time $t$ approach $t_{0}$ from the right of $t_{0}$, that is, we think $t\rightarrow{t_{0}}+$. Denote the different limit points by $c_{1},\cdots,c_{n}$, and classify the indices according to particles colliding each other,let $S_{k}=\\{j\in\\{1,\cdots,N\\}:q_{j}(t)\rightarrow c_{k}~{}{as}~{}t\rightarrow{t_{0}}+\\}$, and assume $S_{k}$ has $n_{k}$ elements for $k=1,\cdots,n$; then we say that every $S_{k}$ is a colliding cluster of particles. Let $M_{k}=\sum_{j\in S_{k}}m_{j}$ be the total mass of particles in cluster $S_{k}$ and $\bar{c}_{k}=\sum_{j\in S_{k}}m_{j}q_{j}/M_{k}$ be the center of mass of the particles in $S_{k}$. When $S_{k}$ has $n_{k}\geq 2$ elements, if $j\in S_{k}$, let $r_{(k)j}(t)=\frac{q_{j}-c_{k}}{(t-t_{0})^{\frac{2}{3}}}$, then we call $r_{(k)}(t)=(r_{(k)l_{1}}(t),\cdots,r_{(k)l_{n_{k}}}(t))$ be the normalized configuration corresponding to the colliding cluster $S_{k}$, where $\\{l_{1},\cdots,l_{n_{k}}\\}=S_{k}$. Let $\textbf{CC}_{k}:=\\{r_{(k)}:\sum_{j\in S_{k},j\neq i}\frac{m_{j}}{\|r_{(k)j}-r_{(k)i}\|^{3}}(r_{(k)j}-r_{(k)i})=-\frac{2}{9}r_{(k)i},i\in S_{k}\\}$ (2.4) be the set of the central configuration corresponding to colliding cluster $S_{k}$, where we assume the value of $\lambda$ which only affects the size of the central configuration to be $\frac{2}{9}$, note that the center of mass of $r_{(k)}$ is zero. Before giving the proofs of the main results of this paper, some lemmas are needed. we recall some classical results concerning a motion $q(t)$ of $N$-body problem in some neighborhood of isolated collision instant $t_{0}$. The first one says that all collision orbits of $N$-body problem in some neighborhood of isolated collision instant $t_{0}$ have the property that $r_{(k)}(t)\rightarrow\textbf{CC}_{k}$ as $t\rightarrow t_{0}$, where $r_{(k)}(t)$ and $\textbf{CC}_{k}$ are respectively the normalized configuration of the collision orbit and the set of the central configuration corresponding to colliding cluster $S_{k}$. ###### Lemma 2.1 Suppose a colliding cluster $S_{k}$ have $n_{k}\geq 2$ elements, let $r_{j}(t)=\frac{q_{j}-c_{k}}{(t-t_{0})^{\frac{2}{3}}}$ for any $j\in S_{k}$, be the normalized configuration. Then for every converging sequence $r(t_{j})=(r_{l_{1}}(t_{j}),\cdots,r_{l_{n_{k}}}(t_{j}))$, where $l_{1},\cdots,l_{n_{k}}\in S_{k}$, $t_{j}$ belong to some neighborhood of $t_{0}$ $(j\in\mathbb{N})$, the limit $\lim_{j\rightarrow\infty}r(t_{j}):=s$ is a central configuration. ###### Remark 2.1 This result is classical(see [18, 12] for a proof). Because of the called ($Painlev\acute{e}$-$Wintner$) infinite spin problem(see [23, 19, 18, 5, 8]et al), in general, one can not get a better result. The second one states the special property, which we need, of the one- dimensional Newtonian $N$-body problem. ###### Lemma 2.2 ([17]) All collinear central configurations are non-degenerate in $\mathbb{R}^{d}$. Then, in the following, we get the important result which says that, for a isolated collision of particles, not only does $r_{(k)}(t)\rightarrow\textbf{CC}_{k}$ as $t\rightarrow t_{0}$, but also there is a central configuration $s\in\textbf{CC}_{k}$ so that $r_{(k)}(t)\rightarrow s$ as $t\rightarrow t_{0}$, so long as all central configurations are non-degenerate. ###### Lemma 2.3 For the one-dimensional $N$-body problem, suppose a colliding cluster $S_{k}$ have $n_{k}\geq 2$ elements, let $r_{j}(t)=\frac{q_{j}-c_{k}}{(t-t_{0})^{\frac{2}{3}}}$ for any $j\in S_{k}$, be the normalized configuration. Then $\lim_{t\rightarrow t_{0}}r(t)$ exists, the limit $s:=\lim_{t\rightarrow t_{0}}r(t)$ is a central configuration, furthermore, $s$ and $r(t)$ have the same order. Proof of Lemma 2.3: It’s similar to a particular case of the results of Saari [18], we can get lemma 2.3 by using the unstable manifold theorem for a normally hyperbolic invariant set (Hirsch et al. [13]) and Lemma 2.2. $~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\Box$ ###### Remark 2.2 There are some methods to study this important problem(see [23, 20, 21, 19, 18, 11, 5, 8, 12]et al). To our knowledge, Lemma 2.3 was not definitely stated. Since all collinear central configurations are non-degenerate, we apply the idea of D.Saari (the unstable manifold theorem for a normally hyperbolic invariant set) to simply get the result. The last lemma is about the existence of isolated collisions for the general $N$-body problem. ###### Lemma 2.4 ([8, 12]) Suppose the action minimizer $q(t)$ of the Newtonian $N$-body problem has collisions in a time interval, then there must exist an isolated collision in this time interval. Using above lemmas, we will give the proofs of our main results in the next section. ## 3 The Proofs of Main Results In this section, we give the proofs of main results in this paper. Proof of Theorem 1.2: This result easily comes from Lemma 2.3. $~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\Box$ First of all, let’s establish a lemma to simplify the proofs of other theorems. ###### Lemma 3.1 Given the initial moment $T_{1}$,the final moment $T_{2}$ $(T_{2}>T_{1})$ and two corresponding N-body configurations $q_{i}=(q_{i1},\cdots,q_{iN})$, $q_{f}=(q_{f1},\cdots,q_{fN})\in\mathcal{X}_{1}\backslash\Delta_{1}$ which have the same order in $\mathbb{R}^{1}$. Suppose a path $q(t)\in\Lambda(q_{i},q_{f})$ has only one collision moment $t_{0}$ in $(T_{1},T_{2})$, then the path $q(t)$ cannot be an action minimizing path of the fixed-ends problem joining $q_{i}$ to $q_{f}$ in time $T_{2}-T_{1}$. Proof of Lemma 3.1: By using reduction to absurdity, assume that the path $q(t)$ is an action minimizing path of the fixed-ends problem joining $q_{i}$ to $q_{f}$ in time $T_{2}-T_{1}$. Without loss of generality, we can assume that $q_{1}(t)<q_{2}(t)<\cdots<q_{N}(t)$ for $t\in[T_{1},T_{2}]\backslash\\{t_{0}\\}$ and $q_{1}(t_{0})\leq q_{2}(t_{0})\leq\cdots\leq q_{N}(t_{0})$. Let $x_{k}(t)=q_{k+1}(t)-q_{k}(t)$ for $k\in\\{1,\cdots,N-1\\}$ and $M=m_{1}+m_{2}+\cdots+m_{N}$, then $x(t)=(x_{1}(t),x_{2}(t),\cdots,x_{N-1}(t))$ is an action minimizing path of the fixed-ends problem joining $x_{i}=x(T_{1})$ to $x_{f}=x(T_{2})$ in time $T_{2}-T_{1}$ for the action functional $\mathcal{{F}}(x)=\int_{\mathbb{T}}{\sum_{1\leq l<k\leq N}\frac{m_{k}m_{l}}{2M}{[{\|\sum_{l\leq j\leq k-1}\dot{x}_{j}\|^{2}}+\frac{2M}{\|\sum_{l\leq j\leq k-1}{x}_{j}\|}]}dt}$ (3.1) In fact, by Lagrangian identity, we have $\displaystyle\mathcal{{A}}(q)$ $\displaystyle=$ $\displaystyle\int_{\mathbb{T}}{L(q(t),\dot{q}(t))dt}$ $\displaystyle=$ $\displaystyle\int_{\mathbb{T}}{[\frac{1}{2(\sum_{1\leq j\leq N})m_{j}}\sum_{1\leq l<k\leq N}m_{k}m_{l}\|\dot{q}_{k}-\dot{q}_{l}\|^{2}+\sum_{1\leq l<k\leq N}\frac{m_{k}m_{l}}{\|{q}_{k}-{q}_{l}\|}]dt}$ $\displaystyle=$ $\displaystyle\int_{\mathbb{T}}{\sum_{1\leq l<k\leq N}\frac{m_{k}m_{l}}{2M}{[{\|\sum_{l\leq j\leq k-1}\dot{x}_{j}\|^{2}}+\frac{2M}{\|\sum_{l\leq j\leq k-1}{x}_{j}\|}]}dt}$ $\displaystyle=$ $\displaystyle\mathcal{{F}}(x)$ In the following, we will construct another path $y(t)$ which satisfies the same boundary conditions with $x(t)$, but the value of $\mathcal{{F}}(y)$ is smaller than the value of $\mathcal{{F}}(x)$. Since we can get similar result by using the following method for any $k\geq 1$ such that $x_{k}(t)\rightarrow 0$ when $t\rightarrow t_{0}$, for the sake of convenience, we only consider that $x_{1}(t)\rightarrow 0$ when $t\rightarrow t_{0}$. Then we have $x_{1}(t)=\alpha(t_{0}-t)^{\frac{2}{3}}+o((t_{0}-t)^{\frac{2}{3}})$ for some left neighborhood of $t_{0}$ and $x_{1}(t)=\beta(t-t_{0})^{\frac{2}{3}}+o((t-t_{0})^{\frac{2}{3}})$ for some right neighborhood of $t_{0}$ from Theorem 1.2, where $\alpha$, $\beta$ are appropriate positive numbers. Let $A=\frac{m_{1}(M-m_{1})}{2M}$ and $B=\sum_{3\leq k\leq N}\frac{m_{1}m_{k}}{M}\sum_{2\leq j\leq k-1}\dot{x}_{j}$, from Theorem 1.2 we know that * • if $x_{j}(t)\rightarrow 0$ when $t\rightarrow t_{1}$ for some $j\in\\{2,\cdots,N-1\\}$, then $B=\frac{d(\tilde{\alpha}(t_{0}-t)^{\frac{2}{3}}+o((t_{0}-t)^{\frac{2}{3}}))}{dt}$ for some left neighborhood of $t_{0}$ and $B=\frac{d(\tilde{\beta}(t-t_{0})^{\frac{2}{3}}+o((t-t_{0})^{\frac{2}{3}}))}{dt}$ for some right neighborhood of $t_{0}$, where $\tilde{\alpha},\tilde{\beta}$ are appropriate positive numbers; * • if $x_{j}(t)>0$ for some neighborhood of $t_{0}$ and any $j\in\\{2,\cdots,N-1\\}$, then $B=\frac{d(a+b(t-t_{0})+o(\|t_{1}-t\|))}{dt}$ for some neighborhood of $t_{0}$, where $a>0,b$ are appropriate real numbers. Then it is easy to know that the inequality $A\dot{x}^{2}_{1}+B\dot{x}_{1}>0$ (3.2) holds in some neighborhood of $t_{0}$. For sufficiently small positive number $\delta$, there are two sufficiently small positive numbers $\epsilon,\varepsilon$ such that $x_{1}(t_{1}-\epsilon)=x_{1}(t_{1}+\varepsilon)=\delta$, ${x}_{1}(t)\leq\delta$ for $t\in[t_{1}-\epsilon,t_{1}+\varepsilon]$ and the interval $[t_{1}-\epsilon,t_{1}+\varepsilon]$ is in this neighborhood of $t_{0}$ for the inequality (3.2) holds. Furthermore, we have the inequalities $\frac{1}{\|x_{1}+\sum_{2\leq j\leq k-1}{x}_{j}\|}\geq\frac{1}{\|\delta+\sum_{2\leq j\leq k-1}{x}_{j}\|}$ (3.3) for $t\in[t_{1}-\epsilon,t_{1}+\varepsilon]$ and any $3\leq k\leq N$. Let $y_{1}(t)=\delta$ for $t\in[t_{1}-\epsilon,t_{1}+\varepsilon]$, $y_{1}(t)=x_{1}(t)$ for $t\in[T_{1},T_{2}]\backslash[t_{1}-\epsilon,t_{1}+\varepsilon]$, and $y_{j}(t)=x_{j}(t)$ for $t\in[T_{1},T_{2}]$ and $2\leq j\leq N-1$. Let $y(t)=(y_{1}(t),y_{2}(t),\cdots,y_{N-1}(t))$, then we know $\displaystyle\mathcal{{F}}(x)-\mathcal{{F}}(y)$ $\displaystyle=$ $\displaystyle\int^{t_{1}+\varepsilon}_{t_{1}-\epsilon}{\sum_{1\leq l<k\leq N}\frac{m_{k}m_{l}}{2M}{[{\|\sum_{l\leq j\leq k-1}\dot{x}_{j}\|^{2}}+\frac{2M}{\|\sum_{l\leq j\leq k-1}{x}_{j}\|}]}dt}$ $\displaystyle-$ $\displaystyle\int^{t_{1}+\varepsilon}_{t_{1}-\epsilon}{\sum_{1\leq l<k\leq N}\frac{m_{k}m_{l}}{2M}{[{\|\sum_{l\leq j\leq k-1}\dot{y}_{j}\|^{2}}+\frac{2M}{\|\sum_{l\leq j\leq k-1}{y}_{j}\|}]}dt}$ $\displaystyle=$ $\displaystyle\int^{t_{1}+\varepsilon}_{t_{1}-\epsilon}{{[A\dot{x}^{2}_{1}+B\dot{x}_{1}+\sum_{3\leq k\leq N}\frac{m_{k}m_{1}}{\|x_{1}+\sum_{2\leq j\leq k-1}{x}_{j}\|}]}dt}$ $\displaystyle-$ $\displaystyle\int^{t_{1}+\varepsilon}_{t_{1}-\epsilon}{\sum_{3\leq k\leq N}\frac{m_{k}m_{1}}{\|\delta+\sum_{2\leq j\leq k-1}{x}_{j}\|}dt}$ $\displaystyle>$ $\displaystyle 0$ Hence the path $q(t)$ is not an action minimizing path of the fixed-ends problem joining $q_{i}$ to $q_{f}$ in time $T_{2}-T_{1}$. $~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\Box$ Henceforth, we think all the particles have equal mass, i.e., we assume $m_{1}=m_{2}=\cdots=m_{N}=m$. Proof of Theorem 1.3: By using reduction to absurdity, without loss of generality, let $t_{0}$ be an instant at which collision times accumulate for some right neighborhood of $t_{0}$. By Lemma 2.4, there are infinite isolated collisions in some right neighborhood of $t_{0}$. Then it’s easy to know that there are three isolated collision moments $t_{1}$, $t_{2}$ and $t_{3}$ ($t_{1}<t_{2}<t_{3}$)such that the collisions at moments $t_{1}$, $t_{2}$ and $t_{3}$ have the same colliding clusters and the same order, i.e., as $t\rightarrow t_{i}(i\in\\{1,2,3\\})$, there exist different limit points $c_{i1},\cdots,c_{in}$ such that $S_{ik}=\\{j\in\\{1,\cdots,N\\}:q_{j}(t)\rightarrow c_{ik}~{}{as}~{}t\rightarrow{t_{i}}\\}$ and $S_{1k}=S_{2k}=S_{3k}$ for $k=1,\cdots,n$, furthermore, (without loss of generality) $c_{i1}<\cdots<c_{in}$ for $i\in\\{1,2,3\\}$. Given $k\in\\{1,\cdots,n\\}$, if the colliding cluster $S_{2k}$ has $n_{k}\geq 2$ elements, suppose the order of the particles in $S_{2k}$ is $({l_{1}},\cdots,l_{n_{k}})$ for some left neighborhood of $t_{2}$ and $({j_{1}},\cdots,j_{n_{k}})$ for some right neighborhood of $t_{2}$, that is, $q_{l_{1}}(t)<\cdots<q_{l_{n_{k}}}(t)$ for some left neighborhood of $t_{2}$ and $q_{j_{1}}(t)<\cdots<q_{j_{n_{k}}}(t)$ for some right neighborhood of $t_{2}$, where $\\{{l_{1}},\cdots,l_{n_{k}}\\}=\\{{j_{1}},\cdots,j_{n_{k}}\\}=S_{2k}$. If $({l_{1}},\cdots,l_{n_{k}})\neq({j_{1}},\cdots,j_{n_{k}})$, assume $\tau_{k}$ is a permutation from $({l_{1}},\cdots,l_{n_{k}})$ to $({j_{1}},\cdots,j_{n_{k}})$, let $(h_{j_{1}}(t),\cdots,h_{j_{n_{k}}}(t))=(q_{\tau_{k}(l_{1})}(t),\cdots,q_{\tau_{k}(l_{n_{k}})}(t))$ (3.4) for $t\in[t_{1},t_{2}]$. If $({l_{1}},\cdots,l_{n_{k}})=({j_{1}},\cdots,j_{n_{k}})$, or if the colliding cluster $S_{2k}$ has $n_{k}=1$ element, that is,the cluster $S_{2k}$ is singleton, thus the body in the cluster is not in a collision, the permutation $\tau_{k}$ can be chosen as unit transformation, then still let $(h_{j_{1}}(t),\cdots,h_{j_{n_{k}}}(t))=(q_{\tau_{k}(l_{1})}(t),\cdots,q_{\tau_{k}(l_{n_{k}})}(t))$ (3.5) for $t\in[t_{1},t_{2}]$. Finally, let $h(t)=(h_{1}(t),\cdots,h_{N}(t))$ for $t\in[t_{1},t_{2}]$ and $h(t)=(q_{1}(t),\cdots,q_{N}(t))$ for $t\in[t_{2},t_{3}]$, then $h(t)$ is a path in the Sobolev space $H^{1}([t_{1},t_{3}],\mathcal{X}_{1})$ with fixed- ends such that $h(t_{1})=q(t_{1})$ and $h(t_{3})=q(t_{3})$. Indeed, by the construction of $h(t)$, the relations $h(t_{1})=q(t_{1})$ and $h(t_{3})=q(t_{3})$ are obvious; by the continuity of $h(t)$ at $t=t_{2}$, it’s easy to know that $h(t)$ has weak derivative $\dot{h}(t)$ in $[t_{1},t_{3}]$, furthermore, $\dot{h}(t)$ is square integrable in $[t_{1},t_{3}]$ by applying the finiteness of the Lagrangian action. Let us recall that, if all the particles have the same masses, there is an obvious fact: suppose $\tau$ is a permutation of $(1,2,\cdots,N)$, let $r(t)=(r_{1}(t),r_{2}(t),\cdots,r_{N}(t))=(q_{\tau(1)}(t),q_{\tau(2)}(t),\cdots,q_{\tau(N)}(t))$, if $m_{1}=m_{2}=\cdots=m_{N}$, then $\int^{T_{2}}_{T_{1}}{L(q(t),\dot{q}(t))dt}=\int^{T_{2}}_{T_{1}}{L(r(t),\dot{r}(t))dt},$ (3.6) Since the path $q(t)$ is an action minimizing path, we know that the path $h(t)$ is an action minimizing path in the Sobolev space $H^{1}([t_{1},t_{3}],\mathcal{X}_{1})$ with fixed-ends $h(t_{1})=q(t_{1})$ and $h(t_{3})=q(t_{3})$. In particular, the path $h(t)$ is an action minimizing path in the Sobolev space $H^{1}([t_{2}-\epsilon,t_{2}+\epsilon],\mathcal{X}_{1})$ with fixed-ends $h(t_{2}-\epsilon)$ and $h(t_{2}+\epsilon)$ for all the sufficiently small $\epsilon>0$. By choosing any sufficiently small $\epsilon>0$, we have a path $h(t)$ such that: the action minimizing path $h(t)\in H^{1}([t_{2}-\epsilon,t_{2}+\epsilon],\mathcal{X}_{1})$ has only one collision moment $t_{2}$ in $(t_{2}-\epsilon,t_{2}+\epsilon)$, the fixed-ends $h(t_{2}-\epsilon)$, $h(t_{2}+\epsilon)\in\mathcal{X}_{1}\backslash\Delta_{1}$ and have the same order in $\mathbb{R}^{1}$. However, this contradicts with Lemma 3.1. In conclusion, if the action minimizing path $q(t)$ of the one-dimensional Newtonian $N$-body problem with equal masses has collisions, then every collision is isolated. Since the set of collision times is closed, we know there are at most finitely many collision moments for the fixed-ends (Bolza) problem. $~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\Box$ Proof of Theorem 1.4: First of all, let’s establish a lemma to simplify the proof. ###### Lemma 3.2 Given the initial moment $T_{1}$,the final moment $T_{2}$ $(T_{2}>T_{1})$ and two corresponding N-body configurations $q_{i}=(q_{i1},\cdots,q_{iN})$, $q_{f}=(q_{f1},\cdots,q_{fN})$ which have the same order in $\mathbb{R}^{1}$, suppose the path $q(t)\in\Lambda(q_{i},q_{f})$ has collision in $(T_{1},T_{2})$, and the collision moments in $(T_{1},T_{2})$ are respectively $t_{1},t_{2},\cdots,t_{n}$ $(T_{1}<t_{1}<\cdots<t_{n}<T_{2})$. Then there is some path $h(t)\in\Lambda(q_{i},q_{f})$ such that $\\{t_{1},\cdots,t_{n}\\}$ are collision moments in $(T_{1},T_{2})$ and the order of $h(t)$ are the same for all the time $t\in[T_{1},T_{2}]$. Furthermore, if all the particles have the same masses, then $\int^{T_{2}}_{T_{1}}{L(q(t),\dot{q}(t))dt}=\int^{T_{2}}_{T_{1}}{L(h(t),\dot{h}(t))dt}.$ (3.7) Proof of Lemma 3.2: It’s easy to know that, there is some path $g(t)$ which has the same order with $q_{i}$ and $q_{f}$ in $\mathbb{R}^{1}$ and $g(t)$ is collision-free for $t\in(T_{1},T_{2})$. Suppose the order of the orbit $g(t)$ for $t\in(T_{1},T_{2})$ is $(j_{1},\cdots,j_{N})$, that is, $g_{j_{1}}(t)<\cdots<g_{j_{N}}(t)$, where $\\{j_{1},\cdots,j_{N}\\}=\\{1,\cdots,N\\}$. Without loss of generality, we can assume that $(j_{1},j_{2},\cdots,j_{N})=(1,2,\cdots,N)$. Let $t_{0}=T_{1}$ and $t_{n+1}=T_{2}$, suppose the order of the orbit $q(t)$ for $t\in(t_{k},t_{k+1})$ is $(j_{k1},\cdots,j_{kN})$, that is, $q_{j_{k1}}(t)<\cdots<q_{j_{kN}}(t)$, where $k\in\\{0,\cdots,n\\}$. Suppose $\tau_{k}$ is a permutation from $(j_{k1},\cdots,j_{kN})$ to $(1,2,\cdots,N)$, let $h^{(k)}(t)=(h^{(k)}_{1}(t),h^{(k)}_{2}(t),\cdots,h^{(k)}_{N}(t))=(q_{\tau_{k}(1)}(t),q_{\tau_{k}(2)}(t),\cdots,q_{\tau_{k}(N)}(t))$ (3.8) for $t\in(t_{k},t_{k+1})$. Firstly, it is easy to know that $\lim_{t\rightarrow{t^{+}_{0}}}h^{(0)}(t)=q_{i},\lim_{t\rightarrow{t^{-}_{n+1}}}h^{(n)}(t)=q_{f}$ (3.9) In the following, we prove that $\lim_{t\rightarrow{t^{-}_{k+1}}}h^{(k)}_{j}(t)=\lim_{t\rightarrow{t^{+}_{k+1}}}h^{(k+1)}_{j}(t)$ (3.10) for every $j\in\\{1,\cdots,N\\}$ and $k\in\\{0,\cdots,n-1\\}$. In fact, from $h^{(k)}_{j}(t)=q_{\tau_{k}(j)}(t)$ for $t\in(t_{k},t_{k+1})$ and $h^{(k+1)}_{j}(t)=q_{\tau_{k+1}(j)}(t)$ for $t\in(t_{k+1},t_{k+2})$, it is easy to know that we only need to prove the relation $q_{\tau_{k}(j)}(t_{k+1})=q_{\tau_{k+1}(j)}(t_{k+1})$. For the sake of a contradiction, we can suppose that $q_{\tau_{k}(j)}(t_{k+1})>q_{\tau_{k+1}(j)}(t_{k+1})$ or $q_{\tau_{k}(j)}(t_{k+1})<q_{\tau_{k+1}(j)}(t_{k+1})$. If $q_{\tau_{k}(j)}(t_{k+1})>q_{\tau_{k+1}(j)}(t_{k+1})$, from $h^{(k)}_{l}(t)>h^{(k)}_{j}(t)$ for $N\geq l>j$,$t\in(t_{k},t_{k+1})$, we have $q_{\tau_{k}(l)}(t_{k+1})=\lim_{t\rightarrow{t^{-}_{k+1}}}h^{(k)}_{l}(t)\geq\lim_{t\rightarrow{t^{-}_{k+1}}}h^{(k)}_{j}(t)=q_{\tau_{k}(j)}(t_{k+1})>q_{\tau_{k+1}(j)}(t_{k+1})$. Hence $h^{(k+1)}_{\tau^{-1}_{k+1}\tau_{k}(l)}(t)=q_{\tau_{k}(l)}(t)>q_{\tau_{k+1}(j)}(t)=h^{(k+1)}_{j}(t)$ for every $l$ such that $N\geq l\geq j$,$t\in(t_{k+1},t_{k+1}+\epsilon)$, where $\epsilon$ is some sufficiently small positive number. So we have $\tau^{-1}_{k+1}\tau_{k}(l)>j$ for for every $l$ such that $N\geq l\geq j$, but there are at most $N-j$ number larger than $j$ in $\\{1,2,\cdots,N\\}$, this is a contradiction. If $q_{\tau_{k}(j)}(t_{k+1})<q_{\tau_{k+1}(j)}(t_{k+1})$, it is similar to get a contradiction. So we have $\lim_{t\rightarrow{t^{-}_{k+1}}}h^{(k)}_{j}(t)=\lim_{t\rightarrow{t^{+}_{k+1}}}h^{(k+1)}_{j}(t)$ (3.11) for every $j\in\\{1,\cdots,N\\}$ and $k\in\\{0,\cdots,n-1\\}$. Let $h(t)=h^{(k)}(t)$ for $t\in(t_{k},t_{k+1})$, $h(t_{k})=\lim_{t\rightarrow{t^{+}_{k}}}h^{(k)}(t)$ for $1\leq k\leq n$, $h(T_{1})=\lim_{t\rightarrow{t^{+}_{0}}}h^{(0)}(t)$, $h(T_{2})=\lim_{t\rightarrow{t^{-}_{n+1}}}h^{(n)}(t)$, then $h(t)\in\Lambda(q_{i},q_{f})$ and $\\{t_{1},\cdots,t_{n}\\}$ are collision moments in $(T_{1},T_{2})$ and the order of $h(t)$ are the same for all the time $t\in[T_{1},T_{2}]$. Furthermore, since all the particles have the same masses, we have $\int^{T_{2}}_{T_{1}}{L(q(t),\dot{q}(t))dt}=\int^{T_{2}}_{T_{1}}{L(h(t),\dot{h}(t))dt}.$ (3.12) From the above, Lemma 3.2 holds. In the following, we prove Theorem 1.4 by using Lemma 3.2. By using reduction to absurdity, suppose the action minimizing path $q(t)$ has collision moments in $(T_{1},T_{2})$, the collision moments in $(T_{1},T_{2})$ are respectively $t_{1},t_{2},\cdots,t_{n}$ $(T_{1}<t_{1}<\cdots<t_{n}<T_{2})$. Furthermore, we can assume that $q_{1}(t)<q_{2}(t)<\cdots<q_{N}(t)$ for $t\in(T_{1},T_{2})\backslash\\{t_{1},\cdots,t_{n}\\}$ and $q_{1}(t_{k})\leq q_{2}(t_{k})\leq\cdots\leq q_{N}(t_{k})$ for $k\in\\{1,\cdots,n\\}$ by using Lemma 3.2. Then we can find a path $q(t)\in H^{1}([t_{1}-\epsilon,t_{1}+\epsilon],\mathcal{X}_{1})$ which has only one collision moment $t_{1}$ in $(t_{1}-\epsilon,t_{1}+\epsilon)$, and the fixed- ends $q(t_{1}-\epsilon)$, $q(t_{1}+\epsilon)\in\mathcal{X}_{1}\backslash\Delta_{1}$ have the same order in $\mathbb{R}^{1}$, so long as the positive number $\epsilon$ is sufficiently small. However, this contradicts with Lemma 3.1. So we know that, for the N-body problem with equal masses, given two moments and corresponding configurations which have the same order in $\mathbb{R}^{1}$, the action minimizing path of the fixed-ends problem joining two configurations is collision-free for $t\in(T_{1},T_{2})$. $~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\Box$ Proof of Theorem 1.5: Suppose the action minimizing orbit $q(t)$ has collision in $(T_{1},T_{2})$, the collision moments in $(T_{1},T_{2})$ are respectively $t_{1},t_{2},\cdots,t_{n}$ $(T_{1}<t_{1}<\cdots<t_{n}<T_{2})$, let $t_{0}=T_{1}$ and $t_{n+1}=T_{2}$. Let us investigate $n+1$ collision-free path sections: $q(t),t\in(t_{k},t_{k+1})$, $0\leq k\leq n$. If $n>N!-1$, then there are two sections which have the same order, suppose the corresponding time intervals are respectively $(t_{j},t_{j+1})$ and $(t_{l},t_{l+1})$, $j<l$. Let us choose two moments $s_{1}\in(t_{j},t_{j+1})$ and $s_{2}\in(t_{l},t_{l+1})$, then it is easy to know that the path $q(t),t\in[s_{1},s_{2}]$ is an action minimizing orbit of the fixed-ends problem for two moments $s_{1},s_{2}$ and corresponding configurations $q(s_{1}),q(s_{2})$. However, from Theorem 1.4, $q(t)$ is collision-free in $(s_{1},s_{2})$, this contradicts with $t_{j+1},t_{l}\in(s_{1},s_{2})$. $~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\Box$ ## Acknowledgements The authors sincerely thank an anonymous expert for his/her many valuable comments and suggestions. ## References * [1] Vivina Barutello and Susanna Terracini. Action minimizing orbits in the n-body problem with simple choreography constraint. Nonlinearity, 17(6):2015, 2004. * [2] Kuo-Chang Chen. Action-minimizing orbits in the parallelogram four-body problem with equal masses. Archive for Rational Mechanics and Analysis, 158(4):293–318, 2001\. * [3] Kuo-Chang Chen. Binary decompositions for planar n-body problems and symmetric periodic solutions. Archive for Rational Mechanics and Analysis, 170(3):247–276, 2003\. * [4] Kuo-Chang Chen. Existence and minimizing properties of retrograde orbits to the three-body problem with various choices of masses. Annals of Math, 167:325–348, 2008. * [5] A Chenciner. Collisions totales, mouvements completement paraboliques et reduction des homotheties dans le probleme des n corps. Regular and chaotic dynamics, 3(3):93–106, 1998. * [6] Alain Chenciner. Action minimizing periodic orbits in the Newtonian n-body problem. In Celestial Mechanics, dedicated to Donald Saari for his 60th Birthday, volume 1, page 71, 2002. * [7] Alain Chenciner. Simple non-planar periodic solutions of the n-body problem. In Proceedings of the NDDS Conference, Kyoto, 2002. * [8] Alain Chenciner. Action minimizing solutions of the newtonian n-body problem: from homology to symmetry. arXiv preprint math/0304449, 2003. * [9] Alain Chenciner and Richard Montgomery. A remarkable periodic solution of the three-body problem in the case of equal masses. Annals of Mathematics-Second Series, 152(3):881–902, 2000. * [10] Alain Chenciner and Andrea Venturelli. Minima de l’intégrale d’action du problème newtoniende 4 corps de masses égales dans r3: Orbites’ hip-hop’. Celestial Mechanics and Dynamical Astronomy, 77(2):139–151, 2000\. * [11] Mohamed Sami ElBialy. Collision singularities in celestial mechanics. SIAM Journal on Mathematical Analysis, 21(6):1563–1593, 1990. * [12] Davide L Ferrario and Susanna Terracini. On the existence of collisionless equivariant minimizers for the classical n-body problem. Inventiones Mathematicae, 155(2):305–362, 2004. * [13] Morris W Hirsch, Charles C Pugh, and Michael Shub. Invariant manifolds. 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On the manifolds of total collapse orbits and of completely parabolic orbits for the n-body problem. Journal of Differential Equations, 41(1):27–43, 1981. * [20] Carl L Siegel and Jürgen K Moser. Lectures on celestial mechanics. Springer, 1971. * [21] Hans J Sperling. On the real singularities of the N-body problem. Journal für die reine und angewandte Mathematik, 245:15–40, 1970. * [22] Andrea Venturelli. Application de la minimisation de l’action au Problème des N corps dans le plan et dans l’espace. PhD thesis, 2002. * [23] Aurel Wintner. The analytical foundations of celestial mechanics. Princeton, NJ, Princeton university press; London, H. Milford, Oxford university press, 1941., 1, 1941. * [24] Xiang Yu and Shiqing Zhang. Saari’s conjecture for elliptical type $n$-body problem and an application. arXiv preprint arXiv:1308.2376, 2013. * [25] Shiqing Zhang and Qing Zhou. Variational methods for the choreography solution to the three-body problem. Science in China Series A: Mathematics, 45(5):594–597, 2002. * [26] Shiqing Zhang and Qing Zhou. Nonplanar and noncollision periodic solutions for n-body problems. Discrete and Continuous Dynamical Systems-A, 10(3):679–686, 2004\.	arxiv-papers	2014-01-06T11:57:43	2024-09-04T02:49:56.322540	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Xiang Yu and Shiqing Zhang", "submitter": "Shiqing Zhang", "url": "https://arxiv.org/abs/1401.1051" }
1401.1086	# Power Grid Defense Against Malicious Cascading Failure Paulo Shakarian Hansheng Lei Roy Lindelauf Dept. EECS and Network Science Center U.S. Military Academy West Point, NY, 10996 paulo[at]shakarian.net Dept. EECS and Network Science Center U.S. Military Academy West Point, NY, 10996 hansheng.lei[at]usma.edu Netherlands Defence Academy Faculty of Military Science Military Operational Art and Science rha.lindelauf.01[at]nlda.nl ###### Abstract An adversary looking to disrupt a power grid may look to target certain substations and sources of power generation to initiate a cascading failure that maximizes the number of customers without electricity. This is particularly an important concern when the enemy has the capability to launch cyber-attacks as practical concerns (i.e. avoiding disruption of service, presence of legacy systems, etc.) may hinder security. Hence, a defender can harden the security posture at certain power stations but may lack the time and resources to do this for the entire power grid. We model a power grid as a graph and introduce the cascading failure game in which both the defender and attacker choose a subset of power stations such as to minimize (maximize) the number of consumers having access to producers of power. We formalize problems for identifying both mixed and deterministic strategies for both players, prove complexity results under a variety of different scenarios, identify tractable cases, and develop algorithms for these problems. We also perform an experimental evaluation of the model and game on a real-world power grid network. Empirically, we noted that the game favors the attacker as he benefits more from increased resources than the defender. Further, the minimax defense produces roughly the same expected payoff as an easy-to-compute deterministic load based (DLB) defense when played against a minimax attack strategy. However, DLB performs more poorly than minimax defense when faced with the attacker’s best response to DLB. This is likely due to the presence of low-load yet high-payoff nodes, which we also found in our empirical analysis. ###### category: I.2.11 Artificial Intelligence Distributed Artificial Intelligence ###### keywords: power grid defense, game theory, complex networks ††terms: Algorithms Security ## 1 Introduction Rapid cascading failure in a power grid caused by a succession of overloading lines can lead to very large outages, as observed in the United States in 2003 [1]. Studies on cascading failure [7, 8, 16] have illustrated that such a failure can be initiated with only a small number of initial node failures. Further, power grid infrastructure is often particularly vulnerable with respect to cyber-security due to a variety of issues, including the use of legacy and proprietary computer hardware and software [26]. In this paper, we extend the work on cascading failure models to a two-player game where an attacker attempts to create a cascade that maximizes the number of customers without power while the defender defends key nodes to avoid a major outage. In Section 2, we introduce an extension to the failure model of [8] to not only consider the attacker and defender, but also the different types of nodes in the power grid (i.e. power generation vs. power consumers). In Section 3, we explore the computational complexity of finding deterministic best-response strategies for the attacker and defender under several different scenarios depending on the relative number of resources each player has and whether the opponent has a deterministic or mixed strategy. Here we found that, in general, these problems are NP-hard, though we do identify some tractable cases. In Section 4, we explore heuristic algorithms for finding determinsitic “best responses” as well as minimax mixed strategies. We introduce a “high-load” strategy for defense (based on the observations of [8]), greedy heuristics for deterministic strategies, and a double-oracle approach based on [15] for finding a mixed strategy. In Section 5 we perform experiments on a real-world dataset of a power grid [20] and find that this game seems to favor the attacker as he benefits more from increased resources than the defender. Further, our experiments revealed that the minimax defense produces roughly the same expected payoff as an easy-to-compute deterministic load based (DLB) defense when played against a minimax attack strategy, though the load based defense does more poorly than minimax when faced with the attacker’s best response to DLB. This is likely due to the presence of low- load yet high-payoff nodes, which we also found in our empirical analysis of the model. Finally, related work is discussed in Section 6. ## 2 Technical Preliminaries Consider a power-grid network modeled as an undirected graph $G=(V,E)$. Let $V_{src},V_{ld}\subseteq V$ be source (producers of power) and load (consumers of power) on the network. We shall use the notation $\textsf{disc}_{V_{ld},V_{src}}(G)$ to denote the number of nodes in $V_{ld}$ which are not connected to any node in $V_{src}$ in graph $G$. Let G be the set of all subgraphs of $G$. For a given node $i$, let $\mathcal{N}_{G}(i)$ be the set of nodes in $V_{src}-\\{i\\}$ that are closest to that node (based on path length in $G$). From this, we define edge load (similar to the idea of edge betweenness [25]). ###### Definition 2.1 (Edge Load) Given edge $ij\in E$, the edge load, $\textit{load}_{G}(ij)$ is defined as follows: $\textit{load}_{G}(ij)=\sum_{t\in V_{ld}}\sum_{s\in\mathcal{N}_{G}(t)}\frac{\sigma_{G}(s,t\|ij)}{\|\mathcal{N}_{G}(t)\|\,\sigma_{G}(s,t)},$ where $\sigma_{G}(s,t)$ is the number of shortest paths between $s,t\in V$ and $\sigma_{G}(s,t\|ij)$ is the subset of these paths that pass through edge $ij\in E$. Starting from initial network $G_{0}=(V_{0},E_{0})$ we use $c_{ij}$ to denote the capacity edge $ij\in E_{0}$. In a real-world setting, we would expect to have this information. However, in this paper, we use the following proxy (similar to [8]). $c_{ij}(G_{0})=(1+\alpha){\textit{load}_{G_{0}}}(ij)$ where $\alpha$ is a non-negative real that specifies the excess capacity available on that line. We shall refer to $\alpha$ as the capacity margin. We assume that an edge $ij\in E$ fails in $G=(V,E)$, with $E\subset E_{0}$, if $\textit{load}_{G}(ij)>c_{ij}(G_{0})$. Once nodes (and adjacent edges) in $V_{0}$ are removed from $G_{0}$, this results in a change of shortest paths between sources and loads, hence more edges will potentially fail. This cascading power failure is modeled by a “failure” operator denoted with F (based on the failure model of [8] \- though we note that our model is a new contribution due to the consideration of source and load nodes) that maps networks to networks. We define it as follows. ###### Definition 2.2 (Failure Operator) The failure operator, $\textbf{{F}}:\textbf{{{G}}}\rightarrow\textbf{{{G}}}$, is defined as follows: $\textbf{{F}}((V,E))=(V,\\{ij\in E\|\textit{load}_{(V,E)}(ij)\leq c_{ij}(G_{0})\\})$ Intuitively, one application of the failure operator removes all edges that have exceeded their maximum capacity. We can define multiple applications of this operator as follows: $\textbf{{F}}^{i}(G)=\begin{cases}G&\text{if $i=0$}\\\ \textbf{{F}}(\textbf{{F}}^{i-1}(G))&\text{otherwise}\end{cases}$ Clearly, there must exist a fixed point that is reached in no more than $\|E\|+1$ applications of F. Hence, we shall use the following notation: $\textbf{{F}}^{}(G)=\textbf{{F}}^{i}(G)\textit{ s.t. }\textbf{{F}}^{i}(G)=\textbf{{F}}^{i+1}(G)$ We now consider two agents: an attacker and a defender. The attacker’s strategy is to destroy nodes (and their adjacent edges) in an effort to cause a cascading failure that maximizes the number of load nodes ($V_{ld}$) that are disconnected from all source nodes ($V_{src}$). Meanwhile, the defender’s strategy is to harden certain nodes such that the attacker is unable to destroy them - though these nodes can be taken offline as a result of the cascading failure111Note that this would likely be the case where the attack and defense occurs in cyber-space, while the cascade occurs in the physical world.. The attacker can destroy $k_{a}$ nodes while the defender can harden $k_{d}$ nodes. Thus the strategy space of both the attacker and defender consists of all subsets $V_{a},V_{d}\subseteq V$ of size $\|V_{a}\|\leq k_{a}$ ($\|V_{d}\|\leq k_{d}$ respectively). We denote these strategy spaces by $ATK$ ($DEF$ respectively), i.e., if we allow the attacker to consider all strategies of size $k_{a}$ or less we have: $ATK=\\{S\in 2^{V}:\|S\|\leq k_{a}\\}$ We now have all of the components to define the payoff function. ###### Definition 2.3 (Payoff Function) Given initial network $G=(V,E)$ with edge capacities $c_{ij}(G)$, attack (defend) strategy $V_{a}(V_{d})$, the payoff function is defined by $p_{G}(V_{a},V_{d})=\textsf{disc}_{V_{ld},V_{src}}(\textbf{{F}}^{}((V-(V_{a}-V_{d}),E)).$ Now, in reality, the defender will have real-world limitations on the number of nodes (i.e. substations) he may harden. For instance, with regard to smart grid defense, applying the most up-to-date patches on all systems may not be realistic as it could potentially require system down-time - affecting customer service. Further, it would also likely not make sense for the defender to only harden certain nodes and ignore others. Hence, it is reasonable to consider a situation where the defender can only harden certain nodes against attack (and may do so probabilistically - i.e. applying hardware or software updates according to a schedule). Therefore, we study mixed strategies. Such strategies will be specified by probability distributions $\textbf{Pr}_{a},\textbf{Pr}_{d}$ for the attacker and defender respectively. We shall denote the number of strategies assigned a non-zero probability as $\|\textbf{Pr}_{a}\|,\|\textbf{Pr}_{d}\|$. We can define expected payoff as follows. ###### Definition 2.4 (Expected Payoff) Let $\textbf{Pr}_{a},\textbf{Pr}_{d}$ be probability distributions over all subsets of $V$ of sizes $k_{a}$ (resp. $k_{d}$) or less. These probability distributions correspond to a mixed strategy for the attacker and defender respectively. Hence, given such probability distributions, the expected payoff can be computed as follows: $\textbf{ExP}(\textbf{Pr}_{a},\textbf{Pr}_{d})=\sum_{V_{a}\in 2^{V}}\textbf{Pr}_{a}(V_{a})\sum_{V_{d}\in 2^{V}}\textbf{Pr}_{d}(V_{d})p_{G}(V_{a},V_{d})$ In this work our goal is to find the minimax strategy for the defender - that is the mixed strategy for the defender that minimizes the attacker’s maximum expected payoff - as well as deterministic “best responses” for both players given the other’s strategy. ## 3 Computational Complexity In this section, we analyze the computational complexity of determining the best response for each of the agents to a strategy of its opponent. First, we shall discuss the case for finding a deterministic strategy for the defender and attacker. Then we shall explore the computational complexity of finding a mixed strategy. We summarize our complexity results in Table 3. Opponent Strategy \| Attacker \| Defender ---\|---\|--- Mixed w. $1$ resource \| NP-Compl. \| PTIME \| Thm. 8 \| Prop. 4 Det. w. fewer resources \| NP-Compl. \| PTIME \| Thm. 8 \| Prop. 3.1 Det. w. greater resources \| NP-Compl. \| NP-Compl. \| Thm. 8 \| Thm. 2 Mixed w. fewer resources \| NP-Compl. \| NP-Compl. \| Thm. 8 \| Thm. 6 Mixed w. greater resources \| NP-Compl. \| NP-Compl. \| Thm. 8 \| Thm. 2 Table 1: Complexity Results for Finding a Deterministic Best Response We frame the formal combinatorial problem of finding the best-response for the defender as follows: Grid-Defend Deterministic Best Response (GD-DBR) INPUT: Network $G=(V,E)$, attacker mixed strategy $\textbf{Pr}_{a}$ (where each option is of size no greater than $k_{a}$), natural number $k_{d}$, real numbers $X,\alpha$ OUTPUT: “Yes” if there exists a set $V_{d}\subseteq V$ s.t. $\|V_{d}\|\leq k_{d}$ and $\sum_{V_{a}\in ATK}\textbf{Pr}_{a}(V_{a})p_{G}(V_{a},V_{d})\leq X$ and “no” otherwise. We shall study this case under several conditions. The first, and easiest case is when $\textbf{Pr}_{a}=1$ (the attacker uses a deterministic strategy) and $k_{a}\leq k_{d}$. ###### Proposition 3.1 When $k_{a}\leq k_{d}$ and $\|\textbf{Pr}_{a}\|=1$ then GD-DBR is solvable in polynomial time. ###### Proof 3.1. As the attacker plays only one strategy and the defender can defend at least as many nodes as are being attacked, the defender simply defends all the nodes in the attacker’s strategy. However, even with $\|\textbf{Pr}_{a}\|=1$, the problem becomes NP-hard in the case where $k_{a}>k_{d}$. ###### Theorem 2. When $k_{a}>k_{d}$ then GD-DBR is NP-complete, even when $\|\textbf{Pr}_{a}\|=1$ and $X$ is an integer. ###### Proof 3. Clearly, checking if a given deterministic defender strategy $V_{d}$ meets the requirements of the “output” of GD-DBR can be completed in polynomial-time, providing membership in the class NP. For NP-hardness consider the known NP-hard “set cover” problem [11] that takes as input a natural number $k$, set of elements $S=\\{s_{1},\ldots,s_{n}\\}$, family of subsets of $S$, $H=\\{h_{1},\ldots,h_{m}\\}$ and returns “yes” if there is a $k$-sized (or smaller) subset of $H$ s.t. their union is equal to $S$. We can embed Set Cover into an instance of GD-DBR in polynomial time with the following embedding: set $k_{a}=\|H\|$, $k_{d}=k$, $X=0$, $\alpha=\|H\|+\|S\|$, create $G=(V,E)$ as follows: * • For each $h\in H$ create a node $v_{h}$ and for each $s\in S$ create node $v_{s}$ * • If $s\in h$, create edge $(v_{h},v_{s})$, for each $ij\in E$ * • Set $V_{src}=\\{v_{h}\|h\in H\\}$, $V_{ld}=\\{v_{s}\|s\in S\\}$, $V_{a}=V-V_{ld}$ Suppose, by way of contradiction (BWOC), that there is a “yes” answer to Set Cover but a “no” answer to GD-DBR. Consider set $H^{\prime}$ a subset of $H$ that is the certificate for Set Cover and the corresponding set $V^{\prime}=\\{v_{h}\|h\in H^{\prime}\\}$ in the instance of GD-DBR. Suppose the defender utilizes this as a strategy. The attacker then effectively attacks the set $V-V_{ld}-V^{\prime}$. Note that as the graph is bi-bipartite, this does not cause any cascading failure. By the construction, each load node must be connected to a source node, hence the number of offline load nodes is $X$. This gives us a contradiction. Suppose, BWOC, that there is a “yes” answer to GD-DBR but a “no” answer to the corresponding instance of Set Cover. Let $V^{\prime}$ be the certificate for GD-DBR. We note that any element of $V_{ld}\cap V^{\prime}$ in $V^{\prime}$ can be replaced by a neighboring node from $V_{src}$ without changing the size of this set and that such a set would still allow for all load nodes to remain online, let $V^{\prime\prime}$ be this new set. Consider the set $\\{h\|v_{h}\in V^{\prime\prime}\\}$. By the contra-positive of the claim, this cannot be a cover of all elements of $S$. However, this would also imply that there is some element $v_{s}\in V_{ld}$ that is not connected to $V^{\prime\prime}$ meaning that it fails (as the attacker successfully destroys all its neighbors). This means that the adversary has a payoff greater than $0$ (which is what $X$ was set to) – hence a contradiction. Hence, the presence of a more advantageous attacker is a source of complexity. The next question would be if the attacker’s behavior, i.e. deterministic vs. non-deterministic, also affects the complexity of the problem, even if the defender has the advantage. First, let us examine the case where the attacker has a mixed strategy with $k_{a}=1$. ###### Proposition 4. When $k_{a}=1$ then GD-DBR is solvable in polynomial time (w.r.t. $\|\textbf{Pr}_{a}\|$), even when $\|\textbf{Pr}_{a}\|\geq 0$. ###### Proof 5. In this case, we can re-write the payoff function as $p_{G}(\\{v\\},V_{d})=0$ if $v\in V_{d}$ and $p_{G}(\\{v\\},V_{d})=p_{G}(\\{v\\},\emptyset)$ otherwise. Let $V^{\prime}=\cup\\{V_{a}\in ATK\|\textbf{Pr}_{a}(V_{a})>0\\}$. Note that each element of $V^{\prime}$ is also a strategy the attacker plays with a non- zero probability (as the attacker only plays singletons). Hence, the expected payoff can be re-written as $\sum_{v\in V^{\prime}-V_{d}}\textbf{Pr}_{a}(\\{v\\})p_{G}(\\{v\\},\emptyset)$. Therefore, the best a defender can do is defend the top $k_{d}$ nodes in $V^{\prime}$ where $\textbf{Pr}_{a}(\\{v\\})p_{G}(\\{v\\},\emptyset)$ is the greatest - which can be easily computed in polynomial time and allows us to determine the answer to GD-DBR. However, if the defender is playing a mixed strategy with $k_{a}>1$, then the problem again becomes NP-complete. ###### Theorem 6. When $\|\textbf{Pr}_{a}\|>1$ and $k_{a}>1$, GD-DBR is NP-complete, even when $k_{d}>k_{a}$ and $X$ is an integer. ###### Proof 7. NP-completeness mirrors that of Theorem 2. For NP-hardness, we again consider a reduction from set-cover (defined in the proof of Theorem 2. The embedding can again be performed in polynomial time as follows: set $k_{a}=\max_{s\in S}\|\\{h\|s\in h\\}\|$, set $k_{d}=k$, $X=0$, $\alpha=\|H\|+\|S\|$, create $G=(V,E)$, $V_{src}$, and $V_{ld}$ as per the construction in Theorem 2. We then set up the mixed strategy as follows: for each $s\in S$, let $V_{a}^{s}=\\{h\|s\in h\\}$ and $\textbf{Pr}_{a}(V_{a}^{s})=1/\|S\|$. Suppose, BWOC, that there is a “yes” answer to set cover and a “no” answer to the instance of GD-DBR. Consider set cover solution $H^{}$ and set $V_{d}=\\{v_{h}\|h\in H^{}\\}$. Note that $V_{d}$ meets the cardinality requirement. Note that by the construction, a source node becomes disconnected only if all of the load nodes connected to it are attacked, hence there is some node in the set $V_{ld}$ that is totally disconnected under at least one attacker strategy - let $v_{s}$ be this node. However, as set $H^{}$ covers $S$, then regardless of the attacker strategy, there is always some node $v_{h}$ that is connected and never attacked (giving the attacker a payoff of zero) - hence a contradiction. Suppose, BWOC, that there is a “yes” answer to GD-DBR and a “no” answer to the instance of set cover. Consider GD-DBR solution $V^{\prime}$. We note that any element of $V_{ld}\cap V^{\prime}$ in $V^{\prime}$ can be replaced by a neighboring node from $V_{src}$ without changing the size of this set and that such a set would still allow for all load nodes to remain online, let $V^{\prime\prime}$ be this new set. Consider the set $H^{}=\\{h\|v_{h}\in V^{\prime\prime}\\}$. Note that $\|H^{}\|\leq k$. By the contra-positive, there must be at least one element of $S$ not covered by $H^{}$. Let node $v_{s}$ be a node associated with uncovered element $s$. As GD-DBR returned “yes” then there is no attacker strategy where $v_{s}$ becomes disconnected from some node in $V_{src}$. As attack strategy $V_{a}^{s}$ includes all nodes that are connected to $v_{s}$, then at least one of these nodes must be included in $V^{\prime\prime}$. Therefore, for every node $v_{s}\in V_{ld}$ there is some node $v_{h}\in V_{ld}\cap V^{\prime\prime}$ that is connected to it, which means, by the construction, that $H^{}$ must cover all elements of $S$ \- a contradiction. We now frame the formal problem for finding a deterministic best-response for the attacker below. Grid-Attack Deterministic Best Response (GA-DBR) INPUT: Network $G=(V,E)$, defender mixed strategy $\textbf{Pr}_{d}$ (where each option is of size no greater than $k_{d}$), natural number $k_{a}$, real numbers $X,\alpha$ OUTPUT: “Yes” if there exists a set $V_{a}\subseteq V$ s.t. $\|V_{a}\|\leq k_{a}$ and $\sum_{V_{d}\in DEF}\textbf{Pr}_{d}(V_{d})p_{G}(V_{a},V_{d})\geq X$ and “no” otherwise. In the case of $k_{a}=1$, this problem is solvable in polynomial time: simply consider each $v\in V$. The attacker computes $\sum_{V_{d}\in DEF}\textbf{Pr}_{d}(V_{d})p_{G}(\\{v\\},V_{d})$ until one is found that causes the payoff to exceed or be equal to $X$. However, for strategies of larger size, the problem becomes NP-hard, regardless of the size of the defender strategy. ###### Fact 1. When $k_{a}=1$, GA-DBR is solvable in polynomial time (w.r.t. $\|\textbf{Pr}_{d}\|$). ###### Theorem 8. GA-DBR is NP-complete. ###### Proof 9. Clearly, a certificate consisting of a set $V_{a}\subseteq V$ can be verified in polynomial time, giving us membership in NP. For NP-hardness consider the known NP-hard “vertex cover” problem [11] that takes as input a graph $G^{\prime}=(V^{\prime},E^{\prime})$ (with no self-loops) and natural number $k$ and returns “yes” iff there is a set of $k$ or fewer vertices that are adjacent to each edge in $E$. We can embed vertex cover into an instance of GD-DBR in polynomial time with the following embedding: set $k_{a}=k$, $k_{d}=0$, $V_{d}=\emptyset$, $X=\|V^{\prime}\|$, $\alpha=\|E\|$, $G=G^{\prime}$, and $V_{src}=V_{ld}=V^{\prime}$. Suppose, BWOC, the above problem instance provides a “yes” answer to the vertex cover problem but a “no” answer to GA-DBR. Let $V^{\prime\prime}$ be a vertex cover of size $k$ or less for $G^{\prime}$. Consider the corresponding set of vertices in $G$ (we shall call this $V^{}$). Note that $\|V^{}\|\leq k_{a}$. As an attacker attacking $V^{}$ disconnects those nodes from the network, all edges adjacent to $V^{}$ fail. As $V^{}$ is a vertex cover for $G$, this means that there are no edges in the graph once $V^{}$ is removed. Hence, no load node is connected to any source node - giving the attacker a payoff of at least $X$ – hence a contradiction. Suppose, BWOC, the above problem instance provides a “yes” answer to GA-DBR but a “no” answer to the vertex cover problem. Let $V_{a}$ be the set of nodes the attacker attacks in GA-DBR. As $\alpha=\|E\|$ and as $V_{src}=V$, nodes only fail in a cascade if they are either targeted by the attacker or become totally disconnected. Further, as $X=\|V\|$, all nodes in $G$ are either in $V_{a}$ or disconnected - meaning that $V_{a}$ must be a vertex cover of size $k_{a}$ or less. As $k_{a}=k$ we have a contradiction. Due to the use of covering problems for the complexity results in Theorems 2, 6, and 8, it may seem reasonable to frame the problem as a sub- or super- modularity optimization where the objective function is monotonic. However, here we show (unfortunately) that these properties do not hold for either player. First, we shall make statements regarding the monotonicity of the payoff function. ###### Proposition 10. Iff $\forall V_{d}^{}$, $V_{a}\subseteq V_{a}^{\prime}$: $p_{G}(V_{a},V_{d}^{})\leq p_{G}(V_{a}^{\prime},V_{d}^{})$ then $\forall V_{a}^{}$, $V_{d}\subseteq V_{d}^{\prime}$: $p_{G}(V_{a}^{},V_{d})\geq p_{G}(V_{a}^{},V_{d}^{\prime})$. The idea of submodularity can be thought of as “diminishing returns.” Given a set of elements $S$ and a function $f:2^{S}\rightarrow\Re^{+}$, we say a $f$ is submodular if for any sets $S_{1}\subseteq S_{2}$ and element $s\notin S_{2}$, we have the following relationship: $\displaystyle f(S_{1}\cup\\{s\\})-F(S_{1})\geq f(S_{2}\cup\\{s\\})-F(S_{2})$ A complementary idea of supermodularity is also often studied - in this case the inequality is reversed. Unfortunately, when we fix the strategy for the defender, the attacker strategy is neither submodular nor supermodular - making the dynamics of this model significantly different from others (i.e. [24]). Let consider strategies $V_{a},V_{d}$ where $V_{a}$ causes some load node $v\notin(V_{a}\cup V_{d})\cap V_{ld}$ to disconnect and any node the strategy $\\{v\\}$ causes to disconnect will also become disconnected with strategy $V_{a}$ (such a case is easy to contrive, particularly with a bi- partite network). Therefore, we get the following relationship: $\displaystyle p_{G}(V_{a}\cup\\{v\\},V_{d})-p_{G}(V_{a},V_{d})<p_{G}(\\{v\\},V_{d})-p_{G}(\emptyset,V_{d})$ This arises from the fact that the left-hand side of the above equation becomes zero and the right hand side of the equation is equal to $p_{G}(\\{v\\},V_{d})$ which must be at least one. Now consider another example. Suppose we have a simple V-shaped network of three nodes. The angle of the V is a load node, while the other two nodes are source nodes. With $\alpha=1$, the load node receives power if at least one of the source nodes is connected to it. However, it does not require both. Let $V_{a}$ be a strategy consisting of one source node and $v$ be the other source node, and $V_{d}$ consist of the load node. From this, we have the following relationship: $\displaystyle p_{G}(V_{a}\cup\\{v\\},V_{d})-p_{G}(V_{a},V_{d})>p_{G}(\\{v\\},V_{d})-p_{G}(\emptyset,V_{d})$ In this case, the right-hand side becomes zero while the left hand side becomes one. This leads us to the following fact: ###### Fact 2. When $V_{d}$ is fixed, $p_{G}$ is neither submodular nor supermodular. Now let us consider when we fix the attacker’s strategy. If the payoff is submodular when the attacker’s strategy is fixed, then we have the following for $V_{d}\subseteq V_{d}^{\prime}$ and $v\notin V_{d}^{\prime}$ if the payoff subtracted from the number of nodes is submodular: $\displaystyle p_{G}(V_{a},V_{d}^{\prime}\cup\\{v\\})-p_{G}(V_{a},V_{d}^{\prime})\geq p_{G}(V_{a},V_{d}\cup\\{v\\})-p_{G}(V_{a},V_{d})$ This is equivalent to the following: $\displaystyle p_{G}(V_{a}-(V_{d}^{\prime}\cup\\{v\\}),\emptyset)-p_{G}(V_{a}-V_{d}^{\prime},\emptyset)\geq$ $\displaystyle\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,p_{G}(V_{a}-(V_{d}\cup\\{v\\}),\emptyset)-p_{G}(V_{a}-V_{d},\emptyset)$ Now let $V_{a}^{\prime}=V_{a}-(V_{d}^{\prime}\cup\\{v\\})$ and $V_{a}^{\prime\prime}=V_{a}^{\prime}\cup(V_{d}^{\prime}-V_{d})$. Clearly $V_{a}^{\prime\prime}\supseteq V_{a}^{\prime}$ and $v\notin V_{a}^{\prime\prime}$. Now we get the following: $\displaystyle p_{G}(V_{a}^{\prime},\emptyset)-p_{G}(V_{a}^{\prime}\cup\\{v\\},\emptyset)$ $\displaystyle\geq$ $\displaystyle p_{G}(V_{a}^{\prime\prime},\emptyset)-p_{G}(V_{a}^{\prime\prime}\cup\\{v\\},\emptyset)$ $\displaystyle p_{G}(V_{a}^{\prime}\cup\\{v\\},\emptyset)-p_{G}(V_{a}^{\prime},\emptyset)$ $\displaystyle\leq$ $\displaystyle p_{G}(V_{a}^{\prime\prime}\cup\\{v\\},\emptyset)-p_{G}(V_{a}^{\prime\prime},\emptyset)$ Hence, submodualrity of the payoff function when the attacker’s strategy is fixed would give us supermodualrity of the payoff function when the defender’s strategy is fixed at the empty set. However, this clearly violates Fact 2 and gives rise to the following: ###### Fact 3. When $V_{a}$ is fixed, $p_{G}$ is neither submodular nor supermodular. ## 4 Algorithms In this section, we present heuristic algorithms for finding the deterministic best response of each player as the results of the previous section generally preclude a polynomial time algorithm for an exact solution. We first introduce a version of a “high load” strategy for the defender based on the ideas of [8]. Then we introduce a greedy heuristic for each player. This is followed by our approach for finding mixed strategies based on the double-oracle algorithm of [15]. Hi-Load Node Approach. In [8], the authors study “high load” nodes: nodes through which the greatest number of shortest paths pass. They show that attacks on these nodes tend to initiate cascading failures – suggesting that they should be a priority for defense. We formalize the definition of nodal load in our framework (essentially an extended definition of node betweenness [25]) by extending our function $\textit{load}_{G}$ for nodes as follows. ###### Definition 1 (Nodal Load). For a given node, the nodal load is defined as the sum of the fraction of shortest paths for each pair that pass through that node. Formally: $\textit{load}_{G}(i)=\sum_{s\in V_{src},t\in V_{ld}}\frac{\sigma_{G}(s,t\|i)}{\sigma_{G}(s,t)},$ where $\sigma_{G}(s,t\|i)$ is the number of shortest paths between $s,t\in V$ that pass through node $i$. Hence, we shall refer to the Deterministic Load-Based or DLB strategy for the defender as one in which he deterministically protects the $k_{d}$ nodes with the greatest load. We note that this is not necessarily a “best response” but the intuition is that defense will occur at nodes that are perceived to be critical to the adversary. This intuition is similar to that of the “most vital arc” idea seen in other failure model games [2, 21]. Greedy Heuristics for Finding Deterministic Strategies. Here we present a simple greedy heuristic to find the defender’s best-response (GREEDY_DEFENDER_RESP). The analogous heuristic for the attacker is not shown due to space constraints, but we shall refer to it as GREEDY_ATTACKER_RESP. We note that while we do not make general approximation guarantees (due to the results in Section 3), we note that by Proposition 10, that nodes added in step 18 will always cause an increase in payoff to the defender (and in the analogous greedy approach for the attacker, this holds true as well). Further, by Proposition 4, when $k_{a}=1$, we can be sure that GREEDY_DEFENDER_RESP returns an exact solution, even when the attacker has a mixed strategy. Unfortunately, by Theorem 8, the same cannot be said if the greedy heuristic is used for the attacker’s best response. Algorithm 1 GREEDY_DEFENDER_RESP 0: Mixed strategy $\textbf{Pr}_{a}$, Natural number $k_{d}$ 0: Set of nodes $V_{d}$ 1: $V_{d}=\emptyset$ 2: Let $ATK$ be the set of strategies associated with $\textbf{Pr}_{a}$ 3: Set $flag=\textsf{True}$, $p^{}=-\infty$ 4: while $\|V_{d}\|\leq k_{d}$ and $flag$ and $p^{}<0$ do 5: $p^{}=-\sum_{V_{a}\in ATK}\textbf{Pr}_{d}(V_{a})p_{G}(V_{a},V_{d})$ 6: $curBest=null$, $curBestScore=0$, $haveValidScore=\textsf{False}$ 7: for $i\in V-V_{d}$ do 8: $curScore=p^{}-\sum_{V_{a}\in ATK}\textbf{Pr}_{d}(V_{a})p_{G}(V_{a},V_{d}\cup\\{i\\})$ 9: if $curScore\geq curBestScore$ then 10: $curBest=i$ 11: $curBestScore=curScore$ 12: $haveValidScore=\textsf{True}$ 13: end if 14: end for 15: if $haveValidScore=\textsf{False}$ then 16: $flag=\textsf{False}$ 17: else 18: $V_{d}=V_{d}\cup\\{curBest\\}$ 19: end if 20: end while 21: return $V_{d}$. Finding Mixed Strategies. If the attacker uses a mixed strategy that consists of uniformly attacking elements of $\\{S\subset V_{ld}:\|S\|=k_{a}\\}$ then the best any pure defender strategy can do is defending $V_{d}\subset V_{ld}$. The attacker’s strategy implies that any node in $V_{ld}$ is attacked with probability $\frac{k_{a}}{\|V_{ld}\|}$. Each of the $\|V_{ld}\|-k_{a}$ remaining nodes in $V_{ld}$ is then disconnected with probability $\frac{k_{a}}{\|V_{ld}\|}$, i.e., $x\geq k_{a}(1-\frac{k_{d}}{\|V_{ld}\|})$. Clearly due to the cascading the value of the game will probably be higher, illustrating the disadvantage the defender has in this game. To determine both player’s optimal strategies and the value of the game we resort to an algorithmic approach. We find the defender’s optimal strategy with the following linear program. We can find minimax strategy for the defender with the following linear program. It simply assigns a probability to each of the defenders strategies in a manner that minimizes the maximum payoff for the adversary. As a consequence, the solution to the following linear program, DEF_LP can provide the mixed minimax strategy for the defender. An analogous linear program, ATK_LP (not shown), which mirrors DEF_LP, will provide that result for the attacker. ###### Definition 2 (DEF_LP). $\displaystyle\min p^{}$ (1) $\displaystyle subj.to$ $\displaystyle p^{}\geq\sum_{V_{d}\in DEF}X_{V_{d}}p_{G}(V_{a},V_{d})$ $\displaystyle\forall V_{a}\in ATK$ (2) $\displaystyle 1=\sum_{V_{d}\in DEF}X_{V_{d}}$ (3) $\displaystyle X_{V_{d}}\in[0,1]$ $\displaystyle\forall V_{d}\in DEF$ (4) Note that the above linear program requires one variable for each of the defender’s strategies and one constraint for each of the attacker’s strategies. However, as there are a combinatorial number of strategies, even writing down such a linear program is not practical except for very small problem instances. To address this issue of intractability, we employ the double-oracle framework for zero-sum games introduced in [15] and has been applied in more recent work as well [5, 12]. We present the algorithm DOUBLE_ORACLE as follows: Algorithm 2 DOUBLE_ORACLE 0: Network $G=(V,E)$, natural number $maxIters$ 0: Mixed defender strategy $\textbf{Pr}_{d}$ 1: Initialize $numIters=0$, $flag=\textsf{True}$ 2: Initialize the sets of strategies $ATK,DEF$ to both be $\\{\emptyset\\}$ 3: while $flag$ and $numIters\leq maxIters$ do 4: Create $\textbf{Pr}_{a},\textbf{Pr}_{d}$ based on the solutions to ATK_LP and DEF_LP respectively. 5: IF $numIters<maxIters$ THEN let $V_{a}$ be the attacker’s best response to $\textbf{Pr}_{d}$ and $V_{d}$ be the defender’s best response to $\textbf{Pr}_{a}$ 6: IF $V_{a}\in ATK$ and $V_{d}\in DEF$ THEN $flag=\textsf{False}$ ELSE $ATK=ATK\cup\\{V_{a}\\}$, $DEF=DEF\cup\\{V_{d}\\}$ 7: $numIters+=1$ 8: end while 9: return $\textbf{Pr}_{a}$. The intuition behind the above algorithm is that it iteratively creates mixed strategies for both the attacker and defender based on a solution to a linear program over the sets of current possible strategies for both players ($ATK,DEF$). This is followed by finding (for each player) the best deterministic response to it’s opponent’s strategy. If these new strategies are both already in the set of possible strategies for the respective players, the algorithm terminates. Otherwise, they are added to $ATK,DEF$ respectively. We note that by Theorem 1 of [15] that the above algorithm will guarantee an exact solution if $maxIters$ is set to the number of possible strategies. In practice, [15] demonstrates that the algorithm converges much faster. In DOUBLE_ORACLE, the finding the solutions to DEF_LP, ATK_LP will be tractable provided that the algorithm converges in a polynomial number of steps (either through convergence or after the specified $maxIters$). However, as we have shown, computing the best responses is usually computationally difficult. Although, we note in the case where $k_{a}=1$, that by Proposition 4 and Fact 1, the double oracle algorithm will return an optimal solution, even if greedy approximations are used for the oracles (provided it runs until convergence). ## 5 Experimental Evaluation All experiments were run on a computer equipped with an Intel X5677 Xeon Processor operating at 3.46 GHz with a 12 MB Cache and 288 GB of physical memory. The machine was running Red Hat Enterprise Linux version 6.1. Only one core was used for experiments. All algorithms were coded using Python 2.7 and leveraged the NetworkX library222http://networkx.lanl.gov/ as well as the PuLP library for linear programming333http://pythonhosted.org/PuLP/. All statistics presented in this section were calculated using the R statistics software. In our experiments, we utilized a dataset of an Italian 380 kV power transmission grid [20]. This power grid network consisted of $310$ nodes of which $113$ were source, $96$ were load, and the remainder were transmission nodes. The nodes were connected with $361$ edges representing the power lines. In our initial experiments, we examined the properties of the model when no defense is employed. In Figure 1 (left) we show results concerning nodal load vs. the payoff achieved by the adversary if that node is attacked (and no others). Interestingly, we noticed a significant number of nodes with low nodal load yet high-payoff if attacked (see nodes in dashed box). This may suggest that the DLB strategy may be insufficient in some cases. Later we see how DLB fails to provide adequate in a defense against the attacker best response to DLB. This is likely due to these hi-payoff, low-load nodes. In Figure 1 (right) we examine $\alpha$ (capacity margin) vs. attacker payoff for various settings of $k_{a}$ (using the GREEDY_ATTACKER_RESP heuristic). Here we found that, in general, payoff decreases linearly with capacity margin ($R^{2}\geq 0.84$ for each trial). Figure 1: Left: Nodal load vs. payoff (note hi-payoff, low-load nodes in the dashed box), Right: Capacity margin ($\alpha$) vs. payoff Figure 2: Minimax and DLB defense strategies vs. minimax attack strategy (left) and the attacker’s greedy best response to DLB (right). Examined are the cases where $k_{a}=k_{d}$ (top), $k_{a}=1$, $k_{d}$ varies (middle) and $k_{d}=1$, $k_{a}$ varies (bottom). Next, we examined the relative performance of the minimax (mixed) defense strategy and the DLB strategy under different resource constraints and against the minimax (mixed) attack strategy as well as the attacker’s (deterministic) greedy response to the DLB defense. In these experiments, we considered the case where both players have equal resources, the attacker has one resource (which by Proposition 4 and Fact 1 we are guaranteed an optimal solution), and the defender has one resource. These results are displayed in Figure 2. In these trials we set the capacity margin $\alpha=0.5$, meaning that all edges had an excess capacity of $50\%$. We did not use the $maxIters$ parameter of the DOUBLE_ORACLE algorithm, but instead allowed it to run until convergence. With regard to the comparison between DLB and minimax defense, both performed comparably against the minimax attack strategy. In fact, an analysis of variance (ANOVA) indicated little variance between the two when faced with the minimax attacker ($p\geq 0.74$ for these trials). Yet, a defender known to be playing a single strategy would likely not face an attacker who plays the minimax strategy, but rather the best response to the DLB. In this case, DLB play resulted in significantly greater payoff to the attacker than the defender ($p\leq 0.29$ for these trials, the DLB defense results in $15.6$ more disconnected nodes on average). This failure of the DLB strategy to perform well against a deterministic attacker best response is likely due to the presence of low-load yet high-payoff nodes as shown in Figure 1. We also noticed that an increase in resources seems to favor the attacker more than the defender. When both players played their respective minimax strategy, the expected payoff for the attacker increased monotonically with the cardinality of the strategies. Further, when $k_{d}=1$ and $k_{a}$ was greater, the attacker’s payoff tripled when his resources increased from $1$ to $6$. However, when $k_{a}=1$ and $k_{d}$ was greater, the defender’s payoff only increased by a factor of $1.7$. Hence, the attacker can cause more damage than the defender can mitigate with the same amount of extra resources. We suspect that this is likely because a defended node can still fail during a cascade - which would likely be the case if the attack and defense operations are restricted to cyber-space, where physical system failure may still be possible as the result of a cascade initiated by virtual means. We also examined the run-time of our approach, as displayed in Figure 3 (left). Though run-time did seem to scale linearly with strategy size ($R^{2}=0.90\pm 0.2$ for each experiment), it appears that run-time will in general prohibit the study of larger strategies or networks (our longest experiment ran for 12 days). In examining the iterations of the DOUBLE_ORACLE algorithm, Figure 3 (left), we find that run-time of an iteration of the algorithm progressively increases (note that this figure is showing the run- time for each iteration, not a cumulative time). This increase is likely the combined result of the growing linear program and the growing size of the mixed strategies considered by the greedy approximation sub-routines. We are currently exploring reliable methods to limit the number of iterations while maintaining defender payoff. Figure 3: Strategy size vs. run-time in hours (left) and the run-time of each iteration for the experiments where $k_{a}=k_{d}$ ## 6 Related Work Network security has received much attention from the research community in the past two decades. Recent incidents have shown that due to their internet connectedness such networks can come under cyber attack, causing severe problems444http://www.wired.com/threatlevel/2009/10/smartgrid/. See [26] for a discussion of cyber-security issues relevant to smart grid grids. The utilization of game theory in designing defense solutions seems ubiquitous. For instance [13] model the interaction between a DDoS attacker and the network administrator while [14] considers a game theoretic formulation for intrusion detection. Other formulations consist include stochastic games [17], signaling games [19], allocation games [4] and repeated games [3]. Game theory is also being used in monitoring and decision making in smart grids, see for instance [9] or the survey by Fadlullah et al. [10]. However to date no game theoretic approach has been given for the specific problem where the attacker explicitly sets of a cascading power failure to maximize the damage to the defender. Cascading failure models applied to power grid infrastructure have been studied in the past [7, 8, 16]. The model of [8] introduces the idea of edge failure based on excessive loads. The goal of the research presented in these papers was to illustrate properties of the cascade, rather than explore strategies for attack and defense as this work does. There has been work on attack and defense of a power-grid network under the DC power-flow mode [2, 21, 20, 6]. However, the DC power flow model is not designed to model the more rapid cascading failures (i.e. the 2003 cascading failure in the eastern United States [1]). The application of game theory to security situations was made popular by [18] where it used for airport security patrol scheduling. Since then, other applications have emerged including port protection [23], finding weapons caches [22], and security checkpoint placement [12]. One that bears similarity to this work is [24] \- studying games for controlling contagions on a network. However, as previously discussed, that model operates under very different dynamics. ## 7 Conclusion In this paper, we explored complexity, algorithmic, and implementation issues in a two-player security game where the attacker/defender look to create/mitigate cascading failure on a power grid. Future work includes an examination of scalability issues (larger networks and strategies), adding uncertainty to the model, and the consideration of more real-world information about the power grid network (i.e. actual line capacities, etc.) in order to create a richer model. ## 8 Acknowledgments We would like to thank D. Alderson for his input on related work and V. Rosato for providing us the power grid dataset. Some of the authors are supported by ARO project 2GDATXR042. The opinions in this paper are those of the authors and do not necessarily reflect the opinions of the funders, the U.S. Military Academy, or the U.S. Army. ## References [1] Final Report on the August 14, 2003 Blackout in the United States and Canada: Causes and Recommendations. U.S.-Canada Power System Outage Task Force, April 2004. * [2] D. L. Alderson, G. G. Brown, M. W. Carlyle, and L. Anthony Cox. Sometimes there is no ”most-vital” arc: Assessing and improving the operational resilience of systems. 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Security games for controlling contagion. In J. Hoffmann and B. Selman, editors, AAAI. AAAI Press, 2012. * [25] S. Wasserman and K. Faust. Social Network Analysis: Methods and Applications. Number 8 in Structural analysis in the social sciences. Cambridge University Press, 1 edition, 1994. * [26] D. Wei, Y. Lu, M. Jafari, P. Skare, and K. Rohde. Protecting smart grid automation systems against cyberattacks. Smart Grid, IEEE Transactions on, 2(4):782–795, 2011.	arxiv-papers	2014-01-06T14:08:09	2024-09-04T02:49:56.331659	{ "license": "Public Domain", "authors": "Paulo Shakarian, Hansheng Lei, Roy Lindelauf", "submitter": "Paulo Shakarian", "url": "https://arxiv.org/abs/1401.1086" }
1401.1153	# Ab initio lattice dynamical studies of silicon clathrate frameworks and their negative thermal expansion Ville J. Härkönen Antti J. Karttunen [email protected] University of Jyväskylä, PO Box 35, FI-40014, Finland ###### Abstract The thermal and lattice dynamical properties of seven silicon clathrate framework structures are investigated with ab initio density functional methods (frameworks I, II, IV, V, VII, VIII, and H). The negative thermal expansion (NTE) phenomenon is investigated by means of quasiharmonic approximation and applying it to equal time displacement correlation functions. The thermal properties of the studied clathrate frameworks, excluding the VII framework, resemble those of the crystalline silicon diamond structure. The clathrate framework VII was found to have anomalous NTE temperature range up to 300 K and it is suitable for further studies of the mechanisms of NTE. Investigation of the displacement correlation functions revealed that in NTE, the volume derivatives of the mean square displacements and mean square relative displacements of atoms behave similarly to the vibrational entropy volume derivatives and consequently to the coefficients of thermal expansion as a function of temperature. All studied clathrate frameworks, excluding the VII framework, possess a phonon band gap or even two in the case of the framework V. negative thermal expansion, phonon dispersion, clathrate, ab initio lattice dynamics, DFTP, displacement correlation function, momentum correlation function, mean square displacement, vibrational entropy, silicon. ###### pacs: 63.20.dk, 63.70.+h, 65.40.De, 65.60.+a, 65.80.-g ## I Introduction In addition to the crystalline silicon diamond structure (d-Si), various other crystal structures contain tetrahedrally bonded silicon atoms. Examples of such structures are the experimentally known semiconducting clathrates,Kasper _et al._ (1965) which are typically obtained as host-guest compounds such as $\textrm{Na}_{24}\textrm{Si}_{136}$, where a porous Si clathrate II framework (sometimes denoted as Si34 or Si136) is partially filled by Na guest atoms.Shevelkov and Kovnir (2011) However, silicon clathrates with a (practically) guest-free Si framework structure have been prepared, as well.Gryko _et al._ (2000); Ammar _et al._ (2004) The semiconducting clathrates have been investigated intensively due to their interesting properties such as anomalous electronic structure (differing from that of d-Si),Adams _et al._ (1994); Saito and Oshiyama (1995) superconductivityKawaji _et al._ (1995) and relatively high thermoelectric efficiency.Nolas _et al._ (1998); Sootsman _et al._ (2009); Christensen _et al._ (2010) In particular, the most intense research efforts on the thermoelectric properties of clathrates have so far been directed at germanium-based clathrates such as $\textrm{Ba}_{8}\textrm{Ga}_{16}\textrm{Ge}_{30}$.Kuznetsov _et al._ (2000); Toberer _et al._ (2008) Tang et al. have investigated the thermal properties of the clathrate II framework by means of ab initio calculations and experimental measurements and have shown that the clathrate II framework exhibits negative thermal expansion (NTE) in the temperature range of 10–140 K.Tang _et al._ (2006) NTE is a phenomenon in which a material contracts instead of expanding when heated to higher temperatures. From the point of view of practical applications, NTE materials have been used to design composite materials with zero thermal expansion.Evans _et al._ (1997) The NTE materials have been studied actively after the discovery of large temperature range NTE materials such as $\textrm{ZrW}_{2}\textrm{O}_{8}$.Evans _et al._ (1996); Mary _et al._ (1996), which shows NTE in a wide temperature range from nearly 0 K up to 1080 K. The underlying mechanisms causing the NTE behavior are not fully understood. The NTE phenomenon and the different NTE mechanisms have been reviewed for example by Barrera et al.Barrera _et al._ (2005) d-Si also shows NTE within temperature range from 25 K up to 120 K.Reeber and Wang (1996) However, to our knowledge, there are no elemental semiconductors with anomalously large NTE temperature ranges comparable to the oxides such as $\textrm{ZrW}_{2}\textrm{O}_{8}$. The NTE behavior of materials can be experimentally studied by means of Extended X-Ray Absorption Fine Structure (EXAFS) spectroscopy, which enables the measurement of local dynamics of the crystal lattice.Sayers _et al._ (1971); Lee and Pendry (1975); Hayes _et al._ (1976); Tröger _et al._ (1994) Quantities which can be obtained from EXAFS spectra are, for example, mean square displacements (MSD) and mean square relative displacements (MSRD). Consequently, EXAFS has also been used to study the NTE phenomenon.Sanson _et al._ (2006); el All _et al._ (2012) However, computational studies of the NTE phenomenon from the point of view of MSRD are not common. To our knowledge, there are no previous studies where the NTE behavior of materials is investigated by means of full ab initio calculation of MSRDs. In this paper we investigate the thermal properties and negative thermal expansion of seven different silicon clathrate frameworks by means of ab initio lattice dynamics. The thermal properties of the materials are determined within quasiharmonic approximation (QHA).Huang and Born (1954) The results are compared with available experimental data and previous computational predictions for d-Si to establish the accuracy of the computational methods. We use equal time displacement correlation functions to investigate the mechanisms of the thermal expansion and to elucidate the negative thermal expansion behavior of the silicon clathrate frameworks. ## II Theory, computational methods and studied structures ### II.1 Lattice dynamics and thermal expansion To obtain the various thermal properties, the phonon eigenvalues and phonon eigenvectors have been extracted with ab initio methods, by applying density functional perturbation theory (DFPT).Baroni _et al._ (2001) The DFTP formalism enables the calculation of dynamical matrix (introduced by the relations below), after which the phonon eigenvalues and eigenvectors can be solved from an eigenvalue equation written within the harmonic approximation asMaradudin _et al._ (1971) $\omega^{2}_{j}\left(\mathbf{q}\right)e_{\alpha}\left(\kappa\|\mathbf{q}j\right)=\sum_{\kappa^{\prime},\beta}D_{\alpha\beta}\left(\kappa\kappa^{\prime}\|\mathbf{q}\right)e_{\beta}\left(\kappa^{\prime}\|\mathbf{q}j\right),$ (1) where $\alpha$ and $\beta$ are cartesian indices, $\kappa$ and $\kappa^{\prime}$ are atom indices within the primitive unit cell, $\omega_{j}\left(\mathbf{q}\right)$ is phonon eigenvalue (referred as phonon frequency from now on) for phonon branch $j$ and wave vector $\mathbf{q}$, $e_{\alpha}\left(\kappa\|\mathbf{q}j\right)$ are the components of the phonon eigenvector and $D_{\alpha\beta}\left(\kappa\kappa^{\prime}\|\mathbf{q}\right)$ are the components of the dynamical matrix. The dynamical matrix is the Fourier transform of the real space force constant matrix $\Phi_{\alpha\beta}\left(l\kappa;l^{,}\kappa^{,}\right)$ $\displaystyle D_{\alpha\beta}\left(\kappa\kappa^{\prime}\|\mathbf{q}\right)=\left(M_{\kappa}M_{\kappa^{\prime}}\right)^{-1/2}\sum_{l^{\prime}}\Phi_{\alpha\beta}\left(l\kappa;l^{\prime}\kappa^{\prime}\right)$ $\displaystyle\times{}e^{-i\mathbf{q}\cdot\left[\mathbf{x}\left(l\right)-\mathbf{x}\left(l^{\prime}\right)\right]},$ (2) where $M_{\kappa}$ and $M_{\kappa^{\prime}}$ are the atomic masses of atoms $\kappa$ and $\kappa^{\prime}$ respectively, and $\mathbf{x}\left(l\right)$ is the position vector of the $l_{\textrm{th}}$ unit cell. The components of the eigenvector $\mathbf{e}\left(\kappa\|\mathbf{q}j\right)$ are chosen to satisfy the orthonormality and closure conditionsHuang and Born (1954) $\sum_{\kappa,\alpha}e_{\alpha}\left(\kappa\|\mathbf{q}j^{\prime}\right)e^{}_{\alpha}\left(\kappa\|\mathbf{q}j\right)=\delta_{jj^{\prime}},$ (3) $\sum_{j}e_{\alpha}\left(\kappa\|\mathbf{q}j\right)e^{}_{\beta}\left(\kappa^{\prime}\|\mathbf{q}j\right)=\delta_{\alpha\beta}\delta_{\kappa\kappa^{\prime}},$ (4) where $\delta_{\alpha\beta}$ is the Kronecker delta. The volumetric coefficient of thermal expansion (CTE) $\alpha_{V}$ is defined as $\alpha_{V}=\frac{1}{V_{0}}\frac{\partial{V}}{\partial{T}},$ (5) where $V$ is the volume, $V_{0}$ is the equilibrium volume and $T$ the temperature. The linear CTE is a measure of the change in particular dimension of the crystal with respect to temperature. In the cubic crystal system, linear CTE $\alpha_{L}$ in the direction of the principal axis can be written as $\alpha_{L}=\frac{1}{L_{0}}\frac{\partial{L}}{\partial{T}}=\frac{1}{3V_{0}}\frac{\partial{V}}{\partial{T}}=\frac{1}{3}\alpha_{V}.$ (6) In non-cubic crystals, the anisotropy of the crystal structure must be taken in account. That is, the different components of the thermal expansion along different principal axes must be calculated separately to obtain the linear CTEs. In this work only volumetric CTEs have been calculated for non-cubic structures. Using QHA, the expression for the volumetric CTE can be written in terms of isothermal bulk modulus $B$, phonon mode Grüneisen parameters $\gamma_{j}\left(\mathbf{q}\right)$ and phonon mode heat capacities at constant volume $c_{v,j}\left(\mathbf{q}\right)$Ashcroft and Mermin (1976) $\alpha_{V}=\frac{1}{B}\sum_{\mathbf{q},j}c_{v,j}\left(\mathbf{q}\right)\gamma_{j}\left(\mathbf{q}\right),$ (7) The heat capacity can be expressed in terms of Bose-Einstein distribution function $\overline{n}_{\mathbf{q},j}$ and phonon frequencies $\omega_{j}\left(\mathbf{q}\right)$ $c_{v,j}\left(\mathbf{q}\right)=\frac{\omega_{j}\left(\mathbf{q}\right)}{V_{0}}\frac{\partial}{\partial{T}}\overline{n}_{\mathbf{q},j},$ (8) and the Grüneisen parameters can be expressed as $\gamma_{j}\left(\mathbf{q}\right)=-\frac{V_{0}}{\omega_{0,j}\left(\mathbf{q}\right)}\frac{\partial{\omega_{j}}\left(\mathbf{q}\right)}{\partial{V}},$ (9) where $\omega_{0,j}\left(\mathbf{q}\right)$ is the phonon frequency for the phonon mode $j$ and wave vector $\mathbf{q}$ at the equilibrium volume $V_{0}$. The Grüneisen parameters can also be expressed asTaylor (1998) $\displaystyle\gamma_{j}\left(\mathbf{q}\right)=-\frac{1}{2}\frac{V_{0}}{\omega^{2}_{0,j}\left(\mathbf{q}\right)}\sum_{\alpha,\beta}\sum_{\kappa,\kappa^{\prime}}e^{}_{\alpha}\left(\kappa\|\mathbf{q}j\right)$ $\displaystyle\times{}\frac{\partial D_{\alpha\beta}\left(\kappa\kappa^{\prime}\|\mathbf{q}\right)}{\partial{V}}e_{\beta}\left(\kappa^{\prime}\|\mathbf{q}j\right).$ (10) Eq. 7 shows that when NTE occurs, the Grüneisen parameters must have negative values since the bulk modulus and the heat capacities have positive values. Furthermore, from Eq. 9 it can be seen that to have negative Grüneisen parameter values, the phonon frequencies must decrease with decreasing volume. Analogously, Eq. 10 implies that in NTE the components of the dynamical matrix have smaller values with decreasing volume. Finally, an alternative way to express $\alpha_{V}$ is to make use of thermodynamical relations and to write its expression in terms of vibrational entropyBarrera _et al._ (2005) $\alpha_{V}=\frac{1}{B}\sum_{\mathbf{q},j}\frac{\partial{S_{j}\left(\mathbf{q}\right)}}{\partial{V}},$ (11) From Eq. 11 it can be seen that the vibrational entropy $S$ must increase with decreasing volume for NTE to occur. ### II.2 Correlation functions Mean square displacements (MSD) are a special case of the displacement time correlation functions derived by for example Maradudin.Maradudin _et al._ (1971) Equal time displacement correlation function can be written as $\displaystyle\left\langle u_{\alpha}\left(l\kappa\right)u_{\beta}\left(l^{\prime}\kappa^{\prime}\right)\right\rangle=\frac{\hbar}{2N\left(M_{\kappa}M_{\kappa^{\prime}}\right)^{1/2}}$ $\displaystyle\times{}\sum_{\mathbf{q},j}\frac{e_{\alpha}\left(\kappa\|\mathbf{q}j\right)e^{}_{\beta}\left(\kappa^{\prime}\|\mathbf{q}j\right)}{\omega_{j}\left(\mathbf{q}\right)}$ $\displaystyle\times{}e^{i\mathbf{q}\cdot\left[\mathbf{x}\left(l\right)-\mathbf{x}\left(l^{\prime}\right)\right]}\coth\left[\frac{\hbar\omega_{j}\left(\mathbf{q}\right)}{2k_{B}T}\right],$ (12) where $N$ is the number of unit cells (and $\mathbf{q}$-points) and $k_{B}$ is the Boltzmann constant. In the special case of $l=l^{\prime}$ and $\kappa=\kappa^{\prime}$, Eq. 12 is called the equal time displacement autocorrelation function, vibrational amplitude, or MSD. From now on, all discussed displacement correlation functions are equal time displacement correlation functions and corresponding autocorrelation functions, if not otherwise mentioned. To analyze the NTE phenomenon, it is useful to know how the atoms move relative to each other. Mean square relative displacement (MSRD) in the interatomic direction between atoms $\left(l\kappa\right)$ and $\left(l^{\prime}\kappa^{\prime}\right)$, also known as the parallel MSRD, is defined asBeni and Platzman (1976) $\left\langle u^{2}_{\parallel}\right\rangle=\left\langle\left[\left(\mathbf{u}\left(l\kappa\right)-\mathbf{u}\left(l^{\prime}\kappa^{\prime}\right)\right)\cdot\mathbf{\hat{r}}\right]^{2}\right\rangle,$ (13) where $\mathbf{\hat{r}}$ is unit vector in the direction of vector between atoms $\left(l\kappa\right)$ and $\left(l^{\prime}\kappa^{\prime}\right)$. Only the difference of $l$ and $l^{\prime}$ matters, not their absolute values, thus $l=0$ is chosen. Writing out the component form of Eq. 13, the parallel MSRD is $\displaystyle\left\langle u^{2}_{\parallel}\right\rangle=\sum_{\alpha,\beta}r_{\alpha}r_{\beta}\left.[\left\langle u_{\alpha}\left(0\kappa\right)u_{\beta}\left(0\kappa\right)\right\rangle\right.$ $\displaystyle\left.+\left\langle u_{\alpha}\left(l^{\prime}\kappa^{\prime}\right)u_{\beta}\left(l^{\prime}\kappa^{\prime}\right)\right\rangle-2\left\langle u_{\alpha}\left(0\kappa\right)u_{\beta}\left(l^{\prime}\kappa^{\prime}\right)\right\rangle\right.].$ (14) Perpendicular MSRD is defined asFornasini (2001) $\left\langle u^{2}_{\perp}\right\rangle=\left\langle u^{2}_{R}\right\rangle-\left\langle u^{2}_{\parallel}\right\rangle,$ (15) where $\left\langle u^{2}_{R}\right\rangle=\left\langle\left\\|\mathbf{u}\left(0\kappa\right)-\mathbf{u}\left(l^{\prime}\kappa^{\prime}\right)\right\\|^{2}\right\rangle,$ (16) which is the total MSRD of atoms $\left(l\kappa\right)$ and $\left(l^{\prime}\kappa^{\prime}\right)$. Writing Eq. 16 in the component form and substituting to Eq. 15, perpendicular MSRD can be written as $\displaystyle\left\langle u^{2}_{\perp}\right\rangle=\sum_{\alpha,\beta}\left(\delta_{\alpha\beta}-r_{\alpha}r_{\beta}\right)\left.[\left\langle u_{\alpha}\left(0\kappa\right)u_{\beta}\left(0\kappa\right)\right\rangle\right.$ $\displaystyle\left.+\left\langle u_{\alpha}\left(l^{\prime}\kappa^{\prime}\right)u_{\beta}\left(l^{\prime}\kappa^{\prime}\right)\right\rangle-2\left\langle u_{\alpha}\left(0\kappa\right)u_{\beta}\left(l^{\prime}\kappa^{\prime}\right)\right\rangle\right.].$ (17) Similar calculations involving the displacement correlation functions have been previously carried out by Nielsen and Weber, who used empirical interatomic potentials to investigate the correlation functions for d-Si.Nielsen and Weber (1980); Weber (1987) Finally, the total MSD of atom $\left(l\kappa\right)$ is the sum of the autocorrelation functions along the different cartesian components $\left\langle u^{2}\right\rangle=\left\langle\left\\|\mathbf{u}\left(l\kappa\right)\right\\|^{2}\right\rangle=\sum_{\alpha}\left\langle u_{\alpha}\left(l\kappa\right)u_{\alpha}\left(l\kappa\right)\right\rangle.$ (18) In a cubic crystal, the total MSD (within numerical accuracy) is due to symmetry reasons equal to $3\left\langle u_{\alpha}\left(l\kappa\right)u_{\alpha}\left(l\kappa\right)\right\rangle$, for any $\alpha=1,2,3$. Because the preceding displacement correlation functions are valid only within harmonic approximation, they cannot be used to describe the thermal expansion phenomenon as such. If QHA is assumed to be valid, the relation (Eq. 9) for the volume dependence of the phonon frequencies can be used. Inspection of Eq. 12 shows that the two terms including the hyperbolic cotangent and the inverse of the frequency are always positive. As in Sec. II.1, from Eq. 7 and Eq. 9 it can be seen that in NTE, the change in the frequencies must be in the same direction as the change in the volume. In general, the change of Eq. 12 with respect to volume also depends on the behaviour of the phonon eigenvector and its components $e_{\alpha}\left(\kappa\|\mathbf{q}j\right)$ as a function of volume. We show first with heuristic deduction that the volume dependence of Eq. 18 in the case of d-Si is only due to the volume dependence of the frequencies. Eq. 18 reads $\left\langle u^{2}\right\rangle=C\sum_{\alpha}\sum_{\mathbf{q},j}\left\|e_{\alpha}\left(\kappa\|\mathbf{q}j\right)\right\|^{2}\xi\left[\omega_{j}\left(\mathbf{q}\right),T\right],$ (19) where $C=\frac{\hbar}{2NM_{\kappa}},$ (20) and $\xi\left[\omega_{j}\left(\mathbf{q}\right),T\right]=\omega_{j}\left(\mathbf{q}\right)^{-1}\coth\left[\frac{\hbar\omega_{j}\left(\mathbf{q}\right)}{2k_{B}T}\right].$ (21) The frequencies are not dependent on the cartesian index $\alpha$, and $\xi\left(\omega_{j}\left(\mathbf{q}\right)\right)$ can be moved out of the sum over $\alpha$ $\left\langle u^{2}\right\rangle=C\sum_{\mathbf{q},j}\xi\left[\omega_{j}\left(\mathbf{q}\right),T\right]\sum_{\alpha}\left\|e_{\alpha}\left(\kappa\|\mathbf{q}j\right)\right\|^{2}.$ (22) From Eq. 3 it follows that the sum over $\alpha$ in Eq. 22 with any chosen $j$ and $\mathbf{q}$ must have value $1/2$, in the case of d-Si with two equivalent atoms in the primitive unit cell. Eq. 18 for d-Si is then $\left\langle u^{2}\right\rangle_{\textit{d}-Si}=\frac{C}{2}\sum_{\mathbf{q},j}\xi\left[\omega_{j}\left(\mathbf{q}\right),T\right].$ (23) Preceding is also true for structures where the lattice constants have been displaced in a way that preserves the original symmetry and it is also confirmed for d-Si by means of a direct calculation. This implies that in NTE, the MSD of a particular atom $\left(l\kappa\right)$ and a particular mode $j$ increases with decreasing volume. The reason for this behaviour becomes clearer when the hyperbolic cotangent is written in terms of the Bose-Einstein distribution function $\overline{n}_{\mathbf{q}j}$. Using the definition of the hyperbolic cotangent, the last term in Eq. 23 is $\xi\left[\omega_{j}\left(\mathbf{q}\right),T\right]=\omega_{j}\left(\mathbf{q}\right)^{-1}\left(2\overline{n}_{\mathbf{q},j}+1\right),$ (24) In NTE, Eq. 11 and Eq. 23 actually have their values because of the same underlying reason. In the following, the similarity between these quantities, MSD for d-Si and $S$, in NTE is considered when the temperature increases. If NTE occurs, the phonon frequency of the state $\left(\mathbf{q}j\right)$ contributing to NTE must have lower values when the volume decreases. The same effect occurs, when the considered phonon mode corresponds to a positive CTE and the volume increases. For the phonon modes corresponding to NTE, the preceding implies that there are more lower frequency states for phonons to occupy ($\overline{n}_{\mathbf{q}j}$ have higher values for lower $\omega_{j}\left(\mathbf{q}\right)$ in a particular temperature $T$). Therefore, the entropy increases and according to Eq. 11, NTE occurs when the opposite effect of the phonon modes corresponding to positive CTE is weaker. Similar conclusion applies to Eq. 23 in NTE. If the state $\left(\mathbf{q}j\right)$ has lower values of frequency when the volume decreases, there are more low frequency states that the phonons can occupy and $\xi\left[\omega_{j}\left(\mathbf{q}\right),T\right]$ has larger values, resulting in larger MSD values for the phonon modes contributing to NTE. The opposite is true for the phonon modes contributing to positive CTE. For both negative and positive CTE, the total MSD increases with increasing temperature. As a summary, in NTE, the MSD and entropy of the phonon modes corresponding to positive CTE increase less than the MSD and entropy of the phonon modes corresponding to NTE. To study the effect of thermal expansion phenomenon to the MSD as a function of temperature, we define the following dimensionless quantity $\left\langle u^{2}\right\rangle_{r}\equiv\frac{1}{N_{a}}\left(\frac{V_{0}}{\left\langle u^{2}\right\rangle_{V_{0}}}\frac{\partial\left\langle u^{2}\right\rangle}{\partial V}-\left\langle u^{2}\right\rangle_{r,T_{0}}\right),$ (25) which includes the volume derivatives of the total MSD, and where $\left\langle u^{2}\right\rangle_{r,T_{0}}=\frac{V_{0}}{\left\langle u^{2}\right\rangle_{V_{0},T_{0}}}\frac{\partial\left\langle u^{2}\right\rangle_{T_{0}}}{\partial V}.$ (26) In Eq. 25, $N_{a}$ is the number of atoms within the primitive unit cell, $V_{0}$ is the volume of the equilibrium structure, $\left\langle u^{2}\right\rangle_{V_{0}}$ is the total MSD for the equilibrium structure, and $T_{0}$ indicates the value of the function at $T=0$K. Eq. 25 measures the product of the total MSD volume derivative and the ratio of the equilibrium volume and the equilibrium values of the total MSD. The values of Eq. 25 are therefore negative for modes for which the values of the frequency decrease as the volume decreases, at a particular temperature $T$. This is based on the discussions above and the fact that the MSD (Eq. 18) have only positive values for all values of $T$. The values of the first term within the brackets in Eq. 25 differ from zero when $T\rightarrow 0$ K. For clarity, a normalization factor is defined (the second term within the brackets in Eq. 25), which has a constant value for any chosen state $\left(\mathbf{q}j\right)$. The normalization factor ensures that Eq. 25 is equal to zero when $T\rightarrow 0$ K. The preceding normalization works properly since the first term within the brackets in Eq. 25 increases (positive CTE) or decreases (negative CTE) with all relevant $T>0$. In addition, a second normalizing factor $N_{a}$ is introduced to facilitate comparisons between structures with different-sized primitive unit cells. $N_{a}$ is related to the number of phonon modes $N_{j}$ by the relation $N_{j}=3N_{a}$. The same scheme is implemented for the parallel and perpendicular MSRD to study changes in the relative displacements with respect to volume and their role in the NTE phenomenon. They are defined as $\left\langle u^{2}_{\parallel}\right\rangle_{r}\equiv\frac{1}{N_{a}}\left(\frac{V_{0}}{\left\langle u^{2}_{\parallel}\right\rangle_{V_{0}}}\frac{\partial\left\langle u^{2}_{\parallel}\right\rangle}{\partial V}-\left\langle u^{2}_{\parallel}\right\rangle_{r,T_{0}}\right),$ (27) and $\left\langle u^{2}_{\perp}\right\rangle_{r}\equiv\frac{1}{N_{a}}\left(\frac{V_{0}}{\left\langle u^{2}_{\perp}\right\rangle_{V_{0}}}\frac{\partial\left\langle u^{2}_{\perp}\right\rangle}{\partial V}-\left\langle u^{2}_{\perp}\right\rangle_{r,T_{0}}\right).$ (28) The preceding quantities in Eqs. 25 and 27–28 are named as relative MSD and MSRD volume derivatives, and these quantities are denoted as MSD-VD and MSRD- VD, respectively. Properties analogous to the MSD-VD and MSRD-VD introduced above can be derived also for equal time momentum correlation functions, which can be written within harmonic approximation asMaradudin _et al._ (1971) $\displaystyle\left\langle p_{\alpha}\left(l\kappa\right)p_{\beta}\left(l^{\prime}\kappa^{\prime}\right)\right\rangle=\frac{\hbar}{2N}\left(M_{\kappa}M_{\kappa^{\prime}}\right)^{1/2}$ $\displaystyle\times{}\sum_{\mathbf{q},j}e_{\alpha}\left(\kappa\|\mathbf{q}j\right)e^{}_{\beta}\left(\kappa^{\prime}\|\mathbf{q}j\right)\omega_{j}\left(\mathbf{q}\right)$ $\displaystyle\times{}e^{i\mathbf{q}\cdot\left[\mathbf{x}\left(l\right)-\mathbf{x}\left(l^{\prime}\right)\right]}\coth\left[\frac{\hbar\omega_{j}\left(\mathbf{q}\right)}{2k_{B}T}\right]=$ $\displaystyle M_{\kappa}M_{\kappa^{\prime}}\omega^{2}_{j}\left(\mathbf{q}\right)\left\langle u_{\alpha}\left(l\kappa\right)u_{\beta}\left(l^{\prime}\kappa^{\prime}\right)\right\rangle,$ (29) where $p_{\alpha}\left(l\kappa\right)$ is the momentum for the atom $\left(l\kappa\right)$ in the direction $\alpha$. The difference is that the change of the mean square momentum (MSM) with respect to the volume, derived from Eq. 29, is opposite in direction to the MSD-VD and MSRD-VD. This means that if the volume decreases and the temperature increases for example by 1 K (NTE occurs), the phonon modes contributing to NTE (positive CTE) resist (assist) the increase of the MSM, but since the changes in the frequencies are small ( $\left\|\Delta\omega_{j}\left(\mathbf{q}\right)\right\|/\omega_{j}\left(\mathbf{q}\right)<<1$), MSM increases in both cases. Finally, few remarks about the method of calculating CTE from Eq. 7 or 11 and the related properties calculated from Eq. 25 are necessary. In general, the Grüneisen parameters are not independent of the temperature, which also means that $\omega_{j}\left[\mathbf{q},V\left(T\right),T\right]$. When the temperature independence is assumed it follows from Eq. 7 and 11 that the phonon modes corresponding to positive CTE (NTE) have positive (negative) contribution to CTE at all finite $T>0$. The same features result for the MSD- VD and MSRD-VD and the related quantities derived for momentum. ### II.3 Studied structures and computational details The structural characteristics of the studied seven silicon clathrate frameworks are illustrated in Fig. 1 and described in Table 1. Figure 1: Schematic figures of the silicon clathrate structures included in this work. The vertices of the polyhedral cages present silicon atoms. Crystallographic unit cell edges are drawn in black. For a more comprehensive description of the structural characteristics, see.Karttunen _et al._ (2010) (Color online) Table 1: Structural data and computational details for the studied structures. Structure \| Space Group \| Atoms/cell111The number of atoms in the primitive cell. \| Elect. (k1,k2,k3)222The mesh used for the electronic $\mathbf{k}$-sampling. \| Phon. (q1,q2,q3)333The meshes used for phonon calculations, PDOS calculations, and correlation function calculations, respectively. The meshes for the PDOS and correlation function calculations were Fourier interpolated from the meshes used in the phonon calculations. \| PDOS (q1,q2,q3)c \| Corr. (q1,q2,q3)c ---\|---\|---\|---\|---\|---\|--- d-Si \| $Fd\bar{3}m\left(227\right)$ \| 2 \| 16,16,16 \| 8,8,8 \| 80,80,80 \| 20,20,20 I \| $Pm\bar{3}n\left(223\right)$ \| 46 \| 4,4,4 \| 4,4,4 \| 30,30,30 \| - II \| $Fd\bar{3}m\left(227\right)$ \| 34 \| 4,4,4 \| 4,4,4 \| 30,30,30 \| 8,8,8 IV \| $P6/mmm\left(191\right)$ \| 40 \| 4,4,4 \| 3,3,3 \| 30,30,30 \| - V \| $P6_{3}/mmc\left(194\right)$ \| 68 \| 4,4,3 \| 4,4,3 \| 30,30,30 \| - VII \| $Im\bar{3}m\left(229\right)$ \| 6 \| 8,8,8 \| 8,8,8 \| 60,60,60 \| - VIII \| $I\bar{4}3m\left(217\right)$ \| 23 \| 6,6,6 \| 4,4,4 \| 40,40,40 \| - H \| $P6/mmm\left(191\right)$ \| 34 \| 4,4,4 \| 3,3,3 \| 30,30,30 \| - The semiconducting clathrate structures are classified according to the polyhedral cages they are composed of. The structural properties of the clathrates have been reviewed in e.g. Rogl (2006); Kovnir and Shevelkov (2004); Shevelkov and Kovnir (2011) The ab initio density functional calculations to optimize the structures and to calculate the phonon dispersion relations were carried out with the Quantum Espresso program package (QE, version 5.0.3).Giannozzi _et al._ (2009) The Si atoms were described by Martins-Troullier-type norm-conserving pseudopotentials Troullier and Martins (1991); Dal Corso (2012) and the Local Density Approximation (LDA) was used as the exchange-correlation energy functional.Perdew and Zunger (1981) In all calculations, the following kinetic energy cutoffs have been used: 40 Ry for wavefunctions and 160 Ry for charge densities and potentials. The applied $\mathbf{k}$\- and $\mathbf{q}$-sampling for each studied structure is listed in Table 1. The $\mathbf{q}$-meshes for the phonon density of states (PDOS) and correlation function calculations were Fourier interpolated from the mesh used in the corresponding phonon calculation (QE module matdyn.x).Baroni _et al._ (2001) We carried out convergence tests for both SCF and phonon frequency calculations with different $\mathbf{k}$-meshes. The lattice constant for the optimized structure was practically identical for (8,8,8) and (16,16,16) $\mathbf{k}$-meshes and the final energy was about 510-3 eV lower with the (16,16,16) mesh. The maximum differences between the phonon frequencies obtained with the (8,8,8) and (16,16,16) $\mathbf{k}$-meshes are about 1.5%. Since a reasonable convergence was found for the (8,8,8) $\mathbf{k}$-mesh in the case of d-Si, the $\mathbf{k}$-meshes listed in in Table 1 for the clathrate frameworks were chosen as a compromise between accuracy and computational cost. For d-Si, the use of the denser (16,16,16) $\mathbf{k}$-mesh was still computationally feasible. We have also confirmed the convergence of the calculated phonon frequencies with respect to the applied $\mathbf{q}$-meshes. Both the lattice constants and the atomic positions of the studied structures were fully optimized (applying the space group symmetries listed in Table 1). The Grüneisen parameters were calculated by displacing the structures from the equilibrium lattice constant by $\pm$0.5%, optimizing the atomic positions, and calculating the numerical derivatives using central differences. Bulk moduli were calculated by displacing the structures from the equilibrium lattice constant up to $\pm$2% with a step size of $\pm$0.5% and calculating the bulk modulus using the Murnaghan’s equation of state (QE module ev.x.). Phonon density of states values were calculated using the tetrahedron method (QE module matdyn.x).Blöchl _et al._ (1994) In all cases where a summation over $\mathbf{q}$ is involved, the appropriate weights to normalize the sums have been used. The algorithms for calculating the thermodynamic quantities and the correlation functions were implemented as Matlab routines. Calculation of the correlation functions involves some numerical issues, which need to be discussed. To our knowledge, there is no way to label the phonon modes uniquely in a point of degeneracy when the diagonalization of the dynamical matrix in Eq. 1 is done numerically. This is true also for the QE program package and probably results in minor numerical inaccuracies in the calculation of the correlation functions. Some phonon labeling could be carried out by continuity arguments for the phonon eigenvectors in particular direction when using relatively dense $\mathbf{q}$-meshes, but in the absence of a rigorous general approach, the labeling of the phonon modes is left to the default algorithm in QE. The numerical accuracy is also sensitive to the applied Fast Fourier Transform (FFT) mesh. In this work, the default settings of QE were used to choose the FFT grid. ## III Results and discussion ### III.1 Optimized structures and phonon dispersion relations Table 2 lists the lattice constants, the relative energies $\Delta E$ with respect to d-Si, and the bulk moduli for the optimized silicon clathrate frameworks. Table 2: Lattice constants $a,c$, relative energies ($\Delta E$), and bulk moduli ($B$) for the optimized structures. Structure \| $a$(Å) \| $c$(Å) \| $\Delta E$ (eV/atom) \| $B$ (GPa) ---\|---\|---\|---\|--- d-Si \| 5.38 \| $-$ \| 0.00 \| 96.5 I \| 10.08 \| $-$ \| 0.09 \| 84.1 II \| 14.52 \| $-$ \| 0.08 \| 82.4 IV \| 10.07 \| 10.23 \| 0.11 \| 78.5 V \| 10.26 \| 16.80 \| 0.08 \| 82.1 VII \| 6.63 \| $-$ \| 0.27 \| 72.6 VIII \| 9.95 \| $-$ \| 0.10 \| 86.6 H \| 10.36 \| 8.43 \| 0.11 \| 79.6 As could be expected, the structure II, which has been synthetized as a nearly guest-free silicon framework, shows the lowest relative energy among the studied clathrate frameworks. It is followed closely by structure V, which is a hypothetical hexagonal modification of the structure II.Karttunen _et al._ (2010) The predicted relative energies are in good agreement with previous results obtained with the PBE0 hybrid density functional method and localized Gaussian type basis sets (GTO).Karttunen _et al._ (2010) The structure VII shows the largest difference in $\Delta E$ in comparison to the PBE0 results (0.03 eV/atom), while for the other structures the difference is only 0.01 eV/atom. The structure VII is a rather strained one with a high relative energy and it is not experimentally as relevant as the other studied structures. The predicted bulk moduli are systematically somewhat smaller in comparison to previous results obtained with the PBE0 functional,Karttunen _et al._ (2011) but the result for d-Si is similar to previous LDA- predictions.Nielsen and Martin (1985) The previous LDA predictions for the clathrate frameworks I and II (87 and 87.5 GPa, respectively) are also comparable to the present results.San-Miguel _et al._ (2002) LDA slightly underestimates the bulk modulus ($\approx$1.3% in the case of d-Si) in comparison to experiment, the experimental value for d-Si being 97.8 GPa.Hall (1967) The difference between calculated and experimental lattice constant value is approximapdftely 1%.Basile _et al._ (1994) The experimental value of the bulk modulus for the silicon clathrate framework II has been found to be $90\pm 5$ GPa.San-Miguel _et al._ (1999) Figures 2 and 3 illustrate the phonon dispersion relations in the studied structures and the corresponding phonon density of states (PDOS). Figure 2: Phonon dispersion relations along high symmetry paths in the first Brillouin zone for a$)$ d-Si (experimental values indicated by diamondsNilsson and Nelin (1972)) and b$)$ for structure VII. Figure 3: Phonon dispersion relations along high symmetry paths in the first Brillouin zone for structures I, II, IV, V, VIII, and H. Fig. 2 also shows the experimental phonon dispersion data for d-Si.Nilsson and Nelin (1972) The LDA results for d-Si are in relatively good agreement with the experiment, the differences being the largest for large wave vectors (the max. difference is approximately 7%). The estimated experimental error is $\pm$1 cm-1.Nilsson and Nelin (1972) The LDA phonon dispersion data for the clathrate framework II are also in agreement with previous computational results.Tang _et al._ (2006) The PDOS of the silicon clathrate frameworks I and II have also been determined experimentally using inelastic neutron scattering for structures containing some potassium and sodium impurities (e.g. $\textrm{Na}:\textrm{Si}$ ratio 1:34). Despite the impurities, the experimental results show similar main features when compared to the results shown in Fig. 3, such as the high PDOS values at the frequencies of approximately 175 cm-1.Mélinon _et al._ (1999) In the structure VII (Fig. 2b), a significant decrease of the lowest optical modes towards the acoustic modes can be observed, in particular for small wave vectors. This indicates strong anomalies in the interatomic forces. The other clathrate frameworks show very similar phonon dispersion relations compared to each other. The structures other than d-Si and structure VII have a phonon band gap within the frequency range of 400–475 cm-1, the structure V even showing two phonon band gaps at frequencies of about 440 and 390 cm-1 (Fig. 3). ### III.2 Thermal properties from the quasiharmonic approximation The calculated heat capacities at constant volume are shown in Fig. 4. Figure 4: Heat capacities at constant volume for studied structures. Only the structure VII is slightly different from the other structures (indicated with a dash-dotted line). The results are very similar for all structures, the strained structure VII showing the largest difference to the other structures. The results for the structure II are in agreement with previous experimental and computational results.Biswas _et al._ (2008) The reason for the deviation observed for the heat capacity of structure VII can be seen from Fig. 5, where Eq. 8 has been plotted. Figure 5: Heat capacity at constant volume as a function of temperature and frequency in QHA. Every separate line represents the value of Eq. 8 for a particular phonon frequency as a function of the temperature. At low $T$, only the low frequency modes affect the heat capacity significantly. When $T$ increases, the contribution of the low frequency modes does not increase as rapidly as the contribution of the higher frequency modes. Since the structure VII has more low frequency modes and less higher frequency modes in comparison to the other structures, it has a slightly higher heat capacity at low temperature and lower heat capacity at higher temperatures. This also affects the CTE of the structure VII. The very similar heat capacities suggest that in the studied structures, and within QHA, the Grüneisen parameters and the differences in the values of the bulk moduli have the greatest effect on the magnitude of CTE. Fig. 6 shows the Grüneisen parameters for the studied structures as a function of the phonon frequency. Figure 6: Grüneisen parameters for the studied structures, each dot represents one mode at a particular $\mathbf{q}$-point. For every structure, approximately 75000 frequency values have been used to generate the Grüneisen parameter data. The Grüneisen parameters predicted for d-Si and the structure II are in agreement with previous computational results.Tang _et al._ (2006) The structure VII has distinctly different Grüneisen parameters from the other structures, showing negative values that are approximately 10 times larger than for the other structures. An interesting feature is that the Grüneisen parameters only have negative values when the phonon frequency is under a certain threshold that is 220 cm-1 for d-Si and VII and 190 cm-1 for the other structures. However, in every structure, the acoustic modes with frequencies significantly below 100 cm-1 do have positive Grüneisen parameter values at some $\mathbf{q}$. For the longitudal phonon mode of d-Si, the Grüneisen parameter values are positive for all wave vectors. On the other hand, the two transverse acoustic phonon modes have positive values of Grüneisen parameter only at few wave vectors. The Grüneisen parameters of the clathrate frameworks show features similar to amorphous silicon,Fabian and Allen (1997) where the Grüneisen parameter values are spread out at low frequency values. This is in contrast to d-Si, where the Grüneisen parameters show a rather symmetric shape as a function of frequency in comparison to the studied clathrate frameworks and amorphous silicon. The results in Figures 4 and 6 suggest that only the structure VII should show significantly different CTE in comparison to the other studied structures. This is confirmed by the linear and volumetric CTE values shown in Fig. 7 for all studied structures. Figure 7: Top: Linear CTE for the studied cubic structures. The experimental values for d-Si are indicated with circles.Reeber and Wang (1996) Bottom: Volumetric CTE for the studied hexagonal structures, d-Si is included as a reference. For d-Si, the experimental values for the linear CTE are also shown.Reeber and Wang (1996) The CTE values obtained from Eq. 7 and Eq. 11 are identical. In comparison to experiment, LDA overestimates the magnitude of the NTE for d-Si and underestimates the CTE at positive values. In all studied structures, excluding the structure VII, the CTE values are very similar. The structure VII with distinctly different Grüneisen parameters shows NTE behavior up to approximately 300 K and the NTE is nearly five times larger than for d-Si. As discussed above, the structure VII has a very high relative energy and it is not experimentally as relevant as the other structures. However, it is still an interesting test case for detailed studies on the NTE phenomenon due to the small size of its primitive unit cell and its anomalous lattice dynamical properties. Fig. 8 illustrates the change in entropy $\Delta S$ with respect to the change in the volume $\Delta V$ for each phonon mode as a function of temperature for d-Si, and the clathrate frameworks II and VIII. Figure 8: $\Delta S/\Delta V$ for each phonon mode in the d-Si, II, and VIII structures as a function of the temperature. The zero level is at the point where the lines representing the phonon modes coincide at $T=0$ K. The acoustic modes are indicated with dash-dotted lines. In d-Si, the longitudinal acoustic mode has a positive $\Delta S/\Delta V$ at all temperatures, having a positive contribution to CTE (Eq. 11). The transverse acoustic modes have a negative contribution to CTE, giving rise to the experimentally confirmed NTE behavior of d-Si. In the structures II and VIII, according to QHA, all acoustic modes have a negative contribution to CTE at all considered temperatures. The reason for this was discussed at the end of Sec. II.2. An interesting feature of the structures II and VIII is that several optical modes (all optical modes in the structure II) have larger contributions to NTE than the acoustic modes when $T>$100 K. Similar crossing occurs between optical phonon modes having a positive contribution to CTE, as shown in Fig. 8. The difference is that the crossing occurs at $T\approx 200$K. For the studied structures, the maximum value of $\Delta S/\Delta V$, when the temperature changes by $\pm 1$K, is six orders of magnitude smaller than the absolute value of $S$ at the same temperature. ### III.3 Thermal expansion and displacement correlation functions Fig. 9 shows the displacement autocorrelation functions (Eq. 12) for all atoms within the primitive unit cells of d-Si and clathrate II. Figure 9: Displacement autocorrelation functions for d-Si and the clathrate framework II (Eq. 12). Each line represents the displacement autocorrelation function for one atom in the primitive unit cell. The results for d-Si are in good agreement with previous results of Nielsen and Weber obtained using empirical potentials,Nielsen and Weber (1980) and with the results of Strauch et al. obtained using ab initio methods.Strauch _et al._ (1996) The differences between d-Si and the clathrate framework II are small. In the case of the structure II, the values are separated in three different groups corresponding to the three different Wyckoff positions in the crystal structure. The primitive unit cell of the structure II is shown in Fig. 10 to facilitate the discussion on the results related to the displacement correlation functions. Figure 10: The primitive unit cell of the clathrate framework II shown in a) [100] b) [111] and c) general direction. Unit cell edges are drawn in black. The highest MSD values correspond to atoms 1 and 2 in Fig. 10 (Wyckoff position 8a), the intermediate values to atoms 3-10 (32e), and the smallest values to atoms 11-34 (96g). The MSD values calculated for the symmetry equivalent atoms in the structure II show a minor deviation from each other. The highest deviations correspond to atoms 11-34, when the max. difference within this group is approximately 1.8%, which occurs at $T=500$ K. One possible reason for this is the issue of phonon mode labeling for degenerate modes during the diagonalization of the dynamical matrix, as discussed in the end of Sec. II.3. The total MSD (Eq. 18) calculated for the structure VII is approximately 25% higher than the total MSDs calculated for d-Si and the structure II. The parallel and perpendicular displacement correlation functions between the nearest neighbours in d-Si ($l\kappa=l1;l\kappa^{\prime}=l2$) are presented in Fig. 11. Figure 11: Parallel and perpendicular correlation functions for d-Si within same unit cell, i.e. for the nearest neigbours. Figure 12: Parallel and perpendicular MSRD within the same unit cell for the structure II. a) Normalized parallel MSRD. b) Parallel MSRD and c) perpendicular MSRD. Different lines represent different atoms $\kappa$, the second atom $\kappa^{\prime}=2$ in all cases (see Fig. 10). The perpendicular MSRDs have larger values than the parallel MSRDs. Similar behavior has been confirmed with experimental EXAFS studies and computational results for d-Ge,Dalba _et al._ (1999),Nielsen and Weber (1980) and also for other materials such as CdTe with zincblende structure using EXAFS.el All _et al._ (2012) In cubic crystal structures, the perpendicular MSRD values must be larger than the parallel MSRD values due to symmetry reasons. If a three- dimensional Cartesian coordinate system is considered to have one of its orthonormal basis vectors along the vector between the nearest neighbour atoms in d-Si, the difference in the magnitudes of perpendicular and parallel MSRD becomes more evident, as the perpendicular vibrations can be represented as linear combinations of the two other basis vectors. Because the MSDs of the atoms have the same values for each cartesian component, it is expected that the relative displacements are distributed among the perpendicular and parallel MSRD as shown in Fig. 11. Fig. 12 shows the normalized parallel and perpendicular MSRD values for the clathrate framework II. All results presented in Fig. 12 are for $\kappa^{\prime}=2$ and $\kappa$ runs over all atoms within the same unit cell. The normalized parallel correlation functions have again similar values as in the previous works for d-Si obtained using empirical potentials.Nielsen and Weber (1980) Both the parallel and perpendicular MSRD have similar characteristics, that is, the nearest atoms have the smallest MSRD values and vice versa. The atoms $\kappa=7-10$ have the smallest values and the atoms $\kappa=1,3$ the largest values in all cases in Fig. 12. The atoms $\kappa=1,3$ are on the opposite side of the cavity with respect to atom $\kappa^{\prime}=2$ (see Fig. 10), which may partly explain the higher values of the MSRD in comparison to the other atoms. The perpendicular MSRD values do not form as distinct groups for the three symmetry equivalent atoms as in the case of the parallel MSRD. Fig. 13 shows the MSD-VD and MSRD-VD values for d-Si. The values in Fig. 13 for the different modes and for the total volume derivatives have similar characteristics as CTE calculated from Eq. 11. Figure 13: MSD-VD- and MSRD-VD values for d-Si calculated from Eq. 25, Eq. 27, and Eq. 28. The MSRD-VD values are for the nearest neighbours. The acoustic modes are indicated with a dash-dotted line. The reason for this similarity was discussed at the end of Sec. II.2. The MSD- VD values for the phonon modes corresponding to NTE are negative within the whole temperature range similar to the entropies shown in Fig. 8. Like in the case of the entropy, the MSD-VD values are negative for two acoustic phonon modes, while one acoustic mode and the optical modes have a positive contribution to CTE. The similarity of the MSRD-VD and CTE results could be due to the fact that in d-Si only the relative motion of the two atoms within the primitive unit cell atoms determines the CTE. According to our results, the parallel and perpendicular MSRD-VD have nearly identical values. The MSD-VD values for the clathrate framework II shown in Fig. 14 correspond relatively well to the values for d-Si. Figure 14: MSD-VD for three different atoms in the structure II calculated from Eq. 25. Acoustic modes are indicated with a dash-dotted line. Only three atoms representing the different symmetry independent positions in the crystal structure are shown. Comparison of Fig. 8 and Fig. 14 implies that the MSD-VD and $\Delta S/\Delta V$ behave similarly as a function of temperature also for the structure II. The MSD-VD results also agree with the results obtained for CTE in the sense that the structure II has slightly lower MSD-VD values in both positive CTE and NTE temperature ranges when comparing it to d-Si. The proper treatment of MSRD-VD values in larger structures requires symmetry considerations. ## IV Conclusions We have investigated the thermal properties of silicon clathrate frameworks by combining ab initio DFT lattice dynamics with QHA. The computational results for d-Si were in relatively good agreement with the experimental values for the phonon dispersion relations, heat capacities, and CTE. Studied clathrate frameworks have similar phonon spectra, only the structure VII differing from the others significantly. All studied clathrate frameworks were found to possess a phonon band gap and similar threshold frequencies confining the phonons that can have negative Grüneisen parameter values. A similar threshold frequency was also found for d-Si. Furthermore, the clathrate framework V has two phonon band gaps. The NTE temperature range of the studied clathrate frameworks is slightly wider than for d-Si. The MSD and MSRD values calculated for d-Si were in agreement with the previous computational results. The results obtained from the displacement correlation functions for the clathrate framework II were similar to d-Si. Equal time displacement correlation functions were used to study the NTE phenomenon. The MSD-VD and MSRD-VD quantities defined here behave almost identically to the vibrational entropy volume derivatives in the studied structures. The parallel and perpendicular MSRD-VD results for d-Si indicate that both quantities behave in similar fashion as a function of temperature. Further application of the volume derivatives of the displacement correlation functions may produce useful information on the mechanisms of the NTE phenomenon. ###### Acknowledgements. We thank Prof. Andrea Dal Corso (SISSA, Trieste) for providing modified QE routines to enable efficient distributed calculation of the phonon dispersion relations. 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Purans, Phys. Rev. Lett. 82, 4240 (1999).	arxiv-papers	2014-01-06T17:43:00	2024-09-04T02:49:56.343189	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Ville J. H\\\"ark\\\"onen and Antti J. Karttunen", "submitter": "Ville H\\\"ark\\\"onen Mr.", "url": "https://arxiv.org/abs/1401.1153" }
1401.1329	# Ends, fundamental tones, and capacities of minimal submanifolds via extrinsic comparison theory Vicent Gimeno Departament de Matemàtiques- INIT-IMAC, Universitat Jaume I, Castelló, Spain. [email protected] and S. Markvorsen DTU Compute, Mathematics, Technical University of Denmark. [email protected] ###### Abstract. We study the volume of extrinsic balls and the capacity of extrinsic annuli in minimal submanifolds which are properly immersed with controlled radial sectional curvatures into an ambient manifold with a pole. The key results are concerned with the comparison of those volumes and capacities with the corresponding entities in a rotationally symmetric model manifold. Using the asymptotic behavior of the volumes and capacities we then obtain upper bounds for the number of ends as well as estimates for the fundamental tone of the submanifolds in question. ###### Key words and phrases: First Dirichlet eigenvalue, Capacity, Effective resistance, Minimal submanifolds ###### 2010 Mathematics Subject Classification: Primary 53A, 53C Work partially supported by DGI grant MTM2010-21206-C02-02. ## 1\. Introduction Let $M$ be a complete non-compact Riemannian manifold. Let $K\subset M$ be a compact set with non-empty interior and smooth boundary. We denote by $\mathcal{E}_{K}(M)$ the number of connected components $E_{1},\cdots,E_{\mathcal{E}_{K}(M)}$ of $M\setminus K$ with non-compact closure. Then $M$ has $\mathcal{E}_{K}(M)$ ends $\\{E_{i}\\}_{i=1}^{\mathcal{E}_{K}(M)}$ with respect to $K$ (see e.g. [GSC09]), and the _global_ number of ends $\mathcal{E}(M)$ is given by (1.1) $\mathcal{E}(M)=\sup_{K\subset M}\mathcal{E}_{K}(M)\quad,$ where $K$ ranges on the compact sets of $M$ with non-empty interior and smooth boundary. The number of ends of a manifold can be bounded by geometric restrictions. For example, in the particular setting of an $m-$dimensional minimal submanifold $P$ which is properly immersed into Euclidean space $\mathbb{R}^{n}$, the number of ends $\mathcal{E}(P)$ is known to be related to the extrinsic properties of the immersion. Indeed, V. G. Tkachev proved in [Tka94, Theorem 2] (see also [Che95]) that for any properly immersed $m-$dimensional minimal submanifold $P$ in $\mathbb{R}^{n}$ with finite volume growth $V_{w_{0}}(P)<\infty$ the number of ends is bounded from above by (1.2) $\mathcal{E}(P)\leq C_{m}V_{w_{0}}(P)\quad,$ where $C_{m}=1$ ($C_{m}={2^{m}}$ in the original [Tka94]) and the volume growth $V_{w_{0}}(P)$ is (1.3) $V_{w_{0}}(P)=\lim_{R\to\infty}\frac{\operatorname{Vol}\left(P\cap B_{R}^{\mathbb{R}^{n}}\left(o\right)\right)}{\operatorname{Vol}\left(B_{R}^{\mathbb{R}^{m}}\left(o\right)\right)}\quad.$ Here $\operatorname{Vol}\left(B_{R}^{\mathbb{R}^{m}}\left(o\right)\right)$ is the volume of a geodesic ball $B_{R}^{\mathbb{R}^{m}}\left(o\right)$ of radius $R$ centered at $o$ in $\mathbb{R}^{m}$. The inequality (1.2) thus shows a significant relation between the number of ends (i.e. a topological property) and the behavior of a quotient of volumes (i.e. a metric property). Motivated by Tkachev’s application of the _volume quotient_ appearing in equation (1.3), we will consider the corresponding _flux quotient_ and _capacity quotient_ of the minimal submanifolds. These quotients are constructed in the same way as indicated by the _volume quotient_ but here we generalize the setting as well as Tkachev’s result to minimal submanifolds in more general ambient spaces as alluded to in the abstract. Specifically we assume that the minimal immersion goes into an ambient manifold $N$ with a pole and with sectional curvatures $K_{N}$ bounded from above by the radial curvatures $K_{w}$ of a rotationally symmetric model space $M_{w}^{n}=\mathbb{R}^{+}\times\mathbb{S}_{1}^{n-1}$, with warped metric tensor $g_{M_{w}^{n}}$ constructed using a positive warping function $w:\mathbb{R}^{+}\to\mathbb{R}^{+}$ in such a way that $g_{M_{w}^{n}}=dr^{2}+w(r)^{2}g_{\mathbb{S}_{1}^{n-1}}$ is also balanced from below (see [MP06] and §3 for a precise definitions). Our generalization of inequality (1.2) is thence the following: ###### Theorem 1.1. Let $\varphi:P^{m}\to N^{n}$ be a proper minimal and complete immersion into a $n-$dimensional ambient manifold $N^{n}$ which possesses a pole $o\in N^{n}$ and its sectional curvatures $K_{N}$ at any point $p\in N$ are bounded by above by the radial curvatures $K_{w}$ of a balanced from below model space $M_{w}^{n}$ (1.4) $K_{N}\left(p\right)\leq K_{M_{w}^{n}}\left(r\left(p\right)\right)=-\frac{w^{\prime\prime}}{w}\left(r\left(p\right)\right)\quad.$ Suppose moreover, that $w^{\prime}>0$ and there exist $R_{0}$ such that $K_{M_{w}^{n}}(R)\leq 0$ for any $R>R_{0}$. Then, the number of ends $\mathcal{E}_{D_{R}}(P)$ with respect to the extrinsic ball $D_{R}=P\cap B_{R}^{N}(o)$ for $R>R_{0}$ is bounded from above by (1.5) $\mathcal{E}_{D_{R}}(P)\leq\left(\frac{2}{1-\frac{R}{t}}\right)^{m}\left(\frac{\int_{0}^{t}w(s)^{m-1}ds}{t^{m}/m}\right)\frac{\operatorname{Vol}(D_{t})}{\operatorname{Vol}(B^{w}_{t})}\quad,$ for any $t>R$. Using the above theorem we can estimate the global number of ends as follows: ###### Corollary 1.2. Under the assumptions of theorem 1.1, suppose moreover (1.6) $\limsup_{t\to\infty}\left(\frac{\int_{0}^{t}w(s)^{m-1}ds}{t^{m}/m}\right)=C_{w}<\infty\quad,$ and suppose also that the submanifold has finite volume growth, namely (1.7) $\operatorname{Vol}_{w}(P)=\lim_{t\to\infty}\frac{\operatorname{Vol}(D_{t})}{\operatorname{Vol}(B^{w}_{t})}<\infty\quad.$ Then (1.8) $\mathcal{E}(P)\leq 2^{m}C_{w}\operatorname{Vol}_{w}(P)\quad.$ ###### Remark a. If we choose $w(t)=w_{0}(t)=t$, the model space becomes $\mathbb{R}^{m}$, which is balanced from below, and the hypothesis of theorem 1.1 are therefore automatically fulfilled for any complete minimal submanifold properly immersed in a Cartan-Hadamard ambient manifold. Inequality (1.5) becomes (1.9) $\mathcal{E}_{D_{R}}(P)\leq\left(\frac{2}{1-\frac{R}{t}}\right)^{m}\frac{\operatorname{Vol}(D_{t})}{V_{m}t^{m}}\quad,$ For any $R>0$ and any $t>R$, being $V_{m}$ the volume of a geodesic ball of radius $1$ in $\mathbb{R}^{m}$. From inequality (1.6) we get (1.10) $C_{w_{0}}=1\quad.$ Thus inequality (1.8) becomes (1.11) $\mathcal{E}(P)\leq 2^{m}\lim_{t\to\infty}\frac{\operatorname{Vol}(D_{t})}{V_{m}t^{m}}\quad,$ which is the original inequality obtained by Tkachev (inequality (1.2)), but now inequality (1.11) is valid for any minimal submanifold properly immersed in a Cartan-Hadamard ambient manifold with finite volume growth. In [GP12, Che95] are also obtained lower bounds for the number of ends, but we note that those lower bounds seem to need stronger assumptions: Dimension greater than $2$, or embeddedness of the ends and codimension $1$, decay on the second fundamental form, and a rotationally symmetric ambient manifold. As a counterpart, those lower bounds are associated to the so-called gap type theorems. Combining the results of [GP12, Theorem 3.5] and corollary 1.2, and taking into account the role of sectional curvatures of the model space (see [GP12, Proposition 2.6]) we have ###### Corollary 1.3. Let $\varphi:P^{m}\to M^{n}_{w}$ be a minimal and proper immersion into a balanced from below model space $M^{n}_{w}$ with increasing warping function $w$ satisfying the following conditions: (1.12) $\displaystyle\limsup_{t\to\infty}\left(\frac{\int_{0}^{t}w(s)^{m-1}ds}{t^{m}/m}\right)=C_{w}<\infty\quad,$ $\displaystyle\text{ there exist }R_{0}\text{ such that for any }R>R_{0}$ $\displaystyle\begin{cases}-\frac{w^{\prime\prime}(R)}{w(R)}\leq 0\\\ \frac{1-\left(w^{\prime}(R)\right)^{2}}{w(R)^{2}}\leq 0\end{cases}$ Suppose moreover that $m>2$, that the center $o_{w}$ of $M_{w}^{n}$ satisfies $\varphi^{-1}(o_{w})\neq\emptyset$ and that the norm of the second fundamental form $\\|B^{P}\\|$ of the immersion is bounded for large $r$ by (1.13) $\\|B^{P}\\|\leq\frac{\epsilon(r)}{w^{\prime}(r)w(r)}\quad,$ where $\epsilon$ is a positive function such that $\epsilon(r)\to 0$ when $r\to\infty$. Then the number of ends is bounded from below and from above by (1.14) $\operatorname{Vol}_{w}(P)\leq\mathcal{E}(P)\leq 2^{m}C_{w}\operatorname{Vol}_{w}(P)\quad.$ Figure 1. Two examples of extrinsic annuli in $\mathbb{R}^{3}$: A catenoid on the left and the singly periodic Scherk surface on the right. The extrinsic annuli are constructed by cutting the surfaces with two spheres (with the same center but of different radii) in the ambient manifold ($\mathbb{R}^{3}$). The catenoid has two ends and finite total curvature. Hence, by theorem 1.4, the capacity of the extrinsic annulus of the catenoid is greater than the capacity of the corresponding annulus of the Euclidean $2$-plane but is smaller than two times that capacity. The same is true for the extrinsic annulus of the singly periodic Scherk surface (we refer the reader to the introduction of [MW07] for the area growth of the singly periodic Scherk surface). By using our results about the behavior of the comparison quotients we can also estimate the capacity of an extrinsic annulus $\text{A}_{\rho,R}=P\cap\left(B_{R}^{N}(o)\setminus B_{\rho}^{N}(o)\right)$ (see figure 1 and, §2 and §3 for a precise definition of _capacity_ and _extrinsic annulus_): ###### Theorem 1.4. Let $\varphi:P^{m}\to\mathbb{R}^{n}$ denote a complete and proper minimal immersion into the Euclidean space $\mathbb{R}^{n}$. Then, for any $R>\rho>0$, the capacity of the extrinsic annulus $A_{\rho,R}$ is bounded by (1.15) $\frac{\operatorname{Vol}(D_{\rho})}{V_{m}\rho^{m}}\leq\frac{\operatorname{Cap}(A_{\rho,R})}{\operatorname{Cap}(A^{\mathbb{R}^{m}}_{\rho,R})}\leq\frac{\operatorname{Vol}(D_{R})}{V_{m}R^{m}}\quad,$ where $\operatorname{Cap}(A^{\mathbb{R}^{m}}_{\rho,R})$ is the capacity of the geodesic annulus $A^{\mathbb{R}^{m}}_{\rho,R}$ in $\mathbb{R}^{m}$ of inner radius $\rho$ and outer radius $R$. ###### Remark b. Since, from Theorem 2.1, the quotient $\frac{\operatorname{Vol}(D_{s})}{V_{m}s^{m}}$ is a non-decreasing function on $s$, we can state Theorem 1.4 in the limit case ( $\rho\to 0$ and $R\to\infty$) and inequality (1.15) there becomes (1.16) $1\leq\frac{\operatorname{Cap}(A_{\rho,R})}{\operatorname{Cap}(A^{\mathbb{R}^{m}}_{\rho,R})}\leq\lim_{R\to\infty}\frac{\operatorname{Vol}(D_{R})}{V_{m}R^{m}}=V_{w_{0}}(P)\quad.$ When we deal with a minimal surface $\Sigma\subset\mathbb{R}^{3}$ which is properly _embedded_ into the Euclidean space $\mathbb{R}^{3}$ the limit (1.17) $V_{w_{0}}(\Sigma)=\lim_{R\to\infty}\frac{\operatorname{Vol}(\Sigma\cap B_{R}^{\mathbb{R}^{3}}(o))}{\pi R^{2}}$ is well understood. For instance, the above limit corresponds to the number of ends if the surface $\Sigma$ has finite total curvature. ###### Remark c. In order to bound the capacity quotient, our theorems do not make use of the volume quotient as in Theorem 1.4, but instead they make use of the flux quotient (see Theorems 2.2 and 2.3). In the special case when the ambient manifold is $\mathbb{R}^{n}$ (such as in Theorem 1.4) the volume quotient agrees however with the flux quotient (see equation (6.28) and theorem 6.1). ### 1.1. Outline of the paper In §2 we show our main theorems concerning the flux quotients, the volume quotients, and the capacity quotients. In §3 we state the preliminary concepts in order to prove the main theorems of §2 in §4. This allows us then to prove Theorem 1.1 and Corollary 1.2 in §5. Finally, in §6, we present several corollaries and examples of applications of the extrinsic theory and results which have been established in §2. ## 2\. Extrinsic theory: Flux, Capacity and Volume comparison for extrinsic balls Let $(M^{n},g)$ be a Riemannian manifold. For any oriented hypersurface $\Sigma\subset M$ with unit normal vector field $\nu$, we define the flux $F_{X}(\Sigma)$ of the vector field $X$ through $\Sigma$ by (2.1) $F_{X}(\Sigma):=\int_{\Sigma}\langle X,\nu\rangle d\mu_{\Sigma}\quad,$ where $d\mu_{\Sigma}$ is the associated Riemannian density determined by the metric $g_{\Sigma}=i^{}g$ (being $i:\Sigma\to M$ the inclusion map). By the divergence theorem (see [Cha93] for instance), if one has an oriented domain $\Omega$ in $M$ with smooth boundary $\partial\Omega$, and the vector field $X$ is $C^{1}$ in $\overline{\Omega}$ and with compact support in $\overline{\Omega}$, the flux of $X$ through $\partial\Omega$ is related to the divergence of $X$ by (2.2) $\int_{\Omega}\text{div}Xd\mu=\int_{\partial\Omega}\langle X,\nu\rangle d\mu_{\partial\Omega}=F_{X}(\partial\Omega)\quad.$ Given a smooth function $u:M\to\mathbb{R}$, we can also define the flux of a function $u$, but then the flux $J_{u}(t)$ is the flux of the gradient $\nabla u$ (i.e. the metric dual vector to $du$, $du(X)=\langle\nabla u,X\rangle$) through the level set $\Sigma^{u}_{t}:=\\{x\in M\,\|\,u(x)=t\\}$ so that: (2.3) $J_{u}(t):=F_{\nabla u}(\Sigma_{t}^{u})\quad.$ Taking into account that the unit normal vector field $\nu$ of $\Sigma^{u}_{t}$ is $\nu=\frac{\nabla u}{\|\nabla u\|}$, it is easy to see that (2.4) $J_{u}(t)=\int_{\Sigma_{t}}\|\nabla u\|d\mu_{\Sigma_{t}^{u}}\quad.$ Observe moreover, that by the Sard theorem and by the regular set theorem we need no further restrictions on the smoothness of $\Sigma_{t}^{u}$ and on the smoothness of the unit normal vector field $\nu$. The overall goal of this work is to characterize the _isoperimetric inequalities for extrinsic balls_ , and the _capacity of minimal submanifolds_ in terms of the flux of extrinsic distance functions. Actually we are interested on the flux of the extrinsic distance function on minimal submanifolds in an ambient manifold $N$ which possesses a pole and has the radial curvatures bounded form above by the radial curvatures of rotationally symmetric model space $K_{N}\leq K_{M_{w}^{n}}=-\frac{w^{\prime\prime}}{w}$, see [MP06] or section 3 of this paper for precise definitions. It is the behavior of this particular flux that allows us to study the mean exit time function, the capacity, the conformal type, the fundamental tone, and in special cases also the number of ends of the submanifold. ### 2.1. Flux and volume comparison: isoperimetric inequalities and the mean exit time function Given an isometric immersion $\varphi:P\to(N,o)$ into a manifold with a pole $o\in N$, the flux $J_{r}$ of the extrinsic distance function $r_{o}$ (i.e., the restriction by the immersion of the ambient distance function to the submanifold) is given by $J_{r}(R)=\int_{\partial D_{R}}\|\nabla^{P}r_{o}\|d\mu\quad,$ where $\partial D_{R}$ is the level set $\partial D_{R}=r_{o}^{-1}(R)$, and therefore, $D_{R}=r_{o}^{-1}([0,R))$ is the extrinsic ball of radius $R$. When the immersion is minimal and the ambient manifold has its radial sectional curvatures $K_{N}$ bounded from above by the radial sectional curvatures of a rotationally symmetric model space $M_{w}^{n}$ that is balanced from below (see [MP06] and section 3), $K_{N}\leq K_{M_{w}^{n}}$, we can compare the volume quotient $\frac{\operatorname{Vol}(D_{R})}{\operatorname{Vol}(B_{R}^{w})}$ and the flux quotient $\frac{J_{r}(R)}{J_{r}^{w}(R)}$ . The volume quotient is the quotient between the volume of a extrinsic ball $D_{R}$ of radius $R$ in $P^{m}$ and the volume of a geodesic ball $B_{R}^{w}$ of the same radius $R$ in $M_{w}^{m}$. The flux quotient is the quotient between the flux of the extrinsic distance in $P^{m}$ and the flux of the geodesic distance in $M_{w}^{m}$. These two quotients are related by the following theorem ###### Theorem 2.1. Let $\varphi:P^{m}\longrightarrow N^{n}$ be an isometric, proper, and minimal immersion of a complete non-compact Riemannian $m$-manifold $P^{m}$ into a complete Riemannian manifold $N^{n}$ with a pole $o\in N$ . Let us suppose that the $o-$radial sectional curvatures of $N$ are bounded from above by $K_{o,N}(\sigma_{x})\leq-\frac{w^{\prime\prime}(r)}{w(r)}(\varphi(x))\,\,\,\forall x\in P\,,$ and the model space $M_{w}^{m}$ is balanced from bellow. Then 1. (1) $J_{r}(R)$ is related with $\operatorname{Vol}(D_{R})$ by (2.5) $\frac{\operatorname{Vol}(D_{R})}{\operatorname{Vol}(B_{R}^{w})}\leq\frac{J_{r}(R)}{J_{r}^{w}(R)}.$ 2. (2) The functions $\frac{\operatorname{Vol}(D_{R})}{\operatorname{Vol}(B_{R}^{w})}$ and $\frac{J_{r}(R)}{J_{r}^{w}(R)}$ are non decreasing functions on $R$. 3. (3) Denoting by $E_{R}^{P}(x)$ the mean time for the first exit from the extrinsic ball $D_{R}(o)$ for a Brownnian particle starting at $o\in P^{m}$, and denoting by $E_{R}^{w}$ the mean exit time function for the $R-$ball $B_{R}^{w}$ in the model space $M_{w}^{m}$, if equality holds in (2.5) for some fixed radius $R>0$, then for any $x\in D_{R}$, $E_{R}^{P}(x)=E_{R}^{w}(r(x))$, where $r(x)$ the extrinsic distance from $o$ to the point $x\in P$. ### 2.2. Capacity and flux comparison: conformal type Given a compact set $F\subset M$ in a Riemannian manifold $M$ and an open set $G\subset M$ containing $F$, we call the couple $(F,G)$ a _capacitor_. Each capacitor then has its capacity defined by (2.6) $\operatorname{Cap}(F,G):=\inf_{u}\int_{G\setminus F}\\|\nabla u\\|d\mu\quad,$ where the $\inf$ is taken over all Lipschitz functions $u$ with compact support in $G$ such that $u=1$ on $F$. When $G$ is precompact, the infimum is attained for the function $u=\Psi$ which is the solution of the following Dirichlet problem in $G-F$: (2.7) $\begin{cases}\Delta\Psi=0\\\ \Psi\|_{\partial F}=0\\\ \Psi\|_{\partial G}=1\end{cases}$ From a physical point of view, the capacity of the capacitor $(F,G)$ represents the total electric charge (generated by the electrostatic potential $\Psi$) flowing into the domain $G-F$ through the interior boundary $\partial F$. Since the total current stems from a potential difference of $1$ between $\partial F$ and $\partial G$, we get from Ohm’s Law that the effective resistance of the domain $G-F$ is (2.8) $R_{\text{eff}}(G-F)=\frac{1}{\operatorname{Cap}(F,G)}\quad.$ The exact value of the capacity of a set is known only in a few cases, and so its estimation in geometrical terms is of great interest, not only in electrostatic, but in many physical descriptions of flows, fluids, heat, or generally where the Laplace operator plays a key role, see [CFG05, HPR12]. Given a capacitor $(F,G)$, if we have a smooth function $u$ with $u=a$ on $\partial F$ and $u=b$ on $\partial G$. The capacity and the flux are then related by (see [Gri99b]): (2.9) $\operatorname{Cap}(F,G)\leq\left(\int_{a}^{b}\frac{ds}{J_{u}(s)}\right)^{-1}\quad.$ In this paper we are interested on the $o$-centered _extrinsic annulus_ $A_{\rho,R}(o)\subset P^{m}$ for $0<\rho<R$ given by (2.10) $A_{\rho,R}(o):=D_{R}(o)-D_{\rho}(o)\quad.$ To be more precise, we are interested on the behavior of the flux and the capacity of those extrinsic domains. In the following theorems we provide upper and lower bounds for the capacity quotient in terms of the flux quotient. ###### Theorem 2.2. Let $\varphi:P^{m}\longrightarrow N^{n}$ be an isometric, proper, and minimal immersion of a complete non-compact Riemannian $m$-manifold $P^{m}$ into a complete Riemannian manifold $N^{n}$ with a pole $o\in N$ and satisfying $\varphi^{-1}(o)\neq\emptyset$. Let us suppose that the $o-$radial sectional curvatures of $N$ are bounded from above by $K_{o,N}(\sigma_{x})\leq-\frac{w^{\prime\prime}(r)}{w(r)}(\varphi(x))\,\,\,\forall x\in P\quad,$ and the warping function $w$ satisfies $w^{\prime}\geq 0\quad.$ Then (2.11) $\frac{J_{r}(\rho)}{J_{r}^{w}(\rho)}\leq\frac{\operatorname{Cap}(A_{\rho,R})}{\operatorname{Cap}(A_{\rho,R}^{w})}\quad,$ where $A^{w}_{\rho,R}$ is the intrinsic annulus in $M_{w}^{m}$. ###### Theorem 2.3. Let $\varphi:P^{m}\longrightarrow N^{n}$ be an isometric, proper, and minimal immersion of a complete non-compact Riemannian $m$-manifold $P^{m}$ into a complete Riemannian manifold $N^{n}$ with a pole $o\in N$ . Let us suppose that the $o-$radial sectional curvatures of $N$ are bounded from above by $K_{o,N}(\sigma_{x})\leq-\frac{w^{\prime\prime}(r)}{w(r)}(\varphi(x))\,\,\,\forall x\in P\,,$ and the model space $M_{w}^{m}$ is balanced from bellow. Then (2.12) $\frac{\operatorname{Cap}(A_{\rho,R})}{\operatorname{Cap}(A_{\rho,R}^{w})}\leq\frac{J_{r}(R)}{J_{r}^{w}(R)}\quad,$ where $A^{w}_{\rho,R}$ is the intrinsic annulus in $M_{w}^{m}$. Moreover, if equality holds in (2.12) for some fixed $R>0$, then $D_{R}$ is a minimal cone in $N^{n}$. Geometric estimates of the capacity are sufficient to obtain large scale consequences such as as the parabolic or hyperbolic character of the manifold, [Ich82b, Ich82a, MP03, MP05]. We note here the following important equivalent conditions about the conformal type: ###### Theorem A. Let (M,g) be a given Riemannian manifold, Then the following conditions are equivalent • There is a precompact open domain $K$ in $M$, such that the Brownian motion $X_{t}$ starting from $K$ does not return to $K$ with probability $1$, i.e. (2.13) $P_{x}\left\\{\omega\|X_{t}(\omega)\in K\text{ for some }t>0\right\\}<1$ * • $M$ has positive capacity: There exist in $M$ a compact domain $K$, such that (2.14) $\operatorname{Cap}(K,M)>0$ * • $M$ has finite resistance to infinity: There exist in $M$ a compact domain $K$, such that (2.15) $R_{\text{eff}}(M-K)<\infty$ A manifold satisfying the conditions of the above theorem will be called a hyperbolic manifold, otherwise it is called a parabolic manifold. As a consequence of the above theorem we can state the following corollary for minimal submanifolds: ###### Corollary 2.4. Let $\varphi:P^{m}\longrightarrow N^{n}$ be an isometric, proper, and minimal immersion of a complete non-compact Riemannian $m$-manifold $P^{m}$ into a complete Riemannian manifold $N^{n}$ with a pole $o\in N$ . Let us suppose that the $o-$radial sectional curvatures of $N$ are bounded from above by $K_{o,N}(\sigma_{x})\leq-\frac{w^{\prime\prime}(r)}{w(r)}(\varphi(x))\,\,\,\forall x\in P\quad,$ and the warping function $w$ satisfies $w^{\prime}\geq 0\quad.$ Then 1. (1) If $M_{w}^{m}$ is a hyperbolic manifold, then $P$ is a hyperbolic manifold. 2. (2) In consequence, if $P$ is parabolic, then $M_{w}^{m}$ is also parabolic. Since $\frac{J_{r}(R)}{J_{r}^{w}(R)}$ and $\frac{\operatorname{Vol}(D_{R})}{\operatorname{Vol}(B_{R}^{w})}$ are non- decreasing functions under our hypothesis, we can define two expressions which are analogous to the projective volume defined by V. G. Tkachev in [Tka94] ###### Definition 2.5. Given $\varphi:P^{m}\to N^{n}$ an immersion into a manifold $N$ with a pole $o\in N$. The $w$-_flux_ $\text{Flux}_{w}(P)$ and the $w$-_volume_ $\operatorname{Vol}_{w}(P)$ of the submanifold $P$ are defined by : (2.16) $\displaystyle\text{Flux}_{w}(P)$ $\displaystyle:=\sup_{R\in\mathbb{R}^{+}}\frac{J_{r}(R)}{J_{r}^{w}(R)}\quad,$ $\displaystyle\operatorname{Vol}_{w}(P)$ $\displaystyle:=\sup_{R\in\mathbb{R}^{+}}\frac{\operatorname{Vol}(D_{R})}{\operatorname{Vol}(B^{w}_{R})}\quad.$ We will say that $P$ has _finite_ $w-$ _flux_ ( resp. _finite_ $w-$ _volume_) if and only if $\text{Flux}_{w}(P)<\infty$ ( or $\operatorname{Vol}_{w}(P)<\infty$). We refer to theorem 6.1 for the relation between the $w-$flux and the $w-$volume of a submanifold. From theorem A and theorem 2.3 we can now state that for minimal submanifolds with finite $w-$flux we have: ###### Corollary 2.6. Let $\varphi:P^{m}\longrightarrow N^{n}$ be an isometric, proper and minimal immersion of a complete non-compact Riemannian $m$-manifold $P^{m}$ into a complete Riemannian manifold $N^{n}$ with a pole $o\in N^{n}$. Let us suppose that the $o-$radial sectional curvatures of $N^{n}$ are bounded from above as follows $K_{o,N}(\sigma_{x})\leq-\frac{w^{\prime\prime}(r)}{w(r)}(\varphi(x))\,\,\,\forall x\in P\quad,$ and that the model space $M_{w}^{m}$ is balanced from below. Suppose moreover that $P$ has finite $w-$flux. Then 1. (1) If $M_{w}^{m}$ is a parabolic manifold, then $P$ is a parabolic manifold. 2. (2) If $P$ is an hyperbolic manifold, then $M_{w}^{n}$ is an hyperbolic manifold. Joining the previous two corollaries together we get: ###### Corollary 2.7. Let $\varphi:P^{m}\longrightarrow N^{n}$ be an isometric, proper and minimal immersion of a complete non-compact Riemannian $m$-manifold $P^{m}$ into a complete Riemannian manifold $N^{n}$ with a pole $o\in N^{n}$ . Let us suppose that the $o-$radial sectional curvatures of $N^{n}$ are bounded from above, $K_{o,N}(\sigma_{x})\leq-\frac{w^{\prime\prime}(r)}{w(r)}(\varphi(x))\,\,\,\forall x\in P\quad,$ that the warping function $w$ satisfies $w^{\prime}\geq 0\quad,$ and that the model space $M_{w}^{m}$ is balanced from below, and that $P$ has finite $w-$flux. Then $P$ is hyperbolic (parabolic) if and only if $M_{w}^{m}$ is hyperbolic (parabolic). ## 3\. Preliminaires We assume throughout the paper that $\varphi:P^{m}\longrightarrow N^{n}$ is an isometric immersion of a complete non-compact Riemannian $m$-manifold $P^{m}$ into a complete Riemannian manifold $N^{n}$ with a pole $o\in N^{n}$ . Recall that a pole is a point $o$ such that the exponential map $\exp_{o}\colon T_{o}N^{n}\to N^{n}$ is a diffeomorphism. For every $x\in N^{n}-\\{o\\}$ we define $r(x)=r_{o}(x)=\operatorname{dist}_{N}(o,x)$, since $o$ is a pole this distance is realized by the length of a unique geodesic from $o$ to $x$, which is the radial geodesic from $o$. We also denote by $r\|_{P}$ or by $r$ the composition $r\circ\varphi:P\to\mathbb{R}_{+}\cup\\{0\\}$. This composition is called the extrinsic distance function from $o$ in $P^{m}$. With the extrinsic distance we can construct the extrinsic ball $D_{R}(o)$ of radius $R$ centered at $o$ as $D_{R}(o):=\\{x\in P:r(\varphi(x))<R\\}\quad.$ Since $\partial D_{t}(o)=\Sigma_{t}^{r}$, the flux of the extrinsic distance function $r$ on $P$ is $J_{r}(t)=\int_{\partial D_{t}}\|\nabla^{P}r\|d\rho\quad,$ where the gradients of $r$ in $N$ and $r\|_{P}$ in $P$ are denoted by $\nabla^{N}r$ and $\nabla^{P}r$, respectively. These two gradients have the following basic relation, by virtue of the identification, given any point $x\in P$, between the tangent vector fields $X\in T_{x}P$ and $\varphi_{_{x}}(X)\in T_{\varphi(x)}N$ (3.1) $\nabla^{N}r=\nabla^{P}r+(\nabla^{N}r)^{\bot},$ where $(\nabla^{N}r)^{\bot}(\varphi(x))=\nabla^{\bot}r(\varphi(x))$ is perpendicular to $T_{x}P$ for all $x\in P$. We now present the curvature restrictions which constitute the geometric framework of the present study. ###### Definition 3.1. Let $o$ be a point in a Riemannian manifold $N$ and let $x\in N-\\{o\\}$. The sectional curvature $K_{N}(\sigma_{x})$ of the two-plane $\sigma_{x}\in T_{x}N$ is then called a $o$-radial sectional curvature of $N$ at $x$ if $\sigma_{x}$ contains the tangent vector to a minimal geodesic from $o$ to $x$. We denote these curvatures by $K_{o,N}(\sigma_{x})$. ### 3.1. Model spaces Throughout this paper we shall assume that the ambient manifold $N^{n}$ has its $o$-radial sectional curvatures $K_{o,N}(x)$ bounded from above by the expression $K_{w}(r(x))=-w^{\prime\prime}(r(x))/w(r(x))$, which are precisely the radial sectional curvatures of the $w$-model space $\,M^{m}_{w}\,$ we are going to define. ###### Definition 3.2 (See [O’N83, Gri99a, GW79]). A $w-$model $M_{w}^{m}$ is a smooth warped product with base $B^{1}=[0,\Lambda[\,\subset\mathbb{R}$ (where $0<\Lambda\leq\infty$), fiber $F^{m-1}=\mathbb{S}^{m-1}_{1}$ (i.e. the unit $(m-1)$-sphere with standard metric), and warping function $w\colon[0,\Lambda[\to\mathbb{R}_{+}\cup\\{0\\}$, with $w(0)=0$, $w^{\prime}(0)=1$, and $w(r)>0$ for all $r>0$. The point $o_{w}=\pi^{-1}(0)$, where $\pi$ denotes the projection onto $B^{1}$, is called the center point of the model space. If $\Lambda=\infty$, then $o_{w}$ is a pole of $M_{w}^{m}$. ###### Proposition 3.3. The simply connected space forms $\mathbb{K}^{m}(b)$ of constant curvature $b$ are $w-$models with warping functions $w_{b}(r)=\begin{cases}\frac{1}{\sqrt{b}}\sin(\sqrt{b}\,r)&\text{if $b>0$}\\\ \phantom{\frac{1}{\sqrt{b}}}r&\text{if $b=0$}\\\ \frac{1}{\sqrt{-b}}\sinh(\sqrt{-b}\,r)&\text{if $b<0$}.\end{cases}$ Note that for $b>0$ the function $w_{b}(r)$ admits a smooth extension to $r=\pi/\sqrt{b}$. ###### Proposition 3.4 (See [O’N83, GW79, Gri99a]). Let $M_{w}^{m}$ be a $w-$model space with warping function $w(r)$ and center $o_{w}$. The distance sphere of radius $r$ and center $o_{w}$ in $M_{w}^{m}$ is the fiber $\pi^{-1}(r)$. This distance sphere has the constant mean curvature $\eta_{w}(r)=\frac{w^{\prime}(r)}{w(r)}$ On the other hand, the $o_{w}$-radial sectional curvatures of $M_{w}^{m}$ at every $x\in\pi^{-1}(r)$ (for $r>0$) are all identical and determined by $K_{o_{w},M_{w}}(\sigma_{x})=-\frac{w^{\prime\prime}(r)}{w(r)}.$ ###### Remark d. The $w-$model spaces are completely determined via $w$ by the mean curvatures of the spherical fibers $S^{w}_{r}$: $\,\eta_{w}(r)=w^{\prime}(r)/w(r)\,\quad,$ by the volume of the fiber $\,\operatorname{Vol}(S^{w}_{r})\,=V_{0}\,w^{m-1}(r)\,\quad,$ and by the volume of the corresponding ball, for which the fiber is the boundary $\,\operatorname{Vol}(B^{w}_{r})\,=\,V_{0}\,\int_{0}^{r}\,w^{m-1}(t)\,dt\,\quad.$ Here $V_{0}$ denotes the volume of the unit sphere $S^{0,m-1}_{1}$, (we denote in general as $S^{b,m-1}_{r}$ the sphere of radius $r$ in the real space form $\mathbb{K}^{m}(b)$) . The latter two functions define the isoperimetric quotient function as follows $\,q_{w}(r)\,=\,\operatorname{Vol}(B^{w}_{r})/\operatorname{Vol}(S^{w}_{r})\quad.$ We observe moreover that the flux of the geodesic distance function $r_{o}$ from the center to the model space is $J^{w}_{r}(R)=\int_{S^{w}_{R}}\|\nabla r\|d\sigma=\operatorname{Vol}(S^{w}_{R})\quad.$ Besides the already defined comparison controllers for the radial sectional curvatures of $N^{n}$, we shall need two further purely intrinsic conditions on the model spaces: ###### Definition 3.5. A given $w-$model space $\,M^{m}_{w}\,$ is called balanced from below and balanced from above, respectively, if the following weighted isoperimetric conditions are satisfied: $\displaystyle\text{Balance from below:}\quad q_{w}(r)\,\eta_{w}(r)$ $\displaystyle\geq 1/m\quad\text{for all}\quad r\geq 0\quad;$ $\displaystyle\text{Balance from above:}\quad q_{w}(r)\,\eta_{w}(r)$ $\displaystyle\leq 1/(m-1)\quad\text{for all}\quad r\geq 0\quad.$ A model space is called totally balanced if it is balanced both from below and from above. ### 3.2. Laplacian comparison for radial functions Let us recall the expression of the Laplacian on model spaces for radial functions ###### Proposition 3.6 (See [O’N83], [GW79] and [Gri99a]). Let $M_{w}^{n}$ be a model space, denote by $r:M_{w}^{n}-\\{o_{w}\\}\to\mathbb{R}^{+}$ the geodesic distance from the center $o_{w}$, let $f:\mathbb{R}\to\mathbb{R}$ be a smooth function, then (3.2) $\Delta^{M_{w}^{n}}\left(f\circ r\right)=f^{\prime\prime}\circ r+\left(n-1\right)\left(f^{\prime}\cdot\eta_{w}\right)\circ r\quad.$ Applying the Hessian comparison theorems given in [GW79] we can obtain (see [MP06] for instance) ###### Proposition 3.7. Let $\varphi:P^{m}\to N^{n}$ be an immersion into a manifold $N$ with a pole. Suppose the the radial sectional curvatures $K_{N}$ of $N$ are bounded from above by the radial sectional curvatures of a model space $M_{w}^{m}$ as follows: (3.3) $K_{N}\leq-\frac{w^{\prime\prime}}{w}\circ r\quad.$ Let $f:\mathbb{R}\to\mathbb{R}$ be a smooth function with $f^{\prime}\geq 0$, and dennote by $r:P\to\mathbb{R}^{+}$ the extrinsic distance function. Then (3.4) $\displaystyle\Delta^{P}\left(f\circ r\right)\geq$ $\displaystyle\|\nabla^{P}r\|\left(f^{\prime\prime}-f^{\prime}\cdot\eta_{w}\right)\circ r$ $\displaystyle+m\left(f^{\prime}\cdot\eta_{w}\right)\circ r+m\langle\nabla^{N}r,H_{P}\rangle f^{\prime}\circ r\quad,$ where $H_{P}$ denotes the mean curvature vector of $P$ in $N$. #### 3.2.1. Capacity and the Mean Exit Time function on Model spaces One key purpose of this paper is to compare the capacity of extrinsic annuli of an immersed minimal submanifold with the capacity in an adequate model space. In the model space we can obtain the value of the capacity directly: ###### Proposition 3.8 (See [Gri99b]). Let $M_{w}^{n}$ be a model space. Then (3.5) $\operatorname{Cap}(A^{w}_{\rho,R})=\left(\int_{\rho}^{R}\frac{ds}{\operatorname{Vol}(S_{s}^{w})}\right)^{-1}=V_{n}\left(\int_{\rho}^{R}\frac{ds}{w^{n-1}}\right)^{-1}\quad.$ We note that the radial function $\Psi:M_{w}^{m}\to\mathbb{R}$ given by (3.6) $\Psi(p):=\Psi^{w}_{\rho,R}(r(p))\quad,$ being (3.7) $\Psi^{w}_{\rho,R}(t)=\int_{\rho}^{t}\frac{\operatorname{Cap}(A^{w}_{\rho,R})}{\operatorname{Vol}(S^{w}_{s})}ds\quad,$ is the solution to the Dirichlet problem given in 2.7 for the annular region $A^{w}_{\rho,R}$, namely (3.8) $\begin{cases}\Delta^{M_{w}^{m}}\Psi=0\\\ \Psi\|_{S^{w}_{\rho}}=0\\\ \Psi\|_{S^{w}_{R}}=1\end{cases}$ Another important tool in this paper is the comparison result for the mean exit time. Let now $E_{R}^{w}$ denote the mean time of the first exit from $B_{R}^{w}$ for a Brownnian particle starting at $o_{w}$. A remark due to Dynkin in [Dyn65] claims that $E_{R}^{w}$ is the continuous solution to the following Poisson equation with Dirichlet boundary data, (3.9) $\displaystyle\Delta^{M_{w}^{n}}E_{R}^{w}=$ $\displaystyle-1$ $\displaystyle E_{R}^{w}\|_{S_{R}^{w}}=$ $\displaystyle 0.$ Since the ball $B_{R}^{w}$ has maximal isotropy at the center $o_{w}$, so we have that $E_{R}^{w}$ only depends on the extrinsic distance $r$. Therefore, we will write $E_{R}^{w}=E_{R}^{w}(r)$ and ###### Proposition 3.9 (See [MP06]). Let $M_{w}^{n}$ be a model space of dimension $n$ then (3.10) $E^{w}_{R}(r)=\int_{r}^{R}q_{w}(t)dt$ ## 4\. Proof of the main theorems of §2 ### 4.1. Proof of theorem 2.1 Since the mean time function $E_{R}^{w}$ is a radial function, we can transplant it to $P$ using the extrinsic distance, hence, we also denote as $E_{R}^{w}:P\to\mathbb{R}$ the function given by $E_{R}^{w}(x)=E_{R}^{w}(r(x))$. To compare the mean exit time function, we need the following comparison for the mean exit time ###### Proposition 4.1. ([MP06]) Let $\varphi:P^{m}\longrightarrow N^{n}$ be an isometric, proper and minimal immersion of a complete non-compact Riemannian $m$-manifold $P^{m}$ into a complete Riemannian manifold $N^{n}$ with a pole $o\in N$. Let us suppose that the $o-$radial sectional curvatures of $N$ are bounded from above by $K_{o,N}(\sigma_{x})\leq-\frac{w^{\prime\prime}(r)}{w(r)}(\varphi(x))\,\,\,\forall x\in P\,,$ and the model space $M_{w}^{m}$ is balanced from below, then (4.1) $\Delta^{P}E^{w}_{R}\leq-1=\Delta^{P}E_{R}.