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+{"text":"\\section{Introduction} \\label{Introduction}\n\nSolar flares have been studied in detail both observationally and theoretically ever since their discovery by \\citet{Ca859}. Although increasingly more sophisticated instrumentation provides ever more detailed data, we still lack the basic understanding of many processes at work in a solar flare. \n\nThe common flare picture as deduced from hard X-ray (HXR) observations features an HXR source in the corona~\\citep[coronal or loop-top source,][]{Fr71,Hu78}, and two or more HXR sources (footpoints) in the chromosphere \\citep{Hoy81}. These sources are thought to be due to bremsstrahlung emission produced by fast electrons accelerated somewhere above the loop. If we assume that a single particle beam creates both coronal and footpoint emission, the most basic model would involve thin target emission at the top of the coronal loop and thick target emission from the footpoints, which both produce characteristic spectra. \n\\citet{Wh95} developed a more sophisticated model (intermediate thin-thick target or ITTT model) to fit observations by Yohkoh. They based their model on observations by \\citet{Fe94}, who found high column densities at the loop top, which might act as a thick target below a certain electron energy. In the ITTT model, the shape of the coronal and footpoint non-thermal spectra and the relation between them, observed by Yohkoh, can be explained. The column density in the coronal source determines a critical energy level for the electrons. Electrons that have an energy below this critical energy are stopped in the coronal region. Consequently, the distribution of electron energies measured at the footpoints is depleted in low energy electrons. If the column density is high, the coronal source may act as a thick target to electrons of energies as high as 60 keV, which would leave almost no footpoint emission. Observational evidence for such coronal thick targets were found in RHESSI observations \\citep[eg.][]{Ve04}.\n\nLess extreme cases, flares with one or more footpoints, have frequently been observed by RHESSI.\nTo study the spectral time evolution of individual sources, five well-observed events were analyzed by \\citet{Ba06} who focused on the differences between the spectral indices of coronal and footpoint spectra. They found that in two of those events, the differences at specific times as well as the time-averaged difference was significantly larger than two, ruling out a simple thin-thick target interpretation. In \\citet{Ba07}, the spectra of the five events were compared with the predictions of the ITTT model. The authors exploited the order of magnitude improvement in spectral resolution of RHESSI over the 4-point Yohkoh spectra and showed that most RHESSI observations could not be explained by the ITTT model. \n\n\\citet{Ba07} proposed that by considering non-collisional energy loss inside the loop this inconsistency could be resolved.  \nA possible mechanism that causes non-collisional energy loss is an electric field. Accelerating electrons out of the coronal source region drives a return current to maintain charge neutrality in the whole loop. For finite conductivity, Ohm's law implies that an electric field must be present. The beam electrons lose energy because of work expended in moving inside the electric potential. This produces a change in the shape of the electron spectrum at the footpoints. The formation and evolution of these return currents were studied by various authors \\cite[e.g.][]{Kn77,SS84,La89,Oo90}. \\citet{Zh06} proposed that return currents could explain the high energy break observed in flare HXR spectra. \nMost studies have, however, been theoretical proposals or numerical simulations, based on standard flare values, that do not attempt to explain or reproduce true solar flare observations. \n\n\\citet{Ba07} compared RHESSI spectra to the ITTT-model of \\citet{Wh95}, demonstrating that the qualitative shape and relations between coronal and footpoint-spectrum often do not agree with the model predictions. In this study, we take an additional step by completing a quantitative analysis of the relation linking coronal and footpoint spectra in the context of the thin-thick target model; we demonstrate that, in some cases, electric fields related to return currents can indeed explain the relation between coronal and footpoint spectra.\n\nIn Sect.~\\ref{Theory} we summarize the basic physical concepts applied in the paper. Section~\\ref{evdescription} provides a brief overview of the analyzed events and a description of the spectral analysis. In Sect. \\ref{Method}, we describe our calculation of the energy loss required to reproduce the observed footpoint spectrum, constrained by the coronal emission. Our results are presented in Sect.~\\ref{results}. In Sect. \\ref{retcurrnefield}, we link those results to the concept of return currents. \n\n\\section{Thin and Thick target emission} \\label{Theory}\n\nTwo types of bremsstrahlung emission are distinguished. If the electrons pass a target without losing a significant amount of energy, the corresponding emission is referred to as thin target \\citep{Da73}. This situation is expected to occur in coronal regions when electrons pass through a target of insufficient column density to stop them. If the electrons are fully stopped inside the target, the resultant emission is called thick target emission \\citep{Br71}. This is the case for the dense chromospheric material at the footpoints. \n\nFor an input power-law electron distribution of the shape $F(E)=A_{E}E^{-\\delta}$, the non-relativistic bremsstrahlung theory predicts power-law photon spectra \n\\[I(\\epsilon)\\propto \\epsilon^{-\\gamma}\\quad\\mbox{where}\\quad\\left\\{\\begin{array}{l} \\gamma=\\delta+1\\quad\\mbox{in the thin target case} \\\\ \\gamma=\\delta-1\\quad\\mbox{in the thick target case} \\end{array} \\right. \\] \nThe observable distinction between the two emission mechanisms is a difference $\\Delta \\gamma$ of value 2 in their observed photon spectral indices. \n\nAssuming a final column density in the coronal source, the coronal source spectrum is a thick target at low energies and a thin target at high energies with a break at some critical energy. The footpoint spectrum is depleted at low energies, as low energy electrons do not reach the chromosphere. An illustration of this can be found in \\citet{Wh95} or \\citet{Ba07}.\n\nIn the events that we analyze here, a thermal component is present in all observations. Observation of the non-thermal emission is therefore only possible at photon energies higher than 15 keV.  For these energies we can assume that the coronal source is a pure thin target and the footpoints are a pure thick target.\n\n\n\\section{Event description and spectral analysis} \\label{evdescription}\nThe two events that we analyze were described in detail by \\citet{Ba06, Ba07}. They were selected because of the significant differences between the footpoint and coronal non-thermal spectral index.  The first event occurred on 24 October 2003 around 02:00 UT (GOES M7.7), the second on 13 July 2005 around 14:15 UT (GOES M5.1). Both events occurred close to the limb but were not occulted; two footpoints were therefore fully observed in both cases. RHESSI light curves from the time of main emission are shown in Fig.~\\ref{2flimages}. An EIT image of the 24 October 2003 event and a GOES SXI image for the event of 13 July 2005 are presented for orientation. The contours of the coronal source and the footpoints from RHESSI images are overlaid. \n\\begin{figure*} \n\\resizebox{\\hsize}{!}{\\includegraphics{9418fig1.eps}}\n\\caption {\\textit{Top}: RHESSI light curves in the 6-12 and 25-50 keV energy band. The analyzed time interval is indicated by the \\textit{vertical bar}. \\textit{Bottom}: SOHO\/EIT image at 24 October 2003 02:47:31 (\\textit{left}), GOES SXI image at 13 July 2005 14:19:04 (\\textit{right}). The 30 \\%, 50 \\% and 70 \\% contours from RHESSI Pixon images of the coronal source at 12-16 keV (\\textit{solid contours}) and 20-24 keV (\\textit{dashed contours}, mainly non-thermal emission) are overlaid along with the 50\\% contour of the footpoint sources at 25-50 keV.}\n\\label{2flimages}\n\\end{figure*}\n\n\n\\subsection{Spectral fitting and analysis} \\label{fitting}\n\nThe two events were analyzed using imaging spectroscopy with the PIXON  algorithm \\citep{Me96, Hur02}. Images were made in a 30 second time interval during which the flux was sufficiently high for good images but pile-up was low. The image times are given in Table~\\ref{fitpartable} and shown in Fig.~\\ref{2flimages}. The spectra of the footpoints and the coronal source were measured and fitted. The regions of interest from which the spectra were computed were chosen to be a circle around the coronal source and a polygon around the footpoints to include all of the emission at all energies. The effects of this method of region selection are discussed in \\cite{Ba05}. As a simplification, the footpoints were treated as one region and the spectrum was fitted with a single power-law. In the presence of the footpoints, the non-thermal emission in coronal sources is difficult to observe. Therefore, two methods of fitting the coronal source were used. First, a thermal component was fitted to the spectrum at low energies and a single power-law to the energies higher than about 25 keV. As a second method, the full sun thermal spectrum was fitted. As shown in \\cite{Ba05}, the thermal emission observed in full sun spectra is mostly coronal emission. We therefore used the thermal full sun fit as an approximation to the coronal thermal component and completed a power-law fit at the higher energies, while the thermal emission was fixed. This supports the idea that non-thermal emission exists in the coronal source and provides an estimate of the accuracy of the non-thermal coronal fit. The energy ranges for the fits were 8-36 keV for the coronal source and 24-80 keV for the footpoints. All fit parameters are provided in Table~\\ref{fitpartable}.\n\nIn the thin-thick target model an electron beam is assumed to be injected into the center of the coronal source.\nThe column depth that the electrons travel through in the corona is then $\\Delta N=n_e\\cdot l$, where the path length \\textit{l} is half the coronal source length. \nFrom RHESSI images in the 10-12 keV band, the source area \\textit{A} was measured to be the 50\\% contour of the maximum emission. We approximate the source volume to be $V=A^{3\/2}$ and the path length to be $l=\\sqrt{A}\/2$. Using the observed emission measure EM, the particle density is computed to be $n_e=\\sqrt{EM\/V}$ which corresponds to a column depth of $\\Delta N=\\sqrt{EM}A^{-1\/4}\/2$ expressed in observable terms.\nA volume filling factor of  1 was assumed for the computation of the density, which will be improved in Sect.~\\ref{expfemissionloss}.\nThe emission measures were taken from the spectral fits to the coronal source and to full sun spectra. Additionally, temperatures and emission measures observed by GOES were included. This provides a range for the emission measures, temperatures, and column depths, and an estimate of their uncertainty.  \\\\\n\n\n\\section{Method}\\label{Method}\n\nStarting from the assumption that the observed coronal spectrum at high photon energies is caused by thin target emission, we compute the electron distribution and therefore the expected footpoint photon spectrum. \nThis is completed via the following steps.\n\\begin{enumerate}\n\\item We assume that the observed coronal photon spectrum can be fitted by a power law:\n\\begin{equation}\n F^{cs}_{obs}(\\epsilon)=A_{\\epsilon}^{cs}\\epsilon^{-\\gamma^{cs}},\n\\end{equation} where A$_{\\epsilon}^{cs}$ is the normalization and $\\gamma^{cs}$ the photon spectral index.\n\\item Using thin target emission, the injected electron spectrum F(E) is then proportional to  \\label{itemelsp}\n\\begin{equation} \\label{bla}\nF(E)=A_E E^{-\\delta}\\sim\\frac{A_{\\epsilon}^{cs}}{\\Delta N}E^{-\\gamma^{cs}+1}\n\\end{equation} where $\\Delta N$ is the column depth the electrons travel through inside the coronal target \\citep{Da73}. \n\\item The expected thick target emission $F_{exp}^{fp}(\\epsilon)$ caused by this electron distribution in the footpoints can be computed as follows \\citep{Br71}:\n\\begin{equation}\n F^{fp}_{exp}(\\epsilon)=A_{\\epsilon,exp}^{fp}\\epsilon^{-\\gamma^{fp}_{exp}}\\sim\\frac{A_{\\epsilon}^{cs}}{\\Delta N}\\epsilon^{-(\\gamma^{cs}-2)}\n\\end{equation}\nThe superscripts fp and cs denote the footpoint and coronal source values, respectively.\n\\item The normalization and spectral index of $F^{fp}_{exp}(\\epsilon)$ is compared to the observed footpoint spectrum $F^{fp}_{obs}(\\epsilon)$.\n\\end{enumerate}\nIn thin-thick target models, the difference in spectral index $\\Delta \\gamma=|\\gamma^{cs}-\\gamma^{fp}|$ is 2. As the observed difference is larger than 2 in the selected events, a mechanism has to be found that causes the electron spectrum to harden while the beam passes down the loop. We present a mechanism that assumes an electric field which causes electrons to lose the energy $\\mathcal{E}_{loss}$ independently of the initial electron energy. The resulting spectrum is flatter, although not strictly a power-law function anymore (Fig.~\\ref{elsp}a). The deviation becomes substantial below 2 $\\mathcal{E}_{loss}$.\n\n\nThe necessary energy loss is determined as follows:\n\\begin{enumerate}\n\\item We start with the coronal electron distribution as found from Point~\\ref{itemelsp} in the above list.\n\\item By assuming a thin target, the electron distribution leaves the coronal source and propagates down the loop. A constant energy loss $\\mathcal{E}_{loss}$ is subtracted from the electron energies as the energy loss is independent of the electron energy.\n\\item We compute the expected thick target photon spectrum $F^{fp}_{exp}(\\epsilon)$ from this altered electron spectrum.\n\\item A power law is fitted to  $F^{fp}_{exp}(\\epsilon)$. The fitted energy range is 30-80~keV. This is the range for which footpoint emission is typically observed. \n\\end{enumerate}\nThe relation between the energy loss experienced  and the corresponding photon spectral index of the best-fit power law function depends on the initial electron spectral index $\\delta$ and the energy loss $\\mathcal{E}_{loss}$. It is equivalent to the elementary charge times the electric potential between the coronal source and the footpoints. If the initial electron spectral index is 8 for instance, the thick target photon spectral index without energy loss is 7. With increasing energy loss, this value decreases rapidly. The effect is less pronounced when the initial electron spectrum is harder. This is shown in Fig.~\\ref{elsp}b for several values of $\\delta$ and $\\mathcal{E}_{loss}$. Using the curves in this figure, we can easily determine the energy loss that causes an electron spectrum of spectral index $\\delta$ to result in a fitted photon spectral index $\\gamma^{fp}$. \n\n\\begin{figure*} \n\\resizebox{\\hsize}{!}{\\includegraphics{9418fig2.eps}}\n\\caption {a) Normalized electron power-law spectrum (\\textit{solid}) and altered spectrum due to a constant energy loss of 30 keV (\\textit{dashed}). b) Relation between loss energy $\\mathcal{E}_{loss}$ and fitted thick target photon power-law spectral index $\\gamma^{fp}$ for initial electron spectral index $\\delta=3,4,5,6,7,8$ in the accelerator.}\n\\label{elsp}\n\\end{figure*}\n\n\n\n\n\n\n\\begin{table*}\n\n\\begin{minipage}[t][]{2\\columnwidth}\n\n\n\\caption{Overview of main event properties and fit parameters }\n\\renewcommand{\\footnoterule}{} \n\\begin{tabular}{l|lll|lll}  \n\\hline \\hline\nTime interval &\\multicolumn{3}{c}{24 October 2003 02:48:20-02:48:50 }& \\multicolumn{3}{c}{13 July 2005 14:15:00-14:15:30 }\\\\\n\\hline\nArea [cm$^2$] \/ Volume [cm$^3$]&\\multicolumn{3}{c}{$7.9\\cdot 10^{18}$ \/ $2.2\\cdot 10^{28}$}&\\multicolumn{3}{c}{$1.7\\cdot 10^{18}$ \/ $2.2\\cdot 10^{27}$} \\\\\n\\hline\n&full sun&imspec& GOES&full sun&imspec&GOES\\\\\n\\hline\nTemperature [MK]\\footnote{ Thermal parameters for three different measuring methods (RHESSI full sun fit, RHESSI imaging spectroscopy, GOES). See also comment in Sect~\\ref{obsspectra}. }&21.6&23.2&15.4&23.8&22.3&18.1\\\\\nEmission measure [$10^{49}$cm$^{-3}$]&0.98&0.44&3.4&0.22&0.22&0.47\\\\\nElectron density [$10^{10}$cm$^{-3}$]&2.1&1.4&3.9&3.2&3.2&4.6\\\\\nColumn density [$10^{19}$cm$^{-2}$]&2.9&2.0&5.5&2.1&2.1&3.0 \\\\\n\\hline\n&footpoints&cs fit 1&cs fit2&footpoints&cs fit 1&cs fit 2\\\\\n$\\gamma$\\footnote{ Two different values (cs fit 1, cs fit 2) for the non-thermal coronal fit, distinguishing the two fitting methods used (compare Sect~\\ref{fitting}).}&2.6&6.2&6.1&2.9&5.1&5.6 \\\\\n$F_{50}[\\mathrm{photons\\, cm^{-2}s^{-1}keV^{-1}}]$&2.0&0.07&0.07&0.88&0.06&0.05 \\\\\n\n\\hline\n\\end{tabular}\n\\label{fitpartable}\n\\end{minipage}\n\n\\end{table*}\n\n\n\n\n\\section{Results}\\label{results}\n\n\\subsection{Observed spectra} \\label{obsspectra}\n\nFigure~\\ref{spectra} shows the observed spectra overlaid with the spectral fits. As indicated in Sect. \\ref{fitting}, the thermal fits differ slightly from each-other. The main reason why the fits do not agree is the wider energy binning adopted by imaging spectroscopy. With this binning, the atomic lines are not resolved, contrary to full sun spectroscopy.\n\\begin{figure*} \n\\resizebox{\\hsize}{!}{\\includegraphics{9418fig3.eps}}\n\\caption {Coronal and footpoint source spectra overlaid with the according fits. \\textit{Dots} are the measured footpoint spectrum, the \\textit{dashed-dotted} line indicates the fit to this spectrum in the range 30-80 keV. \\textit{Squares} indicate the observed coronal source spectrum. The \\textit{solid} lines provide the thermal and non-thermal fits as found from imaging spectroscopy. The \\textit{dotted} lines give the thermal fit to the full sun spectrum and the resulting non-thermal fit in imaging spectroscopy.}\n\\label{spectra}\n\\end{figure*}\n\nUsing the different fitting methods as an estimate of the uncertainty, an average difference in spectral index of $\\Delta \\gamma = 3.55 \\pm 0.07$ for the event of 24 October 2003 and $\\Delta \\gamma = 2.45 \\pm 0.35$ for the event of 13 July 2005 is found between the coronal source and footpoints. \n\\subsection{Expected footpoint emission and energy loss} \\label{expfemissionloss}\nAs described in Sect.~\\ref{Method}, we computed the electron distribution from the coronal source photon spectrum and the expected thick target emission (footpoint spectrum) caused by this electron distribution. Figure~\\ref{efieldspec} shows the measured spectra, the expected footpoint spectrum from a pure thick target, and the footpoint spectrum when introducing energy loss. \n\nTo compute the electron flux, the coronal column density is required. As given in Table~\\ref{fitpartable}, three different values of the column density were estimated. Using those, we are able to reproduce a range of possible electron spectra (Table \\ref{beamvalues}) and, therefore footpoint spectra. The confidence range of the footpoint spectra is indicated by a light gray (green) area in Fig.~\\ref{efieldspec}. \n\n\n\\begin{figure*} \n\\resizebox{\\hsize}{!}{\\includegraphics{9418fig4.eps}}\n\\caption {Observed (fitted) power-laws of the non-thermal coronal source and footpoints (\\textit{solid}). The \\textit{light-gray (green)} area indicates the range of expected footpoint spectra without energy loss. The \\textit{dark-gray (red)} area marks the range of expected footpoint spectra when energy loss is applied to the electrons to find the same spectral index as the observed footpoint spectrum. The \\textit{dash-dotted} lines give the expected spectra at the footpoints if the only transport effect was Coulomb collisions of the beam electrons (cf. Sect.~\\ref{collisions}).}\n\\label{efieldspec}\n\\end{figure*}\n\n\nThe derived energy loss depends on the fitted coronal and footpoint spectra. For the two different coronal fitting methods, a range of loss energies $\\mathcal{E}_{loss}=[58.0,59.4]$ keV  is found for the event of 24 October 2003 and $\\mathcal{E}_{loss}=[8.7,26]$ keV for the event of 13 July 2005. The normalization of the new spectrum depends on the initial electron distribution. The initial electron distribution is computed according to Eq.~(\\ref{bla}). It depends on the column depth. If the column depth is lower, more electrons are needed to produce the same X-ray intensity. The resulting range of possible spectra is shaded in dark-gray in Fig.~\\ref{efieldspec}. As shown in the figure, the footpoint spectrum with energy loss reproduces well the observed footpoint spectrum for the event of 13 July 2005. \n\nDuring the event of 24 October 2003, the predicted footpoint spectrum is, however, an order of magnitude less intense than observed. In the context of the ITTT model, this implies that the electron flux density emanating from the coronal region is higher than predicted. The discrepancy may be explained by density inhomogeneities in the coronal source resulting in a smaller effective column density. The electron flux is underestimated if the non-thermal X-ray emission originates in regions that are less dense than average. In the following we therefore assume that the coronal source has an inhomogeneous density; this is represented by dense regions with filling factor smaller than 1 for thermal emission and, for the non-thermal coronal source in the 24 October 2003 event, a density that is lower by an order of magnitude. The observations do not allow to determine the filling factor.\nAs can be seen from Fig.~\\ref{2flimages}, the coronal source in the event of 13 July 2005 is very compact, while the source in the event of 24 October 2003 is more extended, showing isolated intense regions. This supports the assumption that the density is inhomogeneous in the 24 October 2003 coronal source and the true column density in the X-ray emitting plasma might be smaller than deduced from the measurements. Since $F_{exp}^{fp}(\\epsilon)$ is proportional to $1\/\\Delta N$, an effective column density of an order of magnitude less than the observed could produce the observed footpoint spectrum. \nIn the computations presented in Sect.~\\ref{retcurrnefield}, we assume an effective column density $\\Delta N_{eff} = \\Delta N\/14$ for the event of 24 October 2003.\n\n\n\\section{Return current and electric field} \\label{retcurrnefield}\nIn the above analysis, we assumed that the electrons experienced a constant energy loss while streaming down the loop. We now demonstrate that this energy loss could be caused by an electric potential in the loop that drives a return current. \nThere was much controversy surrounding the precise physical mechanism that generates the return current \\citep[eg.][]{Kn77, SS84, Oo90}. The basic scenario is the following: We assume that the electrons are accelerated in the coronal source region. When a beam of accelerated electrons, which is not balanced by an equal beam of ions, leaves this region, a return current prohibits charge build-up and the induction of a beam-associated magnetic field. In the return current, thermal electrons move towards the coronal source. Since their velocity is relatively small, they collide with background ions and cause resistivity. Ohm's law then implies the presence of an electric field in the downward direction.  \nThe return current density $j_{ret}$ can be derived from the equation of motion for the background electrons \\citep{Bebook}.\n\\begin{equation}\n\\frac{\\partial \\vec{v}}{\\partial t}+(\\vec{v}\\cdot \\triangledown)\\vec{v}=-\\frac{e}{m}\\vec{E_{ind}}-\\frac{e}{mc}(\\vec{v}\\times \\vec{B})-\\nu_{e,i}\\vec{v}\n\\end{equation}\nwhere $\\vec{E_{ind}}$ is the electric field induced by the return current, $\\vec{v}$ is the mean velocity of the electrons that represent the return current, and $\\nu_{e,i}$ is the electron-ion collision frequency. Using $\\vec{j_{ret}}=-en\\vec{v}$, this expression can be written as\n\\begin{equation} \\label{fullohm}\n\\left(\\frac{\\partial}{\\partial t}+\\nu_{e,i}\\right)\\vec{j_{ret}}=\\frac{(\\omega_p^e)^2}{4\\pi}\\vec{E_{ind}}+\\frac{e}{mc}[\\vec{j_{ret}}\\times (\\vec{B_0}+\\vec{B_{ind}})],\n\\end{equation}\nwhere $B_0$ is the guiding magnetic field and $B_{ind}$ the field induced by the beam. We neglect the last term on the right side of Eq.~\\ref{fullohm} by assuming that the beam and return currents are anti-parallel, oriented along the guiding magnetic field, and that the perpendicular component of $B_{ind}$ vanishes. We then obtain\n\\begin{equation}\n\\left(\\frac{\\partial}{\\partial t}+\\nu_{e,i}\\right)\\vec{j_{ret}}=\\frac{(\\omega_p^e)^2}{4\\pi}\\vec{E_{ind}}.\n\\end{equation}\n\nSince we consider a fixed time interval that is far longer than the collision time, we assume a steady state, neglecting the time derivative of the return current. The equation then takes the form of the classical Ohm's law:\n\\begin{equation}\n\\vec{j_{ret}}=\\frac{(\\omega_p^e)^2}{4\\pi \\nu_{e,i}}\\vec{E_{ind}}=\\sigma \\vec{E_{ind}}.\n\\end{equation}\n\n\nWe estimate whether the energy loss computed in Sect.~\\ref{results} is caused by this electric field.\nFrom the observations, we estimated the electron loss energy $\\mathcal{E}_{loss}$ that the electrons experience in the loop (Sect.~\\ref{expfemissionloss}). Assuming this loss is caused by the induced electric field $E_{ind}$ and across the distance \\textit{s} from the coronal source to the footpoints (i.e. half the loop length), we compute the electric field to be\n\\begin{equation}\nE_{ind}=\\frac{\\mathcal{E}_{loss}}{e\\cdot s}.\n\\end{equation}\n\nUsing Spitzer conductivity~\\citep{Spbook}, the term for the return current is related to the observed loss energy by:\n\\begin{equation} \\label{returneq}\nj_{ret}=6.9\\cdot 10^6T_{loop}^{3\/2}\\frac{\\mathcal{E}_{loss}}{e\\cdot s}\\quad \\mathrm{[statamp\/cm^2]},\n\\end{equation}\nwhere $T_{loop}$ is the temperature in the loop.\n\nOn the other hand, the beam current density can be written as\n\\begin{equation} \\label{jbeam}\nj_{beam}=\\frac{F_{tot}(E)}{A_{fp}}\\cdot e \\quad \\mathrm{[statamp\/cm^2]},\n\\end{equation}\nwhere $A_{fp}$ is the total footpoint area. The total electron flux per second $F^{tot}(E)$ is computed from the observed electron spectrum as follows: Let the electron spectrum be $F(E)=A_eE^{-\\delta}$. The total flux of streaming electrons per second above a cutoff energy $E_{cut}$ is then:\n\\begin{equation} \nF_{tot}(E)=\\int_{E_{cut}}^{\\infty}F(E) \\mathrm{d}E=\\frac{A_e}{\\delta-1}E_{cut}^{-(\\delta-1)}.\n\\end{equation}\n\nIn a steady state, the relation \n\\begin{equation}\nj_{beam}=j_{ret}\n\\end{equation}\n is valid. \\\\\n\nComparing the beam current as described in Eq.~(\\ref{jbeam}) with the return current from the observed energy loss according to Eq.~(\\ref{returneq}), we test whether the assumption of Spitzer conductivity holds.\n\n\n\\subsection{Results} \\label{retresults}\nTable \\ref{beamvalues} presents the relevant physical parameters necessary for the derivation of the beam- and return currents.\nFor Spitzer conductivity, the loop temperature $T_{loop}$ is required. Its is expected to have a value between the coronal source temperature and the footpoint temperature (see Table~\\ref{beamvalues}). As a first assumption, a mean temperature of $T_{loop}=15$ MK is chosen. The loop length is evaluated from RHESSI images, approximating the distance between the sources from the centroid positions and assuming a symmetrical loop structure. This provides a typical half loop length of $4\\cdot 10^9$ cm. \nThe footpoint area is measured from the 50\\% contour in RHESSI images in the 25-50 keV energy range, yielding a total footpoint area of $\\approx (6-7)\\cdot 10^{17}$ cm$^2$. \nThe beam current density depends critically on the electron cut off energy $E_{cut}$. We use a value of 20 keV. This gives an approximate lower limit to the total amount of streaming electrons.  \n\nUsing the presented observations, Eq.(\\ref{jbeam}) and by assuming Spitzer conductivity (Eq. \\ref{returneq}), the return current results to be of an order of magnitude higher than the beam current. This contradicts the assumptions of a steady state, and is also unphysical. \n\\begin{table*}\n\\caption{Values used for the computation of the beam and return currents and computed currents. } \n\\begin{tabular}{lrr}  \n\\hline \\hline\nEvent & 24 October 2003 & 13 July 2005 \\\\\n\\hline\nAssumed loop temperature $T_{loop}$ [MK] & 15 & 15 \\\\\n1\/2 Loop length s [cm] & $3.2\\cdot 10^9$&$4.3\\cdot 10^9$     \\\\\nElectron flux $F(E)$ [s$^{-1}$ keV$^{-1}$]&6.33$\\cdot 10^{41}E^{-5.2}$-1.5$\\cdot 10d^{42}E^{-5.2}$&1.7$\\cdot10^{40}E^{-4.1}-8.4\\cdot 10^{40}E^{-4.6}$\\\\\nElectron cutoff energy [keV] & 20 & 20  \\\\\nTotal footpoint area [cm$^2$] & $7.2\\cdot10^{17}$& $6.2\\cdot10^{17}$ \\\\\n$E_{loss}$ [keV] & 58-59.4 & 8.7-26 \\\\\nElectric field strength [statvolt\/cm] &$(6-6.3) \\cdot 10^{-8}$&$(6.7-20.3)\\cdot 10^{-9}$ \\\\\n\\hline\n$j_{ret}$[$statamp\/cm^2$]&$(2.4-2.5) \\cdot 10^{10}$& $(2.7-8.1)\\cdot 10^9$\\\\\n$j_{beam}$ [$statamp\/cm^2$]&$(5.1-14.4) \\cdot 10^9$&$(1.9-3.6) \\cdot 10^8$ \\\\\n\\hline\n\\end{tabular}\n\n\\label{beamvalues}\n\\end{table*}\n\n\n\\section{Discussion} \\label{discussion}\n\n\\subsection{Instability}\nIn Sect.~\\ref{retresults}, we assumed Spitzer conductivity when computing the return current which produced the unphysical result of $j_{ret}>j_{beam}$. Using Eq.~(\\ref{returneq}), the loop temperature required to maintain equality between the return current and beam current ($j_{ret}=j_{beam}$) can be computed. In the 24 October 2003 event, the loop temperature $T_{loop}$ would need to be smaller than 10 MK; in the 13 July 2005 event, $T_{loop}$ should be less than 3.9 MK. Such low loop temperatures are highly unlikely.