$ Applying now the divergence theorem to inequality (4.1) we obtain (4.2) $\displaystyle-\operatorname{Vol}(D_{R})=$ $\displaystyle\int_{D_{R}}\Delta^{P}E^{P}_{R}(r)d\mu\geq\int_{D_{R}}\Delta^{P}E^{w}_{R}(r)d\mu$ $\displaystyle=$ $\displaystyle\int_{\partial D_{R}}E^{w}_{R}(r)^{\prime}\langle\nabla^{P}r,\nu\rangle d\sigma=-q_{w}(R)\int_{\partial D_{R}}\\|\nabla^{P}r\\|d\sigma$ Therefore, (4.3) $\frac{\operatorname{Vol}(D_{R})}{\operatorname{Vol}(B^{w}_{R})}\leq\frac{J_{r}(R)}{\operatorname{Vol}(S^{w}_{R})}=\frac{J_{r}(R)}{J_{r}^{w}(R)}\quad.$ Observe that equality in inequality (4.3) implies equality in inequality (4.2) and therefore, in inequality (4.1). Taking, thus, into account that $E_{R}^{P}=E_{R}^{w}$ in $x\in\partial D_{R}$, $\Delta E_{R}^{P}=\Delta E_{R}^{w}$ in $x\in D_{R}$, and the maximum principle, we obtain that equality in (4.3) implies (4.4) $E_{R}^{P}=E_{R}^{w}\quad,$ for all $x\in D_{R}$. In order to obtain the monotonicity of the quotient $\frac{\operatorname{Vol}(D_{R})}{\operatorname{Vol}(B^{w}_{R})}$, we note that by the co-area formula we get: (4.5) $\displaystyle\left(\ln\frac{\operatorname{Vol}(D_{R})}{\operatorname{Vol}(B^{w}_{R})}\right)^{\prime}=$ $\displaystyle\frac{\int_{\partial D_{R}}\frac{1}{\\|\nabla^{P}r\\|}d\sigma}{\operatorname{Vol}(D_{R})}-\frac{\operatorname{Vol}(S_{R}^{w})}{\operatorname{Vol}(B_{R}^{w})}$ $\displaystyle\geq$ $\displaystyle\frac{\int_{\partial D_{R}}\\|\nabla^{P}r\\|d\sigma}{\operatorname{Vol}(D_{R})}-\frac{\operatorname{Vol}(S_{R}^{w})}{\operatorname{Vol}(B_{R}^{w})}$ $\displaystyle=$ $\displaystyle\frac{\operatorname{Vol}(S_{R}^{w})}{\operatorname{Vol}(D_{R})}\left(\frac{J_{r}(R)}{\operatorname{Vol}(S_{R}^{w})}-\frac{\operatorname{Vol}(D_{R})}{\operatorname{Vol}(B_{R}^{w})}\right)$ $\displaystyle\geq$ $\displaystyle 0\quad.$ Hence $\frac{\operatorname{Vol}(D_{R})}{\operatorname{Vol}(B^{w}_{R})}$ is a monotone non-decreasing function. To prove that also $\frac{J_{r}(R)}{J_{r}^{w}(R)}$ is a monotone nondecreasing function we need the following lemma ###### Lemma 4.2. (4.6) $\text{div }\left(\frac{\nabla^{P}E^{w}_{R}(r)}{\operatorname{Vol}(B_{r}^{w})}\right)\leq 0\quad.$ ###### Proof. Taking into account the product rule for the divergence and the mean exit time comparison result (4.7) $\displaystyle\text{div }\left(\frac{\nabla^{P}E^{w}_{R}(r)}{\operatorname{Vol}(B_{r}^{w})}\right)=$ $\displaystyle\frac{\Delta^{P}E_{R}^{w}(r)}{\operatorname{Vol}(B_{r}^{w})}-\frac{\operatorname{Vol}(B_{r}^{w})^{\prime}}{\operatorname{Vol}(B_{r}^{w})^{2}}\langle\nabla^{P}r,\nabla^{P}E_{R}^{w}(r)\rangle$ $\displaystyle=$ $\displaystyle\frac{\Delta^{P}E_{R}^{w}(r)}{\operatorname{Vol}(B_{r}^{w})}-\frac{\operatorname{Vol}(S_{r}^{w})}{\operatorname{Vol}(B_{r}^{w})^{2}}E_{R}^{w}(r)^{\prime}\\|\nabla^{P}r\\|^{2}$ $\displaystyle\leq$ $\displaystyle\frac{-1}{\operatorname{Vol}(B_{r}^{w})}+\frac{\\|\nabla^{P}r\\|^{2}}{\operatorname{Vol}(B_{r}^{w})}\leq 0\quad.$ ∎ Using now this lemma and the divergence theorem in the extrinsic annulus $A_{\rho,R}$ for $\rho<R$ (4.8) $\displaystyle 0\geq$ $\displaystyle\int_{A_{\rho,R}}\text{div }\left(\frac{\nabla^{P}E^{w}_{R}(r)}{\operatorname{Vol}(B_{r}^{w})}\right)d\mu$ $\displaystyle=$ $\displaystyle\int_{\partial D_{R}}\frac{E^{w}_{R}(r)^{\prime}\\|\nabla^{P}r\\|}{\operatorname{Vol}(B_{r}^{w})}d\sigma-\int_{\partial D_{\rho}}\frac{E^{w}_{R}(r)^{\prime}\\|\nabla^{P}r\\|}{\operatorname{Vol}(B_{r}^{w})}d\sigma$ $\displaystyle=$ $\displaystyle-\frac{J_{r}(R)}{\operatorname{Vol}(S_{R}^{w})}+\frac{J_{r}(\rho)}{\operatorname{Vol}(S_{\rho}^{w})}\quad.$ Therefore, (4.9) $\frac{J_{r}(R)}{J^{w}_{r}(R)}\geq\frac{J_{r}(\rho)}{J_{r}^{w}(\rho)}\quad,$ for any $R>\rho$, and the theorem is proven. ### 4.2. Proof of theorem 2.2 The corresponding Dirichlet problem for the capacity of the extrinsic annulus $A_{\rho,R}$ is (4.10) $\begin{cases}\Delta^{P}\Psi=0\\\ \Psi\|_{\partial D_{\rho}}=0\\\ \Psi\|_{\partial D_{R}}=1\end{cases}$ Let us transplant the function $\Psi^{w}_{\rho,R}$ with the extrinsic distance function $r$: (4.11) $\Psi^{w}(p):A_{\rho,R}\to\mathbb{R},\quad p\to\Psi^{w}(p):=\Psi^{w}_{\rho,R}(r(p))\quad.$ Then, applying proposition 3.7 (4.12) $\Delta^{P}\Psi^{w}\geq m\left(1-\|\nabla^{P}r\|\right)\left((\Psi^{w}_{\rho,R})^{\prime}\cdot\eta_{w}\right)\circ r\quad.$ Taking into account that $\eta_{w}\geq 0$ (4.13) $\Delta^{P}\Psi^{w}\geq 0=\Delta^{P}\Psi\quad.$ Since $\Delta^{P}\left(\Psi^{w}-\Psi\right)\geq 0$ and since $\Psi_{\partial A_{\rho,R}}=\Psi^{w}_{\partial A_{\rho,R}}$, we have by the Maximum Principle that $\Psi^{w}\leq\Psi$ on $A_{\rho,R}$, and, since $\Psi_{\partial D_{\rho}}=\Psi^{w}_{\partial D_{\rho}}=0$, we obtain (4.14) $\|\nabla^{P}\Psi^{w}\|\leq\|\nabla^{P}\Psi\|\quad\text{ on }\partial D_{\rho}\quad.$ Finally, we can estimate the capacity (4.15) $\displaystyle\operatorname{Cap}(A_{\rho,R})$ $\displaystyle=\int_{\partial D_{\rho}}\|\nabla^{P}\Psi\|d\sigma$ $\displaystyle\geq\int_{\partial D_{\rho}}\|\nabla^{P}\Psi^{w}\|d\sigma$ $\displaystyle=(\Psi^{w}(\rho))^{\prime}\int_{\partial D_{\rho}}\|\nabla^{P}r\|d\sigma$ $\displaystyle=\operatorname{Cap}(A_{\rho,R}^{w})\frac{J_{r}(\rho)}{J^{w}_{r}(\rho)}\quad,$ and the theorem follows. ### 4.3. Proof of theorem 2.3 With the flux we can provide an upper bound for the capacity (see inequality (2.9) ). Using theorem 2.1 we obtain that (4.16) $\displaystyle\operatorname{Cap}(A_{\rho,R})\leq$ $\displaystyle\frac{1}{\int_{\rho}^{R}\frac{ds}{\int_{\partial D_{s}}\\|\nabla^{P}r\\|d\sigma}}=\frac{1}{\int_{\rho}^{R}\frac{ds}{\frac{J_{r}(s)}{\operatorname{Vol}(S_{s}^{w})}\operatorname{Vol}(S_{s}^{w})}}$ $\displaystyle\leq$ $\displaystyle\frac{\frac{J_{r}(R)}{\operatorname{Vol}(S_{R}^{w})}}{\int_{\rho}^{R}\frac{ds}{\operatorname{Vol}(S_{s}^{w})}}=\frac{J_{r}(R)}{\operatorname{Vol}(S_{R}^{w})}\operatorname{Cap}(A^{w}_{\rho,R}).$ For the bounds from below, see [MP03]. Observe moreover that equality in the above inequality implies that (4.17) $\int_{t}^{R}\left(\frac{\frac{J_{r}(R)}{\operatorname{Vol}(S^{w}_{R})}}{\frac{J_{r}(s)}{\operatorname{Vol}(S^{w}_{s})}}-1\right)\frac{1}{\operatorname{Vol}(S_{s}^{w})}ds=0.$ Therefore (4.18) $\frac{J_{r}(R)}{\operatorname{Vol}(S^{w}_{R})}=\frac{J_{r}(s)}{\operatorname{Vol}(S^{w}_{s})},$ for any $s\in[\rho,R]$. Then, by inequality (4.8) (4.19) $\text{div }\left(\frac{\nabla^{P}E^{w}_{R}(r)}{\operatorname{Vol}(B_{r}^{w})}\right)=0,$ for any $p\in A_{\rho,R}$. From inequality (4.7) (4.20) $\\|\nabla^{P}r\\|=1,$ for any $p\in A_{\rho,R}$, and hence, $D_{R}$ is a minimal cone. ## 5\. Proof of Theorem 1.1 and Corollary 1.2 This proof mimics the argument given in [Tka94, Theorem 2], so we merely give a sketch emphasizing the points where the line of reasoning from [Tka94] is modified to hold in the present more general setting. First of all, note that we can construct the following order-preserving bijection $F:\mathbb{R}^{+}\to\mathbb{R}^{+},\quad F(t)=\int_{0}^{t}w(s)ds\quad.$ Since $\varphi:P^{m}\to N^{n}$ is a complete proper and minimal immersion within a manifold with a pole $N^{n}$, applying proposition 3.7 we have (5.1) $\Delta^{P}F\circ r\geq mw^{\prime}\circ r\quad.$ Hence, by using the assumption $w^{\prime}>0$, the extrinsic distance has no local maximum. Therefore for any $R$, $P^{m}\setminus D_{R}$ has no bounded components, being each component of $P^{m}\setminus D_{R}$ non compact, and the number of ends $\mathcal{E}_{D_{R}}(P)$ with respect to $D_{R}$ is the number of connected components of $P^{m}\setminus D_{R}$. Let us denote by $\displaystyle\left\\{\Omega^{i}\right\\}_{i=1}^{\mathcal{E}_{D_{R}}(P)}$ the set of $\mathcal{E}_{D_{R}}(P)$ connected components of $P^{m}\setminus D_{R}$ (every one of them is a minimal submanifold with boundary). Now we need the following lemma ###### Lemma 5.1. For any connected component $\Omega_{i}$ of $P^{m}\setminus D_{R}$ the volume of the set $D_{t}^{\Omega_{i}}=D_{t}\cap\Omega_{i}\quad.$ is bounded from below by (5.2) $\operatorname{Vol}(D_{t}^{\Omega_{i}})\geq V_{m}\left(\frac{t-R}{2}\right)^{m}\quad.$ ###### Proof. Now pick a point $o^{\prime}\in D_{t}^{\Omega_{i}}$ such that its extrinsic distance is $r_{o}(o^{\prime})=\frac{R+t}{2}$, then the extrinsic ball $D^{\Omega_{i}}_{\frac{t-R}{2}}(o^{\prime})$ in $\Omega_{i}$ centered at $o^{\prime}$ with radius $\frac{t-R}{2}$ satisfies (5.3) $D^{\Omega_{i}}_{\frac{t-R}{2}}(o^{\prime})\subset D_{t}^{\Omega_{i}}\quad.$ Hence, (5.4) $\operatorname{Vol}(D_{t}^{\Omega_{i}})\geq\operatorname{Vol}(D^{\Omega_{i}}_{\frac{t-R}{2}}(o^{\prime}))\quad.$ Since $r(o^{\prime})>R$ and the sectional curvatures of any tangent $2-$plane of the tangent space at every point in the geodesic ball $B^{N}_{\frac{t-R_{0}}{2}}(\varphi(o^{\prime}))$ of the ambient manifold are non-positive, we can make use of the behavior of the volume quotient (claim (2) in theorem 2.1) for extrinsic balls to the immersion $\varphi:D^{\Omega_{i}}_{\frac{t-R_{0}}{2}}(o^{\prime})\to B^{N}_{\frac{t-R_{0}}{2}}(\varphi(o^{\prime}))$ with the new model comparison $w(r)=w_{0}(r)=r$ (namely $M_{w}^{m}=\mathbb{R}^{m}$) (5.5) $\frac{\operatorname{Vol}(D^{\Omega_{i}}_{\frac{t-R}{2}}(o^{\prime}))}{V_{m}\left(\frac{t-R}{2}\right)^{m}}\geq\lim_{s\to 0}\frac{\operatorname{Vol}(D^{\Omega_{i}}_{s}(o^{\prime}))}{V_{m}s^{m}}\geq 1\quad.$ And the lemma is proved. ∎ Summing now in inequality (5.2) we obtain (5.6) $\operatorname{Vol}(A_{R,t})=\sum_{i=1}^{\mathcal{E}_{D_{R}}(P)}\operatorname{Vol}(D_{t}^{\Omega_{i}})\geq\mathcal{E}_{D_{R}}(P)V_{m}\left(\frac{t-R}{2}\right)^{m}\quad.$ Taking into account that $\operatorname{Vol}(A_{R,t})\leq\operatorname{Vol}(D_{t})$ and dividing by $\operatorname{Vol}(B_{t}^{w})$ we obtain (5.7) $\frac{\operatorname{Vol}(D_{t})}{\operatorname{Vol}(B_{t}^{w})}\geq\mathcal{E}_{D_{R}}(P)V_{m}\frac{\left(\frac{t-R}{2}\right)^{m}}{\operatorname{Vol}(B_{t}^{w})}\quad.$ We can split the last quotient by division and multiplication by $t^{m}$ (5.8) $\frac{\operatorname{Vol}(D_{t})}{\operatorname{Vol}(B_{t}^{w})}\geq\mathcal{E}_{D_{R}}(P)V_{m}\left(\frac{1-R/t}{2}\right)^{m}\frac{t^{m}}{\operatorname{Vol}(B_{t}^{w})}\quad.$ Hence, finally, using the explicit expression for $\operatorname{Vol}(B_{t}^{w})$ the theorem follows. In order to prove corollary 1.2, note that by the maximum principle $\mathcal{E}_{D_{R}}^{P}$ is a non-decreasing function with respect to $R$. By inequality (1.5) and the assumptions of the corollary we can conclude that $\mathcal{E}_{D_{R}}^{P}$ is stabilized, i.e. $\mathcal{E}_{D_{R}}^{P}=\text{ constant for sufficient large }R$. Now let $F\subset P$ be an arbitrary compact subset. Using again the maximum principle of the immersion, we conclude that $\mathcal{E}_{F}(P)$ is a non- decreasing function of the compact set $F$ (namely, if $F_{1}\subset F_{2}$ then $\mathcal{E}_{F_{1}}(P)\leq\mathcal{E}_{F_{2}}(P)$). Taking into account that for any compact set $K$ there exist $R_{K}$ such that $K\subset D_{R_{K}}$, we finally obtain (5.9) $\mathcal{E}(P)=\lim_{R\to\infty}\mathcal{E}_{D_{R}}(P)\quad,$ and the corollary follows. ## 6\. Corollaries and application of the extrinsic comparison theory ### 6.1. Relation between $w-$volume and $w-$flux of submanifolds Under the hypotesis of theorem 2.1, if the submanifold has finite $w$-flux, the submanifold has finite $w$-volume. But in particular settings we can also state a reverse: ###### Theorem 6.1. Let $\varphi:P^{m}\longrightarrow N^{n}$ be an isometric, proper, and minimal immersion of a complete non-compact Riemannian $m$-manifold $P^{m}$ into a complete Riemannian manifold $N^{n}$ with a pole $o\in N$ . Let us suppose that the $o-$radial sectional curvatures of $N$ are bounded from above by $K_{o,N}(\sigma_{x})\leq-\frac{w^{\prime\prime}(r)}{w(r)}(\varphi(x))\,\,\,\forall x\in P\,.$ Suppose that the model space $M_{w}^{m}$ is balanced from bellow with warping function satisfying $w^{\prime}(r)\geq 0\quad\forall r\in\mathbb{R}_{+}.$ Then, if the submanifold has finite $w$-volume, we have: 1. (1) The submanifold has finite $w$-flux. 2. (2) $\text{Flux}_{w}(P)=\operatorname{Vol}_{w}(P)$. ###### Proof. To prove the theorem let us state the following metric property for geodesic balls and geodesic spheres in a rotationally symmetric model space ###### Lemma 6.2. Let $M^{m}_{w}$ be a model space with $w^{\prime}(r)\geq 0\quad\forall r\in\mathbb{R}_{+}.$ Then $q_{w}(s)=\frac{\operatorname{Vol}(B^{w}_{S})}{\operatorname{Vol}(S^{w}_{s})}\leq s.$ ###### Proof. Observe that (6.1) $q_{w}(0)=0,$ and, since $w^{\prime}\geq 0$ , (6.2) $q_{w}^{\prime}(t)\leq 1,\,\forall t\geq 0.$ Hence, by integrating the above inequality, the lemma follows. ∎ Now, since $P$ has finite $w$-volume, then there exists $S\in\mathbb{R}^{+}$ such that (6.3) $\frac{\operatorname{Vol}(D_{R})}{\operatorname{Vol}(B_{R}^{w})}\leq\lim_{t\to\infty}\frac{\operatorname{Vol}(D_{t})}{\operatorname{Vol}(B_{t}^{w})}=S<\infty\quad.$ By inequality (4.5) (6.4) $\displaystyle\left(\ln\frac{\operatorname{Vol}(D_{s})}{\operatorname{Vol}(B^{w}_{s})}\right)^{\prime}\geq$ $\displaystyle\frac{\operatorname{Vol}(S_{s}^{w})}{S\operatorname{Vol}(B_{s}^{w})}\left(\frac{J_{r}(s)}{J_{r}^{w}(s)}-\frac{\operatorname{Vol}(D_{s})}{\operatorname{Vol}(B_{s}^{w})}\right)\geq 0\quad.$ Therefore, taking lemma 6.2 into acount we get: (6.5) $\displaystyle 0\leq\left(\frac{J_{r}(s)}{J_{r}^{w}(s)}-\frac{\operatorname{Vol}(D_{s})}{\operatorname{Vol}(B_{s}^{w})}\right)\leq S\left(\ln\frac{\operatorname{Vol}(D_{s})}{\operatorname{Vol}(B^{w}_{s})}\right)^{\prime}s\quad.$ But since (6.6) $\lim_{t\to\infty}\frac{\operatorname{Vol}(D_{t})}{\operatorname{Vol}(B_{t}^{w})}=\frac{\operatorname{Vol}(D_{R_{1}})}{\operatorname{Vol}(B_{R_{1}}^{w_{b}})}e^{\int_{R_{1}}^{\infty}\left(\ln\frac{\operatorname{Vol}(D_{s})}{\operatorname{Vol}(B^{w}_{s})}\right)^{\prime}ds}=S<\infty\quad,$ then, for any $\epsilon>0$ there exist a sequence $\\{t_{i}\\}_{i=1}^{\infty}$ with $t_{i}\to\infty$ when $i\to\infty$, and $R_{\epsilon}$, such that (6.7) $\left(\ln\frac{\operatorname{Vol}(D_{t_{i}})}{\operatorname{Vol}(B^{w}_{t_{i}})}\right)^{\prime}t_{i}<\epsilon\quad.$ This holds for any $t_{i}>R_{\epsilon}$. Applying inequality (6.5) taking into account the monotonicity of the flux and volume comparison quotients (6.8) $0\leq\text{Flux}_{w_{b}}(P)-\operatorname{Vol}_{w_{b}}(P)\leq\epsilon\quad,$ for any $\epsilon>0$ . Letting $\epsilon$ tend to $0$, the theorem is proven. ∎ ### 6.2. Intrinsic versions In this subsection we consider the intrinsic versions of Theorems 2.1 and 2.3 assuming that $P^{m}=N^{n}$. In this case, the extrinsic distance to the pole $p$ becomes the intrinsic distance in $N^{n}$, hence, for all $R$ the extrinsic domains $D_{R}$ become the geodesic balls $B^{N}_{R}$ of the ambient manifold $N^{n}$. Then, for all $x\in P$ $\displaystyle\nabla^{P}r(x)$ $\displaystyle=$ $\displaystyle\nabla^{N}r(x).$ As a consequence, $\\|\nabla^{P}r\\|=1$. From this intrinsic viewpoint, we have the following isoperimetric and volume comparison inequalities. ###### Theorem 6.3. Let $N^{n}$ denote a complete Riemannian manifold with a pole $p$. Suppose that the $p$-radial sectional curvatures of $N^{n}$ are bounded from above by the $p_{w}$-radial sectional curvatures of a $w$-model space $M^{n}_{w}$. Assume that (6.9) $w^{\prime}\geq 0\quad.$ Then the capacity of the intrinsic annulus $A_{\rho,R}$ is bounded from below by $\frac{\operatorname{Vol}(\partial B^{N}_{\rho})}{\operatorname{Vol}(S_{\rho}^{w})}\leq\frac{\operatorname{Cap}(A_{\rho,R})}{\operatorname{Cap}(A_{\rho,R}^{w})}$ And, furthermore, if $M_{w}^{n}$ is hyperbolic, then $N^{n}$ is also hyperbolic. ###### Theorem 6.4. Let $N^{n}$ denote a complete Riemannian manifold with a pole $p$. Suppose that the $p$-radial sectional curvatures of $N^{n}$ are bounded from above by the $p_{w}$-radial sectional curvatures of a $w$-model space $M^{n}_{w}$. Assume that $M_{w}^{n}$ is balanced from below. Then, 1. (1) for all $R>0$ (6.10) $\frac{\operatorname{Vol}(B^{N}_{R})}{\operatorname{Vol}(B^{w}_{R})}\leq\frac{\operatorname{Vol}(\partial B^{N}_{R})}{\operatorname{Vol}(S^{w}_{R})}\quad.$ 2. (2) The functions $\frac{\operatorname{Vol}(B^{N}_{R})}{\operatorname{Vol}(B^{w}_{R})}$ and $\frac{\operatorname{Vol}(\partial B^{N}_{R})}{\operatorname{Vol}(S^{w}_{R})}$ are non decreasing on $R$. 3. (3) Denoting by $E_{R}^{N}(x)$ the mean exit time function for the geodesic ball $B^{N}_{R}$ in $N$ and denoting by $E_{R}^{w}$ the mean exit time function in the $R-$ball $B_{R}^{w}$ in the model space $M_{w}^{n}$. If equality holds in (6.10) for some fixed $R>0$ then for any $x\in B^{N}_{R}$, $E_{R}^{N}(x)=E_{R}^{w}(r(x))$. 4. (4) The capacity of the intrinsic annulus $A_{\rho,R}$ is bounded from above by $\frac{\operatorname{Cap}(A_{\rho,R})}{\operatorname{Cap}(A_{\rho,R}^{w})}\leq\frac{\operatorname{Vol}(\partial B^{N}_{R})}{\operatorname{Vol}(S_{R}^{w})}$ Furthermore, if we suppose that there exist a finite real constant $C<\infty$ such that $\frac{\operatorname{Vol}(B^{N}_{R})}{\operatorname{Vol}(B_{R}^{w})}<C$ (or $\frac{\operatorname{Vol}(\partial B^{N}_{R})}{\operatorname{Vol}(S_{R}^{w})}<C$) then if $M_{w}^{n}$ is parabolic, $N$ is parabolic, and $\lim_{R\to\infty}\frac{\operatorname{Vol}(B^{N}_{R})}{\operatorname{Vol}(B_{R}^{w})}=\lim_{R\to\infty}\frac{\operatorname{Vol}(\partial B^{N}_{R})}{\operatorname{Vol}(S_{R}^{w})}.$ ### 6.3. Upper bounds for the fundamental tone S.T Yau suggested in [Yau00] the “very interesting” question to find an upper estimate to the first Dirichlet eigenvalue of minimal surfaces. Recall that for any precompact region $\Omega\subset M$ in a Riemannian manifold $M$, the first eigenvalue $\lambda_{1}(\Omega)$ of the Dirichlet problem in $\Omega$ for the Laplace operator is defined by the variational property (6.11) $\lambda_{1}(\Omega)=\inf_{u}\frac{\int\\|\nabla u\\|^{2}d\mu}{\int\\|u\\|^{2}}$ where the $\inf$ is taken over all Lipschitz functions $u\neq 0$ compactly supported in $\Omega$. The fundamental tone $\lambda^{}(M)$ of a complete Riemannian manifold can be obtained as the limit of the first Dirichlet eigenvalues of the precompact open sets in any exhaution sequence $\\{\Omega_{n}\\}_{n\in\mathbb{N}}$ for $M$, see [Gri99a] (6.12) $\lambda^{}(M)=\lim_{n\to\infty}\lambda_{1}(\Omega_{n})\quad.$ In this section, we shall impose flux and volume restrictions not on the submanifold $P$ but on one end $V$ of the submanifold with respect to the extrinsic ball $D_{R_{0}}$. Let us denote $D_{R}^{V}$ the intersection of the extrinsic ball $D_{R}$ with the end $V$ with respect to $D_{R_{0}}$ (6.13) $D_{R}^{V}=D_{R}\cap V\quad.$ Let us denote $J^{V}_{r}(R)$ the flux of the extrinsic distance in the end $V$, namely (6.14) $J^{V}_{r}(R)=\int_{\partial D_{R}\cap V}\|\nabla^{P}r\|d\sigma\quad.$ With this setting we then have: ###### Theorem 6.5. Let $\varphi:P^{m}\longrightarrow N^{n}$ be an isometric, proper and minimal immersion of a complete non-compact Riemannian $m$-manifold $P^{m}$ into a complete Riemannian manifold $N^{n}$ with a pole $o\in N$. Let us suppose that the $o-$radial sectional curvatures of $N$ are bounded from above by $K_{o,N}(\sigma_{x})\leq-\frac{w^{\prime\prime}(r)}{w(r)}(\varphi(x))\,\,\,\forall x\in P\,,$ and the model space $M_{w}^{m}$ is balanced from below. Suppose moreover that there exists an end $V$ with respect to an extrinsic ball $D_{R_{0}}$ with finite $w$-flux. Then (6.15) $\lambda^{}(P)\leq\frac{\text{Flux}_{w}(V)}{\operatorname{Vol}_{w}(V)}\limsup_{t\to\infty}\left(\frac{1}{\operatorname{Vol}(B_{t}^{w})\int_{t}^{\infty}\frac{ds}{\operatorname{Vol}(S^{w}_{s})}}\right)\quad.$ ###### Proof. Due to the relation between the first Dirichlet eigenvalue and the capacity given in [Gri99b] we can conclude for the extrinsic ball $D_{R}^{V}$ that (6.16) $\lambda_{1}(D_{R}^{V})\leq\frac{\operatorname{Cap}(A^{V}_{t,R})}{\operatorname{Vol}(D_{t}^{V})}\quad.$ Being $t<R$ and $A^{V}_{t,R}$ the extrinsic annulus in $V$. Hence, by the theorem 2.3 (6.17) $\lambda_{1}(D_{R}^{V})\leq\frac{\frac{J_{r}^{V}(R)}{J_{r}^{w}(R)}}{\frac{\operatorname{Vol}(D_{t}^{V})}{\operatorname{Vol}(B_{t}^{w})}}\frac{\operatorname{Cap}(A^{w}_{t,R})}{\operatorname{Vol}(B^{w}_{t})}\quad.$ For any $t<R$. Finally, taking into account that $\lambda(D_{R})\leq\lambda_{1}(D_{R}^{V})$ (by the monotonicity of the first eigenvalue), and letting $R$ tend to infinity we have (6.18) $\lambda^{}(P)\leq\frac{\text{Flux}_{w}(V)}{\frac{\operatorname{Vol}(D_{t}^{V})}{\operatorname{Vol}(B_{t}^{w})}}\frac{1}{\operatorname{Vol}(B^{w}_{t})\int_{t}^{\infty}\frac{ds}{\operatorname{Vol}(S_{s}^{w})}}\quad.$ Taking limits, the theorem follows. ∎ Obviously, by using theorem 6.1 we also have the following: ###### Corollary 6.6. Under the assumptions of theorem 6.5 suppose moreover $w^{\prime}\geq 0.$ Then, (6.19) $\lambda^{}(P)\leq\limsup_{t\to\infty}\left(\frac{1}{\operatorname{Vol}(B_{t}^{w})\int_{t}^{\infty}\frac{ds}{\operatorname{Vol}(S^{w}_{s})}}\right)\quad.$ Using the Cheeger isoperimetric constant we can deduce the following lower bounds ###### Theorem 6.7. Let $\varphi:P^{m}\longrightarrow N^{n}$ be an isometric, proper and minimal immersion of a complete non-compact Riemannian $m$-manifold $P^{m}$ into a complete Riemannian manifold $N^{n}$ with a pole $o\in N^{n}$ . Let us suppose that the $o-$radial sectional curvatures of $N^{n}$ are bounded from above by $K_{o,N}(\sigma_{x})\leq-\frac{w^{\prime\prime}(r)}{w(r)}(\varphi(x))\,\,\,\forall x\in P\,,$ and the model space $M_{w}^{m}$ is balanced from below. Suppose moreover that $L:=\sup_{t\in\mathbb{R}^{+}}q_{w}(t)<\infty.$ Then (6.20) $\frac{1}{4L^{2}}\leq\lambda^{}(P)\quad.$ ###### Proof. Consider $\Omega\subset P^{m}$ a smooth domain with smooth boundary $\partial\Omega$. Using the transplanted mean exit function in a similar way as in the proof of theorem 2.1 we obtain: (6.21) $\displaystyle-\operatorname{Vol}(\Omega)=$ $\displaystyle\int_{\Omega}\Delta^{P}E_{R}d\mu\geq\int_{\Omega}\Delta^{P}E_{R}^{w}d\mu=\int_{\partial\Omega}E_{R}^{w}(r)^{\prime}\langle\nabla^{P}r,\nu\rangle d\sigma$ $\displaystyle\geq$ $\displaystyle-\int_{\partial\Omega}q_{w}(r)\langle\nabla^{P}r,\nu\rangle d\sigma\geq-\int_{\partial\Omega}q_{w}(r)d\sigma$ $\displaystyle\geq$ $\displaystyle-L\operatorname{Vol}(\partial\Omega)\quad.$ Hence, for any $\Omega\subset P$, (6.22) $\frac{\operatorname{Vol}(\partial\Omega)}{\operatorname{Vol}(\Omega)}\geq\frac{1}{L}\quad.$ Thence the Cheeger constant $h(P)$ (see [Cha84]) satisfies (6.23) $h(P)\geq\frac{1}{L}\quad.$ Taking into account that (6.24) $\lambda^{}(P)\geq\frac{1}{4}\left(h(P)\right)^{2}\quad,$ the theorem follows. ∎ As an immediate consequence of the previous theorems and corollaries in the particular setting of a minimal submanifold within a Cartan-Hadamard ambient manifold is the following: ###### Corollary 6.8. Let $\varphi:P^{m}\longrightarrow N^{n}$ be a complete minimal immersion into a simply connected Cartan-Hadamard manifold $N^{n}$ with sectional curvatures $K_{N}\leq b\leq 0$. Suppose moreover that there exists an end $V$ with respect to an extrinsic ball $D_{R_{0}}$ with finite $w_{b}$-volume. Then (6.25) $\frac{-(m-1)^{2}b}{4}\leq\lambda^{}(P)\leq-(m-1)^{2}b\quad.$ ###### Remark e. Note that if $b=0$ in the above theorem, $\lambda^{}(P)=0$. See also [Gim13]. ### 6.4. Applications to minimal submanifolds in $\mathbb{R}^{n}$ If $P^{m}$ is a minimal submanifold in $\mathbb{R}^{n}$, it is well known that the extrinsic distance $r$ satisfies (6.26) $\Delta^{P}r^{2}=2m$ Applying the divergence theorem (6.27) $\displaystyle 2m\operatorname{Vol}(D_{R})=$ $\displaystyle\int_{D_{R}}\Delta^{p}r^{2}d\mu=\int_{\partial D_{R}}2r\langle\nabla r,\nu\rangle d\sigma$ $\displaystyle=$ $\displaystyle 2R\int_{\partial D_{R}}\langle\nabla r,\frac{\nabla r}{\|\nabla r\|}\rangle d\sigma=2R\int_{\partial D_{R}}\|\nabla r\|d\sigma$ $\displaystyle=$ $\displaystyle 2m\frac{\operatorname{Vol}(B_{R}^{w_{0}})}{\operatorname{Vol}(S_{R}^{w_{0}})}\int_{\partial D_{R}}\|\nabla r\|d\sigma$ hence, the volume comparison quotient $\frac{\operatorname{Vol}(D_{R})}{\operatorname{Vol}(B_{R}^{w_{0}})}$ is just (6.28) $\frac{\operatorname{Vol}(D_{R})}{\operatorname{Vol}(B_{R}^{w_{0}})}=\frac{J_{r}(R)}{J^{w_{0}}_{r}(R)}\quad.$ And therefore, we can state that ###### Corollary 6.9. Let $P^{m}$ be a minimal submanifold properly immersed in the Euclidean space $\mathbb{R}^{n}$. Then $E^{P}_{R}(x)=E^{\mathbb{R}^{m}}_{R}(r(x))\quad,$ where $E^{P}_{R}(x)$ denotes the first exit from $D_{R}$ for a Brownian particle starting at $x\in D_{R}$, and $E_{R}^{w}(r)$ denotes the (rotationally symmetric) mean exit time function for the $R-$ball $B_{R}^{w}$ in the model space $M_{w}^{m}$ If we have finite $w_{0}$-volume ($\sup_{R\in\mathbb{R}^{+}}\frac{\operatorname{Vol}(D_{R})}{\operatorname{Vol}(B_{R}^{w_{0}})}<\infty$) we also get: ###### Corollary 6.10. Let $P^{m}$ be a minimal submanifold immersed in $\mathbb{R}^{n}$, suppose moreover that $P$ has finite $w_{0}$-volume then: 1. (1) $P$ is parabolic if $m=2$ and if $m\geq 3$, $P$ is hyperbolic. 2. (2) $\lambda^{}(P)=0$. On the other hand, in special geometric settings the finiteness of the $w_{0}$-volume is related to the number of ends ###### Theorem B. (See [And84] and [Che95]) Let $P^{m}$ be a minimal submanifold properly immersed in $\mathbb{R}^{n}$ with finite total scalar curvature i.e. $\int_{P}\\|B^{P}\\|^{m}d\mu<\infty$ where $\\|B^{P}\\|$ denotes the norm of the second fundamental form in $P$, then (6.29) $\frac{J_{r}(R)}{J_{r}^{w_{0}}(R)}\leq\mathcal{E}(P),$ provided either of the following two conditions hold 1. (1) $m=2$, $n=3$ and each end of $P$ is embedded. 2. (2) $m\geq 3$. Where $\mathcal{E}(P)$ denotes the finite number of ends of $P$. This relation between the number of ends and the flux quotient allow us to state ###### Corollary 6.11. Let $P^{m}$ be a minimal submanifold properly immersed in $\mathbb{R}^{n}$ with finite total scalar curvature and either $m\geq 3$, or $m=2$ $n=3$ and each end of $P$ is embedded, then for any $\rho>0$ and any $R>\rho$ (6.30) $1\leq\frac{\operatorname{Cap}(A_{\rho,R})}{\operatorname{Cap}(A_{\rho,R}^{w})}\leq\mathcal{E}(P)\quad.$ And for the fundamental tone (6.31) $\lambda^{}(P)=0\quad.$ ## References * [And84] Michael T. Anderson, _The compactification of a minimal submanifold in euclidean space by the gauss map_ , 1984. * [CFG05] Graziano Crasta, Ilaria Fragalà, and Filippo Gazzola, _On a long-standing conjecture by Pólya-Szegö and related topics_ , Z. Angew. Math. Phys. 56 (2005), no. 5, 763–782. MR 2184904 (2006i:31006) * [Cha84] Isaac Chavel, _Eigenvalues in Riemannian geometry_ , Pure and Applied Mathematics, vol. 115, Academic Press Inc., Orlando, FL, 1984, Including a chapter by Burton Randol, With an appendix by Jozef Dodziuk. 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MR 1803723 (2002e:53002)	arxiv-papers	2014-01-07T10:09:36	2024-09-04T02:49:56.359910	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Vicent Gimeno and Steen Markvorsen", "submitter": "Vicent Gimeno", "url": "https://arxiv.org/abs/1401.1329" }
1401.1450	# A Recursive Algorithmic Approach to the Finding of Permutations for the Combination of Any Two Sets Diego Fernando C. Carrión L (July 18th, 2013) ###### Abstract In this paper I present a conjecture for a recursive algorithm that finds each permutation of combining two sets of objects (AKA the Shuffle Product). This algorithm provides an efficient way to navigate this problem, as each atomic operation yields a permutation of the union. The permutations of the union of the two sets are represented as binary integers which are then manipulated mathematically to find the next permutation. The routes taken to find each of the permutations then form a series of associations or adjacencies which can be represented in a tree graph which appears to possess some properties of a fractal. This algorithm was discovered while attempting to identify every possible end- state of a Tic-Tac-Toe (Naughts and Crosses) board. It was found to be a viable and efficient solution to the problem, and now—in its more generalized state—it is my belief that it may find applications among a wide range of theoretical and applied sciences. I hypothesize that, due to the fractal-like nature of the tree it traverses, this algorithm sheds light on a more generic principle of combinatorics and as such could be further generalized to perhaps be applied to the union of any number of sets. ## 1 Introduction For at least the last century, mathematicians have dedicated significant energy to studying the process of combining sets of objects together, and over many years numerous algorithms have been conceived for performing such unions. However, few—if indeed any—specialized algorithms exist which provide a way to identify and describe the permutations of a union of this sort. An individual permutation can easily be obtained through randomization, or even careful arrangement of the components. However an exhaustive list of all possible configurations—without duplicates—cannot obtained without a thorough algorithm—and to do so efficiently is another problem in of itself. This problem directly pertains to a branch of mathematics called _Shuffle Algebras_ which was developed in the first half of the 20th Century, initially for the probabilistic study of the shuffling of cards[2]. In their 1952 paper, _On the Groups H(II, n), I_[1], Samuel Eilenberg and Saunders MacLane defined the sum over the permutations of a union of two sets (or in other words: every possible way two sets could be shuffled together) as the _shuffle product_ (represented by the Cyrillic symbol: $\shuffle$) of those two sets[3, p.126]. The shuffle product is therefore a central principle of study within shuffle algebras and, although treated abstractly within pure combinatorics, is concretely applicable to many sciences. While finding the shuffle product for two small sets is relatively easy, doing so for larger sets rapidly increases in complexity as the size of the problem set grows. The number of individual permutations which compose a shuffle product may be determined by the formula: $g(x,y)=\frac{(x+y)!}{x!y!}$ where $x$ and $y$ are the number of elements in each set, respectively. As can be seen in _figs. 2 & 2_, the factorial growth rate of this function causes the result set to become rapidly unmanageable as the size of both sets grows; thus illustrating the need for an efficient algorithmic solution, with little or no effort expended on invalid permutations and with minimal overhead. Figure 1: The rate of growth as $x$ and $y$ increase Figure 2: _fig.1_ without clipping, and scaled So far as I have been able to determine, there does not currently exist an algorithm specialized for solving this problem in either an efficient or an inefficient way. This may be due to the fact that within the realm of Shuffle Algebras there has up until now, not been a pressing need to know what each permutation is for large sets, since the principles could be explored and defined with more manageable sizes. However the explicit definition of each permutation may prove more useful to applications where concrete data is needed instead of broad abstractions. The approach described in this paper provides precisely the sort of efficient solution needed to solve this problem. This is done by simplifying the encoding of the problem set to the most elementary format possible. By representing the elements of each set as, respectively, 1s and 0s each permutation can be represented as a binary integer. Once converted to binary format, a standard CPU is now well suited for calculating an exhaustive list of the shuffle product’s elements. Then, by recursively doubling and subtracting from the integer, each permutation can be found through an atomic operation. The resulting collection of integers then represents each permutation, or element which makes up the shuffle product, which for the sake of simplicity we will merely refer to as: the shuffle product. ### 1.1 Summary of Contributions * • _The need for and conception of the algorithm._ Circumstances surrounding the initial need for finding this algorithm and the process of devising it. * • _Definition of the algorithm’s design and functionality._ Detailed explanation of the algorithm; its structure and how it is used. * • _Observations on the resulting tree graph traversed by the algorithm._ Data collected on the resulting information from the algorithm, and observations on its properties. * • _Hypothesis on the generic nature and future applications to n-dimensional problem sets._ Conclusions drawn from this work. ## 2 The Algorithm Referred to here simply as: the algorithm (for lack of a better name); the method described in this paper was developed by myself in consultation with Prof. Lee Barney (of Brigham Young University—Idaho) between May 2012 and July 2013, with invaluable contributions made by numerous other colleagues and professors (_see Acknowledgments_). It provides a wholly specialized approach to mathematically discovering the complete shuffle product of two sets, and should provide a new perspective on this area of study. ### 2.1 Tic-Tac-Toe and the Union of Xs and Os The development of the algorithm had its inception with the consideration of a seemingly simple problem: what are all the possible configurations for the end-state of a Tic-Tac-Toe board? As we considered this exercise, we had to first define what it was we really wanted. Since Tic-Tac-Toe (AKA Naughts and Crosses) is played on a $3\times 3$ board (thus having an odd number of spaces) the first player must naturally be able to place one more mark on the board than the second, resulting in five total moves possible for player one and four total moves possible for player two. A Tic-Tac-Toe game, however, is only played until one player gets three marks in a row or all the spaces on the board are occupied (resulting in a tie or _cat’s game_); yet it is still possible for someone to win and also have filled every space on the board in doing so. Because of this we could assume that any game which did not fill all the spaces was only a super-set of other filled boards—since we did not care how the game was played, rather only how the marks on the board were configured in the end. Therefore, all five of player one’s marks would be present in every configuration as well as all four of player two’s marks. Furthermore, we decided that the arrangement of the spaces in a $3\times 3$ grid was arbitrary (albeit very useful for game-play) and that the spaces could just as easily be arranged in a line, so long as each was treated as having a unique identity. At this point, our question could be rephrased to be: what are all the possible ways a homogeneous set of five objects can be interlaced (or shuffled) with a homogeneous set of four objects? What we had in this form was essentially a question of _Combinatorics on Words_ ; the bread and butter of Shuffle Algebras. $A=OOOO$ $B=XXXXX$ $A\shuffle B=\emph{?}$ Figure 3: Finding the Shuffle Product of Tic-Tac-Toe’s marks We knew that we could expect to find 126 unique permutations for the union of sets A and B or, in other words, 126 elements in the shuffle product of sets A and B (see _fig. 3_) (from the previously stated formula of $g(x,y)=\frac{(x+y)!}{x!y!}$). However with a result set of that size it became obvious that an algorithm would have to be devised to methodically identify each permutation. The first question then was how to efficiently encode the data for computation. While the natural choice for representing sets in a computer program is with a data collection of some sort, the two homogeneous sets seemed to naturally relate to a binary integer where the 1s and 0s represented Xs and Os, respectively. Once encoded as such, an entire permutation could be represented as a single integer, and it was hypothesized that a mathematical pattern could be found which would allow us to predict the next integer which represented a valid permutation—starting with the smallest valid integer (000011111 or 31). A valid integer was defined as any which had the correct number of 1s and 0s within the first 9 bits (and no 1s beyond that). An early approach involved attempting to find a pattern in the intervals between valid integers as one counted from the lowest to the highest, however no consistent pattern could be observed between 000011111 (31) and 111110000 (496) and any patterns that were observed seemed to be increasing infinitely in complexity. The alternate approach involved manipulating the bits in the integers through successive shifts and subtractions, searching for patterns which would avoid duplicates and invalid integers. Following this approach I was able to design and successfully implement the algorithm which is the subject of this paper. While its initial design was still constrained to the set sizes of the Tic- Tac-Toe problem, it later (after further analysis) was generalized to support any sized sets. ### 2.2 Design and Implementation The algorithm consists of two main operations: shifts and subtractions. These are performed on binary integers (representing permutations of union of the two sets) recursively through _Mutual Recursion_ with each shift leading to a subtraction which in turn leads to a shift until the base cases are reached. Each atomic operation (be it a shift or a subtraction) will yield a valid permutation or element of the shuffle product, which can be either printed to the console or stored for later access. These binary integers can then be used as logical mappings or masks for actual permutations which can be made from the two sets. To begin, the sets are encoded as a binary integer, with all the elements of the first set represented by 1s and all the elements of the second set represented by 0s. As will be shown later, it does not matter which set is represented by the 1s and which is represented by the 0s, since the results of either representation will be isomorphic to each other. When the sets are encoded they are configured into the first (and smallest) valid integer: a permutation with all the 1s grouped to the right and all 0s to the left (in the Tic-Tac-Toe example this would be 000011111). Once encoded as such it is passed into the algorithm so that each successive permutation can then be identified. The algorithm begins by doubling the integer (in other words: shifting the bits) until all the 1s reside on the left side of the significant region (see 2nd Caveat). $000011111\ll 1\Longrightarrow 000111110\ll 1\Longrightarrow 001111100\ll 1\Longrightarrow 011111000\ll 1\Longrightarrow 111110000$ Figure 4: Successive bit shifts After each one of these shifts is performed, the resulting integer is then subjected to series of subtractions which will cause the grouping of 0s on the right-hand side to cascade through the contiguous 1s until a single 1 remains between it and the left-hand 0s (so as to prevent duplicates). In order to do this, the amount subtracted doubles each time, and the initial amount subtracted is doubled plus 1 for each successive shift result (starting with 0). $000111110-1\Longrightarrow 000111101-10\Longrightarrow 000111011-100\Longrightarrow 000110111-1000\Longrightarrow 000101111$ Figure 5: Successive subtractions on first shift result $001111100-11\Longrightarrow 001111001-110\Longrightarrow 001110011-1100\Longrightarrow 001100111-11000\Longrightarrow 001001111$ Figure 6: Successive subtractions on second shift result Each of these subtraction results is then shifted again, until all the 1s reside on the left side of the significant region. $001111001\ll 1\Longrightarrow 011110010\ll 1\Longrightarrow 111100100$ Figure 7: Successive bit shifts on first subtract result of _fig. 6_ The subtractions which follow these are then performed according to the same rules which governed the previous set of subtractions. The initial amount is also doubled plus one for each successive shift result (starting with double the previous subtraction value in its lineage on the recursive tree). $011110010-1101\Longrightarrow 011100101-11010\Longrightarrow 011001011-110100\Longrightarrow 010010111-1101000$ Figure 8: Successive subtractions on first shift result of _fig. 7_ These shifts and subtractions are repeatedly performed on each number so as to find every possible permutation; and as can be seen in _algorithm 1_: when put together these steps form a simple and elegant process for finding the shuffle product. The base cases which prevent the recursion depth from increasing infinitely (and thus also prevent duplicates and overflow) are based on the size of the sets passed in. Any given branch in the recursive tree is constrained to only have so many shifts and so many subtractions before it cannot go deeper (without generating duplicates). The maximum number of shifts allowed is equal to the size of the first set (represented by 0s) and the maximum number of subtractions allowed is equal to the size of the second set (represented by 1s) minus 1. Without these controls the shifts would cause the bits to spill outside out the significant region (or even the allocated bits of the data- type), and the subtractions would eventually begin to yield duplicate values. As shown in _algorithm 1_, the design for the algorithm uses Tail-Recursion in conjunction with its Mutually-Recursive architecture. Depending on the language in which it is implemented, this may or may not provide some improvement to overall performance. This depends on whether the language used provides any sort of _tail optimization_ so as to conserve stack space and related overhead. More on the algorithms execution and subsequent overhead will be covered in Section 2.3 _Execution_. Algorithm 1 Pseudo-Code representation of the algorithm. Key: $p$ = initial valid integer/permutation $\|A\|$ = size of set $A$ $\|B\|$ = size of set $B$ $v$ = subtraction value $i$ = current shift count $j$ = current subtraction count $C$ = set for storing permutations $\|A\|>0$ $\|B\|>0$ $v=0$ $i=0$ $j=0$ $C$ already contains initial value of $p$ function shift($p$, $\|A\|$, $\|B\|$, $v$, $i$, $j$, $C$) $\triangleright$ the _shift_ function is always called first for all $\|A\|$ do if $i\geq\|A\|$ then $\triangleright$ test for base-case return end if $p\leftarrow p\ll 1$ $\triangleright$ perform shift $i\leftarrow i+1$ Store $p$ in $C$ $v\leftarrow(v\ll 1)+1$ subtract($p$, $\|A\|$, $\|B\|$, $v$, $i$, $j$, $C$) end for end function function subtract($p$, $\|A\|$, $\|B\|$, $v$, $i$, $j$, $C$) for all $\|B\|-1$ do if $j\geq\|B\|-1$ then $\triangleright$ test for base-case return end if $p\leftarrow p-v$ $\triangleright$ perform subtraction $j\leftarrow j+1$ Store $p$ in $C$ $v\leftarrow(v\ll 1)+1$ shift($p$, $\|A\|$, $\|B\|$, $v$, $i$, $j$, $C$) end for end function As mentioned previously, it does not matter which of the sets is represented by the 1s and which is represented by 0s. This can be proven by first: running the algorithm twice, with the inputs swapped; then performing a bitwise XOR operation on each of the elements in one of the outputs against a mask of 1s the size of the significant region; when the untouched output and the XORed output are sorted it becomes readily apparent that they are identical. (see _fig. 9_) $\|A\|=3$ $\|B\|=2$ $A\shuffle B=C=\\{3,6,5,10,20,12,9,18,24,17\\}$ $B\shuffle A=D=\\{7,14,13,26,21,11,22,28,25,19\\}$ $E=C_{n}\oplus 31=\\{28,25,26,21,11,19,22,13,7,14\\}$ $sort(E)=\\{7,11,13,14,19,21,22,25,26,28\\}$ $sort(D)=\\{7,11,13,14,19,21,22,25,26,28\\}$ Figure 9: Example showing the isomorphic nature between the outputs of the algorithm called with swapped inputs. #### 2.2.1 Caveats 1. 1. The term _Set_ is used here loosely. While in Set Theory a set is strictly a collection of unique objects, for our purposes they may or may not be unique. The algorithm is only concerned with how the elements of two sets may be interlaced; irrespective of what each element is or its unique identity. Although, it may be possible that the algorithm can be modified to support permutations within a set, through treating each unique or individual object as a further dimension of the problem set (see Section 5 _Future Work_). 2. 2. Due to the nature of computer data-types, it is likely that the total elements of the combined sets is not equal to the amount of bits in any available type. Because of this it is necessary to use a data-type larger than the needed bits. The left most bits totaling the amount needed to encode both sets is then treated as the _Significant Region_ and all other bits must be turned off. Because of this, it is important to only use unsigned data-types so as to avoid increasing the needed type-size before absolutely necessary. It is also possible for the total elements of the combined sets to be larger than an available data-type or the processor’s word-size. For those cases a special library will have to be used to compensate for the platform’s limitations. However, as will be explained in Section 2.3 _Execution_ , one is likely to encounter storage limitations long before this. ### 2.3 Execution As was explained in the introduction and _figs. 1 & 2_, the number of elements (or permutations) in the shuffle product grows factorially as the size of both sets grow (so long as a consistent ratio is maintained between them). $(\|A\|=$ \| 1 \| & \| $\|B\|=1)$ \| $\rightarrow\|\shuffle\|=2$ ---\|---\|---\|---\|--- $(\|A\|=$ \| 5 \| & \| $\|B\|=5)$ \| $\rightarrow\|\shuffle\|=252$ $(\|A\|=$ \| 10 \| & \| $\|B\|=10)$ \| $\rightarrow\|\shuffle\|=184,756$ $(\|A\|=$ \| 20 \| & \| $\|B\|=20)$ \| $\rightarrow\|\shuffle\|=137,846,528,820$ $(\|A\|=$ \| 40 \| & \| $\|B\|=40)$ \| $\rightarrow\|\shuffle\|=107,507,208,733,336,176,461,620$ $(\|A\|=$ \| 2 \| & \| $\|B\|=5)$ \| $\rightarrow\|\shuffle\|=21$ $(\|A\|=$ \| 4 \| & \| $\|B\|=10)$ \| $\rightarrow\|\shuffle\|=1,001$ $(\|A\|=$ \| 8 \| & \| $\|B\|=20)$ \| $\rightarrow\|\shuffle\|=3,108,105$ $(\|A\|=$ \| 16 \| & \| $\|B\|=40)$ \| $\rightarrow\|\shuffle\|=41,648,951,840,265$ $(\|A\|=$ \| 32 \| & \| $\|B\|=80)$ \| $\rightarrow\|\shuffle\|=10,484,776,488,844,408,407,191,115,273$ ($>10$ octillion) Figure 10: Example of increasing result-set sizes showing factorial growth, with $\shuffle$ representing the shuffle product set Since the number of steps required in the algorithm has a direct correlation to the number of elements in the result set, we can conclude that the algorithm possesses a cost of O$(\frac{(n+m)!}{n!m!})$. Although being a very rapid growth rate for execution time, it is none the less significantly optimized for the problem it solves. Additionally, despite the rapid growth of the result set, the algorithm benefits from a relatively shallow recursion depth; with a maximum depth of $\|A\|+\|B\|-1$. However, the factorial growth in the result set raises serious issues regarding the storage and memory space required to handle all of the permutations. Despite the efficient encoding of each permutation, the amount of storage required becomes very difficult to manage, very quickly. While the permutations for two sets of 10 would only require just over 5 Megabytes of space (given an optimum implementation), the permutations for two sets of 32 would require over 100 Exabytes of storage. The amount of storage required for the complete shuffle product (or permutations) of any two sets, can be determined through the following formula (given an optimum implementation): $f(x,y)=\left(\frac{(x+y)!}{x!y!}\right)\left(\Big{\lceil}\frac{2^{\lceil log_{2}{(x+y)}\rceil}}{8}\Big{\rceil}\right)$ where $x$ and $y$ are the number of elements in each set, respectively. Considering current hardware limitations, these issues present significant challenges for execution as the sets grow in size, and must be taken into account when implementing the algorithm towards most any application. ### 2.4 The Tree Graph and its Properties While the algorithm was conceptualized and later developed, an edge & vertex graph was used to represent the paths leading to each permutation in the shuffle product. This graph was used in the analysis of patterns which were emerging as we tried different methods to manipulate the data in search of the next permutation. Vertical lines (or edges) were used between vertexes to represent bit shifts, while horizontal lines were used to represent subtractions, providing a simple way to keep track of each path. The direction of each of these edges alternated in order to conserve space. Figure 11: Fragment of a utilitarian graph used in the design of the algorithm. Eventually, as the complexity of the graph increased, we began to notice the emergence of a larger pattern which appeared fractal-like in its nature. In order to more easily study this pattern, the graph was converted to a more traditional graph with dots for each vertex. Once organized in this way a design that was both complex and very elegant became clearly visible. Figure 12: Formal graph of the algorithm’s entire execution on the Tic-Tac-Toe example Additionally, this layout explicitly demonstrates how the algorithm categorizes each permutation it finds. Every permutation belongs to a discrete branch of the tree and therefore possesses a unique relationship with permutations in that same branch. The recursive depth of a permutation—which can be identified by how many turns exist in its path from the initial permutation—also provides a metric which might be used to help calculate the level of shuffling within a permutation in relation to all others in the shuffle product (see Section 5 _Future Work_). Traditionally in Graph Theory, of course, the layout of the graph does not matter as a graph of any physical layout can be proven to be isomorphic to any other graph with an identical adjacency-matrix. The precise layout of this graph then is not necessary to the meaning of its information. However one’s understanding and analysis of this graph is greatly assisted by applying some set of rules to distinguish shifts from subtractions—although the specific rules used in this paper are arbitrary and may be replaced by any rules considered useful to one’s analysis. Due to the observations we can make by representing the algorithm and its result set in this way, it is possible that it is showing some hint towards a broader principle of combinatorics which likely merits further study. ## 3 Performance Experiments Real-world benchmarks were taken for the algorithm, using an x86_64 processor clocked at 2.2 GHz. The algorithm was implemented in ANSI C and clocked in CPU time. Speeds were taken for symetric sets of 1, 5, 10 & 11 (sets larger than this were excluded since implementations which supported larger pairs required additional overhead which would skew the data). The sets of 1 took, on average, 0.000002 seconds of CPU time to compute. At 5 the average climbed to 0.000006 seconds; at 10 it climbed to approximately 0.002750 seconds, and at 11: approximately 0.009950 seconds. A more thorough examination of the execution time under various conditions can and should be made; however, it exceeds the scope of this paper. It suffices to say that the observed data is in line with the predictions made about the algorithm’s cost in execution time. ## 4 Conclusions The algorithm presented in this paper has been shown to be a viable solution for a number of different inputs and, as such, I present it to the scientific community as a conjecture for how to find the elements of the shuffle product of any two sets. I hypothesize that because of the fractal-like nature of the tree graph which represents the algorithm—and since natural phenomena are often accompanied by fractals—the algorithm begins to shed light on a more generic principle of Shuffle Algebras and Combinatorics, which may increase our understanding of this area thorough further analysis. Additional study on this algorithm is still needed in order to construct a rigorous proof and to understand the full extent of its applications. However I trust that, regardless of what findings may yet be made, it will prove to be a valuable discovery for the study of Shuffle Algebras and Combinatorics. ## 5 Future Work Suggestions for future work which may be done with the findings of this paper include: 1. 1. _Construct a rigorous proof for the conjectured algorithm_. 2. 2. _Generalization to handle finding the shuffle product for $n$ sets_. If the tree for the shuffle product of two sets is two dimensional, we can assume that the tree for the shuffle product of $n$ sets will be $n$-dimensional. In order to manage these additional sets, it may be necessary to find a new way to encode the data and adjust the operations accordingly. A mask might also be used to keep track of the elements of additionals sets. On the other hand, it is possible that a series of successive shuffle products could be found for each dimension. 3. 3. _Design of a formula to produce a metric for the level of shuffling within a permutation_. Assuming that having all elements of both sets on either side of the string of elements is the least shuffled state for a permutation, it may be possible to design a formula, using the recursion depth at which a permutation is found, to calculate the level of shuffling a permutation possesses in relation to all other permutations in the shuffle product. 4. 4. _Further analysis of the properties of the tree graph_. Using graph analysis there is likely room for additional study of the fractal—and other—properties of the graph and what significance they might have on Shuffle Algebras and other branches of Combinatorics. ## 6 Acknowledgments I would like to publicly acknowledge the following people and thank them for their invaluable help, support and contributions: * Lee Barney (Professor of Computer Information Technology, BYU-Idaho) * Rick Neff (Professor of Computer Science, BYU-Idaho) * Dave Brown (Professor of Mathematics, BYU-Idaho) * Kory Godfrey (Professor of Computer Information Technology, BYU-Idaho) * Michael MacLochlain (Professor of Computer Information Technology, BYU-Idaho) * Rex Barzee (Professor of Computer Information Techonology, BYU-Idaho) * Will Graham (Undergraduate, BYU-Idaho) * Ben Barreto (Undergraduate, BYU-Idaho) * Ben Williams (Undergraduate, BYU-Idaho) * Aaron Andrews (Undergraduate, BYU-Idaho) ## References * [1] Eilenberg, S., and MacLane, S. On the groups h(ii, n), i. Annals of Mathematics 58, 1 (Jul 1953), 55–106. * [2] Foissy, L., and Patras, F. Natural endomorphisms of shuffle algebras. arXiv (May 2012). arXiv Identifier: arXiv:1205.2986v1 [math.RA]. * [3] Lothaire, M. Combinatorics on Words. Cambridge University Press, 1983.	arxiv-papers	2014-01-07T17:30:05	2024-09-04T02:49:56.376147	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Diego Fernando C. Carri\\'on L", "submitter": "Diego Fernando C. Carri\\'on L.", "url": "https://arxiv.org/abs/1401.1450" }
1401.1460	λ$\lambda$ Maximal Sharing in the Lambda Calculus with letrec Clemens Grabmayer Dept. of Computer Science, VU University Amsterdam de Boelelaan 1081a, 1081 HV Amsterdam [email protected] Jan Rochel Dept. of Computing Sciences, Utrecht University Princetonplein 5, 3584 CC Utrecht, The Netherlands [email protected] [COPYRIGHTDATA] # Maximal Sharing in the Lambda Calculus with letrec 111 This work was supported by NWO in the framework of the project _Realising Optimal Sharing (ROS)_ , project number 612.000.935. (2014) ###### Abstract Increasing sharing in programs is desirable to compactify the code, and to avoid duplication of reduction work at run-time, thereby speeding up execution. We show how a maximal degree of sharing can be obtained for programs expressed as terms in the lambda calculus with letrec. We introduce a notion of ‘maximal compactness’ for $\lambda_{\text{\sf letrec}}$-terms among all terms with the same infinite unfolding. Instead of defined purely syntactically, this notion is based on a graph semantics. $\lambda_{\text{\sf letrec}}$-terms are interpreted as first-order term graphs so that unfolding equivalence between terms is preserved and reflected through bisimilarity of the term graph interpretations. Compactness of the term graphs can then be compared via functional bisimulation. We describe practical and efficient methods for the following two problems: transforming a $\lambda_{\text{\sf letrec}}$-term into a maximally compact form; and deciding whether two $\lambda_{\text{\sf letrec}}$-terms are unfolding-equivalent. The transformation of a $\lambda_{\text{\sf letrec}}$-term $L$ into maximally compact form $L_{0}$ proceeds in three steps: (i) translate $L$ into its term graph $G=\llbracket{L}\rrbracket\,$; (ii) compute the maximally shared form of $G$ as its bisimulation collapse $G_{0}\,$; (iii) read back a $\lambda_{\text{\sf letrec}}$-term $L_{0}$ from the term graph $G_{0}$ with the property $\llbracket{L_{0}}\rrbracket=G_{0}$. This guarantees that $L_{0}$ and $L$ have the same unfolding, and that $L_{0}$ exhibits maximal sharing. The procedure for deciding whether two given $\lambda_{\text{\sf letrec}}$-terms $L_{1}$ and $L_{2}$ are unfolding-equivalent computes their term graph interpretations $\llbracket{L_{1}}\rrbracket$ and $\llbracket{L_{2}}\rrbracket$, and checks whether these term graphs are bisimilar. For illustration, we also provide a readily usable implementation. ###### category: D.3.3 Language constructs and features Recursion ###### category: F.3.3 Studies of Programming Constructs Functional constructs ††conference: ICFP 2014 September 1–3 2014, Gothenburg, Sweden.††terms: f unctional programming, compiler optimisation ## 1 Introduction Explicit sharing in pure functional programming languages is typically expressed by means of the letrec construct, which facilitates cyclic definitions. The $\lambda$-calculus with letrec, $\lambda_{\text{\sf letrec}}$ forms a syntactic core of these languages, and it can be viewed as their abstraction. As such $\lambda_{\text{\sf letrec}}$ is well-suited as a test bed for developing program transformations in functional programming languages. This certainly holds for the transformation presented here that has a strong conceptual motivation, is justified by a form of semantic reasoning, and is best described first for an expressive, yet minimal language. ### 1.1 Expressing sharing and infinite $\lambda$-terms For the programmer the letrec-construct offers the possibility to write a program more compactly by utilising subterm sharing. letrec-expressions bind subterms to variables; these variables then denote occurrences of the respective subterms and can be used anywhere inside of the letrec-expression (also recursively). In this way, instead of repeating a subterm multiple times, a single definition can be given which is then referenced from multiple positions. We will denote the letrec-construct here by let as in Haskell. ###### Example 1.1. Consider the $\lambda$-term ${(\lambda{x}.\,{x})}\hskip 1.5pt{(\lambda{x}.\,{x})}$ with two occurrences of the subterm $\lambda{x}.\,{x}$. These occurrences can be shared with as result the $\lambda_{\text{\sf letrec}}$-term $({\text{\sf let}}\;{id=\lambda{x}.\,{x}}\;{\text{\sf in}}\;{{id}\hskip 1.5pt{\leavevmode\nobreak\ id}})$. As let-bindings permit definitions with cyclic dependencies, terms in $\lambda_{\text{\sf letrec}}$ are able to finitely denote infinite $\lambda$-terms (for short: $\lambda^{\infty}$-terms). The $\lambda^{\infty}$-term $M$ represented by a $\lambda_{\text{\sf letrec}}$-term $L$ can be obtained by a typically infinite process in which the let-bindings in $L$ are unfolded continually with $M$ as result in the limit. Then we say that $M$ is the _infinite unfolding_ of $L$, or that $M$ is the denotation of $L$ in the _unfolding semantics_ , indicated symbolically by $M=\llbracket{L}\rrbracket_{{\lambda^{\infty}}}$. ###### Example 1.2. For the $\lambda_{\text{\sf letrec}}$-terms $L$ and $P$ and the $\lambda^{\infty}$-term $M$: $\displaystyle\begin{aligned} L&\mathrel{{:=}}\lambda{f}.\,{{\text{\sf let}}\;{r={f}\hskip 1.5pt{r}}\;{\text{\sf in}}\;{r}}\\\ P&\mathrel{{:=}}\lambda{f}.\,{{\text{\sf let}}\;{r={f}\hskip 1.5pt{({f}\hskip 1.5pt{r})}}\;{\text{\sf in}}\;{r}}\end{aligned}$ $\displaystyle M$ $\displaystyle\mathrel{{:=}}\lambda{f}.\,{{f}\hskip 1.5pt{({f}\hskip 1.5pt{(\dots)})}}$ it holds that both $L$ and $P$ (which represent fixed-point combinators) have $M$ as their infinite unfolding: $\llbracket{L}\rrbracket_{{\lambda^{\infty}}}=\llbracket{P}\rrbracket_{{\lambda^{\infty}}}=M$. $L$ and $P$ in this example are ‘unfolding equivalent’. Note that $L$ represents $M$ in a more compact way than $P$. It is intuitively clear that there is no $\lambda_{\text{\sf letrec}}$-term that represents $M$ more compactly than $L$. So $L$ can be called a ‘maximally shared form’ of $P$ (and of $M$). We address, and efficiently solve, the problems of computing the maximally shared form of a $\lambda_{\text{\sf letrec}}$-term, and of determining whether two $\lambda_{\text{\sf letrec}}$-terms are unfolding-equivalent. Note that these notions are based on the static unfolding semantics. We _do not consider_ any dynamic semantics based on evaluation by $\beta$-reduction or otherwise. ### 1.2 Recognising potential for sharing A general risk for compilers of functional programs is “[to construct] multiple instances of the same expression, rather than sharing a single copy of them. This wastes space because each instance occupies separate storage, and it wastes time because the instances will be reduced separately. This waste can be arbitrarily large, […]” ([Peyton Jones, 1987, p.243]). Therefore practical compilers increase sharing, and do so typically for supercombinator translations of programs (such as fully-lazy lambda-lifting). Thereby two goals are addressed: to increase sharing based on a syntactical analysis of the ‘static’ form of the program; and to prevent splits into too many supercombinators when an anticipation of the program’s ‘dynamic’ behaviour is able to conclude that no sharing at run-time will be gained. A well-known method for the ‘static’ part is common subexpression elimination (CSE) Chitil [1998]. For the ‘dynamic’ part, a predictive syntactic program analysis has been proposed for fine-tuning sharing of partial applications in supercombinator translations Goldberg and Hudak [1987]. We focus primarily on the ‘static’ aspect of introducing sharing. We provide a conceptual solution that substantially extends CSE. But instead of maximising sharing for a supercombinator translation of a program, we carry out the optimisation on the program itself (the $\lambda_{\text{\sf letrec}}$-term). And instead of applying a purely syntactical program analysis, we use a term graph semantics for $\lambda_{\text{\sf letrec}}$-terms. ### 1.3 Approach based on a term graph semantics We develop a combination of techniques for realising maximal sharing in $\lambda_{\text{\sf letrec}}$-terms. For this we proceed in four steps: $\lambda_{\text{\sf letrec}}$-terms are interpreted as higher-order term graphs; the higher-order term graphs are implemented as first-order term graphs; maximally compact versions of such term graphs can be computed by standard algorithms; $\lambda_{\text{\sf letrec}}$-terms that represent compacted term graphs (or in fact arbitrary ones) can be retrieved by a ‘readback’ operation. In more detail, the four essential ingredients are the following: (1) A _semantics_ $\llbracket{\cdot}\rrbracket_{{\cal H}}$ for interpreting $\lambda_{\text{\sf letrec}}$-terms as _higher-order term graphs_ , which are first-order term graphs enriched with a feature for describing binding and scopes. We call this specific kind of higher-order term graphs ‘$\lambda$-ho- term-graphs’. The variable binding structure is recorded in this term graph concept because it must be respected by any addition of sharing. The term graph interpretation adequately represents sharing as expressed by a $\lambda_{\text{\sf letrec}}$-term. It is not injective: a $\lambda$-ho-term-graph typically is the interpretation of various $\lambda_{\text{\sf letrec}}$-terms. Different degrees of sharing as expressed by $\lambda_{\text{\sf letrec}}$-terms can be compared via the $\lambda$-ho-term-graph interpretations by a sharing preorder, which is defined as the existence of a homomorphism (functional bisimulation). While comparing higher-order term graphs via this preorder is computable in principle, standard algorithms do not apply. Therefore efficient solvability of the compactification problem and the comparison problem is, from the outset, not guaranteed. For this reason we devise a first-order implementation of $\lambda$-ho-term-graphs: (2) An _interpretation_ ${\cal HT}$ of $\lambda$-ho-term-graphs into a specific kind of _first-order term graphs_ , which we call ‘$\lambda$-term-graphs’. It preserves and reflects the sharing preorder. ${\cal HT}$ reduces bisimilarity between $\lambda$-ho-term-graphs (higher- order) to bisimilarity between $\lambda$-term-graphs (first-order), and facilitates: (3) The use of standard methods for _checking_ bisimilarity and for computing the bisimulation _collapse_ of $\lambda$-term-graphs. Via ${\cal HT}$ also the analogous problems for $\lambda$-ho-term-graphs can be solved. Term graphs can be represented as deterministic process graphs (labelled transition systems), and even as deterministic finite-state automata (DFAs). That is why it is possible to apply efficient algorithms for state minimisation and language equivalence of DFAs. Finally, an operation to return from term graphs to $\lambda_{\text{\sf letrec}}$-terms: (4) A _readback_ function rb from $\lambda$-term-graphs to $\lambda_{\text{\sf letrec}}$-terms that, for every $\lambda$-term-graph $G$, computes a $\lambda_{\text{\sf letrec}}$-term $L$ from the set of $\lambda_{\text{\sf letrec}}$-terms that have $G$ as their interpretation via $\llbracket{\cdot}\rrbracket_{{\cal H}}$ and ${\cal HT}$ (i.e. a $\lambda_{\text{\sf letrec}}$-term for which it holds that ${\cal HT}({\llbracket{L}\rrbracket_{{\cal H}}})=G$). ### 1.4 Methods and their correctness On the basis of the concepts above we develop efficient methods for introducing maximal sharing, and for checking unfolding equivalence, of $\lambda_{\text{\sf letrec}}$-terms, as sketched below. In describing these methods, we use the following notation: ${\cal H}\>$: class of $\lambda$-ho-term-graphs, the image of the semantics $\llbracket{\cdot}\rrbracket_{{\cal H}}\,$; ${\cal T}\>$: class of $\lambda$-term-graphs, the image of the interpretation ${\cal HT}\,$; $\llbracket{\cdot}\rrbracket_{{\cal T}}\mathrel{{:=}}{{\cal HT}}\mathrel{\circ}{\llbracket{\cdot}\rrbracket_{{\cal H}}}\>$: first-order term graph semantics for $\lambda_{\text{\sf letrec}}$-terms; ${{\text{\small\textbar}}\downarrow}\>$: bisimulation collapse on ${\cal H}$ and ${\cal T}$; rb : readback mapping from $\lambda$-term-graphs to $\lambda_{\text{\sf letrec}}$-terms. We obtain the following methods (for illustrations, see Fig. 1.4): * $\smalltriangleright$ _Maximal sharing_ : for a given $\lambda_{\text{\sf letrec}}$-term, a maximally shared form can be obtained by collapsing its first-order term graph interpretation, and then reading back the collapse: ${{{\textsf{rb}}}\mathrel{\circ}{{{\text{\small\textbar}}\downarrow}}}\mathrel{\circ}{\llbracket{\cdot}\rrbracket_{{\cal T}}}\,$ * $\smalltriangleright$ _Unfolding equivalence_ : for given $\lambda_{\text{\sf letrec}}$-terms $L$ and $P$, it can be decided whether $\llbracket{L}\rrbracket_{{\lambda^{\infty}}}=\llbracket{P}\rrbracket_{{\lambda^{\infty}}}$ by checking whether their term graph interpretations $\llbracket{L}\rrbracket_{{\cal T}}$ and $\llbracket{P}\rrbracket_{{\cal T}}$ are bisimilar. Figure 1: Component-step build-up of the methods for computing a maximally shared form $L_{0}$ of a $\lambda_{\text{\sf letrec}}$-term $L$ (left), and for deciding unfolding equivalence of $\lambda_{\text{\sf letrec}}$-terms $L_{1}$ and $L_{2}$ via bisimilarity $\mathrel{\hbox{\kern 0.43057pt\vbox{\hbox{\raise 0.43057pt\hbox{\kern-0.43057pt{$\leftrightarrow$}\kern-0.43057pt}}\hrule}\kern 0.43057pt}}$ (right). $\lambda{f}.\,{{\text{\sf let}}\;{r={f}\hskip 1.5pt{r}}\;{\text{\sf in}}\;{r}}$$\lambda{f}.\,{{\text{\sf let}}\;{r={f}\hskip 1.5pt{({f}\hskip 1.5pt{r})}}\;{\text{\sf in}}\;{r}}$$\lambda{f}.\,{{f}\hskip 1.5pt{({f}\hskip 1.5pt{(\dots)})}}$${{\text{\small\textbar}}\downarrow}$$\llbracket{\cdot}\rrbracket_{{\lambda^{\infty}}}$rb$\llbracket{\cdot}\rrbracket_{{\lambda^{\infty}}}$$\llbracket{\cdot}\rrbracket_{{\cal T}}$$\llbracket{\cdot}\rrbracket_{{\cal T}}$ Figure 2: Computing a maximally compact version of the term $P$ from Ex. 1.2 (right) by using composition of term graph semantics $\llbracket{\cdot}\rrbracket_{{\cal T}}$, collapse ${{\text{\small\textbar}}\downarrow}$, and readback rb, yielding the term $L$ (left). See Fig. 2 for an illustration of the application of the maximal sharing method to the $\lambda_{\text{\sf letrec}}$-terms $L$ and $P$ from Ex. 1.2. The correctness of these methods hinges on the fact that the term graph translation and the readback satisfy the following properties: 1. (P1) $\lambda_{\text{\sf letrec}}$-terms $L$ and $P$ have the same infinite unfolding if and only if the term graphs $\llbracket{L}\rrbracket_{{\cal T}}$ and $\llbracket{P}\rrbracket_{{\cal T}}$ are bisimilar. 2. (P2) The class ${\cal T}$ of $\lambda$-term-graphs is closed under homomorphism. 3. (P3) The readback rb is a right inverse of $\llbracket{\cdot}\rrbracket_{{\cal T}}$ up to isomorphism ${\simeq}$, that is, for all term graphs $G\,{\in}\,{\cal T}$ it holds: ${(\llbracket{\cdot}\rrbracket_{{\cal T}}}\mathrel{\circ}{{\textsf{rb}})}({G})\mathrel{{\simeq}}G$. _Note_ : (P2) and (P3) will be established only for a subclass ${\cal T}_{\text{eag}}$ of ${\cal T}$. Furthermore, practicality of these methods depends on the property: 1. (P4) Translation $\llbracket{\cdot}\rrbracket_{{\cal T}}$ and readback rb are efficiently computable. ### 1.5 Overview of the development In the Preliminaries (Section 2) we fix basic notions and notations for first- order term graphs. $\lambda_{\text{\sf letrec}}$-terms and their unfolding semantics are defined in Section 3. In Section 4 we develop the concept of ‘$\lambda$-ho-term-graph’, which gives rise to the class ${\cal H}$, and the higher-order term graph semantics $\llbracket{\cdot}\rrbracket_{{\cal H}}$ for $\lambda_{\text{\sf letrec}}$-terms. In Section 5 we develop the concept of first-order ‘$\lambda$-term-graph’ in the class ${\cal T}$, and define the interpretation ${\cal HT}$ of $\lambda$-ho-term-graphs into $\lambda$-term-graphs as a mapping from ${\cal H}$ to ${\cal T}$. This induces the first-order term graph semantics $\llbracket{\cdot}\rrbracket_{{\cal T}}\mathrel{{:=}}{{\cal HT}}\mathrel{\circ}{\llbracket{\cdot}\rrbracket_{{\cal H}}}$, for which we also provide a direct inductive definition. In Section 6 we define the readback rb with the desired property as a function from $\lambda$-term-graphs to $\lambda_{\text{\sf letrec}}$-terms. Subsequently in Section 7 we report on the complexity of the described methods, individually, and in total for the methods described in Subsection 1.4. In Section 8 we link to our implementation of the presented methods. Finally in Section 9 we explain easy modifications, describe possible extensions, and sketch potential practical applications. ### 1.6 Applications and scalability While our contribution is at first a conceptual one, it holds the promise for a number of practical applications: * • Increasing the efficiency of the execution of programs by transforming them into their maximally shared form at compile-time. * • Increasing the efficiency of the execution of programs by repeatedly compactifying the program at run time. * • Improving systems for recognising program equivalence. * • Providing feedback to the programmer, along the lines: ‘This code has identical fragments and can be written more compactly.’ These and a number of other potential applications are discussed in more detail in Section 9. The presented methods scale well to larger inputs, due to the quadratic bound on their runtime complexity (see Section 7). ### 1.7 Relationship with other concepts of sharing The maximal sharing method is targeted at increasing ‘static’ sharing: in the sense that a program is transformed at compile time into a version with a higher degree of sharing. It is not (at least not a priori) a method for ‘dynamic’ sharing, i.e. for an evaluator that maintains a certain degree of sharing at run time, such as graph rewrite mechanisms for fully-lazy Wadsworth [1971] or optimal evaluation Asperti and Guerrini [1998] of the $\lambda$-calculus. However, we envisage run-time collapsing of the program’s graph interpretation integrated with the evaluator (see Section 9). The term ‘maximal sharing’ stems from work on the ATERM library Brand and Klint [2007]. It describes a technique for minimising memory usage when representing a set of terms in a first-order term rewrite system (TRS). The terms are kept in an aggregate directed acyclic graph by which their syntax trees are shared as much as possible. Thereby terms are created only if they are entirely new; otherwise they are referenced by pointers to roots of sub- dags. Our use of the expression ‘maximal sharing’ is inspired by that work, but our results generalise that approach in the following ways: * • Instead of first-order terms we consider terms in a higher-order language with the letrec-construct for expressing sharing. * • Since letrec typically defines cyclic sharing dependencies, we interpret terms as cyclic graphs instead of just dags. * • We are interested in increasing sharing by bisimulation collapse instead of by identifying isomorphic sub-dags. ATERM only checks for equality of subexpressions. Therefore it only introduces horizontal sharing and implements a form of _common subexpression elimination (CSE)_ [Peyton Jones, 1987, p. 241]. Our approach is stronger than CSE: while Ex. 1.1 can be handled by CSE, this is not the case for Ex. 1.2. In contrast to CSE, our approach increases also vertical and twisted sharing 222For definitions of horizontal, vertical, and twisted sharing we refer to Blom [2001].. (see also Blom [2001]). ### 1.8 Contribution of this paper in context Blom introduces _higher-order term graphs_ Blom [2001], which are extensions of first-order term graphs by adding a scope function that assigns a set of vertices, its _scope_ , to every abstraction vertex. As a stepping stone for the methods we develop here, we use concepts and results that we described in an earlier paper Grabmayer and Rochel [2013a]. There, for interpreting $\lambda_{\text{\sf letrec}}$-terms, a modification of Blom’s higher-order term graphs (the $\lambda$-ho-term-graphsof the class ${\cal H}$) in which scopes are represented by means of ‘abstraction prefix functions’. We also investigated first-order $\lambda$-term-graphs with scope- delimiter vertices (corresponding to the class ${\cal T}$ here). In particular we examined which specific class of first-order $\lambda$-term-graphs can faithfully represent the higher-order $\lambda$-ho-term-graphs in such a way that compactification of the latter can be realised through bisimulation collapse of the former (this led to the $\lambda$-term-graphs of the class ${\cal T}$). Whereas in the paper Grabmayer and Rochel [2013a] we exclusively focused on the graph formalisms, and investigated them in their own right, here we connect the results obtained there to the language $\lambda_{\text{\sf letrec}}$ for expressing sharing and cyclicity. Since the methods presented here are based on the graph formalisms, and rely on their properties for correctness, we recapitulate the concepts and the relevant results in Sec. 4 and 5. The translation $\llbracket{\cdot}\rrbracket_{{\cal T}}$ of $\lambda_{\text{\sf letrec}}$-terms into first-order term graphs was inspired by related representations that use scope delimiters to indicate end of scopes. Such representations are generalisations of a de Bruijn index notation for $\lambda$-terms de Bruijn [1972] in which the de Bruijn indexes are numerals of the form $\mathsf{S}\hskip 0.5pt{(\ldots(\mathsf{S}\hskip 0.5pt{(\mathsf{0})})\ldots)}$. In the generalised form, due to Patterson and Bird Bird and Patterson [1999], the symbol $\mathsf{S}$ can occur anywhere between a variable occurrence and its binding abstraction. The idea to view $\mathsf{S}$ as a scope delimiter was employed by Hendriks and van Oostrom, who defined an end-of-scope symbol $\textstyle{\lambda}$ Hendriks and van Oostrom [2003]. This approach is also used in the translation of pure $\lambda$-terms (without letrec) into Lambdascope-graphs (interaction nets) on which van Oostrom defines an optimal evaluator for the $\lambda$-calculus Oostrom et al. [2004]. We have also used these first-order representations of $\lambda$-terms with scope delimiters for studying the limits of an optimising program transformation that, for a given $\lambda_{\text{\sf letrec}}$-term, contracts directly visible redexes, and, whenever possible, also contracts such redexes that are concealed by recursion Rochel and Grabmayer [2011]. The result of the optimisation should again be a $\lambda_{\text{\sf letrec}}$-term. Since this program transformation can best be defined, for a given $\lambda_{\text{\sf letrec}}$-term $L$, on the infinite $\lambda$-term $M$ that is the unfolding semantics (see Section 3) of $L$, it is crucial to know when the result of contracting a development of redexes in $M$ that corresponds to a visible or a concealed redex in $L$ can again be written as a $\lambda_{\text{\sf letrec}}$-terms. This suggested the question: how can those infinite $\lambda$-terms be characterised that are expressible by $\lambda_{\text{\sf letrec}}$-terms in the sense that they arise as the unfolding semantics of a $\lambda_{\text{\sf letrec}}$-term? We answered this question for $\lambda_{\text{\sf letrec}}$-terms in the report Grabmayer and Rochel [2012], and, obtaining the same answer, for $\lambda_{\mu}$-terms in the article Grabmayer and Rochel [2013b] with accompanying report Grabmayer and Rochel [2013c]. We defined a rewrite system that decomposes $\lambda$-terms by steps that ‘observe’ $\lambda$-abstractions, applications, variable occurrences, and end of scopes. We showed that infinite $\lambda$-terms that are the unfolding semantics of $\lambda_{\text{\sf letrec}}$-terms or of $\lambda_{\mu}$-terms are precisely those that have only finitely many ‘generated subterms’, that is, reducts in the decomposition rewrite system. ## 2 Preliminaries By $\mathbb{N}$ we denote the natural numbers including zero. For words $w$ over an alphabet $A$, the length of $w$ is denoted by ${\left\|{w}\right\|}$. Let $\Sigma$ be a TRS-signature Terese [2003] with arity function ${\mit ar}\mathrel{:}\Sigma\to\mathbb{N}$. A _term graph over $\Sigma$_ (or a _$\Sigma$ -term-graph_) is a tuple $\langle V,\mathit{lab},\mathit{args},\mathit{r}\rangle$ where: $V$ is a set of _vertices_ , $\mathit{lab}\mathrel{:}V\to\Sigma$ the _(vertex) label function_ , $\mathit{args}\mathrel{:}V\to V^{}$ the _argument function_ that maps every vertex $v$ to the word $\mathit{args}({v})$ consisting of the ${\mit ar}({\mathit{lab}({v})})$ successor vertices of $v$ (hence ${\left\|{\mathit{args}({v})}\right\|}={\mit ar}({\mathit{lab}({v})})$), and $\mathit{r}$, the _root_ , is a vertex in $V$. Term graphs may have infinitely many vertices. Let $G$ be a term graph over signature $\Sigma$. As useful notation for picking out an arbitrary vertex, or the $i$-th vertex, from among the ordered successors of a vertex $v$ in $G$, we define for each $i\in\mathbb{N}$ the indexed edge relation ${\rightarrowtail_{i}}\subseteq{V\times V}$, and additionally the (not indexed) edge relation ${{\rightarrowtail}}\subseteq V\times V$, by stipulating for all $w,w^{\prime}\in V$: $\begin{aligned} w\mathrel{\rightarrowtail_{i}}w^{\prime}\,\mathrel{:}&\Leftrightarrow\,\exists{w_{0},\ldots,w_{n}\in V\\!}.\;{\mathit{args}({w})=w_{0}\cdots w_{n}\mathrel{{\wedge}}w^{\prime}=w_{i}}\\\ w\mathrel{{\rightarrowtail}}w^{\prime}\,\mathrel{:}&\Leftrightarrow\,\exists{i\in\mathbb{N}}.\;{\,w\rightarrowtail_{i}w^{\prime}}\end{aligned}$ A _path_ in $G$ is described by $w_{0}\mathrel{\rightarrowtail_{k_{1}}}w_{1}\mathrel{\rightarrowtail_{k_{2}}}{}\cdots{}\mathrel{\rightarrowtail_{k_{n}}}w_{n}$, where $w_{0},w_{1},\ldots,w_{n}\in V$ and $n,k_{1},k_{2},\ldots,k_{n}\in\mathbb{N}$. An _access path_ of a vertex $w$ of $G$ is a path that starts at the root of $G$, ends in $w$, and does not visit any vertex twice. Access paths need not be unique. A term graph is _root- connected_ if every vertex has an access path. _Note:_ By a ‘term graph’ we will, from now on, always mean a root-connected term graph. Let $G_{1}=\langle V_{1},\mathit{lab}_{1},\mathit{args}_{1},\mathit{r}_{1}\rangle$, $G_{2}=\langle V_{2},\mathit{lab}_{2},\mathit{args}_{2},\mathit{r}_{2}\rangle$ be term graphs over signature $\Sigma$, in the sequel. A _bisimulation_ between $G_{1}$ and $G_{2}$ is a relation $R\subseteq V_{1}\times V_{2}$ such that the following conditions hold, for all $\langle w,\hskip 0.5ptw^{\prime}\rangle\in R$: $\left.\begin{array}[]{cr}\langle\mathit{r}_{1},\hskip 0.5pt\mathit{r}_{2}\rangle\in R&(\text{roots})\\\\[1.29167pt] \mathit{lab}_{1}({w})=\mathit{lab}_{2}({w^{\prime}})&(\text{labels})\\\\[1.29167pt] \langle\mathit{args}_{1}({w}),\hskip 0.5pt\mathit{args}_{2}({w^{\prime}})\rangle\in R^{}&\hskip 21.52771pt(\text{arguments})\end{array}\quad\right\\}$ (4) where the extension $R^{}\subseteq{V_{1}^{}}\times{V_{2}^{}}$ of $R$ to a relation between words over $V_{1}$ and words over $V_{2}$ is defined as: $R^{}\\!\mathrel{{:=}}\\!\parbox[t]{205.0pt}{\\{$\langle w_{1}\cdots w_{k},\hskip 0.5ptw^{\prime}_{1}\cdots w^{\prime}_{k}\rangle\;\\!\mid\\!$ \\\ \hskip 25.00003pt $w_{1},\ldots,w_{k}\in V_{1},w^{\prime}_{1},\ldots,w^{\prime}_{k}\in V_{2}$, \\\ $\text{for $k\in\mathbb{N}$ such that\leavevmode\nobreak\ }\langle w_{i},\hskip 0.5ptw^{\prime}_{i}\rangle\in R\text{ for all $1\leq i\leq k$}\\}$.}$ We write $G_{1}\mathrel{\mathrel{\hbox{\kern 0.43057pt\vbox{\hbox{\raise 0.43057pt\hbox{\kern-0.43057pt{$\leftrightarrow$}\kern-0.43057pt}}\hrule}\kern 0.43057pt}}}G_{2}$ if there is a bisimulation between $G_{1}$ and $G_{2}$, and we say, in this case, that $G_{1}$ and $G_{2}$ are _bisimilar_. Bisimilarity $\mathrel{\hbox{\kern 0.43057pt\vbox{\hbox{\raise 0.43057pt\hbox{\kern-0.43057pt{$\leftrightarrow$}\kern-0.43057pt}}\hrule}\kern 0.43057pt}}$ is an equivalence relation on term graphs. A _functional bisimulation_ from $G_{1}$ to $G_{2}$ is a bisimulation that is the graph of a function from $V_{1}$ to $V_{2}$. An alternative characterisation of this concept is that of _homomorphism_ from $G_{1}$ to $G_{2}$: a morphism from the structure $G_{1}$ to the structure $G_{2}$, that is, a function $h\mathrel{:}V_{1}\to V_{2}$ such that, for all $v\in V_{1}$ it holds: $\left.\begin{array}[]{cr}h({\mathit{r}_{1}})=\mathit{r}_{2}&(\text{roots})\\\\[1.29167pt] \mathit{lab}_{1}({v})=\mathit{lab}_{2}({h({v})})&(\text{labels})\\\\[1.29167pt] {h}^{}({\mathit{args}_{1}({v})})=\mathit{args}_{2}({h({v})})&\hskip 21.52771pt(\text{arguments})\end{array}\quad\right\\}$ (8) where ${h}^{}$ is the homomorphic extension ${h}^{}\mathrel{:}V_{1}^{}\to V_{2}^{}$, $v_{1}\hskip 1.0pt\cdots\hskip 1.25ptv_{n}\mapsto h({v_{1}})\hskip 1.0pt\cdots\hskip 1.25pth({v_{n}})$ of $h$ to words over $V_{1}$. We write $G_{1}\mathrel{{{\hbox{\kern 0.43057pt\vbox{\hbox{\raise 0.43057pt\hbox{\kern-0.43057pt{$\rightarrow$}\kern-0.43057pt}}\hrule}\kern 0.43057pt}}}}G_{2}$ if there is a functional bisimulation (a homomorphism) from $G_{1}$ to $G_{2}$. An _isomorphism_ between $G_{1}$ and $G_{2}$ is a bijective homomorphism $i\mathrel{:}V_{1}\to V_{2}$ from $G_{1}$ to $G_{2}$. If there is an isomorphism between $G_{1}$ and $G_{2}$, we write $G_{1}\mathrel{{\simeq}}G_{2}$, and say that $G_{1}$ and $G_{2}$ are _isomorph_. Let $G=\langle V,\mathit{lab},\mathit{args},\mathit{r}\rangle$ be a term graph. A _bisimulation collapse_ of $G$ is a maximal element in the class $\left\\{{G^{\prime}}\mathrel{\|}{G\mathrel{{{\hbox{\kern 0.43057pt\vbox{\hbox{\raise 0.43057pt\hbox{\kern-0.43057pt{$\rightarrow$}\kern-0.43057pt}}\hrule}\kern 0.43057pt}}}}G^{\prime}}\right\\}$ up to ${\simeq}$, that is, a term graph $G^{\prime}_{0}$ with $G\mathrel{{{\hbox{\kern 0.43057pt\vbox{\hbox{\raise 0.43057pt\hbox{\kern-0.43057pt{$\rightarrow$}\kern-0.43057pt}}\hrule}\kern 0.43057pt}}}}G^{\prime}_{0}$ such that if $G^{\prime}_{0}\mathrel{{{\hbox{\kern 0.43057pt\vbox{\hbox{\raise 0.43057pt\hbox{\kern-0.43057pt{$\rightarrow$}\kern-0.43057pt}}\hrule}\kern 0.43057pt}}}}G^{\prime\prime}_{0}$ for some term graph $G^{\prime\prime}_{0}$, then $G^{\prime\prime}_{0}\mathrel{{\simeq}}G^{\prime}_{0}$. The _canonical bisimulation collapse_ ${G}\hskip 0.73193pt{{{\text{\small\textbar}}\downarrow}}$ of $G$ is defined as the root- connected part of the ‘factor term graph’ ${G}/_{{R}}$ of $G$ with respect to the largest bisimulation $R$ between $G$ and $G$ (the largest ‘self- bisimulation’ on $G$), which is an equivalence relation on $V$. The _factor term graph_ ${G}/_{{\sim}}$ of $G$ with respect to an equivalence relation $\sim$ on $V$ is defined as ${G}/_{{\sim}}\mathrel{{:=}}\langle{V}/_{{\sim}},{\mathit{lab}}/_{{\sim}},{\mathit{args}}/_{{\sim}},[{\mathit{r}}]_{\sim}\rangle$ where ${V}/_{{\sim}}$ is the set of $\sim$-equivalence classes of vertices in $V$, $[{\mathit{r}}]_{\sim}$ is the $\sim$-equivalence class of $\mathit{r}$, and ${\mathit{lab}}/_{{\sim}}$ and ${\mathit{args}}/_{{\sim}}$ are the mappings on ${V}/_{{\sim}}$ that are induced by $\mathit{lab}$ and $\mathit{args}$, respectively. Every two bisimulation collapses of $G$ are isomorphic. This justifies the common abbreviation of saying that ‘the bisimulation collapse’ of $G$ is unique up to isomorphism. ## 3 Unfolding Semantics of $\lambda_{\text{\sf letrec}}$-terms Informally, we regard _$\lambda_{\text{\sf letrec}}$ -terms_ as being defined defined by the following grammar: $\begin{array}[]{lllll}L&::=&\lambda{x}.\,{L}&(\textit{abstraction})\\\ &\|&{L}\hskip 1.5pt{L}&(\textit{application})\\\ &\|&x&(\textit{variable})\\\ &\|&{\text{\sf let}}\;{B}\;{\text{\sf in}}\;{L}&(\textit{letrec})\\\\[3.22916pt] \mathit{B}&::=&{f_{1}=L},\,{\dots},\,{f_{n}=L}&(\textit{equations})\\\ &&(f_{1},\dots,f_{n}\in{\cal R}\leavevmode\nobreak\ \text{all distinct})\end{array}$ Formally, we consider $\lambda_{\text{\sf letrec}}$-terms to be defined correspondingly as higher-order terms in the formalism of Combinatory Reduction Systems (CRS) Terese [2003]. CRSs are a higher-order term rewriting framework tailor-made for formalising and manipulating expressions in higher- order languages (i.e. languages with binding constructs like $\lambda$-abstractions and let-bindings). They provide a sound basis for defining our language and for reasoning with letrec-expressions. By formalising a system of unfolding rules as a CRS we conveniently externalise issues like name capturing and $\alpha$-renaming, which otherwise would have to be handled by a calculus of explicit substitution. Also, we can lean on the rewriting theory of CRSs for the proofs. As CRS-signature we use $\Sigma_{\lambda_{\text{\sf letrec}}}=\Sigma_{\lambda}\cup\left\\{{\textsf{let}_{n},\textsf{rec- in}_{n}}\mathrel{\|}{n\in\mathbb{N}}\right\\}$ with $\Sigma_{\lambda}=\left\\{{\mathsf{abs},\mathsf{app}}\right\\}$, where the unary symbol $\mathsf{abs}$ and the binary symbol $\mathsf{app}$ represent $\lambda$-abstraction and application, respectively; the symbols $\textsf{let}_{n}$ of arity one, and $\textsf{rec-in}_{n}$ of arity $n+1$ together formalise let-expressions with $n$ bindings. By ${\left\|{L}\right\|}$ we denote the size (number of symbols) of a $\lambda_{\text{\sf letrec}}$-term $L$. By $\mathit{Ter}({\lambda_{\text{\sf letrec}}})$ we denote the set of CRS-terms over $\Sigma_{\lambda_{\text{\sf letrec}}}$. For readability, we will rely on the informal first-order notation. _Infinite $\lambda$-terms_ are formalised as iCRS-terms (terms in an infinitary CRS Ketema and Simonsen [2011]) over $\Sigma_{\lambda}$, forming the set $\mathit{Ter}({\lambda^{\infty}})$. Informally, infinite $\lambda$-terms are generated co-inductively by the alternatives (abstraction), (application), and (variable) of the grammar above. In order to formally define the infinite unfolding of $\lambda_{\text{\sf letrec}}$-terms we utilise a CRS whose rewrite rules formalise unfolding steps Grabmayer and Rochel [2012]. Every $\lambda_{\text{\sf letrec}}$-term $L$ that represents an infinite $\lambda$-term $M$ can be rewritten by a typically infinite rewrite sequence that converges to $M$ in the limit. However, not every $\lambda_{\text{\sf letrec}}$-term represents an $\lambda^{\infty}$-term. For instance the $\lambda_{\text{\sf letrec}}$-term $Q=\lambda{x}.\,{{\text{\sf let}}\;{f=f}\;{\text{\sf in}}\;{{f}\hskip 1.5pt{x}}}$ with a meaningless let-binding for $f$ does not unfold to a $\lambda^{\infty}$-term. Therefore we introduce a constant symbol $\bullet$, called ‘black hole’, for expressing meaningless bindings, in order to define the unfolding operation as a total function. The unfolding semantics of $Q$ will then be $\lambda{x}.\,{{\bullet}\hskip 1.5pt{x}}$. So we extend the signature $\Sigma_{\lambda}$ to $\Sigma_{\lambda_{\bullet}}$ including $\bullet$, and denote the set of infinite $\lambda$-terms over $\Sigma_{\lambda}$ by $\mathit{Ter}({\lambda^{\infty}_{\bullet}})$. Similarly, the rules below are defined for terms in $\mathit{Ter}({\lambda_{\text{\sf letrec},\bullet}})$ based on signature $\Sigma_{\lambda_{\text{\sf letrec},\bullet}}$ that extends $\Sigma_{\lambda_{\text{\sf letrec}}}$ by the blackhole constant. ###### Definition 3.1 (unfolding CRS for $\lambda_{\text{\sf letrec}}$-terms). The rules $\displaystyle(@)$ $\displaystyle{\text{\sf let}}\;{B}\;{\text{\sf in}}\;{{L_{0}}\hskip 1.5pt{L_{1}}}\;\mathrel{\to}\;{({\text{\sf let}}\;{B}\;{\text{\sf in}}\;{L_{0}})}\hskip 1.5pt{({\text{\sf let}}\;{B}\;{\text{\sf in}}\;{L_{1}})}$ $\displaystyle(\lambda)$ $\displaystyle{\text{\sf let}}\;{B}\;{\text{\sf in}}\;{\lambda{x}.\,{L_{0}}}\;\mathrel{\to}\;\lambda{x}.\,{{\text{\sf let}}\;{B}\;{\text{\sf in}}\;{L_{0}}}$ $\displaystyle(\mathsf{let}\\_\mathsf{in})$ $\displaystyle{\text{\sf let}}\;{B_{0}}\;{\text{\sf in}}\;{{\text{\sf let}}\;{B_{1}}\;{\text{\sf in}}\;{L}}\;\mathrel{\to}\;{\text{\sf let}}\;{B_{0},B_{1}}\;{\text{\sf in}}\;{L}$ $\displaystyle(\mathsf{let}\text{-rec})$ $\displaystyle{\text{\sf let}}\;{{B_{1}},\,{f=L},\,{B_{2}}}\;{\text{\sf in}}\;{f}\;\mathrel{\to}\;{\text{\sf let}}\;{{B_{1}},\,{f=L},\,{B_{2}}}\;{\text{\sf in}}\;{L}$ $\displaystyle(\text{gc})$ $\displaystyle{\text{\sf let}}\;{{f_{1}=L_{1}},\,{\ldots},\,{f_{n}=L_{n}}}\;{\text{\sf in}}\;{P}\;\mathrel{\to}\;P$ (if $f_{1},\ldots,f_{n}$ do not occur in $P$) $\displaystyle(\text{tighten})$ $\displaystyle{\text{\sf let}}\;{{B_{1}},\,{f=g},\,{B_{2}}}\;{\text{\sf in}}\;{L}$ $\displaystyle\hskip 6.45831pt\mathrel{\to}\;{\text{\sf let}}\;{{{B_{1}}[{f}\mathrel{{:=}}{g}]},\,{{B_{2}}[{f}\mathrel{{:=}}{g}]}}\;{\text{\sf in}}\;{{L}[{f}\mathrel{{:=}}{g}]}$ (where $g$ with $g\neq f$ a recursion variable in $B_{1}$ or $B_{2}$) $\displaystyle(\bullet)$ $\displaystyle{\text{\sf let}}\;{{B_{1}},\,{f=f},\,{B_{2}}}\;{\text{\sf in}}\;{L}\;\mathrel{\to}\;{\text{\sf let}}\;{{B_{1}},\,{f=\bullet},\,{B_{2}}}\;{\text{\sf in}}\;{L}$ define, in informal notation, the _unfolding CRS_ for $\lambda_{\text{\sf letrec}}$-terms with rewrite relation ${\to}_{\text{unf}}$. See Fig. 3 for the formulation of these rules in explicit CRS-notation. $\displaystyle(@)$ $\displaystyle\textsf{let}_{n}({[{\vec{f}}]\hskip 1.0pt{\textsf{rec- in}_{n}({X_{1}({\vec{f}}),\ldots,X_{n}({\vec{f}}),\mathsf{app}({Z_{0}({\vec{f}}),Z_{1}({\vec{f}})})})}})$ $\displaystyle{}\hskip 12.91663pt\mathrel{\to}\;\mathsf{app}{((\textsf{let}_{n}({[{\vec{f}}]\hskip 1.0pt{\textsf{rec- in}_{n}({\ldots,X_{n}({\vec{f}}),Z_{0}({\vec{f}})})}})),(\textsf{let}_{n}({[{\vec{f}}]\hskip 1.0pt{\textsf{rec-in}_{n}({\ldots,X_{n}({\vec{f}}),Z_{1}({\vec{f}})})}})))}$ $\displaystyle(\mathsf{let}\text{-rec})$ $\displaystyle\textsf{let}_{n}({[{\vec{f}}]\hskip 1.0pt{\textsf{rec- in}_{n}({X_{1}({\vec{f}}),\ldots,X_{n}({\vec{f}}),f_{i}})}})\;\mathrel{\to}\;\textsf{let}_{n}({[{\vec{f}}]\hskip 1.0pt{\textsf{rec- in}_{n}({X_{1}({\vec{f}}),\ldots,X_{n}({\vec{f}}),X_{i}({\vec{f}})})}})$ $\displaystyle(\text{gc})$ $\displaystyle\textsf{let}_{n}({[{\vec{f}}]\hskip 1.0pt{\textsf{rec- in}_{n}({X_{1}({\vec{f}}),\ldots,X_{n}({\vec{f}}),Z})}})\;\mathrel{\to}\;Z$ $\displaystyle(\text{tighten})$ $\displaystyle\textsf{let}_{n}({[{\vec{f}}]\hskip 1.0pt{\textsf{rec- in}_{n}({X_{1}({\vec{f}}),\ldots,X_{i-1}({\vec{f}}),f_{j},X_{i+1}({\vec{f}}),\ldots,X_{n}({\vec{f}}),Z({\vec{f}})})}})$ $\displaystyle\hskip 12.91663pt\mathrel{\to}\;\textsf{let}_{n-1}({[{\vec{g}}]\hskip 1.0pt{\textsf{rec- in}_{n-1}({X_{1}({\vec{g}^{\prime}}),\ldots,X_{i-1}({\vec{g}^{\prime}}),X_{i+1}({\vec{g}^{\prime}}),\ldots,X_{n}({\vec{g}^{\prime}}),Z({\vec{g}^{\prime}})})}})$ where: $i,j\in\left\\{{1,\ldots,n}\right\\}$ with $i\neq j$, and $\vec{g}^{\prime}=\langle g_{1},\ldots,g_{i-1},g_{j},g_{i+1},\ldots,g_{n}\rangle$ $\displaystyle(\bullet)$ $\displaystyle\textsf{let}_{n}({[{\vec{f}}]\hskip 1.0pt{\textsf{rec- in}_{n}({X_{1}({\vec{f}}),\ldots,X_{i-1}({\vec{f}}),f_{i},X_{i+1}({\vec{f}}),\ldots,X_{n}({\vec{f}}),f_{i}})}})$ $\displaystyle\hskip 12.91663pt\mathrel{\to}\;\textsf{let}_{n}({[{\vec{f}}]\hskip 1.0pt{\textsf{rec- in}_{n}({X_{1}({\vec{f}}),\ldots,X_{i-1}({\vec{f}}),\bullet,X_{i+1}({\vec{f}}),\ldots,X_{n}({\vec{f}}),f_{i}})}})$ Figure 3: The rules for unfolding $\lambda_{\text{\sf letrec}}$-terms in explicit CRS-notation. ###### Example 3.2 (Unfolding derivation of $L$ from Ex. 1.2). $\lambda{f}.\,{{\text{\sf let}}\;{r={f}\hskip 1.5pt{r}}\;{\text{\sf in}}\;{r}}\leavevmode\nobreak\ {\to}_{\text{unf}}^{(\mathsf{let}\text{-rec})}\leavevmode\nobreak\ \lambda{f}.\,{{\text{\sf let}}\;{r={f}\hskip 1.5pt{r}}\;{\text{\sf in}}\;{{f}\hskip 1.5pt{r}}}\leavevmode\nobreak\ {\to}_{\text{unf}}^{(@)}$ $\lambda{f}.\,{{({\text{\sf let}}\;{r={f}\hskip 1.5pt{r}}\;{\text{\sf in}}\;{f})}\hskip 1.5pt{({\text{\sf let}}\;{r={f}\hskip 1.5pt{r}}\;{\text{\sf in}}\;{r})}}\leavevmode\nobreak\ {\to}_{\text{unf}}^{(\text{gc})}$ $\lambda{f}.\,{{f}\hskip 1.5pt{({\text{\sf let}}\;{r={f}\hskip 1.5pt{r}}\;{\text{\sf in}}\;{r}})}\leavevmode\nobreak\ {\to}_{\text{unf}}^{(\mathsf{let}\text{-rec})}\leavevmode\nobreak\ \dots$ We say that a $\lambda_{\text{\sf letrec}}$-term $L$ _unfolds to_ an $\lambda_{\bullet}^{\infty}$-term $M$, or that $L$ _expresses_ $M$, if there is a (typically) infinite ${\to}_{\text{unf}}$-rewrite sequence from $L$ that converges to $M$, symbolically $L\mathrel{\twoheadrightarrow\twoheadrightarrow_{\text{unf}}}M$. Note that any such rewrite sequence is strongly convergent (the depth of the contracted redexes tends to infinity), because the resulting term does not contain any let-expressions. ###### Lemma 3.3. Every $\lambda_{\text{\sf letrec}}$-term unfolds to precisely one $\lambda_{\bullet}^{\infty}$-term. ###### Proof (Outline). Infinite normal forms of ${\to}_{\text{unf}}$ are $\lambda_{\bullet}^{\infty}$-terms since: every occurrence of a let-expression in a $\lambda_{\text{\sf letrec},\bullet}$ gives rise to a redex; and infinite terms over $\Sigma_{\lambda_{\text{\sf letrec},\bullet}}$ without let- expressions are $\lambda_{\bullet}^{\infty}$-terms. Also, outermost-fair rewrite sequences in which the rules (tighten) and ($\bullet$) are applied eagerly are (strongly) convergent. Unique infinite normalisation of ${\to}_{\text{unf}}$ follows from finitary confluence of ${\to}_{\text{unf}}$. In previous work Grabmayer and Rochel [2012] we proved confluence for the slightly simpler CRS that does not contain the final two rules, which together introduce black holes in terms with meaningless bindings. That confluence proof can be adapted by extending the argumentation to deal with the additional critical pairs. ∎ ###### Definition 3.4. The _unfolding semantics_ for $\lambda_{\text{\sf letrec}}$-terms is defined by the function $\llbracket{\cdot}\rrbracket_{{\lambda^{\infty}_{\bullet}}}\mathrel{:}\mathit{Ter}({\lambda_{\text{\sf letrec}}})\to\mathit{Ter}({\lambda^{\infty}_{\bullet}})$, where $L\mapsto\llbracket{L}\rrbracket_{{\lambda^{\infty}_{\bullet}}}\mathrel{{:=}}\,$ the infinite unfolding of $L$. ###### Remark 3.5 (Regular and strongly regular $\lambda^{\infty}$-terms). $\lambda^{\infty}$-terms that arise as infinite unfoldings of $\lambda_{\text{\sf letrec}}$-terms form a proper subclass of those $\lambda^{\infty}$-terms that have a regular term structure Grabmayer and Rochel [2012]. $\lambda^{\infty}$-terms that belong to this subclass are called ‘strongly regular’, and can be characterised by means of a decomposition rewrite system, and as those that contain only finite ‘binding–capturing chains’ Grabmayer and Rochel [2012, 2013b]. ## 4 Lambda higher-order term graphs In this section we motivate the use of higher-order term graphs as a semantics for $\lambda_{\text{\sf letrec}}$-terms; we introduce the class ${\cal H}$ of ‘$\lambda$-ho-term-graphs’ and define the semantics $\llbracket{\cdot}\rrbracket_{{\cal H}}$ for interpreting $\lambda_{\text{\sf letrec}}$-terms as $\lambda$-ho-term-graphs. Finally, we sketch a proof of the correctness of $\llbracket{\cdot}\rrbracket_{{\cal H}}$ with respect to unfolding equivalence (the property (P1)). We start out from a natural interpretation of $\lambda_{\text{\sf letrec}}$-terms as first-order term graphs: occurrences of abstraction variables are resolved as edges pointing to the corresponding abstraction; occurrences of recursion variables as edges to the subgraph belonging to the respective binding. We therefore consider term graphs over the signature ${\Sigma}^{\lambda}_{\bullet}=\left\\{{@,\lambda,\mathsf{0},\bullet}\right\\}$ with arities ${\mit ar}({@})=2$, ${\mit ar}({\lambda})=1$, ${\mit ar}({\mathsf{0}})=1$, and ${\mit ar}({\bullet})=0$. These function symbols represent applications, $\lambda$-abstractions, abstraction variables, and black holes. We will later define a subclass of these term graphs that excludes meaningless graphs. In line with the choice to regard all terms as higher-order terms (thus modulo $\alpha$-conversion), we consider a nameless graph representation, so that $\alpha$-equivalence of two terms can be recognised as their graph interpretations being isomorphic. For a term graph $G$ over ${\Sigma}^{\lambda}_{\bullet}$ with set $V$ of vertices we will henceforth denote by $V\\!({@})$, $V\\!({\lambda})$, $V\\!({\mathsf{0}})$, and $V\\!({\bullet})$ the sets of _application vertices_ , _abstraction vertices_ , _variable vertices_ , and _blackhole vertices_ , that is, those with label $@$, $\lambda$, $\mathsf{0}$, $\bullet$, respectively. ###### Example 4.1 (Natural first-order interpretation). The $\lambda_{\text{\sf letrec}}$-terms $L$ and $P$ in Ex. 1.2 can be represented as the term graphs in Fig. 2. These two graphs are bisimilar, which suggests that $L$ and $P$ are unfolding equivalent. Moreover, there is a functional bisimulation from the larger term graph to the smaller one, indicating that $L$ expresses more sharing than $P$, or in other words: $L$ is more compact. Also, there is no smaller term graph that is bisimilar to $L$ and $P$. We conclude that $L$ is a maximally shared form of $P$. However, this translation is incorrect in the sense that bisimilarity does not in general guarantee unfolding equivalence, the desired property (P1). This is witnessed by the following counterexample. ###### Example 4.2 (Incorrectness of the natural first-order interpretation). $\displaystyle L_{1}$ $\displaystyle\;\;=\;\;{\text{\sf let}}\;{f=\lambda{x}.\,{{(\lambda{y}.\,{{f}\hskip 1.5pt{y}})}\hskip 1.5pt{x}}}\;{\text{\sf in}}\;{f}$ $\displaystyle L$ $\displaystyle\;\;=\;\;{\text{\sf let}}\;{f=\lambda{x}.\,{{f}\hskip 1.5pt{x}}}\;{\text{\sf in}}\;{f}$ $\displaystyle L_{2}$ $\displaystyle\;\;=\;\;{\text{\sf let}}\;{f=\lambda{x}.\,{{(\lambda{y}.\,{{f}\hskip 1.5pt{x}})}\hskip 1.5pt{x}}}\;{\text{\sf in}}\;{f}$ While $\llbracket{L_{1}}\rrbracket_{{\lambda^{\infty}}}=\llbracket{L}\rrbracket_{{\lambda^{\infty}}}$ and $\llbracket{L}\rrbracket_{{\lambda^{\infty}}}\neq\llbracket{L_{2}}\rrbracket_{{\lambda^{\infty}}}$, all of their term graphs $G_{1}$, $G$, $G$ are bisimilar (please ignore the shading for now): $\rightarrow$ $\leftarrow$ $G_{1}$ $G$ $G_{2}$ Consequently this interpretation lacks the necessary structure for correctly modelling compactification via bisimulation collapse. We therefore impose additional structure on the term graphs. This is indicated by the shading in the picture above, and in the graphs throughout this paper. A shaded area depicts the _scope_ of an abstraction: it comprises all positions between the abstraction and its bound variable occurrences as well as the scope of any abstraction on these positions. By this stipulation, scopes are properly nested. Now note that the functional bisimulation on the right in the picture in Ex. 4.2 does _not_ respect the scopes: The scope of the topmost abstraction vertex in the term graph $G_{2}$ interpreting $L_{2}$ contains another $\lambda$-abstraction; hence the image of this scope under the functional bisimulation cannot fit into, and is not contained in, the single scope in the term graph $G$ of $L$. Also, the trivial scope of the vacuous abstraction in $G_{2}$ is not mapped to a scope in $G$. Thus the natural first-order interpretation is incorrect, in the sense that functional bisimulation does not preserve scopes on the first-order term graphs that are interpretations of $\lambda_{\text{\sf letrec}}$-terms. To prevent that interpretations of not unfolding-equivalent terms like $L_{1}$ and $L_{2}$ in Ex. 4.2 become bisimilar, we enrich first-order term graphs by a formal concept of scope. More precisely, _abstraction prefixes_ are added as vertex labels. They also serve the purpose of defining the subclass of meaningful term graphs over ${\Sigma}^{\lambda}_{\bullet}$ that sensibly represent cyclic $\lambda$-terms. In the enriched term graphs, each vertex $v$ is annotated with a label $P({v})$, the _abstraction prefix_ of $v$, which is a list of vertex names that identifies the abstraction vertices in whose scope $v$ resides. Alternatively scopes can be represented by a scope function (as in Blom [2001]) that assigns to every abstraction vertex the set of vertices in its scope. In the article Grabmayer and Rochel [2013a] we show that higher- order term graphs with scope functions correspond bijectively to those with abstraction prefix functions. Abstraction prefixes can be determined by traversing over the graph and recording every binding encountered. When passing an abstraction vertex $v$ while descending into the subgraph representing the body of the abstraction, one enters or opens the scope of $v$. This is recorded by appending $v$ to the abstraction prefix of $v$’s successor. $v$ is removed from the prefix at positions under which the abstraction variable is no longer used, but not before any other variable that was added to the prefix in the meantime has itself been removed. In other words, the abstraction prefix behaves like a stack. We call term graphs for representing $\lambda_{\text{\sf letrec}}$-terms that are equipped with abstraction-prefixes ‘$\lambda$-higher- order term graphs’ ($\lambda$-ho-term-graphs). ###### Example 4.3 (The $\lambda$-ho-term-graphs of the terms in Ex. 4.2). $\begin{aligned} \includegraphics[scale={0.81}]{figs/counterex_onlyvarbl_y_prefixed.pdf}\end{aligned}\hfill{\hbox{\kern 0.43057pt\vbox{\hbox{\raise 0.43057pt\hbox{\kern-0.43057pt{$\rightarrow$}\kern-0.43057pt}}\hrule}\kern 0.43057pt}}\hfill\begin{aligned} \includegraphics[scale={0.81}]{figs/counterex_onlyvarbl_collapse_prefixed.pdf}\end{aligned}\hfill\not{\hbox{\kern 0.43057pt\vbox{\hbox{\raise 0.43057pt\hbox{\kern-0.43057pt{$\leftarrow$}\kern-0.43057pt}}\hrule}\kern 0.43057pt}}\hfill\begin{aligned} \includegraphics[scale={0.81}]{figs/counterex_onlyvarbl_x_prefixed.pdf}\end{aligned}$ The superscripts of abstraction vertices indicate their names. The abstraction prefix of a vertex is annotated to its top left. Note that abstraction vertices themselves are not included in their own prefix. We define $\lambda$-ho-term-graphs as term graphs over ${\Sigma}^{\lambda}_{\bullet}$ together with an abstraction-prefix function that assigns to each vertex an abstraction prefix. It has to respect certain correctness conditions restricting the $\lambda$-ho-term-graphs to exclude meaningless term graphs. ###### Definition 4.4 (correct abstraction-prefix function for term graphs over ${\Sigma}^{\lambda}_{\bullet}$). Let $G=\langle V,\mathit{lab},\mathit{args},\mathit{r}\rangle$ be a ${\Sigma}^{\lambda}_{\bullet}$-term-graph. An _abstraction-prefix function_ for $G$ is a function $P\mathrel{:}V\to V^{}$ from vertices of $G$ to words of vertices. Such a function is called _correct_ if for all $w,w_{0},w_{1}\in V$ and $k\in\left\\{{0,1}\right\\}$ it holds: $\displaystyle P({\mathit{r}})=\epsilon$ $\displaystyle(\text{root})$ $\displaystyle P({\bullet})=\epsilon$ $\displaystyle(\text{black hole})$ $\displaystyle w\in V\\!({\lambda})\;\mathrel{{\wedge}}\;w\mathrel{\rightarrowtail_{0}}w_{0}\;\;$ $\displaystyle\Rightarrow\;\;P({w_{0}})\leq P({w})w$ $\displaystyle(\lambda)$ $\displaystyle w\in V\\!({@})\;\mathrel{{\wedge}}\;w\mathrel{\rightarrowtail_{k}}w_{k}\;\;$ $\displaystyle\Rightarrow\;\;P({w_{k}})\leq P({w})$ $\displaystyle(@)$ $\displaystyle w\in V\\!({\mathsf{0}})\;\mathrel{{\wedge}}\;w\mathrel{\rightarrowtail_{0}}w_{0}\;\;$ $\displaystyle\Rightarrow\;\;\left\\{\hskip 1.0pt\begin{aligned} &w_{0}\in V\\!({\lambda})\\\\[-2.15277pt] &\;\mathrel{{\wedge}}\;P({w_{0}})w_{0}=P({w})\end{aligned}\right.$ $\displaystyle(\mathsf{0})$ Here and later we denote by $\leq$ the ‘is-prefix-of’ relation. ###### Definition 4.5 ($\lambda$-ho-term-graph). A _$\lambda$ -ho-term-graph_ over ${\Sigma}^{\lambda}_{\bullet}$ is a five- tuple ${\cal G}=\langle V,\mathit{lab},\mathit{args},\mathit{r},P\rangle$ where $G_{{\cal G}}=\langle V,\mathit{lab},\mathit{args},\mathit{r}\rangle$ is a term graph over ${\Sigma}^{\lambda}_{\bullet}$, called the term graph _underlying_ ${\cal G}$, and $P$ is a correct abstraction-prefix function for $G_{{\cal G}}$. The class of $\lambda$-ho-term-graphs over ${\Sigma}^{\lambda}_{\bullet}$ is denoted by ${\cal H}$. ###### Definition 4.6 (homomorphism, bisimulation for $\lambda$-ho-term- graphs). Let ${\cal G}_{1}=\langle V_{1},\mathit{lab}_{1},\mathit{args}_{1},\mathit{r}_{1},P_{1}\rangle$ and ${\cal G}_{2}=\langle V_{2},\mathit{lab}_{2},\mathit{args}_{2},\mathit{r}_{2},P_{2}\rangle$ be $\lambda$-ho-term-graphs over ${\Sigma}^{\lambda}_{\bullet}$. A _bisimulation_ between ${\cal G}_{1}$ and ${\cal G}_{2}$ is a relation $R\subseteq V_{1}\times V_{2}$ such that for all $\langle w,\hskip 0.5ptw^{\prime}\rangle\in R$ the conditions (4), and additionally: A _bisimulation_ between ${\cal G}_{1}$ and ${\cal G}_{2}$ is a relation $R\subseteq V_{1}\times V_{2}$ that is a bisimulation between the term graphs $G_{{\cal G}_{1}}$ and $G_{{\cal G}_{2}}$ underlying ${\cal G}_{1}$ and ${\cal G}_{2}$, respectively, and for which also the following condition: $\displaystyle\langle P_{1}({w}),\hskip 0.5ptP_{2}({w^{\prime}})\rangle$ $\displaystyle\in R^{}$ $\displaystyle(\text{abstraction-prefix functions})$ (9) (for $R^{}$ see p. 4 below (4)) is satisfied for all $w\in V_{1}$ and all $w^{\prime}\in V_{2}$. If there is a such bisimulation, then ${\cal G}_{1}$ and ${\cal G}_{2}$ are _bisimilar_ , denoted by ${\cal G}_{1}\mathrel{\mathrel{\hbox{\kern 0.43057pt\vbox{\hbox{\raise 0.43057pt\hbox{\kern-0.43057pt{$\leftrightarrow$}\kern-0.43057pt}}\hrule}\kern 0.43057pt}}}{\cal G}_{2}$. A _homomorphism_ (a _functional bisimulation_) from ${\cal G}_{1}$ to ${\cal G}_{2}$ is a morphism from the structure ${\cal G}_{1}$ to the structure ${\cal G}_{2}$, or more explicitly, it is a homomorphism $h\mathrel{:}V_{1}\to V_{2}$ from $G_{{\cal G}_{1}}$ to $G_{{\cal G}_{2}}$ that additionally satisfies, for all $w\in V_{1}$, the following condition: $\displaystyle\bar{h}({P_{1}({w})})$ $\displaystyle=P_{2}({h({w})})$ $\displaystyle(\text{abstraction-prefix functions})$ (10) for all $w\in V_{1}$, where ${\bar{h}}$ is the homomorphic extension of $h$ to words over $V_{1}$. We write ${\cal G}_{1}\mathrel{{{\hbox{\kern 0.43057pt\vbox{\hbox{\raise 0.43057pt\hbox{\kern-0.43057pt{$\rightarrow$}\kern-0.43057pt}}\hrule}\kern 0.43057pt}}}}{\cal G}_{2}$ if there is a homomorphism between ${\cal G}_{1}$ and ${\cal G}_{2}$. ### 4.1 Interpretion of $\lambda_{\text{\sf letrec}}$-terms as $\lambda$-ho- term-graphs $\lambda$: $(\vec{p})\hskip 1.0pt{\lambda{x}.\,{L}}$ $\overset{}{\underset{}{\implies}}$ $(\vec{p}\;x^{v}[{\hskip 1.0pt}])\hskip 1.0pt{L}$$\lambda$$\scriptstyle v$$\scriptstyle(\mathit{vs}({\vec{p}}))$ $@$: $(\vec{p})\hskip 1.0pt{{L_{0}}\hskip 1.5pt{L_{1}}}$ $\overset{}{\underset{}{\implies}}$ $@$$(\vec{p})\hskip 1.0pt{L_{0}}$$(\vec{p})\hskip 1.0pt{L_{1}}$$\scriptstyle(\mathit{vs}({\vec{p}}))$ $f$: $(\vec{p}\;x^{v}[{\ldots,f^{w}=L,\ldots}])\hskip 1.0pt{f}$ $\overset{}{\underset{}{\implies}}$ \|$\scriptstyle w$$\scriptstyle(\mathit{vs}({\vec{p}})\;v)$ Figure 4: Translation rules ${\cal R}$ for interpreting $\lambda_{\text{\sf letrec}}$-terms as $\lambda$-ho-term-graphs. See Section 4.1 for explanations. $\mathsf{0}$: $(x_{0}^{v_{0}}[{B_{0}}]\;\mathrel{\cdots}\;x_{n}^{v_{n}}[{B_{n}}])\hskip 1.0pt{x_{n}}$ $\overset{}{\underset{}{\implies}}$ $\mathsf{0}$$\scriptstyle(v_{1}\dots v_{n})$$\lambda$$\scriptstyle v_{n}$ $\mathsf{S}$: $(\vec{p}\;x^{v}[{f_{1}^{v_{1}}=L_{1},\dots,f_{n}^{v_{n}}=L_{n}}])\hskip 1.0pt{L}$ $\overset{\text{\small\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $x\not\in\textit{FV}({L})$\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ }}{\underset{\text{\small$f_{i}\not\in\textit{FV}({L})$}}{\implies}}$ $(\vec{p})\hskip 1.0pt{L}$ $\mathsf{let}$: $\begin{aligned} (x_{0}^{v_{0}}[{B_{0}}]\;\mathrel{\cdots}\;x_{n}^{v_{n}}[{B_{n}}])\\\ {{\text{\sf let}}\;{B}\;{\text{\sf in}}\;{L_{0}}}\end{aligned}$($B$ stands for $f_{1}=L_{1},\dots,f_{k}=L_{k}$) $\overset{}{\underset{}{\implies}}$ $(x_{0}^{v_{0}}[{B^{\prime}_{0}}]\;\mathrel{\cdots}\;x_{n}^{v_{n}}[{B^{\prime}_{n}}])\hskip 1.0pt{L_{0}}$…$(x_{0}^{v_{0}}[{B^{\prime}_{0}}]\;\mathrel{\cdots}\;x_{l_{1}}^{v_{l_{1}}}[{B^{\prime}_{l_{1}}}])\hskip 1.0pt{L_{1}}$$(x_{0}^{v_{0}}[{B^{\prime}_{0}}]\;\mathrel{\cdots}\;x_{l_{k}}^{v_{l_{k}}}[{B^{\prime}_{l_{k}}}])\hskip 1.0pt{L_{k}}$\|$\scriptstyle(v_{1}\dots v_{l_{1}})$$\scriptstyle w_{1}$\|$\scriptstyle(v_{1}\dots v_{l_{n}})$$\scriptstyle w_{k}$($B^{\prime}_{i}=B_{i},\\{f_{j}^{w_{j}}=L_{j}\leavevmode\nobreak\ \|\leavevmode\nobreak\ l_{j}=i,\leavevmode\nobreak\ 1\leq j\leq k\\}$)$l_{1},\ldots,\l_{k}\leq n$ such that $\forall i,j\leq k\leavevmode\nobreak\ :\leavevmode\nobreak\ l_{i}<l_{j}\Rightarrow\forall f=L\in B^{\prime}_{l_{i}},\,g=P\in B^{\prime}_{l_{j}}:g\not\in FV(L)$ $\text{and }\forall i\leq k\>\left\\{{y}\mathrel{\|}{\text{$y$ is required variable of $f_{i}$}}\right\\}\subseteq\left\\{{x_{0},\ldots,x_{l_{i}}}\right\\}$ In order to interpret a $\lambda_{\text{\sf letrec}}$-term $L$ as $\lambda$-ho-term-graph, the translation rules ${\cal R}$ from Fig. 4.1 are applied to a ‘translation box’ $({[]})\hskip 0.5pt{L}$ . It contains $L$ furnished with a prefix consisting of a dummy variable $$, and an empty set $[]$ of binding equations. The translation process proceeds by induction on the syntactical structure of the prefixed $\lambda_{\text{\sf letrec}}$-expression’s body. Ultimately, a term graph $G$ over ${\Sigma}^{\lambda}_{\bullet}$ is produced, together with a correct abstraction-prefix function for $G$. For reading the rules ${\cal R}$ in Fig. 4.1 correctly, observe the details as described here below. For illustration of their application, please refer to Appendix A where several $\lambda_{\text{\sf letrec}}$-terms are translated into $\lambda$-ho-term-graphs. * • A translation box $(\vec{p})\hskip 1.0pt{L}$ contains a prefixed, partially decomposed $\lambda_{\text{\sf letrec}}$-term $L$. The prefix contains a vector $\vec{p}$ of annotated $\lambda$-abstractions that have already been translated and whose scope typically extends into $L$. Every prefix abstraction is annotated with a set of binding equations that are defined at its level. There is special dummy variable denoted by $$ at the left of the prefix that carries top-level function bindings, i.e. binding equations that are not defined under any enclosing $\lambda$-abstraction. The $\lambda$-rule strips off an abstraction from the body of the expression, and pushes the abstraction variable into the prefix, which initially contains an empty set of function bindings. • Names of abstraction vertices are indicated to the right, and abstraction- prefixes to the left of the created vertices. In order to refer to the vertices in the prefix we use the following notation: $\mathit{vs}({\vec{p}})=v_{1}\,\cdots\,v_{n}$ if $\vec{p}=[B_{0}]\;x_{1}^{v_{1}}[B_{1}]\;\dots\;x_{n}^{v_{n}}[B_{n}]$. • Vertices drawn with dashed lines have been created in earlier translation steps, and in the current step are referenced by edges in the current step. * • In the $\mathsf{S}$-rule, which takes care of closing scopes, $FV(L)$ stands for the set of free variables in $L$. * • The $\mathsf{let}$-rule for translating let-expressions creates a box for the in-part as well as for each binding equation. The translation of each of the bindings starts with an _indirection vertex_. These vertices guarantee the well-definedness of the process when it translates meaningless bindings such as $f=f$, or $g=h,\,h=g$, which would otherwise give rise to loops without vertices. The $\mathsf{let}$-rule pushes the function bindings into the abstraction prefix, associating each function binding with one of the variables in the abstraction prefix. There is some freedom as to which variable a function binding is assigned to. This freedom is limited by scoping conditions that ensure that the prefixed term is a valid CRS-term: function bindings may only depend on variables and functions that occur further to the left in the prefix. The chosen association also directly determines the prefix lengths used in the translation boxes for the function bindings. * • Indirection vertices are eliminated by an erasure process at the end: Every indirection vertex that does not point to itself is removed, redirecting all incoming edges to the successor vertex. Finally every loop on a single indirection vertex is replaced by a _black hole_ vertex that represents a meaningless binding. Abstraction prefixes for such black holes are defined to be empty. Figure 5: Translation of $\lambda{a}.\,{\lambda{b}.\,{{\text{\sf let}}\;{f=a}\;{\text{\sf in}}\;{{{{a}\hskip 1.5pt{a}}\hskip 1.5pt{({f}\hskip 1.5pt{a})}}\hskip 1.5pt{b}}}}$ with equal (left) and with minimal prefix lengths (right) in the let-rule. ###### Definition 4.7. We say that a term graph $G$ over ${\Sigma}^{\lambda}_{\bullet}$ and an abstraction-prefix function $P$ is _${\cal R}$ -generated from_ a $\lambda_{\text{\sf letrec}}$-term $L$ if $G$ and $P$ are obtained by applying the rules ${\cal R}$ from Fig. 4.1 to $({[]})\hskip 0.5pt{L}$ . ###### Remark 4.8 (Inference rule formulation of ${\cal R}$). See also Fig. 6 for inference rules that correspond to the deconstruction of prefixed terms in ${\cal R}$. $\displaystyle\begin{aligned} &\mbox{ \ignorespaces\ignorespaces \ignorespaces\leavevmode\lower 26.0pt\hbox{\vbox{\hbox{\hskip 0.34787pt\hbox{\hskip 4.0pt\hbox{$(\vec{p}\;x^{v}[{}])\hskip 1.0pt{L}$}\hskip 4.0pt}}\vskip-1.27223pt\nointerlineskip\hbox{\hskip 0.0pt\lower-0.2pt\hbox{}\hbox to47.78123pt{\xleaders\hrule\hfill}\lower 3.27222pt\hbox{\hskip 3.0pt$\lambda$}}\vskip-1.27222pt\nointerlineskip\hbox{\hbox{\hskip 4.0pt\hbox{$(\vec{p})\hskip 1.0pt{\lambda{x}.\,{L}}$}\hskip 4.0pt}}}} \ignorespaces}&\hskip 25.83325pt&\mbox{ \ignorespaces \ignorespaces\ignorespaces\ignorespaces\leavevmode\lower 26.0pt\hbox{\vbox{\hbox{\hbox{\hskip 4.0pt\hbox{$(\vec{p})\hskip 1.0pt{L_{0}}$}\hskip 4.0pt}\hbox{\hskip 14.45377pt}\hbox{\hskip 4.0pt\hbox{$(\vec{p})\hskip 1.0pt{L_{1}}$}\hskip 4.0pt}}\vskip-1.27223pt\nointerlineskip\hbox{\hskip 0.0pt\lower-0.2pt\hbox{}\hbox to80.2969pt{\xleaders\hrule\hfill}\lower 3.27222pt\hbox{\hskip 3.0pt$@$}}\vskip-1.27222pt\nointerlineskip\hbox{\hskip 18.13489pt\hbox{\hbox{\hskip 4.0pt\hbox{$(\vec{p})\hskip 1.0pt{{L_{0}}\hskip 1.5pt{L_{1}}}$}\hskip 4.0pt}}}}} \ignorespaces}&\hskip 25.83325pt&\mbox{ \ignorespaces\ignorespaces \ignorespaces\leavevmode\lower 26.325pt\hbox{\vbox{\hbox{\hskip 4.0pt\hbox{$\phantom{(\vec{p}\;x^{v}[{\ldots,f^{w}=L,\ldots}])\hskip 1.0pt{f}}$}\hskip 4.0pt}\vskip 0.04723pt\nointerlineskip\hbox{\hskip 0.0pt\lower-0.2pt\hbox{}\hbox to95.96754pt{\xleaders\hrule\hfill}\lower 1.95277pt\hbox{\hskip 3.0pt$\text{\sf rec}$}}\vskip 0.04723pt\nointerlineskip\hbox{\hskip 0.0pt\hbox{\hbox{\hskip 4.0pt\hbox{$(\vec{p}\;x^{v}[{\ldots,f^{w}=L,\ldots}])\hskip 1.0pt{f}$}\hskip 4.0pt}}}}} \ignorespaces}\end{aligned}$ $\displaystyle\begin{aligned} &\mbox{ \ignorespaces\ignorespaces \ignorespaces\leavevmode\lower 26.0pt\hbox{\vbox{\hbox{\hskip 4.0pt\hbox{$\phantom{(x_{0}^{v_{0}}[{B_{0}}]\;\mathrel{\cdots}\;x_{n}^{v_{n}}[{B_{n}}])\hskip 1.0pt{x_{n}}}$}\hskip 4.0pt}\vskip-1.02223pt\nointerlineskip\hbox{\hskip 0.0pt\lower-0.2pt\hbox{}\hbox to89.94774pt{\xleaders\hrule\hfill}\lower 3.02222pt\hbox{\hskip 3.0pt$\mathsf{0}$}}\vskip-1.02222pt\nointerlineskip\hbox{\hskip 0.0pt\hbox{\hbox{\hskip 4.0pt\hbox{$(x_{0}^{v_{0}}[{B_{0}}]\;\mathrel{\cdots}\;x_{n}^{v_{n}}[{B_{n}}])\hskip 1.0pt{x_{n}}$}\hskip 4.0pt}}}}} \ignorespaces}&\hskip 25.83325pt&\mbox{ \ignorespaces\ignorespaces \ignorespaces\leavevmode\lower 26.325pt\hbox{\vbox{\hbox{\hskip 46.17104pt\hbox{\hskip 4.0pt\hbox{$(\vec{p})\hskip 1.0pt{L}$}\hskip 4.0pt}}\vskip-2.8pt\nointerlineskip\hbox{\hskip 0.0pt\lower-0.2pt\hbox{}\hbox to122.46364pt{\xleaders\hrule\hfill}\lower 2.3pt\hbox{\hskip 3.0pt$\mathsf{S}$ \mbox{ } (if $x\not\in\textit{FV}({L})$ and $f_{i}\not\in\textit{FV}({L})$)}}\vskip-2.8pt\nointerlineskip\hbox{\hbox{\hskip 4.0pt\hbox{$(\vec{p}\;x^{v}[{f_{1}^{v_{1}}=L_{1},\dots,f_{n}^{v_{n}}=L_{n}}])\hskip 1.0pt{L}$}\hskip 4.0pt}}}} \ignorespaces}\end{aligned}$ $(x_{0}^{v_{0}}[{B^{\prime}_{0}}]\;\mathrel{\cdots}\;x_{l_{1}}^{v_{l_{1}}}[{B^{\prime}_{l_{1}}}])\hskip 1.0pt{L_{1}}$ … $(x_{0}^{v_{0}}[{B^{\prime}_{0}}]\;\mathrel{\cdots}\;x_{l_{k}}^{v{l_{k}}}[{B^{\prime}_{l_{k}}}])\hskip 1.0pt{L_{k}}$ $(x_{0}^{v_{0}}[{B^{\prime}_{0}}]\;\mathrel{\cdots}\;x_{n}^{v_{n}}[{B^{\prime}_{n}}])\hskip 1.0pt{L_{0}}$ let (where $B^{\prime}_{0},\ldots,B^{\prime}_{n}$ and $l_{1},\ldots,l_{k}$ as in rule $\mathsf{let}$ in Fig. 4.1) $(x_{0}^{v_{0}}[{B_{0}}]\;\mathrel{\cdots}\;x_{n}^{v_{n}}[{B_{n}}])\hskip 1.0pt{{\text{\sf let}}\;{f_{1}=L_{1},\dots,f_{k}=L_{k}}\;{\text{\sf in}}\;{L_{0}}}$ Figure 6: Alternative formulation as inference rules of the translation rules in Fig. 4.1 for the interpretation of $\lambda_{\text{\sf letrec}}$-terms as $\lambda$-ho-term-graphs. ###### Proposition 4.9. Let $L$ be a $\lambda_{\text{\sf letrec}}$-term. Suppose that a term graph $G$ over ${\Sigma}^{\lambda}_{\bullet}$, and an abstraction-prefix function $P$ are ${\cal R}$-generated from $L$. Then $P$ is a correct abstraction-prefix function for $G$, and consequently, $G$ and $P$ together form a $\lambda$-ho- term-graph in ${\cal H}$. There are two sources of non-determinism in this translation: The $\mathsf{S}$-rule for shortening prefixes can be applicable at the same time as other rules. And the let-rule does not fix the lengths $l_{1},\dots,l_{k}$ of the abstraction prefixes for the translations of the binding equations, but admits various choices of prefixes that are shorter than the prefix of the left-hand side. Neither kind of non-determinism affects the term graph that is produced, but in general several abstraction-prefix functions, and thus different $\lambda$-ho-term-graphs, can be obtained. Figure 7: Translation of $\lambda{a}.\,{{(\lambda{b}.\,{\lambda{c}.\,{{a}\hskip 1.5pt{c}}})}\hskip 1.5pt{(\lambda{d}.\,{{a}\hskip 1.5pt{d}})}}$ with eager scope-closure (left), and with lazy scope-closure (right). While in the left term graph four vertices can be shared, with as result the translation of the term $\lambda{a}.\,{{\text{\sf let}}\;{f=\lambda{c}.\,{{a}\hskip 1.5pt{c}}}\;{\text{\sf in}}\;{{(\lambda{b}.\,{f})}\hskip 1.5pt{f}}}$, in the right term graph only a single variable occurrence can be shared. ### 4.2 Interpretation as eager-scope $\lambda$-ho-term-graphs Of the different translations of a $\lambda_{\text{\sf letrec}}$-term into $\lambda$-ho-term-graphs we are most interested in the one with the shortest possible abstraction prefixes. We say that such a term graph has ‘eager scope- closure‘, or that it is ‘eager-scope’.The reason for this choice is illustrated in Fig. 7: eager-scope closure allows for more sharing. ###### Definition 4.10 (eager scope). Let ${\cal G}=\langle V,\mathit{lab},\mathit{args},\mathit{r},P\rangle$ be a $\lambda$-ho-term-graph. ${\cal G}$ is called _eager-scope_ if for every $w\in V$ with $P({w})=pv$ for $p\in V^{}$ and $v\in V$, there is a path $w=w_{0}\mathrel{{\rightarrowtail}}w_{1}\mathrel{{\rightarrowtail}}\cdots\mathrel{{\rightarrowtail}}w_{m}\mathrel{\rightarrowtail_{0}}v$ in ${\cal G}$ from $w$ to $v$ with $P({w})\leq P({w_{i}})$ for all $i\in\left\\{{1,\ldots,m}\right\\}$, and (this follows) $w_{m}\in V\\!({\mathsf{0}})$ and $v\in V\\!({\lambda})$. Hence if a $\lambda$-ho-term-graph is not eager-scope, then it contains a vertex $w$ with abstraction-prefix $v_{1}\dots v_{n}$ from which $v_{n}$ is only reachable, if at all, by leaving the scope of $v_{n}$. It can be shown that in this case another abstraction-prefix function with shorter prefixes exists, and in which $v_{n}$ has been removed from the prefix of $w$. ###### Proposition 4.11 (eager-scope = minimal scope; uniqueness of eager- scope $\lambda$-ho-term-graphs). Let ${\cal G}_{i}=\langle V,\mathit{lab},\mathit{args},\mathit{r},P_{i}\rangle$ for $i\in\left\\{{1,2}\right\\}$ be $\lambda$-ho-term-graphs with the same underlying term graph. If ${\cal G}_{1}$ is eager-scope, then ${\left\|{P_{1}({w})}\right\|}\leq{\left\|{P_{2}({w)})}\right\|}$ for all $w\in V$. If, in addition, also ${\cal G}_{2}$ is eager-scope, then $P_{1}=P_{2}$. Hence eager-scope $\lambda$-ho-term-graphs over the same underlying term graph are unique. Also, we will call a translation process ‘eager-scope’ if it resolves the non- determinism in ${\cal R}$ in such a way that it always yields eager-scope $\lambda$-ho-term-graphs. In order to obtain an eager-scope translation we have to consider the following aspects. _Garbage removal._ In the presence of _garbage_ , unused function bindings, a translation cannot be eager-scope. Consider the term $\lambda{x}.\,{\lambda{y}.\,{{\text{\sf let}}\;{f=x}\;{\text{\sf in}}\;{y}}}$. The expendable binding $f=x$ prevents the application of the $\mathsf{S}$-rule, and hence the closure of the scope of $\lambda{x}$, directly below $\lambda{x}$. Therefore we henceforth assume that _all unused function bindings are removed_ prior to applying the rules ${\cal R}$. A $\lambda_{\text{\sf letrec}}$-term without garbage will be called _garbage- free_. _Short enough prefix lengths in the $\mathsf{let}$-rule._ For obtaining an eager-scope translation, we will usually stipulate that the $\mathsf{S}$-rule is applied eagerly, i.e. it is given precedence over the other rules. This is clearly necessary for keeping the abstraction prefixes minimal. But how do we choose the prefix lengths $l_{1},\dots,l_{k}$ in the let-rule? The prefix lengths $l_{i}$ determine at which position a binding $f_{i}=L_{i}$ is inserted into the abstraction prefixes. Therefore $l_{i}$ may not be chosen too short; otherwise a function $f$ depending on a function $g$ may end up to the right of $g$, and hence may be removed from the prefix by the $\mathsf{S}$-rule prematurely. preventing completion of the translation. Yet simply choosing $l_{i}=n$ may prevent scopes from being minimal. For example, when translating the term $\lambda{a}.\,{\lambda{b}.\,{{\text{\sf let}}\;{f=a}\;{\text{\sf in}}\;{{{{a}\hskip 1.5pt{a}}\hskip 1.5pt{({f}\hskip 1.5pt{a})}}\hskip 1.5pt{b}}}}$, it is crucial to allow shorter prefixes for the binding than for the in-part. As shown in Fig. 5 the graph on the left does not have eager scope-closure even if the $\mathsf{S}$-rule is applied eagerly. Consequently the opportunity for sharing the lower application vertices is lost. _Required variable analysis._ For choosing the prefixes in the let-rule correctly, the translation process must know for each function binding which $\lambda$-variables are ‘required’ on the right-hand side of its definition. For this we use an analysis obtaining the required variables for positions in a $\lambda_{\text{\sf letrec}}$-term as employed by algorithms for lambda- lifting Johnsson [1985]; Danvy and Schultz [2004]. The term ‘required variables’ was coined by Morazán and Schultz Morazán and Schultz [2008]. A $\lambda$-variable $x$ is called _required at a position $p$_ in a $\lambda_{\text{\sf letrec}}$-term $L$ if $x$ is bound by an abstraction above $p$, and has a free occurrence in the complete unfolding of $L$ below $p$ (also recursion variables from above $p$ are unfolded). The required variables at position $p$ in $L$ can be computed as those $\lambda$-variables with free occurrences that are reachable from $p$ by a downwards traversal with the stipulations: on encountering a let-binding the in-part is entered; when encountering a recursion variable the traversal continues at the right-hand side of the corresponding function binding (even if it is defined above $p$). With the result of the required variable analysis at hand, we now define properties of the translation process that can guarantee that the resulting $\lambda$-ho-term-graph is eager-scope. ###### Definition 4.12 (eager-scope and minimal-prefix generated). Let $L$ be a $\lambda_{\text{\sf letrec}}$-term, and let ${\cal G}$ be a $\lambda$-ho-term-graph. We say that ${\cal G}$ is _eager-scope_ ${\cal R}$-generated from $L$ if ${\cal G}$ is ${\cal R}$-generated from $L$ by a translation process with the following property: for every translation box reached during the process with label $(\vec{p}\;x^{v}[{B}])\hskip 1.0pt{P}$, where $P$ is a subterm of $L$ at position $q$, it holds that if $x$ is not a required variable at $q$ in $L$, then in the next translation step performed to this box either one of the rules $f$ or $\mathsf{let}$ is applied, or the prefix is shortened by the $\mathsf{S}$-rule. We say that ${\cal G}$ is ${\cal R}$-generated _with minimal prefixes_ from $L$ if ${\cal G}$ is ${\cal R}$-generated from $L$ by a translation process in which minimal prefix lengths are achieved by giving applications of the $\mathsf{S}$-rule precedence over applications of all other rules, and by always choosing prefixes minimally in applications of the $\mathsf{let}$-rule. ###### Proposition 4.13. Let ${\cal G}$ be a $\lambda$-ho-term-graph that is ${\cal R}$-generated from a garbage-free $\lambda_{\text{\sf letrec}}$-term $L$. The following statements hold: 1. (i) If ${\cal G}$ is eager-scope ${\cal R}$-generated from $L$, then ${\cal G}$ is eager-scope. 2. (ii) If ${\cal G}$ is ${\cal R}$-generated with minimal prefixes from $L$, then ${\cal G}$ is eager-scope ${\cal R}$-generated from $L$, hence by (i) ${\cal G}$ is eager-scope. ###### Proposition 4.14. For every $\lambda_{\text{\sf letrec}}$-term $L$, $\llbracket{L}\rrbracket_{{\cal H}}$ is eager-scope. ### 4.3 Correctness of $\llbracket{\cdot}\rrbracket_{{\cal H}}$ with respect to unfolding semantics In preparation of establishing the desired property (P1) in Sect. 5, we formulate, and outline the proof of, the fact that the semantics $\llbracket{\cdot}\rrbracket_{{\cal H}}$ is correct with respect to the unfolding semantics on $\lambda_{\text{\sf letrec}}$-terms. ###### Theorem 4.15. $\llbracket{L_{1}}\rrbracket_{{\lambda^{\infty}}}=\llbracket{L_{2}}\rrbracket_{{\lambda^{\infty}}}$ if and only if $\llbracket{L_{1}}\rrbracket_{{{\cal H}}}\mathrel{\mathrel{\hbox{\kern 0.43057pt\vbox{\hbox{\raise 0.43057pt\hbox{\kern-0.43057pt{$\leftrightarrow$}\kern-0.43057pt}}\hrule}\kern 0.43057pt}}}\llbracket{L_{2}}\rrbracket_{{{\cal H}}}$, for all $\lambda_{\text{\sf letrec}}$-terms $L_{1}$ and $L_{2}$. ###### Sketch of Proof. Central for the proof are $\lambda$-ho-term-graphs that have tree form and only contain variable backlinks, but no recursive backlinks. They form the class ${\cal H}_{T}\subsetneqq{\cal H}$. Every ${\cal G}\in{\cal H}$ has a unique ‘tree unfolding’ $\mathit{Tree}({{\cal G}})\in{\cal H}_{T}$. We make use of the following statements. For all $L,L_{1},L_{2}\in\mathit{Ter}({\lambda_{\text{\sf letrec}}})$, $M,M_{1},M_{2}\in\mathit{Ter}({\lambda^{\infty}_{\bullet}})$, ${\cal G},{\cal G}_{1},{\cal G}_{2}\in{\cal H}$, and ${\cal T}r,{\cal T}r_{1},{\cal T}r_{2}\in{\cal H}_{T}$ it can be shown that: $\displaystyle L_{1}\mathrel{{\to}_{\text{unf}}}L_{2}\;\;\Rightarrow\;\;\llbracket{L_{1}}\rrbracket_{{\cal H}}\mathrel{\mathrel{\hbox{\kern 0.43057pt\vbox{\hbox{\raise 0.43057pt\hbox{\kern-0.43057pt{$\leftarrow$}\kern-0.43057pt}}\hrule}\kern 0.43057pt}}}\llbracket{L_{2}}\rrbracket_{{\cal H}}$ (11) $\displaystyle L\mathrel{\twoheadrightarrow\twoheadrightarrow_{\text{unf}}}M\;\;(\text{hence}\;\llbracket{L}\rrbracket_{{\lambda^{\infty}}}=M)\;\;\Rightarrow\;\;\llbracket{L}\rrbracket_{{\cal H}}\mathrel{\mathrel{\hbox{\kern 0.43057pt\vbox{\hbox{\raise 0.43057pt\hbox{\kern-0.43057pt{$\leftarrow$}\kern-0.43057pt}}\hrule}\kern 0.43057pt}}}\llbracket{M}\rrbracket_{{\cal H}}$ (12) $\displaystyle\llbracket{M}\rrbracket_{{\cal H}}\in{\cal H}_{T}$ (13) $\displaystyle\llbracket{M_{1}}\rrbracket_{{\cal H}}\mathrel{{\simeq}}\llbracket{M_{2}}\rrbracket_{{\cal H}}\;\;\Rightarrow\;\;M_{1}=M_{2}$ (14) $\displaystyle{\cal G}\mathrel{\mathrel{\hbox{\kern 0.43057pt\vbox{\hbox{\raise 0.43057pt\hbox{\kern-0.43057pt{$\leftarrow$}\kern-0.43057pt}}\hrule}\kern 0.43057pt}}}\mathit{Tree}({{\cal G}})$ (15) $\displaystyle{\cal T}r_{1}\mathrel{\mathrel{\hbox{\kern 0.43057pt\vbox{\hbox{\raise 0.43057pt\hbox{\kern-0.43057pt{$\leftrightarrow$}\kern-0.43057pt}}\hrule}\kern 0.43057pt}}}{\cal T}r_{2}\;\;\Rightarrow\;\;{\cal T}r_{1}\mathrel{{\simeq}}{\cal T}r_{2}$ (16) $\displaystyle{\cal G}_{1}\mathrel{\mathrel{\hbox{\kern 0.43057pt\vbox{\hbox{\raise 0.43057pt\hbox{\kern-0.43057pt{$\leftrightarrow$}\kern-0.43057pt}}\hrule}\kern 0.43057pt}}}{\cal G}_{2}\;\;\Rightarrow\;\;\mathit{Tree}({{\cal G}_{1}})\mathrel{{\simeq}}\mathit{Tree}({{\cal G}_{2}})$ (17) Hereby (11) is used for proving (12), and (15) with (16) for (17). Now for proving the theorem, let $L_{1}$ and $L_{2}$ be arbitrary $\lambda_{\text{\sf letrec}}$-terms. “$\Rightarrow$”: Suppose $\llbracket{L_{1}}\rrbracket_{{\lambda^{\infty}}}=\llbracket{L_{2}}\rrbracket_{{\lambda^{\infty}}}$. Let $M$ be the infinite unfolding of $L_{1}$ and $L_{2}$, i.e., $\llbracket{L_{1}}\rrbracket_{{\cal H}}=M=\llbracket{L_{2}}\rrbracket_{{\cal H}}$. Then by (12) it follows $\llbracket{L_{1}}\rrbracket_{{\cal H}}\mathrel{\mathrel{\hbox{\kern 0.43057pt\vbox{\hbox{\raise 0.43057pt\hbox{\kern-0.43057pt{$\leftarrow$}\kern-0.43057pt}}\hrule}\kern 0.43057pt}}}\llbracket{M}\rrbracket_{{\cal H}}\mathrel{{{\hbox{\kern 0.43057pt\vbox{\hbox{\raise 0.43057pt\hbox{\kern-0.43057pt{$\rightarrow$}\kern-0.43057pt}}\hrule}\kern 0.43057pt}}}}\llbracket{L_{2}}\rrbracket_{{\cal H}}$, and hence $\llbracket{L_{1}}\rrbracket_{{\cal H}}\mathrel{\mathrel{\hbox{\kern 0.43057pt\vbox{\hbox{\raise 0.43057pt\hbox{\kern-0.43057pt{$\leftrightarrow$}\kern-0.43057pt}}\hrule}\kern 0.43057pt}}}\llbracket{L_{2}}\rrbracket_{{\cal H}}$. “$\Leftarrow$”: Suppose $\llbracket{L_{1}}\rrbracket_{{\cal H}}\mathrel{\mathrel{\hbox{\kern 0.43057pt\vbox{\hbox{\raise 0.43057pt\hbox{\kern-0.43057pt{$\leftrightarrow$}\kern-0.43057pt}}\hrule}\kern 0.43057pt}}}\llbracket{L_{2}}\rrbracket_{{\cal H}}$. Then by (17) it follows that $\mathit{Tree}({\llbracket{L_{1}}\rrbracket_{{\cal H}}})\mathrel{{\simeq}}\mathit{Tree}({\llbracket{L_{2}}\rrbracket_{{\cal H}}})$. Let $M_{1},M_{2}\in\mathit{Ter}({\lambda^{\infty}_{\bullet}})$ be the infinite unfoldings of $L_{1}$ and $L_{2}$, i.e. $M_{1}=\llbracket{L_{1}}\rrbracket_{{\lambda^{\infty}}}$, and $M_{2}=\llbracket{L_{2}}\rrbracket_{{\lambda^{\infty}}}$. Then (12) together with the assumption entails $\llbracket{M_{1}}\rrbracket_{{\cal H}}\mathrel{\mathrel{\hbox{\kern 0.43057pt\vbox{\hbox{\raise 0.43057pt\hbox{\kern-0.43057pt{$\leftrightarrow$}\kern-0.43057pt}}\hrule}\kern 0.43057pt}}}\llbracket{M_{2}}\rrbracket_{{\cal H}}$. Since $\llbracket{M_{1}}\rrbracket_{{\cal H}},\llbracket{M_{2}}\rrbracket_{{\cal H}}\in{\cal H}_{T}$ by (13), it follows by (16) that $\llbracket{M_{1}}\rrbracket_{{\cal H}}\mathrel{{\simeq}}\llbracket{M_{2}}\rrbracket_{{\cal H}}$. Finally, by using (14) we get $M_{1}=M_{2}$, and hence $\llbracket{L_{1}}\rrbracket_{{\lambda^{\infty}}}=M_{1}=M_{2}=\llbracket{L_{2}}\rrbracket_{{\lambda^{\infty}}}$.∎ ## 5 Lambda term graphs While modelling sharing expressed by $\lambda_{\text{\sf letrec}}$-terms through $\lambda$-ho-term-graphs facilitates comparisons via bisimilarity, it is not immediately clear how the compactification of $\lambda$-ho-term-graphs via the bisimulation collapse ${{\text{\small\textbar}}\downarrow}$ for $\lambda$-ho-term-graphs (which has to respect scopes in the form of the abstraction-prefix functions) can be computed efficiently. We therefore develop an implementation as first-order term graphs, for which standard methods are available. Due to Ex. 4.2, the scoping information cannot just be discarded, as functional bisimilarity on the underlying term graphs does not faithfully implement functional bisimilarity on $\lambda$-ho-term-graphs. Therefore the scoping information has to be incorporated in the first-order interpretation in some way. We accomplish this by extending ${\Sigma}^{\lambda}_{\bullet}$ with $\mathsf{S}$-vertices, scope delimiters, that signify the end of scopes. When translating a $\lambda$-ho-term-graph into a first-order term graph, $\mathsf{S}$-vertices are placed along those edges in the underlying term graph at which the abstraction prefix decreases in the $\lambda$-ho-term- graph. ###### Example 5.1 (Adding $\mathsf{S}$-vertices). Consider the terms in Ex. 4.2 and their $\lambda$-ho-term-graphs in Ex. 4.3. In the first-order interpretation below, the shading is just for illustration purposes; it is _not_ part of the structure, and does _not directly_ impair functional bisimulation. $\begin{aligned} \includegraphics[scale={0.78}]{figs/counterex_onlyvarbl_y_scoped.pdf}\end{aligned}\hskip 8.61108pt{\hbox{\kern 0.43057pt\vbox{\hbox{\raise 0.43057pt\hbox{\kern-0.43057pt{$\rightarrow$}\kern-0.43057pt}}\hrule}\kern 0.43057pt}}\hskip 8.61108pt\begin{aligned} \includegraphics[scale={0.78}]{figs/counterex_onlyvarbl_collapse_scoped.pdf}\end{aligned}\hskip 8.61108pt\not{\hbox{\kern 0.43057pt\vbox{\hbox{\raise 0.43057pt\hbox{\kern-0.43057pt{$\leftarrow$}\kern-0.43057pt}}\hrule}\kern 0.43057pt}}\hskip 8.61108pt\begin{aligned} \includegraphics[scale={0.78}]{figs/counterex_onlyvarbl_x_scoped.pdf}\end{aligned}$ The addition of scope delimiters resolves the problem of Ex.4.2. They adequately represent the scoping information. As for $\lambda$-ho-term-graphs, we will define correctness conditions by means of an abstraction-prefix function. However, the current approach with unary delimiter vertices leads to a problem. ###### Example 5.2 ($\mathsf{S}$-backlinks). The term graph with scope delimiters on the left admits a functional bisimulation that fuses two $\mathsf{S}$-vertices that close different scopes. We cannot hope to find a unique abstraction prefix for the resulting fused $\mathsf{S}$-vertex. This is remedied on the right by using a variant representation that requires backlinks from each $\mathsf{S}$-vertex to the abstraction vertex whose scope it closes. Then $\mathsf{S}$-vertices can only be fused if the corresponding abstractions have already been merged. Hence in the presence of $\mathsf{S}$-backlinks, as in the right illustration below, only the variable vertex can be shared. $\rightarrow$ $\rightarrow$ Therefore we consider term graphs over the extension ${\Sigma}^{\lambda}_{\mathsf{S},\bullet}$ of ${\Sigma}^{\lambda}_{\bullet}$ with a symbol $\mathsf{S}$ of arity $2$; one edge targets the successor vertex, the other is a backlink. We give correctness conditions, similar as for $\lambda$-ho-term-graphs, and define the arising class of ‘$\lambda$-term- graphs’. ###### Definition 5.3 (correct abstraction-prefix function for term graphs over ${\Sigma}^{\lambda}_{\mathsf{S},\bullet}$). Let $G=\langle V,\mathit{lab},\mathit{args},\mathit{r}\rangle$ be a ${\Sigma}^{\lambda}_{\mathsf{S},\bullet}$-term-graph. An _abstraction-prefix function_ $P\mathrel{:}V\to V^{}$ on $G$ is called _correct_ if for all $w,w_{0},w_{1}\in V$ and $k\in\left\\{{0,1}\right\\}$ it holds: $\displaystyle P({\mathit{r}})=\epsilon$ $\displaystyle(\text{root})$ $\displaystyle P({\bullet})=\epsilon$ $\displaystyle(\text{black hole})$ $\displaystyle w\in V\\!({\lambda})\,\mathrel{{\wedge}}\,w\mathrel{\rightarrowtail_{0}}{w}_{0}\;$ $\displaystyle\Rightarrow\;P({{w}_{0}})=P({w})w$ $\displaystyle(\lambda)$ $\displaystyle w\in V\\!({@})\,\mathrel{{\wedge}}\,w\mathrel{\rightarrowtail_{k}}{w}_{k}\;$ $\displaystyle\Rightarrow\;\;P({{w}_{k}})=P({w})$ $\displaystyle(@)$ $\displaystyle w\in V\\!({\mathsf{0}})\,\mathrel{{\wedge}}\,w\mathrel{\rightarrowtail_{0}}{w}_{0}\;$ $\displaystyle\Rightarrow\;\;\left\\{\hskip 1.0pt\begin{aligned} &{w}_{0}\in V\\!({\lambda})\\\\[-2.15277pt] &\,\mathrel{{\wedge}}\,P({{w}_{0}}){w}_{0}=P({w})\end{aligned}\right.$ $\displaystyle(\mathsf{0})_{1}$ $\displaystyle w\in V\\!({\mathsf{S}})\,\mathrel{{\wedge}}\,w\mathrel{\rightarrowtail_{0}}{w}_{0}\;$ $\displaystyle\Rightarrow\;\;\left\\{\hskip 1.0pt\begin{aligned} &P({{w}_{0}})v=P({w})\\\\[-2.15277pt] &\hskip 25.83325pt\text{for some $v\in V$}\end{aligned}\right.$ $\displaystyle(\mathsf{S})_{1}$ $\displaystyle w\in V\\!({\mathsf{S}})\,\mathrel{{\wedge}}\,w\mathrel{\rightarrowtail_{1}}{w}_{1}\;$ $\displaystyle\Rightarrow\;\;\left\\{\hskip 1.0pt\begin{aligned} &{w}_{1}\in V\\!({\lambda})\\\\[-2.15277pt] &\,\mathrel{{\wedge}}\,P({{w}_{1}}){w}_{1}=P({w})\end{aligned}\right.$ $\displaystyle(\mathsf{S})_{2}$ While in $\lambda$-ho-term-graphs the abstraction prefix can shrink by several vertices along an edge (cf. Def. 4.4), here the situation is strictly regulated: the prefix can only shrink by one variable, and only along the outgoing edge of a delimiter vertex. ###### Proposition 5.4 (uniqueness of the abstraction prefix function). Let $G$ be a term graph over the signature ${\Sigma}^{\lambda}_{\mathsf{S},\bullet}$. If $P_{1}$ and $P_{2}$ are correct abstraction prefix functions of $G$, then $P_{1}=P_{2}$. ###### Definition 5.5 ($\lambda$-term-graph). A _$\lambda$ -term-graph_ is a term graph $G=\langle V,\mathit{lab},\mathit{args},\mathit{r}\rangle$ over ${\Sigma}^{\lambda}_{\mathsf{S},\bullet}$ that has a correct abstraction- prefix function (which is not a part of $G$). The class of $\lambda$-term- graphs is ${\cal T}$. ###### Definition 5.6 (eager scope). A $\lambda$-term-graph $G$ is called _eager-scope_ if together with its abstraction-prefix function it meets the condition in Def. 4.10. ${\cal T}_{\text{eag}}$ denotes the class of eager-scope graphs. ### 5.1 Correspondence between $\lambda$-ho- and $\lambda$-term-graphs The correspondences between $\lambda$-ho-term-graphs and $\lambda$-term- graphs: $\displaystyle{\cal HT}\mathrel{:}{\cal H}\to{\cal T}$ $\displaystyle\hskip 12.91663pt{\cal TH}\mathrel{:}{\cal T}\to{\cal H}$ are defined as follows: For obtaining ${\cal HT}({{\cal G}})$ for a ${\cal G}\in{\cal H}$, insert scope-delimiters wherever the prefix decreases, as illustrated in Fig. 8. For obtaining ${\cal TH}({G})$ for a $G\in{\cal T}$, retain the abstraction-prefix function, and remove every delimiter vertex from $G$, thereby connecting its incoming edge with its outgoing edge. For formal definitions and well-definedness of ${\cal TH}$ and ${\cal HT}$, see Grabmayer and Rochel [2013a]. $a$$\scriptstyle(v_{1}\dots v_{n})$$b$$\scriptstyle(v_{1}\dots v_{m})$ $\overset{m<n}{\implies}$ $a$$\scriptstyle(v_{1}\dots v_{n})$$\mathsf{S}$$\scriptstyle(v_{1}\dots v_{n-1})$$\vdots$$\mathsf{S}$$\scriptstyle(v_{1}\dots v_{m+1})$$b$$\scriptstyle(v_{1}\dots v_{m})$ Figure 8: Left: definition of ${\cal HT}$ by inserting $\mathsf{S}$-vertices, between edge-connected vertices of a $\lambda$-ho-term-graph. Right: interpretation ${\cal HT}({{\cal G}})$ of the eager-scope $\lambda$-ho-term- graph ${\cal G}$ in Fig. 7. Note that a $\lambda$-ho-term-graph may have multiple corresponding $\lambda$-term-graphs that differ only with respect to their ‘degree’ of $\mathsf{S}$-sharing (the extent to which $\mathsf{S}$-vertices occur shared). ${\cal HT}$ maps to a $\lambda$-term-graph with no sharing of $\mathsf{S}$-vertices at all. The proposition below guarantees the usefulness of the translation ${\cal HT}$ for implementing functional bisimulation on $\lambda$-ho-term-graphs. In particular, this is due to items (iii) and (iv). As formulated by item (i), ${\cal TH}$ is a retraction of ${\cal HT}$ (and ${\cal HT}$ a section of ${\cal TH}$). The converse is not the case, yet it holds up to $\mathsf{S}$-sharing by item (ii). For the proof, we refer to our article Grabmayer and Rochel [2013a]. ###### Proposition 5.7 (correspondence with $\lambda$-ho-term-graphs). 1. (i) ${{\cal TH}}\mathrel{\circ}{{\cal HT}}=\text{id}_{{\cal H}}$. 2. (ii) . $({{\cal HT}}\mathrel{\circ}{{\cal TH})}({G})\mathrel{{\hbox{\kern 0.43057pt\vbox{\hbox{\raise 0.43057pt\hbox{\kern-0.43057pt{$\rightarrow$}\kern-0.43057pt}}\hrule}\kern 0.43057pt}}^{\mathsf{S}}}G$ holds for all $G\in{\cal T}\,$. 3. (iii) ${\cal TH}$ and ${\cal HT}$ preserve and reflect functional bisimulation $\rightarrow$ and bisimulation $\mathrel{\hbox{\kern 0.43057pt\vbox{\hbox{\raise 0.43057pt\hbox{\kern-0.43057pt{$\leftrightarrow$}\kern-0.43057pt}}\hrule}\kern 0.43057pt}}$ on ${\cal H}$ and ${\cal T}$. 4. (iv) ${\cal TH}$ and ${\cal HT}$ preserve and reflect the property eager-scope. 5. (v) ${\cal T}$ is closed under ${\hbox{\kern 0.43057pt\vbox{\hbox{\raise 0.43057pt\hbox{\kern-0.43057pt{$\rightarrow$}\kern-0.43057pt}}\hrule}\kern 0.43057pt}}^{\mathsf{S}}$, $\mathrel{\hbox{\kern 0.43057pt\vbox{\hbox{\raise 0.43057pt\hbox{\kern-0.43057pt{$\leftarrow$}\kern-0.43057pt}}\hrule}\kern 0.43057pt}}^{\mathsf{S}}$, and $\mathrel{\hbox{\kern 0.43057pt\vbox{\hbox{\raise 0.43057pt\hbox{\kern-0.43057pt{$\leftrightarrow$}\kern-0.43057pt}}\hrule}\kern 0.43057pt}}^{\mathsf{S}}$. 6. (vi) ${\cal HT}$ and ${\cal TH}$ induce isomorphisms between ${\cal H}$ and ${{\cal T}}/_{{\mathrel{\hbox{\scalebox{0.75}{\vbox{\hbox{\raise 0.3014pt\hbox{\kern-0.3014pt{$\leftrightarrow$}\kern-0.3014pt}}\hrule}}}}^{\mathsf{S}}}}$. ### 5.2 Closedness of ${\cal T}$ under functional bisimulation While preservation of $\rightarrow$ by ${\cal HT}$ is necessary for its implementation via $\rightarrow$ on ${\cal T}$, the practicality of the interpretation ${\cal HT}$ also depends on the closedness of ${\cal T}$ under $\rightarrow$ . Namely, if the bisimulation collapse $G={{\cal HT}({{\cal G}})}\hskip 0.73193pt{{{\text{\small\textbar}}\downarrow}}$ of the interpretation of some ${\cal G}\in{\cal H}$ were not contained in ${\cal T}$, then the converse interpretation ${\cal TH}$ could not be applied to $G$ in order to obtain the bisimulation collapse of ${\cal G}$. A subclass ${\cal K}$ of the term graphs over a signature $\Sigma$ is called _closed under functional bisimulation_ if, for all term graphs $G$, $G^{\prime}$ over $\Sigma$, whenever $G\in{\cal K}$ and $G\mathrel{{{\hbox{\kern 0.43057pt\vbox{\hbox{\raise 0.43057pt\hbox{\kern-0.43057pt{$\rightarrow$}\kern-0.43057pt}}\hrule}\kern 0.43057pt}}}}G^{\prime}$, then also $G^{\prime}\in{\cal K}$. Note that for obtaining this property the use of variable backlinks, and backlinks for delimiter vertices is crucial (cf. Ex. 5.2). Yet the class ${\cal T}$ is actually not closed under $\rightarrow$ : See Fig. 5.2 at the top for a homomorphism from a non-eager-scope $\lambda$-term- graph to a term graph over ${\Sigma}^{\lambda}_{\mathsf{S},\bullet}$ that is not a $\lambda$-term-graph (as suggested by the overlapping scopes). The use of eager scope-closure remedies the situation, see at the bottom: then the bisimulation collapse is a $\lambda$-term-graph. This motivates the following theorem, which is proved in the extended report of Grabmayer and Rochel [2013a]. It justifies property (P2) with ${\cal T}_{\text{eag}}$ for ${\cal T}$. $\mathrel{{{\hbox{\kern 0.43057pt\vbox{\hbox{\raise 0.43057pt\hbox{\kern-0.43057pt{$\rightarrow$}\kern-0.43057pt}}\hrule}\kern 0.43057pt}}}}$ Figure 9: ${\cal T}$ is not closed under functional bisimulation, yet ${\cal T}_{\text{eag}}$ is. $\mathrel{{{\hbox{\kern 0.43057pt\vbox{\hbox{\raise 0.43057pt\hbox{\kern-0.43057pt{$\rightarrow$}\kern-0.43057pt}}\hrule}\kern 0.43057pt}}}}$ ###### Theorem 5.8. The class ${\cal T}_{\text{eag}}$ of eager-scope $\lambda$-term-graphs is closed under functional bisimulation $\rightarrow$ . ### 5.3 $\lambda$-term-graph semantics for $\lambda_{\text{\sf letrec}}$-terms We will consider in fact two interpretations of $\lambda_{\text{\sf letrec}}$-terms as $\lambda$-term-graphs: first we define $\llbracket{\cdot}\rrbracket_{{\cal T}}^{\textsf{min}}$ as the composition of $\llbracket{\cdot}\rrbracket_{{\cal H}}$ and ${\cal HT}$; then we define the semantics $\llbracket{\cdot}\rrbracket_{{\cal T}}$ with more fine-grained $\mathsf{S}$-sharing, which is necessary for defining a readback with the property (P3). By composing the interpretation ${\cal HT}$ of $\lambda$-ho-term-graphs as $\lambda$-term-graphs with the $\lambda$-ho-term-graph semantics $\llbracket{\cdot}\rrbracket_{{\cal H}}$, a semantics of $\lambda_{\text{\sf letrec}}$-terms as $\lambda$-term-graphs is obtained. There is, however, a more direct way to define this semantics: by using an adaptation of the translation rules ${\cal R}$ in Fig. 4.1, on which $\llbracket{\cdot}\rrbracket_{{\cal H}}$ is based. For this, let ${\cal R}_{\mathsf{S}}$ be the result of replacing the rule $\mathsf{S}$ in ${\cal R}$ by the version in Fig. 10. While applications of this variant of the $\mathsf{S}$-rule also shorten the abstraction-prefix, they additionally produce a delimiter vertex. $\mathsf{S}$: $(\vec{p}\;x^{v}[f_{1}^{v_{1}}=L_{1},\dots,f_{n}^{v_{n}}=L_{n}])\hskip 1.0pt{L}$ $\overset{\text{\small\leavevmode\nobreak\ $x\not\in\textit{FV}({L})$}}{\underset{\text{\small\leavevmode\nobreak\ $f_{i}\not\in\textit{FV}({L})$}}{\implies}}$ $(\vec{p})\hskip 1.0pt{L}$$\mathsf{S}$$\scriptstyle(\mathit{vs}({\vec{p}})\;v)$$\lambda$$\scriptstyle v$$\scriptstyle v$ Figure 10: Delimiter-vertex producing version of the $\mathsf{S}$-rule in Fig. 4.1 Here, at the end of the translation process, every loop on an indirection vertex with a prefix of length $n$ is replaced by a chain of $n$ $\mathsf{S}$-vertices followed by a black hole vertex.Note that, while the system ${\cal R}_{\mathsf{S}}$ inherits all of the non-determinism of ${\cal R}$, the possible degrees of freedom have additional impact on the result, because now they also determine the precise degree of $\mathsf{S}$-vertex sharing. By analogous stipulations as in Def. 4.12 we define the conditions under which a $\lambda$-term-graph is called _eager-scope_ ${\cal R}_{\mathsf{S}}$-generated, or ${\cal R}_{\mathsf{S}}$-generated _with minimal prefixes_ , from a $\lambda_{\text{\sf letrec}}$-term. For these notions, statements entirely analogous to Prop. 4.13 hold. ###### Definition 5.9. The _semantics_ $\llbracket{\cdot}\rrbracket_{{\cal T}}^{\textsf{min}}$ for $\lambda_{\text{\sf letrec}}$-terms as $\lambda$-term-graphs is defined as $\llbracket{\cdot}\rrbracket_{{\cal T}}^{\textsf{min}}\mathrel{:}\mathit{Ter}({\lambda_{\text{\sf letrec}}})\to{\cal T}_{\text{eag}}$, $L\mapsto\llbracket{L}\rrbracket_{{\cal T}}^{\textsf{min}}\mathrel{{:=}}\,$ the eager-scope term graph that is ${\cal R}_{\mathsf{S}}$-generated with minimal prefixes from a garbage-free version $L^{\prime}$ of $L$. For an example, see Ex. 5.14 below. In $\llbracket{\cdot}\rrbracket_{{\cal T}}^{\textsf{min}}$, ‘min’ also indicates that $\lambda$-term-graphs obtained via this semantics exhibit minimal (in fact no) sharing (two or more incoming edges) of $\mathsf{S}$-vertices. This is substantiated by the next proposition, in the light of the fact that ${\cal HT}$ does not create any shared $\mathsf{S}$-vertices. ###### Proposition 5.10. $\llbracket{\cdot}\rrbracket_{{\cal T}}^{\textsf{min}}={{\cal HT}}\mathrel{\circ}{\llbracket{\cdot}\rrbracket_{{\cal H}}}$. Hence $\llbracket{\cdot}\rrbracket_{{\cal T}}^{\textsf{min}}$ only yields $\lambda$-term-graphs without sharing of $\mathsf{S}$-vertices, and therefore its image cannot be all of ${\cal T}_{\text{eag}}$. As a consequence, we cannot hope to define a readback function rb with respect to $\llbracket{\cdot}\rrbracket_{{\cal T}}^{\textsf{min}}$ that has the desired property (P3), because that requires that the image of the semantics is ${\cal T}_{\text{eag}}$ in its entirety. Therefore we modify the definition of $\llbracket{\cdot}\rrbracket_{{\cal T}}^{\textsf{min}}$ to obtain another $\lambda$-term-graph semantics $\llbracket{\cdot}\rrbracket_{{\cal T}}$ with image $\textrm{im}({\llbracket{\cdot}\rrbracket_{{\cal T}}})={\cal T}_{\text{eag}}$. This is achieved by letting the let-binding-structure of the $\lambda_{\text{\sf letrec}}$-term influence the degree of $\mathsf{S}$-sharing as much as possible, while staying eager-scope. We say that a $\lambda$-ho-term-graph ${\cal G}$ is _eager-scope ${\cal R}$-generated with maximal prefixes_ from a $\lambda_{\text{\sf letrec}}$-term $L$ if ${\cal G}$ is ${\cal R}$-generated from $L$ by a translation process in which in applications of the $\mathsf{let}$-rule the prefixes are chosen maximally, but so that the eager-scope property of the process is not compromised. It can be shown that this condition fixes the prefix lengths per application of the $\mathsf{let}$-rule. ###### Definition 5.11. The _semantics_ $\llbracket{\cdot}\rrbracket_{{\cal T}}$ for $\lambda_{\text{\sf letrec}}$-terms as $\lambda$-term-graphs is defined as $\llbracket{\cdot}\rrbracket_{{\cal T}}\mathrel{:}\mathit{Ter}({\lambda_{\text{\sf letrec}}})\to{\cal T}_{\text{eag}}$, $L\mapsto\llbracket{L}\rrbracket_{{\cal T}}\mathrel{{:=}}\,$ the $\lambda$-term-graph that is eager-scope ${\cal R}_{\mathsf{S}}$-generated with maximal prefixes from a garbage-free version $L^{\prime}$ of $L$. $\llbracket{L_{1}}\rrbracket_{{\cal T}}$ $=\llbracket{L_{1}}\rrbracket_{{\cal T}}^{\textsf{min}}=\llbracket{L_{2}}\rrbracket_{{\cal T}}^{\textsf{min}}$ $=\llbracket{L_{3}}\rrbracket_{{\cal T}}^{\textsf{min}}=\llbracket{L^{\prime}_{3}}\rrbracket_{{\cal T}}^{\textsf{min}}$ $\llbracket{L_{2}}\rrbracket_{{\cal T}}$ $\llbracket{L_{3}}\rrbracket_{{\cal T}}=\llbracket{L^{\prime}_{3}}\rrbracket_{{\cal T}}$ Figure 11: Translation of the $\lambda_{\text{\sf letrec}}$-terms from Ex. 5.14 with the semantics $\llbracket{\cdot}\rrbracket_{{\cal T}}^{\textsf{min}}$ and $\llbracket{\cdot}\rrbracket_{{\cal T}}$. For legibility some backlinks are merged. ###### Proposition 5.12. $\llbracket{L}\rrbracket_{{\cal T}}^{\textsf{min}}\mathrel{{\hbox{\kern 0.43057pt\vbox{\hbox{\raise 0.43057pt\hbox{\kern-0.43057pt{$\rightarrow$}\kern-0.43057pt}}\hrule}\kern 0.43057pt}}^{\mathsf{S}}}\llbracket{L}\rrbracket_{{\cal T}}$ holds for all $\lambda_{\text{\sf letrec}}$-terms $L\,$. Now due to this, and due to Prop. 5.7, (iii), the statement of Thm. 4.15 can be transferred to ${\cal T}$, yielding property (P1) for $\llbracket{\cdot}\rrbracket_{{\cal T}}$. ###### Theorem 5.13. For all $\lambda_{\text{\sf letrec}}$-terms $L_{1}$ and $L_{2}$ the following holds: $\llbracket{L_{1}}\rrbracket_{{\lambda^{\infty}}}=\llbracket{L_{2}}\rrbracket_{{\lambda^{\infty}}}$ if and only if $\llbracket{L_{1}}\rrbracket_{{{\cal T}}}\mathrel{\mathrel{\hbox{\kern 0.43057pt\vbox{\hbox{\raise 0.43057pt\hbox{\kern-0.43057pt{$\leftrightarrow$}\kern-0.43057pt}}\hrule}\kern 0.43057pt}}}\llbracket{L_{2}}\rrbracket_{{{\cal T}}}$. ###### Example 5.14. Consider the following four $\lambda_{\text{\sf letrec}}$-terms: $\displaystyle L_{1}$ $\displaystyle={\text{\sf let}}\;{I=\lambda{z}.\,{z}}\;{\text{\sf in}}\;{\lambda{x}.\,{\lambda{y}.\,{{\text{\sf let}}\;{f=x}\;{\text{\sf in}}\;{{({({y}\hskip 1.5pt{I})}\hskip 1.5pt{({I}\hskip 1.5pt{y})})}\hskip 1.5pt{({f}\hskip 1.5pt{f})}}}}}$ $\displaystyle L_{2}$ $\displaystyle=\lambda{x}.\,{{\text{\sf let}}\;{I=\lambda{z}.\,{z}}\;{\text{\sf in}}\;{\lambda{y}.\,{{\text{\sf let}}\;{f=x}\;{\text{\sf in}}\;{{({({y}\hskip 1.5pt{I})}\hskip 1.5pt{({I}\hskip 1.5pt{y})})}\hskip 1.5pt{({f}\hskip 1.5pt{f})}}}}}$ $\displaystyle L_{3}$ $\displaystyle=\lambda{x}.\,{\lambda{y}.\,{{\text{\sf let}}\;{I=\lambda{z}.\,{z},\,f=x}\;{\text{\sf in}}\;{{({({y}\hskip 1.5pt{I})}\hskip 1.5pt{({I}\hskip 1.5pt{y})})}\hskip 1.5pt{({f}\hskip 1.5pt{f})}}}}$ $\displaystyle L^{\prime}_{3}$ $\displaystyle=\lambda{x}.\,{{\text{\sf let}}\;{I=\lambda{z}.\,{z}}\;{\text{\sf in}}\;{\lambda{y}.\,{{\text{\sf let}}\;{f=x,\ g=I}\;{\text{\sf in}}\;{{({({y}\hskip 1.5pt{g})}\hskip 1.5pt{({g}\hskip 1.5pt{y})})}\hskip 1.5pt{({f}\hskip 1.5pt{f})}}}}}$ The three possible fillings of the dashed area in Fig. 11 depict the translations $\llbracket{L_{1}}\rrbracket_{{\cal T}}$, $\llbracket{L_{2}}\rrbracket_{{\cal T}}$, and $\llbracket{L_{3}}\rrbracket_{{\cal T}}=\llbracket{L^{\prime}_{3}}\rrbracket_{{\cal T}}$. The translations of the four terms with $\llbracket{\cdot}\rrbracket^{\textsf{min}}$ are identical $\llbracket{L_{1}}\rrbracket_{{\cal T}}^{\textsf{min}}=\llbracket{L_{2}}\rrbracket_{{\cal T}}^{\textsf{min}}=\llbracket{L_{3}}\rrbracket_{{\cal T}}^{\textsf{min}}=\llbracket{L^{\prime}_{3}}\rrbracket_{{\cal T}}^{\textsf{min}}=\llbracket{L_{1}}\rrbracket_{{\cal T}}$. ## 6 Readback of $\lambda$-term-graphs In this section we describe how from a given $\lambda$-term-graph $G$ a $\lambda_{\text{\sf letrec}}$-term $L$ that represents $G$ (i.e. for which $\llbracket{L}\rrbracket_{{\cal T}}=G$ holds) can be ‘read back’. For this purpose we define a process based on term synthesis rules. It defines a readback function from $\lambda$-term-graphs to $\lambda_{\text{\sf letrec}}$-terms. We illustrate this process by an example, formulate its most important properties, and sketch the proof of (P3). The idea underlying the definition of the readback procedure is the following: For a given $\lambda$-term-graph $G$, a spanning tree $T$ for $G$ (augmented with a dedicated root node) is constructed that severs cycles of $G$ at (some) recursive bindings or variable $\mathsf{S}$-backlinks. Now the spanning tree $T$ facilitates an inductive bottom-up (from the leafs upwards) synthesis process along $T$, which labels the edges of $G$ (except for variable backlinks) with prefixed $\lambda_{\text{\sf letrec}}$-terms. For this process we use local rules (see Fig. 12) that synthesise labels for incoming edges of a vertex from the labels of its outgoing edges. Eventually the readback of $G$ is obtained as the label for the edge that singles out the root of term graph. The design of the readback rules is based on a decision about where let- bindings are placed in the synthesised term. Namely there exists some freedom for these placements, as certain kind of shifts of let-expressions (let- floating steps Grabmayer and Rochel [2013d]) preserve the $\lambda$-term-graph interpretation. Here, let-bindings will always be declared in a let-expression that is placed as high up in the term as possible: a binding arising from the term synthesised for a shared vertex $w$ is placed in a let-expression that is created at the enclosing $\lambda$-abstraction of $w$ (the leftmost vertex in the abstraction-prefix $P({w})$ of $w$). $({[]})\hskip 0.5pt{\lambda{x}.\,{{\text{\sf let}}\;{f=\lambda{y}.\,{{{f}\hskip 1.5pt{x}}\hskip 1.5pt{y}}}\;{\text{\sf in}}\;{{f}\hskip 1.5pt{f}}}}$$\top$$\lambda$$\scriptstyle x$$([])\hskip 1.0pt{\lambda{x}.\,{{\text{\sf let}}\;{f=\lambda{y}.\,{{{f}\hskip 1.5pt{x}}\hskip 1.5pt{y}}}\;{\text{\sf in}}\;{{f}\hskip 1.5pt{f}}}}$$\scriptstyle()$$@$$([]\;x[f=\lambda{y}.\,{{{f}\hskip 1.5pt{x}}\hskip 1.5pt{y}}])\hskip 1.0pt{{f}\hskip 1.5pt{f}}$$\scriptstyle()$\|$([]\;x[f=\lambda{y}.\,{{{f}\hskip 1.5pt{x}}\hskip 1.5pt{y}}])\hskip 1.0pt{f}$$\scriptstyle(x)$$\scriptstyle\;\;f$$([]\;x[f=\text{?}])\hskip 1.0pt{f}$$\lambda$$\scriptstyle y$$([]\;x[f=\text{?}])\hskip 1.0pt{\lambda{y}.\,{{{f}\hskip 1.5pt{x}}\hskip 1.5pt{y}}}$$\scriptstyle(x)$$@$$([]\;x[f=\text{?}]\;y[])\hskip 1.0pt{{{f}\hskip 1.5pt{x}}\hskip 1.5pt{y}}$$\scriptstyle(x)$$\mathsf{S}$$([]\;x[f=\text{?}]\;y[])\hskip 1.0pt{{f}\hskip 1.5pt{x}}$$\scriptstyle(x\;y)$$@$$([]\;x[f=\text{?}])\hskip 1.0pt{{f}\hskip 1.5pt{x}}$$\scriptstyle(x\;y)$$\mathsf{0}$$\scriptstyle(x)$$([]\;x[])\hskip 1.0pt{x}$$\scriptstyle(x)$$\mathsf{0}$$\scriptstyle(x\;y)$$([]\;x[]\;y[])\hskip 1.0pt{y}$$([]\;x[f=\text{?}])\hskip 1.0pt{f}$ Figure 12: Example of the readback synthesis from a $\lambda$-term-graph. Figure 13: Readback synthesis rules for computing a representing $\lambda_{\text{\sf letrec}}$-term from a $\lambda$-term-graph. The rules for $\top$\- and $\lambda$-vertices have variants for the case that $B$ is empty. For explanations, see Def. 6.1, (Rb-5). ###### Definition 6.1 (readback of $\lambda$-term-graphs). Let $G\in{\cal T}$ be a $\lambda$-term-graph. The process of computing the readback of $G$ (a $\lambda_{\text{\sf letrec}}$-term) consists of the following five steps, starting on $G\,$: (Rb-1) Determine the abstraction-prefix function $P$ for $G$ by performing a traversal over $G$, and associate with every vertex $w$ of $G$ its abstraction-prefix $P({w})$. (Rb-2) Add a new vertex on top with label $\top$, arity 1, and empty abstraction prefix. Let $G^{\prime}$ be the resulting term graph, and $P^{\prime}$ its abstraction-prefix function. (Rb-3) Introduce indirection vertices to organise sharing: For every vertex $w$ of $G^{\prime}$ with two or more incoming non-variable-backlink edges, add an indirection vertex $w_{0}$, redirect the incoming edges of $w$ that are not variable backlinks to $w_{0}$, and direct the outgoing edge from $w_{0}$ to $w$. In the resulting term graph $G^{\prime\prime}$ only indirection vertices are shared333Incoming variable backlinks are not counted as sharing here.; their names will be used. Extend $P^{\prime}$ to an abstraction-prefix function $P^{\prime\prime}$ for $G^{\prime\prime}$ so that every indirection vertex $w_{0}$ gets the prefix of its successor $w$. (Rb-4) Construct a spanning tree $T^{\prime\prime}$ of $G^{\prime\prime}$ by using a depth-first search (DFS) on $G^{\prime\prime}$. Note that all variable backlinks, and $\mathsf{S}$-backlinks, and some of the recursive back- bindings, of $G^{\prime\prime}$, are not contained in $T^{\prime\prime}$, because they are back-edges of the DFS. (Rb-5) Apply the readback synthesis rules from Fig. 12 to $G^{\prime\prime}$ with respect to $T^{\prime\prime}$. By this a complete labelling of the edges of $G^{\prime\prime}$ by prefixed $\lambda_{\text{\sf letrec}}$-terms is constructed. The rules define how the labelling for an incoming edge (on top) of a vertex $w$ is synthesised under the assumption of an already determined labelling of an outgoing edge of (and below) $w$. If the outgoing edge in the rule does not carry a label, then the labelling of the incoming edge can happen regardless. Note that in these rules: • full line (dotted line) edges indicate spanning tree (non-spanning tree) edges, broken line edges either of these sorts; * • abstraction prefixes of vertices are crucial for the $\mathsf{0}$-vertex, and the second indirection vertex rule, where the prefixes in the synthesised terms are created; in the other rules the prefix of the assumed term is used; for indicating a correspondence between a term’s and a vertex’s abstraction prefix we denote by $\mathit{v}({\vec{p}})$ the word of vertices occurring in a term’s prefix $\vec{p}$; * • the rule for indirection vertices with incoming non-spanning tree edge introduces an unfinished binding $f\mathrel{=}{?}$ for $f$; unfinished bindings are completed in the course of the process; * • the $@$-vertex rule applies only if $\mathit{v}({\vec{p}_{0}})=\mathit{v}({\vec{p}_{1}})$; the operation $\vec{\cup}$ used in the synthesised term’s prefix builds the union per prefix variable of the pertaining bindings; if the prefixed terms $(\vec{p}_{0})\hskip 1.0pt{L_{0}}$ and $(\vec{p}_{1})\hskip 1.0pt{L_{1}}$ assumed in this rule contain both a yet unfinished binding equation $f={?}$ and a completed equation $f=P$ at a $\lambda$-variable $z$, then the synthesised term contains the completed binding $f=P$ for $f$ at $z\,$; * • not depicted in Fig. 12 are variants of $\top$\- and $\lambda$-vertices rules for the cases with empty $B$: then no let-binding is introduced in the synthesised term, but the term from the in-part is used. $\lambda$$\scriptstyle v_{n}$$\scriptstyle(\mathit{vs}({\vec{p}}))$$(\vec{p})\hskip 1.0pt{\lambda{v_{n}}.\,{{\text{\sf let}}\;{B}\;{\text{\sf in}}\;{L}}}$$(\vec{p}\;v_{n}[B])\hskip 1.0pt{L}$ $\mathsf{0}$$\scriptstyle(v_{1}\cdots v_{n})$$({}[{}]\;v_{1}[]\mathrel{\cdots}v_{n}[])\hskip 1.0pt{v_{n}}$$\lambda$$\scriptstyle v_{n}$ $\mathsf{S}$$\scriptstyle(\mathit{vs}({\vec{p}})v_{n})$$(\vec{p}\;v_{n}[])\hskip 1.0pt{L}$$\lambda$$\scriptstyle v_{n}$$(\vec{p})\hskip 1.0pt{L}$ . \|$\scriptstyle(\mathit{vs}({\vec{p}})\;v_{n+1})$$\scriptstyle f$$(\vec{p}\;{v_{n+1}}[{B,f=L}])\hskip 1.0pt{f}$$(\vec{p}\;{v_{n+1}}[{B,(f=\text{?})}])\hskip 1.0pt{L}$ \|$\scriptstyle(v_{1}\cdots v_{n}v_{n+1})$$\scriptstyle f$$({}[{}]\;v_{1}[]\mathrel{\cdots}{v_{n}}[{}]\;v_{n+1}[f=\text{?}])\hskip 1.0pt{f}$$(\vec{p}\;{v_{n+1}}[{B}])\hskip 1.0pt{L}$ . If this process yields the label $({[]})\hskip 0.5pt{L}$ for the (root-)edge pointing to the new top vertex of $G^{\prime\prime}$, where $L$ is a $\lambda_{\text{\sf letrec}}$-term, then we call $L$ the _readback of $G$_. Note that firing of the rules in step (Rb-5) of the readback process proceeds in bottom-up direction in the spanning tree, starting from the back-edges, with some room for parallelism concerning work in different subtrees. Furthermore observe that on all directed edges $e$ (spanning tree edges or back edges) the rule applied to derive the edge label is uniquely determined by (is tied to) the label of the target vertex $v$ of $e$, with the single exception of $v$ being an indirection vertex. In that case one of the two indirection vertex rules applies, depending on whether $e$ is a spanning-tree edge or a back-edge. ###### Proposition 6.2. Let $G$ be a $\lambda$-term-graph. The process described in Def. 6.1 produces a complete edge labelling of the (modified) term graph, with label $({[]})\hskip 0.5pt{L}$ for the topmost edge, where $L$ is a $\lambda_{\text{\sf letrec}}$-term. Hence it yields $L$ as the readback of $G$. Thus Def. 6.1 defines a function ${\textsf{rb}}\mathrel{:}{\cal T}\to\mathit{Ter}({\lambda_{\text{\sf letrec}}})$, the _readback function_. ###### Example 6.3. See Fig. 12 for the illustration of the synthesis of the readback from an exemplary $\lambda$-term-graph. Full line edges are in the spanning tree, dotted line edges are not. Note that at the top vertex, no empty let-binding is created since the variant of the $\top$-vertex rule for empty binding groups is applied. The following theorem validates property (P3), with ${\cal T}_{\text{eag}}$ for ${\cal T}$. ###### Theorem 6.4. For all $G\in{\cal T}_{\text{eag}}$: ${(\llbracket{\cdot}\rrbracket_{{\cal T}}}\mathrel{\circ}{{\textsf{rb}})}({G})=\llbracket{{\textsf{rb}}({G})}\rrbracket_{{\cal T}}\mathrel{{\simeq}}G$, i.e., rb is a right-inverse of $\llbracket{\cdot}\rrbracket_{{\cal T}}$, and $\llbracket{\cdot}\rrbracket_{{\cal T}}$ a left-inverse of rb, up to ${\simeq}$. Hence rb is injective, and $\llbracket{\cdot}\rrbracket_{{\cal T}}$ is surjective, thus $\textrm{im}({\llbracket{\cdot}\rrbracket_{{\cal T}}})={\cal T}_{\text{eag}}$. $f$: \|$\scriptstyle w$$\scriptstyle(\mathit{vs}({\vec{p}})\;v)$ $\overset{\text{$w$ already has an outgoing edge}}{\Longleftarrow}$ $(\vec{p}\;x^{v}[{\ldots,f^{w}=L,\ldots}])\hskip 1.0pt{f}$ $\overset{\text{no outgoing edge yet for $w$}}{\implies}$ \|$\scriptstyle w$$\scriptstyle(\mathit{vs}({\vec{p}})\;v)$$(\vec{p}\;x^{v}[{\ldots,f^{w}=L,\ldots}])\hskip 1.0pt{L}$ $\mathsf{let}$: $(\vec{p}\;x^{v}[{B}])\hskip 1.0pt{{\text{\sf let}}\;{f_{1}=L_{1},\dots,f_{k}=L_{k}}\;{\text{\sf in}}\;{L_{0}}}$ $\overset{\text{$w_{1}$,\ldots,$w_{k}$ fresh names}}{\underset{}{\hskip 25.83325pt}\implies{\hskip 25.83325pt}}$ $(\vec{p}\;x^{v}[{B,f_{1}^{w_{1}}=L_{1},\dots,f_{k}=L_{k}^{w_{k}}}])\hskip 1.0pt{L_{0}}$\|$\scriptstyle w_{1}$$\scriptstyle(v_{1}\dots v_{l_{1}})$$\cdots$\|$\scriptstyle w_{k}$$\scriptstyle(v_{1}\dots v_{l_{k}})$ Figure 14: Modification of (two of) the translation rules in Fig. 4.1 for a variant definition of the $\lambda$-term-graph interpretation of $\lambda_{\text{\sf letrec}}$-terms. Here the translation of a let-expression does not directly spawn translations for the binding equations, but the in- part has to be translated first. rec $(\vec{p}\;x^{v}[{\ldots,f^{w}=L,\ldots}])\hskip 1.0pt{f}$ $(\vec{p}\;x^{v}[{\ldots,f^{w}=L,\ldots}])\hskip 1.0pt{L}$ rec (name $w$ is fresh) $(\vec{p}\;x^{v}[{\ldots,f=L,\ldots}])\hskip 1.0pt{f}$ Figure 15: Formulation of the local translation rules in Fig. 14 in the form of inference rules. $(\vec{p}\;x^{v}[{B,f_{1}=L_{1},\dots,f_{k}=L_{k}}])\hskip 1.0pt{L_{0}}$ let $(\vec{p}\;x^{v}[{B}])\hskip 1.0pt{{\text{\sf let}}\;{f_{1}=L_{1},\dots,f_{k}=L_{k}}\;{\text{\sf in}}\;{L_{0}}}$ ###### Sketch of the Proof. Graph translation steps can be linked with corresponding readback steps in order to establish that the former roughly reverse the latter. Roughly, because e.g. reversing a $\lambda$-readback step necessitates both a $\lambda$\- and a let-translation step. However, this holds only for a modification of the translation rules ${\cal R}_{\mathsf{S}}$ from Fig. 4.1, Fig. 10 where the rules $\mathsf{let}$ (for let-expressions) and $f$ (for occurrences of recursion variables) are replaced by the locally-operating versions in Fig. 14, and a initiating rule: $\top$: $\overset{}{\underset{}{\implies}}$ $\top$$\scriptstyle$$({[]})\hskip 0.5pt{L}$ (start of translation of $\lambda_{\text{\sf letrec}}$-term $L$) for creating a top vertex is added. Now the translation of a let-expression does no longer directly spawn translations of the bindings, but the bindings will only be translated later once their calls have been reached during the translation process of the in-part, or of the definitions of other already translated bindings. Note that in the $\mathsf{let}$-rule in Fig. 14 function bindings are associated with the rightmost variable in the prefix, which corresponds to choosing $l_{i}=n$ in the $\mathsf{let}$-rule in Fig. 4.1. While such a stipulation does not guarantee the eager-scope translation of every term, it actually does so for all $\lambda_{\text{\sf letrec}}$-terms that are obtained by the readback Please find in Fig. 16 on page 16 and in Fig. 17) on page 17 graphical arguments for the stepwise reversal of readback steps through translation steps. This establishes that graph translations steps reverse readback steps, which is the crucial step in the proof of the theorem. 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{{}\pgfsys@rect{-43.7504pt}{-8.33301pt}{87.5008pt}{16.66602pt}\pgfsys@stroke\pgfsys@invoke{ } }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-40.41739pt}{-2.5pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$({v_{0}}[{\bar{B}_{0}}]\mathrel{\cdots}{v_{n}}[{\bar{B}_{n}}])\hskip 1.0pt{{L_{0}}\hskip 1.5pt{L_{1}}}$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} { {}{}{}}{}{{}}{}{{{}{}}{}} {}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{10.53297pt}\pgfsys@lineto{0.0pt}{19.91411pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{0.0pt}{10.53297pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\\\\[-7.5347pt] \rotatebox{270.0}{$\Rightarrow$}\\\\[3.22916pt] \raisebox{-0.5pt}{\scalebox{1.0}{ \leavevmode\hbox to166.35pt{\vbox to49.65pt{\pgfpicture\makeatletter\hbox{\hskip 83.17348pt\lower-30.7603pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ 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}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} { {}{}{}}{{{{}{}{}}}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {{}\pgfsys@rect{6.57828pt}{-30.5603pt}{76.3952pt}{16.66602pt}\pgfsys@stroke\pgfsys@invoke{ } }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{9.91129pt}{-24.7273pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$({v_{0}}[{\bar{B}_{0}}]\mathrel{\cdots}{v_{n}}[{\bar{B}_{n}}])\hskip 1.0pt{L_{1}}$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{{}{}}}{{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-30.42873pt}{5.56319pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptstyle(v_{1}\mathrel{\cdots\,}v_{n})$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{}{ {}{}{}} {}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}{}{}{{}}\pgfsys@moveto{-6.99347pt}{-2.13884pt}\pgfsys@lineto{-42.86337pt}{-13.10938pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{-0.95627}{-0.29247}{0.29247}{-0.95627}{-42.86337pt}{-13.10938pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}} {{}}{}{ {}{}{}} {}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}{}{}{{}}\pgfsys@moveto{6.99347pt}{-2.13884pt}\pgfsys@lineto{42.86337pt}{-13.10938pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ 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to66.32pt{\pgfpicture\makeatletter\hbox{\hskip 95.01291pt\lower-30.75696pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{7.11319pt}{0.0pt}\pgfsys@curveto{7.11319pt}{3.92854pt}{3.92854pt}{7.11319pt}{0.0pt}{7.11319pt}\pgfsys@curveto{-3.92854pt}{7.11319pt}{-7.11319pt}{3.92854pt}{-7.11319pt}{0.0pt}\pgfsys@curveto{-7.11319pt}{-3.92854pt}{-3.92854pt}{-7.11319pt}{0.0pt}{-7.11319pt}\pgfsys@curveto{3.92854pt}{-7.11319pt}{7.11319pt}{-3.92854pt}{7.11319pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.66667pt}{-4.16666pt}\pgfsys@invoke{ 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}\pgfsys@endscope{{}}{}{{}}{}{{{}{}}{}} {}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-7.31319pt}\pgfsys@lineto{0.0pt}{-16.69432pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{0.0pt}{-16.69432pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\end{gathered}$ $\displaystyle\hskip 12.91663pt\Longleftarrow$ \|$\scriptstyle(v_{1}\mathrel{\cdots\,}v_{n})$$\scriptstyle f$$({}[{}]\;{v_{1}}[{}]\mathrel{\cdots}{v_{n-1}}[{}]\;{v_{n}}[{f=\text{?}}])\hskip 1.0pt{f}$$(\vec{p}\;v_{n+1}[B])\hskip 1.0pt{L}$ Figure 17: Reversal of readback steps for variable vertices, black-hole vertices, and indirection vertices by corresponding translation steps. ## 7 Complexity analysis Here we report on a complexity analysis for the individual operations from the previous sections, for the used standard algorithms, and overall, for compactification and unfolding-equivalence. In the lemma below, (ii) and (v) justify the property (P4) of our methods. Items (iii) and (iv) detail the complexity of standard methods when used for computing bisimulation collapse and bisimilarity of $\lambda$-term-graphs. For this note that first-order term graphs can be modelled by deterministic process graphs, and hence by DFAs. Therefore bisimilarity of term graphs can be computed via language equivalence of corresponding DFAs Hopcroft and Karp [1971] (in time $O({n\alpha({n})})$ Norton [2009], where $\alpha$ is the quasi-constant _inverse Ackermann function_) and bisimulation collapse through state minimisation of DFAs (in time $O({n\log n})$) Hopcroft [1971]. ###### Lemma 7.1. 1. (i) $\text{size}({\llbracket{L}\rrbracket_{{\cal T}}\\!})\in O({{\left\|{L}\right\|}^{2}})$ for $L\in\mathit{Ter}({\lambda_{\text{\sf letrec}}})$. 2. (ii) Translating $L\in\mathit{Ter}({\lambda_{\text{\sf letrec}}})$ into $\llbracket{L}\rrbracket_{{\cal T}}\in{\cal T}$ takes time $O({{\left\|{L}\right\|}^{2}})$. 3. (iii) Collapsing $G\in{\cal T}$ to ${G}\hskip 0.73193pt{{{\text{\small\textbar}}\downarrow}}$ is in $O({\text{size}({G})\log\text{size}({G})})$. 4. (iv) Deciding bisimilarity of $G_{1},G_{2}\in{\cal T}$ requires time $O({n\alpha({n})})$ for $n=\max\left\\{{\text{size}({G_{1}}),\text{size}({G_{2}})}\right\\}$. 5. (v) Computing the readback ${\textsf{rb}}({G})$ for a given $G\in{\cal T}$ requires time $O({n\log n})$, for $n=\text{size}({G})$. See Fig. 18 for an example that the size bound in item (i) of the lemma is tight. ###### Proposition 7.2. $\text{size}({\llbracket{L}\rrbracket_{{\cal T}}\\!})\in O({{\left\|{L}\right\|}^{2}})$ for $\lambda_{\text{\sf letrec}}$-terms $L$. Consider the finite $\lambda$-terms $M_{n}$ with $n$ occurrences of bindings $\lambda{x_{2}}$: $\lambda{x_{0}x_{1}}.\,{{{x_{0}}\hskip 1.5pt{x_{1}}}\hskip 1.5pt{\lambda{x_{2}}.\,{{{x_{0}}\hskip 1.5pt{x_{1}}}\hskip 1.5pt{\lambda{x_{1}}.\,{{{x_{0}}\hskip 1.5pt{x_{2}}}\hskip 1.5pt{\lambda{x_{2}}.\,{{{x_{0}}\hskip 1.5pt{x_{1}}}\hskip 1.5pt{\ldots{\lambda{x_{2}}.\,{{{x_{0}}\hskip 1.5pt{x_{1}}}\hskip 1.5pt{x_{2}}}}}}}}}}}}$ Then ${\left\|{M_{n}}\right\|}\in O({n})$. But both the transformation of $M_{n}$ into a de-Bruijn index representation: $\lambda({\lambda({{@}({{@}({\mathsf{S}({\mathsf{0}})},\hskip 0.5pt{\mathsf{0}})},\hskip 0.5pt{\lambda({{@}({{@}({\mathsf{S}^{2}({\mathsf{0}})},\hskip 0.5pt{\mathsf{S}({\mathsf{0}})})},\hskip 0.5pt{\lambda({{@}({{@}({\mathsf{S}^{3}({\mathsf{0}})},\hskip 0.5pt{\mathsf{S}({\mathsf{0}})})},\hskip 0.5pt{\lambda({{@}({{@}({\mathsf{S}^{4}({\mathsf{0}})},\hskip 0.5pt{\mathsf{S}({\mathsf{0}})})},\hskip 0.5pt{\ldots\lambda({{@}({{@}({\mathsf{S}^{2n}({\mathsf{0}})},\hskip 0.5pt{\mathsf{S}({\mathsf{0}})})},\hskip 0.5pt{\mathsf{0}})})})})})})})})})})})$ and the rendering of $M_{n}$ with respect to the eager scope-delimiting strategy: $\lambda({\lambda({{@}({{@}({\mathsf{S}({\mathsf{0}})},\hskip 0.5pt{\mathsf{0}})},\hskip 0.5pt{\lambda({{@}({\mathsf{S}({{@}({\mathsf{S}^{1}({\mathsf{0}})},\hskip 0.5pt{\mathsf{0}})})},\hskip 0.5pt{\lambda({{@}({\mathsf{S}({{@}({\mathsf{S}^{2}({\mathsf{0}})},\hskip 0.5pt{\mathsf{0}})})},\hskip 0.5pt{\lambda({{@}({\mathsf{S}({{@}({\mathsf{S}^{3}({\mathsf{0}})},\hskip 0.5pt{\mathsf{0}})})},\hskip 0.5pt{\ldots\lambda({{@}({\mathsf{S}({{@}({\mathsf{S}^{2n-1}({\mathsf{0}})},\hskip 0.5pt{\mathsf{0}})})},\hskip 0.5pt{\mathsf{0}})})})})})})})})})})})$ have size $O({n^{2}})$. Figure 18: Example of a sequence $\\{M_{n}\\}_{n}$ of finite $\lambda$-terms $M_{n}$ whose translation into $\lambda$-term-graphs grows quadratically in the size of $M_{n}$. Based on this lemma, and on further considerations, we obtain the following complexity statements for our methods. ###### Theorem 7.3. 1. (i) The computation for a $\lambda_{\text{\sf letrec}}$-term $L$ with ${\left\|{L}\right\|}=n$, of a maximally compactified form $({\textsf{rb}}\circ{{\text{\small\textbar}}\downarrow}\circ\llbracket{\cdot}\rrbracket_{{\cal T}})({L})$ of a $\lambda_{\text{\sf letrec}}$-term $L$ requires time $O({n^{2}\log n})$. By using an $\mathsf{S}$-unsharing operation $\textsf{unsh}_{\mathsf{S}}$, a (typically smaller) $\lambda_{\text{\sf letrec}}$-term ${({{\textsf{rb}}}\mathrel{\circ}{{\textsf{unsh}_{\mathsf{S}}}\mathrel{\circ}{{{\text{\small\textbar}}\downarrow}}}}\mathrel{\circ}{\llbracket{\cdot}\rrbracket_{{\cal T}})}({L})$ of size $O({n\log n})$ can be obtained, with the same time complexity. 2. (ii) The decision of whether two $\lambda_{\text{\sf letrec}}$-terms $L_{1}$ and $L_{2}$ are unfolding equivalent requires time $O({n^{2}\alpha({n})})$ for $n=\max\left\\{{{\left\|{L_{1}}\right\|},{\left\|{L_{2}}\right\|}}\right\\}$. ## 8 Implementation We have implemented our methods in Haskell using the Utrecht University Attribute Grammar System. The implementation is available at http://hackage.haskell.org/package/maxsharing/. Output produced for three examples from this paper, and explanations for it, can be found in Appendix B; for all examples in Grabmayer and Rochel [2014]. ## 9 Modifications, extensions and applications We have described an adaptation of the bisimulation proof method for $\lambda_{\text{\sf letrec}}$-terms. Recognising unfolding equivalence and increasing sharing are reduced to problems involving first-order term graphs. The principal idea is to use the nested scope structure of higher-order terms for an interpretation by term graphs with scope delimiters. We conclude by describing easy modifications, rather direct extensions, and finally, promising areas of application for our methods. ### 9.1 Modifications _Implicit sharing of $\lambda$-variables_. Multiple occurrences of the same $\lambda$-variable in a $\lambda_{\text{\sf letrec}}$-term $L$ are not shared (represented by a shared variable vertex) in the graph interpretation $\llbracket{L}\rrbracket_{{\cal H}}$. Consequently, our method compactifies the term $\lambda{x}.\,{{x}\hskip 1.5pt{x}}$ into $\lambda{x}.\,{{\text{\sf let}}\;{f=x}\;{\text{\sf in}}\;{{f}\hskip 1.5pt{f}}}$. Such explicit sharing of variables is excessive for many applications. It can be remedied easily, namely by unsharing variable vertices before applying the readback, or by preventing the readback from introducing let-bindings when only a variable vertex is shared. _Avoiding aliases produced by the readback_. The readback function in Section 6 is sensitive to the degree of sharing of $\mathsf{S}$-vertices in the given $\lambda$-term-graph: it maps two $\lambda$-term-graphs that only differ in what concerns sharing of $\mathsf{S}$-vertices to different $\lambda_{\text{\sf letrec}}$-terms. Typically, for $\lambda$-term-graphs with maximal sharing of $\mathsf{S}$-vertices this can produce let-bindings that are just ‘aliases’, such as $g$ is alias for $I$ in $L^{\prime}_{3}$ from Ex. 5.14. This can be avoided in two ways: by slightly adapting the readback function, or by performing maximal unsharing of $\mathsf{S}$-vertices before applying the readback as defined. _Preventing disadvantageous sharing_. Introducing sharing at compile-time can cause ‘space leaks’, i.e. a needlessly high memory footprint, at run-time, because ‘a large data structure becomes shared […], and therefore its space which before was reclaimed by garbage collection now cannot be reclaimed until its last reference is used’ de Medeiros Santos [1995]. For this reason, realisations of CSE Chitil [1998] restrict the locally operating rewrite rules employed for introducing sharing by suitable conditions that account for the type of potentially shared subexpressions, and their strictness in the program. For our global method of introducing sharing via the bisimulation collapse, a different approach is needed. Here the bisimulation collapse can be restricted so that sharing is not introduced at vertices that should not be shared. More precisely, it can be prevented that any unshared vertex (in-degree one) from a pre-determined set of ‘sharing-unfit’ vertices would have a shared vertex (in-degree greater than one) as its image in the bisimulation collapse. This can be achieved by modifying the graph interpretation $\llbracket{\cdot}\rrbracket_{{\cal T}}$. Any set of sharing-unfit positions in $L$ gives rise to a set of sharing-unfit vertices in $\llbracket{L}\rrbracket_{{\cal T}}$. In the modification of $\llbracket{L}\rrbracket_{{\cal T}}\,$, special back-links are added from every sharing-unfit vertex with in-degree one to its immediate successor. These back-links prevent that such a sharing-unfit vertex $v$ can collapse with another vertex $v^{\prime}$ without that also the predecessors of $v$ and $v^{\prime}$ would collapse as well. _A more general notion of readback_. Condition (P3) is rather rigorous in that it imposes sharing structure on $\lambda_{\text{\sf letrec}}$ that is specific to $\lambda$-term-graphs (degrees of $\mathsf{S}$-sharing). For a weaker version of (P3) with ${\mathrel{\hbox{\scalebox{0.75}{\vbox{\hbox{\raise 0.43057pt\hbox{\kern-0.43057pt{$\leftrightarrow$}\kern-0.43057pt}}\hrule}}}}^{\mathsf{S}}}$ in place of isomorphism, a readback does not have to be injective, and, independently of how much $\mathsf{S}$-sharing a translation into $\lambda$-term-graphs introduces, a readback function always exists. ### 9.2 Extensions _Full functional languages_. In order to support programming languages that are based on $\lambda_{\text{\sf letrec}}$ like Haskell, additional language constructs need to be supported. Such languages can typically be desugared into a core language, which comprises only a small subset of language constructs such as constructors, case statements, and primitives. These constructs can be represented in an extension of $\lambda_{\text{\sf letrec}}$ by additional function symbols. In conjunction with a desugarer our methods are applicable to full programming languages. _Other programming languages, and calculi with binding constructs_. Most programming languages feature constructs for grouping definitions that are similar to letrec. We therefore expect that our methods can be adapted to many imperative languages in particular, and may turn out to be fruitful for optimising compilers. Our methods for achieving maximal sharing certainly generalise to theoretical frameworks, and calculi with binding constructs, such as the $\pi$-calculus Milner [1999], and higher-order rewrite systems (e.g. CRSs and HRSs, Terese [2003]) as used here for the formalisation of $\lambda_{\text{\sf letrec}}$. _Fully-lazy lambda-lifting_. There is a close connection between our methods and fully-lazy lambda-lifting Hughes [1982]; Peyton Jones [1987]. In particular, the required-variable and scope analysis of a $\lambda_{\text{\sf letrec}}$-term $L$ on which the $\lambda$-term-graph-translation $\llbracket{L}\rrbracket_{{\cal T}}$ is based is closely analogous to the one needed for extracting from $L$ the supercombinators in the result $\hat{L}$ of fully-lazy lambda-lifting $L$. Moreover, the fully-lazy lambda-lifting transformation can even be implemented in a natural way on the basis of our methods. Namely as the composition ${\textsf{rb}_{\text{\it LL}}}\mathrel{\circ}{\llbracket{\cdot}\rrbracket_{{\cal T}}}$ of the translation $\llbracket{\cdot}\rrbracket_{{\cal T}}$ into $\lambda$-term- graphs, where $\textsf{rb}_{\text{\it LL}}$ is a variant readback function that, for a given $\lambda$-term-graph, synthesises the system $\hat{L}$ of supercombinators, instead of the $\lambda_{\text{\sf letrec}}$-term ${\textsf{rb}}({L})$. _Maximal sharing on supercombinator translations of $\lambda_{\text{\sf letrec}}$-terms_. $\lambda_{\text{\sf letrec}}$-terms $L$ correspond to supercombinator systems $\hat{L}$, the result of fully-lazy lambda-lifting $L$: the combinators in $\hat{L}$ correspond to ‘extended scopes’ Grabmayer and Rochel [2012] (or ‘skeletons’ Balabonski [2012]) in $L$, and supercombinator reduction steps on $\hat{L}$ correspond to weak $\beta$-reduction steps $L$. In the case of $\lambda$-calculus this has been established by Balabonski Balabonski [2012]. Via this correspondence the maximal-sharing method for $\lambda_{\text{\sf letrec}}$-terms can be lifted to obtain a maximal-sharing method systems of supercombinators obtained by fully-lazy lambda-lifting. _Non-eager scope-closure strategies_. We focused on eager-scope translations, because they facilitate maximal sharing, and guarantee that interpretations of unfolding-equivalent $\lambda_{\text{\sf letrec}}$-terms are bisimilar. Yet every scope-closure strategy Grabmayer and Rochel [2012] induces a translation and its own notion of maximal sharing. For adapting our maximal sharing method it is necessary to modify the translation into first-order term graphs in such a way that the image class obtained is closed under homomorphism (${\cal T}$ is not closed under $\rightarrow$ , unlike its subclass ${\cal T}_{\text{eag}}$). This can be achieved by using delimiter vertices also below variable vertices to close scopes that are still open [Grabmayer and Rochel, 2013a, report]. _Weaker notions of sharing_. The presented methods deal with sharing as expressed by letrec that is horizontal, vertical, or twisted Blom [2001]. By contrast, the construct $\mu$ Blom [2001]; Grabmayer and Rochel [2013b] expresses only vertical, and the non-recursive let only horizontal, sharing. By restricting bisimulation, our methods can be adapted to the $\lambda$-calculus with $\mu$, or with let. _Nested term graphs_. The nested scope structure of a $\lambda_{\text{\sf letrec}}$-term can also be represented by a nested structure of term graphs. The representation of a $\lambda_{\text{\sf letrec}}$-term as a ‘nested term graph’ Grabmayer and van Oostrom [2014] starts with an ordinary term graph in which some of the vertices are labelled by ‘nested’ symbols that designate outermost bindings together with their scope. Any such vertex is additionally associated with a usual term graph that specifies the subterm context describing the scope, where any inner scopes are again expressed by nested symbols. The association between nested symbols and their term graph specifications is required to be tree-like. The implementation result developed here can be generalised to show that nested term graphs can be implemented faithfully by first-order term graphs Grabmayer and van Oostrom [2014]. ### 9.3 Applications _Maximal sharing at run-time_. Maximal sharing can be applied repeatedly at run-time in order to regain a maximally shared form, thereby speeding up evaluation. This is reminiscent of ‘collapsed tree rewriting’ Plump [1993] for evaluating first-order term graphs represented as maximally shared dags. Since the state of a program in the memory at run-time is typically represented as a supercombinator graph, compactification by bisimulation collapse can take place directly on that graph (see Sec. 9.2), no translation is needed. Compactification can be coupled with garbage collection as bisimulation collapse subsumes some of the work required for a mark and sweep garbage collector. However, a compromise needs to be found between the costs for the optimisation and the gained efficiency. _Compile-time optimisation phase_. Increasing sharing facilitates potential gains in efficiency. Our method generalises common subexpression elimination, but therefore it also inherits its shortcomings: the cost of sharing (e.g. of very small functions) might exceed the gain. In non-strict functional languages, sharing can cause ‘memory leaks’ Chitil [1998]. Therefore, similar as for CSE, additional dynamic analyses like binding-time analysis Palsberg and Schwartzbach [1994], and heuristics to restrict sharing in cases when it is disadvantageous Peyton Jones [1987]; Goldberg and Hudak [1987] are advisable. _Additional prevention of disadvantageous sharing_. While static analysis methods for preventing sharing that may be disadvantageous at run-time can be adapted from CSE to the maximal-sharing method (see Sec. 9.1), this has yet to be investigated for binding-time analysis Palsberg and Schwartzbach [1994] and a sharing analysis of partial applications Goldberg and Hudak [1987]. _Code improvement_. In programming it is generally desirable to avoid duplication of code. As extension of CSE, our method is able to detect code duplication. The bisimulation collapse of the term graph interpretation of a program can, together with the readback, provide guidance on how code can be refactored into a more compact form. This application requires some fine- tuning to avoid excessive behaviour like the explicit sharing of variable occurrences (see Sec. 9.1). Yet for this only lightweight additional machinery is needed, such as size constraints or annotations to restrict the bisimulation collapse. _Function equivalence_. Recognising whether two programs implement the same function is undecidable. Still, this problem is tackled by proof assistants, and by automated theorem provers used in type-checkers of compilers for dependently-typed programming languages such as Agda. For such systems co- inductive proofs are more difficult to find than inductive ones, and require more effort by the user. Our method for deciding unfolding-equivalence could help to develop new approaches to finding co-inductive proofs. #### Acknowledgment We want to thank Vincent van Oostrom for extensive feedback on a draft, Doaitse Swierstra and Dimitri Hendriks for helpful comments, and Jeroen Keiren for a suggestion concerning restricting the bisimulation collapse. We also thank the anonymous reviewers for their comments, and a number of stimulating questions. ## References * Asperti and Guerrini [1998] A. Asperti and S. Guerrini. _The Optimal Implementation of Functional Programming Languages_. Cambridge University Press, 1998. * Balabonski [2012] T. Balabonski. A unified approach to fully lazy sharing. In _Proceedings of POPL ’12_ , pages 469–480, New York, NY, USA, 2012\. ACM. * Bird and Patterson [1999] R. S. Bird and R. Patterson. de Bruijn notation as a nested datatype. _Journal of Functional Programming_ , 9(1):77–91, 1999. * Blom [2001] S. Blom. _Term Graph Rewriting – Syntax and Semantics_. 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Echahed, editor, _Proceedings of TERMGRAPH 2011_ , volume 48 of _EPTCS, on arXiv:1102.2268_ , pages 85–100. arxiv.org, 2011. * Terese [2003] Terese. _Term Rewriting Systems_ , volume 55 of _Cambridge Tracts in Theoretical Computer Science_. Cambridge University Press, 2003. * Wadsworth [1971] C. P. Wadsworth. _Semantics and Pragmatics of the Lambda-Calculus_. PhD thesis, University of Oxford, 1971. ## Appendix A Example for the translation $\lambda_{\text{\sf letrec}}$-terms into $\lambda$-ho-term-graphs and into $\lambda$-term-graphs For two terms from the paper we provide the stepwise translation of $\lambda_{\text{\sf letrec}}$-terms into $\lambda$-ho-term-graphs. We start off by a simple example, namely the translation of the term $P$ from Example 1.2 on page 1.2. There is no application of the $\mathsf{S}$-rule, thus it yields the same sequence of graphs regardless of whether the rules ${\cal R}$ from Fig. 4.1 are used, by which a $\lambda$-ho-term-graph is produced, or the modified rules ${\cal R}_{\mathsf{S}}$ (the result of dropping the $\mathsf{S}$-rule from ${\cal R}$, and using the $\mathsf{S}$-rule in Fig. 10 instead), by which $\lambda$-term-graphs with an abstraction-prefix function are produced. $({[]})\hskip 0.5pt{\lambda{f}.\,{{\text{\sf let}}\;{r={f}\hskip 1.5pt{({f}\hskip 1.5pt{r})}}\;{\text{\sf in}}\;{r}}}$ $\implies_{\lambda\hskip 17.07164pt}$ $([]\;f^{v}[])\hskip 1.0pt{{\text{\sf let}}\;{r={f}\hskip 1.5pt{({f}\hskip 1.5pt{r})}}\;{\text{\sf in}}\;{r}}$$\lambda$$\scriptstyle v$$\scriptstyle()$ $\implies_{\mathsf{let}\hskip 17.07164pt}$ $([]\;f^{v}[r^{w}={f}\hskip 1.5pt{({f}\hskip 1.5pt{r})}])\hskip 1.0pt{r}$$\lambda$$\scriptstyle v$$\scriptstyle()$$([]\;f^{v}[r^{w}={f}\hskip 1.5pt{({f}\hskip 1.5pt{r})}])\hskip 1.0pt{{f}\hskip 1.5pt{({f}\hskip 1.5pt{r})}}$\|$\scriptstyle w$ $\implies_{@,f}$ $\lambda$$\scriptstyle v$$\scriptstyle()$\|$\scriptstyle w$$@$$\scriptstyle(v)$$([]\;f^{v}[r^{w}={f}\hskip 1.5pt{({f}\hskip 1.5pt{r})}])\hskip 1.0pt{f}$$([]\;f^{v}[r^{w}={f}\hskip 1.5pt{({f}\hskip 1.5pt{r})}])\hskip 1.0pt{{f}\hskip 1.5pt{r}}$ $\implies_{\mathsf{0},@}$ $\lambda$$\scriptstyle v$$\scriptstyle()$\|$\scriptstyle w$$@$$\scriptstyle(v)$$\mathsf{0}$$\scriptstyle(v)$$@$$\scriptstyle(v)$$([]\;f^{v}[r^{w}={f}\hskip 1.5pt{({f}\hskip 1.5pt{r})}])\hskip 1.0pt{f}$$([]\;f^{v}[r^{w}={f}\hskip 1.5pt{({f}\hskip 1.5pt{r})}])\hskip 1.0pt{r}$ $\implies_{\mathsf{0},f}$ $\lambda$$\scriptstyle v$$\scriptstyle()$\|$\scriptstyle w$$@$$\scriptstyle(v)$$\mathsf{0}$$\scriptstyle(v)$$@$$\scriptstyle(v)$$\mathsf{0}$$\scriptstyle(v)$ $\implies_{\text{eliminate indirection vertices}}$ $\lambda$$\scriptstyle v$$\scriptstyle()$$@$$\scriptstyle(v)$$\mathsf{0}$$\scriptstyle(v)$$@$$\scriptstyle(v)$$\mathsf{0}$$\scriptstyle(v)$ We continue with the term $L_{2}$ from Example 5.14 on page 5.14. We translate it in two different ways correspoding to the first-order term graph semantics $\llbracket{\cdot}\rrbracket_{{\cal T}}^{\textsf{min}}$ from Def. 5.9 and $\llbracket{\cdot}\rrbracket_{{\cal T}}$ from Def. 5.11, respectively. Both sequences of translation steps yield the same $\lambda$-ho-term-graph $\llbracket{L_{2}}\rrbracket_{{\cal H}}$, but obtain the different $\lambda$-term-graphs $\llbracket{L_{2}}\rrbracket_{{\cal T}}^{\textsf{min}}$ and $\llbracket{L_{2}}\rrbracket_{{\cal T}}$ in Fig. 11. These two stepwise translation processes exemplify, on the one hand (when ignoring the dotted $\mathsf{S}$-vertices that do not occur in $\lambda$-ho- term-graphs) the translation into $\lambda$-ho-term-graphs according to the rules ${\cal R}$ from Fig. 4.1, and on the other hand (now taking the dotted delimiter $\mathsf{S}$-vertices into account) the translation into $\lambda$-term-graphs according to the modified rules ${\cal R}_{\mathsf{S}}$. In each step, one or more translation rules (whose names are indicated as subscripts in the steps) are applied to the translation boxes in the graph. When no more rules are applicable, indirection vertices are erased. Both translations are eager-scope (i.e. applications of $\mathsf{S}$-rules are given priority, and prefixes lengths are chosen too be small enough in the $\mathsf{let}$-rule, see Sec. 4.2) but differ in how they resolve the non- determinism due to different choices for the prefix lengths $l_{1},\dots,l_{k}$ in the $\mathsf{let}$-rule. First we consider the translations of $L_{2}$ with $\llbracket{\cdot}\rrbracket_{{\cal H}}$ and $\llbracket{\cdot}\rrbracket_{{\cal T}}^{\textsf{min}}$, i.e. the translation process in which prefix lengths are chosen minimally when applying the $\mathsf{let}$-rule. $({[]})\hskip 0.5pt{\lambda{x}.\,{\lambda{y}.\,{{\text{\sf let}}\;{I=\lambda{z}.\,{z},f=x}\;{\text{\sf in}}\;{{{{y}\hskip 1.5pt{I}}\hskip 1.5pt{({I}\hskip 1.5pt{y})}}\hskip 1.5pt{({f}\hskip 1.5pt{f})}}}}}$ $\implies_{\lambda}$ $([]\;x^{u}[])\hskip 1.0pt{\lambda{y}.\,{{\text{\sf let}}\;{I=\lambda{z}.\,{z},f=x}\;{\text{\sf in}}\;{{{{y}\hskip 1.5pt{I}}\hskip 1.5pt{({I}\hskip 1.5pt{y})}}\hskip 1.5pt{({f}\hskip 1.5pt{f})}}}}$$\lambda$$\scriptstyle u$$\scriptstyle()$ $\implies_{\lambda}$ $([]\;x^{u}[]\;y^{v}[])\hskip 1.0pt{{\text{\sf let}}\;{I=\lambda{z}.\,{z},f=x}\;{\text{\sf in}}\;{{{{y}\hskip 1.5pt{I}}\hskip 1.5pt{({I}\hskip 1.5pt{y})}}\hskip 1.5pt{({f}\hskip 1.5pt{f})}}}$$\lambda$$\scriptstyle v$$\scriptstyle(u)$$\lambda$$\scriptstyle u$$\scriptstyle()$ $\implies_{\mathsf{let}}$ $([I^{s}=\lambda{z}.\,{z}]\;x^{u}[f^{t}=x]\;y^{v}[])\hskip 1.0pt{{{{y}\hskip 1.5pt{I}}\hskip 1.5pt{({I}\hskip 1.5pt{y})}}\hskip 1.5pt{({f}\hskip 1.5pt{f})}}$$\lambda$$\scriptstyle v$$\scriptstyle(v)$$\lambda$$\scriptstyle u$$\scriptstyle()$\|$\scriptstyle s$$([I^{s}={\lambda{z}.\,{z}}])\hskip 1.0pt{\lambda{z}.\,{z}}$\|$\scriptstyle t$$([I^{s}={\lambda{z}.\,{z}}]\;x^{u}[f^{t}=x])\hskip 1.0pt{x}$ $\implies_{\lambda,@,\mathsf{0}}$ $([I^{s}=\lambda{z}.\,{z}]\;x^{u}[f^{t}=x]\;y^{v}[])\hskip 1.0pt{{{y}\hskip 1.5pt{I}}\hskip 1.5pt{({I}\hskip 1.5pt{y})}}$$([I^{s}=\lambda{z}.\,{z}]\;x^{u}[f^{t}=x]\;y^{v}[])\hskip 1.0pt{{f}\hskip 1.5pt{f}}$$@$$\scriptstyle(u\;v)$$\lambda$$\scriptstyle v$$\scriptstyle(u)$$\lambda$$\scriptstyle u$$\scriptstyle()$$\lambda$$\scriptstyle w$$\scriptstyle()$\|$\scriptstyle s$$([I^{s}={\lambda{z}.\,{z}}]\;z^{w}[])\hskip 1.0pt{z}$$\mathsf{0}$$\scriptstyle(u)$\|$\scriptstyle t$ $\implies_{\mathsf{0},@,@}$ $@$$\scriptstyle(u\;v)$$([I^{s}=\lambda{z}.\,{z}]\;x^{u}[f^{t}=x]\;y^{v}[])\hskip 1.0pt{{y}\hskip 1.5pt{I}}$$([I^{s}=\lambda{z}.\,{z}]\;x^{u}[f^{t}=x]\;y^{v}[])\hskip 1.0pt{{I}\hskip 1.5pt{y}}$$\mathsf{S}$$\scriptstyle(u\;v)$$([I^{s}=\lambda{z}.\,{z}]\;x^{u}[f^{t}=x])\hskip 1.0pt{{f}\hskip 1.5pt{f}}$$@$$\scriptstyle(u\;v)$$\lambda$$\scriptstyle v$$\scriptstyle(u)$$\lambda$$\scriptstyle u$$\scriptstyle()$\|$\scriptstyle s$$\lambda$$\scriptstyle w$$\scriptstyle()$$\mathsf{0}$$\scriptstyle(w)$\|$\scriptstyle t$$\mathsf{0}$$\scriptstyle(u)$ $\implies_{@,@,@}$ $@$$\scriptstyle(u\;v)$$@$$\scriptstyle(u\;v)$$([I^{s}=\lambda{z}.\,{z}]\;x^{u}[f^{t}=x]\;y^{v}[])\hskip 1.0pt{I}$$([I^{s}=\lambda{z}.\,{z}]\;x^{u}[f^{t}=x]\;y^{v}[])\hskip 1.0pt{y}$$@$$\scriptstyle(u\;v)$$([I^{s}=\lambda{z}.\,{z}]\;x^{u}[f^{t}=x]\;y^{v}[])\hskip 1.0pt{y}$$([I^{s}=\lambda{z}.\,{z}]\;x^{u}[f^{t}=x]\;y^{v}[])\hskip 1.0pt{I}$$\mathsf{S}$$\scriptstyle(u\;v)$$@$$\scriptstyle(u\;v)$$([I^{s}=\lambda{z}.\,{z}]\;x^{u}[f^{t}=x])\hskip 1.0pt{f}$$([I^{s}=\lambda{z}.\,{z}]\;x^{u}[f^{t}=x])\hskip 1.0pt{f}$$@$$\scriptstyle(u\;v)$$\lambda$$\scriptstyle v$$\scriptstyle(u)$$\lambda$$\scriptstyle u$$\scriptstyle()$\|$\scriptstyle s$$\lambda$$\scriptstyle w$$\scriptstyle()$$\mathsf{0}$$\scriptstyle(w)$\|$\scriptstyle t$$\mathsf{0}$$\scriptstyle(u)$ $\implies_{\mathsf{0},\mathsf{S},\mathsf{S},\mathsf{0},f,f}$ $@$$\scriptstyle(u\;v)$$@$$\scriptstyle(u\;v)$$\mathsf{S}$$\scriptstyle(u\;v)$$([I^{s}=\lambda{z}.\,{z}]\;x^{u}[f^{t}=x])\hskip 1.0pt{I}$$\mathsf{0}$$\scriptstyle(u\;v)$$@$$\scriptstyle(u\;v)$$\mathsf{0}$$\scriptstyle(u\;v)$$\mathsf{S}$$\scriptstyle(u\;v)$$([I^{s}=\lambda{z}.\,{z}]\;x^{u}[f^{t}=x])\hskip 1.0pt{I}$$\mathsf{S}$$\scriptstyle(u\;v)$$@$$\scriptstyle(u\;v)$$@$$\scriptstyle(u\;v)$$\lambda$$\scriptstyle v$$\scriptstyle(u)$$\lambda$$\scriptstyle u$$\scriptstyle()$\|$\scriptstyle s$$\lambda$$\scriptstyle w$$\scriptstyle()$$\mathsf{0}$$\scriptstyle(w)$\|$\scriptstyle t$$\mathsf{0}$$\scriptstyle(u)$ $\implies_{\mathsf{S},\mathsf{S}}$ $@$$\scriptstyle(u\;v)$$@$$\scriptstyle(u\;v)$$\mathsf{S}$$\scriptstyle(u\;v)$$\mathsf{S}$$\scriptstyle(u)$$([I^{s}=\lambda{z}.\,{z}])\hskip 1.0pt{I}$$\mathsf{0}$$\scriptstyle(u)$$@$$\scriptstyle(u\;v)$$\mathsf{0}$$\scriptstyle(u\;v)$$\mathsf{S}$$\scriptstyle(u\;v)$$\mathsf{S}$$\scriptstyle(u)$$([I^{s}=\lambda{z}.\,{z}])\hskip 1.0pt{I}$$\mathsf{S}$$\scriptstyle(u)$$@$$\scriptstyle(u\;v)$$@$$\scriptstyle(u\;v)$$\lambda$$\scriptstyle v$$\scriptstyle(u)$$\lambda$$\scriptstyle u$$\scriptstyle()$\|$\scriptstyle s$$\lambda$$\scriptstyle w$$\scriptstyle()$$\mathsf{0}$$\scriptstyle(w)$\|$\scriptstyle t$$\mathsf{0}$$\scriptstyle(u)$ $\implies_{f,f}$ $@$$\scriptstyle(u\;v)$$@$$\scriptstyle(u\;v)$$\mathsf{S}$$\scriptstyle(u\;v)$$\mathsf{S}$$\scriptstyle(u)$$\mathsf{0}$$\scriptstyle(u)$$@$$\scriptstyle(u\;v)$$\mathsf{0}$$\scriptstyle(u\;v)$$\mathsf{S}$$\scriptstyle(u\;v)$$\mathsf{S}$$\scriptstyle(u)$$\mathsf{S}$$\scriptstyle(u)$$@$$\scriptstyle(u\;v)$$@$$\scriptstyle(u\;v)$$\lambda$$\scriptstyle v$$\scriptstyle(u)$$\lambda$$\scriptstyle u$$\scriptstyle()$\|$\scriptstyle s$$\lambda$$\scriptstyle w$$\scriptstyle()$$\mathsf{0}$$\scriptstyle(w)$\|$\scriptstyle t$$\mathsf{0}$$\scriptstyle(u)$ $\implies_{\text{eliminate indirection vertices}}$ $@$$\scriptstyle(u\;v)$$@$$\scriptstyle(u\;v)$$\mathsf{S}$$\scriptstyle(u\;v)$$\mathsf{S}$$\scriptstyle(u)$$\mathsf{0}$$\scriptstyle(u)$$@$$\scriptstyle(u\;v)$$\mathsf{0}$$\scriptstyle(u\;v)$$\mathsf{S}$$\scriptstyle(u\;v)$$\mathsf{S}$$\scriptstyle(u)$$\mathsf{S}$$\scriptstyle(u)$$@$$\scriptstyle(u\;v)$$@$$\scriptstyle(u\;v)$$\lambda$$\scriptstyle v$$\scriptstyle(u)$$\lambda$$\scriptstyle u$$\scriptstyle()$$\lambda$$\scriptstyle w$$\scriptstyle()$$\mathsf{0}$$\scriptstyle(w)$$\mathsf{0}$$\scriptstyle(u)$ Second, we give the translation of the same term $L_{2}$ (from Example 5.14 on page 5.14) with the process needed for the first-order term graph semantics $\llbracket{\cdot}\rrbracket_{{\cal T}}$, yielding $\llbracket{L_{2}}\rrbracket_{{\cal T}}$ in Fig. 4.1, and, at first sight incidentally44footnotemark: 4, the corresponding $\lambda$-ho-term-graph $\llbracket{L_{2}}\rrbracket_{{\cal H}}$. Note that the resulting $\lambda$-ho-term-graph (ignore the dotted delimiter $\mathsf{S}$-vertices) is again $\llbracket{L_{2}}\rrbracket_{{\cal H}}$, that is, it is identical444The fact that this is actually not just a coincidence in this specific example is an easy consequence of Prop. 5.7, (vi), Prop. 5.10, and Prop. 5.12. with the one that was produced by the translation process above. Yet the obtained $\lambda$-term-graph (now taking the dotted $\mathsf{S}$-vertices into account) $\llbracket{L_{2}}\rrbracket_{{\cal T}}$ differs from the $\lambda$-term-graph $\llbracket{L_{2}}\rrbracket_{{\cal T}}^{\textsf{min}}$ obtained above by exhibiting a higher degree of $\mathsf{S}$-sharing. $({[]})\hskip 0.5pt{\lambda{x}.\,{\lambda{y}.\,{{\text{\sf let}}\;{I=\lambda{z}.\,{z},f=x}\;{\text{\sf in}}\;{{{{y}\hskip 1.5pt{I}}\hskip 1.5pt{({I}\hskip 1.5pt{y})}}\hskip 1.5pt{({f}\hskip 1.5pt{f})}}}}}$ $\implies_{\lambda\hskip 28.45274pt}$ $([]\;x^{u}[])\hskip 1.0pt{\lambda{y}.\,{{\text{\sf let}}\;{I=\lambda{z}.\,{z},f=x}\;{\text{\sf in}}\;{{{{y}\hskip 1.5pt{I}}\hskip 1.5pt{({I}\hskip 1.5pt{y})}}\hskip 1.5pt{({f}\hskip 1.5pt{f})}}}}$$\lambda$$\scriptstyle u$$\scriptstyle()$ $\implies_{\lambda}$ $([]\;x^{u}[]\;y^{v}[])\hskip 1.0pt{{\text{\sf let}}\;{I=\lambda{z}.\,{z},f=x}\;{\text{\sf in}}\;{{{{y}\hskip 1.5pt{I}}\hskip 1.5pt{({I}\hskip 1.5pt{y})}}\hskip 1.5pt{({f}\hskip 1.5pt{f})}}}$$\lambda$$\scriptstyle v$$\scriptstyle(u)$$\lambda$$\scriptstyle u$$\scriptstyle()$ $\implies_{\mathsf{let}}$ $([]\;x^{u}[f^{t}=x]\;y^{v}[I^{s}=\lambda{z}.\,{z}])\hskip 1.0pt{{{{y}\hskip 1.5pt{I}}\hskip 1.5pt{({I}\hskip 1.5pt{y})}}\hskip 1.5pt{({f}\hskip 1.5pt{f})}}$$\lambda$$\scriptstyle v$$\scriptstyle(v)$$\lambda$$\scriptstyle u$$\scriptstyle()$\|$\scriptstyle s$$([]\;x^{u}[f^{t}=x]\;y^{v}[I^{s}=\lambda{z}.\,{z}])\hskip 1.0pt{\lambda{z}.\,{z}}$\|$\scriptstyle t$$([]\;x^{u}[f^{t}=x])\hskip 1.0pt{x}$ $\implies_{\mathsf{S},@,\mathsf{0}}$ $([]\;x^{u}[f^{t}=x]\;y^{v}[I^{s}=\lambda{z}.\,{z}])\hskip 1.0pt{{{y}\hskip 1.5pt{I}}\hskip 1.5pt{({I}\hskip 1.5pt{y})}}$$([]\;x^{u}[f^{t}=x]\;y^{v}[I^{s}=\lambda{z}.\,{z}])\hskip 1.0pt{{f}\hskip 1.5pt{f}}$$@$$\scriptstyle(u\;v)$$\lambda$$\scriptstyle v$$\scriptstyle(u)$$\lambda$$\scriptstyle u$$\scriptstyle()$$\mathsf{S}$$\scriptstyle(u\;v)$\|$\scriptstyle s$$([]\;x^{u}[f^{t}=x])\hskip 1.0pt{\lambda{z}.\,{z}}$$\mathsf{0}$$\scriptstyle(u)$\|$\scriptstyle t$ $\implies_{\mathsf{S},@,\mathsf{S}}$ $@$$\scriptstyle(u\;v)$$([]\;x^{u}[f^{t}=x]\;y^{v}[I^{s}=\lambda{z}.\,{z}])\hskip 1.0pt{{y}\hskip 1.5pt{I}}$$([]\;x^{u}[f^{t}=x]\;y^{v}[I^{s}=\lambda{z}.\,{z}])\hskip 1.0pt{{I}\hskip 1.5pt{y}}$$\mathsf{S}$$\scriptstyle(u\;v)$$([]\;x^{u}[f^{t}=x])\hskip 1.0pt{{f}\hskip 1.5pt{f}}$$@$$\scriptstyle(u\;v)$$\lambda$$\scriptstyle v$$\scriptstyle(u)$$\lambda$$\scriptstyle u$$\scriptstyle()$$\mathsf{S}$$\scriptstyle(u\;v)$\|$\scriptstyle s$$\mathsf{S}$$\scriptstyle(u)$$([])\hskip 1.0pt{\lambda{z}.\,{z}}$\|$\scriptstyle t$$\mathsf{0}$$\scriptstyle(u)$ $\implies_{\lambda,@,@,@}$ $@$$\scriptstyle(u\;v)$$@$$\scriptstyle(u\;v)$$([]\;x^{u}[f^{t}=x]\;y^{v}[I^{s}=\lambda{z}.\,{z}])\hskip 1.0pt{I}$$([]\;x^{u}[f^{t}=x]\;y^{v}[I^{s}=\lambda{z}.\,{z}])\hskip 1.0pt{y}$$@$$\scriptstyle(u\;v)$$([]\;x^{u}[f^{t}=x]\;y^{v}[I^{s}=\lambda{z}.\,{z}])\hskip 1.0pt{y}$$([]\;x^{u}[f^{t}=x]\;y^{v}[I^{s}=\lambda{z}.\,{z}])\hskip 1.0pt{I}$$\mathsf{S}$$\scriptstyle(u\;v)$$@$$\scriptstyle(u\;v)$$([]\;x^{u}[f^{t}=x])\hskip 1.0pt{f}$$([]\;x^{u}[f^{t}=x])\hskip 1.0pt{f}$$@$$\scriptstyle(u\;v)$$\lambda$$\scriptstyle v$$\scriptstyle(u)$$\lambda$$\scriptstyle u$$\scriptstyle()$\|$\scriptstyle s$$\mathsf{S}$$\scriptstyle(u\;v)$$\mathsf{S}$$\scriptstyle(u)$$\lambda$$\scriptstyle w$$\scriptstyle()$$([]\;z^{w}[])\hskip 1.0pt{z}$\|$\scriptstyle t$$\mathsf{0}$$\scriptstyle(u)$ $\implies_{\mathsf{0},\mathsf{0},f,f,\mathsf{0},f,f}$ $@$$\scriptstyle(u\;v)$$@$$\scriptstyle(u\;v)$$\mathsf{0}$$\scriptstyle(u\;v)$$@$$\scriptstyle(u\;v)$$\mathsf{0}$$\scriptstyle(u\;v)$$\mathsf{S}$$\scriptstyle(u\;v)$$@$$\scriptstyle(u\;v)$$@$$\scriptstyle(u\;v)$$\lambda$$\scriptstyle v$$\scriptstyle(u)$$\lambda$$\scriptstyle u$$\scriptstyle()$\|$\scriptstyle s$$\mathsf{S}$$\scriptstyle(u\;v)$$\mathsf{S}$$\scriptstyle(u)$$\lambda$$\scriptstyle w$$\scriptstyle()$$\mathsf{0}$$\scriptstyle(w)$\|$\scriptstyle t$$\mathsf{0}$$\scriptstyle(u)$ $\implies_{\text{erasure of indirection vertices}}$ $@$$\scriptstyle(u\;v)$$@$$\scriptstyle(u\;v)$$\mathsf{0}$$\scriptstyle(u\;v)$$@$$\scriptstyle(u\;v)$$\mathsf{0}$$\scriptstyle(u\;v)$$\mathsf{S}$$\scriptstyle(u\;v)$$@$$\scriptstyle(u\;v)$$@$$\scriptstyle(u\;v)$$\lambda$$\scriptstyle v$$\scriptstyle(u)$$\lambda$$\scriptstyle u$$\scriptstyle()$$\mathsf{S}$$\scriptstyle(u\;v)$$\mathsf{S}$$\scriptstyle(u)$$\lambda$$\scriptstyle w$$\scriptstyle()$$\mathsf{0}$$\scriptstyle(w)$$\mathsf{0}$$\scriptstyle(u)$ ## Appendix B Implementation Showcase To demonstrate the realisability of our method, and for further illustration, we include the output of our implementation for the examples used in the paper. The implementation is called maxsharing and is available on Hackage. It is written in Haskell and therefore requires the Haskell Platform to be installed. Then, maxsharing can be installed via cabal-install using the commands cabal update and and cabal install maxsharing from the terminal. Invoke the executable maxsharing in your cabal-directory with a file as an argument that contains a $\lambda_{\text{\sf letrec}}$-term. Run maxsharing -h for help on run-time flags. ### B.1 Example 1.1 15(5.525,0) the original term as recognised by the parser 15(5.2,0.5) The user can specify which term graph semantics shall be used in processing the term. All further output is with respect to that translation. The two options are: • $\llbracket{\cdot}\rrbracket_{{\cal T}}$ is indicated by: maximal prefix lengths while maintaining eager scope-closure • $\llbracket{\cdot}\rrbracket_{{\cal T}}^{\textsf{min}}$ is indicated by: minimal prefix lengths. For this example both semantics yield the same output. 15(5.2,2.3) term with scope delimiters; see Example 4.2 for a more interesting case and more explanations 15(5.2,3.2) Derivation according to the proof system in Fig. 6 that shows the translation as a stepwise process and includes all subterms with their abstraction prefixes. Note that in the notation of the prefixed terms the binding annotation of a variable is omitted if it is empty. Also, it only includes the names of the function variables, not their entire definition. Even though the correspondence between the derivations in the proof system in Fig. 6 and the translations $\llbracket{\cdot}\rrbracket_{{\cal T}}$ and $\llbracket{\cdot}\rrbracket_{{\cal T}}^{\textsf{min}}$ is not provided here, we think that the derivation can help as an illustration of the translation process and its result. 15(5.2,4.9) The implementation produces a graphical depiction for the term’s $\lambda$-term-graph in DFA form, as well as for the minimised form of the DFA; see below for pictures. 15(5.2,5.7) A textual representation of the minimised DFA’s spanning tree, used for readback. It is displayed in first-order term rewriting notation, i.e. with a unary function symbols L, S for abstraction and scope delimiters, a binary function symbol A for application and a nullary symbol 0 for abstraction variable occurrences. Furthermore there is a class of function symbols written as a vertical bar followed by an upper-case variable name \|F, \|G, etc. which signify a vertex with multiple incoming non-backlink edges, and therefore a vertex that will be the root of a shared subterm. There is also a class of corresponding nullary function symbols F, G, etc. which represent non-backlink, non-spanning-tree edges to these shared vertices. 15(5.2,7.8) Readback of the minimised DFA. 15(0.5,9.0) 15(2.5,9.0) λ-letrec-term: (λx. x) (λx. x) translation used: minimal prefix lengths scoped (with adbmals): (λx. x) (λx. x) scoped (with scope delimiters and nameless abstractions): (λ. 0) (λ. 0) derivation: ------- 0 ------- 0 (* x) x (* x) x --------- λ --------- λ () λx. x () λx. x ------------------------ @ () (λx. x) (λx. x) DFA: writing to file minimised DFA: writing to file spanning tree: A(\|F(L(0)), F) readback: let F = λx. x in F F ### B.2 Example 1.2 Also for the terms $L$ and $P$ of Ex. 1.2 on page 1.2 the translations are identical for $\llbracket{\cdot}\rrbracket_{{\cal H}}$ and $\llbracket{\cdot}\rrbracket_{{\cal T}}^{\textsf{min}}$. 15(4.5,0) 15(4.5,2) λ-letrec-term: λf. let r = f r in r translation used: minimal prefix lengths scoped (with adbmals): λf. let r = f r in r scoped (with scope delimiters and nameless abstractions): λ. let r = 0 r in r derivation: ---------- 0 ( f[r]) f (* f[r]) r ------------------------ @ (* f[r]) f r (* f[r]) r -------------------------------------- let (* f) let r = f r in r ------------------------------------------ λ () λf. let r = f r in r DFA: writing to file minimised DFA: writing to file spanning tree: L(\|F(A(0, F))) readback: λx. let F = x F in F 15(4.7,0) 15(4.7,3) λ-letrec-term: λf. let r = f (f r) in r translation used: minimal prefix lengths scoped (with adbmals): λf. let r = f (f r) in r scoped (with scope delimiters and nameless abstractions): λ. let r = 0 (0 r) in r derivation: ---------- 0 ( f[r]) f (* f[r]) r ---------- 0 ------------------------ @ (* f[r]) f (* f[r]) f r ---------------------------------------- @ (* f[r]) f (f r) (* f[r]) r ------------------------------------------------------ let (* f) let r = f (f r) in r ---------------------------------------------------------- λ () λf. let r = f (f r) in r DFA: writing to file minimised DFA: writing to file spanning tree: L(\|F(A(0, F))) readback: λx. let F = x F in F ### B.3 Example 4.2 Again, for the terms $L_{1}$ and $L_{2}$ of Ex. 4.2 on page 4.2 the translations are identical for $\llbracket{\cdot}\rrbracket_{{\cal H}}$ and $\llbracket{\cdot}\rrbracket_{{\cal T}}^{\textsf{min}}$. 15(5.0,0.9) The term enriched by abdmals Hendriks and van Oostrom [2003]. The adbmal ( $\lambda$ ) is to be read as a scope delimiter that explicitly includes the name of the $\lambda$-variable whose scope it delimits. The adbmals are placed in accordance to the translation used. 15(5.0,1.7) A nameless scoped representation, where the names of abstraction variables are omitted for lambdas as well as for abstraction variable occurrences, shown as a $0$-symbol. The scoping is expressed by scope-delimiters in the shape of an $\mathsf{S}$-symbol. Note that these terms can be obtained by a simple syntactical transformation from the adbmal-terms. 15(0,7.5) 15(3,7.5) λ-letrec-term: let f = λx. (λy. f y) x in f translation used: minimal prefix lengths scoped (with adbmals): let f = λx. /x. λy. /y. f y x in f scoped (with scope delimiters and nameless abstractions): let f = λ. S((λ. S(f) 0)) 0 in f derivation: ([f]) f ---------- S ---------- 0 ([f] y) f ([f] y) y -------------------------- @ ([f] y) f y ---------------------------- λ ([f]) λy. f y ------------------------------ S ---------- 0 ([f] x) λy. f y ([f] x) x ---------------------------------------------- @ ([f] x) (λy. f y) x ------------------------------------------------ λ ([f]) λx. (λy. f y) x ([f]) f ------------------------------------------------------------ let () let f = λx. (λy. f y) x in f DFA: writing to file minimised DFA: writing to file spanning tree: \|F(L(A(S(F), 0))) readback: let F = λx. F x in F 15(4.4,0) 15(4.5,4.8) λ-letrec-term: let f = λx. (λy. f x) x in f translation used: minimal prefix lengths scoped (with adbmals): let f = λx. (λy. /y. /x. f x) x in f scoped (with scope delimiters and nameless abstractions): let f = λ. (λ. S((S(f) 0))) 0 in f derivation: ([f]) f ---------- S ---------- 0 ([f] x) f ([f] x) x -------------------------- @ ([f] x) f x ---------------------------- S ([f] x y) f x ------------------------------ λ ---------- 0 ([f] x) λy. f x ([f] x) x ---------------------------------------------- @ ([f] x) (λy. f x) x ------------------------------------------------ λ ([f]) λx. (λy. f x) x ([f]) f ------------------------------------------------------------ let () let f = λx. (λy. f x) x in f DFA: writing to file minimised DFA: writing to file spanning tree: \|F(L(A(L(S(A(S(F), \|G(0)))), G))) readback: let F = λx. let G = x in (λy. F G) G in F ### B.4 Figure 5 Also for the term $\lambda{a}.\,{\lambda{b}.\,{{\text{\sf let}}\;{f=a}\;{\text{\sf in}}\;{{{{a}\hskip 1.5pt{a}}\hskip 1.5pt{({f}\hskip 1.5pt{a})}}\hskip 1.5pt{b}}}}$ from Fig. 5 on page 5 the translations are identical for $\llbracket{\cdot}\rrbracket_{{\cal H}}$ and $\llbracket{\cdot}\rrbracket_{{\cal T}}^{\textsf{min}}$. 15(9,0) 15(9,5) λ-letrec-term: λa. λb. let f = a in a a (f a) b translation used: minimal prefix lengths scoped (with adbmals): λa. λb. let f = a in /b. a a (f a) b scoped (with scope delimiters and nameless abstractions): λ. λ. let f = 0 in S((0 0 (f 0))) 0 derivation: ---------- 0 ---------- 0 ---------- 0 ( a[f]) a (* a[f]) a (* a[f]) f (* a[f]) a -------------------------- @ ------------------------ @ (* a[f]) a a (* a[f]) f a -------------------------------------------------------- @ (* a[f]) a a (f a) ---------------------------------------------------------- S ------------ 0 (* a[f] b) a a (f a) (* a[f] b) b ---------- 0 ---------------------------------------------------------------------------- @ (* a[f]) a (* a[f] b) a a (f a) b -------------------------------------------------------------------------------------------- let (* a b) let f = a in a a (f a) b ------------------------------------------------------------------------------------------------ λ (* a) λb. let f = a in a a (f a) b -------------------------------------------------------------------------------------------------- λ () λa. λb. let f = a in a a (f a) b DFA: writing to file minimised DFA: writing to file spanning tree: L(L(A(S(A(\|F(A(\|G(0), G)), F)), 0))) readback: λx. let F = G G G = x in λy. F F y ### B.5 Example 5.14 For the terms $L_{1}$, $L_{2}$, and $L_{3}$ from Ex. 5.14 on page 5.14 the translations differ for $\llbracket{\cdot}\rrbracket_{{\cal H}}$ and $\llbracket{\cdot}\rrbracket_{{\cal T}}^{\textsf{min}}$. Thus on the follwing pages we provide the output for both translations for each of the terms. 15(3.6,7.7) 15(8.7,6.3) λ-letrec-term: λx. let I = λz. z in λy. let f = x in y I (I y) (f f) translation used: minimal prefix lengths scoped (with adbmals): λx. let I = λz. z in λy. let f = x in y /y. /x. I (/y. /x. I y) /y. f f scoped (with scope delimiters and nameless abstractions): λ. let I = λ. 0 in λ. let f = 0 in 0 S(S(I)) (S(S(I)) 0) S((f f)) derivation: ([I]) I ([I]) I ------------- S ------------- S ([I] x[f]) I ([I] x[f]) I --------------- 0 --------------- S --------------- S --------------- 0 ([I] x[f] y) y ([I] x[f] y) I ([I] x[f] y) I ([I] x[f] y) y ([I] x[f]) f ([I] x[f]) f ------------------------------------ @ ------------------------------------ @ ---------------------------- @ ([I] x[f] y) y I ([I] x[f] y) I y ([I] x[f]) f f ------------------------------------------------------------------------------ @ ------------------------------ S ([I] x[f] y) y I (I y) ([I] x[f] y) f f ------------- 0 ------------------------------------------------------------------------------------------------------------------ @ ([I] x[f]) x ([I] x[f] y) y I (I y) (f f) ------------------------------------------------------------------------------------------------------------------------------------- let ([I] x y) let f = x ---------- 0 in y I (I y) (f f) ([I] z) z ----------------------------------------------------------------------------------------------------------------------------------------- λ ------------ λ ([I] x) λy. let f = x ([I]) λz. z in y I (I y) (f f) ----------------------------------------------------------------------------------------------------------------------------------------------------------- let (* x) let I = λz. z in λy. let f = x in y I (I y) (f f) --------------------------------------------------------------------------------------------------------------------------------------------------------------- λ () λx. let I = λz. z in λy. let f = x in y I (I y) (f f) DFA: writing to file minimised DFA: writing to file spanning tree: L(L(A(A(A(\|H(0), \|G(S(S(L(0))))), A(G, H)), S(A(\|F(0), F))))) readback: λx. let F = x in λy. let H = y G = λz. z in H G (G H) (F F) 15(4,7.2) 15(9,6.3) λ-letrec-term: λx. let I = λz. z in λy. let f = x in y I (I y) (f f) translation used: maximal prefix lengths while maintaining eager scope-closure scoped (with adbmals): λx. let I = /x. λz. z in λy. let f = x in y /y. I (/y. I y) /y. f f scoped (with scope delimiters and nameless abstractions): λ. let I = S((λ. 0)) in λ. let f = 0 in 0 S(I) (S(I) 0) S((f f)) derivation: ( x[I f]) I (* x[I f]) I -------------- 0 -------------- S -------------- S -------------- 0 (* x[I f] y) y (* x[I f] y) I (* x[I f] y) I (* x[I f] y) y (* x[I f]) f (* x[I f]) f ---------------------------------- @ ---------------------------------- @ -------------------------- @ (* x[I f] y) y I (* x[I f] y) I y (* x[I f]) f f -------------------------------------------------------------------------- @ ---------------------------- S (* x[I f] y) y I (I y) (* x[I f] y) f f ------------ 0 ------------------------------------------------------------------------------------------------------------ @ (* x[I f]) x (* x[I f] y) y I (I y) (f f) ------- 0 ------------------------------------------------------------------------------------------------------------------------------ let (* z) z (* x[I] y) let f = x --------- λ in y I (I y) (f f) () λz. z ---------------------------------------------------------------------------------------------------------------------------------- λ -------------- S ( x[I]) λy. let f = x (* x[I]) λz. z in y I (I y) (f f) ------------------------------------------------------------------------------------------------------------------------------------------------------ let (* x) let I = λz. z in λy. let f = x in y I (I y) (f f) ---------------------------------------------------------------------------------------------------------------------------------------------------------- λ () λx. let I = λz. z in λy. let f = x in y I (I y) (f f) DFA: writing to file minimised DFA: writing to file spanning tree: L(L(A(A(A(\|H(0), \|G(S(S(L(0))))), A(G, H)), S(A(\|F(0), F))))) readback: λx. let F = x in λy. let H = y G = λz. z in H G (G H) (F F) 15(3,8) 15(9,6.2) λ-letrec-term: λx. λy. let I = λz. z f = x in y I (I y) (f f) translation used: minimal prefix lengths scoped (with adbmals): λx. λy. let I = λz. z f = x in y /y. /x. I (/y. /x. I y) /y. f f scoped (with scope delimiters and nameless abstractions): λ. λ. let I = λ. 0 f = 0 in 0 S(S(I)) (S(S(I)) 0) S((f f)) derivation: ([I]) I ([I]) I ------------- S ------------- S ([I] x[f]) I ([I] x[f]) I --------------- 0 --------------- S --------------- S --------------- 0 ([I] x[f] y) y ([I] x[f] y) I ([I] x[f] y) I ([I] x[f] y) y ([I] x[f]) f ([I] x[f]) f ------------------------------------ @ ------------------------------------ @ ---------------------------- @ ([I] x[f] y) y I ([I] x[f] y) I y ([I] x[f]) f f ---------- 0 ------------------------------------------------------------------------------ @ ------------------------------ S ([I] z) z ([I] x[f] y) y I (I y) ([I] x[f] y) f f ------------ λ ------------- 0 ------------------------------------------------------------------------------------------------------------------ @ ([I]) λz. z ([I] x[f]) x ([I] x[f] y) y I (I y) (f f) ----------------------------------------------------------------------------------------------------------------------------------------------------- let (* x y) let I = λz. z f = x in y I (I y) (f f) --------------------------------------------------------------------------------------------------------------------------------------------------------- λ (* x) λy. let I = λz. z f = x in y I (I y) (f f) ----------------------------------------------------------------------------------------------------------------------------------------------------------- λ () λx. λy. let I = λz. z f = x in y I (I y) (f f) DFA: writing to file minimised DFA: writing to file spanning tree: L(L(A(A(A(\|H(0), \|G(S(S(L(0))))), A(G, H)), S(A(\|F(0), F))))) readback: λx. let F = x in λy. let H = y G = λz. z in H G (G H) (F F) 15(4,7) 15(9,6) λ-letrec-term: λx. λy. let I = λz. z f = x in y I (I y) (f f) translation used: maximal prefix lengths while maintaining eager scope-closure scoped (with adbmals): λx. λy. let I = /y. /x. λz. z f = x in y I (I y) /y. f f scoped (with scope delimiters and nameless abstractions): λ. λ. let I = S(S((λ. 0))) f = 0 in 0 I (I 0) S((f f)) derivation: ------- 0 --------------- 0 --------------- 0 ( z) z (* x[f] y[I]) y (* x[f] y[I]) I (* x[f] y[I]) I (* x[f] y[I]) y (* x[f]) f (* x[f]) f --------- λ ---------------------------------- @ ---------------------------------- @ ---------------------- @ () λz. z ( x[f] y[I]) y I (* x[f] y[I]) I y (* x[f]) f f -------------- S -------------------------------------------------------------------------- @ ------------------------ S (* x[f]) λz. z (* x[f] y[I]) y I (I y) (* x[f] y[I]) f f ------------------- S ---------- 0 -------------------------------------------------------------------------------------------------------- @ (* x[f] y[I]) λz. z (* x[f]) x (* x[f] y[I]) y I (I y) (f f) ----------------------------------------------------------------------------------------------------------------------------------------------- let (* x y) let I = λz. z f = x in y I (I y) (f f) --------------------------------------------------------------------------------------------------------------------------------------------------- λ (* x) λy. let I = λz. z f = x in y I (I y) (f f) ----------------------------------------------------------------------------------------------------------------------------------------------------- λ () λx. λy. let I = λz. z f = x in y I (I y) (f f) DFA: writing to file minimised DFA: writing to file spanning tree: L(L(A(A(A(\|H(0), \|F(S(S(L(0))))), A(F, H)), S(A(\|G(0), G))))) readback: λx. let G = x in λy. let H = y F = λz. z in H F (F H) (G G) 15(5,8) 15(10,7.25) λ-letrec-term: λx. let I = λz. z in λy. let f = x g = I in y g (g y) (f f) translation used: minimal prefix lengths scoped (with adbmals): λx. let I = λz. z in λy. let f = x g = I in y /y. /x. g (/y. /x. g y) /y. f f scoped (with scope delimiters and nameless abstractions): λ. let I = λ. 0 in λ. let f = 0 g = I in 0 S(S(g)) (S(S(g)) 0) S((f f)) derivation: ([I g]) g ([I g]) g --------------- S --------------- S ([I g] x[f]) g ([I g] x[f]) g ----------------- 0 ----------------- S ----------------- S ----------------- 0 ([I g] x[f] y) y ([I g] x[f] y) g ([I g] x[f] y) g ([I g] x[f] y) y ([I g] x[f]) f ([I g] x[f]) f ---------------------------------------- @ ---------------------------------------- @ -------------------------------- @ ([I g] x[f] y) y g ([I g] x[f] y) g y ([I g] x[f]) f f -------------------------------------------------------------------------------------- @ ---------------------------------- S ([I g] x[f] y) y g (g y) ([I g] x[f] y) f f --------------- 0 ------------------------------------------------------------------------------------------------------------------------------ @ ([I g] x[f]) x ([I g]) I ([I g] x[f] y) y g (g y) (f f) --------------------------------------------------------------------------------------------------------------------------------------------------------------- let ([I] x y) let f = x g = I in y g (g y) (f f) ---------- 0 ------------------------------------------------------------------------------------------------------------------------------------------------------------------- λ ([I] z) z ([I] x) λy. let f = x ------------ λ g = I ([I]) λz. z in y g (g y) (f f) ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- let ( x) let I = λz. z in λy. let f = x g = I in y g (g y) (f f) ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- λ () λx. let I = λz. z in λy. let f = x g = I in y g (g y) (f f) DFA: writing to file minimised DFA: writing to file spanning tree: L(L(A(A(A(\|H(0), \|G(S(S(L(0))))), A(G, H)), S(A(\|F(0), F))))) readback: λx. let F = x in λy. let H = y G = λz. z in H G (G H) (F F) 15(5,8) 15(9.5,7) λ-letrec-term: λx. let I = λz. z in λy. let f = x g = I in y g (g y) (f f) translation used: maximal prefix lengths while maintaining eager scope-closure scoped (with adbmals): λx. let I = /x. λz. z in λy. let f = x g = /y. I in y g (g y) /y. f f scoped (with scope delimiters and nameless abstractions): λ. let I = S((λ. 0)) in λ. let f = 0 g = S(I) in 0 g (g 0) S((f f)) derivation: ----------------- 0 ----------------- 0 ( x[I f] y[g]) y (* x[I f] y[g]) g (* x[I f] y[g]) g (* x[I f] y[g]) y (* x[I f]) f (* x[I f]) f -------------------------------------- @ -------------------------------------- @ -------------------------- @ (* x[I f] y[g]) y g (* x[I f] y[g]) g y (* x[I f]) f f ---------------------------------------------------------------------------------- @ ---------------------------- S (* x[I f]) I (* x[I f] y[g]) y g (g y) (* x[I f] y[g]) f f ------------ 0 ----------------- S -------------------------------------------------------------------------------------------------------------------- @ (* x[I f]) x (* x[I f] y[g]) I (* x[I f] y[g]) y g (g y) (f f) ----------------------------------------------------------------------------------------------------------------------------------------------------------- let (* x[I] y) let f = x ------- 0 g = I (* z) z in y g (g y) (f f) --------- λ --------------------------------------------------------------------------------------------------------------------------------------------------------------- λ () λz. z ( x[I]) λy. let f = x -------------- S g = I (* x[I]) λz. z in y g (g y) (f f) ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- let (* x) let I = λz. z in λy. let f = x g = I in y g (g y) (f f) --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- λ () λx. let I = λz. z in λy. let f = x g = I in y g (g y) (f f) DFA: writing to file minimised DFA: writing to file spanning tree: L(L(A(A(A(\|H(0), \|G(S(S(L(0))))), A(G, H)), S(A(\|F(0), F))))) readback: λx. let F = x in λy. let H = y G = λz. z in H G (G H) (F F) For the term $L^{\prime}$ from the same example (Ex. 5.14, page 5.14) the translations for $\llbracket{\cdot}\rrbracket_{{\cal H}}$ and $\llbracket{\cdot}\rrbracket_{{\cal T}}^{\textsf{min}}$ are identical again. 15(3,7.9) 15(9,7) λ-letrec-term: let I = λz. z in λx. λy. let f = x in y I (I y) (f f) translation used: minimal prefix lengths scoped (with adbmals): let I = λz. z in λx. λy. let f = x in y /y. /x. I (/y. /x. I y) /y. f f scoped (with scope delimiters and nameless abstractions): let I = λ. 0 in λ. λ. let f = 0 in 0 S(S(I)) (S(S(I)) 0) S((f f)) derivation: ([I]) I ([I]) I ------------- S ------------- S ([I] x[f]) I ([I] x[f]) I --------------- 0 --------------- S --------------- S --------------- 0 ([I] x[f] y) y ([I] x[f] y) I ([I] x[f] y) I ([I] x[f] y) y ([I] x[f]) f ([I] x[f]) f ------------------------------------ @ ------------------------------------ @ ---------------------------- @ ([I] x[f] y) y I ([I] x[f] y) I y ([I] x[f]) f f ------------------------------------------------------------------------------ @ ------------------------------ S ([I] x[f] y) y I (I y) ([I] x[f] y) f f ------------- 0 ------------------------------------------------------------------------------------------------------------------ @ ([I] x[f]) x ([I] x[f] y) y I (I y) (f f) ------------------------------------------------------------------------------------------------------------------------------------- let ([I] x y) let f = x in y I (I y) (f f) ----------------------------------------------------------------------------------------------------------------------------------------- λ ([I] x) λy. let f = x ---------- 0 in y I (I y) (f f) ([I] z) z ------------------------------------------------------------------------------------------------------------------------------------------- λ ------------ λ ([I]) λx. λy. let f = x ([I]) λz. z in y I (I y) (f f) ------------------------------------------------------------------------------------------------------------------------------------------------------------- let () let I = λz. z in λx. λy. let f = x in y I (I y) (f f) DFA: writing to file minimised DFA: writing to file spanning tree: L(L(A(A(A(\|H(0), \|G(S(S(L(0))))), A(G, H)), S(A(\|F(0), F))))) readback: λx. let F = x in λy. let H = y G = λz. z in H G (G H) (F F) ### B.6 Term $L_{2}$ from Example 5.14 (page 5.14) This term has different translations for $\llbracket{\cdot}\rrbracket_{{\cal T}}$ and $\llbracket{\cdot}\rrbracket_{{\cal T}}^{\textsf{min}}$. Thus on the following two pages we provide the output for both translations. See also, and compare with, the stepwise translations in Appendix A. 15(4,6.8) 15(9,5.8) λ-letrec-term: λx. λy. let I = λz. z f = x in y I (I y) (f f) translation used: maximal prefix lengths while maintaining eager scope-closure scoped (with adbmals): λx. λy. let I = /y. /x. λz. z f = x in y I (I y) /y. f f scoped (with scope delimiters and nameless abstractions): λ. λ. let I = S(S((λ. 0))) f = 0 in 0 I (I 0) S((f f)) derivation: ------- 0 --------------- 0 --------------- 0 (* z) z (* x[f] y[I]) y (* x[f] y[I]) I (* x[f] y[I]) I (* x[f] y[I]) y (* x[f]) f (* x[f]) f --------- λ ---------------------------------- @ ---------------------------------- @ ---------------------- @ () λz. z ( x[f] y[I]) y I (* x[f] y[I]) I y (* x[f]) f f -------------- S -------------------------------------------------------------------------- @ ------------------------ S (* x[f]) λz. z (* x[f] y[I]) y I (I y) (* x[f] y[I]) f f ------------------- S ---------- 0 -------------------------------------------------------------------------------------------------------- @ (* x[f] y[I]) λz. z (* x[f]) x (* x[f] y[I]) y I (I y) (f f) ----------------------------------------------------------------------------------------------------------------------------------------------- let (* x y) let I = λz. z f = x in y I (I y) (f f) --------------------------------------------------------------------------------------------------------------------------------------------------- λ (* x) λy. let I = λz. z f = x in y I (I y) (f f) ----------------------------------------------------------------------------------------------------------------------------------------------------- λ () λx. λy. let I = λz. z f = x in y I (I y) (f f) DFA: writing to file minimised DFA: writing to file spanning tree: L(L(A(A(A(\|H(0), \|F(S(S(L(0))))), A(F, H)), S(A(\|G(0), G))))) readback: λx. let G = x in λy. let H = y F = λz. z in H F (F H) (G G) 15(3,7.5) 15(9,6.2) λ-letrec-term: λx. λy. let I = λz. z f = x in y I (I y) (f f) translation used: minimal prefix lengths scoped (with adbmals): λx. λy. let I = λz. z f = x in y /y. /x. I (/y. /x. I y) /y. f f scoped (with scope delimiters and nameless abstractions): λ. λ. let I = λ. 0 f = 0 in 0 S(S(I)) (S(S(I)) 0) S((f f)) derivation: ([I]) I ([I]) I ------------- S ------------- S ([I] x[f]) I ([I] x[f]) I --------------- 0 --------------- S --------------- S --------------- 0 ([I] x[f] y) y ([I] x[f] y) I ([I] x[f] y) I ([I] x[f] y) y ([I] x[f]) f ([I] x[f]) f ------------------------------------ @ ------------------------------------ @ ---------------------------- @ ([I] x[f] y) y I ([I] x[f] y) I y ([I] x[f]) f f ---------- 0 ------------------------------------------------------------------------------ @ ------------------------------ S ([I] z) z ([I] x[f] y) y I (I y) ([I] x[f] y) f f ------------ λ ------------- 0 ------------------------------------------------------------------------------------------------------------------ @ ([I]) λz. z ([I] x[f]) x ([I] x[f] y) y I (I y) (f f) ----------------------------------------------------------------------------------------------------------------------------------------------------- let (* x y) let I = λz. z f = x in y I (I y) (f f) --------------------------------------------------------------------------------------------------------------------------------------------------------- λ (* x) λy. let I = λz. z f = x in y I (I y) (f f) ----------------------------------------------------------------------------------------------------------------------------------------------------------- λ (*) λx. λy. let I = λz. z f = x in y I (I y) (f f) DFA: writing to file minimised DFA: writing to file spanning tree: L(L(A(A(A(\|H(0), \|G(S(S(L(0))))), A(G, H)), S(A(\|F(0), F))))) readback: λx. let F = x in λy. let H = y G = λz. z in H G (G H) (F F)	arxiv-papers	2014-01-07T18:01:41	2024-09-04T02:49:56.392246	{ "license": "Public Domain", "authors": "Clemens Grabmayer and Jan Rochel", "submitter": "Jan Rochel", "url": "https://arxiv.org/abs/1401.1460" }
1401.1475	11institutetext: Department of Electrical Engineering and Computer Science U.S. Military Academy, West Point, NY, USA 11email: [email protected] 22institutetext: Department of Computer Science, University of Oxford, United Kingdom 22email: [email protected] 33institutetext: Departamento de Ciencias e Ingeniería de la Computación Universidad Nacional del Sur, Bahía Blanca, Argentina 33email: [email protected] # Belief Revision in Structured Probabilistic Argumentation Paulo Shakarian 11 Gerardo I. Simari 22 Marcelo A. Falappa 33 ###### Abstract In real-world applications, knowledge bases consisting of all the information at hand for a specific domain, along with the current state of affairs, are bound to contain contradictory data coming from different sources, as well as data with varying degrees of uncertainty attached. Likewise, an important aspect of the effort associated with maintaining knowledge bases is deciding what information is no longer useful; pieces of information (such as intelligence reports) may be outdated, may come from sources that have recently been discovered to be of low quality, or abundant evidence may be available that contradicts them. In this paper, we propose a probabilistic structured argumentation framework that arises from the extension of Presumptive Defeasible Logic Programming (PreDeLP) with probabilistic models, and argue that this formalism is capable of addressing the basic issues of handling contradictory and uncertain data. Then, to address the last issue, we focus on the study of non-prioritized belief revision operations over probabilistic PreDeLP programs. We propose a set of rationality postulates – based on well-known ones developed for classical knowledge bases – that characterize how such operations should behave, and study a class of operators along with theoretical relationships with the proposed postulates, including a representation theorem stating the equivalence between this class and the class of operators characterized by the postulates. ## 1 Introduction and Related Work Decision-support systems that are part of virtually any kind of real-world application must be part of a framework that is rich enough to deal with several basic problems: (i) handling contradictory information; (ii) answering abductive queries; (iii) managing uncertainty; and (iv) updating beliefs. Presumptions come into play as key components of answers to abductive queries, and must be maintained as elements of the knowledge base; therefore, whenever candidate answers to these queries are evaluated, the (in)consistency of the knowledge base together with the presumptions being made needs to be addressed via belief revision operations. In this paper, we begin by proposing a framework that addresses items (i)–(iii) by extending Presumptive DeLP [1] (PreDeLP, for short) with probabilistic models in order to model uncertainty in the application domain; the resulting framework is a general-purpose probabilistic argumentation language that we will refer to as Probabilistic PreDeLP(P-PreDeLP, for short). In the second part of this paper, we address the problem of updating beliefs – item (iv) above – in P-PreDeLP knowledge bases, focusing on the study of non- prioritized belief revision operations. We propose a set of rationality postulates characterizing how such operations should behave – these postulates are based on the well-known postulates proposed in [2] for non-prioritized belief revision in classical knowledge bases. We then study a class of operators and their theoretical relationships with the proposed postulates, concluding with a representation theorem. Related Work. Belief revision studies changes to knowledge bases as a response to epistemic inputs. Traditionally, such knowledge bases can be either belief sets (sets of formulas closed under consequence) [3, 4] or belief bases [5, 2] (which are not closed); since our end goal is to apply the results we obtain to real-world domains, here we focus on belief bases. In particular, as motivated by requirements (i)–(iv) above, our knowledge bases consist of logical formulas over which we apply argumentation-based reasoning and to which we couple a probabilistic model. The connection between belief revision and argumentation was first studied in [6]; since then, the work that is most closely related to our approach is the development of the explanation-based operators of [7]. The study of argumentation systems together with probabilistic reasoning has recently received a lot attention, though a significant part has been in the combination between the two has been in the form of probabilistic abstract argumentation [8, 9, 10, 11]. There have, however, been several approaches that combine structured argumentation with models for reasoning under uncertainty; the first of such approaches to be proposed was [12], and several others followed, such as the possibilistic approach of [13], and the probabilistic logic-based approach of [14]. The main difference between these works and our own is that here we adopt a bipartite knowledge base, where one part models the knowledge that is not inherently probabilistic – uncertain knowledge is modeled separately, thus allowing a clear separation of interests between the two kinds of models. This approach is based on a similar one developed for ontological languages in the Semantic Web (see [15], and references within). Finally, to the best of our knowledge, this is the first paper in which the combination of structured argumentation, probabilistic models, and belief revision has been addressed in conjunction. ## 2 Preliminaries The Probabilistic PreDeLP (P-PreDeLP, for short) framework is composed of two separate models of the world. The first is called the environmental model (referred to as “EM”), and is used to describe the probabilistic knowledge that we have about the domain. The second one is called the analytical model (referred to as “AM”), and is used to analyze competing hypotheses that can account for a given phenomenon – what we will generally call queries. The AM is composed of a classical (that is, non-probabilistic) PreDeLP program in order to allow for contradictory information, giving the system the capability to model competing explanations for a given query. Two Kinds of Uncertainty. In general, the EM contains knowledge such as evidence, uncertain facts, or knowledge about agents and systems. The AM, on the other hand, contains ideas that a user may conclude based on the information in the EM. Table 1 gives some examples of the types of information that could appear in each of the two models in a cyber-security application. Note that a knowledge engineer (or automated system) could assign a probability to statements in the EM column, whereas statements in the AM column can be either true or false depending on a certain combination (or several possible combinations) of statements from the EM. There are thus two kinds of uncertainty that need to be modeled: probabilistic uncertainty and uncertainty arising from defeasible knowledge. As we will see, our model allows both kinds of uncertainty to coexist, and also allows for the combination of the two since defeasible rules and presumptions (that is, defeasible facts) can also be annotated with probabilistic events. In the rest of this section, we formally describe these two models, as well as how knowledge in the AM can be annotated with information from the EM – these annotations specify the conditions under which the various statements in the AM can potentially be true. Probabilistic Model (EM) \| Analytical Model (AM) ---\|--- “Malware X was compiled on a system \| “Malware X was compiled on a system in using the English language.” \| English-speaking country Y.” “County Y and country Z are \| “Country Y has a motive to launch a currently at war.” \| cyber-attack against country Z “Malware W and malware X were created \| “Malware W and malware X are related. in a similar coding style.” \| Table 1: Examples of the kind of information that could be represented in the two different models in a cyber-security application domain. Basic Language. We assume sets of variable and constant symbols, denoted with V and C, respectively. In the rest of this paper, we will use capital letters to represent variables (e.g., $X,Y,Z$), while lowercase letters represent constants. The next component of the language is a set of $n$-ary predicate symbols; the EM and AM use separate sets of predicate symbols, denoted with $\textsf{{{P}}}_{\textit{{EM}}},\textsf{{{P}}}_{\textit{{AM}}}$, respectively – the two models can, however, share variables and constants. As usual, a term is composed of either a variable or constant. Given terms $t_{1},...,t_{n}$ and $n$-ary predicate symbol $p$, $p(t_{1},...,t_{n})$ is called an atom; if $t_{1},...,t_{n}$ are constants, then the atom is said to be ground. The sets of all ground atoms for EM and AM are denoted with $\textsf{{{G}}}_{\textit{{EM}}}$ and $\textsf{{{G}}}_{\textit{{AM}}}$, respectively. Given set of ground atoms, a world is any subset of atoms – those that belong to the set are said to be true in the world, while those that do not are false. Therefore, there are $2^{\|\textsf{{{G}}}_{\textit{{EM}}}\|}$ possible worlds in the EM and $2^{\|\textsf{{{G}}}_{\textit{{AM}}}\|}$ worlds in the AM. These sets are denoted with $\mathcal{W}_{\textit{{EM}}}$ and $\mathcal{W}_{\textit{{AM}}}$, respectively. In order to avoid worlds that do not model possible situations given a particular domain, we include integrity constraints of the form $\textsf{oneOf}(\mathcal{A}^{\prime})$, where $\mathcal{A}^{\prime}$ is a subset of ground atoms. Intuitively, such a constraint states that any world where more than one of the atoms from set $\mathcal{A}^{\prime}$ appears is invalid. We use $\textsf{{{IC}}}_{\textit{{EM}}}$ and $\textsf{{{IC}}}_{\textit{{AM}}}$ to denote the sets of integrity constraints for the EM and AM, respectively, and the sets of worlds that conform to these constraints is denoted with $\mathcal{W}_{\textit{{EM}}}(\textsf{{{IC}}}_{\textit{{EM}}}),\mathcal{W}_{\textit{{AM}}}(\textsf{{{IC}}}_{\textit{{AM}}})$, respectively. Finally, logical formulas arise from the combination of atoms using the traditional connectives ($\wedge$, $\vee$, and $\neg$). As usual, we say a world $w$ satisfies formula ($f$), written $w\models f$, iff: (i) If $f$ is an atom, then $w\models f$ iff $f\in w$; (ii) if $f=\neg f^{\prime}$ then $w\models f$ iff $w\not\models f^{\prime}$; (iii) if $f=f^{\prime}\wedge f^{\prime\prime}$ then $w\models f$ iff $w\models f^{\prime}$ and $w\models f^{\prime\prime}$; and (iv) if $f=f^{\prime}\vee f^{\prime\prime}$ then $w\models f$ iff $w\models f^{\prime}$ or $w\models f^{\prime\prime}$. We use the notation $\textit{form}_{EM},\textit{form}_{AM}$ to denote the set of all possible (ground) formulas in the EM and AM, respectively. ### 2.1 Probabilistic Model The EM or environmental model is largely based on the probabilistic logic of [16], which we now briefly review. ###### Definition 1 Let $f$ be a formula over $\textsf{{{P}}}_{\textit{{EM}}}$, V, and C, $p\in[0,1]$, and $\epsilon\in[0,\min(p,1-p)]$. A probabilistic formula is of the form $f:p\pm\epsilon$. A set $\mathcal{K}_{\textit{{EM}}}$ of probabilistic formulas is called a probabilistic knowledge base. In the above definition, the number $\epsilon$ is referred to as an error tolerance. Intuitively, probabilistic formulas are interpreted as “formula $f$ is true with probability between $p-\epsilon$ and $p+\epsilon$” – note that there are no further constraints over this interval apart from those imposed by other probabilistic formulas in the knowledge base. The uncertainty regarding the probability values stems from the fact that certain assumptions (such as probabilistic independence) may not be suitable in the environment being modeled. ###### Example 1 Consider the following set $\mathcal{K}_{\textit{{EM}}}$: $\begin{array}[]{llllllllllll}f_{1}&=&a&:0.8\pm 0.1&f_{4}&=&d\wedge e&:0.7\pm 0.2&f_{7}&=&k&:1\pm 0\\\ f_{2}&=&b&:0.2\pm 0.1&f_{5}&=&f\wedge g\wedge h&:0.6\pm 0.1&&&&\\\ f_{3}&=&c&:0.8\pm 0.1&f_{6}&=&i\vee\neg j&:0.9\pm 0.1&&&&\\\ \end{array}$ Throughout the paper, we also use $\mathcal{K}_{\textit{{EM}}}^{\prime}=\\{f_{1},f_{2},f_{3}\\}$ $\blacksquare$ A set of probabilistic formulas describes a set of possible probability distributions Pr over the set $\mathcal{W}_{\textit{{EM}}}(\textsf{{{IC}}}_{\textit{{EM}}})$. We say that probability distribution Pr satisfies probabilistic formula $f:p\pm\epsilon$ iff: $p-\epsilon\leq\sum_{w\in\mathcal{W}_{\textit{{EM}}}(\textsf{{{IC}}}_{\textit{{EM}}})}\textsf{{Pr}}(w)\leq p+\epsilon.$ We say that a probability distribution over $\mathcal{W}_{\textit{{EM}}}(\textsf{{{IC}}}_{\textit{{EM}}})$ _satisfies_ $\mathcal{K}_{\textit{{EM}}}$ iff it satisfies all probabilistic formulas in $\mathcal{K}_{\textit{{EM}}}$. Given a probabilistic knowledge base and a (non-probabilistic) formula $q$, the maximum entailment problem seeks to identify real numbers $p,\epsilon$ such that all valid probability distributions Pr that satisfy $\mathcal{K}_{\textit{{EM}}}$ also satisfy $q:p\pm\epsilon$, and there does not exist $p^{\prime},\epsilon^{\prime}$ s.t. $[p-\epsilon,p+\epsilon]\supset[p^{\prime}-\epsilon^{\prime},p^{\prime}+\epsilon^{\prime}]$, where all probability distributions Pr that satisfy $\mathcal{K}_{\textit{{EM}}}$ also satisfy $q:p^{\prime}\pm\epsilon^{\prime}$. In order to solve this problem we must solve the linear program defined below. ###### Definition 2 Given a knowledge base $\mathcal{K}_{\textit{{EM}}}$ and a formula $q$, we have a variable $x_{i}$ for each $w_{i}\in\mathcal{W}_{\textit{{EM}}}(\textsf{{{IC}}}_{\textit{{EM}}})$. * • For each $f_{j}:p_{j}\pm\epsilon_{j}\in\mathcal{K}_{\textit{{EM}}}$, there is a constraint of the form: > > $p_{j}-\epsilon_{j}\leq\sum_{w_{i}\in\mathcal{W}_{\textit{{EM}}}(\textsf{{{IC}}}_{\textit{{EM}}})\textit{ > s.t.\ }w_{i}\models f_{j}}x_{i}\leq p_{j}+\epsilon_{j}.$ * • We also have the constraint: $\sum_{w_{i}\in\mathcal{W}_{\textit{{EM}}}(\textsf{{{IC}}}_{\textit{{EM}}})}x_{i}=1.$ * • The objective is to minimize the function: $\sum_{w_{i}\in\mathcal{W}_{\textit{{EM}}}(\textsf{{{IC}}}_{\textit{{EM}}})\textit{ s.t.\ }w_{i}\models q}x_{i}.$ We use the notation $\textsf{EP-LP-MIN}(\mathcal{K}_{\textit{{EM}}},q)$ to refer to the value of the objective function in the solution to the EM-LP-MIN constraints. The next step is to solve the linear program a second time, but instead maximizing the objective function (we shall refer to this as EM-LP-MAX) – let $\ell$ and $u$ be the results of these operations, respectively. In [16], it is shown that $\epsilon=\frac{u-\ell}{2}$ and $p=\ell+\epsilon$ is the solution to the maximum entailment problem. We note that although the above linear program has an exponential number of variables in the worst case (i.e., no integrity constraints), the presence of constraints has the potential to greatly reduce this space. Further, there are also good heuristics (cf. [17, 18]) that have been shown to provide highly accurate approximations with a reduced-size linear program. ###### Example 2 Consider KB $\mathcal{K}_{\textit{{EM}}}^{\prime}$ from Example 1 and a set of ground atoms restricted to those that appear in that program; we have the following worlds: $\begin{array}[]{lllllllllllllll}w_{1}&=&\\{a,b,c\\}&w_{2}&=&\\{a,b\\}&w_{3}&=&\\{a,c\\}&w_{4}&=&\\{b,c\\}\\\ w_{5}&=&\\{b\\}&w_{6}&=&\\{a\\}&w_{7}&=&\\{c\\}&w_{8}&=&\emptyset&&&\\\ \end{array}$ and suppose we wish to compute the probability for formula $q=a\vee c$. For each formula in $\mathcal{K}_{\textit{{EM}}}$ we have a constraint, and for each world above we have a variable. An objective function is created based on the worlds that satisfy the query formula (in this case, worlds $w_{1},w_{2},w_{3},w_{4},w_{6},w_{7}$). Solving $\textsf{EP-LP- MAX}(\mathcal{K}_{\textit{{EM}}}^{\prime},q)$ and $\textsf{EP-LP- MIN}(\mathcal{K}_{\textit{{EM}}}^{\prime},q)$, we obtain the solution $0.9\pm 0.1$. $\blacksquare$ ## 3 Argumentation Model For the analytical model (AM), we choose a structured argumentation framework [19] due to several characteristics that make such frameworks highly applicable to many domains. Unlike the EM, which describes probabilistic information about the state of the real world, the AM must allow for competing ideas. Therefore, it must be able to represent contradictory information. The algorithmic approach we shall later describe allows for the creation of arguments based on the AM that may “compete” with each other to answer a given query. In this competition – known as a dialectical process – one argument may defeat another based on a comparison criterion that determines the prevailing argument. Resulting from this process, certain arguments are warranted (those that are not _defeated_ by other arguments) thereby providing a suitable explanation for the answer to a given query. The transparency provided by the system can allow knowledge engineers to identify potentially incorrect input information and fine-tune the models or, alternatively, collect more information. In short, argumentation-based reasoning has been studied as a natural way to manage a set of inconsistent information – it is the way humans settle disputes. As we will see, another desirable characteristic of (structured) argumentation frameworks is that, once a conclusion is reached, we are left with an explanation of how we arrived at it and information about why a given argument is warranted; this is very important information for users to have. In the following, we first recall the basics of the underlying argumentation framework used, and then go on to introduce the analytical model (AM). ### 3.1 Defeasible Logic Programming with Presumptions (PreDeLP) Defeasible Logic Programming with Presumptions (PreDeLP) [1] is a formalism combining logic programming with defeasible argumentation; it arises as an extension of classical DeLP [20] with the possibility of having presumptions, as described below – since this capability is useful in many applications, we adopt this extended version in this paper. In this section, we briefly recall the basics of PreDeLP; we refer the reader to [20, 1] for the complete presentation. The formalism contains several different constructs: facts, presumptions, strict rules, and defeasible rules. Facts are statements about the analysis that can always be considered to be true, while presumptions are statements that may or may not be true. Strict rules specify logical consequences of a set of facts or presumptions (similar to an implication, though not the same) that must always occur, while defeasible rules specify logical consequences that may be assumed to be true when no contradicting information is present. These building blocks are used in the construction of _arguments_ , and are part of a PreDeLP program, which is a set of facts, strict rules, presumptions, and defeasible rules. Formally, we use the notation $\Pi_{\textit{AM}}=(\mbox{$\Theta$},\mbox{$\Omega$},\mbox{$\Phi$},\mbox{$\Delta$})$ to denote a PreDeLP program, where $\Omega$ is the set of strict rules, $\Theta$ is the set of facts, $\Delta$ is the set of defeasible rules, and $\Phi$ is the set of presumptions. In Figure 1, we provide an example $\Pi_{\textit{AM}}$. We now define these constructs formally. $\mbox{$\Theta$}:$ ${\theta}_{1a}=$ $p$ ${\theta}_{1b}=$ $q$ ${\theta}_{2}=$ $r$ $\mbox{$\Omega$}:$ ${\omega}_{1a}=$ $\neg s\leftarrow t$ ${\omega}_{1b}=$ $\neg t\leftarrow s$ ${\omega}_{2a}=$ $s\leftarrow p,u,r,v$ ${\omega}_{2b}=$ $t\leftarrow q,w,x,v$ $\mbox{$\Phi$}:$ ${{\phi}}_{1}=$ $y\;{\raise 1.5pt\hbox{\tiny\mbox{\bf--\raise 0.1185pt\hbox{$\prec$} }}}$ ${{\phi}}_{2}=$ $v\;{\raise 1.5pt\hbox{\tiny\mbox{\bf--\raise 0.1185pt\hbox{$\prec$} }}}$ ${{\phi}}_{3}=$ $\neg z\;{\raise 1.5pt\hbox{\tiny\mbox{\bf--\raise 0.1185pt\hbox{$\prec$} }}}$ $\mbox{$\Delta$}:$ ${{\delta}}_{1a}=$ $s\;{\raise 1.5pt\hbox{\tiny\mbox{\bf--\raise 0.1185pt\hbox{$\prec$} }}}p$ ${{\delta}}_{1b}=$ $t\;{\raise 1.5pt\hbox{\tiny\mbox{\bf--\raise 0.1185pt\hbox{$\prec$} }}}q$ ${{\delta}}_{2}=$ $s\;{\raise 1.5pt\hbox{\tiny\mbox{\bf--\raise 0.1185pt\hbox{$\prec$} }}}u$ ${{\delta}}_{3}=$ $s\;{\raise 1.5pt\hbox{\tiny\mbox{\bf--\raise 0.1185pt\hbox{$\prec$} }}}r,v$ ${{\delta}}_{4}=$ $u\;{\raise 1.5pt\hbox{\tiny\mbox{\bf--\raise 0.1185pt\hbox{$\prec$} }}}y$ ${{\delta}}_{5a}=$ $\neg u\;{\raise 1.5pt\hbox{\tiny\mbox{\bf--\raise 0.1185pt\hbox{$\prec$} }}}\neg z$ ${{\delta}}_{5b}=$ $\neg w\;{\raise 1.5pt\hbox{\tiny\mbox{\bf--\raise 0.1185pt\hbox{$\prec$} }}}\neg n$ Figure 1: An example (propositional) argumentation framework. Facts ($\Theta$) are ground literals representing atomic information or its negation, using strong negation “$\neg$”. Note that all of the literals in our framework must be formed with a predicate from the set $\textsf{{{P}}}_{\textit{{AM}}}$. Note that information in the form of facts cannot be contradicted. We will use the notation $[\mbox{$\Theta$}]$ to denote the set of all possible facts. Strict Rules ($\Omega$) represent non-defeasible cause-and-effect information that resembles an implication (though the semantics is different since the contrapositive does not hold) and are of the form $L_{0}\\!\leftarrow L_{1},\ldots,L_{n}$, where $L_{0}$ is a ground literal and $\\{L_{i}\\}_{i>0}$ is a set of ground literals. We will use the notation $[\mbox{$\Omega$}]$ to denote the set of all possible strict rules. Presumptions ($\Phi$) are ground literals of the same form as facts, except that they are not taken as being true but rather defeasible, which means that they can be contradicted. Presumptions are denoted in the same manner as facts, except that the symbol –$\prec$ is added. Defeasible Rules ($\Delta$) represent tentative knowledge that can be used if nothing can be posed against it. Just as presumptions are the defeasible counterpart of facts, defeasible rules are the defeasible counterpart of strict rules. They are of the form $L_{0}\;{\raise 1.5pt\hbox{\tiny\mbox{\bf--\raise 0.1185pt\hbox{$\prec$} }}}L_{1},\ldots,L_{n}$, where $L_{0}$ is a ground literal and $\\{L_{i}\\}_{i>0}$ is a set of ground literals. In both strict and defeasible rules, strong negation is allowed in the head of rules, and hence may be used to represent contradictory knowledge. Even though the above constructs are ground, we allow for schematic versions with variables that are used to represent sets of ground rules. We denote variables with strings starting with an uppercase letter. Arguments. Given a query in the form of a ground atom, the goal is to derive arguments for and against it’s validity – derivation follows the same mechanism of logic programming [21]. Since rule heads can contain strong negation, it is possible to defeasibly derive contradictory literals from a program. For the treatment of contradictory knowledge, PreDeLP incorporates a defeasible argumentation formalism that allows the identification of the pieces of knowledge that are in conflict and, through the previously mentioned dialectical process, decides which information prevails as warranted. This dialectical process involves the construction and evaluation of arguments, building a _dialectical tree_ in the process. Arguments are formally defined next. ###### Definition 3 An _argument_ $\langle\mbox{$\mathcal{A}$},L\rangle$ for a literal $L$ is a pair of the literal and a (possibly empty) set of the EM ($\mbox{$\mathcal{A}$}\subseteq\Pi_{\textit{AM}}$) that provides a minimal proof for $L$ meeting the following requirements: (i) $L$ is defeasibly derived from $\mathcal{A}$; (ii) $\mbox{$\Omega$}\cup\mbox{$\Theta$}\cup\mbox{$\mathcal{A}$}$ is not contradictory; and (iii) $\mathcal{A}$ is a minimal subset of $\mbox{$\Delta$}\cup\mbox{$\Phi$}$ satisfying 1 and 2, denoted $\langle\mbox{$\mathcal{A}$},L\rangle$. Literal $L$ is called the conclusion supported by the argument, and $\mathcal{A}$ is the _support_ of the argument. An argument $\langle\mathcal{B},L\rangle$ is a subargument of $\langle\mathcal{A},L^{\prime}\rangle$ iff $\mathcal{B}\subseteq\mathcal{A}$. An argument $\langle\mathcal{A},L\rangle$ is presumptive iff $\mathcal{A}\cap\mbox{$\Phi$}$ is not empty. We will also use $\mbox{$\Omega$}(\mathcal{A})=\mathcal{A}\cap\mbox{$\Omega$}$, $\mbox{$\Theta$}(\mathcal{A})=\mathcal{A}\cap\mbox{$\Theta$}$, $\mbox{$\Delta$}(\mathcal{A})=\mathcal{A}\cap\mbox{$\Delta$}$, and $\mbox{$\Phi$}(\mathcal{A})=\mathcal{A}\cap\mbox{$\Phi$}$. Our definition differs slightly from that of [22], where DeLP is introduced, as we include strict rules and facts as part of arguments – the reason for this will become clear in Section 4. Arguments for our scenario are shown next. ###### Example 3 Figure 2 shows example arguments based on the knowledge base from Figure 1. Note that $\langle\mathcal{A}_{5},u\rangle$ is a sub-argument of $\langle\mathcal{A}_{2},s\rangle$ and $\langle\mathcal{A}_{3},s\rangle$. $\blacksquare$ $\langle\mathcal{A}_{1},s\rangle$ $\mathcal{A}_{1}=\\{{\theta}_{1a},{{\delta}}_{1a}\\}$ $\langle\mathcal{A}_{2},s\rangle$ $\mathcal{A}_{2}=\\{{{\phi}}_{1},{{\phi}}_{2},{{\delta}}_{4},{\omega}_{2a},{\theta}_{1a},{\theta}_{2}\\}$ $\langle\mathcal{A}_{3},s\rangle$ $\mathcal{A}_{3}=\\{{{\phi}}_{1},{{\delta}}_{2},{{\delta}}_{4}\\}$ $\langle\mathcal{A}_{4},s\rangle$ $\mathcal{A}_{4}=\\{{{\phi}}_{2},{{\delta}}_{3},{\theta}_{2}\\}$ $\langle\mathcal{A}_{5},u\rangle$ $\mathcal{A}_{5}=\\{{{\phi}}_{1},{{\delta}}_{4}\\}$ $\langle\mathcal{A}_{6},\neg s\rangle$ $\mathcal{A}_{6}=\\{{{\delta}}_{1b},{\theta}_{1b},{\omega}_{1a}\\}$ $\langle\mathcal{A}_{7},\neg u\rangle$ $\mathcal{A}_{7}=\\{{{\phi}}_{3},{{\delta}}_{5a}\\}$ Figure 2: Example ground arguments from the framework of Figure 1. Given an argument $\langle\mathcal{A}_{1},L_{1}\rangle$, counter-arguments are arguments that contradict it. Argument $\langle\mathcal{A}_{2},L_{2}\rangle$ is said to counterargue or attack $\langle\mathcal{A}_{1},L_{1}\rangle$ at a literal $L^{\prime}$ iff there exists a subargument $\langle\mathcal{A},L^{\prime\prime}\rangle$ of $\langle\mathcal{A}_{1},L_{1}\rangle$ such that the set $\mbox{$\Omega$}(\mathcal{A}_{1})\cup\mbox{$\Omega$}(\mathcal{A}_{2})\cup\mbox{$\Theta$}(\mathcal{A}_{1})\cup\mbox{$\Theta$}(\mathcal{A}_{2})\cup\\{L_{2},L^{\prime\prime}\\}$ is contradictory. ###### Example 4 Consider the arguments from Example 3. The following are some of the attack relationships between them: $\mathcal{A}_{1}$, $\mathcal{A}_{2}$, $\mathcal{A}_{3}$, and $\mathcal{A}_{4}$ all attack $\mathcal{A}_{6}$; $\mathcal{A}_{5}$ attacks $\mathcal{A}_{7}$; and $\mathcal{A}_{7}$ attacks $\mathcal{A}_{2}$. $\blacksquare$ A proper defeater of an argument $\langle A,L\rangle$ is a counter-argument that – by some criterion – is considered to be better than $\langle A,L\rangle$; if the two are incomparable according to this criterion, the counterargument is said to be a blocking defeater. An important characteristic of PreDeLP is that the argument comparison criterion is modular, and thus the most appropriate criterion for the domain that is being represented can be selected; the default criterion used in classical defeasible logic programming (from which PreDeLP is derived) is _generalized specificity_ [23], though an extension of this criterion is required for arguments using presumptions [1]. We briefly recall this criterion next – the first definition is for generalized specificity, which is subsequently used in the definition of presumption-enabled specificity. ###### Definition 4 Let $\mbox{$\Pi_{\textit{AM}}$}=(\mbox{$\Theta$},\mbox{$\Omega$},\mbox{$\Phi$},\mbox{$\Delta$})$ be a PreDeLP program and let $\mathcal{F}$ be the set of all literals that have a defeasible derivation from $\Pi_{\textit{AM}}$. An argument $\langle\mathcal{A}_{1},L_{1}\rangle$ is _preferred to_ $\langle\mathcal{A}_{2},L_{2}\rangle$, denoted with $\mathcal{A}_{1}\succ_{PS}\mathcal{A}_{2}$ iff: $(1)$ For all $H\subseteq\mathcal{F}$, $\mbox{$\Omega$}(\mathcal{A}_{1})\cup\mbox{$\Omega$}(\mathcal{A}_{2})\cup H$ is non-contradictory: if there is a derivation for $L_{1}$ from $\mbox{$\Omega$}(\mathcal{A}_{2})\cup\mbox{$\Omega$}(\mathcal{A}_{1})\cup\mbox{$\Delta$}(\mathcal{A}_{1})\cup H$, and there is no derivation for $L_{1}$ from $\mbox{$\Omega$}(\mathcal{A}_{1})\cup\mbox{$\Omega$}(\mathcal{A}_{2})\cup H$, then there is a derivation for $L_{2}$ from $\mbox{$\Omega$}(\mathcal{A}_{1})\cup\mbox{$\Omega$}(\mathcal{A}_{2})\cup\mbox{$\Delta$}(\mathcal{A}_{2})\cup H$; and $(2)$ there is at least one set $H^{\prime}\subseteq\mathcal{F}$, $\mbox{$\Omega$}(\mathcal{A}_{1})\cup\mbox{$\Omega$}(\mathcal{A}_{2})\cup H^{\prime}$ is non-contradictory, such that there is a derivation for $L_{2}$ from $\mbox{$\Omega$}(\mathcal{A}_{1})\cup\mbox{$\Omega$}(\mathcal{A}_{2})\cup H^{\prime}\cup\mbox{$\Delta$}(\mathcal{A}_{2})$, there is no derivation for $L_{2}$ from $\mbox{$\Omega$}(\mathcal{A}_{1})\cup\mbox{$\Omega$}(\mathcal{A}_{2})\cup H^{\prime}$, and there is no derivation for $L_{1}$ from $\mbox{$\Omega$}(\mathcal{A}_{1})\cup\mbox{$\Omega$}(\mathcal{A}_{2})\cup H^{\prime}\cup\mbox{$\Delta$}(\mathcal{A}_{1})$. Intuitively, the principle of specificity says that, in the presence of two conflicting lines of argument about a proposition, the one that uses more of the available information is more convincing. A classic example involves a bird, Tweety, and arguments stating that it both flies (because it is a bird) and doesn’t fly (because it is a penguin). The latter argument uses more information about Tweety – it is more specific – and is thus the stronger of the two. ###### Definition 5 ([1]) Let $\mbox{$\Pi_{\textit{AM}}$}=(\mbox{$\Theta$},\mbox{$\Omega$},\mbox{$\Phi$},\mbox{$\Delta$})$ be a PreDeLP program. An argument $\langle\mathcal{A}_{1},L_{1}\rangle$ is _preferred to_ $\langle\mathcal{A}_{2},L_{2}\rangle$, denoted with $\mathcal{A}_{1}\succ\mathcal{A}_{2}$ iff any of the following conditions hold: $(1)$ $\langle\mathcal{A}_{1},L_{1}\rangle$ and $\langle\mathcal{A}_{2},L_{2}\rangle$ are both factual arguments and $\langle\mathcal{A}_{1},L_{1}\rangle\succ_{PS}\langle\mathcal{A}_{2},L_{2}\rangle$. $(2)$ $\langle\mathcal{A}_{1},L_{1}\rangle$ is a factual argument and $\langle\mathcal{A}_{2},L_{2}\rangle$ is a presumptive argument. $(3)$ $\langle\mathcal{A}_{1},L_{1}\rangle$ and $\langle\mathcal{A}_{2},L_{2}\rangle$ are presumptive arguments, and $(a)$ $\mbox{$\Phi$}(\mathcal{A}_{1})\subsetneq\mbox{$\Phi$}(\mathcal{A}_{2})$ or, $(b)$ $\mbox{$\Phi$}(\mathcal{A}_{1})=\mbox{$\Phi$}(\mathcal{A}_{2})$ and $\langle\mathcal{A}_{1},L_{1}\rangle\succ_{PS}\langle\mathcal{A}_{2},L_{2}\rangle$. Generally, if $\mathcal{A},\mathcal{B}$ are arguments with rules $X$ and $Y$, resp., and $X\subset Y$, then $\mathcal{A}$ is stronger than $\mathcal{B}$. This also holds when $\mathcal{A}$ and $\mathcal{B}$ use presumptions $P_{1}$ and $P_{2}$, resp., and $P_{1}\subset P_{2}$. ###### Example 5 The following are some relationships between arguments from Example 3, based on Definitions 4 and 5. > $\mathcal{A}_{1}$ and $\mathcal{A}_{6}$ are incomparable (blocking > defeaters); > $\mathcal{A}_{6}\succ\mathcal{A}_{2}$, and thus $\mathcal{A}_{6}$ defeats > $\mathcal{A}_{2}$; > $\mathcal{A}_{5}$ and $\mathcal{A}_{7}$ are incomparable (blocking > defeaters). $\blacksquare$ A sequence of arguments called an _argumentation line_ thus arises from this attack relation, where each argument defeats its predecessor. To avoid undesirable sequences, which may represent circular argumentation lines, in DeLP an _argumentation line_ is _acceptable_ if it satisfies certain constraints (see [20]). A literal $L$ is _warranted_ if there exists a non- defeated argument $\mathcal{A}$ supporting $L$. Clearly, there can be more than one defeater for a particular argument $\langle\mathcal{A},L\rangle$. Therefore, many acceptable argumentation lines could arise from $\langle\mathcal{A},L\rangle$, leading to a tree structure. The tree is built from the set of all argumentation lines rooted in the initial argument. In a dialectical tree, every node (except the root) represents a defeater of its parent, and leaves correspond to undefeated arguments. Each path from the root to a leaf corresponds to a different acceptable argumentation line. A dialectical tree provides a structure for considering all the possible acceptable argumentation lines that can be generated for deciding whether an argument is defeated. We call this tree _dialectical_ because it represents an exhaustive dialectical111In the sense of providing reasons for and against a position. analysis for the argument in its root. For a given argument $\langle\mathcal{A},L\rangle$, we denote the corresponding dialectical tree as ${\mathcal{T}}({\small\langle\mbox{$\mathcal{A}$},L\rangle})$. Given a literal $L$ and an argument $\langle\mbox{$\mathcal{A}$},L\rangle$, in order to decide whether or not a literal $L$ is warranted, every node in the dialectical tree ${\mathcal{T}}({\small\langle\mbox{$\mathcal{A}$},L\rangle})$ is recursively marked as “D” (_defeated_) or “U” (_undefeated_), obtaining a marked dialectical tree ${\mathcal{T}^{}}({\small\langle\mbox{$\mathcal{A}$},L\rangle})$ as follows: 1. 1. All leaves in ${\mathcal{T}^{}}({\small\langle\mbox{$\mathcal{A}$},L\rangle})$ are marked as “U”s, and 2. 2. Let $\langle\mbox{${\mathcal{B}}$},q\rangle$ be an inner node of ${\mathcal{T}^{}}({\small\langle\mbox{$\mathcal{A}$},L\rangle})$. Then $\langle\mbox{${\mathcal{B}}$},q\rangle$ will be marked as “U” iff every child of $\langle\mbox{${\mathcal{B}}$},q\rangle$ is marked as “D”. The node $\langle\mbox{${\mathcal{B}}$},q\rangle$ will be marked as “D” iff it has at least a child marked as “U”. Given an argument $\langle\mbox{$\mathcal{A}$},L\rangle$ obtained from $\Pi_{\textit{AM}}$, if the root of ${\mathcal{T}^{}}({\small\langle\mbox{$\mathcal{A}$},L\rangle})$ is marked as “U”, then we will say that ${\mathcal{T}^{}}({\small\mbox{$\langle\mbox{$\mathcal{A}$},h\rangle$}})$ _warrants_ $L$ and that $L$ is _warranted_ from $\Pi_{\textit{AM}}$. (Warranted arguments correspond to those in the grounded extension of a Dung argumentation system [24].) There is a further requirement when the arguments in the dialectical tree contains presumptions – the conjunction of all presumptions used in even (respectively, odd) levels of the tree must be consistent. This can give rise to multiple trees for a given literal, as there can potentially be different arguments that make contradictory assumptions. We can then extend the idea of a dialectical tree to a dialectical forest. For a given literal $L$, a dialectical forest $\mathcal{F}(L)$ consists of the set of dialectical trees for all arguments for $L$. We shall denote a marked dialectical forest, the set of all marked dialectical trees for arguments for $L$, as $\mathcal{F}^{}(L)$. Hence, for a literal $L$, we say it is warranted if there is at least one argument for that literal in the dialectical forest $\mathcal{F}^{}(L)$ that is labeled as “U”, not warranted if there is at least one argument for the literal $\neg L$ in the dialectical forest $\mathcal{F}^{}(\neg L)$ that is labeled as “U”, and undecided otherwise. ## 4 Probabilistic PreDeLP Probabilistic PreDeLP arises from the combination of the environmental and analytical models ($\Pi_{\textit{EM}}$ and $\Pi_{\textit{AM}}$, respectively). Intuitively, given $\Pi_{\textit{AM}}$, every element of $\mbox{$\Omega$}\cup\mbox{$\Theta$}\cup\mbox{$\mbox{$\Delta$}\cup\mbox{$\Phi$}$}$ might only hold in certain worlds in the set $\mathcal{W}_{\textit{{EM}}}$ – that is, they are subject to probabilistic events. Therefore, we associate elements of $\mbox{$\Omega$}\cup\mbox{$\Theta$}\cup\mbox{$\mbox{$\Delta$}\cup\mbox{$\Phi$}$}$ with a formula from $\textit{form}_{EM}$. For instance, we could associate formula rainy to fact umbrella to state that the latter only holds when the probabilistic event rainy holds; since weather is uncertain in nature, it has been modeled as part of the EM. We can then compute the probabilities of subsets of $\mbox{$\Omega$}\cup\mbox{$\Theta$}\cup\mbox{$\mbox{$\Delta$}\cup\mbox{$\Phi$}$}$ using the information contained in $\Pi_{\textit{EM}}$, as we describe shortly. The notion of an annotation function associates elements of $\mbox{$\Omega$}\cup\mbox{$\Theta$}\cup\mbox{$\mbox{$\Delta$}\cup\mbox{$\Phi$}$}$ with elements of $\textit{form}_{EM}$. ###### Definition 6 An annotation function is any function $\textit{af{\,}}:\mbox{$\Omega$}\cup\mbox{$\Theta$}\cup\mbox{$\mbox{$\Delta$}\cup\mbox{$\Phi$}$}\rightarrow\textit{form}_{EM}$. We shall use $[\textit{af{\,}}]$ to denote the set of all annotation functions. We will sometimes denote annotation functions as sets of pairs $(f,\textit{af}(f))$ in order to simplify the presentation. Figure 3 shows an example of an annotation function for our running example. $\textit{af}({\theta}_{1a})=\textit{af}({\theta}_{1b})$ $=k\vee\big{(}f\wedge\big{(}h\vee(e\wedge l)\big{)}\big{)}$ $\textit{af}({{\phi}}_{3})$ $=b$ $\textit{af}({\theta}_{2})$ $=i$ $\textit{af}({{\delta}}_{1a})=\textit{af}({{\delta}}_{1b})$ $=\textsf{True}$ $\textit{af}({\omega}_{1a})=\textit{af}({\omega}_{1b})$ $=\textsf{True}$ $\textit{af}({{\delta}}_{2})$ $=\textsf{True}$ $\textit{af}({\omega}_{2a})=\textit{af}({\omega}_{2b})$ $=\textsf{True}$ $\textit{af}({{\delta}}_{3})$ $=\textsf{True}$ $\textit{af}({{\phi}}_{1})$ $=c\vee a$ $\textit{af}({{\delta}}_{4})$ $=\textsf{True}$ $\textit{af}({{\phi}}_{2})$ $=f\wedge m$ $\textit{af}({{\delta}}_{5a})=\textit{af}({{\delta}}_{5b})$ $=\textsf{True}$ Figure 3: Example annotation function. We now have all the components to formally define Probabilistic PreDeLP programs (P-PreDeLP for short). ###### Definition 7 Given environmental model $\Pi_{\textit{EM}}$, analytical model $\Pi_{\textit{AM}}$, and annotation function af , a probabilistic PreDeLP program is of the form $\mathcal{I}=(\Pi_{\textit{EM}},\Pi_{\textit{AM}},\textit{af{\,}})$. We use notation $[\mathcal{I}]$ to denote the set of all possible programs. Given this setup, we can consider a world-based approach; that is, the defeat relationship among arguments depends on the current state of the (EM) world. ###### Definition 8 Let $\mathcal{I}=(\Pi_{\textit{EM}},\Pi_{\textit{AM}},\textit{af{\,}})$ be a P-PreDeLP program, argument $\langle\mathcal{A},L\rangle$ is valid w.r.t. world $w\in\mathcal{W}_{\textit{{EM}}}$ iff $\forall c\in\mathcal{A},w\models\textit{af}(c)$. We extend the notion of validity to argumentation lines, dialectical trees, and dialectical forests in the expected way (for instance, an argumentation line is valid w.r.t. $w$ iff all arguments that comprise that line are valid w.r.t. $w$). We also extend the idea of a dialectical tree w.r.t. worlds; so, for a given world $w\in\mathcal{W}_{\textit{{EM}}}$, the dialectical (resp., marked dialectical) tree induced by $w$ is denoted with $\mathcal{T}_{w}{\langle\mbox{$\mathcal{A}$},L\rangle}$ (resp., $\mathcal{T}^{}_{w}{\langle\mbox{$\mathcal{A}$},L\rangle}$). We require that all arguments and defeaters in these trees to be valid with respect to $w$. Likewise, we extend the notion of dialectical forests in the same manner (denoted with $\mathcal{F}_{w}(L)$ and $\mathcal{F}^{}_{w}(L)$, resp.). Based on these concepts we introduce the notion of warranting scenario. ###### Definition 9 Let $\mathcal{I}=(\Pi_{\textit{EM}},\Pi_{\textit{AM}},\textit{af{\,}})$ be a P-PreDeLP program and $L$ be a literal formed with a ground atom from $\textsf{{{G}}}_{\textit{{AM}}}$; a world $w\in\mathcal{W}_{\textit{{EM}}}$ is said to be a warranting scenario for $L$ (denoted $w\vdash_{\textsf{war}}L$) iff there is a dialectical forest $\mathcal{F}^{}_{w}(L)$ in which $L$ is warranted and $\mathcal{F}^{}_{w}(L)$ is valid w.r.t. $w$. Hence, the set of worlds in the EM where a literal $L$ in the AM must be true is exactly the set of warranting scenarios – these are the “necessary” worlds: $nec(L)=\\{w\in\mathcal{W}_{\textit{{EM}}}\;\|\;(w\vdash_{\textsf{war}}L)\\}$. Now, the set of worlds in the EM where AM literal $L$ can be true is the following – these are the “possible” worlds: $poss(L)=\\{w\in\mathcal{W}_{\textit{{EM}}}\;\|\;w\not\vdash_{\textsf{war}}\neg L\\}.$ The probability distribution Pr defined over the worlds in the EM induces an upper and lower bound on the probability of literal $L$ (denoted $\textbf{{P}}_{L,\textsf{{Pr}},\mathcal{I}}$) as follows: $\ell_{L,\textsf{{Pr}},\mathcal{I}}=\sum_{w\in nec(L)}\textsf{{Pr}}(w),\ \ \ \ \ u_{L,\textsf{{Pr}},\mathcal{I}}=\sum_{w\in poss(L)}\textsf{{Pr}}(w)$ $\ell_{L,\textsf{{Pr}},\mathcal{I}}\leq\textbf{{P}}_{L,\textsf{{Pr}},\mathcal{I}}\leq u_{L,\textsf{{Pr}},\mathcal{I}}$ Since the EM in general does not define a single probability distribution, the above computations should be done using linear programs EP-LP-MIN and EP-LP- MAX, as described above. ### 4.1 Sources of Inconsistency We use the following notion of (classical) consistency of PreDeLP programs: $\Pi$ is said to be consistent if there does not exist ground literal $a$ s.t. $\Pi\vdash a$ and $\Pi\vdash\neg a$. For P-PreDeLP programs, there are two main kinds of inconsistency that can be present; the first is what we refer to as EM, or Type I, (in)consistency. ###### Definition 10 Environmental model $\Pi_{\textit{EM}}$ is Type I consistent iff there exists a probability distribution Pr over the set of worlds $\mathcal{W}_{\textit{{EM}}}$ that satisfies $\Pi_{\textit{EM}}$. We illustrate this type of consistency in the following example. ###### Example 6 The following formula is a simple example of an EM for which there is no satisfying probability distribution: $\displaystyle rain$ $\displaystyle\vee$ $\displaystyle hail:0.3\pm 0;$ $\displaystyle rain$ $\displaystyle\wedge$ $\displaystyle hail:0.5\pm 0.1.$ A P-PreDeLP program using such an EM gives rise to an example of Type I inconsistency, as it arises from the fact that there is no satisfying interpretation for the EM knowledge base. $\blacksquare$ Assuming a consistent EM, inconsistencies can still arise through the interaction between the annotation function and facts and strict rules. We will refer to this as combined, or Type II, (in)consistency. ###### Definition 11 A P-PreDeLP program $\mathcal{I}=(\Pi_{\textit{EM}},\Pi_{\textit{AM}},\textit{af{\,}})$, with $\Pi_{\textit{AM}}$ $=$ $\langle\Theta,\Omega,\Phi,\Delta\rangle$, is Type II consistent iff: given any probability distribution Pr that satisfies $\Pi_{\textit{EM}}$, if there exists a world $w\in\mathcal{W}_{\textit{{EM}}}$ such that $\bigcup_{x\in\Theta\cup\Omega\,\|\,w\models\textit{af}(x)}\\{x\\}$ is inconsistent, then we have $\textsf{{Pr}}(w)=0$. Thus, any EM world in which the set of associated facts and strict rules are inconsistent (we refer to this as “classical consistency”) must always be assigned a zero probability. The following is an example of this other type of inconsistency. ###### Example 7 Consider the EM knowledge base from Example 1, the AM presented in Figure 1 and the annotation function from Figure 3. Suppose the following fact is added to the argumentation model: ${\theta}_{3}=\neg p,$ and that the annotation function is expanded as follows: $\textit{af{\,}}({\theta}_{3})=\neg k.$ Clearly, fact ${\theta}_{3}$ is in direct conflict with fact ${\theta}_{1a}$ – this does not necessarily mean that there is an inconsistency. For instance, by the annotation function, ${\theta}_{1a}$ holds in the world $\\{k\\}$ while ${\theta}_{3}$ does not. However, if we consider the world: $w=\\{f,h)$ Note that $w\models\textit{af{\,}}({\theta}_{3})$ and $w\models\textit{af{\,}}({\theta}_{2})$, which means that, in this world, two contradictory facts can occur. Since the environmental model indicates that this world can be assigned a non-zero probability, we have a Type II inconsist program. $\blacksquare$ Another example (perhaps easier to visualize) in the rain/hail scenario discussed above, is as follows: suppose we have facts $f=umbrella$ and $g=\neg umbrella$, and annotation function $\textit{af{\,}}(f)=rain\vee hail$ and $\textit{af{\,}}(g)=wind$. Intuitively, the first fact states that an umbrella should be carried if it either rains or hails, while the second states that an umbrella should not be carried if it is windy. If the EM assigns a non-zero probability to formula $(rain\vee hail)\wedge wind$, then we have Type II inconsistency. In the following, we say that a P-PreDeLP program is consistent if and only if it is both Type I and Type II consistent. However, in this paper, we focus on Type II consistency and assume that the program is Type I consistent. ### 4.2 Basic Operations for Restoring Consistency Given a P-PreDeLP program that is Type II inconsistent, there are two basic strategies that can be used to restore consistency: Revise the EM: the probabilistic model can be changed in order to force the worlds that induce contradicting strict knowledge to have probability zero. Revise the annotation function: The annotations involved in the inconsistency can be changed so that the conflicting information in the AM does not become induced under any possible world. It may also appear that a third option would be to adjust the AM – this is, however, equivalent to modifying the annotation function. Consider the presence of two facts in the AM: $a,\neg a$. Assuming that this causes an inconsistency (that is, there is at least one world in which they both hold), one way to resolve it would be to remove one of these two literals. Suppose $\neg a$ is removed; this would be equivalent to setting $\textit{af}(\neg a)=\bot$ (where $\bot$ represents a contradiction in the language of the EM). In this paper, we often refer to “removing elements of $\Pi_{\textit{AM}}$” to refer to changes to the annotation function that cause certain elements of the $\Pi_{\textit{AM}}$ to not have their annotations satisfied in certain EM worlds. Now, suppose that $\Pi_{\textit{EM}}$ is consistent, but that the overall program is Type II inconsistent. Then, there must exist a set of worlds in the EM where there is a probability distribution that assigns each of them a non- zero probability. This gives rise to the following result. ###### Proposition 1 If there exists a probability distribution Pr that satisfies $\Pi_{\textit{EM}}$ s.t. there exists a world $w\in\mathcal{W}_{\textit{{EM}}}$ where $\textsf{{Pr}}(w)>0$ and $\bigcup_{x\in\Theta\cup\Omega\,\|\,w\models\textit{af}(x)}\\{x\\}$ is inconsistent (Type II inconsistency), then any change made in order to resolve this inconsistency by modifying only $\Pi_{\textit{EM}}$ yields a new EM $\Pi_{\textit{EM}}^{\prime}$ such that $\big{(}\bigwedge_{a\in w}a\wedge\bigwedge_{a\notin w}\neg a\big{)}:0\pm 0$ is entailed by $\Pi_{\textit{EM}}^{\prime}$. Proposition 1 seems to imply an easy strategy of adding formulas to $\Pi_{\textit{EM}}$ causing certain worlds to have a zero probability. However, this may lead to Type I inconsistencies in the resulting model $\Pi_{\textit{EM}}^{\prime}$. If we are applying an EM-only strategy to resolve inconsistencies, this would then lead to further adjustments to $\Pi_{\textit{EM}}^{\prime}$ in order to restore Type I consistency. However, such changes could potentially lead to Type II inconsistency in the overall P-PreDeLP program (by either removing elements of $\Pi_{\textit{EM}}^{\prime}$ or loosening probability bounds of the sentences in $\Pi_{\textit{EM}}^{\prime}$), which would lead to setting more EM worlds to a probability of zero. It is easy to devise an example of a situation in which the probability mass cannot be accommodated given the constraints imposed by the AM and EM together – in such cases, it would be impossible to restore consistency by only modifying $\Pi_{\textit{EM}}$. We thus arrive at the following observation: ###### Observation 1 Given a Type II inconsistent P-PreDeLP program, consistency cannot always be restored via modifications to $\Pi_{\textit{EM}}$ alone. Therefore, due to this line of reasoning, in this paper we focus our efforts on modifications to the annotation function only. However, in the future, we intend to explore belief revision operators that consider both the annotation function (which, as we saw, captures changes to the AM) along with changes to the EM, as well as combinations of the two. ## 5 Revising Probabilistic PreDeLP Programs Given a P-PreDeLP program $\mathcal{I}=(\Pi_{\textit{EM}},\Pi_{\textit{AM}},\textit{af{\,}})$, with $\Pi_{\textit{AM}}=\mbox{$\Omega$}\cup\mbox{$\Theta$}\cup\mbox{$\mbox{$\Delta$}\cup\mbox{$\Phi$}$}$, we are interested in solving the problem of incorporating an epistemic input $(f,\textit{af{\,}}^{\prime})$ into $\mathcal{I}$, where $f$ is either an atom or a rule and $\textit{af{\,}}^{\prime}$ is equivalent to af , except for its expansion to include $f$. For ease of presentation, we assume that $f$ is to be incorporated as a fact or strict rule, since incorporating defeasible knowledge can never lead to inconsistency. As we are only conducting annotation function revisions, for $\mathcal{I}=(\Pi_{\textit{EM}},\Pi_{\textit{AM}},\textit{af{\,}})$ and input $(f,\textit{af{\,}}^{\prime})$ we denote the revision as follows: $\mathcal{I}\bullet(f,\textit{af{\,}}^{\prime})=(\Pi_{\textit{EM}},\Pi_{\textit{AM}}^{\prime},\textit{af{\,}}^{\prime\prime})$ where $\Pi_{\textit{AM}}^{\prime}=\Pi_{\textit{AM}}\cup\\{f\\}$ and $\textit{af{\,}}^{\prime\prime}$ is the revised annotation function. Notation. We use the symbol “$\bullet$” to denote the revision operator. We also slightly abuse notation for the sake of presentation, as well as introduce notation to convert sets of worlds to/from formulas. * • $\mathcal{I}\cup(f,\textit{af{\,}}^{\prime})$ to denote $\mathcal{I}^{\prime}=(\Pi_{\textit{EM}},\Pi_{\textit{AM}}\cup\\{f\\},\textit{af{\,}}^{\prime})$. * • $(f,\textit{af{\,}}^{\prime})\in\mathcal{I}=(\Pi_{\textit{AM}},\Pi_{\textit{EM}},\textit{af{\,}})$ to denote $f\in\Pi_{\textit{AM}}$ and $\textit{af{\,}}=\textit{af{\,}}^{\prime}$. * • $wld(f)=\\{w\;\|\;w\models f\\}$ – the set of worlds that satisfy formula $f$; and * • $for(w)=\bigwedge_{a\in w}a\wedge\bigwedge_{a\notin w}\neg a$ – the formula that has $w$ as its only model. * • ${\Pi_{\textit{AM}}^{\mathcal{I}}}(w)=\\{f\in\Theta\cup\Omega\;\|\;w\models\textit{af}(f)\\}$ * • $\mathcal{W}_{\textit{{EM}}}^{0}(\mathcal{I})=\\{w\in\mathcal{W}_{\textit{{EM}}}\;\|\;\Pi_{\textit{AM}}^{\mathcal{I}}(w)\;\text{is inconsistent}\\}$ * • $\mathcal{W}_{\textit{{EM}}}^{I}(\mathcal{I})=\\{w\in\mathcal{W}_{\textit{{EM}}}^{0}\;\|\;\exists\textsf{{Pr}}\textit{ s.t. }\textsf{{Pr}}\models\Pi_{\textit{EM}}\wedge\textsf{{Pr}}(w)>0\\}$ Intuitively, $\Pi_{\textit{AM}}^{\mathcal{I}}(w)$ is the subset of facts and strict rules in $\Pi_{\textit{AM}}$ whose annotations are true in EM world $w$. The set $\mathcal{W}_{\textit{{EM}}}^{0}(\mathcal{I})$ contains all the EM worlds for a given program where the corresponding knowledge base in the AM is classically inconsistent and $\mathcal{W}_{\textit{{EM}}}^{I}(\mathcal{I})$ is a subset of these that can be assigned a non-zero probability – the latter are the worlds where inconsistency in the AM can arise. ### 5.1 Postulates for Revising the Annotation Function We now analyze the rationality postulates for non-prioritized revision of belief bases first introduced in [2] and later generalized in [25], in the context of P-PreDeLP programs. These postulates are chosen due to the fact that they are well studied in the literature for non-prioritized belief revision. Inclusion: For $\mathcal{I}\bullet(f,\textit{af{\,}}^{\prime})=(\Pi_{\textit{EM}},\Pi_{\textit{AM}}\cup\\{f\\},\textit{af{\,}}^{\prime\prime})$, $\forall g\in\Pi_{\textit{AM}}$, $wld\big{(}\textit{af{\,}}^{\prime\prime}(g)\big{)}\subseteq wld(\textit{af{\,}}^{\prime}(g))$. This postulate states that, for any element in the AM, the worlds that satisfy its annotation after the revision are a subset of the original set of worlds satisfying the annotation for that element. Vacuity: If $\mathcal{I}\cup(f,\textit{af{\,}}^{\prime})$ is consistent, then $\mathcal{I}\bullet(f,\textit{af{\,}}^{\prime})=\mathcal{I}\cup(f,\textit{af{\,}}^{\prime})$ Consistency Preservation: If $\mathcal{I}$ is consistent, then $\mathcal{I}\bullet(f,\textit{af{\,}}^{\prime})$ is also consistent. Weak Success: If $\mathcal{I}\cup(f,\textit{af{\,}}^{\prime})$ is consistent, then $(f,\textit{af{\,}}^{\prime})\in\mathcal{I}\bullet(f,\textit{af{\,}}^{\prime})$. Whenever the simple addition of the input doesn’t cause inconsistencies to arise, the result will contain the input. Core Retainment: For $\mathcal{I}\bullet(f,\textit{af{\,}}^{\prime})=(\Pi_{\textit{EM}},\Pi_{\textit{AM}}\cup\\{f\\},\textit{af{\,}}^{\prime\prime})$, for each $w\in\mathcal{W}_{\textit{{EM}}}^{I}(\mathcal{I}\cup(f,\textit{af{\,}}^{\prime}))$, we have $X_{w}=\\{h\in\Theta\cup\Omega\;\|\;w\models\textit{af{\,}}^{\prime\prime}(h)\\}$; for each $g\in\Pi_{\textit{AM}}(w)\setminus X_{w}$ there exists $Y_{w}\subseteq X_{w}\cup\\{f\\}$ s.t. $Y_{w}$ is consistent and $Y_{w}\cup\\{g\\}$ is inconsistent. For a given EM world, if a portion of the associated AM knowledge base is removed by the operator, then there exists a subset of the remaining knowledge base that is not consistent with the removed element and $f$. Relevance: For $\mathcal{I}\bullet(f,\textit{af{\,}}^{\prime})=(\Pi_{\textit{EM}},\Pi_{\textit{AM}}\cup\\{f\\},\textit{af{\,}}^{\prime\prime})$, for each $w\in\mathcal{W}_{\textit{{EM}}}^{I}(\mathcal{I}\cup(f,\textit{af{\,}}^{\prime}))$, we have $X_{w}=\\{h\in\Theta\cup\Omega\;\|\;w\models\textit{af{\,}}^{\prime\prime}(h)\\}$; for each $g\in\Pi_{\textit{AM}}(w)\setminus X_{w}$ there exists $Y_{w}\supseteq X_{w}\cup\\{f\\}$ s.t. $Y_{w}$ is consistent and $Y_{w}\cup\\{g\\}$ is inconsistent. For a given EM world, if a portion of the associated AM knowledge base is removed by the operator, then there exists a superset of the remaining knowledge base that is not consistent with the removed element and $f$. Uniformity 1: Let $(f,\textit{af{\,}}^{\prime}_{1}),(g,\textit{af{\,}}^{\prime}_{2})$ be two inputs where $\mathcal{W}_{\textit{{EM}}}^{I}(\mathcal{I}\cup(f,\textit{af{\,}}^{\prime}_{1}))=\mathcal{W}_{\textit{{EM}}}^{I}(\mathcal{I}\cup(g,\textit{af{\,}}^{\prime}_{2}))$; for all $w\in\mathcal{W}_{\textit{{EM}}}^{I}(\mathcal{I}\cup(f,\textit{af{\,}}^{\prime}))$ and for all $X\subseteq\Pi_{\textit{AM}}(w)$; if $\\{x\;\|\;x\in X\cup\\{f\\},w\models\textit{af{\,}}^{\prime}_{1}(x)\\}$ is inconsistent iff $\\{x\;\|\;x\in X\cup\\{g\\},w\models\textit{af{\,}}^{\prime}_{2}(x)\\}$ is inconsistent, then for each $h\in\Pi_{\textit{AM}}$, we have that: $\\{w\in\mathcal{W}_{\textit{{EM}}}^{I}(\mathcal{I}\cup(f,\textit{af{\,}}^{\prime}_{1}))\;\|\;w\models\textit{af{\,}}^{\prime}_{1}(h)\wedge\neg\textit{af{\,}}^{\prime\prime}_{1}(h)\\}=$ $\\{w\in\mathcal{W}_{\textit{{EM}}}^{I}(\mathcal{I}\cup(g,\textit{af{\,}}^{\prime}_{2}))\;\|\;w\models\textit{af{\,}}^{\prime}_{2}(h)\wedge\neg\textit{af{\,}}^{\prime\prime}_{2}(h)\\}.$ If two inputs result in the same set of EM worlds leading to inconsistencies in an AM knowledge base, and the consistency between analogous subsets (when joined with the respective input) are the same, then the models removed from the annotation of a given strict rule or fact are the same for both inputs. Uniformity 2: Let $(f,\textit{af{\,}}^{\prime}_{1}),(g,\textit{af{\,}}^{\prime}_{2})$ be two inputs where $\mathcal{W}_{\textit{{EM}}}^{I}(\mathcal{I}\cup(f,\textit{af{\,}}^{\prime}_{1}))=\mathcal{W}_{\textit{{EM}}}^{I}(\mathcal{I}\cup(g,\textit{af{\,}}^{\prime}_{2}))$; for all $w\in\mathcal{W}_{\textit{{EM}}}^{I}(\mathcal{I}\cup(f,\textit{af{\,}}^{\prime}))$and for all $X\subseteq\Pi_{\textit{AM}}(w)$; if $\\{x\;\|\;x\in X\cup\\{f\\},w\models\textit{af{\,}}^{\prime}_{1}(x)\\}$ is inconsistent iff $\\{x\;\|\;x\in X\cup\\{g\\},w\models\textit{af{\,}}^{\prime}_{2}(x)\\}$ is inconsistent, then $\\{w\in\mathcal{W}_{\textit{{EM}}}^{I}(\mathcal{I}\cup(f,\textit{af{\,}}^{\prime}_{1}))\;\|\;w\models\textit{af{\,}}^{\prime}_{1}(h)\wedge\textit{af{\,}}^{\prime\prime}_{1}(h)\\}=$ $\\{w\in\mathcal{W}_{\textit{{EM}}}^{I}(\mathcal{I}\cup(g,\textit{af{\,}}^{\prime}_{2}))\;\|\;w\models\textit{af{\,}}^{\prime}_{2}(h)\wedge\textit{af{\,}}^{\prime\prime}_{2}(h)\\}.$ If two inputs result in the same set of EM worlds leading to inconsistencies in an AM knowledge base, and the consistency between analogous subsets (when joined with the respective input) are the same, then the models retained in the the annotation of a given strict rule or fact are the same for both inputs. Relationships between Postulates. There are a couple of interesting relationships among the postulates. The first is a sufficient condition for Core Retainment to be implied by Relevance. ###### Proposition 2 Let $\bullet$ be an operator such that $\mathcal{I}\bullet(f,\textit{af{\,}}^{\prime})=(\Pi_{\textit{EM}},\Pi_{\textit{AM}}\cup\\{f\\},\textit{af{\,}}^{\prime\prime})$, where $\forall w\in\mathcal{W}_{\textit{{EM}}}^{I}(\mathcal{I}\cup(f,\textit{af{\,}}^{\prime}))$, $\Pi_{\textit{AM}}^{\mathcal{I}\bullet(f,\textit{af{\,}}^{\prime})}(w)$ is a maximal consistent subset of $\Pi_{\textit{AM}}^{\mathcal{I}\cup(f,\textit{af{\,}}^{\prime})}(w)$. If $\bullet$ satisfies Relevance then it also satisfies Core Retainment. Similarly, we can show the equivalence between the two Uniformity postulates under certain conditions. ###### Proposition 3 Let $\bullet$ be an operator such that $\mathcal{I}\bullet(f,\textit{af{\,}}^{\prime})=(\Pi_{\textit{EM}},\Pi_{\textit{AM}}\cup\\{f\\},\textit{af{\,}}^{\prime\prime})$ and $\forall w$, $\Pi_{\textit{AM}}^{\mathcal{I}\bullet(f,\textit{af{\,}}^{\prime})}(w)\subseteq\Pi_{\textit{AM}}^{\mathcal{I}\cup(f,\textit{af{\,}}^{\prime})}(w)$. Operator $\bullet$ satisfies Uniformity 1 iff it satisfies Uniformity 2. Given the results of Propositions 2 and 3, we will not study Core Retainment and Uniformity 2 with respect to the construction of a belief revision operator in the next section. ### 5.2 An Operator for P-PreDeLP Revision In this section, we introduce an operator for revising a P-PreDeLP program. As stated earlier, any subset of $\Pi_{\textit{AM}}$ associated with a world in $\mathcal{W}_{\textit{{EM}}}^{I}(\mathcal{I}\cup(f,\textit{af{\,}}^{\prime}))$ must be modified by the operator in order to remain consistent. So, for such a world $w$, we introduce a set of candidate replacement programs for $\Pi_{\textit{AM}}(w)$ in order to maintain consistency and satisfy the Inclusion postulate. $\displaystyle candPgm(w,\mathcal{I})$ $\displaystyle=$ $\displaystyle\\{\Pi_{\textit{AM}}^{\prime}\;\|\;\Pi_{\textit{AM}}^{\prime}\subseteq\Pi_{\textit{AM}}(w)\;\text{s.t.}\;\Pi_{\textit{AM}}^{\prime}\;\text{is consistent and}\;$ $\displaystyle\nexists\Pi_{\textit{AM}}^{\prime\prime}\subseteq\Pi_{\textit{AM}}(w)\textit{ s.t. }\Pi_{\textit{AM}}^{\prime\prime}\supset\Pi_{\textit{AM}}^{\prime}\;\text{s.t.}\;\Pi_{\textit{AM}}^{\prime\prime}\;\text{is consistent}\\}$ Intuitively, $candPgm(w,\mathcal{I})$ is the set of maximal consistent subsets of $\Pi_{\textit{AM}}(w)$. Coming back to the rain/hail example presented above, we have: ###### Example 8 Consider the P-PreDeLP program $\mathcal{I}$ presented right after Example 7, and the following EM knowledge base: $\displaystyle rain\vee hail$ $\displaystyle:$ $\displaystyle 0.5\pm 0.1;$ $\displaystyle rain\wedge hail$ $\displaystyle:$ $\displaystyle 0.3\pm 0.1;$ $\displaystyle wind$ $\displaystyle:$ $\displaystyle 0.2\pm 0.$ Given this setup, we have, for instance: $candPgm(\\{rain,hail,wind\\},\mathcal{I})=\Big{\\{}\big{\\{}umbrella\big{\\}},\big{\\{}\neg umbrella\big{\\}}\Big{\\}}.$ Intuitively, this means that, since the world where $rain$, $hail$, and $wind$ are all true can be assigned a non-zero probability by the EM, we must choose either $umbrella$ or $\neg umbrella$ in order to recover consistency. $\blacksquare$ We now show a series of intermediate results that lead up to the representation theorem (Theorem 5.1). First, we show how this set plays a role in showing a necessary and sufficient requirement for Inclusion and Consistency Preservation to hold together. ###### Lemma 1 Given program $\mathcal{I}$ and input $(f,\textit{af{\,}}^{\prime})$, operator $\bullet$ satisfies Inclusion and Consistency Preservation iff for $\mathcal{I}\bullet(f,\textit{af{\,}}^{\prime})=(\Pi_{\textit{EM}},\Pi_{\textit{AM}},\textit{af{\,}}^{\prime\prime})$, for all $w\in\mathcal{W}_{\textit{{EM}}}^{I}(\mathcal{I}\cup(f,\textit{af{\,}}^{\prime}))$, there exists an element $X\in candPgm(w,\mathcal{I}\cup(f,\textit{af{\,}}^{\prime}))$ s.t. $\\{h\in\Theta\cup\Omega\cup\\{f\\}\;\|\;w\models\textit{af{\,}}^{\prime\prime}(h)\\}\subseteq X$. Next, we investigate the role that the set $candPgm$ plays in showing the necessary and sufficient requirement for satisfying Inclusion, Consistency Preservation, and Relevance all at once. ###### Lemma 2 Given program $\mathcal{I}$ and input $(f,\textit{af{\,}}^{\prime})$, operator $\bullet$ satisfies Inclusion, Consistency Preservation, and Relevance iff for $\mathcal{I}\bullet(f,\textit{af{\,}}^{\prime})=(\Pi_{\textit{EM}},\Pi_{\textit{AM}},\textit{af{\,}}^{\prime\prime})$, for all $w\in\mathcal{W}_{\textit{{EM}}}^{I}(\mathcal{I}\cup(f,\textit{af{\,}}^{\prime}))$ we have $\\{h\in\Theta\cup\Omega\cup\\{f\\}\;\|\;w\models\textit{af{\,}}^{\prime\prime}(h)\\}\in candPgm(w,\mathcal{I}\cup(f,\textit{af{\,}}^{\prime}))$. The last of the intermediate results shows that if there is a consistent program where two inputs cause inconsistencies to arise in the same way, then for each world the set of candidate replacement programs (minus the added AM formula) is the same. This result will be used as a support of the satisfaction of the first Uniformity postulate. ###### Lemma 3 Let $\mathcal{I}=(\Pi_{\textit{EM}},\Pi_{\textit{AM}},\textit{af{\,}})$ be a consistent program, $(f_{1},\textit{af{\,}}^{\prime}_{1})$, $(f_{2},\textit{af{\,}}^{\prime}_{2})$ be two inputs, and $\mathcal{I}_{i}=(\Pi_{\textit{EM}},\Pi_{\textit{AM}}\cup\\{f_{i}\\},\textit{af{\,}}^{\prime}_{i})$. If $\mathcal{W}_{\textit{{EM}}}^{I}(\mathcal{I}_{1})=\mathcal{W}_{\textit{{EM}}}^{I}(\mathcal{I}_{2})$, then for all $w\in\mathcal{W}_{\textit{{EM}}}^{I}(\mathcal{I}_{1})$ and all $X\subseteq\Pi_{\textit{AM}}(w)$ we have that: 1. 1. If $\\{x\;\|\;x\in X\cup\\{f_{1}\\},w\models\textit{af{\,}}^{\prime}_{1}(x)\\}$ is inconsistent $\Leftrightarrow$ $\\{x\;\|\;x\in X\cup\\{f_{2}\\},w\models\textit{af{\,}}^{\prime}_{2}(x)\\}$ is inconsistent, then $\\{X\setminus\\{f_{1}\\}\;\|\;X\in candPgm(w,\mathcal{I}_{1})\\}=\\{X\setminus\\{f_{2}\\}\;\|\;X\in candPgm(w,\mathcal{I}_{2})\\}$. 2. 2. If $\\{X\setminus\\{f_{1}\\}\;\|\;X\in candPgm(w,\mathcal{I}_{1})\\}=\\{X\setminus\\{f_{2}\\}\;\|\;X\in candPgm(w,\mathcal{I}_{2})\\}$ then $\\{x\;\|\;x\in X\cup\\{f_{1}\\},w\models\textit{af{\,}}^{\prime}_{1}(x)\\}$ is inconsistent $\Leftrightarrow$ $\\{x\;\|\;x\in X\cup\\{f_{2}\\},w\models\textit{af{\,}}^{\prime}_{2}(x)\\}$ is inconsistent. We now have the necessary tools to present the construction of our non- prioritized belief revision operator. Construction. Before introducing the construction, we define some preliminary notation. Let $\Phi:\mathcal{W}_{\textit{{EM}}}\rightarrow 2^{[\mbox{$\Theta$}]\cup[\mbox{$\Omega$}]}$. For each $h$ there is a formula in $\Pi_{\textit{AM}}\cup\\{f\\}$, where $f$ is part of the input. Given these elements, we define: $\displaystyle\textit{newFor}(h,\Phi,\mathcal{I},(f,\textit{af{\,}}^{\prime}))$ $\displaystyle=$ $\displaystyle\textit{af{\,}}^{\prime}(h)\wedge\bigwedge_{w\in\mathcal{W}_{\textit{{EM}}}^{I}(\mathcal{I}\cup(f,\textit{af{\,}}^{\prime}))\;\|\;h\notin\Phi(w)}\neg for(w_{i})$ The following definition then characterizes the class of operators called AFO (annotation function-based operators). ###### Definition 12 (AF-based Operators) A belief revision operator $\bullet$ is an “annotation function-based” (or af- based) operator ($\bullet\in\textbf{{AFO}}$) iff given program $\mathcal{I}=(\Pi_{\textit{EM}},\Pi_{\textit{AM}},\textit{af{\,}})$ and input $(f,\textit{af{\,}}^{\prime})$, the revision is defined as $\mathcal{I}\bullet(f,\textit{af{\,}}^{\prime})=(\Pi_{\textit{EM}},\Pi_{\textit{AM}}\cup\\{f\\},\textit{af{\,}}^{\prime\prime})$, where: $\forall h,\textit{af{\,}}^{\prime\prime}(h)=\textit{newFor}(h,\Phi,\mathcal{I},(f,\textit{af{\,}}^{\prime}))$ where $\forall w\in\mathcal{W}_{\textit{{EM}}}$, $\Phi(w)\in CandPgm_{af}(w,\mathcal{I}\cup(f,\textit{af{\,}}^{\prime}))$. As the main result of the paper, we now show that satisfying a key set of postulates is a necessary and sufficient condition for membership in AFO. ###### Theorem 5.1 (Representation Theorem) An operator $\bullet$ belongs to class AFO iff it satisfies Inclusion, Vacuity, Consistency Preservation, Weak Success, Relevance, and Uniformity 1. ###### Proof. (Sketch) (If) By the fact that formulas associated with worlds in the set $\mathcal{W}_{\textit{{EM}}}^{I}(\mathcal{I}\cup(f,\textit{af{\,}}^{\prime}))$ are considered in the change of the annotation function, Vacuity and Weak Success follow trivially. Further, Lemma 2 shows that Inclusion, Consistency Preservation, and Relevance are satisfied while Lemma 3 shows that Uniformity 1 is satisfied. (Only-If) Suppose BWOC that an operator $\bullet$ satisfies all postulates and $\bullet\notin\textbf{{AFO}}$. Then, one of four conditions must hold: (i) it does not satisfy Lemma 2 or (ii) it does not satisfy Lemma 3. However, by those previous arguments, if it satisfies all postulates, these arguments must be true as well – hence a contradiction. $\hfill\Box$ ∎ ## 6 Conclusions We have proposed an extension of the PreDeLP language that allows sentences to be annotated with probabilistic events; such events are connected to a probabilistic model, allowing a clear separation of interests between certain and uncertain knowledge. After presenting the language, we focused on characterizing belief revision operations over P-PreDeLP KBs. We presented a set of postulates inspired in the ones presented for non-prioritized revision of classical belief bases, and then proceeded to study a construction based on these postulates and prove that the two characterizations are equivalent. As future work, we plan to study other kinds of operators, such as more general ones that allow the modification of the EM, as well as others that operate at different levels of granularity. Finally, we are studying the application of P-PreDeLP to real-world problems in cyber security and cyber warfare domains. Acknowledgments. The authors are partially supported by UK EPSRC grant EP/J008346/1 (“PrOQAW”), ERC grant 246858 (“DIADEM”), ARO project 2GDATXR042, DARPA project R.0004972.001, Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET) and Universidad Nacional del Sur (Argentina). The opinions in this paper are those of the authors and do not necessarily reflect the opinions of the funders, the U.S. Military Academy, or the U.S. Army. ## References * [1] Martinez, M.V., García, A.J., Simari, G.R.: On the use of presumptions in structured defeasible reasoning. In: Proc. of COMMA. (2012) 185–196 * [2] Hansson, S.: Semi-revision. J. of App. Non-Classical Logics 7(1-2) (1997) 151–175 * [3] Alchourrón, C.E., Gärdenfors, P., Makinson, D.: On the logic of theory change: Partial meet contraction and revision functions. J. Sym. 