\nHowever, it is possible that the return current is unstable to wave growth. For an extended discussion of instabilities in parallel electric currents, see e.g. \\citet{Bebook}. Instability causes an enhanced effective collision frequency of electrons in the return current and therefore a lower effective conductivity. The ion cyclotron instability develops if the drift velocity of the beam particles $V_d$ exceeds the thermal ion velocity $v_{th}^{ion}$ as follows: \n\n\\begin{equation} \\label{instcond}\nV_d\\ge15\\frac{T_i}{T_e}v_{th}^{ion}\n\\end{equation}\nwith $v_{th}^{ion}=\\sqrt{\\frac{k_BT_i}{m_i}}$ and $T_e$ and $T_i$ being the electron and ion temperatures, respectively. \n\nWe assume a steady state for which $j_{beam}=j_{ret}=n_e eV_d$, where $V_d$ is the mean drift velocity of the electrons constituting the return current. We therefore express $V_d$ as\n\\begin{equation}\nV_d=\\frac{j_{beam}}{n_e e}\n\\end{equation}\nand substitute this expression and that for $v_{th}^{ion}$ in Eq. (\\ref{instcond}). Assuming $T_e = T_i=T_{loop}$ and solving Eq.~(\\ref{instcond}) for $T_{loop}$ the instability condition holds\n\\begin{equation} \\label{tcond}\nT_{loop}\\le 2.3\\cdot 10^8 \\left(\\frac{j_{beam}}{n_e}\\right)^2 \\quad \\mathrm{[K]}.\n\\end{equation}\n\nSince the loop temperature and density are not known exactly, \nthis relation is illustrated in Fig.~\\ref{instab} for several values of $n_e$ and $T_{loop}$ typical in flare loops. For the values of $j_{beam}$ found for the observations of the two events, we find that instability occurs in the 24 October 2003 event for all values of $n_e$ and $T_{loop}$ in Fig.~\\ref{instab}.\nFor the 13 July 2005 flare, three distinct regions in the diagram can be found. The solid line indicates the relation of Eq. (\\ref{tcond}). Below this line, the return current is unstable. At high densities and low temperatures (lower right), the return current is stable and $j_{beam}=j_{ret}$ with Spitzer conductivity. The range of beam currents $j_{beam}$ deduced from the data allows for loop temperatures $< 3.9$ MK. \nIn the upper right quadrangle, Spitzer conductivity would imply $j_{ret}>j_{beam}$, which is unphysical. If the loop was in this parameter range, the current instability would be most likely saturated and $T_e>T_i$. This would shift the instability threshold in Fig.~\\ref{instab} to the right. Further, a loop in the state presented by the uppermost part of the figure (temperature above 10 MK, high density) would be detectable by the RHESSI satellite even in the presence of the coronal source. Since no loop emission is observed, we conclude that the loop is either less dense, cooler or both. The values in the upper right quadrangle are therefore unlikely. \n\n\n\\begin{figure} \n\\resizebox{\\hsize}{!}{\\includegraphics{9418fig5.eps}}\n\\caption {Region of instability in the density\/temperature space for the event of 13 July 2005. The \\textit{grey} region indicates the densities and temperatures $T_{loop}$ for which the return current is unstable (Eq.~\\ref{instcond}). In the lower right part, the return current is stable and $j_{beam}=j_{ret}$ with Spitzer conductivity. The current in the event of 24 October 2003 is unstable for all values of density and temperature in the Figure.}\n\\label{instab}\n\\end{figure}\n\n\n\\subsection{Low energy electron cutoff}\nIn the above computations, a value of 20 keV for the electron cutoff energy $E_{cut}$ was assumed. This value is within the range for which the thermal and non-thermal components of the spectrum intersect. Values around 20 keV or higher are also supported by detailed studies of the exact determination of low-energy cutoffs \\citep[eg.][]{Sa05,Ve05,Sui07}. What if the cutoff energy were substantially lower than 20 keV? A cutoff energy as low as 10 keV would increase the total electron flux and therefore the beam current by an order of magnitude, leading to $j_{ret}\\approx j_{beam}$ for Spitzer conductivity. Conductivity could then not be reduced significantly by wave turbulence, and instability would be marginal. If the low energy cutoff were even lower than 10 keV, we would find that $j_{ret}< j_{beam}$. This could not be explained in terms of the model used here.\n\\subsection{Source inhomogeneity and filling factor} \\label{filling}\nIn the above paragraphs, it was demonstrated that the energy loss for the electrons due to an electric field could resolve the inconsistency in the difference between footpoint and coronal source spectral indices (Sect. \\ref{results}). For the event of the 24 October 2003, this produces footpoint emission that is lower than the observed emission (Fig. \\ref{efieldspec}). Images show that the coronal source in this event is not compact, but extended with brighter and darker regions. It is therefore possible that the standard density estimate, which favors high density regions, produces higher densities than average and that the effective column density of the regions, where the largest part of the non-thermal emission originates, is lower. This would provide a higher expected footpoint emission, in closer agreement with observations. \n\nAs mentioned in Sect.~\\ref{fitting}, a filling factor of 1 was used for the computation of the column density. A filling factor smaller than 1 would lead to even higher densities of the SXR emitting plasma. However, this would not affect the lower effective column density of the regions, where the HXR emission originates when assuming source inhomogeneity. \n\n\\subsection{Collisions and other possible scenarios} \\label{collisions}\nWhile return currents may not be the only means of attaining non-collisional energy loss, they are the most obvious and best studied. However, other scenarios are conceivable, which could produce a harder footpoint spectrum (or a softer coronal spectrum). \nIn the model presented here, collisional energy loss of beam electrons is neglected. This is a valid assumption for the following reasons: If collisions of the beam electrons in the loop were to play an important role, significant HXR emission should originate in the loop. Within the dynamical range limitations of RHESSI, this is not the case. However, GOES SXI and SOHO\/EIT images imply that the loop is filled with hot material. To study possible effects of collisions, we compared the change in the electron spectrum and the resulting footpoint spectrum for collisional energy loss and energy loss due to the electric field. The change in the electron spectrum due to collisions depends on the column depth through which the beam passes and was computed by \\citet{Le81} and \\citet{Br75}. We assume a column depth derived from the density in the loop times half the loop length. Assuming the same density as in the coronal source, we derive an upper limit to the collisional effects. The expected footpoint spectrum from purely collisional losses is indicated in Fig.~\\ref{efieldspec} as dash-dotted lines. Collisions affect the low energetic electrons most where a significant change in the spectrum is found. At the higher energies observed in this study, the spectrum does not change significantly. The neglect of collisional effects is therefore justified.\n\\citet{Br08} showed that in certain cases, emission from non-thermal recombination can be important, generating a coronal spectrum that is steeper than expected by the thin-target model. This could also produce a difference in the spectral index that is larger than two. Acceleration over an extended region \\citep[as proposed by][]{Xu08} could alter the electron distribution at the footpoints. If the distribution was harder at the edge of the region, a spectral index difference larger than 2 would result. A thorough comparison of such models with observations may be the scope of future work. \n\\section{Conclusions} \\label{conclusions}\nThe spectral relations between coronal and footpoint HXR-sources provide information about electron transport processes in the coronal loop between the coronal source and the footpoints. Most models neglect these processes in the prediction of the shape and quantitative differences between the source spectra. As shown by \\citet{Ba07}, the observations of some solar flares do not fit the predictions of such models, in particular the intermediate thin-thick target model by \\citet{Wh95}: there is a discrepancy concerning the difference in coronal and footpoint spectral indices, which is expected to be 2. \n\nWe have analyzed the two out of five events that display a spectral index difference larger than two in more detail. Such a behavior can be attributed to energy loss during transport that is not proportional to electron energy, but $\\mathcal{E}_{loss}\/E$ is larger at low energies. Such an energy loss causes the footpoint spectrum to flatten, which increases the difference in spectral indices. Two loss mechanisms come to mind immediately: Coulomb collisions and an electric potential. Figure 4 demonstrates that the assumption of an electric potential reproduces the observations more accurately.\n\nIn one of the two events, there remains a discrepancy between the observed and expected footpoint emission, such that the electron flux at the footpoints is larger than predicted. This flux was estimated from the observed non-thermal HXR (photon) flux and the observed thermal emission of the coronal source. We attribute the discrepancy to propagation or acceleration in low density plasma, which also heats the adjacent high-density regions. \n\nThe energy loss can therefore be explained by an electric field in the loop associated to the return current, which builds up as a reaction to the electrons streaming down the loop and the associated beam current. In a steady state ($j_{beam}=j_{ret}$), the return current is unstable to wave growth in one event for all realistic temperature and density parameters in the loop. The kinetic current instability drives a wave turbulence that enhances the electric resistivity by many orders of magnitude. This anomalous resistivity in turn significantly enhances the electric field. In the event of 13 July 2005, the return current may be stable if the loop density is high and the temperature is low, and Spitzer conductivity is applied. Both cases (out of five) present strong evidence for a return current in flares for the first time.\n\nTransport effects by return currents constitute a considerable energy input by Ohmic heating into the loop outside the acceleration region. It may be observable in EUV. Comprehensive MHD modeling including the coronal source, the footpoints, and the region in-between, may be the goal of future theoretical work.\n\n\n\n\n\n\\begin{acknowledgements}\nRHESSI data analysis at ETH Z\\\"urich is supported by ETH grant TH-1\/04-2 and the Swiss National Science Foundation (grant 20-105366). \nWe thank S\\\"am Krucker for helpful comments and discussions.\n\\end{acknowledgements}\n\n\n\\bibliographystyle{aa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe ability to design nanostructures which accurately self-assemble from \nsimple units is central to the goal of engineering objects and machines \non the nanoscale. \nWithout self-assembly, structures must be laboriously constructed in a step \nby step fashion.  Double-stranded DNA (dsDNA) has the ideal properties for \na nanoscale building block,\\cite{Seeman2003,Pitchiaya2006} \nwith structural length scales determined by the \nseparation of base pairs, the helical pitch and its persistence length \n(approximately 0.33\\,nm, 3.4\\,nm (Ref.\\ \\onlinecite{Saenger1984}) and \n50\\,nm,\\cite{Hagerman1988} respectively). \nOver these distances, dsDNA acts as an almost rigid \nrod and so it is capable of forming well-defined three dimensional structures.\n\nIt is the selectivity of base pairing between single strands, however, that makes DNA ideal for controlled self-assembly. \nBy designing sections of different strands to be complementary, a certain configuration of a system of oligonucleotides can be specified as the global minimum of the energy landscape. \nIn this way the target structure (usually consisting of branched double helices) can be `programmed' into the sequences. \nThis approach was initially demonstrated for a four-armed junction by the Seeman\ngroup in 1983.\\cite{Kallenbach83}\nSuch junctions and more rigid double crossover motifs\\cite{Fu93}\ncan then be used to create two-dimensional lattices.\\cite{Winfree98,Malo2005}\nYan \\it et al. \\rm  \\cite{Yan2003} have also constructed ribbons and two dimensional lattices from larger four-armed structures, each arm consisting of a junction of four strands.  \nFurthermore, using Rothemund's DNA ``origami'' approach an almost arbitrary \nvariety of two-dimensional shapes can be created.\\cite{Rothemund06}\n\nProgress in forming three-dimensional DNA nanostructures was initially much \nslower. The Seeman goup managed to synthesize a DNA cube\\cite{Chen91} and \na truncated octahedron,\\cite{Zhang94} but only after a long series of steps and \nwith a low final yield.\nMore recently, approaches have been developed that allow polyhedral cages, \nsuch as tetrahedra,\\cite{Goodman2005} trigonal bipyramids,\\cite{Erben07} \noctahedra,\\cite{Shih04,Anderson08} dodecahedra and truncated icosahedra,\n\\cite{He2008} to been obtained in high yields simply by cooling \nsolutions of appropriately designed oligonucleotides from high temperature. \nAdditional structures have also been produced using pre-assembled modular building blocks incorporating other organic molecules.\\cite{Aldaye2007,Zimmermann08} \n\nIn designing strand sequences, it is important to minimize the stability of competing structures with respect to the stability of the target configuration. \nIn addition, if systems can be designed to follow certain routes through configuration space---for example, by the hierarchical assembly of simple motifs \\cite{Pistol2006}---the target can potentially be reached more efficiently. \nA standard approach to hierarchical assembly, such as that described by \nHe {\\it et al.},\\cite{He2008}\ninvolves choosing sequences so that bonds between different pairs of oligonucleotides become stable at different temperatures. \nThis allows certain motifs to form in isolation at high temperatures before bonding to each other as the solution is cooled. \nAn alternative, elegant system for programming assembly pathways has been proposed by Yin \\it et al. \\rm \\cite{Yin2008}, which relies on the metastability of single stranded loop structures and the possibility of catalyzing their interactions using other oligonucleotides.\n\nGiven these recent experimental advances in creating DNA nanostructures, it would be useful to have\ntheoretical models that allow further insights into the self-assembly process.\nIn particular, a successful model would be able to provide information on the \nformation pathways and free energy landscape associated with the self-assembly, \nand as such would be of use to experimentalists wishing to consider increasingly more complex designs. \nAtomistic simulations of DNA would offer potentially the most spatially-detailed descriptions of the self-assembly.  \nHowever, they are computationally very expensive, and are generally restricted to time scales that are \ntoo short to study self-assembly.