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1401.1513	# On the Stability of Random Multiple Access with Feedback Exploitation and Queue Priority Karim G. Seddik Electronics Engineering Department, American University in Cairo AUC Avenue, New Cairo 11835, Egypt. email: [email protected] ###### Abstract In this paper, we study the stability of two interacting queues under random multiple access in which the queues leverage the feedback information. We derive the stability region under random multiple access where one of the two queues exploits the feedback information and backs off under negative acknowledgement (NACK) and the other, higher priority, queue will access the channel with probability one. We characterize the stability region of this feedback-based random access protocol and prove that this derived stability region encloses the stability region of the conventional random access (RA) scheme that does not exploit the feedback information. ## I Introduction The stability of interacting queues has been extensively considered in literature. Several works have considered the characterization of the stability region of interacting queues under random access protocols. The stability region is characterized for the case $M=2$ and $M=3$ interacting queues as well as the case of $M>3$ with symmetric arrivals. The stability region for the general case of $M>3$ with asymmetric arrivals is still an open problem and only inner achievable bounds are known. Recently, many papers have considered the problem of interacting queues in different contexts. For example, [1] considers the problem of interacting queues in a TDMA system where a relay is used to help the source nodes in forwarding their lost packets. In [2], the stability of interacting queues under a random access protocol in the context of Cognitive Radio Network was derived. In [3], the stability region of two interacting queues under random access protocol where the two queues harvest energy was characterized. Other works can be found in [4, 5], where derivations of the stability regions in the context of different cognitive radio networks were considered. In this paper, we derive the stability region of a two-queue random access (RA) protocol with priorities. The queues will apply the conventional RA protocol but in the case of packet loss due to collision the two queues will exploit the feedback information to provide some level of coordination. We set a priority to one of the two queues as follows. In the case of a negative acknowledgement, the queue with the higher priority will attempt transmission in the following time slot with probability one and the other queue will back off to allow for collision-free transmission of the higher priority queue. Clearly, this will enhance the service rate for the higher priority queue but more interestingly it will also improve the service rate for the other, less priority queue as will be explained later. We derive an expression for the boundary of the stability region and prove that the RA with priority scheme encloses the stability region of the conventional RA scheme. To the best of our knowledge, the problem of characterizing the stability region of the random access protocol with feedback leveraging has not been considered before. We will characterize the stable arrival rates region and prove that it contains that of the conventional random multiple access scheme (with no feedback exploitation). The rest of the paper is organized as follows. The system model is presented in Section II. The performance of the proposed scheme is investigated in Section III. The paper is concluded in Section IV. We have moved most of the proofs to the appendices to preserve the flow of ideas in the paper. ## II System Model The system model is shown in Fig. 1. We consider the case of two interacting packet queues, namely $Q_{1}$ and $Q_{2}$. $Q_{1}$ and $Q_{2}$ have infinite buffers for storing fixed length packets. The channel is slotted in time and any slot duration equals one packet transmission time. The arrival processes at the two queues, $Q_{1}$ and $Q_{2}$, are modeled as Bernoulli arrival processes with means $\lambda_{1}$ and $\lambda_{2}$, respectively [3]. Under our system model assumptions, the average arrival rates are $\lambda_{1}$ and $\lambda_{2}$ packets per time slot, and are bounded as $0<\lambda_{i}<1$, $i=1,2$111The maximum service rate in our model is 1 packet/slot, since the slot duration equals one packet transmission time, then the arrival rates must be less than 1 otherwise the system will be unstable [3].. We can assume that the packets arrive at the start of the time slot. Figure 1: The system model. The channel is modeled as a collision channel, where packet loss results only in the case of simultaneous transmissions from the two queues. If only one queue attempts transmitting at a given time slot, the packet is considered to be correctly received [6, 3]. In the random access phase, the first queue accesses the channel with probability $p_{1}$ whenever it has packets to send and the second queue will access the channel with probability $p_{2}$ whenever it has packets to send. If at any time slot some queue is empty, it will not attempt any channel access. In this paper, we will consider the use of the feedback information that is leveraged at the queues in the case of collision. In the conventional random multiple access system and in the case of collision, the collided packets stay on the head of the queues and retransmissions are attempted employing the same random multiple access scheme. In this paper, we consider a system where the feedback information is leveraged at the queues and a priority is set to the first queue; in the next time slot after collision, queue 2 ($Q_{2}$) will back off and queue 1 ($Q_{1}$) will retransmit its collided packet to allow for collision-free transmission of $Q_{1}$; after that the two queues return to the conventional random multiple access scheme. The priority set to queue 1 can be due to some quality of service (QoS) requirement that is different from the QoS requirement of queue 2. The interesting result is that although the feedback will enhance the service of queue 1 by setting a higher priority to it, the service will be enhanced as well for queue 2 as will be explained later. ## III The Stability Region for the Feedback-Based Random Access Protocol with Priorities In this section, we will characterize the stability region for the feedback- based random access scheme. Stability can be loosely defined as having a certain quantity of interest kept bounded. In our case, we are interested in the queue size being bounded. For an irreducible and aperiodic Markov chain with countable number of states, the chain is stable if and only if it is positive recurrent, which implies the existence of its stationary distribution. For a rigorous definition of stability under more general scenarios see [6] and [7]. If the arrival and service processes of a queueing system are strictly stationary, then one can apply Loynes’s theorem to check for stability conditions [8]. This theorem states that if the arrival process and the service process of a queueing system are strictly stationary, and the average arrival rate is less than the average service rate, then the queue is stable, otherwise it is unstable. Characterizing the stability region will be a difficult problem due to the interaction of the two queues and due to the fact that the service for one queue will depend on the state of the other queue. We will consider the use of the Dominant System concept that was proposed in [6] to characterize the stability region of the conventional RA scheme. We will define two dominant systems tailored to match our feedback-based random access scheme and in each of the two systems we will determine the boundaries of the stability region. ### III-A Dominant System 1 In any dominant system, we define a system that “stochastically dominates” our system, that is the queues lengths in the dominant system are always larger than the queues lengths in our system if both, the dominant system and our system, start from the same initial state and have the same arrivals and encounter the same packet collisions. Figure 2: Queue 1, $Q_{1}$, Markov chain model for Dominant System 1. For the first Dominant System, we assume that queue 2 will always have packets to transmit; even if the queue was empty dummy packets will be transmitted from queue 2. Clearly this will set a dominant system to our system since the transmission of dummy packets can only result in more collisions and packet losses. If for a given arrival rate pair ($\lambda_{1}$, $\lambda_{2}$) the first dominant system is stable then clearly our system will be stable. Therefore, the stability region of the first dominant system will provide an inner bound for our system stability region. For queue 1, the Markov chain describing the evolution of the queue is shown in Fig. 2. Note that the Markov chain has two classes of states, namely, $k_{F}$ and $k_{R}$ and $k=0,1,2,\cdots$. The subscript $F$ denotes first transmission states and the subscript $R$ denotes retransmission states. Note that in the retransmission states, queue 1 packet will always be delivered since there is no collisions in these states (queue 2 is backing off); in these states, either queue 1 length decreases by 1 if no arrival occurs or the queue length will remain the same if an arrival occurs while being in these retransmission states since the packet on the head of the queue is successfully transmitted with probability 1. The stability condition for queue 1 in Dominant System 1 is given in the following lemma, which is proved in Appendix A. ###### Lemma III.1 The arrival rates for queue 1 and queue 2 in Dominant System 1 must satisfy the following two conditions, respectively, $\begin{split}\lambda_{1}&<\frac{p_{1}}{1+p_{1}p_{2}}\\\ \lambda_{2}&<p_{2}(1-\lambda_{1}-\lambda_{1}p_{2})\end{split}$ (1) for the system to be stable. ### III-B Dominant System 2 In the second Dominant System, we assume that queue 1 always has packets to send (dummy packets are sent if the queue decides to transmit while being empty). Again, this will decouple the interaction of the two queues since the service rate of queue 2 will be independent of the state of queue 1. The Markov chain for the evolution of queue 2 is shown in Fig. 3. Two classes of states are defined in Fig. 3 and denoted by the subscripts ON and OFF. The ON states denote the states where queue 2 can access the channel. The OFF states denote the back off states where queue 1 is retransmitting its collided packets. Note that the transitions from the $k_{\text{OFF}}$ state can be either to the $k_{\text{ON}}$ state, if no arrival occurs in the slot, or to the $(k+1)_{\text{ON}}$ state, if one arrival occurs in the slot. The OFF states can never make a transition to a state with a lower number of packets since in the OFF states queue 2 is in the back off mode and no access is attempted. The stability condition for queue 2 in Dominant System 2 is given in the following lemma, which is proved in Appendix B (the analysis in Appendix B will be based on the theory of homogeneous quasi birth-and-death (QBD) Markov chains [9]). ###### Lemma III.2 The arrival rates for queue 1 and queue 2 in Dominant System 2 must satisfy the following two conditions, respectively, $\begin{split}\lambda_{1}&<\frac{p_{1}(1-p_{1}-\lambda_{2}p_{1})}{(1-p_{1})}\\\ \lambda_{2}&<\frac{p_{2}(1-p_{1})}{1+p_{1}p_{2}}\end{split}$ (2) for the system to be stable. Figure 3: Queue 2, $Q_{2}$, Markov chain model for Dominant System 2. Note that the intersection of the two stability regions described in Lemma III.1 and Lemma III.2 for a given access vector $\mathbf{p}=[p_{1}\;p_{2}]^{T}$ (grey area in Fig. 4) can be interpreted as follows. Define a new Dominant System (Dominant System 3) in which every queue has always a packet to transmit. In this case, the transmission state of queue 1 can be represented by the two-state Markov chain model shown in Fig 5(a); note that in this case queue 1 will be either in the “Transmission” state denoted by $F$ or in the “Retransmission” state denoted by $R$ in Fig. 5(a). Fig. 5(b) shows the Markov chain model for queue 2. Queue 2 will have two states denoted by ON when queue 1 is in the F state and OFF when queue 1 is the $R$ state (when queue 1 is in the $R$ state queue 2 will be in the back off, OFF state). It is straightforward to show that the steady state distributions for the two Markov chains shown in Fig. 5 are given by $\begin{split}\pi_{F}&=\pi_{\text{ON}}=\frac{1}{1+p_{1}p_{2}}\\\ \pi_{R}&=\pi_{\text{OFF}}=\frac{p_{1}p_{2}}{1+p_{1}p_{2}}.\end{split}$ (3) Figure 4: The union of the stability regions for the two dominant systems for fixed access probabilities $p_{1}$ and $p_{2}$. The service rate for queue 1 in Dominant System 3, $\mu_{1}^{\prime\prime}$, is given by $\begin{split}\mu_{1}^{\prime\prime}=p_{1}(1-p_{2})\pi_{F}+\pi_{R}=\frac{p_{1}}{1+p_{1}p_{2}},\end{split}$ (4) where queue 1 is served with probability $p_{1}(1-p_{2})$ in the $F$ state and with probability 1 in the $R$ state. The service rate for queue 2 in Dominant System 3, $\mu_{2}^{\prime\prime}$, is given by $\begin{split}\mu_{2}^{\prime\prime}=p_{2}(1-p_{1})\pi_{\text{ON}}+0\times\pi_{\text{OFF}}=\frac{p_{2}(1-p_{1})}{1+p_{1}p_{2}},\end{split}$ (5) where queue 2 is served with probability $p_{2}(1-p_{1})$ in the ON state and with probability 0 in the OFF state. (a) The two-state Markov chain model for queue 1 transmission state in Dominant System 3. (b) The two-state Markov chain model for queue 2 transmission state in Dominant System 3. Figure 5: Dominant System 3 Markov chain model. ### III-C The Stability Region of the Random Access Protocol with Priorities In this section, we derive the expression for the stability region of the random access scheme with feedback exploitation where a priority is set to one of the two queues. The following Lemma characterizes the stability region for fixed random access probabilities, $p_{1}$ and $p_{2}$, for queue 1 and queue 2, respectively. ###### Lemma III.3 For a fixed random access probability vector $\mathbf{p}=[p_{1}\;p_{2}]^{T}$, the stability region $\mathcal{R}(\mathbf{p})$ of the random access with priorities is the union of the two regions described by $\begin{split}\lambda_{2}<p_{2}(1-\lambda_{1}-\lambda_{1}p_{2})\end{split}$ (6) when $\begin{split}\lambda_{1}<\frac{p_{1}}{1+p_{1}p_{2}}\end{split}$ (7) and $\begin{split}\lambda_{1}<\frac{p_{1}(1-p_{1}-\lambda_{2}p_{1})}{(1-p_{1})}\end{split}$ (8) when $\begin{split}\lambda_{2}<\frac{p_{2}(1-p_{1})}{1+p_{1}p_{2}}.\end{split}$ (9) for the system to be stable. ###### Proof: The result in Lemma III.3 can be proved using the tool of stochastic dominance presented in [6]. The indistinguishability argument at the stability region boundary states that if the original system is unstable then its queues will saturate and they will always have packets to transmit; therefore at the boundaries of the stability region of the original system, the original system will be indistinguishable from the dominant system and thus has the same stability region boundaries [6]. ∎ The next theorem characterizes the entire stability region for the random access protocol with priorities. ###### Theorem III.4 The boundary of the stability region, $\mathcal{R}$, of the random access protocol with priorities, which is defined as the union of the $\mathcal{R}(\mathbf{p})$ regions for the different $\mathbf{p}=[p_{1}\;p_{2}]^{T}$ as $\begin{split}\mathcal{R}=\bigcup_{\mathbf{p}\in[0,1]^{2}}\mathcal{R}(\mathbf{p})\end{split}$ (10) can be characterized as $\begin{split}\lambda_{2}=\left\\{\begin{array}[]{ll}1-2\lambda_{1}&\lambda_{1}\leq\frac{1}{3}\\\ \frac{(1-\lambda_{1})^{2}}{4\lambda_{1}}&\lambda_{1}>\frac{1}{3}.\end{array}\right.\end{split}$ (11) ###### Proof: First, we will derive the boundary of the stability region defined in lemma III.1, which can be found as $\begin{split}\lambda_{2}^{}(\lambda_{1})=&\mathbf{max}_{p_{1},p_{2}}\;p_{2}(1-\lambda_{1}-\lambda_{1}p_{2})\\\ &\text{{subject to }}0\leq p_{1}\leq 1,\;0\leq p_{2}\leq 1,\;\lambda_{1}<\frac{p_{1}}{1+p_{1}p_{2}}.\end{split}$ (12) Ignoring the constraints in the last optimization problem and differentiating the cost function in the last expression with respect to $p_{2}$ and equating the derivative to 0 we can get the optimal value for $p_{2}$, denoted by $p_{2}^{}$, as222it is straightforward to prove that the cost function is concave in $p_{2}$. $p_{2}^{}=\frac{1-\lambda_{1}}{2\lambda_{1}}.$ (13) Note that for $\lambda_{1}\geq\frac{1}{3}$, we have $p_{2}^{}\leq 1$. Also, for $p_{1}=1$ and $p_{2}^{}=\frac{1-\lambda_{1}}{2\lambda_{1}}$, the maximum value for the first queue arrival rate is $\frac{p_{1}}{1+p_{1}p_{2}}=\frac{2\lambda_{1}}{1+\lambda_{1}}>\lambda_{1}$ (i.e., the last constraint, $\lambda_{1}<\frac{p_{1}}{1+p_{1}p_{2}}$ is satisfied with $p_{1}=1$), which means that for $\lambda_{1}\geq\frac{1}{3}$, the value for $p_{2}$ that maximizes $\lambda_{2}$ for a given $\lambda_{1}$ is given by $p_{2}^{}=\frac{1-\lambda_{1}}{2\lambda_{1}}$, with all the constraints in (12) not being violated. For $\lambda_{1}<\frac{1}{3}$, following similar steps to the $\lambda_{1}\geq\frac{1}{3}$ case, we can easily prove that the value for $p_{2}$ that maximizes $\lambda_{2}$ is giving by $p_{2}^{}=1$; clearly the values of $p_{1}=1$ and $p_{2}^{}=1$ can be easily checked to satisfy the constraints in (12) for $\lambda_{1}<\frac{1}{3}$. Substituting the optimal values for $p_{2}$ for the different ranges of $\lambda_{1}$ we can easily get the boundary of the stability region spanned by the expression in lemma III.1 to be given by $\begin{split}\lambda_{2}=\left\\{\begin{array}[]{ll}1-2\lambda_{1}&\lambda_{1}\leq\frac{1}{3}\\\ \frac{(1-\lambda_{1})^{2}}{4\lambda_{1}}&\lambda_{1}>\frac{1}{3}.\end{array}\right.\end{split}$ (14) Finally, following a similar approach to that considered here it is straightforward to show that the boundary derived in (14) is the boundary of the stability regions defined in lemma III.2, which completes the proof. ∎ In Fig. 6, we have plotted the regions $\mathcal{R}(\mathbf{p})$, for $p_{1}$ and $p_{2}$ ranging from 0 to 1 with a step of 0.01, along with the derived stability region boundary given in the previous theorem. Fig. 6 also shows the stability region of the random access scheme, whose boundary is given by the following relation [6] $\sqrt{\lambda_{1}}+\sqrt{\lambda_{2}}=1.$ (15) In Fig. 6, we also show the boundary of the stability region for the time division (TD) based scheme (genie-aided), which serves as the stability region upper bound, given by333Time Division (TD) corresponds to full coordination between the two queues and requires knowledge of the queues arrival rates a priori before dividing the resources (time slots in this case). $\lambda_{1}+\lambda_{2}=1.$ (16) Figure 6: The stability regions for the Random Access, Random Access with Priorities, and Time Division schemes. It is clear, and straightforward to analytically prove from the closed-form stability region boundary expressions, that the stability region for the RA scheme with priorities encloses the stability region of the RA scheme. This can explained as follows. For a given arrival rate at the first queue, $\lambda_{1}$, the RA with priority scheme will provide a better service rate to that queue if compared to the RA scheme and this means that queue 1 will be empty with a higher probability and this means that queue 2 will have a higher service rate as well under the RA with priority scheme as compared to the RA scheme. So setting a priority to the first queue in the retransmission will also result in a service rate improvement for the second queue; this is because the RA with priority scheme has some form of coordination between the two queues in the retransmission stage. Allowing for collision free retransmission from the first queue will decrease the amount of expected collisions between the transmissions of the two queues and this will result in better service rates for the two queues. ## IV Conclusions In this paper, we consider the problem of deriving the stability region for random access protocol with feedback exploitation. We consider the case of two interacting queue with priority set to one of the two queues. The two queues will access the channel through a conventional random access protocol and in the case of collision the higher priority queue will access the channel in the next slot with probability 1 while the other queue will back off. We derive the stability region for the random access with priorities protocol and prove that it contains the stability region for the conventional random access protocol. We show that not only the service rate for the higher priority queue is enhanced but also the service rate for the other queue is improved if compared to the conventional random access protocol. ## Appendix A Proof of Lemma III.1 In this Appendix, we provide a proof for Lemma III.1. We start by calculating the steady state distribution for the Markov chain shown in Fig. 2. First, it is clear that $\epsilon_{0}=0$ since the queue can never be in a retransmission state while being empty. Writing the balance equation around $1_{R}$, we have $\epsilon_{1}=\lambda_{1}p_{1}p_{2}\pi_{0}+\left(1-\lambda_{1}\right)p_{1}p_{2}\pi_{1}.$ (17) Then around $0_{F}$, we have $(\lambda_{1}p_{1}p_{2}+\lambda_{1}(1-p_{1}))\pi_{0}=\left(1-\lambda_{1}\right)\epsilon_{1}+\left(1-\lambda_{1}\right)p_{1}(1-p_{2})\pi_{1}.$ (18) Substituting for $\epsilon_{1}$ from (17) into (18), and after some manipulations, we can get $\pi_{1}=\frac{\lambda_{1}\left(1-p_{1}+\lambda_{1}p_{1}p_{2}\right)}{p_{1}\left(1-\lambda_{1}\right)\left(1-\lambda_{1}p_{2}\right)}\pi_{0}.$ (19) Substituting from (19) into (17), we get $\epsilon_{1}=\frac{\lambda_{1}p_{2}}{1-\lambda_{1}p_{2}}\pi_{0}.$ (20) Writing the balance equation around $1_{F}$, we have $\begin{split}&\left(1-\lambda_{1}p_{1}\left(1-p_{2}\right)-\left(1-\lambda_{1}\right)(1-p_{1})\right)\pi_{1}=\\\ &\quad\lambda_{1}\pi_{0}+\lambda_{1}\epsilon_{1}+\left(1-\lambda_{1}\right)\epsilon_{2}+\left(1-\lambda_{1}\right)p_{1}\left(1-p_{2}\right)\pi_{2}.\end{split}$ (21) Around $2_{R}$, we have $\epsilon_{2}=\lambda_{1}p_{1}p_{2}\pi_{1}+\left(1-\lambda_{1}\right)p_{1}p_{2}\pi_{2}.$ (22) To get the relation between $\pi_{1}$ and $\pi_{2}$, we can substitute for the values of $\epsilon_{1}$, $\pi_{0}$ and $\epsilon_{2}$ from equations (17), (18) and (22), respectively in equation (21); after some tedious manipulation, we get $\pi_{2}=\frac{\lambda_{1}\left(1-p_{1}+\lambda_{1}p_{1}p_{2}\right)}{p_{1}\left(1-\lambda_{1}\right)\left(1-\lambda_{1}p_{2}\right)}\pi_{1}.$ (23) Substituting from (23) into (22), we get $\epsilon_{2}=\frac{\lambda_{1}p_{2}}{1-\lambda_{1}p_{2}}\pi_{1}.$ (24) Note that the Markov chain is repeating from stage 2 till the end. For $k\geq 2$, we have the following relations. $\pi_{k}=\frac{\lambda_{1}\left(1-p_{1}+\lambda_{1}p_{1}p_{2}\right)}{p_{1}\left(1-\lambda_{1}\right)\left(1-\lambda_{1}p_{2}\right)}\pi_{k-1}.$ (25) $\epsilon_{k}=\frac{\lambda_{1}p_{2}}{1-\lambda_{1}p_{2}}\pi_{k-1}.$ (26) The last relation can be used to prove the following relation between $\epsilon_{k}$ and $\epsilon_{k-1}$. $\epsilon_{k}=\frac{\lambda_{1}\left(1-p_{1}+\lambda_{1}p_{1}p_{2}\right)}{p_{1}\left(1-\lambda_{1}\right)\left(1-\lambda_{1}p_{2}\right)}\epsilon_{k-1}.$ (27) The steady state distribution can now be written as follows. * • $\epsilon_{0}=0$. * • $\pi_{k}=\rho^{k}\pi_{0}$, $k\geq 1$ and $\rho=\frac{\lambda_{1}\left(1-p_{1}+\lambda_{1}p_{1}p_{2}\right)}{p_{1}\left(1-\lambda_{1}\right)\left(1-\lambda_{1}p_{2}\right)}$. * • $\epsilon_{1}=\frac{\lambda_{1}p_{2}}{1-\lambda_{1}p_{2}}\pi_{0}$. * • $\epsilon_{k}=\rho^{k-1}\epsilon_{1}$, $k\geq 2$. This steady state distribution can be easily checked to satisfy the balance equation at any general state (details are omitted since it is a rather straightforward, yet very tedious, procedure). To get the value of the steady state probabilities, we apply the following normalization requirement. $\begin{split}&\sum_{k=0}^{\infty}(\pi_{k}+\epsilon_{k})=1\\\ &\quad\quad\rightarrow\pi_{0}+\sum_{k=1}^{\infty}(\pi_{k}+\epsilon_{k})=\pi_{0}\left(1+\frac{\lambda_{1}p_{2}}{1-\lambda_{1}p_{2}}\right)\sum_{k=0}^{\infty}\rho^{k}=1,\end{split}$ (28) where $\rho=\frac{\lambda_{1}\left(1-p_{1}+\lambda_{1}p_{1}p_{2}\right)}{p_{1}\left(1-\lambda_{1}\right)\left(1-\lambda_{1}p_{2}\right)}$ as defined above. Note that for the steady state distribution to exist, i.e. to have $\pi_{0}$ to be non zero, then we must have $\rho<1$, which is the stability condition for queue 1 in this dominant system. Therefore, the stability condition can be stated as $\rho<1\;\rightarrow\lambda_{1}<\frac{p_{1}}{1+p_{1}p_{2}}.$ (29) From the normalization condition in (28), we can get the value of $\pi_{0}$ as $\pi_{0}=\frac{p_{1}-\lambda_{1}(1+p_{1}p_{2})}{p_{1}(1-\lambda_{1})}.$ (30) In Dominant System 1, queue 2 will be served only in the states denoted by the subscript $F$ in Fig. 2 since in the retransmission states, denoted by the subscript $R$ in Fig. 2, queue 2 will be in the back off mode. Hence, the service rate, $\mu_{2}$, for queue 2 in Dominant System 1 is given by $\begin{split}\mu_{2}&=p_{2}(1-\lambda_{1})\pi_{0}+p_{2}(1-p_{1})\lambda_{1}\pi_{0}+\sum_{k=1}^{\infty}p_{2}(1-p_{1})\pi_{k}\\\ &=p_{2}(1-p_{1}\lambda_{1})\pi_{0}+\sum_{k=1}^{\infty}p_{2}(1-p_{1})\pi_{k},\end{split}$ (31) where in the $0_{F}$ state, and with the arrival at the beginning of the slot assumption, queue 2 is served with a rate of $p_{2}(1-\lambda_{1})\pi_{0}$ with no arrival at the beginning of the slot since queue 1 will not attempt any random access since it is empty, and $p_{2}(1-p_{1})\lambda_{1}\pi_{0}$ with arrival at the slot beginning; for the other first transmission states, queue 2 will be served if it decides to access the medium, which occurs with probability $p_{2}$, and queue 1 decides not to access the medium, which occurs with probability $(1-p_{1})$. After some manipulation, we can write the expression for $\mu_{2}$ as $\mu_{2}=p_{2}(1-\lambda_{1}-\lambda_{1}p_{2}).$ (32) For the stability of queue 2, we must have $\lambda_{2}<\mu_{2}=p_{2}(1-\lambda_{1}-\lambda_{1}p_{2}).$ (33) ## Appendix B Proof of Lemma III.2 In this Appendix, we provide a proof for Lemma III.2. We start by calculating the steady state distribution for the Markov chain shown in Fig. 3. The state transition matrix, $\mathbf{\Phi}$, of the Markov chain shown in Fig. 3 can be written as $\mathbf{\Phi}=\left(\begin{array}[]{ ccccc}\mathbf{B}&\mathbf{A}_{0}&\mathbf{0}&\mathbf{0}&\cdots\\\ \mathbf{A}_{2}&\mathbf{A}_{1}&\mathbf{A}_{0}&\mathbf{0}&\cdots\\\ \mathbf{0}&\mathbf{A}_{2}&\mathbf{A}_{1}&\mathbf{A}_{0}&\cdots\\\ \mathbf{0}&\mathbf{0}&\mathbf{A}_{2}&\mathbf{A}_{1}&\cdots\\\ \vdots&\vdots&\vdots&\vdots&\ddots\end{array}\right)$ (34) where $\begin{split}\mathbf{B}&=\left(\begin{array}[]{ cc}(1-\lambda_{2})+\lambda_{2}(1-p_{1})p_{2}&0\\\ 0&0\end{array}\right),\\\ \mathbf{A}_{0}&=\left(\begin{array}[]{ cc}(1-\lambda_{2})(1-p_{1})p_{2}&0\\\ 0&0\end{array}\right),\\\ \mathbf{A}_{1}&=\left(\begin{array}[]{ cc}\lambda_{2}p_{2}(1-p_{1})+(1-\lambda_{2})(1-p_{2})&1-\lambda_{2}\\\ (1-\lambda_{2})p_{1}p_{2}&0\end{array}\right),\\\ \mathbf{A}_{2}&=\left(\begin{array}[]{ cc}\lambda_{2}&\lambda_{2}\\\ \lambda_{2}p_{1}p_{2}&\lambda_{2}\end{array}\right).\end{split}$ The steady state distribution vector is given by $\mathbf{v}=[\pi_{0}^{\prime}\;\epsilon_{0}^{\prime}\;\pi_{1}^{\prime}\;\epsilon_{1}^{\prime}\;\pi_{2}^{\prime}\;\epsilon_{2}^{\prime}\;\cdots]^{T}$ and $\mathbf{v}=\mathbf{\Phi}\mathbf{v}$. The state transition matrix $\mathbf{\Phi}$ is a block-tridiagonal matrix; therefore the Markov chain shown in Fig. 3 is a homogeneous quasi birth-and- death (QBD) Markov chain [9]. Note that to make the state transition matrix a block-tridiagonal matrix we have added a transition from the $0_{\text{OFF}}$ state to the $1_{\text{ON}}$ state as shown in Fig. 7 and this will preserve the structure of the state transitions between the different stages in the Markov chain. Note that adding this transition will not affect the stationary state distribution of the Markov chain as well as the balance equations since $\epsilon_{0}^{\prime}=0$ even with the added transition since the Markov chain will never enter the $0_{\text{OFF}}$ state444The analysis presented here could have been used for analyzing the Markov chain shown in Fig. 2; however, the structure of this Markov chain allowed for the use of a simpler approach that was adopted in Appendix A. Figure 7: The queue 2 Markov chain with added transition between $0_{\text{OFF}}$ and $1_{\text{ON}}$ to make the state transition matrix a block-tridiagonal matrix. Define the vector $\mathbf{v}_{k}^{\prime}=[\pi_{k}^{\prime}\;\epsilon_{k}^{\prime}]^{T}$. Note that $\mathbf{v}_{0}^{\prime}=[\pi_{0}^{\prime}\;0]^{T}$. The steady state distribution of the Markov chain shown in Fig. 3 satisfies the following equation [9] $\mathbf{v}^{\prime}_{k}=\mathbf{R}^{k}\mathbf{v}_{0}^{\prime},\quad k\geq 1,$ (35) where the $2\times 2$ matrix $\mathbf{R}$ is given by the solution to the following equation. $\mathbf{A}_{0}+\mathbf{R}(\mathbf{A_{1}}-\mathbf{I}_{2})+\mathbf{R}^{2}\mathbf{A_{2}}=\mathbf{0}_{2\times 2},$ (36) where $\mathbf{I}_{2}$ is the $2\times 2$ identity matrix and $\mathbf{0}_{2\times 2}$ is the all zeros $2\times 2$ matrix. To get the stationary distribution, we have to find the matrix $\mathbf{R}=\left(\begin{array}[]{ cc}r_{11}&r_{12}\\\ r_{21}&r_{22}\end{array}\right).$ Note that for $\mathbf{v}_{1}^{\prime}=\mathbf{R}\mathbf{v}_{0}^{\prime}$, where $\mathbf{v}_{0}^{\prime}=[\pi_{0}^{\prime}\;0]^{T}$ and $\mathbf{v}_{1}^{\prime}=[\pi_{1}^{\prime}\;\epsilon_{1}^{\prime}]^{T}$. Therefore, we have $\begin{split}r_{11}=\frac{\pi_{1}^{\prime}}{\pi_{0}^{\prime}}\text{ and }r_{21}=\frac{\epsilon_{1}^{\prime}}{\pi_{0}^{\prime}}.\end{split}$ (37) Writing the balance equation around the $0_{\text{ON}}$ in Fig. 3, we have $\begin{split}&(\lambda_{2}p_{1}p_{2}+\lambda_{2}(1-p_{2}))\pi_{0}^{\prime}=(1-\lambda_{2})(1-p_{1})p_{2}\pi_{1}^{\prime}\\\ &\quad\rightarrow\pi_{1}^{\prime}=\frac{\lambda_{2}(1-p_{2}+p_{1}p_{2})}{(1-\lambda_{2})(1-p_{1})p_{2}}\pi_{0}^{\prime}.\end{split}$ (38) Therefore, we have $\begin{split}r_{11}=\frac{\lambda_{2}(1-p_{2}+p_{1}p_{2})}{(1-\lambda_{2})(1-p_{1})p_{2}}.\end{split}$ (39) Writing the balance equation around $1_{\text{OFF}}$, we have $\epsilon_{1}^{\prime}=\lambda_{2}p_{1}p_{2}\pi_{0}^{\prime}+(1-\lambda_{2})p_{1}p_{2}\pi_{1}^{\prime}\rightarrow\epsilon_{1}^{\prime}=\frac{\lambda_{2}p_{1}}{1-p_{1}}\pi_{0}^{\prime}.$ (40) Therefore, we have $r_{21}=\frac{\lambda_{2}p_{1}}{1-p_{1}}.$ (41) To get the values of $r_{12}$ and $r_{22}$, we consider the transition across the border shown in Fig. 8. For the Markov chain to be positive recurrent then the probability of going across the border in both directions must be the same [10]; hence, we have $\begin{split}(1-\lambda_{2})(1-p_{1})p_{2}\pi_{2}^{\prime}=(\lambda_{2}p_{1}p_{2}+\lambda_{2}(1-p_{2}))\pi_{1}^{\prime}+\lambda_{2}\epsilon_{1}^{\prime}.\end{split}$ (42) But we have $\mathbf{v}_{2}^{\prime}=\mathbf{R}\mathbf{v}_{1}^{\prime}$, from which we have $\pi_{2}^{\prime}=r_{11}\pi_{1}^{\prime}+r_{12}\epsilon_{1}^{\prime}$; using (39) and (42), we can easily show that $\begin{split}r_{12}=\frac{\lambda_{2}}{(1-\lambda_{2})(1-p_{1})p_{2}}.\end{split}$ (43) Figure 8: The segment of the Markov chain used to calculate the values of $r_{12}$ and $r_{22}$. Finally, the balance equation around $2_{\text{OFF}}$ can be written as $\epsilon_{2}^{\prime}=\lambda_{2}p_{1}p_{2}\pi_{1}^{\prime}+(1-\lambda_{2})p_{1}p_{2}\pi_{2}^{\prime}=r_{21}\pi_{1}^{\prime}+r_{22}\epsilon_{2}^{\prime}.$ (41) Substituting for $\pi_{2}^{\prime}$ from (42), we can easily show that $r_{22}=\frac{\lambda_{2}p_{1}}{1-p_{1}}.$ (42) Now the matrix $\mathbf{R}$ is given by $\mathbf{R}=\left(\begin{array}[]{cc}\frac{\lambda_{2}(1-p_{2}+p_{1}p_{2})}{(1-\lambda_{2})(1-p_{1})p_{2}}&\frac{\lambda_{2}}{(1-\lambda_{2})(1-p_{1})p_{2}}\\\ \frac{\lambda_{2}p_{1}}{1-p_{1}}&\frac{\lambda_{2}p_{1}}{1-p_{1}}\end{array}\right),$ (43) which can be easily checked to satisfy the balance equation given by (36). To get the stationary distribution of the Markov chain shown in Fig. 3, we apply the following normalization requirement. $\sum_{k=0}^{\infty}(\pi_{k}^{\prime}+\epsilon_{k}^{\prime})=1\rightarrow[1\;1]\left(\sum_{k=0}^{\infty}\mathbf{R}^{k}\right)\mathbf{v}_{0}^{\prime}=1.$ (44) For the summation $\left(\sum_{k=0}^{\infty}\mathbf{R}^{k}\right)$ to converge we must have the spectral radius of the matrix $\mathbf{R}$, $\textbf{sp}(\mathbf{R})$, to be less than one [9]555The spectral radius of a matrix is the maximum over the magnitudes of its eigenvalues.. From (43), we can easily get $\textbf{sp}(\mathbf{R})$ to be given by $\begin{split}\textbf{sp}(\mathbf{R})=\frac{\lambda_{2}\left(1-p_{2}-\lambda_{2}p_{1}p_{2}+2p_{1}p_{2}+\sqrt{1-2p_{2}+p_{2}^{2}+4p_{1}p_{2}-2\lambda_{2}p_{1}p_{2}-2\lambda_{2}p_{1}p_{2}^{2}+\lambda_{2}^{2}p_{1}^{2}p_{2}^{2}}\right)}{2p_{2}\left(1-\lambda_{2}-p_{1}+\lambda_{2}p_{1}\right)}.\end{split}$ (45) The requirement that $\textbf{sp}(\mathbf{R})<1$ can be used in the last expression to get the stability condition of the second queue arrival rate $\lambda_{2}$ as $\begin{split}\lambda_{2}<\frac{p_{2}(1-p_{1})}{1+p_{1}p_{2}}.\end{split}$ (46) Going back to the normalization requirement in (44), we have $[1\;1]\left(\sum_{k=0}^{\infty}\mathbf{R}^{k}\right)\mathbf{v}_{0}^{\prime}=[1\;1]\left(\mathbf{I}_{2}-\mathbf{R}\right)^{-1}\left[\begin{array}[]{c}\pi_{0}^{\prime}\\\ 0\end{array}\right]=1.$ (47) From the last expression, we can easily prove that $\pi_{0}^{\prime}$ is given by $\pi_{0}^{\prime}=\frac{p_{2}-\lambda_{2}-p_{1}p_{2}-\lambda_{2}p_{1}p_{2}}{(1-\lambda_{2})(1-p_{1})p_{2}}.$ (48) Note that the requirement that $\pi_{0}^{\prime}>0$, i.e. a non-zero empty queue probability, is satisfied if $\lambda_{2}<\frac{p_{2}(1-p_{1})}{1+p_{1}p_{2}}$, which is the queue stability condition. The service rate, $\mu_{1}^{\prime}$, for queue 1 in Dominant System 2 can now be expressed as $\begin{split}\mu_{1}^{\prime}&=p_{1}(1-\lambda_{2})\pi_{0}^{\prime}+p_{1}(1-p_{2})\lambda_{2}\pi_{0}^{\prime}+p_{1}(1-p_{2})\sum_{k=2}^{\infty}\pi_{k}^{\prime}+\sum_{k=2}^{\infty}\epsilon_{k}^{\prime}\\\ &=p_{1}(1-\lambda_{2}p_{2})\pi_{0}^{\prime}+[p_{1}(1-p_{2})\;1]\left(\sum_{k=1}^{\infty}\mathbf{R}^{k}\right)\left[\begin{array}[]{c}\pi_{0}^{\prime}\\\ 0\end{array}\right]\\\ &=p_{1}(1-\lambda_{2}p_{2})\pi_{0}^{\prime}+[p_{1}(1-p_{2})\;1]\;\mathbf{R}\left(\mathbf{I}_{2}-\mathbf{R}\right)^{-1}\left[\begin{array}[]{c}\pi_{0}^{\prime}\\\ 0\end{array}\right]\\\ &=\frac{p_{1}(1-p_{1}-\lambda_{2}p_{1})}{(1-p_{1})},\end{split}$ (49) where in the OFF states, queue 1 is served with probability 1 since queue 2 will be in the back off mode. For the stability of queue 1 in Dominant System 2 we must have $\lambda_{1}<\mu_{1}^{\prime}=\frac{p_{1}(1-p_{1}-\lambda_{2}p_{1})}{(1-p_{1})},$ (50) which completes the proof. ## References * [1] A.K. Sadek, K.J.R. Liu, and Anthony Ephremides, “Cognitive multiple access via cooperation: Protocol design and performance analysis,” Information Theory, IEEE Transactions on, vol. 53, no. 10, pp. 3677–3696, 2007. * [2] S. Kompella, Gam D. Nguyen, J.E. Wieselthier, and Anthony Ephremides, “Stable throughput tradeoffs in cognitive shared channels with cooperative relaying,” in INFOCOM, 2011 Proceedings IEEE, 2011, pp. 1961–1969. * [3] Jeongho Jeon and Anthony Ephremides, “The stability region of random multiple access under stochastic energy harvesting,” in Information Theory Proceedings (ISIT), 2011 IEEE International Symposium on, 2011, pp. 1796–1800. * [4] A. Fanous and Anthony Ephremides, “Effect of secondary nodes on the primary’s stable throughput in a cognitive wireless network,” in Information Theory Proceedings (ISIT), 2012 IEEE International Symposium on, 2012, pp. 1807–1811. * [5] A. Fanous and A. Ephremides, “Stable throughput in a cognitive wireless network,” Selected Areas in Communications, IEEE Journal on, vol. 31, no. 3, pp. 523–533, 2013. * [6] R. Rao and A. Ephremides, “On the Stability of Interacting Queues in a Multi-Access System,” IEEE Trans. Info. Theory, vol. 34, pp. 918–930, Sept. 1988. * [7] W. Szpankowski, “Stability Conditions for Some Multiqueue Distributed System: Buffered Random Access Systems,” Adv. Appl. Probab., vol. 26, pp. 498–515, 1994. * [8] R. M. Loynes, “The Stability of a Queue with Non-Independent Interarrival and Service Times,” Proc. Cambridge Philos. Soc., pp. 497–520, 1962. * [9] G. Latouche and V. Ramaswami, Introduction to Matrix Analytic Methods in Stochastic Modeling, ASA-SIAM Series on Statistics and Applied Probability. Society for Industrial and Applied Mathematics, 1999. * [10] R.G. Gallager, Discrete stochastic processes, vol. 101, Kluwer Academic Publishers, 1996.	arxiv-papers	2014-01-06T20:11:21	2024-09-04T02:49:56.429944	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Karim G. Seddik", "submitter": "Karim Seddik", "url": "https://arxiv.org/abs/1401.1513" }
1401.1542	# Optical phase-noise dynamics of Titanium:sapphire optical frequency combs Qudsia Quraishi,1,4∗ Scott Diddams,2 and Leo Hollberg3 1 Army Research Laboratory, Adelphi, MD 20783 2National Institute of Standards and Technology, Boulder, Colorado 80305 3Department of Physics, Stanford University, Stanford, California 94305 4Formerly with Department of Physics, University of Colorado, Boulder, CO 80309 ∗Corresponding author: [email protected] ###### Abstract Stabilized optical frequency combs (OFC) can have remarkable levels of coherence across their broad spectral bandwidth. We study the scaling of the optical noise across hundreds of nanometers of optical spectra. We measure the residual phase noise between two OFC’s (having offset frequencies $f^{(1)}_{0}$ and $f^{(2)}_{0}$) referenced to a common cavity-stabilized narrow linewidth CW laser. Their relative offset frequency $\Delta f_{0}$ = $f^{(2)}_{0}$ \- $f^{(1)}_{0}$, which appears across their entire spectra, provides a convenient measure of the phase noise. By comparing $\Delta f_{0}$ at different spectral regions, we demonstrate that the observed scaling of the residual phase noise is in very good agreement with the noise predicted from the standard frequency comb equation. ###### keywords: ultrafast , phase noise , optical reference , frequency comb equation The development of stabilized optical frequency combs (OFC) allows for straightforward schemes for comparisons of optical frequencies which are separated by 100‘s of terahertz. When OFC’s are referenced to stable atomic transitions, the OFC provides a bridge to evaluate relative instabilities based on different atomic frequency standards Diddams et al. (2001); Gerginov et al. (2006); Ma et al. (2004); Newbury (2011). In such an atomic-optical ‘clockwork’, the fractional instability of the (stabilized) OFC must be sufficiently low that it does not degrade the measurement of the stability of the atomic transition. In fact, recent work has shown that the frequency instability of OFC can achieve levels of 10-19 at averaging times of 500 seconds (when compared against another stabilized OFC Ma et al. (2004)). Building on previous work with Titanium:sapphire Bartels et al. (2004); Schibli et al. (2008); Stenger et al. (2002), which showed phase-coherence across the optical spectrum, here, we focus on the optical phase-noise dynamics on shorter times scales (100 ns $\leq\tau\leq$ 1 s). We explore factors that limit the noise floor and demonstrate that the measured scaling of the phase-noise exhibits the scaling expected from the simple frequency comb equation. Figure 1: (a) Schematic of the optical frequency comb (OFC) output of two 1 GHz repetition rate TiS mode-locked lasers. One optical mode of each OFC is referenced, using a phase-locked loop (PLL), to a cavity-stabilized narrow linewidth CW source (REF) at 657 nm. (b) Schematic of the setup used to compare $\Delta f_{0}$ = $f^{(2)}_{0}$ \- $f^{(1)}_{0}$ derived from two spectral regions, $\nu_{n}$ and $\nu_{q}$. Polarizing beam splitters (PBS) and a waveplate (W) are used to obtain spatial and polarization overlap. (c) Photodetected heterodyne beat of $\Delta f_{0}$ from one spectral region. To determine the optical phase-noise attributable to the OFCs we lock two independent OFCs to the same cavity-stabilized CW optical frequency reference and measure the residual phase-noise between the two combs [Fig. 1(a)]. Alternatively, one could use two highly-stabilized optical frequency references to measure the phase fluctuations of individual comb modes. That approach would add a source of phase noise not attributable to the combs. Such a scheme is used for comparisons of optical atomic clocks Rosenband et al. (2008), spectroscopy Cingoz et al. (2012) and for frequency comparisons Coddington et al. (2007), in which high levels of long-term coherence have already been demonstrated. While comparing two free-running frequency combs Schlatter et al. (2007) give some measure of phase-noise, locking the frequency combs allows one to robustly quantify the noise across a broad spectral region and compare with theoretical predictions. The excellent stability of the comb Ma et al. (2004) makes it an excellent reference tool in spectral regions from microwave and terahertz to optical domains Quraishi et al. (2005). Our apparatus consists of passive two mode-locked titanium:sapphire (TiS) ring lasers having a pulse repetition rate of 1 GHz. These self-referenced lasers have been discussed in detail elsewhere Bartels et al. (2004) and employ piezoelectric actuators for cavity repetition rate stabilization. The offset frequency of both OFC’s stabilized with an 2f-to-3f technique Ramond et al. (2002) which requires less than an octave of optical bandwidth, in our case, from approximately 600 nm to 1150 nm. The $n^{\rm th}$ optical frequency mode is identified as $\\!\\!\\!\\!\\!\\!\nu_{n}=nf_{\rm rep}+f_{0}$ (1) where $f_{\rm rep}$ is the repetition rate and $f_{0}$ is the offset frequency Udem et al. (2002) and $n$ indexes an optical mode and is an element of the integers of order $10^{5}$. However, when an OFC is referenced to a stable optical source with frequency $\nu_{0}$ (at 657nm), we can also identify the optical mode as $\nu_{n}=\nu_{0}+f^{i}_{\rm beat}$, where the RF heterodyne beatnote between a mode from OFC1 (OFC2) and the CW reference is denoted by $f^{(1)}_{\rm beat}$ ($f^{(2)}_{\rm beat}$) or $f^{i}_{\rm beat}$ (where $i=1,2$). Solving for the repetition rate we have, $f^{i}_{\rm rep}=(\nu_{0}+f^{i}_{\rm beat}-f^{i}_{0})/n$. Figure 2: Measured residual phase noise (double sideband) between $\Delta f_{0}$ from two spectral regions, 800 nm and 621 nm (bandwidth $<$10 nm) as compared to the spectral region at 657 nm (bandwidth $<$3 nm), the left axis units are rad/$\sqrt{\rm Hz}$. The prediction is for the residual phase noise at 800 nm, obtained from Eqn. 3. When we tune the repetition rates of two OFC’s to be equal, we have a uniform frequency shift ($\Delta f_{0}$ $=f^{(2)}_{0}-f^{(1)}_{0}$) between all of the modes of the two combs [Fig. 1(a)]. Here, we are interested in the scaling of the phase noise away from the optical lock point at 657 nm (that is, from an imposed fixed point Benkler et al. (2005) of the comb). At the lock point, we measure $\Delta f_{0}$ within a small bandwidth about 657 nm and then compare it with $\Delta f_{0}$ measured at another spectral region [Fig. 1(b)]. We observe excellent signal-to-noise ratios (SNR) on the $\Delta f_{0}$ signal because 1 nm of optical bandwidth corresponds to $\sim$1000 modes contributing to the measured beatnote. Figure 1(c) shows the measured RF heterodyne signal from OFC1 and OFC2 which are spatially, spectrally and temporally combined. To calculate the residual phase noise, we begin by expressing each optical mode in terms of the laser’s free parameters, $f_{0}$ and $f_{\rm beat}$, to obtain: $\nu^{i}_{n}=r_{n}(\nu_{0}+f^{i}_{\rm beat})+(1-r_{n})f^{i}_{0}$ (where $r_{n}=n/n_{\rm lock}$ and $n_{\rm lock}$ indexes the mode nearest the CW optical reference) Newbury and Swann (2007). The optical phase-noise signal between OFC1 and OFC2 for the spectral region $n$ may be expressed as $\delta\nu_{n}=\nu^{(2)}_{n}-\nu^{(1)}_{n}$, $\delta\nu_{n}=r_{n}(\Delta f_{\rm beat}-\Delta f_{0})+\Delta f_{0}=r_{n}\Delta f_{\rm rep}+\Delta f_{0}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!$ (2) where $\Delta f_{\rm rep}=f^{(2)}_{\rm rep}-f^{(1)}_{\rm rep}$ = 0. A similar equation may be written for the spectral region around the lock point at 657 nm, and this is denoted by $\delta\nu_{q}$. The relative phase noise power spectral density (PSD) $S_{\Phi}(f)$ away from the lock point is calculated from the variance of $\delta\nu_{n}-\delta\nu_{q}$, scaled with respect to the noise bandwidth, giving $\displaystyle S_{\Phi,n,q}(f)=(r_{n}-r_{q})^{2}[S_{\Phi,f^{(2)}_{\rm beat}}(f)+S_{\Phi,f^{(2)}_{0}}(f)$ $\displaystyle+S_{\Phi,f^{(1)}_{\rm beat}}(f)+S_{\Phi,f^{(1)}_{0}}(f)],$ (3) where $S_{\Phi,f^{(1)}_{\rm beat}}(f)$ and $S_{\Phi,f^{(1)}_{0}}(f)$ denote the PSD of the electronic locks and where we have neglected all cross terms [such as $S_{\Phi,f^{i}_{0}}(f)\bigotimes S_{\Phi,f^{j}_{\rm beat}}(f)$]. Phase noise on the relative offset frequency $\Delta f_{0}$ is a measure of the combs’ residual optical phase noise. The residual phase noise is the remaining phase noise between the two combs after common mode noise has been subtracted. Notably, both OFCs are locked to a common CW reference to better ensure that noise on the beat between the two OFCs (Eqn. 2) yields noise of one comb with respect to the other and not simply a measure of different CW references. Regarding coupling between an individual OFC’s two parameters of $f^{i}_{0}$ and $f^{i}_{beat}$, we note that the two OFCs have independent pump lasers and phase-locked loops so ideally there should be little correlation between the two noise sources on each OFCs. Some coupling between $f^{i}_{0}$ and $f^{i}_{beat}$ is evident and enhanced by amplitude noise on the pump source of the Ti:S modelocked laser. Phase-locking $f_{0}$ with an acousto-optic or electro-optic transducer in the pump beam path significantly reduces the coupling Quraishi (Ph.D. dissertation, University of Colorado, 2007). Figure 3: In-loop phase noise of the phase-locks of each optical frequency comb obtained by comparing each beatnote against a synthesizer (left axis units, rad/$\sqrt{\rm Hz}$). The noise from the offset frequencies $f^{(1)}_{0}$ and $f^{(2)}_{0}$ as well as the noise of the comb mode locked to the optical reference are shown (each is filtered by a 1 MHz low pass). The integrated phase on $f^{(1)}_{\rm beat}$ ($f^{(1)}_{0}$) corresponds to 0.11 fs (0.074 fs) of timing jitter. The integrated phase noise for the phase-locks of OFC2 for $f^{(2)}_{\rm beat}$ ($f^{(2)}_{0}$) is 0.30 fs (0.15 fs). We measure the relative optical phase noise across the comb (Eqn. 3) by using the setup shown in Fig. 1(b) to extract $\Delta f_{0}$ from the spectral region around 800 nm and compare that against $\Delta f_{0}$ extracted from near the lock point at 657 nm. We observe that the optical phase noise increases with increasing frequency range away from the lock point (Fig. 2). The data in Fig. 2 is integrated from 1 Hz to 1 MHz, where we obtain 1.4 radians for the 800 nm data (the lock point contribution to this value is 0.9 radians). In order to obtain high resolution across all the Fourier frequencies from 0 Hz to 1 MHz, the phase noise traces shown in this work were taken by stitching together traces from smaller frequency spans where the spans initially extended less than 1 kHz and ended at a 1 MHz span. We can also simply measure the noise of each lasers’ lock to the reference separately (left hand side of Eqn. 3). Once we measure $f^{i}_{0}$ and $f^{i}_{beat}$, we can use Eqn. 3 to predict the phase noise of comb modes ($\delta\nu_{n}$) away from the lock point at 657 nm. Using the results shown in Fig. 3 for the in-loop PSD for all the phase-locks, we plot one typical predicted phase noise in Fig.2. The predicted phase noise (Fig.2) is in good agreement with the measured spectral densities across the frequency range. We find comparable agreement for all of our data sets. Discrepancies arising above 55 kHz are due to the prevalence of noise outside our PLL bandwidth. Comparing Fig. 2 and Fig. 3, it is clear that features in the relative noise traces can be ascribed to features in each of the phase-locks. For instance, we see that $f^{(1)}_{\rm beat}$ ($f^{(2)}_{0}$, $f^{(2)}_{\rm beat}$) dominates the noise in the vicinity of 1 kHz (5 kHz, 46 kHz). Equation 3 gives equal relative weight to contributions from each of the phase-locks. The phase noise profiles provide useful information about the laser’s phase-noise dynamics Paschotta et al. (2006). Not surprisingly, tighter phase-locks would lead to reduced optical phase noise ($\Delta f_{0}$ is a measure of the optical phase noise). The two lasers having differing noise profiles seen in Fig. 2 and Fig. 3 because they are pumped with different pump sources, have different electronic locking bandwidths and are situated in different parts of the laboratory. Ideally, the phase noise would be reduced to the level of the shot noise, which is already the case for frequencies above $\sim$1 MHz. The majority of the noise we observe is in the frequency range below 1 MHz range. In Ti:sapphire lasers there is a strong correlation between the pump laser’s amplitude noise and the optical phase noise, particularly on $f_{0}$ Holman et al. (2003). Indeed, for the data presented in Fig. 3, the phase noise is relatively high near 50 kHz due to increased amplitude noise present on the pump source of both OFC’s Quraishi (Ph.D. dissertation, University of Colorado, 2007). However, other noise contributions are also important including: photodetection noise (which can limit our SNR), electrical noise (ground loops, in-loop electronic noise), vibrational effects (prominent near 30 Hz), servo system gain-bandwidths (60 kHz, limited by the PZT for locking $f_{\rm beat}$ ) and thermal fluctuations (both within the Ti:S laser cavities and in their non-common-mode optical paths between the lasers). Excessive noise on $f_{0}$ causes optical phase noise and hence, can effect the $f_{\rm beat}$ lock. To characterize the scaling of the optical phase noise over almost 300 nm, we integrated the phase noise at discrete points in the spectrum using optical bandwidths of a few nm and RF bandwidths of 1 MHz. For each of the traces shown in Fig. 2 and 631 nm, 704 nm, 716 nm and 900 nm, we obtain the integrated noise values, as shown in Fig. 4. Since one comb mode is locked to the reference point at 657 nm, the noise at this lock point is measured by integrating the PLL error signals of $f_{0}$ and $f_{\rm beat}$ . A linear trend, as predicated by Eqn. 3, is clearly evident [because the integrated noise $\sim[S_{\Phi,m,q}(f)]^{1/2}$ is proportional to $(r_{n}-r_{q})$]. A synthesizer was used to mix the lock signals to DC for the phase noise measurements and its noise level is below the measured noise levels shown here. The good agreement between the predicted and measured values demonstrate that the frequency comb equation can predict the integrated phase noise dynamics across the entire spectral range of the frequency comb. These results were reproducible even when measurements are separated by several months. Data for shorter wavelengths ($<$621 nm) are limited by the spectral bandwidth of our OFC (which extend down to approximately 600 nm). As seen in Fig. 4, the phase noise at the lock point is 0.9 rad and increases by approximately 0.45 rad at 800 nm (meaning a culmulative phase noise value of approximately 1.35 rad). The 2f-to-3f technique which measures the offset frequency and locks this RF frequency, uses two vastly different spectral regions to derive a feedback signal (namely a $\sim$2 nm spectral region around both 600 nm and 900 nm). The phase noise from this lock point ($f_{0}$ ) contributes experimentally to the optical phase noise between comb modes, (see the discussion of Fig. 2). That is, when we use the frequency comb equation (Eqn. 1) to predict the phase noise (Eqn. 3), as one moves away from the lock-point, we observe increased phase-noise which is well matched to the prediction, as shown in Fig. 4. We see the residual phase coherence is maintained up to $\sim$30 nm away from the lock point, however we can still predict the scaling of the noise with Eqn. 3 hundreds of nm away. This demonstrates that the frequency comb equation can also accurately predict the phase-noise. Certainly, to improve phase coherence one could use high performance phase-locks, with broadband widths and fast locking actuator feedback, such as acousto-optic modulators Koke et al. (2010). However, for applications of the frequency comb where two vastly different spectral lines are used, such as in the case of comparing optical clocks Rosenband et al. (2008), averaging the measured beat with the comb produces a highly frequency stable signal Rosenband et al. (2008), notwithstanding the lack of phase coherence across the comb. Since the scaling is predicted to be independent of the lock points (that is $f_{0}$ and $f_{\rm beat}$ ), a future study could investigate the scaling with respect to the lock point. Figure 4: Scaling of the integrated (from 1 Hz to 1 MHz) phase-noise away from +243 nm (-124 THz) to -36 nm (+26 THz) from the optical lock point at 657 nm (where the lock-point phase noise of 0.9 radians is subtracted from all the data). The error bars are not visible on the scale shown. The line is a least- squares fit to the data. In summary, we have measured the scaling of the phase noise away from the comb’s optical lock point. We observe that the majority of the measured phase noise is attributable to technical noise associated with the comb’s pump source, servo system gain-bandwidths and environmental effects, which are all most significant below 1 MHz. Yet, by implementing a scheme whereby we measure the difference in phase noise between two frequency comb’s locked to a common reference, we are able to measure the optical phase noise dynamics over the comb bandwidth of almost 300 nm. The results do not show any fundamental limit to achieving shot noise performance over a broad range of the measured Fourier frequencies. Finally, we have shown that the standard frequency comb equation ($\nu_{n}=nf_{\rm rep}+f_{0}$) can predict not only the frequency modes of the comb but the phase noise scaling as well and we are able to measure the optical phase noise dynamics over the comb bandwidth of almost 300 nm. footnote: This work is based on experiments performed in the Time and Frequency Division at the National Institute of Standards and Technology, Boulder, CO. This work is a contribution of the US government and is not subject to copyright in the US. ## References * Bartels et al. (2004) Bartels, A., Oates, C.W., Hollberg, L., Diddams, S.A., 2004\. Stabilization of femtosecond laser frequency combs with subhertz residual linewidths. Opt. Lett. 29, 1081\. * Benkler et al. (2005) Benkler, E., Telle, H.R., Zach, A., Tauser, F., 2005\. Circumvention of noise contributions in fiber laser based frequency combs. Opt. Exp. 13, 5662\. * Cingoz et al. (2012) Cingoz, A., Yost, D.C., Allison, T.K., Ruehl, A., Fermann, M.E., Hartl, I., Ye, J., 2012. Direct frequency comb spectroscopy in the extreme ultraviolet. Nat. 68, 482. * Coddington et al. (2007) Coddington, I., Swann, W.C., Lorini, L., Bergquist, J.C., Coq, Y.L., Oates, C.W., Quraishi, Q., Feder, K.S., Nicholsen, J.W., Westbrook, P.S., Diddams, S.A., Newbury, J.R., 2007\. Coherent optical link over hundreds of metres and hundreds of terahertz with subfemtosecond timing jitter. Nat. Photonics 1, 283\. * Diddams et al. (2001) Diddams, S.A., Udem, T., Bergquist, J.C., Curtis, E.A., Drullinger, R.E., Hollberg, L., Itano, W.M., Lee, W.D., Oates, C.W., Vogel, K.R., Wineland, D.J., 2001. An optical clock based on a single trapped 199hg+ ion. Science 293, 825\. * Gerginov et al. (2006) Gerginov, V., Calkins, K., Tanner, C.E., McFerran, J.J., Diddams, S.A., Bartels, A., Hollberg, L., 2006. Optical frequency measurements of 6s2s1/2-6p2p1/2 (d1) transitions in 133cs and their impact on the fine-structure constant. Phys. Rev. A 73, 032504\. * Holman et al. (2003) Holman, K.W., Jones, R.J., Marian, A., Cundiff, S.T., Ye, J., 2003. Intensity-related dynamics of femtosecond frequency combs. Opt. Lett. 28, 851\. * Koke et al. (2010) Koke, S., Grebing, C., Fei, H., Anderson, A., Assion, A., Steinmeyer, G., 2010\. Direct frequency comb synthesis with arbitrary offset and shot-noise-limited phase noise. Nat. Photon. 4, 462\. * Ma et al. (2004) Ma, L.S., Bi, Z., Bartels, A., Robertsson, L., Zucco, M., Windeler, R.S., Wilpers, G., Oates, C., Hollberg, L., Diddams, S.A., 2004. Optical frequency synthesis and comparison with uncertainty at the 10${}^{-}19$ level. Science 303, 1843\. * Newbury (2011) Newbury, N., 2011. Searching for applications with a fine-tooth comb. Nat. Photon. 5, 186\. * Newbury and Swann (2007) Newbury, N.R., Swann, W.C., 2007\. Low-noise fiber-laser frequency combs. J. Opt. Soc. B 24, 1756\. * Paschotta et al. (2006) Paschotta, R., Schlatter, A., Zeller, S.C., Telle, H.R., Keller, U., 2006. Optical phase noise and carrier-envelope offset noise of mode-locked lasers. Appl. Phys. B 82, 265\. * Quraishi (Ph.D. dissertation, University of Colorado, 2007) Quraishi, Q., Ph.D. dissertation, University of Colorado, 2007. Optical Frequency Combs: Properties and Applications. * Quraishi et al. (2005) Quraishi, Q., Griebel, M., Kleine-Ostmann, T., Bratschitsch, R., 2005\. Generation of phase-locked and tunable continuous-wave radiation in the terahertz regime. Opt. Lett. 30, 3231\. * Ramond et al. (2002) Ramond, T.M., Diddams, S.A., Hollberg, L., Bartels, A., 2002\. Phase-coherent link from optical to microwave frequencies by means of the broadband continuum from a 1-ghz ti:sapphire femtosecond oscillator. Opt. Lett. 27, 1842\. * Rosenband et al. (2008) Rosenband, T., Hume, D.B., Schmidt, P.O., Chou, C.W., Brusch, A., Lorini, L., Oskay, W.H., Drullinger, R.E., Fortier, T.M., Stalnaker, J.E., Diddams, S.A., Swann, W.C., Newbury, N.R., Itano, W.M., Wineland, D.J., Bergquist, J.C., 2008\. Frequency ratio of al+ and hg+ single-ion optical clocks; metrology at the 17th decimal place. Science 319, 1808\. * Schibli et al. (2008) Schibli, T.R., Hartl, I., Yost, D.C., Martin, M.J., Marcinkevicius, A., Fermann, M.E., Ye, J., 2008. Optical frequency comb with submillihertz linewidth and more than 10 w average power. Nat. Photon. 2, 355\. * Schlatter et al. (2007) Schlatter, Z., Zeller, S.C., Paschotta, R., Keller, U., 2007\. Simultaneous measurement of the phase noise on all optical modes of a mode-locked laser. Appl. Phys. B 88, 385\. * Stenger et al. (2002) Stenger, J., Schnatz, H., Tamm, C., Telle, H.R., 2002\. Ultraprecise measurement of optical frequency ratios. Phys. Rev. Lett. 88, 073601\. * Udem et al. (2002) Udem, T., Holzwarth, R., Hänsch, T.W., 2002. Optical frequency metrology. Nature 416, 233\.	arxiv-papers	2014-01-07T23:21:09	2024-09-04T02:49:56.438611	{ "license": "Public Domain", "authors": "Qudsia Quraishi and Scott Diddams and Leo Hollberg", "submitter": "Qudsia Quraishi", "url": "https://arxiv.org/abs/1401.1542" }
1401.1643	# Primordial Black Hole Clusters and their Evolution M.Yu. Khlopov Centre for Cosmoparticle Physics ”Cosmion” National Research Nuclear University ”Moscow Engineering Physics Institute”, 115409 Moscow, Russia APC laboratory 10, rue Alice Domon et Léonie Duquet 75205 Paris Cedex 13, France [email protected] N.A. Chasnikov National Research Nuclear University ”Moscow Engineering Physics Institute”, 115409 Moscow, Russia [email protected] ###### Abstract A possibility of pregalactic seeds of the Active Galactic Nuclei can be a nontrivial cosmological consequence of particle theory. Such seeds can appear as Primordial Black Hole (PBH) clusters, formed in the succession of phase transitions with spontaneous and then manifest breaking of the global U(1) symmetry. If the first phase transition takes place at the inflationary stage, a set of massive closed walls may be formed at the second phase transition and the collapse of these closed walls can result in formation of PBH clusters. We present the results of our studies of the evolution of such PBH Clusters. ## 1 Introduction Primordial Black Holes (PBHs) are a very sensitive cosmological probe for physics phenomena occurring in the early Universe. They could be formed by many different mechanisms, reflecting the fundamental structure of particle theory and nonhomogeneity of very early Universe. Here after a brief review of mechanisms of PBH formation we consider a nontrivial possibility of clusters of massive PBHs. The evolution of such clusters can provide pregalactic seeds of Active Galactic Nuclei (AGN) and we discuss various aspects of this evolution in the present paper. ## 2 PBH Formation Primordial Black Holes can be formed in many different ways [1, 2], such as initial density inhomogeneities, first order and non-equilibrium second order phase transitions, etc. Let us give a brief review of these possibilities. ### 2.1 PHB formation in initial density inhomogeneities A probability for fluctuation of 1 for metric fluctuations distributed according to Gaussian law with dispersion $\left\langle\delta^{2}\right\rangle\ll 1$ is determined by exponentially small tail of high amplitude part of this distribution. In non-Gaussian fluctuations, this process can be more suppressed [3]. In the space, described by $p=\gamma\epsilon,0\leq\gamma\leq 1$ (1) equation of state a probability to form black hole from fluctuation within cosmological horizon is given by [4, 5] $W_{PBH}=e^{-\frac{\gamma^{2}}{2\left\langle\delta^{2}\right\rangle}}$ (2) It provides exponential sensitivity of PBH spectrum to softening of equation of state in early Universe ($\gamma\rightarrow 0$) or to increase of ultraviolet part of spectrum of density fluctuations ($\delta^{2}\rightarrow 1$). These phenomena can appear as cosmological consequence of particle theory (see [4, 5] for review of this and some other mechanisms of PBH formation and for references). ### 2.2 PBH from non-equilibrium second order phase transition The mechanism of PBH formation in the non-equilibrium second order phase transition is of special interest, since it can provide formation of massive and even Supermassive PBHs. PBHs are produced in this mechanism by self- collapsing of closed domain walls. If there are two vacuum states of a system, there are two possibilities to populate that states in the early Universe: under the usual circumstances of thermal phase transition the Universe contains both states populated with equal probability. The other possibility is beyond the pure thermodynamical equilibrium condition, when the two vacuum states are populated with islands of the less probable vacuum, surrounded by the sea of another, more preferable, vacuum. It is necessary to redefine effectively the correlation length of the scalar field that drives a phase transition and consequently the formation of topological defects and the only necessary ingredient for that is the existence of an effectively flat direction(s), along which the scalar potential vanishes during inflation. The background deSitter fluctuations of such effectively massless scalar field could provide non-equilibrium redefinition of correlation length and give rise to the islands of one vacuum in the sea of another one. In spite of such redefinition the phase transition itself takes place deeply in the Friedman- Robertson-Walker (FRW) epoch. After the phase transition two vacua are separated by a wall, and such a closed wall, separating the island with the less probable vacuum, can be very large. At some moment after crossing horizon the walls start shrinking due to surface tension. As a result, if the wall does not release the significant fraction of its energy in the form of outward scalar waves, almost the whole energy of such closed wall can be concentrated in a small volume within its gravitational radius what is the necessary condition for PBH formation. The mass spectrum of the PBHs which can be created by such a way depends on the scalar field potential which parameterizes the flat direction during inflation and triggers the phase transition at the FRW stage. We consider the Universe that, due to the existence of an inflaton, goes through a period of inflation and then settles down to the standard FRW geometry. Then we introduce a complex scalar field $\varphi$, not the inflaton, with a large radial mass $\sqrt{\lambda f}>H_{i}$ that has got Mexican hat potential $V(\varphi)=\lambda\left(\left\|\varphi\right\|^{2}-\frac{f^{2}}{2}\right),$ (3) which provides the U(1) symmetry spontaneous breaking in the period of inflation, corresponding to the scales of the modern cosmological horizon. Therefore we deal only with the phase of that complex field $\theta=\frac{\varphi}{f}$, which parameterizes potential $V=\Lambda^{4}\left(1-cos\frac{\phi}{f}\right)$ (4) Under this condition we come to the conclusion, that the correlation length of second order phase transition with spontaneously broken U(1) symmetry exceeds the present cosmological horizon, and all global U(1) strings are beyond our horizon. If we assume $m\ll H_{i}$ then this implies that during inflation the potential energy of field $\varphi$ is much smaller than the cosmological friction term what justifies neglecting the potential until the Universe goes deeply into the FRW phase. During inflation and long afterward, $H_{i}$ is very large (by assumption) compared to the potential (2). It follows that we can drop the gradient term in the equation of motion [6] $\ddot{\theta}+3H\dot{\theta}+\frac{dV}{d\theta}=0$ (5) and resulting equation is solved by $\theta_{0}=\theta_{Nmax}$ , where $\theta_{Nmax}$ is an arbitrary constant. In the standard assumption, our present horizon has been nucleated at the $N_{max}$ e-folds before the end of inflationary epoch, being embedded in an enormous inflation horizon, created by exponential blow up of a single casual horizon. It follows that $\theta_{Nmax}$ will be the same over the inter inflationary horizon. Without loss of generality, we put $\theta_{Nmax}<\pi$ and considering the quantum fluctuations of the phase $\theta$ at the deSitter background. There are quantum fluctuations produced on the vacuum state of $\theta$ due to the boundary conditions of deSitter space. These fluctuations are sometimes referred to as contribution to the “Hawking temperature” of deSitter space but, there are no true thermal effects. It makes the dynamics of phase $\theta$ strongly non-equilibrium leading to the non-thermal distribution of scales populated with different vacuums in the postinflationary Universe. The average amplitude of such fluctuations for massless field generated during each time interval $H_{{}_{i}}^{-1}$ is $\delta\theta=\frac{H_{i}}{2\pi f}$. The total number of steps during time interval $\Delta t$ is given by $N=H_{i}\Delta t$ \- looks like the one-dimensional Brownian motion. Each domain is characterized by average phase value $\theta_{Nmax-1}=\theta_{Nmax}\pm\delta\theta$. In the half of these domains the phases evolve toward $\pi$ while in the other domains they move toward zero. This process is duplicated in each volume of size $H^{-1}$ during next e-fold. Now at any given scale $l=k^{-1}$ the size of distribution of the phase value $\theta$ can be described by Gaussian law [7] $P(\theta_{l},l)=\frac{1}{\sqrt{2\pi}\sigma_{l}}exp\left(-\frac{(\theta_{Nmax}-\theta_{l})^{2}}{2\sigma_{l}^{2}}\right)$ (6) It is recommended for more information to address the papers [1, 6, 8, 9]. ### 2.3 Initial PBH Mass spectrum Initial mass spectrum $n\left(m,t=0\right)$ depends of parameter $f$ and $\Lambda$. [6] In addition, is a numerical solution. There is another way: one can describe this system by Ito’s equation. One-dimensional Brownian motion in the terms of stochastic equations is an Ornstein-Uhlenbeck process. [10, 11] Using this mathematical framework, one can find the analytical solution of Ito’s equation and obtain initial mass spectrum as an analytical formula $n_{0}=n_{f,\Lambda}\left(m,t=0\right)$. This method is in the process of development. ## 3 Clusters of PHBs According to the $2^{nd}$ order phase transitions mechanism, PBH appears as a sufficiently large cluster, which could collapse into one large Super Massive Black Hole (SMBH) - the Active Galactic Nucleus (AGN) of the future galaxy. ### 3.1 PBH Cluster dynamics By analogy with the star cluster [12] with the difference that the black holes can merge into one, the following processes are significant: * • BH collisions $\rightarrow BH$ merging and as a result $N_{BH}\rightarrow 1$ * • Flying-out BH from cluster $\rightarrow$ reducing the mass of the cluster [13, 14] * • Dynamical friction $\rightarrow$ lower Maxwell distribution [15] Dynamical friction mostly contributes into the $\left\langle\sigma\nu\right\rangle$ of the collision process. [15] The equation describing the dynamics of the BH Cluster is a modification of Smoluchowski (or Kolmogorov-Feller) equation and runs as follows $\begin{split}n{}^{\prime}=\int_{0}^{M}n\left(m-\mu,t\right)\left\langle\sigma\nu\right\rangle_{m-\mu,m}d\mu\\\ -n\left(m,t\right)\left(\int_{0}^{\infty}n\left(\mu,t\right)\left\langle\sigma\nu\right\rangle_{\mu,m}d\mu+\int_{0}^{\infty}n\left(\mu,t\right)\Lambda\left(m,\mu\right)\mu^{2}d\mu\right),\end{split}$ (7) where M is total initial cluster mass, $\Lambda\left(m,\mu\right)$ one can find in [14]. Let us consider numerical solution of that equation for arbitrary initial parameters: Numerical solution of that equation shows, that there is the “trend” to the decrease of BHs with larger masses. However, the numerical solution is indispensable because of the unknown initial parameters $f$ and $\Lambda$ of the model. Exact analytical solution of that equation is overly precise and is a very nontrivial exercise. If a single SMBH is supposed to be the result, one needs to get a solution of that equation as $n(m,t)=\delta(m-m_{SMBH})\chi(t-t_{gen})$ (8) Substituting this partial solution into the equation of the PBH cluster dynamics, one can obtain the timescale of the process and the mass of the resulting SMBH as a functionals of initial conditions: $m_{SMBH}=F\left[n\left(m,t=0\right)\right]$ (9) $t_{gen}=G\left[n(m,t=0)\right]$ (10) The calculated values of $t_{gen}$ and $m_{SMBH}$ can be confronted with the observational data, putting constraints on the fundamental physical scales $f$ and $\Lambda$. ## Acknowledgements We are grateful to the Organizers of XVI Bled Workshop for hospitality and fruitful and creative atmosphere of discussions. ## References * [1] M. Yu. Khlopov: Res.Astron.Astrophys. 10, 495, 2010. * [2] M. Yu. Khlopov: Int.J.Mod.Phys.A, 28, 1330042, 2013. * [3] Bullock, J.S., and Primack, J.R. Phys. Rev. D 55, 7423, arXiv:astro-ph/9611106 * [4] Khlopov, M.Yu.: _Basics of Cosmoparticle physics_ , URSS Publishing, 2011. * [5] Maxim Khlopov, Fundamentals of Cosmic Particle physics, CISP-SPRINGER, Cambridge 2012. * [6] S.G. Rubin, M.Yu. Khlopov, A.S. Sakharov:Primordial Black Holes from Non-Equilibrium Second Order Phase Transition arXiv:hep-ph/0005271 * [7] A. Vilenkin and L. Ford, Phys.Rev D26 A.D. Linde. Phys.Lett 116B, 335 (1982), A. Starobinsky, ibid 117B, 175 (1982) * [8] V. Dokuchaev, Yu. Eroshenko, and S. Rubin _Origin of supermassive black holes_ , arXiv:0709.0070v2 (2007) * [9] M.Yu. Khlopov, S.G. Rubin, AlS. Sakharov _Primordial Structure of Massive Black Hole Clusters_ , arXiv:astro-ph/0401532 * [10] Uhlenbeck, G. E.; Ornstein, L. S: On the theory of Brownian Motion, Phys. Rev. 36: 823–841, 1930 * [11] Frank G. Ball, Ian L. Dryden, Mousa Golalizadeh: Brownian Motion and Ornstein–Uhlenbeck Processes in Planar Shape Space, doi:10.1007/s11009-007-9042-6 * [12] Ayven R. King: _Introduction to classical stellar dynamics_ , URSS Publishing, 2002 * [13] H´enon, M. 1969, A-A 2 151 * [14] J. M. Diederik Kruijssen, arXiv:0910.4579v1 * [15] A. Just, F. M. Khan, P. Berczik, A. Ernst, R. Spurzem _Dynamical friction of massive objects in galactic centres_ , arXiv:1009.2455	arxiv-papers	2014-01-08T10:16:12	2024-09-04T02:49:56.447412	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "M.Yu. Khlopov, N.A. Chasnikov", "submitter": "Nikita Chasnikov", "url": "https://arxiv.org/abs/1401.1643" }
1401.1753		arxiv-papers	2014-01-08T17:05:41	2024-09-04T02:49:56.457909	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Sounak Sadhukhan, Samar Sen Sarma", "submitter": "Sounak Sadhukhan", "url": "https://arxiv.org/abs/1401.1753" }
1401.1766		arxiv-papers	2014-01-08T18:15:31	2024-09-04T02:49:56.462279	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "James Z. Wang, Yuanyuan Zhang, Liang Dong, Lin Li, Pradip K Srimani,\n Philip S. Yu", "submitter": "Yuanyuan Zhang", "url": "https://arxiv.org/abs/1401.1766" }
1401.1949	# Mean value property associated with the Dunkl Laplacian Kods Hassine Department of Mathematics, Faculty of Sciences, University of Monastir 5019 Monastir, Tunisia E-mail: [email protected] ###### Abstract Let $\Delta_{k}$ be the Dunkl Laplacian on $\mathbb{R}^{d}$. The main goal of this paper is to characterize $\Delta_{k}$-harmonic functions by means of a mean value property. ### Keywords : Dunkl Laplacian, Mean value property, $\Delta_{k}$-harmonic functions. ### MSC (2010): 31A05, 51F15, 42B99. ## 1 Introduction Let $R$ be a root system of $\mathbb{R}^{d}$, $d\geq 1$, $k:R\rightarrow\mathbb{R}_{+}$ be a multiplicity function and $W$ be the group generated by the reflections $\sigma_{\alpha}$, $\alpha\in R$. The Dunkl Laplacian is defined in [1] for every function $f\in C^{2}(\mathbb{R}^{d})$ by $\Delta_{k}f(x)=\Delta f(x)+2\sum_{\alpha\in R_{+}}k(\alpha)\left(\frac{<\nabla f(x),\alpha>}{<\alpha,x>}-\frac{\|\alpha\|^{2}}{2}\frac{f(x)-f(\sigma_{\alpha}(x))}{<\alpha,x>^{2}}\right),$ where $\Delta$ and $\nabla$ denote respectively the usual Laplacian and gradient on $\mathbb{R}^{d}$ and $R_{+}$ is a positive subsystem of $R$. Clearly, if $k$ is the identically vanishing function, then $\Delta_{k}$ is reduced to $\Delta$. It is well known that a locally bounded function $f$ on an open subset $D$ of $\mathbb{R}^{d}$, is $\Delta$-harmonic (i.e., $f\in C^{2}(D)$ and $\Delta f=0$ on $D$) if and only if $f(x)=\frac{1}{\sigma_{x,r(S(x,r))}}\int_{S(x,r)}f(y)d\sigma_{x,r}(y),$ for every $x\in D$ and every $r>0$ such that the closed ball $\overline{B}(x,r)$ of center $x$ and radius $r$ is contained in $D$. Here $\sigma_{x,r}$ is the surface area measure on the sphere $S(x,r)$ with center $x$ and radius $r$. H. Mejjaolli and K. Trimèche showed in [4] that every infinitely differentiable function $f$ on $\mathbb{R}^{d}$ is $\Delta_{k}$-harmonic on $\mathbb{R}^{d}$ if and only if for all $x\in\mathbb{R}^{d}$ and $r>0$, $f(x)=\frac{1}{d_{k}}\int_{S(0,1)}\tau_{x}f(ry)\left(\prod_{\alpha\in R_{+}}\|\langle y,\alpha\rangle\|^{2k(\alpha)}\right)d\sigma_{0,1}(y),$ (1) where $d_{k}$ is a normalized constant and $\tau_{x}$ is the Dunkl translation. The main goal of this paper is to investigate a mean value property which characterizes the $\Delta_{k}$-harmonicity of locally bounded functions on an open subset of $\mathbb{R}^{d}$. Let $D\subset\mathbb{R}^{d}$ be an open set which is $W$-invariant. We shall say that a function $f:D\rightarrow\mathbb{R}$ satisfies the mean value property on $D$ if for every $x\in D$ and $r>0$ such that $\overline{B}(x,r)\subset D$, $f(x)=\int_{\mathbb{R}^{d}}f(y)d\sigma_{x,r}^{k}(y),$ where $\sigma_{x,r}^{k}$ (see [7]) is the unique probability measure on $\mathbb{R}^{d}$ such that the right hand side of (1) coincides with $\int_{\mathbb{R}^{d}}f(y)d\sigma_{x,r}^{k}(y).$ We shall prove that every locally bounded function $f$ on $D$ is $\Delta_{k}$-harmonic if and only if it satisfies the mean value property on $D$. To that end, we prove first the equivalence for infinitely differentiable functions on $D$. Next, we show that for a locally bounded function $f$ on $D$, if $f$ satisfies the mean value property then $f$ is infinitely differentiable on $D$. Thus, $f$ is $\Delta_{k}$-harmonic provided it satisfies the mean value property on $D$. To prove the converse, we need only show that if $f$ is $\Delta_{k}$-harmonic then it is infinitely differentiable on $D$. This will be proved once we have shown that the operator $\Delta_{k}$ is hypoelliptic. Thus, by means of convergence property of $\Delta_{k}$-harmonic functions, we prove that the operator $\Delta_{k}$ is hypoelliptic on $D$. Note that the condition that $D$ is $W$-invariant is nearly optimal. In fact, in the case where $d=1$, for every open set $D\subset\mathbb{R}$ which is not $W$-invariant, we can always construct a $\Delta_{k}$-harmonic function function $f$ on $D$ which does not satisfy the mean value property on $D$. ## 2 Preliminaries and some lemmas Let $S(\mathbb{R}^{d})$ be the Schwartz space and $C_{0}(\mathbb{R}^{d})$ be the set of all continuous functions on $\mathbb{R}^{d}$ vanishing at infinity. For every open set $U\subset\mathbb{R}^{d}$, $C(U)$ and $C_{c}(U)$ will denote respectively the set of all continuous functions on $U$ and the set of all continuous functions with compact support on $U$. The set of all bounded functions in $C(U)$ will be denoted by $C_{b}(U)$. For every $\alpha\in\mathbb{R}^{d}\backslash\\{0\\}$, let $H_{\alpha}$ be the hyperplane of $\mathbb{R}^{d}$ orthogonal to $\alpha$ and let $\sigma_{\alpha}$ be the reflection in $H_{\alpha}$, i.e., $\sigma_{\alpha}(x):=x-2\frac{<\alpha,x>}{\|\alpha\|^{2}}\alpha,$ where $\langle x,y\rangle=\sum_{i=1}^{d}x_{i}y_{i}$ and $\|x\|:=\sqrt{\langle x,x\rangle}.$ A finite subset $R$ of $\mathbb{R}^{d}\setminus\\{0\\}$ is called a _root system_ if $R\cap\mathbb{R}\alpha=\\{\pm\alpha\\}$ and $\sigma_{\alpha}(R)=R$ for all $\alpha\in R$. For a given root system $R$, we denote by $W$ the finite group generated by all refections $\sigma_{\alpha},\,\alpha\in R$. A function $k:R\rightarrow\mathbb{R}_{+}$ is called a _multiplicity function_ if it satisfies $k(w\alpha)=k(\alpha)$, for every $w\in W$ and every $\alpha\in R$. Throughout this paper we fix a root system $R$, a multiplicity function $k$ and a _$W$ -invariant open_ subset $D$ of $\mathbb{R}^{d}$, that is, $w(D)\subset D$ for all $w\in W$. Let $w_{k}$ be the _weight function_ on $\mathbb{R}^{d}$ defined by, $w_{k}(x):=\prod_{\alpha\in R_{+}}\|\langle x,\alpha\rangle\|^{2k(\alpha)},$ where $R_{+}:=\\{\alpha\in R:\langle\alpha,\beta\rangle>0\\}$ for some $\beta\in\mathbb{R}^{d}\setminus\cup_{\alpha\in R}H_{\alpha}$. Note that $w_{k}$ is homogeneous of degree $2\gamma$, with $\gamma:=\sum_{\alpha\in R_{+}}k(\alpha).$ From now on, we assume that $\lambda:=\gamma+\frac{d}{2}-1>0.$ The Dunkl Laplacian associated with the root system $R$ and the multiplicity function $k$ is the operator $\Delta_{k}:=\sum_{i=1}^{d}T_{i}^{2},$ where for every $1\leq i\leq d$ and $f\in C^{1}(D)$, $T_{i}f(x):=\partial_{i}f(x)+\sum_{\alpha\in R_{+}}k(\alpha)\alpha_{i}\frac{f(x)-f(\sigma_{\alpha}(x))}{<\alpha,x>},\quad x\in D.$ By [2], there exists a unique linear isomorphism $V_{k}$ from the space of homogenous polynomials of degree $n$ on $\mathbb{R}^{d}$ into it self such that $V_{k}1=1$ and $T_{i}V_{k}=V_{k}\partial_{i}$. Later, it was shown in [9] that the intertwining operator $V_{k}$ has an homeomorphism extension to $C^{\infty}(\mathbb{R}^{d})$. The positivity of $V_{k}$ (see [6]) yields the existence of a family of probability measures $(\mu_{x}^{k})_{x}$ such that for every $x\in\mathbb{R}^{d}$ and every $f\in C^{\infty}(\mathbb{R}^{d})$, $V_{k}f(x)=\int_{\mathbb{R}^{d}}f(y)d\mu_{x}^{k}(y).$ The support of $\mu_{x}^{k}$ is contained in the convex hull $C(x)$ of the orbit of $x$ under the reflection group $W$, $C(x):=co\\{wx,\quad w\in W\\}.$ _The Dunkl kernel_ associated with $R$ and $k$ is defined on $\mathbb{R}^{d}\times\mathbb{R}^{d}$ by $E_{k}(x,y):=\int_{\mathbb{R}^{d}}e^{\langle y,\xi\rangle}d\mu_{x}^{k}(\xi).$ It is well known that $E_{k}$ is positive, symmetric and admits a unique holomorphic extension to $\mathbb{C}^{d}\times\mathbb{C}^{d}$ satisfying $E_{k}(\xi z,\omega)=E_{k}(z,\xi\omega)$ for every $z,\omega\in\mathbb{C}^{d}$ and every $\xi\in\mathbb{C}$. The corresponding _Dunkl transform_ is then given for every bounded measure $\mu$ on $\mathbb{R}^{d}$ by $\mathcal{F}_{D}(\mu)(x):=c_{k}\int_{\mathbb{R}^{d}}E_{k}(-i\xi,x)d\mu(\xi),\quad x\in\mathbb{R}^{d},$ where $c_{k}:=\left(\int_{\mathbb{R}^{d}}e^{-\frac{\|y\|^{2}}{2}}w_{k}(y)dy\right)^{-1}.$ If $\mu=fw_{k}dx$ where $f\in S(\mathbb{R}^{d})$ and $dx$ is the Lebesgue measure on $\mathbb{R}^{d}$, then we shall write $\mathcal{F}_{D}(f)$ instead of $\mathcal{F}_{D}(\mu)$. Note that $\mathcal{F}_{D}$ is injective on the space of all bounded Borel measures $\mathcal{M}_{b}(\mathbb{R}^{d})$ on $\mathbb{R}^{d}$ (see [8]) and is a topological isomorphism from $S(\mathbb{R}^{d})$ into it self (see [3]). For each $x\in\mathbb{R}^{d}$, _the Dunkl translation_ $\tau_{x}$ is defined for every $f\in S(\mathbb{R}^{d})$ by $\tau_{x}f={\mathcal{F}}_{D}^{-1}(E_{k}(ix,\cdot)\mathcal{F}_{D}f),$ where ${\mathcal{F}}_{D}^{-1}$ denotes the inverse of $\mathcal{F}_{D}$ on $S(\mathbb{R}^{d})$. In [10], this translation was extended to $C^{\infty}(\mathbb{R}^{d})$ by $\tau_{x}f(y)=\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}V_{k}^{-1}f(z+\eta)d\mu_{x}^{k}(z)d\mu_{y}^{k}(\eta),$ where $V_{k}^{-1}$ is the inverse of $V_{k}$ on $C^{\infty}(\mathbb{R}^{d})$. It was shown that, for every $f\leavevmode\nobreak\ \in\leavevmode\nobreak\ C^{\infty}(\mathbb{R}^{d})$, the function $u:(x,y)\mapsto\tau_{x}f(y)$ is symmetric, infinitely differentiable on $\mathbb{R}^{d}\times\mathbb{R}^{d}$ and for every $x,y\in\mathbb{R}^{d}$, $(T_{i})_{x}u(x,y)=(T_{i})_{y}u(x,y).$ (2) Moreover, $\tau_{x}f(0)=f(0)$, $T_{i}\tau_{x}f=\tau_{x}T_{i}f$ and $\tau_{x}E_{k}(z,\cdot)(y)=E_{k}(x,z)E_{k}(y,z)$ for every $z\in\mathbb{C}^{d}$. Further, if the support of $f$ (noted $\mbox{ supp\,}f$) is in $B(0,r)$ for some $r>0$, then $\mbox{ supp }\,\tau_{x}f\subset B(0,r+\|x\|)$. According to [7], for each $x\in\mathbb{R}^{d}$ and $r>0$, there exists a unique probability measure $\sigma_{x,r}^{k}$ on $\mathbb{R}^{d}$ which is supported by $\bigcup_{w\in W}\overline{B}(wx,r)\setminus B(0,\|\|x\|-r\|)$ such that for every $f\in C^{\infty}(\mathbb{R}^{d})$, $\frac{1}{d_{k}}\int_{S(0,1)}\tau_{x}f(ry)w_{k}(y)d\sigma_{0,1}(y)=\int_{\mathbb{R}^{d}}f(y)d\sigma^{k}_{x,r}(y).$ (3) where, $d_{k}:=\int_{S(0,1)}w_{k}(y)d\sigma_{0,1}(y)=\frac{1}{c_{k}2^{\lambda}\Gamma(\lambda+1)}.$ ###### Lemma 2.1. Let $\varphi\in S(\mathbb{R}^{d})$ be radial. Then for every Borel set $A\subset\mathbb{R}^{d}$ and every $x\in\mathbb{R}^{d}$, $\int_{A}\tau_{-x}\varphi(y)w_{k}(y)dy=d_{k}\int_{0}^{\infty}{\varphi}(t)t^{2\lambda+1}\left(\int_{A}d\sigma^{k}_{x,t}(y)\right)dt.$ (4) ###### Proof. Let $x\in\mathbb{R}^{d}$ and denote by $\mu(A)$ and $\nu(A)$ the left hand side and the right hand side respectively of (4). Clearly, both $\mu$ and $\nu$ are bounded measures on $\mathbb{R}^{d}$. For every $y\in\mathbb{R}^{d}$, $\displaystyle{\mathcal{F}}_{D}(\mu)(y)$ $\displaystyle=$ $\displaystyle{\mathcal{F}}_{D}(\tau_{-x}\varphi)(y)$ $\displaystyle=$ $\displaystyle{\mathcal{F}}_{D}\varphi(y)E_{k}(-iy,x)$ $\displaystyle=$ $\displaystyle c_{k}E_{k}(-iy,x)\int_{\mathbb{R}^{d}}\varphi(z)E_{k}(-iy,z)w_{k}(z)dz$ $\displaystyle=$ $\displaystyle c_{k}\int_{\mathbb{R}^{d}}\tau_{x}E_{k}(-iy,\cdot)(z)\varphi(z)w_{k}(z)dz$ Using spherical coordinates and (3), we deduce that, $\displaystyle{\mathcal{F}}_{D}(\mu)(y)$ $\displaystyle=$ $\displaystyle c_{k}d_{k}\int_{0}^{\infty}t^{2\lambda+1}{\varphi}(t)\int_{\mathbb{R}^{d}}E_{k}(\xi,-iy)d\sigma_{x,t}^{k}(\xi)dt$ $\displaystyle=$ $\displaystyle{\mathcal{F}}_{D}(\nu)(y).$ Finally, we use the injectivity of $\mathcal{F}_{D}$ on $\mathcal{M}_{b}(\mathbb{R}^{d})$ to conclude.∎ Let $\varphi\in S(\mathbb{R}^{d})$ be a radial function with support in $\overline{B}(0,r)$, $r>0$. We claim that for every $x\in\mathbb{R}^{d}$, $\mbox{ supp\,}\tau_{x}\varphi\subset\bigcup_{w\in W}\overline{B}(wx,r).$ (5) Indeed, let $A$ be a Borel subset of $\mathbb{R}^{d}\setminus\cup_{w\in W}\overline{B}(wx,r)$. Then by (4), $\int_{A}\tau_{-x}\varphi(y)w_{k}(y)dy=d_{k}\int_{0}^{r}{\varphi}(t)t^{2\lambda+1}\left(\int_{A}d\sigma^{k}_{x,t}(y)\right)dt.$ Since for every $0<t<r$, $\mbox{{ supp\,} }\sigma_{x,t}^{k}\subset\cup_{w\in W}\overline{B}(wx,r)$ we deduce that, $\int_{A}\tau_{-x}\varphi(y)w_{k}(y)dy=0.$ This proves the claim. In the sequel we shall write $M_{x,r}(f)=\int_{\mathbb{R}^{d}}f(y)d\sigma_{x,r}^{k}(y),$ whenever the integral makes sense. A Borel function $f:D\rightarrow\mathbb{R}$ is said to satisfy _the mean value property_ on $D$ if $M_{x,r}(f)=f(x)$ for every $x\in\mathbb{R}^{d}$ and $r>0$ such that $\overline{B}(x,r)\subset D$. ###### Lemma 2.2. Let $f$ be a locally bounded function on $D$. If $f$ satisfies the mean value property on D, then $f\in C^{\infty}(D)$. ###### Proof. Without loss of generality we suppose that $f$ is bounded on $D$. Let $\phi$ be the function defined for every $t\in\mathbb{R}$ by $\phi(t):=ce^{-\frac{1}{t}}{\chi}_{]0,\infty[}(t),$ where ${\chi}_{]0,\infty[}$ is the indicator function of $]0,\infty[$ and the constant $c$ is chosen so that $cd_{k}\int_{0}^{1}\phi(1-t^{2})t^{2\lambda+1}dt=1.$ For every $n\geq 1$ we define the function $\phi_{n}$ by, $\phi_{n}(x)=n^{2\lambda+2}\phi(1-n^{2}\|x\|^{2}),\quad x\in\mathbb{R}^{d}.$ (6) Obviously $\phi_{n}$ is infinitely differentiable on $\mathbb{R}^{d}$ with support in $\overline{B}(0,\frac{1}{n})$. Thus, by (5), for every $x\in\mathbb{R}^{d}$, $\mbox{ supp\,}\tau_{x}\phi_{n}\subset\cup_{w\in W}\overline{B}(wx,\frac{1}{n}).$ Let $D_{n}:=\\{x\in D:\;\overline{B}(x,\frac{1}{n})\subset D\\}$ and let $f_{n}(x):=\int_{D}f(y)\tau_{-x}\phi_{n}(y)w_{k}(y)dy,\quad x\in\mathbb{R}^{d}.$ Then $f_{n}\in C^{\infty}(D_{n})$. On the other hand, it follows from (4) that for every $x\in D_{n}$, $\displaystyle f_{n}(x)$ $\displaystyle=$ $\displaystyle d_{k}\int_{0}^{\frac{1}{n}}{\phi_{n}}(t)t^{2\lambda+1}M_{x,t}(f)dt=f(x).$ Hence $f\in C^{\infty}(D_{n})$ and consequently $f\in C^{\infty}(D)$ as desired.∎ ## 3 Main result $D$ will always denotes a $W$-invariant open subset of $\mathbb{R}^{d}$. Our main result is the following: ###### Theorem 3.1. Let $f$ be a locally bounded function on D. The following statements are equivalent: (a) $f\in C^{2}(D)\mbox{ and }\Delta_{k}f=0\mbox{ on }D$. (b) $M_{x,r}(f)=f(x)$ for every $x\in D$ and $r>0$ such that $\overline{B}(x,r)\subset D$ . The following proposition shows the equivalence between (a) and (b) whenever $f$ is infinitely differentiable on $D$. First, let us recall the Green formula associated with the Dunkl Laplacian (see [4]): For every $f\in C^{2}(\overline{B}(0,t))$, $t>0$, $\int_{B(0,t)}\Delta_{k}f(y)w_{k}(y)dy=\int_{S(0,t)}\frac{\partial}{\partial n}f(y)w_{k}(y)d\sigma_{0,t}(y),$ (7) where $\frac{\partial}{\partial n}$ is the partial derivation operator in the direction of the exterior unit normal. ###### Proposition 3.2. Assume that $f$ is infinitely differentiable on $D$. Then $f$ is $\Delta_{k}$-harmonic on $D$ if and only if $f$ satisfies the mean value property on $D$. ###### Proof. Let $x\in D$ and $r>0$ such that $\overline{B}(x,r)\subset D$. We claim that $t\leavevmode\nobreak\ \mapsto\leavevmode\nobreak\ M_{x,t}(f)$ is derivable on $]0,r[$ and for every $t\in]0,r[$, $\frac{d}{dt}M_{x,t}(f)=\frac{1}{t^{2\lambda+1}}\int_{0}^{t}s^{2\lambda+1}M_{x,s}(\Delta_{k}f)ds.$ (8) Indeed, since for every $s\in]0,r[$, the support of $\sigma_{x,s}^{k}$ is contained in $\cup_{w\in W}B(wx,r),$ it suffices to prove (8) replacing $f$ by a function $h\in C^{\infty}(\mathbb{R}^{d})$ such that $h=f\mbox{ on }\cup_{w\in W}B(wx,r).$ It is easily seen from (3) that for every $t\in]0,r[$, $\displaystyle\frac{d}{dt}M_{x,t}(h)$ $\displaystyle=$ $\displaystyle\frac{1}{d_{k}}\int_{S(0,1)}\langle\nabla(\tau_{x}h)(ty),y\rangle w_{k}(y)d\sigma_{0,1}(y)$ $\displaystyle=$ $\displaystyle\frac{1}{d_{k}t^{2\lambda+1}}\int_{S(0,t)}\langle\nabla(\tau_{x}h)(u),\frac{u}{t}\rangle w_{k}(u)d\sigma_{0,t}(u)$ $\displaystyle=$ $\displaystyle\frac{1}{d_{k}t^{2\lambda+1}}\int_{S(0,t)}\frac{\partial}{\partial n}(\tau_{x}h)(u)w_{k}(u)d\sigma_{0,t}(u).$ Therefore, by the Green formula (7) and the fact that $\Delta_{k}\tau_{x}=\tau_{x}\Delta_{k}$, $\displaystyle\frac{d}{dt}M_{x,t}(h)$ $\displaystyle=$ $\displaystyle\frac{1}{d_{k}t^{2\lambda+1}}\int_{B(0,t)}\tau_{x}(\Delta_{k}h)(u)w_{k}(u)du.$ Hence, using spherical coordinates we deduce that, $\frac{d}{dt}M_{x,t}(h)=\frac{1}{t^{2\lambda+1}}\int_{0}^{t}s^{2\lambda+1}M_{x,s}(\Delta_{k}h)ds.$ Thus the claim is proved. Now assume that $\Delta_{k}f=0$ on $D$. Then for all $t\in]0,r[$, $\frac{d}{dt}M_{x,t}(f)=0$, by (8). This yields that $M_{x,t}(f)=\lim_{s\rightarrow 0}M_{x,s}(f)$. On the other hand, it is known from [7] that the map $(x,s)\mapsto\sigma_{x,s}^{k}$ is continuous with respect to the weak topology on $\mathcal{M}_{b}(\mathbb{R}^{d})$. Thus, $\lim_{s\rightarrow 0}M_{x,s}(f)=f(x).$ (9) Whence $M_{x,t}(f)=f(x)$ which yields the necessity . Conversely, assume that $f$ satisfies the mean value property on $D$. Then, using (8) we deduce that $M_{x,t}(\Delta_{k}f)=0$ for all $t\in]0,r[$. Letting $t$ tend to 0 we obtain that $\Delta_{k}f(x)=0$.∎ We then conclude, in virtue of Lemma 2.2, that every locally bounded function $f$ on $D$ which satisfies the mean value property on $D$ is necessarily $\Delta_{k}$-harmonic on $D$. The converse statement will be proved in the remainder of this section. ###### Lemma 3.3. Let $(h_{n})_{n\geq 1}\subset C^{\infty}(D)$ be a locally uniformly bounded sequence of $\Delta_{k}$-harmonic functions on $D$ with pointwise limit $h$. Then $h\leavevmode\nobreak\ \in\leavevmode\nobreak\ C^{\infty}(D)$ and $\Delta_{k}h=0$ on $D$. ###### Proof. Let $x\in D$ and let $r>0$ such that $\overline{B}(x,r)\subset D$. Since for every $n\geq 1$ the function $h_{n}$ is $\Delta_{k}$-harmonic on $D$, it follows from Proposition 3.2 that, $h_{n}(x)=\int_{\mathbb{R}^{d}}h_{n}(y)d\sigma_{x,r}^{k}(y).$ Applying the dominated convergence theorem, we get $h(x)=M_{x,r}(h)$. Whence $h$ satisfies the mean value property on $D$ which finishes the proof, by Lemma 2.2 and Proposition 3.2.∎ Let $g_{k}$ be _the fundamental solution of the Dunkl Laplacian_. That is, for every $\varphi\in C_{c}^{\infty}(\mathbb{R}^{d})$, $\int_{\mathbb{R}^{d}}g_{k}(y)\Delta_{k}\varphi(y)w_{k}(y)dy=-\varphi(0).$ (10) It is well known from [4] that, $g_{k}(y)=c_{k}\Gamma(\lambda)2^{\lambda-1}\|y\|^{-2\lambda}.$ (11) ###### Theorem 3.4. Let $h\in C(D)$ and $f\in C^{\infty}(D)$. Assume that for every $\varphi\leavevmode\nobreak\ \in\leavevmode\nobreak\ C_{c}^{\infty}(D)$, $\int_{D}h(x)\Delta_{k}\varphi(x)w_{k}(x)dx=\int_{D}f(x)\varphi(x)w_{k}(x)dx.$ Then $h\in C^{\infty}(D)$. ###### Proof. It suffices to prove that $h\in C^{\infty}(U)$, for every $W$\- invariant open set $U$ such that $\overline{U}\subset D$. Step 1. Assume first that $f=0$ on $D$. Choose $n_{0}\geq 1$ such that for every $x\in U$, $\overline{B}(x,\frac{1}{n_{0}})\subset D$. For every $n\geq n_{0}$, let $\phi_{n}$ be as in (6). Then, the function $h_{n}$ defined on $U$ by $h_{n}(x):=\int_{D}h(y)\tau_{-x}\phi_{n}(y)w_{k}(y)dy,$ is infinitely differentiable on $U$ and by (2) for every $x\in U$, $\Delta_{k}h_{n}=\int_{D}h(y)\Delta_{k}(\tau_{-x}\varphi_{n})(y)w_{k}(y)dy.$ On the other hand, it follows from (4) that for every $x\in U$, $\displaystyle h_{n}(x)$ $\displaystyle=$ $\displaystyle d_{k}\int_{0}^{\frac{1}{n}}{\phi_{n}}(t)t^{2\lambda+1}M_{x,t}(h)dt$ $\displaystyle=$ $\displaystyle cd_{k}\int_{0}^{1}\phi(1-u^{2})u^{2\lambda+1}M_{x,\frac{u}{n}}(h)du.$ This yields that $(h_{n})_{n\geq n_{0}}$ is uniformly bounded on $U$ and converges pointwise to $h$ on $U$. Hence, in view of Lemma 3.3, $h\in C^{\infty}(U)$ and $\Delta_{k}h=0$ on $U$. Step 2. We now turn to the general case where $f$ is not trivial. Let $v\leavevmode\nobreak\ \in\leavevmode\nobreak\ C^{\infty}_{c}(\mathbb{R}^{d})$ such that $v=f$ on $U$ and define $\psi$ on $\mathbb{R}^{d}$ by $\psi(x):=\int_{\mathbb{R}^{d}}g_{k}(y)\tau_{x}v(y)w_{k}(y)dy,$ where $g_{k}$ is given by (11). Using spherical coordinates, it easily seen that the function $g_{k}w_{k}$ is locally Lebesgue integrable on $\mathbb{R}^{d}$. Thus, $\psi\leavevmode\nobreak\ \in\leavevmode\nobreak\ C^{\infty}(\mathbb{R}^{d})$. Furthermore, it follows from (2) and (10) that $\Delta_{k}\psi=-f$ on $U$. Then, for every $\varphi\in C^{\infty}_{c}(U)$, $\int_{\mathbb{R}^{d}}(h(x)+\psi(x))\Delta_{k}\varphi(x)w_{k}(x)dx=\int_{\mathbb{R}^{d}}(f(x)+\Delta_{k}\psi(x))\varphi(x)w_{k}(x)dx=0.$ Whence, the first step yields that $h+\psi$ is infinitely differentiable on $U$ which finishes the proof.∎ We note that the previous theorem was already proved by H. Mejjaolli and K. Trimèche [5] using Sobolev spaces associated with the Dunkl operators. Proof of Theorem 3.1 Statement $\it{(a)}$ follows from $\it{(b)}$ by means of Lemma 2.2 and Proposition 3.2. Assume now that $\it{(a)}$ holds. Then, by [dunkl3], for every $\varphi\leavevmode\nobreak\ \in\leavevmode\nobreak\ C_{c}^{\infty}(D)$, $\int_{D}f(x)\Delta_{k}\varphi(x)w_{k}(x)dx=\int_{D}\Delta_{k}f(x)\varphi(x)w_{k}(x)dx=0.$ Use now Theorem 3.4 and Proposition 3.2 to finish the proof. $\Box$ In the following we shall give a counterexample proving that Theorem 3.1 does not hold true if the open set $D$ is not $W$-invariant. To that end, let $d=1$ and consider the root system $R=\leavevmode\nobreak\ \\{\pm\sqrt{2}\\}$. Then, the corresponding reflection group is given by $W=\\{\pm id_{\mathbb{R}}\\}$. Therfore, an open set $U\subset\mathbb{R}$ is $W$-invariant if and only if it is symmetric. ###### Proposition 3.5. For every non symmetric open set $U\subset\mathbb{R}$ there exists a function $h:\mathbb{R}\rightarrow\mathbb{R}$ which is $\Delta_{k}$-harmonic on $U$ but does not satisfy the mean value property on $U$. ###### Proof. To abbreviate the notation we write $I_{x,r}:=]x-r,x+r[$. Let $x\in U$ and $r>0$ such that $\overline{I}_{x,r}\subset U$ and $\overline{I}_{-x,r}\cap\overline{U}=\emptyset$. Choose $f\in C_{c}^{\infty}(\mathbb{R}^{d})$ such that $f=-1$ on $\overline{I}_{-x,r}$ and $f=0$ on $U$. Since $g_{k}w_{k}$ is locally Lebesgue integrable, we deduce that the function $h$ defined on $\mathbb{R}$ by $h(z)=\int_{\mathbb{R}}g_{k}(y)\tau_{z}f(y)w_{k}(y)dy,$ is infinitely differentiable on $\mathbb{R}$. Moreover, by (10), for every $z\in\mathbb{R}$, $\Delta_{k}h(z)=-\tau_{z}f(0)=-f(z).$ Hence, $\Delta_{k}h=0$ on $U$. On the other hand, for every $t\in]0,r[$, $M_{x,t}(\Delta_{k}h)=-\int_{\mathbb{R}}f(y)d\sigma^{k}_{x,t}(y)=\sigma_{x,t}^{k}(\overline{I}_{-x,t}).$ Moreover, it follows from [7, Remarks 4.2] that $\mbox{ supp\, }\sigma_{x,t}^{k}=\overline{I}_{x,t}\cup\overline{I}_{-x,t}.$ Thus $\sigma_{x,t}^{k}(\overline{I}_{-x,t})>0$ and consequently $M_{x,t}(\Delta_{k}h)>0$. Whence, by (8) the function $t\mapsto\frac{d}{dt}M_{x,t}(h)$ is positive on $]0,r[$. Hence, $M_{x,t}(h)\neq h(x)$ for every $t\in]0,r[$, which means that $h$ does not satisfy the mean value property on $U$.∎ ## References * [1] C.F. Dunkl, Differential-difference operators associated to reflection groups. Trans. Amer. Math. Soc. 311(1989) 167-183. * [2] C.F. Dunkl, Integrals kernels with reflection group invariance, Canad. J. Math, 43 (1991) 1213-1227. * [3] M.F. de Jeu, The Dunkl transform. Invent. Math, 113 (1993) 147-162. * [4] H. Mejjaoli, K. Trimèche, On a mean value property associated with the dunkl Laplacian operator and applications. Integral transform Spec. Funct., 12 (2001) 279-302. * [5] H. Mejjaoli, K. Trimèche, Hypoellipticity and hypoanalyticity of the Dunkl Laplacian operator. Integral transforms Spec. Funct., 15 (2004) 523-548. * [6] M. Rösler, Positivity of Dunkl’s intertwining operator. Duke Math. J. 98 (1999) 445-463. * [7] M. Rösler, A positive radial product formula for Dunkl kernel. Trans. Amer. Math. Soc. 355 (2003) 2413-2438. * [8] M. Rösler, M. Voit, Markov processes related with Dunkl operators. Advances in Applied Mathematics. 21 (1998) 575-643 . * [9] K. Trimèche, The Dunkl intertwining operator on spaces of functions and distributions and integral representation of its dual. Integral Transforms Spec. Funct., 12 (2001) 349-374. * [10] K. Trimèche, Paley-Wiener theorems for the Dunkl transform and Dunkl translation operators. Integral Transforms Spec. Funct., 13 (2002) 17-38.	arxiv-papers	2014-01-09T10:44:49	2024-09-04T02:49:56.473704	{ "license": "Public Domain", "authors": "Kods Hassine", "submitter": "Khalifa El Mabrouk", "url": "https://arxiv.org/abs/1401.1949" }
1401.2054	Zhiyong Zhang, Kaifeng Jiang, Haiyan LiuIn-Sue Oh University of Notre DameFox School of Business, Temple University Correspondance should be sent to Zhiyong Zhang, 118 Haggar Hall, Univeristy of Notre Dame, IN 46556. Email: [email protected]. # Bayesian meta-analysis of correlation coefficients through power prior ###### Abstract To answer the call of introducing more Bayesian techniques to organizational research (e.g., Kruschke, Aguinis, & Joo, 2012; Zyphur & Oswald, 2013), we propose a Bayesian approach for meta-analysis with power prior in this article. The primary purpose of this method is to allow meta-analytic researchers to control the contribution of each individual study to an estimated overall effect size though power prior. This is due to the consideration that not all studies included in a meta-analysis should be viewed as equally reliable, and that by assigning more weights to reliable studies with power prior, researchers may obtain an overall effect size that reflects the population effect size more accurately. We use the relationship between high-performance work systems and financial performance as an example to illustrate how to apply this method to organizational research. We also provide free online software that can be used to conduct Bayesian meta- analysis proposed in this study. Research implications and future directions are discussed. ## 1 INTRODUCTION Meta-analysis is a statistical method of combining findings from multiple studies to get a more comprehensive understanding of the population (Hunter & Schmidt, 2004). A simple way to combine studies is to calculate the weighted average of correlations between two variables (or differences between two treatments) with the sample size being the weight (e.g., Hunter & Schmidt, 2004). In addition to the averaged effect size, the variation of it can also be investigated to see how the effect size changes from one study to another. This method has become increasingly popular in management research in recent years. According to a review by Aguinis, Dalton, Bosco, Pierce, and Dalton (2011), thousands of meta-analyses have been conducted and published in five major management journals from 1982 to 2009. The number of annually published meta-analytically derived effect sizes is also expected to keep growing in the future. Both fixed-effects and random-effects models have been used in meta-analysis (e.g., Field, 2001; Hedges & Olkin, 1985; Hedges & Vevea, 1998; Hunter & Schmidt, 2004). Fixed-effects models assume the population under study is fixed and homogenous and the finding from each study provides an estimate, ideally unbiased or consistent, of the population effect. For example, if one wants to study the relationship between job satisfaction and job performance, he or she may believe the relationship between the two variables is universal in the population. The differences in the reported correlations in identified studies simply result from sampling variation. Random-effects models assume the population is variable and heterogeneous and can show different effects according to the distinct features that characterize it. For example, for different types of measures, research designs, and research samples, the relationship between job satisfaction and job performance can be quite different (Judge, Thoresen, Bono, & Patton, 2001). Therefore, the differences in the reported correlations reflect the heterogeneous effect sizes in the population. Meta-analysis has been conducted within both the frequentist and Bayesian frameworks although arguably meta-analysis can naturally be viewed as a Bayesian method in general. The frequentist methods for meta-analysis can be found in many places such as Hedges and Olkin (1985), Hunter and Schmidt (2004), and Rosenthal (1991). Relatively few studies have discussed Bayesian meta-analysis (e.g., Carlin, 1992; Morris, 1992; Smith, Spiegelhalter, & Thomas, 1995; Steele & Kammeyer-Mueller, 2008), which has been considered as having several advantages, such as “full allowance for all parameter uncertainty in the model, the ability to include other pertinent information that would otherwise be excluded, and the ability to extend the models to accommodate more complex, but frequently occurring, scenarios” (Sutton & Abrams, 2001, p. 277). Traditional meta-analysis, using either the frequentist or Bayesian approach, typically treats each study equivalently. In other words, each study contributes equally to estimated overall effect size after considering the weights proportional to sample sizes. However, in many cases, not all studies included in a meta-analysis should make equal contribution to the overall effect size; treating them equivalently might cause unexpected consequences in meta-analysis. For example, strategic management scholars may be interested in the relationships between financial performance and its antecedents, such as human resource management (HRM) practices (Combs, Liu, Hall, & Ketchen, 2006) and human capital (Crook, Todd, Combs, Woehr, & Ketchen, 2011). Financial performance can be measured objectively using data from archival data or subjectively using survey data. Although both objective and subjective measures are widely adopted in the literature, objective information may reflect a firm’s financial status more accurately than subjective ratings because the latter involves more cognitively demanding assessments and the informants may not always have the best knowledge of the information. Therefore, those using objective measures may provide more reliable information of the relationships between financial performance and other variables than those based on subjective measures. For another example, due to the difficulty of collecting longitudinal data, longitudinal studies often result in a relatively small sample size compared with cross-sectional studies obtaining all information from a single source. Even though longitudinal designs may help avoid common method bias and reduce inflation of correlations (Podsakoff, MacKenzie, Lee, & Podsakoff, 2003), their small sample sizes make them contribute less to the final result. Instead, the cross-sectional studies with inflated relationships may easily dominate the overall effect size because of their large sample sizes. As illustrated in the two examples, treating individual studies equivalently may produce potential misleading results. However, not much attention has been paid to this issue when estimating overall effect size in traditional meta-analysis. To address this research need, this study proposes a Bayesian method for meta-analysis that can control the contribution of each individual study through power prior. As we discuss below, this method can allow meta-analysis researchers more flexibility to estimate overall effect size by specifying power parameters for individual studies. For illustration, the current study focuses on the meta-analysis of sample correlation although the same method can be applied for other effect size measures. In the following, we first demonstrate the use of power prior through a fixed-effects model and then we extend our method to random-effect models and meta-regression. Free online software is introduced to carry out the Bayesian meta-analysis discussed in this study. The use of Bayesian meta- analysis is further demonstrated through a real meta-analysis example. ## 2 BAYESIAN META-ANALYSIS THROUGH POWER PRIOR The proposed method is derived based on the Fisher z-transformation of correlation. Suppose $\rho$ is the population correlation of two variables that follow a bivariate normal distribution. For a given sample correlation $r$ from a sample of $n$ independent subjects, its Fisher z-transformation, denoted by $z$, is defined as $z=\frac{1}{2}\ln\frac{1+r}{1-r}.$ $z$ approximately follows a normal distribution with mean $\frac{1}{2}\ln\frac{1+\rho}{1-\rho}$ and variance $\phi=\frac{1}{n-3}$ (Fisher, 1921). Meta-analysis of correlation concerns the analysis of correlation between two variables when a set of studies regarding the relationship between the two variables are available. Suppose there are $m$ studies that report the sample correlation between two variables. Each study reports a sample correlation $r_{i}$ with the corresponding sample size $n_{i}$. Let $z_{i}=\frac{1}{2}\ln\frac{1+r_{i}}{1-r_{i}}$ denote the Fisher z-transformation of $r_{i}$ and $\zeta_{i}=\frac{1}{2}\ln\frac{1+\rho_{i}}{1-\rho_{i}}$ be the Fisher z-transformation of the population correlation. Then, $z_{i}\sim N(\zeta_{i},\phi_{i})$ with $\phi_{i}=(n_{i}-3)^{-1}$. ### 2.1 Fixed-effects Models We first investigate the situation where the population can be considered as homogeneous and, therefore, a fixed-effects model can be used. In this case, the population correlation is $\zeta_{i}\equiv\zeta=\frac{1}{2}\ln\frac{1+\rho}{1-\rho}$ and $z_{i}\sim N(\zeta,\phi_{i})$. The use of Bayesian methods requires the specification of priors (e.g., Gelman, Carlin, Stern, & Rubin, 2003), which provides a perfect way to conduct meta-analysis. A prior represents information on the population correlation, or its Fisher z-transformation, without any data collection. Although a prior is required, it may consist of “no” information through certain types of prior such as Jeffreys’ prior (e.g., Gill, 2002; Jeffreys, 1946). Suppose the prior for $\zeta$ follows a normal distribution $N(\zeta_{0},\psi_{0})$ where $\zeta_{0}$ and $\psi_{0}$ are pre-determined values. For example, $\zeta$ could have a prior N(0,1), which means a researcher initially believe the mean value of $\zeta$ is 0, corresponding a correlation 0, with variance 1. If little to none information is available, the so-called diffuse prior can be used by specifying a large variance such as $\psi_{0}=10^{8}$. After collecting data, in the framework of meta-analysis, with the availability of a study, one can get a better picture about the population correlation. Bayesian methods provide a way to update the information on the population correlation through Bayes’ Theorem. Let $z_{1}$ denote the new information on the correlation after Fisher z-transformation and $z_{1}\sim N(\zeta,\phi_{1})$. The distribution of the population correlation $\zeta$ by combining the prior and the study is $\displaystyle p(\zeta\|z_{1})$ $\displaystyle=$ $\displaystyle\frac{p(\zeta)p(z_{1}\|\zeta)}{p(z_{1})},$ where $p(\zeta\|z_{1})$ is called the posterior of $\zeta$ after considering $z_{1}$. From Appendix A, we can conclude that the posterior distribution is also a normal distribution $N(\zeta_{1},\psi_{1})$ where $\displaystyle\zeta_{1}$ $\displaystyle=$ $\displaystyle\frac{\frac{1}{\psi_{0}}\zeta_{0}+\frac{1}{\phi_{1}}z_{1}}{\frac{1}{\psi_{0}}+\frac{1}{\phi_{1}}}$ (1) $\displaystyle\psi_{1}$ $\displaystyle=$ $\displaystyle\frac{1}{\frac{1}{\psi_{0}}+\frac{1}{\phi_{1}}}.$ (2) Therefore, the posterior mean $\zeta_{1}$ is the weighted average of prior mean $\zeta_{0}$ and $z_{1}$ where the weights are the inverse of variances of prior and data. If the prior is very accurate, e.g., with a small variance, the prior mean will exert a big effect on the posterior. For an extreme case, if $\psi_{0}=0$, the posterior mean is $\zeta_{0}$, which is also the prior mean. On the other hand, if only little prior information is available, reflected by large variance of prior, the prior mean has little influence on the posterior. For a special case where $\psi_{0}=+\infty$, the posterior mean is $z_{1}$ and therefore, the posterior is fully determined by data. The above analysis assumes that $z_{1}$ is fully reliable or the researcher wants to utilize full information from $z_{1}$. However, if, for practical reason, the information in $z_{1}$ is not accurate enough (e.g., obtained from a flawed research design), it might distort the posterior. In this situation, a researcher might prefer using only partial information from $z_{1}$. Using the power prior idea developed by Ibriham and Chen (2000), we can get the posterior $\displaystyle p(\zeta\|z_{1},\alpha_{1})$ $\displaystyle=$ $\displaystyle\frac{p(\zeta)[p(z_{1}\|\zeta)]^{\alpha_{1}}}{p(z_{1})},$ (3) where $\alpha_{1}$ is a power parameter. Note that if $\alpha_{1}=0$, no information from $z_{1}$ is used while when $\alpha_{1}=1$, full information of $z_{1}$ is used. Partial information of $z_{1}$ can be utilized by setting $\alpha_{1}$ to be a value between 0 and 1. It can be shown (see Appendix B) that the posterior is still a normal distribution with $N(\zeta_{1}^{},\psi_{1}^{})$ where $\displaystyle\zeta_{1}^{}$ $\displaystyle=$ $\displaystyle\frac{\frac{1}{\psi_{0}}\zeta_{0}+\frac{\alpha_{1}}{\phi_{1}}z_{1}}{\frac{1}{\psi_{0}}+\frac{\alpha_{1}}{\phi_{1}}}$ $\displaystyle\psi_{1}^{}$ $\displaystyle=$ $\displaystyle\frac{1}{\frac{1}{\psi_{0}}+\frac{\alpha_{1}}{\phi_{1}}}.$ Again the posterior mean is a weighted average of the prior mean and $z_{1}$. However, note that the weight is different from the previous situation because it is related to the power $\alpha_{1}$. If $\alpha_{1}<1$, then the weight for $z_{1}$ is smaller than the previous results in Equation 1. This means posterior will rely more on the prior. Note that this is equivalent to let $z_{1}\sim N(\zeta,\frac{\phi_{1}}{\alpha_{1}})$. The information is passed through a normal distribution with the same mean but enlarged variance. Suppose without data collection, a researcher’s prior information on $\zeta$ is $N(0,1)$. One study in the literature reported a correlation 0.5 with the sample size 28 and, therefore, $z_{1}=0.549$ with variance 0.04. Table 1 shows the posterior mean and variance for $\zeta$ with power $\alpha_{1}$ ranges from 0 to 1. When $\alpha_{1}=0$, the posterior is the same as the prior. When $\alpha_{1}$ increases from 0.1 to 1, the posterior mean changes towards to $z_{1}$ because more information from $z_{1}$ is included in the posterior. Furthermore, the posterior variance is also becoming smaller. In summary, the use of power $\alpha_{1}$ influences both the posterior mean and posterior variance and can control the contribution of data to the posterior. Table 1: The influence of the selection of power parameters for a single study Data \| \| z-transformation \| Variance ---\|---\|---\|--- $r_{1}=0.5$ \| \| 0.549 \| 0.04 Prior \| \| 0 \| 1 Power \| \| Posterior $\alpha_{1}$ \| \| Mean \| Variance 0 \| \| 0 \| 1 0.1 \| \| 0.392 \| 0.286 0.2 \| \| 0.458 \| 0.167 0.3 \| \| 0.485 \| 0.118 0.4 \| \| 0.499 \| 0.091 0.5 \| \| 0.509 \| 0.074 0.6 \| \| 0.515 \| 0.063 0.7 \| \| 0.520 \| 0.054 0.8 \| \| 0.523 \| 0.048 0.9 \| \| 0.526 \| 0.043 1 \| \| 0.528 \| 0.038 In meta-analysis, data from multiple studies are available. Bayesian methods provide a natural way to combine the data together. For example, suppose we have another study with transformed correlation $z_{2}$ and its variance $\phi_{2}$ as well as the sample size $n_{2}$. Furthermore, the power $\alpha_{2}$ is used when combining this study. We have already obtained the posterior of $\zeta$ with the first study in Equation 3. To get the posterior by combining $z_{2}$, we can simply view the posterior in Equation 3 as a new prior. Then, the posterior of $\zeta$ with both $z_{1}$ and $z_{2}$ is $p(\zeta\|z_{1},z_{2},\alpha_{1},\alpha_{2})=\frac{p(\zeta\|z_{1},\alpha_{1})[p(z_{2}\|\zeta)]^{\alpha_{2}}}{p(z_{2})}.$ From Appendix C, we know posterior distribution is a normal distribution $N(\zeta_{2}^{},\psi_{2}^{})$ where $\displaystyle\zeta_{2}^{}$ $\displaystyle=$ $\displaystyle\frac{\frac{1}{\psi_{0}}\zeta_{0}+\frac{\alpha_{1}}{\phi_{1}}z_{1}+\frac{\alpha_{2}}{\phi_{2}}z_{2}}{\frac{1}{\psi_{0}}+\frac{\alpha_{1}}{\phi_{1}}+\frac{\alpha_{2}}{\phi_{2}}}$ $\displaystyle\psi_{2}^{}$ $\displaystyle=$ $\displaystyle\frac{1}{\frac{1}{\psi_{0}}+\frac{\alpha_{1}}{\phi_{1}}+\frac{\alpha_{2}}{\phi_{2}}}.$ Clearly, the posterior mean is a weighted average of prior and the two studies. More generally, if we have $m$ studies with $z_{i}$, $n_{i}$, and $\alpha_{i}$, the posterior distribution of $\zeta$ is $N(\zeta_{m}^{},\psi_{m}^{})$ with $\displaystyle\zeta_{m}^{}$ $\displaystyle=$ $\displaystyle\frac{\frac{1}{\psi_{0}}\zeta_{0}+\sum_{i=1}^{m}\frac{a_{i}}{\phi_{i}}z_{i}}{\frac{1}{\psi_{0}}+\sum_{i=1}^{m}\frac{a_{i}}{\phi_{i}}}$ $\displaystyle\psi_{m}^{}$ $\displaystyle=$ $\displaystyle\frac{1}{\frac{1}{\psi_{0}}+\sum_{i=1}^{m}\frac{a_{i}}{\phi_{i}}}.$ For illustration, we show the combination of two studies where the first study reported a correlation 0.5 with the sample size 28 and the second study reported a correlation 0 with the sample size 103. Therefore, $z_{1}=0.549$ with variance 0.04 and $z_{2}=0$ with variance 0.01. A diffuse prior $N(0,100)$ is used here so that the effect of prior is minimized. Table 2 presents the posterior mean and variance for the population correlation with different combinations of power for the two studies. First, when no information from the two studies is utilized ($\alpha_{1}=\alpha_{2}=0$), the posterior is just the prior. Second, when only the information of Study 1 is fully used ($\alpha_{1}=1$, $\alpha_{2}=0$), the posterior mean and variance are essentially the same as the Fisher z-transformation and its variance of Study 1 because of the use of the diffuse prior. Similarly, one can solely use the information from Study 2 by setting $\alpha_{1}=0$ and $\alpha_{2}=1$. Third, when the information of the two studies are used fully ($\alpha_{1}=\alpha_{2}=1$), the posterior mean is about 0.110, the weighted average of 0.549 and 0 but leaning towards 0 because the second study has a smaller variance. When setting $\alpha_{1}=\alpha_{2}=0.5$, the posterior mean is still 0.110 but the variance is about 0.016, twice of that when $\alpha_{1}=\alpha_{2}=1$. This is because only partial information is used from the two studies. Similar results can be seen from the table when other combination of power is used. In summary, by controlling the power parameter, one can control the contribution of each study to meta-analysis. Table 2: The influence of the selection of power parameters for combining two studies Data \| \| z-transformation \| Variance ---\|---\|---\|--- $r_{1}=0.5$ \| \| 0.549 \| 0.04 $r_{2}=0$ \| \| 0 \| .01 Prior \| \| 0 \| 100 Power \| \| Posterior $\alpha_{1}$ \| $\alpha_{2}$ \| \| Mean \| Variance 0 \| 0 \| \| 0 \| 100 1 \| 0 \| \| 0.549 \| 0.040 0 \| 1 \| \| 0.000 \| 0.010 0.1 \| 1 \| \| 0.013 \| 0.010 1 \| 0.1 \| \| 0.392 \| 0.029 0.5 \| 0.5 \| \| 0.110 \| 0.016 0.2 \| 1 \| \| 0.026 \| 0.010 1 \| 0.2 \| \| 0.305 \| 0.022 0.2 \| 0.8 \| \| 0.032 \| 0.012 0.8 \| 0.2 \| \| 0.275 \| 0.025 1 \| 1 \| \| 0.110 \| 0.008 ### 2.2 Random-effects Models When the population is not homogeneous, it is not reasonable to assume that $z_{i}$ has the same mean $\zeta$. Therefore, we discuss the random-effects models in the Bayesian framework. A random-effects model can be written as a two-level model, $\left\\{\begin{array}[]{l}z_{i}=\zeta_{i}+e_{i}\\\ \zeta_{i}=\zeta+v_{i}\end{array}\right.$ (4) where $Var(e_{i})=\phi_{i}$ and $Var(v_{i})=\tau$. In the model, each $z_{i}$ has its mean $\zeta_{i}$ and the grand mean of $\zeta_{i}$ is $\zeta$. Based on Fisher z-transformation, $z_{i}\sim N(\zeta_{i},\phi_{i})$. It is often assumed that $v_{i}$ has a normal distribution and, therefore, $\zeta_{i}\sim N(\zeta,\tau)$. For the random-effects model, we have the fixed-effects parameter $\zeta$ and the random-effects parameter $\tau$. The parameter $\tau$ represents the between-study variability. The parameter $\zeta$ can be transformed back to correlation that represents the overall correlation across all studies. In addition, we can also estimate the random effects $\zeta_{i}$, which can be transformed back to correlations for individual studies. As for the fixed-effects models, to estimate model parameters for the random- effects models, we need to specify priors. In this study, the normal prior $N(\zeta_{0},\psi_{0})$ is used for $\zeta$ and the inverse gamma prior $IG(\delta_{0},\gamma_{0})$ is used for $\tau$ with $\zeta_{0}$, $\psi_{0}$, $\delta_{0}$ and $\gamma_{0}$ denoting known constants. In practice, $\zeta_{0}=0$, $\psi_{0}=10^{6}$, $\delta_{0}=10^{-3}$ and $\gamma_{0}=10^{-3}$ are often used to reduce the influence of priors. With the priors, the conditional posteriors for $\zeta$, $\tau$, and $\zeta_{i}$ can be obtained as in Appendix D. Then, the following Gibbs sampling procedure can be used to get a Markov chain for each parameter. Choose a set initial values for $\zeta$ and $\tau$, e.g., $\zeta^{(0)}=0$ and $\tau^{(0)}=1$. Generate $\zeta_{i}^{(1)},i=1,\ldots,m$ from the normal distribution $N\left(\frac{\frac{\zeta^{(0)}}{\tau^{(0)}}+\frac{z_{i}\alpha_{i}}{\phi_{i}}}{\frac{1}{\tau^{(0)}}+\frac{\alpha_{i}}{\phi_{i}}},\frac{1}{\frac{1}{\tau^{(0)}}+\frac{\alpha_{i}}{\phi_{i}}}\right).$ Generate $\tau^{(1)}$ from the inverse Gamma distribution IG($\delta_{0}+m/2,\gamma_{0}+[\sum_{i=1}^{m}(\zeta_{i}^{(1)}-\zeta^{(0)})^{2}]/2$). Generate $\zeta^{(1)}$ from the normal distribution $N\left(\frac{\frac{\sum_{i=1}^{m}\zeta_{i}^{(1)}}{\tau^{(1)}}+\frac{\zeta_{0}}{\psi_{0}}}{\frac{m}{\tau^{(1)}}+\frac{1}{\psi_{0}}},\frac{1}{\frac{m}{\tau^{(1)}}+\frac{1}{\psi_{0}}}\right).$ Let $\zeta^{(0)}=\zeta^{(1)}$ and $\tau^{(0)}=\tau^{(1)}$ and repeat Steps 2-4 to get $\zeta^{(2)}$, $\tau^{(2)}$ and $\zeta_{i}^{(2)},i=1,\ldots,m$. The above algorithm can be repeated for $R$ times to get a Markov chain for $\zeta$, $\tau$, and $\zeta_{i}$. It can be shown that the Markov chains converge to their marginal distributions after a certain period and therefore can be used to infer on the parameters (e.g., Gelman et al, 2003). The period for the Markov chains to converge is called the burn-in period. Suppose the burn-in period is $k$. Then the rest of the Markov chain from $(k+1)$th iteration to the $R$th iteration can be used to get the mean and variance of $\zeta$, $\tau$, and $\zeta_{i}$. Because a research is ultimately interested in the correlation, we can also get the Markov chain for $\rho=\frac{\exp(2\zeta)-1}{\exp(2\zeta)+1}$ and for $\rho_{i}=\frac{\exp(2\zeta_{i})-1}{\exp(2\zeta_{i})+1}$. To illustrate the influence of power parameters on the random-effects meta- analysis, we consider a simple example with three studies that report correlations 0.5, 0 and -0.5 with sample sizes 103, 28 and 103\. The Fisher z-transformed data and their variances are given in Table 3. Table 3 also reports the estimated overall correlation $\rho$ and individual correlation $\rho_{i},i=1,2,3$. When the power is 1 for all three studies, the estimated $\rho$ is approximately 0. Note that the estimated individual population correlations for the first and third studies are smaller than the observed ones. This is called “shrinkage” or “multilevel averaging” effect of multilevel analysis (e.g., Greenland, 2000). The estimated random effects are pulled towards the average effects. If, based on expert opinions or other information, we suspect the reported negative correlation could be because of unreliable study, we might assign it a different weight. For example, if we give the third study a power 0.1, the estimated overall correlation becomes 0.061. Furthermore, if we assign a power 0.01, the overall population becomes 0.215. Therefore, the effect of the observed unreliable negative correlation can be controlled through chosen power parameters. Table 3: The influence of the use of power parameters on random-effects meta-analysis1 Data \| \| z-transformation \| Variance ---\|---\|---\|--- $r_{1}=0.5$ \| \| 0.549 \| 0.01 $r_{2}=0$ \| \| 0 \| 0.04 $r_{3}=-0.5$ \| \| -0.549 \| 0.01 Prior \| \| 0 \| 100 Power \| \| Posterior mean $\alpha_{1}$ \| $\alpha_{2}$ \| $\alpha_{3}$ \| \| $\rho$ \| $\rho_{1}$ \| $\rho_{2}$ \| $\rho_{3}$ 1 \| 1 \| 1 \| \| -0.002 \| 0.482 \| -0.001 \| -0.482 1 \| 1 \| 0.1 \| \| 0.061 \| 0.476 \| 0.022 \| -0.305 1 \| 1 \| 0.01 \| \| 0.215 \| 0.469 \| 0.099 \| 0.099 #### 2.2.1 Selection between fixed-effects and random-effects models The choice of the fixed-effects or random-effects models is often a subjective decision. However, we suggest using two methods to assist such a decision. A random-effects model is only beneficial when there is significant variation in the population effect sizes. Therefore, for the first method, we can test whether the variance of $\zeta_{i}$, $\tau$, is significant. If it is significant, it suggests that a random-effects model is preferred. Otherwise, the fixed-effects model, as a special case of random-effects models, can be used. Some scholars have argued that the significance test may not have enough power to detect variation in population values due to the small number of studies (Hunter & Schmidt, 2004). Therefore, we recommend another method to compare the fixed-effects model and the random-effects model through the deviance information criterion (DIC, Spiegelhalter et al., 2002). If the fixed-effects model has the smaller DIC, it is preferred. Otherwise, the random-effects model is better used. ### 2.3 Meta-regression Models When a random-effects model is suggested, it often indicates possible heterogeneity in the population. Therefore, predictors or covariates can be identified to explain such a heterogeneity. Suppose a set of $p$ covariates are available, denoted by $x_{1},x_{2},\ldots,x_{p}$. Then, a meta-regression model can be constructed as below $\left\\{\begin{array}[]{l}z_{i}=\zeta_{i}+e_{i}\\\ \zeta_{i}=\beta_{1}+\beta_{2}x_{1i}+\cdots+\beta_{p+1}x_{pi}+v_{i}=\mathbf{x}_{i}\bm{\beta}+v_{i}\end{array}\right.,$ (5) where $\bm{\beta}=(\beta_{1},\beta_{2},\ldots,\beta_{p+1})^{\prime}$, $\mathbf{x}_{i}=(1,x_{1i},x_{2i},\ldots,x_{pi})$, and $v_{i}\sim N(0,\tau)$. If a coefficient $\beta_{i}$ is significant, $x_{p}$ is a significant predictor that might be related to the heterogeneity of the population correlation. To estimate $\bm{\beta}$ and $\phi$, we specify the multivariate normal prior for $\bm{\beta}$ as $N(\bm{\zeta}_{0},\mathbf{\Psi}_{0})$ and the inverse Gamma prior $IG(\delta_{0},\gamma_{0})$ for $\tau$. Typically, we use the following hyper-parameters for the priors: $\bm{\zeta}_{0}=\mathbf{0}_{(p+1)\times 1}$, $\mathbf{\Psi}_{0}=10^{6}\mathbf{I}$ with $\mathbf{I}$ denoting a $(p+1)\times(p+1)$ identity matrix, and $\delta_{0}=\gamma_{0}=10^{-3}$. With the prior, the conditional posteriors for $\bm{\beta}$, $\tau$, and $\zeta_{i}$ can be obtained as shown in Appendix E. The conditional posterior distribution of $\tau$ is an inverse Gamma distribution $\tau\|\bm{\beta},\zeta_{i}\sim IG(\delta_{0}+m/2,\gamma_{0}+\sum_{i=1}^{m}(\zeta_{i}-\mathbf{x}_{i}\bm{\beta})^{2}/2)$. The conditional posterior distribution for $\beta$ is still a multivariate normal distribution $N\Bigg{(}\bigg{(}\mathbf{\Psi}_{0}^{-1}+\frac{\mathbf{X}^{\prime}\mathbf{X}}{\tau}\bigg{)}^{-1}\bigg{(}\mathbf{\Psi}_{0}^{-1}\bm{\zeta}_{0}+\frac{\mathbf{X}^{\prime}\mathbf{X}}{\tau}\hat{\bm{\beta}}\bigg{)},(\mathbf{\Psi}_{0}^{-1}+\frac{\mathbf{X}^{\prime}\mathbf{X}}{\tau})^{-1}\Bigg{)}$ where $\hat{\bm{\beta}}$ is the least square estimate of $\bm{\beta}$ such that $\hat{\bm{\beta}}=(\mathbf{X}^{\prime}\mathbf{X})^{-1}\mathbf{X}^{\prime}\bm{\zeta}$ with $\mathbf{X}=(\mathbf{x}_{1},\ldots,\mathbf{x}_{m})^{\prime}$ as the design matrix and $\bm{\zeta}=(\zeta_{1},\zeta_{2},\ldots,\zeta_{m})^{\prime}$. The conditional posterior for $\zeta_{i}$ is $N\left(\frac{\frac{\alpha_{i}z_{i}}{\phi_{i}}+\frac{X_{i}\bm{\beta}}{\tau}}{\frac{\alpha_{i}}{\phi_{i}}+\frac{1}{\tau}},\frac{1}{\frac{\alpha_{i}}{\phi_{i}}+\frac{1}{\tau}}\right).$ With the set of conditional posteriors, the Gibbs sampling algorithm can be used to generate Markov chain for each unknown parameter as for the random- effects meta-analysis. ## 3 SOFTWARE To facilitate the use of Bayesian meta-analysis method through power prior, we developed a free online program that can be accessed with the URL http://webbugs.psychstat.org/modules/metacorr/. The online program can be used within a typical Web browser. It has an interface shown in Figure 1. To use the program, one needs either to upload a new data file or select an existing file. Note names of the existing files are shown in the drop down menu. The existing file has to be a text file in which the data values are separated by one or more white spaces. The first line of the data file will be the variable names, which will be used in the model. Figure 1: The interface of the online software metacorr Next, a user chooses the model to use. For example, the user can choose to use either the random-effects model (default option) or the fixed-effects model. Then, information on the model can be provided. Both the _Correlation_ and _Sample size_ are required for all analysis, which can be specified using the variable names in the data set. For example, if we use “fi” to represent the correlation between financial performance and another variable in the data set, then “fi” should be input in the field of _Correlation_ in the interface. Similarly, “n” is used in the Sample size field because in the data set, “n” is also the variable name for sample size. In addition, a user can also specify the variables for power and covariates used in the model. Finally, one can elect to control Markov chain Monte Carlo (MCMC) method and output of the meta-analysis. For example, the total number of Monte Carlo iteration and the burn-in period can be specified. In the output, one can require the output of the estimates for random effects $\zeta_{i}$, DIC, and diagnostic plots for all model parameters including the random effects. If one checks the option _Email notification_ , an email will be sent to the user once the analysis is completed. ## 4 AN EXAMPLE We use the relationship between high-performance work systems (HPWS) and financial performance as an example to illustrate the use of Bayesian meta- analysis with power prior. HPWS refers to a bundle of human resource management (HRM) practices that are intended to enhance employees’ abilities, motivation, and opportunity to make contribution to organizational effectiveness, including practices such as selective hiring, extensive training, internal promotion, developmental performance appraisal, performance-based compensation, flexible job design, and participation in decision making (Lepak, Liao, Chung, & Harden, 2006). Strategic HRM scholars have devoted considerable effort to studying the influence of HPWS on firm performance in the past three decades and consistently found that the use of HPWS is positively related to employee and firm performance (Paauwe, Wright, & Guest, 2013). Indeed, recent meta-analyses have demonstrated the positive relationships between HPWS and a variety of performance outcomes (Combs et al., 2006; Jiang, Lepak, Hu, & Baer, 2012; Subramony, 2009), including employee outcomes (e.g., human capital, employee motivation), operational outcomes (e.g., productivity, service quality, and innovation), and financial outcomes (e.g., profit, return on assets, and sales growth). The purpose of this study is not to compare the results obtained from Bayesian meta-analysis to those of previous research. Instead, we just use the research on HPWS as an example and specifically focus on the relationship between HPWS and financial performance, which is one of the most important considerations of strategic HRM research. We used several search techniques to identify previous empirical studies that examined the relationship between HPWS and financial performance. First, we searched for published studies in the databases _Business Source Premier_ , _Google Scholar_ , _Web of Science_ , and _PsycINFO_ by combining keywords associated with HPWS (e.g., HRM systems, high-performance work systems, high- commitment HRM practices, and high-involvement HRM practices) and with keywords related to financial performance (e.g., financial performance, profit, return on assets, return on equity, Tobin’s Q, and sales growth). Second, we checked the references of previous reviews on HPWS (e.g., Combs et al., 2006; Jiang et al., 2012; Subramony, 2009) and added the articles that were missed in the database searches. Third, we conducted a manual search of major management journals that often publish strategic HRM research, including _Academy of Management Journal, Journal of Management, Journal of Applied Psychology, Personnel Psychology, Strategic Management Journal, Organization Science, Journal of Management Studies, Human Resource Management, Human Resource Management Journal, and International Journal of Human Resource Management_ , to locate studies that were not included in the previous searches. Finally, we searched ProQuest Digital Dissertations and conference proceedings for the annual meetings of the Academy of Management for unpublished dissertations and conference papers from 2008 to 2013. We used four criteria to include studies in the following meta-analysis. First, we only included studies that examined the relationship between HPWS and financial performance at the unit-level of analysis (e.g., teams, stores, business units, and firms). Studies conducted at the individual level of analysis were excluded from the analysis. Second, we only included studies that examined HRM practices as a system. Those examining the relationships between individual HRM practices and financial performance were not considered in this research. Third, consistent with previous meta-analyses (e.g., Combs et al., 2006; Jiang et al., 2012; Subramony, 2009), we limited financial performance to variables indicating financial or accounting outcomes, such as profit, return on assets, return on invested capital, return on equity, Tobin’s Q, sales growth, and perceived financial performance. Those only reporting the relationships between HPWS and other types of outcomes (e.g., employee outcomes and operational outcomes) were not included in the analysis. Fourth, the studies need to report at least two pieces of information in order to be included in the meta-analysis – correlation coefficient between HPWS and financial performance and sample size. This procedure resulted in 56 independent studies that were entered in the following analysis. Before conducting Bayesian meta-analysis, we first corrected the observed correlation from each sample for unreliability by following the procedure outlined by Hunter and Schmidt (2004). Because HPWS has been considered as a formative construct (Delery, 1998) for which a high internal reliability (e.g., Cronbach’s alpha) is not required, we used a reliability of 1 for the measure of HPWS. Similarly, we used a reliability of 1 for the objective measures of financial performance and used Cronbach’s alpha as the reliability of the subjective measures of financial performance. In addition, we consider firm size as a potential moderator of the relationship between HPWS and financial performance in order to test the meta- regression model of this study. Firm size is commonly included as a control variable in strategic HRM research, but its moderating effect has rarely been explored in either primary studies or a meta-analysis. Two competing hypotheses can be proposed in terms of its moderating role. On the one hand, some researchers have suggested that large organizations are likley to use more sophisticated HRM practices (e.g., HPWS) compared with small and mediem enterprises (e.g., Guthrie, 2001; Jackson & Schuler, 1995). As firm size increases, firms may also have more advantages such as economy of scale (e.g., Pfeffer & Salancik, 1978) and thus be more likely to gain benefit from their investment in HRM practices. On the other hand, large firms’ financial performance may be more affected by other factors beyond human resources (Capon, Farley, & Hoenig, 1990). In this case, the role of HPWS in enhancing financial performance may be limited in large firms than in small and medium firms. Taking these considerations together, we expect that firm size may moderate the relationship between HPWS and financial performance but make no directional prediction of this effect. Firm size is usually indicated by the number of employees. Studies with average number of employees greater than 250 were coded as 1 (i.e., large firms) and the others were coded as 0 (i.e., small and medium firms). Table 4 shows the summary statistics of the data used in this example. Among the total of 56 studies, 46 measured financial performance using the archival data (i.e., objective performance) and 10 used subjective measures of financial performance (i.e., subjective performance). In addition, 37 studies were coded as large firms and 19 were coded as small and medium firms. The observed correlations ranged from 0.01 to 0.52 with sample sizes ranging from 50 to 2136. Table 4: Summary statistics \| Minmum \| Mean \| Median \| Maximum \| Standard deviation ---\|---\|---\|---\|---\|--- Correlation \| 0.01 \| 0.22 \| 0.200 \| 0.52 \| 0.13 Sample size \| 50 \| 281 \| 191 \| 2136 \| 325 Reliability \| 0.74 \| 0.97 \| 1 \| 1 \| 0.07 \| Small & Medium: 19 \| Large: 37 \| Objective studies: 46 \| Subjective studies: 10 Four power schemes are considered in the meta-analysis. First, every study is given the power of 1. In this case, every study contributes to the meta- analysis result fully and equally. This is equivalent to conduct traditional meta-analysis using Bayesian methods. Second, the reliability of financial performance of each study is used as power. The reason for this choice is that, if a measure is not reliable, only partial information will be used in meta-analysis. Third, two studies have sample sizes larger than 1000 (1212 and 2136, respectively). In order to avoid the dominant influence of the two studies on the final result, we assign them a power of 0.1 and the rest of studies a power of 1 in meta-analysis. Fourth, arguably a study with a large effect size is more likely to be published, which might cause publication bias. Therefore, reducing the influence of the studies with large effect sizes might be helpful in reducing publication bias. In this power scheme, we set the power at 0.5 for studies with correlations larger than 0.2. For the power schemes 3 and 4, the choice of power is rather liberal. A more serious analysis might consider different levels of power. ### 4.1 Results of Fixed-effects Meta-analysis We first apply the fixed-effects meta-analysis model to the example data. Table 5 shows the results using the four different power schemes. When every study is assigned the equal power of 1 (Power 1), the estimated population correlation $\rho$ is 0.263 ($\zeta$ is the Fisher z-transformed estimate). If the reliability of financial performance is used as power (Power 2), the estimated correlation is about 0.264. However, when the two studies with the largest sample size are assigned a power of 0.5 (Power 3), the estimated correlation becomes 0.226. Note the estimated correlation in this condition is significantly different from the other two correlation estimates simply based on the credible interval estimates. In the observed studies, the correlations for the two study are 0.34 and 0.45, respectively, both of which are larger than the estimated fixed-effect correlation. When no power is used, the two studies pull the estimates close to them because their large sample sizes lead to big weights in the estimated correlation. Under the situation where the studies with large correlation are assigned a weight 0.5 (Power 4), the estimated correlation is 0.22, which is even smaller than the situation of Power 3\. This is because the large correlations are downweighted. Table 5: Results from fixed-effects meta-analysis2,3 \| \| Estimate \| sd \| CI \| DIC ---\|---\|---\|---\|---\|--- Power 1 \| $\zeta$ \| 0.27* \| 0.008 \| 0.254 \| 0.285 \| 184 $\rho$ \| 0.26* \| 0.007 \| 0.249 \| 0.278 Power 2 \| $\zeta$ \| 0.27* \| 0.008 \| 0.255 \| 0.286 \| 175.9 $\rho$ \| 0.26* \| 0.008 \| 0.249 \| 0.279 Power 3 \| $\zeta$ \| 0.23* \| 0.009 \| 0.212 \| 0.247 \| 72.11 $\rho$ \| 0.23* \| 0.008 \| 0.209 \| 0.242 Power 4 \| $\zeta$ \| 0.22* \| 0.009 \| 0.205 \| 0.242 \| 77.44 $\rho$ \| 0.22* \| 0.009 \| 0.202 \| 0.237 ### 4.2 Results of Random-effects Meta-analysis Table 6 shows the results from the random-effects meta-analysis. First, the estimated correlations from the random-effects and fixed-effects methods are quite different (0.23 vs. 0.27) when the power is not considered. This is because for the random-effects method, the between-study variability is considered. Therefore, extreme studies (e.g., those with unusual large sample sizes) are shrunk towards the average. Furthermore, within the random-effects method, there is not much difference in the estimated correlation. Second, only for power scheme 4, the estimated correlation shows notable difference from the rest of the power schemes. The reason is because studies with large correlations are downweighted. Third, in all situation, the variance estimate of $\tau$ is significant. This indicates there is sufficient variability in the studies to consider a random-effects meta-analysis to model the heterogeneity in the population. Table 6: Results from random-effects meta-analysis2,3 \| \| Estimate \| sd \| CI \| DIC ---\|---\|---\|---\|---\|--- Power 1 \| $\zeta$ \| 0.23* \| 0.02 \| 0.191 \| 0.269 \| -98.45 $\tau$ \| 0.016* \| 0.004 \| 0.01 \| 0.026 $\rho$ \| 0.226* \| 0.019 \| 0.189 \| 0.263 Power 2 \| $\zeta$ \| 0.23* \| 0.02 \| 0.191 \| 0.27 \| -97.18 $\tau$ \| 0.016* \| 0.004 \| 0.01 \| 0.026 $\rho$ \| 0.226* \| 0.019 \| 0.189 \| 0.263 Power 3 \| $\zeta$ \| 0.228* \| 0.02 \| 0.19 \| 0.267 \| -93.99 $\tau$ \| 0.016* \| 0.004 \| 0.009 \| 0.025 $\rho$ \| 0.224* \| 0.019 \| 0.187 \| 0.261 Power 4 \| $\zeta$ \| 0.218* \| 0.02 \| 0.178 \| 0.259 \| -85.4 $\tau$ \| 0.015* \| 0.004 \| 0.008 \| 0.024 $\rho$ \| 0.214* \| 0.019 \| 0.177 \| 0.253 ### 4.3 Results of Meta-regression From the random-effects meta-analysis, we concluded that the population should be considered as heterogeneous.Through meta-regression analysis, we investigate whether the heterogeneity is related to firm size of different studies. Based on the results in Table 7, firm size is not significantly related to the individual differences in the population correlation because the slope parameter $\beta_{2}$ is not significant regardless of the choice of power. Furthermore, the results from the first three power schemes are very close. Comparing all four power schemes, power scheme 4 has smaller intercept while larger absolute slope. Combined, the results do not suggest the moderating effect of firm size on the relationship between HPWS and financial performance. It implies that HPWS used in both large firms and small and medium firms are salutary for enhancing financial performance. Table 7: Results from meta-regression2,3 \| \| Estimate \| sd \| CI \| DIC ---\|---\|---\|---\|---\|--- Power 1 \| $\beta_{1}$(intercept) \| 0.248* \| 0.034 \| 0.181 \| 0.316 \| -97.9 $\beta_{2}$(size) \| -0.028 \| 0.042 \| -0.113 \| 0.053 $\tau$ \| 0.017* \| 0.004 \| 0.01 \| 0.026 Power 2 \| $\beta_{1}$(intercept) \| 0.249* \| 0.034 \| 0.181 \| 0.317 \| -96.63 $\beta_{2}$(size) \| -0.029 \| 0.042 \| -0.113 \| 0.053 $\tau$ \| 0.016* \| 0.004 \| 0.01 \| 0.026 Power 3 \| $\beta_{1}$(intercept) \| 0.245* \| 0.034 \| 0.179 \| 0.312 \| -93.5 $\beta_{2}$(size) \| -0.027 \| 0.042 \| -0.111 \| 0.054 $\tau$ \| 0.016* \| 0.004 \| 0.009 \| 0.025 \| $\beta_{1}$(intercept) \| 0.24* \| 0.035 \| 0.172 \| 0.31 \| -84.79 Power 4 \| $\beta_{2}$(size) \| -0.034 \| 0.043 \| -0.121 \| 0.048 \| $\tau$ \| 0.015* \| 0.004 \| 0.008 \| 0.025 ### 4.4 Fixed-effects Meta-analysis, Random-effects Meta-analysis, or Meta- regression? Selection among different methods deserves much more investigation. Here, we just illustrate several possibilities using the example above, which certainly have their limitations. In choosing between fixed-effects and random-effects meta-analysis, one can check whether the variance parameter $\tau$ from the random-effects meta-analysis is significant. If it is significant, random- effects meta-analysis can be used. Our example showed that random-effects meta-analysis might be preferred. For the choice between random-effects meta- analysis and meta-regression, one can focus on the significance of the regression coefficients for predictors. If the coefficients are not significant, it might suggest that there is no need to include the proposed predictor in meta-regression. We can also directly compare fixed-effects meta-analysis, random-effects meta- analysis, and meta-regression using DIC. The model with the smallest DIC indicates it fits the data best. However, DIC should only be used to compare models under the same power scheme. The calculation of DIC across power schemes would utilize different information and therefore is not valid. For example, under power scheme 1, DICs for the three models are 184, -98.45, and -97.9. This suggests that the meta-regression fits the current data best. However, there is no given cut-off on when a model can be considered as fitting data significantly better. ## 5 DISCUSSION The current study presents a Bayesian method for meta-analysis. A unique feature of our method is to enable researchers to control the contribution of individual studies included in a meta-analysis through power prior. The motivation of this approach comes from the notion that not all studies should be treated equivalently when estimating the overall effect size in a meta- analysis. By developing an online program and using the example of the relationship between HPWS and financial performance, we have shown how to apply this method into management research. In the rest of this article, we briefly summarize the example results derived from the method we proposed. And then we discuss some implications of this method to meta-analysis in the field of management. In the example study, we use four power schemes to assign powers to individual studies included in the meta-analysis. As shown in fixed-effects, random- effects, and meta-regression models, using the reliability of financial performance as power does not dramatically change the results obtained from regular meta-analysis that uses full information provided by each study. This is because that only ten studies used subjective measures of financial performance and the use of reliability as power would only influence how the ten out of 56 studies contribute to the final results. Moreover, the reliabilities for the subjective measures are typically high, so the vast majority of the information they provide still contributes to the overall effect size. If one uses another example with more subjective measures, the difference in effect size between regular meta-analysis and meta-analysis using reliability as power may be more obvious. Either way, our method provides a way to evaluate whether reliability influences meta-analysis results. When power is used to reduce the influence of two studies with large sample sizes, the overall effect size in fixed-effects model becomes significantly different from what is obtained in the regular model, and the change is less obvious in random-effects and meta-regression models. This is because between- study variability is taken into account in random-effects model, which can shrink extreme effect sizes towards the average. However, this does not mean that using power to modify the impact of extremely large samples always has a larger impact on fixed-effects model than on random-effects model. It may also depend on the observed correlations of studies with large sample sizes. For example, if the correlation of a large sample is similar to the weighted average of the rest of the studies, assigning a small power to the large sample may not significantly change the overall effect size in either fixed- effects model or random-effects model. The influence of power becomes more salient under power scheme 4 where studies with correlations larger than 0.2 are assigned a power of 0.5. We argue that this setting can potentially be used to deal with publication bias. For example, if we believe the studies with large effect sizes are over-sampled, we can assign them power smaller than 1\. On the other hand, if one believes the studies with small effect sizes are under-sampled, power larger than 1 can also be used. Certainly the choice of power needs careful consideration. Combined, the example of the relationship between HPWS and financial performance provides an initial illustration of our Bayesian approach of meta- analysis. We encourage researchers who are interested in this approach to test their data using the developed software and compare the results derived from different power schemes. In the following, we shift attention to the implications of this method to meta-analysis in management research. First of all, we want to make it clear that it is not necessary to apply power prior to all meta-analyses. However, if researchers believe certain factors may impact the credibility of research findings and they can distinguish between more reliable and less reliable studies, we would recommend them to estimate the overall effect size with power prior, at least for the purpose of comparison. For example, reliability has been commonly used to correct for measurement error in meta-analysis (Hunter & Schmidt, 2004). As reliability decreases, the corrected effect size is more likely to be enlarged. However, a low reliability often indicates poor quality of a measure, which may not accurately reflect the intended construct. Therefore, the study using the measure with low reliability may be less likely to represent the true relationship of interest. In addition, research design (e.g., cross-sectional vs. longitudinal, single-source data vs. multiple-source data, self-report ratings vs. observer ratings, field studies vs. experimental studies) may also affect the extent to which a study can provide reliable information of the relationship between two variables. Combining study findings without considering the credibility of the information may lead to misleading results. Although researchers can summarize studies of different characteristics separately (e.g., Judge, Colbert, & Ilies, 2004; Oh, Wang, & Mount, 2011; Sin, Nahrgang, & Morgeson, 2009), they may still need to combine all studies to yield overall effect sizes that can be used as inputs of other analyses, such as meta-analytic structural equation modeling (MASEM; Viswesvaran & Ones, 1995). In these cases, researchers may choose to use more information from reliable studies through power prior in order to obtain overall effect sizes that can better represent the true relationships. Bayesian meta-analysis with power prior can also be used to deal with outliers, including outliers of observed correlations and outliers of sample sizes. Traditionally, researchers often eliminate the most extreme data points to attenuate the influence of outliers on overall effect sizes (e.g., Hedges, 1987, Huber, 1980, Tukey, 1960). This is similar to assigning a power of 0 to studies considered as outliers and using no information of the eliminated studies in analysis. However, rather than deleting the data points completely, researchers can also choose to use only a small part of their information by assigning a small non-zero power to those studies. One important issue that is out of the discussion of this article is what power value should be assigned to each study in meta-analysis with power prior. The method proposed in this study cannot determine whether a power prior scheme is realistic or not to reflect the contribution of each study to the final results. It is more reasonable for researchers who are familiar with the nature of the included studies to make the decisions. The general guideline is to identify the criteria that can indicate the credibility of research findings and use it to guide power prior decision in meta-analysis. One attempt of this study is to use reliability as a power for studies relying on subjective measures, which may reduce the overcorrection for unreliability due to extremely low reliabilities. In addition, we recommend that one should always compare the results from the analysis with and without power priors to inform the influence of the use of power priors. We encourage more efforts to further explore this issue in the future. This study can be improved and extended in many ways. First, in both random- effects meta-analysis and meta-regression, we assume that the random effects follow a normal distribution. This assumption might not be valid when there are extreme values. Further study can incorporate robust Bayesian analysis to deal with the problem (e.g., Zhang, Lai, Lu, & Tong, 2013). Second, the current study has focused on the development of the method for correlation. However, the method can be equally applied to other effect sizes such as mean differences and odds ratios. 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Bayesian estimation and inference: A user’s guide. _Journal of Management_. ## 7 APPENDIX A With the prior and the information from the first study, the posterior, based on Bayes’ Theorem, is $\displaystyle p(\zeta\|z_{1})$ $\displaystyle=$ $\displaystyle\frac{p(\zeta)p(z_{1}\|\zeta)}{p(z_{1})}$ $\displaystyle=$ $\displaystyle\frac{\frac{1}{\sqrt{2\pi\psi_{0}}}\exp\left[-\frac{(\zeta-\zeta_{0})^{2}}{2\psi_{0}}\right]\frac{1}{\sqrt{2\pi\phi_{1}}}\exp\left[-\frac{(z_{1}-\zeta)^{2}}{2\phi_{1}}\right]}{p(z_{1})}$ $\displaystyle=$ $\displaystyle\frac{\frac{1}{\sqrt{2\pi\psi_{0}}}\frac{1}{\sqrt{2\pi\phi_{1}}}\exp\left[-(\frac{1}{2\psi_{0}}+\frac{1}{2\phi_{1}})\zeta^{2}+2(\frac{1}{2\psi_{0}}\zeta_{0}+\frac{1}{2\phi_{1}}z_{1})\zeta-(\frac{\zeta_{0}^{2}}{2\psi_{0}}+\frac{z_{1}^{2}}{2\phi_{1}})\right]}{p(z_{1})},$ $\displaystyle=$ $\displaystyle\frac{D\exp\left[-\frac{1}{2}(A\zeta^{2}+2B\zeta+C)\right]}{p(z_{1})}$ $\displaystyle=$ $\displaystyle\frac{D\exp\left[-\frac{(\zeta-\frac{B}{A})^{2}}{2\frac{1}{A}}-\frac{1}{2}(C-\frac{B^{2}}{A})\right]}{p(z_{1})}$ where $\displaystyle A$ $\displaystyle=$ $\displaystyle\frac{1}{\psi_{0}}+\frac{1}{\phi_{1}}$ $\displaystyle B$ $\displaystyle=$ $\displaystyle\frac{1}{\psi_{0}}\zeta_{0}+\frac{1}{\phi_{1}}z_{1}.$ $\displaystyle C$ $\displaystyle=$ $\displaystyle\frac{\zeta_{0}^{2}}{\psi_{0}}+\frac{z_{1}^{2}}{\phi_{1}}$ The denominator is $\displaystyle p(z_{1})$ $\displaystyle=$ $\displaystyle\int_{-\infty}^{+\infty}(D\exp\left[-\frac{(\zeta-\frac{B}{A})^{2}}{2\frac{1}{A}}-\frac{1}{2}(C-\frac{B^{2}}{A})\right])d\zeta$ $\displaystyle=$ $\displaystyle D\exp\left[-\frac{1}{2}(C-\frac{B^{2}}{A})\right]\times\sqrt{2\pi\frac{1}{A}}$ Therefore, the posterior is $p(\zeta\|z_{1})=\frac{1}{\sqrt{2\pi\frac{1}{A}}}\exp\left[-\frac{(\zeta-\frac{B}{A})^{2}}{2\frac{1}{A}}\right],$ a normal distribution with mean $\displaystyle B/A$ $\displaystyle=$ $\displaystyle\frac{\frac{1}{\psi_{0}}\zeta_{0}+\frac{1}{\phi_{1}}z_{1}}{\frac{1}{\psi_{0}}+\frac{1}{\phi_{1}}}=\frac{\phi_{1}\zeta_{0}+\psi_{0}z_{1}}{\phi_{1}+\psi_{0}}$ and variance $1/A=\frac{1}{\frac{1}{\psi_{0}}+\frac{1}{\phi_{1}}}.$ ## 8 APPENDIX B With the power parameter $\alpha_{1}$, the posterior $\displaystyle p(\zeta\|z_{1})$ $\displaystyle=$ $\displaystyle\frac{p(\zeta)[p(z_{1}\|\zeta)]^{\alpha_{1}}}{p(z_{1})}$ $\displaystyle=$ $\displaystyle\frac{\frac{1}{\sqrt{2\pi\psi_{0}}}\exp\left[-\frac{(\zeta-\zeta_{0})^{2}}{2\psi_{0}}\right]\left\\{\frac{1}{\sqrt{2\pi\phi_{1}}}\exp\left[-\frac{(z_{1}-\zeta)^{2}}{2\phi_{1}}\right]\right\\}^{\alpha_{1}}}{p(z_{1})}$ $\displaystyle=$ $\displaystyle\frac{\frac{1}{\sqrt{2\pi\psi_{0}}}\left(\frac{1}{\sqrt{2\pi\phi_{1}}}\right)^{\alpha_{1}}\exp\left[-\frac{(\zeta-\zeta_{0})^{2}}{2\psi_{0}}-\exp\left[-\frac{(z_{1}-\zeta)^{2}}{2\phi_{1}/\alpha_{1}}\right]\right]}{}$ $\displaystyle=$ $\displaystyle\frac{D\exp\left[-(\frac{1}{2\psi_{0}}+\frac{1}{2\phi_{1}^{}})\zeta^{2}+2(\frac{1}{2\psi_{0}}\zeta_{0}+\frac{1}{2\phi_{1}^{}}z_{1})\zeta-(\frac{\zeta_{0}^{2}}{2\psi_{0}}+\frac{z_{1}^{2}}{2\phi_{1}^{}})\right]}{p(z_{1})},$ $\displaystyle=$ $\displaystyle\frac{D\exp\left[-\frac{1}{2}(A\zeta^{2}+2B\zeta-C)\right]}{p(z_{1})}$ $\displaystyle=$ $\displaystyle\frac{D\exp\left[-\frac{(\zeta-\frac{B}{A})^{2}}{2\frac{1}{A}}-\frac{1}{2}(C-\frac{B^{2}}{A})\right]}{p(z_{1})}$ where $\displaystyle A$ $\displaystyle=$ $\displaystyle\frac{1}{\psi_{0}}+\frac{1}{\phi_{1}^{}}$ $\displaystyle B$ $\displaystyle=$ $\displaystyle\frac{1}{\psi_{0}}\zeta_{0}+\frac{1}{\phi_{1}^{}}z_{1},$ $\displaystyle C$ $\displaystyle=$ $\displaystyle\frac{\zeta_{0}^{2}}{\psi_{0}}+\frac{z_{1}^{2}}{\phi_{1}^{}}$ and $\phi_{1}^{}=\phi_{1}/\alpha_{1}$. From Appendix A, the posterior is $N(B/A,1/A)$ where $\displaystyle B/A$ $\displaystyle=$ $\displaystyle\frac{\frac{1}{\psi_{0}}\zeta_{0}+\frac{1}{\phi_{1}^{}}z_{1}}{\frac{1}{\psi_{0}}+\frac{1}{\phi_{1}^{}}}=\frac{\phi_{1}^{}\zeta_{0}+\psi_{0}z_{1}}{\phi_{1}^{}+\psi_{0}}=\frac{\frac{\phi_{1}}{\alpha_{1}}\phi_{1}+\psi_{0}z_{1}}{\frac{\phi_{1}}{\alpha_{1}}+\psi_{0}}$ $\displaystyle 1/A$ $\displaystyle=$ $\displaystyle\frac{1}{\frac{1}{\psi_{0}}+\frac{1}{\phi_{1}^{}}}=\frac{1}{\frac{1}{\psi_{0}}+\frac{\alpha_{1}}{\phi_{1}}}.$ ## 9 APPENDIX C Show the posterior $\displaystyle p(\zeta\|z_{1},z_{2},\alpha_{1},\alpha_{2})$ $\displaystyle=$ $\displaystyle\frac{p(\zeta\|z_{1},\alpha_{1})[p(z_{2}\|\zeta)]^{\alpha_{2}}}{p(z_{2})}$ $\displaystyle=$ $\displaystyle\frac{\frac{1}{\sqrt{2\pi\psi_{1}^{}}}\exp\left[-\frac{(\zeta-\zeta_{1}^{})^{2}}{2\psi_{1}^{}}\right]\left\\{\frac{1}{\sqrt{2\pi\phi_{2}}}\exp\left[-\frac{(z_{2}-\zeta)^{2}}{2\phi_{2}}\right]\right\\}^{\alpha_{2}}}{p(z_{2})}$ $\displaystyle=$ $\displaystyle\frac{\frac{1}{\sqrt{2\pi\psi_{1}^{}}}\left(\frac{1}{\sqrt{2\pi\phi_{2}}}\right)^{\alpha_{2}}\exp\left[-\frac{(\zeta-\zeta_{1}^{})^{2}}{2\psi_{1}^{}}-\exp\left[-\frac{(z_{1}-\zeta)^{2}}{2\phi_{1}/\alpha_{1}}\right]\right]}{p(z_{2})}$ $\displaystyle=$ $\displaystyle\frac{D\exp\left[-\frac{1}{2}(A\zeta^{2}+2B\zeta-C)\right]}{p(z_{1})}$ $\displaystyle=$ $\displaystyle\frac{D\exp\left[-\frac{(\zeta-\frac{B}{A})^{2}}{2\frac{1}{A}}-\frac{1}{2}(C-\frac{B^{2}}{A})\right]}{p(z_{1})}$ The denominator is $\displaystyle p(z_{1})$ $\displaystyle=$ $\displaystyle\int_{-\infty}^{+\infty}(D\exp\left[-\frac{(\zeta-\frac{B}{A})^{2}}{2\frac{1}{A}}-\frac{1}{2}(C-\frac{B^{2}}{A})\right])d\zeta$ $\displaystyle=$ $\displaystyle D\exp\left[-\frac{1}{2}(C-\frac{B^{2}}{A})\right]\times\sqrt{2\pi\frac{1}{A}}$ The posterior is $N(B/A,1/A)$ where $\displaystyle B/A$ $\displaystyle=$ $\displaystyle\frac{\frac{1}{\psi_{1}^{}}\zeta_{1}^{}+\frac{1}{\phi_{2}^{}}z_{2}}{\frac{1}{\psi_{1}^{}}+\frac{1}{\phi_{2}^{}}}=\frac{\frac{1}{\psi_{0}}\zeta_{0}+\frac{\alpha_{1}}{\phi_{1}}z_{1}+\frac{\alpha_{2}}{\phi_{2}}z_{2}}{\frac{1}{\psi_{0}}+\frac{\alpha_{1}}{\phi_{1}}+\frac{\alpha_{2}}{\phi_{2}}}$ $\displaystyle 1/A$ $\displaystyle=$ $\displaystyle\frac{1}{\frac{1}{\psi_{1}^{}}+\frac{1}{\phi_{1}^{}}}=\frac{1}{\frac{1}{\psi_{0}}+\frac{\alpha_{1}}{\phi_{1}}+\frac{\alpha_{2}}{\phi_{2}}}$ ## 10 APPENDIX D The joint posterior distribution is $\displaystyle p(\zeta,\tau,\zeta_{i}\|z_{i},\phi_{i},\alpha_{i})$ $\displaystyle\propto$ $\displaystyle p(\zeta)p(\tau)\prod_{i=1}^{m}p_{\alpha_{i}}(z_{i},\zeta_{i}\|\zeta,\tau)$ $\displaystyle=$ $\displaystyle p(\zeta)p(\tau)\prod_{i=1}^{m}\left[p_{\alpha_{i}}(z_{i}\|\zeta_{i},\phi_{i})p(\zeta_{i}\|\zeta,\tau)\right]$ $\displaystyle\propto$ $\displaystyle\frac{1}{\sqrt{2\pi\psi_{0}}}\exp\left[-\frac{(\zeta-\zeta_{0})^{2}}{2\psi_{0}}\right]\tau^{-\delta_{0}-1}\exp\left[-\frac{\gamma_{0}}{\tau}\right]$ $\displaystyle\times\left[\prod_{i=1}^{m}(2\pi\phi_{i})^{-\alpha_{i}/2}\right]\exp\left[-\sum_{i=1}^{m}\frac{(z_{i}-\zeta_{i})^{2}}{2\phi_{i}/\alpha_{i}}\right](2\pi\tau)^{-m/2}\exp\left[-\frac{\sum_{i=1}^{m}(\zeta_{i}-\zeta)^{2}}{2\tau}\right].$ Now we obtain the conditional posterior distributions. First, we get the conditional posterior distribution of $\tau$, which is $p(\tau\|\zeta_{i},\zeta,\alpha_{i})\propto\tau^{-\delta_{0}-1-m/2}\exp\left[-\frac{2\gamma_{0}+\sum(\zeta_{i}-\zeta)^{2}}{2\tau}\right].$ Therefore, the posterior is inverse Gamma distribution IG($\delta_{0}+m/2,\gamma_{0}+[\sum(\zeta_{i}-\zeta)^{2}]/2$). Second, the conditional posterior distribution of $\zeta$ is $p(\zeta\|\zeta_{i},\tau)\propto\exp\left[-\frac{(\zeta-\zeta_{0})^{2}}{2\psi_{0}}-\frac{\sum(\zeta_{i}-\zeta)^{2}}{2\tau}\right].$ Therefore, the conditional posterior is a normal distribution $N\left(\frac{\frac{\sum_{i=1}^{m}\zeta_{i}}{\tau}+\frac{\zeta_{0}}{\psi_{0}}}{\frac{m}{\tau}+\frac{1}{\psi_{0}}},\frac{1}{\frac{m}{\tau}+\frac{1}{\psi_{0}}}\right).$ Third, the conditional posterior distribution of $\zeta_{i}$ is $p(\zeta_{i}\|\zeta,z_{i},\tau,\alpha_{i})\propto\exp\left[-\frac{(z_{i}-\zeta_{i})^{2}}{2\phi_{i}/\alpha_{i}}-\frac{(\zeta_{i}-\zeta)^{2}}{2\tau}\right],$ which is a normal distribution $N\left(\frac{\frac{\zeta}{\tau}+\frac{z_{i}\alpha_{i}}{\phi_{i}}}{\frac{1}{\tau}+\frac{\alpha_{i}}{\phi_{i}}},\frac{1}{\frac{1}{\tau}+\frac{\alpha_{i}}{\phi_{i}}}\right).$ ## 11 APPENDIX E The joint posterior distribution for the meta-regression model is $\displaystyle p(\bm{\beta},\tau\|z_{i},\zeta_{i},\alpha_{i})$ $\displaystyle\propto$ $\displaystyle p(\bm{\beta})p(\tau)\prod p_{\alpha_{i}}(z_{i},\zeta_{i}\|\bm{\beta},\tau)$ $\displaystyle=$ $\displaystyle p(\bm{\beta})p(\tau)\prod p_{\alpha_{i}}(z_{i}\|\zeta_{i},\phi_{i})p(\zeta_{i}\|\bm{\beta},\tau)$ $\displaystyle\propto$ $\displaystyle\|\mathbf{\Psi}_{0}\|^{-1/2}\exp\left[-\frac{1}{2}(\bm{\beta}-\bm{\zeta}_{0})^{\prime}\mathbf{\Psi_{0}}^{-1}(\bm{\beta}-\bm{\zeta}_{0})\right]\tau^{-\delta_{0}-1}\exp\left[-\frac{\gamma_{0}}{\tau}\right]$ $\displaystyle\times\prod\left\\{(2\pi\phi_{i})^{-\alpha_{i}/2}\exp\left[-\sum\frac{(z_{i}-\zeta_{i})^{2}}{2\phi_{i}/\alpha_{i}}\right](2\pi\tau)^{-m/2}\exp\left[-\frac{\sum(\zeta_{i}-\mathbf{x}_{i}\bm{\beta})^{2}}{2\tau}\right]\right\\}.$ The conditional posterior distribution of $\tau$ is $p(\tau\|\bm{\beta},\zeta_{i})\propto\tau^{-(\delta_{0}+m/2)-1}\exp\Bigg{[}-\frac{\gamma_{0}+\sum(\zeta_{i}-\mathbf{x}_{i}\bm{\beta})^{2}/2}{\tau}\Bigg{]}$ (6) Therefore, $\tau\|\bm{\beta},\zeta_{i}\sim IG(\delta_{0}+m/2,\gamma_{0}+\sum(\zeta_{i}-\mathbf{x}_{i}\bm{\beta})^{2}/2)$. The conditional posterior distribution for $\bm{\beta}$ is $p(\bm{\beta}\|\tau,\zeta_{i})\propto\exp\Bigg{[}-\frac{1}{2}(\bm{\beta}-\bm{\zeta}_{0})^{\prime}\mathbf{\Psi}_{0}^{-1}(\bm{\beta}-\bm{\zeta}_{0})\Bigg{]}\exp\Bigg{[}\frac{\sum(\zeta_{i}-\mathbf{x}_{i}\bm{\beta})^{2}}{2\tau}\Bigg{]}.$ (7) Let $\bm{\zeta}=(\zeta_{1},\zeta_{2},\cdots,\zeta_{m})^{\prime}$ be the vector of $\zeta_{i}^{\prime}s$, and $\hat{\beta}$ be the least square estimate such that $\hat{\bm{\beta}}=(\mathbf{X}^{\prime}\mathbf{X})^{-1}\mathbf{X}^{\prime}\bm{\zeta}$ with $\mathbf{X}=(\mathbf{x}_{1},\ldots,\mathbf{x}_{m})^{\prime}$ as the design matrix. Then the conditional posterior distribution of $\beta$ is a multivariate normal distribution $N\Bigg{(}\bigg{(}\mathbf{\Psi}_{0}^{-1}+\frac{\mathbf{X}^{\prime}\mathbf{X}}{\tau}\bigg{)}^{-1}\bigg{(}\mathbf{\Psi}_{0}^{-1}\bm{\zeta}_{0}+\frac{\mathbf{X}^{\prime}\mathbf{X}}{\tau}\hat{\bm{\beta}}\bigg{)},(\mathbf{\Psi}_{0}^{-1}+\frac{\mathbf{X}^{\prime}\mathbf{X}}{\tau})^{-1}\Bigg{)}$ For $\zeta_{i}$, its conditional distribution is $p(\zeta_{i}\|\beta,\tau)\propto\exp\Bigg{[}-\frac{(z_{i}-\zeta_{i})^{2}}{2\phi_{i}/\alpha_{i}}\Bigg{]}\exp\Bigg{[}-\frac{(\zeta_{i}-X_{i}\beta)^{2}}{2\tau}\Bigg{]},$ a normal distribution $N(\frac{\frac{\alpha_{i}z_{i}}{\phi_{i}}+\frac{X_{i}\bm{\beta}}{\tau}}{\frac{\alpha_{i}}{\phi_{i}}+\frac{1}{\tau}},\frac{1}{\frac{\alpha_{i}}{\phi_{i}}+\frac{1}{\tau}}).$ ## 12 FOOTNOTES The results are based on a total of 10,000 iterations with the first 4,000 iterations as burn-in. $p<0.05$. Power 1: each study is given a power of 1. Power 2: the reliability of financial performance is used as power. Power 3: the two studies with the largest sample sizes are given a power of 0.1. Power 4: studies with correlations larger than 0.2 are given a power of 0.5, otherwise, 1.	arxiv-papers	2014-01-09T16:15:41	2024-09-04T02:49:56.484132	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Zhiyong Zhang, Kaifeng Jiang, Haiyan Liu, In-Sue Oh", "submitter": "Zhiyong Zhang", "url": "https://arxiv.org/abs/1401.2054" }
1401.2055	# Approximation by Genuine $q$-Bernstein-Durrmeyer Polynomials in Compact Disks in the case $q>1$ Nazim I. Mahmudov ###### Abstract This paper deals with approximating properties of the newly defined $q$-generalization of the genuine Bernstein-Durrmeyer polynomials in the case $q>1$, whcih are no longer positive linear operators on $C[0,1]$. Quantitative estimates of the convergence, the Voronovskaja type theorem and saturation of convergence for complex genuine $q$-Bernstein-Durrmeyer polynomials attached to analytic functions in compact disks are given. In particular, it is proved that for functions analytic in $\left\\{z\in\mathbb{C}:\left\|z\right\|<R\right\\}$, $R>q,$ the rate of approximation by the genuine $q$-Bernstein-Durrmeyer polynomials ($q>1$) is of order $q^{-n}$ versus $1/n$ for the classical genuine Bernstein-Durrmeyer polynomials. We give explicit formulas of Voronovskaja type for the genuine $q$-Bernstein-Durrmeyer for $q>1$. ## 1 Introduction In several recent papers, convergence properties of complex $q$-Bernstein polynomials, proposed by Phillips [3], attached to an analytic function $f$ in closed disks, were intensively studied. Ostrovska [17], [18], and Wang and Wu [21], [22] have investigated convergence properies of $B_{n,q}$ in the case $q>1.$ In the case $q>1$, the $q$-Bernstein polynomials are no longer positive operators, however, for a function analytic in a disc $\mathbb{D}_{R}:=\left\\{z\in\mathbb{C}:\left\|z\right\|<R\right\\},\ R>q$, it was proved in [17] that the rate of convergence of $\left\\{B_{n,q}\left(f;z\right)\right\\}$ to $f\left(z\right)$ has the order $q^{-n}$ (versus $1/n$ for the classical Bernstein polynomials). Moreover, Ostrovska [18] obtained Voronovskaya type theorem for monomials. If $q\geq 1$ then qualitative Voronovskaja-type and saturation results for complex $q$-Bernstein polynomials were obtained in Wang-Wu [21]. Wu [22] studied saturation of convergence on the interval $[0,1]$ for the $q$-Bernstein polynomials of a continuous function $f$ for arbitrary fixed $q>1$. Genuine Bernstein–Durrmeyer operators were first considered by Chen [10] and Goodman and Sharma [15] around 1987. In recent years, the genuine Bernstein–Durrmeyer operators have been investigated intensively by a number of authors. Among the many articles written on the genuine Bernstein–Durrmeyer operators, we mention here only the ones by Gonska and etc [13], by Parvanov and Popov [5], by Sauer [7], by Waldron [8], and the book of Páltánea [9]. On the other hand, Gal [4] obtained quantitative estimates of the convergence and of the Voronovskaja theorem in compact disks, for the complex genuine Bernstein–Durrmeyer polynomials attached to analytic functions. Besides, in other very recent papers, similar studies were done for complex Bernstein- Durrmeyer operators in Anastassiou-Gal [16], for complex Bernstein-Durrmeyer operators based on Jacobi weights in Gal [23], for complex genuine Bernstein- Durrmeyer operator in Gal [24], for complex $q$-genuine Bernstein-Durrmeyer operator in Mahmudov [25] and for other kinds of complex Durrmeyer operators in Mahmudov [26] and Gal-Gupta-Mahmudov [28]. Also, for the case $q>1,$ exact quantitative estimates and quantitative Voronovskaja-type results for complex $q$-Lorentz polynomials, $q$-Stancu polynomials, $q$-Stancu-Faber polynomials, $q$-Bernstein-Faber polynomials, $q$-Kantorovich polynomials, $q$-Szász- Mirakjan operators obtained by different researchers are collected in the recent book of Gal [29]. In this paper we define the genuine $q$-Bernstein-Durrmeyer polynomials for $q>1.$ Note that similar to the $q$-Bernstein operators the genuine $q$-Bernstein-Durrmeyer operators in the case $q>1$ are not positive operators on $C\left[0,1\right]$. The lack of positivity makes the investigation of convergence in the case $q>1$ essentially more difficult than that for $0<q<1$. We present upper estimates in approximation and we prove the Voronovskaja type convergence theorem in compact disks in $\mathbb{C}$, centered at origin, with quantitative estimate of this convergence. These results allow us to obtain the exact degrees of approximation by complex genuine $q$-Bernstein-Durrmeyer polynomials. Our results show that approximation properties of the complex genuine $q$-Bernstein-Durrmeyer polynomials are better than approximation properties of the complex Bernstein- Durrmeyer polynomials considered in [4]. ## 2 Formulation Let $q>0.$ For any $n\in\mathbb{N}\cup\left\\{0\right\\}$, the $q$-integer $\left[n\right]_{q}$ is defined by $\left[n\right]_{q}:=1+q+...+q^{n-1},\ \ \ \left[0\right]_{q}:=0;$ and the $q$-factorial $\left[n\right]_{q}!$ by $\left[n\right]_{q}!:=\left[1\right]_{q}\left[2\right]_{q}...\left[n\right]_{q},\ \ \ \ \left[0\right]_{q}!:=1.$ For integers $0\leq k\leq n$, the $q$-binomial is defined by $\left[\begin{array}[c]{c}n\\\ k\end{array}\right]_{q}:=\frac{\left[n\right]_{q}!}{\left[k\right]_{q}!\left[n-k\right]_{q}!}.$ For $q=1$ we obviously get $\left[n\right]_{q}=n$, $\left[n\right]_{q}!=n!$, $\left[\begin{array}[c]{c}n\\\ k\end{array}\right]_{q}=\left(\begin{array}[c]{c}n\\\ k\end{array}\right).$ Moreover $\left(1-z\right)_{q}^{n}:={\displaystyle\prod_{s=0}^{n-1}}\left(1-q^{s}z\right),\ \ p_{n,k}\left(q;z\right):=\left[\begin{array}[c]{c}n\\\ k\end{array}\right]_{q}z^{k}\left(1-z\right)_{q}^{n-k},\ \ z\in\mathbb{C}.$ For fixed $q>0,\ q\neq 1$, we denote the $q$-derivative $D_{q}f\left(z\right)$ of $f$ by $D_{q}f\left(z\right)=\left\\{\begin{tabular}[c]{lll}$\frac{f\left(qz\right)-f\left(z\right)}{\left(q-1\right)z},$&$z\neq 0,$&\\\ &&\\\ $f^{\prime}\left(0\right),$&$z=0.$&\end{tabular}\ \ \ \right.$ The $q$-analogue of integration in the interval $[0,A]$ (see [1]) is defined by $\int_{0}^{A}f\left(t\right)d_{q}t:=A\left(1-q\right)\sum_{n=0}^{\infty}f\left(Aq^{n}\right)q^{n},\ \ \ 0<q<1.$ Let $\mathbb{D}_{R}$ be a disc $\mathbb{D}_{R}:=\left\\{z\in\mathbb{C}:\left\|z\right\|<R\right\\}$ in the complex plane $\mathbb{C}$. Denote by $H\left(\mathbb{D}_{R}\right)$ the space of all analytic functions on $\mathbb{D}_{R}$. For $f\in H\left(\mathbb{D}_{R}\right)$ we assume that $f\left(z\right)=\sum_{m=0}^{\infty}a_{m}z^{m}$. ###### Definition 1 For $f:\left[0,1\right]\rightarrow\mathbb{C}$, the genuine $q$-Bernstein- Durrmeyer operator is defined as follows: $U_{n,q}\left(f;z\right):=\left\\{\begin{tabular}[c]{ll}$f\left(0\right)p_{n,0}\left(q;z\right)+f\left(1\right)p_{n,n}\left(q;z\right)$&\\\ $+\left[n-1\right]_{q}{\displaystyle\sum_{k=1}^{n-1}}q^{1-k}p_{n,k}\left(q;z\right)\int_{0}^{1}p_{n-2,k-1}\left(q;qt\right)f\left(t\right)d_{q}t,$&$0<q<1,$\\\ $f\left(0\right)p_{n,0}\left(z\right)+f\left(1\right)p_{n,n}\left(z\right)+\left(n-1\right){\displaystyle\sum_{k=1}^{n-1}}p_{n,k}\left(z\right)\int_{0}^{1}p_{n-2,k-1}\left(t\right)f\left(t\right)dt,$&$q=1,$\\\ $f\left(0\right)p_{n,0}\left(q;z\right)+f\left(1\right)p_{n,n}\left(q;z\right)$&\\\ $+\left[n-1\right]_{q^{-1}}{\displaystyle\sum_{k=1}^{n-1}}q^{k-1}p_{n,k}\left(q;z\right)\int_{0}^{1}p_{n-2,k-1}\left(q^{-1};q^{-1}t\right)f\left(q^{k-n}t\right)d_{q^{-1}}t,$&$q>1,$\end{tabular}\right.$ (1) where for $n=1$ the sum is empty, i.e., equal to $0.$ $U_{n,q}\left(f;z\right)$ are linear operators reproducing linear functions and interpolating every function $f\in C\left[0,1\right]$ at $0$ and $1$. The genuine $q$-Bernstein-Durrmeyer operators are positive operators on $C\left[0,1\right]$ for $0<q\leq 1,$ and they are not positive for $q>1.$ As a consequence, the cases $0<q\leq 1$ and $q>1$ are not similar to each other regarding the convergence. For $q\rightarrow 1^{-}$ and $q\rightarrow 1^{+}$ we recapture the classical ($q=1$) genuine Bernstein-Durrmeyer polynomials. We start with the following quantitative estimates of the convergence for complex $q$-Bernstein-Durrmeyer polynomials attached to an analytic function in a disk of radius $R>1$ and center $0$. ###### Theorem 2 Let $f\in H\left(\mathbb{D}_{R}\right)$, $1\leq r<\dfrac{R}{q}$ and $q>1$. Then for all $\left\|z\right\|\leq r$ we have $\left\|U_{n,q}\left(f;z\right)-f\left(z\right)\right\|\leq\frac{r\left(1+r\right)}{\left[n+1\right]_{q}}{\displaystyle\sum\limits_{m=2}^{\infty}}\left\|a_{m}\right\|m\left(m-1\right)q^{m-2}r^{m-2}.$ Theorem 2 says that for functions analytic in $\mathbb{D}_{R}$, $R>q,$ the rate of approximation by the genuine $q$-Bernstein-Durrmeyer polynomials ($q>1$) is of order $q^{-n}$ versus $1/n$ for the classical genuine Bernstein- Durrmeyer polynomials, see [4]. The Voronovskaja theorem for the real case with a quantitative estimate is obtained by Gonska, Piţul and Raşa [14] in the following form: $\left\|U_{n}\left(f;x\right)-f\left(x\right)-\frac{x\left(1-x\right)}{n+1}f^{\prime\prime}\left(z\right)\right\|\leq\frac{x\left(1-x\right)}{n+1}\omega\left(f^{\prime\prime}\frac{2}{3\sqrt{n+3}}\right),$ for all $n\in\mathbb{N},\ 0\leq x\leq 1.$ For the complex genuine $q$-Bernstein-Durrmeyer ( $0<q\leq 1$) a quantitative estimate is obtained by Gal [4]( $q=1$) and Mahmudov [25]($0<q<1$) in the following form: $\left\|U_{n,q}\left(f;z\right)-f\left(z\right)-\frac{z\left(1-z\right)}{\left[n+1\right]_{q}}f^{\prime\prime}\left(z\right)\right\|\leq\frac{M_{r,f}}{\left[n\right]_{q}^{2}},\ \ 0<q\leq 1,$ for all $n\in\mathbb{N},\ \left\|z\right\|\leq r$. To formulate and prove the Voronovskaja type theorem with a quantitative estimate in the case $q>1$ we introduce a function $L_{q}\left(f;z\right)$. Let $R>q\geq 1$ and let $f\in H\left(\mathbb{D}_{R}\right)$. For $\left\|z\right\|<R/q^{2}$, we define $L_{q}\left(f;z\right):=\frac{\left(1-z\right)q\left(D_{q}f\left(z\right)-D_{q^{-1}}f\left(z\right)\right)}{q-1}\ \ \ \ \ \text{for }q>1$ (2) and for $0<q\leq 1$, $L_{q}\left(f;z\right)=L_{1}\left(f;z\right):=f^{\prime\prime}\left(z\right)z\left(1-z\right).$ The next theorem gives Voronovskaja type result in compact disks, for complex $q$-Bernstein-Durrmeyer polynomials attached to an analytic function in $\mathbb{D}_{R}$, $R>q^{2}>1$ and center $0$ in terms of the function $L_{q}\left(f;z\right)$. ###### Theorem 3 Let $f\in H\left(\mathbb{D}_{R}\right)$, $1\leq r<\dfrac{R}{q^{2}}$ and $q>1$. The following Voronovskaja-type result holds $\left\|U_{n,q}\left(f;z\right)-f\left(z\right)-\frac{1}{\left[n+1\right]_{q}}L_{q}\left(f;z\right)\right\|\leq\frac{4r^{2}\left(1+r\right)^{2}}{\left[n+1\right]_{q}^{2}}{\displaystyle\sum\limits_{m=3}^{\infty}}\left\|a_{m}\right\|\left(m-1\right)^{2}\left(m-2\right)^{2}\left(q^{2}r\right)^{m-2}.$ for all $n\in\mathbb{N}$, $\left\|z\right\|\leq r.$ Now we are in position to prove that the order of approximation in Theorem 2 is exactly $q^{-n}$ versus $1/n$ for the classical genuine Bernstein-Durrmeyer polynomials, see [4]. ###### Theorem 4 Let $1<q<R$, $1\leq r<\frac{R}{q^{2}}$ and $f\in H\left(\mathbb{D}_{R}\right)$. If $f$ is not a polynomial of degree $\leq 1$, the estimate $\left\\|U_{n,q}\left(f\right)-f\right\\|_{r}\geq\frac{1}{\left[n+1\right]_{q}}C_{r,q}\left(f\right),\ \ \ n\in\mathbb{N},$ holds, where the constant $C_{r,q}\left(f\right)$ depends on $f,$ $q$ and $r$ but is independent of $n$. From Theorem 3 we conclude that for $q>1,$ $\left[n+1\right]_{q}\left(U_{n,q}\left(f;z\right)-f\left(z\right)\right)\rightarrow L_{q}\left(f;z\right)$ in $H\left(\mathbb{D}_{R/q}\right)$ and therefore, $L_{q}\left(f;z\right)\in H\left(\mathbb{D}_{R/q}\right)$. Furthermore, we have the following saturation of convergence for the genuine $q$-Bernstein- Durrmeyer polynomials for fixed $q>1$. ###### Theorem 5 Let $1<q<R$, $1\leq r<\frac{R}{q^{2}}$. If a function $f$ is analytic in the disc $\mathbb{D}_{R/q}$, then $\left\|U_{n,q}\left(f;z\right)-f\left(z\right)\right\|=o\left(q^{-n}\right)$ for infinite number of points having an accumulation point on $\mathbb{D}_{R/q}$ if and only if $f$ is linear. The next theorem shows that $L_{q}\left(f;z\right),$ $q\geq 1,$ is continuous in the parameter $q$ for $f\in H\left(\mathbb{D}_{R}\right)$, $R>1$. ###### Theorem 6 Let $R>1$ and $f\in H\left(\mathbb{D}_{R}\right)$. Then for any $r,$ $0<r<R$, $\lim_{q\rightarrow 1+}L_{q}\left(f;z\right)=L_{1}\left(f;z\right)$ uniformly on $\mathbb{D}_{R}$. ## 3 Auxiliary results The $q$-analogue of Beta function for $0<q<1$ (see [1]) is defined as $B_{q}(m,n)=\int_{0}^{1}t^{m-1}(1-qt)_{q}^{n-1}d_{q}t,\,\,\,m,n>0,\ \ 0<q<1.$ Since we consider the case $q>1$, we need to use $B_{q^{-1}}(m,n):$ $B_{q^{-1}}(m,n)=\int_{0}^{1}t^{m-1}(1-q^{-1}t)_{q^{-1}}^{n-1}d_{q^{-1}}t,\,\,\,m,n>0,\ \ 0<q^{-1}<1.$ Also, it is known that $B_{q^{-1}}(m,n)=\frac{[m-1]_{q^{-1}}![n-1]_{q^{-1}}!}{[m+n-1]_{q^{-1}}!},\ \ 0<q^{-1}<1.$ For $m=0,1,...$, we have $\displaystyle\left[n-1\right]_{q^{-1}}q^{k-1}\int_{0}^{1}t^{m}p_{n-2,k-1}\left(q^{-1};q^{-1}t\right)d_{q^{-1}}t$ $\displaystyle=\left[n-1\right]_{q^{-1}}\left[\begin{array}[c]{c}n-2\\\ k-1\end{array}\right]_{q^{-1}}q^{m\left(k-n\right)}\int_{0}^{1}t^{k+m-1}\left(1-q^{-1}t\right)_{q^{-1}}^{n-k-1}d_{q^{-1}}t$ $\displaystyle=q^{m\left(k-n\right)}\frac{\left[n-1\right]_{q^{-1}}!}{\left[k-1\right]_{q^{-1}}!\left[n-k-1\right]_{q^{-1}}!}B_{q^{-1}}(k+m,n-k)$ $\displaystyle=q^{m\left(k-n\right)}\frac{\left[n-1\right]_{q^{-1}}!}{\left[k-1\right]_{q^{-1}}!\left[n-k-1\right]_{q^{-1}}!}\frac{\left[k+m-1\right]_{q^{-1}}!\left[n-k-1\right]_{q^{-1}}!}{\left[k+m+n-k-1\right]_{q^{-1}}!}$ $\displaystyle=\frac{\left[n-1\right]_{q}!\left[k+m-1\right]_{q}!}{\left[k-1\right]_{q}!\left[n+m-1\right]_{q}!}=\frac{\left[k+m-1\right]_{q}...\left[k\right]_{q}}{\left[n+m-1\right]_{q}...\left[n\right]_{q}}.$ Thus, we get the following formula for $U_{n,q}\left(e_{m};z\right):$ $\displaystyle U_{n,q}\left(e_{m};z\right)$ $\displaystyle=f\left(0\right)p_{n,0}\left(q;z\right)+f\left(1\right)p_{n,n}\left(q;z\right)$ $\displaystyle+\left[n-1\right]_{q^{-1}}{\displaystyle\sum_{k=1}^{n-1}}p_{n,k}\left(q;z\right)\int_{0}^{1}p_{n-2,k-1}\left(q^{-1};q^{-1}t\right)f\left(q^{k-n}t\right)d_{q^{-1}}t$ $\displaystyle=z^{n}+{\displaystyle\sum_{k=1}^{n-1}}p_{n,k}\left(q;z\right)\frac{\left[k+m-1\right]_{q}...\left[k\right]_{q}}{\left[n+m-1\right]_{q}...\left[n\right]_{q}}.$ (3) Note for $m=0,1,2$ we have $U_{n,q}\left(e_{0};z\right)=1,\ \ \ U_{n,q}\left(e_{1};z\right)=z,\ \ \ U_{n,q}\left(e_{2};z\right)=z^{2}+\frac{\left(1+q\right)z\left(1-z\right)}{\left[n+1\right]}.$ ###### Lemma 7 $U_{n,q}\left(e_{m};z\right)$ is a polynomial of degree less than or equal to $\min\left(m,n\right)$ and $U_{n,q}\left(e_{m};z\right)=\frac{\left[n-1\right]_{q}!}{\left[n+m-1\right]_{q}!}\sum_{s=1}^{m}S_{q}\left(m,s\right)\left[n\right]_{q}^{s}B_{n,q}\left(e_{s};z\right).$ Proof. From (3) it follows that $\displaystyle U_{n,q}\left(e_{m};z\right)$ $\displaystyle={\displaystyle\sum_{k=1}^{n}}p_{n,k}\left(q;z\right)\frac{\left[k+m-1\right]_{q}...\left[k\right]_{q}}{\left[n+m-1\right]_{q}...\left[n\right]_{q}}$ $\displaystyle=\frac{\left[n-1\right]_{q}!}{\left[n+m-1\right]_{q}!}{\displaystyle\sum_{k=1}^{n}}\left[k\right]_{q}\left[k+1\right]_{q}...\left[k+m-1\right]_{q}p_{n,k}(q;z).$ Now using $\left[k\right]_{q}\left[k+1\right]_{q}...\left[k+m-1\right]_{q}={\displaystyle\prod\limits_{s=0}^{m-1}}\left(q^{s}\left[k\right]_{q}+\left[s\right]_{q}\right)=\sum_{s=1}^{m}S_{q}\left(m,s\right)\left[k\right]_{q}^{s},$ (4) where $S_{q}\left(m,s\right)>0$, $s=1,2,...,m$, are the constants independent of $k,$ we get $\displaystyle U_{n,q}\left(e_{m};z\right)$ $\displaystyle=\frac{\left[n-1\right]_{q}!}{\left[n+m-1\right]_{q}!}{\displaystyle\sum_{k=0}^{n}}\sum_{s=1}^{m}S_{q}\left(m,s\right)\left[k\right]_{q}^{s}p_{n,k}(q;z)$ $\displaystyle=\frac{\left[n-1\right]_{q}!}{\left[n+m-1\right]_{q}!}\sum_{s=1}^{m}S_{q}\left(m,s\right)\left[n\right]_{q}^{s}B_{n,q}\left(e_{s};z\right),$ Since $B_{n,q}(e_{s};z)$ is a polynomial of degree less than or equal to $\min\left(s,n\right)$ and $S_{q}\left(m,s\right)>0$, $s=1,2,...,m$, it follows that $U_{n,q}\left(e_{m};z\right)$ is a polynomial of degree less than or equal to $\min\left(m,n\right)$. ###### Lemma 8 The numbers $S_{q}\left(m,s\right),\ \left(m,s\right)\in\left(\mathbb{N}\cup\left\\{0\right\\}\right)\times\left(\mathbb{N}\cup\left\\{0\right\\}\right),$ given by (4) enjoy the following properties $\displaystyle S_{q}\left(0,0\right)$ $\displaystyle=1,\ \ S_{q}\left(m,0\right)=0,\ \ m\in N,$ $\displaystyle S_{q}\left(m+1,s\right)$ $\displaystyle=\left[m\right]_{q}S_{q}\left(m,s\right)+q^{m}S_{q}\left(m,s-1\right),\ \ \ m\in N_{0},\ s\in N,$ $\displaystyle S_{q}\left(m+1,m+1\right)$ $\displaystyle=q^{m}S_{q}\left(m,m\right),\ \ \ S_{q}\left(m,s\right)=0\ \ \text{for\ }\ s>m.$ Also, the following lemma holds. ###### Lemma 9 For all $m,n\in\mathbb{N}$ the identity $\frac{\left[n-1\right]_{q}!}{\left[n+m-1\right]_{q}!}\sum_{s=1}^{m}S_{q}\left(m,s\right)\left[n\right]_{q}^{s}=1,$ holds. Proof. It follows from end points interpolation property of $U_{n,q}\left(e_{m};z\right)$ and $B_{n,q}\left(e_{s};z\right).$Indeed $1=U_{n,q}\left(e_{m};1\right)=\frac{\left[n-1\right]_{q}!}{\left[n+m-1\right]_{q}!}\sum_{s=1}^{m}S_{q}\left(m,s\right)\left[n\right]_{q}^{s}B_{n,q}\left(e_{s};1\right)=\frac{\left[n-1\right]_{q}!}{\left[n+m-1\right]_{q}!}\sum_{s=1}^{m}S_{q}\left(m,s\right)\left[n\right]_{q}^{s}.$ Lemma 9 implies that for all $m,n\in\mathbb{N}$ and $\left\|z\right\|\leq r$ we have $\displaystyle\left\|U_{n,q}\left(e_{m};z\right)\right\|$ $\displaystyle\leq\frac{\left[n-1\right]_{q}!}{\left[n+m-1\right]_{q}!}\sum_{s=1}^{m}S_{q}\left(m,s\right)\left[n\right]_{q}^{s}\left\|B_{n,q}\left(e_{s};z\right)\right\|$ $\displaystyle\leq\frac{\left[n-1\right]_{q}!}{\left[n+m-1\right]_{q}!}\sum_{s=1}^{m}S_{q}\left(m,s\right)\left[n\right]_{q}^{s}r^{s}\leq r^{m}.$ (5) For our purpose first we need a recurrence formula for $U_{n,q}\left(e_{m};z\right).$ ###### Lemma 10 For all $m,n\in\mathbb{N}\cup\left\\{0\right\\}$ and $z\in\mathbb{C}$ we have $U_{n,q}\left(e_{m+1};z\right)=\frac{q^{m}z\left(1-z\right)}{\left[n+m\right]_{q}}D_{q}U_{n,q}\left(e_{m};z\right)+\frac{q^{m}\left[n\right]z+\left[m\right]_{q}}{\left[n+m\right]_{q}}U_{n,q}\left(e_{m};z\right).$ (6) Proof. By simple calculation we obtain (see [27]) $z\left(1-z\right)D_{q}\left(p_{n,k}\left(q;z\right)\right)=\left(\left[k\right]_{q}-\left[n\right]_{q}z\right)p_{n,k}\left(q;z\right),$ and $\displaystyle U_{n,q}\left(e_{m};z\right)$ $\displaystyle=z^{n}+{\displaystyle\sum_{k=1}^{n-1}}p_{n,k}\left(q;z\right)\frac{\left[k+m-1\right]_{q}...\left[k\right]_{q}}{\left[n+m-1\right]_{q}...\left[n\right]_{q}}$ $\displaystyle=z^{n}+{\displaystyle\sum_{k=1}^{n-1}}p_{n,k}\left(q;z\right)I_{k,m},$ $\displaystyle I_{k,m}$ $\displaystyle:=\frac{\left[k+m-1\right]_{q}...\left[k\right]_{q}}{\left[n+m-1\right]_{q}...\left[n\right]_{q}}.$ It follows that $\displaystyle z\left(1-z\right)D_{q}U_{n,q}\left(e_{m};z\right)$ $\displaystyle=\left[n\right]_{q}z\left(1-z\right)z^{n-1}+{\displaystyle\sum_{k=1}^{n-1}}\left(\left[k\right]_{q}-\left[n\right]_{q}z\right)p_{n,k}\left(q;z\right)I_{k,m}$ $\displaystyle=\left[n\right]_{q}z^{n}+{\displaystyle\sum_{k=1}^{n-1}}\left[k\right]_{q}p_{n,k}\left(q;z\right)I_{k,m}-\left[n\right]_{q}z{\displaystyle\sum_{k=1}^{n-1}}p_{n,k}\left(q;z\right)I_{k,m}-\left[n\right]_{q}z^{n+1}$ $\displaystyle=\left[n\right]_{q}z^{n}+{\displaystyle\sum_{k=1}^{n-1}}\left[k\right]_{q}p_{n,k}\left(q;z\right)I_{k,m}-z\left[n\right]_{q}U_{n,q}\left(e_{m};z\right)$ $\displaystyle=\left[n\right]_{q}z^{n}+q^{-m}{\displaystyle\sum_{k=1}^{n-1}}p_{n,k}\left(q;z\right)\left(q^{m}\left[k\right]_{q}+\left[m\right]_{q}-\left[m\right]_{q}\right)I_{k,m}-z\left[n\right]_{q}U_{n,q}\left(e_{m};z\right)$ $\displaystyle=\left[n\right]_{q}z^{n}+q^{-m}{\displaystyle\sum_{k=1}^{n-1}}p_{n,k}\left(q;z\right)\left(q^{m}\left[k\right]_{q}+\left[m\right]_{q}-\left[m\right]_{q}\right)I_{k,m}-z\left[n\right]_{q}U_{n,q}\left(e_{m};z\right)$ $\displaystyle=q^{-m}\left(q^{m}\left[n\right]_{q}+\left[m\right]_{q}-\left[m\right]_{q}\right)z^{n}+q^{-m}\left[n+m\right]_{q}{\displaystyle\sum_{k=1}^{n-1}}p_{n,k}\left(q;z\right)I_{k,m+1}$ $\displaystyle-q^{-m}\left[m\right]_{q}{\displaystyle\sum_{k=1}^{n-1}}p_{n,k}\left(q;z\right)I_{k,m}-z\left[n\right]_{q}U_{n,q}\left(e_{m};z\right)$ $\displaystyle=q^{-m}\left[n+m\right]_{q}U_{n,q}\left(e_{m+1};z\right)-q^{-m}\left[m\right]_{q}U_{n,q}\left(e_{m};z\right)-z\left[n\right]_{q}U_{n,q}\left(e_{m};z\right),$ (7) which implies the recurrence in the statement. Let $\Theta_{n,m}\left(q;z\right):=U_{n,q}\left(e_{m};z\right)-z^{m}-\frac{1}{\left[n+1\right]_{q}}\left(q{\displaystyle\sum\limits_{i=1}^{m-1}}\left[i\right]_{q}+{\displaystyle\sum\limits_{i=1}^{m-1}}\left[i\right]_{q^{-1}}\right)z^{m-1}\left(1-z\right).$ Using the recurrence formula (6) we prove two more recurrence formulas. ###### Lemma 11 For all $m,n\in\mathbb{N}$ and $z\in\mathbb{C}$ we have $\displaystyle U_{n,q}\left(e_{m};z\right)-z^{m}$ $\displaystyle=\frac{q^{m-1}z\left(1-z\right)}{\left[n+m-1\right]_{q}}D_{q}U_{n,q}\left(e_{m-1};z\right)$ $\displaystyle+\frac{q^{m-1}\left[n\right]z+\left[m-1\right]_{q}}{\left[n+m-1\right]_{q}}\left(U_{n,q}\left(e_{m-1};z\right)-z^{m-1}\right)+\frac{\left[m-1\right]_{q}}{\left[n+m-1\right]_{q}}\left(1-z\right)z^{m-1},$ (8) $\displaystyle\Theta_{n,m}\left(q;z\right)$ $\displaystyle=\frac{q^{m-1}z\left(1-z\right)}{\left[n+m-1\right]_{q}}D_{q}\left(U_{n,q}\left(e_{m-1};z\right)-z^{m-1}\right)$ $\displaystyle+\frac{q^{m-1}\left[n\right]z+\left[m-1\right]_{q}}{\left[n+m-1\right]_{q}}\Theta_{n,m-1}\left(q;z\right)+R_{n,m}\left(q;z\right),$ (9) where $R_{n,m}\left(q;z\right)=\frac{\left[m-1\right]_{q}}{\left[n+m-1\right]_{q}\left[n+1\right]_{q}}\left[\left(1+q^{m-1}\right)+\left(q{\displaystyle\sum\limits_{i=1}^{m-2}}\left[i\right]_{q}+{\displaystyle\sum\limits_{i=1}^{m-2}}\left[i\right]_{q^{-1}}\right)\left(z+1\right)\right]z^{m-2}\left(1-z\right).$ (10) Proof. From the recurrence formula in Lemma 10, for all $m\geq 2$ we get $\displaystyle U_{n,q}\left(e_{m};z\right)-z^{m}$ $\displaystyle=\frac{q^{m-1}z\left(1-z\right)}{\left[n+m-1\right]_{q}}D_{q}U_{n,q}\left(e_{m-1};z\right)+\frac{q^{m-1}\left[n\right]z+\left[m-1\right]_{q}}{\left[n+m-1\right]_{q}}\left(U_{n,q}\left(e_{m-1};z\right)-z^{m-1}\right)$ $\displaystyle+\frac{q^{m-1}\left[n\right]z+\left[m-1\right]_{q}}{\left[n+m-1\right]_{q}}z^{m-1}-z^{m}$ $\displaystyle=\frac{q^{m-1}z\left(1-z\right)}{\left[n+m-1\right]_{q}}D_{q}U_{n,q}\left(e_{m-1};z\right)+\frac{q^{m-1}\left[n\right]z+\left[m-1\right]_{q}}{\left[n+m-1\right]_{q}}\left(U_{n,q}\left(e_{m-1};z\right)-z^{m-1}\right)$ $\displaystyle+\frac{\left[m-1\right]_{q}}{\left[n+m-1\right]_{q}}\left(1-z\right)z^{m-1},$ and $\displaystyle U_{n,q}\left(e_{m};z\right)-z^{m}-\frac{1}{\left[n+1\right]_{q}}\left(q{\displaystyle\sum\limits_{i=1}^{m-1}}\left[i\right]_{q}+{\displaystyle\sum\limits_{i=1}^{m-1}}\left[i\right]_{q^{-1}}\right)z^{m-1}\left(1-z\right)$ $\displaystyle=\frac{q^{m-1}z\left(1-z\right)}{\left[n+m-1\right]_{q}}D_{q}\left(U_{n,q}\left(e_{m-1};z\right)-z^{m-1}\right)$ $\displaystyle+\frac{q^{m-1}\left[n\right]z+\left[m-1\right]_{q}}{\left[n+m-1\right]_{q}}\left(U_{n,q}\left(e_{m};z\right)-z^{m-1}-\frac{1}{\left[n+1\right]_{q}}\left(q{\displaystyle\sum\limits_{i=1}^{m-2}}\left[i\right]_{q}+{\displaystyle\sum\limits_{i=1}^{m-2}}\left[i\right]_{q^{-1}}\right)z^{m-2}\left(1-z\right)\right)$ $\displaystyle+R_{n,m}\left(q;z\right),$ where $\displaystyle R_{n,m}\left(q;z\right)$ $\displaystyle=\frac{\left[m-1\right]_{q}}{\left[n+m-1\right]_{q}}\left(1-z\right)z^{m-1}-\frac{1}{\left[n+1\right]_{q}}\left(q{\displaystyle\sum\limits_{i=1}^{m-1}}\left[i\right]_{q}+{\displaystyle\sum\limits_{i=1}^{m-1}}\left[i\right]_{q^{-1}}\right)z^{m-1}\left(1-z\right)$ $\displaystyle+\frac{q^{m-1}\left[m-1\right]_{q}}{\left[n+m-1\right]_{q}}\left(1-z\right)z^{m-1}$ $\displaystyle+\frac{q^{m-1}\left[n\right]z+\left[m-1\right]_{q}}{\left[n+m-1\right]_{q}}\frac{1}{\left[n+1\right]_{q}}\left(q{\displaystyle\sum\limits_{i=1}^{m-2}}\left[i\right]_{q}+{\displaystyle\sum\limits_{i=1}^{m-2}}\left[i\right]_{q^{-1}}\right)z^{m-2}\left(1-z\right)$ $\displaystyle:=T_{n^{\prime}m}\left(q\right)z^{m-1}\left(1-z\right)+\frac{\left[m-1\right]_{q}}{\left[n+m-1\right]_{q}\left[n+1\right]_{q}}\left(q{\displaystyle\sum\limits_{i=1}^{m-2}}\left[i\right]_{q}+{\displaystyle\sum\limits_{i=1}^{m-2}}\left[i\right]_{q^{-1}}\right)z^{m-2}\left(1-z\right).$ Again by simple calculation we obtain $\displaystyle T_{n,m}\left(q\right)$ $\displaystyle=\frac{\left[m-1\right]_{q}}{\left[n+m-1\right]_{q}}-\frac{1}{\left[n+1\right]_{q}}\left(q{\displaystyle\sum\limits_{i=1}^{m-1}}\left[i\right]_{q}+{\displaystyle\sum\limits_{i=1}^{m-1}}\left[i\right]_{q^{-1}}\right)$ $\displaystyle+\frac{q^{m-1}\left[m-1\right]_{q}}{\left[n+m-1\right]_{q}}+\frac{q^{m-1}\left[n\right]_{q}}{\left[n+m-1\right]_{q}}\frac{1}{\left[n+1\right]_{q}}\left(q{\displaystyle\sum\limits_{i=1}^{m-1}}\left[i\right]_{q}+{\displaystyle\sum\limits_{i=1}^{m-1}}\left[i\right]_{q^{-1}}\right)$ $\displaystyle-\frac{q^{m-1}\left[n\right]_{q}}{\left[n+m-1\right]_{q}}\frac{1}{\left[n+1\right]_{q}}\left(q\left[m-1\right]_{q}+\left[m-1\right]_{q^{-1}}\right)$ $\displaystyle=\left(\frac{\left[m-1\right]_{q}}{\left[n+m-1\right]_{q}}+\frac{q^{m-1}\left[m-1\right]_{q}}{\left[n+m-1\right]_{q}}-\frac{q^{m-1}\left[n\right]_{q}}{\left[n+m-1\right]_{q}}\frac{1}{\left[n+1\right]_{q}}\left(q\left[m-1\right]_{q}+\left[m-1\right]_{q^{-1}}\right)\right)$ $\displaystyle+\left(\frac{q^{m-1}\left[n\right]_{q}}{q^{m-1}\left[n\right]_{q}+\left[m-1\right]_{q}}-1\right)\frac{1}{\left[n+1\right]_{q}}\left(q{\displaystyle\sum\limits_{i=1}^{m-1}}\left[i\right]_{q}+{\displaystyle\sum\limits_{i=1}^{m-1}}\left[i\right]_{q^{-1}}\right)$ $\displaystyle:=T_{n,m}^{1}\left(q\right)+T_{n,m}^{2}\left(q\right),$ where $T_{n,m}^{1}\left(q\right)$ and $T_{n,m}^{2}\left(q\right)$ can be simplified as follows: $\displaystyle T_{n,m}^{2}\left(q\right)$ $\displaystyle=\left(1-\frac{q^{m-1}\left[n\right]_{q}}{\left[n+m-1\right]_{q}}\right)\frac{1}{\left[n+1\right]_{q}}\left(q{\displaystyle\sum\limits_{i=1}^{m-2}}\left[i\right]_{q}+{\displaystyle\sum\limits_{i=1}^{m-2}}\left[i\right]_{q^{-1}}\right)$ $\displaystyle=\frac{\left[m-1\right]_{q}}{\left[n+m-1\right]_{q}\left[n+1\right]_{q}}\left(q{\displaystyle\sum\limits_{i=1}^{m-2}}\left[i\right]_{q}+{\displaystyle\sum\limits_{i=1}^{m-2}}\left[i\right]_{q^{-1}}\right)$ and $\displaystyle T_{n,m}^{1}\left(q\right)=\frac{\left[m-1\right]_{q}}{\left[n+m-1\right]_{q}}+\frac{q^{m-1}\left[m-1\right]_{q}}{\left[n+m-1\right]_{q}}$ $\displaystyle-\frac{q^{m-1}\left[n\right]_{q}}{\left[n+m-1\right]_{q}}\frac{1}{\left[n+1\right]_{q}}\left(q\left[m-1\right]_{q}+\left[m-1\right]_{q^{-1}}\right)$ $\displaystyle=\left[m-1\right]_{q}\left(\frac{1}{\left[n+m-1\right]_{q}}-\frac{q}{\left[n+1\right]_{q}}\frac{q^{m-1}\left[n\right]_{q}}{\left[n+m-1\right]_{q}}\right)$ $\displaystyle+\left[m-1\right]_{q}\left(\frac{q^{m-1}}{\left[n+m-1\right]_{q}}-\frac{1}{\left[n+1\right]_{q}}\frac{q\left[n\right]_{q}}{\left[n+m-1\right]_{q}}\right)$ $\displaystyle=\left[m-1\right]_{q}\frac{\left[n+1\right]_{q}-q^{m}\left[n\right]_{q}}{\left[n+m-1\right]_{q}\left[n+1\right]_{q}}+\left[m-1\right]_{q}\frac{q^{m-1}\left[n+1\right]_{q}-q\left[n\right]_{q}}{\left[n+m-1\right]_{q}\left[n+1\right]_{q}}$ $\displaystyle=\left[m-1\right]_{q}\frac{\left(1+q^{m-1}\right)\left[n+1\right]_{q}-\left(1+q^{m-1}\right)q\left[n\right]_{q}}{\left[n+m-1\right]_{q}\left[n+1\right]_{q}}$ $\displaystyle=\frac{\left[m-1\right]_{q}\left(1+q^{m-1}\right)}{\left[n+m-1\right]_{q}\left[n+1\right]_{q}}.$ ###### Lemma 12 Let $q>1$ and $f\in H\left(\mathbb{D}_{R}\right)$. The function $L_{q}\left(f;z\right)$ has the following representation $L_{q}\left(f;z\right)={\displaystyle\sum\limits_{m=2}^{\infty}}a_{m}\left(q{\displaystyle\sum\limits_{i=1}^{m-1}}\left[i\right]+{\displaystyle\sum\limits_{i=1}^{m-1}}\left[i\right]_{q^{-1}}\right)z^{m-1}\left(1-z\right),\ \ \ z\in\mathbb{D}_{R}.$ Proof. Using the following identity $\displaystyle\left[m\right]_{q}-m$ $\displaystyle=1+q+q^{2}+...q^{m-1}-m$ $\displaystyle=\left(1-1\right)+\left(q-1\right)+\left(q^{2}-1\right)+...+\left(q^{m-1}-1\right)$ $\displaystyle=\left(q-1\right)\left[1\right]_{q}+\left(q-1\right)\left[2\right]_{q}+...+\left(q-1\right)\left[m-1\right]_{q}+$ $\displaystyle=\left(q-1\right)\left(\left[1\right]_{q}+...+\left[m-1\right]_{q}\right)=\left(q-1\right){\displaystyle\sum\limits_{i=1}^{m-1}}\left[i\right]_{q},$ we get $\displaystyle L_{q}\left(f;z\right)$ $\displaystyle=\sum_{m=2}^{\infty}a_{m}\left(\frac{q\left(\left[m\right]_{q}-\left[m\right]_{q^{-1}}\right)}{q-1}\right)z^{m-1}\left(1-z\right)$ $\displaystyle=\sum_{m=2}^{\infty}a_{m}\left(\frac{q\left(\left[m\right]_{q}-m\right)}{q-1}+\frac{\left[m\right]_{q^{-1}}-m}{q^{-1}-1}\right)z^{m-1}\left(1-z\right)$ $\displaystyle={\displaystyle\sum\limits_{m=2}^{\infty}}a_{m}\left(q{\displaystyle\sum\limits_{i=1}^{m-1}}\left[i\right]_{q}+{\displaystyle\sum\limits_{i=1}^{m-1}}\left[i\right]_{q^{-1}}\right)z^{m-1}\left(1-z\right),$ where $f\left(z\right)=\sum_{m=0}^{\infty}a_{m}z^{m}$. ## 4 Proofs of the main results Firstly we prove that $U_{n,q}(f;z)=\sum_{m=0}^{\infty}a_{m}U_{n,q}(e_{m},z)$. Indeed denoting $f_{k}(z)=\sum_{j=0}^{k}a_{j}z^{j},\|z\|\leq r$ with $m\in\mathbb{N}$, by the linearity of $U_{n,q}$, we have $U_{n,q}(f_{k},z)=\sum_{m=0}^{k}a_{m}U_{n,q}(e_{m},z),$ and it is sufficient to show that for any fixed $n\in\mathbb{N}$ and $\|z\|\leq r$ with $r\geq 1$, we have $\lim_{k\rightarrow\infty}U_{n,q}(f_{k},z)=U_{n,q}(f;z)$. But this is immediate from $\lim_{k\rightarrow\infty}\|\|f_{k}-f\|\|_{r}=0$, the norm being the defined as $\|\|f\|\|_{r}=\max\\{\|f(z)\|:{\|z\|\leq r}\\}$ and from the inequality $\displaystyle\|U_{n,q}(f_{k},z)-U_{n,q}(f,z)\|$ $\displaystyle\leq\|f_{k}(0)-f(0)\|\cdot\|(1-z)^{n}\|+\|f_{k}(1)-f(1)\|\cdot\|z^{n}\|$ $\displaystyle+[n+1]_{q^{-1}}\sum_{j=1}^{n-1}\|p_{n,j}(q;z)\|q^{j-1}\int_{0}^{1}p_{n-2,j-1}(q^{-1},q^{-1}t)\|f_{k}(t)-f(t)\|d_{q^{-1}}t$ $\displaystyle\leq C_{r,n}\|\|f_{k}-f\|\|_{r},$ valid for all $\|z\|\leq r$, where $\displaystyle C_{r,n}$ $\displaystyle=(1+r)^{n}+r^{n}+[n+1]_{q^{-1}}\sum_{j=1}^{n-1}\left[\begin{array}[c]{c}n\\\ j\end{array}\right]_{q}(1+r)^{n-j}r^{j}q^{j-1}\int_{0}^{1}p_{n-2,j-1}(q^{-1};q^{-1}t)d_{q^{-1}}t$ $\displaystyle=(1+r)^{n}+r^{n}+\sum_{j=1}^{n-1}\left[\begin{array}[c]{c}n\\\ j\end{array}\right]_{q}(1+r)^{n-j}r^{j}q^{j-1}.$ Therefore we get $\|U_{n,q}(f;z)-f(z)\|\leq\sum_{m=0}^{\infty}\|a_{m}\|\|U_{n,q}(e_{m},z)-e_{m}(z)\|=\sum_{m=2}^{\infty}\|a_{m}\|\|U_{n,q}(e_{m},z)-e_{m}(z)\|,$ as $U_{n,q}(e_{0},z)=e_{0}(z)$ and .$U_{n,q}(e_{1},z)=e_{1}(z).$ Proof of Theorem 2. From the recurrence formula (8) and the inequality (5) for $m\geq 2$ we get $\displaystyle\left\|U_{n,q}\left(e_{m};z\right)-z^{m}\right\|$ $\displaystyle\leq\frac{q^{m-1}z\left(1-z\right)}{q^{m-2}\left[n+1\right]_{q}+\left[m-2\right]_{q}}\left\|D_{q}U_{n,q}\left(e_{m-1};z\right)\right\|$ $\displaystyle+\frac{q^{m-1}\left[n\right]z+\left[m-1\right]_{q}}{q^{m-1}\left[n\right]_{q}+\left[m-1\right]_{q}}\left\|U_{n,q}\left(e_{m-1};z\right)-z^{m-1}\right\|+\frac{\left[m-1\right]_{q}}{q^{m-2}\left[n+1\right]_{q}+\left[m-2\right]_{q}}\left\|1-z\right\|\left\|z\right\|^{m-1}.$ It is known that by a linear transformation, the Bernstein inequality in the closed unit disk becomes $\left\|P_{k}^{\prime}\left(z\right)\right\|\leq\frac{k}{qr_{1}}\left\\|P_{k}\right\\|_{qr},\ \ \ \text{for all\ \ }\left\|z\right\|\leq qr,\ \ r\geq 1,$ which combined with the mean value theorem in complex analysis implies $\left\|D_{q}\left(P_{k};z\right)\right\|\leq\left\\|P_{k}^{\prime}\right\\|_{qr}\leq\frac{k}{qr}\left\\|P_{k}\right\\|_{qr},$ for all $\left\|z\right\|\leq qr$, where $P_{k}\left(z\right)$ is a complex polynomial of degree $\leq k$. It follows that $\displaystyle\left\|U_{n,q}\left(e_{m};z\right)-z^{m}\right\|$ $\displaystyle\leq\frac{q^{m-1}r\left(1+r\right)}{q^{m-2}\left[n+1\right]_{q}+\left[m-2\right]_{q}}\frac{m-1}{qr}\left\\|U_{n,q}\left(e_{m-1}\right)\right\\|_{qr}$ $\displaystyle+r\left\|U_{n,q}\left(e_{m-1};z\right)-z^{m-1}\right\|+\frac{\left[m-1\right]_{1/q}}{\left[n+1\right]_{q}}\left(1+r\right)r^{m-1}$ $\displaystyle\leq\frac{\left(m-1\right)}{\left[n+1\right]_{q}}\left(1+r\right)q^{m-1}r^{m-1}+r\left\|U_{n,q}\left(e_{m-1};z\right)-z^{m-1}\right\|+\frac{\left[m-1\right]_{1/q}}{\left[n+1\right]_{q}}\left(1+r\right)r^{m-1}$ $\displaystyle\leq 2q\left(m-1\right)\frac{r\left(1+r\right)}{\left[n+1\right]_{q}}\left(qr\right)^{m-2}+r\left\|U_{n,q}\left(e_{m-1};z\right)-z^{m-1}\right\|.$ By writing the last inequality for $m=2,3,...,$ we easily obtain, step by step the following $\displaystyle\left\|U_{n,q}\left(e_{m};z\right)-z^{m}\right\|$ $\displaystyle\leq r\left(r\left\|U_{n,q}\left(e_{m-2};z\right)-z^{m-2}\right\|+2\frac{\left(m-2\right)}{\left[n+1\right]_{q}}r\left(1+r\right)\left(qr\right)^{m-3}\right)$ $\displaystyle+2\frac{\left(m-1\right)}{\left[n+1\right]_{q}}r\left(1+r\right)\left(qr\right)^{m-2}$ $\displaystyle=r^{2}\left\|U_{n,q}\left(e_{m-2};z\right)-z^{m-2}\right\|+2\frac{r\left(1+r\right)}{\left[n+1\right]_{q}}r^{m-2}\left(m-1+m-2\right)$ $\displaystyle\leq...\leq\frac{r\left(1+r\right)}{\left[n+1\right]_{q}}m\left(m-1\right)q^{m-2}r^{m-2}.$ It follows that $\left\|U_{n,q}\left(f;z\right)-f\left(z\right)\right\|\leq{\displaystyle\sum\limits_{m=2}^{\infty}}\left\|a_{m}\right\|\left\|U_{n,q}\left(e_{m};z\right)-z^{m}\right\|\leq\frac{r\left(1+r\right)}{\left[n+1\right]_{q}}{\displaystyle\sum\limits_{m=2}^{\infty}}\left\|a_{m}\right\|m\left(m-1\right)q^{m-2}r^{m-2}.$ The second main result of the paper is the Voronovskaja theorem with a quantitative estimate for the complex version of genuine $q$-Bernstein- Durrmeyer polynomials. Proof of Theorem 3. By Lemma 11 we have $\displaystyle\Theta_{n,m}\left(q;z\right)$ $\displaystyle=\frac{q^{m-1}z\left(1-z\right)}{\left[n+m-1\right]_{q}}D_{q}\left(U_{n,q}\left(e_{m-1};z\right)-z^{m-1}\right)$ (11) $\displaystyle+\frac{q^{m-1}\left[n\right]z+\left[m-1\right]_{q}}{\left[n+m-1\right]_{q}}\Theta_{n,m-1}\left(q;z\right)+R_{n,m}\left(q;z\right),$ where $R_{n,m}\left(q;z\right)=\frac{\left[m-1\right]_{q}}{\left[n+m-1\right]_{q}\left[n+1\right]_{q}}\left[\left(1+q^{m-1}\right)+\left(q{\displaystyle\sum\limits_{i=1}^{m-2}}\left[i\right]_{q}+{\displaystyle\sum\limits_{i=1}^{m-2}}\left[i\right]_{q^{-1}}\right)\left(z+1\right)\right]z^{m-2}\left(1-z\right).$ It follows that $\displaystyle\left\|R_{n,m}\left(q;z\right)\right\|$ $\displaystyle\leq\frac{\left[m-1\right]_{q}}{\left[n+1\right]_{q}^{2}}\left(\left(1+q^{m-1}\right)r+\left(q{\displaystyle\sum\limits_{i=1}^{m-2}}\left[i\right]_{q}+{\displaystyle\sum\limits_{i=1}^{m-2}}\left[i\right]_{q^{-1}}\right)\left(1+r\right)\right)\left(1+r\right)r^{m-2}$ $\displaystyle\leq\frac{\left[m-1\right]_{q}}{\left[n+1\right]_{q}^{2}}\left(\left(1+q^{m-1}\right)+\left(q\left(m-2\right)\left[m-2\right]_{q}+\left(m-2\right)^{2}\right)\right)\left(1+r\right)^{2}r^{m-2}$ $\displaystyle=\frac{q^{m-2}\left[m-1\right]_{q^{-1}}}{\left[n+1\right]_{q}^{2}}q^{m-2}\left(\left(\frac{1}{q^{m-2}}+q\right)+\left(m-2\right)\left[m-2\right]_{q^{-1}}+\frac{1}{q^{m-2}}\left(m-2\right)^{2}\right)\left(1+r\right)^{2}r^{m-2}$ $\displaystyle\leq\frac{3}{\left[n+1\right]_{q}^{2}}\left(m-1\right)\left(m-2\right)^{2}\left(1+r\right)^{2}\left(q^{2}r\right)^{m-2}$ for all $m\geq 2$, $n\in\mathbb{N}$ and $z\in\mathbb{C}$. (11) implies that for $\left\|z\right\|\leq r$ $\displaystyle\left\|\Theta_{n,m}\left(q;z\right)\right\|$ $\displaystyle\leq r\left\|\Theta_{n,m-1}\left(q;z\right)\right\|+\frac{q^{m-1}r\left(1+r\right)}{q^{m-2}\left[n+1\right]_{q}}\frac{m-1}{qr}\left\\|U_{n,q}\left(e_{m-1}\right)-e_{m-1}\right\\|_{qr}$ $\displaystyle+\frac{3}{\left[n+1\right]_{q}^{2}}\left(m-1\right)\left(m-2\right)^{2}\left(1+r\right)^{2}\left(q^{2}r\right)^{m-2}$ $\displaystyle\leq r\left\|\Theta_{n,m-1}\left(q;z\right)\right\|+\frac{r^{2}\left(1+r\right)^{2}}{\left[n+1\right]_{q}^{2}}\left(m-1\right)^{2}\left(m-2\right)\left(q^{2}r\right)^{m-3}$ $\displaystyle+\frac{3}{\left[n+1\right]_{q}^{2}}\left(m-1\right)\left(m-2\right)^{2}\left(1+r\right)^{2}\left(q^{2}r\right)^{m-2}$ $\displaystyle\leq r\left\|\Theta_{n,m-1}\left(q;z\right)\right\|+\frac{4r^{2}\left(1+r\right)^{2}}{\left[n+1\right]_{q}^{2}}\left(m-1\right)^{2}\left(m-2\right)\left(q^{2}r\right)^{m-2}.$ By writing the last inequality for $m=3,4,...,$ we easily obtain, step by step the following $\displaystyle\left\|U_{n,q}\left(f;z\right)-f\left(z\right)-\frac{1}{\left[n+1\right]_{q}}L_{q}\left(f;z\right)\right\|$ $\displaystyle\leq\frac{4r^{2}\left(1+r\right)^{2}}{\left[n+1\right]_{q}^{2}}{\displaystyle\sum\limits_{m=2}^{\infty}}\left\|a_{m}\right\|\left(q^{2}r\right)^{m-2}{\displaystyle\sum\limits_{j=2}^{m}}\left(j-1\right)^{2}\left(j-2\right)\leq\frac{4r^{2}\left(1+r\right)^{2}}{\left[n+1\right]_{q}^{2}}{\displaystyle\sum\limits_{m=2}^{\infty}}\left\|a_{m}\right\|\left(m-1\right)^{2}\left(m-2\right)^{2}\left(q^{2}r\right)^{m-2}.$ Proof of Theorem 4. For all $z\in\mathbb{D}_{R}$ and $n\in\mathbb{N}$ we get $U_{n,q}\left(f;z\right)-f\left(z\right)=\frac{1}{\left[n+1\right]_{q}}\left\\{L_{q}\left(f;z\right)+\left[n+1\right]_{q}\left(U_{n,q}\left(f;z\right)-f\left(z\right)-\frac{1}{\left[n+1\right]_{q}}L_{q}\left(f;z\right)\right)\right\\}.$ It follows that $\left\\|U_{n,q}\left(f\right)-f\right\\|_{r}\geq\frac{1}{\left[n+1\right]_{q}}\left\\{\left\\|L_{q}\left(f;z\right)\right\\|_{r}-\left[n+1\right]_{q}\left\\|U_{n,q}\left(f\right)-f-\frac{1}{\left[n+1\right]_{q}}L_{q}\left(f;z\right)\right\\|_{r}\right\\}.$ Because by hypothesis $f$ is not a polynomial of degree $\leq 1$ in $\mathbb{D}_{R}$, it follows $\left\\|L_{q}\left(f;z\right)\right\\|_{r}>0$. Indeed, assuming the contrary it follows that $L_{q}\left(f;z\right)=0$ for all $z\in\mathbb{D}_{r}$ that is $D_{q}f\left(z\right)=D_{q^{-1}}f\left(z\right)$ for all $z\in\mathbb{D}_{r}.$ Thus $a_{m}=0,$ $m=2,3,...$ and, $f$ is linear, which is contradiction with the hypothesis. Now, by Theorem 3 we have $\displaystyle\left[n+1\right]_{q}\left\|U_{n,q}\left(f;z\right)-f\left(z\right)-\frac{1}{\left[n+1\right]_{q}}L_{q}\left(f;z\right)\right\|$ $\displaystyle\leq\frac{4r^{2}\left(1+r\right)^{2}}{\left[n+1\right]_{q}}{\displaystyle\sum\limits_{m=3}^{\infty}}\left\|a_{m}\right\|\left(m-1\right)^{2}\left(m-2\right)^{2}\left(q^{2}r\right)^{m-2}$ $\displaystyle\rightarrow 0\ \ \ \text{as\ \ \ }n\rightarrow\infty.$ Consequently, there exists $n_{1}$ (depending only on $f$ and $r$) such that for all $n\geq n_{1}$ we have $\left\\|L_{q}\left(f;z\right)\right\\|_{r}-\left[n+1\right]_{q}\left\\|U_{n,q}\left(f\right)-f-\frac{1}{\left[n+1\right]_{q}}L_{q}\left(f;z\right)\right\\|_{r}\geq\frac{1}{2}\left\\|L_{q}\left(f;z\right)\right\\|_{r},$ which implies $\left\\|U_{n,q}\left(f\right)-f\right\\|_{r}\geq\frac{1}{2\left[n+1\right]_{q}}\left\\|L_{q}\left(f;z\right)\right\\|_{r},\ \ \ \text{for all }n\geq n_{1}.$ For $1\leq n\leq n_{1}-1$ we have $\left\\|U_{n,q}\left(f\right)-f\right\\|_{r}\geq\frac{1}{\left[n+1\right]_{q}}\left(\left[n+1\right]_{q}\left\\|U_{n,q}\left(f\right)-f\right\\|_{r}\right)=\frac{1}{\left[n+1\right]_{q}}M_{r,n,q}\left(f\right)>0,$ which finally implies that $\left\\|U_{n,q}\left(f\right)-f\right\\|_{r}\geq\frac{1}{\left[n+1\right]_{q}}C_{r,q}\left(f\right),$ for all $n$, with $C_{r,q}\left(f\right)=\min\left\\{M_{r,1,q}\left(f\right),...,M_{r,n_{1}-1,q}\left(f\right),\frac{1}{2}\left\\|L_{q}\left(f;z\right)\right\\|_{r}\right\\}$, which ends the proof. Proof of Theorem 6. Let $1\leq r<R$, let $1<q_{0}<\dfrac{R}{r}$ be fixed. Then by Lemma 12 for any $1\leq q\leq q_{0}$ and $\left\|z\right\|\leq r$, we have $\displaystyle L_{q}\left(f;z\right)$ $\displaystyle={\displaystyle\sum\limits_{m=2}^{\infty}}a_{m}\left(q{\displaystyle\sum\limits_{i=1}^{m-1}}\left[i\right]_{q}+{\displaystyle\sum\limits_{i=1}^{m-1}}\left[i\right]_{q^{-1}}\right)z^{m-1}\left(1-z\right),$ $\displaystyle L_{1}\left(f;z\right)$ $\displaystyle={\displaystyle\sum\limits_{m=2}^{\infty}}a_{m}m\left(m-1\right)z^{m-1}\left(1-z\right).$ Using the inequality $\displaystyle\left\|q{\displaystyle\sum\limits_{i=1}^{m-1}}\left[i\right]_{q}-\dfrac{m\left(m-1\right)}{2}\right\|$ $\displaystyle=q{\displaystyle\sum\limits_{i=2}^{m-1}}\left(\left[i\right]_{q}-i\right)+\left(q-1\right)\dfrac{m\left(m-1\right)}{2}$ $\displaystyle=q\left(q-1\right){\displaystyle\sum\limits_{i=2}^{m-1}}{\displaystyle\sum\limits_{j=1}^{i}}\left[j\right]_{q}+\left(q-1\right)\dfrac{m\left(m-1\right)}{2}$ $\displaystyle\leq q\left(q-1\right)\left[m-1\right]_{q}\dfrac{m\left(m-1\right)}{2}+\left(q-1\right)\dfrac{m\left(m-1\right)}{2}$ $\displaystyle=\left(q-1\right)\dfrac{m\left(m-1\right)}{2}\left(q\left[m-1\right]_{q}+1\right)$ $\displaystyle\leq\left(q-1\right)q^{m-1}\dfrac{m^{2}\left(m-1\right)}{2}$ and $\displaystyle\left\|{\displaystyle\sum\limits_{i=1}^{m-1}}\left[i\right]_{q^{-1}}-\dfrac{m\left(m-1\right)}{2}\right\|$ $\displaystyle={\displaystyle\sum\limits_{i=2}^{m-1}}\left(i-\left[i\right]_{q^{-1}}\right)$ $\displaystyle=\left(1-q^{-1}\right){\displaystyle\sum\limits_{i=2}^{m-1}}{\displaystyle\sum\limits_{j=1}^{i}}\left[j\right]_{q^{-1}}$ $\displaystyle\leq\left(1-q^{-1}\right)\dfrac{m\left(m-1\right)^{2}}{2},$ we get for $1\leq q\leq q_{0}$ and $\left\|z\right\|\leq r$, $\displaystyle\left\|L_{q}\left(f;z\right)-L_{1}\left(f;z\right)\right\|$ $\displaystyle\leq{\displaystyle\sum\limits_{m=2}^{N-1}}\left\|a_{m}\right\|\left\|q{\displaystyle\sum\limits_{i=1}^{m-1}}\left[i\right]_{q}-\dfrac{m\left(m-1\right)}{2}\right\|\left\|z^{m-1}-z^{m}\right\|+{\displaystyle\sum\limits_{m=N}^{\infty}}\left\|a_{m}\right\|\left\|q{\displaystyle\sum\limits_{i=1}^{m-1}}\left[i\right]_{q}-\dfrac{m\left(m-1\right)}{2}\right\|\left\|z^{m-1}-z^{m}\right\|$ $\displaystyle+{\displaystyle\sum\limits_{m=2}^{N-1}}\left\|a_{m}\right\|\left\|{\displaystyle\sum\limits_{i=1}^{m-1}}\left[i\right]_{q^{-1}}-\dfrac{m\left(m-1\right)}{2}\right\|\left\|z^{m-1}-z^{m}\right\|+{\displaystyle\sum\limits_{m=N}^{\infty}}\left\|a_{m}\right\|\left\|{\displaystyle\sum\limits_{i=1}^{m-1}}\left[i\right]_{q^{-1}}-\dfrac{m\left(m-1\right)}{2}\right\|\left\|z^{m-1}-z^{m}\right\|$ $\displaystyle\leq\left(q-1\right){\displaystyle\sum\limits_{m=2}^{N-1}}\left\|a_{m}\right\|m^{2}\left(m-1\right)q_{0}^{m-1}r^{m}+4{\displaystyle\sum\limits_{m=N}^{\infty}}\left\|a_{m}\right\|\left(m-1\right)^{2}q_{0}^{m}r^{m}$ $\displaystyle+\left(1-q^{-1}\right){\displaystyle\sum\limits_{m=2}^{N-1}}\left\|a_{m}\right\|m\left(m-1\right)^{2}r^{m}+2{\displaystyle\sum\limits_{m=N}^{\infty}}\left\|a_{m}\right\|m\left(m-1\right)r^{m}.$ Since $f\in H\left(\mathbb{D}_{R}\right),$ we can find $N=N_{\varepsilon}\in\mathbb{N}$ such that $4{\displaystyle\sum\limits_{m=N}^{\infty}}\left\|a_{m}\right\|\left(m-1\right)^{2}q_{0}^{m}r^{m}+2{\displaystyle\sum\limits_{m=N}^{\infty}}\left\|a_{m}\right\|m\left(m-1\right)r^{m}<\varepsilon/2.$ Thus for $q$ sufficiently close to $1$ from the right, we conclude that $\lim\limits_{q\rightarrow 1^{+1}}L_{q}\left(f;z\right)=L_{1}\left(f;z\right)$ uniformly on $\mathbb{D}_{r}$. The proof is finished. Proof of Theorem 5. Then by Theorem 3, we get $L_{q}\left(f;z\right)=\lim_{n\rightarrow\infty}\left[n+1\right]_{q}\left(U_{n,q}\left(f;z\right)-f\left(z\right)\right)=0$ for infinite number of points having an accumulation point on $\mathbb{D}_{R/q}$. Since $L_{q}\left(f;z\right)\in H\left(\mathbb{D}_{R/q}\right)$, by the Unicity Theorem for analytic functions we get $L_{q}\left(f;z\right)=0$ in $\mathbb{D}_{R/q}$, and therefore, by (2), $a_{m}=0$, $m=2,3,...$ Thus, $f$ is linear. Theorem 5 is proved. ## References * [1] Andrews G E, Askey R, Roy R. Special functions. Cambridge: Cambridge University Press; 1999. * [2] Majid S. Foundations of quantum group theory. Cambridge: Cambridge University Press; 2000. * [3] Phillips, G. M. A survey of results on the $q$-Bernstein polynomials. IMA J. Numer. Anal. 30 (2010), no. 1, 277–288. * [4] S.G. Gal, Approximation by complex genuine Durrmeyer type polynomials in compact disks. Appl. Math. 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Gal, Overconvergence in Complex Approximation, Springer New York Heidelberg Dordrecht London, 2013.	arxiv-papers	2014-01-09T16:19:35	2024-09-04T02:49:56.495718	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Nazim I. Mahmudov", "submitter": "Nazim Mahmudov Idris", "url": "https://arxiv.org/abs/1401.2055" }
1401.2081	Correspondance should be sent to Zhiyong Zhang, 118 Haggar Hall, Univeristy of Notre Dame, IN 46556. Email: [email protected]. # Mediation analysis with missing data through multiple imputation and bootstrap Lijuan Wang Zhiyong Zhang and Xin Tong University of Notre Dame ###### Abstract A method using multiple imputation and bootstrap for dealing with missing data in mediation analysis is introduced and implemented in SAS. Through simulation studies, it is shown that the method performs well for both MCAR and MAR data without and with auxiliary variables. It is also shown that the method works equally well for MNAR data if auxiliary variables related to missingness are included. The application of the method is demonstrated through the analysis of a subset of data from the National Longitudinal Survey of Youth. ###### keywords: Mediation analysis, missing data, multiple imputation, auxiliary variables, bootstrap, SAS ## 1 Introduction Mediation models and mediation analysis are widely used in behavioral and social sciences as well as in health and medical research. The influential article on mediation analysis by Baron and Kenny, (1986) has been cited more than 8,000 times. Mediation models are very useful for theory development and testing as well as for identification of intervention points in applied work. Although mediation models were first developed in psychology (e.g., MacCorquodale and Meehl,, 1948; Woodworth,, 1928), they have been recognized and used in many disciplines where the mediation effect is also known as the indirect effect (Sociology, Alwin and Hauser,, 1975) and the surrogate or intermediate endpoint effect (Epidemiology, Freedman and Schatzkin,, 1992). Figure 1 (after Shrout and Bolger,, 2002) depicts the path diagram of a simple mediation model. In this figure, $X$, $M$, and $Y$ represent the independent or input variable, the mediation variable (mediator), and the dependent or outcome variable, respectively. The $e_{M}$ and $e_{Y}$ are residuals or disturbances with variances $\sigma_{eM}^{2}$ and $\sigma_{eY}^{2}$. $c^{\prime}$ is called the direct effect and the mediation effect or indirect effect is measured by the product term $ab$. The other parameters in this model include the intercepts $i_{M}$ and $i_{Y}$. Figure 1: Path diagram demonstration of a mediation model. Statistical approaches to estimating and testing mediation effects with complete data have been discussed extensively in the psychological literature (e.g., Baron and Kenny,, 1986; Bollen and Stine,, 1990; MacKinnon et al.,, 2002, 2007; Shrout and Bolger,, 2002). One way to test mediation effects is to test $H_{0}:ab=0$. If a large sample is available, the normal approximation method can be used, which constructs the standard error of $ab$ through the delta method so that $s.e.(ab)=\sqrt{\hat{b}^{2}\hat{\sigma_{a}^{2}}+2\hat{a}\hat{b}\hat{\sigma}_{ab}+\hat{a}^{2}\hat{\sigma_{b}^{2}}}$ with parameter estimates $\hat{a}$ and $\hat{b}$, their estimated variances $\hat{\sigma_{a}^{2}}$ and $\hat{\sigma_{b}^{2}}$, and covariance $\hat{\sigma}_{ab}$ (e.g., Sobel,, 1982, 1986). Many researchers suggested that the distribution of $ab$ may not be normal especially when the sample size is small although with large sample sizes the distribution may approach normality (Bollen and Stine,, 1990; MacKinnon et al.,, 2002). Thus, bootstrap methods have been recommended to obtain the empirical distribution and confidence interval of $ab$ (MacKinnon et al.,, 2004; Mallinckrodt et al.,, 2006; Preacher and Hayes,, 2008; Shrout and Bolger,, 2002; Zhang and Wang,, 2008). Mediation analysis can be conducted in a variety of programs and software. Notably, the SAS and SPSS macros by Preacher and Hayes, (2004, 2008) have popularized the application of bootstrap techniques in mediation analysis. Based on search results from Google scholar, Preacher and Hayes, (2004) has been cited more than 900 times and Preacher and Hayes, (2008) has already been cited more than 400 times in less than two years after publication. Missing data problem is continuously a challenge even for a well designed study. Although there are approaches to dealing with missing data for path analysis in general (for a recent review, see Graham,, 2009), there are few studies focusing on the treatment of missing data in mediation analysis. Particularly, mediation analysis is different from typical path analysis because the focus is on the product of two path coefficients. A common practice is to analyze complete data through listwise deletion or pairwise deletion (e.g., Chen et al.,, 2005; Preacher and Hayes,, 2004). However, with the availability of advanced approaches such as multiple imputation (MI), listwise and pairwise deletion is no longer deemed acceptable (Little and Rubin,, 2002; Savalei and Bentler,, 2009; Schafer,, 1997). In this study, we discuss how to deal with missing data for mediation analysis through multiple imputation (MI) and bootstrap using SAS. The rationale of using multiple imputation is that it can be implemented in existing popular statistical software such as SAS and it can deal with different types of missing data. In the following, we will first present the technical backgrounds of multiple imputation for mediation analysis with missing data. Then, we will discuss how to implement the method in SAS. After that, we will present several simulation examples to evaluate the performance of MI for mediation analysis with missing data. Finally, an empirical example will be used to demonstrate the application of the method. ## 2 Method In this section, we present the technical backgrounds of mediation analysis with missing data through multiple imputation and bootstrap. First, we will discuss how to estimate mediation model parameters with complete data. Second, we will reiterate the definition of missing data mechanisms by Little and Rubin, (2002). Third, we will discuss how to apply multiple imputation to mediation analysis. Finally, we will discuss the bootstrap procedure to obtain the bias corrected confident intervals for mediation model parameters. ### 2.1 Complete data mediation analysis In mathematical form, the mediation model displayed in Figure 1 can be expressed using two equations, $\displaystyle M$ $\displaystyle=i_{M}+aX+e_{M}$ $\displaystyle Y$ $\displaystyle=i_{Y}+bM+c^{\prime}X+e_{Y},$ (1) which can be viewed as a collection of two linear regression models. To obtain the parameter estimates in the model, one can maximize the product of the likelihood functions from the two regression models using the maximum likelihood method. Because $e_{M}$ and $e_{Y}$ are assumed to be independent, maximizing the product of the likelihood functions is equivalent to maximizing the likelihood function of each regression model separately. Thus, parameter estimates can be obtained by fitting two separate regression models in Equation 1. Specifically, the mediation effect estimate is $\hat{a}\hat{b}$ with $\displaystyle\hat{a}$ $\displaystyle=s_{XM}/s_{X}^{2}$ $\displaystyle\hat{b}$ $\displaystyle=(s_{MY}s_{X}^{2}-s_{XM}s_{XY})/(s_{X}^{2}s_{M}^{2}-s_{XM}^{2})$ (2) where $s_{X}^{2},s_{M}^{2},s_{Y}^{2},s_{XM},s_{MY},s_{XY}$ are sample variances and covariances of $X,M,Y$, respectively. ### 2.2 Missing mechanisms Little and Rubin, (1987, 2002) have distinguished three types of missing data – missing completely at random (MCAR), missing at random (MAR), and missing not at random (MNAR). Let $D=(X,M,Y)$ denote all data that can be potentially observed in a mediation model. $D_{obs}$ and $D_{miss}$ denote data that are actually observed and data that are not observed, respectively. Let $R$ denote an indicator matrix of zeros and ones. If a datum in $D$ is missing, the corresponding element in $R$ is equal to 1. Otherwise, it is equal to 0. Finally, let $A$ denote the auxiliary variables that are related to the missingness of $D$ but not a component of the mediation model in Equation 1. If the missing mechanism is MCAR, then we have $\Pr(R\|D_{obs},D_{miss},A,\bm{\theta})=\Pr(R\|\bm{\theta}),$ where the vector $\bm{\theta}$ represents all model parameters in the mediation model including $a$, $b$, $ab$, $c^{\prime}$, $i_{M}$, $i_{Y}$, $\sigma_{eM}^{2}$, and $\sigma_{eY}^{2}$. This suggests that missing data $D_{miss}$ are a simple random sample of $D$ and missingness is not related to the data of interest $D$ or auxiliary variables A. If the missing mechanism is MAR, then $\Pr(R\|D_{obs},D_{miss},A,\bm{\theta})=\Pr(R\|D_{obs},\bm{\theta}),$ which indicates that the probability that a datum is missing is related to the observed data $D_{obs}$ but not to the missing data $D_{miss}$. Finally, if the probability that a datum is missing is related to the missing data $D_{miss}$ or auxiliary variables $A$ while $A$ are not considered in the data analysis, the missing mechanism is MNAR. ### 2.3 Multiple imputation for mediation analysis with missing data Most techniques dealing with missing data including multiple imputation in general require missing data to be either MCAR or MAR (see also, e.g., Little and Rubin,, 2002; Schafer,, 1997). For MNAR, the missing mechanism has to be known to correctly recover model parameters. Practically, researchers have suggested including auxiliary variables to facilitate MNAR missing data analysis (Graham,, 2003; Savalei and Bentler,, 2009). Auxiliary variables are variables that are not a component of a model (not model variables) but can explain missingness of variables in the model. After including appropriate auxiliary variables, we may be able to assume that data from both model variables and auxiliary variables are MAR. The setting for mediation analysis with missing data is described below. Assume that a set of $p(p\geq 0)$ auxiliary variables $A_{1},A_{2},\ldots,A_{p}$ are available. These auxiliary variables may or may not be related to missingness of the mediation model variables. Furthermore, there may or may not be missing data in auxiliary variables. By augmenting the auxiliary variables with the mediation model variables, we have a total of $p+3$ variables denoted by $D=(X,M,Y,A_{1},\ldots,A_{p})$. To proceed, we assume that the missing mechanism is MAR after including the auxiliary variables. That is $\Pr(R\|D_{obs},D_{miss},A_{1},\ldots,A_{p},\bm{\theta})=\Pr(R\|D_{obs},A_{1},\ldots,A_{p},\bm{\theta}).$ Multiple imputation (Little and Rubin,, 2002; Rubin,, 1976; Schafer,, 1997) is a procedure to fill each missing value with a set of plausible values. The multiple imputed data sets are then analyzed using standard procedures for complete data and the results from these analyses are combined for obtaining point estimates of model parameters and standard errors of parameter estimates. For mediation analysis with missing data, the following steps can be implemented for obtaining point estimates of mediation model parameters. Assuming that $D=(X,M,Y,A_{1},\ldots,A_{p})$ are from a multivariate normal distribution, generate $K$ ($K$ is the number of multiple imputations) sets of values for each missing value. Combine the generated values with the observed data to produce $K$ sets of complete data (Schafer,, 1997). For each of the $K$ sets of complete data, apply the formula in Equation 2 to obtain a point mediation effect estimate $\hat{a}_{k}\hat{b}_{k}(j=1,\ldots,K)$. The point estimate for the mediation effect through multiple imputation is the average of the $K$ complete data mediation effect estimates: $\hat{a}\hat{b}=\frac{1}{K}\sum_{k=1}^{K}\hat{a}_{k}\hat{b}_{k}.$ Parameter estimates for the other model parameters $a$, $b$, $c^{\prime}$, $i_{M}$, $i_{Y}$, $\sigma_{eM}^{2}$, and $\sigma_{eY}^{2}$ can be obtained in the same way. ### 2.4 Testing mediation effects through the bootstrap method The procedure described above is implemented to obtain point estimates of mediation effects. To test mediation effects, we need to obtain standard errors of the parameter estimates. Because mediation effects are measured by $ab$, researchers suggest using bootstrap to obtain empirical standard errors as mentioned in a previous section. The bootstrap method (Efron,, 1979, 1987) was first employed in mediation analysis by Bollen and Stine, (1990) and has been studied in a variety of research contexts (e.g., MacKinnon et al.,, 2004; Mallinckrodt et al.,, 2006; Preacher and Hayes,, 2008; Shrout and Bolger,, 2002). This method has no distribution assumption on the indirect effect $ab$. Instead, it approximates the distribution of $ab$ using its bootstrap empirical distribution. The bootstrap method used in Bollen and Stine, (1990) can be applied along with multiple imputation to obtain standard errors of mediation effect estimates and confidence intervals for mediation analysis with missing data. Specifically, the following procedure can be used. Using the _original data set_ (Sample size = _N_) as a population, draw a bootstrap sample of _N_ persons randomly with replacement from the original data set. This bootstrap sample generally would contain missing data. With the bootstrap sample, implement the $K$ multiple imputation procedure described in the above section to obtain point estimates of model parameters and a point estimate of the mediation effect . Repeat Steps 1 and 2 for a total of $B$ times. $B$ is called the number of bootstrap samples. Empirical distributions of model parameters and the mediation effect are then obtained using the $B$ sets of bootstrap point estimates. Thus, confidence intervals of model parameters and mediation effect can be constructed. The procedure described above can be considered as a procedure of $K$ multiple imputations nested within $B$ bootstrap samples. Using the $B$ bootstrap sample point estimates, one can obtain bootstrap standard errors and confidence intervals of model parameters and mediation effects conveniently. Let $\bm{\theta}=(iM,iY,a,b,c^{\prime},\sigma_{eM}^{2},\sigma_{eY}^{2},ab)^{t}$ denote a vector of model parameters and the mediation effect $ab$. With data from each bootstrap, we can obtain $\hat{\bm{\theta}}^{b},\ b=1,\ldots,B$. The standard error of the $p$th parameter $\hat{{\theta}}_{p}$ can be calculated as $\widehat{s.e.(\hat{\theta}_{p})}=\sqrt{\sum_{b=1}^{B}(\hat{\theta}_{p}^{b}-\bar{\hat{\theta}}_{p}^{b})^{2}/(B-1)}$ with $\bar{\hat{\theta}}_{p}^{b}=\sum_{b=1}^{B}\hat{\theta}_{p}^{b}/B.$ Many methods for constructing confidence intervals from $\hat{\bm{\theta}}^{b}$ have been proposed such as the percentile interval, the bias-corrected (BC) interval, and the bias-corrected and accelerated (BCa) interval (Efron,, 1987; MacKinnon et al.,, 2004). In the present study, we focus on the BC interval because MacKinnon et al., (2004) showed that the BC confidence intervals have correct Type I error and largest power among many different evaluated confidence intervals. The $1-2\alpha$ BC interval for the $p$th element of $\bm{\theta}$ can be constructed using the percentiles $\hat{\theta}_{p}^{b}(\tilde{\alpha}_{l})$ and $\hat{\theta}_{p}^{b}(\tilde{\alpha}_{u})$ of $\hat{\theta}_{p}^{b}$. Here $\tilde{\alpha}_{l}=\Phi(2z_{0}+z^{(\alpha)})$ and $\tilde{\alpha}_{u}=\Phi(2z_{0}+z^{(1-\alpha)})$ where $\Phi$ is the standard cumulative normal distribution function and $z^{(\alpha)}$ is the $\alpha$ percentile of the standard normal distribution and $z_{0}=\Phi^{-1}\left[\frac{\text{number of times that }\hat{\theta}_{p}^{b}<\hat{\theta}_{p}}{B}\right].$ ## 3 Multiple imputation and bootstrap for mediation analysis with missing data in SAS To facilitate the implementation of the method described in the above section, we have written a SAS program for mediation analysis with missing data using multiple imputation and bootstrap. The complete SAS program scripts are contained in the Appendix. Now we briefly explain the functioning of each part of the SAS program. Lines 3-9 of the SAS program specifies all global parameters that control multiple imputation and bootstrap for mediation analysis. This part is the one that a user needs to modify according to his/her data analysis environment. Line 3 specifies the directory and name of the data file to be used. Line 4 lists the names of the variables in the data file. Line 5 specifies the missing data value indicator. For example, 99999 in the data file represents a missing datum. Line 6 specifies the number of imputations ($K$) for imputing missing data. Line 7 defines the number of bootstrap samples ($B$). A number larger than 1000 is usually recommended. Line 8 and Line 9 specify the confidence level and the random number generator seed, respectively. Lines 15-22 first read data into SAS from the data file specified on line 3 and then change missing data to the SAS missing data format \- a dot. Lines 26-28 impute missing data for the original data set with auxiliary variables and generate $K$ imputed data sets. Lines 30-34 estimate the mediation model parameters for each imputed data set. Lines 37-74 collect the results from the multiple imputed data sets and save the point estimates of model parameters and mediation effect in a SAS data set called “pointest”. The SAS codes in this section produce point parameter estimates for the model parameters and the mediation effect based on the original data after multiple imputation. Lines 77-88 generate $B$ bootstrap samples from the original data set with the same sample size. Lines 91-95 impute each bootstrap sample independently for $K$ times. Lines 98-143 produce point estimates of mediation model parameters and mediation effect for each bootstrap sample and collect the point estimates for all bootstrap samples in the SAS data set named “bootest”. The last part of the SAS program from Line 146 to Line 195 calculates the bootstrap standard errors and the bias-corrected confidence intervals for mediation model parameters and mediation effect. It also generates a table containing the point estimates, standard errors, and confidence intervals in the SAS output window. To use the SAS program, one only needs to first change the global parameters in Lines 3-9, usually only lines 3 and 4, and then run the whole SAS program from the beginning to the end. ## 4 Evaluating the method for mediation analysis with missing data In this section, we conduct several simulation studies to evaluate the performance of the proposed method for mediation analysis with missing data. We first evaluate its performance under different missing data mechanisms including MCAR, MAR, and MNAR without and with auxiliary variables. Then, we investigate how many imputations are needed for mediation analysis with different proportions of missing data. In the following, we first discuss our simulation design and then present the simulation results. ### 4.1 Simulation design For mediation analysis with complete data, simulation studies have been conducted to investigate a variety of features of mediation models (e.g., MacKinnon et al.,, 2002, 2004). For the current study, we follow the parameter setup from previous literature and set the model parameter values to be $a=b=.39$, $c^{\prime}=0$, $i_{M}=i_{Y}=0$, and $\sigma_{eM}^{2}=\sigma_{eY}^{2}=\sigma_{eX}^{2}=1$. Furthermore , we fix the sample size at $N=100$ and consider three proportions of missingness with missing data percentages at 10%, 20%, and 40%, respectively. To facilitate the comparisons among different missing mechanisms, missing data are only allowed in $M$ and $Y$ although our SAS program allows missingness in $X$. Two auxiliary variables ($A_{1}$ and $A_{2}$) are also generated where the correlation between $A_{1}$ and $M$ and the correlation between $A_{2}$ and $Y$ are both $0.5$. For each of the following simulation studies, results are from $R=1,000$ sets of simulated data. For each simulation study, we report point estimate bias, coverage probability, and power or Type I error for evaluations. Let $\theta$ denote the true parameter value in the simulation and $\hat{\theta}_{r}(r=1,\ldots,1000)$ denote the corresponding estimate from the $r$th replication. The bias is calculated as $\text{Bias}=\begin{cases}100\times\left[\frac{\sum_{r=1}^{1000}\hat{\theta}_{r}}{1000\theta}-1\right]&\theta\neq 0\\\ 100\times\left[\frac{\sum_{r=1}^{1000}\hat{\theta}_{r}}{1000}-\theta\right]&\theta=0\end{cases}.$ Note that the bias is rescaled by multiplying 100. Smaller bias indicates the point estimate is less biased. Furthermore, Let $\hat{l_{r}}$ and $\hat{u_{r}}$ denote the lower and upper limits of the $95\%$ confidence interval in the $r$th replication. The coverage probability is calculated by $\text{coverage}=\frac{\\#(\hat{l_{r}}<\gamma<\hat{u}_{r})}{1000}$ where $\\#(\hat{l_{r}}<\gamma<\hat{u}_{r})$ is the total number of replications with confidence intervals covering the true parameter value. Good $95\%$ confidence intervals should give coverage probabilities close to $0.95$. Power or Type I error is calculated by $\text{power}=\frac{\\#(\hat{l_{r}}>0)+\\#(\hat{u}_{r}<0)}{1000}$ where $\\#(\hat{l_{i}}>0)$ is the total number of replications with the lower limits of confidence intervals larger than 0 and $\\#(\hat{u}_{r}<0)$ is the total number of replications with the upper limits smaller than 0. If the population parameter value is not equal to 0, a better method should have greater statistical power. If the population parameter value is equal to 0, a good method should have type I error close to the nominal alpha level. ### 4.2 Simulation 1. Analysis of MCAR data The parameter estimate biases, coverage probabilities, and power/Type I errors for MCAR data with 10%, 20%, and 40% missing data are obtained without and with auxiliary variables and are summarized in Table 1. From the results, we can conclude the following. First, biases of the parameter estimates for all conditions under the studied MCAR conditions are smaller than 1.5%. Second, the coverage probabilities are close to the true value .95 except that the coverage probabilities of variance parameters range from .88 to .94 and are slightly underestimated. Third, the inclusion of auxiliary variables in MCAR data mediation analysis does not seem to influence the accuracy of parameter estimates and coverage probabilities although the auxiliary variables are correlated with $M$ and $Y$($r=.5$). The use of auxiliary variables, however, slightly boosters the power of detecting mediation effect especially when the missing proportion is larger (e.g., 40%). Table 1: Biases, coverage probabilities, and power/type I error under MCAR situations \| \| Without Auxiliary Variables \| With Auxiliary Variables ---\|---\|---\|--- \| \| Bias \| Coverage \| Power/Type I error \| Bias \| Coverage \| Power/Type I error 10% \| $a$ \| 0.595 \| 0.938 \| 0.946 \| 0.861 \| 0.943 \| 0.956 $b$ \| 0.055 \| 0.941 \| 0.920 \| -0.130 \| 0.944 \| 0.927 $c^{\prime}$ \| 0.487 \| 0.953 \| 0.047 \| 0.304 \| 0.945 \| 0.055 $ab$ \| 0.219 \| 0.967 \| 0.900 \| 0.263 \| 0.967 \| 0.920 $i_{Y}$ \| 0.116 \| 0.945 \| 0.055 \| 0.304 \| 0.946 \| 0.054 $i_{M}$ \| 0.065 \| 0.956 \| 0.044 \| -0.072 \| 0.952 \| 0.048 $\sigma_{eY}^{2}$ \| -0.657 \| 0.935 \| 1.000 \| -0.494 \| 0.931 \| 1.000 $\sigma_{eM}^{2}$ \| -0.148 \| 0.931 \| 1.000 \| -0.051 \| 0.930 \| 1.000 20% \| $a$ \| -0.218 \| 0.936 \| 0.907 \| -0.047 \| 0.938 \| 0.920 $b$ \| -0.525 \| 0.940 \| 0.829 \| -0.131 \| 0.937 \| 0.862 $c^{\prime}$ \| 0.430 \| 0.934 \| 0.066 \| 0.266 \| 0.941 \| 0.059 $ab$ \| -1.222 \| 0.966 \| 0.808 \| -0.593 \| 0.963 \| 0.845 $i_{Y}$ \| -0.165 \| 0.946 \| 0.054 \| -0.204 \| 0.944 \| 0.056 $i_{M}$ \| 0.349 \| 0.956 \| 0.044 \| 0.268 \| 0.954 \| 0.046 $\sigma_{eY}^{2}$ \| -0.818 \| 0.920 \| 1.000 \| -0.539 \| 0.918 \| 1.000 $\sigma_{eM}^{2}$ \| -0.244 \| 0.942 \| 1.000 \| -0.105 \| 0.944 \| 1.000 40% \| $a$ \| 0.640 \| 0.938 \| 0.822 \| 0.634 \| 0.930 \| 0.849 $b$ \| -1.310 \| 0.930 \| 0.565 \| -0.593 \| 0.935 \| 0.635 $c^{\prime}$ \| 0.607 \| 0.944 \| 0.056 \| 0.226 \| 0.945 \| 0.055 $ab$ \| -0.716 \| 0.946 \| 0.531 \| 0.112 \| 0.950 \| 0.615 $i_{Y}$ \| -0.007 \| 0.943 \| 0.057 \| -0.044 \| 0.939 \| 0.061 $i_{M}$ \| -0.127 \| 0.966 \| 0.034 \| 0.050 \| 0.963 \| 0.037 $\sigma_{eY}^{2}$ \| -1.484 \| 0.888 \| 1.000 \| -0.860 \| 0.911 \| 1.000 $\sigma_{eM}^{2}$ \| 0.077 \| 0.924 \| 1.000 \| 0.498 \| 0.933 \| 1.000 ### 4.3 Simulation 2. Analysis of MAR data The estimate biases, coverage probabilities, and power for MAR data analysis are summarized in Table 2. The findings from MAR data are similar to those from MCAR data and thus are not repeated here. However, the power of detecting mediation effects from MAR data are smaller than that from MCAR data given the same proportion of missing data. Table 2: Biases, coverage probabilities, and power/type I error under MAR situations \| \| Without Auxiliary Variables \| With Auxiliary Variables ---\|---\|---\|--- \| \| Bias \| Coverage \| Power/type I error \| Bias \| Coverage \| Power/type I error 10% \| $a$ \| 0.716 \| 0.944 \| 0.927 \| 0.439 \| 0.938 \| 0.929 $b$ \| -0.331 \| 0.936 \| 0.896 \| -0.314 \| 0.946 \| 0.917 $c^{\prime}$ \| 0.679 \| 0.954 \| 0.046 \| 0.435 \| 0.947 \| 0.053 $ab$ \| -0.119 \| 0.957 \| 0.870 \| -0.403 \| 0.961 \| 0.893 $i_{Y}$ \| 0.369 \| 0.948 \| 0.052 \| 0.294 \| 0.948 \| 0.052 $i_{M}$ \| -0.010 \| 0.956 \| 0.044 \| -0.084 \| 0.956 \| 0.044 $\sigma_{eY}^{2}$ \| -0.574 \| 0.924 \| 1.000 \| -0.457 \| 0.921 \| 1.000 $\sigma_{eM}^{2}$ \| -0.409 \| 0.932 \| 1.000 \| -0.234 \| 0.936 \| 1.000 20% \| $a$ \| 0.838 \| 0.936 \| 0.862 \| -1.871 \| 0.935 \| 0.862 $b$ \| -0.897 \| 0.935 \| 0.807 \| 0.095 \| 0.933 \| 0.833 $c^{\prime}$ \| 0.320 \| 0.940 \| 0.060 \| 0.027 \| 0.952 \| 0.048 $ab$ \| -0.546 \| 0.962 \| 0.767 \| -1.940 \| 0.958 \| 0.791 $i_{Y}$ \| -0.294 \| 0.945 \| 0.055 \| 0.035 \| 0.952 \| 0.048 $i_{M}$ \| 0.375 \| 0.951 \| 0.049 \| -0.120 \| 0.949 \| 0.051 $\sigma_{eY}^{2}$ \| -0.886 \| 0.918 \| 1.000 \| -0.650 \| 0.920 \| 1.000 $\sigma_{eM}^{2}$ \| -0.109 \| 0.941 \| 1.000 \| -0.063 \| 0.942 \| 1.000 40% \| $a$ \| -0.563 \| 0.937 \| 0.697 \| -0.135 \| 0.942 \| 0.772 $b$ \| -1.863 \| 0.929 \| 0.597 \| -1.372 \| 0.926 \| 0.647 $c^{\prime}$ \| 0.837 \| 0.940 \| 0.060 \| 0.315 \| 0.945 \| 0.055 $ab$ \| -2.932 \| 0.960 \| 0.511 \| -1.747 \| 0.955 \| 0.599 $i_{Y}$ \| 0.220 \| 0.950 \| 0.050 \| 0.004 \| 0.943 \| 0.057 $i_{M}$ \| -0.604 \| 0.945 \| 0.055 \| -0.202 \| 0.955 \| 0.045 $\sigma_{eY}^{2}$ \| -1.137 \| 0.908 \| 1.000 \| -0.452 \| 0.909 \| 1.000 $\sigma_{eM}^{2}$ \| -0.017 \| 0.936 \| 1.000 \| 0.466 \| 0.940 \| 1.000 ### 4.4 Simulation 3. Analysis of MNAR data The results from MNAR data analysis are summarized in Table 3. The results clearly show that when auxiliary variables are not included, parameter estimates are highly biased especially when the missing data proportion is larger, e.g., about 67% bias with 40% missing data for the mediation effect. Correspondingly, coverage probabilities are highly underestimated. For example, with 40% of missing data, the coverage probabilities for intercepts and variance parameters are almost zero. However, with the inclusion of appropriate auxiliary variables, the parameter estimate biases dramatically decrease to 3% or below and the coverage probabilities are close to 95%. Thus, multiple imputation can be used to analyze MNAR data and recover true parameter values by including appropriate auxiliary variables that can explain missingness of the variables in the mediation model. Table 3: Biases, coverage probabilities, and power/type I error under MNAR situations \| \| Without Auxiliary Variables \| With Auxiliary Variables ---\|---\|---\|--- \| \| Bias \| Coverage \| Power/type I error \| Bias \| Coverage \| Power/type I error 10% \| $a$ \| -20.534 \| 0.824 \| 0.891 \| 0.918 \| 0.938 \| 0.956 $b$ \| -15.339 \| 0.888 \| 0.827 \| -1.099 \| 0.933 \| 0.923 $c^{\prime}$ \| 1.930 \| 0.955 \| 0.045 \| 0.468 \| 0.946 \| 0.054 $ab$ \| -32.633 \| 0.831 \| 0.800 \| -0.513 \| 0.951 \| 0.925 $i_{Y}$ \| 11.320 \| 0.739 \| 0.261 \| -0.076 \| 0.951 \| 0.049 $i_{M}$ \| 14.547 \| 0.591 \| 0.409 \| -0.090 \| 0.948 \| 0.052 $\sigma_{eY}^{2}$ \| -13.029 \| 0.532 \| 1.000 \| 0.004 \| 0.939 \| 1.000 $\sigma_{eM}^{2}$ \| -13.121 \| 0.508 \| 1.000 \| 0.248 \| 0.938 \| 1.000 20% \| $a$ \| -29.841 \| 0.728 \| 0.838 \| 0.782 \| 0.941 \| 0.929 $b$ \| -27.443 \| 0.810 \| 0.589 \| -2.856 \| 0.928 \| 0.826 $c^{\prime}$ \| 2.197 \| 0.943 \| 0.057 \| 0.190 \| 0.947 \| 0.053 $ab$ \| -49.117 \| 0.673 \| 0.570 \| -2.583 \| 0.941 \| 0.815 $i_{Y}$ \| 22.228 \| 0.356 \| 0.644 \| -0.001 \| 0.955 \| 0.045 $i_{M}$ \| 27.597 \| 0.145 \| 0.855 \| 0.234 \| 0.956 \| 0.044 $\sigma_{eY}^{2}$ \| -20.661 \| 0.228 \| 1.000 \| -0.494 \| 0.933 \| 1.000 $\sigma_{eM}^{2}$ \| -20.331 \| 0.215 \| 1.000 \| 0.426 \| 0.936 \| 1.000 40% \| $a$ \| -45.357 \| 0.525 \| 0.638 \| -0.044 \| 0.943 \| 0.846 $b$ \| -38.421 \| 0.839 \| 0.355 \| -1.824 \| 0.934 \| 0.666 $c^{\prime}$ \| 3.041 \| 0.936 \| 0.064 \| 1.053 \| 0.947 \| 0.053 $ab$ \| -66.815 \| 0.559 \| 0.305 \| -2.951 \| 0.951 \| 0.642 $i_{Y}$ \| 45.112 \| 0.113 \| 0.887 \| -1.212 \| 0.950 \| 0.050 $i_{M}$ \| 55.439 \| 0.000 \| 1.000 \| -0.055 \| 0.949 \| 0.051 $\sigma_{eY}^{2}$ \| -31.444 \| 0.086 \| 1.000 \| 0.333 \| 0.923 \| 1.000 $\sigma_{eM}^{2}$ \| -31.484 \| 0.048 \| 1.000 \| 1.194 \| 0.921 \| 1.000 ### 4.5 Simulation 4. Impact of the number of imputations A potential difficulty of applying multiple imputation is to make an appropriate decision on how many imputations are needed. For example, Rubin, (1987) has suggested that five imputations are sufficient in the case of 50% missing data for estimating simple mean. But Graham et al., (2007) recommend that many more imputations than that Rubin recommended should be used. Although one may always choose to use a very large number of imputations for mediation analysis with missing data, this may not be practically possible because of the amount of computational time involved (In total, K (number of imputations) x B (number of bootstrap samples) mediation models need to be estimated). In this simulation study, we will briefly investigate the impact of the number of imputations on the point estimates and standard error estimates of mediation effects in mediation analysis with missing data. More specifically, we collect the results from MNAR data analysis with auxiliary variables with the number of imputations from 10 to 100 with an interval of 10. We focus on how the mediation effect estimates and the bootstrap standard error estimates change with the number of imputations. For the purpose of comparison, we calculate the relative deviances of mediation effect estimates and their standard error estimates from those estimates with 100 imputations. Those relative deviances from conditions 10% missing data and 40% missing data are plotted in Figure 2. Figure 2a portrays the relative deviances from results with 10% missing data. Note that with the number of imputations larger than 50, the relative deviances of point estimates are all close to zero and remain unchanged. Thus, 50 imputations seem to be sufficient for mediation analysis with 10% missing data. With 40% missing data, however, the relative deviances of point estimates do not approach zero until the number of imputations is larger than 80 as shown in Figure 2b. Therefore, the number of imputations required is related to the amount of missing data. In our simulation study, the choice of 100 imputations appears to be enough based on this simulation. (a) 10% missing data (b) 40% missing data Figure 2: The impact of different numbers of imputations on the accuracy of point estimates and bootstrap standard error estimates. ## 5 An Empirical Example In this section, we apply the proposed method to analyze a real data set to illustrate its application. Research has found that parents’ education levels can influence adolescent mathematics achievement directly and indirectly. For example, Davis-Kean, (2005) showed that parents’ education levels are related to children’s academic achievement through parents’ beliefs and behaviors. To test a similar hypothesis, we investigate whether home environment is a mediator in the relation between mothers’ education and children’s mathematical achievement . Data used in this example are from the National Longitudinal Survey of Youth, the 1979 cohort (NLSY79, Center for Human Resource Research,, 2006). Data were collected in 1986 from $N=475$ families on mothers’ education level (ME), home environment (HE), children’s mathematical achievement (Math), children’s behavior problem index (BPI), and children’s reading recognition and reading comprehension achievement. For the mediation analysis, mothers’ education is the independent variable, home environment is the mediator, and children’s mathematical achievement is the outcome variable. The missing data patterns and the sample size of each pattern are presented in Table 4. In this data set, 417 families have complete data and 58 families have missing data on at least one of the two model variables: home environment and children’s mathematical achievement. For the purpose of demonstration, children’s behavior problem index (BPI) and children’s reading recognition and reading comprehension achievement- are used as auxiliary variables in the data analysis. Table 4: Missing data patterns of the empirical data set. Pattern \| ME \| HE \| Math \| Sample size ---\|---\|---\|---\|--- 1 \| O \| O \| O \| 417 2 \| O \| X \| O \| 36 3 \| O \| O \| X \| 14 4 \| O \| X \| X \| 8 Total \| \| \| \| 475 _Note_. O: observed; X: missing. ME: mother’s education level; HE: home environments; Math: mathematical achievement. In Table 5, the results from empirical data analysis using the proposed method without and with the auxiliary variables are presented.111For the empirical data analysis, 1000 bootstraps and 100 imputations were used. The results reveal that the inclusion of the auxiliary variable only slightly changed the parameter estimates, standard errors, and the BC confidence intervals. This indicates that the auxiliary variables may not be related to the missingness in the mediation model variables.The results from the analysis with auxiliary variables also show that home environment partially mediates the relationship between mothers’ education and children’s mathematical achievement because both the indirect effect $ab$ and the direct effect $c^{\prime}$ are significant. Table 5: Mediation effect of home environment on the relationship between mothers’ education and children’s mathematical achievement \| Without Auxiliary Variable \| With Auxiliary Variable ---\|---\|--- Parameter \| Estimate \| S.E. \| 95% BC \| Estimate \| S.E. \| 95% BC $a$ \| 0.035 \| 0.049 \| 0.018 \| 0.162 \| 0.036 \| 0.049 \| 0.018 \| 0.163 $b$ \| 0.475 \| 0.126 \| 0.252 \| 0.754 \| 0.458 \| 0.125 \| 0.221 \| 0.711 $c^{\prime}$ \| 0.134 \| 0.191 \| 0.071 \| 0.611 \| 0.134 \| 0.188 \| 0.072 \| 0.609 $ab$ \| 0.017 \| 0.021 \| 0.005 \| 0.071 \| 0.016 \| 0.021 \| 0.005 \| 0.067 $i_{Y}$ \| 7.953 \| 2.047 \| 3.530 \| 9.825 \| 8.045 \| 2.025 \| 3.778 \| 10.006 $i_{M}$ \| 5.330 \| 0.556 \| 3.949 \| 5.641 \| 5.327 \| 0.558 \| 3.945 \| 5.646 $\sigma_{eY}^{2}$ \| 4.532 \| 0.269 \| 4.093 \| 5.211 \| 4.520 \| 0.268 \| 4.075 \| 5.141 $\sigma_{eM}^{2}$ \| 1.660 \| 0.061 \| 1.545 \| 1.789 \| 1.660 \| 0.061 \| 1.542 \| 1.790 _Note_. S.E.: bootstrap standard error. BC: bias-corrected confidence interval. ## 6 Discussion In this study, we discussed how to conduct mediation analysis with missing data through multiple imputation and bootstrap. We implemented the method by using SAS and the program scripts are also provided and easy to use. Through simulation studies, we demonstrated that the proposed method performed well for both MCAR and MAR without and with auxiliary variables. It is also shown that multiple imputation worked equally well for MNAR if auxiliary variables related to missingness were included. The analysis of a subset of data from the NLSY79 revealed that home environment partially mediated the relationship between mothers’ education and children’s mathematical achievement. ### 6.1 Strength of the proposed method The multiple imputation and bootstrap method for mediation analysis with missing data has several advantages. First, the idea of imputation and bootstrap is easy to understand. Second, multiple imputation has been widely implemented in both free and commercial software and thus can be extended to mediation analysis. Third, it is natural and easy to include auxiliary variables in multiple imputation for analyzing MNAR data. Fourth, multiple imputation does not assume a specific model for imputing data. The implementation of multiple imputation and bootstrap in SAS also has its own advantages. First, only a minimum number of parameters usually need to be changed to run the SAS program for mediation analysis with missing data. Second, the SAS program can be easily extended for more complex mediation analysis by taking advantage of available SAS procedures. For example, one can also conduct mediation analysis with moderators through modifying the PROC REG statements. One can conduct mediation analysis with latent variables through the use of SAS PROC CALIS. Third, SAS excels in terms of performance in dealing with large dataset, which is critical for multiple imputation and bootstrap. For example, for a data set with a sample size 100, to generate 1000 bootstrap samples and impute each bootstrap sample 100 times, one needs to deal with a data set with 10,000,000 (ten million) records. Although this seems to be a huge data set, it only took SAS about 7 minutes to conduct such missing data mediation analysis with 20% missing data. ### 6.2 Assumptions and limitations There are several assumptions and limitations of the current study. First, the study only discusses the mediation model with a single mediator. The current SAS program is also based on this model. Second, in applying multiple imputation, we have assumed that all variables are multivariate normally distributed. However, it is possible that one or more variables are not normally distributed. Third, the current mediation model only focuses on the cross-sectional data analysis. 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You can use any names except for x, m, and y for naming the auxiliary variables; 5%LET missing=99999; specify the missing data value; 6%LET nimpute = 100; define the number of imputations K; 7%LET nboot = 1000; define the number of bootstraps B; 8%LET alpha = 0.95; define the confidence level; 9%LET seed = 2010; random number seed; 10/* End of setup of global parameters / 11 12 13/In general, there is no need to change the codes below/ 14/Read data into sas/ 15DATA dset; 16 INFILE &filename; 17 INPUT &varname; 18 ARRAY nvarlist &varname; 19 DO OVER nvarlist; 20 IF nvarlist = &missing THEN nvarlist = .; 21 END; 22RUN; 23 24/Use multiple imputation to obtain point estimates of the model parameters based on the original data set/ 25/Imputing the original data set multipe times/ 26PROC MI DATA=dset SEED=&seed NIMPUTE=&nimpute OUT=imputed NOPRINT; 27 VAR &varname; 28RUN; QUIT; 29/Estimating model parameters for each imputed data set/ 30PROC REG DATA=imputed OUTEST= est NOPRINT; 31 MODEL y = x m; 32 MODEL m = x; 33 BY _Imputation_; 34RUN; QUIT; 35 36/Collecting results from mutiple imputations/ 37DATA temp; 38 SET est; 39 id =INT((_N_-.1)/2)+1; 40 modelnum = MOD(_N_+1, 2)+1; 41RUN; 42 43DATA temp1; 44 SET temp; 45 ARRAY int[2] iY iM; 46 ARRAY xpar[2] c a; 47 ARRAY mpar[2] b tmp1; 48 ARRAY sigma[2] sy sm; 49 RETAIN a b c iY iM sy sm; 50 BY id; 51 IF FIRST.id THEN DO I = 1 to 2; 52 int[I] = .; 53 xpar[I] = .; 54 mpar[I]=.; 55 sigma[I]=.; 56 END; 57 int[modelnum] = intercept; 58 xpar[modelnum] = x; 59 mpar[modelnum] = m; 60 sigma[modelnum] = _RMSE_; 61 IF LAST.id THEN OUTPUT; 62 KEEP _imputation_ a b c iY iM sy sm; 63RUN; 64/Calcuating mediation effects/ 65DATA temp2; 66 SET temp1; 67 ab=ab; 68RUN; 69 70/Saving the point estimates of model parameters and mediation effect from multiple imputation into a data set named ’pointest’/ 71PROC MEANS DATA=temp2 NOPRINT; 72 VAR a b c ab iY iM sy sm; 73 OUTPUT OUT=pointest MEAN(a b c ab iY iM sy sm)=a b c ab iY iM sy sm; 74RUN; 75 76/* Bootstraping data to obtain standard errors and confidence intervals / 77DATA bootsamp; 78 DO sampnum = 1 to &nboot; 79 DO i = 1 TO nobs; 80 ran = ROUND(RANUNI(&seed) nobs); 81 SET dset 82 nobs = nobs 83 point = ran; 84 OUTPUT; 85 END; 86 END; 87 STOP; 88RUN; QUIT; 89 90/* Imputing K data sets for each bootstrap sample / 91PROC MI DATA=bootsamp SEED=&seed NIMPUTE=&nimpute OUT=imputed NOPRINT; 92 EM MAXITER = 500; 93 VAR &varname; 94 BY sampnum; 95RUN; QUIT; 96 97/Estimate model parameters for each imputed data set (in total, there are BK imputed data sets.)/ 98PROC REG DATA=imputed OUTEST= est NOPRINT; 99 MODEL y = x m; 100 MODEL m = x; 101 BY sampnum _Imputation_; 102RUN; QUIT; 103 104/Collecting results from different imputed data sets/ 105DATA temp; 106 SET est; 107 id =INT((_N_-.1)/2)+1; 108 modelnum = MOD(_N_+1, 2)+1; 109RUN; 110 111DATA temp1; 112 SET temp; 113 ARRAY int[2] iY iM; 114 ARRAY xpar[2] c a; 115 ARRAY mpar[2] b tmp1; 116 ARRAY sigma[2] sy sm; 117 RETAIN a b c iY iM sy sm; 118 BY id; 119 IF FIRST.id THEN DO I = 1 to 2; 120 int[I] = .; 121 xpar[I] = .; 122 mpar[I]=.; 123 sigma[I]=.; 124 END; 125 int[modelnum] = intercept; 126 xpar[modelnum] = x; 127 mpar[modelnum] = m; 128 sigma[modelnum] = _RMSE_; 129 IF LAST.id THEN OUTPUT; 130 KEEP sampnum _imputation_ a b c iY iM sy sm; 131RUN; 132 133DATA temp2; 134 SET temp1; 135 ab=ab; 136RUN; 137 138/Compute point estimates of model parameters and mediation effect for each bootstrap sample and the results are saved in the data file named ’bootest’. / 139PROC MEANS DATA=temp2 NOPRINT; 140 BY sampnum; 141 VAR a b c ab iY iM sy sm; 142 OUTPUT OUT=bootest MEAN(a b c ab iY iM sy sm)=a b c ab iY iM sy sm; 143RUN; 144 145/ Calculate the BC intervals based on the point estimates from different bootstrap samples and produce a table containing the points estimates, standard errors, confidence intervals in the output window./ 146PROC IML; 147 START main; 148 USE pointst; 149 READ ALL INTO Y; 150 USE bootest; 151 READ ALL INTO X; 152 153 n=NROW(X); 154 m=NCOL(X); 155 156 bc_lo=J(1,m-3,0); 157 bc_up=J(1,m-3,0); 158 se=J(1,m-3,0); 159 160 alphas=1-(1-&alpha)/2; 161 zcrit = PROBIT(alphas); 162 163 DO j=1 TO m-3; 164 se[j]=SQRT((SSQ(X[,j+3]) -(SUM(X[,j+3]))2/n)/(n-1)); 165 number=0; 166 DO i=1 TO n; 167 IF X[i,j+3]<Y[j+2] THEN number=number+1; 168 END; 169 p=number/n; 170 z0hat=PROBIT(p); 171 172 q1=z0hat+(z0hat-zcrit); 173 q2=z0hat+(z0hat+zcrit); 174 alpha1=PROBNORM(q1); 175 alpha2=PROBNORM(q2); 176 177 vec=X[,j+3]; 178 CALL SORT(vec,{1}); 179 180 low=int(alpha1(n+1)); 181 up=int(alpha2*(n+1)); 182 IF low<1 THEN low=1; 183 IF up>n THEN up=n; 184 bc_lo[j]=vec[low]; 185 bc_up[j]=vec[up]; 186 END; 187 188 result=Y[3:10]\|\|se‘\|\|(bc_lo‘)\|\|(bc_up‘); 189 MATTRIB result ROWNAME=({a, b, c, ab, iy, im, sy, sm}) 190 COLNAME=({estiamtes se CI_lo CI_up}) 191 LABEL=’MEDIATION␣ANALYSIS␣RESULTS’ FORMAT=f10.5; 192 PRINT result; 193 FINISH; 194 RUN main; 195QUIT;	arxiv-papers	2014-01-09T17:11:44	2024-09-04T02:49:56.509982	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Lijuan Wang, Zhiyong Zhang, Xin Tong", "submitter": "Zhiyong Zhang", "url": "https://arxiv.org/abs/1401.2081" }
1401.2096	Closed equations of the two-point functions for tensorial group field theory Dine Ousmane Samary Perimeter Institute for Theoretical Physics 31 Caroline St. N. Waterloo, ON N2L 2Y5, Canada International Chair in Mathematical Physics and Applications (ICMPA-UNESCO Chair), University of Abomey-Calavi, 072B.P.50, Cotonou, Republic of Benin E-mails: [email protected] ###### Abstract In this paper we provide the closed equations that satisfy two-point correlation functions of the rank 3 and 4 tensorial group field theory. The formulation of the present problem extends the method used by Grosse and Wulkenhaar in [arXiv 0909.1389] to the tensor case. Ward-Takahashi identities and Schwinger-Dyson equations are combined to establish a nonlinear integral equation for the two-point functions. In the 3D case the solution of this equation is given perturbatively at second order of the coupling constant. Pacs numbers: 11.10.Gh, 04.60.-m Key words: Renormalization, tensorial group field theory, Ward-Takahashi identities, Schwinger-Dyson equation, two-point correlation functions. ## 1 Introduction Random Tensor Models [1][2][3] extends Matrix Models [4] as promising candidates to understand Quantum Gravity in higher dimension, $D\geq 3$. The formulation of such models is based on a Feynman path integral generating randomly graphs representing simplicial pseudo manifolds of dimension $D$. The equivalent of the t’Hooft large $N$ limit [5][6] for these Tensor Models has been recently discovered by Gurau [7][8][9]. The large $N$ limit behaviour is a powerful tool which allows to understand the continuous limit of these models through, for instance, the study of critical exponents and phase transitions [10][11][12]. With the advent of the field theory formulation of Random Tensor Models, henceforth called Tensorial Group Field Theory (TGFT) [13]–[24], one addresses several different questions such as Renormalizability (for removing divergences) and the study UV behaviour of these models. It turns out that Renormalization can be consistently defined for TGFTs and most of them, for the higher rank $D\geq 3$ are UV asymptotically free [21]. This is of course very encouraging for the Geomeogenesis Scenario [15][23][24]. It becomes more and more convincing that Random Tensors and TGFT’s will take a growing role for giving answers for the Quantum Gravity conundrum. Despite all these results, a lot of questions (both conceptual and technical) arise in this framework for obtaining a final and emergent theory of General Relativity [3]. Among other goals, it would be strongly desirable to establish more connections with other studies and important results around Gravity. This is the purpose of this paper which provides the first glimpses of the extension of the recent full resolution of the correlation functions in the Grosse- Wulkenhaar (GW) model [28][29] [30]. One of the main purpose of a field theory is to find the exact value of the Green’s functions also called correlation functions. Obviously, this can be a highly nontrivial task. In almost scarce cases where this is successfully done, one calls the model exactly solvable. In a recent work, the renormalizable noncommutative scalar field theory called the GW model was solved [29]–[45]. This particular noncommutative field theory projects on a matrix model and then can be seen a model for QG in $2D$. Let us review this model arising in Noncommutative Geometry. Grosse and Wulkenhaar modified the propagator of the noncommutative field theory by adding a harmonic term and showed that the resulting functional action is renormalizable at all orders of pertubation. The proof of this claim was given using the matrix basis dual to the Moyal space of functions. In [35][36][37] a new proof of the renormalizability was given in direct space using multiscale analysis [38]. The GW propagator breaks the $U(N)$ symmetry invariance in the infrared regime, but is asymptotically safe in the ultraviolet regime [39][40][41]. The model is also non invariant under translation and rotation of spacetime. The only known invariance satisfied by the model is the so-called Langman Szabo duality [42]. At the perturbative level, the associated Feynman graphs are ribbon graphs. In a recent remarkable contribution, Grosse and Wulkenhaar solve successfully all correlators in this model. Using both Ward-Takahashi identities and the Schwinger-Dyson equation, these authors provide, via Hilbert transform, a nonlinear integral equation for the two-point functions [30]. From this result, they were able to generate solutions for all correlators. Thus, the GW model is exactly nonperturbatively solvable. The question is whether or not this method may apply to other models, in particular to TGFTs dealing with higher rank tensors. We give a partial positive answer of this question. Indeed, as we will show in the following, the resolution method can be applied to find nonlinear equations for the correlations here as well. Due to the highly nontrivial equations and combinatorics, the full resolution of all correlators deserves more work which should be addressed elsewhere. The present paper is organized as follows. In the section 2, we derive the Ward-Takahashi identities of arbitrary rank $D$ TGFT. In section 3 we give the closed equation of the two-point correlation functions for the rank $3$ TGFT. We also give the solution of this equation at second order of perturbation. In section 4 we provide the closed equation of rank $4$ tensor field. We give a summary of our results and outlook of the paper in section 5. ## 2 Ward-Takahashi identities for arbitrary $D$-tensor field model TGFT’s are generally defined by an action $S[\bar{\varphi},\varphi]$, that depends on the field $\varphi$ and its conjugate $\bar{\varphi}$ defined on the compact Lie group $G$ i.e. $\varphi:G^{D}\longrightarrow{\mathbbm{C}};\,\,\,(g_{1},\cdots,g_{D})\longmapsto\varphi(g_{1},\cdots,g_{D}).$ For simplicity, we will always consider $G=U(1)$. We are using the Fourier transformation of the field and are defining the momentum variable associated to the group elements $[g]=(g_{1},g_{2},\cdots,g_{D})\in U(1)^{D}$ as $[p]=(p_{1},p_{2},\cdots,p_{D})\in\mathbb{Z}^{D}$. Using the parametrization $g_{k}=e^{i\theta_{k}}$ we write $\displaystyle\varphi({g_{1},\cdots,g_{D}})=\sum_{p_{i}\in\mathbb{Z}}\varphi({p_{1},\cdots,p_{D}})e^{i\sum_{k}\theta_{k}p_{k}},\quad\theta_{i}\in[0,2\pi).$ (1) The Fourier transform of the field $\varphi$ is denoted by $\varphi_{12\cdots D}=:\varphi(p_{1},\cdots,p_{D})=:\varphi_{[D]}$ for simplicity. The functional action $S[\bar{\varphi},\varphi]$ is written in general case as $\displaystyle S[\bar{\varphi},\varphi]=\sum_{p_{i}}\bar{\varphi}_{12\cdots D}C^{-1}(p_{1},p_{2},\cdots,p_{D};p^{\prime}_{1},p^{\prime}_{2},\cdots p^{\prime}_{D})\varphi_{12\cdots D}\prod_{i=1}^{D}\delta_{p_{i}p^{\prime}_{i}}+S^{\text{int\,}}$ (2) where $C$ stands for the propagator and $S^{\text{int\,}}$ collects all vertex contributions of the interaction. Let $d\mu_{C}$ be the field measure associated with the covariance $C$, we have the relation $C([p];[p^{\prime}])=\int\,d\mu_{C}\,\varphi_{[p]}\bar{\varphi}_{[p^{\prime}]},\quad d\mu_{C}=\prod_{[p]}d\bar{\varphi}_{[p]}\,d\varphi_{[p]}e^{-\bar{\varphi}_{[p]}\,C^{-1}([p],[p])\,\varphi_{[p]}}.$ (3) The Green’s functions or $N$-point correlation functions are defined by the relation $\displaystyle G([p]_{1},[p]_{2},\cdots[p]_{N})=\frac{1}{\mathcal{Z}}\int\,d\mu_{C}\,\,\varphi_{[p]_{1}}\bar{\varphi}_{[p]_{1}}\cdots\varphi_{[p]_{N}}\bar{\varphi}_{[p]_{N}}e^{-S^{\text{int\,}}},$ (4) where $\mathcal{Z}$ is the normalization factor also called partition function given by $\displaystyle\mathcal{Z}=\int\,d\mu_{C}\,e^{-S^{\text{int\,}}}.$ (5) Let us write the interaction term of the action (2) as $S^{\text{int\,}}=\lambda V[\bar{\varphi},\varphi]=:\sum_{k}\lambda_{k}V_{k}[\bar{\varphi},\varphi]$. The main idea of the pertubative theory is to expand the Green’s functions as $\displaystyle G([p]_{1},[p]_{2},\cdots[p]_{N})$ $\displaystyle=$ $\displaystyle\sum_{n=0}^{\infty}\,\frac{(-\lambda)^{n}}{n!}\int\,d\mu_{C}\,V^{n}[\bar{\varphi},\varphi]\,\varphi_{[p]_{1}}\bar{\varphi}_{[p]_{1}}\cdots\varphi_{[p]_{N}}\bar{\varphi}_{[p]_{N}}$ (6) $\displaystyle=$ $\displaystyle\sum_{n=0}^{\infty}\,\lambda^{n}\,G^{(n)}_{N}.$ (7) Using this formula, the Green’s functions can be computed order by order using Dyson’s theorem. We consider the rank $D$ tensor field $\varphi$ and its conjugate $\bar{\varphi}$, which are transformed under the tensor product of $D$ fundamental representations of the unitary group $\mathcal{U}_{\otimes}^{N_{D}}:=\otimes_{i=1}^{D}U(N_{i})$. Let $U^{(a)}\in U(N_{a})$, $a=1,2,\cdots,D$. The field $\varphi$ and its conjugate $\bar{\varphi}$ are transformed under $U(N_{a})$ as $\displaystyle\varphi_{12\cdots D}\rightarrow[U^{(a)}\varphi]_{12\cdots a\cdots D}=\sum_{p^{\prime}_{a}\in Z}U^{(a)}_{p_{a}p^{\prime}_{a}}\varphi_{12\cdots a^{\prime}\cdots D},$ (8) $\displaystyle\bar{\varphi}_{12\cdots D}\rightarrow[\bar{\varphi}U^{{\dagger}(a)}]_{12\cdots a\cdots D}=\sum_{p^{\prime}_{a}\in Z}\bar{U}^{(a)}_{p_{a}p^{\prime}_{a}}\bar{\varphi}_{12\cdots a^{\prime}\cdots D}.$ (9) $p^{\prime}_{a}$ or simply $a^{\prime}$ is the momentum index at the position $a$ in the expression $\varphi_{12\cdots a^{\prime}\cdots D}$. The kinetic action in (2) is re-expressed as follows $\displaystyle S^{\text{kin\,}}[\bar{\varphi},\varphi]=\sum_{p_{1},\cdots,p_{D}}\varphi_{12\cdots D}M_{12\cdots D}\bar{\varphi}_{12\cdots D},\quad M_{12\cdots D}=C^{-1}_{12\cdots D}.$ (10) $M_{12\cdots D}$ is the inverse of propagator associated to the model. Rank $D$ tensor fields are represented by half lines made with $D$ segments called strands. A propagator is a $D$ stranded line and as usual connects vertices. The variation of the action $S^{\text{kin\,}}$ under infinitesimal $U(N_{a})$ transformation is given by $\displaystyle\delta^{(a)}[S^{\text{kin\,}}]=-i\sum_{p_{1},\cdots,p_{D}}\Big{[}M\Big{(}\varphi[\bar{B}\bar{\varphi}]^{(a)}-[B\varphi]^{(a)}\bar{\varphi}\Big{)}\Big{]}_{12\cdots D}$ (11) where $B$ is the infinitesimal Hermitian operator corresponding to the generator of unitary group $U(N_{a})$ i.e. $\displaystyle U_{pp^{\prime}}^{(a)}=\delta_{pp^{\prime}}^{(a)}+iB_{pp^{\prime}}^{(a)}+O(B^{2}),\quad\bar{U}_{pp^{\prime}}^{(a)}=\delta_{pp^{\prime}}^{(a)}-i\bar{B}_{pp^{\prime}}^{(a)}+O(\bar{B}^{2}),$ (12) with $\bar{B}_{pp^{\prime}}^{(a)}={B}_{p^{\prime}p}^{(a)}$. Consider now the theory defined with external source $F[\varphi,\bar{\varphi};\eta,\bar{\eta}]$ as $\displaystyle F[\eta,\bar{\eta}]=\sum_{p_{1},\cdots,p_{D}}\bar{\varphi}_{12\cdots D}\eta_{12\cdots D}+\bar{\eta}_{12\cdots D}\varphi_{12\cdots D}.$ (13) The partition function of the model is re-expressed as $\displaystyle\mathcal{Z}[\eta,\bar{\eta}]=\int\,d\varphi d\bar{\varphi}\,e^{-S[\varphi,\bar{\varphi}]+F[\varphi,\bar{\varphi};\eta,\bar{\eta}]}.$ (14) Under $U(N_{a})$ infinitesimal transformation $\displaystyle\delta^{(a)}[F]=i\sum_{p_{1},\cdots,p_{D}}\Big{[}\bar{\eta}[B\varphi]^{(a)}-[\bar{B}\bar{\varphi}]^{(a)}\eta\Big{]}_{12\cdots D}.$ (15) Let $\delta^{(\otimes)}$ be the total variation under the action of the group element $U^{(1)}\otimes U^{(2)}\otimes\cdots\otimes U^{(D)}\in\mathcal{U}_{\otimes}^{N_{D}}$. Then we get the following proposition ###### Proposition 1. The kinetic term of the action (2), i.e. $S^{\text{kin\,}}$ and $F$ are respectively transformed linearly as $\displaystyle\delta^{(\otimes)}S^{\text{kin\,}}=\sum_{a=1}^{D}\delta^{(a)}S^{\text{kin\,}},\quad\delta^{(\otimes)}F=\sum_{a=1}^{D}\delta^{(a)}F.$ (16) Then $\delta^{(\otimes)}S=0$ if and only if $\delta^{(a)}S=0$ for all functional quantity $S$, which depends on $\varphi$, $\bar{\varphi}$, $\eta$ and $\bar{\eta}$. We assume that $N_{i}=N,\,\,i=1,2\cdots D$, and we take the interaction terms such that there are invariant under the transformation $U^{(a)}$ i.e. $\delta^{(a)}S^{\text{int\,}}=0.$ This is the new input in TGFT’s: the $U(N_{a})$ tensor invariance must be the one defining the interaction [1]. Note that the measure $d\varphi d\bar{\varphi}$ is also invariant under $U^{(a)}$. The variation of the partition function can be performed for $a=1$ and the results for all value of $a\in\\{1,2,\cdots,D\\}$ may be deduced using proposition (1). We write $\displaystyle\frac{\delta^{(1)}\ln\mathcal{Z}[\eta,\bar{\eta}]}{\delta B_{p_{m}p_{n}}}$ $\displaystyle=$ $\displaystyle\frac{1}{\mathcal{Z}[\eta,\bar{\eta}]}\int\,d\varphi d\bar{\varphi}\,\Big{\\{}i\sum_{p_{2},\cdots,p_{D}}\Big{(}M_{n\,2\cdots D}\varphi_{n\,2\cdots D}\bar{\varphi}_{m\,2\cdots D}-M_{m\,2\cdots D}\bar{\varphi}_{m\,2\cdots D}\varphi_{n\,2\cdots D}\Big{)}$ (17) $\displaystyle+$ $\displaystyle i\sum_{p_{2},\cdots,p_{D}}\Big{(}\bar{\eta}_{m\,2\cdots D}\varphi_{n\,2\cdots D}-\bar{\varphi}_{m\,2\cdots D}\eta_{n\,2\cdots D}\Big{)}\Big{\\}}e^{-S[\varphi,\bar{\varphi}]+F[\varphi,\bar{\varphi};\eta,\bar{\eta}]}=0.$ (18) Now take $\partial_{\bar{\eta}}\partial_{\eta}$ of the above expression, we get only the connected components of the correlation functions as $\displaystyle\sum_{[p]}\Big{(}M_{m\,2\cdots D}-M_{n\,2\cdots D}\Big{)}\langle\Big{[}\frac{\partial(\bar{\eta}\varphi)}{\partial\bar{\eta}}\frac{\partial(\bar{\varphi}\eta)}{\partial\eta}\Big{]}\varphi_{n\,2\cdots D}\bar{\varphi}_{m\,2\cdots D}\rangle_{c}$ (19) $\displaystyle=\sum_{[p]}\langle\frac{\partial(\bar{\eta}_{m\,2\cdots D}\varphi_{n\,2\cdots D})}{\partial\bar{\eta}}\Big{[}\frac{\partial(\bar{\varphi}\eta)}{\partial\eta}\Big{]}-\frac{\partial(\bar{\varphi}_{m\,2\cdots D}\eta_{n\,2\cdots D})}{\partial\eta}\Big{[}\frac{\partial(\bar{\eta}\varphi)}{\partial\bar{\eta}}\Big{]}\rangle_{c},$ (20) which can be simply written as $\displaystyle\sum_{[p]}\Big{(}M_{m}-M_{n}\Big{)}\langle\Big{[}\frac{\partial(\bar{\eta}\varphi)}{\partial\bar{\eta}}\frac{\partial(\bar{\varphi}\eta)}{\partial\eta}\Big{]}\varphi_{n}\bar{\varphi}_{m}\rangle_{c}$ (21) $\displaystyle=\sum_{[p]}\langle\frac{\partial(\bar{\eta}_{m}\varphi_{n})}{\partial\bar{\eta}}\frac{\partial(\bar{\varphi}\eta)}{\partial\eta}\rangle_{c}-\sum_{[p]}\langle\frac{\partial(\bar{\varphi}_{m}\eta_{n})}{\partial\eta}\frac{\partial(\bar{\eta}\varphi)}{\partial\bar{\eta}}\rangle_{c}.$ (22) Note that the equation (21) is valid for all positions indices $a=1,2,\cdots,D$. Let us also remark that for $m=n$ the left hand side (lhs) of the equation (21) vanishes. In the double derivative $\partial_{\bar{\eta}}\partial_{\eta}$, we fix the indices such that $\bar{\eta}_{[\alpha]}\eta_{[\beta]}$. Then comes the following proposition: ###### Proposition 2. For index $a=1$ (corresponding to $U^{(1)}$), we get the Ward-Takahashi identity $\displaystyle\sum_{p_{2},\cdots,p_{D}}\big{(}M_{m2\cdots D}-M_{n2\cdots D}\big{)}\langle\varphi_{[\alpha]}\bar{\varphi}_{[\beta]}\varphi_{n2\cdots D}\bar{\varphi}_{m2\cdots D}\rangle_{c}$ (23) $\displaystyle=\delta_{m\alpha_{1}}\langle\varphi_{n\alpha_{2}\cdots\alpha_{D}}\bar{\varphi}_{\beta_{1}\cdots\beta_{D}}\rangle_{c}-\delta_{n\beta_{1}}\langle\bar{\varphi}_{m\beta_{2}\cdots\beta_{D}}\varphi_{\alpha_{1}\cdots\alpha_{D}}\rangle_{c},$ (24) which can be re-expressed for arbitrary position $a$ taking any value in $\\{1,2,\cdots,D\\}$ as $\displaystyle\big{(}M_{m}-M_{n}\big{)}\langle[\varphi_{m}\bar{\varphi}_{n}]\varphi_{n}\bar{\varphi}_{m}\rangle_{c}=\langle\varphi_{n}\bar{\varphi}_{n}\rangle_{c}-\langle\bar{\varphi}_{m}\varphi_{m}\rangle_{c},\quad[\varphi_{m}\bar{\varphi}_{n}]=\sum_{p_{2},\cdots,p_{D}}\varphi_{n2\cdots D}\bar{\varphi}_{m2\cdots D}.$ (25) We emphasize that the position taken by the indices $m$ and $n$ in the relation (25) are the position of the momentum index $p_{a}$ used in the transformation $U^{(a)}$. In conclusion, there are exactly $D$ Ward-Takahashi identities for the rank $D$ TGFT’s associated with this type of invariance. Note that the Ward-Takahashi identities for Boulatov model can be found in reference [48]. The result obtained therein radically differs from the present identities found in (25). Furthermore, we mention that we are not considering the TGFT with gauge invariance condition on the fields like in the works [18][19]. We consider here the simplest the TGFT as treated in [13][14]. Most of the result of this work might be extended to this different framework with not much work since only the propagator will be modified. Thus, one expects similar Ward identities in that gauge invariant framework . ## 3 Two-point functions of rank $3$ TGFT In this section we consider the just renormalizable rank $3$ TGFT on compact $U(1)$ group, addressed firstly in [15]. The rank $3$ tensor field is defined by $\varphi:U(1)^{3}\longrightarrow\mathbb{C}$, and we expand in Fourier modes as $\varphi(g_{1},g_{2},g_{3})=\sum_{p_{j}\in\mathbb{Z}}\varphi_{123}e^{ip_{1}\theta_{1}}e^{ip_{2}\theta_{2}}e^{ip_{3}\theta_{3}},\quad\theta_{i}\in[0,2\pi).$ (26) We write as usual $\varphi_{123}:=\varphi_{p_{1}p_{2}p_{3}}$. The renormalizable $3D$ tensor model is defined by the action $S_{3D}$, in which the kinetic term take’s the form $S^{\text{kin\,}}_{3D}=\sum_{[p]}\bar{\varphi}_{123}\,C^{-1}_{123}\,\varphi_{123},$ (27) where $C_{123}$ is the propagator. We write the resulting action for the bare quantities which involves the bare mass $m_{bar}$ and the three wave functions renormalizations $Z_{\rho=1,2,3}$, each of which is associated with a strand index $a=1,2,3$. The field strength can be modified as follows: $\varphi\longrightarrow\left(Z_{1}Z_{2}Z_{3}\right)^{\frac{1}{6}}\varphi=Z^{1/2}\varphi,\quad Z_{\rho}=1-\partial_{b_{\rho}}\Gamma_{b_{1}b_{2}b_{3}}\Big{\|}_{b_{1,2,3}=0},$ (28) where $\Gamma_{b_{1}b_{2}b_{3}}$ is the self-energy or one particle irreducible (1PI) two-point functions. Then, the renormalized propagator takes the form $C_{abc}=Z^{-1}(\|a\|+\|b\|+\|c\|+m^{2})^{-1},\quad a,b,c\in\mathbb{Z}.$ (29) $m$ is the renormalized mass parameter. The interaction of the model is defined by the three contributions $V_{1}$, $V_{2}$, and $V_{3}$ expressed in momentum space as $\displaystyle S^{\text{int\,}}_{3D}$ $\displaystyle=$ $\displaystyle\lambda_{1}Z^{2}\sum_{\stackrel{{\scriptstyle 1,2,3}}{{1^{\prime},2^{\prime},3^{\prime}}}}\varphi_{123}\bar{\varphi}_{321^{\prime}}\varphi_{1^{\prime}2^{\prime}3^{\prime}}\bar{\varphi}_{3^{\prime}2^{\prime}1}+\lambda_{2}Z^{2}\sum_{\stackrel{{\scriptstyle 1,2,3}}{{1^{\prime},2^{\prime},3^{\prime}}}}\varphi_{123}\bar{\varphi}_{32^{\prime}1}\varphi_{1^{\prime}2^{\prime}3^{\prime}}\bar{\varphi}_{3^{\prime}21^{\prime}}$ (30) $\displaystyle+$ $\displaystyle\lambda_{3}Z^{2}\sum_{\stackrel{{\scriptstyle 1,2,3}}{{1^{\prime},2^{\prime}3^{\prime}}}}\varphi_{123}\bar{\varphi}_{3^{\prime}21}\varphi_{1^{\prime}2^{\prime}3^{\prime}}\bar{\varphi}_{32^{\prime}1^{\prime}}=\lambda_{1}V_{1}+\lambda_{2}V_{2}+\lambda_{3}V_{3},$ (31) and are represented in the figure 1. $V_{1}$ $V_{2}$ $V_{3}$ Figure 1: The vertices of rank 3 tensor model Figure 2: Ward-Takahashi identities The Ward-Takahashi identities (25) now find the form after reducing some constraints $\displaystyle\sum_{p_{2},p_{3}}\big{(}M_{m23}-M_{n23}\big{)}\langle\varphi_{m23}\bar{\varphi}_{n23}\varphi_{nab}\bar{\varphi}_{mab}\rangle_{c}=\langle\varphi_{nab}\bar{\varphi}_{nab}\rangle_{c}-\langle\bar{\varphi}_{mab}\varphi_{mab}\rangle_{c}$ (32) $\displaystyle\sum_{p_{1},p_{3}}\big{(}M_{1m3}-M_{1n3}\big{)}\langle\varphi_{1m3}\bar{\varphi}_{1n3}\varphi_{anb}\bar{\varphi}_{amb}\rangle_{c}=\langle\varphi_{anb}\bar{\varphi}_{anb}\rangle_{c}-\langle\bar{\varphi}_{amb}\varphi_{amb}\rangle_{c}$ (33) $\displaystyle\sum_{p_{1},p_{2}}\big{(}M_{12m}-M_{12n}\big{)}\langle\varphi_{12m}\bar{\varphi}_{12n}\varphi_{abn}\bar{\varphi}_{abm}\rangle_{c}=\langle\varphi_{abn}\bar{\varphi}_{abn}\rangle_{c}-\langle\bar{\varphi}_{abm}\varphi_{abm}\rangle_{c}$ (34) with $M_{abc}=C_{abc}^{-1}$. Graphically the equations (32), (33) and (34) are given in figure 2. Let $G^{ins}_{[mn]ab}$ be the two-point functions with insertion $(2,3)$ i.e. $\displaystyle G^{ins}_{[mn]ab}=\sum_{p_{2},p_{3}}\langle\varphi_{m23}\bar{\varphi}_{n23}\varphi_{nab}\bar{\varphi}_{mab}\rangle_{c}.$ (35) The rest of this section is devoted to find pertubatively, the exact value of the renormalizable two- and four-point functions. We will use the Schwinger- Dyson equation, and then combine it with Ward-Takahashi identities to yield the closed equation that satisfies the connected two- and four-point functions. The Schwinger-Dyson equation is represented graphically in figure 3. In this figure the quantities $T^{\rho}_{abc}$ and $\Sigma^{\rho}_{abc}$ for $\rho=1,2,3,$ are given in the figures 4 and 5. $=\,\sum_{\rho=1}^{3}\Big{(}T^{\rho}_{abc}+\Sigma_{abc}^{\rho}\Big{)}$$\Gamma_{abc}=$ Figure 3: Schwinger-Dyson equation for 1PI two-point functions $T^{1}_{abc}=$$T^{2}_{abc}=$$T^{3}_{abc}=$ Figure 4: $\Sigma^{1}_{abc}=$$+$ $\Sigma^{2}_{abc}=$$+$ $\Sigma^{3}_{abc}=$$+$ Figure 5: In the figure 3 the quantity $\Gamma_{abc}$ is the self-energy or 1PI two- point functions that expresses as $\displaystyle\Gamma_{abc}=\sum_{\rho=1}^{3}\Gamma_{abc}^{\rho},\quad\mbox{where}\quad\Gamma_{abc}^{\rho}=T^{\rho}_{abc}+\Sigma^{\rho}_{abc}.$ (36) Also, in figures 3, 4 and 5 a single circle represents a connected graph and a double circle stands for a 1PI subgraph. Figure 6: Decomposition of the two-point functions with insertion: Case where $\rho=1$ Let us consider now the decomposition given in figure 6. The lhs of this equation collects all connected graphs having the vertex insertion. Cutting this vertex out one gets a four-point functions, but the four-point functions can either be disconnected (first graph on the right hand side (rhs)), or connected (second graph on the rhs). The connected four-point functions must somewhere have a 1PI four-point functions as its core and then full connected two-point functions attached to its four legs. Now, multiplying this equation by $G_{abc}^{-1}$ means on the rhs to remove in the first graph the upper (bc)-branch attached to the insertion vertex and in the second graph the $(abc)$-branch attached to the 1PI four-point functions. If one now sums over $p$ and uses the fact that the newly created vertex is $\lambda_{1}Z^{2}$ one gets precisely the function $\Sigma^{\rho}_{abc}$. Then the equation (36) can be written explicitly using the decomposition of figure 6 as $\displaystyle\Sigma^{1}_{abc}=Z^{2}\lambda_{1}\sum_{p}G_{abc}^{-1}G_{[ap]bc}^{ins},\quad T^{1}_{abc}=Z^{2}\lambda_{1}\sum_{p,q}G_{apq}.$ (37) In the same manner we can obtain the decomposition of figure 7, Figure 7: Decomposition of the two-point functions with insertion: Case where $\rho=2$ and $\rho=3$ which allows to obtain the relations $\Sigma^{2}_{abc}=Z^{2}\lambda_{2}\sum_{p}G_{abc}^{-1}G_{[bp]ca}^{ins},\quad T^{2}_{abc}=Z^{2}\lambda_{2}\sum_{p,q}G_{pbq}$ (38) and $\Sigma^{3}_{abc}=Z^{2}\lambda_{3}\sum_{p}G_{abc}^{-1}G_{[cp]ab}^{ins},\quad T^{3}_{abc}=Z^{2}\lambda_{3}\sum_{p,q}G_{pqc}.$ (39) Therefore using the last expressions (37), (38) and (39), the 1PI two-point functions take the form $\displaystyle\Gamma_{abc}$ $\displaystyle=$ $\displaystyle Z^{2}\lambda_{1}\sum_{p,q}G_{apq}+Z^{2}\lambda_{2}\sum_{p,q}G_{pbq}+Z^{2}\lambda_{3}\sum_{p,q}G_{pqc}$ (40) $\displaystyle+$ $\displaystyle Z^{2}\lambda_{1}\sum_{p}G_{abc}^{-1}G_{[ap]bc}^{ins}+Z^{2}\lambda_{2}\sum_{p}G_{abc}^{-1}G_{[bp]ca}^{ins}+Z^{2}\lambda_{3}\sum_{p}G_{abc}^{-1}G_{[cp]ab}^{ins}$ (41) $\displaystyle=$ $\displaystyle Z^{2}\lambda_{1}\sum_{p,q}G_{apq}+Z^{2}\lambda_{2}\sum_{p,q}G_{pbq}+Z^{2}\lambda_{3}\sum_{p,q}G_{pqc}+Z\lambda_{1}\sum_{p}G_{abc}^{-1}\frac{G_{abc}-G_{pbc}}{\|p\|-\|a\|}$ (42) $\displaystyle+$ $\displaystyle Z\lambda_{2}\sum_{p}G_{abc}^{-1}\frac{G_{bca}-G_{pca}}{\|p\|-\|b\|}+Z\lambda_{3}\sum_{p}G_{abc}^{-1}\frac{G_{cab}-G_{pab}}{\|p\|-\|c\|}.$ (43) We assume now that the function $G_{abc}$ satisfy the condition $\displaystyle G_{abc}=G_{bca}=G_{cab}$ (44) and then, we get the following proposition: ###### Proposition 3. Symmetry properties: The connected two-point functions $\Gamma_{abc}^{2}$ can be obtained using $\Gamma_{abc}^{1}$ and replace respectively $a\rightarrow b$ and $b\rightarrow c$ and $c\rightarrow a$. In the same manner $\Gamma_{abc}^{3}$ can be obtained using $\Gamma_{abc}^{1}$ and replacing respectively $a\rightarrow c$, $b\rightarrow a$ and $c\rightarrow b$. Now using the relation $G_{abc}^{-1}=M_{abc}-\Gamma_{abc}$ , we get $\displaystyle\Gamma_{abc}^{1}=Z^{2}\lambda_{1}\Big{[}\sum_{pq}\frac{1}{M_{apq}-\Gamma_{apq}}+\sum_{p}\frac{1}{M_{pbc}-\Gamma_{pbc}}-\sum_{p}\frac{1}{M_{pbc}-\Gamma_{pbc}}\frac{\Gamma_{abc}-\Gamma_{pbc}}{Z(\|a\|-\|p\|)}\Big{]},$ (45) $\displaystyle\Gamma_{abc}^{2}=Z^{2}\lambda_{2}\Big{[}\sum_{pq}\frac{1}{M_{pbq}-\Gamma_{pbq}}+\sum_{p}\frac{1}{M_{pca}-\Gamma_{pca}}-\sum_{p}\frac{1}{M_{pca}-\Gamma_{pca}}\frac{\Gamma_{bca}-\Gamma_{pca}}{Z(\|b\|-\|p\|)}\Big{]},$ (46) $\displaystyle\Gamma_{abc}^{3}=Z^{2}\lambda_{3}\Big{[}\sum_{pq}\frac{1}{M_{pqc}-\Gamma_{pqc}}+\sum_{p}\frac{1}{M_{pab}-\Gamma_{pab}}-\sum_{p}\frac{1}{M_{pab}-\Gamma_{pab}}\frac{\Gamma_{cab}-\Gamma_{pab}}{Z(\|c\|-\|p\|)}\Big{]}.$ (47) For the rest of this section we consider the connected two-point functions $\Gamma_{abc}^{1}$ and finally $\Gamma_{abc}^{2}$ and $\Gamma_{abc}^{3}$ will be deduced using the proposition (3). Then we pass to renormalized quantities using the Taylor expansion as $\displaystyle\Gamma_{abc}^{1}=ZM_{abc}^{bar}-M_{abc}^{phys}+\Gamma_{abc}^{phys},\quad\Gamma_{000}^{phys}=0=\partial\Gamma_{000}^{phys}$ (48) such that $M_{abc}^{phys}=\|a\|+\|b\|+\|c\|+m^{2},\quad M_{abc}^{bar}=\|a\|+\|b\|+\|c\|+m_{bar}^{2}.$ (49) We get after replacing the expression of $M_{abc}$, $\displaystyle\Gamma_{abc}^{1}=(Z-1)(\|a\|+\|b\|+\|c\|)+Zm^{2}_{bar}-m^{2}+\Gamma_{abc}^{phys},$ (50) which expresses the relation between renormalized and bare quantities. The equation (40) takes the form (we set $\lambda_{1}=\lambda$) $\displaystyle Zm_{bar}^{2}-m^{2}+(Z-1)(\|a\|+\|b\|+\|c\|)+\Gamma_{abc}^{phys}=Z^{2}\lambda\sum_{p,q}\frac{1}{\|p\|+\|q\|+\|a\|+m^{2}-\Gamma^{phys}_{pqa}}$ (51) $\displaystyle+Z\lambda\Big{[}\sum_{p}\frac{1}{\|p\|+\|b\|+\|c\|+m^{2}-\Gamma^{phys}_{pbc}}-\frac{1}{\|p\|+\|b\|+\|c\|+m^{2}-\Gamma^{phys}_{pbc}}\frac{\Gamma_{abc}^{phys}-\Gamma_{pbc}^{phys}}{(\|a\|-\|p\|)}\Big{]}.$ (52) For $a=b=c=0$ the relation of mass variation after renormalization is written as $\displaystyle Zm^{2}_{bar}-m^{2}$ $\displaystyle=$ $\displaystyle Z^{2}\lambda\sum_{p,q}\frac{1}{\|p\|+\|q\|+m^{2}-\Gamma_{pq0}^{phys}}+Z\lambda\sum_{p}\frac{1}{\|p\|+m^{2}-\Gamma_{p00}^{phys}}$ (53) $\displaystyle-$ $\displaystyle Z\lambda\sum_{p}\frac{1}{\|p\|+m^{2}-\Gamma_{p00}^{phys}}\frac{\Gamma_{p00}^{phys}}{\|p\|}.$ (54) Inserting the equation (53) in (51), we get the closed equation of the two- point functions of renormalizable rank $3$ TGFT as $(Z-1)(\|a\|+\|b\|+\|c\|)+\Gamma_{abc}^{phys}=Z^{2}\lambda\sum_{p,q}\Big{[}\frac{1}{\|p\|+\|q\|+\|a\|+m^{2}-\Gamma^{phys}_{pqa}}-\frac{1}{\|p\|+\|q\|+m^{2}-\Gamma_{pq0}^{phys}}\Big{]}\cr+Z\lambda\sum_{p}\Big{[}\frac{1}{\|p\|+\|b\|+\|c\|+m^{2}-\Gamma^{phys}_{pbc}}-\frac{1}{\|p\|+\|b\|+\|c\|+m^{2}-\Gamma^{phys}_{pbc}}\frac{\Gamma_{abc}^{phys}-\Gamma_{pbc}^{phys}}{(\|a\|-\|p\|)}\cr-\frac{1}{\|p\|+m^{2}-\Gamma_{p00}^{phys}}+\frac{1}{\|p\|+m^{2}-\Gamma_{p00}^{phys}}\frac{\Gamma_{p00}^{phys}}{\|p\|}\Big{]}.$ (55) The equation (55) is still very complicated compared to an equivalent one in [28]. To simplify it and get explicit solution, we pass to the integral transforms. The process is to set $\sum_{p\in\mathbb{Z}}=2\int_{0}^{\infty}\,d\|p\|,\quad\sum_{p,q\in\mathbb{Z}}=2\int_{0}^{\infty}\,\|p\|d\|p\|.$ (56) We also assume that $\Gamma_{abc}=\Gamma_{\|a\|\|b\|\|c\|}$. Then we get the integral equation of (55) as $\displaystyle(Z-1)(\|a\|+\|b\|+\|c\|)+\Gamma_{abc}^{phys}$ (57) $\displaystyle=2Z^{2}\lambda\int_{0}^{\infty}\,\|p\|d\|p\|\Big{[}\frac{1}{2\|p\|+\|a\|+m^{2}-\Gamma^{phys}_{ppa}}-\frac{1}{2\|p\|+m^{2}-\Gamma_{pp0}^{phys}}\Big{]}$ (58) $\displaystyle+2Z\lambda\int_{0}^{\infty}\,d\|p\|\Big{[}\frac{1}{\|p\|+\|b\|+\|c\|+m^{2}-\Gamma^{phys}_{pbc}}-\frac{1}{\|p\|+m^{2}-\Gamma_{p00}^{phys}}$ (59) $\displaystyle-\frac{1}{\|p\|+\|b\|+\|c\|+m^{2}-\Gamma^{phys}_{pbc}}\frac{\Gamma_{abc}^{phys}-\Gamma_{pbc}^{phys}}{(\|a\|-\|p\|)}+\frac{1}{\|p\|+m^{2}-\Gamma_{p00}^{phys}}\frac{\Gamma_{p00}^{phys}}{\|p\|}\Big{]}$ (60) with $p\in\mathbb{R^{+}}.$ We introduce a change of variables $\displaystyle\|a\|=m^{2}\frac{\alpha}{1-\alpha},\quad\|b\|=m^{2}\frac{\beta}{1-\beta},\quad\|c\|=m^{2}\frac{\gamma}{1-\gamma},\quad\|p\|=m^{2}\frac{\rho}{1-\rho},$ (61) $\displaystyle\Gamma_{abc}^{phys}=m^{2}\frac{\Gamma_{\alpha\beta\gamma}}{(1-\alpha)(1-\beta)(1-\gamma)}.$ (62) We also take the cutoff $\Lambda$ such that $p_{\Lambda}=m^{2}\frac{\Lambda}{1-\Lambda}.$ Let us now define the quantity $G_{\alpha\beta\gamma}$ as $1-\alpha\beta-\alpha\gamma-\beta\gamma+2\alpha\beta\gamma-\Gamma_{\alpha\beta\gamma}=\frac{1-\alpha\beta-\alpha\gamma-\beta\gamma+2\alpha\beta\gamma}{G_{\alpha\beta\gamma}}.$ (63) Let $\mathcal{J}_{\alpha\beta\gamma}$, $\mathcal{L}_{\alpha\beta\gamma}$ and $\mathcal{K}_{\alpha}$ are three integrals relation given by $\displaystyle\mathcal{J}_{\alpha\beta\gamma}=\int_{0}^{1}\,\frac{d\rho}{(\alpha-\rho)}\frac{G_{\rho\beta\gamma}}{(1-\beta\rho-\gamma\rho-\gamma\beta+2\rho\gamma\beta)},$ (64) $\displaystyle\mathcal{L}_{\alpha\beta\gamma}=\int_{0}^{1}\,\frac{d\rho}{(1-\rho)}\frac{G_{\rho\beta\gamma}-1}{(\alpha-\rho)},$ (65) $\displaystyle\mathcal{K}_{\alpha}=m^{2}\frac{\int_{0}^{1}\,\frac{\rho d\rho}{(1-\rho)}\Big{(}\frac{(1-\alpha)G_{\rho\rho\alpha}}{1-\rho^{2}-2\alpha\rho+2\alpha\rho^{2}}-\frac{G_{\rho\rho 0}}{1-\rho^{2}}\Big{)}}{1+\frac{2\lambda}{m^{2}}\int_{0}^{1}\,d\rho\Big{(}\frac{G^{\prime}_{\rho 00}}{\rho}+G_{\rho 00}\Big{)}}.$ (66) Then we get the following theorem ###### Theorem 1. The connected two-point functions $G_{\alpha\beta\gamma}$ of the renormalizable rank $3$ TGFT on $U(1)$ satisfies the closed integral equation $\displaystyle G_{\alpha\beta\gamma}$ $\displaystyle=$ $\displaystyle 1+\lambda^{\prime}\Big{\\{}\mathcal{Y}+\int_{0}^{1}\,d\rho\,G_{\rho 00}+(1-\alpha)(1-\beta)(1-\gamma)\mathcal{J}_{\alpha\beta\gamma}$ (68) $\displaystyle+\frac{(1-\alpha)(1-\beta)(1-\gamma)G_{\alpha\beta\gamma}}{1-\alpha\beta-\alpha\gamma-\beta\gamma+2\alpha\beta\gamma}\Big{[}-\mathcal{Y}-\int_{0}^{1}\,d\rho\,G_{\rho 00}+\mathcal{K}_{\alpha}$ $\displaystyle+$ $\displaystyle\int_{0}^{1}\,d\rho\,\frac{G_{\rho\beta\gamma}-G_{\rho 00}}{1-\rho}-\int_{0}^{1}\,d\rho\,\frac{G_{\rho\beta\gamma}}{\alpha-\rho}+(1-\alpha)\mathcal{L}_{\alpha\beta\gamma}-\mathcal{L}_{000}\Big{]}\Big{\\}}$ (69) where $\displaystyle\mathcal{Y}=\lim_{\epsilon\rightarrow 0}\int_{0}^{1}\,d\rho\,\frac{G_{\rho\epsilon 0}-G_{\rho 00}}{\epsilon\rho},\qquad\lambda^{\prime}=\frac{2\lambda}{m^{2}}.$ (70) ###### Proof. Using the transformations given in the equations (61) and (62), the expression (57) takes the form $\displaystyle(Z-1)\Big{(}\frac{\alpha}{1-\alpha}+\frac{\beta}{1-\beta}+\frac{\gamma}{1-\gamma}\Big{)}+\frac{\Gamma_{\alpha\beta\gamma}}{(1-\alpha)(1-\beta)(1-\gamma)}$ (71) $\displaystyle=2Z^{2}\lambda\int_{0}^{\Lambda}\,\frac{\rho d\rho}{(1-\rho)}\Big{[}\frac{(1-\alpha)}{1-\rho^{2}-2\alpha\rho+2\alpha\rho^{2}-\Gamma_{\rho\rho\alpha}}-\frac{1}{1-\rho^{2}-\Gamma_{\rho\rho 0}}\Big{]}$ (72) $\displaystyle+\frac{2Z\lambda}{m^{2}}\int_{0}^{\Lambda}\,\frac{d\rho}{(1-\rho)}\Big{[}\frac{(1-\beta)(1-\gamma)}{1-\beta\rho-\gamma\rho-\gamma\beta+2\rho\gamma\beta-\Gamma_{\rho\beta\gamma}}-\frac{1}{1-\Gamma_{\rho 00}}$ (73) $\displaystyle-\frac{1}{1-\beta\rho-\gamma\rho-\gamma\beta+2\rho\gamma\beta-\Gamma_{\rho\beta\gamma}}\frac{(1-\rho)\Gamma_{\alpha\beta\gamma}-(1-\alpha)\Gamma_{\rho\beta\gamma}}{\alpha-\rho}$ (74) $\displaystyle+\frac{1}{1-\Gamma_{\rho 00}}\frac{\Gamma_{\rho 00}}{\rho}\Big{]}.$ (75) Noting that $\beta$ and $\gamma$ are symmetric parameters in the equation (71). This implies that $\Gamma_{\alpha\beta\gamma}=\Gamma_{\alpha\gamma\beta}$. Let us now take $\frac{\partial}{\partial\alpha}\Big{\|}_{\alpha=\beta=\gamma=0}$ and $\frac{\partial}{\partial\beta}\Big{\|}_{\alpha=\beta=\gamma=0}$ of the above equation. We come to the relations that satisfies the renormalized wave function $Z$: $\displaystyle Z-1=2Z^{2}\lambda\int_{0}^{\Lambda}\,\frac{\rho d\rho}{(1-\rho)}\frac{(-1+2\rho-\rho^{2}+\Gamma^{\prime}_{\rho\rho 0}+\Gamma_{\rho\rho 0})}{(1-\rho^{2}-\Gamma_{\rho\rho 0})^{2}}-\frac{2Z\lambda}{m^{2}}\int_{0}^{\Lambda}\,d\rho\frac{\Gamma_{\rho 00}}{\rho^{2}(1-\Gamma_{\rho 00})},$ (76) and $\displaystyle Z-1$ $\displaystyle=$ $\displaystyle\frac{2Z\lambda}{m^{2}}\int_{0}^{\Lambda}\,\frac{d\rho}{(1-\rho)}\Big{[}\frac{-1+\rho+\Gamma_{\rho 00}+\Gamma^{\prime}_{\rho 00}}{(1-\Gamma_{\rho 00})^{2}}-\frac{(\rho+\Gamma^{\prime}_{\rho 00})\Gamma_{\rho 00}}{\rho(1-\Gamma_{\rho 00})^{2}}-\frac{\Gamma^{\prime}_{\rho 00}}{\rho(1-\Gamma_{\rho 00})}\Big{]},$ (77) where we take $\Gamma^{\prime}_{\rho 00}=:\frac{\partial\Gamma_{\rho\beta\gamma}}{\partial\beta}\Big{\|}_{\beta=\gamma=0}$ or $\Gamma^{\prime}_{\rho 00}=:\frac{\partial\Gamma_{\rho\beta\gamma}}{\partial\gamma}\Big{\|}_{\beta=\gamma=0}$ and $\Gamma^{\prime}_{\rho\rho 0}=:\frac{\partial\Gamma_{\rho\rho\alpha}}{\partial\alpha}\Big{\|}_{\alpha=0}$. Now let us pass to the new function $G_{\alpha\beta\gamma}$ given in (63). We find the following relations $\displaystyle\rho+\Gamma^{\prime}_{\rho 00}=\frac{\rho}{G_{\rho 00}}+\frac{G^{\prime}_{\rho 00}}{G_{\rho 00}^{2}},\quad\,\,2\rho-2\rho^{2}+\Gamma^{\prime}_{\rho\rho 0}=\frac{2\rho(1-\rho)}{G_{\rho\rho 0}}+\frac{(1-\rho^{2})G^{\prime}_{\rho\rho 0}}{G_{\rho\rho 0}}.$ (78) Therefore the equation (77) reduces to $\displaystyle Z^{-1}=1+\frac{2\lambda}{m^{2}}\int_{0}^{\Lambda}\,d\rho\Big{[}\frac{G^{\prime}_{\rho 00}}{\rho}+G_{\rho 00}\Big{]},$ (79) and (71) takes the form $\displaystyle ZG_{\alpha\beta\gamma}-1-(Z-1)\frac{(1-\alpha)(1-\beta)(1-\gamma)}{1-\alpha\beta-\alpha\gamma-\beta\gamma+2\alpha\beta\gamma}G_{\alpha\beta\gamma}$ (80) $\displaystyle=\frac{(1-\alpha)(1-\beta)(1-\gamma)}{1-\alpha\beta-\alpha\gamma-\beta\gamma+2\alpha\beta\gamma}G_{\alpha\beta\gamma}\Big{\\{}2Z^{2}\lambda\int_{0}^{\Lambda}\,\frac{\rho d\rho}{(1-\rho)}\Big{[}\frac{(1-\alpha)G_{\rho\rho\alpha}}{1-\rho^{2}-2\alpha\rho+2\alpha\rho^{2}}-\frac{G_{\rho\rho 0}}{1-\rho^{2}}\Big{]}$ (81) $\displaystyle+\frac{2Z\lambda}{m^{2}}\int_{0}^{\Lambda}\,\frac{d\rho}{(1-\rho)}\Big{[}\frac{(1-\beta)(1-\gamma)G_{\rho\beta\gamma}}{1-\beta\rho-\gamma\rho-\gamma\beta+2\rho\gamma\beta}-G_{\rho 00}+\frac{(1-\alpha)(G_{\rho\beta\gamma}-1)}{(\alpha-\rho)}+\frac{G_{\rho 00}-1}{\rho}$ (82) $\displaystyle-\frac{G_{\rho\beta\gamma}}{1-\beta\rho-\gamma\rho-\gamma\beta+2\rho\gamma\beta}\frac{(1-\rho)(1-\alpha\beta-\alpha\gamma-\beta\gamma+2\alpha\beta\gamma)(G_{\alpha\beta\gamma}-1)}{(\alpha-\rho)G_{\alpha\beta\gamma}}\Big{]}\Big{\\}}.$ (83) Inserting (79) into the left hand side of (80) and dividing by $Z$, one gets $\displaystyle G_{\alpha\beta\gamma}$ $\displaystyle=$ $\displaystyle Z^{-1}-\frac{2\lambda}{m^{2}}\frac{(1-\alpha)(1-\beta)(1-\gamma)}{1-\alpha\beta-\alpha\gamma-\beta\gamma+2\alpha\beta\gamma}G_{\alpha\beta\gamma}\int_{0}^{\Lambda}\,d\rho\Big{(}\frac{G^{\prime}_{\rho 00}}{\rho}+G_{\rho 00}\Big{)}$ (84) $\displaystyle-$ $\displaystyle\frac{2\lambda}{m^{2}}\int_{0}^{\Lambda}\,d\rho\frac{(1-\alpha)(1-\beta)(1-\gamma)G_{\rho\beta\gamma}}{1-\beta\rho-\gamma\rho-\gamma\beta+2\rho\gamma\beta}\cdot\frac{(G_{\alpha\beta\gamma}-1)}{(\alpha-\rho)}$ (85) $\displaystyle+$ $\displaystyle\frac{(1-\alpha)(1-\beta)(1-\gamma)G_{\alpha\beta\gamma}}{1-\alpha\beta-\alpha\gamma-\beta\gamma+2\alpha\beta\gamma}\Big{\\{}\frac{2\lambda}{Z^{-1}}\int_{0}^{\Lambda}\,\frac{\rho d\rho}{(1-\rho)}\Big{[}\frac{(1-\alpha)G_{\rho\rho\alpha}}{1-\rho^{2}-2\alpha\rho+2\alpha\rho^{2}}-\frac{G_{\rho\rho 0}}{1-\rho^{2}}\Big{]}$ (86) $\displaystyle+$ $\displaystyle\frac{2\lambda}{m^{2}}\int_{0}^{\Lambda}\,\frac{d\rho}{(1-\rho)}\Big{[}\frac{(1-\beta)(1-\gamma)G_{\rho\beta\gamma}}{1-\beta\rho-\gamma\rho-\gamma\beta+2\rho\gamma\beta}-G_{\rho 00}+\frac{(1-\alpha)(G_{\rho\beta\gamma}-1)}{(\alpha-\rho)}$ (87) $\displaystyle+$ $\displaystyle\frac{G_{\rho 00}-1}{\rho}\Big{]}\Big{\\}}.$ (88) Replacing (79) (84) yields $\displaystyle G_{\alpha\beta\gamma}$ $\displaystyle=$ $\displaystyle 1+\frac{2\lambda}{m^{2}}\Big{\\{}\int_{0}^{\Lambda}\,d\rho\Big{(}\frac{G^{\prime}_{\rho 00}}{\rho}+G_{\rho 00}\Big{)}-\frac{(1-\alpha)(1-\beta)(1-\gamma)}{1-\alpha\beta-\alpha\gamma-\beta\gamma+2\alpha\beta\gamma}G_{\alpha\beta\gamma}$ (89) $\displaystyle\times$ $\displaystyle\int_{0}^{\Lambda}\,d\rho\Big{(}\frac{G^{\prime}_{\rho 00}}{\rho}+G_{\rho 00}\Big{)}-\int_{0}^{\Lambda}\,d\rho\frac{(1-\alpha)(1-\beta)(1-\gamma)G_{\rho\beta\gamma}}{1-\beta\rho-\gamma\rho-\gamma\beta+2\rho\gamma\beta}\cdot\frac{(G_{\alpha\beta\gamma}-1)}{(\alpha-\rho)}$ (90) $\displaystyle+$ $\displaystyle\frac{(1-\alpha)(1-\beta)(1-\gamma)G_{\alpha\beta\gamma}}{1-\alpha\beta-\alpha\gamma-\beta\gamma+2\alpha\beta\gamma}\Big{[}m^{2}\frac{\int_{0}^{\Lambda}\,\frac{\rho d\rho}{(1-\rho)}\Big{(}\frac{(1-\alpha)G_{\rho\rho\alpha}}{1-\rho^{2}-2\alpha\rho+2\alpha\rho^{2}}-\frac{G_{\rho\rho 0}}{1-\rho^{2}}\Big{)}}{1+\frac{2\lambda}{m^{2}}\int_{0}^{\Lambda}\,d\rho\Big{(}\frac{G^{\prime}_{\rho 00}}{\rho}+G_{\rho 00}\Big{)}}$ (91) $\displaystyle+$ $\displaystyle\int_{0}^{\Lambda}\,\frac{d\rho}{(1-\rho)}\Big{(}\frac{(1-\beta)(1-\gamma)G_{\rho\beta\gamma}}{1-\beta\rho-\gamma\rho-\gamma\beta+2\rho\gamma\beta}-G_{\rho 00}+\frac{(1-\alpha)(G_{\rho\beta\gamma}-1)}{(\alpha-\rho)}$ (92) $\displaystyle+$ $\displaystyle\frac{G_{\rho 00}-1}{\rho}\Big{)}\Big{]}\Big{\\}}.$ (93) Simplifying identical terms we get the result of Theorem 1. ∎ Note that $G_{000}=1$ and $\partial G_{000}=0$. The equation (89) shows the occurrence of the singular integral kernel $\int_{0}^{\Lambda}\frac{d\rho}{\rho-\alpha}$, $\int_{0}^{\Lambda}\frac{d\rho}{1-\rho}$ and $\int_{0}^{\Lambda}\frac{d\rho}{\rho}$ for $\Lambda=1$, which needs to be removed. We will use the Cauchy principal value of the divergent integrals and also take the limit value at points $0$ and $1$ i.e. $\int_{0}^{1}=\lim_{\epsilon\rightarrow 0}\Big{[}\int_{0}^{a-\epsilon}+\int_{a+\epsilon}^{1}\Big{]},\quad a\in(0,1),\quad\int_{0}^{1}=\lim_{\epsilon\rightarrow 0,\epsilon^{\prime}\rightarrow 1}\int_{\epsilon}^{\epsilon^{\prime}}$ (94) The nonlinear integral equation (68) is of the form $\displaystyle G_{\alpha\beta\gamma}=1+\lambda\int_{0}^{1}\,f(G_{\alpha\beta\gamma},G_{\rho\beta\gamma},G_{\rho\alpha 0},G_{\rho 00},\mathcal{Y},\alpha,\beta,\gamma)d\rho.$ (95) Now we can easily see that (68) suffers for the lack of symmetry. This inconvenience is due to the position of parameter $\alpha$. So taken $\alpha=0$ we get the symmetric solution given in the following proposition ###### Proposition 4. At first order in $\lambda$ the solution of the equation (68) for $\alpha=0$ is given by $\displaystyle G_{0\beta\gamma}$ $\displaystyle=$ $\displaystyle 1+\lambda^{\prime}\Big{[}1+\frac{(1-\beta)(1-\gamma)}{1-\beta\gamma}\Big{(}\ln\frac{1+\beta\gamma-\beta-\gamma}{1-\beta\gamma}-1\Big{)}\Big{]}=1+\lambda^{\prime}\mathcal{K}_{0\beta\gamma}.$ (96) Then, using the symmetry properties of proposition 3 we get the symmetric solution $G_{\alpha\beta\gamma}^{sym}$ as $\displaystyle G^{sym}_{\alpha\beta\gamma}=1+\lambda^{\prime}_{1}\mathcal{K}_{0\beta\gamma}+\lambda^{\prime}_{2}\mathcal{K}_{0\gamma\alpha}+\lambda^{\prime}_{3}\mathcal{K}_{0\alpha\beta}$ (97) with $\lambda^{\prime}_{\rho}=\frac{2\lambda_{\rho}}{m^{2}};\,\,\rho=1,2,3$ and $\alpha,\beta,\gamma\in[0,1).$ We use the symmetry relation (44) and get the following result ###### Theorem 2. The closed equation of the symmetric two-point functions $G_{\alpha\beta\gamma}$ satisfies the nonlinear integral equation $\displaystyle G_{\alpha\beta\gamma}$ $\displaystyle=$ $\displaystyle 1+\lambda^{\prime}\Big{\\{}\mathcal{Y}+\int_{0}^{1}d\rho\,G_{\rho 00}+\frac{(1-\alpha)(1-\beta)(1-\gamma)}{1-\alpha\beta-\alpha\gamma-\beta\gamma+2\alpha\beta\gamma}\Big{[}\int_{0}^{1}\,d\rho\Big{(}\frac{G_{\rho\beta\gamma}}{\alpha-\rho}$ (98) $\displaystyle+$ $\displaystyle\frac{(2\beta\gamma-\beta-\gamma)G_{\rho\beta\gamma}}{1-\beta\rho-\gamma\rho-\gamma\beta+2\rho\gamma\beta}\Big{)}+G_{\alpha\beta\gamma}\Big{[}\int_{0}^{1}\,d\rho\,\frac{G_{\rho\alpha 0}-G_{\rho 00}}{1-\rho}+\int_{0}^{1}\,d\rho\,\frac{G_{\rho 00}}{\rho}-\mathcal{Y}$ (99) $\displaystyle-$ $\displaystyle\int_{0}^{1}d\rho\,G_{\rho 00}-G_{0\alpha 0}^{-1}\Big{(}\int_{0}^{1}\,d\rho\,\frac{G_{\rho\alpha 0}}{\rho}+\alpha\int_{0}^{1}\,d\rho\,\frac{G_{\rho\alpha 0}}{1-\alpha\rho}+\int_{0}^{1}\,d\rho\,\frac{G_{\rho 00}}{\alpha-\rho}\Big{)}\Big{]}\Big{\\}}.$ (100) ###### Proof. Using the relation (44) we can extract the quantity $\mathcal{K}_{\alpha}$ after simplification as $\displaystyle\mathcal{K}_{\alpha}$ $\displaystyle=$ $\displaystyle- G_{0\alpha\beta}^{-1}\Big{[}\int_{0}^{1}\,d\rho\,\frac{G_{\rho\alpha\beta}}{\rho}-(2\alpha\beta-\alpha-\beta)\int_{0}^{1}\,d\rho\,\frac{G_{\rho\alpha\beta}}{1-\alpha\rho-\beta\rho-\alpha\beta+2\alpha\beta\rho}$ (101) $\displaystyle+$ $\displaystyle\int_{0}^{1}\,d\rho\,\frac{G_{\rho 0\beta}}{\alpha-\rho}-\beta\int_{0}^{1}\,d\rho\,\frac{G_{\rho 0\beta}}{1-\beta\rho}\Big{]}+\int_{0}^{1}\,d\rho\,\frac{G_{\rho\alpha\beta}-G_{\rho 0\beta}}{1-\rho}+\int_{0}^{1}\,d\rho\,\Big{(}\frac{1}{\rho}+\frac{1}{\alpha-\rho}\Big{)}.$ (102) Then remark that $\mathcal{K}_{\alpha}$ is function of only the parameter $\alpha$. We then take $\beta=0$ in the last equation and we get $\displaystyle\mathcal{K}_{\alpha}$ $\displaystyle=$ $\displaystyle-G_{0\alpha 0}^{-1}\Big{(}\int_{0}^{1}\,d\rho\,\frac{G_{\rho\alpha 0}}{\rho}+\alpha\int_{0}^{1}\,d\rho\,\frac{G_{\rho\alpha 0}}{1-\alpha\rho}+\int_{0}^{1}\,d\rho\,\frac{G_{\rho 00}}{\alpha-\rho}\Big{)}$ (104) $\displaystyle+$ $\displaystyle\int_{0}^{1}\,d\rho\,\frac{G_{\rho\alpha 0}-G_{\rho 00}}{1-\rho}+\int_{0}^{1}\,d\rho\,\Big{(}\frac{1}{\rho}+\frac{1}{\alpha-\rho}\Big{)}.$ (105) By replacing the relation (104) in expression (89) we get the desired result. ∎ Now we are reaching the point where it is possible to give the solution of the equation (98). Let us write the solution of this equation as $G_{\alpha\beta\gamma}=1+\sum_{n=1}^{\infty}\mathcal{(}\lambda^{\prime})^{n}X_{\alpha\beta\gamma}^{(n)}$ (106) The $n$ order terms $X_{\alpha\beta\gamma}^{(n)}$ can be deduced by iteration. We give here the quantities $X_{\alpha\beta\gamma}^{(1)}$ and $X_{\alpha\beta\gamma}^{(2)}$ in the following statement ###### Proposition 5. Pertubatively, at second order in $\lambda$ the symmetry solution of the equation (98) using the Cauchy principal value is given by $\displaystyle G_{\alpha\beta\gamma}$ $\displaystyle=$ $\displaystyle 1+\lambda^{\prime}\mathcal{X}^{(1)}_{\alpha\beta\gamma}+\lambda^{\prime 2}\Big{\\{}\frac{\pi^{2}}{6}-\frac{3}{2}+\frac{(1-\alpha)(1-\beta)(1-\gamma)}{1-\alpha\beta-\alpha\gamma-\beta\gamma+2\alpha\beta\gamma}\Big{[}\mathcal{X}^{(1)}_{\alpha\beta\gamma}\Big{(}\ln\frac{(1-\alpha)^{2}}{\alpha}-1\Big{)}$ (107) $\displaystyle+$ $\displaystyle\int_{0}^{1}\,d\rho\frac{(2\beta\gamma-\beta-\gamma)\mathcal{X}^{(1)}_{\rho\beta\gamma}}{1-\beta\rho-\gamma\rho-\beta\gamma+2\beta\gamma\rho}+\int_{0}^{1}\,d\rho\frac{\mathcal{X}^{(1)}_{\rho\alpha 0}-\mathcal{X}^{(1)}_{\rho 00}}{1-\rho}-\alpha\int_{0}^{1}\,d\rho\,\frac{\mathcal{X}_{\rho\alpha 0}^{(1)}}{1-\alpha\rho}$ (108) $\displaystyle-$ $\displaystyle\frac{\pi^{2}}{6}+\frac{3}{2}-\int_{0}^{1}\,d\rho\frac{\mathcal{X}^{(1)}_{\rho\alpha 0}-\mathcal{X}^{(1)}_{\rho 00}+\mathcal{X}^{(1)}_{0\alpha 0}}{\rho}-\mathcal{X}^{(1)}_{0\alpha 0}\ln\frac{(1-\alpha)^{2}}{\alpha}\Big{]}\Big{\\}}+O(\lambda^{\prime 3}),$ (109) where $G_{000}=1$ and where the first order term $\mathcal{X}^{(1)}_{\alpha\beta\gamma}$ is $\displaystyle\mathcal{X}^{(1)}_{\alpha\beta\gamma}$ $\displaystyle=$ $\displaystyle 1+\frac{(1-\alpha)(1-\beta)(1-\gamma)}{1-\alpha\beta-\alpha\gamma-\beta\gamma+2\alpha\beta\gamma}\Big{(}\ln(1-\alpha)-1+\ln\frac{\beta\gamma-\beta-\gamma+1}{1-\beta\gamma}\Big{)}.$ (110) The exact value of the integrals in the rhs of (107) are given using the following relations $\displaystyle\int_{0}^{1}\,d\rho\,\frac{\mathcal{X}^{(1)}_{\rho\beta\gamma}}{a-\rho}$ $\displaystyle=$ $\displaystyle\ln\frac{a}{1-a}+\frac{(1-\beta)(1-\gamma)}{1-a\beta-a\gamma-\beta\gamma+2a\beta\gamma}\Big{(}-1+\ln\frac{\beta\gamma-\beta-\gamma+1}{1-\beta\gamma}\Big{)}$ (111) $\displaystyle\times$ $\displaystyle\Big{(}(1-a)\ln\frac{a}{1-a}+\frac{\beta\gamma-\beta-\gamma+1}{2\beta\gamma-\beta-\gamma}\ln\frac{\beta\gamma-\beta-\gamma+1}{1-\beta\gamma}\Big{)}$ (112) $\displaystyle+$ $\displaystyle\frac{(1-a)(1-\beta)(1-\gamma)}{1-a\beta-a\gamma-\beta\gamma+2a\beta\gamma}\Big{(}\frac{\pi^{2}}{6}-Li_{2}\frac{-a}{1-a}+\ln(1-a)\ln\frac{a}{1-a}-\frac{1}{1-a}\Big{)}$ (113) $\displaystyle+$ $\displaystyle\frac{(1-\beta\gamma)(1-\beta)(1-\gamma)}{(\beta+\gamma-2\beta\gamma)(1-a\beta-a\gamma-\beta\gamma+2a\beta\gamma)}\Big{(}\frac{\pi^{2}}{6}-Li_{2}\frac{\beta\gamma-\beta-\gamma+1}{1-\beta\gamma}$ (114) $\displaystyle-$ $\displaystyle\ln\frac{\beta+\gamma-2\beta\gamma}{1-\beta\gamma}\ln\frac{\beta\gamma-\beta-\gamma+1}{1-\beta\gamma}\Big{)}$ (115) and $\displaystyle\int_{0}^{1}\,d\rho\frac{\mathcal{X}^{(1)}_{\rho\alpha 0}-\mathcal{X}^{(1)}_{\rho 00}+\mathcal{X}^{(1)}_{0\alpha 0}}{\rho}$ $\displaystyle=$ $\displaystyle\frac{(1-\alpha)^{2}}{\alpha}\ln(1-\alpha)\Big{(}\ln(1-\alpha)-1\Big{)}-(1-\alpha)\Big{(}\frac{\pi^{2}}{6}-1\Big{)}$ (116) $\displaystyle+$ $\displaystyle\frac{1-\alpha}{\alpha}\Big{(}\ln\alpha\ln(1-\alpha)+Li_{2}(1-\alpha)-\frac{\pi^{2}}{6}\Big{)}+\frac{\pi^{2}}{6}-1$ (117) where $Li_{2}(x)=\sum_{k=1}^{\infty}\frac{x^{k}}{k^{2}},\quad Li_{2}(1)=\frac{\pi^{2}}{6},\quad Li_{2}(-1)=-\frac{\pi^{2}}{12},\quad Li_{2}(0)=0.$ (118) Let us immediately emphasize that the above solution is related to the coupling constant $\lambda_{1}$. To establish the full solution of the two- point functions of our model, which takes into account the three coupling constants $\lambda_{\rho},\,\,\rho=1,2,3$ we must use the symmetry condition of proposition 3. The end result is given by the sum of the three equations (45),(46) and (47). Therefore the two-point functions $G^{Sym}_{\alpha\beta\gamma}$ of $3D$ tensor model is given by the relation $\displaystyle G_{\alpha\beta\gamma}^{sym}$ $\displaystyle=$ $\displaystyle 1+\lambda^{\prime}_{1}\mathcal{X}^{(1)}_{\alpha\beta\gamma}+\lambda^{\prime 2}_{1}\Big{\\{}\frac{\pi^{2}}{6}-\frac{3}{2}+\frac{(1-\alpha)(1-\beta)(1-\gamma)}{1-\alpha\beta-\alpha\gamma-\beta\gamma+2\alpha\beta\gamma}\Big{[}\mathcal{X}^{(1)}_{\alpha\beta\gamma}\Big{(}\ln\frac{(1-\alpha)^{2}}{\alpha}-1\Big{)}$ (119) $\displaystyle+$ $\displaystyle\int_{0}^{1}\,d\rho\frac{(2\beta\gamma-\beta-\gamma)\mathcal{X}^{(1)}_{\rho\beta\gamma}}{1-\beta\rho-\gamma\rho-\beta\gamma+2\beta\gamma\rho}+\int_{0}^{1}\,d\rho\frac{\mathcal{X}^{(1)}_{\rho\alpha 0}-\mathcal{X}^{(1)}_{\rho 00}}{1-\rho}-\alpha\int_{0}^{1}\,d\rho\,\frac{\mathcal{X}_{\rho\alpha 0}^{(1)}}{1-\alpha\rho}$ (120) $\displaystyle-$ $\displaystyle\frac{\pi^{2}}{6}+\frac{3}{2}-\int_{0}^{1}\,d\rho\frac{\mathcal{X}^{(1)}_{\rho\alpha 0}-\mathcal{X}^{(1)}_{\rho 00}+\mathcal{X}^{(1)}_{0\alpha 0}}{\rho}-\mathcal{X}^{(1)}_{0\alpha 0}\ln\frac{(1-\alpha)^{2}}{\alpha}\Big{]}\Big{\\}}+O(\lambda_{1}^{\prime 3})$ (121) $\displaystyle+$ $\displaystyle\lambda^{\prime}_{2}\mathcal{X}^{(1)}_{\beta\gamma\alpha}+\lambda^{\prime 2}_{2}\Big{\\{}\frac{\pi^{2}}{6}-\frac{3}{2}+\frac{(1-\alpha)(1-\beta)(1-\gamma)}{1-\alpha\beta-\alpha\gamma-\beta\gamma+2\alpha\beta\gamma}\Big{[}\mathcal{X}^{(1)}_{\beta\gamma\alpha}\Big{(}\ln\frac{(1-\beta)^{2}}{\beta}-1\Big{)}$ (122) $\displaystyle+$ $\displaystyle\int_{0}^{1}\,d\rho\frac{(2\alpha\gamma-\alpha-\gamma)\mathcal{X}^{(1)}_{\rho\gamma\alpha}}{1-\alpha\rho-\gamma\rho-\alpha\gamma+2\alpha\gamma\rho}+\int_{0}^{1}\,d\rho\frac{\mathcal{X}^{(1)}_{\rho\beta 0}-\mathcal{X}^{(1)}_{\rho 00}}{1-\rho}-\beta\int_{0}^{1}\,d\rho\,\frac{\mathcal{X}_{\rho\beta 0}^{(1)}}{1-\beta\rho}$ (123) $\displaystyle-$ $\displaystyle\frac{\pi^{2}}{6}+\frac{3}{2}-\int_{0}^{1}\,d\rho\frac{\mathcal{X}^{(1)}_{\rho\beta 0}-\mathcal{X}^{(1)}_{\rho 00}+\mathcal{X}^{(1)}_{0\beta 0}}{\rho}-\mathcal{X}^{(1)}_{0\beta 0}\ln\frac{(1-\beta)^{2}}{\beta}\Big{]}\Big{\\}}+O(\lambda^{\prime 3}_{2})$ (124) $\displaystyle+$ $\displaystyle\lambda^{\prime}_{3}\mathcal{X}^{(1)}_{\gamma\beta\alpha}+\lambda^{\prime 2}_{3}\Big{\\{}\frac{\pi^{2}}{6}-\frac{3}{2}+\frac{(1-\alpha)(1-\beta)(1-\gamma)}{1-\alpha\beta-\alpha\gamma-\beta\gamma+2\alpha\beta\gamma}\Big{[}\mathcal{X}^{(1)}_{\gamma\beta\alpha}\Big{(}\ln\frac{(1-\gamma)^{2}}{\gamma}-1\Big{)}$ (125) $\displaystyle+$ $\displaystyle\int_{0}^{1}\,d\rho\frac{(2\alpha\beta-\alpha-\beta)\mathcal{X}^{(1)}_{\rho\alpha\beta}}{1-\alpha\rho-\beta\rho-\alpha\beta+2\alpha\beta\rho}+\int_{0}^{1}\,d\rho\frac{\mathcal{X}^{(1)}_{\rho\gamma 0}-\mathcal{X}^{(1)}_{\rho 00}}{1-\rho}-\gamma\int_{0}^{1}\,d\rho\,\frac{\mathcal{X}_{\rho\gamma 0}^{(1)}}{1-\gamma\rho}$ (126) $\displaystyle-$ $\displaystyle\frac{\pi^{2}}{6}+\frac{3}{2}-\int_{0}^{1}\,d\rho\frac{\mathcal{X}^{(1)}_{\rho\gamma 0}-\mathcal{X}^{(1)}_{\rho 00}+\mathcal{X}^{(1)}_{0\gamma 0}}{\rho}-\mathcal{X}^{(1)}_{0\gamma 0}\ln\frac{(1-\gamma)^{2}}{\gamma}\Big{]}\Big{\\}}+O(\lambda^{\prime 3}_{3})$ (127) where $\lambda^{\prime}_{\rho}=2\lambda_{\rho}/m^{2};\,\,\rho=1,2,3$, $\alpha,\beta,\gamma\in(0,1)$ and $G_{000}=1.$ Noting that the solution (119) satisfies the condition (44) if and only if we set $\lambda_{1}^{\prime}=\lambda_{2}^{\prime}=\lambda_{3}^{\prime}$. Let us also emphasize that the higher order solution can be get pertubatively by iteration. ## 4 Closed equation for two-point functions of rank 4 TGFT The same method use in last section will be performed here to establish the renormalized two-point functions of rank $4$ tensor field firstly given in [13]. We provide the master equation of the two-point functions. The action $S_{4D}$ of the model is also subdivided into two terms as $\displaystyle S_{4D}=S_{4D}^{\text{kin\,}}+S_{4D}^{\text{int\,}}.$ (128) The kinetic term $S_{4D}^{\text{kin\,}}$ is given by $\displaystyle S_{4D}^{\text{kin\,}}=\sum_{p_{j}\in\mathbb{Z}}\varphi_{1234}\Big{(}\sum_{i=1}^{4}p_{i}^{2}+m^{2}\Big{)}\bar{\varphi}_{1234}.$ (129) Noting that in four dimensional case the renormalization is guaranteed by the presence of the propagator associated with the heat kernel [27]: $C([p])=\Big{(}\sum_{i=1}^{4}p_{i}^{2}+m^{2}\Big{)}^{-1}=M_{1234}^{-1}.$ (130) $S_{4D}^{\text{int\,}}$ is related to the interaction, which is divided into three fundamental contributions $V_{6,1}$, $V_{6,2}$ and $V_{4,1}$ given by $\displaystyle V_{6;1}$ $\displaystyle=$ $\displaystyle\sum_{p_{j}\in\mathbb{Z}}\varphi_{1234}\,\bar{\varphi}_{1^{\prime}234}\,\varphi_{1^{\prime}2^{\prime}3^{\prime}4^{\prime}}\,\bar{\varphi}_{1^{\prime\prime}2^{\prime}3^{\prime}4^{\prime}}\,\varphi_{1^{\prime\prime}2^{\prime\prime}3^{\prime\prime}4^{\prime\prime}}\,\bar{\varphi}_{12^{\prime\prime}3^{\prime\prime}4^{\prime\prime}}+\text{permutations }$ (131) $\displaystyle V_{6;2}$ $\displaystyle=$ $\displaystyle\sum_{p_{j}\in\mathbb{Z}}\varphi_{1234}\,\bar{\varphi}_{1^{\prime}2^{\prime}3^{\prime}4}\,\varphi_{1^{\prime}2^{\prime}3^{\prime}4^{\prime}}\,\bar{\varphi}_{1^{\prime\prime}234^{\prime}}\,\varphi_{1^{\prime\prime}2^{\prime\prime}3^{\prime\prime}4^{\prime\prime}}\,\bar{\varphi}_{12^{\prime\prime}3^{\prime\prime}4^{\prime\prime}}+\text{permutations }$ (132) $\displaystyle V_{4;1}$ $\displaystyle=$ $\displaystyle\sum_{p_{j}\in\mathbb{Z}}\varphi_{1234}\,\bar{\varphi}_{1^{\prime}234}\,\varphi_{1^{\prime}2^{\prime}3^{\prime}4^{\prime}}\,\bar{\varphi}_{12^{\prime}3^{\prime}4^{\prime}}\,+\text{permutations }$ (133) and an anomalous term, namely $V_{4,2}$ $\displaystyle V_{4;2}$ $\displaystyle=$ $\displaystyle\Big{(}\sum_{p_{j}\in\mathbb{Z}}\bar{\varphi}_{1234}\,\varphi_{1234}\Big{)}\Big{(}\sum_{p_{j}\in\mathbb{Z}}\bar{\varphi}_{1^{\prime}2^{\prime}3^{\prime}4^{\prime}}\,\varphi_{1^{\prime}2^{\prime}3^{\prime}4^{\prime}}\Big{)}.$ (134) This last vertex is not taken into account in the computation of the correlation functions due to the fact that it is disconnected and does not contribute to the melonic Feynman graph of the theory. This vertex could be interpreted as the generation of a scalar matter field out of pure gravity [13]. The vertices are represented in figure 8. Let us immediately emphasize that the vertices of the type $V_{6,1}$ and $V_{4,1}$ are parametrized by four indices $\rho\in\\{1,2,3,4\\}$, and the vertices contributing to $V_{6,2}$ are parametrized by six index values $\rho\rho^{\prime}\in\\{1.2,1.3,1.4,2.3,2.4,3.4\\}$. The couple $\rho\rho^{\prime}$ will be totally symmetric i.e., $\rho\rho^{\prime}=\rho^{\prime}\rho$. $V_{6,1;1}$ $V_{6,2;14}$ $V_{4,1;1}$ $V_{4,2;1}$ Figure 8: Vertex representation of $4D$ tensor model One can check that these interaction are invariant under $U(N_{a})$ transformations. Then the same procedure of finding the Ward-Takahashi identities applies. The Ward-Takahashi identities of the equation (25) is re- expressed as $\displaystyle\big{(}M_{m234}-M_{n234}\big{)}\langle[\varphi_{m}\bar{\varphi}_{n}]_{234}\varphi_{n234}\bar{\varphi}_{m234}\rangle_{c}=\langle\varphi_{n234}\bar{\varphi}_{n234}\rangle_{c}-\langle\bar{\varphi}_{m234}\varphi_{m234}\rangle_{c}$ (135) $\displaystyle\big{(}M_{1m34}-M_{1n34}\big{)}\langle[\varphi_{m}\bar{\varphi}_{n}]_{134}\varphi_{1n34}\bar{\varphi}_{1m34}\rangle_{c}=\langle\varphi_{1n34}\bar{\varphi}_{1n34}\rangle_{c}-\langle\bar{\varphi}_{1m34}\varphi_{1m34}\rangle_{c}$ (136) $\displaystyle\big{(}M_{12m4}-M_{12n4}\big{)}\langle[\varphi_{m}\bar{\varphi}_{n}]_{124}\varphi_{12n4}\bar{\varphi}_{12m4}\rangle_{c}=\langle\varphi_{12n4}\bar{\varphi}_{12n4}\rangle_{c}-\langle\bar{\varphi}_{12m4}\varphi_{12m4}\rangle_{c}$ (137) $\displaystyle\big{(}M_{123m}-M_{123n}\big{)}\langle[\varphi_{m}\bar{\varphi}_{n}]_{123}\varphi_{123n}\bar{\varphi}_{123m}\rangle_{c}=\langle\varphi_{123n}\bar{\varphi}_{123n}\rangle_{c}-\langle\bar{\varphi}_{123m}\varphi_{123m}\rangle_{c}.$ (138) $\Gamma_{abcd}=$ $\sum_{\rho}\Big{(}\Gamma_{abcd}^{6,1;\rho}+\Gamma_{abcd}^{4,1;\rho}\Big{)}+\sum_{\rho\rho^{\prime}}\Gamma_{abcd}^{6,2;\rho\rho^{\prime}}$ Figure 9: Schwinger-Dyson equation of rank $4$ tensor model The figure 9 gives the Schwinger-Dyson equation of the two-point functions. This figures collects the 1PI two-point functions. $3$Permutation $\rho$$\Gamma_{abcd}^{6,1}=$ Figure 10: $5$Permutation $\rho\rho^{\prime}$$\Gamma_{abcd}^{6,2}=$ Figure 11: $3$Permutation $\rho$$\Gamma_{abcd}^{4,1}=$ Figure 12: Let us discuss the contributions in this figure. The graphs of figure 10 are related to the graphs made with vertex $V_{6,1}$. The first graph of this figure is denoted by $T^{6,1}_{abcd}$ and the sum of the other two is $\Sigma^{6,1}_{abcd}$. The graphs of figure 11 are related to the graphs built with the vertex $V_{6,2}$. The first graph of this figure is called $T^{6,2}_{abcd}$ and the sum of the other two is $\Sigma^{6,2}_{abcd}$. In the same manner, the graphs of figure 12 take into account the graphs built with vertex $V_{4,1}$. The first graph is called $\Sigma^{4,1}_{abcd}$ and the sum of the over two is $T^{4,1}_{abcd}$. Then the relations given in figures 10, 11 and 12 are re-expressed simply as $\displaystyle\Gamma_{abcd}^{6,1}=\sum_{\rho}\Gamma_{abcd}^{6,1;\rho},\quad\Gamma_{abcd}^{6,2}=\sum_{\rho\rho^{\prime}}\Gamma_{abcd}^{6,2;\rho\rho^{\prime}}\quad\Gamma_{abcd}^{4,1}=\sum_{\rho}\Gamma_{abcd}^{4,1;\rho}$ (139) with $\displaystyle\Gamma_{abcd}^{6,1;\rho}=T_{abcd}^{6,1;\rho}+\Sigma_{abcd}^{6,1;\rho},\quad\Gamma_{abcd}^{6,2;\rho\rho^{\prime}}=T_{abcd}^{6,2;\rho\rho^{\prime}}+\Sigma_{abcd}^{6,2;\rho\rho^{\prime}},\quad\Gamma_{abcd}^{4,1;\rho}=T_{abcd}^{4,1;\rho}+\Sigma_{abcd}^{4,1;\rho}.$ (140) Therefore the equation on figure 9 takes the form $\displaystyle\Gamma_{abcd}=\Gamma_{abcd}^{6,1}+\Gamma_{abcd}^{4,1}+\Gamma_{abcd}^{6,2}.$ (141) All of the above quantities are obtained by using the following symmetry properties: ###### Proposition 6. * • $\Gamma_{abcd}^{6,1;2}$ can be obtained using $\Gamma_{abcd}^{6,1;1}$ and replaced $a\rightarrow b$ and $b\rightarrow a$. * • $\Gamma_{abcd}^{6,1;3}$ is obtained using $\Gamma_{abcd}^{6,1;1}$ and replaced $a\rightarrow c$, $b\rightarrow a$ and $c\rightarrow b$. * • $\Gamma_{abcd}^{6,1;4}$ is obtained using $\Gamma_{abcd}^{6,1;1}$ and replaced $a\rightarrow d$, $b\rightarrow a$, $c\rightarrow b$ and $d\rightarrow c$. This above symmetries is well satisfied for $\Gamma_{abcd}^{4,1;\rho}$. In the case of $\Gamma_{abcd}^{6,2;\rho\rho^{\prime}}$ we get: * • $\Gamma_{abcd}^{6,2;13}$ can be obtained using $\Gamma_{abcd}^{6,2;14}$ and by replaced $a\rightarrow b$ and $b\rightarrow a$. * • $\Gamma_{abcd}^{6,2;12}$ is obtained by replaced in $\Gamma_{abcd}^{6,2;14}$, $a\rightarrow c$, $b\rightarrow a$ and $c\rightarrow b$. * • $\Gamma_{abcd}^{6,2;23}$ is obtained by replaced in $\Gamma_{abcd}^{6,2;14}$, $a\rightarrow b$, $b\rightarrow a$, $c\rightarrow d$ and $d\rightarrow c$. * • $\Gamma_{abcd}^{6,2;24}$ is obtained by replaced in $\Gamma_{abcd}^{6,2;14}$, $c\rightarrow d$ and $d\rightarrow c$. * • $\Gamma_{abcd}^{6,2;34}$ is obtained by replaced in $\Gamma_{abcd}^{6,2;14}$, $b\rightarrow c$, $c\rightarrow d$ and $d\rightarrow b$. We then focus our attention to $\Gamma_{abcd}^{1}=(\Gamma_{abcd}^{6,1;1}+\Gamma_{abcd}^{4,1;1})+\Gamma_{abcd}^{6,2;14}$. We also call $G_{[mn]abc}^{ins}$ the two-point functions with insertion (1,2,3) wherein the momentum indices $p_{1},p_{2},p_{3}$ are summed i.e. $\displaystyle G_{[mn]abc}^{ins}=\sum_{p_{1},p_{2},p_{3}}\langle\varphi_{m123}\bar{\varphi}_{n123}\varphi_{nabc}\bar{\varphi}_{mabc}\rangle_{c}.$ (142) The following relations are satisfied: $\displaystyle\Sigma_{abcd}^{6,1;1}=Z^{2}\lambda_{6,1;1}C_{abcd}\sum_{p}G_{abcd}^{-1}G_{[ap]bcd}^{ins},\quad T_{abcd}^{6,1;1}=Z^{2}\lambda_{6,1;1}C_{abcd}\sum_{p,q,r}G_{pqra},$ (143) $\displaystyle\Sigma_{abcd}^{6,2;14}=Z^{2}\lambda_{6,2;14}C_{abcd}\sum_{p}G_{abcd}^{-1}G_{[ap]bcd}^{ins},\quad T_{abcd}^{6,2;14}=Z^{2}\lambda_{6,2;14}C_{abcd}\sum_{p,q,r}G_{pqra},$ (144) $\displaystyle\Sigma_{abcd}^{4,1;1}=Z^{2}\lambda_{4,1;1}\sum_{p}G_{abcd}^{-1}G_{[ap]bcd}^{ins},\quad T_{abcd}^{4,1;1}=Z^{2}\lambda_{6,1;1}\sum_{p,q,r}G_{pqra}$ (145) and then $\displaystyle\Gamma_{abcd}^{1}$ $\displaystyle=$ $\displaystyle Z^{2}C_{abcd}\lambda_{6,1;1}\Big{[}\sum_{p}G_{abcd}^{-1}G_{[ap]bcd}^{ins}+\sum_{p,q,r}G_{pqra}\Big{]}+Z^{2}C_{abcd}\lambda_{6,2;14}\Big{[}\sum_{p}G_{abcd}^{-1}G_{[ap]bcd}^{ins}$ (146) $\displaystyle+$ $\displaystyle\sum_{p,q,r}G_{pqra}\Big{]}+Z^{2}\lambda_{4,1;1}\Big{[}\sum_{p}G_{abcd}^{-1}G_{[ap]bcd}^{ins}+\sum_{p,q,r}G_{pqra}\Big{]}.$ (147) We set $\lambda_{6,1;\rho}=\lambda_{6,1}$, $\lambda_{6,2;\rho\rho^{\prime}}=\lambda_{6,2}$ and $\lambda_{4,1;\rho}=\lambda_{4,1}$. Noting that the connected to point function can be expressed as $G_{abcd}^{-1}=M_{abcd}-\Gamma_{abcd}$. Then we get $\displaystyle\Gamma_{abcd}^{1}$ $\displaystyle=$ $\displaystyle Z^{2}M^{-1}_{abcd}\lambda_{6,1}\Big{[}\sum_{p}G_{abcd}^{-1}\frac{G_{pbcd}-G_{abcd}}{Z(a^{2}-p^{2})}+\sum_{p,q,r}G_{pqra}\Big{]}$ (148) $\displaystyle+$ $\displaystyle Z^{2}M^{-1}_{abcd}\lambda_{6,2}\Big{[}\sum_{p}G_{abcd}^{-1}\frac{G_{pbcd}-G_{abcd}}{Z(a^{2}-p^{2})}+\sum_{p,q,r}G_{pqra}\Big{]}$ (149) $\displaystyle+$ $\displaystyle Z^{2}\lambda_{4,1}\Big{[}\sum_{p}G_{abcd}^{-1}\frac{G_{pbcd}-G_{abcd}}{Z(a^{2}-p^{2})}+\sum_{p,q,r}G_{pqra}\Big{]}$ (150) $\displaystyle=$ $\displaystyle Z^{2}M^{-1}_{abcd}\lambda_{6,1}\Big{[}\sum_{p}\Big{(}\frac{1}{M_{pbcd}-\Gamma_{pbcd}}-\frac{1}{M_{pbcd}-\Gamma_{pbcd}}\frac{\Gamma_{abcd}-\Gamma_{pbcd}}{Z(a^{2}-p^{2})}\Big{)}+\sum_{p,q,r}\frac{1}{M_{pqra}-\Gamma_{pqra}}\Big{]}$ (151) $\displaystyle+$ $\displaystyle Z^{2}M^{-1}_{abcd}\lambda_{6,2}\Big{[}\sum_{p}\Big{(}\frac{1}{M_{pbcd}-\Gamma_{pbcd}}-\frac{1}{M_{pbcd}-\Gamma_{pbcd}}\frac{\Gamma_{abcd}-\Gamma_{pbcd}}{Z(a^{2}-p^{2})}\Big{)}+\sum_{p,q,r}\frac{1}{M_{pqra}-\Gamma_{pqra}}\Big{]}$ (152) $\displaystyle+$ $\displaystyle Z^{2}\lambda_{4,1}\Big{[}\sum_{p}\Big{(}\frac{1}{M_{pbcd}-\Gamma_{pbcd}}-\frac{1}{M_{pbcd}-\Gamma_{pbcd}}\frac{\Gamma_{abcd}-\Gamma_{pbcd}}{Z(a^{2}-p^{2})}\Big{)}+\sum_{p,q,r}\frac{1}{M_{pqra}-\Gamma_{pqra}}\Big{]}.$ (153) Now we use the Taylor expansion that allows to pass to the renormalized quantity as $\Gamma_{abcd}^{1}=Zm_{bar}^{2}-m^{2}+(Z-1)(a^{2}+b^{2}+c^{2}+d^{2})+\Gamma_{abcd}^{phys},$ (154) with condition $\Gamma_{0000}=0$ and $\partial\Gamma_{0000}=0$. This implies that $G_{abcd}^{-1}=a^{2}+b^{2}+c^{2}+d^{2}+m^{2}-\Gamma_{abcd}^{ren}.$ (155) Then we get the following proposition ###### Proposition 7. The closed equation of the two-point functions of four dimension tensor model is given by $\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad(Z-1)(a^{2}+b^{2}+c^{2}+d^{2})+\Gamma_{abcd}^{phys}$ (158) $\displaystyle=M^{-1}_{abcd}\lambda_{6,1}\Big{\\{}\sum_{p}\Big{[}\frac{Z}{p^{2}+b^{2}+c^{2}+d^{2}+m^{2}-\Gamma_{pbcd}^{phys}}-\frac{1}{m^{2}}\frac{M_{abcd}}{(p^{2}+m^{2}-\Gamma_{p000}^{phys})}$ $\displaystyle-\frac{Z}{p^{2}+b^{2}+c^{2}+d^{2}+m^{2}-\Gamma_{pbcd}^{phys}}\frac{\Gamma_{abcd}^{phys}-\Gamma_{pbcd}^{phys}}{(a^{2}-p^{2})}+\frac{1}{m^{2}}\frac{M_{abcd}}{(p^{2}+m^{2}-\Gamma_{p000}^{phys})}\frac{\Gamma_{p000}^{phys}}{p^{2}}\Big{]}$ $\displaystyle+$ $\displaystyle\sum_{p,q,r}\Big{[}\frac{Z^{2}}{p^{2}+q^{2}+r^{2}+a^{2}+m^{2}-\Gamma_{pqra}^{phys}}-\frac{1}{m^{2}}\frac{ZM_{abcd}}{(p^{2}+q^{2}+r^{2}+m^{2}-\Gamma_{pqr0}^{phys})}\Big{]}\Big{\\}}$ (160) $\displaystyle+M^{-1}_{abcd}\lambda_{6,2}\Big{\\{}\sum_{p}\Big{[}\frac{Z}{p^{2}+b^{2}+c^{2}+d^{2}+m^{2}-\Gamma_{pbcd}^{phys}}-\frac{1}{m^{2}}\frac{M_{abcd}}{(p^{2}+m^{2}-\Gamma_{p000}^{phys})}$ $\displaystyle-$ $\displaystyle\frac{Z}{p^{2}+b^{2}+c^{2}+d^{2}+m^{2}-\Gamma_{pbcd}^{phys}}\frac{\Gamma_{abcd}^{phys}-\Gamma_{pbcd}^{phys}}{(a^{2}-p^{2})}+\frac{1}{m^{2}}\frac{M_{abcd}}{(p^{2}+m^{2}-\Gamma_{p000}^{phys})}\frac{\Gamma_{p000}^{phys}}{p^{2}}\Big{]}$ (161) $\displaystyle+$ $\displaystyle\sum_{p,q,r}\Big{[}\frac{Z^{2}}{p^{2}+q^{2}+r^{2}+a^{2}+m^{2}-\Gamma_{pqra}^{phys}}-\frac{1}{m^{2}}\frac{ZM_{abcd}}{(p^{2}+q^{2}+r^{2}+m^{2}-\Gamma_{pqr0}^{phys})}\Big{]}\Big{\\}}$ (163) $\displaystyle+\lambda_{4,1}\Big{\\{}\sum_{p}\Big{[}\frac{Z}{p^{2}+b^{2}+c^{2}+d^{2}+m^{2}-\Gamma_{pbcd}^{phys}}-\frac{Z}{p^{2}+m^{2}-\Gamma_{p000}^{phys}}$ $\displaystyle-$ $\displaystyle\frac{Z}{p^{2}+b^{2}+c^{2}+d^{2}+m^{2}-\Gamma_{pbcd}^{phys}}\frac{\Gamma_{abcd}^{phys}-\Gamma_{pbcd}^{phys}}{(a^{2}-p^{2})}+\frac{Z}{p^{2}+m^{2}-\Gamma_{p000}^{phys}}\frac{\Gamma_{p000}^{phys}}{p^{2}}\Big{]}$ (165) $\displaystyle+\sum_{p,q,r}\Big{[}\frac{Z^{2}}{p^{2}+q^{2}+r^{2}+a^{2}+m^{2}-\Gamma_{pqra}^{phys}}-\frac{Z^{2}}{p^{2}+q^{2}+r^{2}+m^{2}-\Gamma_{pqr0}^{phys}}\Big{]}\Big{\\}}.