\\cite{Cheatham2004}\n\nStatistical approaches such as that of Poland and Scheraga \\cite{Poland1970} and the nearest-neighbour model \\cite{Everaers2007} use simple expressions for the free energy of helix and random coil states to obtain equilibrium results for the bonding of two strands. \nWhilst the parameters in these models can be tuned to give very accurate correspondence with experimental data,\\cite{SantaLucia1998,SantaLucia2004} they give no information on the dynamics and formation pathways and hence are only useful for ensuring that the target structure has significantly lower free energy than competing configurations. \nFurthermore, any description purely based on secondary structure (i.e.\\ which bases are paired) is inherently incapable of accounting for topological effects such as linking of looped structures.\\cite{Bois2005} \n\nCoarse-grained or minimal models offer a compromise  between detail and computational simplicity, and are well  suited to the study of hybridization of oligonucleotides.  The aim of these models is to be capable of describing  both the thermodynamic and kinetic behaviour of systems, a vital feature if kinetic metastability is inherent in assembly pathways.\\cite{Yin2008} In developing such minimal models the approach is usually to retain just those physical features of the system that are essential to the behaviour that is of interest. \n\nDauxois, Peyrard and Bishop models,\\cite{Dauxois1993} and modified versions such as that proposed by Buyukdagli, Sanrey, and Joyeux,\\cite{Buyukdagli2005} \nconstitute the simplest class of dynamical models. \nAlthough these are dynamic models in the sense that the energy is a function of the separation between each base, the nucleotides are constrained to move in one dimension. This lack of conformational freedom means that these models are \nincapable of capturing the nuances of the self-assembly from single-stranded DNA (ssDNA).\n\nRecently, models have been proposed which capture the helicity of dsDNA using two \\cite{Drukker2001} or three \\cite{Knotts2007}  interaction sites per nucleotide. \nThese models, however, are optimized for studying deviations from the ideal double-stranded state, and so have not been used to examine self-assembly.\nAlthough they have been used to study the thermal denaturation of dsDNA, \nit is essential for our purposes to be able to simulate the assembly of a \nstructure from ssDNA as it is this process that will reveal the kinetic traps \nand free energy landscape associated with the formation of a particular \nDNA nanostructure.  \n\nSimpler,  linear models, also with two interaction sites per nucleotide, have been used to investigate duplex hybridization,\\cite{Araque2006} hairpin formation \\cite{Sales-Pardo2005} and gelation of colloids functionalized with oligonucleotides.\\cite{Starr2006} \nThese models all use two interaction sites to represent one nucleotide, with backbone sites linked to each other to represent the sugar-phosphate chain, and interaction sites which represent the bases. This work investigates the possibility of extending the use of such coarse-grained models to study the self-assembly of nanostructures that involve multiple strands forming branched duplexes. We hypothesize that the self-assembly properties of DNA are dominated by the fact that ssDNA is a semi-flexible polymer with selective attractive interactions. \nWe introduce an extremely simple model, similar to that of Starr and Sciortino,\nto test this hypothesis. \\cite{Starr2006}  \nThis simplicity enables us to explore the thermodynamics and kinetics of self-assembly in the model in great depth, and hence examine the feasibility of simulating nanostructure formation through minimal models.\n\nWe first describe the model in Section \\ref{methods}, then examine its success in reproducing the general features of hybridization in Section \\ref{Duplex Formation}. \nNext in Section \\ref{Holliday Junction}, we apply it to the formation of a Holliday junction, a simple nanostructure consisting of a four-armed cross.\\cite{Malo2005}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=8.4cm]{Fig1.eps}\n\\end{center}\n\\caption{(Colour online) A schematic representation of the model. \nThe thick lines represent the rigid backbone monomer units and the large circles the repulsive Lennard-Jones interactions at their centres.\nThe smaller, darker circles represent the bases. \nThe panels illustrate the definitions of (a) the bending angle between two units ($\\theta$), and (b) the torsional angle ($\\phi$) which is found after the monomers have been rotated to lie parallel.}\n\\label{model picture}\n\\end{figure}\n\n\\section{Methods}\n\\label{methods}\n\\subsection{Model}\n\\label{model}\n\nWe introduce an off-lattice model inspired by that which Starr and Sciortino \nused to study the gelation of four-armed DNA dendrimers.\\cite{Starr2006} \nAs our aim is to reproduce the basic physics with as simple a \nmodel as possible, we neglect contributions to the interactions\ndue to base stacking, and the charge and asymmetry of the phosphate backbone. \nWe do not attempt to include the detailed geometrical structure \nof DNA, but instead\nrepresent the oligonucleotides as a chain of monomer units,\neach corresponding to one nucleotide (Fig.\\ \\ref{model picture}). \nA monomer consists of a rod (chosen to be rigid for simplicity) of length $l$ \nwith a repulsive backbone interaction site at the centre of the rod.\nIn addition, each unit has a bonding interaction site (or base) at a distance \nof $0.3\\,l$ from the backbone site \n(perpendicular to the rod). Each monomer is also assigned a base type \n(A,G,C,T) to model the selective nature of bonding. In this model we only \nconsider bonds between the complementary pairs A-T and G-C.\n\n\nWe do not explicitly include any solvent molecules in our simulations, but instead use effective potentials\nto describe the interactions between the DNA.\nSites interact through shifted-force \nLennard-Jones (LJ) potentials, where, as well as truncating and shifting the \npotential, an extra term is included to ensure the force goes smoothly to zero \nat the cutoff $r_c$. \nFor $r < r_{c}$\n \\begin{equation} \nV_{\\rm sf}(r)=V_{\\rm LJ }(r) - V_{\\rm LJ}(r_{c}) - \n              (r-r_{c})\\left.\\frac{dV_{\\rm{LJ}}}{dr}\\right|_{r=r_c}\n \\label{potential} \n \\end{equation} \nwhere\n \\begin{equation} \n V_{\\rm LJ}(r) = 4\\epsilon \\left [\\left({\\sigma \\over r} \\right) ^{12} - \\left( {\\sigma \\over r} \\right) ^{6} \\right], \\label{LJ} \n\\end{equation} \nand $V_{\\rm sf}(r) = 0$ for $r\\ge r_{c}$. \nBackbone sites (except adjacent units on the same strand) interact through \nEq.\\ (\\ref{potential}) with $\\sigma = l$ and $r_{c} = 2^{1\/6}\\sigma$. \nThis purely repulsive interaction models the steric repulsion between strands. \nBonding sites (again excluding adjacent units on a strand) interact via \nEq.\\ (\\ref{potential}) with $\\sigma = 0.35\\,l$ and  $r_{c} = 2.5\\,\\sigma$ for complementary bases (to allow for attraction) and $r_{\\rm c} = 2^{1\/6}\\sigma$ for all other pairings. \nThe depth of the resulting potential well between complementary bases, \n$\\epsilon^{\\rm eff}_{\\rm base}$, is $0.396\\epsilon$. In what follows we will measure\nthe temperature in terms of a reduced temperature, \n$T^*=k_B T\/\\epsilon^{\\rm eff}_{\\rm base}$. \nThe above choice of parameters ensures that the attractive interaction between complementary bases is largely shielded by backbone repulsion. Monomers therefore bond selectively and can only bond strongly to one other monomer at a given instant. \nThese are the key features of Watson-Crick base pairing that make DNA so useful for self-assembly.\n\nThe model also includes potentials between consecutive monomers associated with bending and twisting the strand: \n\\begin{equation} \nV_{\\rm bend} = \\begin{cases}\n                k_1 (1-\\cos(\\theta))& \n                        \\text{if $\\theta < {3\\pi\\over4}$},\\\\ \n\t\t\\infty& \\text{otherwise}\n\t\t\t\\end{cases}\n\\label{bend} \n\\end{equation}\nand \n\\begin{equation}\nV_{\\rm twist} = k_2 (1-\\cos(\\phi)).\n\\label{twist}\n\\end{equation}\nWe define $\\theta$ as the angle between the vectors along adjacent monomer rods. As previously mentioned, consecutive backbone sites do not interact via LJ potentials. \nInstead, a hard cutoff is introduced in Eq.\\ (\\ref{bend}) to reflect the fact that an oligonucleotide cannot double back on itself. \n$\\phi$ is taken as the angle between adjacent backbone to bonding site vectors after the monomers have been rotated to lie parallel (Fig.\\ \\ref{model picture}). \nFor simplicity, we choose the torsional potential to have a minimum at $\\phi=0$. Thus, neither ssDNA or dsDNA\nwill be helical in our model.\n\n$k_1$ is chosen to be $0.1\\,\\epsilon$ to give a persistence length, $l_{\\rm ps}=3.149\\,l$ at a reduced temperature of $T^*=0.09677$ for ssDNA. \nWe obtain this result by simulating a single strand 70 bases in length, and using the definition:\\cite{Cifra2004}\n\\begin{equation}\nl_{\\rm{ps}} = \\frac{\\langle {\\bf{L} \\cdot \\bf{l_{\\rm 1}}}\\rangle} {\\langle l_{\\rm 1} \\rangle},\n\\end{equation} \nwhere $\\bf{L}$ is the end to end vector of the strand, $\\bf{l_{\\rm 1}}$ is the vector associated with the first monomer and $\\langle \\rangle$ indicates a thermal average. \nTaking $l$ to be 6.3{\\AA}, $T^*=0.09677$ is mapped to $24^\\circ$C and so the model is consistent with experimental data for ssDNA in 0.445M NaCl solution.\\cite{Murphy2004,lengthscale} $k_2$ is chosen to be $0.4\\,\\epsilon$.\n\nIn neglecting the geometrical structure of a double helix we do not accurately represent certain types of bonding. \n`Bulged' bonding occurs when consecutive bases in one of the strands attach to non-consecutive bases in the other strand. \n`Internal loops' consist of stretches of non-complementary bases (either symmetric or asymmetric in the number of bases involved in each strand). \n`Hairpins' result when a single strand doubles back and bonds to itself. The details of these motifs are complicated but an empirical description of their thermodynamic properties is given in Ref.\\ \\onlinecite{SantaLucia2004}. \nImportantly, they are generally penalized due to the disruption of the geometry of DNA in a way which is not well reproduced by our model. \nIn the case of the short strands we consider, these motifs will only play a small role as the strands are not specifically intended to have stable structures of these forms. \nIn fact, as the base sequences we use were designed to form the Holliday junction, the possibility of forming these motifs at relevant temperatures was deliberately avoided.\\cite{MaloThesis}\n\nFor simplicity we therefore include only two alterations to the model. Firstly, we define `kinked states' as those for which the number of unpaired bases between two bonding pairs on either side of a duplex is not equal (including asymmetric loops and bulges). We impose an infinite energy penalty on the formation of these kinked states if the total number of intermediate bases is less than six. Secondly, we treat complementary units within six bases of each other on the same strand as non-complementary, but allow all other hairpins without penalty. \n\nIt should be also noted that this model neglects the directional asymmetry of the sugar-phosphate backbone. Therefore,\nparallel as well as anti-parallel bonding is possible in our model, whereas parallel bonding does \nnot occur in experiment.    \n\n\\subsection{Monte Carlo Simulation}\n\\label{Monte Carlo Simulation}\n\nIn a fully-atomistic model of DNA the natural way to simulate its dynamics would be to use molecular dynamics. However, the best way to simulate the dynamics\nin a coarse-grained model is an important, but not fully resolved, question, and\none that will depend on the nature of the model. Clearly, for the current model\nstandard molecular dynamics is inappropriate as it will lead to ballistic \nmotion of the strands between collisions because of the absence of explicit \nsolvent particles, whereas DNA in solution undergoes diffusive Brownian motion. \nAn alternative approach is to use Metropolis Monte Carlo (MC) \nalgorithm\\cite{Metropolis1953,Frenkel2001} where the moves \nare restricted to be local, as it has been argued that this can provide a \nreasonable approximation to the dynamics.\\cite{Kikuchi91,Berthier07,Tiana07} \nThis is the approach that we use here to simulate the dynamics of \nself-assembly of DNA duplexes and Holliday junctions. \nIn particular, the local MC moves that we use are translation and \nrotation of whole strands and bending of a strand about a particular monomer,\nthus ensuring that the strands undergo an approximation to diffusive Brownian \nmotion in the simulations.\nTherefore we expect the MC simulations, which are all initiated with free single strands, to mimic the real self-assembly processes in our model. It is \nimportant to note that this will include, as well as successful assembly into the target structure, kinetic trapping in non-equilibrium\nconfigurations and that when the latter occurs this reflects the inefficiency of\nthe self-assembly under those conditions.\n\n\nAlthough a true measure of time is impossible in Monte Carlo simulations, an approximate time scale for diffusion-limited processes can be found by comparing the diffusive properties of objects to experiment. \nBy measuring the diffusion of isolated strands, and assuming diffusion coefficients comparable to those of double strands and hairpin loops of similar length,\\cite{Lapham1997} we conclude that one step per strand corresponds to a time scale of approximately 2\\,ps. \nThus, our model allows our systems to be studied on millisecond time scales. \n\nAt the end of the above MC simulations, our systems will not necessarily have reached equilibrium, \nboth because the energy barriers to escape from misbonded configurations can be difficult to \novercome at low temperature and the low rate of association at higher \ntemperatures.\nTherefore, as a comparison we also compute\nthe equilibrium thermodynamic properties of our systems using umbrella \nsampling.\\cite{Torrie1977,Frenkel2001} \nFormally, we can write the thermal average of a function $B(\\bf{r}^N)$ in the canonical ensemble as:\n\\begin{equation}\n\\langle{B}\\rangle = \\frac\n   {\\int{\\frac{B}{W(Q)} \\left[{W(Q) \\exp(-V\/\\rm{k_B}\\it T)}\\right]} d\\bf{r}^{N}} \n   {\\int{\\frac{1}{W(Q)}\\left[{W(Q) \\exp(-V\/\\rm{k_B} \\it T)}\\right]} d\\bf{r}^{N}} \n\\label{US avg}\n\\end{equation}\nwhere $Q=Q(\\bf{r}^N)$ is an order parameter or reaction coordinate and $V=V(\\bf{r}^N)$ is the potential energy. \nWe are free to choose $W(Q)$, and by taking the term in square brackets as the weighting of states and keeping statistics for $B\/W$ and $1\/W$ at each step we can find $\\langle{B}\\rangle$. \nIn standard Metropolis MC, $W=1$, but by choosing $W(Q)$ in such a way that those states \nwith intermediate values of $Q$ are visited more frequently, the effective free energy barrier between \n(meta)stable states can be lowered allowing the system to pass easily between the free energy minima, and equilibrium to be reached.