$ ###### Proof. The equation (158) can be simply obtained using the relation of $Zm_{bar}^{2}-m^{2}$ in the same way of the last section as $\displaystyle Zm_{bar}^{2}-m^{2}=ZM^{-1}_{0000}\lambda_{6,1}\Big{[}\sum_{p}\Big{(}\frac{1}{p^{2}+m^{2}-\Gamma_{p000}^{phys}}-\frac{1}{p^{2}+m^{2}-\Gamma_{p000}^{phys}}\frac{\Gamma_{p000}^{phys}}{p^{2}}\Big{)}$ (166) $\displaystyle+\sum_{p,q,r}\frac{Z}{p^{2}+q^{2}+r^{2}+m^{2}-\Gamma_{pqr0}^{phys}}\Big{]}+ZM^{-1}_{0000}\lambda_{6,2}\Big{[}\sum_{p}\Big{(}\frac{1}{p^{2}+m^{2}-\Gamma_{p000}^{phys}}$ (167) $\displaystyle-\frac{1}{p^{2}+m^{2}-\Gamma_{p000}^{phys}}\frac{\Gamma_{p000}^{phys}}{p^{2}}\Big{)}+\sum_{p,q,r}\frac{Z}{p^{2}+q^{2}+r^{2}+m^{2}-\Gamma_{pqr0}^{phys}}\Big{]}+Z\lambda_{4,1}\Big{[}\sum_{p}\Big{(}\frac{1}{p^{2}+m^{2}-\Gamma_{p000}^{phys}}$ (168) $\displaystyle-\frac{1}{p^{2}+m^{2}-\Gamma_{p000}^{phys}}\frac{\Gamma_{p000}^{phys}}{p^{2}}\Big{)}+\sum_{p,q,r}\frac{Z}{p^{2}+q^{2}+r^{2}+m^{2}-\Gamma_{pqr0}^{phys}}\Big{]},\quad M_{0000}^{-1}=\frac{1}{Zm^{2}}$ (169) Then (158) takes the form by replacing the relation (166) into the right hand side of equation (148). ∎ Let us remark that the continuous limit of the equation (158) can be built. We identify the sum as $\sum_{p}=2\int_{0}^{\infty}\,dp$ and $\sum_{p,q,r}=2\int_{0}^{\infty}\,p^{2}dp$. We also impose the cutoff $p_{\Lambda}$ in the $UV$ and changing the variables as $\displaystyle a^{2}=m^{2}\frac{\alpha}{1-\alpha},\quad b^{2}=m^{2}\frac{\beta}{1-\beta},\quad c^{2}=m^{2}\frac{\gamma}{1-\gamma},$ (170) $\displaystyle d^{2}=m^{2}\frac{\epsilon}{1-\epsilon},\quad p^{2}=m^{2}\frac{\rho}{1-\rho},\quad p^{2}_{\Lambda}=m^{2}\frac{\Lambda}{1-\Lambda}.$ (171) Now let us define the two quantities $s(\alpha,\beta,\gamma,\epsilon)$ and $p(\alpha,\beta,\gamma,\epsilon)$ as $s(\alpha,\beta,\gamma,\epsilon)=1-\alpha\beta-\alpha\gamma-\alpha\epsilon-\beta\gamma-\beta\epsilon-\gamma\epsilon+2\alpha\beta\gamma+2\alpha\beta\epsilon+2\alpha\gamma\epsilon+2\beta\gamma\epsilon-3\alpha\beta\gamma\epsilon$ (172) and $p(\alpha,\beta,\gamma,\epsilon)=(1-\alpha)(1-\beta)(1-\gamma)(1-\epsilon).$ (173) The equation (158) is re-expressed as $\displaystyle m^{2}(Z-1)\frac{p(\alpha,\beta,\gamma,\epsilon)}{(1-\alpha)(1-\beta)(1-\gamma)(1-\epsilon)}+m^{2}\frac{\Gamma_{\alpha\beta\gamma\epsilon}}{(1-\alpha)(1-\beta)(1-\gamma)(1-\epsilon)}$ (174) $\displaystyle=$ $\displaystyle 2M^{-1}_{\alpha\beta\gamma\epsilon}(\lambda_{6,1}+\lambda_{6,2})\Big{\\{}\int_{0}^{\Lambda}\,\frac{1}{2m}\sqrt{\frac{1-\rho}{m\rho}}\frac{d\rho}{(1-\rho)^{2}}\Big{[}\frac{Z(1-\rho)(1-\beta)(1-\gamma)(1-\epsilon)}{s(\rho,\beta,\gamma,\epsilon)-\Gamma_{\rho\beta\gamma\epsilon}}$ (175) $\displaystyle-$ $\displaystyle\frac{1}{m^{2}}\frac{M_{\alpha\beta\gamma\epsilon}(1-\rho)}{1-\Gamma_{\rho 000}}-\frac{Z(1-\rho)}{(s(\rho,\beta,\gamma,\epsilon)-\Gamma_{\rho\beta\gamma\epsilon})}\frac{(1-\rho)\Gamma_{\alpha\beta\gamma\epsilon}-(1-\alpha)\Gamma_{\rho\beta\gamma\epsilon}}{(\alpha-\rho)}$ (176) $\displaystyle+$ $\displaystyle\frac{1}{m^{2}}\frac{M_{\alpha\beta\gamma\epsilon}(1-\rho)}{(1-\Gamma_{\rho 000})}\frac{\Gamma_{\rho 000}}{\rho}\Big{]}+\int_{0}^{\Lambda}\frac{1}{2m}\sqrt{\frac{m\rho}{1-\rho}}\frac{d\rho}{(1-\rho)^{2}}\Big{[}\frac{Z^{2}(1-\rho)^{3}(1-\alpha)}{s(\rho,\rho,\rho,\alpha)-\Gamma_{\rho\rho\rho\alpha}}$ (177) $\displaystyle-$ $\displaystyle\frac{1}{m^{2}}\frac{ZM_{\alpha\beta\gamma\epsilon}(1-\rho)^{3}}{(2\rho^{3}-3\rho^{2}+1-\Gamma_{\rho\rho\rho 0})}\Big{]}\Big{\\}}$ (178) $\displaystyle+$ $\displaystyle\lambda_{4,1}\Big{\\{}\int_{0}^{\Lambda}\,\frac{1}{2m}\sqrt{\frac{1-\rho}{m\rho}}\frac{d\rho}{(1-\rho)^{2}}\Big{[}\frac{Z(1-\rho)(1-\beta)(1-\gamma)(1-\epsilon)}{s(\rho,\beta,\gamma,\epsilon)-\Gamma_{\rho\beta\gamma\epsilon}}$ (179) $\displaystyle-$ $\displaystyle\frac{Z(1-\rho)}{1-\Gamma_{\rho 000}}-\frac{Z(1-\rho)}{(s(\rho,\beta,\gamma,\epsilon)-\Gamma_{\rho\beta\gamma\epsilon})}\frac{(1-\rho)\Gamma_{\alpha\beta\gamma\epsilon}-(1-\alpha)\Gamma_{\rho\beta\gamma\epsilon}}{(\alpha-\rho)}$ (180) $\displaystyle+$ $\displaystyle\frac{Z(1-\rho)}{(1-\Gamma_{\rho 000})}\frac{\Gamma_{\rho 000}}{\rho}\Big{]}+\int_{0}^{\Lambda}\frac{1}{2m}\sqrt{\frac{m\rho}{1-\rho}}\frac{d\rho}{(1-\rho)^{2}}\Big{[}\frac{Z^{2}(1-\rho)^{3}(1-\alpha)}{s(\rho,\rho,\rho,\alpha)-\Gamma_{\rho\rho\rho\alpha}}$ (181) $\displaystyle-$ $\displaystyle\frac{Z^{2}(1-\rho)^{3}}{(2\rho^{3}-3\rho^{2}+1-\Gamma_{\rho\rho\rho 0})}\Big{]}\Big{\\}}.$ (182) The wave function $Z$ can be also deduced as $\displaystyle Z=\frac{1-\frac{2}{m^{2}}(\lambda_{6,1}+\lambda_{6,2})\int_{0}^{\Lambda}\,\frac{d\rho}{2m^{3}}\sqrt{\frac{1-\rho}{m\rho}}\Big{(}G_{\rho 000}+\frac{G^{\prime}_{\rho 000}}{\rho}\Big{)}}{1+\lambda_{4,1}\int_{0}^{\Lambda}\,\frac{d\rho}{2m^{3}}\sqrt{\frac{1-\rho}{m\rho}}\Big{(}G_{\rho 000}+\frac{G^{\prime}_{\rho 000}}{\rho}\Big{)}},$ (183) where $\displaystyle s(\alpha,\beta,\gamma,\epsilon)-\Gamma_{\alpha\beta\gamma\epsilon}=\frac{s(\alpha,\beta,\gamma,\epsilon)}{G_{\alpha\beta\gamma\epsilon}},$ (184) and $M_{\alpha\beta\gamma\epsilon}=Zm^{2}\frac{s(\alpha,\beta\gamma,\epsilon)}{p(\alpha,\beta\gamma,\epsilon)}.$ (185) Finally by replacing the expressions (183) and (184) in the equation (174), we obtain the closed equation in the continuous limit, which will also be fully addressed in forthcoming work. ## 5 Conclusion In the present paper, we have presented a perturbative calculation of two- point correlation functions of rank $3$ TGFT. As discussed earlier the correlation functions are given by combining Ward-Takahashi identities and Schwinger-Dyson equations that allows to establish the appropriate closed equation. The closed equation in the $4D$ case is also given. In this work, we proved that the nonperturbative techniques as developed in [29][30][28] can be reported to the tensor situation. Indeed, although, we only solve our closed form equations for the two-point functions at initial orders, it is very promising to see that we can obtain even solutions in this highly combinatoric case. As future investigations, we can now undertake a calculation of the general solution at all orders of the coupling constants for both rank 3 and 4 models. ## Acknowledgements Discussions with Joseph Ben Geloun, Raimar Wulkenhaar and Vincent Rivasseau are gratefully acknowledged. This research was supported in part by Perimeter Institute for Theoretical Physics and Fields Institute for Research in Mathematical Sciences (Toronto). 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1401.2143	Drinfel'd basis of Twisted Yangians S. Belliard and V. Regelskis [email protected], [email protected] Laboratoire Charles Coulomb L2C, UMR 5221, CNRS, F-34095 Montpellier, France Department of Mathematics, University of Surrey, Guildford, GU2 7XH, UK We present a quantization of a Lie bi-ideal structure for twisted half-loop algebras of finite dimensional simple complex Lie algebras. We obtain Drinfel'd basis formalism and algebra closure relations of twisted Yangians for all symmetric pairs of simple Lie algebras and for simple twisted even half-loop Lie algebras. We also give an explicit form of twisted Yangians in Drinfel'd basis for the $\fsl_3$ Lie algebra. § INTRODUCTION Yangian $\cY(\fg)$ is a flat quantization of the half-loop Lie algebra $\cL^+\cong\fg[u]$ of a finite dimensional simple complex Lie algebra $\fg$ [9]. The name Yangian is due to V. G. Drinfel'd to honour C. N. Yang who found the simplest solution of the Yang-Baxter equation, the rational $R$ matrix [2] (see also [3, 7]). This $R$ matrix and the Yang-Baxter equation were discovered in the studies of the exactly solvable two dimensional statistical models and quantum integrable systems. One of the most important result was the quantization of the inverse scattering method by Leningrad's school [5] that lead to the formulation of quantum groups in the so-called RTT formalism [13]. These quantum groups are deformations of semi-simple Lie algebras and are closely associated to quantum integrable systems. In particular, the representation theory of the Yangian $\cY(sl_2)$, which is one of the simplest examples of an infinite dimensional quantum groups, is used to solve the rational $6$-vertex statistical model [7], the XXX Heisenberg spin chain [6], the principal chiral field model with the $SU(2)$ symmetry group [10, 28]. The mathematical formalism for quantum groups and for quantization of Lie bi-algebras was presented by Drinfel'd in his seminal work [9] (see also [11]). Drinfel'd gave a quantization procedure for the universal enveloping algebra $\cU(\tilde\fg)$ for any semi-simple Lie algebra $\tilde\fg$. [Throughout this work we will use $\tilde\fg$ to denote any Lie algebra, while the undecorated $\fg$ will be reserved for denoting finite dimensional simple complex Lie algebras.] The quantization is based on the Lie bi-algebra structure on $\tilde\fg$ given by a skew symmetric map $\delta$ : $ \tilde\fg \to \tilde\fg \wedge \tilde\fg $, the cocommutator. A quantization of $(\tilde\fg,\delta)$ is a (topological) Hopf algebra $(\cU_\hbar(\tilde\fg),\Delta_\hbar)$, such that $\cU_\hbar(\tilde\fg)/\hbar \cU_\hbar(\tilde\fg)\cong\cU(\tilde\fg)$ as a Hopf algebra and \delta(x)=\frac{\Dh(X)-\sigma\circ\Dh(X)}{\hbar}\Big\|_{ \hbar\to 0} where $\sigma \circ (a\ot b)=b\ot a$ and $X$ is any lifting of $x\in\tilde\fg$ to $\cU_\hbar(\tilde\fg)$. The Lie bi-algebra structure on $\tilde\fg$ can be constructed from the Manin triple $(\tilde\fg,\tilde\fg_+,\tilde\fg_-)$, where $\tilde\fg_\pm$ are isotropic subalgebras of $\tilde\fg$ such that $\tilde\fg_+\op\tilde\fg_-=\tilde\fg$ as a vector space and $\tilde\fg_-\cong\tilde\fg_+^$, the dual of $\tilde\fg_+$. Then the commutation relations of the quantum group can be obtained by requiring $\Dh$ to be a homomorphism of algebras $ \cU_\hbar(\tilde\fg)\to \cU_\hbar(\tilde\fg)\ot \cU_\hbar(\tilde\fg)$. The question of the existence of such quantization for any Lie bi-algebra was raised by Drinfeld in [15] and was answered by P. Etingof and D. Kazhdan in [20]. They proved that any finite or infinite dimensional Lie bi-algebra admits a quantization. Here we will consider only the Yangian case, $\cU_\hbar(\tilde\fg)=\cY(\fg)$ with $\tilde\fg=\cL^+$. We will use the so-called Drinfel'd basis approach which is very convenient to approach the quantization problem. In physics, quantum groups are related to unbounded quantum integrable models and their extensions to models with boundaries. The underlying symmetry of the models with boundaries is given by coideal subalgebras of quantum groups that were introduced in the context of 1+1 dimensional quantum field theories on a half-line by Cherednick [8] and in the context of one dimensional spin chains with boundaries by Sklyanin [12] in the so-called reflection algebra formalism. Mathematical aspects of reflection algebras in the $RTT$ formalism, called twisted Yangians, were first considered by G. Olshasnkii in [14] and were further explored in [21, 25]. A similar approach for the q-deformed universal enveloping and quantum affine Lie algebras, $\cU_q(\fg)$ and $\cU_q(\hat\fg)$, was considered in [26, 33, 40]. A slightly different approach to coideal subalgebras, using Serre-Chevalley basis of $\cU_q(\fg)$ was surveyed by G. Letzter in [23]. Here coideal subalgebras of $\cU_q(\fg)$ were constructed using quantum symmetric pairs, introduced in [16], and have been classified for all finite dimensional semi-simple Lie algebras. For quantum affine Lie algebras of type $A$ first examples of coideal algebras in the Serre-Chevalley basis were the q-Onsager algebra [29] and the generalized q-Onsager algebras [30]. A generalization of quantum symmetric pairs for infinite dimensional Lie algebras of type $A$ was proposed in [38] and the general theory of quantum symmetric pairs for Kac-Moody Lie algebras was developed in [39]. A remarkably simple approach to coideal subalgebras based on the Drinfel'd original construction of Yangians, usually referred to as Drinfel'd first presentation or simply as Drinfel'd basis, was introduced in [22] (see also [24]) and is now conveniently called the MacKay twisted Yangians [40]. Twisted Yangians, which we denote by $\cY(\fg,\fh)^{tw}$, are in exact correspondence with the symmetric pair decomposition of $\fg$ given by a proper involution $\theta$ [1, 4]. This decomposition is given by $\fg=\fh\op\fm$ with $\theta(\fh)=\fh$, $\theta(\fm)=-\fm$, the positive and negative eigenspaces of $\theta$. For the half-loop Lie algebra this decomposition is given by $\cL^+=\cH^+\op \cM^+$, where the positive eigenspace of $\theta$ is the twisted current Lie algebra $\cH^+\cong(\fh\oplus u\fm)[u^2]$. The reflection algebra formalism provides evidence of an existence of quantization of such a symmetric pair. More generally, P. Etingof and D. Kazhdan have shown that any homogeneous space $G/H$ admits a local quantization [19]. In [37], the notion of left and right Lie bi-ideal structures for the $\theta$-invariant Lie subalgebras, noted $\tau$ and $\tau'$, respectively, was introduced by one of the authors. The Lie bi-ideal structure is in one-to-one correspondence with a twisted version of the Manin triple $(\tilde\fg,\tilde\fg_+,\tilde\fg_-)$ and gives a quantization scheme for the $\theta$-invariant Lie subalgebra leading to a quantum coideal subalgebra. These Lie bi-ideal structures correspond to a splitting $\delta=\tau+\tau'$ of the so-called coisotropic cocommutator $\delta$ of a Lie bi-algebras satisfying $\delta(\cH^+)\in \cM^+ \wedge \cH^+$ [35, 31]. Here we consider the left Lie bi-ideal structure, which allows us to construct a quantization of twisted half-loop Lie algebras leading to quantum coideal subalgebras called twisted Yangians in Drinfel'd basis formalism. A quantization of twisted half-loop Lie algebras for ${\rm rank}(\fg)=1$ case was shown in [37]. In this paper we consider twisted Yangians $\cY(\fg,\fh)^{tw}$ in the Drinfel'd basis formalism for any ${\rm rank}(\fg)\geq2$. We present a quantization procedure which holds for all symmetric pairs of half-loop Lie algebras of finite dimensional simple complex Lie algebras of rank $\fg \geq 2$. These symmetric pairs follow from the ones of simple complex Lie algebras that have been classified by Araki [1] (see also [4]). Such symmetric pairs can by grouped into four classes: $\fh$ is simple, $\fh=\fa\op\fb$, $\fh=\fa\op\fk$ and $\fh=\fa\op\fb\op\fk$, where $\fa$ and $\fb$ are simple Lie subalgebras of $\fh$, and $\fk$ is a one dimensional central extension. We will further refer to all of these cases as the $\theta\neq id$ case. We will also consider the trivial involution $\theta$ of $\fg$. Being trivial at the Lie algebra level, this involution can be extended non-trivially to the half-loop Lie algebra. In this case the $\theta$-fixed subalgebra is the even half-loop Lie algebra $\fg[u^2]$ isomorphic to $\fg[u]$ as a Lie algebra. We will refer to this setup as the $\theta=id$ case. The main results of this paper are Theorems <ref> and <ref>, which are analogues of the Drinfel'd Theorem <ref> ([9], Theorem 2) for the twisted Yangians for $\theta\neq id$ and $\theta= id$ cases, respectively. We call the defining relations of the twisted Yangians the horrific relations due to their complex form and similarity to the Drinfel'd terrific relations. A proof the horrific relations of Theorem <ref> is given in the first part of section <ref>. A proof of the horrific relation of Theorem (<ref>) is presented as an outline in the second part of section <ref>. The results of this paper give a uniform way of constructing twisted Yangians for all simple complex Lie algebras. Moreover, twisted Yangians in Drinfel'd basis have important applications in quantum integrable models with boundaries. They emerge as the non-abelian symmetries of the principal chiral models defined on a half-line and give the corresponding scalar boundary $S$-matrices [22, 24, 28]. The closure relations will allow to classify algebraicly all the scalar and dynamical boundary conditions and to construct the corresponding dynamical boundary $S$-matrices, i.e. in a similar way as it was done for the affine Toda models with boundaries in [33, 36]. Twisted Yangians in Drinfel'd basis can also be used to solve the spectral problem of a semi-infinite XXX spin chain for an arbitrary simple Lie algebra using the `Onsager method' [30]. We also remark, that twisted Yangians of this type were shown to play an important role in quantum integrable systems for which the $RTT$ presentation of the underlying symmetries is not known, for example in the AdS/CFT correspondence [32, 34]. The paper is organized as follows: in section <ref> we recall basic definitions of simple complex Lie algebras and define the symmetric pair decomposition with respect to involution $\theta$. Then, in section <ref>, we recall definitions of a half-loop Lie algebra $\cL^+$ of $\fg$ given in the Drinfel'd basis, and introduce the Drinfel'd basis of a twisted half-loop Lie algebra $\cH^+$ with respect to the symmetric pair decomposition of $\cL^+$. In section <ref> we construct the Lie bi-algebra structure on $\cL^+$ and Lie bi-ideal structure on $\cH^+$ that provide the necessary data to achieve the quantification presented in section <ref>. The special case $\fg=\fsl_3$ is fully considered in section <ref>, where we have presented all the corresponding twisted Yangians. Section <ref> contains the proofs which were omitted in the main part of the paper due to their length and for the convenience of the reader. The authors would like to thank P. Baseilhac, N. Crampé, N. Guay, N. Mackay and J. Ohayon for discussions and their interest for this work. V.R. acknowledges the UK EPSRC for funding under grant EP/K031805/1. § DEFINITIONS AND PRELIMINARIES §.§ Lie algebra Consider a finite dimensional complex simple Lie algebra $\fg$ of dimension ${\rm dim}(\fg)=n$, with a basis $\{x_a\}$ given by [x_a,x_b]= α_ab^c x_c, α_ab^c+α_ba^c=0, α_ab^c α_dc^e+ α_da^c α_bc^e+ α_bd^c α_ac^e=0 . Here $\alpha_{ab}^{\sk c}$ are the structure constants of $\fg$ and the Einstein summation rule of the dummy indices is assumed. We will further always assume $\fg$ to be simple. Let $\eta_{ab}$ denote the non-degenerate invariant bilinear (Cartan–Killing) form of $\fg$ in the $\{x_a\}$ basis (x_a,x_b)_= η_ab= α_ac^d α_bd^c, that can be used to lower indices {$a,b,c,\ldots$} of the structure constants {\alpha}_{ab}^{\sk d} \eta_{dc}={\alpha}_{abc} \qquad \text{with} \qquad {\alpha}_{abc}+{\alpha}_{acb}=0. The inverse of $\eta_{ab}$ is given by $\eta^{ab}$ and satisfies $\eta_{ab}\eta^{bc}=\delta_a^{\; c}$. Set $\{a,b\}=\tfrac{1}{2}(ab+ba)$. Let $C_\fg=\eta^{ab}\{x_a, x_b\}$ denote the second order Casimir operator of $\fg$ and let $\fc_{\fg}$ be its eigenvalue in the adjoint representation. For a simple Lie algebra it is non-zero and is given by _ δ_c^ d=η^abα_ac^eα_be^d=α_c^ ebα_be^d . The elements $\alpha_a^{\;bc}$ satisfy the co-Jacobbi identity, which is obtained by raising one of the lower indices of the Jacobi identity in (<ref>). Moreover, contracting $\al_a^{\;bc}$ with the Lie commutator in (<ref>) gives α_a^ bc[x_c,x_b] = _ x_a . §.§ Symmetric pair decomposition Let $\theta$ be an involution of $\fg$. Then $\fg$ can be decomposed into the positive and negative eigenspaces of $\theta$, i.e. $\fg=\fh\op \fm$ with $\theta(\fh)=\fh$ and $\theta(\fm)=-\fm$, here ${\rm dim}(\fh)=h$, ${\rm dim}(\fm)=m$ satifying $h+m=n$. Numbers $h$ and $m$ correspond to the number of positive and negative eigenvalues of $\theta$. This decomposition leads to the symmetric pair relations [\fh,\fh] \subset \fh, \qquad [\fh,\fm] \subset \fm, \qquad [\fm,\fm] \subset \fh . From the classification of the symmetric pairs for simple complex Lie algebras it follows that the invariant subalgebra $\fh$ is a (semi)simple Lie algebra which can be decomposed into a direct sum of two simple complex Lie algebras $\fa$ and $\fb$, and a one dimensional centre $\fk$ at most. We write $\fh=\fa \op \fb \op \fk$ (see e.g. section 5 in [4]). Set ${\rm dim}(\fa)=a$ and ${\rm dim}(\fb)=b$. Let the elements X_i ∈, X_i'∈, X_z∈, Y_p∈, with i=1…a, i'=1…b and p=1…m , be a basis of $\fg$ such that $\theta(X_\alpha)=X_\alpha$ for any $\alpha\in\{i,i',z\}$, and $\theta(Y_p)=-Y_p$. We will further use indices $i(j,k,...)$ for elements $X_\alpha\in\fa$, primed indices $i'(j',k',...)$ for elements $X_\alpha\in\fb$, index $\alpha=z$ for the central element $X_\alpha\in\fk$, and indices $p(q,r,\ldots)$ for $Y_p\in \fm$, when needed. We will denote the commutators in this basis as follows: [X_α,X_β] = f^γ_αβ X_γwith f^γ_αβ=0 for≠, [X_α,Y_p] = g^q_αp Y_q, [Y_p,Y_q]=∑_αw^α_pq X_α. The structure constants above are obtained from the ones of $\fg$ by restricting to the appropriate elements. Here and further we will use the sum symbol $\sum_{\alpha}$ to denote the summation over all simple subalgebras of $\fh$. The Einstein summation rule for the Greek indices will be used in cases when the sum is over a single simple subalgabera of $\fh$. The notation $\al\neq\ga$ means that indices $\al$ and $\ga$ correspond to different subalgebras of $\fh$. The structure constants given above satisfy the (anti-)symmetry relations f^{\sk \gamma}_{\alpha\beta}+f^{\sk \gamma}_{\beta\alpha} =0, \qquad\quad g^{\sk q}_{\mu p} +g^{\sk q}_{p\mu} =0, \qquad\quad w^{\sk\mu}_{pq} + w^{\sk \mu}_{qp} =0, and the homogeneous and mixed Jacobi identities f_αβ^νf_γν^μ + f_γα^νf_βν^μ + f_βγ^νf_αν^μ = 0, f^μ_αβ g^s_pμ + g^q_βp g^s_αq - g^q_αp g^s_βq = 0, w^β_pq f^μ_αβ + g^r_αp w^μ_qr- g^r_αq w^μ_pr = 0 , with $\alpha(\beta, \gamma,...)=i(j,k,...)\in \fa$ or $\alpha(\beta, \gamma,...)=i'(j',k',...)\in \fb$ and ∑_α(w^α_pq g^s_rα + w^α_qr g^s_pα + w^α_rp g^s_qα) = 0 , g_pα^rw_qr^β - g_qα^rw_pr^β = 0 for ≠. We will further refer to the set $\{X_\al,Y_p\}=\{X_i, X_{i'},X_z, Y_p\}$ given by (<ref>) and satisfying relations (<ref>–<ref>) as to the symmetric space basis for a given Lie algebra $\fg$ and involution $\theta$. The Killing form of $\fg$ has a block diagonal decomposition with respect to the symmetric space basis, namely (X_i,X_j)_\fg=(\kappa_\fa)_{ij}, \quad ({X}_{i'},{X}_{j'})_\fg=(\kappa_\fb)_{i'j'}, \quad (X_z,X_z)_\fg=(\kappa_\fk)_{zz}, \quad (Y_p,Y_q)_\fg=(\kappa_\fm)_{pq}, with the remaining entries being trivial. The Casimir element $C_\fg$ in this basis decomposes as C_ =C_X+C_Y = ∑_,κ^αβ{X_α,X_β}+(κ_)^pq{Y_p,Y_q},C_X =C+C'+C_z= (κ_)^ij{X_i,X_j}+(κ_)^i'j'{X_i',X_j'}+(κ_)^zz { X_z,X_z } . Here $\kappa^{\alpha\beta}\in\{(\kappa_\fa)^{ij},(\kappa_\fb)^{i'j'},(\kappa_\fk)^{zz}\}$. The decomposition of the inverse Killing form can be used to raise the indices of the structure constants. We set f_\alpha^{\;\beta\gamma} = \kappa^{\beta \mu}\, f_{\alpha\mu}^{\sk \gamma},\quad g_p^{\;q\nu} = \kappa^{\nu\rho}\, g_{\rho p}^{\sk q} , \quad w_\nu^{\;pq} = (\kappa_\fm)^{pr} g_{\nu r}^{\sk q}. with $\alpha(\beta, \gamma,\mu...)=i(j,k,...)$ or $i'(j',k',...)$ and $\nu(\rho)= i(j)$ or $i'(j')$ or $z(z)$. Consider the commutation relations. For the generator $Y_p$ we have [Y_p,C_X] =2\sum_{\al}g_{p}^{\;\alpha q}\{Y_q,X_\alpha\}, \qquad [Y_p,C_Y] =2\sum_{\al}g_{p}^{\;q\alpha}\{Y_q,X_\alpha\}. The remaining commutation relations are trivial. Let $\fc_\fa$, $\fc_\fb$, $\fc_\fz$ and $\fc_\fm$ be the eigenvalues of $C, C',C_z$ and $C_{Y}$ respectively. We have $\fc_\fg=\fc_\fa+\fc_\fb+\fc_\fm+\fc_z$. Using (<ref>) we find f_^ ν [X_ν,X_]=_() X_, w_γ^ qp [Y_p,Y_q] = _(γ) X_γ, with $\al=i(i')$, $\gamma=i(i',z)$, $\bar\fc_{(\alpha)}= \fc_\fg-\fc_{(\alpha)}$, $ \fc_{(i)}= \fc_\fa$, $ \fc_{(i')}= \fc_\fb$ and $ \fc_{(z)}=0$. Using (<ref>), (<ref>) we obtain f_^ μνf^_νμ = _() δ_^ , w_^ qp w^_pq = _() δ_^ , w_α^ qp w^γ_pq =0 for ≠, and $\al=i(i',z)$, $\bet=j(j',z)$. Finally, for $Y_q$ we have ∑_α g^ pα_q [X_α,Y_p] = _/2 Y_q, ∑_α g^ rα_p g_αr^q = _/2 δ_p^ q . § SYMMETRIC SPACES AND SIMPLE HALF-LOOP LIE ALGEBRAS §.§ Half-loop Lie algebra Consider a half-loop Lie algebra $\cL^+$ generated by elements $\{x^{(k)}_a\}$ with $k\in\NN$, $a=1,\dots, {\rm dim}(\fg)$. It is a graded algebra with ${\rm deg}(x^{(k)}_a)=k$ and the defining relations [x^(k)_a,x^(l)_b]=α^c_ab x^(k+l)_c. This algebra can be identified with the set of polynomial maps $f : \CC \to \fg$ using the Lie algebra isomorphism $\cL^+\cong\fg[u] = \fg \ot \CC[u]$ with $x^{(k)}_a \cong x_a\ot u^{k}$. The half-loop Lie algebra has another basis conveniently called the Drinfel'd basis: The half-loop Lie algebra $\cL^+$ admits a Drinfel'd basis generated by elements $\{x_a, J(x_b)\}$ satisfying [x_a,x_b]=α^c_ab x_c, J(μx_a+ νx_b) = μJ(x_a)+νJ(x_b), [x_a,J(x_b)]=α^c_ab J(x_c) , [J(x_a),J([x_b,x_c])] + [J(x_b),J([x_c,x_a])] + [J(x_c),J([x_a,x_b])] = 0 , [[J(x_a),J(x_b)],J([x_c,x_d])] + [[J(x_c),J(x_d)],J([x_a,x_b])] = 0 , for any $\mu,\nu\in\CC$. In the ${\rm rank}(\fg)=1$ case the level-2 loop terrific relation <ref> becomes trivial and for the ${\rm rank}(\fg)\geq 2$ case the level-3 loop terrific relation <ref> follows from level-2 loop terrific relation. The isomorphism with the standard loop basis is given by the map x_a \mapsto x^{(0)}_a, \qquad J(x_a) \mapsto x^{(1)}_a . A proof is given in section <ref>. §.§ Twisted half-loop Lie algebra Let us extend the involution $\theta$ of $\fg$ to the whole of $\cL^+$ as follows: θ(x^(k)_a)=(-1)^kθ_a^ b x^(k)_b for all k≥0. The twisted half-loop Lie algebra $\cH^+\cong\fg[u]^{\theta}$ is a fixed-point subalgebra of $\cL^+$ generated by the elements stable under the action of the (extended) involution $\theta$, namely $\cH^+=\{x\in\cL^+\,\|\,\theta(x)=x\}$. In physics literature it is ofted referred to as the twisted current algebra. Consider the symmetric space basis of $\fg$. We write the half-loop Lie algebra $\cL^+$ in terms of the elements $\{X^{(k)}_\alpha, Y^{(k)}_q\}$ satisfying [X^(k)_α,X^(l)_β] = f^γ_αβ X^(k+l)_γ, [X^(k)_,Y^(l)_p]=g^q_p Y^(k+l)_q, [Y^(k)_p,Y^(l)_q] = ∑_αw^α_pq X^(k+l)_α, for all $k,l\geq0$ and $\al\neq\la$. The action of $\theta$ on this basis is given by $\theta(X_\al^{(k)})=(-1)^kX_\al^{(k)}$ and $\theta(Y^{(k)}_p)=(-1)^{k+1}Y^{(k)}_p$. The half-loop Lie algebra decomposes as $\cL^+=\cH^+\op\cM^+$, where $\cH^+=\{X^{(2k)}_\al, Y^{(2k+1)}_q\}$ is the subalgebra of $\cL^+$ generated by $\theta$-invariant elements, and $\cM^+=\{X^{(2k+1)}_\al,Y^{(2k)}_q\}$ is the subset of $\cL^+$ of $\theta$-anti-invariant elements. The twisted half-loop Lie algebra can be defined in terms of the Drinfel'd basis: Let ${\rm rank}(\fg)\geq2$. Then the twisted half-loop Lie algebra admits a Drinfel'd basis generated by elements $\{X_\al, B(Y_p)\}$ satisfying [X_,X_]=f^ _ X_, [X_,B(Y_p)] = g^ q_p B(Y_q), B(a Y_p + b Y_q) = a B(Y_p) + b B(Y_q), for all $a,b\in\CC$ and [B(Y_p),B(Y_q)] + ∑_1/_() w_pq^w_^ rs [B(Y_r),B(Y_s)] = 0 , [[B(Y_p),B(Y_q)],B(Y_r)] + 2/_ ∑_ _^tu w_pq^ g_r^s [[B(Y_s),B(Y_t)],B(Y_u)] = 0 . The isomorphism with the standard twisted half-loop basis is given by the map X_\al\mapsto X^{(0)}_\al, \qquad B(Y_p)\mapsto Y^{(1)}_p. A proof is given in section <ref>. Note that in the contrast to $\cL^+$, the twisted algebra $\cH^+$ for ${\rm rank}(\fg)\geq2$ has level-2 and level-3 higher-order defining relations, which we call horrific relations. This is due to the fact that even and odd levels of $\cH^+$ are not equivalent. The ${\rm rank}(\fg)=1$ case is exceptional. The Drinfel'd presentation in this case has level-$4$ relation instead (see [37], section 4.2). Let ${\rm rank}(\fg)\geq2$. Then the even half-loop Lie algebra admits a Drinfel'd basis generated by elements $\{x_i, G(x_j)\}$ satisfying [x_i,x_j] = ^k_ij x_k, G(x_a + μx_b) = G(x_a) +μG(x_b) , [x_i,G(x_j)]=^k_ij G(x_k) . [G(x_i),G([x_j,x_k])] + [G(x_j),G([x_k,x_i])] + [G(x_k),G([x_i,x_j])] = 0 , for any $\mu,\nu\in\CC$. The isomorphism with the standard half-loop basis is given by the map x_i\mapsto x^{(0)}_i, \qquad G(x_i)\mapsto x^{(2)}_i. The proof follows directly by Proposition <ref>, since $\fg[u^2]\cong \fg[u]$ as Lie algebra. § LIE BI-ALGEBRAS AND BI-IDEALS §.§ Lie bi-algebra structure of a half-loop Lie algebra A Lie bi-algebra structure on $\cL^+$ is a skew-symmetric linear map $ \delta : \cL^+ \to \cL^+ \ot \cL^+$, the cocommutator, such that $ \delta^$ is a Lie bracket and $\delta $ is a 1-cocycle, $\delta([x,y])=x.\delta(y)-y.\delta(x)$, where dot denotes the adjoint action on $\cL^+ \ot \cL^+$. The cocommutator is given for the elements in the Drinfel'd basis of $\cL^+$ by \delta(x_a)=0, \qquad \delta(J(x_a))=[x_a \ot 1, \Omega_\fg], \qquad \Omega_{\fg}=\eta^{ab} x_a \ot x_b. This cocommutator can be constructed from the Manin triple Lie algebra $(\cL, \cL^+, \cL^-)$, with $\cL=\fg((u^{-1}))$ the loop algebra generated by elements $\{x^{(n)}\}$ with $x\in \fg$, $n\in\ZZ$ and defining relations (<ref>) (but with $n,m\in\ZZ$), $\cL^+=\fg[u]$ the positive half-loop algebra (<ref>), and $\cL^-=\fg[[u^{-1}]]$ the negative half-loop algebra (i.e. $n,m<0$). The triple $(\cL, \cL^+, \cL^-)$ satisfies the basic axioms of the Manin triple: A Manin triple is a triple of Lie bi-algebras $(\tilde \fg, \tilde\fg^+,\tilde \fg^-)$ together with a non-degenerate symmetric bilinear form $(\;,\;)_{\tilde \fg}$ on $\tilde \fg$ invariant under the adjoint action of $\tilde \fg$: * ${\tilde \fg}^+$ and ${\tilde \fg}^-$ are Lie subalgebras of ${\tilde \fg}$; * ${\tilde \fg}={\tilde \fg}^+ \op{\tilde \fg}^-$ as a vector space; * $(\;,\;)_{\tilde \fg}$ is isotopic for $ {\tilde \fg}^\pm$ (i.e. $({\tilde \fg}^\pm,{\tilde \fg}^\pm)_\cL=0$); * $({\tilde \fg}^+)^* \cong {\tilde \fg}^-$. Here the bilinear form is given by $(x^{(k)},y^{(l)})_\cL = (x,y)_\fg \,\delta_{k+l+1,0} $. If $({\tilde \fg},{\tilde \fg}^+,{\tilde \fg}^-)$ is a Manin triple for ${\rm dim}({\tilde \fg}^+)=\infty$, then $(\fg_+)^* \cong {\bar \fg}_-$, where ${\bar \fg}_-$ is the completion of ${\tilde \fg}_-$. However in our case $(\cL^+)^* \cong {\cL}^-$, as it is easy to see: (\cL^+)^* \cong \big( \bigoplus_{k\geq0} \fg\ot u^k \big)^* = \prod_{k\geq0} (\fg\ot u^k)^* = \prod_{k\geq1} \fg\ot u^{-k} \cong \cL^- . Here in the second equality we have used the identity (\bigoplus_{i\geq0} V_i)^* = \prod_{i\geq0} V_i^, where $V_i$ denotes a finite dimensional vector space; an equivalent identity is used in the last equality. The cocomutator is obtained using the duality between $\cL^+$ and $\cL^-$. Recall that $\delta^ : \cL^-\ot \cL^- \to \cL^-$ is the Lie bracket of $\cL^-$. We can deduce the cocommutator $\delta$ of $\cL^+$ from the duality relation The cocommutator of the level zero generators $x^{(0)}_a=x_a$ is trivial since the level of $[y,z]\subset \cL^-$ is strictly inferior to $-1$, thus $(x_a,[y,z])_\cL=0$ for all $[y,z]\subset \cL^-$. The simplest solution of $(\delta(x_a),y\ot z)_\cL=0$ for all $y,z\in \cL^-$ is $$ \delta(x_a)=0. $$ The case of level one generators $x^{(1)}_a=J(x_a)$ is considered in a similar way. For this case we have a non trivial pairing $(J(x_a),x^{(-2)}_b)_\cL=\eta_{ab}$. Using $x^{(-2)}_b=\fc_\fg^{-1}\alpha_b^{ji}[x^{(-1)}_i,x^{(-1)}_j]$ and the duality relation (<ref>) we obtain a constraint (\delta(J(x_a)),\alpha_b^{ji} x^{(-1)}_i\ot x^{(-1)}_j)_\cL=\fc_\fg\,\eta_{ab} . Let us consider an ansatz $\delta(J(x_a))=v_a^{\;lk} x_k\ot x_l$ for some $v_a^{\;lk}$. Then we must have $v_a^{\;lk}\alpha_{blk}=\fc_\fg\,\eta_{ab} $. It follows from (<ref>) that \delta(J(x_a))=\alpha_a^{\;lk} x_k\ot x_l=[x_a \ot 1, \Omega_\fg]. §.§ Lie bi-ideal structure of twisted half-loop algebras The Lie bi-ideal structure of twisted half-loop algebras is constructed by employing the anti-invariant Manin triple twist. Here we will consider the left Lie bi-ideal structure. The right Lie bi-ideal is obtained in a similar way. The anti-invariant Manin triple twist $\phi$ of $(\cL, \cL^+, \cL^-)$ is an automorphism of $\cL$ satisfying: * $\phi$ is an involution; * $\phi(\cL^\pm)=\cL^\pm$; * $(\phi(x),y)_\cL=-(x,\phi(y))_\cL$ for all $x,y\in \cL^+$. Here $\phi$ is the natural extension of $\theta$ to $\cL$ (i.e. $\phi(x_a^{(k)})=(-1)^k\theta_a^b x_b^{(k)}$ for all $k\in\ZZ$). The twist $\phi$ gives the symmetric pair decomposition of the Manin triple $(\cL, \cL^+, \cL^-)$: =, ^±=^±^±with ϕ(^±)=^±and ϕ(^±)=-^±. From the anti-invariance of $\phi$ for $(\;,\;)_\cL$ it follows $$ (\cH^-,\cH^+)_\cL=(\cM^-,\cM^+)_\cL=0. $$ Then we must have $(\cH^\pm)^* \cong {\cM}^\mp$. This is easy to check: (^+)^* ≅( ⊕_k≥0 (u)u^2k)^* = ∏_k≥0 ((u)u^2k)^* = ∏_k≥1( (u)u^-2k) ≅^- , (^-)^* ≅( ⊕_k≥1 (u)u^-2k)^* = ∏_k≥1( (u)u^-2k)^* = ∏_k≥0( (u)u^2k) ≅^+ . This decomposition of the Manin triple allows us to construct the Lie bi-ideal structure on $\cH^+$. Let $\phi$ be an anti-invariant Manin triple twist for $(\cL,\cL^+,\cL^-)$ which leads to the symmetric space decomposition (<ref>). Then the linear map $\tau:\cH^+\rightarrow \cM^+ \ot \cH^+$ is a left Lie bi-ideal structure for the couple $(\cH^+,\cM^+)$ if it is the dual of the following action of $\cH^-$ on $\cM^-$, τ^* : ^-^- → _- , xy ↦ [x,y]__- , for all $x\in\cH^-$ and $y\in\cM^-$. We are now ready to define the Lie bi-ideal structure for $(\cL^+,\cH^+)$. The Lie bi-ideal structure of $(\cL^+,\cH^+)$, $\tau : \cH^+ \to \cM^+ \ot \cH^+$ is given by θ≠id : τ(X_α)=0, τ(B(Y_p))=[Y_p 1, Ω_], Ω_=∑_,κ^αβ X_α⊗X_β θ= id : τ(x_a)=0, τ(G(x_a))=[J(x_a) 1, Ω_] . The construction of the Lie bi-ideal structure $ \tau$ from the anti-invariant Manin triple twist is similar to the one of the Lie bi-algebra structure from the Manin triple. We have to consider the duality relation $(\tau(x),y\ot z)_\cL=(x,[y,z])_\cL$ with $x\in \cH^+$, $y\in \cH^-$ and $z\in \cM^-$. Consider the case $\theta \neq id$. For the level zero generators $X_\alpha^{(0)}=X_\alpha$, we have $(X_\alpha,[y,z])_\cL=0$ for all $y\in \cH^-$ and $z\in \cM^-$. This follows by similar arguments as for the level zero generators of the half-loop Lie algebra. Hence we have \tau(X_\alpha)=0. For the level one generators $Y^{(1)}_p=B(Y_p)$, we have a non trivial paring $(B(Y_p), Y^{(-2)}_q)_\cL=(\kappa_\fm)_{pq}$. Then, using relation $\displaystyle Y^{(-2)}_q= \sum_{\alpha} 2\,\fc_\fg^{-1} g_{q}^{\;\alpha p}\, [Y^{(-1)}_p,X^{(-1)}_\alpha]$ and the duality relation we obtain \sum_{\alpha} (\tau(B(Y_p)),g_{q}^{\;r\alpha} Y_r^{(-1)}\ot X^{(-1)}_\alpha)_\cL=\frac{\fc_\fg}{2}(\kappa_\fm)_{pq}. Let us consider an ansatz $\displaystyle \tau(B(Y_p))= \sum_{\beta} v_{p}^{\;\beta s}Y_s\ot X_\beta$. We must have $\displaystyle \sum_{\alpha}v_{p}^{\;\alpha s}g_{qs\alpha}=\frac{\fc_\fg}{2}(\kappa_\fm)_{pq}$. Using (<ref>) we find \tau(B(Y_p))= \sum_{\alpha}g_{p}^{\;\alpha s}Y_s\ot X_\alpha=[Y_p\ot 1,\Omega_{\fh}]. The Lie bi-ideal structure for the $\theta=id$ case follows from the pairing $(G(x_a),x_b^{(-3)})_\cL = (\kappa_\fg)_{ab}$ by similar arguments. For completeness we give a remark which was stated by one of the authors in [37]. The notion of left (right) Lie bi-ideal is related to the notion of co-isotropic subalgebra $\fh$ of a Lie bi-algebra $(\fg, \delta)$. It is a Lie subalgebra which is also a Lie coideal, meaning that $\delta (\fh) \subset \fh \wedge \fg$. We have $\delta(x)=\tau(x) + \tau'(x)$, for $x\in\fh$ with $\tau'=-\sigma\circ\tau$ the right bi-ideal structure. § QUANTIZATION To obtain a quantization of Lie bi-ideal we need to introduce some additional algebraic structures. Recall the definition of a bi-algebra and of a Hopf algebra. A bi-algebra is a quintuple $(A,\mu,\imath,\Delta,\varepsilon)$ such that $(A,\mu,\imath)$ is an algebra and $(A,\Delta,\varepsilon)$ is a coalgebra; here $A$ is a $\CC$-module, $\mu : A \ot A \to A$ is the multiplication, $\Delta:A\to A\ot A$ is the comultiplication (coproduct), $\imath : \CC \to A$ is the unit and $\varepsilon : A \to \CC$ is the counit. A Hopf algebra is a bi-algebra with an antipode $S : A \to A$, an antiautomorphism of algebra. Let $\cA=(A,\Delta,\varepsilon)$ be a coalgebra. Then $\cB=(B,\LD,\epsilon)$ is a left coideal of $\cA$ if: * $B$ is a sumbodule of $A$, i.e. there exists an injective homomorphism $\varphi: B \to A$; * coaction $\LD$ is a coideal map $\LD : B \to A \ot B$ and is a homomorphism of modules; * coalgebra and coideal structures are compatible with each other, i.e. the following identities hold: (Δid)∘=(id )∘ (id φ) ∘= Δ∘φ; * $\epsilon : B \to \CC$ is the counit. Let $\cA=(A,\mu,\eta,\Delta,\varepsilon)$ be a bi-algebra. Then $\cB=(B,m,i,\LD,\epsilon)$ is a left coideal subalgebra of $\cA$ if: * the triple $(B,m,i)$, where $m$ is the multiplication and $i$ is the unit, is an algebra; * $B$ is a subalgebra of $A$, i.e. there exists an injective homomorphism $\varphi: B \to A$; * the triple $(B,\LD,\epsilon)$ is a coideal of $(A,\Delta,\varepsilon)$. The relation (<ref>) is usually referred to as the coideal coassociativity of $B$. We will refer to (<ref>) as the coideal coinvariance. We will refer to the map $\varphi$ as to the natural embedding $\varphi : B \hookrightarrow A$. The next definition, a quantization of a Lie bi-algebra ${\tilde \fg}$, is due to Drinfel'd [11]: Let $({\tilde \fg},\delta)$ be a Lie bi-algebra. We say that a quantized universal enveloping algebra $(\cU_{\hbar}({\tilde \fg}),\Delta_\hbar)$ is a quantization of $({\tilde \fg},\delta)$, or that $({\tilde \fg},\delta)$ is the quasi-classical limit of $(\cU_{\hbar}({\tilde \fg}),\Delta_\hbar)$, if it is a topologically free $\CC[[\hbar]]$ module and: * $\cU_{\hbar}({\tilde \fg})\,/\,\hbar\,\cU_{\hbar}({\tilde \fg})$ is isomorphic to $\cU({\tilde \fg})$ as a Hopf algebra; * for any $x \in {\tilde \fg}$ and any $X \in \cU_{\hbar}({\tilde \fg})$ equal to $x \,(mod \, \hbar)$ one has \Big(\Delta_\hbar(X)-\sigma\circ\Delta_\hbar(X)\Big)/\hbar \sim \delta(x) \quad (mod\, \hbar) with $\sigma$ the permutation map $\sigma(a\ot b)=b\ot a$. Note that $(\cU_{\hbar}({\tilde \fg}),\Delta_\hbar)$ is a topological Hopf algebra over $\CC[[\hbar]]$ and is a topologically free $\CC[[\hbar]]$ module. Drinfel'd also noted that for a given Lie-bialgebra $(\tilde\fg,\delta)$, there exists a unique extension of the map $\delta: \tilde\fg\to\tilde\fg\ot\tilde\fg$ to $\delta : \cU(\tilde\fg) \to \cU(\tilde\fg) \ot \cU(\tilde\fg)$ which turns $\cU(\tilde\fg)$ into a co-Poisson-Hopf algebra. The converse is also true. In such a way $(\cU_{\hbar}({\tilde \fg}),\Delta_\hbar)$ can be viewed as a quantization of $(\cU(\tilde\fg),\delta)$. Next we define a quantization of the Lie bi-ideal $({\tilde \fg}^\theta,\tau)$ following [37]: Let $({\tilde \fg},\delta)$ be a Lie bi-algebra and $({\tilde \fg}^\theta,\tau)$ be a left Lie bi-ideal of $({\tilde \fg},\delta)$ relative to the involution $\theta$ of ${\tilde \fg}$. We say that a left coideal subalgebra $(\cU_{\hbar}({\tilde \fg},{\tilde \fg}^\theta),\LHb)$ is a quantization of $({\tilde \fg}^\theta,\tau)$, or that $({\tilde \fg}^\theta,\tau)$ is the quasi-classical limit of $(\cU_{\hbar}({\tilde \fg},{\tilde \fg}^\theta),\LHb)$, if it is a topologically free $\CC[[\hbar]]$ module and: * $(\cU_{\hbar}({\tilde \fg}),\Delta_\hbar)$ is a quantization of $(\tilde\fg,\delta)$; * $\cU_{\hbar}({\tilde \fg},{\tilde \fg}^\theta)/\,\hbar \,\cU_{\hbar}({\tilde \fg},{\tilde \fg}^\theta)$ is isomorphic to $\cU({\tilde \fg}^\theta)$ as a Lie algebra; * $(\cU_{\hbar}({\tilde \fg},{\tilde \fg}^\theta),\LHb)$ is a left coideal subalgebra of $(\cU_{\hbar}({\tilde \fg}),\Delta_\hbar)$; * for any $x \in {\tilde \fg}^\theta$ and any $X \in \cU_{\hbar}({\tilde \fg},{\tilde \fg}^\theta)$ equal to $x \, (mod \, \hbar)$ one has \Big(\LHb(X) -(\varphi(X) \ot 1 + 1 \ot X)\Big)/\hbar\sim \, \tau(x) \quad (mod\, \hbar) with $\varphi$ the natural embedding $\cU_{\hbar}({\tilde \fg},{\tilde \fg}^\theta) \hookrightarrow \cU_{\hbar}({\tilde \fg})$. The Lie bi-ideal structure $\tau$ can be extended to $\tau : \cU(\tilde\fg^\theta) \to \cU(\tilde\fg) \ot \cU(\tilde\fg^\theta)$ such that $\tau(a_1a_2)=\tau(a_1)\LH(a_2)+\tau(a_2)\LH(a_1)$ [37]. However this does not turn $(\cU(\tilde\fg^\theta),\LH,\tau)$ into a co-Poisson-Hopf structure. Rather it would be a `one sided coideal-Poisson' extension of the one-sided Lie bi-ideal $(\tilde\fg^\theta, \tau)$. In the case of the two-sided coideal structures, the associated Poisson structures are called `Poisson homogeneous spaces'. We refer to [18] and [19] for details on quantization of such structures. In the remaining part of this section we will consider a quantization of symmetric pairs of half-loop Lie algebras $(\tilde \fg,\tilde \fg^\theta)=(\cL^+,\cH^+)$. We will recall the coproduct of the Yangian $\cY(\fg)=\cU_{\hbar}(\cL^+)$ which follows from the Lie bi-algebra structure on $\cL^+$. Then we will construct the coaction of the twisted Yangian $\cY(\fg,\fg^\theta)^{tw}=\cU_{\hbar}(\cL^+,\cH^+)$ which will follow from the Lie bi-ideal structure on $\cH^+$ and the coideal compatibility relations (<ref>) and (<ref>). And finally, we will recall the defining relations of the Drinfel'd Yangian and give the main results of this paper, the defining relations of the twisted Yangians in Drinfel'd basis. §.§ Coproduct on $\cY(\fg)$ The coproduct is given by Δ_h(x_a) = x_a 1 + 1 x_a , Δ_h((x_a)) = (x_a) 1 + 1 (x_a) + ħ/2 [x_a 1, Ω_] . This is the simplest solution of the quantization condition satisfying the coassociativity property (1 Δ_h) ∘Δ_h=(Δ_h 1) ∘Δ_h . The grading on $\cY(\fg)$ is given by (x_a)=0, (ħ)=1, ((x_a))=1. §.§ Coaction on $\cY(\fg,\fg^\theta)^{tw}$ As in the previous section, we need to consider the cases $\theta \neq id$ and $\theta = id$ separately. Let $\theta \neq id$. Then the coideal subalgebra structure is given by (X_α) = X_α1 + 1 X_α, ((Y_p)) =φ((Y_p)) 1 + 1 (Y_p) + ħ [Y_p 1, Ω_X], φ((Y_p)) =(Y_p) + 1/4ħ [Y_p,C_X] . The grading on $\cY(\fg,\fg^\theta)^{tw}$ is given by ${\rm deg}(X_\alpha)=0$, ${\rm deg}(\hbar)=1$ and ${\rm deg}(\B(Y_p))=1$. Let $\theta = id$. Then the coideal subalgebra structure is given by (x_a) = x_a 1 + 1 x_a , ((x_a)) = φ((x_a))1+1(x_a) + ħ [(x_a)1,Ω_] + 1/4ħ^2 ( [[x_a1,Ω_],Ω_] + _^-1 _a^ bc[[x_c1,Ω_],[x_b1,Ω_]] ) , φ((x_a)) = _^-1 _a^bc [(x_c),(x_b)] + 1/4ħ [(x_a),C_]. The grading on $\cY(\fg,\fg)^{tw}$ is given by $\deg(x_a)=0$, ${\rm deg}(\hbar)=1$ and $\deg(\G(x_a))=2$. The proofs of Propositions <ref> and <ref> are stated in section <ref>. The map (<ref>) is the MacKay twisted Yangian formula [22]. The next remark is due to Lemma <ref>: It will be convenient to write the coaction (<ref>) in the following way ((x_a)) = φ((x_a))1+1(x_a) + ħ [(x_a)1,Ω_] + ħ^2 ( h_a^ bcd x_b{x_c,x_d} + h̅_a^ bcd {x_c,x_d} x_b ), h_a^ bcd = ϕ_a^ bcd + 2 ψ_a^ bcd, h̅_a^ bcd = ϕ_a^ bcd - ψ_a^ bcd, ϕ_a^ bcd = 1/24 _∑_π(_a^ jk _j^ π(d)r _k^ π(b)s _sr^π(c)) , ψ_a^ bcd = 1/12(_a^ jd _j^ bc+_a^ jc _j^ bd) . §.§ Yangians and twisted Yangians in Drinfel'd basis For any elements $x_{i_1},x_{i_2},\ldots,x_{i_m}$ of any associative algebra over $\CC$, set {x_i_1,x_i_2,…,x_i_m} = 1/m! ∑_πx_π(i_1)x_π(i_2)⋯x_π(i_m), where the sum is over all permutations $\pi$ of $\{{i_1},{i_2},\ldots,{i_m}\}$ and x_i_1,x_i_2,…,x_i_m _n = 1/m! ∑_πx_π(i_1)⋯x_π(i_n) x_π(i_n+1) ⋯x_π(i_m), such that Δ({x_i_1,x_i_2,…,x_i_m}) = ∑_n=0^m mn x_i_1,x_i_2,…,x_i_m _n, where $\binom{m}{n}$ denotes the binomial coefficient. Moreover, for any set of indices $\{i_1,\dots,i_m\}$, set a_(i_1 a_i_2 ⋯a_i_m ) = ∑_σa_σ(i_1)a_σ(i_2)⋯a_σ(i_m), where the sum is over the cyclic permutations, for example: \al_{(ab}^{\quad d} \al_{c)d}^{\quad e} = \al_{ab}^{\sk d}\, \al_{cd}^{\sk e} + \al_{bc}^{\sk d}\, \al_{ad}^{\sk e} + \al_{ca}^{\sk d}\, \al_{bd}^{\sk e} . Next, let us recall the definition of the Drinfel'd Yangian [9]: Let $\fg$ be a finite dimensional complex simple Lie algebra. Fix a (non-zero) invariant bilinear form on $\fg$ and a basis $\{x_a\}$. There is, up to isomorphism, a unique homogeneous quantization $\cY(\fg):=\cU_\hbar(\fg[u])$ of $(\fg[u],\delta)$. It is topologically generated by elements $x_a$, $\cJ(x_a)$ with the defining relations: [x_a,x_b] = α_ab^c x_c, [x_a,(x_b)] = α_ab^c (x_c), (λ x_a + μ x_b) = λ (x_a) + μ (x_b), [(x_a),([x_b,x_c])] + [(x_b),([x_c,x_a])] + [(x_c),([x_a,x_b])] = 14ħ^2 β_abc^ijk {x_i,x_j,x_k}, [[(x_a),(x_b)],([x_c,x_d])] + [[(x_c),(x_d)],([x_a,x_b])] = 14ħ^2 γ_abcd^ijk {x_i,x_j,(x_k)} , β_abc^ijk = α_a^ ilα_b^ jmα_c^ knα_lmn , γ_abcd^ijk = α_cd^e β_abe^ijk + α_ab^e β_cde^ijk , for all $x_a\in\fg$ and $\lambda,\mu\in\CC$. The coproduct and grading is given by (<ref>), (<ref>) and (<ref>), the antipode is S(x_a)=-x_a, S((x_a))=-(x_a)+1/4 ħ _x_a. The counit is given by $\varepsilon_\hbar(x_a)=\varepsilon_\hbar(\cJ(x_a))=0$. An outline of a proof can be found in chapter 12 of [17]. Let us make a remark on the Drinfel'd terrific relations (<ref>) and (<ref>), which are deformations of the relations (<ref>) and (<ref>), respectively. The right-hand sides (rhs) of the terrific relations are such that $\Delta_\hbar : \cY(\fg)\to \cY(\fg)\ot\cY(\fg)$ is a homomorphism of algebras. Choose any total ordering on the elements $x_a$, $\cJ(x_b)$. Then it is easy to see that the basis of $\cY(\fg)$ is spanned by the totally symmetric polynomials $\{ x_{a_1}, \ldots , x_{a_m}, \cJ(x_{b_1}),\ldots, \cJ(x_{b_n})\}$ with $m+n\geq1$, $m,n\geq0$, and ordering $a_i\preceq\cdots\preceq a_m$, $b_i\preceq\cdots\preceq b_n$. Moreover, the defining relations must be even in $\hbar$. Indeed, consider the coaction on the left-hand side (lhs) of (<ref>). The linear terms in $\hbar$ vanish due to the Jacobi identity. What remains are the $\hbar^2$–order terms cubic and totally symmetric in $x_a$. Hence the rhs of the terrific relation must be of the form $\hbar^2 A_{abc}^{ijk} \{x_i,x_j,x_k\}$ for some set of coefficients $A_{abc}^{ijk}\in\CC$. By comparing the terms on the both sides of the equation and using the Jacobi identity one finds $A_{abc}^{ijk}=\beta_{abc}^{ijk}$. The level three terrific relation (<ref>) is obtained in a similar way. Proving (<ref>) is much more complicated since it requires a heavy usage of (<ref>) and Jacobi identity. We were unable to locate the explicit proof of Drinfel'd terrific relations in the mathematical literature available to us. Let $(\fg,\fg^\theta)$ be a symmetric pair decomposition of a finite dimensional simple complex Lie algebra $\fg$ of ${\rm rank}(\fg)\geq2$ with respect to the involution $\theta$, such that $\fg^\theta$ is the positive eigenspace of $\theta$. Let $\{X_\al,Y_p\}$ be the symmetric space basis of $\fg$ with respect to $\theta$. There is, up to isomorphism, a unique homogeneous quantization $\cY(\fg,\fg^\theta)^{tw}:=\cU_\hbar(\fg[u],\fg[u]^\theta)$ of $(\fg[u],\fg[u]^\theta,\tau)$. It is topologically generated by elements $X_\al$, $\B(Y_p)$ with the defining relations: [X_,X_]=f^_ X_, [X_,(Y_p)] = g^q_p(Y_q) , (a Y_p + b Y_q) = a (Y_p) + b (Y_q), [(Y_p),(Y_q)] + ∑_ 1/_() w_pq^w_^ rs [(Y_r),(Y_s)] = ħ^2 ∑_,μ,νΛ_pq^μν {X_,X_μ,X_ν} , [[(Y_p),(Y_q)],(Y_r)] + 2/_ ∑_ ^tu_ w_pq^ g_r^s [[(Y_s),(Y_t)],(Y_u)] = ħ^2 ∑_,μ,u Υ_pqr^μu {X_,X_μ,(Y_u)} , Λ_pq^μν = 1/3 ( g^μt_p g^u_q + ∑_ (_())^-1 w_pq^ w_^ rs g^μt_r g^u_s ) w_tu^ν , Υ_pqr^μu = 1/4 ∑_ ( w_st^ g_p^ s g_q^ μt g_r^u + ∑_w_pq^ f_^ g_r^ μs g_s^u) + 1/2 _ ∑_, ^vx_ w_pq^ g_r^y ( w_st^ g_y^ s g_v^ μt g_x^u + ∑_w_yv^ f_^ g_x^ μs g_s^u ) . for all $X_\al,Y_p\in\fg$ and $a,b\in\CC$. The coaction and grading is as in Proposition <ref>. The counit is $\epsilon_\hbar(X_\al) = \epsilon_\hbar(\B(Y_p)) = 0$ for all non-central $X_\al$. In the case when $\fg^\theta$ has a non-trivial centre $\fk$ generated by $X_z$, then $\epsilon_\hbar(X_z)=c$ with $c\in\CC$. In the case when $\fh$ has a central element, the one dimensional representation of $\fh$ has a free parameter $c\in\CC$. This parameter corresponds to the free boundary parameter of a quantum integrable model with a twisted Yangian as the underlying symmetry algebra. For Lie algebras of type $A$, this parameter also appears in the solutions of the boundary intertwining equation leading to a one-parameter family of the boundary $S$-matrices satisfying the reflection equation [27]. Let $\fg$ be a finite dimensional simple complex Lie algebra of ${\rm rank}(\fg)\geq2$. Let $\{x_i\}$ be a basis of $\fg$. Fix a (non-zero) invariant bilinear form on $\fg$ and a basis $\{x_i\}$. There is, up to isomorphism, a unique homogeneous quantization $\cY(\fg,\fg)^{tw}:=\cU_\hbar(\fg[u],\fg[u^2])$ of $(\fg[u],\fg[u^2],\tau)$. It is topologically generated by elements $x_i$, $\G(x_i)$ with the defining relations: [x_a,x_b] = α^c_ab x_c, [x_a,(x_b)] = ([x_a,x_b]) =α^c_ab (x_c), ( x_a+μ x_b) = (x_a)+μ (x_b), [(x_a),([x_b,x_c])] + [(x_b),([x_c,x_a])] + [(x_c),([x_a,x_b])] = ħ^2 Ψ_abc^ijk {x_i,x_j,(x_k)} + ħ^4 ( Φ_abc^ijk{x_i,x_j,x_k} + Φ̅_abc^ijklm{x_i,x_j,x_k,x_l,x_m}) , Ψ_abc^ijk = _(ab^ d_c)r^ k h̅_d^rij - _dr^k_(ab^ d h̅_c)^rij , Φ̅_abc^ijklm = 1/5(_rs^i _(ab^ dh_c)^rjk h_d^slm - Ψ_abc^jkr h_r^ilm ) , Φ_abc^ijk=1/9( _(ab^ d W_c)d^ijk + 1/6Φ̅_abc^(ix(yzj))_xy^r_rz^k - (Ψ_abc^xjy h̅_y^kzr _zx^s_rs^i + Ψ_abc^xyz h_z^rsk _rx^i_ys^j ) ) , W_cd^ijk = _rs^i h_c^rxy ( h_d^szk_xt^j _yz^t+ h_d^szt_xt^k _yz^j ) + ( (h̅_c^xyzh_d^efk-h̅_d^xyzh_c^efk)_ye^t _zt^i_xf^j + h̅_c^jxyh̅_d^kzr_xr^s(_zy^t_st^i+_sy^t_zt^i)), for all $x_a\in\fg$ and $\la,\mu\in\CC$. The coaction and grading is as in Proposition <ref>. The co-unit is $\epsilon_\hbar(x_i)=\epsilon_\hbar(\G(x_i))=0$. Theorems <ref> and <ref> can be proven using essentially the same strategy outlined in chapter 12 of [17]. The complicated part is to obtain the horrific relations (<ref>), (<ref>) and (<ref>), which are quantizations of (<ref>), (<ref>) and (<ref>), respectively. A proof of the first two horrific relations is given in the first part of section <ref>. Proving the third horrific relation is substantially more difficult. We have given an outline of a proof in the second part of section <ref>. We believe that coefficients of the horrific relation (<ref>) could be further reduced to a more elegant and compact form. We have succeeded to find such a form for the $\fsl_3$ Lie algebra: For $\fg=\fsl_3$ the coefficients of the horrific relation (<ref>) get simplified to Ψ_abc^ijk = 1/3 β_(abc)^ijk + _(ab^ d _c)l^ k ϕ_d^ lij - _dl^k _(ab^ d ϕ_c)^ lij , Φ_abc^ijk = - 1/6 β_abc^ijk, Φ̅_abc^ijkln = 1/36 _(a^ir_b^ jsβ_c)rs^ klm . The Yangian $\cY(\fg)$ has a one-parameter group of automorphisms $\tau_c$, $c\in\CC$, given by $\tau_c(x_a)=x_a$ and $\tau_c(\J(x_a))=\J(x_a)+\hbar\,c\,x_a$, which is compatible with both algebra and Hopf algebra structure. An analogue of this automorphism for the twisted Yangian $\cY(\fg,\fg^\theta)^{tw}$ is a one-parameter group of automorphism of embeddings (<ref>) given by $\tau_c(\varphi(X_\al))=X_\al$ and $\tau_c(\varphi(\B(Y_p)))=J(Y_p)+\frac{1}{4}\hbar[Y_p,C_X]+\hbar\,c\,Y_p$. There is no analogue of such an automorphism for the twisted Yangian $\cY(\fg,\fg)^{tw}$, since it is not compatible with the relation (<ref>). § COIDEAL SUBALGEBRAS OF THE YANGIAN $\CY(\FSL_3)$ In this section we present three coideal subalgebras $\cY(\fg,\fh)^{tw}$ of $\cY(\fsl_3)$, with $\fh=\fso_3$, $\fh=\fgl_2$ and $\fh=\fsl_3$ ($\theta=id$ case). We will denote generators of the first two algebras by symbols $\Ch,\Ce,\Cf,\Ck$ and $\CH,\CE,\CF$. We will start by recalling the definition of the Lie algebra $\fsl_3$ in the Serre–Chevalley basis and in the Cartan–Chevalley basis. §.§ The $\fsl_3$ Lie algebra Lie algebra $\fsl_3$ in the Serre–Chevalley basis is generated by $\{e_i,f_i,h_i~\|~i=1,2\}$ subject to the defining relations [e_i,f_j]=δ_ijh_i, [h_i,e_j]=a_ij e_j, [h_i,f_j]=-a_ij f_j, [e_i,[e_i,e_i±1]]=0, [f_i,[f_i,f_i±1]]=0, where $a_{ii}=2$ and $a_{12}=-1=a_{21}$ are the matrix elements of the Cartan matrix of $\fsl_3$. The Cartan–Chevalley basis contains 8 elements: there are 6 generators that correspond to the simple roots, and 2 to the non-simple ones, $e_3=[e_1,e_2]$ and $f_3=[f_2,f_1]$. The defining relations are obtained by dropping the Serre relations in the previous definition and adding the commutators [e_1,e_2]=e_3, [e_1,e_3]=[e_2,e_3]=0, [f_1,f_2]=-f_3, [f_1,f_3]=[f_2,f_3]=0, [e_1,f_3]=-f_2, [e_2,f_3]=f_1, [f_1,e_3]=e_2, [f_2,e_3]=-e_1, [h_i,f_3]=-f_3, [h_i,e_3]=e_3, [e_3,f_3]=h_1+h_2. The quadratic Casimir operator of $\cU(\fsl_3)$ is given by ($\fc_\fg=6$) C_{\fsl_3}=\sum_{i=1}^3 \{e_i,f_i\} + \frac{1}{3} \sum_{i,j=1}^{2} \{h_i,h_j\} . §.§ Yangian $\cY(\fsl_3)$ Let $\cY(\fsl_3)$ denote the associative unital algebra with sixteen generators $e_i,f_i,h_j,\cJ(e_i)$, $\cJ(f_i),\cJ(h_j)$ with $i=1,2,3$, $j=1,2$ and the defining relations (<ref>) and \null[x,\cJ(y)] = \cJ([x,y]),\qquad \cJ(a\,x+b\,y)=a\, \cJ(x) + b\,\cJ(y), where $x,y\in\{e_i,f_i,h_i\}$, $a,b\in\CC$, and level-2 terrific relation [(h_1),(h_2)] = 34 ħ^2 ({e_1,e_2,f_3}+{e_3,f_1,f_2}). The remaining terrific relations given by (<ref>) can be obtained by the adjoint action of the level-0 generators on the relation (<ref>). The same applies to the horrific relations in the examples given below. The Hopf algebra structure on $\cY(\fsl_3)$ is given by (x) = x1+1x, ((x)) = (x) 1+1(x)+ 12 ħ [x1,Ω_], S_ħ(x) = -x, S_ħ((x))=-(x)+14 ħ _ x, ϵ_ħ(x) = ϵ_ħ(J(x)) = 0 . §.§ Orthogonal twisted Yangian $\cY(\fsl_3,\fso_3)^{tw}$ Let the involution $\theta$ be given by θ : e_1 ↦-e_2, f_1 ↦- f_2, h_1 ↦h_2 . The action of $\theta$ on the rest of the algebra elements is deduced by the constraint $\theta^2=id$. The symmetric space basis for $\fsl_3$ is given by $\fg^\theta=\{\Ch=h_1+h_2,\Ce=e_1-e_2,\Cf=f_1-f_2\}$ and $\fm=\{h_1-h_2,e_1+e_2,f_1+f_2,e_3,f_3\}$. The positive eigenspace forms the orthogonal subgroup $\fg^\theta=\fso_3\subset\fsl_3$. We will denote the generators of this subgroup by $\{\Ch,\Ce,\Cf\}$. They form the level-0 basis of the twisted Yangian $\cY(\fsl_3,\fso_3)^{tw}$. We will denote the level-1 generators by $\{\CH,\CE,\CF,\CE_2,\CF_2\}$. Let $\cY(\fsl_3,\fso_3)^{tw}$ denote the associative unital algebra with a basis of eight generators $\Ch,\Ce,\Cf,\CH,\CE,\CF,\CE_2,\CF_2$ and the defining level-0 relations (of the $\fso_3$ Lie algebra) [,]=, [,]=, [,]=-, level-1 Lie relations [,]=[,]=, [,]=, [,]=-, [,]=2_2, [,]=2_2, [,_2]=[,_2]=0, [,_2]=, [,_2]=, [,_2]=-2_2, [,_2]=2_2, [,]=3 , [,]=-3 , [,]=0, level-2 horrific relation [,]+[_2,_2] = 14 ħ^2 ( {,,} - 3 {,,} ), level-3 horrific relation [[,],] = 32 ħ^2 ({_2,,} + {_2,,} ) + 154 ħ^2 ({,,} - {,,} ). The algebra $\cY(\fsl_3,\fso_3)^{tw}$ admits a unique left co-action given by () = φ() 1+1 , () = φ()1 +1+ ħ [φ_0()1,Ω_] , for all $\Cx\in\{\Ch,\Ce,\Cf\}$ and $\CY\in\{\CE,\CF,\CH,\CE_2,\CF_2\}$ with φ() =e_1-e_2, φ() =f_1-f_2, φ() =h_1+h_2, φ_0() =e_1+e_2, φ_0() =f_1+f_2, φ_0() =h_1-h_2, φ_0(_2) =e_3, φ_0(_2) =f_3, φ() = (e_1)+(e_2) + 14 ħ [e_1+e_2,C_], φ(_2) = (e_3) + 14 ħ [e_3,C_], φ() = (f_1)+(f_2) + 14 ħ [f_1+f_2,C_], φ(_2) = (f_3) + 14 ħ [f_3,C_] , φ() = (h_1)-(h_2) + 14 ħ [h_1-h_2,C_], C_= 12({,} + ^2)∈, Ω_= 12(φid)∘(+ + 2)∈. The co-unit is ϵ(𝗑)= ϵ(𝖸)=0 . §.§ General twisted Yangian $\cY(\fsl_3,\fgl_2)^{tw}$ Let the involution $\theta$ be given by θ : e_1↦e_1, f_1↦f_1, e_2↦-e_2, f_2↦-f_2, h_i↦h_i, In this case $\fg^\theta = \{\Ch=h_1,\Ck=2h_2+h_1,\Ce=e_1,\Cf=f_1\}\sim \fgl_2 $ and $\fm = \{e_2,f_2,e_3,f_3\}$. We denote the corresponding level-1 generators by $\CE_2, \CF_2, \CE_3, \CF_3$. Let $\cY(\fsl_3,\fgl_2)^{tw}$ denote the associative unital algebra with a basis of eight generators $\Ch, \Ce, \Cf, \Ck$ and $\CE_2, \CF_2, \CE_3, \CF_3$ and the defining level-0 relations (of the $\fgl_2$ Lie algebra) [We have the standard $\fgl_2$ basis $\{e_{ij}\}$ with the defining relations $[e_{ij},e_{kl}]=\delta_{kj}e_{il}-\delta_{il}e_{kj}$ that follow from the identification $e_{11}=-(\Ch+\Ck)/2$, $e_{22}=(\Ch-\Ck)/2$, $e_{12}=\Cf$ and $e_{21}=\Ce$.] [,]=, [,]=2, [,]=-2, [,]= [,]= [,]=0, level-1 Lie relations [,_2]=_3, [,_2]=_3, [,_2]=[,_2]=0, [,_3]=_2, [,_3]=_2, [,_3]=[,_3]=0, [,_2]=-_2, [,_2]=_2, [, _i]= 3_i, [,_3]=_3, [,_3]=-_3, [, _i]= -3_i, level-2 horrific relations [_2,_3] = 0, [_2,_3] = 0, level-3 horrific relations [_2,[_2,_3]] = -2 ħ^2{_2,,}, [_2,[_3,_2]] = -2 ħ^2{_2,,}. The algebra $\cY(\fsl_3,\fgl_2)^{tw}$ admits a unique left co-action given by () = φ() 1+1 , for all $\Cx\in\{\Ch,\Ce,\Cf,\Ck\}$ and (_i) = φ(_i) 1+1 _i + ħ [e_i1,Ω_] , (_i) = φ(_i) 1+1 _i + ħ [f_i1,Ω_] , φ() = h_1 , φ() = 2h_2+h_1 , φ() = e_1 , φ() = f_1 , φ(_i)=(e_i) + 14ħ [e_i,C_] , φ(_i)=(f_i) + 14ħ [f_i,C_] . C_ = {,} + 23(^2++^2)∈, Ω_ = (φid)∘(++23(+)+13(+))∈. The co-unit is given by ϵ()=c∈, and ϵ()=ϵ()=0, for all $\Cx\in\{\Ce,\Cf,\Ch\}$ and $\CY\in\{\CE_i,\CF_i\}$ $(i=1,2)$. §.§ Twisted Yangian of even levels $\cY(\fsl_3,\fsl_3)^{tw}$ The last case contains the trivial involution, $\theta_3=id$, hence $\fg^\theta = \fg = \fsl_3$ and $\fm=\emptyset$. Let $\cY(\fsl_3,\fsl_3)^{tw}$ denote the associative unital algebra with sixteen generators $e_i,f_i, h_j, \G(e_i), $ $\G(f_i),\G(h_j)$ with $i=1,2,3$, $j=1,2$, obeying the standard $\fsl_3$ Lie algebra relations of the Cartan–Chevalley basis and the standard level-2 Lie relations [x,\G(y)] = \G([x,y]), \qquad \G(a\,x+ b\,y) = a\, \G(x) + b\, \G(y), for any $x,y\in\{e_i,f_i,h_j\}$, $a,b\in\CC$, and the following level-4 horrific relation [(h_1),(h_2)] =ħ^2 ({e_1,e_2,(f_3)}+{f_1,f_2,(e_3)}) - ħ^42 ({e_1,e_2,f_3}+{f_1,f_2,e_3}) +ħ^2 ({f_3,e_2,(e_1)}+{f_3,e_1,(e_2)}+{e_3,f_2,(f_1)}+{e_3,f_1,(f_2)}) + ħ^22( {h_1,f_2,(e_2)}-{h_1,e_2,(f_2)}- {h_2,f_1,(e_1)}+ {h_2,e_1,(f_1)}) + ħ^24 ( {h_1,e_1,(f_1)}- {h_1,f_1,(e_1)}- {h_2,e_2,(f_2)}+ {h_2,f_2,(e_2)} +{h_1-h_2,e_3,(f_3)}- {h_1-h_2,f_3,(e_3)}) - ħ^44 ({e_1,e_1,e_2,f_1,f_3}+{e_1,e_2,e_2,f_2,f_3}-{e_1,e_2,e_3,f_3,f_3} +{f_1,f_1,f_2,e_1,e_3}+{f_1,f_2,f_2,e_2,e_3}-{f_1,f_2,f_3,e_3,e_3}) -ħ^412 ({e_1,e_2,h_1,h_1,f_3}+{e_1,e_2,h_1,h_2,f_3}+{e_1,e_2,h_2,h_2,f_3} +{f_1,f_2,h_1,h_1,e_3}+{f_1,f_2,h_1,h_2,e_3}+{f_1,f_2,h_2,h_2,e_3}) . The algebra $\cY(\fsl_3,\fsl_3)^{tw}$ admits a left co-action given by (x_i) = x_i 1+1 x_i , ((x_i)) = φ((x_i)) 1 + 1(x_i) + ħ [(x_i)1,Ω_] + 14_^-1α_i^ jk ħ^2 [[x_k1,Ω_],[x_j1,Ω_]] + 14 ħ^2 [[x_i1,Ω_],Ω_] . where $\varphi(\G(x_i)) = \cK(x_i)+\tfrac{1}{4}\hbar\, [\cJ(x_i),C_\fg]$ and $\cK(x_i)=\fc_\fg^{-1}\alpha_i^{\;kj}[\cJ(x_j),\cJ(x_k)]$ with $\fc_\fg=6$, namely [The non-zero structure constants $\alpha_i^{\;kj}$ for $\fsl_3$ in the Cartan-Chevalley basis can be read from here.] (e_1) = 13([(e_3),(f_2)]+[(h_1),(e_1)]) , (f_1) = 13([(f_3),(e_2)]-[(h_1),(f_1)]) , (e_2) = 13([(f_1),(e_3)]+[(h_2),(e_2)]) , (f_2) = 13([(e_1),(f_3)]-[(h_2),(f_2)]) , (e_3) = 13([(e_1),(e_2)]+[(h_1+h_2),(e_3)]) , (f_3) = 13([(f_1),(f_2)]-[(h_1+h_2),(f_3)]) , (h_1) = 13(2[(e_1),(f_1)]+[(f_2),(e_2)]+[(f_3),(e_3)]) , (h_2) = 13(2[(e_2),(f_2)]+[(f_1),(e_1)]+[(f_3),(e_3)]) . The co-unit is $\epsilon(x_i)= \epsilon(\G(x_i))=0$. § PROOFS §.§ Proofs of isomorphisms for Drinfel'd basis §.§.§ Proof of Proposition <ref> Let ${\rm rank}(\fg)\geq2$. Define level-$n$ Drinfel'd generators of $\cL^+$ by J^(n)(x_a) = _^-1 _a^ cb [J^(1)(x_b),J^(n-1)(x_c)], where $J^{(1)}(x_a)=J^{}(x_a)$ and $J^{(0)}(x_a)=x_a$. We need to show that [J^(n)(x_b),J^(m)(x_c)] = _bc^a J^(n+m)(x_a) holds for all $n+m\geq0$, which is equivalent to the (standard) half-loop Lie algebra basis (<ref>) upon identification $J^{(n)}(x_a)\to x^{(n)}_a$. The cases $n+m=0$ and $n+m=1$ are given by (<ref>). The case $n+m=2$ follows from the level-$2$ Drinfel'd terrific relation (<ref>). Indeed, we can rewrite (<ref>) as \al_c^{\;ad}[J^{(n)}(x_b),[x_d,J^{(m)}(x_a)]] = \al_c^{\;ad}[x_b,[J^{(1)}(x_d),J^{(1)}(x_a)]] , which gives \al_c^{\;ad} ([J^{(n)}(x_b),[x_d,J^{(m)}(x_a)]] + [J^{(1)}(x_a),[x_b,J^{(1)}(x_d)]] + [J^{(1)}(x_d),[x_a,J^{(1)}(x_b)]]) =0 . For $n=m=1$ it is equivalent to the level-$2$ Drinfel'd terrific relation and for $n\neq m$ this equality follows from definition (<ref>) and the Jacobi identity (<ref>). Let us recall that for the ${\rm rank}(\fg)=1$ case, this level-$2$ terrific relation is trivial and one has to consider the level-$3$ terrific relation (<ref>) instead, which can be constructed in a similar way. To prove that (<ref>) holds for all $n+m\geq3$, provided it holds for $0\leq n+m\leq2$ and $n+m\geq0$, we will use the Cartan decomposition of $\fg$. Let $\fl\subset\fg$ be the Cartan subalegbra, $\Delta\subset\fh^*$ the root system, and $\Delta=\Delta^+\cup\Delta^-$ be a polarization of $\Delta$ ($\Delta^-=-\Delta^+$) and $\Gamma=\{\al_1,\ldots,\al_N\}$ be the set of positive simple roots. We have $\fg=\fl\op(\op_{\al} \fg_\al)$ with $\fl=span_\CC\{h_{\al_i}\,\|\,\al_i\in\Gamma\}$ and $\fg_\al=span_\CC\{x\in\fg\,\|\,[h,x]=\al(h)x, \forall h\in\fl \}$. Choose $e_{\pm\al}\in\fg_{\pm\al}$ such that $(e_\al,e_{-\al})_\fg=1$ (note that $(h_\al,e_\bet)_\fg=0$ for all $\al,\bet\in\Delta$, and $(e_\al,e_\bet)_\fg=0$ if $\al+\bet\notin\Delta$) and set $h_\al=[e_\al,e_{-\al}]$ and $\lan\bet,\al\ran=\al(h_\bet)$. Moreover, we have $h_{-\al}=-h_\al$ and $h_{\al+\bet}=h_\al+h_\bet$. Define level-$n$ generators by e^{(n)}_\al = \lan\al,\bet\ran^{-1} \ad h^{(1)}_{\al} (e^{(n-1)}_\al) , \qquad h^{(n)}_\al = \ad e^{(1)}_{\al} (e^{(n-1)}_{-\al}) , where $h^{(0)}_\al=h_\al$, $e^{(0)}_\al=e_\al$ and $h^{(1)}_\al$, $e^{(1)}_\al$ are the level-$0$ and level-$1$ Drinfel'd generators, respectively. Then (<ref>) is equivalent to [h^(n)_,h^(m)_] = 0, [h^(n)_,e^(m)_] = , e^(n+m)_, [e^(n)_,e^(m)_] = N_ e^(n+m)_+ if +∈Δ, h^(n+m)_ if +=0 , 0 if +∉Δ, for some $N_{\al\bet}\in\CC^\times$ satisfying $N_{\al\bet}=-N_{\bet\al}$. We will prove (<ref>) by induction. The base of induction is given by the cases with $0\leq m+n\leq 2$. Now suppose (<ref>) is true for some $n+m=k\geq3$. The action of $\ad h^{(1)}_{\bet} $ on the second relation gives [h^{(n)}_\al,e^{(m+1)}_\bet] = \lan\al,\bet\ran\, e^{(k+1)}_\bet . The action of $\ad e^{(1)}_{-\bet}$ leads to ,[e^(n+1)_-,e^(m)_] - [h^(n)_,h^(m+1)_] = , h^(k+1)_. Let $\al+\bet\in\Delta$. Act with $\ad e^{(1)}_{-\al-\bet}$ on the third relation in (<ref>), use induction hypothesis and require linearity $h^{(k+1)}_{\al+\bet}=h^{(k+1)}_\al+h^{(k+1)}_\bet$. In such a way we obtain N_{-\al-\bet,\al} [e^{(n+1)}_{-\bet},e^{(m)}_\bet] + N_{-\al-\bet,\bet} [e^{(n)}_{\al},e^{(m+1)}_{-\al}] = N_{\al\bet}(h^{(k+1)}_{\al}+h^{(k+1)}_{\bet}) , which gives [e^(n)_,e^(m+1)_-] = h^(k+1)_ and [e^(n+1)_-,e^(m)_] = h^(k+1)_. Here we have used $N_{-\al-\bet,\bet} = -N_{-\al-\bet,\al} = N_{\al\bet}$, which follows from an equivalent calculation for the $m=n=0$ case. Set $\beta=-\al$. Then [e^{(n)}_{\al},e^{(m+1)}_{-\al}] =[e^{(m)}_{\al},e^{(n+1)}_{-\al}] . Combining (<ref>) with (<ref>) we find [h^(n)_,h^(m+1)_] = 0. Now consider the identity [h^{(n+1)}_\al,e^{(m-1)}_\bet] = \lan\al,\bet\ran\, e^{(k)}_\bet . Acting with $\ad h^{(1)}_{\al}$ and using (<ref>) we get [h^{(n+1)}_\al,e^{(m)}_\bet] = \lan\al,\bet\ran\, e^{(k+1)}_\bet . To obtain the remaining relations we act with $\ad h^{(1)}_{\al}$ on the third relation in (<ref>). This gives ,[e^(n+1)_,e^(m)_] + ,[e^(n)_,e^(m+1)_] = N_ ,+e^(k+1)_+ . The same operation with $\ad h^{(1)}_{\bet}$ leads to ,[e^(n+1)_,e^(m)_] + ,[e^(n)_,e^(m+1)_] = N_ ,+e^(k+1)_+ . Multiply (<ref>) by $\lan\al,\bet\ran$ and (<ref>) by $\lan\al,\al\ran$ and take the difference, (\lan\bet,\bet\ran\lan\al,\al\ran-\lan\al,\bet\ran^2)\,[e^{(n)}_\al,e^{(m+1)}_\bet] = (\lan\bet,\al+\bet\ran\lan\al,\al\ran-\lan\al,\al+\bet\ran\lan\al,\bet\ran) \, N_{\al\bet}\, e^{(k+1)}_{\al+\bet} , which gives [e^{(n)}_\al,e^{(m+1)}_\bet] = N_{\al\bet}\, e^{(k+1)}_{\al+\bet} \mb{if} \al+\bet\in\Delta, since $(\lan\bet,\bet\ran\lan\al,\al\ran-\lan\al,\bet\ran^2) = (\lan\bet,\al+\bet\ran\lan\al,\al\ran-\lan\al,\al+\bet\ran\lan\al,\bet\ran) \neq 0$ unless $\bet\neq t \al$ for some $t\in\NN$, but this is excluded. Now suppose $\al+\bet\notin\Delta$. Then a similar calculation gives [e^{(n)}_\al,e^{(m+1)}_\bet] = 0 \mb{if} \al+\bet\notin\Delta. This proves the induction hypothesis and thus (<ref>). The isomorphism with the standard loop basis follows straightforwardly from (<ref>). $\square$ §.§.§ Proof of Proposition <ref> Let ${\rm rank}(\fg)\geq2$. Consider the symmetric space basis of $\fg$ and set $B^{(0)}(Y_p)=Y_p$, $B^{(0)}(X_\al)=X_\al$ and $B^{(1)}(Y_p)=J(Y_p)$. Define the higher level generators by B^(2n+1)(Y_p) = 2 _^-1 ∑_g_p^ q [B^(1)(Y_q),B^(2n)(X_)] , B^(2n+2)(X_) = (_())^-1 g_^ qp [B^(1)(Y_p),B^(2n+1)(X_)] . These generators are required to satisfy the following relations: [B^(2k+1)(Y_p) , B^(2n-2k+1)(Y_q) ] = ∑_w_pq^ B^(2n+2)(X_) , [B^(2k+1)(Y_p) , B^(2n-2k)(X_) ] = g_p^q B^(2n+1)(Y_q) , [B^(2k)(X_) , B^(2n-2k)(X_) ] = f_^ B^(2n)(X_) , for all $0\leq k \leq n$. It is easy to see that the map $B^{(2n+1)}(Y_p) \to Y^{(2n+1)}_p $ and $B^{(2n)}(X_\al) \to X_\al^{(2n)}$ gives the $\theta$-fixed subalgebra $\cH^+$ of $\cL^+$ (<ref>). We need to show that (<ref>) with $k=n=0$, (<ref>) with $k=0$, $n=1$ and (<ref>) with $k=n=0$ together with the induction relations (<ref>) and (<ref>) define $\cH^+$ uniquely. Moreover, we need to show that the first two relations follow from (<ref>) and (<ref>). The level-$2$ generators are defined by _() B^(2)(X_) = w_^ qp[B^(1)(Y_p),B^(1)(Y_q)] . They are required to satisfy the following identities: [B^(0)(X_),B^(2)(X_)] = f_^ B^(2)(X_), [B^(1)(Y_p),B^(1)(Y_q)] = ∑_ w_pq^ B^(2)(X_) . The first identity is the Lie algebra relation for the level-$2$ generators. It follows by a direct calculation: [B^(0)(X_),B^(2)(X_)] = (_())^-1 w_^ qp [X_,[B(Y_p),B(Y_q)]] by (<ref>) = (_())^-1 w_^ qp (g_q^r[B^(1)(Y_p),B^(1)(Y_r)]+g_p^r[B^(1)(Y_q),B^(1)(Y_r)]) by (<ref>) = (_())^-1 w^rp_f^_ [B^(1)(Y_p),B^(1)(Y_r)] = f_^ B^(2)(X_) , by the mixed Jacobi identity $w^{pr}_{\sk \ga}f^{\sk \ga}_{\al\bet}+w_{\bet}^{\;qp}g_{\al q}^{\sk r}+w_{\bet}^{\;qr}g_{q\al}^{\sk p}=0$ and by (<ref>). The second identity ensures that there are exactly $h={\rm dim}(\fg^\theta)$ level-$2$ generators. It gives the level-2 horrific relation (<ref>): [B^{(1)}(Y_p),B^{(1)}(Y_q)] = \sum_{\al} w_{pq}^{\sk \al} B^{(2)}(X_\al) = \sum_{\al} w_{pq}^{\sk \al} (\bar\fc_{(\al)})^{-1} w_{\al}^{\;st} [B^{(1)}(Y_t),B^{(1)}(Y_s)]. Now consider the level-$3$ generator defined by _ B^(3)(Y_p) = 2 ∑_ g_p^ q [B^(1)(Y_q),B^(2)(X_)] . We require $B^{(3)}(Y_p)$ to satisfy the following identities: [B^(0)(X_),B^(3)(Y_p)] = g_p^qB^(3)(Y_q) , [B^(1)(Y_r),B^(2)(X_)] = g_r^sB^(3)(Y_s) . The first identity is the Lie algebra relation for the level-$3$ generators. The second identity ensures that there are exactly $m={\rm dim}(\fm)$ level-$3$ generators. Let us show the first identity. We will need the following mixed Jacobi relation g_p^ q f_^μ-g_p^ μr g_r^q-g_r^ μq g_p^r =0 . Use definition (<ref>). Then _[B^(0)(X_),B^(3)(Y_p)] = 2 ∑_ g_p^ q ( f_^μ[B^(1)(Y_q),B^(2)(X_μ)] - g_q^r[B^(1)(Y_r),B^(2)(X_)]) by (<ref>) = 2 ( g_p^ q f_^μ[B^(1)(Y_q),B^(2)(X_μ)] - g_p^ q g_q^r[B^(1)(Y_r),B^(2)(X_)]) by <ref>/ = 2 ( g_p^ q f_^μ[B^(1)(Y_q),B^(2)(X_μ)] - g_p^ μr g_r^q[B^(1)(Y_q),B^(2)(X_μ)]) by ren. $\bet\to\mu$, $q\leftrightarrow r$/ = 2 g_r^ μq g_p^r [B^(1)(Y_q),B^(2)(X_μ)] = _ g_ip^qB^(3)(Y_q) by (<ref>) and (<ref>). / For the second identity in (<ref>) we have ∑_w_pq^ [B^(2)(X_),B^(1)(Y_r)] = ∑_ (_())^-1 w_pq^ w_^ st [[B^(1)(Y_t),B^(1)(Y_s)],B^(1)(Y_r)] by (<ref>) = [[B^(1)(Y_p),B^(1)(Y_q)],B^(1)(Y_r)] by (<ref>), and ∑_w_pq^ g_r^sB^(3)(Y_s) = ∑_, 2/_ w_pq^ g_r^s g_s^ u[B^(2)(X_),B^(1)(Y_u)] by (<ref>) = ∑_, 2 _^tu/_ _() w_pq^ g_r^ s g_st^ w_^ vx [[B^(1)(Y_x),B^(1)(Y_v)],B^(1)(Y_u)] by (<ref>) = 2 _^-1_^tu w_pq^ g_r^ s [[B^(1)(Y_s),B^(1)(Y_t)],B^(1)(Y_u)] by (<ref>) , which combined with (<ref>) gives the level-$3$ horrific relation (<ref>). For completeness, let us recall that for the ${\rm rank}(\fg)=1$ case, the level-$2$ and level-$3$ horrific relation are trivial and one has to consider a level-$4$ horrific relation instead. This can be shown in a similar way. We need to show that (<ref>) and (<ref>) hold for all $n\geq1$, and that (<ref>) holds for all $n\geq2$ provided they hold for $n=0$ and $n=1$, respectively. We will use the symmetric space basis of the Cartan decomposition of $\fg$ and an induction hypothesis to complete the proof. For simplicity, we will restrict to a special case of a symmetric pair with the Cartan decomposition given by $\fh=\fl_\fh \op (\op_{\al} \fh_\al)$ and $\fm=(\op_{\mu} \fm_\mu)$ with $\al \in \Delta_\fh$ and $\mu \in \Delta_\fm$, the roots of $\fh$ and $\fm$, respectively. This case corresponds to the symmetric pair of type AIII. We will use the lower case letters to denote the generators of even levels and the upper case letters for the odd levels. We define level-$(2k+2)$ generators $\{h^{(2k+2)}_\al,e^{(2k+2)}_\bet\}$ by h_^(2k+2) = E_^(1)( E_-^(2k+1)) , e_+β^(2k+2) = (N_)^-1E_^(1)( E_^(2k+1)), +∈Δ_, and level-$(2k+3)$ generators $\{E^{(2k+3)}_\mu\}$ by where $h_{\al}^{(0)}=h_{\al}$, $e_{\al}^{(0)}=e_{\al}$ and $E^{(1)}_{\mu}=E_{\mu}$ are the level-$0$ and level-$1$ Drinfel'd basis generators. Let $\al,\bet\in\Delta_\fh$, $\mu,\nu,\la\in\Delta_\fm$, and $\ga,\delta\in\Delta$ denote any root. The relations (<ref>), (<ref>) and (<ref>) in this basis are equivalent to [h^(2n)_,h^(2m)_δ] = 0, [h^(2n)_,e^(2m)_] = , e^(2n+2m)_, [e^(2n)_,e^(2m)_] = N_ e^(2n+2m)_+ if +∈Δ_, h^(2n+2m)_ if +=0 , 0 if +∉Δ_, [h^(2n)_,E^(2m+1)_μ] = ,μ E^(2n+2m+1)_μ, [e^(2n)_,E^(2m+1)_μ] = N_μ E^(2n+2m+1)_+μ if +μ∈Δ_, 0 if +μ∉Δ_, [E^(2n+1)_μ,E^(2m+1)_ν] = N_μν e^(2n+2m+2)_μ+ν if μ+ν∈Δ_, h_μ^(2n+2m+2) μ+ν=0, 0 if μ+ν∉Δ_. Suppose that the relations above hold for all levels up to $2k+1\geq3$, the marginal case being the base of induction consisting of (<ref>), (<ref>) and (<ref>) with $m=n=0$, which give level-0, level-1 and level-2 relations, respectively, and (<ref>) with $m=0$ and $n=1$ giving level-3 relations. Consider the action of $-\ad E_{-\mu}$ on the first relation in (<ref>) and of $-\ad E_{-\al-\mu}$ on the second relation in (<ref>) assuming $\al+\mu\in \Delta_\fm$. Use definitions (<ref>) and (<ref>) for level-$(2k+2)$ generators $h^{(2k+2)}_{\mu}$ and $e^{(2k+2)}_{\al}$ and level-$(2k)$ relations (<ref>) and (<ref>), that hold due to the induction hypothesis. In this way we obtain ,μ h^(2k+2)_μ = [h^(2n)_,h_μ^(2m+2)]-,μ [E^(2n+1)_-μ,E^(2m+1)_μ] , N_μ h^(2k+2)_+μ = N_,--μ[E^(2n+1)_-μ,E^(2m+1)_μ]+N_--μ,μ [e^(2n)_,e^(2m+2)_-] . Let $\al=\ga$. Requiring linearity $h^{(2k+2)}_{\al+\mu}=h^{(2k+2)}_{\al}+h^{(2k+2)}_{\mu}$ and using $N_{\al\mu}=N_{-\al-\mu,\mu}=N_{\al,-\al-\mu}$, which follows from the level-$0$ relations of $\fg$, we find , [E^(2n+1)_μ,E^(2m+1)_-μ]=h^(2k+2)_μ . Now let $\ga$ be arbitrary and choose $\mu$ such that $\mu=\nu+\al$ for some $\nu$ and $\al$. This allows us to obtain the required general form for any $\delta$. Next, act with $\ad E_{\bet-\mu}$ on the first relation in (<ref>) and with $\ad E_{-\mu}$ on the second relation in (<ref>). This gives ,μN_-μ,μ e^(2k+2)_ = N_-μ,μ [h^(2n)_,e^(2m+2)_] -,-μ [E^(2n+1)_-μ,E^(2m+1)_μ], N_μ N_-μ,+μ e^(2k+2)_ = -[e^(2n)_,h^(2m+2)_μ]+N_-μ, [E^(2n+1)_-μ,E^(2m+1)_μ]. The action of $\ad e_{\al}$ on (<ref>) gives [h^(2m+2)_,e^(2n)_]=[h^(2n)_,e^(2m+2)_] . Set $\ga=\mu$. Add together (<ref>) and (<ref>) times $-N_{\bet-\mu,\mu}$ and use (<ref>). We obtain (\lan \mu,\mu\ran - N_{\bet\mu}\, N_{-\mu,\bet+\mu}\,)\,N_{\bet-\mu,\mu}\, e^{(2k+2)}_{\bet} = (N_{\bet,-\mu}N_{\bet-\mu,\mu}-\lan\mu,\bet-\mu\ran)\,[E^{(2n+1)}_{\bet-\mu},E^{(2m+1)}_\mu]. Use the corresponding Lie algebra relations to equate the coefficients. In such a way we find [E^(2n+1)_-μ,E^(2m+1)_μ] = N_-μ,μ e^(2k+2)_ . Combining this identity with (<ref>) and (<ref>), and equating the coefficients we find [h^(2m+2)_,e^(2n)_] = [h^(2m)_,e^(2n+2)_] = , e^(2k+2)_. Let $\al+\mu \notin\Delta_\fm$. Act with $\ad E^{(1)}_{\bet-\mu}$ on the second relation in (<ref>), use the induction hypothesis and, without the loss of generality, require $\al+\bet\notin\Delta_\fh$ and $\al+\bet-\mu\notin\Delta_\fm$. This gives the remaining level-$(2k+1)$ relation in (<ref>). Now repeat the same steps but with $\ad E^{(1)}_{\nu-\al}$ instead and require $\nu+\mu-\al\notin\Delta_\fh$. This gives the last level-$(2k+1)$ relation in (<ref>). We have demonstrated that all level-$(2k+2)$ relations hold provided all the defining relation of level no greater than $(2k+1)$ hold. It remains to show that level-$(2k+3)$ relations in (<ref>) also hold. Consider (<ref>) with $2n+2m+2=2k+2$. Acting with $-\ad E^{(1)}_{\mu}$ on the first relation we find ,μ [E_μ^(2n+1),h^(2m)_δ] +δ,μ [h^(2n)_,E_μ^(2m+1)]=0. Set $\delta=\ga$ and use the induction hypothesis and definition (<ref>). Then [h^(2m)_,E_μ^(2n+1)] = [h^(2n)_,E_μ^(2m+1)]=[h^(2k+2)_,E_μ^(1)]= ,μ E_μ^(2k+3) . Next, act with $-\ad E_{\mu}^{(1)}$ on the second relation in (<ref>). We obtain ,μ [E_μ^(2n+1),e^(2m)_] + N_βμ,μ+E_μ+^(2k+3) = , [e^(2k+2)_,E_μ^(1)] if μ+∈Δ_, ,μ [E_μ^(2n+1),e^(2m)_]=, [e^(2k+2)_,E_μ^(1)] if μ+∉Δ_. For $n=0$ and $m=k+1$ the first relation becomes $[e^{(2k+2)}_{\bet},E_{\mu}^{(1)}]=N_{\beta\mu}E_{\mu+\bet}^{(2k+3)}$, which combined with the initial expression gives the required relation for arbitrary $2n+2m=2k+3$, namely [e^(2m)_,E_μ^(2n+1)]=N_β,μE_μ+^(2k+3) . Now consider (<ref>). Set $n=0$ and $m=k+1$. Then $[e^{(2k+2)}_{\bet},E_{\mu}^{(1)}]=0$, which gives $[E_{\mu}^{(2n+1)},e^{(2m)}_\bet]=0$ for all $2m+2n=2k+2$. Hence we have demonstrated that all level-$(2k+3)$ relations in (<ref>) hold. This proves the induction hypothesis and completes proof for the AIII case. The proof of the symmetric space basis for the general case can be shown using the same strategy (see [1] for the complete details of the Cartan decomposition for the general case). $\square$ To end this section we remark that we would welcome a proof of the Drinfel'd basis for the half-loop and twisted half-loop Lie algebras, which would not be based on the Cartan decomposition of the underlying Lie algebra. §.§ Proofs of coactions §.§.§ Proof of Proposition <ref> The Lie bi-ideal structure on $\cH^+$ defines the coaction up to the first order in $\hbar$, \LHb(x) =\varphi(x) \ot 1 + 1 \ot x + \hbar\, \tau(x)+\cO(\hbar^2) , with $x \in \cY(\fg,\fh)^{tw}$ and $\varphi$ the natural embedding $\cY(\fg,\fh)^{tw}\hookrightarrow\cY(\fg)$. For the level zero generators of $\cH^+$ the Lie bi-ideal structure is trivial and the minimal form of the coaction is given by \LHb(X_\alpha) =\varphi(X_\alpha) \ot 1 + 1 \ot X_\alpha . The coideal compatibility relations (<ref>) and (<ref>), and the classical limit requirement $\varphi(X_\alpha)\|_{\hbar\to 0}=X_\alpha$ implies that $\varphi(X_\alpha)=X_\alpha$ is the natural inclusion $\varphi : X_\alpha \in \cY(\fg,\fh)^{tw} \mapsto X_\alpha \in \cY(\fg)$. For the level one generators of $\cH^+$ the Lie bi-ideal structure is non trivial. The simplest coaction is given by \LHb(\B(Y_p)) =\varphi(\B(Y_p)) \ot 1 + 1 \ot \B(Y_p)+\hbar\, [Y_p \ot 1, \Omega_\fh] . As previously, the coaction must satisfy relations (<ref>) and (<ref>), and in the classical limit we must obtain $\varphi(\B(Y_p))\|_{\hbar\to 0}=B(Y_p)$. By (<ref>) it follows \Delta_{\hbar}(\varphi(\B(Y_p)))=\varphi(\B(Y_p)) \ot 1+1\ot \varphi(\B(Y_p))+\hbar\, [Y_q\ot1,\Omega_\fh] . Consider an ansatz $\varphi(\B(Y_p)) =\cJ(Y_p)+\hbar F^{(0)}_p$ with some level zero element $F^{(0)}_p \in \cY(\fg)$. We have $\Dh(\cJ(Y_p)) = \cJ(Y_p)\ot 1+1\ot \cJ(Y_p)+\frac{1}{2}\hbar\,[Y_p\ot1, \Omega_\fg]$. The coinvariance property implies \Delta_{\hbar}(\cJ(Y_p)+\hbar F^{(0)}_p)-\big((\cJ(Y_p)+\hbar F^{(0)}_p)\ot 1+1 \ot (\cJ(Y_p)+\hbar F^{(0)}_p)\big)=\hbar\, [Y_p \ot 1, \Omega_\fh] , Δ_ħ(F^(0)_p)-(F^(0)_p1+1 F^(0)_p) = 1/2[Y_p 1, 2 Ω_- Ω_]=1/2∑_α(g_p^ αq Y_qX_α-w_p^ qα X_αY_q) =1/2∑_αg_p^ αq (Y_qX_α+ X_αY_q) . We choose $F^{(0)}_p=\frac{1}{4}\sum_{\alpha}g_{p}^{\;\alpha q}(Y_q X_\alpha+ X_\alpha Y_q) = \tfrac{1}{4}[Y_q,C_{X}]$ giving \varphi(\B(Y_p)) = \cJ(Y_p) + \tfrac{1}{4}\hbar\, [Y_q,C_X] . It remains to check the coideal coassociativity (<ref>) and the homomorphism property $\varphi([b_i,b_j])=[\varphi(b_i),\varphi(b_j)]$ and $\LHb([b_i,b_j])=[\LHb(b_i),\LHb(b_j)]$ for all $b_i,b_j\in\cY(\fg,\fh)^{tw}$, which follow by a direct calculation. $\square$ §.§.§ Proof of Proposition <ref> In what follows we will need the following auxiliary Lemma: In a simple Lie algebra $\fg$ the following identities hold: [[x_i1,Ω_],Ω_] = 1/2(_i^ jc _j^ ab+_i^ jb _j^ ac)(x_a{x_b,x_c} - {x_b,x_c} x_a) , _i^ jk [[x_k1,Ω_],[x_j1,Ω_]] = _i^ jk _j^ cr _k^ bs _sr^a (x_a{x_b,x_c} + {x_b,x_c} x_a) , 6 _i^ jk _j^ cr _k^ bs _sr^a = ∑_π(_i^ jk _j^ π(c)r _k^ π(b)s _sr^π(a)) + 1/2_(_i^ jc _j^ ab+_i^ jb _j^ ac). Recall that [a\otimes b,c\otimes d]=[a,c]\otimes \{b,d\}+\{a,c\}\otimes [b,d]. For the first identity we have [[x_i1,Ω_],Ω_] = _i^ bj^kc[x_jx_b,x_kx_c] = _i^ bj^kc (_jk^a x_a {x_b,x_c} + _bc^a {x_k,x_j} x_a) = _i^ bj^kc (_jk^a x_a {x_b,x_c} - _jk^a {x_c,x_b} x_a) by ren. k,j↔c,b = _i^ bj _j^ ca (x_a {x_b,x_c} - {x_c,x_b} x_a) . For the second identity we have _i^ jk [[x_k1,Ω_],[x_j1,Ω_]] = _i^ jk _k^ br_j^ cs [x_r x_b, x_s x_c] = _i^ jk _j^ cs _k^ br(_rs^a x_a {x_b,x_c} + _bc^a {x_s ,x_r} x_a ) = _i^ jk _j^ cs _k^ br _rs^a ( x_a {x_b,x_c} + {x_c,x_b} x_a ) by ren. $b,c\leftrightarrow r,s$. The third identity is obtained using the following auxiliary identities 2 _i^ jk _j^ cr _k^ bs _sr^a = _i^ jk _j^ cr(_k^ bs _sr^a + _k^ as _sr^b) + 1/2__i^ jc _j^ ab , 2 _i^ jk _j^ cr _k^ bs _sr^a = _i^ jk _k^ bs (_j^ cr _sr^a + _j^ ar _sr^c) + 1/2__i^ jb _j^ ac . The first auxiliary identity follows by multiple application of the Jacoby identity: \al_i^{\;jk} \al_{j}^{\;cr} (\al_{k}^{\;bs} \al_{sr}^{\sk a}-\al_{k}^{\;as} \al_{sr}^{\sk b}) = \al_i^{\;jk} \al_{j}^{\;cr} \al_{rk}^{\sk s} \al_s^{\; ab} = \frac{1}{2} \al_i^{\;jc} \al_{j}^{\;kr} \al_{rk}^{\sk s} \al_s^{\; ab} = \frac{1}{2} \fc_\fg \al_i^{\;jc} \al_j^{\; ab} . The second auxiliary identity follows by the $b\leftrightarrow c$ symmetry and renaming $j,s\leftrightarrow k,r$. The level zero generators have a trivial bi-ideal structure, hence \LHb(x_i) = \varphi(x_i) \ot 1 + 1 \ot x_i , and $\varphi : x_i\in\cY(\fg,\fg)^{tw} \mapsto x_i\in\cY(\fg)$ is the natural inclusion. For the level two generators we have \LHb(\G(x_i)) = \varphi(\G(x_i))\ot 1+1 \ot \G(x_i)+ \hbar [\cJ(x_i)\ot1,\Omega_{\fg}]+ \hbar^2 W^{(0)}_i , with $W^{(0)}_i$ being a level zero element in $\cY(\fg) \ot \cY(\fg,\fg)^{tw}$. Coinvariance (<ref>) implies Δ_ħ( φ((x_i)))= φ((x_i))1+1 φ((x_i))+ ħ[J(x_i)1,Ω_]+ ħ^2 (1 φ) ( W^(0)_i ) . In the classical limit we must have $\varphi(\G(x_i))\|_{\hbar\to 0}= \fc_\fg^{-1} \al_i^{\;jk} [J(x_k),J(x_j)]$. Define a level two Yangian generator by $K^{(2)}_i = \fc_\fg^{-1} \al_i^{\;jk} [\cJ(x_k),\cJ(x_j)]$ and choose an ansatz $\varphi(\G(x_i)) = K^{(2)}_i + \hbar F^{(1)}_i$ with some $F^{(1)}_i\in \cY(\fg)$ of level one. We have (K^(2)_i) = K^(2)_i 1 + 1 K^(2)_i + 1/2ħ [(x_i) 1 - 1 (x_i), Ω_] + 1/4_^-1ħ^2 _i^ jk[[x_k1,Ω_],[x_j1,Ω_]] . By comparing the $\hbar$-order terms in (<ref>) and (<ref>) we choose $F^{(1)}_i = \tfrac{1}{4}[\cJ(x_i),C_\fg]$. Note that \Dh (F^{(1)}_i) = F^{(1)}_i\ot1 + 1\ot F^{(1)}_i + \frac{1}{2} [\cJ(x_i)\ot1+1\ot \cJ(x_i),\Omega_\fg] + \frac{1}{4}\hbar[[x_i\ot1,\Omega_\fg],\Omega_\fg] . Now (<ref>) gives 4 W^(0)_i = [[x_i1,Ω_],Ω_] + _^-1 _i^ jk [[x_k1,Ω_],[x_j1,Ω_]] . The coassociativity relation (<ref>) reads as ((id)∘-(id)∘) ((x_i))= (Δ_h (φ((x_i)))-(φ((x_i))1+1 φ((x_i))+ħ [(x_i)1,Ω_]))1 + ħ^2((id)(W^(0)_i)-(id)(W^(0)_i) - 1W^(0)_i + 1/2 _i^ jk ([x_k1, Ω_]) x_j) . Using the fact $\varphi(x_i)=x_i$ the relation above gives a constraint (id)(W^(0)_i) -(id )(W^(0)_i) + W^(0)_i1 - 1W^(0)_i + 1/2 _i^ jk ([x_k1, Ω_]) x_j = 0 . Using Lemma (<ref>) the explicit form of $W^{(0)}_i$ is equal to W^(0)_i = 1/24_^-1∑_π(_i^ jk _j^ π(c)r _k^ π(b)s _sr^π(a)) (x_a{x_b,x_c} + {x_b,x_c} x_a) + 1/12(_i^ jc _j^ ab+_i^ jb _j^ ac)( 2 x_a{x_b,x_c} - {x_b,x_c} x_a). Denote $W^{(0)}_i = h_i^{\;abc} x_a\ot \{x_b,x_c\} + \bar{h}_i^{\;abc} \{x_b,x_c\} \ot x_a$. Then (<ref>) is equivalent to 4\bar{h}_i^{\;abc} x_c\ot x_b \ot x_a - 4 h_i^{\;abc} x_a\ot x_b\ot x_c + \al_{i}^{\;jc}\al_j^{\;ab} x_a \ot x_b \ot x_c = 0 , giving $4\bar{h}_i^{\;cba} - 4 h_i^{\;abc} + \al_{i}^{\;jc}\al_j^{\;ab}=0$. Using (<ref>) we find 4h̅_i^ cba - 4 h_i^ abc + _i^ jc_j^ ab = -1/3(_i^ ja_j^ cb+_i^ jb_j^ ca)-2/3(_i^ jc_j^ ab+_i^ jb_j^ ac) + _i^ jc_j^ ab = -1/3 (_i^ ja_j^ cb + _i^ jb_j^ ac) +1/3 _i^ jc_j^ ab = 0 by the co-Jacobi identity. It remains to check the homomorphism properties $\varphi([b_i,b_j])=[\varphi(b_i),\varphi(b_j)]$ and $\LHb([b_i,b_j])=[\LHb(b_i),\LHb(b_j)]$ for all $b_i,b_j\in\cY(\fg,\fg)^{tw}$, which follow by a lengthy but direct calculation. $\square$ §.§ Proofs of horrific relations §.§.§ Proof of the horrific relations of Theorem <ref> Denote the level-2 and level-3 horrific relations (<ref>) and (<ref>) as H^{(2)}_{pq} = \hbar^2 \Lambda_{pq}^{\la\mu\nu}\{X_\la,X_\mu,X_\nu\} \qquad\text{and}\qquad H^{(3)}_{pqr} = \hbar^2 \Upsilon_{pqr}^{\la\mu u}\{X_\la,X_\mu,\B(Y_u)\}, respectively. The defining relations must respect the grading and be even in $\hbar$. The right hand sides are designed to make $\LHb$ extend to a homomorphism of algebras $\cY(\fg,\fg^\theta)^{tw} \to \cY(\fg) \ot \cY(\fg,\fg^\theta)^{tw}$. This can be checked by a direct calculation. §.§.§ Level-2 Set $B_{pq}=[\cB(Y_p),\cB(Y_q)]$ and use Proposition <ref> and the mixed Jacobi identities. Then (B_pq) = φ(B_pq)1 + 1 B_pq + ħ∑_μw_pq^μ ( w_μ^ st Y_t (Y_s) + ∑_, f_μ^ (X_) X_+ ħ/2 ∑_ w_μ^ sug_u^ t {Y_s,Y_t} X_) + ħ^2/2 ∑_,, w_st^ (g_p^ sg_q^ t+g_p^ sg_q^ t) ({X_,X_}X_+ X_{ X_, X_} ) φ(B_pq) = [(Y_p),(Y_q)] + ħ/2 ∑_μ w_pq^μ ( w_μ^ st {(Y_s),Y_t} + ∑_, f_μ^ {(X_),X_} ) + ħ^2/4 g_p^ sg_q^ t [{Y_s,X_},{Y_t,X_}] . By a similar procedure as it was done in the proof of Lemma (<ref>), the last line in (<ref>) can be factorized as ħ^2/6 ∑_,,,,μ w_pq^μ f_μ^ f_^ ({X_,X_}X_- 2 X_{ X_, X_} ) + ħ^2 ∑_,, g_p^ sg_q^ tw_st^ (X_,X_,X__1 + X_,X_,X__2 ) , where we have used the same notation as in (<ref>). The terms in the second line of (<ref>) and in the first line of (<ref>) do not contribute to $\LHb(H^{(2)}_{pq})$, since w_{pq}^{\sk \mu} + \sum_\al (\bar\fc_{(\al)})^{-1} w_{pq}^{\sk \al}w^{\;rs}_{\al} w_{rs}^{\sk \mu} = 0 . What remains are the terms in the second line of (<ref>) giving (H^(2)_pq) = φ(H^(2)_pq)1 + 1 H^(2)_pq +ħ^2 ∑_,, (g_p^ s g_q^ t + ∑_(_())^-1 w_pq^ w_^ us g_u^ s g_s^ t) w_st^ (X_,X_,X__1 + X_,X_,X__2 ) . By (<ref>) we have ({X_,X_,X_}) - ({X_, X_,X_} 1 + 1 {X_,X_μ,X_ν}) = 3(X_,X_,X__1 + X_,X_,X__2 ) . Comparing (<ref>) with (<ref>) gives (<ref>), as required. §.§.§ Level-3 Set $B_{pqr}=[B_{pq},\cB(Y_r)]$. We write (B_pqr)=φ(B_pqr)1+1B_pqr + ħ B_pqr^(2) + ħ^2 B_pqr^(1) + ħ^3 B_pqr^(0), where $B_{pqr}^{(i)}$ denotes elements of grade $i$. By consistency, the elements of grade two in $\LHb(H^{(3)}_{pqr})$ must vanish altogether, since they are of a linear order in $\hbar$. This can be shown explicitly. We have B_pqr^(2) = ∑_ [[(Y_p),(Y_q)]1 + 1 [(Y_p),(Y_q)] , g_r^ t Y_tX_] + ∑_μw_pq^μ [ w_μ^ st Y_t (Y_s) + ∑_, f_μ^ (X_) X_, (X_r)1+1(Y_r) ] . Denote this expression as $B_{pqr}^{(2)}=B_{pqr}^{(2,0)}+B_{pqr}^{(0,2)}+B_{pqr}^{(1,1)}$, where $B_{pqr}^{(i,j)}$ represents elements of grade $i$ in the left tensor factor and of grade $j$ in the right tensor factor. Then B_pqr^(2,0) = ∑_, g_r^ t (w_pt^[(X_),(Y_q)]+w_qt^[(Y_p),(X_)])X_+ ∑_,,μ w_pq^μ f_μ^ [(X_),(Y_r)] X_ = 2∑_,, (g_r^ t (w_pt^g_q^s-w_qt^g_p^s) + ∑_μ w_pq^μ f_μ^ g_r^s ) _^-1 g_s^ u [(Y_u),(X_)] X_ + ħ^2/2_ ∑_,, ( g_r^ t (w_pt^ g_q^u - w_qt^ g_p^u) + ∑_μw_pq^μ f_μ^ g_r^u ) ∑_ijk β_u ^ijk {x_i,x_j,x_k}X_ = 2∑_,, w_pq^ g_r^ t g_t^ s g_s^ u _^-1 [(Y_u),(X_)] X_+ O_1(ħ^2) , where $\cal{O}_1(\hbar^2)$ is a short–hand notation for the terms in the third line, and we have used the mixed Jacobi identities and the Drinfel'd terrific relation (<ref>) in the form [\J(X_\bet),\J(X_q)] + \frac{2}{\fc_\fg}\, g_{\bet q}^{\sk\, s} g_{s}^{\;\ga u} [\J(X_\ga),\J(Y_u)] = \frac {\hbar^2}{2\,\fc_\fg} \sum_{\ga} \sum_{ijk} g_{q}^{\;\ga u} \bet_{\bet u \ga}^{ijk} \{x_i,x_j,x_k\} . Here the sum $\sum_i x_i$ spans all of the symmetric space basis, $x_i=\{Y_p,X_\al\}$; the same applies to $x_j$ and $x_k$. In a similar way we find B_pqr^(0,2) = ∑_g_r^ t Y_t (g_p^u [(Y_u),(Y_q)] + g_q^u [(Y_p),(Y_u)]) + ∑_w_pq^ w_^ st Y_t [(Y_s),(X_r)] = ∑_, (g_r^ t (g_p^u w_uq^ - g_q^u w_up^) + w_pq^ w_^ st w_sr^) (_())^-1 w_^ vs Y_t [(Y_s),(Y_v)] + ħ^2 ∑_,,μ,ν (g_r^ t (g_p^u _uq^μν - g_q^u _up^μν) + w_pq^ w_^ st _sr^μν ) Y_t {X_,X_μ,X_ν} = ∑_, w_pq^ g_r^ t g_t^ t w_^ vs (_())^-1 Y_t [(Y_s),(Y_v)] + O_2(ħ^2) , where we have used the mixed Jacobi identities and the horrific relation (<ref>). The notation for $\cal{O}_2(\hbar^2)$ is the same as before. The remaining elements give B_pqr^(1,1) = ∑_μ,β w_pq^μ ( w_μ^ st w_tr^β + ∑_ f_μ^ g_r^s ) (X_) (Y_s) = ∑_,β w_pq^ g_r^ t g_t^ s (X_) (Y_s) . Now it is an easy calculation to check that the leading terms in (<ref>), (<ref>) and (<ref>) do not contribute to $\LHb(H_{pqr}^{(3)})$, since ∑_w_pq^ g_r^ t + 2 _^-1 ∑_, _^uv w_pq^ g_r^ s w_su^ g_v^ t = 0 , by (<ref>). Next step is to consider the grade one term in (<ref>). We write $B_{pqr}^{(1)}=B_{pqr}^{(1,0)}+B_{pqr}^{(0,1)}$. Using (<ref>), (<ref>) and (<ref>) we find B_pqr^(1,0) = 1/2 ∑_,μ w_pq^μ ( ∑_, f_μ^ g_r^ s [(X_) X_,2Y_sX_+ {Y_s,X_}1] + w_μ^ sug_u^ t [{Y_s,Y_t},(Y_r)] X_) + 1/2∑_,μ w_pq^μ g_r^ u( w_μ^ st [{(Y_s),Y_t}, Y_u ] + ∑_, f_μ^ [{(X_),X_}, Y_u] ) X_ + 1/2 ∑_,, w_st^ (g_p^ sg_q^ t+g_p^ sg_q^ t) [{X_,X_}X_+ X_{ X_, X_},(Y_r)1] = 1/2 ∑_,,μ w_pq^μ ( ∑_, g_r^ s ( 2 f_μ^ f_^ + f_μ^ f_^) + w_tr^(w_μ^ tug_u^ s+ w_μ^ sug_u^ t) + g_r^ u(w_μ^ tsw_tu^+∑_f_μ^ g_u^s)) {Y_s,J(X_)} X_ + 1/2∑_,,,μ ( w_pq^μ f_μ^ g_r^ s g_s^u + w_st^ g_p^ s g_q^ t g_r^u ) ×( 2 (Y_u) {X_,X_} + {X_,(Y_u)}X_+ {X_,(Y_u)}X_) + 1/2∑_,, w_ts^( w_pq^ w_^ ut g_r^ s + g_r^u (g_p^ sg_q^ t+g_p^ s g_q^ t) ) {X_,(Y_u)}X_. The first sum in the last equality can be simplified using the mixed Jacobi identities, giving \frac{1}{2} \sum_{\ga,\la,\mu} w_{pq}^{\sk \mu} g_{\mu r}^{\sk t} \big( g_t^{\;u\la} g_u^{\ga s} + \sum_\al g_t^{\;\al s} f_\al^{\ga\la} \big) \{Y_s,J(X_\la)\} \ot X_\ga . This component does not contribute to $\LHb(H_{pqr}^{(3)})$ due to (<ref>). The second grade one element, $B_{pqr}^{(0,1)}$, is equal to B_pqr^(0,1) = 1/2 ∑_, w_pq^ ( w_^ st g_r^ u [Y_t (Y_s),2 Y_uX_+{Y_u,X_}1] + w_^ sug_u^ t [{Y_s,Y_t} [X_,(Y_r)] ) + 1/2 ∑_,, w_st^ (g_p^ sg_q^ t+g_p^ sg_q^ t) [{X_,X_}X_+ X_{ X_, X_},1(Y_r)] = 1/2 ∑_, w_pq^ ( 2 w_^ st g_r^ u g_s^v + w_^ vs g_r^ u g_s^t + w_^ tsg_s^ u g_r^v ) {Y_t,Y_u} (Y_v) + ∑_,, w_ts^( w_pq^ w_^ ut g_r^ s + g_r^u (g_p^ sg_q^ t+g_p^ s g_q^ t) ) ( X_{X_,(Y_u)} + 1/2 {X_,X_}(Y_u) ) . In the same way as before, the first sum in the last equality above can be simplified to \frac{1}{2} \sum_{\al,\bet} w_{pq}^{\sk \bet} g_{r\bet}^{\sk\, s} g_{s}^{\;\al t} w_{\al}^{\;uv} \{Y_t,Y_u\} \ot \B(Y_v) , and does contribute to $\LHb(H_{pqr}^{(3)})$ due to (<ref>). Set A_pqr^u = 1/2∑_,μ ( w_pq^μ f_μ^ g_r^ s g_s^u + w_st^ g_p^ s g_q^ t g_r^u + (↔)) , B_pqr^u = 1/2∑_ w_ts^( w_pq^ w_^ ut g_r^ s + g_r^u (g_p^ sg_q^ t+g_p^ s g_q^ t) ) . Using the mixed Jacobi identities (<ref>) we find that Υ_pqr^u = ( A_pqr^u + 2/_ ∑_μ_^tu w_pq^μ g_rμ^s A_stu^u = B_pqr^u + 2/_ ∑_μ_^tu w_pq^μ g_rμ^s B_stu^u ) satisfying $\Upsilon_{pqr}^{\al\ga u}=\Upsilon_{pqr}^{\ga\al u}$. The contribution of the remaining sums in (<ref>) and (<ref>) to $\LHb(H_{pqr}^{(3)})$ is ∑_, Υ_pqr^u ( (Y_u) {X_,X_} + 2 {X_,(Y_u)}X_+ 2 X_{X_,(Y_u)} + {X_,X_}(Y_u) ) . By Proposition (<ref>) and (<ref>) we have ({X_,X_,(Y_u)}) - (φ({X_,X_,(Y_u)}) 1 + 1 {X_,X_,(Y_u)} + O_3(ħ)) = (Y_u){X_,X_} + {X_,X_}(Y_u) + ({(Y_u),X_}X_+ X_{X_,(Y_u)} + (↔)) by (<ref>); here $\cal{O}_3(\hbar)$ denotes the $\hbar$–order terms that appear due to (<ref>). Comparing the relation above with (<ref>) and using symmetry $\Upsilon_{pqr}^{\al\ga u}=\Upsilon_{pqr}^{\ga\al u}$ gives (<ref>), as required. The remaining elements in $\LHb(H_{pqr}^{(3)})$ are of cubic order in $\hbar$ and, by summing with $\hbar\,\cal{O}_1(\hbar^2)$ of (<ref>) and $\hbar\,\cal{O}_2(\hbar^2)$ of (<ref>), give precisely the element $\hbar^2\,\cal{O}_3(\hbar)$. [This equality is very large and not illuminative, thus we have chosen not to spell it out explicitly.] This concludes the proof of the horrific relations (<ref>) and (<ref>). $\square$ §.§.§ An outline of a proof of the horrific relation of Theorem <ref> We will need the following Lemma, which follows by a direct calculation: In any associative algebra over $\CC$ the following identities hold: {x_i,{x_j,x_k}} = {x_i,x_j,x_k}+1/12 [x_(j,[x_k),x_i]] , {x_j,x_k,x_l,x_m} = 1/3{x_j,x_(k,{x_l,x_m)}} - 1/36 [[x_j,x_(k],{x_l,x_m)}] , {1x_i,1x_j,x_a{x_b,x_c}} = x_a {x_i,x_j,{x_b,x_c}} , {x_i 1,1x_j,x_a{x_b,x_c}} = {x_i, x_a}{x_j, {x_b,x_c}}-1/12 [x_a,x_i]⊗[x_j, {x_b,x_c}], {x_i 1,1x_j,{x_b,x_c} x_a} = {x_i, {x_b,x_c}}{x_j,x_a}-1/12 [{x_b,x_c},x_i] [x_j, x_a]. Denote the level-4 horrific relation (<ref>) by H^(4)_abc = ħ^2Ψ_abc^ijk {x_i,x_j,(x_k)} + ħ^4 (Φ_abc^ijk {x_i,x_j,x_k} + Φ̅_abc^ijklm {x_i,x_j,x_k,x_l,x_m}). It holds for some set of coefficients $\Psi_{abc}^{ijk}$, $\Phi_{abc}^{ijk}$, $\bar\Phi_{abc}^{ijklm} \in \CC$. We will find these coefficients by acting with coaction $\LHb$ on (<ref>) and equating the numeric factors of the selected algebra elements on both sides of the equation. These elements are \{x_i,x_j\} \otimes \G(x_k), \qquad x_i \otimes \{x_j,x_k\}, \qquad x_i\otimes \{x_j,x_k,\{x_l,x_m\}\} . All the remaining elements will be referred as unwanted terms ($UWT$). Denote $G_{ad}=[\cG(x_a),\cG(x_d)]$. Then $H^{(4)}_{abc}=\al {}_{(ab} {}^{d} G{}_{c)d}$. Using Proposition <ref> we write \LHb(G_{cd})=\varphi(G_{cd})\otimes 1+1\otimes G_{cd} + \hbar \,G^{(3)}_{cd} +\hbar^2 \,G^{(2)}_{cd} +\hbar^3\, G^{(1)}_{cd} +\hbar^4\, G^{(0)}_{cd} , where $G^{(i)}_{cd}$ denote elements of grade $i$ and can be computed explicitly using (<ref>) and (<ref>). For our purpose we only need to consider the selected algebra elements of $G^{(2)}_{cd}$ and $G^{(0)}_{cd}$. For $G^{(2)}_{cd}$ we have G^(2)_cd =(α_cr^kh̅_d^rij-α_dr^kh̅_c^rij) {x_i,x_j} ⊗(x_k)+ UWT, and for $G^{(0)}_{cd}$ we find G^(0)_cd = _rs^i h_c^rjk h_d^slm x_i { { x_j, x_k }, { x_l, x_m }} + h̅_c^jln h̅_d^krs [{x_l,x_n},{x_r,x_s}] { x_j, x_k } +( h̅_c^trs h_d^lmn-h̅_d^trs h_c^lmn ){x_l,{x_r,x_s}} [x_t,{ x_m, x_n }] + UWT. Using Lemma <ref> and symmetries $h_{c}^{rjk}=h_{c}^{rkj}$, $\bar h_{c}^{rjk}=\bar h_{c}^{rkj}$ we reduce the expression above to G^(0)_cd = _rs^i h_c^rjk h_d^slm x_i{x_j,x_k,{x_l,x_m}} + 1/3W_cd^ijk x_i {x_j,x_k}+ UWT, W_cd^ijk = _rs^i h_c^rxy ( h_d^szk_xt^j _yz^t+ h_d^szt_xt^k _yz^j ) + ( (h̅_c^xyzh_d^efk-h̅_d^xyzh_c^efk)_ye^t _zt^i_xf^j Next step is to find the corresponding elements in the rhs of $\LHb(H^{(4)}_{abc})$. Using (<ref>), (<ref>) and Lemma <ref> we have ( {x_i,x_j,(x_k)} ) = {x_i,x_j} (x_k) + ħ^2 h_k^abc x_a {x_i,x_j,{x_b,x_c}} + 1/6 ħ^2 h̅_k^abc (_bi^r_cr^s x_s{x_j,x_a} + _bj^r_cr^s x_s{x_i,x_a} ) -1/6 ħ^2 h_k^abc (α_ai^r_jb^s+_aj^r_ib^s) x_r{x_s,x_c} + UWT, ( {x_i,x_j,x_k} ) = x_(i {x_j,x_k)} + UWT, ( {x_i,x_j,x_k,x_l,x_m} ) = 1/3 x_(i {x_j,x_(k,{x_l,x_m))} } - 1/36 x_(i [[x_j,x_(k],{x_l,x_m))}] +UWT. This gives (H^(4)_abc) = ħ^2 Ψ_abc^ijk {x_i,x_j} (x_k) + ħ^4(1/3 Φ̅_abc^(ij(klm)) + Ψ_abc^jkr h_r^ilm ) x_i{x_j,x_k,{x_l,x_m}} + UWT + ħ^4( Φ_abc^(ijk)-1/36Φ̅_abc^(ix(y(zj)))_xy^r_rz^k + 1/6(Ψ_abc^(xj)y h̅_y^kzr _zx^s_rs^i + Ψ_abc^(xy)z h_z^rsk _rx^i_ys^j )) x_i{x_j,x_k} . Then, by plugging (<ref>) and (<ref>) into $\LHb (\al {}_{(ab} {}^{d} G{}_{c)d})=\LHb (H^{(4)}_{abc})$ and comparing the elements on the both sides of the obtained equation we find Ψ_abc^ijk = _(ab^ d_c)r^ k h̅_d^rij - _dr^k_(ab^ d h̅_c)^rij , Φ̅_abc^(ij(klm)) = 3(_rs^i _(ab^ dh_c)^rjk h_d^slm - Ψ_abc^jkr h_r^ilm ) , Φ_abc^(ijk) = 1/3_(ab^ d W_c)d^ijk + 1/18Φ̅_abc^(ix(yzj))_xy^r_rz^k - 1/3(Ψ_abc^xjy h̅_y^kzr _zx^s_rs^i + Ψ_abc^xyz h_z^rsk _rx^i_ys^j ) , where in the last expression we used the symmetries $\Psi_{abc}^{ijk}=\Psi_{abc}^{jik}$ and $\Phi_{abc}^{(ij(klm))}=\Phi_{abc}^{(ij(kml))}$. 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1401.2205	# Optimal Testing for Planted Satisfiability Problems Quentin Berthetlabel=e1][email protected] [ Department of Computing and Mathematical Sciences California Institute of Technology Pasadena, CA 91125, USA ###### Abstract We study the problem of detecting planted solutions in a random satisfiability formula. Adopting the formalism of hypothesis testing in statistical analysis, we describe the minimax optimal rates of detection. Our analysis relies on the study of the number of satisfying assignments, for which we prove new results. We also address algorithmic issues, and give a computationally efficient test with optimal statistical performance. This result is compared to an average- case hypothesis on the hardness of refuting satisfiability of random formulas. 62C20, 68R01, 60C05, Satisfiability problem, High-dimensional detection, Polynomial-time algorithms, ###### keywords: [class=AMS] ###### keywords: t1The author thanks Philippe Rigollet, Emmanuel Abbé, Dan Vilenchick and Amin Coja-Oghlan for very helpful discussions. t2Partially supported by NSF grant CAREER-DMS-1053987 when the author was at Princeton University, and by AFOSR grant FA9550-14-1-0098. ###### Contents 1. 1 Problem description 2. 2 Optimal testing 1. 2.1 Likelihood-ratio test 2. 2.2 Information-theoretic lower bound 3. 3 Polynomial-time testing 1. 3.1 Variable coupling test 2. 3.2 Hardness hypothesis on random instances 4. 4 Alternative choices for planting distributions 5. A Proofs of technical results ## Introduction We study in this paper the problem of detecting a planted solution in a random $k$-SAT formula of $m$ clauses on $n$ variables. This is formulated as a hypothesis testing problem: Given a formula $\phi$, our goal is to decide whether it is a typical instance, drawn uniformly among all formulas, or if it has been drawn such that it is guaranteed to be satisfiable, by planting a solution. There is a resurgence in statistics of hypothesis testing problems, i.e., distinguishing null hypotheses with pure noise, against the presence of a structured signal in a high-dimensional setting. The seminal work of [Ing82, Ing98, DJ04], on the problem of detecting sparse or weakly sparse signals in high dimension has inspired a wide literature of detection problems. Examples include [ITV10] in the context of sparse linear regression, [ACCD11, BI13, ACV13, MW13] for small cliques or communities in graphs and matrices, [ABBDL10] for general combinatorial structured signals, and [ACBL12, BR12, BR13] for sparse principal components of covariance matrices. These problems are combinatorial in nature, and the complexity of the class of possible signals (sparse vectors, cliques in a graph, small submatrices, or here the $n$-dimensional hypercube) has a direct influence on the statistical and algorithmic difficulties of the detection problem. Minimax theory gives a formal definition of the statistical complexity of a hypothesis testing problem, in terms of the sample size needed to identify with high probability the underlying distribution of given instances. It describes the interplay between the interesting parameters of a problem: sample size, ambient dimension, signal-to-noise ratio, sparsity, underlying dimension, etc. This framework is particularly adapted to the study of random instances of $k$-SAT formulas: a random formula $\phi$ can be interpreted as $m$ independent, identically distributed clauses, each on $k$ of the $n$ variables. The uniform distribution is equivalent to pure noise, the absence of signal. Planting a solution is equivalent to changing the distribution of the clauses, dependent on an assignment $x\in\\{0,1\\}^{n}$. This planted satisfying assignment is the signal whose presence we seek to detect. The optimal rate of detection will describe how large $m$ (the sample size) needs to be for detection to be possible, as a function of $n$ (the ambient dimension), and $k$, treated as a constant. The properties of random instances of uniform $k$-SAT formulas have been widely studied in the probability and statistical physics literature. Particular attention has been paid to the notions of satisfiability thresholds (sharp changes of behavior when the clause-to-variable density ratio $\Delta=m/n$ varies) [AP04, AM06, CO09, COP13, CO13, DSS14], maximum satisfiability [ANP03] geometry of the space of solutions [ANP03, ART06, ACO08, KMRT+06, MRT09], and concentration of specific statistics [AM10, AM13]. The planted distribution has also been studied, often in order to create random instances that are known to be satisfiable, such as in [BHL+01, HJKN06, AGKS00, AJM04, ACO08, JMS05], and at high density in [AMZ06, CoKV07, FMV06]. Methods from statistical physics such as belief and survey propagation have been applied to this problem and rigorously studied [BMZ02, MPZ02, MZ02, CO10]. More recently, the algorithmic complexity (in a specific computational model) of estimating the planted assignment has been studied in [FPV13]. Here, the use of tools from statistical analysis, such as the likelihood ratio and the total variation distance, highlights the importance of a specific statistic: the number of satisfying assignments. More specifically, we study its deviations from its expected value. Optimal rates of detection are obtained by proving new results concerning the concentration (or absence thereof) of this statistic. We address algorithmic issues by showing that the optimal rates of detection can be obtained by a newly introduced polynomial- time test. We also show the effect of choosing a different planting distribution on the detection problem, particularly on the optimal rates of detection. The following subsection introduces notations for $k$-SAT formulas. Our hypothesis testing problem is formally described in Section 1. The optimal rates of detection are derived in Section 2, and the problem of testing in polynomial time is addressed in Section 3. The effect on the detection rates of different choices for the planting distributions is studied in Section 4. ### Notations for $k$-SAT formulas Let $n$ and $m$ be positive integers. For all fixed positive integers $k$, we denote by $\mathcal{F}_{n,m}^{k}$ the set of boolean formulas on $n$ variables that are the conjunction of $m$ disjunctions of $k$ distinct literals. Formally, for all $\phi\in\mathcal{F}_{n,m}^{k}$, we have for all $x\in\\{0,1\\}^{n}$ $\phi(x)=\bigwedge_{i=1}^{m}C_{i}(x)\,,$ where for all $i\in\\{1,\ldots,m\\}$, the clause $C_{i}$ is the disjunction of $k$ literals on $k$ distinct variables, i.e., the value of a variable or its negation $C_{i}(x)=\ell_{i,1}\vee\ldots\vee\ell_{i,k}\,,\;\ell_{i,j}\in\\{x_{1},\bar{x}_{1},\ldots,x_{n},\bar{x}_{n}\\}\,,\text{and }\ell_{i,j}\notin\\{\ell_{i,j^{\prime}},\bar{\ell}_{i,j^{\prime}}\\}.$ The $k$-SAT problem (short for satisfiability) is the decision problem of determining whether a given formula $\phi$ is satisfiable, i.e., if there exists $x\in\\{0,1\\}^{n}$ such that $\phi(x)$ evaluates to ’true’. For a given $k$-SAT formula $\phi$, we denote by $\mathcal{S}(\phi)$ the set of satisfying assignments $\mathcal{S}(\phi)=\big{\\{}x\in\\{0,1\\}^{n}:\phi(x)=\,\text{{\sf'true'}}\big{\\}}\,,$ and by $Z(\phi)=\|\mathcal{S}(\phi)\|$ the number of satisfying assignments for $\phi$. We often write $Z$ when it is not ambiguous. For a subset $S$ of $\\{1,\ldots,m\\}$, we define the sub-formula $\phi_{S}=\bigwedge_{i\in S}C_{i}\,.$ The definition of satisfying assignments extends to single clauses and sub- formulas in general, with the notations $\mathcal{S}(C_{i})$ and $\mathcal{S}(\phi_{S})$ for the set of assignments satisfying respectively, the clause $C_{i}$ or the formula $\phi_{S}$. We denote by ${\sf SAT}$ the set of satisfiable formulas: those with satisfying assignments. ## 1 Problem description We are interested in distinguishing two distributions on $\mathcal{F}^{k}_{m,n}$, the uniform, and planted distributions. The uniform distribution, denoted by $\mathbf{P}_{\text{unif}}$, is generated by independently selecting each clause uniformly from the $2^{k}{n\choose k}$ possible choices. The planted distribution, denoted by $\mathbf{P}_{\text{planted}}$, is generated by randomly selecting an assignment $x^{}$ uniformly among the $2^{n}$ elements of $\\{0,1\\}^{n}$, and then independently selecting all the clauses among the $(2^{k}-1){n\choose k}$ clauses that are satisfied by $x^{}$ (denoted by $\mathbf{P}_{x^{}}$). Each clause is given as $k$ literals, in a uniformly random order. We represent this as a hypothesis testing problem, on the observation $\phi\in\mathcal{F}^{k}_{m,n}$ $\displaystyle H_{0}$ $\displaystyle:$ $\displaystyle\phi\sim\mathbf{P}_{\text{unif}}$ $\displaystyle H_{1}$ $\displaystyle:$ $\displaystyle\phi\sim\mathbf{P}_{\text{planted}}=\frac{1}{2^{n}}\sum_{x\in\\{0,1\\}^{n}}\mathbf{P}_{x}\,.$ It is also possible to consider the detection problem with composite alternative hypothesis over the $\mathbf{P}_{x}$. Our formulation is equivalent to choosing a uniform prior over the planted assignments, and to consider the distribution $\mathbf{P}_{\text{planted}}$, mixture of the $\mathbf{P}_{x}$. We will mention two regimes: the linear regime, when $m=\Delta n$, for some $\Delta>0$, usually the only one considered in the probability theory literature; and the square-root regime, when $m=C\sqrt{n}$, for some $C>0$, particularly relevant to the study of our statistical problem. We will often consider $m,n$ large enough, but will mainly focus on non- asymptotic results. We define a test as a measurable function $\Psi:\mathcal{F}^{k}_{m,n}\rightarrow\\{0,1\\}$, whose goal is to determine the underlying distribution of the observation $\phi$. We define the probability of error as the maximum of the probabilities of type I and type II error, formally $\mathbf{P}_{\text{unif}}(\Psi(\phi)=1)\vee\mathbf{P}_{\text{planted}}(\Psi(\phi)=0)\,.$ This quantity is used here to measure the success of any test $\Psi$. We will consider that a test is successful when its probability of error is smaller than $\delta\in(0,1)$, considered fixed for the whole problem, such as $\delta=0.05$. We can make the simple observation that under the planted distribution, formulas are guaranteed to be satisfiable. This suggests to test satisfiability of the formula in order to solve the hypothesis testing problem. This test has a probability of error of type II equal to zero. Under the uniform distribution, the behavior of $\mathbf{P}_{\text{unif}}(\phi\in\text{\sf SAT})$ has been extensively studied, and a phase transition has been shown to exist in the linear regime of $m=\Delta n$, from satisfiability to unsatisfiability, around some $\Delta_{k}$ close to $2^{k}\log(2)$. We refer to [COP13, CO13] and references therein for more information, as well as [DSS14] for a proof of the sharpness of the phase transition, for $k$ large enough. In this setting, when $\Delta>\Delta_{k}$, the satisfiability test $\Psi_{\text{\sf SAT}}=\mathbf{1}\\{\cdot\in\text{\sf SAT}\\}$ has a probability of error going to 0, and when $\Delta<\Delta_{k}$, the error will converge to 1 (entirely because of the probability of a type I error). When thinking of the formula $\phi$ as a sequence of $m$ i.i.d. clauses, $m$ can be interpreted as the sample size, and the problem becomes easier when $\Delta$ increases. When $\Delta$ is too small, the probability of error of the test $\Psi_{\text{\sf SAT}}$ converges to 1. We see in the following section that this simple rate can be significantly improved. ## 2 Optimal testing In this section, we derive the optimal rate of detection for this problem, i.e., how large $m$ should be for a test to be able to distinguish with high probability the two hypotheses. We prove that the likelihood-ratio test is successful in the square-root regime, and show that it is information- theoretic optimal. ### 2.1 Likelihood-ratio test A test based on the likelihood ratio between the two candidate distributions can distinguish between them with high probability, in the square-root regime. When $m\geq C\sqrt{n}$ for a specific constant $C$, the probability of error of the likelihood-ratio test is smaller than $\delta\in(0,1)$. ###### Theorem 2.1. For all $k\geq 2$, positive $m,n$, denote $\Psi_{\text{\sf LR}}$ the likelihood-ratio test defined by $\Psi_{\text{\sf LR}}(\phi)=\mathbf{1}\\{Z(\phi)>\mathbf{E}_{\text{unif}}[Z]\\}\,.$ (1) For any $\delta\in(0,1)$, there exists $\bar{C}_{k,\delta}>0$ such that for $m\geq\bar{C}_{k,\delta}\sqrt{n}$, for $m,n$ large enough, it holds $\mathbf{P}_{\text{unif}}(\Psi_{\text{\sf LR}}(\phi)=1)\vee\mathbf{P}_{\text{planted}}(\Psi_{\text{\sf LR}}(\phi)=0)\leq\delta\,.$ Proof We first prove that the likelihood-ratio test has indeed form (1). For discrete distributions, the likelihood ratio is simply equal to the ratio of the two distributions. For all $\phi\in\mathcal{F}^{k}_{m,n}$, it holds $\frac{\mathbf{P}_{\text{planted}}(\phi)}{\mathbf{P}_{\text{unif}}(\phi)}=\frac{1}{2^{n}}\sum_{x\in\\{0,1\\}^{n}}\frac{\mathbf{P}_{x}(\phi)}{\mathbf{P}_{\text{unif}}(\phi)}\,.$ To compute the probabilities in the above ratios, we can interpret the drawing of $\phi$ by placing $m$ balls in $N=2^{k}{n\choose k}$ bins independently - if it has distribution $\mathbf{P}_{\text{unif}}$ \- or otherwise in the $N_{k}=(2^{k}-1){n\choose k}$ bins corresponding to clauses that are satisfied by $x$. Therefore, it holds for all $\phi$ $\frac{\mathbf{P}_{x}(\phi)}{\mathbf{P}_{\text{unif}}(\phi)}=\left\\{\begin{array}[]{rl}0&\;\text{if }x\notin\mathcal{S}(\phi)\\\ \Big{(}\frac{N}{N_{k}}\Big{)}^{m}&\;\text{otherwise }\end{array}\right.$ It can then be expressed in terms of $\mathbf{1}\\{x\in\mathcal{S}(\phi)\\}$, and $N/N_{k}=1/(1-2^{-k})$ $\displaystyle\frac{\mathbf{P}_{\text{planted}}}{\mathbf{P}_{\text{unif}}}(\phi)$ $\displaystyle=$ $\displaystyle\frac{1}{2^{n}}\sum_{x\in\\{0,1\\}^{n}}\Big{(}\frac{N}{N_{k}}\Big{)}^{m}\mathbf{1}\\{x\in\mathcal{S}(\phi)\\}$ $\displaystyle=$ $\displaystyle\frac{1}{\mathbf{E}_{\text{unif}}[Z(\phi)]}\sum_{x\in\\{0,1\\}^{n}}\mathbf{1}\\{x\in\mathcal{S}(\phi)\\}=\frac{Z(\phi)}{\mathbf{E}_{\text{unif}}[Z(\phi)]}\,,$ by the known closed form of $\mathbf{E}_{\text{unif}}[Z(\phi)]=2^{n}(1-2^{-k})^{m}$, which can be directly derived by linearity. The likelihood-ratio test is therefore indeed $\Psi_{\text{\sf LR}}(\phi)=\mathbf{1}\\{Z(\phi)>\mathbf{E}_{\text{unif}}[Z(\phi)]\\}$. It is now sufficient to prove $\mathbf{P}_{\text{unif}}(\Psi(\phi)=1)+\mathbf{P}_{\text{planted}}(\Psi(\phi)=0)\leq\delta$, as the maximum of two nonnegative numbers is smaller than their sum. By definition of the likelihood-ratio test, $\mathbf{P}_{\text{unif}}(\Psi_{\text{\sf LR}}(\phi)=1)+\mathbf{P}_{\text{planted}}(\Psi_{\text{\sf LR}}(\phi)=0)=1-d_{TV}(\mathbf{P}_{\text{unif}},\mathbf{P}_{\text{planted}})\,.$ Furthermore, by definition of the total variation distance $\displaystyle d_{TV}(\mathbf{P}_{\text{unif}},\mathbf{P}_{\text{planted}})$ $\displaystyle=$ $\displaystyle\sum_{\begin{subarray}{c}\phi\in\mathcal{F}^{k}_{m,n}\\\ \mathbf{P}_{\text{unif}}(\phi)>\mathbf{P}_{\text{planted}}(\phi)\end{subarray}}\\{\mathbf{P}_{\text{unif}}-\mathbf{P}_{\text{planted}}\\}(\phi)$ $\displaystyle=$ $\displaystyle\sum_{\begin{subarray}{c}\phi\in\mathcal{F}^{k}_{m,n}\\\ Z(\phi)/\mathbf{E}[Z]<1\end{subarray}}\Big{(}1-\frac{Z(\phi)}{\mathbf{E}[Z]}\Big{)}\mathbf{P}_{\text{unif}}(\phi)$ $\displaystyle=$ $\displaystyle\mathbf{E}_{\text{unif}}\Big{[}\Big{(}1-\frac{Z(\phi)}{\mathbf{E}[Z]}\Big{)}_{+}\Big{]}\,.$ The total variation distance between distributions of i.i.d. elements being non-decreasing in the sample size, we obtain by Lemma 2.2 that in the square- root regime, for $C$ large enough and $m\geq C\sqrt{n}$, $d_{TV}(\mathbf{P}_{\text{unif}},\mathbf{P}_{\text{planted}})\geq(1-e^{-\gamma_{k}C^{2}/C_{0}})(1-C_{0}/C^{2})\,.$ This bound yields the desired result for some large enough constant $C_{k,\delta}>0$. The proof of this theorem indicates that it is possible to distinguish the two distributions whenever $Z$ is not concentrated around its expectation under the uniform distribution. Our result is a consequence of the following lemma, that states that in the square-root regime, for a constant $C$ large enough, the ratio $Z/\mathbf{E}[Z]$ is much smaller than 1, with high probability. ###### Lemma 2.2. For all $k\geq 2$, $C_{0}$ an absolute constant, $m=C\sqrt{n}$, and $C,n$ large enough, it holds with probability $1-C_{0}/C^{2}$,for some constant $\gamma_{k}>0$ that $Z<e^{-\gamma_{k}C^{2}/C_{0}}\,\mathbf{E}[Z]\,.$ A stronger result, concerning the linear regime, can be derived similarly in order to answer a question regarding the behavior of $Z$ with respect to its expectation. It is known [AM10] that for $\Delta$ small enough and $n\rightarrow+\infty$, $n^{-1}\log(Z)$ and $n^{-1}\mathbf{E}[\log(Z)]$ have the same limit, called the quenched average. In the following lemma, we prove that this limit is actually different from the constant $n^{-1}\log(\mathbf{E}[Z])$, called the annealed average, for all $\Delta>0$. ###### Lemma 2.3. For all $k\geq 2$, $\Delta>0$, and $m=\Delta n$ large enough, if $\phi\sim\mathbf{P}_{\text{unif}}$, it holds with probability $1-o(1)$, for some constant $c_{k,\Delta}>0$ that $Z<e^{-c_{k,\Delta}n}\,\mathbf{E}[Z]\,.$ This result is tangential to the problem at hand but of interest in and of itself. We show here that the quenched and annealed averages are different for all $\Delta$ and $k$, with a gap greater than $c_{k,\Delta}$, for which we give no explicit formula. This phenomenon is hinted at in [ACO08, CO09], and proven to hold for $\Delta$ large enough in [COP13], with an explicit lower bound for the gap. We provide a proof for Lemma 2.2 and 2.3 in Appendix A. ### 2.2 Information-theoretic lower bound The proof of Theorem 2.1 also hints at a lower bounds for the statistical problem. The total variation distance $d_{TV}$ between the uniform and planted distributions is close to 0 (and the statistical problem is impossible) when $Z(\phi)$ is concentrated around its expectation. The number of satisfying assignments is actually equal to its expectation whenever no variable appears in two different clauses. Indeed, when this is the case, the set of satisfying assignments can be described thus. There are $m$ clauses on $m$ distinct groups of $k$ distinct variables. Each clause allows a specific group of $k$ variables to take $2^{k}-1$ values, and the $n-km$ remaining variables are free. There are therefore $(2^{k}-1)^{m}$ possible values for the constrained variables and $2^{n-km}$ possible values for the $n-km$ remaining. Overall, $Z=(2^{k}-1)^{m}2^{n-km}=2^{n}(1-2^{-k})^{m}=\mathbf{E}[Z]$. This observation yields the following lower bound. ###### Theorem 2.4. For $\nu\in(0,1/2)$, $m\leq 2\sqrt{\nu n}/k$, and $m,n$ large enough, it holds that $\inf_{\Psi}\big{\\{}\mathbf{P}_{\text{unif}}(\Psi(\phi)=1)\vee\mathbf{P}_{\text{planted}}(\Psi(\phi)=0)\big{\\}}\geq\frac{1}{2}-\nu\,.$ Proof We use the total variation bound, for any test $\Psi$ $\displaystyle\mathbf{P}_{\text{unif}}(\Psi(\phi)=1)\vee\mathbf{P}_{\text{planted}}(\Psi(\phi)=0)$ $\displaystyle\geq$ $\displaystyle\frac{1}{2}\big{(}\mathbf{P}_{\text{unif}}(\Psi(\phi)=1)+\mathbf{P}_{\text{planted}}(\Psi(\phi)=0)\big{)}$ $\displaystyle\geq$ $\displaystyle\frac{1-d_{TV}(\mathbf{P}_{\text{unif}},\mathbf{P}_{\text{planted}})}{2}\,.$ We denote by $F$ the set of formulas where no variable appears in two different clauses. $\displaystyle d_{TV}(\mathbf{P}_{\text{unif}},\mathbf{P}_{\text{planted}})$ $\displaystyle=$ $\displaystyle\frac{1}{2}\sum_{\phi\in\mathcal{F}^{k}_{m,n}}\|\mathbf{P}_{\text{unif}}-\mathbf{P}_{\text{planted}}\|(\phi)$ $\displaystyle=$ $\displaystyle\frac{1}{2}\sum_{\phi\in F}\|\mathbf{P}_{\text{unif}}-\mathbf{P}_{\text{planted}}\|(\phi)+\frac{1}{2}\sum_{\phi\in F^{c}}\|\mathbf{P}_{\text{unif}}-\mathbf{P}_{\text{planted}}\|(\phi)$ $\displaystyle=$ $\displaystyle\frac{1}{2}\sum_{\phi\in F}\Big{\|}\frac{Z(\phi)}{\mathbf{E}[Z]}-1\Big{\|}\mathbf{P}_{\text{unif}}(\phi)+\frac{1}{2}\sum_{\phi\in F^{c}}\|\mathbf{P}_{\text{unif}}-\mathbf{P}_{\text{planted}}\|(\phi)$ As noticed above, for all $\phi\in F$, $Z(\phi)=\mathbf{E}[Z]$; the likelihood ratio is equal to 1. The first term of this equation is therefore equal to 0. This also implies that $\mathbf{P}_{\text{unif}}(\phi)=\mathbf{P}_{\text{planted}}(\phi)$ for all $\phi\in F$, and $\mathbf{P}_{\text{unif}}(F)=\mathbf{P}_{\text{planted}}(F)$. The second term is thus upper bounded by $\mathbf{P}_{\text{unif}}(F^{c})=\mathbf{P}_{\text{planted}}(F^{c})$. It is sufficient to prove that $\mathbf{P}_{\text{unif}}(F^{c})\leq 2\nu$, a variant of the “birthday problem”: We place a group of $k$ balls in $n$ distinct bins uniformly at random, $m$ times independently. The probability that none of these $m$ groups intersect is equal to $\mathbf{P}_{\text{unif}}(F)$. When $i$ groups have already been drawn, occupying $ki$ bins, the probability that one of the next $k$ balls falls in an occupied bin is smaller than $k^{2}i/n$ (the expected number of such collisions). As $k^{2}(m-1)/n<1/2$ (for fixed $\nu$ and $n$ large enough) the following holds $\mathbf{P}_{\text{unif}}(F)\geq\prod_{i=1}^{m-1}\Big{(}1-\frac{k^{2}i}{n}\Big{)}>\prod_{i=1}^{m-1}e^{-2k^{2}i/n}=e^{-k^{2}(m-1)(m-2)/n}>1-k^{2}m^{2}/n\,.$ This gives the desired result. From the last two theorems, we can conclude that the optimal rate of detection is $m^{}=\sqrt{n}$. When $m=C\sqrt{n}$, detection is possible with probability of error smaller than $\delta$, for $C$ greater than some constant $\bar{C}_{k,\delta}$, by using the likelihood-ratio test. It is impossible to distinguish the two hypotheses with error probability smaller than $1/2-\nu$ for $C<\underline{C}_{k,\nu}:=2\sqrt{\nu}/k$. No effort has been made to optimize (or even quantify) the constant $\bar{C}_{k,\delta}$, as a function of $k$ and $\delta$. ## 3 Polynomial-time testing For $k\geq 2$, computing the outcome of the likelihood-ratio test involves solving a $\\#$P-complete problem [Val79], and for $k\geq 3$, even computing the outcome of the satisfiability test $\Psi_{\text{\sf SAT}}$ (which is already suboptimal) is equivalent to solving a NP-hard problem. The testing methods described in the previous section are not computationally efficient: determining if a formula is satisfiable is the quintessential hard problem, the first known to be NP-complete [Coo71, Lev73], at the root of the web of problems known to be in the same class [Kar72]. None of the tests described above can be computed in a computationally efficient manner. It is therefore legitimate to examine the performance of tests that can be computed in polynomial time. Finding a satisfying assignment in formulas that are known to be satisfiable has been the focus of substantial efforts [BMZ02, Fla02, KV06, CoKV07]. A polynomial-time algorithm that does so in the linear regime (for a large enough $\Delta$) is presented in [CoKV07], for the case $k=3$ (their results extend to any fixed $k$). A similar problem is studied as well in [FPV13]. This method can be used as a tool for detection: in the unsatisfiable regime (when $\Delta$ is large enough), the existence of a satisfying assignment is a sufficient reason to reject the null. The main issue of this approach is that the regime of detection is not optimal: $m$ needs to be of order $n$ (linear regime), when only $\sqrt{n}$ (square-root regime) is required for the likelihood-ratio test. ### 3.1 Variable coupling test The proof that the likelihood-ratio test has a low probability of error in the optimal regime is based on the fact that there is a large number of variables that appear more than once, and on the fact that under the null distribution, a couple of literals based on the same variable have equal probability to have the same sign or opposite signs. We can use this fact to design a test that runs in polynomial time and achieves the optimal rate of detection. We recall that in each clause, the literals are given in a uniformly random order. Let $T$ be the number of variables (among the $n$ possible) that appear more than once as the first literal of a clause of $\phi$ (according to the random ordering in the data) and $P$ (resp. $D$) the number of those for which the first two occurrences (according to the natural order of the clauses) of the same variable have the same sign (resp. different signs), so that $P+D=T$. The following holds ###### Theorem 3.1. For all $k\geq 2$, $m,n>0$ and $\delta\in(0,1)$, denote $\Psi_{\text{\sf COU}}$ the test defined by $\Psi_{\text{\sf COU}}(\phi)=\mathbf{1}\\{P/T>1/2+1/[2(2^{k}-1)]^{2}\\}\,,$ and $\tilde{C}_{k,\delta}:=[2(2^{k}-1)]^{2}\sqrt{2\log(2/\delta)}\vee\sqrt{1024/\delta}\,.$ For $m\geq\tilde{C}_{k,\delta}\,\sqrt{n}$, it holds $\mathbf{P}_{\text{unif}}(\Psi_{\text{\sf COU}}(\phi)=1)\vee\mathbf{P}_{\text{planted}}(\Psi_{\text{\sf COU}}(\phi)=0)\leq\delta\,.$ Proof For each variable that appears at least twice as the first literal of a clause, consider the probability that the two first occurrences (according to the natural order of the clauses) of a variable as the first literal of a clause (according to the random ordering in the data) have the same value. It is equal to 1/2 under the uniform distribution, and conditionally on the value of $T$, $P\sim\mathcal{B}(T,1/2)$. Under the planted distribution, each literal has independently probability $(1+1/(2^{k}-1))/2$ to have the same value as the corresponding variable in $x_{i}^{}$, and probability $(1-1/(2^{k}-1))/2$ to have a different value. Overall, the probability that these two literals have the same sign under the planted distribution is $\frac{1}{4}\big{(}1+\frac{1}{2^{k}-1}\big{)}^{2}+\frac{1}{4}\big{(}1-\frac{1}{2^{k}-1}\big{)}^{2}=\frac{1}{2}+\frac{1}{2(2^{k}-1)^{2}}\,.$ Therefore, conditionally on the value of $T$, $P$ has distribution $\mathcal{B}(T,1/2+1/[2(2^{k}-1)^{2}])$. By Hoeffding’s inequality, the following holds for all $\varepsilon>0$ $\displaystyle\mathbf{P}_{\text{unif}}\big{(}P/T>1/2+\varepsilon\,\|\,T\big{)}\leq\exp(-2\varepsilon^{2}T)$ $\displaystyle\mathbf{P}_{\text{planted}}\big{(}P/T<1/2+1/[2(2^{k}-1)^{2}]-\varepsilon\,\|\,T\big{)}\leq\exp(-2\varepsilon^{2}T)\,$ By Lemma A.1, and by definition of $\tilde{C}_{k,\delta}$, $T\geq\tilde{C}_{k,\delta}^{2}/4$ with probability at least $1-\delta/2$. Let $\varepsilon=1/[2(2^{k}-1)]^{2}$, and condition on the event $T\geq\tilde{C}_{k,\delta}^{2}/4$. The previous yields, for $C_{k,\delta}\geq\sqrt{2\log(2/\delta)}/\varepsilon$ $\displaystyle\mathbf{P}_{\text{unif}}\big{(}P/T>1/2+1/[2(2^{k}-1)]^{2}\,\|\,T\big{)}\leq\delta/2$ $\displaystyle\mathbf{P}_{\text{planted}}\big{(}P/T<1/2+1/[2(2^{k}-1)]^{2}\,\|\,T\big{)}\leq\delta/2\,.$ Which gives the desired result by a simple union bound. ### 3.2 Hardness hypothesis on random instances The result of Theorem 3.1 can be contrasted with a hypothesis by Feige, formulated in [Fei02], to prove hardness of approximation results in the worst case. We recall the proposed assumption on the hardness of determining the satisfiability of $3$-SAT formulas on average: “Even when $\Delta$ is an arbitrarily large constant independent of $n$, there is no polynomial time algorithm that refutes most 3CNF formulas with $n$ variables and $m=\Delta n$ clauses, and never wrongly refutes a satisfiable formula.” Formally, in a statistical language, it is conjectured in this hypothesis that for all $\Delta>0$, in the linear regime, there is no test $\Psi$ that runs in polynomial time such that $\mathbf{P}_{\text{unif}}(\Psi=1)\leq 1/2$, and $\mathbf{P}_{1}(\Psi=0)=0$, for any distribution $\mathbf{P}_{1}$ supported on SAT. In particular, in our testing problem, this hypothesis states that no test that runs in polynomial time has a type I error smaller than 1/2 and a type II error equal to 0. At first sight, this is in apparent contradiction with theorem 3.1. Interestingly, this result shows that up to the optimal square-root regime it is possible to design a test with small type I and type II errors simultaneously, even though it is conjectured and widely believed that it is impossible to distinguish those distributions with a completely one-sided error. There has been a recent interest in the notions of optimal rates for polynomial-time algorithms. More specifically, there is a growing literature on limitations, beyond those imposed by information theory, to the statistical performance of computationally efficient procedures. Such phenomena have been hinted at [DGR98, Ser00, CJ13, SSST12], and studied in specific computational models, such as in [FGR+13, FPV13]. More recently, these barriers have been proven to hold for various supervised tasks such as in [DLS13], based on a primitive on random 3-SAT instances, and unsupervised problems in statistics in [BR13] and the subsequent [MW13, Che13, WBS14], based on a hardness hypothesis for the planted clique problem. The above discussion shows the difficulty of using Feige’s hypothesis as a primitive to prove computational lower bounds for statistical problems: it does not imply that it is impossible to detect planted distributions in a computationally efficient manner in the linear regime, and is extremely sensitive to the allowed probability of type I and type II errors. ## 4 Alternative choices for planting distributions The tests described in Theorems 2.1 and 3.1 exploit a fundamental difference between the two considered distributions. Planting a satisfying assignment $x^{}\in\\{0,1\\}^{n}$ breaks the symmetry of the uniform distribution. The likelihood ratio $Z/\mathbf{E}[Z]$ is affected by the imbalances in interactions between variables. Similarly, the variable coupling test is based on the bias in the signs of chosen literals, under the planted distribution. This asymmetry is a characteristic of our choice of the planting distribution. In this section, we observe that the rates of detection are different for other natural choices of distribution on SAT, the set of satisfiable formulas. Such an example is $\mathbf{P}_{\text{\sf SAT}}$, the uniform distribution on SAT. In this new statistical problem, the alternative hypothesis becomes $\tilde{H}_{1}:\phi\sim\mathbf{P}_{\text{\sf SAT}}$. It is a fundamentally different statistical problem: its optimal rate of detection is the linear regime $m^{}=n$, achieved by the satisfiability test $\Psi_{\text{\sf SAT}}$. Indeed, as shown in a simple remark in Section 1, this test is successful in the satisfiable part of the linear regime. Furthermore, as $\mathbf{P}_{\text{\sf SAT}}$ is the uniform distribution on SAT, or $\mathbf{P}_{\text{unif}}(\,\cdot\,\|\phi\in\text{\sf SAT})$, the total variation distance $d_{TV}(\mathbf{P}_{\text{unif}},\mathbf{P}_{\text{\sf SAT}})$ is equal to $\mathbf{P}_{\text{unif}}(\phi\notin\text{\sf SAT})$. As explained before, this probability vanishes to 0 for $\Delta$ small enough, which yields the matching lower bound. From a statistical point of view, this modified hypothesis testing problem is a significantly harder task than the detection of planted satisfiability. Among all distributions on satisfiable formulas, the closest in total variation distance to the uniform distribution (and therefore the choice of alternative that yields the hardest statistical problem) is the uniform distribution on SAT. Other distributions used to generate formulas that are hard to solve, with hidden solutions (usually, with no immediate asymmetry) as in [AJM04, BHL+01, JMS05, KMZ12] are candidates to create detection problems with optimal rate of detection in the linear regime. Such an example is the uniform distribution on formulas that are not-all-equal, or NAE satisfiable. ## Appendix A Proofs of technical results Lemma 2.2 and 2.3 are a consequence of the following result on the number of variables that appear at least twice in the formula. For simplicity of the proof, we only consider the first literal of each clause, which is sufficient to our objective. ###### Lemma A.1. Let $\phi$ be a random formula of $\mathcal{F}^{k}_{m,n}$ with distribution $\mathbf{P}_{\text{unif}}$. Let $T$ be the number of variables (among the possible $n$) that appear more than once as the first literal of a clause of $\phi$. • Let $\Delta>0$, and $m=\Delta n$. There exists positive constants $\varepsilon_{\Delta}$ and $r_{\Delta}$ such that $\mathbf{P}(T<\varepsilon_{\Delta}n)\leq\frac{r_{\Delta}}{n}\,.$ * • Let $C>0$, and $m=C\sqrt{n}$. It holds that $\mathbf{P}(T<C^{2}/4)\leq\frac{576}{C^{2}}\,.$ Proof We prove this deviation bounds in the two regimes. Linear regime We first place ourselves in the linear regime $m=\Delta n$. The first literals of the clauses of the random formula can be interpreted as being drawn by independently placing $m$ balls uniformly in $n$ bins, and $T_{i}$ is the indicator of the event “there are at least two balls in bin $i$”. This is the complement of having either one or no ball in bin $i$, which yields $\mathbf{E}[T_{i}]=1-\Big{[}\Big{(}1-\frac{1}{n}\Big{)}^{m}+m\Big{(}1-\frac{1}{n}\Big{)}^{m-1}\frac{1}{n}\Big{]}=1-\Big{[}\Big{(}1-\frac{\Delta}{m}\Big{)}^{m}+\Delta\Big{(}1-\frac{\Delta}{m}\Big{)}^{m-1}\Big{]}\,,$ which has limit $1-(1+\Delta)e^{-\Delta}=2\varepsilon_{\Delta}>0$. Therefore, for $m$ large enough, $\mathbf{E}[T_{i}]>\varepsilon_{\Delta}$. By, definition $T$ and $T_{i}$, we have $T=T_{1}+\ldots+T_{n}\,.$ Therefore, it holds $\mathbf{E}[T]=\mathbf{E}[T_{1}+\ldots+T_{n}]>n\varepsilon_{\Delta}$. These variables are not independent and the variance is less simple ${\bf Var}[T]=n{\bf Var}[T_{1}]+n(n-1)\big{[}\mathbf{E}[T_{1}T_{2}]-\mathbf{E}[T_{1}]\mathbf{E}[T_{2}]\big{]}\,.$ We control the last term $\displaystyle\mathbf{E}[T_{1}T_{2}]$ $\displaystyle=$ $\displaystyle\mathbf{P}[T_{1}=1,T_{2}=1]=\mathbf{P}[T_{1}=1\|T_{2}=1]\mathbf{P}[T_{2}=1]$ $\displaystyle=$ $\displaystyle\mathbf{P}[T_{1}=1\|T_{2}=1]\mathbf{E}[T_{2}]$ $\displaystyle=$ $\displaystyle\Big{[}1-\Big{[}\Big{(}1-\frac{1}{n}\Big{)}^{m-2}+(m-2)\Big{(}1-\frac{1}{n}\Big{)}^{m-3}\frac{1}{n}\Big{]}\Big{]}\mathbf{E}[T_{2}]$ Therefore, we obtain the bound $\mathbf{E}[T_{1}T_{2}]-\mathbf{E}[T_{1}]\mathbf{E}[T_{2}]\leq\Big{[}1-\Big{(}1-\frac{1}{n}\Big{)}^{2}+\Delta\Big{(}1-\Big{(}1-\frac{1}{n}\Big{)}^{2}\Big{)}\Big{]}\mathbf{E}[T_{2}]\leq\frac{3+3\Delta}{n}\,.$ Overall, this yields ${\bf Var}[T]\leq(4+3\Delta)n$. We now apply Chebyshev’s inequality, with $r_{\Delta}=(3+3\Delta)/(\mathbf{E}[T_{1}]-\varepsilon_{\Delta})^{2}$ $\mathbf{P}[T<\varepsilon_{\Delta}n]\leq\frac{{\bf Var}[T]}{(\mathbf{E}[T_{1}]-\varepsilon_{\Delta})^{2}n^{2}}\leq\frac{r_{\Delta}}{n}\,.$ Square-root regime This proof is a simple modification of the proof of the linear regime with the same notations, for $m=C\sqrt{n}$. We derive the expectation and variance of $T$ $\displaystyle\mathbf{E}[T_{i}]$ $\displaystyle=$ $\displaystyle 1-\Big{[}\Big{(}1-\frac{1}{n}\Big{)}^{m}+m\Big{(}1-\frac{1}{n}\Big{)}^{m-1}\frac{1}{n}\Big{]}$ $\displaystyle=$ $\displaystyle 1-\Big{[}\Big{(}1-\frac{1}{n}\Big{)}^{C\sqrt{n}}+\frac{C}{\sqrt{n}}\Big{(}1-\frac{1}{n}\Big{)}^{C\sqrt{n}-1}\Big{]}$ $\displaystyle=$ $\displaystyle 1-\Big{[}1-\frac{C}{\sqrt{n}}+\frac{C^{2}}{2n}+o\Big{(}\frac{1}{n}\Big{)}+\frac{C}{\sqrt{n}}-\frac{C^{2}}{n}+o\Big{(}\frac{1}{n}\Big{)}\Big{]}=\frac{C^{2}}{2n}+o\Big{(}\frac{1}{n}\Big{)}\,.$ Therefore, for $n$ large enough $\mathbf{E}[T_{i}]\in(C^{2}/3n,C^{2}/n)$ and $\mathbf{E}[T_{i}]\in(C^{2}/3,C^{2}$). For the variance, as in the linear regime it holds ${\bf Var}[T]=n{\bf Var}[T_{1}]+n(n-1)\big{[}\mathbf{E}[T_{1}T_{2}]-\mathbf{E}[T_{1}]\mathbf{E}[T_{2}]\big{]}\,.$ We obtain in a similar way the following bound, for $n$ large enough $\mathbf{E}[T_{1}T_{2}]-\mathbf{E}[T_{1}]\mathbf{E}[T_{2}]\leq\Big{[}1-\Big{(}1-\frac{1}{n}\Big{)}^{2}+\frac{C}{\sqrt{n}}\Big{(}1-\Big{(}1-\frac{1}{n}\Big{)}^{2}\Big{)}\Big{]}\mathbf{E}[T_{2}]\leq\frac{3}{n}\times C^{2}/n\,.$ Therefore, ${\bf Var}[T]\leq 4C^{2}$, and we have, using Chebyshev’s inequality $\mathbf{P}[T\geq C^{2}/4]\leq\frac{{\bf Var}[T]}{(C^{2}/3-C^{2}/4)^{2}}\leq\frac{576}{C^{2}}\,.$ Proof [Proof of Lemma 2.2 and 2.3] For all $x\in\\{0,1\\}^{n}$, $x\in\mathcal{S}(\phi)$ if and only if $x$ satisfies all the clauses of $\phi$. We can therefore write $\displaystyle Z=\sum_{x\in\\{0,1\\}^{n}}\prod_{i=1}^{m}\mathbf{1}\\{x\in\mathcal{S}(C_{i})\\}\,.$ We recall that this yields, for $\phi$ drawn uniformly $\mathbf{E}[Z]=2^{n}(1-2^{-k})^{m}$. In the proof of Theorem 2.4, we use that $Z$ is equal to its expectation when the $km$ variables in the formula are distinct. In the linear regime, or in the square-root regime for a large enough constant, it is not the case, with high probability. The interactions between the clauses that share the same variable will create an imbalance between couples of clauses where the same variables appears with the same sign, and those where it appears with a different one. We compute the conditional expectation of $Z$, given the first variable of each clause, and whether the first two occurrences of every variable (when there are two or more) are the same literal or not. Formally, we denote $G=(G_{1},\ldots,G_{n})$ the partition of $\\{1,\ldots,m\\}$ in $n$ sets (allowing some of them to be empty), where $G_{i}=\big{\\{}j\in\\{1,\ldots,m\\}:C_{j}(x)\in\\{x_{i}\wedge\ldots,\bar{x}_{i}\wedge\ldots\\}\big{\\}}\,,$ and $\sigma=(\sigma_{1},\ldots,\sigma_{n})$, where $\sigma_{i}=0$ if there are less than two elements in $G_{i}$, $\sigma_{i}=1$ if the first two elements of $G_{i}$ have the same first literal (either both $x_{i}$ or both $\bar{x}_{i}$), and $\sigma_{i}=-1$ otherwise. By linearity of expectation, it holds $\mathbf{E}[Z\,\|\,(G,\sigma)]=\sum_{x\in\\{0,1\\}^{n}}\mathbf{E}\Big{[}\mathbf{1}\\{x\in\mathcal{S}(\phi)\\}\,\|\,(G,\sigma)\Big{]}\,.$ We now observe that this conditional expectation is constant, for all $x\in\\{0,1\\}^{n}$. Indeed, let $e_{0}$ be the assignment of all zeroes, and $t_{x}$ be the literal-flipping transformation such that $t_{x}(e_{0})=x$, and $T_{x}$ the corresponding literal-flipping transformation on formulas. For all $x$, it holds $\phi(x)=\phi(t_{x}(e_{0}))=(T_{x}\phi)(e_{0})\,.$ For all $x$, $T_{x}\phi$ also has distribution $\mathbf{P}_{\text{unif}}$, and $(G,\sigma)$ is invariant by this transformation. Therefore, it holds $\displaystyle\mathbf{E}[Z\,\|\,(G,\sigma)]$ $\displaystyle=$ $\displaystyle\sum_{x\in\\{0,1\\}^{n}}\mathbf{E}\Big{[}\mathbf{1}\\{x\in\mathcal{S}(\phi)\\}\,\|\,(G,\sigma)\Big{]}$ $\displaystyle=$ $\displaystyle\sum_{x\in\\{0,1\\}^{n}}\mathbf{E}\Big{[}\mathbf{1}\\{e_{0}\in\mathcal{S}(T_{x}\phi)\\}\,\|\,(G,\sigma)\Big{]}$ $\displaystyle=$ $\displaystyle 2^{n}\mathbf{E}\Big{[}\mathbf{1}\\{e_{0}\in\mathcal{S}(\phi)\\}\,\|\,(G,\sigma)\Big{]}\,.$ The assignment $e_{0}$ will satisfy the formula $\phi$ if and only if it satisfies all the sub-formulas $\phi_{G_{1}},\ldots,\phi_{G_{n}}$ (the empty formula is always satisfied). Given $(G,\sigma)$, the events $\\{e_{0}\in\mathcal{S}(\phi_{G_{i}})\\}$ are independent: the sub-formulas are satisfied by $e_{0}$ if and only if every clause contains at least one negated literal, which occurs independently, conditioned on $(G,\sigma)$. We can therefore compute the conditional expectation $\displaystyle\mathbf{E}\Big{[}\mathbf{1}\\{e_{0}\in\mathcal{S}(\phi)\\}\,\|\,(G,\sigma)\Big{]}$ $\displaystyle=$ $\displaystyle\mathbf{E}\Big{[}\prod_{i=1}^{n}\mathbf{1}\\{e_{0}\in\mathcal{S}(\phi_{G_{i}})\\}\,\|\,(G,\sigma)\Big{]}$ $\displaystyle=$ $\displaystyle\prod_{i=1}^{n}\mathbf{E}\Big{[}\mathbf{1}\\{e_{0}\in\mathcal{S}(\phi_{G_{i}})\\}\,\|\,(G,\sigma)\Big{]}$ $\displaystyle=$ $\displaystyle\prod_{i=1}^{n}\mathbf{E}\Big{[}\mathbf{1}\\{e_{0}\in\mathcal{S}(\phi_{G_{i}})\\}\,\|\,(G_{i},\sigma_{i})\Big{]}$ The product terms can be expressed as a function of $g_{i}=\|G_{i}\|$. If $\sigma_{i}=0$, in the case of $g_{i}<2$, treating separately the cases $g_{i}=0$ or $1$, we have $\mathbf{E}\Big{[}\mathbf{1}\\{e_{0}\in\mathcal{S}(\phi_{G_{i}})\\}\,\|\,(G_{i},\sigma_{i}=0)\Big{]}=\Big{(}1-\frac{1}{2^{k}}\Big{)}^{g_{i}}\,.$ If there are at least two elements in $G_{i}$, we have $\displaystyle\mathbf{E}\Big{[}\mathbf{1}\\{e_{0}\in\mathcal{S}(\phi_{G_{i}})\\}\,\|\,(G_{i},\sigma_{i}=1)\Big{]}=\frac{1}{2}\Big{[}1+\Big{(}1-\frac{1}{2^{k-1}}\Big{)}^{2}\Big{]}\Big{(}1-\frac{1}{2^{k}}\Big{)}^{g_{i}-2}$ $\displaystyle\mathbf{E}\Big{[}\mathbf{1}\\{e_{0}\in\mathcal{S}(\phi_{G_{i}})\\}\,\|\,(G_{i},\sigma_{i}=-1)\Big{]}=\Big{(}1-\frac{1}{2^{k-1}}\Big{)}\Big{(}1-\frac{1}{2^{k}}\Big{)}^{g_{i}-2}\,.$ Overall, this yields $\mathbf{E}\Big{[}\mathbf{1}\\{e_{0}\in\mathcal{S}(\phi_{G_{i}})\\}\,\|\,(G_{i},\sigma_{i})\Big{]}=\Big{[}1+\frac{\sigma_{i}}{2^{2k}(1-2^{-k})^{2}}\Big{]}\Big{(}1-\frac{1}{2^{k}}\Big{)}^{g_{i}}\,.$ Recall that we denote $P$ (resp. $D$) the number of groups for which $\sigma_{i}=1$ (resp. $-1$). It holds that $\mathbf{E}[Z\,\|\,(G,\sigma)]=2^{n}\Big{(}1-\frac{1}{2^{k}}\Big{)}^{m}\Big{[}1+\frac{1}{2^{2k}(1-2^{-k})^{2}}\Big{]}^{P}\Big{[}1-\frac{1}{2^{2k}(1-2^{-k})^{2}}\Big{]}^{D}\,.$ It is possible to design a set of $(G,\sigma)$, event of probability close to 1, for which this expectation has the desired value. To do so, we study the behavior of $P$ and $D$, the number of variables that appear at least twice among the first variables of the clauses, for which respectively $\sigma_{i}=1$ or $-1$. Indeed, for a large $T=P+D$, with $P$ and $D$ close to $(P+D)/2$, this expectation is significantly smaller than $\mathbf{E}[Z]$. Indeed, for all $t\in(0,1)$, the function $f_{t}:\alpha\mapsto(1+t)^{1+\alpha}(1-t)^{1-\alpha}$ is continuous and $f_{t}(0)=1-t^{2}$, so there exists $\alpha_{t}\in(0,1)$ such that $f_{t}(\alpha)<1-t^{2}/2$ for all $\|\alpha\|<\alpha_{t}$. Therefore, there exists $\alpha_{k}\in(0,1)$ such that $\Big{[}1+\frac{1}{2^{2k}(1-2^{-k})^{2}}\Big{]}^{1+\alpha}\Big{[}1-\frac{1}{2^{2k}(1-2^{-k})^{2}}\Big{]}^{1-\alpha}<1-\frac{1}{2^{4k+1}(1-2^{-k})^{4}}:=e^{-\gamma_{k}}\,,$ for all $\|\alpha\|<\alpha_{k}$, for some $\gamma_{k}>0$. For every variable, we denote $T_{i}=\|\sigma_{i}\|\in\\{0,1\\}$, and $T=T_{1}+\ldots+T_{n}$. We now prove independently the two lemmas. Linear regime, Lemma 2.3 We control $P$ and $D$ in the regime $m=\Delta n$. By lemma A.1, it holds that $\mathbf{P}[T<\varepsilon_{\Delta}n]\leq\frac{r_{\Delta}}{n}\,.$ Of these $T$ variables, between $T/2(1+\alpha_{k})$ and $T/2(1-\alpha_{k})$ will have their first two occurrences with the same literal, with probability greater than $1-e^{-\alpha_{k}^{2}\varepsilon_{\Delta}n/2}$, by Hoeffding’s inequality. We call $B$ the event $T\geq n\varepsilon_{\Delta}$ and $P\in(T/2(1-\alpha_{k}),T/2(1+\alpha_{k}))$. By the above, $\mathbf{P}(B)=1-o(1)$. For $(G,\sigma)$ in the event $B$, it holds $\displaystyle\mathbf{E}[Z\,\|\,(G,\sigma)]$ $\displaystyle=$ $\displaystyle 2^{n}\Big{(}1-\frac{1}{2^{k}}\Big{)}^{m}\Big{[}1+\frac{1}{2^{2k}(1-2^{-k})^{2}}\Big{]}^{P}\Big{[}1-\frac{1}{2^{2k}(1-2^{-k})^{2}}\Big{]}^{D}$ $\displaystyle<$ $\displaystyle 2^{n}\Big{(}1-\frac{1}{2^{k}}\Big{)}^{m}(e^{-\gamma_{k}})^{T/2}<e^{-\gamma_{k}\varepsilon_{\Delta}n/2}\mathbf{E}[Z]:=e^{-2c_{k,\Delta}n}\,\mathbf{E}[Z]\,.$ Therefore $\mathbf{E}[Z\,\|\,B]<e^{-2c_{k,\Delta}n}\,\mathbf{E}[Z]$. We can now conclude by conditioning on $B$ and using Markov’s inequality $\displaystyle\mathbf{P}(Z>e^{-c_{k,\Delta}n}\,\mathbf{E}[Z])$ $\displaystyle=$ $\displaystyle\mathbf{P}(Z>e^{-c_{k,\Delta}n}\,\mathbf{E}[Z]\,\|\,B)\mathbf{P}(B)+$ $\displaystyle\mathbf{P}(Z>e^{-c_{k,\Delta}n}\,\mathbf{E}[Z]\,\|\,B^{c})\mathbf{P}(B^{c})$ $\displaystyle\leq$ $\displaystyle\mathbf{P}(Z>e^{-c_{k,\Delta}n}\,\mathbf{E}[Z]\,\|\,B)+\mathbf{P}(B^{c})$ $\displaystyle\leq$ $\displaystyle\frac{\mathbf{E}[Z\,\|\,B]}{e^{-c_{k,\Delta}n}\,\mathbf{E}[Z]}+\mathbf{P}(B^{c})$ $\displaystyle\leq$ $\displaystyle e^{-c_{k,\Delta}n}+\mathbf{P}(B^{c})\,.$ Which yields the desired result. Square-root regime, Lemma 2.2 As in the linear regime, we control $P$ and $D$ when $m=C\sqrt{n}$. Lemma A.1 yields $\mathbf{P}[T\geq C^{2}/4]\leq\frac{576}{C^{2}}\,.$ Again, of these $T$ variables, between $T/2(1+\alpha_{k})$ and $T/2(1-\alpha_{k})$ will have their first two occurrences with the same literal, with probability greater than $1-e^{-\alpha_{k}^{2}C^{2}/8}$, by Hoeffding’s inequality. We call $B$ the event $T\geq C^{2}/4$ and $P\in(T/2(1-\alpha_{k}),T/2(1+\alpha_{k}))$. By the above, $\mathbf{P}(B)=1-O(1/C^{2})$. For $(G,\sigma)$ in the event $B$, it holds $\displaystyle\mathbf{E}[Z\,\|\,(G,\sigma)]$ $\displaystyle=$ $\displaystyle 2^{n}\Big{(}1-\frac{1}{2^{k}}\Big{)}^{m}\Big{[}1+\frac{1}{2^{2k}(1-2^{-k})^{2}}\Big{]}^{P}\Big{[}1-\frac{1}{2^{2k}(1-2^{-k})^{2}}\Big{]}^{D}$ $\displaystyle<$ $\displaystyle 2^{n}\Big{(}1-\frac{1}{2^{k}}\Big{)}^{m}(e^{-\gamma_{k}})^{T/2}<e^{-\gamma_{k}C^{2}/8}\mathbf{E}[Z]\,.$ Therefore $\mathbf{E}[Z\,\|\,B]<e^{-\gamma_{k}C^{2}/8}\,\mathbf{E}[Z]$. We can now conclude by conditioning on $B$ and using Markov’s inequality $\displaystyle\mathbf{P}(Z>e^{-\gamma_{k}C^{2}/16}\,\mathbf{E}[Z])$ $\displaystyle=$ $\displaystyle\mathbf{P}(Z>e^{-\gamma_{k}C^{2}/16}\,\mathbf{E}[Z]\,\|\,B)\mathbf{P}(B)+$ $\displaystyle\mathbf{P}(Z>e^{-c_{k,\Delta}n}\,\mathbf{E}[Z]\,\|\,B^{c})\mathbf{P}(B^{c})$ $\displaystyle\leq$ $\displaystyle\mathbf{P}(Z>e^{-\gamma_{k}C^{2}/16}\,\mathbf{E}[Z]\,\|\,B)+\mathbf{P}(B^{c})$ $\displaystyle\leq$ $\displaystyle\frac{\mathbf{E}[Z\,\|\,B]}{e^{-\gamma_{k}C^{2}/16}\,\mathbf{E}[Z]}+\mathbf{P}(B^{c})$ $\displaystyle\leq$ $\displaystyle e^{-\gamma_{k}C^{2}/8}+\mathbf{P}(B^{c})\,.$ This yields the second result, for $C$ large enough, and some absolute constant $C_{0}$. ## References * [ABBDL10] Louigi Addario-Berry, Nicolas Broutin, Luc Devroye, and Gábor Lugosi, _On combinatorial testing problems_ , Ann. Statist. 38 (2010), no. 5, 3063–3092. 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1401.2475	# $p$-Hahn Sequence Space Murat Kirişci Department of Mathematical Education, Hasan Ali Yücel Education Faculty, Istanbul University, Vefa, 34470, Fatih, Istanbul, Turkey [email protected], [email protected] ###### Abstract. The main purpose of the present paper is to introduce the space $h_{p}$ and study of some properties of new sequence space. Also we compute their dual spaces and characterizations of some matrix transformations. ###### Key words and phrases: Matrix transformations, Hahn sequence space, $BK$-space, dual spaces, Schauder basis, $AK$-property, $AD$-property ###### 2010 Mathematics Subject Classification: Primary 46A45; Secondary 46A45, 46A35. This work was supported by Scientific Projects Coordination Unit of Istanbul University. Project number 35565. ## 1\. Introduction By $\omega=\mathbb{C}^{\mathbb{N}}$, we denote the space of all real- or complex-valued sequences, where $\mathbb{C}$ denotes the complex field and $\mathbb{N}=\\{0,1,2,\ldots\\}$. Each linear subspace of $\omega$ is called a _sequence space_. For $x=(x_{k})\in\omega$, we shall employ the sequence spaces $\ell_{\infty}=\\{x:\sup_{k}\|x_{k}\|<\infty\\}$, $c=\\{x:\lim_{k}x_{k}~{}\textrm{ exists}~{}\\}$, $c_{0}=\\{x:\lim_{k}x_{k}=0\\}$, $bs=\\{x:\sup_{n}\|\sum_{k=1}^{n}x_{k}\|<\infty\\}$, $cs=\\{x:(\sum_{k=1}^{n}x_{k})\in c\\}$ and $\ell_{p}=\\{x:\sum_{k}\|x_{k}\|^{p}<\infty,\quad 1\leq p<\infty\\}$ which are Banach space with the following norms; $\\|x\\|_{\ell_{\infty}}=\sup_{k}\|x_{k}\|$, $\\|x\\|_{bs}=\\|x\\|_{cs}=\sup_{n}\|\sum_{k=1}^{n}x_{k}\|$ and $\\|x\\|_{\ell_{p}}=\left(\sum_{k}\|x_{k}\|^{p}\right)^{1/p}$ as usual, respectively. And also $\displaystyle bv^{p}$ $\displaystyle=$ $\displaystyle\left\\{x=(x_{k})\in\omega:\sum_{k=1}^{\infty}\|x_{k}-x_{k-1}\|^{p}<\infty\right\\},$ $\displaystyle\int\lambda$ $\displaystyle=$ $\displaystyle\left\\{x=(x_{k})\in\omega:(kx_{k})\in\lambda\right\\}.$ A sequence, whose $k-th$ term is $x_{k}$, is denoted by $x$ or $(x_{k})$. _A coordinate space_ (or _$K-$ space_) is a vector space of numerical sequences, where addition and scalar multiplication are defined pointwise. That is, a sequence space $\lambda$ with a linear topology is called a $K$-space provided each of the maps $p_{i}:\lambda\rightarrow\mathbb{C}$ defined by $p_{i}(x)=x_{i}$ is continuous for all $i\in\mathbb{N}$. A $BK-$space is a $K-$space, which is also a Banach space with continuous coordinate functionals $f_{k}(x)=x_{k}$, $(k=1,2,...)$.A $K-$space $\lambda$ is called an _$FK-$ space_ provided $\lambda$ is a complete linear metric space. An _$FK-$ space_ whose topology is normable is called a _$BK-$ space_.If a normed sequence space $\lambda$ contains a sequence $(b_{n})$ with the property that for every $x\in\lambda$ there is unique sequence of scalars $(\alpha_{n})$ such that $\displaystyle\lim_{n\rightarrow\infty}\\|x-(\alpha_{0}b_{0}+\alpha_{1}b_{1}+...+\alpha_{n}b_{n})\\|=0$ then $(b_{n})$ is called _Schauder basis_ (or briefly basis) for $\lambda$. The series $\sum\alpha_{k}b_{k}$ which has the sum $x$ is then called the expansion of $x$ with respect to $(b_{n})$, and written as $x=\sum\alpha_{k}b_{k}$. An _$FK-$ space_ $\lambda$ is said to have $AK$ property, if $\phi\subset\lambda$ and $\\{e^{k}\\}$ is a basis for $\lambda$, where $e^{k}$ is a sequence whose only non-zero term is a $1$ in $k^{th}$ place for each $k\in\mathbb{N}$ and $\phi=span\\{e^{k}\\}$, the set of all finitely non-zero sequences. If $\phi$ is dense in $\lambda$, then $\lambda$ is called an $AD$-space, thus $AK$ implies $AD$. Let $\lambda$ and $\mu$ be two sequence spaces, and $A=(a_{nk})$ be an infinite matrix of complex numbers $a_{nk}$, where $k,n\in\mathbb{N}$. Then, we say that $A$ defines a matrix mapping from $\lambda$ into $\mu$, and we denote it by writing $A:\lambda\rightarrow\mu$ if for every sequence $x=(x_{k})\in\lambda$. The sequence $Ax=\\{(Ax)_{n}\\}$, the $A$-transform of $x$, is in $\mu$; where (1.1) $\displaystyle(Ax)_{n}=\sum_{k}a_{nk}x_{k}~{}\textrm{ for each }~{}n\in\mathbb{N}.$ For simplicity in notation, here and in what follows, the summation without limits runs from $0$ to $\infty$. By $(\lambda:\mu)$, we denote the class of all matrices $A$ such that $A:\lambda\rightarrow\mu$. Thus, $A\in(\lambda:\mu)$ if and only if the series on the right side of (1.1) converges for each $n\in\mathbb{N}$ and each $x\in\lambda$ and we have $Ax=\\{(Ax)_{n}\\}_{n\in\mathbb{N}}\in\mu$ for all $x\in\lambda$. A sequence $x$ is said to be $A$-summable to $l$ if $Ax$ converges to $l$ which is called the $A$-limit of $x$. The matrix domain $\lambda_{A}$ of an infinite matrix $A$ in a sequence space $\lambda$ is defined by (1.2) $\displaystyle\lambda_{A}=\\{x=(x_{k})\in\omega:Ax\in\lambda\\}$ which is a sequence space(for several examples of matrix domains, see [2] p. 49-176). In [5], Başar and Altay have defined the sequence space $bv_{p}$ which consists of all sequences such that $\Delta$-transforms of them are in $\ell_{p}$ where $\Delta$ denotes the matrix $\Delta=(\delta_{nk})$ $\displaystyle\Delta=\delta_{nk}=\left\\{\begin{array}[]{ccl}(-1)^{n-k}&,&\quad(n-1\leq k\leq n)\\\ 0&,&\quad(0\leq k<n-1~{}\textrm{ or }~{}k>n)\end{array}\right.$ for all $k,n\in\mathbb{N}$. The space $[\ell(p)]_{A^{u}}=bv(u,p)$ has been studied by Başar et al. [3] where $\displaystyle A^{u}=a_{nk}^{u}=\left\\{\begin{array}[]{ccl}(-1)^{n-k}u_{k}&,&\quad(n-1\leq k\leq n)\\\ 0&,&\quad(0\leq k<n-1~{}\textrm{ or }~{}k>n)\end{array}\right.$ for all $k,n\in\mathbb{N}$. In the present paper, we introduce $p-$Hahn sequence space. We investigate its some properties and compute duals of this space and characterized some matrix transformations. We assume throughout that $p^{-1}+q^{-1}=1$ for $p,q\geq 1$. We denote the collection of all finite subsets of $\mathbb{N}$ be $\mathcal{F}$. ## 2\. New Hahn Sequence Space Hahn [7] introduced the $BK-$space $h$ of all sequences $x=(x_{k})$ such that $\displaystyle h=\left\\{x:\sum_{k=1}^{\infty}k\|\Delta x_{k}\|<\infty~{}\textrm{ and }~{}\lim_{k\rightarrow\infty}x_{k}=0\right\\},$ where $\Delta x_{k}=x_{k}-x_{k+1}$, for all $k\in\mathbb{N}$. The following norm $\displaystyle\\|x\\|_{h}=\sum_{k}k\|\Delta x_{k}\|+\sup_{k}\|x_{k}\|$ was defined on the space $h$ by Hahn [7] (and also [6]). Rao ([11], Proposition 2.1) defined a new norm on $h$ as $\\|x\\|=\sum_{k}k\|\Delta x_{k}\|.$ Goes and Goes [6] proved that the space $h$ is a $BK-$space. Hahn proved following properties of the space $h$: ###### Lemma 2.1. * (i) $h$ is a Banach space. * (ii) $h\subset\ell_{1}\cap\int c_{0}.$ * (iii) $h^{\beta}=\sigma_{\infty}.$ In [6], Goes and Goes studied functional analytic properties of the $BK-$space $bv_{0}\cap d\ell_{1}$. Additionally, Goes and Goes considered the arithmetic means of sequences in $bv_{0}$ and $bv_{0}\cap d\ell_{1}$, and used an important fact which the sequence of arithmetic means $(n^{-1}\sum_{k=1}^{n}x_{k})$ of an $x\in bv_{0}$ is a quasiconvex null sequence. And also Goes and Goes proved that $h=\ell_{1}\cap\int bv=\ell_{1}\cap\int bv_{0}$. Rao [11] studied some geometric properties of Hahn sequence space and gave the characterizations of some classes of matrix transformations. Balasubramanian and Pandiarani[1] defined the new sequence space $h(F)$ called the Hahn sequence space of fuzzy numbers and proved that $\beta-$ and $\gamma-$duals of $h(F)$ is the Cesàro space of the set of all fuzzy bounded sequences. Kirişci [8] compiled to studies on Hahn sequence space and defined a new Hahn sequence space by Cesàro mean in [9]. Now, we introduce the sequence space $h_{p}$ by $\displaystyle h_{p}=\left\\{x:\sum_{k=1}^{\infty}\left(k\|\Delta x_{k}\|\right)^{p}<\infty~{}\textrm{ and }~{}\lim_{k\rightarrow\infty}x_{k}=0\right\\}\quad\quad(1<p<\infty)$ where $\Delta x_{k}=(x_{k}-x_{k+1})$, $(k=1,2,...)$. If we take $p=1$, $h_{p}=h$ which called Hahn sequence spaces. Define the sequence $y=(y_{k})$, which will be frequently used, by the $M$-transform of a sequence $x=(x_{k})$, i.e., (2.1) $\displaystyle y_{k}=(Mx)_{k}=k(x_{k}-x_{k+1}).$ where $M=(m_{nk})$ with (2.5) $\displaystyle m_{nk}=\left\\{\begin{array}[]{ccl}n&,&\quad(n=k)\\\ -n&,&\quad(n+1=k)\\\ 0&,&\quad other\end{array}\right.$ for all $k,n\in\mathbb{N}$. ###### Theorem 2.2. $h_{p}=\ell_{p}\cap\int bv^{p}=\ell_{p}\cap\int bv_{0}^{p}$ ###### Proof. We consider $\displaystyle k\Delta x_{k}\leq x_{k}+\Delta(kx_{k}).$ Then, for $x\in\ell_{p}\cap\int bv^{p}$ $\displaystyle\sum_{k=1}^{n}k\|\Delta x_{k}\|\leq\sum_{k=1}^{n}\|x_{k}\|+\sum_{k=1}^{n}\|\Delta(kx_{k})\|$ and from $\|a+b\|^{p}\leq 2^{p}\left(\|a\|^{p}+\|b\|^{p}\right),(1\leq p<\infty)$, we obtain $\displaystyle\sum_{k=1}^{n}k^{p}\|\Delta x_{k}\|^{p}\leq 2^{p}\left[\sum_{k=1}^{n}\|x_{k}\|^{p}+\sum_{k=1}^{n}\|\Delta(kx_{k})\|^{p}\right].$ For each positive integer $r$, we get $\displaystyle\sum_{k=1}^{r}k^{p}\|\Delta x_{k}\|^{p}\leq 2^{p}\left[\sum_{k=1}^{r}\|x_{k}\|^{p}+\sum_{k=1}^{r}\|\Delta(kx_{k})\|^{p}\right].$ and as $r\rightarrow\infty$ $\displaystyle\sum_{k=1}^{\infty}k^{p}\|\Delta x_{k}\|^{p}\leq 2^{p}\left[\sum_{k=1}^{\infty}\|x_{k}\|^{p}+\sum_{k=1}^{\infty}\|\Delta(kx_{k})\|^{p}\right].$ and $\lim_{k\rightarrow\infty}x_{k}=0$. Then $x\in h_{p}$ and (2.6) $\displaystyle\ell_{p}\cap\int bv^{p}\subset h_{p}.$ Let $x\in h_{p}$ and we consider $\displaystyle\sum_{k=1}^{\infty}\|x_{k+1}\|^{p}-\sum_{k=1}^{\infty}\|\Delta(kx_{k})\|^{p}\leq\sum_{k=1}^{\infty}k^{p}\|\Delta x_{k}\|^{p}.$ The the series $\sum_{k=1}^{\infty}\|x_{k+1}\|^{p}$ is convergent from the definition of $\ell_{p}$. Also $\sum_{k=1}^{\infty}\|\Delta(kx_{k})\|^{p}<\infty$ and therefore $x\in\ell_{p}\cap\int bv^{p}$. Then (2.7) $\displaystyle h_{p}\subset\ell_{p}\cap\int bv^{p}.$ Form (2.6) and (2.7), we obtain $h_{p}=\ell_{p}\cap\int bv^{p}$. ∎ ###### Theorem 2.3. The sequence space $h_{p}$ is a $BK$-space with $AK$. ###### Proof. If $x$ is any sequence, we write $\sigma_{n}(x)=M_{n}x$. Let $\varepsilon>0$ and $x\in h_{p}$ be given. Then there exists $N$ such that (2.8) $\displaystyle\|\sigma_{n}(x)\|<\varepsilon/2$ for all $n\geq N$. Now let $m\geq N$ be given. Then we have for all $n\geq m+1$ by (2.8) $\displaystyle\left\|\sigma_{n}\left(x-x^{[m]}\right)\right\|\leq\left[\sum_{k=m+1}^{\infty}\bigg{\|}k(\Delta x_{k})\bigg{\|}^{p}\right]^{1/p}\leq\|\sigma_{n}(x)\|+\|\sigma_{m}(x)\|<\varepsilon/2+\varepsilon/2=\varepsilon$ whence $\\|x-x^{[m]}\\|_{h_{p}}\leq\varepsilon$ for all $m\geq N$. This shows $x=\lim_{m\rightarrow\infty}x^{[m]}$. ∎ Since $h_{p}$ is an $AK$-space and every $AK$-space is $AD$, we can give the following corollary: ###### Corollary 2.4. The sequence space $h_{p}$ has $AD$. ###### Theorem 2.5. Define a sequence $b^{(k)}=\big{\\{}b_{n}^{(k)}\big{\\}}_{n\in\mathbb{N}}$ of elements of the space $h_{p}$ for every fixed $k\in\mathbb{N}$ by $\displaystyle b_{n}^{(k)}=\left\\{\begin{array}[]{ccl}\frac{1}{k}&,&\quad(n\leq k)\\\ 0&,&\quad(n>k)\end{array}\right.$ Then the sequence $\bigl{\\{}b_{n}^{(k)}\bigr{\\}}_{n\in\mathbb{N}}$ is a basis for the space $h_{p}$, and any $x\in h_{p}$ has a unique representation of the form (2.10) $\displaystyle x=\sum_{k}\lambda_{k}b^{(k)}$ where $\lambda_{k}=(Mx)_{k}$ for all $k\in\mathbb{N}$ and $1\leq p<\infty$. ###### Proof. It is clear that $\\{b^{(k)}\\}\subset h_{p}$, since (2.11) $\displaystyle Mb^{(k)}=e^{k}\in\ell_{1},\quad(k=0,1,2,...).$ $1\leq p<\infty$. Let $x\in h_{p}$ be given. For every non-negative integer $m$, we put (2.12) $\displaystyle x^{[m]}=\sum_{k=0}^{m}\lambda_{k}b^{(k)}.$ Then, we obtain by applying $M$ to (2.12) with (2.11) that $\displaystyle Mx^{[m]}=\sum_{k=0}^{m}\lambda_{k}Mb^{(k)}=\sum_{k=0}^{m}(Mx)_{k}e^{k}$ and $\displaystyle\left\\{M(x-x^{[m])}\right\\}_{i}=\left\\{\begin{array}[]{ccl}0&,&\quad(0\leq i\leq m)\\\ (Mx)_{i}&,&\quad(i>m)\end{array};\quad\quad(i,m\in\mathbb{N}).\right.$ Given $\varepsilon>0$, then there is an integer $m_{0}$ such that $\displaystyle\left[\sum_{i=m}^{\infty}\|i.(\Delta x)_{i}\|^{p}\right]^{1/p}<\frac{\varepsilon}{2}$ for all $m\geq m_{0}$. Hence, $\displaystyle\\|x-x^{[m]}\\|_{h_{p}}=\left[\sum_{i=m}^{\infty}\|i.(\Delta x)_{i}\|^{p}\right]^{1/p}\leq\left[\sum_{i=m_{0}}^{\infty}\|i.(\Delta x)_{i}\|^{p}\right]^{1/p}<\frac{\varepsilon}{2}<\varepsilon$ for all $m\geq m_{0}$ which proves that $x\in h_{p}$ is represented as in (2.10). To show the uniqueness of this representation, we assume that $x=\sum_{k}\mu_{k}b^{(k)}$. Now, we define the transformation $T$ with the notation of (2.1), from $h_{p}$ to $\ell_{p}$ by $x\mapsto y=Tx$. The linearity of $T$ is clear. Since the linear transformation $T$ is continuous we have at this stage that $\displaystyle(Mx)_{n}=\sum_{k}\mu_{k}\\{Mb^{(k)}\\}_{n}=\sum_{k}\mu_{k}e_{n}^{k}=\mu_{n};\quad(n\in\mathbb{N})$ which contradicts the fact that $(Mx)_{n}=\lambda_{n}$ for all $n\in\mathbb{N}$. Hence, the representation (2.10) of $x\in h_{p}$ is unique. ∎ ###### Theorem 2.6. Except the case $p=2$, the space $h_{p}$ is not an inner product space, therefore not a Hilbert space for $1<p<\infty$. ###### Proof. For $p=2$, we will show that the space $h_{2}$ is a Hilbert space. Since the space $h_{p}$ is a $BK$-space from Theorem 2.3, the space $h_{2}$ is a $BK$-space, for $p=2$. Also its norm can be obtained from an inner product, i.e., $\\|x\\|_{h_{2}}=\langle k\Delta x,k\Delta x\rangle^{1/2}$ holds. Then the space $h_{2}$ is a Hilbert space. Now consider the sequences $e_{1}=(1,0,0,0,\cdots)$ and $e_{2}=(0,1,0,0,\cdots)$. Then we see that $\\|e_{1}+e_{2}\\|_{h_{p}}^{2}+\\|e_{1}-e_{2}\\|_{h_{p}}^{2}\neq 2.\big{(}\\|e_{1}\\|_{h_{p}}^{2}+\\|e_{2}\\|_{h_{p}}^{2}\big{)}$, i.e., the norm of the space $h_{p}$ does not satisfy the parallelogram equality, which menas that the norm cannot be obtained from inner product. Hence, the space $h_{p}$ with $p\neq 2$ is a Banach space that is not a Hilbert space. ∎ Now, we give some inclusion relations concerning with the space $h_{p}$. ###### Theorem 2.7. Neither of the spaces $h_{p}$ and $\ell_{\infty}$ includes the other one, where$1<p<\infty$. ###### Proof. Now we choose the sequences $a=(a_{k})$ and $b=(b_{k})$ such that $a=(a_{k})=\\{(-1)^{k}\\}$ and $b=(b_{k})=\sum_{i=1}^{k}1/(i+1)$. The sequence $a=(a_{k})$ is in $\ell_{\infty}\backslash h_{p}$ and the sequence $b=(b_{k})$ is in $h_{p}\backslash\ell_{\infty}$. So, the sequences $h_{p}$ and $\ell_{\infty}$ does not include each other. ∎ ###### Theorem 2.8. If $1\leq p<r$, then $h_{p}\subset h_{r}$. ###### Proof. This can be obtained by analogy with the proof of Theorem 2.6 in [5]. So, we omit the details. ∎ ## 3\. Duals of New Hahn Sequence Space In this section, we state and prove the theorems determining the $\alpha$-, $\beta$\- and $\gamma$-duals of the sequence space $h_{p}$. Let $x$ and $y$ be sequences, $X$ and $Y$ be subsets of $\omega$ and $A=(a_{nk})_{n,k=0}^{\infty}$ be an infinite matrix of complex numbers. We write $xy=(x_{k}y_{k})_{k=0}^{\infty}$, $x^{-1}Y=\\{a\in\omega:ax\in Y\\}$ and $M(X,Y)=\bigcap_{x\in X}x^{-1}Y=\\{a\in\omega:ax\in Y~{}\textrm{ for all }~{}x\in X\\}$ for the _multiplier space_ of $X$ and $Y$. In the special cases of $Y=\\{\ell_{1},cs,bs\\}$, we write $x^{\alpha}=x^{-1}\ell_{1}$, $x^{\beta}=x^{-1}cs$, $x^{\gamma}=x^{-1}bs$ and $X^{\alpha}=M(X,\ell_{1})$, $X^{\beta}=M(X,cs)$, $X^{\gamma}=M(X,bs)$ for the $\alpha-$dual, $\beta-$dual, $\gamma-$dual of $X$. By $A_{n}=(a_{nk})_{k=0}^{\infty}$ we denote the sequence in the $n-$th row of $A$, and we write $A_{n}(x)=\sum_{k=0}^{\infty}a_{nk}x_{k}$ $n=(0,1,...)$ and $A(x)=(A_{n}(x))_{n=0}^{\infty}$, provided $A_{n}\in x^{\beta}$ for all $n$. Given an $FK-$space $X$ containing $\phi$, its conjugate is denoted by $X^{\prime}$ and its $f-$dual or sequential dual is denoted by $X^{f}$ and is given by $X^{f}=\\{$ all sequences $(f(e^{k})):f\in X^{\prime}\\}$. Let $\lambda$ be a sequence space. Then $\lambda$ is called _perfect_ if $\lambda=\lambda^{\alpha\alpha}$; _normal_ if $y\in\lambda$ whenever $\|y_{k}\|\leq\|x_{k}\|,\quad k\geq 1$ for some $x\in\lambda$; monotone if $\lambda$ contains the canonical preimages of all its stepspace. ###### Lemma 3.1. (i). $A\in(h:\ell_{1})$ if and only if (3.1) $\displaystyle\sum_{n=1}^{\infty}\|a_{nk}\|~{}\textrm{ converges, }~{}(k=1,2,...)$ (3.2) $\displaystyle\sup_{k}\frac{1}{k}\sum_{n=1}^{\infty}\bigg{\|}\sum_{\upsilon=1}^{k}a_{n\upsilon}\bigg{\|}<\infty.$ * (ii). $A\in(\ell_{p}:\ell_{1})$ if and only if $\displaystyle\sup_{K\in\mathcal{F}}\sum_{k}\bigg{\|}\sum_{n\in K}a_{nk}\bigg{\|}^{q}<\infty$ ###### Lemma 3.2. * (i). $A\in(h:c)$ if and only if (3.3) $\displaystyle\sup_{n,k}\frac{1}{k}\bigg{\|}\sum_{\upsilon=1}^{k}a_{n\upsilon}\bigg{\|}<\infty$ (3.4) $\displaystyle\lim_{n\rightarrow\infty}a_{nk}~{}\textrm{ exists, }~{}(k=1,2,...)$ * (ii). $A\in(\ell_{p}:c)$ if and only if (3.4) holds and (3.5) $\displaystyle\sup_{n}\sum_{k}\big{\|}a_{nk}\big{\|}^{q}<\infty,\quad 1<p<\infty$ ###### Lemma 3.3. * (i). $A\in(h:\ell_{\infty})$ if and only if (3.3) holds. * (ii). $A\in(\ell_{p}:\ell_{\infty})$ if and only if (3.5) holds with $1<p\leq\infty$. ###### Lemma 3.4. $A\in(h:c_{0})$ if and only if (3.3) holds and (3.6) $\displaystyle\lim_{n\rightarrow\infty}a_{nk}=0$ ###### Lemma 3.5. $A\in(h:h)$ if and only if (3.6) holds and (3.7) $\displaystyle\sum_{n=1}^{\infty}n\|a_{nk}-a_{n+1,k}\|~{}\textrm{ converges, }~{}(k=1,2,...)$ (3.8) $\displaystyle\sup_{k}\frac{1}{k}\sum_{n=1}^{\infty}n\bigg{\|}\sum_{v=1}^{k}(a_{nv}-a_{n+1,v})\bigg{\|}<\infty.$ ###### Theorem 3.6. We define the sets $d_{1}$ and $d_{2}$ as follows: $\displaystyle d_{1}$ $\displaystyle=$ $\displaystyle\\{a=(a_{k})\in\omega:\sup_{K\in\mathcal{F}}\sum_{k}\left\|\sum_{n\in K}\frac{1}{k}a_{n}\right\|^{q}<\infty\\}\quad 1<p<\infty$ $\displaystyle d_{2}$ $\displaystyle=$ $\displaystyle\\{a=(a_{k})\in\omega:\sup_{K\in\mathcal{F}}\sum_{k}\left\|\sum_{n\in K}\frac{1}{k}a_{n}\right\|<\infty\\}.$ Then $[h_{p}]^{\alpha}=d_{1}$ and $[h]^{\alpha}=d_{2}$. ###### Proof. We give the proof only for the case $[h_{p}]^{\alpha}=d_{1}$. Let us take any $a=(a_{k})\in\omega$ and consider the equation (3.9) $\displaystyle a_{n}x_{n}=\sum_{j=n}^{\infty}\frac{a_{n}}{j}y_{j}=(Dy)_{n}\quad(n\in\mathbb{N})$ where $D=(d_{nk})$ is defined by $\displaystyle d_{nk}=\left\\{\begin{array}[]{ccl}\frac{a_{n}}{k}&,&\quad k\geq n\\\ 0&,&\quad k<n\end{array}\right.$ for all $k,n\in\mathbb{N}$. It follows from (3.9) with Lemma 3.1(ii) that $ax=(a_{n}x_{n})\in\ell_{1}$ whenever $x=(x_{k})\in h_{p}$ if and only if $Dy\in\ell_{1}$ whenever $y=(y_{k})\in\ell_{p}$. This means that $a=(a_{n})\in[h_{p}]^{\alpha}$ whenever $x=(x_{n})\in h_{p}$ if and only if $D\in(h_{p}:\ell_{1})$. This gives the result that $[h_{p}]^{\alpha}=d_{1}$. ∎ Hahn[7] proved that $[h]^{\beta}=\sigma_{\infty}$ where $\sigma_{\infty}=\\{a=(a_{k})\in\omega:\sup_{n}\frac{1}{n}\|\sum_{k=1}^{n}a_{k}\|<\infty\\}$. We can give $\beta$-dual of $h_{p}$. ###### Theorem 3.7. Let $1<p<\infty$. Then, $[h_{p}]^{\beta}=d_{3}$ where $\displaystyle d_{3}=\left\\{a=(a_{k})\in\omega:\sup_{n\in\mathbb{N}}(n^{-1})^{q}\sum_{k}\left\|\sum_{j=k}^{n}a_{j}\right\|^{q}<\infty\right\\}$ ###### Proof. Consider the equation (3.11) $\displaystyle\sum_{k=1}^{n}a_{k}x_{k}=\sum_{k=1}^{n}a_{k}\left(\sum_{j=k}^{n}\frac{y_{j}}{j}\right)=\sum_{k=1}^{n}\left(\sum_{j=1}^{k}\frac{a_{j}}{k}\right)y_{k}=(By)_{n}\quad\quad(n\in\mathbb{N});$ where $B=(b_{nk})$ are defined by $\displaystyle b_{nk}=\left\\{\begin{array}[]{ccl}\sum_{j=1}^{k}\frac{a_{j}}{k}&,&\quad(n\leq k)\\\ 0&,&\quad(n>k)\end{array}\right.$ for all $k,n\in\mathbb{N}$. Thus we deduce from Lemma 3.2 (ii) with (3.11) that $ax=(a_{k}x_{k})\in cs$ whenever $x=(x_{k})\in h_{p}$ if and only if $By\in c$ whenever $y=(y_{k})\in\ell_{p}$. Thus, $(a_{k})\in cs$ and $(a_{k})\in d_{3}$ by (3.4) and (3.5), respectively. Nevertheless, the inclusion $d_{3}\subset cs$ holds and, thus, we have $(a_{k})\in d_{3}$ whence $[h_{p}]^{\beta}=d_{3}$. ∎ ###### Lemma 3.8. ([12], Theorem 7.2.7) Let $X$ be an $FK-$space with $X\supset\phi$. Then, * (i) $X^{\beta}\subset X^{\gamma}\subset X^{f}$; * (ii) If $X$ has $AK$, $X^{\beta}=X^{f}$; * (iii) If $X$ has $AD$, $X^{\beta}=X^{\gamma}$. From Theorem 2.3, Corollary 2.4 and Lemma 3.8, we can write the following corollary: ###### Corollary 3.9. * (i) $[h_{p}]^{\beta}=[h_{p}]^{f}$ * (ii) $[h_{p}]^{\beta}=[h_{p}]^{\gamma}$. ###### Lemma 3.10. Let $\lambda$ be a sequence space. Then the following assertions are true: * (i) $\lambda$ is perfect $\Rightarrow$ $\lambda$ is normal $\Rightarrow$ $\lambda$ is monotone; * (ii) $\lambda$ is normal $\Rightarrow$ $\lambda^{\alpha}=\lambda^{\gamma}$; * (iii) $\lambda$ is monotone $\Rightarrow$ $\lambda^{\alpha}=\lambda^{\beta}$. Combining Theorem 3.6, Theorem 3.7 and Lemma 3.10, we can give the following corollary: ###### Corollary 3.11. The space $h_{p}$ is not monotone and so it is neither normal nor perfect. ## 4\. Matrix Transformations In this section, we characterize some matrix transformations on the space $h_{p}$. ###### Lemma 4.1. [5] Let $\lambda,\mu$ be any two sequence spaces, $A$ be an infinite matrix and $U$ a triangle matrix matrix.Then, $A\in(\lambda:\mu_{U})$ if and only if $UA\in(\lambda:\mu)$. If we define $\widetilde{a}_{nk}=n(a_{nk}-a_{n+1,k})$, then we can give following corollary from Lemma 4.1 with $U=M$ defined by (2.5): ###### Corollary 4.2. * (i) $A\in(\ell_{1}:h)$ if and only if $\displaystyle\sup_{k}\sum_{n}\left\|\widetilde{a}_{nk}\right\|<\infty$ * (ii) $A\in(c:h)=(c_{0}:h)=(\ell_{\infty}:h)$ if and only if $\displaystyle\sup_{K\in\mathcal{F}}\sum_{n}\left\|\sum_{k\in K}\widetilde{a}_{nk}\right\|<\infty$ ###### Theorem 4.3. Suppose that the entries of the infinite matrices $A=(a_{nk})$ and $E=(e_{nk})$ are connected with the relation (4.1) $\displaystyle e_{nk}=\overline{a}_{nk}$ for all $k,n\in\mathbb{N}$, where $\overline{a}_{nk}=\sum_{j=k}^{\infty}\frac{a_{nj}}{j}$ and $\mu$ be any sequence space. Then $A\in(h_{p}:\mu)$ if and only if $\\{a_{nk}\\}_{k\in\mathbb{N}}\in[h_{p}]^{\beta}$ for all $n\in\mathbb{N}$ and $E\in(h:\mu)$. ###### Proof. Let $\mu$ be any given sequence spaces. Suppose that (4.1) holds between $A=(a_{nk})$ and $E=(e_{nk})$, and take into account that the spaces $h_{p}$ and $h$ are norm isomorphic. Let $A\in(h_{p}:\mu)$ and take any $y=y_{k}\in h$. Then $EM$ exists and $\\{a_{nk}\\}_{k\in\mathbb{N}}\in[h_{p}]^{\beta}$ which yields that $\\{e_{nk}\\}_{k\in\mathbb{N}}\in\ell_{1}$ for each $n\in\mathbb{N}$. Hence, $Ey$ exists and thus $\displaystyle\sum_{k}e_{nk}y_{k}=\sum_{k}a_{nk}x_{k}$ for all $n\in\mathbb{N}$. We have that $Ey=Ax$ which leads us to the consequence $E\in(h:\mu)$. Conversely, let $\\{a_{nk}\\}_{k\in\mathbb{N}}\in d_{1}$ for all $n\in\mathbb{N}$ and $E\in(h:\mu)$ hold, and take any $x=x_{k}\in h_{p}$. Then, $Ax$ exists. Therefore, we obtain from the equality $\displaystyle\sum_{k}a_{nk}x_{k}=\sum_{k}\left[\sum_{j=k}^{\infty}\frac{a_{nj}}{j}\right]y_{k}$ for all $n\in\mathbb{N}$. Thus $Ax=Ey$ and this shows that $A\in(h_{p}:\mu)$. ∎ If we use the Corollary 4.2 and change the roles of the spaces $h_{p}$ with $\mu$ in Theorem ref4thm1, we can give following theorem: ###### Theorem 4.4. Suppose that the entries of the infinite matrices $A=(a_{nk})$ and $\widetilde{A}=(\widetilde{a}_{nk})$ are connected with the relation $\widetilde{a}_{nk}=n(a_{nk}-a_{n+1,k})$ for all $k,n\in\mathbb{N}$ and $\mu$ be any sequence space. Then $A\in(\mu:h_{p})$ if and only if and $\widetilde{A}\in(\mu:h)$. ###### Proof. Let $z=(z_{k})\in\mu$ and consider the following equality $\displaystyle\sum_{k=0}^{m}\widetilde{a}_{nk}z_{k}=\sum_{k=0}^{m}n(a_{nk}-a_{n+1,k})z_{k}\quad~{}\textrm{ for all, }~{}m,n\in\mathbb{N}$ which yields that as $m\rightarrow\infty$ that $(\widetilde{A}z)_{n}=\\{M(Az)\\}_{n}$ for all $n\in\mathbb{N}$. Therefore, one can observe from here that $Az\in h_{p}$ whenever $z\in\mu$ if and only if $\widetilde{A}z\in h$ whenever $z\in\mu$. ∎ We can give following corollaries from Lemma 3.1-3.5, Corollary 4.2, Theorem 4.3 and Theorem 4.4: ###### Corollary 4.5. * (i) $A\in(h_{p}:\ell_{\infty})$ if and only if $\\{a_{nk}\\}_{k\in\mathbb{N}}\in[h_{p}]^{\beta}$ for all $n\in\mathbb{N}$ and (4.2) $\displaystyle\sup_{k}\left(\frac{1}{k}\left\|\sum_{v=1}^{k}\overline{a}_{nv}\right\|\right)^{q}<\infty$ * (ii) $A\in(h_{p}:c)$ if and only if $\\{a_{nk}\\}_{k\in\mathbb{N}}\in[h_{p}]^{\beta}$, (4.2) holds and (4.3) $\displaystyle\lim_{n\rightarrow\infty}\overline{a}_{nk}=\alpha_{k}\quad(k\in\mathbb{N})$ * (iii) $A\in(h_{p}:c_{0})$ if and only if $\\{a_{nk}\\}_{k\in\mathbb{N}}\in[h_{p}]^{\beta}$, (4.2) holds and (4.3) holds with $\alpha_{k}=0$. * (iv) $A\in(h_{p}:\ell_{1})$ if and only if $\\{a_{nk}\\}_{k\in\mathbb{N}}\in[h_{p}]^{\beta}$ and $\displaystyle\sum_{n=1}^{\infty}\|\overline{a}_{nk}\|^{q}~{}\textrm{ converges, }~{}(k=1,2,...)$ $\displaystyle\sup_{k}\frac{1}{k^{q}}\sum_{n=1}^{\infty}\bigg{\|}\sum_{\upsilon=1}^{k}\overline{a}_{n\upsilon}\bigg{\|}^{q}<\infty.$ ###### Corollary 4.6. * (i) $A\in(\ell:h_{p})$ if and only if $\displaystyle\sup_{K\in\mathcal{F}}\sum_{k}\left\|\sum_{n\in K}\widetilde{a}_{nk}\right\|<\infty$ * (ii) $A\in(c:h_{p})=(c_{0}:h_{p})=(\ell_{\infty}:h_{p})$ if and only if $\displaystyle\sup_{K\in\mathcal{F}}\sum_{n}\left\|\sum_{k\in K}\widetilde{a}_{nk}\right\|<\infty$ ## 5\. Conclusion Hahn [7] defined the space $h$ and gave some properties. Goes and Goes [6] studied its different properties. Rao [11] introduced the Hahn sequence space and investigated some properties in Banach space theory. Kirişci [8] compiled to studies of Hahn sequence space and defined a new Hahn sequence space by Cesàro mean in [9]. In this paper, we defined the space $p-$Hahn sequence spaces and gave some properties. In section 3, we compute the duals of the space $h_{p}$ and characterize some matrix transformations related to this space, in section 4. Finally, we should note that, as a natural continuation of the present paper, one can study the paranormed Hahn sequence space. Also it can be obtained the new Hahn sequence space by using Euler mean, Riesz mean, generalized weighted mean etc. ## References * [1] T. Balasubramanian, A. Pandiarani, The Hahn sequence spaces of fuzzy numbers, Tamsui Oxf. J. Inf. Math. Sci. 27(2), (2011), 213–224. * [2] F. Başar, Summability Theory and its Applications, Bentham Science Publishers, e-books, Monographs, (2011). * [3] F. Başar, B. Altay and M. Mursaleen, Some generalizations of the space $bv_{p}$ of $p$-bounded variation sequences, Nonlinear Analysis, 68 (2008), 273–287. * [4] B. Altay and F. Başar, Certain topological properties and duals of the domain of a triangle matrix in a sequence spaces, J. Math. Analysis and Appl., 336 (2007), 632–645. * [5] F. Başar and B. Altay On the space of sequences of $p$-bounded variation and related matrix mappings, Ukranian Math. J., 55(1) (2003), 136–147. * [6] G. Goes and S., Goes, Sequences of bounded variation and sequences of Fourier coefficients I, Math. Z., 118(1970), 93–102. * [7] H. Hahn, Über Folgen linearer Operationen, Monatsh. Math., 32(1922), 3–88. * [8] M. Kirişci A survey on the Hahn sequence spaces, Gen. Math. Notes, 19(2), 2013. * [9] M. Kirişci The Hahn sequence spaces sefined by Cesàro Mean,Abstract and Applied Analysis, vol. 2013, Article ID 817659, 6 pages, 2013. doi:10.1155/2013/817659 * [10] S. A. Rakov, Banach-Saks property of a Banach space, Mat. Zametki, 26(6)(1979), 823–834. * [11] W. Chandrasekhara Rao, The Hahn sequence spaces I, Bull. Calcutta Math. Soc. 82(1990), 72–78. * [12] A. Wilansky, Summability through Functinal Analysis, North Holland, New York, (1984).	arxiv-papers	2014-01-10T22:05:51	2024-09-04T02:49:56.569600	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Murat Kiri\\c{s}ci", "submitter": "Murat Kiri\\c{s}ci", "url": "https://arxiv.org/abs/1401.2475" }