\n\nTo ensure that each value of $Q$ is equally likely to be sampled in an \numbrella sampling simulation, one would choose $W(Q)=\\exp(\\beta A(Q))$,\nwhere $A(Q)$ is the free energy as a function of the order parameter. \nTo achieve this, however, would require knowledge of $A(Q)$. \nInstead, there are standard methods to \nconstruct $W(Q)$ iteratively, but for the current examples it was possible to\nconstruct $W(Q)$ manually, because of the relative simplicity of the free\nenergy profiles.\n\nTo a first approximation the interaction between fully bonded structures is negligible. \nTherefore, in the umbrella sampling simulations we consider systems containing the minimum number of strands \nrequired to form a given object (two for a duplex and four for a Holliday junction). We then use the relative weight of bound and free states to extrapolate the expected fractional concentrations for larger systems.\\cite{Ouldridgeunpub}\nThe natural choice for the order parameter $Q$ is the number of correct bonds, where \ntwo monomers are defined to be bonded if their energy of interaction is negative.\n\n\\section{Results}\n\\subsection{Duplex Formation}\n\\label{Duplex Formation}\n\nWe test the model by analysing the duplex bonding of two different complementary strands. \nWe simulate systems of ten oligonucleotides, initially not bonded, in a periodic cell with a concentration of $5.49\\times10^{-5}$ molecules\\,$l^{-3}$\n(or $3.65\\times10^{-4}$\\,M). \nWe separately consider strands consisting of 7 and 13 monomers, which correspond to two of the arms of the Holliday junction studied experimentally by Malo {\\it et al.}\\cite{MaloThesis,Malo2005} and which we consider in Section \\ref{Holliday Junction}:\n\\begin{equation}\n\\text{7 bases} \\begin{cases}\n\t\t\t\t\t\\text{G-A-G-T-T-A-G}\\\\\n\t\t\t\t\t\\text{C-T-A-A-C-T-C}\n\t\t\t\t\t\\end{cases}\n\\label{7 bases}\t\t\t\t\t\n\\end{equation}\n\\begin{equation}\t\t\t\t\t\n\\text{13 bases} \\begin{cases}\n\t\t\t\t\t\\text{G-C-G-A-T-G-A-G-C-A-G-G-A}\\\\\n\t\t\t\t\t\\text{T-C-C-T-G-C-T-C-A-T-C-G-C}\n\t\t\t\t\t\\end{cases}\n\\label{13 bases}\n\\end{equation}\nwhere we have listed strands in the 5'--3' sense for consistency with the literature. \nThe yields of correctly-bonded and misbonded structures at the end of the simulations are depicted in Figure \n\\ref{7 & 13 mers} as a function of temperature.\nWe also display the predicted equilibrium fraction of correctly bonded strands for both systems obtained using umbrella sampling. \nFor convenience, we define a correctly bonded structure to have more than 70\\% of the bonds of the complete duplex and no bonds to other strands. Any other structure is recorded as `misbonded'.\n\n\\begin{figure}\n\\includegraphics[width=6.2cm,angle=-90]{Fig2.eps}\n\\caption{(Colour online) Yields of correctly-formed duplexes and misbonded configurations at the end \nof our MC simulations (lines with data points, as labelled) compared to the equilibrium probability of \nthe strands adopting the correct structure as obtained by umbrella sampling.\nThe solid and dashed lines represent results for strands with 7 and 13 monomers, respectively.\nThe MC results are averages over ten runs of length $3\\times10^8$ steps per strand with ten strands in \nthe simulation cell. }\n\\label{7 & 13 mers}\n \\end{figure}\n\nFigure \\ref{7 & 13 mers} shows a maximum in the yield as a function of temperature. \nSuch behaviour is typical of self-assembling systems \n\\cite{Brooks06,Hagan2006,Wilber2007,Rapaport08,Whitelam08} and reflects the \nthermodynamic and dynamic constraints on the self-assembly process. \nFirstly, the yield is zero at high temperature where only ssDNA is stable, and rises just below the expected \nequilibrium value as the temperature is decreased, \nthe deviation arising due to the large number of steps required to reach equilibrium. \nAt low temperatures, the yield falls away due to the presence of kinetic traps which are now stable with respect to isolated strands, as evidenced by the rise in `misbonded structures' in Figure \\ref{7 & 13 mers}. \nThus, there is a non-monotonic dependency of yield on temperature and an optimum region for successful assembly, which corresponds to the region where only the desired structure is stable against thermal fluctuations. \nFigure \\ref{5 13mers} is a snapshot from near the end of a simulation in this regime. \nIt should be noted that neglecting helicity has the effect of increasing the flexibility of dsDNA in the direction \nperpendicular to the plane of bonding \n(a helix cannot bend in any direction without disturbing its internal structure whereas a `ladder' can). \n\n\\begin{figure}\n\\includegraphics[width=8.4cm]{Fig3.eps}\n\\caption{(Colour online) Snapshot of a fully-assembled configuration in a MC simulation of\nten 13-base strands  at $T^*=0.0971$.\nIn this image the colour of the backbone indicates the type of strand: red for G-C-G-A-T-G-A-G-C-A-G-G-A and grey for its complement.\nBackbone sites are indicated by the large spheres, and bases by the small, blue spheres.}\n\\label{5 13mers}\n\\end{figure}\n\nThe heat capacity obtained from umbrella sampling of pair formation is shown in Figure \\ref{thermo_duplex}(a). \nThe heat capacity peaks indicate a transition from single strands to a duplex. \nAs the formation of duplexes is essentially \na chemical equilibrium between monomers and clusters of a definite size \n(in this case two)\nthe width of the peaks will remain finite as the number of strands is increased. \nThe transition does, however, become increasingly narrow as the DNA strands become longer, as is evident from comparing the \nheat capacity peaks for the 7-mer and 13-mers.\n\n\\begin{figure}\n\\includegraphics[width=8.4cm]{Fig4.eps}\n\\caption{Thermodynamics for the formation of a single duplex. \n(a) Heat capacity curves for 7- and 13-base systems, as labelled.\n(b) Free energy profile associated with the formation of a 13 base pair duplex at different temperatures \n(as labelled), where $Q$ represents the number of correctly formed bonds in the duplex.}\n\\label{thermo_duplex}\n\\end{figure} \n\nFigures \\ref{thermo_duplex}(b) shows the free energy profile, $F(Q)$, for the \nformation of a duplex. \nThe initial peak at low $Q$ is accounted for by the entropic cost of bringing two strands together. \nOnce bonds are formed, however, adding extra bonds costs much less entropy whilst providing a significant decrease in energy, explaining the monotonic decrease in $F(Q)$ beyond $Q=2$. \nThe rise between $Q=1$ and $Q=2$ is partly due to the fact that in order to form two bonds between strands the relative orientation of strands must be specified whereas this is not true for $Q=1$: \nhence there is an additional entropy penalty to the formation of the second bond. \nIn addition, there exist structures with only one correct bond that are \nstabilized by additional incorrect bonds and these misbonded configurations \nalso contribute to $F(1)$. \nThe constant gradient above $Q=2$ indicates that the energetic gain and entropic cost of forming an extra bond are approximately constant at a given temperature, which is consistent with the assumptions underlying nearest-neighbour models of DNA melting.\\cite{Everaers2007}\n\n\\begin{figure}\n\\includegraphics[width=6.2cm,angle=-90]{Fig5.eps}\n\\caption{(Colour online) The bulk equilibrium probability of strands being in \na correct duplex extrapolated from our umbrella sampling simulations (solid lines) compared to the predictions of an empirical two-state model (dashed lines).\nResults are presented for strands with 7 and 13 bases, as labelled.\nFor the two-state model, as well as the results for the sequences\nin Eqs.\\ (\\ref{7 bases}) and (\\ref{13 bases}) (lines (a) and (d)),\nsequences corresponding to the other two arms of the Holliday junction\n(Figure \\ref{HJ schematic}) are considered.\nOnly one line is shown for the umbrella sampling results, because A-T and G-C\nhave the same binding energy in our model.\nTemperatures in our model are converted to  ${\\rm^o}$C using the same mapping given in Section II A.}\n\\label{2 state}\n\\end{figure} \n\n In Figure \\ref{2 state}, we compare our melting curves to those predicted by a simple two-state model,\\cite{Everaers2007} using the same mapping of the reduced temperature as in Section \\ref{model}. In the two-state model the molar concentrations of product (AB) and reactants (A,B) are given by the equilibrium relation: \n\\begin{equation}\n\\frac{\\left[{\\rm AB }\\right]}{\\left[{\\rm A} \\right] \\left[{\\rm B}\\right]} = \\exp{\\left({\\frac{-\\Delta H_0 + T \\Delta S_0}{\\rm{k_B}\\it T}}\\right)},\n\\label{two state model}\n\\end{equation}\nwhere $\\Delta H_0$ and $\\Delta S_0$ are assumed to be constants which depend only on the strand sequences and the salt concentration (we take $\\rm{[Na^+}] = 0.445\\rm{M}$ as in Section \\ref{model}). \nWe use the enthalpy and entropy changes of duplex formation calculated by ``HyTher\",\\cite{Hyther} a program that estimates these values using the ``unified oligonucleotide nearest neighbour parameters\".\\cite{SantaLucia1998, Peyret1999} \nThe authors claim that the thermodynamic parameters predicted by HyTher give the melting temperature $T_{\\rm m}$ (the temperature at which the fraction of bonded strands is 1\/2)\nof a duplex to within a standard error of $\\pm2.2^{\\rm{o}}\\rm{C}$.\\cite{SantaLucia1998}\nFigure \\ref{2 state} shows that our system reflects the melting temperatures predicted by the two-state model with reasonable accuracy, excepting sequence dependent effects which are not included in our model, because the interaction energies \nbetween A-T and C-G complementary base pairs have for simplicity been taken to be the same. The widths of transitions are seen to be of the same order, but slightly larger for our model. \nThis feature, which is typical of coarse-grained models,\\cite{Knotts2007} indicates that the degree of entropy loss on hybridization is too small in our model, \nand is due to a failure to accurately incorporate all degrees of freedom which become frozen on hybridization. \nHowever, the agreement is sufficiently good that the basic features of physical DNA assembly should be reproducible.\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=15cm]{Fig6.eps}\n\\end{center}\n\\caption{(Colour online) Snapshots illustrating four stages in the process of displacement. \n(a) A third strand binds to a misbonded pair. \n(b) The third strand is prevented from forming a complete duplex by the misbond. \n(c) Thermal fluctuations cause bonds in the misbonded structure to break and be replaced by the correct duplex. \n(d) The misbonded strand is displaced and the correct duplex is formed.}\n\\label{displacement}\n\\end{figure*}\n\nA further satisfying feature of the model is that `displacement' was observed on several occasions. This process, during which a misbonded pair of strands is broken up by a third strand, is illustrated in Figure \\ref{displacement}. The third strand is able to bond to the pair, as some bases are free in the misbonded structure. \nThermal fluctuations allow the new strand to bond to sites previously involved in misbonding, in a process known as `branch migration'. Eventually one of the misbonded strands is completely displaced, leaving a correct duplex and an isolated single strand. This behaviour is observed in real DNA systems, and is the driving mechanism of some nanomachines \\cite{Yurke2000} and DNA catalyzed reactions.\\cite{Yin2008, Zhang2007}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=8.4cm]{Fig7.eps}\n\\end{center}\n\\caption{(Colour online) A schematic diagram showing the sequences of the strands used in our Holliday junction simulations, and the alternative bound states that are possible: (a) the square planar configuration and (b) the $\\chi$-stacked form.}\n\\label{HJ schematic}\n\\end{figure}\n\n\\subsection{Holliday Junction}\n\\label{Holliday Junction}\nEncouraged by the above results,\nwe next apply the model to the formation of a Holliday junction. \nHolliday junctions consist of four single strands which bind to form a four-armed cross. \nIn our case we consider a Holliday junction with two long arms (13 bases long) and two short arms (7 bases long). \nWe use the experimental base ordering of Malo {\\it et al.}\\cite{MaloThesis} with the `sticky ends' removed. \n(These sticky ends consist of six unpaired bases on the end of arms and their purpose is to allow the Holliday junctions to bond together to form a lattice). \nThe sequences of the four DNA strands and schematic diagrams of the \npossible junctions that they can form are shown in Figure \\ref{HJ schematic}.\n\nInitially we studied a system of 20 strands (five of each type) that has the potential to form five separate junctions. \nWe use a concentration of $1.56 \\times 10^{-5}$ molecules\\,$l^{-3}$\n(which corresponds to $1.04 \\times 10^{-4}$M). \nThe results are displayed in Figure \\ref{20mers}. \n\n\\begin{figure}\n\\includegraphics[width=6.0cm,angle=-90]{Fig8.eps}\n\\caption{(Colour online) \nA comparison of the kinetics and thermodynamics for a system of 20 strands that \ncan potentially form five Holliday junctions, where the MC simulations\nare initiated from a purely single-stranded configuration. \nThe MC results (lines with data points) are the final yield of Holliday junctions, and the fraction of strands involved in a correctly-formed long ($\\alpha$) or \nshort ($\\beta$) arm, or in misbonding, as labelled.\nThe results are averages over five runs of length $10^{9}$ steps per strand.\nFor comparison, the equilibrium probabilities of being in a $\\alpha$-bonded dimer\nand a Holliday junction are also plotted, along with the equilibrium probability of \nbeing in a $\\beta$-bonded dimer if the longer arms are not allowed to hybridize.\n}\n\\label{20mers}\n\\end{figure}\n\nThe results are as expected for the bonding of the longer arms (which we now describe as `$\\alpha$-bonding'). \nThe yield again displays the characteristic non-monotonic dependence on temperature. \nWe obtain very few complete junctions, however, which is due to two effects. \nFirstly, each simulation is performed at constant temperature, which means the hierarchical route to assembly is less favoured than when the system is cooled, as in the experiments.\\cite{MaloThesis,Malo2005}\nWhen the system is gradually cooled, Figure \\ref{20mers} suggests that at around $T_{\\rm m}(\\alpha) =0.111$ we would expect to find a region in which only $\\alpha$-bonded dimers were stable with respect to ssDNA. \nIf the cooling was sufficiently slow on the timescale of bonding, all strands would form $\\alpha$-structures at around $T_{\\rm m}(\\alpha)$. \nAt lower temperatures, when the Holliday junction becomes stable with respect to the $\\alpha$-bonded dimers, many competing minima would then be inaccessible to the system as they would require the disassociation of stable $\\alpha$-bonded pairs. \nThe free energy landscape of two $\\alpha$-structures forming a Holliday junction is consequentially much simpler than that of four single strands forming a junction at a given temperature.  \nTherefore, one expects the yield for self-assembly at constant temperature to be \nlower than when the system is cooled, because there is only a relatively narrow \ntemperature window between where the Holliday junction becomes stable and misbonded\nconfigurations start to appear. Indeed, the shorter arms are only marginally more \nstable than some competing minima, as evidenced by \nthe rise in misbonded structures in Fig.\\ \\ref{20mers} at temperature just below \nwhere $\\beta$-bonded structures first appear.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=8.4cm]{Fig9.eps}\n\\end{center}\n\\caption{(Colour online) Snapshot showing five Holliday junctions formed at $T=0.0842$ after \n$5.67 \\times 10^8$ MC steps per strand. Again, the backbone colour indicates strand type (1: red, 2: grey,\n 3: orange, 4: yellow) where numbers refer to Figure \\ref{HJ schematic}}\n\\label{5 HJs}\n\\end{figure}\n\nSecondly, even in the temperature range where Holliday junction formation is \nnot hindered by the formation of misbonded configurations, the yield is low\nbecause the Metropolis MC algorithm artificially reduces the diffusion of bound \npairs, and hence the likelihood that two pairs of $\\alpha$-bonded strands come together to form a junction is also reduced.\nThis is because the acceptance probability of trial moves for bonded strands is much lower than for isolated strands \\cite{Luijten2006}, due to the energy penalty associated with trying to move a bound pair apart. \n\n\\begin{figure}\n\\includegraphics[width=6.0cm,angle=-90]{Fig10.eps}\n\\caption{(Colour online) The yields of Holliday junctions (HJ) and misbonded configurations \nfor MC simulations, where the initial configuration was a pair of \n$\\alpha$-bonded dimers. For comparison, \nthe equilibrium probabilities of being in a $\\alpha$-bonded dimer \nand a Holliday junction are also plotted.\nThe results are averages over five runs of length \n$7.5 \\times 10^8$ steps per strand.\n}\n\\label{HJ thermo}\n\\end{figure}\n\nInterestingly, examination of the equilibrium lines in Figure \\ref{20mers} shows that the Holliday junctions are actually stable at a higher temperature than the individual shorter arms. \nThis is because the total loss of entropy when two $\\alpha$-bonded dimers bind together is considerably less than that for two short arms in isolation (as fewer translational degrees of freedom are lost), whereas the energy change is comparable. \nThus, there is a small temperature window at $T^*\\approx 0.1$ where hierarchical assembly can occur at constant temperature as the short arms are only stable once $\\alpha$-bonding has taken place. \nHowever, due to the deficiencies in the MC simulations mentioned above, the yield\nof Holliday junctions in this region is practically zero. Instead, the maximum\nyield of Holliday junctions occurs at lower temperatures where non-hierarchical\npathways that proceed by the addition of single strands become feasible.\n\nThe above simulations were only able to successfully model the first stage of \nthe Holliday junction assembly, namely the formation of $\\alpha$-bonded dimers. \nTo probe the second stage of assembly, we must first make two modifications to \nour simulation approach to overcome the two deficiencies mentioned above. \nFirstly, we study systems initially consisting of pairs of $\\alpha$-bonded strands, which we assume have successfully formed at some higher temperature---this is \nreasonable given the results of our earlier simulations.\nSecondly, we also include simple local cluster moves in addition to those which move only one strand, i.e.\\ translations, rotations and bending of pairs of $\\alpha$-bonded strands. \nWith these changes incorporated, we simulate the same system for $7.5 \\times 10^8$ steps per strand at a range of temperatures below $T_{\\rm m}(\\alpha)$. \nIt should be noted that due to a change in the size of typical moves, one move per strand now corresponds to approximately 10ps.\n\nWe find that Holliday junctions form over a wide range of intermediate temperatures, whilst kinetic traps at low temperature lead to incomplete bonding and consequently to the possibility of forming large clusters. A typical result from the high-yield regime is shown in Figure \\ref{5 HJs}. \nAs fully-bonded Holliday junctions are essentially inert, \nit is reasonable to analyse their assembly behaviour by considering only one \njunction.  The smaller system size has the effect of increasing the assembly rate, because the strands have less distance to diffuse, but does not affect the basic assembly mechanism. \nWe therefore simulated systems consisting of two $\\alpha$-bonded pairs with the same concentration as above. \n\nWe also introduced some modifications to the the umbrella sampling scheme in order\nto more efficiently compute the thermodynamics of the second-stage of \nHolliday junction formation. \nAs well as cluster moves, we also introduced a `tethering' component in the \nweighting function $W(Q)$. \nWe introduce a length $r_{\\rm min}$ that corresponds to the shortest distance between any pair of backbone sites on different strands. \nWe then split $Q=0$ into two regions: we weight those states with  $r_{\\rm min} < 3l$ with $W=1$ but for  $r_{\\rm min} \\geq 3l$ we use $W=0.1$. \nThis enables us to increase the rate of transitions between $Q=0$ and 1, and reduces the time spent \nsimply simulating the diffusion of $\\alpha$-bonded dimers waiting for a collision to occur.\n\nThe MC results are plotted in Figure \\ref{HJ thermo} along with \nthe equilibrium results obtained from umbrella sampling.\nWith the cluster moves in place, we now see a high yield of Holliday junctions\nand a broad maximum in the yield as a function of temperature.\nThe hierarchical pathway has the effect suggested earlier. \nNamely, the temperature window over which correct formation can occur is \nvastly increased, as the most significant competing minima are inaccessible because their formation would require dissociation of the $\\alpha$-bonded pairs. \n\nThe model is therefore consistent with the experimentally observed hierarchical assembly of Holliday junctions as the system is cooled.\\cite{Malo2005, MaloThesis} \nIt should be noted, however, that the junctions in our model usually form in the `square planar' as opposed to the `$\\chi$-stacked' shape (Figure \\ref{HJ schematic}) that is observed under normal experimental conditions. \nThe preference for one structure is a subtle consequence of the concentration of cations and the precise helical geometry of DNA.\\cite{Ortiz-Lombardia1999} \nThis level of detail is not included in our coarse-grained model, so it is not surprising that it cannot reproduce the preference for $\\chi$-stacked structures.\nMoreover, it is relatively easy to see why in our model, which forms \n`ladders' rather than helices, a planar geometry is preferred for the junction.\n\n\\begin{figure}\n\\includegraphics[width=8.4cm]{Fig11.eps}\n\\caption{Thermodynamics for the formation of a single Holliday junction.\n(a) Heat capacity curves for two $\\alpha$-bonded dimers forming a Holliday junction (HJ) \nand four strands forming two $\\alpha$-bonded dimers, as labelled.\n(b) Free energy profiles for the formation of a Holliday junction from two $\\alpha$-bonded dimers \nat different temperatures, as labelled. \n$Q$ represents the total number of correct bonds in the short arms of the junction.\n}\n\\label{thermo_HJ}\n\\end{figure}\n\nSome of the equilibrium thermodynamic properties associated with the formation\nof a Holliday junction are shown in Figure \\ref{thermo_HJ}. \nIn particular, Fig.\\ \\ref{thermo_HJ}(b) shows the free energy profile for the \nformation of a Holliday junction from two $\\alpha$-bonded pairs. \nThe initial peak and subsequent drop is very similar to that for the duplexes and can be accounted for in the same way. \nHowever, the formation of the two arms is not like the zipping up of a \n14-base duplex, because there is much more relative freedom of movement for \nthe bases on either side of the $\\alpha$-bonded sections in the dimers\nthan for consecutive bases on single-stranded DNA.\nThus, there is a rise between $M=7$ and $M=8$ that is a result of the entropy \npenalty of bringing together the two ends to make the second short arm. \nWe note that the penalty is much smaller than the initial cost of bringing the two $\\alpha$-bonded pairs together, and as a result, \nthe value of $T_{\\rm{m}}$ for the junction is higher than for the short arms \nin isolation, as noted earlier.\n\nAn interesting feature of Figure \\ref{thermo_HJ}(b) is the plateau between $Q=6$ and $Q=7$. \nIn general, when two $\\alpha$-bonded pairs meet to form one short arm, there is an entropic penalty associated with the excluded volume that the remaining bases in the $\\alpha$-structures represent to each other. \nThis excluded volume is a large fraction of the total available space if one complete short arm is formed, so that there are no free monomers between the short arm and the $\\alpha$-bonded sections. As a consequence, there is not the usual free energy benefit from forming the final bond in the short arm (the one closest to the centre of the Holliday junction), as the excluded volume penalty is large and those states that are allowed involve distortion of the backbones and bonds near the centre of the Holliday junction. \nAlthough the details of this free energy penalty and the other features in \nFig.\\ \\ref{thermo_HJ}(b) will depend on the exact geometry of the system, \nwe expect the calculated free energy profile to be representative of that \nfor real DNA.\n\nIt is possible to extend the two-state model discussed in Section \\ref{Duplex Formation} to the formation of a Holliday junction by considering the concentrations of all four isolated strands, the two $\\alpha$-bonded intermediates and the junction itself. \nWe assume Eq.\\ (\\ref{two state model}) holds for every possible transition, use the same thermodynamic parameters as before and apply conservation of total strand number. To estimate $\\Delta H_0$ and $\\Delta S_0$  associated with the formation of a Holliday junction, we construct a single strand by linking the ends of the oligonucleotides together with four non-bonding bases. The thermodynamic parameters associated with the folding of this structure are predicted by ``UNAFold''.\\cite{Markham2008} \nThe correction for the fact that our strands are not connected by loops is discussed by Zuker.\\cite{Zuker2003} \nThis leaves five simultaneous equations (assuming perfect stochiometry) which can be solved numerically.\n\n\\begin{figure}\n\\includegraphics[width=6.2cm,angle=-90]{Fig12.eps}\n\\caption{(Colour online) Bulk equilibrium probability of strands being in a Holliday junction (HJ) or\nan $\\alpha$-bonded dimer computed by umbrella sampling (solid lines) and by \nthe extended two-state model (dashed lines), where lines (a) and (b) represent the two possible $\\alpha$-bonded dimers.\n}\n\\label{etsm HJ}\n\\end{figure}\n\nFigure \\ref{etsm HJ} compares this extended two-state model (ETSM) with the bulk thermodynamics predicted by umbrella sampling (using the same temperature scaling as before). \nETSM predictions for both stages of Holliday junction formation agree well with our results, which again supports our hypothesis that much of the physics of self-assembly can be reproduced by a simple coarse-grained model. The extra width of the transitions in our model occurs for the same reasons as mentioned in Section \\ref{Duplex Formation} when discussing Fig.\\ \\ref{2 state}. \n\n\\begin{figure}\n\\includegraphics[width=6.1cm,angle=-90]{Fig13.eps}\n\\caption{(Colour online) Simulation results for the badly-designed Holliday \njunction of Section \\ref{negative}, where the initial configuration was a pair \nof $\\alpha$-bonded dimers. The lines with data points give the yield of \ncorrectly-formed junctions and misbonded configurations, as labelled.\nFor comparison the solid line gives the equilibrium probability that the \nwell-designed sequences of Section \\ref{Holliday Junction} adopt a \nHolliday junction.}\n\\label{thermo neg}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=7.4cm]{Fig14.eps}\n\\end{center}\n\\caption{(Colour online) Example of a competing minimum for a badly designed Holliday junction. The snapshot is \ntaken from a simulation at $T^*=0.0936$. \n}\n\\label{misbond neg}\n\\end{figure}\n\n\\subsection{Negative Design}\n\\label{negative}\nThe hierarchical pathway for the formation of the Holliday junction is one \naspect of the sequence design that aids the formation of the correct structure.\nThe experimental base ordering of the Holliday junction, however, was also chosen to minimize the number of competing structures---a typical example of `negative design'.\\cite{Doye04b} \nWe illustrate the importance of such negative design by considering a badly-designed junction, where the complementary seven-base sections consist of just one base type each. \nWe simulate a system of two $\\alpha$-bonded pairs with the short arms so modified \nunder the same conditions as for Fig.\\ \\ref{HJ thermo} and \nthe results are shown in Figure \\ref{thermo neg}. \nAlthough there is some probability of forming the correct junction, the simulations\nare dominated by misbonded junctions, such as the one depicted in Figure \\ref{misbond neg}.\nAlthough these competing structures are energetically less stable than the target\njunction because of the presence of unpaired bases at the `dangling' ends,\nthey are readily accessible, because the likelihood that the first bonds formed\nbetween two $\\alpha$-bonded pairs are in the same registry as the target structure  \nis low. The yield of the correct Holliday junctions will then depend upon how\nreadily the system is able escape from these malformed junctions. Clearly,\nthis process is slow on the time scales of the current simulations, and is \nalso likely to hinder the location of the target structure in experiment.\n\n\\section{Discussion}\nIn this paper we have introduced a simple coarse-grained model of DNA in order to test the feasibility of modeling the self-assembly of DNA nanostructure by \nMonte Carlo simulations. Any such model involves a trade-off between detail and \ncomputational simplicity, and here we deliberately chose to keep the model as simple\nas possible in order to give us the best chance of being able to probe the time\nscales relevant to self-assembly. The model involves just two interaction sites\nper nucleotide. \n\nThe results from our model are very encouraging. Firstly, we have shown that\nusing our model it is feasible to model the self-assembly of both DNA duplexes\nand a Holliday junction. The latter represents, to the best of our knowledge, the\nfirst example of the simulation of the self-assembly of a DNA structure beyond a \nduplex. Secondly, the model succeeds in reproducing many of the known thermodynamic\nand dynamic features of this self-assembly. \nFor example, the equilibrium melting curves agree well with those predicted by \nthe nearest-neighbour two-state model,\\cite{SantaLucia1998} which is known\nto predict melting temperatures very accurately. \nThe model is also able to capture important dynamical \nphenomena such as displacement.