1401.2481	# Photon and dilepton production in high energy heavy ion collisions Takao Sakaguchi [email protected] Physics Department, Brookhaven National Laboratory, Upton, NY 11973-5000, USA ###### Abstract The recent results on direct photons and dileptons in high energy heavy ion collisions, obtained particularly at RHIC and LHC are reviewed. The results are new not only in terms of the probes, but also in terms of the precision. We will discuss the physics learned from the results. ###### keywords: Photons, Dileptons, RHIC, LHC, QGP ###### pacs: 25.75.-q, 25.75.Bh, 25.75.Cj, 25.75.Nq ## 1 Introduction Electromagnetic radiations are an excellent probe for extracting thermodynamical information of a matter produced in nucleus-nucleus collisions, as they are emitted from all the stages of collisions with wide $Q^{2}$, and don’t interact strongly with medium once produced [1]. They appear in two figures, namely, photons ($\gamma$) that have zero-mass and virtual photons ($\gamma^{}$) that have finite mass. Experimentalists often refer virtual photons as dileptons since they have been measuring virtual photons through lepton-pair channels ($\gamma^{}\rightarrow ee,\mu\mu$), by which a wide kinematic range in invariant mass and transverse momentum ($p_{T}$) can be explored. Photons are produced through a Compton scattering of quarks and gluons ($qg\rightarrow q\gamma$) or an annihilation of quarks and anti-quarks ($q\overline{q}\rightarrow g\gamma$) as leading-order (LO) processes, and the next-to-leading order (NLO) process is dominated by Bremsstrahlung and fragmentation ($qg\rightarrow qg\gamma$), as depicted in Figure 1. Figure 1: Production processes of direct photons. Their yields are proportional to $\alpha\alpha_{s}$, which are $\sim$40 times lower than those of hadrons that are produced in strong interactions. Virtual photons are produced by annihilation of quarks and anti-quarks. The initial hard scattering process that takes place in the beginning of the collisions produces relatively high $p_{T}$ photons, often referred as hard photons. They have same LO and NLO processes, and are called prompt photons and fragment photons, respectively. The production rate of these photons are well described by a NLO pQCD calculation [2]. Photons will be emitted from the hot and dense medium (quark-gluon plasma: QGP) in high energy nucleus collisions and manifest at moderate $p_{T}$(1$<p_{T}<$3 GeV/$c$) if the QGP is formed [3]. We often call these photons as thermal photons. The thermal photons are of interest in exploring thermodynamical nature of the QGP, such as temperature. One can also obtain the degree of freedom of the system by combining the temperature with measurement of the energy density of the system as $g\propto\epsilon/T^{4}$. For $p_{T}<$1 GeV, the photons are predominantly contributed from hadron gas state via the processes of $\pi\pi(\rho)\rightarrow\gamma\rho(\pi)$, $\pi K^{}\rightarrow K\gamma$ and etc., which are no longer the quark-gluon level interaction. We often refer these photons as hadron-gas photons. Photons from Compton scattering of hard- scattered partons and partons in the medium (jet-photon conversion), or Bremsstrahlung of the hard scattered partons in the medium have been predicted to contribute in the $p_{T}$ range of $p_{T}>$2.5 GeV/$c$ if QGP is created [4]. Figure 2 shows a landscape of photon sources as a function of formation time and $p_{T}$. Figure 2: Manifestation of photons from various sources as a function of formation time and $p_{T}$. As described above, photons have rich information on the state they are emitted. However these photons are overwhelmed by the large background coming from the decay of hadrons. $\pi^{0}\rightarrow\gamma\gamma$ and $\eta\rightarrow\gamma\gamma$ are the major contributors to the background photons ($\sim$95 %). The signal to background ratio at 1-3 GeV/$c$ is of the order $\sim$10 %. One can understand the difficulty of the measurement by comparing the uncertainty of the best $\pi^{0}$ measurement at RHIC [5] which directly relates to the precision of background determination. There is a certain probability that photons produced by the same process acquire virtual mass and decay into lepton-pairs (shown as one another degree of freedom in Figure 1(b)). This process is called internal conversion process and is different from the virtual photon production described above. We will explain these photons in detail in a later section. There have been several attempts on measuring thermal photons in high energy nuclear collisions as an evidence of QGP formation. The first sizable signal was reported by the WA98 experiments at SPS in 1$<p_{T}<$3 GeV/$c$ where a calculation predicts that QGP photons manifest [6]. At that time, the hard photons were not measured because the statistics ran out at the $p_{T}$ where the hard photons start arising. Estimating the hard photon contribution had to rely on a theoretical guidance that had large ambiguity, therefore the measurement was not able to exhibit the thermal photon contribution [3]. The source of dileptons depend on their mass and $p_{T}$ of the measurement. The low mass region ($m_{ee}<$1 GeV/$c^{2}$) at low $p_{T}$ is predominantly from the in-medium decay of $\rho$ mesons and/or thermal radiation as shown in Figure 3(a) [7]. Figure 3: (a, left) A calculation of dilepton source in low mass region. (b, right) Outline of background components and resonances in dilepton invariant mass spectra. The same mass region at higher $p_{T}$ or higher mass region at lower $p_{T}$ are from mainly from the thermal radiation. The dileptons of interest are obtained after subtracting large combinatoric background arising from Dalitz decays of $\pi^{0}$ or $\eta$ (e.g., $\pi^{0},\eta\rightarrow e^{+}e^{-}\gamma$). The outline of the background components and resonances in dilepton mass spectra is shown in Figure 3(b) [8]. In this paper, we review the recent measurements of photons and dileptons from RHIC and LHC experiments and discuss what we have learned from the data. ## 2 Hard production of photons and dileptons One of the big success in electromagnetic radiation measurements in relativistic heavy ion collisions is the observation of high $p_{T}$ direct photons that are produced in initial hard scattering [9]. Figure 4(a) and (b) show the latest direct photon $p_{T}$ spectra in Au+Au collisions at $\sqrt{s_{NN}}$=200 GeV for various centralities, and the nuclear modification factor ($R_{AA}$) for the 0-10 % centrality, respectively [10]. Figure 4: (a, left) Direct photon $p_{T}$ spectra in Au+Au collisions at $\sqrt{s_{NN}}$=200 GeV measured by the PHENIX experiment. (b, right) Nuclear modification factor ($R_{AA}$) for direct photons in 10 % central Au+Au collisions [10]. The $R_{AA}$ is consistent with unity within quoted uncertainty, implying that the photons from Au+Au collisions are consistent with ones expected from $p+p$ collisions. This result is not very trivial since photons from hard scattering include both prompt and fragment components. The fragment photons may be reduced in central Au+Au collisions due to the parton energy loss. The data is compared with several model predictions. It is interesting to note that a model that includes reduction of fragment photon due to energy loss of hard scattered partons and increase of jet-photon conversion photons (coherent + conversion +$\Delta E$ ) is not consistent with the data. The small suppression in $R_{AA}$ seen in the highest $p_{T}$ is likely due to the fact that the ratio of the yields in Au+Au to $p+p$ was computed without taking the isospin dependence of direct photon production into account [11]. The LHC heavy ion runs have the cms energy of 2.76 TeV where the hard photon production is copious. One can make a photon isolation cut to enrich the prompt photon component even in heavy ion collisions. Figure 5(a) shows the isolated photon $p_{T}$ distributions in Pb+Pb collisions at $\sqrt{s_{NN}}$=2.76 TeV measured by the ATLAS experiment at LHC, together with ones by the CMS experiments [12, 13]. The lines show the expected values from JetPHOX and PYTHIA event generators. Figure 5: (a, left) Prompt photon $p_{T}$ spectra measured in Pb+Pb collisions at $\sqrt{s_{NN}}$=2.76 TeV by the ATLAS and CMS experiments together with JETPHOX and PYTHIA simulation [12, 13]. (b, right) Yield of $Z$-bosons scaled by the nuclear thickness function ($T_{AB}$) as a function of centrality measured by the CMS experiment [15]. The prompt photon component should be unaffected by the medium created in heavy ion collisions, and was experimentally confirmed by these measurements. Considering the fact that neither prompt photons from LHC nor prompt+fragment photons from RHIC are suppressed, the contribution of photons from fragmentation and jet-photon conversion could be smaller than predicted. Several measurements benefited from the fact that these hard electromagnetic radiations are well under control. For instance, direct photons are utilized to quantify jet energy loss as they carry the initial momenta of jets [14, 12]. A new hard probe that became available at the LHC energy is $Z$-bosons. The $Z$-bosons have a peak mass of $\sim$80 GeV/$c^{2}$, and are produced in the medium with a lifetime of 0.1 fm/$c$. Therefore, they carry information on the initial states of the collisions, and decay before they are affected by the medium. The $Z$-boson yields scaled by nuclear thickness function ($T_{AB}$) as a function of $N_{part}$ (centrality) published by the CMS experiments are shown in Figure 5(b) [15]. The similar results are recently published by the ATLAS experiments [16]. The $Z$-boson yields are found to follow the scaling of the initial hard scattering process. $Z$-boson is also ideal for serving as a reference in many similar measurements, such as $Z$-jet correlations [17]. We point out that $Z$-bosons can serve as one of the most reliable tools to normalize the dilepton spectra between $p+p$ and A+A collisions. ## 3 Thermal photon production ### 3.1 Spectra In the conventional real photon measurement, single photons can be observed after a huge amount of background photons coming from hadron decays ($\pi^{0}$, $\eta$, $\eta^{\prime}$ and $\omega$, etc.) are subtracted off from the inclusive photon distributions. This fact makes it very difficult to look at the signal at low $p_{T}$, where thermal photons from QGP manifest. A breakthrough was made by utilizing internal conversion of photons [18]. Because these photons decay into $e^{+}e^{-}$, one can use measurement technique for dileptons. The relation between real photon production and the $e^{+}e^{-}$ pairs decaying from associated internal conversion photon production can be described as follow [19]: $\frac{d^{2}n_{ee}}{dm_{ee}}=\frac{2\alpha}{3\pi}\frac{1}{m_{ee}}\sqrt{1-\frac{4m_{e}^{2}}{m_{ee}^{2}}}\Bigl{(}1+\frac{2m_{e}^{2}}{m_{ee}^{2}}\Bigr{)}Sdn_{\gamma}$ where $\alpha$ is the fine structure constant, $m_{e}$ and $m_{ee}$ are the masses of the electron and the $e^{+}e^{-}$ pair respectively, and $S$ is a process dependent factor that goes to 1 when $m_{ee}\rightarrow 0$ or $m_{ee}\ll p_{T}$. This equation also applies to the relation between the photons from hadron decays (e.g. $\pi^{0}\rightarrow\gamma\gamma$) and the $e^{+}e^{-}$ pairs from Dalitz decays ($\pi^{0}\rightarrow e^{+}e^{-}\gamma$). For $\pi^{0}$ and $\eta$, the factor $S$ is given as $S=\|F(m_{ee}^{2})\|^{2}(1-m_{ee}^{2}/M_{h}^{2})^{3}$, where $M_{h}$ is the meson mass and $F(m_{ee}^{2})$ is the form factor. The analysis assumes that the form factor for direct photons is $F(m_{ee}^{2})=1$ similar to a purely point-like process. We can select the higher $e+e-$ invariant mass region where $\pi^{0}$ contribution becomes off. This will eliminate the large background coming from $\pi^{0}$ Dalitz decay. Figure 6 shows the $e^{+}e^{-}$ invariant mass distribution in minimum bias Au+Au collisions for 1.0$<p_{T}<$1.5 GeV/$c$ measured by the PHENIX experiment [18]. Figure 6: $e^{+}e^{-}$ invariant mass distribution at 1.0$<p_{T}<$1.5 GeV/$c$ together with the cocktail calculation of known sources and direct photon internal conversion [18]. The $e^{+}e^{-}$ mass spectra were fit with the function that have terms of the cocktail calculation of known sources ($e^{+}e^{-}$ from various hadron Dalitz decays) and the direct photon internal conversion: $f(m_{ee})=(1-r)f_{c}(m_{ee})+r~{}f_{\rm dir}(m_{ee})$ where $f_{c}(m_{ee})$ is the shape of the cocktail mass distribution, $f_{\rm dir}(m_{ee})$ is the expected shape of the direct photon internal conversion. One can obtain the signal to background ratio for a given mass window ($r$ in the plot) at a given $p_{T}$. Using the Kroll-Wada formula [20], $r$ is associated with the ratio at zero-mass and thus is converted to the ratio of direct to inclusive photons: $r=\frac{\gamma^{}_{\rm dir}(m_{ee}>0.15)}{\gamma^{}_{\rm inc}(m_{ee}>0.15)}\propto\ \frac{\gamma^{}_{\rm dir}(m_{ee}\approx 0)}{\gamma^{}_{\rm inc}(m_{ee}\approx 0)}\ =\frac{\gamma_{\rm dir}}{\gamma_{\rm inc}}\equiv r_{\gamma}$ Finally, the direct photon $p_{T}$ spectra is calculated as $\gamma_{\rm inc}\times r_{\gamma}$. Figure 7(a) shows the direct photon $p_{T}$ spectra in Au+Au and $p+p$ collisions as obtained by this procedure. Figure 7: (a, left) PHENIX results on invariant yields of direct photons in Au+Au collisions at various centralities together with that in $p+p$ collisions. Lines show the $N_{coll}$-scaled $p+p$ fit functions with exponential functions [18]. (b, right) ALICE results on invariant yields of direct photons in 2.76 TeV Pb+Pb collisions [21] The spectra were fit with $N_{coll}$-scaled $p+p$ fit function with exponential function, and the slope parameters of $\sim$220 MeV, which is almost independent of centrality, were obtained. The ALICE experiment recently measured real photons using external conversion technique, namely, by looking at the conversion of real photons into $e^{+}e^{-}$ in the material of the inner detectors. Figure 7(b) shows the direct photon $p_{T}$ spectrum measured by the ALICE experiment in 0-40 % Pb+Pb collisions at $\sqrt{s_{NN}}$=2.76 TeV, together with the exponential fit to the low $p_{T}$ region [21]. The ratio of the slope parameter from PHENIX and ALICE measurements is 1.38. From the published result on the Bjorken energy density at Pb+Pb top centrality from the CMS experiment [22] and one at Au+Au from the PHENIX experiment [23], one can find that the ratio of Bjorken energy density of the LHC Pb+Pb collisions ($\sim$14 GeV/fm3) to that of RHIC Au+Au ($\sim$5.7 GeV/fm3) is 2.6, which is smaller than the one expected the ratio of slope parameters (1.384 = 3.65). This is because the photons measured experimentally are a sum of the photons from all the stages from the initial to the final state of collisions and their slope parameters reflect ”average” temperature, while the energy density is measured at the thermalization. In order to obtain the temperature at all the stages as well as the initial energy density, one has to run a hydrodynamical simulation that has a realistic time profile of the system. One might ask if the excess of photons over the initial hard scattering process is due to cold nuclear matter effect (CNM) such as $p_{T}$-broadening. PHENIX has recently measured the direct photon production in $d$+Au collisions to quantify the CNM effect, using the same internal conversion technique applied to Au+Au collisions. Figure 8 shows the $R_{AA}$ of the direct photons in $d$+Au and Au+Au collisions along with a model on the CNM and parton energy loss effect [24]. It was found that the CNM effect to the direct photon production is negligible compared to the large excess seen in Au+Au collisions. Figure 8: PHENIX results of direct photon $R_{AA}$ in $d$+Au and Au+Au collisions at $\sqrt{s_{NN}}$=200 GeV [24]. ### 3.2 Flow After the system reached to local equilibrium, the system proceeds to hydrodynamic expansion and its interaction with hard scattered partons, which results in the anisotropic emission of particles. The magnitude of the anisotropy can be studied from the azimuthal angle distribution of particles relative to the second order event plane angle ($v_{2}$, called as elliptic flow). For particles with low $p_{T}$ ($p_{T}<3$ GeV/$c$), the $v_{2}$ are understood in terms of pressure-gradient anisotropy in an initial ”almond- shaped” collision zone produced in non-central collisions. Recently, a large $v_{2}$ of particles and its scaling in terms of kinetic energy have been found for identified charged hadrons at RHIC [25]. It suggested that the system is locally in equilibrium as early as 0.4 fm/$c$, and the flow occurs at the partonic level [26]. There are predictions that photons also have a collective motion and their $v_{2}$ show different signs and/or magnitudes depending on the production processes [27, 28, 29]. The observable is useful to disentangle the contributions from various photon sources in the $p_{T}$ region where they intermix. The photons from hadron-gas interaction and thermal radiation may follow the collective expansion of a system, and give a positive $v_{2}$. Those photons produced by jet-photon conversion or in-medium Bremsstrahlung will increase as the size of the medium to traverse increases and result in a negative $v_{2}$. The fragment photons will give positive $v_{2}$ since larger energy loss of jets is expected in the orthogonal direction to the event plane. PHENIX has measured the $v_{2}$ of direct photons by subtracting the $v_{2}$ of hadron decay photons off from that of the inclusive photons, following the formula below: ${v_{2}}^{dir.}=(R\times{v_{2}}^{incl.}-{v_{2}}^{bkgd.})/(R-1),\ \ \ R=(\gamma/\pi^{0})_{meas}/(\gamma/\pi^{0})_{bkgd}$ Here, $R$ is obtained either from internal conversion or external conversion method, and $v_{2}^{incl.}$ is obtained from real photons or their external conversions. Figure 9(a) shows the direct photon $v_{2}$ in minimum bias Au+Au collisions at $\sqrt{s_{NN}}$=200 GeV measured by the PHENIX experiment using both internal conversion and external conversion technique [30, 31]. Figure 9: (a, left) Direct photon elliptic flow ($v_{2}$) in minimum bias 200 GeV Au+Au collisions measured by the PHENIX experiment, using internal conversion (marked as arXiv:1105.4126) and external conversion technique [30, 31]. (b, right) Direct photon $v_{2}$ in 0-40 % 2.76 TeV Pb+Pb collisions measured by the ALICE experiment using external conversion technique [32]. The $v_{2}$ of direct photons is sizable and positive, and comparable to the flow of hadrons for $p_{T}<$3 GeV/$c$. The ALICE experiments also obtained the direct photon $v_{2}$ in 0-40% Pb+Pb collisions, using external conversion technique recently [32]. Although the energy density and the temperature of the two systems are very different, the $v_{2}$ at LHC is surprisingly similar to what PHENIX has found at RHIC. There are many models that tried explaining the RHIC result. Several models predicted the positive flow of the photons assuming the photons are boosted with hydrodynamic expansion of the system, but the magnitudes from these models are significantly lower than the measurement [33]. There are two models that give relatively large flows. Figure 10: Direct photon $v_{2}$ from a theoretical model increasing hadron- gas interaction contribution [34]. (a, left) $v_{2}$ from the model calculation, and (b, right) corresponding $p_{T}$ spectra. Figure 10 shows a model calculation with increasing photon contribution from hadron-gas interaction [34]. Since the hadrons have a large positive flow as we observed, the photons produced by the interaction with these hadrons result in a large flow. The models (a) and (b) correspond to two ways of incorporating hard photon contribution, namely, a pQCD parametrization and a fit to PHENIX p+p data, respectively. The spectra in low $p_{T}$ (1$<p_{T}<$3 GeV/$c$) region in this model, where QGP photons are said to dominate, is overwhelmed by the hadron-gas interaction, and QGP contribution is hardly seen. Figure 11 shows a model calculation for direct photon higher order flow ($v_{n}$) for two initial conditions, Glauber-based (MCGlb) and CGC-based (MCKLN) conditions, and the corresponding $p_{T}$ spectrum for the MCGlb case [35, 36]. Figure 11: Direct photon $v_{2}$ from a theoretical model with including initial state fluctuation and shear viscosity [35, 36]. (a, left) $v_{2}$ from the model calculation, and (b, right) corresponding $p_{T}$ spectra for MCGlb case. The shear viscosity is increased from $\eta/s$=0.08 to 0.20 when switching from MCGlb to MCKLN, in order that the model still describes the flow of hadrons. The dashed and sold lines are before and after viscous corrections are applied on the rates. The reason that this model gives higher values of $v_{n}$ for MCKLN is that the rate of QGP photons are reduced in order to compensate the viscous entropy production. For a reference, a recent 3+1D hydrodynamic calculation with a new CGC-inspired initial state (IP-Glasma) gives a good description of $v_{n}$ for charged hadrons at RHIC with shear viscosity of $\eta/s$=0.08 [37]. This implies that the determination of $\eta/s$ is significantly affected by initial conditions. Both models effectively assume a reduction of QGP photons and call for the hadron-gas photon contribution. Further development at the theory side to explain the data is clearly deserved. An another model tries to explain the large flow by the interaction of photons with the strong magnetic field existing in the non- central collisions [38]. The measurement of triangular flow ($v_{3}$) may be useful to discriminate the models; for instance, the strong magnetic field scenario gives $v_{3}\sim 0$ while hydrodynamical expansion scenario gives sizable positive $v_{3}$ [39]. ## 4 Low mass dileptons The PHENIX experiment has performed the first measurement of the $e^{+}e^{-}$ invariant mass spectra in $p+p$ and Au+Au collisions at $\sqrt{s_{NN}}$=200 GeV at RHIC as shown in Figure 12 [19]. Figure 12: (a, left) $e^{+}e^{-}$ mass spectra in minimum bias 200 GeV Au+Au collisions measured by the PHENIX experiments together with cocktail calculation of known sources. (b, right) Centrality dependence of $e^{+}e^{-}$ mass spectra and corresponding cocktail calculations for the same dataset [19]. Figure 13: (a, left) $e^{+}e^{-}$ mass spectra in various centralities in 200 GeV Au+Au collisions measured by the STAR experiment. (b, right) Ratios of data to cocktail calculation for each centralities [40]. Figure 14: (a, left) $e^{+}e^{-}$ mass spectra in 20-40 % Au+Au collisions measured by the PHENIX experiment in 2010, with a hadron blind detector (HBD) installed. (b, right) Comparison of the ratio of the integrated yields in the LMR to the cocktail calculations in 2004 and 2010 dataset [31]. The left panel shows the minimum bias mass spectra with hadronic cocktail components. The right panel shows the centrality dependence of mass spectra with cocktail calculations. The $p+p$ results are well reproduced by the cocktail of known sources of $e^{+}e^{-}$, whereas the Au+Au data show a strong enhancement at low mass region (LMR, $0.15\\!<M_{ee}\\!<\\!0.75$ GeV/$c^{2}$) compared to cocktail calculations. The enhancement for the 0-10 % central collisions is a factor of $7.6\pm 0.5^{stat}\pm 1.3^{syst}\pm 1.5^{model}$ and that for minimum bias collisions is $4.7\pm 0.4^{stat}\pm 1.5^{syst}\pm 0.9^{model}$, respectively. The STAR experiment also recently obtained $e^{+}e^{-}$ mass spectra in $p+p$ and Au+Au collisions at $\sqrt{s_{NN}}=200$ GeV as shown in Figure 13 [40, 41]. In the same LMR, the enhancement for 0-10 % central collisions is $1.72\pm 0.10\pm 0.50$, and that for minimum bias collisions is $1.53\pm 0.07\pm 0.41$, respectively. Clearly, there is a discrepancy in the magnitudes of the excess by a factor of 3-4 between two experiments. PHENIX has installed a hadron blind detector (HBD) in 2010 run in order to reduce systematic uncertainty of the measurement and also to confirm the previous result. The HBD is a Cherenkov detector with CF4 gas and rejects the $e^{+}e^{-}$ tracks from photon conversions and Dalitz decay of $\pi^{0}$’s, which are major background in LMR measurement, by looking at the opening angle of pair tracks in a magnetic field free region [42]. Figure 14(a) shows the $e^{+}e^{-}$ mass spectra measured in 20-40 % Au+Au collisions in the 2010 run with the HBD [31]. The analysis for the most central collisions are still in progress. PHENIX measurements of 2004 and 2010 have several differences in both data (magnetic field and detector material budget), and cocktail calculations (MC@NLO is used for charm contribution estimate in 2010). PHENIX has made a comparison of the data/cocktail ratio for the integrated yield in LMR obtained in the 2004 and 2010 runs for the three centrality bins as shown in Figure 14(b). It is seen that the two runs give consistent results within uncertainties. It should be noted that the previous PHENIX data showed most of the excess is seen in most central events (0-20 %), whose analysis is still in progress. The discrepancy between PHENIX and STAR results persists until PHENIX comes up with the new result with HBD. As for the STAR data, some concern is put on the fact that the cocktail calculation in $p+p$ collisions over-predicts data points in the LMR, though they are still consistent within systematic errors. If one correct for the over- prediction, it is possible STAR and PHENIX see the same amount of excess. Further effort to understand the discrepancy is deserved. As measured for photons, the measurement of dilepton flow is useful. There is a radial flow measurement of dileptons by the NA60 experiment [43], and an elliptic flow measurement by the STAR experiment [44, 45] that could help disentangling the source of the dileptons that contribute to this particular mass region. A theory study is also in progress [46]. ### 4.1 Energy dependence of LMR dilepton production Figure 15: $e^{+}e^{-}$ mass spectra measured by the STAR experiment in 19.6, 62.4 and 200 GeV Au+Au collisions, together with cocktail calculation and a theoretical model assuming in-medium modification of $\rho$ spectral function [40]. The excess in LMR has been explained by various models including in-medium broadening of the $\rho$ mesons or their mass shift [47]. The measurement of the energy dependence of the excess may provide a discrimination of these models. The STAR experiment recently came up with the $e^{+}e^{-}$ invariant mass spectra for minimum bias Au+Au collisions at $\sqrt{s_{NN}}=$ 19.6, 62.4, and 200 GeV as shown in Figure 15 [40]. STAR observed a qualitatively similar excess as observed by the CERES measurements in the Pb$+$Au at $\sqrt{s_{\mathrm{NN}}}=$17.2 GeV [48]. In each of the three panels the hadron cocktail simulation includes contributions from Dalitz decays, photon conversions (19.6 GeV only), and the dielectron decay of the $\omega$ and $\phi$ vector mesons. The cocktail simulations purposely exclude contributions from $\rho$ mesons. Instead, these are explicitly included in the model calculation by Rapp [47] which involve in-medium modifications of the $\rho$ meson spectral shape in the isentropic fireball evolution. The LMR enhancement measured by STAR are consistently agreeing with these model calculations within the quoted errors. One should be careful on taking the absolute magnitude of the excess in all energies, provided that there is still the issue of inconsistency between STAR and PHENIX. Nonetheless, it is interesting to see that the relative change as a function cms energy is well described by this model calculation. If it is the case, the flow of this mass region may exhibit the $KE_{T}$ scaling with assuming they are all $\rho$. ## 5 Summary The recent results on direct photons and dileptons in high energy heavy ion collisions, obtained particularly at RHIC and LHC are reviewed. The results are new not only in terms of the probes, but also in terms of the precision. Much progress has been made in understanding high $p_{T}$ direct photons as well as $Z$-bosons with the latest RHIC and LHC results. 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C 41, 475 (2005).	arxiv-papers	2014-01-10T23:30:11	2024-09-04T02:49:56.577193	{ "license": "Public Domain", "authors": "Takao Sakaguchi", "submitter": "Takao Sakaguchi", "url": "https://arxiv.org/abs/1401.2481" }
1401.2517	[table]font=small # The Semantic Similarity Ensemble111This article extends work presented at the 6th International Workshop on Semantics and Conceptual Issues in Geographical Information Systems (SeCoGIS 2012) [7] Andrea Ballatore School of Computer Science and Informatics, University College Dublin, Ireland Michela Bertolotto School of Computer Science and Informatics, University College Dublin, Ireland David C. Wilson Department of Software and Information Systems, University of North Carolina, USA ###### Abstract Computational measures of semantic similarity between geographic terms provide valuable support across geographic information retrieval, data mining, and information integration. To date, a wide variety of approaches to geo-semantic similarity have been devised. A judgement of similarity is not intrinsically right or wrong, but obtains a certain degree of cognitive plausibility, depending on how closely it mimics human behaviour. Thus selecting the most appropriate measure for a specific task is a significant challenge. To address this issue, we make an analogy between computational similarity measures and soliciting domain expert opinions, which incorporate a subjective set of beliefs, perceptions, hypotheses, and epistemic biases. Following this analogy, we define the _semantic similarity ensemble (SSE)_ as a composition of different similarity measures, acting as a panel of experts having to reach a decision on the semantic similarity of a set of geographic terms. The approach is evaluated in comparison to human judgements, and results indicate that an SSE performs better than the average of its parts. Although the best member tends to outperform the ensemble, all ensembles outperform the average performance of each ensemble’s member. Hence, in contexts where the best measure is unknown, the ensemble provides a more cognitively plausible approach. ###### keywords: Semantic similarity ensemble; SSE; Lexical similarity; Semantic similarity; Ensemble modelling; Geo-semantics; Expert disagreement; WordNet ## 1 Introduction The importance of semantic similarity in geographical information science (GIScience) is widely acknowledged [21]. As diverse information communities generate increasingly large and complex geo-datasets, semantics play an essential role to constrain the meaning of the terms being defined. The automatic assessment of the semantic similarity of terms, such as _river_ and _stream_ , enables practical applications in data mining, geographic information retrieval, and information integration. Research in natural language processing and computational linguistics has produced a wide variety of approaches, classifiable as knowledge-based (structural similarity is computed in expert-authored ontologies), corpus-based (similarity is extracted from statistical patterns in large text corpora), or hybrid (combining knowledge and corpus-based approaches) [29, 32]. Several similarity techniques have been tailored specifically to geographic information [35]. In general, a judgement on semantic similarity is not simply right or wrong, but rather shows a certain degree of cognitive plausibility, i.e. a correlation with human behaviour. Hence, selecting the most appropriate measure for a specific task is non-trivial, and represents in itself a challenge. From this perspective, a semantic similarity measure bears resemblance with a human expert being summoned to give her opinion on a complex semantic problem. In domains such as medicine and economic policy, critical choices have be made in uncertain, complex scenarios. However, disagreement among experts occurs very often, and equally credible and trustworthy experts can hold divergent opinions about a given problem [25]. To overcome decisional deadlocks, an effective solution consists of combining diverse opinions into a representative average. Instead of identifying a supposedly ‘best’ expert in a domain, an opinion is gathered from a panel of experts, extracting a representative average from their diverging opinions [10]. Similarly, complex computational problems in machine learning are often tackled with _ensemble methods_ , which achieve higher accuracy by combining of heterogeneous models, regressors, or classifiers [34]. This idea was first explored in our previous work under the analogy of the _similarity jury_ [7]. Rather than developing a new measure for geo-semantic similarity, we explore the idea of combining existing measures into a _semantic similarity ensemble (SSE)_. In order to gain insight about the merits and limitations of the SSE, we conducted a large empirical evaluation, selecting ten WordNet-based similarity measures as a case study. The ten measures were combined into all of the possible 1,012 ensembles, exploring the entire combinatorial space. To measure the cognitive plausibility of each measure and ensemble, a set of 50 geographic term pairs including 97 unique terms, selected from OpenStreetMap and ranked by 203 human subjects, was adopted as ground truth. The results of this evaluation confirms that, in absence of knowledge about the performance of the similarity measures, the ensemble approach tends to provide more cognitively plausible results than any individual measure. The remainder of this paper is organised as follows. Section 2 reviews relevant related work in the areas of geo-semantic similarity and ensemble methods. Section 3 describes the WordNet-based similarity measures selected as a case study. The SSE is defined in Section 4, while Section 5 presents and discusses the empirical evaluation. Finally, Section 6 draws conclusions about the SSE, and indicates directions for future work. ## 2 Related work The ability to assess similarity between stimuli is considered a central characteristic of human psychology. Hence, it should not come as a surprise that semantic similarity is widely studied in psychology, cognitive science, and natural language processing. Over the past ten years, a scientific literature on the semantic similarity has emerged in the context of GIScience [17, 6, 3]. Schwering [35] surveyed and classified semantic similarity techniques for geographic terms, including network-based, set-theoretical, and geometric approaches. Notably, Rodríguez and Egenhofer [33] have developed the Matching-Distance Similarity Measure (MDSM) by extending Tversky’s set- theoretical similarity for geographic terms. In the area of the Semantic Web, SIM-DL is a semantic similarity measure for spatial terms expressed in description logic (DL) [16]. As these measures are tailored to specific formalisms and data, we selected WordNet-based measures as a more generic case study (see Section 3). A key element in this article is the combination of different semantic similarity measures, relying on the analogy between computable measures and domain experts. The idea of combining divergent opinions is not new. Indeed, expert disagreement is not an exceptional state of affairs, but rather the norm in human activities characterised by uncertainty, complexity, and trade- offs between multiple criteria [25]. As Mumpower and Stewart [26] put it, the “character and fallibilities of the human judgement process itself lead to persistent disagreements even among competent, honest, and disinterested experts” (p. 191). From a psychological perspective, in cases of high uncertainty and risk (e.g. choosing medical treatments and long term investments), decision makers consult multiple experts, and try to obtain a representative average of divergent expert judgements [10]. In the context of risk analysis, mathematical and behavioural models have been devised to elicit judgements from experts, suggesting that simple mathematical methods such as the average perform quite well [11]. The underlying intuition has been controversially labelled as ‘wisdom of crowds,’ and can account for the success of some crowdsourcing applications [37]. In complex domains such as econometrics, genetics, and meteorology, _ensemble methods_ aggregate different models of the same phenomenon, trying to overcome the limitations of each model. In the context of machine learning, a wide variety of ensemble methods have been devised and evaluated [34]. Such methods aim at generating a single classifier from a set of classifiers applied to the same problem, maximising its overall accuracy and robustness [27]. Similarly, clustering ensembles obtain a single partitioning of a set of objects by aggregating several partitionings returned by different clustering techniques [36]. In computational biology, ensemble approaches are currently being used to compute the similarity of proteins [19]. Forecasting complex phenomena can also benefit from ensemble methods. Armstrong [2] pointed out that “combining forecasts is especially useful when you are uncertain about the situation, uncertain about which method is most accurate, and when you want to avoid large errors” (p. 417). Notably, a study of the Blue Chip Economic Indicators survey indicates that forecasts issued by a panel of seventy economists tended to outperform all the seventy individual forecasts [9]. To date, we are not aware of studies that explores systematically the possibility of combining semantic similarity measures through an ensemble method. The next section describes in detail the similarity measures that we selected as a case study. ## 3 WordNet similarity measures In this study, we selected WordNet-based semantic similarity measures as a case study for our ensemble technique, the semantic similarity ensemble (SSE). In the context of natural language processing, WordNet [13] is a well-known knowledge base for the computation of semantic similarity. Numerous knowledge- based approaches exploit its deep taxonomic structure for nouns and verbs [22, 32, 23, 38, 8]. From a geo-semantic viewpoint, WordNet terms have been mapped to OpenStreetMap [5]. Table 1 summarises the salient characteristics of ten popular WordNet-based measures. In order to compute the similarity scores, each measure adopts a different strategy. Seven measures relies on the _shortest path_ between terms in the noun/verb taxonomy, assuming that the number of edges is inversely proportional to the similarity of terms. This approach is limited by the variability in the path lengths in the different semantic areas of WordNet, determined by arbitrary choices and biases of the knowledge base’s owners. Paths in dense, well-developed parts of the taxonomy tend to be longer than those in shallow, sparse areas, making the direct comparison of term pairs from different areas problematic. Missing edges between terms make the score drop to $0$. To overcome these limitations, three measures include the _information content_ of the two terms and that of the _least-common subsumer_ , i.e. the more specific term that is an ancestor to both target terms [e.g. 32]. Hence, at the same path length, terms with a very specific subsumer (‘building’) are considered to be more similar than terms with a generic subsumer (‘thing’). Although this approach mitigates the issues of the shortest paths, a new issue lies in the extraction of the information content from a text corpus. Text corpora tend to be biased towards specific semantic fields, underestimating the specificity of terms contained in those fields, resulting in skewed similarity scores. An alternative approach that do not rely on taxonomy paths consists of comparing the term _glosses_ , i.e. the lexical definition of terms. Definitions can be compared in terms of word overlap (terms that are defined with the same words tend to be similar), or with co-occurrence patterns in a text corpus (terms that are defined with co-occurring words tend to be similar) [28]. The results of this approach are sensitive to noise in the definitions (e.g. very frequent or rare words that skew the scores), and to the arbitrary nature of definitions, which can under- or over-specified. Empirical research suggests that the performance of these measures largely depends on the specific ground-truth dataset utilised in the evaluation [24]. Therefore, these measures constitute a striking example of alternative models of the same phenomenon, none of which can be considered to be uncontroversially better than the others. Each measure is sensitive to specific biases in the knowledge base, and tends to reflect these biases in the similarity scores. For this reason, we consider these measures to be a suitable case study for the ensemble approach, formally defined in the next section. Name \| Reference \| Description \| SPath \| Gloss \| InfoC ---\|---\|---\|---\|---\|--- path \| Rada _et al._ [30] \| Edge count in the semantic network \| $\surd$ \| \| lch \| Leacock and Chodorow [22] \| Edge count scaled by depth \| $\surd$ \| \| res \| Resnik [32] \| Information content of $lcs$ \| $\surd$ \| \| $\surd$ jcn \| Jiang and Conrath [18] \| Information content of $lcs$ and terms \| $\surd$ \| \| $\surd$ lin \| Lin [23] \| Ratio of information content of $lcs$ and terms \| $\surd$ \| \| $\surd$ wup \| Wu and Palmer [38] \| Edge count between $lcs$ and terms \| $\surd$ \| \| hso \| Hirst and St-Onge [15] \| Paths in lexical chains \| $\surd$ \| \| lesk \| Banerjee and Pedersen [8] \| Extended gloss overlap \| \| $\surd$ \| vector \| Patwardhan and Pedersen [28] \| Second order co-occurrence vectors \| \| $\surd$ \| vectorp \| Patwardhan and Pedersen [28] \| Pairwise second order co-occurrence vectors \| \| $\surd$ \| Table 1: WordNet-based similarity measures. _SPath_ : shortest path; _Gloss_ : lexical definitions (glosses); _InfoC_ : information content; _lcs_ : least common subsumer. ## 4 The semantic similarity ensemble (SSE) A computable measure of semantic similarity can be seen as a human domain expert summoned to rank pairs of terms, according to her subjective set of beliefs, perceptions, hypotheses, and epistemic biases. When the performance of an expert can be compared against a gold standard, it is a reasonable policy to trust the expert showing the best performance. Unfortunately, such gold standards are difficult to construct and validate, and the choice of most appropriate expert remains highly problematic in many contexts. To overcome this issue, we propose the _semantic similarity ensemble (SSE)_ , a technique to combine different semantic similarity measures on the same set of terms. This ensemble of measures can be intuitively seen as a jury or a panel of human experts deliberating on a complex case [7]. Formally, the similarity function $sim$ quantifies the semantic similarity of a pair of geographic terms $t_{\\_}a$ and $t_{\\_}b$ ($sim(t_{\\_}a,t_{\\_}b)\in[0,1]$). Set $P$ contains all term pairs whose similarity needs to be assessed, while set $M$ contains a set of selected semantic similarity measures from which the ensembles will be formed: $\displaystyle P=\\{\langle t_{\\_}{a1}t_{\\_}{b1}\rangle,\langle t_{\\_}{a2}t_{\\_}{b2}\rangle~{}\ldots~{}\langle t_{\\_}{an}t_{\\_}{bn}\rangle\\}$ (1) $\displaystyle M=\\{sim_{\\_}{1},sim_{\\_}{2}~{}\ldots~{}sim_{\\_}{m}\\}$ A measure $sim$ from $M$ applied to $P$ maps the set of pairs to a set of scores $S_{\\_}{sc}$, which can then be converted into rankings $S_{\\_}{rk}$, from the most similar (e.g. _stream_ and _river_) to the least similar (e.g. _stream_ and _restaurant_): $\displaystyle sim(P)\rightarrow S_{\\_}{sc}=\\{s_{\\_}1,s_{\\_}2\ldots s_{\\_}n\\}~{}~{}~{}~{}s\in\mathbb{R}_{\\_}{\geq 0}$ (2) $\displaystyle rank(S_{\\_}{sc})\rightarrow S_{\\_}{rk}=\\{r_{\\_}1,r_{\\_}2\ldots r_{\\_}n\\}$ For example, a measure $sim\in M$ applied to a set of three pairs $P$ might return $S_{\\_}{sc}=\\{.45,.13,.91\\}$, corresponding to rankings $S_{\\_}{rk}=\\{2,3,1\\}$. The rankings $S_{\\_}{rk}(P)$ can be used to assess the cognitive plausibility of $sim$ against a human-generated rankings $H(P)$. The cognitive plausibility of $sim$ can be estimated with the Spearman’s correlation $\rho\in[-1,1]$ between $S_{\\_}{rk}(P)$ and $H_{\\_}{rk}(P)$. If $\rho$ is close to 1 or -1, $sim$ is highly plausible, while if $\rho$ is close to 0, $sim$ shows no correlation with human behaviour. In this context, a _semantic similarity ensemble (SSE)_ is defined as a set $E$ of unique semantic similarity measures: $\displaystyle E=\\{sim_{\\_}{1},sim_{\\_}{2}~{}\ldots~{}sim_{\\_}{k}\\},~{}~{}\forall j\in\\{1,2\ldots k\\}:sim_{\\_}j\in M$ (3) $\displaystyle\forall i\in\\{1,2\ldots\|M\|-1\\}:~{}sim_{\\_}i\neq sim_{\\_}{i+1},~{}~{}~{}~{}~{}~{}~{}k\leq m,\|E\|\leq\|M\|$ For example, considering the ten measures in Table 1, ensemble $E_{\\_}a$ has two members $\\{jcn,lesk\\}$, while ensemble $E_{\\_}b$ has three members $\\{jcn,res,wup\\}$. Several techniques have been discussed to aggregate rankings, using either unsupervised or supervised methods. Clemen and Winkler [11] stated that simple mathematical methods, such as the average, tend to perform quite well to combine expert judgements in risk assessment. Hence, we define two aggregation approaches $A$ to compute the rankings of ensemble $E$: 1. 1. Mean of the similarity scores: $A_{\\_}{s}=rank(mean(S_{\\_}{sc1},S_{\\_}{sc2}\ldots S_{\\_}{scn}))$ 2. 2. Mean of the similarity rankings: $A_{\\_}{r}=rank(mean(S_{\\_}{rk1},S_{\\_}{rk2}\ldots S_{\\_}{rkn}))$ The first approach, $A_{\\_}{s}$, combines directly the similarity scores, while the second approach flattens the scores into equidistant rankings. Rankings contain less information than scores: for example, scores $\\{.01,.02,.98,.99\\}$ and $\\{.51,.52,.53,.54\\}$ have very different distributions, but result in the same rankings $\\{1,2,3,4\\}$. For this reason, in some cases, $A_{\\_}s\neq A_{\\_}r$. If two measures on five term pairs generate the scores $S_{\\_}{sc1}=\\{.9,.9,.38,.44,.31\\}$ and $S_{\\_}{sc2}=\\{.28,.47,.14,.61,.36\\}$, the resulting $A_{\\_}s$ is $\\{4,5,1,3,2\\}$, whilst $A_{\\_}r$ is $\\{3,5,1,4,2\\}$. A given similarity measure has a cognitive plausibility, i.e. the ability to approximate human judgement. A traditional approach to quantify the cognitive plausibility of a measure consists of comparing rankings against a human- generated ground truth [14]. The ranked similarity scores are compared with the rankings or ratings returned by human subjects on the same set of term pairs. Following this approach, we define $\rho_{\\_}{sim}$ as the correlation of an individual measure $sim$ (i.e. an ensemble of size one) with human- generated rankings $H_{\\_}{rk}$, while $\rho_{\\_}{E}$ the correlation of the judgement obtained from an ensemble $E$. When knowledge of $\rho_{\\_}{sim}$ is available for the current task, the optimal $sim\in M$ can be simply the $sim$ having highest $\rho_{\\_}{sim}$. However, in real settings this knowledge is often absent, or incomplete, or unreliable. The same semantic similarity measure can obtain considerably different degrees of cognitive plausibility based on the specific dataset in consideration. In such contexts of limited information, the SSE offers a viable alternative to an arbitrary selection of a $sim$ from $M$. The empirical evidence discussed in the next section supports this claim. ## 5 Evaluation This section discusses an empirical evaluation conducted on the SSE in real settings. The purpose of this evaluation is to assess the performance of the SSE in detail, highlighting strengths and weaknesses. Ten semantic similarity measures are tested on a set of pairs of geographic terms utilised in OpenStreetMap. A preliminary evaluation of an analogous technique on a small scale was conducted in [7]. Ensembles of cardinalities 2,3, and 4 were generated from eight similarity measures, for a total of 154 ensembles. The evaluation described below is conducted on a larger scale, adopting a larger set of geographic terms, ranked by 203 human subjects as ground truth. To obtain a complete picture of ensemble’s performance, the entire combinatorial space is considered, for a total of 1,012 unique ensembles. The remainder of this section outlines the evaluation criteria by which the performance of the SSE is assessed (Section 5.1), the human-generated ground truth (Section 5.2), the experiment set-up (Section 5.3), and the empirical results obtained, including a comparison with the preliminary evaluation (Section 5.4). ### 5.1 Evaluation criteria The performance of an ensemble $E$ is measured on its cognitive plausibility $\rho_{\\_}{E}$, with respect to the plausibility of its individual members $\rho_{\\_}{sim}$. Intuitively, an ensemble succeeds when it provides rankings that are more cognitively plausible than those of its members. Four criteria are formally defined in this evaluation: * $-$ Total success. The plausibility of the ensemble is strictly greater than all of its members: $\forall sim\in E:\rho_{\\_}{E}>\rho_{\\_}{sim}$ * $-$ Partial success. The plausibility of the ensemble is strictly greater than a member: $\exists sim\in E:\rho_{\\_}{E}>\rho_{\\_}{sim}$ * $-$ Success over mean. The plausibility of the ensemble is strictly greater than the mean plausibility of its members: $\rho_{\\_}{E}>mean(\rho_{\\_}{sim_{\\_}1},\rho_{\\_}{sim_{\\_}2}\ldots\rho_{\\_}{sim_{\\_}n})$ * $-$ Success over median. The plausibility of the ensemble is strictly greater than the median plausibility of its members: $\rho_{\\_}{E}>median(\rho_{\\_}{sim_{\\_}1},\rho_{\\_}{sim_{\\_}2}\ldots\rho_{\\_}{sim_{\\_}n})$ ### 5.2 Ground truth In order to assess the cognitive plausibility of the similarity measures and the ensembles, a human-generated ground truth has to be selected. In the preliminary evaluation described, a human-generated set of similarity rankings was extracted from an existing dataset [7]. That dataset contains similarity rankings of 50 term pairs, over on 29 geographic terms, originally collected by Rodríguez and Egenhofer [33], and is available online.222http://github.com/ucd-spatial/Datasets In order to provide a thorough assessment of the SSE in the present article, a new and larger human- generated dataset was adopted as ground truth. As part of a wider study on geo-semantic similarity, we selected 50 pairs of geographic terms commonly used in OpenStreetMap, including 97 man-made and natural features. The terms were subsequently mapped to the corresponding terms in WordNet, as exemplified in Table 2. A Web-based survey was subsequently prepared on the set of 50 term pairs, asking human subjects to rate the pairs’ similarity on a five-point Likert scale, from _very dissimilar_ to _very similar_. In order to be understandable by any native speaker of English, regardless of knowledge of the geographic domain, the survey only included common and non-technical terms, aiming to collect a generic set of geo-semantic judgements. The survey was disseminated online through mailing lists, and obtained valid responses from 203 human subjects. The subjects’ ratings for each pair were normalised on a $[0,1]$ interval and averaged, obtaining human-generated similarity scores $H_{\\_}{sc}$, then ranked as $H_{\\_}{rk}$. Table 3 outlines a sample of term pairs, with the similarity score and ranking assigned by the 203 human subjects. This dataset was utilised as ground truth in the experiment outlined in the next section. Term \| OpenStreetMap tag \| WordNet synset ---\|---\|--- bay \| natural=bay \| bay#n#1 canal \| waterway=canal \| canal#n#3 city \| place=city \| city#n#1 post box \| amenity=post_box \| postbox#n#1 floodplain \| natural=floodplain \| floodplain#n#1 historic castle \| historic=castle \| castle#n#2 motel \| tourism=motel \| motel#n#1 supermarket \| shop=supermarket \| supermarket#n#1 … \| … \| … Table 2: Sample of the 97 terms extracted from OpenStreetMap and mapped to WordNet. Term A \| Term B \| $~{}~{}H_{\\_}{sc}~{}~{}$ \| $~{}~{}H_{\\_}{rk}~{}~{}$ ---\|---\|---\|--- motel \| hotel \| .90 \| 1 public transport station \| railway platform \| .81 \| 2 stadium \| athletics track \| .76 \| 3 theatre \| cinema \| .87 \| 4 art shop \| art gallery \| .75 \| 5 … \| … \| … \| … water ski facility \| office furniture shop \| .05 \| 46 greengrocer \| aqueduct \| .03 \| 47 interior decoration shop \| tomb \| .05 \| 48 political boundary \| women’s clothes shop \| .02 \| 49 nursing home \| continent \| .02 \| 50 Table 3: Human-generated similarity scores ($H_{\\_}{sc}$) and rankings ($H_{\\_}{rk}$) on 50 term pairs ### 5.3 Experiment setup To explore the performance of an SSE versus individual measures, we selected a set of ten WordNet-based similarity measures as a case study. Table 4 summarises the resources involved in this experiment. The ten similarity measures were not applied directly to the term pairs, but they were applied to the their lexical definitions, using a paraphrase-detection technique [4].333The _WordNet::Similarity_ tool [29] was used to compute the similarity scores. 10 similarity measures $sim\in M$: \| \| $\\{jcn,lch,hso,lesk,lin,path,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$ ---\|---\|--- \| \| $~{}~{}res,vector,vectorp,wup\\}$ (see Table 1) 9 ensemble cardinalities $\|E\|$: \| \| $\\{2,3,4,5,6,7,8,9,10\\}$ Number of unique ensembles $E$: \| \| {$45,120,$ $210,$ $252,$ $210,$ $120,45,9,1$}; Total: 1,012 2 types of ensembles $E$: \| \| ensemble of scores $E_{\\_}s$ and ensemble of rankings $E_{\\_}r$ Ground truth: \| \| 50 term pairs ranked by \| \| 203 human subjects by semantic similarity 4 evaluation criteria: \| \| (a) total success; (b) partial success; \| \| (c) success over mean; (d) success over median Table 4: Experiment setup In order to explore the space of all the possible ensembles, we considered the entire range of ensemble sizes $\|E\|\in\\{2,3\ldots 10\\}$ for $M$. The entire power set of $M$ was computed. Increasing the ensemble cardinality from 2 to 10, respectively $45,120,$ $210,$ $252,$ $210,$ $120,45,9,1$ ensembles were generated, for a total 1,012 ensembles. The experiment was carried out through the following steps: 1. 1. Compute $S_{\\_}{sc}$ and $S_{\\_}{rk}$ for each of the ten measures on the 50 term pairs from OpenStreetMap. 2. 2. Generate 1,012 ensembles, combining the measures on either similarity scores ($E_{\\_}s$) or rankings ($E_{\\_}r$). 3. 3. For each of the ten measures, compute the cognitive plausibility $\rho_{\\_}{sim}$ against human-generated rankings $H_{\\_}{rk}$. 4. 4. For each of the 1,012 ensembles, compute the cognitive plausibility $\rho_{\\_}E$ against $H_{\\_}{rk}$. 5. 5. Compute the four evaluation criteria (total success, partial success, success over mean, success over median) for each measure and ensemble. \| $\|E\|$ \| vector \| lch \| path \| hso \| wup \| vecp \| res \| lesk \| jcn \| lin \| mean ---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|--- $\rho_{\\_}{sim}$ \| $-$ \| .737 \| .727 \| .727 \| .708 \| .663 \| .641 \| .635 \| .628 \| .588 \| .562 \| .662 Total \| 2 \| 33.3 \| 22.2 \| 22.2 \| 11.1 \| 22.2 \| 33.3 \| 22.2 \| 55.6 \| 11.1 \| 11.1 \| 24.4 success \| 3 \| 27.8 \| 22.2 \| 27.8 \| 27.8 \| 36.1 \| 19.4 \| 36.1 \| 41.7 \| 11.1 \| 8.3 \| 25.8 (%) \| 4 \| 11.9 \| 17.9 \| 16.7 \| 17.9 \| 22.6 \| 10.7 \| 23.8 \| 20.2 \| 6.0 \| 4.8 \| 15.2 \| 5 \| 8.7 \| 11.1 \| 13.5 \| 11.9 \| 12.7 \| 7.1 \| 16.7 \| 12.7 \| 2.4 \| 2.4 \| 9.9 \| 6 \| 8.7 \| 8.7 \| 7.9 \| 9.5 \| 6.3 \| 4.0 \| 9.5 \| 6.3 \| 0.0 \| 0.8 \| 6.2 \| 7 \| 3.6 \| 4.8 \| 4.8 \| 4.8 \| 3.6 \| 3.6 \| 4.8 \| 3.6 \| 0.0 \| 0.0 \| 3.3 \| 8 \| 2.8 \| 2.8 \| 2.8 \| 2.8 \| 2.8 \| 0.0 \| 2.8 \| 2.8 \| 0.0 \| 2.8 \| 2.2 \| 9 \| 0.0 \| 0.0 \| 0.0 \| 0.0 \| 0.0 \| 0.0 \| 0.0 \| 0.0 \| 0.0 \| 0.0 \| 0.0 \| 10 \| 0.0 \| 0.0 \| 0.0 \| 0.0 \| 0.0 \| 0.0 \| 0.0 \| 0.0 \| 0.0 \| 0.0 \| 0.0 Mean \| All \| 10.8 \| 10.0 \| 10.6 \| 9.5 \| 11.8 \| 8.7 \| 12.9 \| 15.9 \| 3.4 \| 3.4 \| 9.7 Partial \| 2 \| 33.3 \| 22.2 \| 22.2 \| 33.3 \| 66.7 \| 77.8 \| 77.8 \| 100.0 \| 88.9 \| 100.0 \| 62.2 success \| 3 \| 27.8 \| 25.0 \| 30.6 \| 58.3 \| 86.1 \| 94.4 \| 97.2 \| 97.2 \| 100.0 \| 100.0 \| 71.7 (%) \| 4 \| 11.9 \| 26.2 \| 23.8 \| 50.0 \| 95.2 \| 97.6 \| 98.8 \| 100.0 \| 100.0 \| 100.0 \| 70.3 \| 5 \| 8.7 \| 19.8 \| 22.2 \| 58.7 \| 95.2 \| 100.0 \| 100.0 \| 100.0 \| 100.0 \| 100.0 \| 70.5 \| 6 \| 8.7 \| 18.3 \| 17.5 \| 56.3 \| 98.4 \| 100.0 \| 100.0 \| 100.0 \| 100.0 \| 100.0 \| 69.9 \| 7 \| 3.6 \| 19.0 \| 19.0 \| 67.9 \| 100.0 \| 100.0 \| 100.0 \| 100.0 \| 100.0 \| 100.0 \| 71.0 \| 8 \| 2.8 \| 11.1 \| 11.1 \| 63.9 \| 100.0 \| 100.0 \| 100.0 \| 100.0 \| 100.0 \| 100.0 \| 68.9 \| 9 \| 0.0 \| 22.2 \| 22.2 \| 55.6 \| 100.0 \| 100.0 \| 100.0 \| 100.0 \| 100.0 \| 100.0 \| 70.0 \| 10 \| 0.0 \| 0.0 \| 0.0 \| 100.0 \| 100.0 \| 100.0 \| 100.0 \| 100.0 \| 100.0 \| 100.0 \| 70.0 Mean \| All \| 10.8 \| 18.2 \| 18.7 \| 60.4 \| 93.5 \| 96.6 \| 97.1 \| 99.7 \| 98.8 \| 100.0 \| 69.4 Succ. mean \| All \| 100.0 \| 100.0 \| 100.0 \| 100.0 \| 100.0 \| 100.0 \| 100.0 \| 100.0 \| 100.0 \| 100.0 \| 100.0 Succ. med. \| All \| 96.6 \| 95.6 \| 96.2 \| 97.9 \| 99.7 \| 98.9 \| 99.9 \| 99.7 \| 97.8 \| 97.1 \| 98.0 Table 5: Overall results of the experiment, including cognitive plausibility $\rho_{\\_}{sim}$, and the four evaluation criteria. _Succ. mean_ : success over mean; _Succ. med._ : success over median. Figure 1: The four evaluation criteria w.r.t. ensemble cardinality ### 5.4 Experiment results The experiment was carried out on two types of ensemble, once with $A_{\\_}s$ (mean of scores), and once with $A_{\\_}r$ (mean of rankings). These two approaches obtained very close results, with a slightly better performance for $A_{\\_}r$, with each evaluation criterion always within a $5\%$ distance from $A_{\\_}s$. To avoid repetition, only cases with $A_{\\_}r$ are included in the discussion. All the cognitive plausibility correlations obtained statistically significant results at $p<.01$. The experiment results are summarised in Table 5, showing the cognitive plausibility of each measure, and the four evaluation criteria across all the ensemble cardinalities. For example, the ensembles of cardinality $2$ containing measure $wup$ obtains partial success in $86.1\%$ of the cases. The cognitive plausibility of the ten measures are in range $\rho\in[.562,.737]$, where $vector$ is the best measure, and $lin$ the worst. Whilst total and partial success change considerably and are fully reported, the success over mean and median obtain homogenous results and only the means are included in the table. The general trends followed by the evaluation criteria are depicted in Figure 1. Figure 2: Ensemble total and partial success w.r.t. similarity measures $sim$ #### Total success. The total success for the 1,012 ensembles falls in interval $[0,55.6]$ percent, with a mean of $9.7\%$. On average, small cardinalities (2 and 3) obtain the best total success rate ($\approx 25\%$). As the cardinality increases, the total success decreases rapidly, dropping below $10\%$ with cardinality greater than 4. This makes sense intuitively, as the larger the ensemble, the less likely the ensemble can outperform every single member. The total success varies across the different measures too, falling in interval $[3.4,15.9]$. No statistically significant correlation exists between a measure’s cognitive plausibility and its rate of total success. In other words, ensembles containing the best measures do not necessarily have better or worse total success rate. Although ensembles do not tend to outperform all of their members, the plausibility of an ensemble is never lower than that of all of its members, $~{}\exists sim\in E:\rho_{\\_}{E}>\rho_{\\_}{sim}$. #### Partial success. Partial success rate is considerably greater than that of total success. Over the entire space of ensembles, the partial success rate varies widely between $0\%$ and $100\%$, with an global average of $\approx 70\%$. The ensembles’ cardinality has no clear impact on the mean partial success rate, which remains in the interval $[62.2,71.7]$ both with small and large ensembles. Unlike total success, partial success rate is affected by each measure’s cognitive plausibility $\rho_{\\_}{sim}$. The top measures in $M$ ($vector$, $lch$, and $path$) obtain low partial success ($<20\%$), whereas ensembles consistently outperform the bottom measures ($100\%$). The average partial success rates bear strong inverse correlation with the measures’ plausibility, i.e. $\rho=-.87~{}(p<.05)$. Ensembles tend to outperform the worst measures, and tend to be outperformed by the top measures. The total and partial success of each measure is displayed in Figure 2. We note that the three top measures do not benefit from being aggregated within the ensemble, whereas all the others do. While in this experiment a ground truth is given, in many real-world settings the best measures are unknown, and therefore the SSE constitutes a viable alternative to the arbitrary selection of a measure. In particular, ensembles of cardinality 3 obtain optimal results over other cardinalities. #### Success over mean and median. Unlike total and partial success, the success of ensembles over the mean and median of their members’ plausibilities is consistent. All 1,012 ensembles obtain higher plausibility than the mean of their members’ plausibilities ($100\%$). Similarly, $98\%$ of the ensembles are more plausible than the median of their members’ plausibilities. Hence, an ensemble is more than the mean (or the median) of its parts. In order to quantify more precisely the advantage of the ensembles over the mean of their members’ plausibilities, we computed the difference between the ensemble’s plausibility $\rho_{\\_}{E}$ and the mean (or median) of all the $\rho_{\\_}{sim}$, where $sim\in E$. On average, the ensembles’ plausibility is $.042$ higher than the mean of their members ($+4.2\%$), and $.046$ over the median ($+4.6\%$). Figure 3 depicts the advantage of the ensemble in terms of cognitive plausibility over mean and median, with respect with the cardinality of the ensemble. The advantage is directly proportional to the ensemble’s size, i.e. the larger the ensemble, the larger the improvement over mean and median. Figure 3: Improvement in cognitive plausibility of the ensembles over the mean and median of their members’ plausibility In other words, by combining the rankings, the ensemble reduces the weight of individual bias, converging towards a shared judgement. Such shared judgement is not necessarily the best fit in absolute terms, but tends to be more reliable than most individual judgements. #### Comparison with preliminary experiment. To further assess the SSE, the empirical evidence described above can be compared with the preliminary evaluation we conducted in [7], discussing their commonalities and differences. That evaluation included only eight of the ten WordNet-based similarity measures, on ensembles of cardinality 2, 3, and 4, called _similarity juries_. These measures and ensembles were compared against an existing similarity dataset, originally collected by Rodríguez and Egenhofer [33]. The salient characteristics of the two evaluations are summarised in Table 6. The comparison of the two evaluations reveals that the same general trends are observable across the board. The total success of the current evaluation appears to be lower than in the preliminary evaluation, and this is because the current evaluation includes larger ensembles, which tend to have lower total success than the small ensembles of cardinality smaller than 5. On average, the partial success rates are very similar in both evaluations ($\approx 70\%$). The success over mean is very high in both evaluations, consistently falling between from $93\%$ to $100\%$. Although the mean plausibility of the measures is consistent across the two evaluations, the relative performances of the individual measures vary widely. Notably, the measure $jcn$ is the most plausible measure in the preliminary evaluation, while being the second-last in the current evaluation. Similarly, $vector$ is the top measure in the current evaluation, and ranks among the worst in the preliminary evaluation. By contrast, $lch$, $wup$, and $lesk$ maintain almost the same relative position in terms of cognitive plausibility. The two sets of plausibilities do not show any statistically significant correlation (Spearman’s $\rho\approx.1$). Although the measures fall within a similar range in both evaluations, it is difficult to identify measures that are always optimal or inadequate. These results confirm the difficulty of identifying optimal semantic similarity measures, suggesting that the SSE offers a way to proceed in a context of limited and uncertain information. Input and output \| Preliminary \| Current ---\|---\|--- parameters \| evaluation \| evaluation Ground truth: geographic terms \| 29 \| 97 Ground truth: term pairs \| 50 \| 50 Ground truth: human subjects \| 72 \| 203 Similarity measures \| 8 \| 10 Similarity ensembles \| 154 \| 1,012 Cardinalities \| $\\{2,3,4\\}$ \| $\\{2,3\ldots 10\\}$ Measures’ plausibility (mean $\rho$) \| $.62$ \| $.66$ Measures’ plausibility (range $\rho$) \| $[.45,.72]$ \| $[.56,.74]$ Total success (range $\%$) \| $[28.6,46.1]$ \| $[0,55.6]$ Total success (mean $\%$) \| $34.8$ \| $9.7$ Partial success (range $\%$) \| $[55,87.2]$ \| $[0,100]$ Partial success (mean $\%$) \| $73.3$ \| $69.4$ Success over mean (mean $\%$) \| $93.2$ \| $100$ Table 6: Comparison between the preliminary evaluation in [7] and the evaluation in this article. ## 6 Conclusions In this paper we have outlined, formalised, and evaluated the _semantic similarity ensemble (SSE)_ , a combination technique for semantic similarity measures. In the SSE, a computational measure of semantic similarity is seen as a human expert giving a judgement on the similarity of two given pairs. Like human experts, similarity measures often disagree, and it is often difficult to identify unequivocally the best measure for a given context. The ensemble approach is inspired by findings in risk management, machine learning, biology, and econometrics, which indicate that analyses that aggregate expert opinions from different experts tend to outperform analyses from single experts [11, 2, 34]. Based on empirical results collected on WordNet-based similarity measures in the context of geographic terms, the following conclusions can be drawn: * $-$ An ensemble $E$, whose members are semantic similarity measures, is generally less cognitively plausible than the best of its members, i.e. $max(\rho_{\\_}{sim})>\rho_{\\_}E$. In $\approx 9\%$ of cases, the ensemble obtains total success, i.e. it outperforms the most plausible measure. The larger the ensemble, the less frequently the ensemble outperforms its best member. * $-$ On average, similarity ensembles $E$ tend to be more cognitively plausible than any of its individual measures $sim$ in isolation (mean of partial success ratio $\approx 70\%$). In our evaluation, ensembles with 3 members are the most successful. * $-$ The SSE confirms what Cooke and Goossens [12] pointed out in the context of risk assessment: “a group of experts tends to perform better than the average solitary expert, but the best individual in the group often outperforms the group as a whole” (p. 644). * $-$ In the vast majority of cases ($\geq 98\%$), the cognitive plausibility of an SSE is higher than the mean and median of its members’ plausibilities. An ensemble is more plausible than the mean (or median) of its parts. These results are overall consistent with a preliminary evaluation [7]. * $-$ Individual similarity measures obtain widely different cognitive plausibility on different ground truths and contexts. In a context of limited information in which the optimal measure is unknown, we believe that the SSE should be favoured over any individual similarity measure. Several issues should be considered for future work. This study focused exclusively on ten WordNet-based similarity measures and, to gather more empirical evidence, the ensemble approach should be extended to different similarity measures. Moreover, to aggregate the similarity scores, we have adopted two simple ensemble methods (the mean of scores and the mean of rankings). More sophisticated ensemble techniques based on machine learning could be explored to increase the ensemble’s performance [31]. Furthermore, the empirical evidence presented in this paper was limited to the geographic context. General-purpose semantic similarity datasets, such as that devised by Agirre _et al._ [1], could be used to further evaluate the ensemble across various semantic domains. The evaluation utilised in this study is based on ranking comparison, which allows to quantify the cognitive plausibility of semantic similarity measure directly. Although this approach is the most popular in the literature, it has several drawbacks, as extensively discussed by Ferrara and Tasso [14]. Alternatively, task-based evaluations could be used to assess the cognitive plausibility of measures indirectly by observing their ability to support a specific task. Suitable tasks in geographic information retrieval and natural language processing, such as geographic query expansion, could be devised and deployed to evaluate the SSE further. In this study, similarity is modelled as a continuous score, but it can also be represented as a set of discrete classes. More importantly, the evaluation discussed in this article focuses on _acontextual_ judgements of similarity of geographic terms. Context, however, has been identified as a crucial component of similarity [20], and the SSE should extended to capture specific facets of the observed terms. The effectiveness of the ensemble should be assessed when observing either the affordances, the size or the physical structure of geospatial entities. The importance of semantic similarity measures in information retrieval, natural language processing, and data mining can hardly be underestimated [17, 21]. In this article, we have shown that a scientific contribution can be given not only by devising new similarity measures, but also by studying the combination of existing measures. The SSE provides a general approach to obtain more cognitively plausible results in settings where the ground truth is unstable and shifting. #### Acknowledgements. 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Wilson", "submitter": "Andrea Ballatore", "url": "https://arxiv.org/abs/1401.2517" }
1401.2521	# Limits of random trees††thanks: MSC2010 Subject Classification: 05C80 Attila Deák [email protected] MTA-ELTE ”Numerical Analysis and Large Networks” Research Group, 1117 Budapest, Pázmány P. s. 1/c., Hungary ###### Abstract Local convergence of bounded degree graphs was introduced by Benjamini and Schramm [2]. This result was extended further by Lyons [6] to bounded average degree graphs. In this paper, we study the convergence of a random tree sequence $(T_{n})$, where the probability of a given tree $T$ is proportional to $\prod_{v_{i}\in V(T)}d(v_{i})!$. We show that this sequence is convergent and describe the limit object, which is a random infinite rooted tree. Keywords: sparse graph limits, random trees, degree sequence ††Research was supported by the ERC Grant Nr.: 227701††The final publication is available at http://link.springer.com/article/10.1007%2Fs10474-013-0321-0 ## 1 Introduction Limits of graph sequences with bounded degree have been studied extensively over the last decade. A natural extension is to study the case when we only require bounded average degree. For instance trees have average degree less than $2$, but in general a sequence of trees can have unbounded maximum degree. A limit theory for trees has been established by Aldous [1] and also by Elek and Tardos [5]. We do not follow their path, but use the limit theory for bounded average degree graphs, described by Lyons [6]. Define $T_{n}$ as the random tree on the nodes $\\{1,2,\cdots,n\\}$ so that for a given tree $T$ with degrees $d_{i}$ we have $\mathds{P}(T_{n}=T)={\prod_{i=1}^{n}d_{i}!\over(n-2)!{3n-3\choose n-2}}.$ We will show that $T_{n}$ converges and has a limit, a random infinite rooted tree. Let ${\cal A}_{n}$ denote the set of trees with $n$ nodes. For motivation consider the following process on ${\cal A}_{n}$: * • Choose a random edge and also one of its endpoints uniformly $(X,V_{old})$. * • Take a uniformly chosen neighbor of $V_{old}$: $V_{new}$. * • If $X=V_{new}$, then do nothing, or else remove the edge $(X,V_{old})$ and add a new edge $(X,V_{new})$. This clearly defines a Markov chain $(A_{t}^{n})$ on ${\cal A}_{n}$. Let $\Pi_{n}$ denote the stationary distribution of the process defined above. It is easy to prove that the Markov chain defined above is reversible and aperiodic. From the reversibility we can also compute the stationary distribution: $\Pi_{n}(T)=\frac{\prod_{i=1}^{n}d_{i}!}{C},$ where $C=(n-2)!{3n-3\choose n-2}$, see Remark 1 below. The distribution of $T_{n}$ depends only on the degrees of its vertices and not on the exact structure of the tree itself. The natural question is: Considering only trees, can the degree sequence alone determine the convergence of a tree sequence? We show that with the above distribution the random tree sequence converges. We do not know how much this result can be extended by defining other distributions, which ensure random local convergence. Our paper is organized as follows. In Section 2 we give the basic definitions and state the main theorem of this paper. Then in Section 3 we prove basic bounds and asymptotics for the degrees of $T_{n}$, such as the expected number of degree $d$ vertices and the expected value of the maximum degree. In Section 4 we investigate subgraph densities and in Section 5 we describe the limit object. ## 2 Basic definitions For a finite simple graph $G$, let $B_{G}(v,R)$ be the rooted $R$-ball around the node $v$, that is the subgraph induced by the nodes at distance at most $R$ from $v$. Given a positive integer $R$, a finite rooted graph and a probability distribution $\rho$ on rooted graphs, let $p(R,F,\rho)$ denote the probability that the graph $F$ is rooted isomorphic to the $R$-ball around the root of a rooted graph chosen with distribution $\rho$. For a finite graph $G$, let $U(G)$ denote the distribution of rooted graphs obtained by choosing a uniform random node of $G$ as the root. ###### Definition 1. Let $(G_{n})$ be a sequence of random finite graphs, $\rho$ a probability distribution on rooted graphs. We say that the random local limit of $G_{n}$ is $\rho$, if for any positive integer $R$ and finite rooted graph $F$, we have $\lim_{n\rightarrow\infty}\mathds{P}(\|p(R,F,U(G_{n}))-p(R,F,\rho)\|>\epsilon)=0.$ ###### Theorem 1. Let $X_{n}$ be a random tree from the distribution $\Pi_{n}$. $X_{n}$ has a random local limit, which is an infinite rooted random tree. Let $T$ be a tree on $n$ nodes and denote by $X_{n}$ a random tree with distribution $\Pi_{n}$. We know that $\mathds{P}(X_{n}=T)={1\over C}\prod_{i=1}^{n}d_{i}!,$ where $d_{i}$ is the degree of the $i$th vertex of the tree $T$. It is easy to see that there are ${n-2\choose d_{1}-1,d_{2}-1,\cdots,d_{n}-1}$ trees that realize the same degree sequence. From this it follows that for a given degree sequence $(d_{i})_{i=1}^{n}$, we get $\mathds{P}((d_{i})_{i=1}^{n})=\frac{1}{C}\prod_{i=1}^{n}d_{i}!{n-2\choose d_{1}-1,d_{2}-1,\cdots,d_{n}-1}=\\\ =\frac{1}{C}\prod_{i=1}^{n}d_{i}!\frac{(n-2)!}{\prod_{i=1}^{n}(d_{i}-1)!}=\frac{(n-2)!}{C}\prod_{i=1}^{n}d_{i}.$ (1) To be able to compute probabilities about the degree sequence, we need to calculate the sum of the product of possible degree sequences. The following lemma states an easy result about this. ###### Lemma 1. $\sum_{\begin{subarray}{c}\sum_{i=1}^{n}d_{i}=m\\\ d_{i}>0\end{subarray}}\prod_{i=1}^{n}d_{i}={n+m-1\choose m-n}.$ Proof. Let $f(x)=\left(\sum_{i=1}^{\infty}ix^{i}\right)^{n}$ and denote the sum in the lemma by $M$. It is easy to see that $M$ is the coefficient of $x^{m}$ in: $f(x)=\left(\sum_{i=1}^{\infty}ix^{i}\right)^{n}=\left({x\over(1-x)^{2}}\right)^{n}={x^{n}\over(1-x)^{2n}},$ hence it follows that $M=(-1)^{m-n}{-2n\choose m-n}={m+n-1\choose m-n}$. ∎ ###### Remark 1. From Lemma 1, it follows that $\sum_{(d_{i})}\prod_{i}d_{i}={3n-3\choose n-2}$, where the summation is over the possible degree sequences. From this using (1) $C=(n-2)!{3n-3\choose n-2}$ also follows. ## 3 Degree distribution and the maximum degree Using Lemma 1 we can compute the expected number of degree $d$ vertices and the maximum degree. ###### Lemma 2. Let $M=\sum_{i=1}^{k}x_{i}$ and assume that $M\leq n+k-2$. Then the probability that the $i$th vertex has degree $x_{i}$ ($i=1,\ldots,k$) is $\mathds{P}(d_{i}=x_{i};\,i=1,\cdots,k)=\frac{{3n-M-k-3\choose n-M+k-2}}{{3n-3\choose n-2}}\prod_{i=1}^{k}x_{i}.$ Proof. Using the above notions with (1) we get: ${3n-3\choose n-2}\mathds{P}(d_{i}=x_{i}\,;i=1,\cdots,k)=\sum_{\begin{subarray}{c}\sum d_{i}=2n-2\\\ d_{j}=x_{j}\,j\leq k\end{subarray}}\prod_{i}d_{i}=$ $\prod_{i=1}^{k}x_{i}\sum_{\begin{subarray}{c}\sum_{j=k+1}^{n}d_{j}=\\\ =2n-M-2\end{subarray}}\prod_{j=k+1}^{n}d_{j}={3n-M-k-3\choose n-M+k-2}\prod_{i=1}^{k}x_{i},$ where the last equation follows from Lemma 1 and this is what we wanted to prove. ∎ Let us denote the maximum degree in a tree $T$ by $D(T)=\max_{i}d_{i}$. Now the following theorem is true: ###### Theorem 2. For every $\epsilon>0$, we have $(1-\epsilon-o(1))\log_{3}n\leq\mathds{E}(D(X_{n}))\leq(1+\epsilon+o(1))\log_{3}n,\textrm{ as }n\rightarrow\infty.$ We bound the expected value by bounding the probabilities $\mathds{P}(D>k)$ and $\mathds{P}(D\leq k)$ with Lemma 3 and Lemma 4 respectively. To simplify the notation, we let $D=D(X_{n})$. ###### Lemma 3. For every $\delta>0$, there exists $n_{0}$, such that $\forall n>n_{0}$ $\mathds{P}(D>k)<(1+\delta)\frac{4}{3}\frac{k}{3^{k}}n.$ Proof: From Lemma 2 it follows that $\mathds{P}(d_{i}=k)=k\frac{{3n-k-4\choose n-k-1}}{{3n-3\choose n-2}}<{4\over 3}{(1+\delta)k\over 3^{k}},\ \forall\delta>0,n>n_{0}(\delta),$ and so $\mathds{P}(d_{i}>k)<\sum_{j=k+1}^{n}{4\over 3}{(1+\delta)j\over 3^{j}}<\sum_{j=k+1}^{\infty}{4\over 3}{(1+\delta)j\over 3^{j}}=(1+\delta){2k+3\over 3^{k+1}}.$ Now the statement of the lemma follows, as $\mathds{P}(D>k)<n\mathds{P}(d_{i}>k)$. ∎ ###### Lemma 4. $\mathds{P}(D\leq k)<{1\over 9}\sqrt{n}\,\exp\left\\{-n{4\over 9}{(k+1)\over 3^{k}}\right\\}$ Proof: It is easy to see that $\mathds{P}(D\leq k)={1\over{3n-3\choose n-2}}\sum_{\begin{subarray}{c}d_{i}\leq k\\\ \sum_{i}d_{i}=2n-2\end{subarray}}\prod_{i=1}^{n}d_{i}.$ Here the sum is just the coefficient of $x^{n-2}$ in $P(x)=\Bigl{(}1+2x+3x^{2}+\ldots+kx^{k-1}\Bigr{)}^{n}$. As $P(x)=\sum a_{i}x^{i}$ and $\forall i,\ a_{i}\geq 0$, we get an upper bound on $a_{i}$: $a_{i}\leq{P(x_{0})\over x_{0}^{i}},$ for any $x_{0}>0$. Hence it follows that ${3n-3\choose n-2}\mathds{P}(D\leq k)\leq P({1\over 3})3^{n-2}=3^{n-2}\left(\sum_{i=1}^{k}{i\over 3^{i-1}}\right)^{n}\leq 3^{n-2}\left({9\over 4}-{k+1\over 3^{k}}\right)^{n}=$ $=3^{n-2}\left({9\over 4}\right)^{n}\left(1-{4\over 9}{k+1\over 3^{k}}\right)^{n}<{1\over 9}\left({27\over 4}\right)^{n}\exp\left\\{-n{4\over 9}{k+1\over 3^{k}}\right\\}.$ By using Stirling’s formula, we get $\mathds{P}(D\leq k)<{1\over 9}\sqrt{n}\,\exp\left\\{-n{4\over 9}{k+1\over 3^{k}}\right\\},$ the desired inequality. ∎ Now we can upper bound the expected value as follows: $\mathds{E}(D)\leq k_{1}\mathds{P}(D\leq k_{1})+k_{2}\mathds{P}(D>k_{1})+(n-1)\mathds{P}(D>k_{2}),$ where $k_{1}=(1+\epsilon)\log_{3}n\textrm{ and }k_{2}=\log^{3}_{3}n,$ Therefore it follows from Lemmas 3 and 4 that $\mathds{E}(D)\leq(1+\epsilon)\log_{3}n+{\log^{3}_{3}n\over n^{\epsilon+o(1)}}+{\log^{3}_{3}n\over n}=(1+\epsilon+o(1))\log_{3}n,\textrm{ as }n\rightarrow\infty.$ Also $\mathds{E}(D)\geq(1-\epsilon)\log_{3}n\mathds{P}\left(D>(1-\epsilon)\log_{3}n\right)\geq(1-\epsilon-o(1))\log_{3}n,\textrm{ as }n\rightarrow\infty,$ which proves our theorem. ∎ ## 4 Labeled subgraph densities Form now on let $T$ be a fixed tree on $k$ nodes. Assign values $r_{i}$ to each node $i$ of $T$ and call it the remainder degree of node $i$. Let $S=\\{s_{1},s_{2},\ldots,s_{k}\\}$ be an ordered subset of $[n]=\\{1,2,\ldots,n\\}$ $(s_{i}\neq s_{j},\,i\neq j)$. Now by $T_{S}$ we denote the tree $T$, with label $s_{i}$ at node $i$. Similarly let $F$ be a forest with $m_{F}$ nodes and $c_{F}$ connected components. Denote these components by $C_{1},\cdots C_{c_{F}}$. As above, define the remainder degrees for $F$, and denote them by $(r_{1},\ldots r_{m_{F}})$. Denote by $F_{S}$ the labeled forest with label $s_{i}$ at node $i$. For two labeled trees $T_{S},T^{\prime}_{S^{\prime}}$, with remainder degrees $r,r^{\prime}$ we define the operation gluing in the usual way. We identify nodes $i\in V(T),j\in V(T^{\prime})$ if $s_{i}=s^{\prime}_{j}$ and keep the label $s_{i}(=s^{\prime}_{j})$. For nodes $i\ (j)$, where $s_{i}\notin S^{\prime}(s^{\prime}_{j}\notin S)$, we do nothing. If $S\cap S^{\prime}=\emptyset$, then the resulting graph is just the disjoint union. We say that a gluing is valid, if it results in a forest and $s_{i}=s^{\prime}_{j}\Rightarrow r_{i}=r^{\prime}_{j}\,\forall i,j$, and denote it by $g(T_{S},T^{\prime}_{S^{\prime}})$. We define the gluing of two labeled forests similarly. Let $X_{S}^{T}=I\\{\phi:i\mapsto s_{i}\text{ is a homomorphism from }T\text{ to }X_{n}\text{, and }\forall i,\ d_{s_{i}}=d^{T}_{i}+r_{i}\\}$ $X_{n}^{T}=\sum_{S\subseteq[n]}X_{S}^{T}=inj(T,X_{n}).$ We define $X^{F}_{S},X^{F}_{n}$ similarly for a forest $F$. Also let $A_{S}^{T}=X^{T}_{S}-\mathds{E}(X^{T}_{S})$. If it does not make any confusion, we use $m_{1},c_{1},X_{S},A_{S}$ instead of $m_{F_{1}},c_{F_{1}},X_{S}^{T},A_{S}^{T}$. ###### Theorem 3. For an arbitrary tree $T$ on $k$ nodes with remainder degrees $(r_{i})_{i=1}^{k}$ and $\forall\epsilon>0$, we have $\mathds{P}\left({\|X^{T}_{n}-\mathds{E}(X^{T}_{n})\|\over n}>\epsilon\right)\leq{c_{k,r}\over n^{2}\epsilon^{4}}.$ We want to bound the deviation from the expectation by bounding the $4$th moment of $X_{n}^{T}$. ###### Lemma 5. With the above notions $\mathds{E}\left(\left(X_{n}^{T}-\mathds{E}(X^{T}_{n})\right)^{4}\right)=O(n^{2}).$ ###### Remark 2. Theorem 3 is a direct corollary of Lemma 5 as $\mathds{P}\left({\|X^{T}_{n}-\mathds{E}(X^{T}_{n})\|\over n}>\epsilon\right)\leq{\mathds{E}\left(\left(X_{n}^{T}-\mathds{E}(X^{T}_{n})\right)^{4}\right)\over n^{4}\epsilon^{4}}={c_{k,r}\over n^{2}\epsilon^{4}}.$ To get this bound we regard $X_{n}^{T}$ as a sum of indicator variables $X_{S}^{T}$, so we need to compute only probabilities $\mathds{P}(X_{S}^{T}=1)$ and $\mathds{P}(X_{S}^{F}=1\|X_{S^{\prime}}^{F^{\prime}}=1)$ for special forests $F,F^{\prime}$. ###### Lemma 6. Let $F$ be an arbitrary forest on $m$ nodes with remainder degrees $r=(r_{1},\cdots,r_{m})$. Let $R=\sum_{i}r_{i}$ and denote by $c$ the number of connected components of $F$. The probability that on an ordered subset $S=(s_{1},\cdots,s_{m})$ of the nodes of $X_{n}$ we see the forest $F$ and $\forall i,\ d_{s_{i}}=d^{F}_{i}+r_{i}$ is $\mathds{P}(X_{S}^{F}=1)=\frac{(n-m+c-2)!}{(n-2)!}{{3n-R-3m+2c-3\choose n-R-m+2c-2}\over{3n-3\choose n-2}}H(r,F),$ where $H(r,F)$ is a constant depending only on $F$ and $r$. Proof: First we want to compute $\mathds{P}(X_{S}^{F}=1\,\big{\|}\,(d_{i})_{i=1}^{n})$. Given the degree sequence $(d_{i})$, the distribution of $X_{n}$ is uniform on the possible trees realizing $(d_{i})$. It is easy to check (simply by contracting the connected components to one node, counting the trees and blowing back the components) that the number of trees, with degree sequence $(d_{i})$ and having $F$ on the first $m$ vertices and having $r_{i}$ edges going out from the forest at the $i$th node is $\displaystyle\prod\limits_{i=1}^{c}\left[{R_{i}!\over\prod\limits_{j\in C_{i}}r_{j}!}\right]{n-m+c_{F}-2\choose R_{1}-1,\cdots,R_{c}-1,d_{m+1}-1,\cdots,d_{n}-1}.$ (2) Here $R_{i}=\sum_{j\in C_{i}}r_{j}$. Since the number of trees realizing $(d_{i})$ is ${n-2\choose d_{1}-1,d_{2}-1,\cdots,d_{n}-1}$, it follows that $\mathds{P}(X_{\\{1,\cdots,m\\}}^{F}=1\|d_{i})=\\\ ={\displaystyle{n-m+c-2\choose R_{1}-1,\cdots,R_{c}-1,d_{m+1}-1,\cdots,d_{n}-1}\over\displaystyle{n-2\choose d_{1}-1,d_{2}-1,\cdots,d_{n}-1}}\displaystyle\prod\limits_{i=1}^{c}{R_{i}!\over\prod\limits_{j\in C_{i}}r_{j}!}.$ After simplifying, and using symmetry, we get the following equation: $\mathds{P}(X_{S}^{F}=1\|d_{i})=\\\ ={(n-m+c-2)!\over(n-2)!}\prod_{i=1}^{c}\left[R_{i}\prod_{j\in C_{i}}(d_{j}^{F}+r_{j}-1)\cdot\ldots\cdot(r_{j}+1)\right].$ (3) Now to get the desired probability, we use Lemma 2 with $k=m$ and $M=(2m-c)+R$: $\mathds{P}(X_{S}^{F}=1)=\sum_{d_{i}}\mathds{P}(X_{S}^{F}=1\|d_{i}){\prod_{i=1}^{n}d_{i}\over{3n-3\choose n-2}}=\\\ ={(n-m+c-2)!\over(n-2)!}{{3n-R-3m+2c-3\choose n-R-m+2c-2}\over{3n-3\choose n-2}}\prod_{i=1}^{c}\left[R_{i}\prod_{j\in C_{i}}\left[(d_{j}^{F}+r_{j})\cdots(r_{j}+1)\right]\right]=\\\ ={(n-m+c-2)!\over(n-2)!}{{3n-R-3m+2c-3\choose n-R-m+2c-2}\over{3n-3\choose n-2}}H(r,F),$ which is the claimed equation. ∎ ###### Lemma 7. Let $F_{1},F_{2}$ be two forests with remainder degrees $r,r^{\prime}$ and labels $S_{1},S_{2}$. If there is a valid gluing of the labeled forests $F_{1}$ and $F_{2}$ then let $F_{1,2}=g(F_{1},F_{2})$. Also let $m_{1,2}=\|V(F_{1,2})\|$, $c_{1,2}=\\{$the number of components of $F_{1,2}\\}$, $R_{1,2}=\sum_{V(F_{1,2})}r_{i}$. We have $\mathds{P}(X_{S_{1}}^{F_{1}}=1\|X_{S_{2}}^{F_{2}}=1)=\\\ {(n-m_{1,2}+c_{1,2}-2)!\over(n-m_{2}+c_{2}-2)!}{{3n-R_{1,2}-3m_{1,2}+2c_{1,2}-3\choose n-R_{1,2}-m_{1,2}+2c_{1,2}-2}\over{3n-R_{2}-3m_{2}+2c_{2}-3\choose n-R_{2}-m_{2}+2c_{2}-2}}{H(r_{1,2},F_{1,2})\over H(r^{\prime},F_{2})},$ (4) Proof: The proof follows in a straightforward way from the definition of the conditional probability. ∎ ###### Remark 3. It is easy to see that ${H(r_{1,2},F_{1,2})\over H(r^{\prime},F_{2})}=H(r,F_{1})$ if $S_{1}\cap S_{2}=\emptyset$. ###### Remark 4. We can rewrite equation (4) as a rational polynomial in $n$ as follows. $\mathds{P}(X_{S_{1}}^{F_{1}}=1\|X_{S_{2}}^{F_{2}}=1){H(r^{\prime},F_{2})\over H(r_{1,2},F_{1,2})}=\\\ ={(n-m_{1,2}+c_{1,2}-2)!\over(n-m_{2}+c_{2}-2)!}{{3n-R_{1,2}-3m_{1,2}+2c_{1,2}-3\choose n-R_{1,2}-m_{1,2}+c_{1,2}-2}\over{3n-R_{2}-3m_{2}+2c_{2}-3\choose n-R_{2}-m_{2}+c_{2}-2}}=\\\ {2^{2(m_{1,2}-m_{2})}\over 3^{R_{1,2}-R_{2}+3(m_{1,2}-m_{2})-2(c_{1,2}-c_{2})}}{P(n;R_{1,2},m_{1,2},c_{1,2},R_{2},m_{2},c_{2})\over Q(n;R_{1,2},m_{1,2},c_{1,2},R_{2},m_{2},c_{2})}$ (5) Where $P(n;x_{1},x_{2},x_{3},x_{4},x_{5},x_{6})=\\\ =n^{x_{1}-x_{4}+3(x_{2}-x_{5})-2(x_{3}-x_{6})}-n^{x_{1}-x_{4}+3(x_{2}-x_{5})-2(x_{3}-x_{6})-1}\\\ \left[(x_{1}+x_{4}+x_{2}+x_{5}-x_{3}-x_{6}+3){(x_{1}-x_{4}+x_{2}-x_{5}-x_{3}+x_{6})\over 2}+\right.\\\ \left.+(2(x_{2}+x_{5})-x_{3}-x_{6}+1){2(x_{2}-x_{5})-x_{3}+x_{6}\over 4}\right]+O(n^{x_{1}-x_{4}+3(x_{2}-x_{5})-2(x_{3}-x_{6})-2})$ and $Q(n;x_{1},x_{2},x_{3},x_{4},x_{5},x_{6})=\\\ =n^{x_{1}-x_{4}+4(x_{2}-x_{5})-3(x_{3}-x_{6})}-n^{x_{1}-x_{4}+4(x_{2}-x_{5})-3(x_{3}-x_{6})-1}\\\ \left[(x_{2}+x_{5}-x_{3}-x_{6}+3){(x_{2}-x_{5}-x_{3}+x_{6})\over 2}+\right.\\\ \left.+(x_{1}+x_{4}+3(x_{2}+x_{5})-2(x_{3}+x_{6})+5){x_{1}-x_{4}+3(x_{2}-x_{5})-2(x_{3}-x_{6})\over 6}\right]+\\\ +O(n^{x_{1}-x_{4}+4(x_{2}-x_{5})-3(x_{3}-x_{6})-2}).$ From this it also follows that $\mathds{E}(X_{S_{1}}^{F_{1}}\|X_{S_{2}}^{F_{2}}=1)=O(n^{m_{2}-m_{1,2}+c_{1,2}-c_{2}})$ (6) and $\mathds{E}(X_{S_{1}}^{F_{1}}\|X_{S_{2}}^{F_{2}}=1)-\mathds{E}(X_{S_{1}}^{F_{1}})=O(n^{m_{2}-m_{1,2}+c_{1,2}-c_{2}-1})=O(n^{-m_{1}+c_{1}-1}),$ (7) if $S_{1}\cap S_{2}=\emptyset$ (since in this case $m_{1,2}-m_{2}=m_{1}$ and $c_{1,2}-c_{2}=c_{1}$). Now we have arrived at the main lemma of Chapter 4. Here we want to bound the following sum $\sum_{S_{1},S_{2},S_{3},S_{4}}\mathds{E}(A^{T}_{S_{1}}A^{T}_{S_{2}}A^{T}_{S_{3}}A^{T}_{S_{4}}).$ We split this sum in 5 parts, depending on the sizes of the intersections $S_{i}\cap S_{j},\ (1\leq i<j\leq 4)$. We can change the indeces so that one of the following holds: * • Case I: $S_{i}$’s are disjoint * • Case II: $\|S_{3}\cap S_{4}\|=i,\ i\leq k$ and $S_{1},S_{2},S_{3}\cup S_{4}$ are disjoint * • Case III: $\|S_{1}\cap S_{2}\|=i,\ \|S_{3}\cap S_{4}\|=j$ and $(S_{1}\cup S_{2})\cap(S_{3}\cup S_{4})=\emptyset$ * • Case IV: $\|S_{2}\cap S_{3}\|=i,\ \|S_{3}\cap S_{4}\|=j,\ \|S_{2}\cap S_{3}\cap S_{4}\|=l,\ i,j\geq 1,l\geq 0$ and $\|S_{1}\cap\\{S_{2}\cup S_{3}\cup S_{4}\\}\|=\emptyset$ * • Case V.: $\|S_{i}\cap S_{i+1}\|\geq 1,\ i=1,2,3$ When $S_{i}$’s are disjoint then the variables $X_{S_{1}},X_{S_{2}},X_{S_{3}},X_{S_{4}}$ are not independent for a fixed $n$, but if $n$ tends to infinity, then they get independent, that is $\|\mathds{P}(X_{S_{i}}=1\|X_{S_{j}}=1)-\mathds{P}(X_{S_{i}}=1)\|\leq o(1)\mathds{P}(X_{S_{i}}=1)$. So in Case I we can bound the sum because disjoint $S_{i}$’s are asymptotically independent. In cases II-V, we can use the fact that the number of intersecting quadruples decreases as we increase the size of the intersection. Proof of Lemma 5: Let us fix $T$, the tree on $k$ nodes, with $r=(r_{1},\ldots r_{k})$. Denote the gluing (considering the remainder degrees $r_{i}$ also) of two copies of $T$ along the set $S_{i},S_{j}$ by $T_{i,j}=g(T_{S_{i}},T_{S_{j}})$ (if there is a valid gluing resulting in a forest). $m_{i,j}=\|V(T_{i,j})\|$ and $c_{i,j}$ is just the number of components of $T_{i,j}$. $R_{i,j}=2(\sum_{l=1}^{k}r_{l})-\sum_{l\in S_{i}\cap S_{j}}r_{l}.$ We define similarly $T_{i,j,h},T_{1,2,3,4},m_{i,j,h},m_{1,2,3,4},R_{i,j,h},R_{1,2,3,4}\ (i\neq j\neq h,i\neq h)$. $\mathds{E}(A_{S_{1}}A_{S_{2}}A_{S_{3}}A_{S_{4}})=\\\ =\mathds{E}(A_{S_{1}}A_{S_{2}}A_{S_{3}}X_{S_{4}})-\mathds{E}(A_{S_{1}}A_{S_{2}}A_{S_{3}})\mathds{P}(X_{S_{4}}=1)=\\\ =\mathds{P}(X_{S_{4}}=1)\Bigl{(}\mathds{E}(A_{S_{1}}A_{S_{2}}A_{S_{3}}\|X_{S_{4}}=1)-\mathds{E}(A_{S_{1}}A_{S_{2}}A_{S_{3}})\Bigr{)}=\\\ =\mathds{P}(X_{S_{4}}=1)\Bigl{(}\mathds{P}(X_{S_{3}}=1)(\mathds{E}(A_{S_{1}}A_{S_{2}}\|X_{S_{3}}=1,X_{S_{4}}=1)-\mathds{E}(A_{S_{1}}A_{S_{2}}\|X_{S_{4}}=1))-\\\ -\mathds{P}(X_{S_{3}}=1)(\mathds{E}(A_{S_{1}}A_{S_{2}}\|X_{S_{3}}=1)-\mathds{E}(A_{S_{1}}A_{S_{2}}))\Bigr{)}=\\\ =\mathds{P}(X_{S_{4}}=1)\mathds{P}(X_{S_{3}}=1)\Bigl{(}\mathds{E}(A_{S_{1}}A_{S_{2}}\|X_{S_{3}}=1,X_{S_{4}}=1)-\mathds{E}(A_{S_{1}}A_{S_{2}}\|X_{S_{4}}=1)-\\\ -\mathds{E}(A_{S_{1}}A_{S_{2}}\|X_{S_{3}}=1)+\mathds{E}(A_{S_{1}}A_{S_{2}})\Bigr{)}$ Let $f_{S_{3},S_{4}}(X,Y)=\mathds{E}(XY\|X_{S_{3}}=1,X_{S_{4}}=1)-\mathds{E}(XY\|X_{S_{4}}=1)-\\\ -\mathds{E}(XY\|X_{S_{3}}=1)+\mathds{E}(XY),$ $f_{S_{4}}(X,Y,Z)=\mathds{E}(XYZ\|X_{S_{4}}=1)-\mathds{E}(XYZ)\text{ and}$ $\mathds{E}(A_{S_{1}}A_{S_{2}}A_{S_{3}}A_{S_{4}})=\mathds{P}(X_{S_{4}}=1)\mathds{P}(X_{S_{3}}=1)(f_{S_{3},S_{4}}(X_{S_{1}},X_{S_{2}})-\\\ -f_{S_{3},S_{4}}(\mathds{E}X_{S_{1}},X_{S_{2}})-f_{S_{3},S_{4}}(X_{S_{1}},\mathds{E}X_{S_{2}})+f_{S_{3},S_{4}}(\mathds{E}X_{S_{1}},\mathds{E}X_{S_{2}}))$ Case I.: $S_{i}$’s are disjoint When $S_{i}$’s are disjoint, we have $R_{1,2,3,4}=4R,\,R_{3,4}=2R,\,m_{1,2,3,4}=4k,\,m_{3,4}=2k,\,c_{1,2,3,4}=4,\,c_{3,4}=2$. Using Equation 5 $\mathds{E}(X_{S_{1}}X_{S_{2}}\|X_{S_{3}}=1,X_{S_{4}}=1){H(r_{3,4},T_{3,4})\over H(r_{1,2,3,4},T_{1,2,3,4})}={2^{4k}\over 3^{2R+6k-4}}\\\ {n^{2R+6k-4}-n^{2R+6k-5}(6R^{2}+12kR-9R+18k^{2}-20k+{11\over 2})+O(n^{2R+6k-6})\over n^{2R+8k-6}-n^{2R+8k-7}(2R^{2}+12kR-{19\over 3}R+24k^{2}-28k+{23\over 3})+O(n^{2R+8k-8})}$ (8) $\mathds{E}(X_{S_{1}}X_{S_{2}}\|X_{S_{3}}=1){H(r_{3},T_{3})\over H(r_{1,2,3},T_{1,2,3})}={2^{4k}\over 3^{2R+6k-4}}\\\ {n^{2R+6k-4}-n^{2R-6k-5}(4R^{2}+8kR-5R+12k^{2}-12k+{5\over 2})+O(n^{2R-6k-6})\over n^{2R+8k-6}-n^{2R+8k-7}({4\over 3}R^{2}+8kR-{11\over 3}R+16k^{2}-16k+3)+O(n^{2R-8k-8})}$ (9) ${\mathds{E}(X_{S_{1}}X_{S_{2}})\over H(r_{1,2},T_{1,2})}={2^{4k}\over 3^{2R+6k-4}}\\\ {n^{2R+6k-4}-n^{2R+6k-5}(2R^{2}+4kR-1R+6k^{2}-4k-{1\over 2})+O(n^{2R+6k-6})\over n^{2R+8k-6}-n^{2R+8k-7}({2\over 3}R^{2}+4kR-R+8k^{2}-4k-{5\over 3})+O(n^{2R+8k-8})}$ (10) $\mathds{E}(\mathds{E}(X_{S_{1}})X_{S_{2}}\|X_{S_{3}}=1,X_{S_{4}}=1)=\mathds{P}(X_{S_{1}}=1)\mathds{E}(X_{S_{2}}\|X_{S_{3}}=1,X_{S_{4}}=1)$ $\mathds{E}(X_{S_{2}}\|X_{S_{3}}=1,X_{S_{4}}=1){H(r_{3,4},T_{3,4})\over H(r_{2,3,4},T_{2,3,4})}={2^{2k}\over 3^{R+3k-2}}\\\ {n^{R+3k-2}-n^{R+3k-3}({5\over 2}R^{2}+5kR-{7\over 2}R+{25\over 2}k^{2}-{25\over 2}k+3)+O(n^{R+3k-4})\over n^{R+4k-3}-n^{R+4k-4}({5\over 6}R^{2}+5kR-{5\over 2}R+10k^{2}-11k+{8\over 3})+O(n^{R+4k-5})}$ (11) $\mathds{E}(X_{S_{1}}\|X_{S_{3}}=1){H(r,T)\over H(r_{1,3},T_{1,3})}={2^{2k}\over 3^{R+3k-2}}\\\ {n^{R+3k-2}-n^{R+3k-3}({3\over 2}R^{2}+3kR-{3\over 2}R+{15\over 2}k^{2}-{13\over 2}k+1)+O(n^{R+3k-4})\over n^{R+4k-3}-n^{R+4k-4}({1\over 2}R^{2}+3kR-{7\over 6}R+6k^{2}-5k+{1\over 3})+O(n^{R+4k-5})}$ (12) ${\mathds{E}(X_{S_{1}})\over H(r,t)}={2^{2k}\over 3^{R+3k-2}}\\\ {n^{R+3k-2}-n^{R+3k-3}({1\over 2}R^{2}+kR+{1\over 2}R+{5\over 2}k^{2}-{1\over 2}k-1)+O(n^{R+3k-4})\over n^{R+4k-3}-n^{R+4k-4}({1\over 6}R^{2}+kR+{1\over 6}R+2k^{2}+k-2)+O(n^{R+4k-5})}$ (13) It is easy to verify that from (8)-(10) we get that $f_{S_{3},S_{4}}(X_{S_{1}},X_{S_{2}})=O(n^{-2k}),$ and using (11)-(13) $f_{S_{3},S_{4}}(\mathds{E}(X_{S_{1}}),X_{S_{2}})=f_{S_{3},S_{4}}(X_{S_{1}},\mathds{E}(X_{S_{2}}))=O(n^{-2k}),\ \mathrm{and}$ $f_{S_{3},S_{4}}(\mathds{E}(X_{S_{1}}),\mathds{E}(X_{S_{2}}))=0.$ Hence $\mathds{E}(A_{S_{1}}A_{S_{2}}A_{S_{3}}A_{S_{4}})=O(n^{-4k+2})$ and since the number of disjoint $S_{i}$ quadruples is just $n^{4k}$, $\sum_{S_{i},\,S_{i}\cap S_{j}=\emptyset,\,i\neq j}\mathds{E}(A_{S_{1}}A_{S_{2}}A_{S_{3}}A_{S_{4}})=O(n^{2}).$ Case II.: $\|S_{3}\cap S_{4}\|=i,\ i\leq k$ and $S_{1},S_{2},S_{3}\cup S_{4}$ are disjoint The number of such quadruples is $n^{4k-i}$. Similarly as above, we want to bound $F(S_{1},S_{2},S_{3},S_{4})$. From (7) we get $\mathds{E}(X_{S_{1}}X_{S_{2}}\|X_{S_{3}}=1,X_{S_{4}}=1)-\mathds{E}(X_{S_{1}}X_{S_{2}})=\\\ =O(n^{m_{3,4}-m_{1,2,3,4}+c_{1,2,3,4}-c_{3,4}-1})=O(n^{2k-i-(4k-i)+3-1-1})=O(n^{-2k+1})$ (14) $\mathds{E}(X_{S_{1}}X_{S_{2}}\|X_{S_{3}}=1)-\mathds{E}(X_{S_{1}}X_{S_{2}})=\\\ =O(n^{m_{3}-m_{1,2,3}+c_{1,2,3}-c_{3}-1})=O(n^{k-3k+3-1-1})=O(n^{-2k+1})$ (15) $\mathds{E}(X_{S_{1}}\|X_{S_{3}}=1,X_{S_{4}}=1)-\mathds{E}(X_{S_{1}})=\\\ =O(n^{m_{3,4}-m_{1,3,4}+c_{1,3,4}-c_{3,4}-1})=O(n^{2k-i-(3k-i)+2-1-1})=O(n^{-k})$ (16) $\mathds{E}(X_{S_{1}}\|X_{S_{3}}=1)-\mathds{E}(X_{S_{1}})=\\\ =O(n^{m_{3}-m_{1,3}+c_{1,3}-c_{3}-1})=O(n^{k-2k+2-1-1})=O(n^{-k})$ (17) From (14)-(17) it follows that $\mathds{E}(A_{S_{1}}A_{S_{2}}A_{S_{3}}A_{S_{4}})=O(n^{-4k+3})$ and so $\sum_{S_{i},\,\|S_{1}\cap S_{2}\|=i}\mathds{E}(A_{S_{1}}A_{S_{2}}A_{S_{3}}A_{S_{4}})=O(n^{2}).$ Case III.: $\|S_{1}\cap S_{2}\|=i,\|S_{3}\cap S_{4}\|=j$ and $(S_{1}\cup S_{2})\cap(S_{3}\cup S_{4})=\emptyset$. As before, using (7) we get: $\mathds{E}(X_{S_{1}},X_{S_{2}}\|X_{S_{3}}=1,X_{S_{4}}=1)-\mathds{E}(X_{S_{1}},X_{S_{2}})=\\\ =O(n^{m_{3,4}-m_{1,2,3,4}+c_{1,2,3,4}-c_{3,4}-1})=O(n^{2k-j-(4k-i-j)+2-1-1})=O(n^{-2k+i})$ (18) $\mathds{E}(X_{S_{1}},X_{S_{2}}\|X_{S_{3}}=1)-\mathds{E}(X_{S_{1}},X_{S_{2}})=\\\ =O(n^{m_{3}-m_{1,2,3}+c_{1,2,3}-c_{3}-1})=O(n^{k-(3k-i)+2-1-1})=O(n^{-2k+i})$ (19) Now it follows from (16)-(19) that $\mathds{E}(A_{S_{1}}A_{S_{2}}A_{S_{3}}A_{S_{4}})=O(n^{-4k+i+2})$ and so $\sum_{\|S_{1}\cap S_{2}\|=i,\|S_{3}\cap S_{4}\|=j}\mathds{E}(A_{S_{1}}A_{S_{2}}A_{S_{3}}A_{S_{4}})=O(n^{2}).$ Case IV.: $\|S_{1}\cap\\{S_{2}\cup S_{3}\cup S_{4}\\}\|=\emptyset,\|S_{2}\cap S_{3}\|=i,\|S_{3}\cap S_{4}\|=j,\|S_{2}\cap S_{3}\cap S_{4}\|=l,$ with $i,j\geq 1,l\geq 0$. Note, that $S_{2}\cup S_{4}\setminus S_{3}=\emptyset$, because otherwise there will be no valid gluing of the trees $T$ along the $S_{i}$’s ($g(T_{S_{1}}T_{S_{2}}T_{S_{3}}T_{S_{4}})$ is not a tree). The number of such $S_{i}$ quadruples is $n^{4k-i-j+l}$. We want to bound $\mathds{E}(A_{S_{1}}A_{S_{2}}A_{S_{3}}A_{S_{4}})=\mathds{P}(X_{S_{1}}=1)\left(\mathds{E}(A_{S_{2}}A_{S_{3}}A_{S_{4}}\|X_{S_{1}}=1)-\mathds{E}(A_{S_{2}}A_{S_{3}}A_{S_{4}})\right).$ (20) Using (7) we get: $\mathds{E}(X_{S_{2}}X_{S_{3}}X_{S_{4}}\|X_{S_{1}}=1)-\mathds{E}(X_{S_{2}}X_{S_{3}}X_{S_{4}})=\\\ =O(n^{m_{1}-m_{1,2,3,4}+c_{1,2,3,4}-c_{1}-1})=O(n^{k-(4k-i-j+l)+2-1-1})=O(n^{-3k+i+j-l})$ (21) $\mathds{E}(X_{S_{2}}X_{S_{3}}\|X_{S_{1}}=1)-\mathds{E}(X_{S_{2}}X_{S_{3}})=\\\ =O(n^{m_{1}-m_{1,2,3}+c_{1,2,3}-c_{1}-1})=O(n^{k-(3k-i)+2-1-1})=O(n^{-2k+i})$ (22) $\mathds{E}(X_{S_{3}}X_{S_{4}}\|X_{S_{1}}=1)-\mathds{E}(X_{S_{3}}X_{S_{4}})=\\\ =O(n^{m_{1}-m_{1,3,4}+c_{1,3,4}-c_{1}-1})=O(n^{k-(3k-j)+2-1-1})=O(n^{-2k+j})$ (23) $\mathds{E}(X_{S_{2}}X_{S_{4}}\|X_{S_{1}}=1)-\mathds{E}(X_{S_{2}}X_{S_{4}})=O(n^{m_{1}-m_{1,2,4}+c_{1,2,4}-c_{1}-1})=\\\ =O(n^{k-(3k-l)+c_{1,2,4}-1-1})=\left\\{\begin{array}[]{ll}O(n^{-2k+l})&l\geq 1\\\ O(n^{-2k+1})&l=0\end{array}\right.$ (24) Now from (17) and (21)-(24) we get that $\sum_{S_{1},S_{2},S_{3},S_{4}}\mathds{E}(A_{S_{1}}A_{S_{2}}A_{S_{3}}A_{S_{4}})=O(n^{2}),$ where the summation is over all quadruples $S_{i}$ satisfying the conditions of Case IV. Case V.: $\|S_{i}\cap S_{j}\|=s_{i,j}$ and $s_{i,i+1}\geq 1\ i=1,2,3$ $\mathds{E}(X_{S_{1}}X_{S_{2}}X_{S_{3}}X_{S_{4}})=O({1\over n^{\|S_{1}\cup S_{2}\cup S_{3}\cup S_{4}\|-1}})$ (25) $\mathds{E}(X_{S_{i}}X_{S_{j}}X_{S_{l}})=O({1\over n^{\|S_{i}\cup S_{j}\cup S_{l}\|-1}})$ (26) $\mathds{E}(X_{S_{i}}X_{S_{j}})=O({1\over n^{\|S_{i}\cup S_{j}\|-c_{i,j}}})$ (27) From the above it follows that $\sum_{S_{1},S_{2},S_{3},S_{4}}\mathds{E}(A_{S_{1}}A_{S_{2}}A_{S_{3}}A_{S_{4}})=O(n^{2}),$ (28) where the summation is over all quadruples $S_{i}$ satisfying the conditions of Case V. From the above the claim of Lemma 5 follows ∎ ## 5 The limit of $X_{n}$ Let $U^{l}$ denote the set of all finite $l$-deep rooted tree. Consider an $l$-deep rooted tree with root $x$: $T^{l}_{x}\in U^{l}$, with $\|T^{l}_{x}\|=k$. Let us denote the nodes at the $i$th level with $T_{i}$, and $\|T_{i}\|=t_{i}$ ($t_{0}$ is just $1$, $t_{1}$ is the degree of the root and $t_{l}$ is the number of leafs). As before $B_{G}(v,l)$ is the rooted $l$-ball around $v$ in $G$ and $X_{n}$ is a random tree on $n$ nodes with distribution $\Pi_{n}$. If we assign remainder degrees $r$ to the rooted tree $T_{x}^{l}$ and forget the root, then using Lemma 6 we get $\mathds{E}(X_{n}^{T})=\mathds{E}\left(\sum_{S\subseteq[n]}X_{S}^{T}\right)={n!\over(n-k)!}\mathds{P}(X_{S}^{T}=1)=n{n-1\over n-k}{{3n-R-3k-1\choose n-R-k}\over{3n-3\choose n-2}}H(r,T).$ (29) Now the number of vertices $v\in X_{n}$ for which $B_{X_{n}}(v,l)\cong T_{x}^{l}$ is just ${1\over\|Aut(T^{l}_{x})\|}\sum_{r_{k}=1}^{n}\sum_{r_{k-1}=1}^{n}\cdots\sum_{r_{k-t_{l}+1}=1}^{n}X_{n}^{T}$ (30) Where the remainder degrees are $0$ except for the leafs of $T^{l}_{x}$ ($r_{i}=0,\ \forall i\notin T_{l}$) and $Aut(T^{l}_{x})$ means the set of all rooted automorphisms of $T^{l}_{x}$. ###### Remark 5. Using lemma 3 we have that the maximum degree is almost surely less than $3\log_{3}n$. For a tree $T$ with remainder degree $r$, if there exists $i$, s.t. $r_{i}>3\log_{3}n$, then $X_{n}^{T}=0$ almost surely. So we can omit the terms with $r_{i}>3\log_{3}n$ in 30. Now if $v$ is a uniform random vertex of $X_{n}$, then using remark 5 we have $\mathds{P}(B_{X_{n}}(v,l)=T_{x}^{l}\|X_{n})={1\over\|Aut(T^{l}_{x})\|}\sum_{r_{k}=1}^{3\log_{3}n}\sum_{r_{k-1}=1}^{3\log_{3}n}\cdots\sum_{r_{k-t_{l}+1}=1}^{3\log_{3}n}{X_{n}^{T}\over n},$ (31) Now from Theorem 3, we get that $\mathds{P}(B_{X_{n}}(v,l)=T_{x}^{l})={1\over\|Aut(T^{l}_{x})\|}\sum_{r_{k}=1}^{3\log_{3}n}\sum_{r_{k-1}=1}^{3\log_{3}n}\cdots\sum_{r_{k-t_{l}+1}=1}^{3\log_{3}n}\left({\mathds{E}(X_{n}^{T})\over n}+o({1\over n^{{1\over 8}}})\right)=\\\ =o(1)+{1\over\|Aut(T^{l}_{x})\|}\sum_{r_{k}=1}^{3\log_{3}n}\sum_{r_{k-1}=1}^{3\log_{3}n}\cdots\sum_{r_{k-t_{l}+1}=1}^{3\log_{3}n}{\mathds{E}(X_{n}^{T})\over n}.$ $\lim_{n\rightarrow\infty}{n-1\over n-k}{{3n-R-3k-1\choose n-R-k}\over{3n-3\choose n-2}}H(r,T)=9\left({4\over 27}\right)^{\sum_{0}^{l}t_{j}}{\sum_{i\in T_{l}}r_{i}\prod_{i\in T_{l}}(r_{i}+1)\over 3^{\sum_{i\in T_{l}}r_{i}}}\prod_{j\notin T_{l}}d_{j}!,$ so it follows that $\lim_{n\rightarrow\infty}\mathds{P}(B_{X_{n}}(v,l)=T_{x}^{l})=\\\ =9\left({4\over 27}\right)^{\sum_{0}^{l-1}t_{j}}{t_{l}\over 3^{t_{l}}}{1\over\|Aut(T_{x}^{l})\|}\prod_{j\notin T_{l}}d_{j}!=p(T_{x}^{l}).$ (32) Let $\cal{G}$ denote the set of all countable connected rooted trees. Let ${\cal T}({\cal G},T_{x}^{l}):=\\{(G,v)\in{\cal G}:B_{G}(v,l)\cong T_{x}^{l}\\}$. Now consider a measure $\mu$ on the sets ${\cal T}({\cal G},T_{x}^{l})$: $\mu({\cal T}({\cal G},T_{x}^{l})=p(T_{x}^{l})$ ###### Lemma 8. $\mu$ extends to a probability measure on $\cal G$. Proof: We only need to show that $p(T_{x}^{l-1})=\sum_{\displaystyle T_{x}^{l}:B_{T_{x}^{l}}(x,l)\cong T_{x}^{l-1}}p(T_{x}^{l})$ (33) Let $\sigma,\rho\in Aut(T_{x}^{l})$ be two automorphisms of the rooted tree $T_{x}^{l}$. We say that $\sigma\sim\rho$ if and only if there exists $\tau\in Aut(T_{x}^{l})$, such that $\tau$ fixes every vertex not in $T_{l}$ and $\sigma\circ\tau=\rho$. $\sim$ is an equivalence. The equivalence classes have $\prod_{i\in T_{l}}(d_{i}-1)!$ elements, hence it follows $\|Aut(T_{x}^{l})\|=\|Aut(T_{x}^{l}\setminus T_{l})\|\prod_{i\in T_{l}}(d_{i}-1)!.$ Now we get equation 33 easily from the following: $\sum_{d_{n-t_{l}-t_{l-1}+1}=1}^{\infty}\sum_{d_{n-t_{l}-t_{l-1}+2}=1}^{\infty}\cdots\sum_{d_{n-t_{l}}=1}^{\infty}{9\left({4\over 27}\right)^{\sum_{0}^{l-1}t_{j}}{t_{l}\over 3^{t_{l}}}\prod_{j\notin T_{l}}d_{j}!\over\|Aut(T_{x}^{l})\|}=\\\ ={9\left({4\over 27}\right)^{\sum_{0}^{l-1}t_{j}}\left({9\over 4}\right)^{t_{l-1}}t_{l-1}\prod_{j\notin T_{l}\cup T_{l-1}}d_{j}!\over\|Aut(T_{x}^{l}\setminus T_{l})\|}.$ ∎ From Lemma 8 it follows that $\mu$ is indeed the limit of the random tree sequence $X_{n}$. This completes the proof of Theorem 1. ## References * [1] D. Aldous, _The Continuum Random Tree III_ , Ann. Probab. Volume 21, Number 1 (1993), 248-289. * [2] I. Benjamini, O. Schramm, _Recurrence of Distributional Limits of Finite Planar Graphs_ , Electronic j. Probab. 6 (2001), paper no. 23, 1-13 * [3] I. Benjamini, O. Schramm, A. Shapira, _Every Minor-Closed Property of Sparse Graphs is Testable_ , 40th Ann. ACM Symp. on Th. Comp. (2008), 393-402. * [4] G. Elek, _On Limits of Finite Graphs_ , Combinatorica 27 (2007), 503-507. * [5] G. Elek, G. Tardos, _Limits of Trees_ , Oberwolfach Report No. 11/2010, 566-568 * [6] R. Lyons, _Asymptotic Enumeration of Spanning Trees_ , Combinatorics, Probability and Computing 14 (2005), 491-522	arxiv-papers	2014-01-11T11:18:55	2024-09-04T02:49:56.602198	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Attila De\\'ak", "submitter": "Attila De\\'ak", "url": "https://arxiv.org/abs/1401.2521" }
1401.2559	A Compact Orbital Angular Momentum Spectrometer Using Quantum Zeno Interrogation Paul Bierdz, Hui Deng* Department of Physics, University of Michigan, Ann Arbor, USA [email protected], [email protected] ###### Abstract We present a scheme to measure the orbital angular momentum spectrum of light using a precisely timed optical loop and quantum non-demolition measurements. We also discuss the influence of imperfect optical components. OCIS codes: 050.4865 Optical vortices, 120.4290 Nondestructive testing. ## References and links * [1] L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Physical Review A 45, 8185 (1992). * [2] G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat Phys 3, 305–310 (2007). * [3] S. Franke-Arnold, L. Allen, and M. J. Padgett, “Advances in optical angular momentum,” Laser & Photonics Review 2, 299–313 (2008). * [4] G. 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White, “Generation of optical phase singularities by computer-generated holograms,” Optics Letters 17, 221–223 (1992). * [15] J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “The production of multiringed Laguerre–Gaussian modes by computer-generated holograms,” Journal of Modern Optics 45, 1231 (1998). * [16] J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon,” Physical Review Letters 92, 013,601 (2004). * [17] G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, “Efficient sorting of orbital angular momentum states of light,” Physical Review Letters 105, 153,601 (2010). * [18] A. Peres, “Zeno paradox in quantum theory,” American Journal of Physics 48, 931 (1980). * [19] A. C. Elitzur and L. Vaidman, “Quantum mechanical interaction-free measurements,” Foundations of Physics 7, 987–997 (1993). * [20] P. G. Kwiat, A. G. White, J. R. Mitchell, O. Nairz, G. Weihs, H. Weinfurter, and A. Zeilinger, “High-Efficiency quantum interrogation measurements via the quantum Zeno effect,” Physical Review Letters 83, 4725 (1999). * [21] S0 and S1 are optical switches that can be switched from been transmittive to reflective. S0 needs to transmit the initial incident light, and be switched to be reflective by the end of the first outer-loop cycle. A high repetition rate is not required and it can be implemeted either mechanically or opto-electrically. Alternatively it could also be a static high-reflectance mirror, if the one-time transmission loss at the very beginning can be tolerated. S1 needs to be switched every QZI loop cycle ($\Delta T$) and need to be polarization insensitive. The fastest switches are Pockels cells, which can operate at 10-100 GHz with $99\%$ transmission. One scheme, similar to the one implemented in Ref. [20], is to an interferometer with Pockels cells in one of the arms. The Pockels cells introduce $\pi$ phase shift in the arm when activated, and thus switch the beam between two output ports of the interferometer. Using two Pockels cells rotated relative to each other will cancel the birefringent effect. * [22] M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Optics Communications 112, 321–327 (1994). * [23] B. Misra and E. C. G. Sudarshan, “The Zeno’s paradox in quantum theory,” Journal of Mathematical Physics 18, 756 (1977). * [24] Technically, each $\|\alpha\|^{2}$ is slightly different, but the difference is well within 1%. The $\|\alpha\|^{2}$ that matters most is the one corresponding to the QZI loop (including S1), which, without any optimization, consists of 4 beam splitters, 5 mirrors, 1 waveplate and up to three Pockels cells. Assuming all optics are anti-reflection coated so that loss is 1% at each Pockels cell and 0.1% at each other component, we have $\|\alpha\|^{2}\geq 0.96$. * [25] J. Jang, “Optical interaction-free measurement of semitransparent objects,” Physical Review A 59, 2322 (1999). Light carries quantized orbital angular momentum (OAM) of $l\hbar$ per photon when the electric field has an overall azimuthal dependence on phase, $e^{-il\phi}$, where $\phi$ is the azimuthal angle about the beam propagation axis [1, 2, 3]. The OAM quantum number $l$ takes integer values from $-\infty$ to $+\infty$. With an infinite number of states available, OAM can be utilized for qudit ($d$-dimensional, $d>2$) systems [4] that allow, for example, higher dimensional entanglement [5], quantum coin-tossing [6], increased violations of local realism [7], improved security for quantum key distribution [8], simplified quantum gates [9] and superdense coding for quantum communications with increased channel capacity [10]. Determining OAM of light in an unknown state, however, is more challenging than measuring different polarizations or frequencies. Eigenstates of OAM can be deduced from the diffraction interference pattern with judiciously chosen apertures [11, 12]. But the method requires a large collection of photons to develop the pattern. Moreover it may become very complicated for superpositions of OAM states. An $l$-fold fork diffraction grating [13, 14, 15] can separate a pre-determined OAM component from others at the single photon level, but cannot be readily applied to determine arbitrary OAM state of light. A cascade Mach-Zehnder interferometer setup, using Dove prisms to introduce an $l$-dependent phase shift, can separate different OAM components of light into different output ports of the interferometers, even at the single-photon level [16]. But the setup requires $N-1$ mutually stabilized interferometers to detect $N$ OAM-modes. A recently proposed scheme [17] requires only a spatial light modulator (or a specially designed hologram), which converts the twisting phase structure of OAM states into a linear phase gradient, and a lens, which focuses different OAM components to different spatial locations on the focal plane. However, the method relies on intricate spatial modulation of the phase, and thus may have limited applicability to broadband ultrafast pulses. Moreover, the extinction ratio between different OAM states is limited to about $10$. To increase the extinction ratio or to detect higher order OAM states will require larger and increasingly complex holograms. In this paper, we present a compact OAM-spectrometer comprising of only one interferometer nested within an optical loop (Fig. 1). It uses a Quantum Zeno Interrogator (QZI) [18, 19, 20] (shaded region in Fig. 1) to perform counterfactual measurements on the OAM state, and thus maps different OAM components of an arbitrary input light pulse into different time bins at the output. It can achieve very high extinction ratios between different OAM states and can work for arbitrarily high OAM orders limited mainly by optical losses. Fig. 1: A schematic of the compact OAM spectrometer. The Quantum Zeno Interrogator (shaded region) distinguishes between zero and nonzero OAM states. The outer loop decreases the OAM value of light by one per round trip. All the beam splitters are polarizing beam splitters (PBSs) that transmits horizontally polarized light and reflects vertically polarized light. The OAM filter transmits states with zero OAM, but blocks states with non-zero OAM. S0 and S1 are switching mirrors that either transmits or reflects incident light [21]. R1 and R2 are fixed polarization rotators, which can be half wave plates. P1 and P2 are fast polarization switches, such as Pockels cells. When activated, P1 and P2 switches horizontal polarization to vertical and vice versa. When de-activated, they are transparent to light. The shaded region is a Quantum Zeno Interrogator [20] which separates OAM components with $l=0$ and $l\neq 0$ into different polarizations. Hence at PBS3, zero OAM component is sent to the detector while the none-zero OAM component is sent back into the outer-loop. The outer loop decreased OAM by one per round trip via, for example, a vortex phase plate (VPP) [22]. We illustrate now how the spectrometer works by tracing, as an example, a horizontally polarized input pulse with an OAM value $l=l_{0}\geq 0$, noted as $\|\psi^{(0)}\rangle=\|H,l_{0}\rangle$. The input pulse first transmits through optical switches S0 and S1 [21], and enters the QZI. The polarization rotator R1 rotates its polarization by $\Delta\theta=\pi/(2N)$, and the state becomes $\|\psi^{(0)}_{1}\rangle=\cos\left(\frac{\pi}{2N}\right)\|H,l_{0}\rangle+\sin\left(\frac{\pi}{2N}\right)\|V,l_{0}\rangle$. If $l=0$, the horizontal and vertical components of $\|\psi^{(0)}_{1}\rangle$ passes through the lower and upper arms of the interferometer, respectively. They recombine into the same state $\|\psi^{(0)}_{1}\rangle$ at the polarizing beam splitter PBS2 (neglecting an overall phase factor). S1 is switched to be reflective at the end of the first QZI loop, and the combined beam continues to loop in the QZI. The polarization is rotated by $\Delta\theta=\pi/(2N)$ each loop. After $N$ loops, the light becomes vertically polarized and enters only the upper path of the interferometer. At this point, the polarization switch P1 is activated and switches the polarization into horizontal. Hence the light transmits through both PBS2 and PBS3, and arrives at the detector at time $T_{0}$. If $l_{0}\neq 0$, however, the vertical component is sent to the upper path at PBS1, and is then blocked by the OAM filter. Only the horizontal component emerges after PBS2, the state collapses into $\|H,l_{0}\rangle$ with a probability $\cos^{2}\left(\frac{\pi}{2N}\right)$. After $N$ loops, a fraction $p=\cos(\pi/(2N))^{2N}$ of the light remains in the horizontal polarization in the lower arm of the interferometer, while a fraction $1-p$ of the light is lost (blocked by OAM filter). At this point, the polarization switch P2 is activated and switches the polarization to vertical, and the light reflects off both PBS2 and PBS3, and enters the outer loop. By this time, S0 is switched to be reflective. As the light cycles in the outer loop, the polarization in rotated back to horizontal by R2, and the OAM value is decreased by $\Delta l=1$ per cycle by a vortex phase plate (VPP) [22]. After $l_{0}$ cycles, $l=0$. When the light enters the QZI again, it will exit the spectrometer to the detector, at a time $T(l_{0})=T_{0}+l_{0}(NL_{QZI}+L_{out})/c$. Here $L_{QZI}$ and $L_{out}$ are the optical path lengths of the QZI loop (from S1 to PBS2 back to S1) and the outer-loop (from S1 to PBS2, to PBS3, to S0, back to S1). The detected fraction of the light intensity is $P(l_{0})=p^{l_{0}}=\cos\left(\frac{\pi}{2N}\right)^{2Nl_{0}}$. In short, the OAM spectrometer sorts different OAM components into different time intervals separated by $\Delta T=(NL_{QZI}+L_{out})/c$ with a _perfect_ extinction ratio. The total transmission efficiency of the spectrometer is $P(l_{0})$ for the component with OAM of $l_{0}\hbar$. $P(l_{0})\rightarrow 1$ for all $l_{0}$ as $N\rightarrow\infty$ due to the quantum Zeno effect [23], as shown in the Fig. 2(a). Fig. 2: The probability of detecting the correct OAM value as a function of the number of loops ($N$) in the QZI using a perfect OAM filter. (a) Neglect optical loss. (b) Assume $\|\alpha\|^{2}=0.96$ based on commercially available optics. When optical loss is included, there exists an optimal $N$ for higher order OAM states, due to the compromise between the quantum Zeno enhancement and optical loss. In practice, optical components introduce loss. Assuming high quality, but commercially available optical components, we estimate a round trip transmission of $\|\alpha\|^{2}\sim 0.96$ [24] per cycle for both the outer loop ($\alpha_{out}$), the QZI loop ($\alpha_{QZI}$) and initial and final optics ($\alpha_{init,final}$). Hence the total transmission efficiency of the OAM spectrometer becomes $P(l_{0})\approx\alpha^{(2N+2)(l_{0}+1)}\cos\left(\frac{\pi}{2N}\right)^{2Nl_{0}}$ for the $l_{0}$-th order OAM component. We plot in Fig. 2(b) the $P(l_{0})$ vs. $N$ for OAM components $l_{0}=0-10$. With increasing $N$, the quantum Zeno effect leads to an increase in $P(l_{0})$, while loss leads to a decrease in $P(l_{0})$. As a result, an optimal $N$ is found at about $7-8$ for high order OAM components. Note that the extinction ratio between different OAM states remains infinite even in the presence of loss. Crosstalk would only take place when the OAM filter is not completely opaque to nonzero OAM states. To take into account imperfect OAM filters, we derive below the general expression for the transmission efficiency and extinction ratio, with finite $N$ and optical loss. We consider the OAM filter having a complex transmission coefficient $\sqrt{T(l)}e^{i\phi(l)}$ for the $l$th OAM component. If the state $\|\psi\rangle=\|H,l_{0}\rangle$ enters the QZI, after $N$ cycles, it exits the QZI loop in a polarization superposition state $p_{H}\|H\rangle+p_{V}\|V\rangle$ [25], where $p_{H}$ and $p_{V}$ are given by: $\left(\begin{array}[]{c}p_{H}\\\ p_{V}\\\ \end{array}\right)=\alpha_{QZI}^{N}\left[\left(\begin{array}[]{cc}1&0\\\ 0&\sqrt{T(l)}e^{i\phi(l)}\\\ \end{array}\right)\left(\begin{array}[]{cc}\cos\left(\frac{\pi}{2N}\right)&\sin\left(\frac{\pi}{2N}\right)\\\ \sin\left(\frac{\pi}{2N}\right)&-\cos\left(\frac{\pi}{2N}\right)\\\ \end{array}\right)\right]^{N}\left(\begin{array}[]{c}1\\\ 0\\\ \end{array}\right).$ (1) The pulse re-enters the outer loop at PBS3 with probability $\|p_{H}\|^{2}$, corresponding to a successful interrogation by the QZI (if $l_{0}\neq 0$). With probably $\|p_{V}\|^{2}$, the pulse exits toward the detector, corresponding to an error (if $l_{0}\neq 0$). The total loss of this QZI interrogation is $\|loss\|^{2}=1-\|p_{H}\|^{2}-\|p_{V}\|^{2}$. In the outer loop, the OAM value of the pulse is lowered by $1$ via the VPP, and the intensity of the pulse is reduced by a factor $\|\alpha_{out}\|^{2}$ per loop. Therefore, the probability of detecting the OAM eigenstate $l_{0}$ in the $l$th time interval (or, measured as with OAM $l\hbar$) is given by: $P(l;l_{0})=\|\alpha_{init,final}\|^{2}\|p_{V}(l_{0}-l)\|^{2}\prod_{m=l_{0}-l+1}^{l_{0}}\left(\|\alpha_{out}\|^{2}\|p_{H}(m)\|^{2}\right).$ (2) And we define the extinction ratio $\eta$ as: $\eta(l_{0})=P(l_{0};l_{0})/\sum_{l\neq l_{0}}P(l;l_{0}).$ (3) With an imperfect OAM filter, light with nonzero OAM has a finite probability of transmitting through the filter in vertical polarization after the $N$th QZI-loop. It will then be switched to horizontal polarization by P1 and exit at a time interval corresponding to components with a lower OAM. Consequently, the extinction ratio is reduced. If the light is transmitted through the filter before the $N$th loop, it will results in a larger loss. An imperfect OAM filter may also partially block light with zero OAM, which which also results in loss. Figure 3(a) shows, per quantum Zeno interrogation of light with OAM of $l\hbar$, the probabilities of the light exiting toward the detector ($\|p_{V}\|^{2}$), re-entering the outer-loop ($\|p_{H}\|^{2}$) and being lost ($1-\|p_{V}\|^{2}-\|p_{H}\|^{2}$). These probabilities are plotted as a function of transmission $T(l,a_{0})$ and $N$. The crossing of $\|p_{H}\|^{2}$ and $\|p_{V}\|^{2}$ separates the regimes when the interrogation result is more likely (to the right side) or less likely (to the left side) to be correct than incorrect. Fig. 3: The probabilities of different outcomes of a QZI interrogation as a function of the transmission of the OAM filter, neglecting optical loss. The blue solid line represents detecting OAM=0, the red dashed line is detecting OAM$\neq 0$, and the orange dotted line, loss. (a) $N=8$. (b) $N=2-10$. Fig. 4: Transmission of the pinhole spatial filter (a) as a function of the normalized aperture size $a_{0}$, for OAM components with $l_{0}=0-3$ and (b) as a function of $l_{0}$ with $a_{0}=0.8$. Fig. 5: (a) Extinction ratio $\eta$ as a function of the number of loops $N$ for various losses $\|\alpha\|^{2}$. Solid symbols are for $l_{0}=1$ and open symbols are for $l_{0}=3$. $l_{0}>3$ are essentially indistinguishable from $l_{0}=3$. For the $l_{0}=0$ case, the extinction ratio is over a 1000 for all $\|\alpha\|^{2}$ values because no premature measurements are possible. The additional green crosses labeled as $\|\alpha\|^{2}=0.95^{*}$ represents $\|\alpha\|^{2}=0.96$ but including misalignment of the OAM filter and VPP as discussed in the text. (b) Extinction ratio $\eta$ as a function of the normalized aperture size $a_{0}$ for $l_{0}=6$, $\Delta l=1-3$, $N=8$, and $\|\alpha\|^{2}=0.96$. Skipping OAM states increases the extinction ratio by orders of magnitude. As a practical example of an imperfect OAM filter, we consider a pinhole spatial filter. Light with OAM of $l\hbar\neq 0$ has zero intensity at the center of the beam, while light without OAM has maximum intensity at the center. Hence a very simple pinhole efficiently distinguishes light with and without OAM. The intensity distribution of a Laguerre-Gaussian beam, a paraxial beam possessing OAM $l\hbar$, is given by [1]: $I_{LG}(l;\rho)=\frac{I_{0}}{\int_{0}^{\infty}duu^{\|l\|}e^{-u}L_{\|l\|}(u)}\left(\frac{\sqrt{2}\rho}{w_{0}}\right)^{\|l\|}L_{\|l\|}\left(\frac{2\rho}{w_{0}^{2}}\right)e^{-\frac{\rho^{2}}{w_{0}^{2}}}$ (4) Where $L_{l}(x)$ is the $l$th order Laguerre Polynomial. Thus the transmission $T(l)$ through a pinhole with a radius $a_{0}$ (normalized by the waist of the $l_{0}=0$ Gaussian beam) is: $T(l,a_{0})=\left.{\int_{0}^{a_{0}}\int_{0}^{2\pi}\rho d\rho d\phi I_{LG}(l;\rho)}\right/{\int_{0}^{\infty}\int_{0}^{2\pi}\rho d\rho d\phi I_{LG}(l;\rho)}$ (5) Figure 4(a) shows $T(l,a_{0})$ vs. $a_{0}$ for $l=0-3$. The transmission decreases sharply with increasing $l$ when $a_{0}$ is smaller than $\sim 0.8$. Choosing $a_{0}=0.8$, we show in Fig. 4(b) the nearly exponential decrease of $T(l,a_{0})$ with $l$. These values are also marked by the red vertical lines in Fig. 3(a). Due to the fast decrease of $T(l,a_{0})$ from $l=0$ to $l\geq 1$, a large extinction ratio is readily achieved, which is very well approximated by: $\eta(l_{0})=P(l_{0};l_{0})/\sum P(l\neq l_{0};l_{0})\approx\frac{\alpha_{out}\|p_{V}(0)\|^{2}\|p_{H}(1)\|^{2}}{\|p_{V}(1)\|^{2}}.$ (6) $\eta$ is essentially the same for all OAM components, and it is mainly determined by how well the QZI can distinguish between states with OAM values $l_{0}=0$ and $l_{0}=1$. We plot in Fig. 5(a) $\eta$ vs. $N$ for $\|\alpha\|^{2}=0.9-1$. In general, $\eta$ increases with $N$ but decreases with $\|\alpha\|^{2}$, resulting in an optimal $N$ for each $\|\alpha\|^{2}<1$. Even for $\|\alpha\|^{2}=0.9$, $\eta>70$ can be reached with $N=7$. For $\|\alpha\|^{2}=0.96$, $\eta$ peaks at $\sim 180$. Fig. 6: (a)The probability of measuring an OAM value $l$ for a given input state $l_{0}$ (Equation 2), using pinhole as the OAM filter, $N=8$, $\|\alpha\|^{2}=0.96$, and misalignment of 10% and 1%, respectively, at the pinhole filter and VPP. Despite the decrease in probability for the diagonal elements at large $l_{0}$, the off diagonal elements decrease much faster, as implied by the large extinction ratios. (b) The diagonal elements of (a) as a function of $N$ for $l_{0}=0-10$. An additional source of error is due to the misalignment of the beam through two OAM-sensitive components: the OAM filter (e.g. a pinhole) and the VPP. Misalignment at the pinhole filter leads to reduced coupling efficiency of the zero OAM state, increased transmission of non-zero OAM states, and thus reduced extinction ratio. Misalignment on the VPP changes the desired OAM state into a superposition with neighboring OAM orders. However, these neighboring orders have very small amplitudes (e.g. $<1\%$ with $1\%$ misalignment) [4], and they are further filtered out through the QZI loop, resulting in negligible reduction in the extinction ratio. The main effect of misalignment at VPP is the slightly reduced transmission of the correct OAM state, hence reduced overall detection probability. We illustrate the effects of misalignment on the extinction ratio in Fig. 5(a) (the green crosses), assuming conservatively $10\%$ misalignment of the focused beam waist at the pinhole and $1\%$ misalignment of the collimated beam waist at the VPP. Extinction ratios over 100 are still readily achieved. The extinction ratio can be increased by many orders of magnitude if we only need to measure every other order, or every third order of OAM (Figure 5(b)). Correspondingly, we can choose smaller aperture sizes and a VPP that reduces the $l$ by $\Delta l=2$ or $3$ per passing. A smaller aperture size also introduces extra loss, but only in the final QZI on the zero OAM state, and thus only decreases the detection probability by about a factor of two. To evaluate the overall performance of the OAM spectrometer, we plot in Fig. 6(a) $P(l;l_{0})$ vs. $l$ and $l_{0}$ on the log scale, including loss and misalignment. The diagonal elements $P(l_{0};l_{0})$ correspond to correctly detecting an OAM component. They are two orders of magnitude higher than neighboring off diagonal elements, consistent with the high extinction ratios calculated before. In Fig. 6(b), we show $P(l_{0};l_{0})$ as a function of $N$ for different $l_{0}$. $N\sim 8$ gives the highest probability for detecting high order OAM components, while still maintaining an extinction ratio of above 100. In summary, we present a compact OAM spectrometer that disperses light of different OAM values in time. Loss is significant for high order OAM components with commercially available optical components. However, the high loss doesn’t have an appreciable effect on the signal to noise ratio; extinction ratios of $>\\!100$ are readily achieved even after taking into account optical loss and misalignments. The extinction ratio can be further improved by many orders of magnitude by skipping OAM orders, or by using a better OAM filter than a simple pinhole.	arxiv-papers	2014-01-11T19:46:03	2024-09-04T02:49:56.610401	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Paul Bierdz, Hui Deng", "submitter": "Paul Bierdz", "url": "https://arxiv.org/abs/1401.2559" }
1401.2645	General Mathematics Vol. xx, No. x (201x), xx–xx A Note On Multi Poly-Euler Numbers And Bernoulli Polynomials 111Received 08 Jun, 2009 Accepted for publication (in revised form) 29 November, 2013 Hassan Jolany, Mohsen Aliabadi, Roberto B. Corcino, and M.R.Darafsheh ###### Abstract In this paper we introduce the generalization of Multi Poly-Euler polynomials and we investigate some relationship involving Multi Poly-Euler polynomials. Obtaining a closed formula for generalization of Multi Poly-Euler numbers therefore seems to be a natural and important problem. 2010 Mathematics Subject Classification: 11B73, 11A07 Key words and phrases: Euler numbers, Bernoulli numbers, Poly-Bernoulli numbers, Poly-Euler numbers, Multi Poly-Euler numbers and polynomials ## 1 Introduction In the 17th century a topic of mathematical interst was finite sums of powers of integers such as the series $1+2+...+(n-1)$ or the series $1^{2}+2^{2}+...+(n-1)^{2}$.The closed form for these finite sums were known ,but the sum of the more general series $1^{k}+2^{k}+...+(n-1)^{k}$was not.It was the mathematician Jacob Bernoulli who would solve this problem.Bernoulli numbers arise in Taylor series in the expansion (1) $\begin{array}[]{c}\frac{x}{e^{x}-1}=\sum\limits_{n=0}^{\infty}B_{n}\frac{x^{n}}{n!}\end{array}.$ and we have, (2) $\begin{array}[]{c}S_{m}(n)=\sum_{k=1}^{n}k^{m}=1^{m}+2^{m}+\cdots+n^{m}={1\over{m+1}}\sum_{k=0}^{m}{m+1\choose{k}}B_{k}\;n^{m+1-k}\end{array}.$ and we have following matrix representation for Bernoulli numbers(for $n\in\mathbf{N}$),[1-4]. (3) $\displaystyle B_{n}$ $\displaystyle=\frac{(-1)^{n}}{(n-1)!}~{}\begin{vmatrix}\frac{1}{2}&\frac{1}{3}&\frac{1}{4}&\cdots\frac{1}{n}&~{}\frac{1}{n+1}\\\ 1&1&1&\cdots 1&1\\\ 0&2&3&\cdots{n-1}&n\\\ 0&0&\binom{3}{2}&\cdots\binom{n-1}{2}&\binom{n}{2}\\\ \vdots&~{}\vdots&~{}\vdots&\ddots~{}~{}\vdots&\vdots&\\\ 0&0&0&\cdots\binom{n-1}{n-2}&\binom{n}{n-2}\\\ \end{vmatrix}.$ Euler on page 499 of [5], introduced Euler polynomials, to evaluate the alternating sum (4) $\begin{array}[]{c}A_{n}(m)=\sum\limits_{k=1}^{m}(-1)^{m-k}k^{n}=m^{n}-(m-1)^{n}+...+(-1)^{m-1}1^{n}\end{array}.$ The Euler numbers may be defined by the following generating functions (5) $\begin{array}[]{c}\frac{2}{e^{t}\\!+\\!1}\;=\;\sum\limits_{{n=0}}^{\infty}E_{n}\frac{t^{n}}{n!}\end{array}.$ and we have following folowing matrix representation for Euler numbers, [1,2,3,4]. (6) $\displaystyle E_{2n}$ $\displaystyle=(-1)^{n}(2n)!~{}\begin{vmatrix}\frac{1}{2!}&1&~{}&~{}&~{}\\\ \frac{1}{4!}&\frac{1}{2!}&1&~{}&~{}\\\ \vdots&~{}&\ddots~{}~{}&\ddots~{}~{}&~{}\\\ \frac{1}{(2n-2)!}&\frac{1}{(2n-4)!}&~{}&\frac{1}{2!}&1\\\ \frac{1}{(2n)!}&\frac{1}{(2n-2)!}&\cdots&\frac{1}{4!}&\frac{1}{2!}\end{vmatrix}.$ The poly-Bernoulli polynomials have been studied by many researchers in recent decade. The history of these polynomials goes back to Kaneko. The poly- Bernoulli polynomials have wide-ranging application from number theory and combinatorics and other fields of applied mathematics. One of applications of poly-Bernoulli numbers that was investigated by Chad Brewbaker in [6,7,8,9], is about the number of $(0,1)$-matrices with $n$-rows and $k$ columns. He showed the number of $(0,1)$-matrices with $n$-rows and $k$ columns uniquely reconstructable from their row and column sums are the poly-Bernoulli numbers of negative index $B_{n}^{(-k)}$. Let us briefly recall poly-Bernoulli numbers and polynomials. For an integer $k\in\mathbf{Z}$, put (7) $\begin{array}[]{c}\operatorname{Li}_{k}(z)=\sum_{n=1}^{\infty}{z^{n}\over n^{k}}.\end{array}.$ which is the $k$-th polylogarithm if $k\geq 1$ , and a rational function if $k\leq 0$. The name of the function come from the fact that it may alternatively be defined as the repeated integral of itself . The formal power series can be used to define Poly-Bernoulli numbers and polynomials. The polynomials $B_{n}^{(k)}(x)$ are said to be poly-Bernoulli polynomials if they satisfy, (8) $\begin{array}[]{c}{Li_{k}(1-e^{-t})\over 1-e^{-t}}e^{xt}=\sum\limits_{n=0}^{\infty}B_{n}^{(k)}(x){t^{n}\over n!}\end{array}.$ In fact, Poly-Bernoulli polynomials are generalization of Bernoulli polynomials, because for $n\leq 0$, we have, (9) $\begin{array}[]{c}(-1)^{n}B_{n}^{(1)}(-x)=B_{n}(x)\end{array}.$ Sasaki,[10], Japanese mathematician, found the Euler type version of these polynomials, In fact, he by using the following relation for Euler numbers, (10) $\begin{array}[]{c}cosht=\sum\limits_{n=0}^{\infty}\frac{E_{n}}{n!}t^{n}\end{array}.$ found a poly-Euler version as follows (11) $\begin{array}[]{c}\frac{Li_{k}(1-e^{-4t})}{4tcosht}=\sum\limits_{n=0}^{\infty}E_{n}^{(k)}{t^{n}\over n!}\end{array}.$ Moreover, he by defining the following $L$-function, interpolated his definition about Poly-Euler numbers. (12) $\begin{array}[]{c}L_{k}(s)=\frac{1}{\Gamma(s)}\int_{0}^{\infty}t^{s-1}\frac{Li_{k}(1-e^{-4t})}{4(e^{t}+e^{-t})}dt\end{array}.$ and Sasaki showed that (13) $\begin{array}[]{c}L_{k}(-n)=(-1)^{n}n\frac{E_{n-1}^{(k)}}{2}\end{array}.$ But the fact is that working on such type of generating function for finding some identities is not so easy. So by inspiration of the definitions of Euler numbers and Bernoulli numbers, we can define Poly-Euler numbers and polynomials as follows which also A.Bayad [11], defined it by following method in same times. ###### Definition 1 (Poly-Euler polynomials):The Poly-Euler polynomials may be defined by using the following generating function, (14) $\begin{array}[]{c}\frac{2Li_{k}(1-e^{-t})}{1+e^{t}}e^{xt}=\sum\limits_{n=0}^{\infty}\mathbf{E}_{n}^{(k)}{t^{n}\over n!}\end{array}.$ If we replace $t$ by $4t$ and take $x=1/2$ and using the definition $cosht=\frac{e^{t}+e^{-t}}{2}$, we get the Poly-Euler numbers which was introduced by Sasaki and Bayad and also we can find same interpolating function for them (with some additional constant coefficient). The generalization of poly-logarithm is defined by the following infinite series (15) $\begin{array}[]{c}Li_{(k_{1},k_{2},...,k_{r})}(z)=\sum\limits_{m_{1},m_{2},...,m_{r}}\frac{z^{m_{r}}}{m_{1}^{k_{1}}...m_{r}^{k_{r}}}\end{array}.$ which here in summation ($0<m_{1}<m_{2}<...m_{r}$). Kim-Kim [12], one of student of Taekyun Kim introduced the Multi poly- Bernoulli numbers and proved that special values of certain zeta functions at non-positive integers can be described in terms of these numbers. The study of Multi poly-Bernoulli numbers and their combinatorial relations has received much attention in [6-13]. The Multi Poly-Bernoulli numbers may be defined as follows (16) $\begin{array}[]{c}\frac{Li_{(k_{1},k_{2},...,k_{r})}(1-e^{-t})}{(1-e^{-t})^{r}}=\sum\limits_{n=0}^{\infty}B_{n}^{(k_{1},k_{2},...,k_{r})}\frac{t^{n}}{n!}\end{array}.$ So by inspiration of this definition we can define the Multi Poly-Euler numbers and polynomials . ###### Definition 2 Multi Poly-Euler polynomials $\mathbf{E}_{n}^{(k_{1},...,k_{r})}(x)$, $(n=0,1,2,...)$ are defined for each integer $k_{1},k_{2},...,k_{r}$ by the generating series (17) $\begin{array}[]{c}\frac{2Li_{(k_{1},...,k_{r})}(1-e^{-t})}{(1+e^{t})^{r}}e^{rxt}=\sum\limits_{n=0}^{\infty}\mathbf{E}_{n}^{(k_{1},...,k_{r})}(x){t^{n}\over n!}\end{array}.$ and if $x=0$, then we can define Multi Poly-Euler numbers $\mathbf{E}_{n}^{(k_{1},...,k_{r})}=\mathbf{E}_{n}^{(k_{1},...,k_{r})}(0)$ Now we define three parameters $a,b,c$, for Multi Poly-Euler polynomials and Multi Poly-Euler numbers as follows. ###### Definition 3 Multi Poly-Euler polynomials $\mathbf{E}_{n}^{(k_{1},...,k_{r})}(x,a,b)$, $(n=0,1,2,...)$ are defined for each integer $k_{1},k_{2},...,k_{r}$ by the generating series (18) $\begin{array}[]{c}\frac{2Li_{(k_{1},...,k_{r})}(1-(ab)^{-t})}{(a^{-t}+b^{t})^{r}}e^{rxt}=\sum\limits_{n=0}^{\infty}\mathbf{E}_{n}^{{(k_{1},...,k_{r})}}(x,a,b){t^{n}\over n!}\end{array}.$ In the same way, and if $x=0$, then we can define Multi Poly-Euler numbers with $a,b$ parameters $\mathbf{E}_{n}^{(k_{1},...,k_{r})}(a,b)=\mathbf{E}_{n}^{(k_{1},...,k_{r})}(0,a,b)$. In the following theorem, we find a relation between $\mathbf{E}_{n}^{(k_{1},...,k_{r})}(a,b)$ and $\mathbf{E}_{n}^{(k_{1},...,k_{r})}(x)$ ###### Theorem 1 Let $a,b>0$, $ab\neq\pm 1$ then we have (19) $\begin{array}[]{c}\mathbf{E}_{n}^{(k_{1},k_{2},...,k_{r})}(a,b)=\mathbf{E}_{n}^{(k_{1},k_{2},...,k_{r})}\left(\frac{lna}{lna+lnb}\right)(lna+lnb)^{n}\end{array}.$ Proof.By applying the Definition 2 and Definition 3,we have $\displaystyle\frac{2Li_{(k_{1},...,k_{r})}(1-(ab)^{-t})}{(a^{-t}+b^{t})^{r}}$ $\displaystyle=\sum\limits_{n=0}^{\infty}\mathbf{E}_{n}^{{(k_{1},...,k_{r})}}(a,b){t^{n}\over n!}$ $\displaystyle=e^{rt\ln a}\frac{2Li_{(k_{1},...,k_{r})}(1-e^{-t\ln ab})}{(1+e^{t\ln ab})^{r}}$ So, we get $\displaystyle\frac{2Li_{(k_{1},...,k_{r})}(1-(ab)^{-t})}{(a^{-t}+b^{t})^{r}}=\sum\limits_{n=0}^{\infty}\mathbf{E}_{n}^{{(k_{1},...,k_{r})}}\left(\frac{\ln a}{\ln a+\ln b}\right)(\ln a+\ln b)^{n}{t^{n}\over n!}$ Therefore, by comparing the coefficients of $t^{n}$ on both sides, we get the desired result. $\square$ Now, In next theorem, we show a shortest relationship between $\mathbf{E}_{n}^{(k_{1},k_{2},...,k_{r})}(a,b)$ and $\mathbf{E}_{n}^{(k_{1},k_{2},...,k_{r})}$. ###### Theorem 2 Let $a,b>0$, $ab\neq\pm 1$ then we have (20) $\begin{array}[]{c}\mathbf{E}_{n}^{(k_{1},k_{2},...,k_{r})}(a,b)=\sum\limits_{i=0}^{n}r^{n-i}(\ln a+\ln b)^{i}(\ln a)^{n-i}\binom{n}{i}\mathbf{E}_{i}^{(k_{1},k_{2},...,k_{r})}\end{array}.$ Proof. By applying the Definition 2, we have, $\displaystyle\sum\limits_{n=0}^{\infty}\mathbf{E}_{n}^{{(k_{1},...,k_{r})}}(a,b){t^{n}\over n!}$ $\displaystyle=\frac{2Li_{(k_{1},...,k_{r})}(1-(ab)^{-t})}{(a^{-t}+b^{t})^{r}}$ $\displaystyle=e^{rt\ln a}\frac{2Li_{(k_{1},...,k_{r})}(1-e^{-t\ln ab})}{(1+e^{t\ln ab})^{r}}$ $\displaystyle=\left(\sum\limits_{k=0}^{\infty}\frac{r^{k}t^{k}(\ln a)^{k}}{k!}\right)\left(\sum\limits_{n=0}^{\infty}\mathbf{E}_{n}^{{(k_{1},...,k_{r})}}(\ln a+\ln b)^{n}\frac{t^{n}}{n!}\right)$ $\displaystyle=\sum\limits_{j=0}^{\infty}\left(\sum\limits_{i=0}^{j}r^{j-i}\frac{\mathbf{E}_{j}^{(k_{1},...,k_{r})}(\ln a+\ln b)^{i}(\ln a)^{j-i}}{i!(j-i)!}t^{j}\right)$ So, by comparing the coefficients of $t^{n}$ on both sides , we get the desired result. $\square$ By applying the definition 2, by simple manipulation, we get the following corollary ###### Corollary 1 For non-zero numbers $a,b$, with $ab\neq-1$ we have (21) $\begin{array}[]{c}\mathbf{E}_{n}^{(k_{1},...,k_{r})}(x;a,b)=\sum\limits_{i=0}^{n}\binom{n}{i}r^{n-i}\mathbf{E}_{i}^{(k_{1},...,k_{r})}(a,b)x^{n-i}\end{array}.$ Furthermore, by combinig the results of Theorem 2, and Corollary 1, we get the following relation between generalization of Multi Poly-Euler polynomials with $a,b$ parameters $\mathbf{E}_{n}^{{(k_{1},...,k_{r})}}(x;a,b)$, and Multi Poly-Euler numbers $\mathbf{E}_{n}^{{(k_{1},...,k_{r})}}$. (22) $\begin{array}[]{c}\mathbf{E}_{n}^{{(k_{1},...,k_{r})}}(x;a,b)=\sum\limits_{k=0}^{n}\sum\limits_{j=0}^{k}r^{n-k}\binom{n}{k}\binom{k}{j}(\ln a)^{k-j}(\ln a+\ln b)^{j}\mathbf{E}_{j}^{{(k_{1},...,k_{r})}}x^{n-k}\end{array}.$ Now, we state the ”Addition formula” for generalized Multi Poly-Euler polynomials ###### Corollary 2 (Addition formula) For non-zero numbers $a,b$, with $ab\neq-1$ we have (23) $\begin{array}[]{c}\mathbf{E}_{n}^{(k_{1},...,k_{r})}(x+y;a,b)=\sum\limits_{k=0}^{n}\binom{n}{k}r^{n-k}\mathbf{E}_{k}^{(k_{1},...,k_{r})}(x;a,b)y^{n-k}\end{array}.$ Proof. We can write $\displaystyle\sum\limits_{n=0}^{\infty}\mathbf{E}_{n}^{{(k_{1},...,k_{r})}}(x+y;a,b){t^{n}\over n!}$ $\displaystyle=\frac{2Li_{(k_{1},...,k_{r})}(1-(ab)^{-t})}{(a^{-t}+b^{t})^{r}}e^{(x+y)rt}$ $\displaystyle=\frac{2Li_{(k_{1},...,k_{r})}(1-(ab)^{-t})}{(a^{-t}+b^{t})^{r}}e^{xrt}e^{yrt}$ $\displaystyle=\left(\sum\limits_{n=0}^{\infty}\mathbf{E}_{n}^{{(k_{1},...,k_{r})}}(x;a,b){t^{n}\over n!}\right)\left(\sum\limits_{i=0}^{n}\frac{y^{i}r^{i}}{i!}t^{i}\right)$ $\displaystyle=\sum\limits_{n=0}^{\infty}\left(\sum\limits_{k=0}^{n}\binom{n}{k}r^{n-k}y^{n-k}\mathbf{E}_{k}^{{(k_{1},...,k_{r})}}(x;a,b)\right)\frac{t^{n}}{n!}$ So, by comparing the coefficients of $t^{n}$ on both sides , we get the desired result. $\square$ ## 2 Explicit formula for Multi Poly-Euler polynomials Here we present an explicit formula for Multi Poly-Euler polynomials. ###### Theorem 3 The Multi Poly-Euler polynomials have the following explicit formula (24) $\begin{array}[]{c}\mathbf{E}^{(k_{1},k_{2},\ldots,k_{r})}_{n}(x)=\sum\limits_{i=0}^{n}\sum\limits_{0\leq m_{1}\leq m_{2}\leq\ldots\leq m_{r}\atop c_{1}+c_{2}+\ldots=r}\sum\limits_{j=0}^{m_{r}}\frac{2(rx-j)^{n-i}r!(-1)^{j+c_{1}+2c_{2}+\ldots}(c_{1}+2c_{2}+\ldots)^{i}\binom{m_{r}}{j}\binom{n}{i}}{(c_{1}!c_{2}!\ldots)(m_{1}^{k_{1}}m_{2}^{k_{2}}\ldots m_{r}^{k_{r}})}.\end{array}.$ Proof. We have $Li_{(k_{1},k_{2},\ldots,k_{r})}(1-e^{-t})e^{rxt}=\sum_{0\leq m_{1}\leq m_{2}\leq\ldots\leq m_{r}}\frac{(1-e^{-t})^{m_{r}}}{m_{1}^{k_{1}}m_{2}^{k_{2}}\ldots m_{r}^{k_{r}}}e^{rxt}\qquad\qquad\qquad\qquad\qquad\qquad$ $\displaystyle=$ $\displaystyle\sum_{0\leq m_{1}\leq m_{2}\leq\ldots\leq m_{r}}\frac{1}{m_{1}^{k_{1}}m_{2}^{k_{2}}\ldots m_{r}^{k_{r}}}\sum_{j=0}^{m_{r}}(-1)^{j}\binom{m_{r}}{j}\sum_{n\geq 0}(rx-j)^{n}\frac{t^{n}}{n!}$ $\displaystyle=$ $\displaystyle\sum_{n\geq 0}\left(\sum_{0\leq m_{1}\leq m_{2}\leq\ldots\leq m_{r}}\sum_{j=0}^{m_{r}}\frac{(-1)^{j}(rx-j)^{n}\binom{m_{r}}{j}}{m_{1}^{k_{1}}m_{2}^{k_{2}}\ldots m_{r}^{k_{r}}}\right)\frac{t^{n}}{n!}.$ On the other hand, $\displaystyle\left(\frac{1}{1+e^{t}}\right)^{r}=$ $\displaystyle\left(\sum_{n\geq 0}(-1)^{n}e^{nt}\right)^{r}$ $\displaystyle=$ $\displaystyle\sum_{c_{1}+c_{2}+\ldots=r}\frac{r!(-1)^{c_{1}+2c_{2}+\ldots}}{c_{1}!c_{2}!\ldots}e^{t(c_{1}+2c_{2}+\ldots)}$ $\displaystyle=$ $\displaystyle\sum_{c_{1}+c_{2}+\ldots=r}\frac{r!(-1)^{c_{1}+2c_{2}+\ldots}}{c_{1}!c_{2}!\ldots}\sum_{n\geq 0}(c_{1}+2c_{2}+\ldots)^{n}\frac{t^{n}}{n!}$ $\displaystyle=$ $\displaystyle\sum_{n\geq 0}\left(\sum_{c_{1}+c_{2}+\ldots=r}\frac{r!(-1)^{c_{1}+2c_{2}+\ldots}(c_{1}+2c_{2}+\ldots)^{n}}{c_{1}!c_{2}!\ldots}\right)\frac{t^{n}}{n!}.$ Hence, $\frac{2Li_{(k_{1},k_{2},\ldots,k_{r})}(1-e^{-t})}{(1+e^{t})^{r}}e^{rxt}=2Li_{(k_{1},k_{2},\ldots,k_{r})}(1-e^{-t})e^{rxt}\left(\frac{1}{1+e^{t}}\right)^{r}\qquad\qquad\qquad\qquad\qquad\qquad$ $\displaystyle=\left(\sum_{n\geq 0}\left(\sum_{0\leq m_{1}\leq m_{2}\leq\ldots\leq m_{r}}\sum_{j=0}^{m_{r}}\frac{(-1)^{j}(rx-j)^{n}\binom{m_{r}}{j}}{m_{1}^{k_{1}}m_{2}^{k_{2}}\ldots m_{r}^{k_{r}}}\right)\frac{t^{n}}{n!}\right)\times$ $\displaystyle\;\;\;\;\times\left(\sum_{n\geq 0}\left(\sum_{c_{1}+c_{2}+\ldots=r}\frac{r!(-1)^{c_{1}+2c_{2}+\ldots}(c_{1}+2c_{2}+\ldots)^{n}}{c_{1}!c_{2}!\ldots}\right)\frac{t^{n}}{n!}\right)$ $\displaystyle=2\sum_{n\geq 0}\sum_{i=0}^{n}\left(\sum_{0\leq m_{1}\leq m_{2}\leq\ldots\leq m_{r}}\sum_{j=0}^{m_{r}}\frac{(-1)^{j}(rx-j)^{n-i}\binom{m_{r}}{j}}{m_{1}^{k_{1}}m_{2}^{k_{2}}\ldots m_{r}^{k_{r}}}\right)\frac{t^{n-i}}{(n-i)!}\times$ $\displaystyle\;\;\;\;\times\left(\sum_{c_{1}+c_{2}+\ldots=r}\frac{r!(-1)^{c_{1}+2c_{2}+\ldots}(c_{1}+2c_{2}+\ldots)^{i}}{c_{1}!c_{2}!\ldots}\right)\frac{t^{i}}{i!}$ $\displaystyle=2\sum_{n\geq 0}\sum_{i=0}^{n}\sum_{0\leq m_{1}\leq m_{2}\leq\ldots\leq m_{r}\atop c_{1}+c_{2}+\ldots=r}\sum_{j=0}^{m_{r}}\frac{(rx-j)^{n-i}r!(-1)^{j+c_{1}+2c_{2}+\ldots}(c_{1}+2c_{2}+\ldots)^{i}\binom{m_{r}}{j}\binom{n}{i}}{(c_{1}!c_{2}!\ldots)(m_{1}^{k_{1}}m_{2}^{k_{2}}\ldots m_{r}^{k_{r}})}\frac{t^{n}}{n!}$ By comparing the coefficient of $t^{n}/n!$, we obtain the desired explicit formula. ###### Definition 4 (Poly-Euler polynomials with $a,b,c$ parameters):The Poly-Euler polynomials with $a,b,c$ parameters may be defined by using the following generating function, (25) $\begin{array}[]{c}\frac{2Li_{k}(1-(ab)^{-t})}{a^{-t}+b^{t}}c^{xt}=\sum\limits_{n=0}^{\infty}\mathbf{E}_{n}^{(k)}(x;a,b,c){t^{n}\over n!}\end{array}.$ Now, in next theorem, we give an explicit formula for Poly-Euler polynomials with $a,b,c$ parameters. ###### Theorem 4 The generalized Poly-Euler polynomials with $a,b,c$ parameters have the following explicit formula (26) $\begin{array}[]{c}\mathbf{E}^{(k)}_{n}(x;a,b,c)=\\\ \sum\limits_{m=0}^{n}\sum\limits_{j=0}^{m}\sum\limits_{i=0}^{j}\frac{2(-1)^{m-j+i}}{j^{k}}\binom{j}{i}(x\ln c-(m-j+i+1)\ln a-(m-j+i+1)\ln b)^{n}.\end{array}.$ Proof. We can write $\displaystyle\sum_{n\geq 0}\mathbf{E}^{(k)}_{n}(x;a,b,c)\frac{t^{n}}{n!}=\frac{2Li_{k}(1-(ab)^{-t})}{a^{-t}((ab)^{-t}+1)}c^{xt}=2a^{-t}\left(\sum_{n\geq 0}(-1)^{n}(ab)^{-nt}\right)\left(\sum_{n\geq 0}\frac{\left(1-(ab)^{-t}\right)^{m}}{m^{k}}\right)c^{xt}.$ $\displaystyle=$ $\displaystyle a^{-t}\sum_{m\geq 0}\sum_{j=0}^{m}\sum_{i=0}^{j}\frac{2(-1)^{m-j+i}}{j^{k}}\binom{j}{i}(ab)^{-t(x+m-j+i)}c^{xt}$ $\displaystyle=$ $\displaystyle\sum_{m\geq 0}\sum_{j=0}^{m}\sum_{i=0}^{j}\frac{2(-1)^{m-j+i}}{j^{k}}\binom{j}{i}e^{-t(x+m-j+i)\ln(ab)}e^{-t\ln a}e^{xt\ln c}$ $\displaystyle=$ $\displaystyle\sum_{n\geq 0}\sum_{m\geq 0}\sum_{j=0}^{m}\sum_{i=0}^{j}\frac{2(-1)^{m-j+i}}{j^{k}}\binom{j}{i}\sum_{n\geq 0}(x\ln c-(m-j+i+1)\ln a-(m-j+i)\ln b)^{n}\frac{t^{n}}{n!}$ $\displaystyle=$ $\displaystyle\sum_{n\geq 0}\sum_{m=0}^{n}\sum_{j=0}^{m}\sum_{i=0}^{j}\frac{2(-1)^{m-j+i}}{j^{k}}\binom{j}{i}(x\ln c-(m-j+i+1)\ln a-(m-j+i)\ln b)^{n}\frac{t^{n}}{n!}.$ By comparing the coefficient of $t^{n}/n!$, we obtain the desired explicit formula.$\square$ ## References * [1] T. M. Apostol, On the Lerch Zeta function, Pacific. J. Math. no. 1, 1951, 161-167. * [2] G. Dattoli, S. Lorenzutta and C. Cesarano, Bernoulli numbers and polynomials from a more general point of view, Rend. Mat. Appl. Vol. 22, No.7, 2002, 193- 202\. * [3] H. Jolany, R. E. Alikelaye and S. S. Mohamad, Some results on the generalization of Bernoulli, Euler and Genocchi polynomials, Acta Universitatis Apulensis,No. 27,2011, pp. 299-306. * [4] S. Araci, M. Acikgoz and E. Şen, On the extended Kim’s p-adic q-deformed fermionic integrals in the p-adic integer ring, Journal of Number Theory 133 (2013) 3348-3361 * [5] L.Euler, Institutiones Calculi Differentialis, Petersberg,1755 * [6] C. Brewbaker, Lonesum (0,1)-matrices and poly-Bernoulli numbers of negative index, Master’s thesis, Iowa State University, 2005. * [7] M. Kaneko, Poly-Bernoulli numbers, J. Théorie de Nombres 9 (1997) 221–228. * [8] Y. Hamahata and H. Masubuchi, Recurrence formulae for multi-poly-Bernoulli numbers, Integers 7 (2007), A46. * [9] Hassan Jolany, Explicit formula for generalization of Poly-Bernoulli numbers and polynomials with a,b,c parameters, arXiv:1109.1387 * [10] Y. Ohno and Y. Sasaki, On poly-Euler numbers, preprint. * [11] A. Bayad, Y. Hamahata, Poly-Euler polynomials and Arakawa-Kaneko type zeta functions, preprint * [12] M.-S. Kim and T. Kim, An explicit formula on the generalized Bernoulli number with order n, Indian J. Pure Appl. Math. 31 (2000), 1455–1461. * [13] H. Jolany, M.R. Darafsheh, R.E. Alikelaye, Generalizations of Poly-Bernoulli Numbers and Polynomials, Int. J. Math. Comb. 2010, No. 2, 7-14 $\begin{array}[]{ll}\textrm{\bf Hassan Jolany}\\\ \textrm{Université des Sciences et Technologies de Lille}\\\ \textrm{UFR de Mathématiques}\\\ \textrm{Laboratoire Paul Painlevé}\\\ \textrm{CNRS-UMR 8524 59655 Villeneuve d'Ascq Cedex/France}\\\ \textrm{e-mail: [email protected] lille1.fr}\end{array}$ $\begin{array}[]{ll}\textrm{\bf Mohsen Aliabadi}\\\ \textrm{Department of Mathematics, Statistics and Computer Science,}\\\ \textrm{University of Illinois at Chicago, USA}\\\ \textrm{e-mail: [email protected]}\end{array}$ $\begin{array}[]{ll}\textrm{\bf Roberto B. Corcino}\\\ \textrm{Department of Mathematics}\\\ \textrm{Mindanao State University, Marawi City, 9700 Philippines}\\\ \textrm{e-mail: [email protected]}\end{array}$ $\begin{array}[]{ll}\textrm{\bf M.R.Darafsheh}\\\ \textrm{Department of Mathematics, Statistics and Computer Science }\\\ \textrm{Faculty of Science}\\\ \textrm{University of Tehran, Iran}\\\ \textrm{e-mail: [email protected]}\end{array}$	arxiv-papers	2014-01-12T17:22:46	2024-09-04T02:49:56.622269	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Hassan Jolany, Mohsen Aliabadi, Roberto B. Corcino, and M.R.Darafsheh", "submitter": "Hassan Jolany", "url": "https://arxiv.org/abs/1401.2645" }
1401.2656	# Linear Perturbation constraints on Multi-coupled Dark Energy Arpine Piloyan Valerio Marra Marco Baldi and Luca Amendola ###### Abstract The Multi-coupled Dark Energy (McDE) scenario has been recently proposed as a specific example of a cosmological model characterized by a non-standard physics of the dark sector of the universe that nevertheless gives an expansion history which does not significantly differ from the one of the standard $\Lambda$CDM model. Thanks to a dynamical screening mechanism, in fact, the interaction between the Dark Energy field and the Dark Matter sector is effectively suppressed at the background level during matter domination. As a consequence, background observables cannot discriminate a McDE cosmology from $\Lambda$CDM for a wide range of model parameters. On the other hand, linear perturbations are expected to provide tighter bounds due to the existence of attractive and repulsive fifth-forces associated with the dark interactions. In this work, we present the first constraints on the McDE scenario obtained by comparing the predicted evolution of linear density perturbations with a large compilation of recent data sets for the growth rate $f\sigma_{8}$, including 6dFGS, LRG, BOSS, WiggleZ and VIPERS. Confirming qualitative expectations, growth rate data provide much tighter bounds on the model parameters as compared to the extremely loose bounds that can be obtained when only the background expansion history is considered. In particular, the 95% confidence level on the coupling strength $\|\beta\|$ is reduced from $\|\beta\|\leq 83$ (background constraints only) to $\|\beta\|\leq 0.88$ (background and linear perturbation constraints). We also investigate how these constraints further improve when using data from future wide-field surveys such as supernova data from LSST and growth rate data from Euclid-type missions. In this case the 95% confidence level on the coupling further reduce to $\|\beta\|\leq 0.85$. Such constraints are in any case still consistent with a scalar fifth-force of gravitational strength, and we foresee that tighter bounds might be possibly obtained from the investigation of nonlinear structure formation in McDE cosmologies. ## 1 Introduction Understanding the fundamental origin of the observed accelerated expansion of the Universe [1, 2, 3, 4] represents the driving scientific case for a large number of complex and ambitious international initiatives planned for the next decade of cosmological observations, including e.g. the Baryon Oscillation Spectroscopic Survey [BOSS, 5], the Dark Energy Survey [DES, 6], the Large Synoptic Space Telescope [LSST, 7] and the ESA satellite mission Euclid111www.euclid-ec.org [8]. Besides an exquisite quality of observational data and a rigorous control of any possible systematics, such a challenging task will also require reliable predictions of how different theoretical scenarios might be scrutinised and possibly disentangled by efficiently combining different observational probes. More specifically, in order to discriminate a wide variety of Dark Energy (DE) or Modified Gravity (MG) models from the standard cosmological constant – presently assumed as the fiducial scenario – an appropriate combination of geometrical and dynamical observables is generally required, as competing models often feature strong degeneracies with the standard cosmological parameters when single observational probes are considered. In this respect, it is particularly instructive to investigate cosmological scenarios that are practically indistinguishable from the fiducial $\Lambda$CDM model as far as some particular observational probes are concerned, while showing characteristic features through other observational channels. While the most widely investigated alternatives to the cosmological constant such as Quintessence [9, 10], k-essence [11], phantom [12] and quintom [13] DE models, or more complex scenarios like interacting DE [14, 15, 16, 17], the Growing Neutrino model [18] and generic MG theories [as proposed e.g. by 19, 20, 21, 22, 23, 24] generally affect both the background and linear perturbations evolution of the universe, some specific realisations of these models were found to have the appealing (and challenging) feature of sharing the same expansion history of the $\Lambda$CDM cosmology to an extreme level of precision. Such models – which include e.g. the Hu & Sawicki realisation of $f(R)$ theories [21] and the Multi-coupled Dark Energy (McDE) scenario [25] discussed in the present work – represent an ideal benchmark to test the predictive power of future multi-probe observational surveys. In particular, the McDE model was proposed as a particular realisation of the more general framework discussed by Ref. [26] with the aim of providing an extension to the standard coupled Dark Energy (cDE) scenario in terms of a multi-particle nature of the CDM field, without introducing additional free parameters. In fact, the McDE model discussed in the present work is characterised by a hidden symmetry in the CDM sector associated with two distinct particle species with opposite couplings to a single DE scalar field, thereby requiring the same number of free parameters (a self-interaction potential slope $\alpha$ and a coupling constant $\beta$) as the widely investigated coupled Quintessence models. However, differently from the latter, the CDM internal symmetry that characterises McDE scenarios has been shown to provide a self-regulating mechanism of the effective interaction strength, thereby very effectively suppressing the DE-CDM interaction at the background level. Such background screening of the DE-CDM interaction has been qualitatively demonstrated by [25] and subsequently investigated in full detail by our team in [27]. In the latter paper, we provided the first direct comparison of McDE cosmologies with real observational data consisting of the supernova luminosities of the publicly available Union2.1 sample [28]. Confirming the previous qualitative results of [25], our analysis directly showed how present observational data on the background expansion history of the Universe are fully consistent with McDE scenarios even for very large values of the DE coupling, up to three orders of magnitude larger than the present bounds on the coupling for standard cDE models ($\beta\lesssim 0.1$, see e.g. [29, 30]). In fact, the conclusion of our previous paper was that the background dynamics could hardly offer any significant constraint on the McDE model and anticipated that more severe constraints could be obtained by working out the behavior of perturbations. This paper is devoted to such a task. Our primary goal is to derive observational constraints on the main parameters of the McDE model based on the latest available data on the growth of linear density perturbations, with the aim to significantly improve the extremely loose bounds derived through background observables, thereby reducing the viable parameter space. This task is also particularly relevant in view of the further extension of the investigation of the McDE scenario to the nonlinear regime of structure formation by means of dedicated N-body simulations. As the latter are in general quite computationally expensive, especially for large values of the coupling, constraining the viable region of the model’s parameter space through linear observables will avoid wasting precious computational resources. To this end, we will first derive the full set of linear perturbation equations in the McDE model and we will analytically solve them for the simplified cases of the background critical points in phase space. Then, we will integrate numerically the equations to obtain the full solution and check the analytical results. With the full numerical evolution of the linear perturbation growth at hand, we will finally compare the predictions obtained for different choices of model parameters with our sample of observational data by performing a detailed sampling of the parameter space. Our final result will be marginalised posterior bounds. As we will discuss below, such a procedure provides the most stringent constraints to date on this type of cosmological models. The paper is organised as follows. In Section 2 we introduce the full system of linear perturbation equations for the McDE scenario, in Section 3 we derive analytical solutions for the background critical points that determine a viable cosmological expansion history, and in Section 4 we compute the full numerical solutions of the system. In Section 5 we present the datasets adopted for our analysis and we describe the procedure for the direct comparison with observational data. In Section 6 we discuss the main results of our work and provide observational constraints on the McDE parameters. Finally, in Section 7 we conclude. Furthermore, in Appendix A we consider the effect of the uncoupled baryonic component showing that this does not significantly affect our main results. ## 2 Linear perturbation equations The McDE model, proposed and investigated by [25, 27, 31], is characterised by the existence of two different species of CDM particles with opposite couplings to the same classical DE scalar field. The system is therefore described by the following Lagrangian: $\displaystyle S=\int d^{4}x\sqrt{-g}\biggl{[}\frac{M_{Pl}^{2}}{2}R-\frac{1}{2}{\phi}^{;\alpha}\phi_{;\alpha}-\underset{\pm}{\sum}m_{\pm}e^{\pm\sqrt{\frac{2}{3}}\frac{\beta}{M_{Pl}}\phi}\bar{\psi}_{\pm}\psi_{\pm}-V(\phi)+\mathcal{L}_{r}\biggl{]}\,,$ (2.1) where $\phi$ is the dark energy scalar field, $\psi_{\pm}$ represent the two CDM fields, $\beta$ is a dimensionless parameter defining the strength of the interaction, and $\mathcal{L}_{r}$ is the radiation Lagrangian. In eq. (2.1) we have discarded the uncoupled baryonic component as its contribution does not significantly alter the results of our analysis. However, in the Appendix A we will drop this assumption and properly quantify the effect of baryons on our results. As we will see, these have a rather small impact, due to the low baryonic density observed today, which makes our simplified setup fully justified. We restrict our attention to a spatially flat FLRW metric. Perturbation equations corresponding to the model of eq. (2.1) on sub-horizon scales for each species of dark matter have been derived in [25] and read: $\displaystyle\ddot{\delta}_{-}$ $\displaystyle=-2H(1+\beta\frac{\dot{\phi}}{\sqrt{6}H})\dot{\delta}_{-}+4\pi G(\rho_{-}\Gamma_{A}\delta_{-}+\rho_{+}\Gamma_{R}\delta_{+})\,,$ (2.2) $\displaystyle\ddot{\delta}_{+}$ $\displaystyle=-2H(1-\beta\frac{\dot{\phi}}{\sqrt{6}H})\dot{\delta}_{+}+4\pi G(\rho_{-}\delta_{-}\Gamma_{R}+\rho_{+}\delta_{+}\Gamma_{A})\,,$ (2.3) where $\rho_{\pm}$ are the energy densities of the two CDM species, $\delta_{\pm}$ their respective density contrasts, and where the $\Gamma$ factors $\displaystyle\Gamma_{R}$ $\displaystyle=1-\frac{4}{3}\beta^{2}\,,$ (2.4) $\displaystyle\Gamma_{A}$ $\displaystyle=1+\frac{4}{3}\beta^{2}\,$ (2.5) encode the effects of repulsive ($R$) and attractive ($A$) fifth-forces. It is convenient to rewrite these equations employing the $e$-folding $N$ as the time variable, $N\equiv\ln a$, and to introduce the following dimensionless quantities: $\displaystyle x^{2}\equiv\frac{\dot{\phi}^{2}}{6M_{{\rm Pl}}^{2}H^{2}}\,,$ $\displaystyle\qquad y^{2}\equiv\frac{V}{3M_{{\rm Pl}}^{2}H^{2}}\,,$ (2.6) $\displaystyle z_{\pm}^{2}\equiv\frac{\rho_{\pm}}{3M_{{\rm Pl}}^{2}H^{2}}\,,$ $\displaystyle\qquad r^{2}\equiv\frac{\rho_{r}}{3M_{{\rm Pl}}^{2}H^{2}}\,.$ (2.7) The linear perturbations equations with respect to $N$ then read $\displaystyle\delta_{-}^{{}^{\prime\prime}}+\left[2(1+\beta x)-\frac{1}{2}(3-3y^{2}+3x^{2}+r^{2})\right]\delta^{\prime}_{-}$ $\displaystyle=\frac{3}{2}(z_{-}^{2}\Gamma_{A}\delta_{-}+z_{+}^{2}\Gamma_{R}\delta_{+})\,,$ (2.8) $\displaystyle\delta_{+}^{{}^{\prime\prime}}+\left[2(1-\beta x)-\frac{1}{2}(3-3y^{2}+3x^{2}+r^{2})\right]\delta^{\prime}_{+}$ $\displaystyle=\frac{3}{2}(z_{-}^{2}\Gamma_{R}\delta_{-}+z_{+}^{2}\delta_{+}\Gamma_{A})\,,$ (2.9) where we have made use of the background equation $\frac{H^{\prime}}{H}=-\frac{1}{2}(3-3y^{2}+3x^{2}+r^{2})\,.$ (2.10) The background behavior was studied and compared to observations in [27], where it was shown that the background evolution on a spatially flat FLRW metric is characterized by several critical-points defined as solutions for which $x,y,z_{\pm}$ are constant. These are summarised in Table 1. Among these, two critical points (point 2 and point 5) represent accelerated stable solutions, and two are metastable (saddle points) matter-dominated solutions (point 3 and point 4): viable cosmologies should connect one of the two matter eras to one of the two accelerated regimes. As points 3 and 4 are metastable points, their occurrence strongly depends on the initial conditions; the occurrence of the stable points, on the contrary, depends only on the values of the parameters $\alpha,\beta$. Therefore, by choosing the parameters in the appropriate range we will obtain either point 2 (in which dark energy dominates) or 5 (in which dark energy and dark matter coexist) as final state. The metastable matter eras will instead generally both occur for the same parameters, one after the other. However, point 3 will always be the last point before dark energy domination, since the matter density dilutes more slowly for point 3 than for point 4 (i.e. the equation of state is smaller for point 3, see Table 1). Therefore, if the initial conditions are set far enough in the past we expect the phase relative to point 4 to end very early and not to affect the late-time observations (supernovae and growth rate) considered in this work. In the following we will therefore always assume that we can neglect the possible point 4 matter era. In the next Sections we will first solve the perturbation equations analytically on the background critical points, and then numerically along the full trajectory. ## 3 Analytical solutions of the perturbation equations Point \| $x$ \| $y$ \| $z_{+}$ \| $z_{-}$ \| $\Omega_{{\rm DE}}$ \| $w_{eff}$ \| $\mu$ ---\|---\|---\|---\|---\|---\|---\|--- 1 \| $\pm 1$ \| 0 \| 0 \| 0 \| 1 \| 1 \| 0 2 \| $\frac{\alpha}{3}$ \| $\frac{1}{3}\sqrt{9-\alpha^{2}}$ \| 0 \| 0 \| 1 \| $-1+\frac{2\alpha^{2}}{9}$ \| 0 3 \| 0 \| 0 \| $\frac{1}{\sqrt{2}}$ \| $\frac{1}{\sqrt{2}}$ \| 0 \| 0 \| $0$ 4 \| $-\frac{2\beta}{3}$ \| 0 \| $\sqrt{1-\frac{4\beta^{2}}{9}}$ \| 0 \| $\frac{4\beta^{2}}{9}$ \| $\frac{4\beta^{2}}{9}$ \| 1 5 \| $\frac{3}{2(\alpha+\beta)}$ \| $\frac{\sqrt{9+4\alpha\beta+4\beta^{2}}}{2\|\alpha+\beta\|}$ \| $\frac{\sqrt{-9+2\alpha\beta+2\alpha^{2}}}{\sqrt{2}\|\alpha+\beta\|}$ \| 0 \| $\frac{9+2\alpha\beta+2\beta^{2}}{2(\alpha+\beta)^{2}}$ \| $\frac{-\beta}{(\alpha+\beta)}$ \| $1$ Table 1: Critical points for background equations provided in [27]. Only the physical solutions for $x,y,z_{+},z_{-}$ are selected. Only points 2 and 5 can have accelerated expansion ($w_{{\rm eff}}<-1/3$). On the particular background solutions corresponding to the critical points of Table 1, the perturbation equations become constant-coefficient linear equations and, therefore, exactly solvable. Thus, the analytical solutions of perturbations on the background critical points will approximate the full solutions in the matter era (point 3) and final accelerated stage (point 2 or point 5), respectively, as shown in the following sections. We define the total matter perturbation as: $\delta=\frac{\Omega_{-}\delta_{-}+\Omega_{+}\delta_{+}}{\Omega_{-}+\Omega_{+}}\,,$ (3.1) and, correspondingly, the total and partial growth rates as: $f=\frac{\delta^{{}^{\prime}}}{\delta}\,,\qquad f_{\pm}=\frac{\delta_{\pm}^{{}^{\prime}}}{\delta_{\pm}}\,.$ (3.2) We consider only one quadrant of the parameter plane, $\left\\{\alpha>0,\beta>0\right\\}$, since due to the symmetry of the system it is sufficient to solve equations with positive values of parameters. Indeed, the sign of the coupling is completely irrelevant and all our constraints will refer to its absolute value $\|\beta\|$. The analytical solutions of eq. (2.8) are presented in Table 2 for each of the five critical points. Only the growing solutions of the dark matter components and total density perturbation, $f_{\pm}$, $f$, respectively, are selected. For point 3 the perturbations $\delta_{+}$ and $\delta_{-}$ have opposite signs and compensate each other so that the total growth function $f$ is always unity. Nonetheless, near the transition between point 3 and point 2 (or point 5) the symmetry between the background densities of the two CDM species, that is ensured by the matter-dominated attractor, starts to be violated, which gives rise to the solution $f=f_{-}+3$ in this region. This can be justified if we write $f$ in more detail, as: $\displaystyle f=\frac{\delta^{{}^{\prime}}}{\delta}=\left[\frac{\Omega_{-}\delta_{-}+\Omega_{+}\delta_{+}}{\Omega_{-}+\Omega_{+}}\right]^{{}^{\prime}}/\left[\frac{\Omega_{-}\delta_{-}+\Omega_{+}\delta_{+}}{\Omega_{-}+\Omega_{+}}\right]\,$ $\displaystyle=\frac{\Omega_{-}\delta_{-}^{{}^{\prime}}+\Omega_{+}\delta_{+}^{{}^{\prime}}}{\Omega_{-}\delta_{-}+\Omega_{+}\delta_{+}}+\frac{\Omega_{-}^{{}^{\prime}}\delta_{-}+\Omega_{+}^{{}^{\prime}}\delta_{+}}{\Omega_{-}\delta_{-}+\Omega_{+}\delta_{+}}-\frac{\Omega_{-}^{{}^{\prime}}+\Omega_{+}^{{}^{\prime}}}{\Omega_{-}+\Omega_{+}}\,.$ (3.3) At point 3, the first term on the second line of eq. (3.3) is unity as long as $\Omega_{-}\delta_{-}$ and $\Omega_{+}\delta_{+}$ exactly compensate each other. However, as soon as $\Omega_{-}\neq\Omega_{+}$ (i.e. when the system is about to move out of point 3) it behaves as $f_{-}$. In this case the second term is also different from zero. In fact, analytical calculations show that $\Omega_{-}-\Omega_{+}\sim e^{3N}$ when the system is about to go out of point 3, which leads to: $\frac{\Omega_{-}^{{}^{\prime}}\delta_{-}+\Omega_{+}^{{}^{\prime}}\delta_{+}}{\Omega_{-}\delta_{-}+\Omega_{+}\delta_{+}}\rightarrow 3\,,$ (3.4) as $\delta_{-}\approx-\delta_{+}$. Finally, the last term is zero at each critical point. It is important to mention that the growth function $f$ passes through a singularity as the total perturbation $\delta$ goes through zero. However, the observable is $\delta^{\prime}/\delta_{0}$ (see eq. (5.6)), which is never singular. For a more complete picture, we present the contour plots of the growth functions for point 2 and point 5 in the left and right panels of Fig. 1, respectively. From these contour plots we conclude that larger values of the McDE characteristic parameters correspond to larger growth rates; moreover, for a significant portion of the parameter space, the growth rate on the background critical point 2 is zero. Figure 1: Left panel: Contour plots of the analytical solution for $f_{+}$ for parameters in the stability region of point 2. $f_{-}$ is always zero in this region. Right panel: The same plot but for the stability region of point 5. Point \| $f_{+}$ \| $f_{-}$ \| $f$ ---\|---\|---\|--- 1 \| max$[-\frac{1}{2}-2\beta\>,\>0]$ \| $-\frac{1}{2}+2\beta$ or $0$ \| $f_{-}$ 2 \| max$[\frac{1}{3}(-6+\alpha^{2}+2\alpha\beta)\>,\>0]$ \| max$[\frac{1}{3}(-6+\alpha^{2}-2\alpha\beta)\>,\>0]$ \| $f_{+}$ 3 \| max$[\frac{1}{4}\left(-1+\sqrt{1+32\beta^{2}}\right),1]$ \| max$[\frac{1}{4}\left(-1+\sqrt{1+32\beta^{2}}\right),1]$ \| $f_{-}+3$ 4 \| $\frac{1}{12}\left(-3-4\beta^{2}+\sqrt{225-216\beta^{2}-112\beta^{4}}\right)$ \| max$[f_{+},-\frac{1}{2}+2\beta^{2},0]$ \| $f_{-}$ 5 \| $\frac{1}{4}(-1-\frac{3\beta}{(\alpha+\beta)}+\Delta)$ \| max$[f_{+},-5+\frac{9\alpha}{2(\alpha+\beta)},0]$ \| $f_{-}$ Table 2: Growth functions for each species and total growth function at the background critical points. Here $\Delta=\sqrt{\frac{4\alpha\beta(5+8\beta^{2})+\alpha^{2}(25+32\beta^{2})-4(27+35\beta^{2})}{(\alpha+\beta)^{2}}}$. ## 4 Numerical solutions of the perturbation equations In this section we will illustrate the numerical solutions of the perturbed system of equations (2.8) and (2.9) defined in Section 2 together with the background equations. Our numerical results show a very good agreement with the analysis of Section 3. We will restrict the range of the model’s parameters to the stability and acceleration regions of the background critical points of Table 1 [see Ref. 27]. As a first test of our numerical integration, we display in Fig. 2 the total growth function of McDE with $\alpha=0.1$ and $\beta=0$ (i.e. for the uncoupled case, solid curve) and the growth function $f_{\Lambda CDM}$ for the standard $\Lambda$CDM model given by $f_{\Lambda CDM}=\Omega_{m}^{\gamma}$ with $\gamma=0.54$ (dashed curve). The two curves show very good agreement, as expected for a standard Quintessence model with a shallow self-interaction potential (indeed the limiting case of $\alpha=0$, $\beta=0$ exactly corresponds to a $\Lambda$CDM cosmology). Figure 2: Plot of $f$ (solid curve) when $\alpha=0.1$ and $\beta=0$, and of $f_{\Lambda\,CDM}=\Omega_{m}^{0.54}$ (dashed curve). Figure 3: The left panel is the plot of $\Omega_{i}$ when $\alpha=0.1$ and $\beta=0.5$. In this panel, solid curves correspond to $\Omega_{DE}$ and dashed curves correspond to $\Omega_{-}$ and $\Omega_{+}$. The right panel is the plot of $f$ for $\alpha=0.1$ and $\beta=$0.5 (blue curve), 3.5 (orange curve), 4 (red curve). The horizontal lines are the analytical values for the matter era (dot-dashed) and the accelerated point 2 (dashed). Figure 4: The top-left panel is the plot of $\Omega_{i}$ when $\alpha=0.9$ and $\beta=3$. In this panel, solid curves correspond to $\Omega_{DE}$ and dashed curves correspond to $\Omega_{-}$ and $\Omega_{+}$. The other panels are plots of $f_{-},f_{+},f$ for $\alpha=0.9$ and $\beta=$0.5 (blue solid curve), 3.5 (orange solid curve), 4 (red solid curve). The dashed horizontal lines are the analytical values for the matter era (dot-dashed) and the accelerated point 2 (dashed). As mentioned earlier, we set initial conditions far enough into the past such that the point 4 matter era has already decayed away, leaving only point 3. As we can see in Table 2, on point 3 there are two possible growth rates, unity and $\frac{1}{4}\left(-1+\sqrt{1+32\beta^{2}}\right)$. Let us now define the initial adiabaticity parameter: $A_{{\rm ic}}=\frac{\Omega_{-}\delta_{-i}}{\Omega_{+}\delta_{+i}}=\frac{1-\mu}{1+\mu}\frac{\delta_{-i}}{\delta_{+i}}\,,$ (4.1) where $\mu$ is the asymmetry parameter [see Ref. 25] defined as $\mu=\frac{\Omega_{+}-\Omega_{-}}{\Omega_{+}+\Omega_{-}}\,,$ (4.2) and the contrasts $\delta_{\pm i}$ are evaluated at the initial time. We see that $A_{{\rm ic}}=1$ implies adiabatic initial conditions. We find that if one starts with $A_{{\rm ic}}=1$ at very high redshifts, then initially the growth rate equals unity. However, this trajectory is unstable for $\|\beta\|\geq\beta_{\rm G}=\sqrt{3}/2$ such that soon the growth rate moves to the second value $\frac{1}{4}\left(-1+\sqrt{1+32\beta^{2}}\right)$. If $\|\beta\|<\beta_{\rm G}$, instead, the growth rate remains stably at unity. That is, adiabatic fluctuations are unstable, as already found in Ref. [25], for $\|\beta\|\geq\beta_{\rm G}$. If instead one starts with $A_{{\rm ic}}$ substantially different from unity, then the growth rate goes directly to max$[1,\frac{1}{4}\left(-1+\sqrt{1+32\beta^{2}}\right)]$. It is instructive to find the transition point analytically. The full perturbation solutions of point 3 are the following: $\displaystyle\delta_{+}$ $\displaystyle=$ $\displaystyle\frac{\delta_{+i}}{2}\left[(1+A_{\rm ic})e^{f_{1}(N-N_{i})}+(1-A_{\rm ic})e^{f_{2}(N-N_{i})}\right]\,,$ (4.3) $\displaystyle\delta_{-}$ $\displaystyle=$ $\displaystyle\frac{\delta_{+i}}{2}\left[(1+A_{\rm ic})e^{f_{1}(N-N_{i})}-(1-A_{\rm ic})e^{f_{2}(N-N_{i})}\right]\,,$ (4.4) where $A_{\rm ic}$ and $\delta_{+i}$ are the initial adiabaticity and initial perturbation for the CDM species with positive coupling at $N_{i}$, respectively, and $f_{1}=1\,,\quad f_{2}=\frac{1}{4}(-1+\sqrt{1+32\beta^{2}})\,.$ (4.5) We can now find the point at which the second term starts to dominate, i.e. the transition time $N_{0}$ at which the two terms in eqs. (4.3) and (4.4) are equal. This is given by: $N_{0}-N_{i}=\frac{1}{f_{2}-f_{1}}\ln\left\|\frac{1+A_{\rm ic}}{1-A_{\rm ic}}\right\|\,.$ (4.6) Therefore, as expected, the transition point $N_{0}$ is very close to the initial time $N_{i}$, unless $A_{{\rm ic}}$ is extremely close to unity. For instance, when $\beta=1$ and the initial adiabaticity is $A_{\rm ic}=2$ the transition point occurs $0.5$ e-foldings after the initial time and if $A_{\rm ic}=1.01$ it occurs after $2.5$ e-foldings. The above derivation explicitly shows that adiabatic initial conditions will rapidly evolve into a non- adiabatic state for coupling values $\|\beta\|\geq\beta_{\rm G}$, while for $\|\beta\|<\beta_{\rm G}$ the initial adiabaticity is preserved. As a consequence, possible observational effects associated with the transition between adiabatic and non-adiabatic initial conditions would be relevant only if the transition occurred in the redshift range covered by observational data, which is a condition that requires a high level of fine-tuning of the model parameters. Therefore, in the following we will restrict to the case of non-adiabatic initial conditions and we will assume $A_{{\rm ic}}=2$, without loss of generality. The total growth rate $f$ is presented in the right panel of Fig. 3 for some values of parameters that lie within the stable range of point 2, comparing with the exact solutions on the critical points. We also present the evolution of background fractional densities for the same values of parameters in the left panel of Fig. 3. For the same values of $\beta$ but different $\alpha=0.9$ we plot the different growth rates $f_{-},f_{+},f$ in the last three panels of Fig. 4. Again, in the first panel of Fig. 4, we illustrate the corresponding evolution of background fractional densities $\Omega_{i}$. We stress here that plots for fractional densities are made for one set of parameters only as the other cases do not differ at the background level. In these two figures, parameters are chosen in the stable range of point 2. Similarly, for parameters within the stable region of point 5, we illustrate the growth rate and the background fractional density evolution in Fig. 5. The numerical integration of equations (2.8) and (2.9) together with the background equations shows a very interesting effect: during the observationally relevant range $z\approx 1$, the growth rate gets strongly enhanced when $\beta$ grows larger than $\beta_{\rm G}$, before being driven back to a small value when dark energy fully dominates. In Fig. 6 we illustrate this behavior plotting the quantity $\delta^{\prime}/\delta_{0}=f(z)G(z)$ (where $G(z)$ is the growth factor normalized to unity today), since this is the observational quantity we will compare our model to in the next section. As can be seen, the McDE behavior suddenly deviates from the $\Lambda$CDM case as $\|\beta\|$ grows larger than $\beta_{\rm G}$. This leads us to expect that for $\|\beta\|$ larger than $\beta_{\rm G}$ the model becomes rapidly inconsistent with observations and that growth rate data can place tight constraints on the coupling value, as indeed we are going to find in the next Section. Figure 5: As Fig. 4 but for the stable region of point 5. Figure 6: Left Panel: The ratio $\Xi\equiv\dfrac{f(z)G(z)\|_{{\rm McDE}}}{f(z)G(z)\|_{{\rm\Lambda CDM}}}$ as a function of the coupling $\beta$ at $z=\left\\{0.5\,,1\,,1.5\right\\}$ and for parameters $\alpha=0.1$ and $\Omega_{{\rm DE},0}=0.692$. As can be seen, the McDE behavior suddenly deviates from the $\Lambda$CDM case as $\|\beta\|$ grows larger than $\beta_{\rm G}$. Right Panel: evolution with respect to redshift of $\delta^{\prime}/\delta_{0}=f(z)G(z)$ for the McDE model with $\beta=1.1$, $\alpha=0.1$ and $\Omega_{{\rm DE},0}=0.692$ (red solid line) and for the $\Lambda$CDM model with the same dark-energy content (blue dotted line). The relative trend of the two curves at different redshifts explains the opposite behavior between the $z=0.5$ and the $z\geq 1$ curves shown in the left panel. ## 5 Comparison to observations In our previous work Ref. [27] we employed the Union2.1 Compilation [28] of Type Ia supernovae to constrain the background behavior of the McDE model. We found that supernova data can constrain the slope of the self-interaction potential $\alpha$, which is found to be bound to values $\leq 1.5$ at the $3\sigma$ confidence level. On the other hand, we found a flat posterior likelihood for the initial asymmetry parameter, $\mu_{in}$, which is therefore completely unconstrained by the data, and we derived an extremely loose bound $\|\beta\|\lesssim 83$ (at the $2\sigma$ confidence level) on the coupling parameter. This showed how efficient is the McDE model in mimicking the $\Lambda$CDM model at the background level. Here we will extend the previous analysis by confronting the McDE model with present growth rate data as well as forecasted future data from upcoming wide-field surveys: as we will see the study of linear perturbations in the McDE model will allow us to put much tighter constraints on the coupling parameter. We proceed by describing the different data sets entering our investigation, and the likelihood estimator adopted for the comparison with the theoretical predictions. ### 5.1 SN data At the background level, we will make use of two different SN datasets. The first is the Union2.1 Compilation [28] of 580 Type Ia SNe in the redshift range $z=0.015-1.414$. More precisely, we use the magnitude vs. redshift table (without systematic errors) publicly available at the Supernova Cosmology Project webpage. The second dataset corresponds instead to the forecasted sample of two years of observations by the Large Synoptic Survey Telescope and features a total of $10^{5}$ supernovae in the redshift range $z=0.1-1.0$ with the redshift distribution as given in [32]. We will refer to this dataset at the “LSST 100k” catalog. The predicted theoretical magnitudes are related to the luminosity distance $d_{L}$ by: $m(z)=5\log_{10}\frac{d_{L}(z)}{10\,\textrm{pc}}\,,$ (5.1) which is computed under the assumption of spatial flatness: $d_{L}(z)=(1+z)\int_{0}^{z}\frac{{\rm d}\tilde{z}}{H(\tilde{z})}\,.$ (5.2) The luminosity distance $d_{L}(z)$ is obtained by integrating numerically the background McDE equations as explained in Ref. [27]. The $\chi^{2}$ function, on which the likelihood analysis will be based, is then: $\chi_{SNIa}^{{}^{\prime}2}=\sum_{i}\frac{[m_{i}-m(z_{i})+\xi]^{2}}{\sigma_{i}^{2}}\,,$ (5.3) where the index $i$ labels the elements of the supernova dataset. The parameter $\xi$ is an unknown offset sum of the supernova absolute magnitudes, of $k$-corrections and other possible systematics. As usual, we marginalize the likelihood $L^{\prime}_{SNIa}=\exp(-\chi_{SNIa}^{{}^{\prime}2}/2)$ over $\xi$, such that $L_{SNIa}=\int{\rm d}\xi\,L^{\prime}_{SNIa}$, leading to a new marginalized $\chi^{2}$ function: $\chi_{SNIa}^{2}=S_{2}-\frac{S_{1}^{2}}{S_{0}}\,,$ (5.4) where we neglected a cosmology-independent normalizing constant, and the auxiliary quantities $S_{n}$ are defined as: $S_{n}\equiv\sum_{i}\frac{\left[m_{i}-m(z_{i})\right]^{n}}{\sigma_{i}^{2}}\,.$ (5.5) As $\xi$ is degenerate with $\log_{10}H_{0}$, we are effectively marginalizing also over the Hubble constant. ### 5.2 $f\sigma_{8}(z)$ data At the linear perturbation level, we will build the growth-rate likelihood using two different datasets. The first contains the latest data [see 33] from 6dFGS [34], LRG [35], BOSS [36], WiggleZ [37] and VIPERS [38]. The second dataset approximates instead the forecasted accuracy of a future Euclid-like mission and it has been obtained in Ref. [39]. These different growth-rate data are given as a set of values $d_{i}$ where $i=\left\\{\right.$6dFGS, LRG, BOSS, WiggleZ, VIPERS, Euclid$\left.\right\\}$ and where $d=f\sigma_{8}(z)=f(z)\sigma_{8}G(z)=\sigma_{8}\delta^{\prime}/\delta_{0}\,.$ (5.6) Let us denote our theoretical estimates as $t_{i}=\delta^{\prime}_{i}/\delta_{0}$, where $\delta$ indicates the total density perturbation. We can then build a $\chi^{2}$ function that reads: $\chi_{f\sigma_{8}}^{{}^{\prime}2}=\left(d_{i}-\sigma_{8}t_{i}\right)C_{ij}^{-1}\left(d_{j}-\sigma_{8}t_{j}\right)\,,$ (5.7) where $C_{ij}$ is the covariance matrix of the data. Since we do not know $\sigma_{8}$ and cannot use the standard estimates because they have been obtained assuming the standard $\Lambda$CDM model, we need to marginalize the likelihood $L^{\prime}_{f\sigma_{8}}=\exp(-\chi_{f\sigma_{8}}^{{}^{\prime}2}/2)$ over $\sigma_{8}$, such that $L_{f\sigma_{8}}=\int{\rm d}\sigma_{8}\,L^{\prime}_{f\sigma_{8}}$, leading to a new marginalized $\chi^{2}$ function: $\chi_{f\sigma_{8}}^{2}=S_{20}-\frac{S_{11}^{2}}{S_{02}}\,,$ (5.8) where we neglected a cosmology-independent normalizing constant, and the auxiliary quantities $S_{nm}$ are defined as: $\displaystyle S_{11}$ $\displaystyle=d_{i}C_{ij}^{-1}t_{j}\,,$ (5.9) $\displaystyle S_{20}$ $\displaystyle=d_{i}C_{ij}^{-1}d_{j}\,,$ (5.10) $\displaystyle S_{02}$ $\displaystyle=t_{i}C_{ij}^{-1}t_{j}\,.$ (5.11) We are effectively marginalizing also over the initial value of $\delta_{0}$ as the latter is degenerate with $\sigma_{8}$. ### 5.3 Full likelihood The full likelihood is based on the total $\chi^{2}$: $\chi_{{\rm tot}}^{2}=\chi_{SNIa}^{2}+\chi_{f\sigma_{8}}^{2}\,,$ (5.12) which depends on the three main parameters $\Omega_{{\rm DE},0},\alpha,\|\beta\|$ plus other parameters specifying the initial conditions, $A_{{\rm ic}},\mu_{{\rm in}},\delta_{\pm,{\rm in}},\delta^{\prime}_{\pm,{\rm in}}$. As discussed above, we will solve the perturbation equations with non-adiabatic initial conditions at very early times: $A_{{\rm ic}}=2$. This choice is conservative for coupling values $\|\beta\|\geq\beta_{\rm G}$ since adiabatic initial conditions would affect the observable quantities only in the highly fine-tuned case where the departure from adiabaticity occurs at very recent times, while for $\|\beta\|<\beta_{\rm G}$ any choice of $A_{{\rm ic}}$ is equivalent since adiabaticity is conserved. This together with the fact that we are effectively marginalizing over $\delta_{\pm,{\rm in}}$ implies that the likelihood $L_{{\rm tot}}=\exp(-\chi_{{\rm tot}}^{2}/2)$ depends very weakly on the initial conditions parameters, which we have therefore fixed for convenience to $A_{{\rm ic}}=2$, $\mu_{{\rm in}}=0$, $\delta^{\prime}_{+,{\rm in}}=\delta_{+,{\rm in}}=\exp N_{{\rm in}}$, $\delta_{-,{\rm in}}=A_{{\rm ic}}\frac{1+\mu}{1-\mu}\delta_{+,{\rm in}}$, $\delta^{\prime}_{-,{\rm in}}=\delta_{-,{\rm in}}$ , consistently with the evolution of perturbations during matter domination. ## 6 Results Figure 7: Best-fit McDE model (second and fourth columns of Table 3 for left and right panel, respectively) together with $f\sigma_{8}$ data points for present (left panel) and future (right panel) data. The $\Lambda$CDM curve is displayed for comparison and is relative to a $\Lambda$CDM model with the same $\Omega_{{\rm DE},0}$. As the likelihood is marginalized over $\sigma_{8}$, a possible vertical shift is inconsequential. Parameter \| Best Fit (SN+$f\sigma_{8}$) \| 95% c.l. \| Best Fit (LSST+Euclid) \| 95% c.l. \| Best Fit (SN) \| 95% c.l. ---\|---\|---\|---\|---\|---\|--- $\Omega_{{\rm DE},0}$ \| 0.734 \| $[0.684,0.824]$ \| 0.692 \| $[0.688,0.698]$ \| 0.719 \| $[0.680,0.765]$ $\alpha$ \| 0.66 \| $[0,1.36]$ \| 0.12 \| $[0,0.54]$ \| 0.62 \| $[0,1.01]$ $\beta$ \| 0.79 \| $[0,0.88]$ \| 0.03 \| $[0,0.85]$ \| 6.4 \| $[0,83]$ Table 3: Best-fit values and 95% confidence intervals for the parameters of the model discussed in this paper (for point 2 and point 5 together) when using the combined Union2.1 supernova dataset and the latest $f\sigma_{8}$ data (2st and 3nd columns), or using the combined LSST 100k supernova dataset and the forecasted Euclid-like $f\sigma_{8}$ data (4rd and 5th columns), while the 6th and 7th columns report the results from [27] when using only background observables and analyzing only point 2. Figure 8: On the Left: 1-dimensional marginalized posterior distributions (for point 2 and point 5 together) on the parameters $\\{\Omega_{{\rm DE},0},\alpha,\beta\\}$ when fitting the model of this paper to the Union2.1 SN Compilation (see Section 5.1) and the latest $f\sigma_{8}$ data (see Section 5.2). See second and third columns of Table 3 for best-fit values with 95% confidence intervals. On the Right: $1\sigma$, $2\sigma$ and $3\sigma$ confidence-level contours for the relevant 2-dimensional marginalized posterior distributions. The black squares mark the best-fit values. The degeneracy between $\Omega_{{\rm DE},0}$ and the parameters $\alpha,\beta$ makes values of $\Omega_{{\rm DE},0}\simeq 0.90$ possible at the $3\sigma$ level. The results obtained with the Union2.1 supernova dataset (see Section 5.1) and the latest $f\sigma_{8}$ data (see Section 5.2) are shown in Fig. 7 (left panel), Fig. 8 and in the 2nd and 3rd columns of Table 3. The best-fit values reported in the latter are relative to the full 3-dimensional likelihood. The last two columns of Table 3 report the constraints obtained in [27] when using only background data. As we can see, the effect of including growth rate data in the analysis is dramatic: the 2$\sigma$ confidence region for the coupling $\|\beta\|$ shrinks from $\|\beta\|\lesssim 83$ when using supernovae only to $\|\beta\|\lesssim 0.88$ when present growth rate data is included. This is indeed expected, since one of the main motivations for the introduction of the McDE model was to explore a case in which the $\Lambda$CDM expansion is followed closely while the perturbations deviate significantly and show new effects. Figure 9: On the Left: 1-dimensional marginalized posterior distributions (for point 2 and point 5 together) on the parameters $\\{\Omega_{{\rm DE},0},\alpha,\beta\\}$ when fitting the model of this paper to the LSST 100k SN dataset (see Section 5.1) and the forecasted Euclid-like $f\sigma_{8}$ data (see Section 5.2). See 4th and 5th columns of Table 3 for best-fit values with 95% confidence intervals. On the Right: $1\sigma$, $2\sigma$ and $3\sigma$ confidence-level contours for the relevant 2-dimensional marginalized posterior distributions. The black squares mark the best-fit values. The results of the combined LSST 100k supernova dataset (see Section 5.1) and the forecasted Euclid-like $f\sigma_{8}$ data (see Section 5.2) are shown in Fig. 7 (right panel), Fig. 9 and in the 4th and 5th columns of Table 3. These two forecasted catalogs are relative to a $\Lambda$CDM fiducial cosmology, using the best-fit cosmological parameters from the Planck Collaboration [40, Table 5, last column]. As we can see in the plots of Fig. 9, with such kind of data we expect to improve constraints on all parameters with respect to present-day data. The 95% limit on $\alpha$ is reduced by more than a factor of 2. The 95% constraint on $\|\beta\|$ is instead only marginally improved. This shows how McDE models might be difficult to constrain even with the exquisite quality of future Euclid-like data. The constraints on $\|\beta\|$ are not strongly improved because the deviation of the McDE growth rate with respect to the $\Lambda$CDM one does not scale linearly with $\beta$, but it has a step-like behavior, as displayed in Fig. 6, which shows indeed that the McDE growth rate mimics the $\Lambda$CDM model for $\|\beta\|\lesssim\beta_{\rm G}$, but departs from it for larger $\|\beta\|$ values. It is therefore very difficult to constrain the coupling parameter to values smaller than $\|\beta\|\sim\beta_{\rm G}$. Nonlinear clustering data may prove necessary to further constrain McDE models. In particular, a first analysis of the nonlinear evolution of structures within a McDE scenario has been attempted in [31], highlighting for the first time very specific effects like the halo fragmentation process and a peculiar shape of the distortion of the matter power spectrum at small scales. A more detailed investigation of such effects with higher-resolution N-body simulations is presently ongoing, and will be discussed in an upcoming paper. ## 7 Conclusions The Multi-coupled Dark Energy model has been recently proposed as a simple extension of the standard coupled Quintessence scenario with the intriguing feature of showing an effective screening of the interaction between Dark Energy and Cold Dark Matter particles, without requiring additional free parameters. As a consequence of such screening, the background evolution of the universe closely follows the standard $\Lambda$CDM expansion history even for very large values of the coupling constant. This effect makes the Multi- coupled Dark Energy scenario an ideal benchmark to test the discriminating power of present and future multi-probe observational surveys since it maximises the degeneracy with the standard cosmological model in all probes that test only the background cosmic evolution. In a previous paper [27] we have quantified such degeneracy by comparing the predicted expansion history of Multi-coupled Dark Energy models with real observational data consisting of the recent Union2.1 Compilation [28] of Type Ia supernovae, confirming that the background expansion history has a very low constraining power with respect to this scenario. The present paper represents the natural extension of the analysis performed in our previous work to the linear evolution of density perturbations, which are expected to show new physics because of the attractive and repulsive fifth-forces acting between Cold Dark Matter particles, a consequence of their individual coupling to the Dark Energy field. In order to compare the predicted behavior of linear density perturbations with both presently available and future forecasted data on the growth of structures, we have first derived the full system of perturbed dynamical equations at first order for a generic Multi-coupled Dark Energy cosmology, and analytically solved such set of equations on the few particular background solutions that we had identified as phase-space critical points of the system in our previous work. This has allowed us to obtain the exact solution for the linear growth rate in both the past matter-dominated epoch and future Dark Energy-dominated regime. Then, we have numerically computed the full solution of linear perturbation equations along the whole expansion history of the universe, for a wide range of model parameters, and compared the numerical solutions with the analytical ones in the appropriate regimes, finding excellent matching between the two. With our numerical solver at hand we have then performed a likelihood analysis by sampling the model’s parameter space and comparing the derived evolution with recent observational data of the growth rate, including data sets from 6dFGRS, LRG, BOSS, WiggleZ and VIPERS, as well as with future data consistent with the forecasted accuracy of the Euclid satellite. Our analysis has shown that – as expected – the growth of density perturbations can strongly constrain Multi-coupled Dark Energy scenarios, putting tight bounds on the coupling constant which is constrained to $\|\beta\|\lesssim 0.88$ and $\|\beta\|\lesssim 0.85$ at 95% confidence level when considering present and future data sets, respectively. Interestingly, we have also found that the evolution of linear density perturbations encoded by the growth rate shows a sharp deviation from the standard $\Lambda$CDM evolution when the coupling constant $\|\beta\|$ approaches and overcomes a critical value $\beta_{\rm G}=\sqrt{3}/2$, corresponding to the coupling value that determines fifth-forces with the same strength as standard gravitational interactions. Nonetheless, as our 95% confidence levels on the coupling directly show, presently available data at the linear level are not yet capable of excluding a coupling value of $\|\beta\|=\beta_{\rm G}$, and therefore cannot rule out scalar interactions of gravitational strength in the context of a Multi-coupled Dark Energy framework. The natural further extension of this analysis is then to investigate the effects of the Multi-coupled Dark Energy scenario in the nonlinear regime of structure formation by means of dedicated high-resolution N-body simulations, in order to highlight possible characteristic footprints of the model that might allow to further tighten its viable parameter space. Such task is ongoing and will be discussed in an upcoming dedicated paper. ## Acknowledgements L.A. and V.M. acknowledge financial support from DFG through the project TRR33 "The Dark Universe". M.B. is supported by the Marie Curie Intra European Fellowship “SIDUN" within the 7th Framework Programme of the European Commission. A.P. thanks DAAD for support. ## Appendix A Including a baryonic component ### A.1 Equations Figure 10: Left panel: The fractional densities $\Omega_{i}$ for $\alpha=0.1$ and $\beta=0.5$. Solid curves correspond to $\Omega_{DE}$,dot-dashed curves correspond to $\Omega_{b}$, dashed curves correspond to $\Omega_{-}$ and $\Omega_{+}$. Parameters are chosen in the stable range of point 2. Right panel: The evolution of $f$ for $\|\beta\|=0.5$ (blue), $3.5$ (orange), and $4$ (red). The dashed horizontal lines are the analytical values for the matter era and the accelerated point 2. We will now consider the case in which, besides dark matter, there is also a baryonic component which does _not_ couple to dark energy. We will fix the baryonic content of the universe to the best-fit value $\Omega_{b,0}=0.048$ from the Planck Collaboration [40, Table 5, last column]. The background equations will be modified as follows: $\displaystyle 2\frac{H^{\prime}}{H}$ $\displaystyle=-(3-3y^{2}+3x^{2}+r^{2})\,,$ (A.1) $\displaystyle x^{\prime}$ $\displaystyle=-x\frac{H^{\prime}}{H}-3x+\alpha y^{2}+\beta(z_{2}^{2}-z_{1}^{2})\,,$ (A.2) $\displaystyle y^{\prime}$ $\displaystyle=-y\frac{H^{\prime}}{H}-\alpha xy\,,$ (A.3) $\displaystyle z_{+}^{\prime}$ $\displaystyle=-z_{+}\frac{H^{\prime}}{H}-\frac{3}{2}z_{+}+\beta xz_{+}\,,$ (A.4) $\displaystyle z_{-}^{\prime}$ $\displaystyle=-z_{-}\frac{H^{\prime}}{H}-\frac{3}{2}z_{-}-\beta xz_{-}\,,$ (A.5) $\displaystyle r^{\prime}$ $\displaystyle=-r\frac{H^{\prime}}{H}-2r\,,$ (A.6) $\displaystyle z_{b}^{\prime}$ $\displaystyle=-z_{b}\frac{H^{\prime}}{H}-\frac{3}{2}z_{b}\,,$ (A.7) where $z_{b}^{2}\equiv\frac{\rho_{b}}{3M_{{\rm Pl}}^{2}H^{2}}\,.$ (A.8) The additional critical points, present when $\Omega_{b}\not=0$, are listed in Table 4. The linear perturbation equations, including baryons, read now: $\displaystyle\delta_{-}^{{}^{\prime\prime}}+(2(1+\beta x)-\frac{1}{2}(3-3y^{2}+3x^{2}+r^{2}))\delta^{\prime}_{-}$ $\displaystyle=\frac{3}{2}(z_{-}^{2}\Gamma_{A}\delta_{-}+z_{+}^{2}\Gamma_{R}\delta_{+}+z_{b}^{2}\delta_{b})\,,$ (A.9) $\displaystyle\delta_{+}^{{}^{\prime\prime}}+(2(1-\beta x)-\frac{1}{2}(3-3y^{2}+3x^{2}+r^{2}))\delta^{\prime}_{+}$ $\displaystyle=\frac{3}{2}(z_{-}^{2}\Gamma_{R}\delta_{-}+z_{+}^{2}\delta_{+}\Gamma_{A}+z_{b}^{2}\delta_{b})\,,$ (A.10) $\displaystyle\delta_{b}^{{}^{\prime\prime}}+(2-\frac{1}{2}(3-3y^{2}+3x^{2}+r^{2}))\delta^{\prime}_{b}$ $\displaystyle=\frac{3}{2}(z_{-}^{2}\delta_{-}+z_{+}^{2}\delta_{+}+z_{b}^{2}\delta_{b})\,.$ (A.11) The analytical solutions for the new background critical points are listed in Table 5. We plot the fractional densities for the new point 2 in the left panel of Fig.10, while in the right panel we display the numerical solutions for $f$ (solid curves) and its analytical predictions in matter domination (dot-dashed) and DE domination (dashed). The matter point 3 is modified by the additional non-zero $\Omega_{b}$. For instance, in the matter era the dot- dashed curve shows $\Omega_{b}=0.2$ and $\Omega_{\pm}=0.4$ (dashed curve) instead of $\Omega_{\pm}=0.5$. The growing solution found for the new critical point 6 in the matter era provided in Table 4 is $f=1\,{\rm or}\,\frac{1}{4}\left(-1+\sqrt{1+64z_{cr}^{2}\beta^{2}}\right)+3$ where $z_{cr}$, defined as $z_{cr}=\sqrt{\frac{1-z_{bI}^{2}}{2}}$, corresponds to the fractional density of each CDM species in the matter era, with $z_{bI}$ the initial baryonic fractional density. Note that the additional 3 in the expression $\frac{1}{4}\left(-1+\sqrt{1+64z_{cr}^{2}\beta^{2}}\right)+3$ appears for the same reason as explained in Section 3. This growing rate, in the previous case with no baryonic component (corresponding to $z_{cr}=\frac{\sqrt{2}}{2}$) goes back to the old solution (i.e. the present point 6 in Table 4 goes back to the previous point 3 in Table 1 in the appropriate limit). One can also see the differences between the numerical solutions with and without baryons by comparing Fig. 10 with Fig. 3. Point \| $x$ \| $y$ \| $z_{+}$ \| $z_{-}$ \| $z_{b}$ \| $\Omega_{{\rm DE}}$ \| $w_{eff}$ \| $\mu$ ---\|---\|---\|---\|---\|---\|---\|---\|--- 6 \| 0 \| 0 \| $\sqrt{\frac{1-z_{b}^{2}}{2}}$ \| $\sqrt{\frac{1-z_{b}^{2}}{2}}$ \| $z_{b}$ \| 0 \| 0 \| 0 7 \| $\frac{3}{2\alpha}$ \| $\frac{3}{2\alpha}$ \| 0 \| 0 \| $\sqrt{1-\frac{9}{2\alpha^{2}}}$ \| $\frac{9}{2\alpha^{2}}$ \| 0 \| 0 Table 4: Additional critical points for background equations (A.1)-(A.7) when uncoupled baryons are included. Only the physical solutions for $x,y,z_{+},z_{-},z_{b}$ are selected. Point \| $f_{+}$ \| $f_{-}$ \| $f$ \| $f_{b}$ ---\|---\|---\|---\|--- 6 \| $f$ \| $f$ \| $\frac{1}{4}\left(-1+\sqrt{1+64z_{cr}^{2}\beta^{2}}\right)+3$ \| 1 7 \| – \| – \| $\frac{1}{4}\left(-1+\sqrt{25+\frac{108}{\alpha^{2}}}\right)$ \| $\frac{1}{4}\left(-1+\sqrt{25+\frac{108}{\alpha^{2}}}\right)$ Table 5: Growth functions for additional critical points presented in Table 4. ### A.2 Results Parameter \| Best Fit (SN+$f\sigma_{8}$) \| 95% c.l. \| Best Fit (LSST+Euclid) \| 95% c.l. \| Best Fit (SN) \| 95% c.l. ---\|---\|---\|---\|---\|---\|--- $\Omega_{{\rm DE},0}$ \| 0.734 \| $[0.684,0.826]$ \| 0.692 \| $[0.688,0.698]$ \| 0.719 \| $[0.680,0.765]$ $\alpha$ \| 0.66 \| $[0,1.37]$ \| 0.12 \| $[0,0.54]$ \| 0.62 \| $[0,1.01]$ $\beta$ \| 0.88 \| $[0,0.98]$ \| 0.03 \| $[0.,0.93]$ \| 6.4 \| $[0,83]$ Table 6: As Table 3 but for the case where the uncoupled baryonic fraction is also included. The results of the combined Union2.1 supernova dataset (see Section 5.1) and the latest $f\sigma_{8}$ data (see Section 5.2) for the case of $\Omega_{b,0}=0.048$ are shown in Fig 11 and in Table 6 (2nd and 3rd columns). The results of the combined LSST 100k supernova dataset (see Section 5.1) and the forecasted Euclid-like $f\sigma_{8}$ data (see Section 5.2) for the case of $\Omega_{b,0}=0.048$ are shown in Fig. 12 and in Table 6 (4th and 5th columns). The last two columns of Table 6 report the constraints from [27] obtained by using only background data. In both cases, the constraints on the model parameters $\Omega_{{\rm DE},0},\alpha$ are basically unchanged, while the constraints on the coupling $\beta$ are slightly weakened, but without substantial modification. This is expected since in the limit where all matter is composed of uncoupled baryons, the value of $\|\beta\|$ becomes obviously irrelevant. Figure 11: On the Left: 1-dimensional marginalized posterior distributions (for point 2 and point 5 together) on the parameters $\\{\Omega_{{\rm DE},0},\alpha,\beta\\}$ when fitting the model of this paper with $\Omega_{b,0}=0.048$ to the Union2.1 SN Compilation (see Section 5.1) and the latest $f\sigma_{8}$ data (see Section 5.2). See Table 6 (2nd and 3rd columns) for best-fit values with 95% confidence intervals. On the Right: $1\sigma$, $2\sigma$ and $3\sigma$ confidence-level contours for the relevant 2-dimensional marginalized posterior distributions. The black squares mark the best-fit values. This plot should be compared to Fig. 8 where the baryonic content has been neglected. 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Cosmological parameters, arXiv:1303.5076.	arxiv-papers	2014-01-12T18:24:26	2024-09-04T02:49:56.629376	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Arpine Piloyan, Valerio Marra, Marco Baldi and Luca Amendola", "submitter": "Arpine Piloyan", "url": "https://arxiv.org/abs/1401.2656" }
1401.2768	$Id:espcrc2.tex,v1.22004/02/2411:22:11speppingExp$ International Journal of Information Processing M. C Hanumantharaju, et al., # Design of Novel Architectures and FPGA Implementation of 2D Gaussian Surround Function M. C Hanumantharaju and M. T Gopalakrishna Department of Information Science and Engineering, Dayananda Sagar College of Engineering, Bangalore 560 078, India, Contact: [email protected], [email protected] ###### Abstract A new design and novel architecture suitable for FPGA/ASIC implementation of a 2D Gaussian surround function for image processing application is presented in this paper. The proposed scheme results in enormous savings of memory normally required for 2D Gaussian function implementation. In the present work, the Gaussian symmetric characteristics which quickly falls off toward plus/minus infinity has been used in order to save the memory. The 2D Gaussian function implementation is presented for use in applications such as image enhancement, smoothing, edge detection and filtering etc. The FPGA implementation of the proposed 2D Gaussian function is capable of processing (blurring, smoothing, and convolution) high resolution color pictures of size up to $1600\times 1200$ pixels at the real time video rate of 30 frames/sec. The Gaussian design exploited here has been used in the core part of retinex based color image enhancement. Therefore, the design presented produces Gaussian output with three different scales, namely, 16, 64 and 128. The design was coded in Verilog, a popular hardware design language used in industries, conforming to RTL coding guidelines and fits onto a single chip with a gate count utilization of 89,213 gates. Experimental results presented confirms that the proposed method offers a new approach for development of large sized Gaussian pyramid while reducing the on-chip memory utilization. Keywords : Gaussian Surround Function, Hardware Architecture, FPGA, Verilog. ## 1 Introduction With the widespread use of technologies like digital television, internet streaming video and DVD video, Gaussian function has become an inevitable component of image/video processing [10] and pattern recognition. Hardware realization of 2D Gaussian surround function for image processing applications demands huge on-chip memory requirement, with massive computations. Further, these functions might not fit on a single FPGA device. This is due to the reason that Gaussian function has an exponential distribution with maximum entropy and implementation such functions using software schemes are complex from computation point of view. In addition, Gaussian function is a non- causal, which implies that function is symmetric about the origin in time domain. Gaussian filter implementation with smaller kernel size in order to blur an image have been reported by number of researchers [8]. However, design of Gaussian function for large kernel size requires enormous amount of resources on FPGA with incredible raise in the total equivalent gate count. The new approach for 2D Gaussian function design provides a technical solution appropriate for the broad range of applications, from image pre-processing to pattern recognition. It simplifies the computational complexity with considerable saving in the memory requirement. The work proposed here utilizes the symmetry property of Gaussian surround function in order to save hardware resource, particularly memory requirement on FPGA. The technical design presented in this work is highly focused on providing huge throughput to process images as well as motion pictures. The Gaussian surround function is designed to process high quality images with picture size exceeding $1600\times 1200$ using reduced memory and high speed computation. The rest of the paper is organized as follows: Section 2 presents the review of related work. Section 3 describes the proposed 2D Gaussian surround function design. Section 4 gives brief details of architecture development suitable for FPGA/ASIC implementation. Section 5 provides experimental results and discussions. Finally conclusion arrived at is presented in Section 6. ## 2 Related Work Image and Video Processing has been a very active field of research and development for over 20 years and many different systems and algorithms for image enhancement, restoration, filtering and smoothing have been proposed and developed. The core part in all these image processing techniques is the Gaussian function. Although a lot of research work has progressed in the development of Gaussian design for image processing application using software and hardware schemes, very little work seems to have been carried out in large sized Gaussian function design. In addition, there are no full-fledged implementations reported for 2D Gaussian surround function using FPGA or ASIC that demands the development of novel algorithms for high speed processing and the best possible memory utilization. The market demands for 2D Gaussian function design are very high. The design of 2D Gaussian function and architecture development [7] is not only challenging but also intellectually stimulating. These are the primary reasons why this work was undertaken by the present researcher. Jobson et al. [1] described the properties of center or surround retinex. In this work authors have claimed that the design of a Gaussian surround function with a space constant of 80 pixels is a reasonable compromise between dynamic range and rendition. Gaussian function design and its application to retinex based image enhancent with DSP implementation has been proposed by Glenn et al. [2]. DSPs can be employed for enhancement of images which provides some improvement compared to general purpose computers. DSPs can be programmed in different languages, such as assembly codes and C language. The Hardware knowledge is required for programming DSPs, However, it is much easier for designers to learn DSP programming compared with the other design choices. However, image enhancement algorithms involving Gaussian function developed for DSP implementation may not be parallelized without using multiple DSPs. The DSPs are well suited for implementation of floating point systems, while for ASICs and FPGAs, floating point operations are difficult to implement. The enhancement of 25-30 frames per second of large size video frames with $1024\times 1024$ pixel resolution is still not possible with DSPs. The image enhancement technique requires massive parallel processing capability in order to support real time enhancement of large video stream. Hiroshi et al. [4] proposed a real time retinex video image enhancement algorithm and its FPGA implementation. Although the retinex algorithm [9] has been used for video enhancement the Gaussian pyramid designed in this work is of size $3\times 3$. The authors have claimed that the architectures developed in this scheme are efficient and can handle color picture of size $1900\times 1200$ pixels at the real time video rate of 60 frames per sec. In this scheme, the computational cost of the algorithm depends on the number of processing layers while the maximum layers and iterations used are 5 and 30 respectively. The authors have not justified how high throughput has been achieved in spite of time consuming iterations to the tune of 30. Further, the algorithm uses HSV color space for the enhancement process. This leads to an additional computational cost with the maximum conversion error occurring in the conversion process from RGB to HSV. Tsung et al. [5] proposed algorithm and architecture design of human machine interaction in foreground detection of dynamic scene. The Gaussian function in this work is computed based on fixed point data. This method adapts 27 bits for the variance computation of which 7 bits represents integer portion and 20 bits are used for fraction part. Three look-up tables are used to index the exponential and divide value. However, the Gaussian density function developed here uses large memory and is not efficient from computation point of view. Design of novel algorithm and architecture for Gaussian based color image enhancement for real time applications have been proposed by Hanumantharaju et al. [3]. The $5\times 5$ Gaussian kernel exploited in this work performs the image enhancement operation efficiently. Although the medium sized kernel employed in this work operates on picture size of $1600\times 1200$ pixels based on window operation, resulting images are not satisfactory due to presence of halo artifacts in the reconstructed images. Design of large sized Gaussian function in place of $5\times 5$ kernel improves visual quality of the enhanced image. Further, this approach consumes enormous amount of FPGA resources. The proposed approach of Gaussian surround function design with reduced on-chip memory utilization is an ideal choice compared to kernel based implementations. In addition, design of large size Gaussian surround function and development of architecture suitable for FPGA/ASIC implementation is the first-of-its-kind in the literature. ## 3 Design of 2D Gaussian Surround Function The analog Gaussian function in 1D, 2D and ND is expressed as follows: $G_{1D}(x)=\frac{1}{\sqrt{2\Pi}\sigma}e^{-\frac{x^{2}}{2\sigma^{2}}}$ (1) $G_{2D}(x,y)=\frac{1}{\sqrt{2\Pi}\sigma}e^{-\frac{x^{2}+y^{2}}{2\sigma^{2}}}$ (2) $G_{ND}(\vec{x})=\frac{1}{\sqrt{2\Pi}\sigma}e^{-\frac{\left\|\vec{x}\right\|^{2}}{2\sigma^{2}}}$ (3) $\sigma$ indicates the width of the Gaussian function. According to statistics, if Gaussian probability density function is considered, than $\sigma$ is referred to as standard deviation and the square of it ($\sigma^{2}$) is called variance. The discrete version of 2D Gaussian surround function is given by Eqn. (4). $G_{n}(x,y)=K_{n}\times e^{-\frac{(x^{2}+y^{2})}{2\sigma^{2}}}$ (4) and $K_{n}$ is given by the Eqn. (5) $K_{n}=\frac{1}{\sum_{i=1}^{M}\sum_{j=1}^{N}{e^{-\frac{x^{2}+y^{2}}{2\sigma^{2}}}}}$ (5) where x and y signify the spatial coordinates, $M\times N$ represents the kernel size, n is preferred as 1, 2 and 3 since the Gaussian function is designed with three scales, namely, 16, 64, and 128. The spatial co-ordinate ’x’ shown in Eqn. (4) are derived from the vector x which is presented in Figure 1. The spatial co-ordinate ’y’ is obtained similar to that of co-ordinate ’x’. It may be observed from the Figures 1 and 2 that the co-ordinates of Gaussian surround function exhibit symmetry property around the origin. Therefore, the spatial co-ordinate for implementation of Gaussian surround functions can be easily stored in the memory. The graphical plots of Gaussian surround function with the scales of 16, 64 and 128 are shown in Figures 3(a), (b) and (c), respectively. Figure 1: Domain Specified by ’x’ Vector Transformed into Array ’x’ Figure 2: Domain Specified by ’y’ Vector Transformed into Array ’y’ (a) Gaussian Function with Small Scale (b) Gaussian Function with Medium Scale (c) Gaussian Function with Large Scale Figure 3: 2D Gaussian Surround Functions with Different Scales ## 4 Architectures of 2D Gaussian Surround Function The top level architecture of Gaussian surround function comprises counter, dual port ROM, multiplier, scaler and exponential. The overall architecture of 2D Gaussian surround function is presented in Figure 4. The signal description of complete design is shown in Table 1. The top design is called as ”gauss2D” its block diagram consists signals ”clk”, ”reset_n”, ”start” and ”gout [7:0]”. The signal ”gout [7:0]” is the Gaussian output which is produced at every rising edge of ”clk” signal. The signal ”reset_n” is the global reset signal which is used to reset the system during power on conditions. The hardware realization of ”gauss2D” functional modules in this work is based on adapting enormous amount of pipeline stages mainly to reduce the computation time. In addition to pipelining technique, parallel processing also has been employed to accelerate the Gaussian function. The Table 2, provides the memory specification of the dual port ROM for the Gaussian surround function of size $256\times 256$. The ROM address ranges between 00000000 (corresponds to decimal 0) and 11111111 (corresponds to decimal 255). The contents of the ROM for each location are specified in the second column of Table 2. As is evident from the Table 2, the ROM stores the first row of the spatial co-ordinate ’x’ presented in Figure 1. It is possible to produce the array ’x’, from the ROM contents since the elements are repeated in the second and subsequent rows of ’x’ co-ordinate. Similarly, it is easy to construct the array ’y’ presented in Figure 2 by incrementing the address (addr2 [7:0]) of ROM every 256 clock cycle. Table 1: Signal Description for the Gaussian Surround Function Design Signals \| Description ---\|--- clk \| This is the global clock signal reset_n \| Active low system reset start \| Asserted to initiate Gaussian function gout [7:0] \| Gaussian Output enable \| Asserted to initiate Counting cnt_out1 [7:0] and cnt_out2 [7:0] \| Counter Outputs addr1 [7:0] and addr2 [7:0] \| Address for Data selection in ROM dout1 [7:0] and dout2 [7:0] \| Output of Dual port ROM n1 [7:0] and n2 [7:0] \| Multiplier inputs of 8-bit result [15:0] \| Multiplier Output of 16-bit Table 2: Memory Specification for 2D Gaussian Surround Function Address \| Data ---\|--- 00000000 \| 10000000 (-128) 00000001 \| 10000001 (-127) 00000010 \| 10000010 (-126) 00000011 \| 10000011 (-125) \| \| \| 11111110 \| 01111110 (126) 11111111 \| 01111111 (127) Figure 4: Top Architecture of 2D Gaussian Surround Function The counters used in the present work is of width 8-bits and is shown in Figure 5. The output of these counters are fed as address input for ROM since the ROM expect the 8-bit address. The ROM address (addr1 [7:0]) increments at every rising edge of clock cycle. However, the other ROM address (addr2 [7:0]) increments while the addr1 reaches its maximum value. This approach produces the matrix ’x’ and ’y’ as described earlier. The architecture of the dual port ROM is shown in Figure 6 and its signal description is provided in Table 1. The output of ROM generates matrix ’x’ and ’y’ in a raster scan order. Figure 5: Design of 8-bit Counter with its Output as Address for Dual Port ROM Figure 6: Architecture of Dual Port ROM Figure 7: Top Architecture of $8\times 8$ Multiplier The multiplier design [6] presented in this work incorporates a high degree of parallel circuits and pipelining of five levels. The multiplier performs the multiplication of two 8-bits unsigned numbers n1 and n2 as shown Figure 7 with its signal description in Table 1. The multiplier of width $16\times 16$ may also employed for Gaussian function of larger size. The multiplier result is of width 16-bits. The detailed architecture for the multiplier is shown in Figure 8. The architecture utilizes many pipelined registers internally. Five pipelined stages are exploited in order to increase the processing speed. Figure 8: Detailed Architecture of Multiplier Design with Eight Pipeline Stages ## 5 Experimental Results and Discussions The proposed FPGA implementation of Gaussian surround function has been coded and tested in Matlab (Version 8.1) first in order to ensure the correct working of the algorithm. Subsequently, the complete system has been coded in Verilog HDL so that it may be implemented on an FPGA or ASIC. The proposed scheme has been coded in RTL compliant Verilog and the hardware simulation results have been obtained. The system simulation has been done using ModelSim (Version SE 6.4) and Synthesized using Xilinx ISE 9.1i. The algorithm has been implemented on Xilinx Virtex-II XC2VP40-7FG676 FPGA device. In the proposed work, Gaussian design developed is of size $256\times 256$ and further can be upgraded to any size without appreciable increase in the hardware. The functional modules comprising of Gaussian control, ROM, Multiplier, Adder and exponential of Gaussian surround function simulated using ModelSim is presented in Figure 9. (a) Start of Simulation for 2D Gaussian Surround Function (b) Waveforms for Multiplier Output (c) Waveforms for Adder Output (d) Waveforms for Gaussian Surround Function Output with Three Scales Figure 9: ModelSim Simulation Waveforms for 2D Gaussian Surround Function The Xilinx generated RTL schematic view of top module of ”gauss” is shown in Figure 10. The output of this schematic consists of g1 [7:0], g2 [7:0] and g3 [7:0] which is the Gaussian output with three scales namely, 16, 64 and 128. The detailed schematic produced by this top schematic is presented in Figure 11. There are seven modules of which multiplier, and exponent are replicated. The input to the exponent module is in the range of 0 to 4. Therefore, the exponent module is designed with the look-up table technique instead of designing it. Figure 10: RTL View of the Top Module ”gauss” Figure 11: Zoomed View of the Top Module ”gauss” Note: The module names are not readable in the zoomed view of the Xilinx RTL view. Therefore, in the generated Figure the module names are marked as follows: U1: Gaussian control, U2: Gaussian Memory (ROM), U3 and U4: Multiplier, U5: Adder, U6, U7 and U8: Scale Down Unit, U9, U10 and U11: Exponent The synthesis as well as place and route results of the Gaussian function is presented in Table 3. The Xilinx place route report for the entire Gaussian function comprising counter, ROM, Multiplier, adder and exponent is shown in Figure 12. Table 3: Device Utilization Summary Selected Device : XC2VP40-7FG676 --- Number of Slices \| 725 out of 19,392 \| 3% Number of Slice Flip Flops \| 1224 out of 38,784 \| 3% Number of 4 input LUTs \| 825 out of 38,784 \| 2% Number used as logic \| 713 \| Number used as Shift registers \| 94 \| Number of IOs \| 51 \| Number of bonded IOBs \| 51 out of 416 \| 12% Number of GCLKs \| 1 out of 16 \| 6% The timing summary for the design as reported by Xilinx ISE tool is as follows: 1\. Speed Grade : -7 2\. Minimum period: 4.483 ns (Maximum Frequency: 223.04 MHz) 3\. Minimum input arrival time before clock: 1.473 ns 4\. Maximum output required time after clock: 6.880 ns 5\. Clock period: 4.483 ns (frequency: 223.04 MHz) 6\. Total number of paths /destination ports: 26126 / 1922 7\. Delay: 4.483 ns (Levels of Logic = 18) Figure 12: Device Utilization Summary as Reported by Xilinx Tool Table 4: Design Mapped on Various FPGA Devices Device \| LUTs \| Gates \| Utilization \| Frequency ---\|---\|---\|---\|--- XC2S50-6TQ144 \| 825 out of 1536 \| 62,455 \| 53% \| 61.992 MHz XC2S200-6PQ208 \| 825 out of 4704 \| 62,455 \| 17% \| 61.992 MHz XC3S1600E-5FG484 \| 825 out of 29504 \| 89,213 \| 5% \| 160.458 MHz XC2VP40-7FG676 \| 825 out of 38784 \| 89,213 \| 2% \| 223.04 MHz ## 6 Conclusion The 2D Gaussian surround function has been designed for processing high resolution pictures of size $1600\times 1200$ at real time rate of 30 frames per second. The Gaussian function design developed is based on the symmetry property which consumes low on-chip memory. The 2D Gaussian function implementation is presented for use in applications such as image enhancement, smoothing, edge detection and filtering etc. The architectures developed was coded in Verilog, conforming to RTL coding guidelines used in industries and fits onto a single chip with a gate count utilization of 89,213 gates. Research work is in progress for employing the Gaussian surround function for Multiscale Retinex based color image enhancement. ## References * [1] Daniel J. Jobson, Zia-ur Rahman and Glenn A. Woodell, _A Multiscale Retinex for Bridging the Gap Between Color Images and the Human Observation of Scenes_ , IEEE Transactions on Image Processing, Vol. 6, No. 7, pp. 965-976, 1997. * [2] Glenn Hines, Zia-ur Rahman, Daniel Jobson and Glenn Woodell, _DSP Implementation of the Retinex Image Enhancement Algorithm_ , In Defense and Security, International Society for Optics and Photonics, pp. 13-24, 2004. * [3] M. C Hanumantharaju, M. Ravishankar, and D. R. Rameshbabu _Design of Novel Algorithm and Architecture for Gaussian Based Color Image Enhancement System for Real Time Applications_ , In Proceedings of International Conference on Advances in Computing, Communication, and Control (ICAC3), Vol. 361, pp. 595-608, 2013. * [4] Hiroshi Tsutsui, Hideyuki Nakamura, Ryoji Hashimoto, Hiroyuki Okuhata, and Takao Onoye, _An FPGA implementation of real-time retinex video image enhancement_ , In IEEE World Automation Congress (WAC), pp. 1-6, 2010. * [5] Tsung Han Tsai, Chung-Yuan Lin, and Sz-Yan Li, _Algorithm and Architecture Design of Human-Machine Interaction in Foreground Object Detection with Dynamic Scene_ , 2013. * [6] Seetharaman Ramachandran, _Digital VLSI Systems Design: A Design Manual for Implementation of Projects on FPGAs and ASICs Using Verilog_ , Springer, 2007. * [7] Donald G. Bailey, _Design for Embedded Image Processing on FPGAs_ , Wiley-IEEE Press, 2011. * [8] H. T Ngo, M.Z. Zhang, L. Tao, and V.K. Asari, _Design of a Digital Architecture for Real-time video, Enhancement based on Illuminance-Reflectance Model_ , In Circuits and Systems, 2006. MWSCAS 06. 49th IEEE International Midwest Symposium on, volume 1, pages 286 290, 2006. * [9] L. Tao and V. Asari, _Modified Luminance based MSR for Fast and Efficient Image Enhancement_ , In Applied Imagery Pattern Recognition Workshop, 2003, Proceedings. 32nd, pages 174 179, 2003. * [10] Ajoy K. Ray Tinku Acharya _Image Processing: Principles and Applications_ , Wiley-Interscience, 2005. M. C Hanumantharaju is currently a Associate Professor in the Department of Information Science & Engineering, Dayananda Sagar College of Engineering, Bangalore. He is currently persuing Ph.D at Visvesvaraya Technological University,Bel- gaum His research interests includes VLSI Architecture Development for Signal & Image Processing Applications, Synthesis & Optimization of ICs, DSP with FPGAs etc. M. T Gopalakrishna is currently a Associate Professor in the Department of Information Science & Engineering, Dayananda Sagar College of Engineering, Bangalore. He is currently persuing his Ph.D at Visvesvaraya Technological University, Belgaum. His Research interests includes Digital Image Processing & Computer Vision, Video Surveillance and Document Image Processing.	arxiv-papers	2014-01-13T09:52:08	2024-09-04T02:49:56.641881	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "M. C. Hanumantharaju and M. T. Gopalakrishna", "submitter": "Hanumantha Raju MC", "url": "https://arxiv.org/abs/1401.2768" }
1401.2864	# Quantum Theory in Design Zhengtang Tan, Shouchuan Zhang Department of Mathematics, Hunan University Changsha 410082, P.R. China, Emails: [email protected] ###### Abstract In this paper, we use braiding diagrams to present rules of shapes and designs. That is, we design colour, design size, design brightness, design codes by means of braiding. 2000 Mathematics Subject Classification: 16B50. Keywords: code, braiding, design. ## 0 Introduction The Yang-Baxter equation first came up in a paper by Yang as factorization condition of the scattering S-matrix in the many-body problem in one dimension and in the work of Baxter on exactly solvable models in statistical mechanics. It has been playing an important role in mathematics and physics ( see [BD82] , [YG89] ). Attempts to find solutions of The Yang-Baxter equation in a systematic way have led to the theory of quantum groups [Ka95]. Shape grammars have been used as a computational design tool for over two decades. Shape grammars are a production system created by taking a sample of the whole for which one is trying to write a language [St06]. From this sample a vocabulary of shapes can be written that represent all the basic forms of that sample. By defining the spatial relationships between those forms and how the forms are related to each other, shape rules can be written. A shape rule consists of a left and right side. If the shape in the left side matches a shape in a drawing then the rule can be applied, and the matching shape changes to match the right side of the rule. The shape rules allow the addition and subtraction of shapes, which in the end are perceived as shape modifications. These shape rules, combined with an initial shape, produce a shape grammar that represents the language of the design [St06]. Shapes themselves can exist as points, lines, planes, volumes, or any combination thereof [St06]. All shape generation must start with an initial shape: a point, a coordinate axis, or some foundation from which to start the shape grammar. If the grammar is going to end, it can end with a terminal rule, which prevents any other rules from being applied after it. This forces there to be closure in the rule sequence. Alternatively, a design sequence can continue indefinitely and designs could be chosen at any point in the design process. The method discussed here fundamentally changes the method of developing the shape grammar. Mackenzie [Ma69] demonstrates in a simplified case that if the fundamental shapes in the language are defined, the relationship between the shapes can be inferred through examples. These inferred relationships can then be mapped to trees which in turn can be used to automatically create shape grammar rules for that language. While this was shown effective in a particular use, it is more common in practice that the vocabulary of the design to be used as the foundation of a shape grammar is determined by the creator of the grammar. The creator of the grammar looks at the sample and subjectively derive the vocabulary. From that vocabulary, the rules are formed based upon the creator s experience and intention. It is quite possible that two different persons looking at the same sample of shapes would create two very different shape grammars. In [OCB08], the results from the principal component analysis are used to create a new coupe shape grammar based upon these discovered shape relationships. The shape grammar is then used to create new coupe vehicles. Although the focus of this paper is on vehicle design, the methods developed here are applicable to any class of physical products based on a consistent form language. In [Ho10], visual 3D spatial grammars are studied. In this paper we use braiding diagrams to represent rule of shapes and designs. This simplifies the expression of designs. The shape rules allow braiding of two shapes, which in the end are perceived as shape modifications. We represent new design by means of restricted PBW of Nichols algebras. That is, we designs in braided tensor categories. We design colour, design size, design brightness, design codes by means of braiding. This completely is a new method and will be applied more and more in various designs. ## 1 Preliminaries We begin with the tensor category (see [Ma98], [Ka95] and [Zh99]). We define the product ${\cal C}\times{\cal D}$ of two category ${\cal C}$ and ${\cal D}$ whose objects are pairs of objects $(U,V)\in(ob{\cal C},ob{\cal D})$ and whose morphisms are given by $Hom_{{\cal C}\times{\cal D}}((V,W)(V^{\prime},W^{\prime}))=Hom_{\cal C}(V,V^{\prime})\times Hom_{\cal D}(W,W^{\prime}).$ Let ${\cal C}$ be a category and $\otimes$ be a functor from ${\cal C}\times{\cal C}$ to ${\cal C}$. This means (i) we have object $V\otimes W$ for any $V,W\in ob{\cal C};$ (ii) we have morphism $f\otimes g$ from $U\otimes V$ to $X\otimes Y$ for any morphisms $f$ and $g$ from $U$ to $X$ and from $V$ to $Y$; (iii) we have $(f\otimes g)(f^{\prime}\otimes g^{\prime})=(ff^{\prime})\otimes(gg^{\prime})$ for any morphisms $f:U\rightarrow X,g:V\rightarrow Y,f^{\prime}:U^{\prime}\rightarrow V$ and $g^{\prime}:V^{\prime}\rightarrow V;$ (iv) $id_{U\otimes V}=id_{U}\otimes id_{V}.$ Let $\otimes\tau$ denote the functor from ${\cal C}\times{\cal C}$ to ${\cal C}$ such that $(\otimes\tau)(U,V)=(V\otimes U)$ and $(\otimes\tau)(f,g)=g\otimes f$, for any objects $U,V,X,Y$ in ${\cal C},$ and for any morphisms $f:U\rightarrow X$ and $g:V\rightarrow Y.$ An associativity constraint $a$ for tensor $\otimes$ is a natural isomorphism $a:\otimes(\otimes\times id)\rightarrow\otimes(id\times\otimes).$ This means that, for any triple $(U,V,W)$ of objects of ${\cal C}$, there is a morphism $a_{U,V,W}:(U\otimes V)\otimes W\rightarrow U\otimes(V\otimes W)$ such that $a_{U^{\prime},V^{\prime},W^{\prime}}((f\otimes g)\otimes h)=(f\otimes(g\otimes h))a_{U,V,W}$ for any morphisms $f,g$ and $h$ from $U$ to $U^{\prime}$, from $V$ to $V^{\prime}$ and from $W$ to $W^{\prime}$ respectively. Let $I$ be an object of ${\cal C}$. If there exist natural isomorphisms $l:\otimes(I\times id)\rightarrow id\hbox{ \ \ \ and \ \ \ }r:\otimes(id\times I)\rightarrow id\ ,$ then $I$ is called the unit object of ${\cal C}$ with left unit constraint $l$ and right unit constraint $r$. ###### Definition 1.1. $({\cal C},\otimes,I,a,l,r)$ is called a tensor category if ${\cal C}$ is equipped with a tensor product $\otimes$, with a unit object $I$, an associativity constraint $a$ , a left unit constraint $l$ and a right unit constraint $r$ such that the Pentagon Axiom and the Triangle Axiom are satisfied, i.e. $(id_{U}\otimes a_{V,W,X})a_{U,V\otimes W,X}(a_{U,V,W}\otimes id_{X})=a_{U,V,W\otimes X}a_{U\otimes V,W,X}$ and $(id_{V}\otimes l_{W})a_{V,I,W}=r_{V}\otimes id_{W}$ for any $U,V,W,X\in ob{\cal C}.$ Furthermore, if there exists a natural isomorphism $C:\otimes\rightarrow\otimes\tau$ such that the Hexagon Axiom holds, i.e. $a_{V,W,U}C_{U,V\otimes W}a_{U,V,W}=(id_{V}\otimes C_{U,W})a_{V,U,W}(C_{U,V}\otimes id_{W})$ and $a^{-1}_{W,U,V}C_{U\otimes V,W}a^{-1}_{U,V,W}=(C_{U,W}\otimes id_{V})a^{-1}_{U,W,V}(id_{U}\otimes C_{V,W}),$ for any $U,V,W\in ob{\cal C},$ then $({\cal C},\otimes,I,a,l,r,C)$ is called a braided tensor category. In this case, $C$ is called a braiding of ${\cal C}.$ If $C_{U,V}=C_{V,U}^{-1}$ for any $U,V\in ob{\cal C},$ then $({\cal C},C)$ is called a symmetric braided tensor category or a symmetric tensor category. Here functor $\tau:{\cal C}\times{\cal C}\rightarrow{\cal C}\times{\cal C}$ is the flip functor defined by $\tau(U\times V)=V\times U$ and $\tau(f\times g)=g\times f,$ for any $U,V\in{\cal C},$ and morphisms $f$ and $g.$ Note that we denote the braiding $C$ in braided tensor category $({\cal C},\otimes,I,a,l,r,C)$ by ${}^{\cal C}C$ sometimes. ###### Example 1.2. (The tensor category of vector spaces ) The most fundamental example of a tensor category is given by the category ${\cal C}={\cal V}ect(k)$ of vector spaces over field $k$. ${\cal V}ect(k)$ is equipped with tensor structure for which $\otimes$ is the tensor product of the vector spaces over $k$, the unit object $I$ is the ground field $k$ itself, and the associativity constraint and unit constraint are the natural isomorphisms $a_{U,V,W}((u\otimes v)\otimes w):=u\otimes(v\otimes w)\hbox{ \ \ and \ \ }l_{V}(1\otimes v):=v:=r_{V}(v\otimes 1)$ for any vector space $U,V,W$ and $u\in U,v\in V,w\in W.$ Furthermore, the most fundamental example of a braided tensor category is given by the tensor category ${\cal V}ect(k)$, whose braiding is usual twist map from $U\otimes V$ to $V\otimes U$ defined by sending $a\otimes b$ to $b\otimes a$ for any $a\in U,b\in V$. Now we define some notations. If $f$ is a morphism from $U$ to $V$ and $g$ is a morphism from $V$ to $W$, we denote the composition $gf$ by $\begin{array}[c]{l}{\hbox{\kern-3.95901pt${U}$}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){1000.0}}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){200.0}}\put(0.0,800.0){\line(0,1){200.0}}\put(0.0,500.0){\oval(1200.0,600.0)}\put(0.0,500.0){\makebox(0.0,0.0)[cc]{\scriptsize$gf$}}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){1000.0}}}\@arraycr{\hbox{\kern-5.41667pt${W}$}}\@arraycr\end{array}\kern 9.9945pt=\kern 9.9945pt\begin{array}[c]{l}{\hbox{\kern-3.95901pt${U}$}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){400.0}}\put(0.0,1600.0){\line(0,1){400.0}}\put(0.0,1000.0){\oval(1200.0,1200.0)}\put(0.0,1000.0){\makebox(0.0,0.0)[cc]{\scriptsize$f$}}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){400.0}}\put(0.0,1600.0){\line(0,1){400.0}}\put(0.0,1000.0){\oval(1200.0,1200.0)}\put(0.0,1000.0){\makebox(0.0,0.0)[cc]{\scriptsize$g$}}}\@arraycr{\hbox{\kern-5.41667pt${W}$}}\@arraycr\end{array}\ \ \ .$ We usually omit $I$ and the morphism $id_{I}$ in any diagrams. In particular, If $f$ is a morphism from $I$ to $V$, $g$ is a morphism from $V$ to $I$,we denote $f$ and $g$ by $\begin{array}[c]{l}\hbox{\put(0.0,0.0){\line(0,1){400.0}}\put(0.0,1000.0){\oval(1200.0,1200.0)}\put(0.0,1000.0){\makebox(0.0,0.0)[cc]{\scriptsize$f$}}}\@arraycr{\hbox{\kern-4.02777pt${V}$}}\@arraycr\end{array}\kern 9.9945pt\hbox{ , }\kern 9.9945pt\begin{array}[c]{l}{\hbox{\kern-4.02777pt${V}$}}\@arraycr\hbox{\put(0.0,1600.0){\line(0,1){400.0}}\put(0.0,1000.0){\oval(1200.0,1200.0)}\put(0.0,1000.0){\makebox(0.0,0.0)[cc]{\scriptsize$g$}}}\@arraycr\end{array}\ \ \ \ .$ If $f$ is a morphism from $U\otimes V$ to $P$, $g$ is a morphism from $U\otimes V$ to $I$ and $\zeta$ is a morphism from $U\otimes V$ to $V\otimes U$, we denote $f$, $g$ and $\zeta$ by $\begin{array}[c]{l}{\hbox{\kern-3.95901pt${U}$}}\kern 19.98901pt{\hbox{\kern-4.02777pt${V}$}}\@arraycr\hbox{\put(400.0,1500.0){\oval(800.0,800.0)[lb]}\put(1600.0,1500.0){\oval(800.0,800.0)[rb]}\put(0.0,1500.0){\line(0,1){500.0}}\put(2000.0,1500.0){\line(0,1){500.0}}\put(1000.0,0.0){\line(0,1){500.0}}\put(1000.0,1100.0){\oval(1200.0,1200.0)}\put(1000.0,1100.0){\makebox(0.0,0.0)[cc]{\scriptsize$f$}}}\@arraycr\kern 9.9945pt{\hbox{\kern-3.90451pt${P}$}}\@arraycr\end{array}\kern 9.9945pt,\kern 9.9945pt\begin{array}[c]{l}{\hbox{\kern-3.95901pt${U}$}}\kern 19.98901pt{\hbox{\kern-4.02777pt${V}$}}\@arraycr\hbox{\put(0.0,1600.0){\line(0,1){400.0}}\put(2000.0,1600.0){\line(0,1){400.0}}\put(400.0,1600.0){\oval(800.0,800.0)[lb]}\put(1600.0,1600.0){\oval(800.0,800.0)[rb]}\put(1000.0,1200.0){\oval(1200.0,1200.0)}\put(1000.0,1200.0){\makebox(0.0,0.0)[cc]{\scriptsize$g$}}}\@arraycr\end{array}\kern 9.9945pt,\kern 9.9945pt\begin{array}[c]{l}{\hbox{\kern-3.95901pt${U}$}}\kern 19.98901pt{\hbox{\kern-4.02777pt${V}$}}\@arraycr\hbox{\put(1000.0,1000.0){\oval(1400.0,1400.0)}\put(1000.0,1000.0){\makebox(0.0,0.0)[cc]{\scriptsize$\zeta$}} }\@arraycr{\hbox{\kern-4.02777pt${V}$}}\kern 19.98901pt{\hbox{\kern-3.95901pt${U}$}}\@arraycr\end{array}\kern 9.9945pt.$ In particular, we denote the braiding $C_{U,V}$ and its inverse $C_{U,V}^{-1}$ by $\begin{array}[c]{l}{\hbox{\kern-3.95901pt${U}$}}\kern 19.98901pt{\hbox{\kern-4.02777pt${V}$}}\@arraycr\hbox{}\@arraycr{\hbox{\kern-4.02777pt${V}$}}\kern 19.98901pt{\hbox{\kern-3.95901pt${U}$}}\@arraycr\end{array}\kern 9.9945pt,\kern 9.9945pt\begin{array}[c]{l}{\hbox{\kern-4.02777pt${V}$}}\kern 19.98901pt{\hbox{\kern-3.95901pt${U}$}}\@arraycr\hbox{}\@arraycr{\hbox{\kern-3.95901pt${U}$}}\kern 19.98901pt{\hbox{\kern-4.02777pt${V}$}}\@arraycr\end{array}\kern 9.9945pt.\\\ $ $\xi$ is called an $R$-matrix of ${\cal C}$ if $\xi$ is a natural isomorphism from $\otimes$ to $\otimes\tau$ and for any $U,V,W\in ob{\cal C}$, the Yang- Baxter equation of ${\cal C}$: (YBE): $\begin{array}[c]{l}{\hbox{\kern-3.95901pt${U}$}}\kern 19.98901pt{\hbox{\kern-4.02777pt${V}$}}\kern 19.98901pt{\hbox{\kern-5.41667pt${W}$}}\@arraycr\hbox{\put(1000.0,1000.0){\oval(1400.0,1400.0)}\put(1000.0,1000.0){\makebox(0.0,0.0)[cc]{\scriptsize$\xi$}} }\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\hbox{\put(1000.0,1000.0){\oval(1400.0,1400.0)}\put(1000.0,1000.0){\makebox(0.0,0.0)[cc]{\scriptsize$\xi$}} }\@arraycr\hbox{\put(1000.0,1000.0){\oval(1400.0,1400.0)}\put(1000.0,1000.0){\makebox(0.0,0.0)[cc]{\scriptsize$\xi$}} }\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr{\hbox{\kern-5.41667pt${W}$}}\kern 19.98901pt{\hbox{\kern-4.02777pt${V}$}}\kern 19.98901pt{\hbox{\kern-3.95901pt${U}$}}\@arraycr\end{array}\ \ =\ \begin{array}[c]{l}{\hbox{\kern-3.95901pt${U}$}}\kern 19.98901pt{\hbox{\kern-4.02777pt${V}$}}\kern 19.98901pt{\hbox{\kern-5.41667pt${W}$}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\hbox{\put(1000.0,1000.0){\oval(1400.0,1400.0)}\put(1000.0,1000.0){\makebox(0.0,0.0)[cc]{\scriptsize$\xi$}} }\@arraycr\hbox{\put(1000.0,1000.0){\oval(1400.0,1400.0)}\put(1000.0,1000.0){\makebox(0.0,0.0)[cc]{\scriptsize$\xi$}} }\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\hbox{\put(1000.0,1000.0){\oval(1400.0,1400.0)}\put(1000.0,1000.0){\makebox(0.0,0.0)[cc]{\scriptsize$\xi$}} }\@arraycr{\hbox{\kern-5.41667pt${W}$}}\kern 19.98901pt{\hbox{\kern-4.02777pt${V}$}}\kern 19.98901pt{\hbox{\kern-3.95901pt${U}$}}\@arraycr\end{array}\ $ holds. In particular, the above equation is called the Yang-Baxter equation on $V$ when $U=V=W.$ ###### Lemma 1.3. (i) The braiding $C$ of braided tensor category $({\cal C},C)$ is an $R$-matrix of ${\cal C}$, i.e. $\begin{array}[c]{l}{\hbox{\kern-3.95901pt${U}$}}\kern 19.98901pt{\hbox{\kern-4.02777pt${V}$}}\kern 19.98901pt{\hbox{\kern-5.41667pt${W}$}}\@arraycr\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\hbox{}\@arraycr\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr{\hbox{\kern-5.41667pt${W}$}}\kern 19.98901pt{\hbox{\kern-4.02777pt${V}$}}\kern 19.98901pt{\hbox{\kern-3.95901pt${U}$}}\@arraycr\end{array}\ \ =\ \begin{array}[c]{l}{\hbox{\kern-3.95901pt${U}$}}\kern 19.98901pt{\hbox{\kern-4.02777pt${V}$}}\kern 19.98901pt{\hbox{\kern-5.41667pt${W}$}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\hbox{}\@arraycr\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\hbox{}\@arraycr{\hbox{\kern-5.41667pt${W}$}}\kern 19.98901pt{\hbox{\kern-4.02777pt${V}$}}\kern 19.98901pt{\hbox{\kern-3.95901pt${U}$}}\@arraycr\end{array}\ $ holds. (ii) $\begin{array}[c]{l}{\hbox{\kern-3.95901pt${U}$}}\kern 19.98901pt{\hbox{\kern-4.02777pt${V}$}}\kern 19.98901pt{\hbox{\kern-5.41667pt${W}$}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\hbox{}\@arraycr\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\hbox{\put(400.0,1500.0){\oval(800.0,800.0)[lb]}\put(1600.0,1500.0){\oval(800.0,800.0)[rb]}\put(0.0,1500.0){\line(0,1){500.0}}\put(2000.0,1500.0){\line(0,1){500.0}}\put(1000.0,0.0){\line(0,1){500.0}}\put(1000.0,1100.0){\oval(1200.0,1200.0)}\put(1000.0,1100.0){\makebox(0.0,0.0)[cc]{\scriptsize$f$}}}\@arraycr{\hbox{\kern-5.41667pt${W}$}}\kern 29.98352pt{\hbox{\kern-3.90451pt${P}$}}\@arraycr\end{array}\kern 9.9945pt=\kern 9.9945pt\begin{array}[c]{l}{\hbox{\kern-3.95901pt${U}$}}\kern 19.98901pt{\hbox{\kern-4.02777pt${V}$}}\kern 9.9945pt{\hbox{\kern-5.41667pt${W}$}}\@arraycr\hbox{\put(400.0,1500.0){\oval(800.0,800.0)[lb]}\put(1600.0,1500.0){\oval(800.0,800.0)[rb]}\put(0.0,1500.0){\line(0,1){500.0}}\put(2000.0,1500.0){\line(0,1){500.0}}\put(1000.0,0.0){\line(0,1){500.0}}\put(1000.0,1100.0){\oval(1200.0,1200.0)}\put(1000.0,1100.0){\makebox(0.0,0.0)[cc]{\scriptsize$f$}}}\kern 9.9945pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\kern 9.9945pt\hbox{}\@arraycr\kern 9.9945pt{\hbox{\kern-5.41667pt${W}$}}\kern 19.98901pt{\hbox{\kern-3.90451pt${P}$}}\@arraycr\end{array}\kern 9.9945pt,\kern 9.9945pt\begin{array}[c]{l}{\hbox{\kern-3.95901pt${U}$}}\kern 19.98901pt{\hbox{\kern-5.41667pt${W}$}}\kern 19.98901pt{\hbox{\kern-4.02777pt${V}$}}\@arraycr\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\hbox{\put(400.0,1500.0){\oval(800.0,800.0)[lb]}\put(1600.0,1500.0){\oval(800.0,800.0)[rb]}\put(0.0,1500.0){\line(0,1){500.0}}\put(2000.0,1500.0){\line(0,1){500.0}}\put(1000.0,0.0){\line(0,1){500.0}}\put(1000.0,1100.0){\oval(1200.0,1200.0)}\put(1000.0,1100.0){\makebox(0.0,0.0)[cc]{\scriptsize$f$}}}\@arraycr{\hbox{\kern-5.41667pt${W}$}}\kern 29.98352pt{\hbox{\kern-3.90451pt${P}$}}\@arraycr\end{array}\kern 9.9945pt=\kern 9.9945pt\begin{array}[c]{l}{\hbox{\kern-3.95901pt${U}$}}\kern 19.98901pt{\hbox{\kern-5.41667pt${W}$}}\kern 9.9945pt{\hbox{\kern-4.02777pt${V}$}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\hbox{}\@arraycr\hbox{\put(400.0,1500.0){\oval(800.0,800.0)[lb]}\put(1600.0,1500.0){\oval(800.0,800.0)[rb]}\put(0.0,1500.0){\line(0,1){500.0}}\put(2000.0,1500.0){\line(0,1){500.0}}\put(1000.0,0.0){\line(0,1){500.0}}\put(1000.0,1100.0){\oval(1200.0,1200.0)}\put(1000.0,1100.0){\makebox(0.0,0.0)[cc]{\scriptsize$f$}}}\kern 9.9945pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\kern 9.9945pt\hbox{}\@arraycr\kern 9.9945pt{\hbox{\kern-5.41667pt${W}$}}\kern 19.98901pt{\hbox{\kern-3.90451pt${P}$}}\@arraycr\end{array}\ \ .$ Proof. (i) $\begin{array}[c]{l}{\hbox{\kern-3.95901pt${U}$}}\kern 19.98901pt{\hbox{\kern-4.02777pt${V}$}}\kern 19.98901pt{\hbox{\kern-5.41667pt${W}$}}\@arraycr\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\hbox{}\@arraycr\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr{\hbox{\kern-5.41667pt${W}$}}\kern 19.98901pt{\hbox{\kern-4.02777pt${V}$}}\kern 19.98901pt{\hbox{\kern-3.95901pt${U}$}}\@arraycr\end{array}\ \ =\ \ \begin{array}[c]{l}\par\par\kern 9.9945pt{\hbox{\kern-3.95901pt${U}$}}\kern 19.98901pt{\hbox{\kern-4.02777pt${V}$}}\kern 19.98901pt{\hbox{\kern-5.41667pt${W}$}}\@arraycr\kern 9.9945pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\par\par\par\hbox{\put(1500.0,1000.0){\oval(3000.0,2000.0)}\put(0.0,0.0){\makebox(3000.0,2000.0)[cc]{\scriptsize$C_{V,W}$}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\par\kern 9.9945pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\par\kern 9.9945pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\hbox{\put(1500.0,1000.0){\oval(3000.0,2000.0)}\put(0.0,0.0){\makebox(3000.0,2000.0)[cc]{\scriptsize$C_{U,W}$}}}\@arraycr\par\par\kern 9.9945pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\par\hbox{\put(1500.0,1000.0){\oval(3000.0,2000.0)}\put(0.0,0.0){\makebox(3000.0,2000.0)[cc]{\scriptsize$C_{V,W}$}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\par\par\kern 9.9945pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\kern 9.9945pt{\hbox{\kern-5.41667pt${W}$}}\kern 19.98901pt{\hbox{\kern-4.02777pt${V}$}}\kern 19.98901pt{\hbox{\kern-3.95901pt${U}$}}\@arraycr\end{array}\ \ \stackrel{{\scriptstyle\mbox{by Hexagon Axiom }}}{{=}}\begin{array}[c]{l}\par\par\kern 9.9945pt{\hbox{\kern-14.09785pt${U\otimes V}$}}\kern 39.97803pt{\hbox{\kern-5.41667pt${W}$}}\@arraycr\kern 9.9945pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 39.97803pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\par\par\par\hbox{\put(1500.0,1000.0){\oval(3000.0,2000.0)}\put(0.0,0.0){\makebox(3000.0,2000.0)[cc]{\scriptsize$C_{V,W}$}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\par\kern 9.9945pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 29.98352pt\kern 19.98901pt\@arraycr\par\hbox{\put(2000.0,1000.0){\oval(4000.0,2000.0)}\put(0.0,0.0){\makebox(4000.0,2000.0)[cc]{\scriptsize$C_{V\otimes U,W}$}}}\@arraycr\par\par\par\par\par\kern 9.9945pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\@arraycr\kern 9.9945pt{\hbox{\kern-5.41667pt${W}$}}\kern 29.98352pt{\hbox{\kern-14.09785pt${V\otimes U}$}}\kern 19.98901pt\@arraycr\end{array}\ \ $ $\ \ \stackrel{{\scriptstyle\mbox{by naturality }}}{{=}}\begin{array}[c]{l}\par\par\kern 9.9945pt{\hbox{\kern-14.09785pt${U\otimes V}$}}\kern 29.98352pt{\hbox{\kern-5.41667pt${W}$}}\@arraycr\kern 9.9945pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\hbox{}\@arraycr\par\par\hbox{\put(2000.0,1000.0){\oval(4000.0,2000.0)}\put(0.0,0.0){\makebox(4000.0,2000.0)[cc]{\scriptsize$C_{U\otimes V,W}$}}}\kern 19.98901pt\@arraycr\par\kern 9.9945pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\@arraycr\par\kern 9.9945pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 9.9945pt\hbox{\put(1500.0,1000.0){\oval(3000.0,2000.0)}\put(0.0,0.0){\makebox(3000.0,2000.0)[cc]{\scriptsize$C_{U,V}$}}}\@arraycr\par\par\par\par\par\kern 9.9945pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\@arraycr\kern 9.9945pt{\hbox{\kern-5.41667pt${W}$}}\kern 29.98352pt{\hbox{\kern-14.09785pt${V\otimes U}$}}\kern 19.98901pt\@arraycr\end{array}\ \ \stackrel{{\scriptstyle\mbox{by Hexagon Axiom }}}{{=}}\begin{array}[c]{l}{\hbox{\kern-3.95901pt${U}$}}\kern 19.98901pt{\hbox{\kern-4.02777pt${V}$}}\kern 19.98901pt{\hbox{\kern-5.41667pt${W}$}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\hbox{}\@arraycr\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\hbox{}\@arraycr{\hbox{\kern-5.41667pt${W}$}}\kern 19.98901pt{\hbox{\kern-4.02777pt${V}$}}\kern 19.98901pt{\hbox{\kern-3.95901pt${U}$}}\@arraycr\end{array}\ \ \ .$ (ii) We have $\begin{array}[c]{l}{\hbox{\kern-3.95901pt${U}$}}\kern 19.98901pt{\hbox{\kern-4.02777pt${V}$}}\kern 9.9945pt{\hbox{\kern-5.41667pt${W}$}}\@arraycr\hbox{\put(400.0,1500.0){\oval(800.0,800.0)[lb]}\put(1600.0,1500.0){\oval(800.0,800.0)[rb]}\put(0.0,1500.0){\line(0,1){500.0}}\put(2000.0,1500.0){\line(0,1){500.0}}\put(1000.0,0.0){\line(0,1){500.0}}\put(1000.0,1100.0){\oval(1200.0,1200.0)}\put(1000.0,1100.0){\makebox(0.0,0.0)[cc]{\scriptsize$f$}}}\kern 9.9945pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\kern 9.9945pt\hbox{}\@arraycr\kern 9.9945pt{\hbox{\kern-5.41667pt${W}$}}\kern 19.98901pt{\hbox{\kern-3.90451pt${P}$}}\@arraycr\end{array}\kern 9.9945pt=\kern 9.9945pt\begin{array}[c]{l}{\hbox{\kern-14.09785pt${U\otimes V}$}}\kern 19.98901pt{\hbox{\kern-5.41667pt${W}$}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){400.0}}\put(0.0,1600.0){\line(0,1){400.0}}\put(0.0,1000.0){\oval(1200.0,1200.0)}\put(0.0,1000.0){\makebox(0.0,0.0)[cc]{\scriptsize$f$}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\hbox{}\@arraycr{\hbox{\kern-5.41667pt${W}$}}\kern 19.98901pt{\hbox{\kern-3.90451pt${P}$}}\@arraycr\end{array}\ \ \stackrel{{\scriptstyle\mbox{by naturality }}}{{=}}\kern 9.9945pt\begin{array}[c]{l}{\hbox{\kern-14.09785pt${U\otimes V}$}}\kern 19.98901pt{\hbox{\kern-5.41667pt${W}$}}\@arraycr\hbox{}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){400.0}}\put(0.0,1600.0){\line(0,1){400.0}}\put(0.0,1000.0){\oval(1200.0,1200.0)}\put(0.0,1000.0){\makebox(0.0,0.0)[cc]{\scriptsize$f$}}}\@arraycr{\hbox{\kern-5.41667pt${W}$}}\kern 19.98901pt{\hbox{\kern-3.90451pt${P}$}}\@arraycr\end{array}$ $\stackrel{{\scriptstyle\mbox{by Hexagon Axiom }}}{{=}}\begin{array}[c]{l}{\hbox{\kern-3.95901pt${U}$}}\kern 19.98901pt{\hbox{\kern-4.02777pt${V}$}}\kern 19.98901pt{\hbox{\kern-5.41667pt${W}$}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\hbox{}\@arraycr\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\hbox{\put(400.0,1500.0){\oval(800.0,800.0)[lb]}\put(1600.0,1500.0){\oval(800.0,800.0)[rb]}\put(0.0,1500.0){\line(0,1){500.0}}\put(2000.0,1500.0){\line(0,1){500.0}}\put(1000.0,0.0){\line(0,1){500.0}}\put(1000.0,1100.0){\oval(1200.0,1200.0)}\put(1000.0,1100.0){\makebox(0.0,0.0)[cc]{\scriptsize$f$}}}\@arraycr{\hbox{\kern-5.41667pt${W}$}}\kern 29.98352pt{\hbox{\kern-3.90451pt${P}$}}\@arraycr\end{array}\ \ \ .$ Similarly, we can show the second equation. $\Box$ Now we give some concepts as follows: Assume that $H,A\in ob\ {\cal C},$ and $\displaystyle\alpha:H\otimes A\rightarrow$ $\displaystyle A$ $\displaystyle,\hbox{ \ \ \ \ }\beta:H\otimes A\rightarrow H,$ $\displaystyle\phi:A\rightarrow H\otimes$ $\displaystyle A$ $\displaystyle,\hbox{ \ \ \ \ }\psi:H\rightarrow H\otimes A,$ $\displaystyle m_{H}:H\otimes H\rightarrow$ $\displaystyle H$ $\displaystyle,\hbox{ \ \ \ \ }m_{A}:A\otimes A\rightarrow A,$ $\displaystyle\Delta_{H}:H\rightarrow H\otimes$ $\displaystyle H$ $\displaystyle,\hbox{ \ \ \ \ }\Delta_{A}:A\rightarrow A\otimes A,$ $\displaystyle\eta_{H}:I\rightarrow$ $\displaystyle H$ $\displaystyle,\hbox{ \ \ \ \ }\eta_{A}:I\rightarrow A,$ $\displaystyle\epsilon_{H}:H\rightarrow$ $\displaystyle I$ $\displaystyle,\hbox{ \ \ \ \ }\epsilon_{A}:A\rightarrow I.$ are morphisms in ${\cal C}$. $(A,m_{A},\eta_{A})$ is called an algebra living in ${\cal C}$, if the following conditions are satisfied: $\begin{array}[c]{l}{\hbox{\kern-3.75pt${A}$}}\kern 9.9945pt{\hbox{\kern-3.75pt${A}$}}\kern 19.98901pt{\hbox{\kern-3.75pt${A}$}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 9.9945pt\hbox{\put(400.0,1500.0){\oval(800.0,800.0)[lb]}\put(1600.0,1500.0){\oval(800.0,800.0)[rb]}\put(0.0,1500.0){\line(0,1){500.0}}\put(2000.0,1500.0){\line(0,1){500.0}}\put(1000.0,0.0){\line(0,1){500.0}}\put(1000.0,1100.0){\oval(1200.0,1200.0)}\put(1000.0,1100.0){\makebox(0.0,0.0)[cc]{\scriptsize$m$}}}\@arraycr\hbox{\put(400.0,1500.0){\oval(800.0,800.0)[lb]}\put(1600.0,1500.0){\oval(800.0,800.0)[rb]}\put(0.0,1500.0){\line(0,1){500.0}}\put(2000.0,1500.0){\line(0,1){500.0}}\put(1000.0,0.0){\line(0,1){500.0}}\put(1000.0,1100.0){\oval(1200.0,1200.0)}\put(1000.0,1100.0){\makebox(0.0,0.0)[cc]{\scriptsize$m$}}}\@arraycr\kern 9.9945pt{\hbox{\kern-3.75pt${A}$}}\@arraycr\end{array}\kern 9.9945pt=\kern 9.9945pt\begin{array}[c]{l}{\hbox{\kern-3.75pt${A}$}}\kern 19.98901pt{\hbox{\kern-3.75pt${A}$}}\kern 9.9945pt{\hbox{\kern-3.75pt${A}$}}\@arraycr\hbox{\put(400.0,1500.0){\oval(800.0,800.0)[lb]}\put(1600.0,1500.0){\oval(800.0,800.0)[rb]}\put(0.0,1500.0){\line(0,1){500.0}}\put(2000.0,1500.0){\line(0,1){500.0}}\put(1000.0,0.0){\line(0,1){500.0}}\put(1000.0,1100.0){\oval(1200.0,1200.0)}\put(1000.0,1100.0){\makebox(0.0,0.0)[cc]{\scriptsize$m$}}}\kern 9.9945pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\kern 9.9945pt\hbox{\put(400.0,1500.0){\oval(800.0,800.0)[lb]}\put(1600.0,1500.0){\oval(800.0,800.0)[rb]}\put(0.0,1500.0){\line(0,1){500.0}}\put(2000.0,1500.0){\line(0,1){500.0}}\put(1000.0,0.0){\line(0,1){500.0}}\put(1000.0,1100.0){\oval(1200.0,1200.0)}\put(1000.0,1100.0){\makebox(0.0,0.0)[cc]{\scriptsize$m$}}}\@arraycr\kern 19.98901pt{\hbox{\kern-3.75pt${A}$}}\@arraycr\end{array}\kern 9.9945pt\kern 9.9945pt,\kern 9.9945pt\kern 9.9945pt\begin{array}[c]{l}\kern 19.98901pt{\hbox{\kern-3.75pt${A}$}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){400.0}}\put(0.0,1000.0){\oval(1200.0,1200.0)}\put(0.0,1000.0){\makebox(0.0,0.0)[cc]{\scriptsize$\eta_{A}$}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\hbox{\put(400.0,1500.0){\oval(800.0,800.0)[lb]}\put(1600.0,1500.0){\oval(800.0,800.0)[rb]}\put(0.0,1500.0){\line(0,1){500.0}}\put(2000.0,1500.0){\line(0,1){500.0}}\put(1000.0,0.0){\line(0,1){500.0}}\put(1000.0,1100.0){\oval(1200.0,1200.0)}\put(1000.0,1100.0){\makebox(0.0,0.0)[cc]{\scriptsize$m$}}}\@arraycr\kern 9.9945pt{\hbox{\kern-3.75pt${A}$}}\@arraycr\end{array}\kern 9.9945pt=\kern 9.9945pt\begin{array}[c]{l}{\hbox{\kern-3.75pt${A}$}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){400.0}}\put(0.0,1000.0){\oval(1200.0,1200.0)}\put(0.0,1000.0){\makebox(0.0,0.0)[cc]{\scriptsize$\eta_{A}$}}}\@arraycr\hbox{\put(400.0,1500.0){\oval(800.0,800.0)[lb]}\put(1600.0,1500.0){\oval(800.0,800.0)[rb]}\put(0.0,1500.0){\line(0,1){500.0}}\put(2000.0,1500.0){\line(0,1){500.0}}\put(1000.0,0.0){\line(0,1){500.0}}\put(1000.0,1100.0){\oval(1200.0,1200.0)}\put(1000.0,1100.0){\makebox(0.0,0.0)[cc]{\scriptsize$m$}}}\@arraycr\kern 9.9945pt{\hbox{\kern-3.75pt${A}$}}\@arraycr\end{array}\kern 9.9945pt=\kern 9.9945pt\begin{array}[c]{l}{\hbox{\kern-3.75pt${A}$}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr{\hbox{\kern-3.75pt${A}$}}\@arraycr\end{array}\quad.$ In this case, $\eta_{A}$ and $m_{A}$ are called unit and multiplication of $A$ respectively. $(H,\Delta_{H},\epsilon_{H})$ is called a coalgebra living in ${\cal C},$ if the following conditions are satisfied: $\begin{array}[c]{l}\kern 9.9945pt{\hbox{\kern-4.56248pt${H}$}}\@arraycr\hbox{\put(400.0,500.0){\oval(800.0,800.0)[lt]}\put(1600.0,500.0){\oval(800.0,800.0)[rt]}\put(0.0,500.0){\line(0,-1){500.0}}\put(2000.0,500.0){\line(0,-1){500.0}}\put(1000.0,2000.0){\line(0,-1){500.0}}\put(1000.0,900.0){\oval(1200.0,1200.0)}\put(1000.0,900.0){\makebox(0.0,0.0)[cc]{\scriptsize$\Delta$}}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 9.9945pt\hbox{\put(400.0,500.0){\oval(800.0,800.0)[lt]}\put(1600.0,500.0){\oval(800.0,800.0)[rt]}\put(0.0,500.0){\line(0,-1){500.0}}\put(2000.0,500.0){\line(0,-1){500.0}}\put(1000.0,2000.0){\line(0,-1){500.0}}\put(1000.0,900.0){\oval(1200.0,1200.0)}\put(1000.0,900.0){\makebox(0.0,0.0)[cc]{\scriptsize$\Delta$}}}\@arraycr{\hbox{\kern-4.56248pt${H}$}}\kern 9.9945pt{\hbox{\kern-4.56248pt${H}$}}\kern 19.98901pt{\hbox{\kern-4.56248pt${H}$}}\@arraycr\end{array}\kern 9.9945pt=\kern 9.9945pt\begin{array}[c]{l}\kern 19.98901pt{\hbox{\kern-4.56248pt${H}$}}\@arraycr\kern 9.9945pt\hbox{\put(400.0,500.0){\oval(800.0,800.0)[lt]}\put(1600.0,500.0){\oval(800.0,800.0)[rt]}\put(0.0,500.0){\line(0,-1){500.0}}\put(2000.0,500.0){\line(0,-1){500.0}}\put(1000.0,2000.0){\line(0,-1){500.0}}\put(1000.0,900.0){\oval(1200.0,1200.0)}\put(1000.0,900.0){\makebox(0.0,0.0)[cc]{\scriptsize$\Delta$}}}\@arraycr\hbox{\put(400.0,500.0){\oval(800.0,800.0)[lt]}\put(1600.0,500.0){\oval(800.0,800.0)[rt]}\put(0.0,500.0){\line(0,-1){500.0}}\put(2000.0,500.0){\line(0,-1){500.0}}\put(1000.0,2000.0){\line(0,-1){500.0}}\put(1000.0,900.0){\oval(1200.0,1200.0)}\put(1000.0,900.0){\makebox(0.0,0.0)[cc]{\scriptsize$\Delta$}}}\kern 9.9945pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr{\hbox{\kern-4.56248pt${H}$}}\kern 19.98901pt{\hbox{\kern-4.56248pt${H}$}}\kern 9.9945pt{\hbox{\kern-4.56248pt${H}$}}\@arraycr\end{array}\kern 9.9945pt\kern 9.9945pt,\kern 9.9945pt\kern 9.9945pt\begin{array}[c]{l}\kern 9.9945pt{\hbox{\kern-4.56248pt${H}$}}\@arraycr\hbox{\put(400.0,500.0){\oval(800.0,800.0)[lt]}\put(1600.0,500.0){\oval(800.0,800.0)[rt]}\put(0.0,500.0){\line(0,-1){500.0}}\put(2000.0,500.0){\line(0,-1){500.0}}\put(1000.0,2000.0){\line(0,-1){500.0}}\put(1000.0,900.0){\oval(1200.0,1200.0)}\put(1000.0,900.0){\makebox(0.0,0.0)[cc]{\scriptsize$\Delta$}}}\@arraycr\hbox{\put(0.0,1600.0){\line(0,1){400.0}}\put(0.0,1000.0){\oval(1200.0,1200.0)}\put(0.0,1000.0){\makebox(0.0,0.0)[cc]{\scriptsize$\varepsilon_{H}$}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\kern 19.98901pt{\hbox{\kern-4.56248pt${H}$}}\@arraycr\end{array}\kern 9.9945pt=\kern 9.9945pt\begin{array}[c]{l}\kern 9.9945pt{\hbox{\kern-4.56248pt${H}$}}\@arraycr\hbox{\put(400.0,500.0){\oval(800.0,800.0)[lt]}\put(1600.0,500.0){\oval(800.0,800.0)[rt]}\put(0.0,500.0){\line(0,-1){500.0}}\put(2000.0,500.0){\line(0,-1){500.0}}\put(1000.0,2000.0){\line(0,-1){500.0}}\put(1000.0,900.0){\oval(1200.0,1200.0)}\put(1000.0,900.0){\makebox(0.0,0.0)[cc]{\scriptsize$\Delta$}}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\hbox{\put(0.0,1600.0){\line(0,1){400.0}}\put(0.0,1000.0){\oval(1200.0,1200.0)}\put(0.0,1000.0){\makebox(0.0,0.0)[cc]{\scriptsize$\varepsilon_{H}$}}}\@arraycr{\hbox{\kern-4.56248pt${H}$}}\@arraycr\end{array}\kern 9.9945pt=\kern 9.9945pt\begin{array}[c]{l}{\hbox{\kern-4.56248pt${H}$}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr{\hbox{\kern-4.56248pt${H}$}}\@arraycr\end{array}\quad.$ In this case, $\epsilon_{H}$ and $\Delta_{H}$ are called counit and comultiplication of $H$ respectively. If $A$ is an algebra and $H$ is a coalgebra, then $Hom_{\cal C}(H,A)$ becomes an algebra under the convolution product $fg=\quad\begin{array}[c]{l}\kern 9.9945pt{\hbox{\kern-4.56248pt${H}$}}\@arraycr\hbox{\put(400.0,500.0){\oval(800.0,800.0)[lt]}\put(1600.0,500.0){\oval(800.0,800.0)[rt]}\put(0.0,500.0){\line(0,-1){500.0}}\put(2000.0,500.0){\line(0,-1){500.0}}\put(1000.0,2000.0){\line(0,-1){500.0}}\put(1000.0,900.0){\oval(1200.0,1200.0)}\put(1000.0,900.0){\makebox(0.0,0.0)[cc]{\scriptsize$\Delta$}}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){400.0}}\put(0.0,1600.0){\line(0,1){400.0}}\put(0.0,1000.0){\oval(1200.0,1200.0)}\put(0.0,1000.0){\makebox(0.0,0.0)[cc]{\scriptsize$f$}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){400.0}}\put(0.0,1600.0){\line(0,1){400.0}}\put(0.0,1000.0){\oval(1200.0,1200.0)}\put(0.0,1000.0){\makebox(0.0,0.0)[cc]{\scriptsize$g$}}}\@arraycr\hbox{\put(400.0,1500.0){\oval(800.0,800.0)[lb]}\put(1600.0,1500.0){\oval(800.0,800.0)[rb]}\put(0.0,1500.0){\line(0,1){500.0}}\put(2000.0,1500.0){\line(0,1){500.0}}\put(1000.0,0.0){\line(0,1){500.0}}\put(1000.0,1100.0){\oval(1200.0,1200.0)}\put(1000.0,1100.0){\makebox(0.0,0.0)[cc]{\scriptsize$m$}}}\@arraycr\kern 9.9945pt{\hbox{\kern-3.75pt${A}$}}\@arraycr\end{array}\ \ \ .$ and its unit element $\eta=\eta_{A}\epsilon_{H}.$ If $S$ is the inverse of $id_{H}$ in $Hom_{\cal C}(H,H)$, then $S$ is called antipode of $H$. If $(H,m_{H},\eta_{H})$ is an algebra, and $(H,\Delta_{H},\epsilon_{H})$ is a coalgebra living in ${\cal C}$, and the following condition is satisfied: $\begin{array}[c]{l}{\hbox{\kern-4.56248pt${H}$}}\kern 19.98901pt{\hbox{\kern-4.56248pt${H}$}}\@arraycr\hbox{\put(400.0,1500.0){\oval(800.0,800.0)[lb]}\put(1600.0,1500.0){\oval(800.0,800.0)[rb]}\put(0.0,1500.0){\line(0,1){500.0}}\put(2000.0,1500.0){\line(0,1){500.0}}\put(1000.0,0.0){\line(0,1){500.0}}\put(1000.0,1100.0){\oval(1200.0,1200.0)}\put(1000.0,1100.0){\makebox(0.0,0.0)[cc]{\scriptsize$m$}}}\@arraycr\hbox{\put(400.0,500.0){\oval(800.0,800.0)[lt]}\put(1600.0,500.0){\oval(800.0,800.0)[rt]}\put(0.0,500.0){\line(0,-1){500.0}}\put(2000.0,500.0){\line(0,-1){500.0}}\put(1000.0,2000.0){\line(0,-1){500.0}}\put(1000.0,900.0){\oval(1200.0,1200.0)}\put(1000.0,900.0){\makebox(0.0,0.0)[cc]{\scriptsize$\Delta$}}}\@arraycr{\hbox{\kern-4.56248pt${H}$}}\kern 19.98901pt{\hbox{\kern-4.56248pt${H}$}}\@arraycr\end{array}\kern 9.9945pt=\kern 9.9945pt\begin{array}[c]{l}\kern 9.9945pt{\hbox{\kern-4.56248pt${H}$}}\kern 29.98352pt{\hbox{\kern-4.56248pt${H}$}}\@arraycr\hbox{\put(400.0,500.0){\oval(800.0,800.0)[lt]}\put(1600.0,500.0){\oval(800.0,800.0)[rt]}\put(0.0,500.0){\line(0,-1){500.0}}\put(2000.0,500.0){\line(0,-1){500.0}}\put(1000.0,2000.0){\line(0,-1){500.0}}\put(1000.0,900.0){\oval(1200.0,1200.0)}\put(1000.0,900.0){\makebox(0.0,0.0)[cc]{\scriptsize$\Delta$}}}\kern 9.9945pt\hbox{\put(400.0,500.0){\oval(800.0,800.0)[lt]}\put(1600.0,500.0){\oval(800.0,800.0)[rt]}\put(0.0,500.0){\line(0,-1){500.0}}\put(2000.0,500.0){\line(0,-1){500.0}}\put(1000.0,2000.0){\line(0,-1){500.0}}\put(1000.0,900.0){\oval(1200.0,1200.0)}\put(1000.0,900.0){\makebox(0.0,0.0)[cc]{\scriptsize$\Delta$}}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\hbox{\put(400.0,1500.0){\oval(800.0,800.0)[lb]}\put(1600.0,1500.0){\oval(800.0,800.0)[rb]}\put(0.0,1500.0){\line(0,1){500.0}}\put(2000.0,1500.0){\line(0,1){500.0}}\put(1000.0,0.0){\line(0,1){500.0}}\put(1000.0,1100.0){\oval(1200.0,1200.0)}\put(1000.0,1100.0){\makebox(0.0,0.0)[cc]{\scriptsize$m$}}}\kern 9.9945pt\hbox{\put(400.0,1500.0){\oval(800.0,800.0)[lb]}\put(1600.0,1500.0){\oval(800.0,800.0)[rb]}\put(0.0,1500.0){\line(0,1){500.0}}\put(2000.0,1500.0){\line(0,1){500.0}}\put(1000.0,0.0){\line(0,1){500.0}}\put(1000.0,1100.0){\oval(1200.0,1200.0)}\put(1000.0,1100.0){\makebox(0.0,0.0)[cc]{\scriptsize$m$}}}\@arraycr\kern 9.9945pt{\hbox{\kern-4.56248pt${H}$}}\kern 29.98352pt{\hbox{\kern-4.56248pt${H}$}}\@arraycr\end{array}\kern 9.9945pt,\kern 9.9945pt\begin{array}[c]{l}{\hbox{\kern-4.56248pt${H}$}}\kern 19.98901pt{\hbox{\kern-4.56248pt${H}$}}\@arraycr\hbox{\put(400.0,1500.0){\oval(800.0,800.0)[lb]}\put(1600.0,1500.0){\oval(800.0,800.0)[rb]}\put(0.0,1500.0){\line(0,1){500.0}}\put(2000.0,1500.0){\line(0,1){500.0}}\put(1000.0,0.0){\line(0,1){500.0}}\put(1000.0,1100.0){\oval(1200.0,1200.0)}\put(1000.0,1100.0){\makebox(0.0,0.0)[cc]{\scriptsize$m$}}}\@arraycr\kern 9.9945pt\hbox{\put(0.0,1600.0){\line(0,1){400.0}}\put(0.0,1000.0){\oval(1200.0,1200.0)}\put(0.0,1000.0){\makebox(0.0,0.0)[cc]{\scriptsize$\varepsilon$}}}\@arraycr\end{array}\kern 9.9945pt=\kern 9.9945pt\begin{array}[c]{l}{\hbox{\kern-4.56248pt${H}$}}\kern 19.98901pt{\hbox{\kern-4.56248pt${H}$}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\hbox{\put(0.0,1600.0){\line(0,1){400.0}}\put(0.0,1000.0){\oval(1200.0,1200.0)}\put(0.0,1000.0){\makebox(0.0,0.0)[cc]{\scriptsize$\varepsilon$}}}\kern 19.98901pt\hbox{\put(0.0,1600.0){\line(0,1){400.0}}\put(0.0,1000.0){\oval(1200.0,1200.0)}\put(0.0,1000.0){\makebox(0.0,0.0)[cc]{\scriptsize$\varepsilon$}}}\@arraycr\end{array}\kern 9.9945pt,\kern 9.9945pt\begin{array}[c]{l}\kern 9.9945pt\hbox{\put(0.0,0.0){\line(0,1){400.0}}\put(0.0,1000.0){\oval(1200.0,1200.0)}\put(0.0,1000.0){\makebox(0.0,0.0)[cc]{\scriptsize$\eta$}}}\@arraycr\hbox{\put(400.0,500.0){\oval(800.0,800.0)[lt]}\put(1600.0,500.0){\oval(800.0,800.0)[rt]}\put(0.0,500.0){\line(0,-1){500.0}}\put(2000.0,500.0){\line(0,-1){500.0}}\put(1000.0,2000.0){\line(0,-1){500.0}}\put(1000.0,900.0){\oval(1200.0,1200.0)}\put(1000.0,900.0){\makebox(0.0,0.0)[cc]{\scriptsize$\Delta$}}}\@arraycr\end{array}\kern 9.9945pt=\kern 9.9945pt\begin{array}[c]{l}\hbox{\put(0.0,0.0){\line(0,1){400.0}}\put(0.0,1000.0){\oval(1200.0,1200.0)}\put(0.0,1000.0){\makebox(0.0,0.0)[cc]{\scriptsize$\eta$}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){400.0}}\put(0.0,1000.0){\oval(1200.0,1200.0)}\put(0.0,1000.0){\makebox(0.0,0.0)[cc]{\scriptsize$\eta$}}}\@arraycr\end{array}\kern 9.9945pt,\kern 9.9945pt\begin{array}[c]{l}\hbox{\put(0.0,0.0){\line(0,1){400.0}}\put(0.0,1000.0){\oval(1200.0,1200.0)}\put(0.0,1000.0){\makebox(0.0,0.0)[cc]{\scriptsize$\eta$}}}\@arraycr\hbox{\put(0.0,1600.0){\line(0,1){400.0}}\put(0.0,1000.0){\oval(1200.0,1200.0)}\put(0.0,1000.0){\makebox(0.0,0.0)[cc]{\scriptsize$\varepsilon$}}}\@arraycr\end{array}\kern 9.9945pt=\kern 9.9945pt\begin{array}[c]{l}{\hbox{\kern-2.59027pt${I}$}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr{\hbox{\kern-2.59027pt${I}$}}\end{array}\kern 9.9945pt.\kern 9.9945pt$ then $H$ is called a bialgebra living in ${\cal C}$. If $H$ is a bialgebra and there is an inverse $S$ of $id_{H}$ under convolution product in $Hom_{\cal C}(H,H)$, then $H$ is called a Hopf algebra living in ${\cal C},$ or a braided Hopf algebra. When $H$ is a Hopf algebra algebra in braided tensor category $({\mathcal{C}},C)$, then the condition above is equivalent to (YD): $\phi\alpha=\begin{array}[c]{l}\kern 29.98352pt{\hbox{\kern-4.56248pt${H}$}}\kern 69.96155pt{\hbox{\kern-5.39583pt${M}$}}\@arraycr\par\kern 9.9945pt\hbox{\put(2000.0,0.0){\oval(4000.0,2000.0)[ct]}\put(2000.0,1000.0){\line(0,1){1000.0}}}\kern 39.97803pt\hbox{\put(400.0,500.0){\oval(800.0,800.0)[lt]}\put(1600.0,500.0){\oval(800.0,800.0)[rt]}\put(0.0,500.0){\line(0,-1){500.0}}\put(2000.0,500.0){\line(0,-1){500.0}}\put(1000.0,2000.0){\line(0,-1){500.0}}\put(1000.0,900.0){\oval(1200.0,1200.0)}\put(1000.0,900.0){\makebox(0.0,0.0)[cc]{\scriptsize$\phi$}}}\@arraycr\par\hbox{\put(1000.0,0.0){\oval(2000.0,2000.0)[ct]}\put(1000.0,1000.0){\line(0,1){1000.0}}}\kern 29.98352pt\hbox{\put(0.0,0.0){\line(0,1){400.0}}\put(0.0,1600.0){\line(0,1){400.0}}\put(0.0,1000.0){\oval(1200.0,1200.0)}\put(0.0,1000.0){\makebox(0.0,0.0)[cc]{\scriptsize$S$}}}\kern 29.98352pt\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\par\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\kern 19.98901pt\kern 39.97803pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\par\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\kern 9.9945pt\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 39.97803pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\par\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\kern 19.98901pt\kern 29.98352pt\hbox{}\@arraycr\par\hbox{\put(1000.0,2000.0){\oval(2000.0,2000.0)[cb]}\put(1000.0,0.0){\line(0,1){1000.0}}}\kern 19.98901pt\kern 19.98901pt\hbox{\put(400.0,1500.0){\oval(800.0,800.0)[lb]}\put(1600.0,1500.0){\oval(800.0,800.0)[rb]}\put(0.0,1500.0){\line(0,1){500.0}}\put(2000.0,1500.0){\line(0,1){500.0}}\put(1000.0,0.0){\line(0,1){500.0}}\put(1000.0,1100.0){\oval(1200.0,1200.0)}\put(1000.0,1100.0){\makebox(0.0,0.0)[cc]{\scriptsize$\alpha$}}}\@arraycr\par\kern 9.9945pt\hbox{\put(1000.0,2000.0){\oval(2000.0,2000.0)[cb]}\put(1000.0,0.0){\line(0,1){1000.0}}}\kern 39.97803pt\kern 9.9945pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\kern 19.98901pt{\hbox{\kern-4.56248pt${H}$}}\kern 59.96704pt{\hbox{\kern-5.39583pt${M}$}}\end{array}\ \ \ \ .$ Let ${}^{H}_{H}{\mathcal{Y}D(C)}$ denote the category of all Yetter-Drinfeld $H$-modules in ${\mathcal{C}}$. If $({\mathcal{C}},C)={\mathcal{V}}ect(k)$, we write ${}^{H}_{H}{\mathcal{Y}D(C)}=$ ${}^{H}_{H}{\mathcal{Y}D}$, called Yetter-Drinfeld category. It follows from [RT93] and [BD98, Theorem 4.1.1] that ${}^{H}_{H}{\mathcal{Y}D}({\mathcal{C}})$ is a braided tensor category with ${}^{YD}C_{U,V}=(\alpha_{V}\otimes id_{U})(id_{H}\otimes C_{U,V})(\phi_{U}\otimes id_{V})$ for any two Yetter-Drinfeld modules $(U,\phi_{U},\alpha_{U})$ and $(V,\phi_{V},\alpha_{V})$ when $H$ has an invertible antipode. In this case, ${}^{YD}C_{U,V}^{-1}=(id_{V}\otimes\alpha_{U})(C_{H,V}^{-1}\otimes id_{U})(S^{-1}\otimes C_{U,V}^{-1})(C_{U,H}^{-1}\otimes id_{V})(id_{U}\otimes\phi_{V}).$ Algebras, coalgebras and Hopf algebras and so on in ${}^{H}_{H}{\mathcal{Y}D}$ are called Yetter-Drinfeld ones or YD ones in short. Let $\mathbb{Z},$ $\mathbb{N}$ and $\mathbb{C}$ denote integer set, natural number set and complex field, respectively. Throughout basic field is a complex field $\mathbb{C}$, which is denoted by $k$ sometimes. If $V$ is a vector space with a basis $v_{1},v_{2},\cdots,v_{n}$ and $q_{ij}\in\mathbb{C}^{}$ for $1\leq i,j\leq n$ such that map $C:\left\\{\begin{array}[]{lll}V\otimes V&\rightarrow&V\otimes V\\\ v_{i}\otimes v_{j}&\mapsto&q_{ij}v_{j}\otimes v_{i}\end{array}\right.,$ then $(V,C)$ is called a braided vector space of diagonal type. Denote by $(q_{ij})_{n\times n}$ the braiding matrix of $(V,C)$ under the basis $v_{1},v_{2},\cdots,v_{n}$. Then $(V,C)$ is also written as $(V,(q_{ij})_{n\times n}).$ We can get Nichols algebra $\mathfrak{B}(V)$ (see [He06b, He06a] and [ZZC04]). Let $\mathcal{B}$ denote the set of all generators of restricted PBW basis and $\mathcal{P}$ denote restricted PBW basis of $\mathfrak{B}(V)$. Let $v_{1},v_{2},\cdots,v_{n}$ be a basis of $V\in^{kG}_{kG}{\mathcal{Y}D}$ with comodule operation and module operation $\delta(v_{i})=g_{i}\otimes v_{i},$ $h\cdot v_{i}=\chi_{i}(h)v_{i}$, where $\chi_{i}$ is a multiplication character of $G$, i.e. a homomorphism from $G$ to $\mathbb{C}^{}$, for $1\leq i\leq n$. It is clear that $(V,C)$ is a braided vector space of diagonal type. Otherwise, for any $q_{ij}\in\mathbb{C}$, $1\leq i,j\leq n$ and a basis $v_{1},v_{2},\cdots,v_{n}$ of $V$, $V$ can become a Yetter-Drinfeld module over group $\mathbb{Z}^{n}$ by defining $\delta(v_{i})=e_{i}\otimes v_{i}$ and $e_{j}\cdot v_{i}=q_{ji}v_{i}$, $1\leq i,j\leq n$, where $e_{1}=(1,0,\cdots,0),\cdots,e_{n}=(0,0,\cdots,1)\in\mathbb{Z}^{n}.$ ## 2 Quantum Design of Industry In this section we give some examples to design by means of braiding. That is, we design by means of quantum theory. ###### Example 2.1. (Category of quantum design grammars) Let $\\{D_{i}\mid 1\leq i\leq m\\}$ be the set of all shapes in some design $D$. Let $v_{1},v_{2},\cdots,v_{n}$ be a basis of $V\in^{kG}_{kG}{\mathcal{Y}D}$ with $\delta(v_{i})=g_{i}\otimes v_{i},$ $h\cdot v_{i}=\chi_{i}(h)v_{i}$. Assume that $g_{i}=g_{j}$ and $\chi_{i}=\chi_{j}$ when there exists rule $R_{ij}:D_{i}\rightarrow D_{j}$. Define a morphism $f_{ij}$ from $kv_{i}$ to $kv_{j}$ in ${}^{kG}_{kG}{\mathcal{Y}D}$ such that $f_{ij}(v_{i})=R_{ij}(D_{i})$. It is clear that $f_{ij}$ is a morphism in ${}^{kG}_{kG}{\mathcal{Y}D}$ and $kv_{i}$ and $V$, are in ${}^{kG}_{kG}{\mathcal{Y}D}$. Define a map $\psi:\mathcal{P}\rightarrow\\{D_{i}\mid 1\leq i\leq m\\}$ such that $\psi(v_{i})=D_{i}$. $\psi(u)$ denote the combination of shapes of $D_{i_{1}}$, $D_{i_{1}}$, $\cdots,$ $D_{i_{r}}$ when $u=v_{i_{1}}v_{i_{2}}\cdots v_{i_{r}}\in\mathcal{P}.$ ###### Example 2.2. ( Quantum design grammars of Coca-Cola) Let $D_{1},D_{2}$, $D_{3}$ and $D_{4}$ be initial shapes; $D_{5},D_{6}$, $D_{7}$ and $D_{8}$ be cap shape, above shape, middle shape and below shape of battle of Coca-Cola; Rule $R_{15}:D_{1}\rightarrow D_{5}$, Rule $R_{26}:D_{2}\rightarrow D_{6}$, Rule $R_{37}:D_{3}\rightarrow D_{7}$ and $R_{48}:D_{4}\rightarrow D_{8}$. (i) Let $G=\mathbb{Z}_{7}$ be cycle group and $C(v_{s}\otimes v_{t})=e^{\frac{2\pi ig_{s}g_{t}}{7}}=\omega^{g_{s}g_{t}}$, where $\omega:=e^{\frac{2\pi i}{7}}$ and $i:=\sqrt{-1}$. Assume that $g_{s}=s$, $\chi_{s}(g_{t})=\omega^{st}$ when $1\leq s,t\leq 4;$ $g_{4+p}=g_{p}$, $\chi_{4+p}=\chi_{p}$ for $1\leq p\leq 4.$ $\begin{array}[c]{clr}\kern 9.9945pt{\hbox{\kern-4.003pt${v_{1}}$}}\kern 19.98901pt{\hbox{\kern-4.003pt${v_{2}}$}}\kern 19.98901pt{\hbox{\kern-4.003pt${v_{3}}$}}\kern 19.98901pt{\hbox{\kern-4.003pt${v_{4}}$}}\@arraycr\kern 9.9945pt\hbox{\put(0.0,0.0){\line(0,1){400.0}}\put(0.0,1600.0){\line(0,1){400.0}}\put(0.0,1000.0){\oval(1200.0,1200.0)}\put(0.0,1000.0){\makebox(0.0,0.0)[cc]{\scriptsize$f_{15}$}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){400.0}}\put(0.0,1600.0){\line(0,1){400.0}}\put(0.0,1000.0){\oval(1200.0,1200.0)}\put(0.0,1000.0){\makebox(0.0,0.0)[cc]{\scriptsize$f_{26}$}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){400.0}}\put(0.0,1600.0){\line(0,1){400.0}}\put(0.0,1000.0){\oval(1200.0,1200.0)}\put(0.0,1000.0){\makebox(0.0,0.0)[cc]{\scriptsize$f_{37}$}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){400.0}}\put(0.0,1600.0){\line(0,1){400.0}}\put(0.0,1000.0){\oval(1200.0,1200.0)}\put(0.0,1000.0){\makebox(0.0,0.0)[cc]{\scriptsize$f_{48}$}}}\@arraycr\kern 9.9945pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\kern 9.9945pt\hbox{\put(1000.0,2000.0){\oval(2000.0,2000.0)[cb]}\put(1000.0,0.0){\line(0,1){1000.0}}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 29.98352pt\hbox{\put(1000.0,2000.0){\oval(2000.0,2000.0)[cb]}\put(1000.0,0.0){\line(0,1){1000.0}}}\@arraycr{\hbox{\kern-4.003pt${v_{1}}$}}\kern 39.97803pt{\hbox{\kern-2.86229pt${u}$}}\@arraycr\kern 9.9945pt{\hbox{\kern-16.5417pt${\hbox{figure }1}$}}\@arraycr\end{array};\ \ \ \begin{array}[c]{clr}\kern 9.9945pt{\hbox{\kern-4.003pt${v_{1}}$}}\kern 19.98901pt{\hbox{\kern-4.003pt${v_{2}}$}}\kern 19.98901pt{\hbox{\kern-4.003pt${v_{3}}$}}\kern 19.98901pt{\hbox{\kern-4.003pt${v_{4}}$}}\@arraycr\kern 9.9945pt\hbox{\put(0.0,0.0){\line(0,1){400.0}}\put(0.0,1600.0){\line(0,1){400.0}}\put(0.0,1000.0){\oval(1200.0,1200.0)}\put(0.0,1000.0){\makebox(0.0,0.0)[cc]{\scriptsize$f_{15}$}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){400.0}}\put(0.0,1600.0){\line(0,1){400.0}}\put(0.0,1000.0){\oval(1200.0,1200.0)}\put(0.0,1000.0){\makebox(0.0,0.0)[cc]{\scriptsize$f_{26}$}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){400.0}}\put(0.0,1600.0){\line(0,1){400.0}}\put(0.0,1000.0){\oval(1200.0,1200.0)}\put(0.0,1000.0){\makebox(0.0,0.0)[cc]{\scriptsize$f_{37}$}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){400.0}}\put(0.0,1600.0){\line(0,1){400.0}}\put(0.0,1000.0){\oval(1200.0,1200.0)}\put(0.0,1000.0){\makebox(0.0,0.0)[cc]{\scriptsize$f_{48}$}}}\@arraycr\kern 9.9945pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\kern 9.9945pt\hbox{\put(1000.0,2000.0){\oval(2000.0,2000.0)[cb]}\put(1000.0,0.0){\line(0,1){1000.0}}}\@arraycr\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 29.98352pt\hbox{\put(1000.0,2000.0){\oval(2000.0,2000.0)[cb]}\put(1000.0,0.0){\line(0,1){1000.0}}}\@arraycr\kern 29.98352pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 9.9945pt\@arraycr\kern 9.9945pt\hbox{}\@arraycr\kern 9.9945pt\hbox{}\@arraycr\kern 9.9945pt\hbox{\put(1000.0,2000.0){\oval(2000.0,2000.0)[cb]}\put(1000.0,0.0){\line(0,1){1000.0}}}\@arraycr\kern 9.9945pt{\hbox{\kern-9.54456pt${\omega^{18}u^{\prime}}$}}\@arraycr\kern 9.9945pt{\hbox{\kern-16.5417pt${\hbox{figure }2}$}}\@arraycr\end{array}\ \ \ \ \ .$ Here $u=v_{6}v_{7}v_{8}$ and $\psi(u)$ is the battle of Coca-Cola without cap, $u^{\prime}=v_{5}v_{6}v_{7}v_{8}$ and $\psi(u^{\prime})$ is the battle of Coca-Cola with cap. Figure 1 is an ordinary design and Figure 2 is a quantum design. Define that $\omega^{1}u^{\prime},\omega^{2}u^{\prime},\cdots,\omega^{7}u^{\prime}$ means that the color of the battle of Coca-Cola is Red, orange, yellow, green, cyan, blue, purple, respectively. Therefore, $\omega^{18}u^{\prime}=\omega^{4}u^{\prime}$ and the color of the battle of Coca-Cola is green. (ii) Let $G=\mathbb{Z}$ and $\chi_{1}(g_{1})=q$; $1\not=q$ be a positive real number; $g_{i}=g_{1}$ and $\chi_{i}=\chi_{1}$ for $1\leq i\leq 8$. Then $C(v_{s}\otimes v_{t})=q(v_{t}\otimes v_{s})$. Similarly, we can get figure 3 by replacing $\omega^{18}$ by $q^{6}$ in figure 2. Define that $q^{6}u^{\prime}$ means that the size of the battle of Coca-Cola is $q^{6}$ times of original battle $\psi(u^{\prime})$ of Coca-Cola. ###### Example 2.3. Let $D_{0},D_{1},D_{3}$, $D_{5}$, $D_{7}$ and $D_{9}$ be frame, left steering wheel, left front gate, left back gate, left front lamp and left back lamp of car. We shall obtain a design of car by means of these left components of car as follows. Let $G=\mathbb{Z}^{10}$ and $C(v_{0}\otimes v_{t})=qv_{t}\otimes v_{0}$, $C(v_{s}\otimes v_{t})=v_{t}\otimes v_{s}$, where $q\not=1$ is positive real number for $s\not=0$, $s,t=1,3,5,7,9.$ $\begin{array}[c]{lll}\kern 9.9945pt{\hbox{\kern-4.003pt${v_{9}}$}}\kern 19.98901pt{\hbox{\kern-4.003pt${v_{7}}$}}\kern 19.98901pt{\hbox{\kern-4.003pt${v_{5}}$}}\kern 19.98901pt{\hbox{\kern-4.003pt${v_{3}}$}}\kern 19.98901pt{\hbox{\kern-4.003pt${v_{0}}$}}\kern 19.98901pt{\hbox{\kern-4.003pt${v_{1}}$}}\kern 19.98901pt{\hbox{\kern-4.003pt${v_{3}}$}}\kern 19.98901pt{\hbox{\kern-4.003pt${v_{5}}$}}\kern 19.98901pt{\hbox{\kern-4.003pt${v_{7}}$}}\kern 19.98901pt{\hbox{\kern-4.003pt${v_{9}}$}}\@arraycr\par\kern 9.9945pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\par\kern 9.9945pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\kern 19.98901pt\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\par\kern 9.9945pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\kern 19.98901pt\kern 19.98901pt\hbox{\put(1000.0,2000.0){\oval(2000.0,2000.0)[cb]}\put(1000.0,0.0){\line(0,1){1000.0}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\par\kern 9.9945pt\kern 19.98901pt\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\kern 19.98901pt\kern 9.9945pt\kern 9.9945pt\hbox{}\@arraycr\par\kern 9.9945pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\kern 19.98901pt\kern 9.9945pt\kern 9.9945pt\hbox{}\@arraycr\@arraycr\par\kern 9.9945pt\hbox{\put(1000.0,2000.0){\oval(2000.0,2000.0)[cb]}\put(1000.0,0.0){\line(0,1){1000.0}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\kern 19.98901pt\hbox{\put(1000.0,2000.0){\oval(2000.0,2000.0)[cb]}\put(1000.0,0.0){\line(0,1){1000.0}}}\kern 9.9945pt\kern 9.9945pt\hbox{}\@arraycr\par\par\kern 19.98901pt\kern 19.98901pt\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\kern 9.9945pt\kern 9.9945pt\hbox{}\@arraycr\par\kern 29.98352pt\hbox{\put(1000.0,2000.0){\oval(2000.0,2000.0)[cb]}\put(1000.0,0.0){\line(0,1){1000.0}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\par\kern 39.97803pt\kern 19.98901pt\kern 19.98901pt\hbox{\put(1000.0,2000.0){\oval(2000.0,2000.0)[cb]}\put(1000.0,0.0){\line(0,1){1000.0}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\par\kern 49.97253pt\hbox{\put(1000.0,2000.0){\oval(2000.0,2000.0)[cb]}\put(1000.0,0.0){\line(0,1){1000.0}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\kern 19.98901pt\hbox{}\@arraycr\par\kern 59.96704pt\kern 19.98901pt\kern 19.98901pt\hbox{}\@arraycr\par\kern 69.96155pt\hbox{\put(1000.0,2000.0){\oval(2000.0,2000.0)[cb]}\put(1000.0,0.0){\line(0,1){1000.0}}}\kern 9.9945pt\kern 9.9945pt\hbox{}\@arraycr\par\kern 79.95605pt\hbox{\put(1000.0,2000.0){\oval(2000.0,2000.0)[cb]}\put(1000.0,0.0){\line(0,1){1000.0}}}\kern 9.9945pt\hbox{}\@arraycr\par\kern 89.95056pt\hbox{\put(1000.0,2000.0){\oval(2000.0,2000.0)[cb]}\put(1000.0,0.0){\line(0,1){1000.0}}}\@arraycr\par\kern 99.94507pt{\hbox{\kern-6.67374pt${q^{4}u}$}}\@arraycr\par\kern 99.94507pt{\hbox{\kern-16.5417pt${\hbox{figure }4}$}}\@arraycr\par\end{array},$ where $u=:v_{0}v_{1}v_{3}^{2}v_{5}^{2}v_{7}^{2}v_{9}^{2}$ and $q^{4}u=v_{0}v_{1}(qv_{3})v_{3}(qv_{5})v_{5}(qv_{7})v_{7}(qv_{9})v_{9}$. Define that $(qv_{1}),$ $(qv_{3}),$ $(qv_{5}),$ $(qv_{7}),$ $(qv_{9})$ denote right steering wheel, right front gate, right back gate, right front lamp and right back lamp of car, respectively. Figure 4 is a braiding diagram which represents a combination with left components of car and whole car consists of them. Notice that the steering wheel is in left side. If we require that the steering wheel is in right side of car, we must place the $v_{1}$ in the left hand side of $v_{0}$ in braiding diagram and have braiding $C(v_{1}\otimes v_{0}).$ ###### Example 2.4. Let $D_{4},D_{1}$, $D_{2}$ and $D_{3}$ be the first floor, the second floor and the third floor of ship; $D_{5}$ the brightness of ship. We shall obtain a design about the brightness of ship by means of braiding diagram as follows. Let $G=\mathbb{Z}^{5}$ and $C(v_{s}\otimes v_{t})=q_{st}(v_{t}\otimes v_{s})$, where $q_{st}$ is positive real number for $1\leq s,t\leq 5.$ $\begin{array}[c]{lll}\kern 9.9945pt{\hbox{\kern-4.003pt${v_{3}}$}}\kern 19.98901pt{\hbox{\kern-4.003pt${v_{2}}$}}\kern 19.98901pt{\hbox{\kern-4.003pt${v_{1}}$}}\kern 19.98901pt{\hbox{\kern-4.003pt${v_{4}}$}}\kern 19.98901pt{\hbox{\kern-4.003pt${v_{5}}$}}\@arraycr\par\kern 9.9945pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\hbox{}\@arraycr\par\kern 9.9945pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\par\kern 9.9945pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\par\kern 9.9945pt\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\par\kern 9.9945pt{\hbox{\kern-4.003pt${v_{5}}$}}\kern 19.98901pt{\hbox{\kern-4.26228pt${u_{3}}$}}\kern 19.98901pt{\hbox{\kern-4.26228pt${u_{2}}$}}\kern 19.98901pt{\hbox{\kern-4.26228pt${u_{1}}$}}\kern 19.98901pt{\hbox{\kern-4.26228pt${u_{4}}$}}\@arraycr\par\kern 49.97253pt{\hbox{\kern-16.5417pt${\hbox{figure 5}}$}}\@arraycr\par\end{array}\ \ \ \ ,$ where $u_{3}:=q_{35}v_{3}$, $u_{2}:=q_{25}v_{2}$, $u_{1}:=q_{15}v_{1}$, $u_{4}:=q_{45}v_{4}.$ Define that the brightness in the first floor, the second floor, the third floor and negative first floor are $q_{15}$ unit, $q_{25}$ unit, $q_{35}$ unit and $q_{45}$ unit according to $q_{35}v_{1}$, $q_{25}v_{2}$, $q_{35}v_{2}$ and $q_{45}v_{4}$ in braiding diagram. We can choose the value of $q_{st}$ according to the physical truth and requirement of customer. In fact, we also provide the brightness of every components of car be means of braiding diagram. ## 3 Quantum Design of Codes In this section we obtain a code by means of braiding. ###### Example 3.1. We need send a word $w$ from $X$ to $Y$ by internet and require to keep secret, where $X$ and $Y$ are two persons or two companies. Let $\\{D_{i}\mid 1\leq i\leq n\\}$ be a vocabulary and $Q=(q_{ij})_{n\times n}$ its quantum matrix. $\begin{array}[c]{lll}\kern 39.97803pt{\hbox{\kern-4.003pt${v_{1}}$}}\kern 19.98901pt{\hbox{\kern-4.003pt${v_{2}}$}}\kern 19.98901pt{\hbox{\kern-4.003pt${v_{3}}$}}\kern 19.98901pt{\hbox{\kern-4.003pt${v_{4}}$}}\kern 19.98901pt{\hbox{\kern-4.003pt${v_{5}}$}}\kern 19.98901pt{\hbox{\kern-4.003pt${v_{6}}$}}\kern 19.98901pt{\hbox{\kern-4.003pt${v_{7}}$}}\kern 19.98901pt{\hbox{\kern-4.003pt${v_{8}}$}}\kern 19.98901pt{\hbox{\kern-4.003pt${v_{9}}$}}\kern 19.98901pt{\hbox{\kern-5.403pt${v_{10}}$}}\@arraycr\par{\hbox{\kern-2.5pt${1}$}}\kern 39.97803pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\par{\hbox{\kern-2.5pt${2}$}}\kern 39.97803pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\kern 19.98901pt\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\par{\hbox{\kern-2.5pt${3}$}}\kern 39.97803pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\kern 19.98901pt\kern 19.98901pt\hbox{\put(1000.0,2000.0){\oval(2000.0,2000.0)[cb]}\put(1000.0,0.0){\line(0,1){1000.0}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\par{\hbox{\kern-2.5pt${4}$}}\kern 39.97803pt\kern 19.98901pt\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\kern 19.98901pt\kern 9.9945pt\kern 9.9945pt\hbox{}\@arraycr\par{\hbox{\kern-2.5pt${5}$}}\kern 39.97803pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\kern 19.98901pt\kern 9.9945pt\kern 9.9945pt\hbox{}\@arraycr\@arraycr\par{\hbox{\kern-2.5pt${6}$}}\kern 39.97803pt\hbox{\put(1000.0,2000.0){\oval(2000.0,2000.0)[cb]}\put(1000.0,0.0){\line(0,1){1000.0}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\kern 19.98901pt\hbox{\put(1000.0,2000.0){\oval(2000.0,2000.0)[cb]}\put(1000.0,0.0){\line(0,1){1000.0}}}\kern 9.9945pt\kern 9.9945pt\hbox{}\@arraycr\par\par{\hbox{\kern-2.5pt${7}$}}\kern 49.97253pt\kern 19.98901pt\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\kern 9.9945pt\kern 9.9945pt\hbox{}\@arraycr\par{\hbox{\kern-2.5pt${8}$}}\kern 59.96704pt\hbox{\put(1000.0,2000.0){\oval(2000.0,2000.0)[cb]}\put(1000.0,0.0){\line(0,1){1000.0}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\par{\hbox{\kern-2.5pt${9}$}}\kern 69.96155pt\kern 19.98901pt\kern 19.98901pt\hbox{\put(1000.0,2000.0){\oval(2000.0,2000.0)[cb]}\put(1000.0,0.0){\line(0,1){1000.0}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\@arraycr\par{\hbox{\kern-5.00002pt${10}$}}\kern 79.95605pt\hbox{\put(1000.0,2000.0){\oval(2000.0,2000.0)[cb]}\put(1000.0,0.0){\line(0,1){1000.0}}}\kern 19.98901pt\hbox{\put(0.0,0.0){\line(0,1){2000.0}}}\kern 19.98901pt\kern 19.98901pt\hbox{}\@arraycr\par{\hbox{\kern-5.00002pt${11}$}}\kern 89.95056pt\kern 19.98901pt\kern 19.98901pt\hbox{}\@arraycr\par{\hbox{\kern-5.00002pt${12}$}}\kern 99.94507pt\hbox{\put(1000.0,2000.0){\oval(2000.0,2000.0)[cb]}\put(1000.0,0.0){\line(0,1){1000.0}}}\kern 9.9945pt\kern 9.9945pt\hbox{}\@arraycr\par{\hbox{\kern-5.00002pt${13}$}}\kern 109.93958pt\hbox{\put(1000.0,2000.0){\oval(2000.0,2000.0)[cb]}\put(1000.0,0.0){\line(0,1){1000.0}}}\kern 9.9945pt\hbox{}\@arraycr\par{\hbox{\kern-5.00002pt${14}$}}\kern 119.93408pt\hbox{\put(1000.0,2000.0){\oval(2000.0,2000.0)[cb]}\put(1000.0,0.0){\line(0,1){1000.0}}}\@arraycr\par\kern 129.92859pt{\hbox{\kern-5.27374pt${qu}$}}\@arraycr\par\kern 119.93408pt{\hbox{\kern-16.5417pt${\hbox{figure 6}}$}}\@arraycr\par\end{array}\ \ \ \ ,$ where $u=v_{5}v_{6}v_{4}v_{7}v_{3}v_{8}v_{2}v_{9}v_{1}v_{10}$. Let $Q$ and figure 6 are the private key cryptography. i.e. $X$ and $Y$ have $Q$ and figure 6 without using internet. Assume $w=D_{s}$. We choose $p$ such that $s\in\\{p,p+1,\cdots,p+9\\}$ and positive real numbers $a_{1},\cdots,a_{10}$ randomly with $a_{1}\not=1,\cdots,a_{10}\not=1$. Let $b_{i}=a_{i}$ when $i\not=s$ and $b_{s}=1.$ Let $v_{i}:=D_{p+i-1}$ and $\widetilde{q_{ij}}:=q_{p+i-1,p+j-1}$ for $1\leq i,j\leq 10.$ It is clear $C^{-1}(v_{i}\otimes v_{j})=\widetilde{q_{ji}}^{-1}(v_{j}\otimes v_{i})$ and $C^{-1}(v_{i}\otimes u)=\prod_{j=1}^{10}\widetilde{q_{ji}}^{-1}(u\otimes v_{i})$. Compute the value $c_{i}$ of $b_{i}\prod_{j=1}^{10}\widetilde{q_{ji}}^{-1}$ and send $c_{i}$ to $Y$ for $i=1,2,\cdots,10$ and send $p$ to $Y$. In $Y$ compute the value $d_{i}$ of $c_{i}\prod_{j=1}^{10}\widetilde{q_{ij}}$ and $d_{i}=1$ if and only if $s=i+p-1$ since coefficient of $C^{-1}(v_{i}\otimes u)$ is $\prod_{j=1}^{10}\widetilde{q_{ji}}^{-1}$ and coefficient of $c(u\otimes v_{i})$ is $\prod_{j=1}^{10}\widetilde{q_{ji}}$. Consequently, we can get the word $w$ in $Y.$ Furthermore, we can send a article from $X$ to $Y$ by means of internet and keeping secret because they consist of some single words. We also can send file from $X$ to $Y$ by means of internet and keeping secret because the password consist of some single words. Otherwise, we can modify the method above as follows. Remark. (i) In case above, we choose the last layer 14-th layer and left line $u$. In fact, we also can choose $t$-th layer with $1\leq t\leq 14$ and left line $v$ which need be sent to $Y$. That is, private key cryptography contains $Q$, figure 6, $p$, $t$-th layer and left line $v$. For example, set $t=7$, i.e. $v=v_{5}v_{6}.$ It is clear $C^{-1}(v_{i}\otimes v)=\prod_{j=5}^{6}\widetilde{q_{ji}}^{-1}(u\otimes v_{i})$ and $c(v\otimes v_{i})=\prod_{j=5}^{6}\widetilde{q_{ji}}(v_{i}\otimes v)$. Compute the value $c_{i}$ of $b_{i}\prod_{j=5}^{6}\widetilde{q_{ji}}^{-1}$ and send $c_{i}$ to $Y$ for $i=1,2,\cdots,10$. In $Y$ compute the value $d_{i}$ of $c_{i}\prod_{j=5}^{6}\widetilde{q_{ij}}$ and $d_{i}=1$ if and only if $s=i$. (ii) figure 6 and vocabulary $\\{D_{i}\mid 1\leq i\leq n\\}$ can become public key cryptography. The order of vocabulary $\\{D_{i}\mid 1\leq i\leq n\\}$ can be the same as Xinhua dictionary. (iii) Vocabulary $\\{D_{r+i}\mid 1\leq i\leq n\\}$ can become public key cryptography and $r$ can become the private key cryptography. ## References [BD82] A. A. Belavin and V. G. Drinfel’d. Solutions of the classical Yang–Baxter equations for simple Lie algebras. Functional Anal. Appl, 16 (1982)3, 159–180. * [BD98] Y.Bespalov, B.Drabant, Hopf (bi-)modules and crossed modules in braided monoidal categories, J. Pure and Applied Algebra, 123(1998), 105-129. * [He06b] I. Heckenberger, The Weyl-Brandt groupoid of a Nichols algebra of diagonal type, Invent. Math. 164 (2006), 175–188. * [He06a] I. Heckenberger, Classification of arithmetic root systems, Adv. Math. 220 (2009), 59-124. * [Ho10] F. Hoisl, An interactive 3D spatial grammar system, Design Computing and Cognition DCC 08. J.S. Gero (eds), Press Springer 2010, 643-662. * [Ka95] C. Kassel. Quantum Groups. Graduate Texts in Mathematics 155, Springer-Verlag, 1995. * [Ma69] C. A Mackenzie, Inferring relational design grammars. Environment and Planning B 16(3): 253-287 * [Ma98] S. MacLane, Categories for the Working Mathematician, Graduate Texts in Mathematics 5 (2nd ed.), Springer-Verlag, 1998 . * [OCB08] S. Orsborn, J. Cagan , P. Boatwright, A methodology for creating a statistically derived shape grammar composed of non-obvious shape chunks. Research in Engineering Design 18(2008)4: 181-196. * [OCP08] S. Orsborn, J. Cagan and P. Boatwright, Automating the Creation of Shape Grammar Rules, Design Computing and Cognition DCC 08. J.S. Gero and A. Goel (eds), Press Springer 2008, pp. 3-22. * [RT93] D.E.Radford, J.Towber, Yetter-Drinfeld categories associated to an arbitrary bialgebra, J. Pure and Applied Algebra, 87(1993) 259-279. * [St06] G. Stiny, Shape, talking about seeing and doing. MIT Press, Cambridge, Massachusetts, 2006. * [YG89] C. N. Yang and M. L. Ge. Braid group, Knot theory and Statistical Mechanics. World scientific , Singapore, 1989. * [Zh99] Zhang S. C., Braided Hopf Algebras. Changsha: Hunan Normal University Press, Second edition 2005. Also in math.RA/0511251. * [ZZC04] S. Zhang, Y-Z Zhang and H.-X. Chen, Classification of PM quiver Hopf algebras, J. Algebra Appl., 6 (2007)(6), 919-950.	arxiv-papers	2014-01-13T15:12:32	2024-09-04T02:49:56.651233	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Zhengtang Tan, Shouchuan Zhang", "submitter": "Shouchuan Zhang", "url": "https://arxiv.org/abs/1401.2864" }
1401.2950	On a problem of A. Nagy concerning permutable semigroups satisfying a non-trivial permutation identity111Research supported by the Hungarian NFSR grant No T042481 AMS Subject classification number: 20M10 Attila Deák Communicated by M. B. Szendrei ###### Abstract We say that a semigroup $S$ is a permutable semigroup if, for every congruences $\alpha$ and $\beta$ of $S$, $\alpha\circ\beta=\beta\circ\alpha$. In [4], A. Nagy showed that every permutable semigroup satisfying an arbitrary non-trivial permutation identity is medial or an ideal extension of a rectangular band by a non-trivial commutative nil semigroup. The author raised the following problem: Is every permutable semigroup satisfying a non-trivial permutation identity medial? In present paper we give a positive answer for this problem. A semigroup $S$ is called a permutable semigroup if the congruences of $S$ commute with each other, that is, $\alpha\circ\beta=\beta\circ\alpha$ is satisfied for every congruences $\alpha$ and $\beta$ of $S$. ###### Theorem 1 Let $S$ be an ideal extension of a rectangular band $K$ by a non trivial commutative nil semigroup $N$. If $S$ is permutable and satisfies a non-trivial permutation identity then either $S$ is medial or $N$ is nilpotent. Proof. Assume that a permutable semigroup $S$ satisfies a permutation identity $x_{1}x_{2}\ldots x_{n}=x_{\sigma(1)}x_{\sigma(2)}\ldots x_{\sigma(n)},\ (\sigma\neq id).$ We can suppose that $n>2$. Let $l$ and $k$ be non-negative integers such that $\sigma(1)=1,\sigma(2)=2,\cdots,\sigma(l)=l,\sigma(l+1)\neq l+1$ and $\sigma(n-k)\neq n-k,\sigma(n-k+1)=n-k+1,\cdots,\sigma(n)=n.$ If $l=k=0$ then, by Theorem 1 of [4], $S$ is commutative. Consider the case when $l>0$ or $k>0$. Let $j$ be $\max\\{l,k\\}$. Assume that $j=1$. Then let $\beta$ be a relation on $S$ such that $a\,\beta\,b\Leftrightarrow uabv=ubav$ for every $u,v\in S$. It is clear that $\beta$ is reflexive and symmetric. We show that $\beta$ is transitive. Let $u,v,a,b,c\in S$ be arbitrary elements such that $a\,\beta\,b$ and $b\,\beta\,c$. If $ac\in S-K$ then $ac=ca$, because $N$ is commutative. Thus $a\,\beta\,c$. Assume that $ac\in K$. Then $ca\in K$ and $ac=ac(xac)ac=acxac,\ ca=ca(xca)ca=caxca$ for every $x\in S$. Thus $uacv=(ua)cb(acv)=uab(cacv)=ubacacv=ubacv=uabcv=uacbv.$ Similarly, $ucav=ubcav=ucbav=ucabv.$ From these results it follows that $uacv=ub^{p}ab^{q}cb^{r}v$ for every non-negative integers $p,q,r$. Similarly, $ucav=ub^{p}cb^{q}ab^{r}v$ for every non-negative integers $p,q,r$. Let $\hat{l},\ \hat{k}$ be such that $\sigma(l+1)=\hat{l},\ \sigma(n-k)=\hat{k}$. It is obvious that $\hat{l}\geq l+2$ and $\hat{k}\leq n-k-1$. So $uacv=uacb^{n-2}v=ub^{l}ab^{\hat{l}-l-2}cb^{n-\hat{l}}v=ub^{l}cb^{t}ab^{s}v=ucav,$ where $s,t$ are non-negative integers such that $s+t+l=n-2$. Hence $\beta$ is an equivalence on $S$. Let $x,y\in K$. We will show that $x\,\beta\,y$. Let $u,v\in S$ be arbitrary elements. Then $uxyv=ux^{\hat{l}-l-1}y^{n-\hat{l}}yv=uyxyv=uyv=uy^{\hat{k}}xy^{n-\hat{k}-k-1}v=uy^{n-k-1}xv=uyxv.$ Since, for every $a\in N$, there is a positive integer $m$ such that $a^{m}\in K,\ a\,\beta\,a^{m}$, we get that $S$ is a $\beta$ class. Hence $S$ is medial. Consider the case when $j>1$. Let $\alpha$ be an equivalence of $S$ such that $a\,\alpha\,b\Leftrightarrow\forall x\in K\ xa=xb$ and $ax=bx$. It is obvious that $x\,\alpha\,y$ implies $x=y$ for every $x,y\in K$. We show that $\alpha$ is also a congruence. Let $y,a,b\in S$ be arbitrary elements. Assume $a\,\alpha\,b$ then $x(ay)=(xa)y=(xb)y=x(by)$ and $(ay)x=a(yx)=b(yx)=(by)x$ and so $ay\,\alpha\,by$. We can prove $ya\,\alpha\,yb$ in a similar way. Assume that $N$ is not nilpotent. Let $a_{1},a_{2},\cdots,a_{j}\in S-K$ be such that $a_{1}a_{2}\ldots a_{j}\in S-K$. We prove that $a_{1}a_{2}\ldots a_{j}\,\alpha\,a_{1}a_{2}\ldots a_{j}^{m}$ where $m$ is the least positive integer such that $a_{j}^{m}\in K$. Since $a_{1}a_{2}\ldots a_{j}\in S-K$ then $a_{i_{1}}a_{i_{2}}=a_{i_{2}}a_{i_{1}}$ for every $i_{1},i_{2}\in\\{1,2,\cdots,j\\}$. Let $l^{{}^{\prime}}$ and $k^{{}^{\prime}}$ be such that $\sigma(l+1)=l^{{}^{\prime}}$ and $\sigma(n-k)=k^{{}^{\prime}}$. Then, for every $x\in K$, $a_{1}a_{2}\ldots a_{j}x=a_{1}a_{2}\ldots a_{j}x^{l^{{}^{\prime}}-l-1}a_{j}x^{n-l^{{}^{\prime}}}x=a_{1}a_{2}\ldots a_{j}a_{j}x^{n-l-1}x=a_{1}a_{2}\ldots a_{j}^{2}x$ and so $a_{1}a_{2}\ldots a_{j}x=a_{1}a_{2}\ldots a_{j}^{m}x.$ Using also $a_{i_{1}}a_{i_{2}}=a_{i_{2}}a_{i_{1}}\ (i_{1},i_{2}\in\\{1,2,\cdots,j\\})$, we have $xa_{1}a_{2}\ldots a_{j}=xa_{1}a_{2}\ldots a_{j}^{m}$ in a similar way. From these results we have $a_{1}a_{2}\ldots a_{j}\,\alpha\,a_{1}a_{2}\ldots a_{j}^{m}.$ From (2) of Lemma 3 of [3], we get $\mid K\mid=1$, and so $S$ is commutative. $\begin{array}[]{c}\sqcap\\\\[-3.99994pt] \hline\cr\lx@intercol\hfil\hfil\lx@intercol\\\\[-8.00003pt] \end{array}$ ###### Theorem 2 If $S$ is an ideal extension of a rectangular band $K$ by a non trivial commutative nilpotent semigroup $N$, and $S$ is permutable then S is commutative. Proof. Assume that a permutable semigroup $S$ is an ideal extension of a rectangular band $K$ by a non trivial commutative nilpotent semigroup $N$. By Lemma 2 of [4], $N$ is a $\Delta$-semigroup. Then, by Lemma 2 of [5], $N$ is finite ciclyc. Let $N=\\{a,a^{2},\cdots,a^{n}\\}$ where $a^{n}\in K$. Assume $a^{n}=(1,1)$ and $K=L\times R$, where $L$ is a leftzero, $R$ is a rightzero semigroup. For every $(x_{1},x_{2})\in K$, $a(x_{1},x_{2})=(\phi_{a}(x_{1},x_{2}),x_{2})$ and $(x_{1},x_{2})a=(x_{1},\psi_{a}(x_{1},x_{2}))$. If $(x_{1},x_{3})\in K$, then $a(x_{1},x_{3})=a(x_{1},x_{2})(x_{2},x_{3})=(\phi_{a}(x_{1},x_{2}),x_{2})(x_{2},x_{3})=(\phi_{a}(x_{1},x_{2}),x_{3})$ and so $\phi_{a}(x_{1},x_{2})=\phi_{a}(x_{1},x_{3})$. So $\phi_{a}$ is independent from the second variable. Similarly, $\psi_{a}$ is independent from the first variable. From these results we get that, for every $(x_{1},x_{2})\in K$, $a(x_{1},x_{2})=(\phi_{a}(x_{1}),x_{2})$ and $(x_{1},x_{2})a=(x_{1},\psi_{a}(x_{2}))$. We will prove that $\phi_{a}(x)=1$ for every $x\in L$. It is obvious that $\underbrace{\phi_{a}\circ\ldots\circ\phi_{a}}_{\mbox{n}}(x)=\phi_{a}^{n}(x)=1$ for every $x\in L$. Let $j$ be such that $\phi_{a}^{j}(x)=\phi_{a}^{j+1}(x)=\ldots=\phi_{a}^{n}(x)=1\ \forall x\in L$. Assume that $\exists x\in L$ such that $\phi_{a}^{j-1}(x)\neq 1$. Let $P:=\\{x\in L:\,\phi_{a}^{j-1}(x)\neq 1\\}$ and $Q:=\\{(x_{1},x_{2})\in K:x_{1}\in P\\}.$ It is clear that, for every $(x_{1},x_{2})\in K$, $a^{r}(x_{1},x_{2})\notin Q$, for all positive integers $r$, because $a^{r}(x_{1},x_{2})=(\phi_{a}^{r}(x_{1}),x_{2})$ and $\phi_{a}^{j-1}(\phi_{a}^{r}(x_{1}))=\phi_{a}^{j+r-1}(x_{1})=1$. Since $\phi_{a}(1)=1$ then $Q\neq K$. Let $\alpha$ be an equivalence on $S$ such that, for arbitrary $x,y\in S$, $x\,\alpha\,y\Leftrightarrow x,y\in Q$ or $x,y\in S-Q$. We show that $\alpha$ is a congruence of $S$. Let $x,y,z\in S$ be arbitrary elements with $x\,\alpha\,y$. Assume that $x,y\in K$. It is clear that if $z\in K$ then $xz\,\alpha\,yz$ and $zx\,\alpha\,zy$. Assume $z=a^{l}\in S-K$. It is obvious that $xa^{l}\,\alpha\,ya^{l}$. Since $a^{l}x,a^{l}y\notin Q$ then $a^{l}x\,\alpha\,a^{l}y$ is obvious. Consider the case when $x=a^{m}\in S-K$ and $y\in K$ where $m$ is a positive integer less than $n$. For every $z\in K\ zx\,\alpha\,zy$, is obvious. Since $xz,y\notin Q$, we get that $xz\,\alpha\,yz$. Consider the case when $x,y\in S-K$, that is $x=a^{p}$ and $y=a^{q}$, where $p,q$ are positive integers less than $n$. For all $z\in S$, $zx\,\alpha\,zy$ and $xz\,\alpha\,yz$ are obvious. So $\alpha$ is a congruence, indeed. There are two $\alpha$ classes on $S$: $Q$ and $S-Q$. It is clear that $a^{n}\in K-Q$ and so $Q\neq K$. From (2) of Lemma 3 of [3], $K$ have to be union of $\alpha$ classes or subset of an $\alpha$ class. Since $(S-Q)\cap K\neq 0$, $(S-Q)\cap(S-K)\neq 0$ and $Q\subset K$, then $Q$ have to be an empty set! Since $\mid Q\mid\geq\mid P\mid$, then $0=\mid Q\mid\geq\mid P\mid\geq 0$. That means $\mid P\mid=0$, that is $\phi_{a}(L)=\\{1\\}$. We can prove $\psi_{a}(R)=\\{1\\}$ in a similar way. Consequently, for every $x\in K$, $ax=a^{2}x=\ldots=a^{n}x$ and $xa=xa^{2}=\ldots=xa^{n}$. Let $\beta$ be an equivalence of $S$ such that $x\,\beta\,y\Leftrightarrow x=y$ or $x=a^{l},\ y=a^{k}$ where $l,k$ are positive integers. Let $x,y,z\in S$ be arbitrary elements with $x\,\beta\,y$. If $x=y$ then $xz\,\beta\,yz$ and $zx\,\beta\,zy$ are obivous. If $x\neq y$ then, with $x\,\beta\,y$, we get that $x=a^{l}$ and $y=a^{k}$, for some positive integers $l,k$. If $z=a^{m}$ then $a^{m}a^{l}\,\alpha\,a^{m}a^{k}$ and $a^{l}a^{m}\,\alpha\,a^{k}a^{m}$ are obvious. Let $x\in K$, then $xa^{l}=xa^{n}=xa^{k}$ and $a^{l}x=a^{n}x=a^{k}x$. Hence $\beta$ is a congruence. It is clear that, for all $x,y\in K$, $x\,\beta\,y\Leftrightarrow x=y$ and, for all $x,y\in\\{a,a^{2},\cdots,a^{n}\\}$, $x\,\beta\,y$. So the $\beta$ classes of $S$ are: $\\{a,a^{2},\cdots,a^{n}\\}=N$ and for every $x\in S-N$, $x$ is a $\beta$ class. From (2) of Lemma 3 of [3] $K$ have to be union of $\beta$ classes, or subset of a $\beta$ class. Since $\\{a,a^{2},\cdots,a^{n}\\}\cap K=a^{n}$ and $S=\\{a,a^{2},\cdots,a^{n-1}\\}\cup K$, then $K={a^{n}}$. That is $\mid K\mid=1$. And so $S$ is commutative. $\begin{array}[]{c}\sqcap\\\\[-3.99994pt] \hline\cr\lx@intercol\hfil\hfil\lx@intercol\\\\[-8.00003pt] \end{array}$ ###### Corollary 1 Every permutable semigroup satisfying a non-trivial permutation identity is medial. Proof. From Theorem 2 of [4] and Theorem 1 and Theorem 2 of this paper it is obvious. $\begin{array}[]{c}\sqcap\\\\[-3.99994pt] \hline\cr\lx@intercol\hfil\hfil\lx@intercol\\\\[-8.00003pt] \end{array}$ ## References * [1] Clifford, A.H. and G.B. Preston, The Algebraic Theory of Semigroups, Amer. Math. Soc., Providence, R.I., I(1961), II(1967) * [2] Hamilton, H., Permutability of congruences on commutative semigroups, Semigroup Forum, 10(1975), 55-66 * [3] Jiang, Z., LC-commutative permutable semigroups, Semigroup Forum, 52(1995), 191-196 * [4] Nagy, A., Permutative semigroups satisfying a non-trivial permutation identity, Acta Sci. Math. (Szeged), 71(2005), 37-43 * [5] Nagy, A. and P.R. Jones, Permutative semigroups whose congruences form a chain, Semigroup Forum (to appear) Attila Deák, Department of Algebra, Institute of Mathematics, Budapest University of Technology and Economics; e-mail: [email protected]	arxiv-papers	2014-01-10T11:28:41	2024-09-04T02:49:56.661556	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Attila De\\'ak", "submitter": "Attila De\\'ak", "url": "https://arxiv.org/abs/1401.2950" }

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