\n\nThirdly, by analysing the thermodynamic and dynamic constraints on assembly, we \nhave been able to gain some important physical insights into the nature of DNA\nself-assembly and how to control it. For example, the optimal conditions for \nself-assembly are in the temperature range just below the melting temperature of the\nthe target structure, where this structure is the only one stable with respect to \nthe precursors, be they ssDNA or some intermediate in a hierarchical assembly \npathway.  At lower temperatures, misbonded configurations can be formed that act \nas kinetic traps and reduce the assembly yield.\nSimilar trade-offs between the thermodynamic driving force and \nkinetic accessibility have been previously seen in a variety of self-assembling\nsystems,\\cite{Brooks06,Hagan2006,Wilber2007,Rapaport08,Whitelam08} \nand also give rise to a maximum in the yield near to and below\nthe temperature at which the target structure becomes stable.\n\nWe have also seen how hierarchical self-assembly through cooling can be a \nparticularly useful strategy to aid self-assembly, because the formation of stable\nintermediates at higher temperatures simplifies the free energy landscape for the\nassembly of the next stage in the hierarchy by reducing the number of misbonded \nconfigurations available to the system. This simplification of the energy\nlandscape is likely to be a general feature of hierarchical self-assembly.\n\nThus, our results have confirmed the utility of using coarse-grained DNA models to study the self-assembly of DNA nanostructures, and supported our hypothesis that much of the physics can be explained by describing DNA as a semi-flexible polymer with selective attractive interactions. The model's success in forming junctions in reasonable computational time suggests that it will be possible to develop further models that have an increased level of detail, but which can still access the time scales relevant to self-assembly.\n\nThe model has also highlighted some features which it would be advantageous to include in such models. For example, greater accuracy in the details of oligonucleotide geometry, particularly the helicity of dsDNA, would allow features such as the characteristically long persistence length of hybridized strands to be reproduced and give the appropriate degree of rigidity to simulated nanostructures. Such improvement might also allow more complicated motifs to be accounted for, such as the preference for $\\chi$-stacked Holliday junctions that the current model could not reproduce. \n\nIt should be noted that if one is to introduce helicity in a physically\nreasonable way it should also allow for ssDNA to undergo a stacking transition \nto a helical form. \nThis transition may play a significant role in the thermodynamics and kinetics of self-assembly.\\cite{Holbrook1999}\nPreviously proposed coarse-grained DNA models that incorporate helicity have not been designed to accurately reproduce this feature. \nIncorporating extra degrees of freedom which are relevant to the stacking transition, such as the rotation of the base with respect to the sugar-base bond, may also help to increase the entropy change on hybridization and hence make the transition narrower as required.\n\nThe approximation to diffusive dynamics provided by the local move Metropolis Monte Carlo algorithm could also be improved. \nCurrently the `local' moves involve displacing, rotating or bending entire strands or pairs of strands---these effectively constitute cluster moves of groups of strongly bound nucleotides, and result in slow relaxation and translation times within bound structures. \nMore realistic dynamics may be achievable by considering trial moves of individual nucleotides, and incorporating cluster moves in a more systematic fashion, such as in the `virtual move' MC algorithm proposed by Whitelam and Geissler.\\cite{Whitelam2007,Whitelam08} \n\nOne potential issue with any coarse-graining is how it preserves the different\ntime scales in a system.\nIn Section \\ref{Monte Carlo Simulation} we assigned an approximate mapping \nbetween the number of Monte Carlo steps and physical time based upon comparison \nof diffusion coefficients. There are, however, other important time scales in the \nsystem, such as the time scale for the internal dynamics of an isolated strand and \nthe time scale over which the `zipping-up' of two strands occurs after a bond has been formed. \nComparisons of experimental diffusion coefficients \\cite{Lapham1997} and \nmelting and bubble formation from molecular dynamics \nsimulations \\cite{Drukker2001,Knotts2007} suggest a large separation in time scale \nbetween diffusion-limited processes and those that rely on the dynamics of \nindividual nucleotides. \nEncouragingly, we observe a similar time scale separation in our model:\nzipping-up and thermal relaxation of isolated strands occur over times scales \nshorter than $10^5$ steps per strand, whereas association typically required on the \norder of $10^7$ to $10^8$ steps per strand near the melting temperature \n(corresponding to tens or hundreds of microseconds). Furthermore, we would\nargue that it is this time scale separation, and not the precise ratios of the\nrelevant rate constants, that it is important to reproduce in self-assembly \nsimulations.\n\nWe should also note that the mapping of the diffusion constants between the model\nand experiment will not necessarily ensure that the rate of association is \naccurate in our model, because although the frequency of collisions in our model \nshould be correct, \nthere is also the contribution\nto the association rate from the probability that a collision will lead to \nsuccessful association. That we can reproduce the thermodynamics of the DNA melting transitions implies that the rates of association and disassociation have the right\nratio, but not that they necessarily have the correct absolute value. \nFor example, it is conceivable that helicity (both in dsDNA and possibly in ssDNA),\nwhich is not included in the current model, will influence the likelihood that a \ncollision is successful.\n\n\n\\begin{acknowledgments}\nThe authors are grateful for financial support from the EPSRC \nand the Royal Society. We also wish to acknowledge helpful discussions\nwith Jonathan Malo, John Santalucia Jr and Michael Zuker.\n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\n\nLet $M$ be a closed oriented smooth $d$-manifold. Let $D: H_*(M)\n\\xrightarrow{\\cong} H^{d-*}(M)$ be the Poincar\\'e duality map. Following a practice in string topology, we shift the homology grading downward by $d$ and let $\\mathbb H_{-*}(M)=H_{d-*}(M)$. The Poincar\\'e duality now takes the form $D:\\mathbb H_{-*}(M) \\xrightarrow{\\cong} H^{*}(M)$. For a homology element $a$, let $|a|$ denote its \n$\\mathbb H_*$-grading of $a$. \n\nThe intersection product $\\cdot$ in homology is defined as the Poincar\\'e dual of the cup product. Namely, for $a,b\\in \\mathbb H_*(M)$, $D(a\\cdot b)=D(a)\\cup D(b)$. If $\\alpha\\in H^*(M)$ is dual to $a$, then $\\alpha\\cap b=a\\cdot b$, and its Poincar\\'e dual is $\\alpha\\cup D(b)$. Thus, through Poincar\\'e duality, the intersection product, the cap product, and the cup product are all the same. In particular, the cap product and the intersection product commute:\n\\begin{equation}\n\\alpha\\cap (b\\cdot c)=(\\alpha\\cap b)\\cdot c\n=(-1)^{|\\alpha||b|}b\\cdot(\\alpha\\cap c). \n\\end{equation}\nIn fact, the direct sum $H^*(M)\\oplus \\mathbb H_*(M)$ can be made into a graded commutative associative algebra with unit, given by $1\\in H^0(M)$, using the cap and the cup product.\n\nFor an infinite dimensional manifold $N$, there is no longer\nPoincar\\'e duality, and geometric intersections of finite dimensional\ncycles are all trivial. However, cap products can still be nontrivial\nand the homology $H_*(N)$ is a module over the cohomology ring\n$H^*(N)$.\n\nWhen the infinite dimensional manifold $N$ is a free loop space $LM$\nof continuous maps from the circle $S^1=\\mathbb R\/\\mathbb Z$ to $M$,\nthe homology $\\mathbb H_*(LM)=H_{*+d}(LM)$ has a great deal more structure. As before, $|a|$ denotes the $\\mathbb H_*$-grading of a homology element $a$ of $LM$. Chas and\nSullivan \\cite{CS} showed that $\\mathbb H_*(LM)$ has a degree preserving associative graded\ncommutative product $\\cdot$ called the loop product, a\nLie bracket $\\{\\ ,\\ \\}$ of degree $1$ called the loop bracket\ncompatible with the loop product, and the BV operator $\\Delta$ of\ndegree $1$ coming from the homology $S^1$ action. These structures turn\n$\\mathbb H_*(LM)$ into a Batalin-Vilkovisky (BV) algebra. The purpose of this\npaper is to clarify the interplay between the cap product with cohomology elements and the BV structure in $\\mathbb H_*(LM)$.\n\n\n\nLet $p: LM \\rightarrow M$ be the base point map\n$p(\\gamma)=\\gamma(0)$ for $\\gamma\\in LM$. For a cohomology class\n$\\alpha\\in H^*(M)$ in the base manifold, its pull-back $p^*(\\alpha)\\in\nH^*(LM)$ is also denoted by $\\alpha$. Let $\\Delta: S^1\\times LM\n\\longrightarrow LM$ be the $S^1$-action map. This map induces a degree $1$ map $\\Delta$ in homology given by $\\Delta a=\\Delta_*([S^1]\\times a)$\nfor $a\\in \\mathbb H_*(LM)$. For a cohomology class $\\beta\\in H^*(LM)$, the formula \n$\\Delta^*(\\beta)=1\\times\\beta+\\{S^1\\}\\times\\Delta\\beta$ defines a degree $-1$ map $\\Delta$ in cohomology, where $\\{S^1\\}$ is the fundamental cohomology class. Although we use the same notation $\\Delta$ in three different but closely related situations, what is meant by $\\Delta$ should be clear in the context.  \n\n\\begin{Theorem A} Let $b,c\\in \\mathbb H_*(LM)$. \nThe cap product with $\\alpha\\in H^*(M)$ graded commutes with the loop product. Namely \n\\begin{equation}\n\\alpha\\cap(b\\cdot c)=(\\alpha\\cap b)\\cdot\nc=(-1)^{|\\alpha||b|}b\\cdot(\\alpha\\cap c).\n\\end{equation}\n\nFor $\\alpha\\in H^*(M)$, the cap product with $\\Delta\\alpha\\in H^*(LM)$\nacts as a derivation on the loop product and the\nloop bracket\\textup{:}\n\\begin{align}\n(\\Delta\\alpha)\\cap(b\\cdot c)=(\\Delta\\alpha\\cap b)\\cdot c+\n(-1)^{(|\\alpha|-1)|b|}b\\cdot (\\Delta\\alpha\\cap c), \\\\\n(\\Delta\\alpha)\\cap\\{b,c\\}=\\{\\Delta\\alpha\\cap b,c\\}+\n(-1)^{|\\alpha|-1)(|b|+1)}\\{b,\\Delta\\alpha\\cap c\\}.\n\\end{align}\n\nThe operator $\\Delta$ acts as a derivation on the cap product. Namely, for $\\alpha\\in H^*(M)$ and $b\\in \\mathbb H_*(LM)$.\n\\begin{equation}\n\\Delta(\\alpha\\cap b)=\\Delta\\alpha\\cap b + (-1)^{|\\alpha|} \\alpha\\cap\n\\Delta b.\n\\end{equation}\n\\end{Theorem A}\n \nWe recall that in the BV algebra $\\mathbb H_*(LM)$, the following identities\nare valid for $a,b,c\\in \\mathbb H_*(LM)$ \\cite{CS}:\n\\begin{gather}\n\\Delta(a\\cdot b)=(\\Delta a)\\cdot b + (-1)^{|a|}a\\cdot \\Delta b +\n(-1)^{|a|}\\{a,b\\} \n\\tag{BV identity} \\\\ \n\\{a, b\\cdot c\\} =\\{a,b\\}\\cdot c + (-1)^{|b|(|a|+1)}b\\cdot\\{a,c\\} \n\\tag{Poisson identity} \\\\ \na\\cdot b=(-1)^{|a||b|}b\\cdot a,\\qquad\n\\{a,b\\}=-(-1)^{(|a|+1)(|b|+1)}\\{b,a\\}\n\\tag{Commutativity} \\\\\n\\{a,\\{b,c\\}\\}=\\{\\{a,b\\},c\\}+(-1)^{(|a|+1)(|b|+1)}\\{b,\\{a,c\\}\\}\n\\tag{Jacobi identity}\n\\end{gather} \nHere, $\\deg a\\cdot b=|a|+|b|, \\deg \\Delta a=|a|+1$, and $\\deg\\{a,b\\}=|a|+|b|+1$. \n\nWe can extend the loop product and the loop bracket in $\\mathbb H_*(LM)$ to\ninclude $H^*(M)$ in the following way. For $\\alpha\\in H^*(M)$ and\n$b\\in \\mathbb H_*(LM)$, we define the loop product and the loop bracket of \n$\\alpha$ and $b$ by \n\\begin{equation}\n\\alpha\\cdot b=\\alpha\\cap b, \\qquad\n\\{\\alpha,b\\}=(-1)^{|\\alpha|}(\\Delta\\alpha)\\cap b.\n\\end{equation} \n\nFurthermore, the BV structure in $\\mathbb H_*(LM)$ can be extended to the direct sum $A_*=H^*(M)\\oplus \\mathbb H_*(LM)$ by defining the BV operator $\\boldsymbol\\Delta$ on $A_*$ to be trivial on $H^*(M)$ and to be the usual homological $S^1$ action $\\Delta$ on $\\mathbb H_*(LM)$. Here in $A_*$, elements in $H^k(M)$ are regarded as having homological degree $-k$. \n\n\\begin{Theorem B} The direct sum $H^*(M)\\oplus \\mathbb H_*(LM)$ has a structure of a BV algebra. In particular, for $\\alpha\\in H^*(M)$ and $b,c\\in \\mathbb H_*(LM)$, the following form of\nPoisson identity and the Jacobi identity hold\\textup{:}\n\\begin{gather} \n\\begin{split}\n\\{\\alpha\\cdot b,c\\}&=\\alpha\\cdot\\{b,c\\}+\n(-1)^{|b|(|c|+1)}\\{\\alpha,c\\}\\cdot b \\\\ &=\\alpha\\cdot\\{b,c\\} +\n(-1)^{|\\alpha||b|} b\\cdot\\{\\alpha,c\\},\n\\end{split} \\\\\n\\{\\alpha,\\{b,c\\}\\}=\\{\\{\\alpha,b\\},c\\} +\n(-1)^{(|\\alpha|+1)(|b|+1)}\\{b,\\{\\alpha,c\\}\\}.\n\\end{gather}\n\\end{Theorem B}\n\nAll the other possible forms of Poisson and Jacobi identities are\nalso valid, and the above two identities are the most nontrivial\nones. These identities are proved by using standard properties of the\ncap product and the BV identity above in $\\mathbb H_*(LM)$ relating the BV operator $\\Delta$ and the loop bracket $\\{\\,,\\,\\}$, but without using Poisson identities nor Jacobi identities in the BV algebra $\\mathbb H_*(LM)$.\n\nThe above identities may seem rather surprising, but they become\ntransparent once we prove the following result.\n\n\\begin{Theorem C} For $\\alpha\\in H^*(M)$, \nlet $a=\\alpha\\cap [M]\\in \\mathbb H_*(M)$ be its Poincar\\'e dual. Then for $b\\in \\mathbb H_*(LM)$,\n\\begin{equation}\\label{cap}\n\\alpha\\cap b=a\\cdot b,\\qquad (-1)^{|\\alpha|}\\Delta\\alpha\\cap\nb=\\{a,b\\}.\n\\end{equation} \nMore generally, for cohomology elements $\\alpha_0,\\alpha_1,\\dotsc\n\\alpha_r\\in H^*(M)$, let $a_0, a_1,\\dotsc a_r\\in \\mathbb H_*(M)$ be their Poincar\\'e duals. Then for $b\\in \\mathbb H_*(LM)$, we have\n\\begin{equation}\\label{composition of derivations}\n(\\alpha_0\\cup \\Delta\\alpha_1\\cup\\dotsm \\cup \\Delta\\alpha_r)\\cap b\n=(-1)^{|a_1|+\\dotsb+|a_r|}a_0\\cdot\\{a_1,\\{a_2,\\dotsc \\{a_r,b\\}\\dotsb \\}\\}.\n\\end{equation}\n\\end{Theorem C}\n\nSince the cohomology $H^*(M)$ and the homology $\\mathbb H_*(M)$ are isomorphic through Poincar\\'e duality and $\\mathbb H_*(M)$ is a subring of $\\mathbb H_*(LM)$, the first formula in \\eqref{cap} is not surprising. However, the main difference between $H^*(M)$ and $\\mathbb H_*(M)$ in our context is that the homology $S^1$ action $\\Delta$ is trivial on $\\mathbb H_*(M)\\subset \\mathbb H_*(LM)$, although cohomology $S^1$ action $\\Delta$ is nontrivial on $H^*(M)$ and is related to loop bracket as in \\eqref{cap}.  \n\nTheorem A and Theorem C describes the cap product action of the cohomology $H^*(LM)$ on the BV algebra $\\mathbb H_*(LM)$ for most elements in $H^*(LM)$. For example, for $\\alpha\\in H^*(M)$, the cap product with $\\Delta\\alpha$ is a derivation on the loop algebra $\\mathbb H_*(LM)$ given by a loop bracket, and consequently the cap product with a cup product $\\Delta\\alpha_1\\cup\\cdots\\cup\\Delta\\alpha_r$ acts on the loop algebra as a composition of derivations, which is equal to a composition of loop brackets, according to \\eqref{composition of derivations}. If $H^*(LM)$ is generated by elements $\\alpha$ and $\\Delta\\alpha$ for $\\alpha\\in H^*(M)$ (for example, this is the case when $H^*(M)$ is an exterior algebra, see Remark \\ref{exterior algebra}), then Theorem C gives a complete description of the cap product with arbitrary elements in $H^*(LM)$ in terms of the BV algebra structure in $\\mathbb H_*(LM)$. However, $H^*(LM)$ is general bigger than the subalgebra generated by $H^*(M)$ and $\\Delta H^*(M)$. \n\nSince $\\mathbb H_*(LM)$ is a BV algebra, in view of Theorem C, the validity of Theorem B may seem obvious. However, in the proof of Theorem B, we only used standard properties of the cap product and the BV identity. In fact, Theorem B gives an alternate elementary and purely homotopy theoretic proof of Poisson and Jacobi identities in $\\mathbb H_*(LM)$, when at least one of the elements $a,b,c$ are in $\\mathbb H_*(M)$. Similarly, Theorem C gives a purely homotopy theoretic interpretation of the loop product and the loop bracket if one of the elements are in $\\mathbb H_*(M)$.\n\nOur interest in cap products in string topology comes from an\nintuitive geometric picture that cohomology classes in $LM$ are dual\nto finite codimension submanifolds of $LM$ consisting of certain loop\nconfigurations. We can consider configurations of loops\nintersecting in particular ways (for example, two loops having their base points in common), or we can consider a family of loops\nintersecting transversally with submanifolds of $M$ at\ncertain points of loops. In a given family of loops, taking the cap product with a cohomology class selects a subfamily of a certain loop configuration, which are ready for certain loop interactions. In this context, roughly speaking, composition of two interactions of loops correspond to the cup product of corresponding cohomology classes.\n\nThe organization of this paper is as follows. In section 2, we\ndescribe a geometric problem of describing certain family of  intersection configuration of loops in terms of cap products. This gives a geometric motivation for the remainder of the paper. In section 3, we review the loop product in $\\mathbb H_*(LM)$ in detail from the point of view of the intersection product in $\\mathbb H_*(M)$. Here we pay careful attention to signs. In particular, we give a homotopy theoretic proof of graded commutativity in the BV algebra $\\mathbb H_*(LM)$, which turned out to be not so trivial. In section 4, we prove compatibility\nrelations between the cap product and the BV algebra structure, and\nprove Theorems A and B. In the last section, we prove Theorem C.\n\nWe thank the referee for numerous suggestions which lead to clarification and improvement of exposition. \n\n\n\n\n\n\\section{Cap products and intersections of loops}\n\nLet $A_1,A_2,\\dotsc A_r$ and $B_1,B_2,\\dotsc B_s$ be oriented closed\nsubmanifolds of $M^d$. Let $F\\subset LM$ be a compact family of\nloops. We consider the following question. \n\n\\smallskip\n\n\\textbf{Question} : Fix $r$ points $0\\le t_1^*,t_2^*,\\dotsc,t_r^*\\le\n1$ in $S^1=\\mathbb R\/\\mathbb Z$. Describe the homology class of the subset $I$ of the compact family $F$\nconsisting of loops $\\gamma$ in $F$ such that $\\gamma$ intersects\nsubmanifolds $A_1,\\dotsc A_r$ at time $t_1^*,\\dotsc t_r^*$ and\nintersects $B_1,\\dotsc B_s$ at some unspecified time.\n\n\\smallskip\n\nThis subset $I\\subset F$ can be described as follows. We consider the following diagram of an evaluation map and a projection map: \n\\begin{equation} \\label{eval}\n\\begin{CD}\n\\overset{s}{\\overbrace{(S^1\\times\\dotsb\\times S^1)}}\\times LM @>{e}>>\n  \\overset{r}{\\overbrace{M\\times \\dotsb\\times M}}\\times\n\\overset{s}{\\overbrace{M\\times\\dotsb\\times M}}  \\\\\n@V{\\pi_2}VV @.   \\\\\nLM  @. \n\\end{CD}\n\\end{equation}\ngiven by $e\\bigl((t_1,\\dotsc t_s),\\gamma\\bigr)\n=\\bigl(\\gamma(t_1^*),\\dotsc \\gamma(t_r^*), \\gamma(t_1), \\dotsc\n\\gamma(t_s)\\bigr)$. Then the pull-back set\n$e^{-1}(\\prod_iA_i\\times\\prod_jB_j)$ is a closed \nsubset of $S^1\\times\\dotsb\\times S^1\\times LM$. Let\n\\begin{equation*}\n\\tilde{I}=e^{-1}(\\prod_iA_i\\times \\prod_jB_j)\\cap \n(S^1\\times\\dotsb\\times S^1\\times F). \n\\end{equation*} \nThe set $I$ in\nquestion is given by $I=\\pi_2(\\tilde{I})$. We want to understand this set $I$ homologically, including multiplicity. Although $e^{-1}(\\prod_iA_i\\times\\prod_jB_j)$ is infinite dimensional, it has finite codimension in $(S^1)^r\\times LM$. So we work cohomologically.\n\nLet $\\alpha_i, \\beta_j\\in H^*(M)$ be cohomology classes dual to $[A_i], [B_j]$ for $1\\le i\\le r$ and $1\\le j\\le s$. Then the subset $e^{-1}(\\prod_iA_i\\times \\prod_jB_j)$ is dual to the cohomology class  $e^*(\\prod_i\\alpha_i\\times\\prod_j\\beta_j)\\in H^*((S^1)^s\\times LM)$. Suppose the family $F$ is parametrized by a closed oriented manifold $K$ by an onto map $\\lambda:K \\longrightarrow F$ and let $b=\\lambda_*([K])\\in \\mathbb H_*(LM)$ be the homology class of $F$ in $LM$. Then the homology class of $\\tilde{I}$ in $(S^1)^s\\times LM$ is given by \n\\begin{equation}\n[\\tilde{I}]=e^*(\\prod_i\\alpha_i\\times\\prod_j\\beta_j)\\cap ([S^1\\times\\dotsb\\times S^1]\\times b). \n\\end{equation} \nNote that the homology class $(\\pi_2)_*([\\tilde{I}])$ represents the homology class of $I$ with multiplicity. \n\n\\begin{proposition}\\label{loop intersection} With the above notation, $(\\pi_2)_*([\\tilde{I}])$ is given by the following formula in terms of the cap product or in terms of the BV structure\\textup{:}\n\\begin{equation}\n\\begin{aligned}\n(\\pi_2)_*([\\tilde{I}])&=(-1)^{\\sum_jj|\\beta_j|-s}\n\\bigl(\\alpha_1\\dotsm\\alpha_r(\\Delta\\beta_1)\\dotsm(\\Delta\\beta_s)\\bigr)\n\\cap b \\\\\n&=(-1)^{\\sum_jj|\\beta_j|-s}[A_1]\\cdots[A_s]\\cdot\\{[B_1],\\{\\cdots\\{[B_s],b\\}\\cdots\\}\n\\in \\mathbb H_*(LM). \n\\end{aligned}\n\\end{equation}\n\\end{proposition} \n\\begin{proof} The evaluation map $e$ in \\eqref{eval} is \ngiven by the following composition.\n\\begin{multline*} \n\\overset{s}{\\overbrace{S^1\\times\\dotsb\\times S^1}}\\times LM \n\\xrightarrow{1\\times\\phi} \n(S^1\\times\\dotsb\\times S^1) \\times\n\\overset{r+s}{\\overbrace{LM\\times\\dotsb\\times LM}} \\\\\n\\xrightarrow{T} \n\\overset{r}{\\overbrace{LM\\times\\dotsb\\times LM}}\\times\n\\overset{s}{\\overbrace{(S^1\\times LM)\\times\\dotsb\\times(S^1\\times LM)}}  \\\\\n\\xrightarrow{1^r\\times\\Delta^s}\n(LM\\times\\dotsb\\times LM) \\times\n(LM\\times\\dotsb\\times LM)\n\\xrightarrow{p^r\\times p^s} \n(M\\times\\dotsb\\times M) \\times\n(M\\times\\dotsb\\times M),\n\\end{multline*} \nwhere $\\phi$ is a diagonal map, $T$ moves $S^1$ factors.  Since we\napply $(\\pi_2)_*$ later, we only need terms in $e^*(\\prod A_i\\times\n\\prod B_j)$ containing the factor\n$\\{S^1\\}\\times\\dotsb\\times\\{S^1\\}$. Since\n$\\Delta^*p^*(\\beta_j)=1\\times p^*(\\beta_j)\n+\\{S^1\\}\\times\\Delta\\beta_j$ for $1\\le j\\le s$, following the above\ndecomposition of $e$, we have\n\\begin{equation*}\ne^*(\\alpha_1\\times\\dotsb\\times\\alpha_r\\times\\beta_1\\times\\dotsb\\times\\beta_s)\n=\\varepsilon\n\\{S^1\\}^s\\times \\bigl(\\alpha_1\\dotsm\\alpha_r(\\Delta\\beta_1)\\dotsm\n(\\Delta\\beta_s)\\bigr) + \\text{ other terms},\n\\end{equation*}\nwhere the sign $\\varepsilon$ is given by $\\varepsilon=\n(-1)^{\\sum_{\\ell=1}^s(s-\\ell)(|\\beta_{\\ell}|-1)\n+s\\sum_{\\ell=1}^r|\\alpha_{\\ell}|}$.  Thus, taking the cap product with\n$[S^1]^s\\times b$ and applying $(\\pi_2)_*$, we get\n\\begin{multline*}\n{\\pi_2}_*\n\\bigl(e^*(\\alpha_1\\times\\dotsb\\times\\alpha_r\\times\n\\beta_1\\times\\dotsb\\times\\beta_s)\n\\cap ([S^1]\\times\\dotsb\\times[S^1]\\times b)\\bigr) \\\\\n=(-1)^{\\sum_{\\ell=1}^{s}\\ell|\\beta_{\\ell}|-s}\n\\alpha_1\\dotsm\\alpha_r(\\Delta\\beta_1)\\dotsm(\\Delta\\beta_s)\\cap b.\n\\end{multline*}\nThe second equality follows from the formula \\eqref{composition of derivations}. \n\\end{proof}\n\n\\begin{remark} In the diagram \\eqref{eval}, \nin terms of cohomology transfer ${\\pi_2}^!$ we have \n\\begin{equation}\n{\\pi_2}^!\ne^*(\\alpha_1\\times\\dotsb\\times\\alpha_r\\times\\beta_1\\times\\dotsb\\times\\beta_s)\n=\\pm\n\\alpha_1\\dotsm\\alpha_r(\\Delta\\beta_1)\\dotsm(\\Delta\\beta_s), \n\\end{equation}\nwhere ${\\pi_2}^!(\\alpha)\\cap\nb=(-1)^{s|\\alpha|}{\\pi_2}_*\\bigl(\\alpha\\cap{\\pi_2}_!(b)\\bigr)$ for any\n$\\alpha\\in H^*((S^1)^s\\times LM)$ and $b\\in \\mathbb H_*(LM)$. Here\n${\\pi_2}_!(b)=[S^1]^s\\times b$.\n\\end{remark}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{The intersection product and the loop product}\n\n\n\n\n\n\n\nLet $M$ be a closed oriented smooth $d$-manifold.  The loop product in\n$\\mathbb H_*(LM)$ was discovered by Chas and Sullivan \\cite{CS}, in terms of\ntransversal chains. Later, Cohen and Jones \\cite{CJ} gave a homotopy\ntheoretic description of the loop product. The loop product is a\nhybrid of the intersection product in $\\mathbb H_*(M)$ and the Pontrjagin\nproduct in the homology of the based loop spaces $H_*(\\Omega M)$. In\nthis section, we review and prove some properties of the loop product\nin preparation for the next section. Our treatment of the loop product\nfollows \\cite{CJ}. However, we will be precise with signs and give a\nhomotopy theoretic proof of the graded commutativity of the loop\nproduct, which \\cite{CJ} did not include. For the Frobenius compatibility formula with careful discussion of signs, see \\cite{T2}. For homotopy theoretic deduction of the BV identity, see \\cite{T3}. \n\nFor our purpose, the free loop space $LM$ is the space of {\\it continuous} maps from $S^1=\\mathbb R\/\\mathbb Z$ to $M$. Our discussion is homotopy theoretic and does not require smoothness of loops, although we do need smoothness of $M$ which is enough to allows us to have tubular neighborhoods for certain submanifolds in the space of continuous loops. Recall that the space $LM$ of continuous loops can be given a structure of a smooth manifold. See the discussion before Definition \\ref{definition of loop product}. \n\nLet $p: LM \\longrightarrow M$ be the base\npoint map given by $p(\\gamma)=\\gamma(0)$. Let $s: M \\longrightarrow\nLM$ be the constant loop map given by $s(x)=c_x$, where $c_x$ is the\nconstant loop at $x\\in M$. Since $p_*\\circ s_*=1$, $\\mathbb H_*(M)$ is\ncontained in $\\mathbb H_*(LM)$ through $s_*$ and we often regard $\\mathbb H_*(M)$ as a subset of $\\mathbb H_*(LM)$.\n\nWe start with a discussion on the intersection ring $\\mathbb H_*(M)$ and later we compare it with the loop homology algebra $\\mathbb H_*(LM)$. An exposition on intersection products in homology of manifolds can be found on Dold's book \\cite{D}, Chapter VIII, \\S13. Our sign convention (which follows Milnor \\cite{M}) is slightly different from Dold's. We give a fairly detailed discussion of the intersection ring $\\mathbb H_*(M)$ because the\ndiscussion for the loop homology algebra goes almost in parallel, and\nbecause our choice of the sign for the loop product comes from and is\ncompatible with the intersection product in $\\mathbb H_*(M)$. Compare formulas \\eqref{intersection product} and \\eqref{loop product}.\n\nThose who are familiar with intersection product and loop products can skip this section after checking Definition \\ref{definition of loop product}. \n\nLet $D: \\mathbb H_*(M) \\xrightarrow{\\cong} H^{d-*}(M)$ be the Poincar\\'e duality map such that $D(a)\\cap[M]=a$ for $a\\in \\mathbb H_*(M)$.  We discuss two ways to define\nintersection product in $\\mathbb H_*(M)$. The first method is the official one\nand we simply define the intersection product as the Poincar\\'e dual\nof the cohomology cup product. Thus, $D(a\\cdot b)=D(a)\\cup D(b)$ for\n$a,b\\in \\mathbb H_*(M)$. For example, we have $a\\cdot\nb=(-1)^{|a||b|}b\\cdot a$.\n\nThe second method uses the transfer map induced from the diagonal map\n$\\phi: M \\longrightarrow M\\times M$. Let $\\nu$ be the normal bundle to\n$\\phi(M)$ in $M\\times M$, and we orient $\\nu$ by $\\nu\\oplus\n\\phi_*(TM)\\cong T(M\\times M)|_{\\phi(M)}$. Let $u'\\in\nH^d(\\phi(M)^{\\nu})$ be the Thom class of $\\nu$. Let $N$ be a closed\ntubular neighborhood of $\\phi(M)$ in $M\\times M$ so that $D(\\nu)\\cong\nN$, where $D(\\nu)$ is the associated closed disc bundle of $\\nu$. Let\n$\\pi: N \\longrightarrow M$ be the projection map. Then the above Thom\nclass can be thought of as $u'\\in \\tilde{H}^d(N\/\\partial N)$, and we\nhave the following commutative diagram, where $c: M\\times M\n\\longrightarrow N\/\\partial N$ is the Thom collapse map, and $\\iota_N$\nand $j$ are obvious maps.\n\\begin{equation}\n\\begin{CD} \nH^d(N, N-\\phi(M)) @>{\\cong}>> H^d(N,\\partial N)\\ni u' \\\\\n@A{\\cong}A{\\iota_N^*}A   @VV{c^*}V   \\\\\nu''\\in H^d\\bigl(M\\times M, M\\times M-\\phi(M)\\bigr) @>{j^*}>>  H^d(M\\times M)\\ni u\n\\end{CD}\n\\end{equation} \nLet $u''\\in H^d\\bigl(M\\times M, M\\times M-\\phi(M)\\bigr)$ and $u\\in\nH^d(M\\times M)$ be the classes corresponding to the Thom class. We\nhave $u=c^*(u')=j^*(u'')$. This class $u$ is characterized by the\nproperty $u\\cap[M\\times M]=\\phi_*([M])$, and $\\phi^*(u)=e_M\\in H^d(M)$\nis the Euler class of $M$. See for example section 11 of \\cite{M}. The\ntransfer map $\\phi_!$ is defined as the following composition:\n\\begin{equation}\n\\phi_!: H_*(M\\times M) \\xrightarrow{c_*} \\tilde{H}_*(N\/\\partial N)\n\\xrightarrow[\\cong]{u'\\cap(\\ )} H_{*-d}(N) \\xrightarrow[\\cong]{\\pi_*}\nH_{*-d}(M).\n\\end{equation} \n\nFor a homology element $a$, let $|a|'$ denote its regular homology degree of $a$, so that we have $a\\in H_{|a|}(M)$ and $|a|'=|a|+d$. \n\n\\begin{proposition}\\label{properties of phi} Suppose $M$ is a connected \noriented closed $d$-manifold with a base point $x_0$. The transfer map\n$\\phi_!: H_*(M\\times M) \\longrightarrow H_{*-d}(M)$ satisfies the\nfollowing properties. For $a,b\\in H_*(M)$,\n\\begin{align}\n\\phi_*\\phi_!(a\\times b)&=u\\cap(a\\times b) \\\\ \\phi_!\\phi_*(a\\times\nb)&=\\chi(M)[x_0]\n\\end{align}\nFor $\\alpha\\in H^*(M\\times M)$ and $b,c\\in H_*(M)$, we have \n\\begin{equation}\n\\phi_!\\bigl(\\alpha\\cap(b\\times c)\\bigr)\n=(-1)^{d|\\alpha|}\\phi^*(\\alpha)\\cap\\phi_!(b\\times c).\n\\end{equation} \nThe intersection product and the transfer map coincide up to a sign. \n\\begin{equation}\\label{intersection product}\na\\cdot b=(-1)^{d(|a|'-d)}\\phi_!(a\\times b).\n\\end{equation}\n\\end{proposition} \n\\begin{proof} For the first identity, we consider the following \ncommutative diagram, where $M^2$ denotes $M\\times M$. \n\\begin{equation*}\n\\begin{CD}\nH_*(M^2) @>{c_*}>> H_*(N,\\partial N) @>{u'\\cap (\\ )}>{\\cong}> H_{*-d}(N) \n@>{\\pi_*}>{\\cong}>  H_{*-d}(M)  \\\\\n@|   @V{\\cong}V{{\\iota_N}_*}V  @VV{{\\iota_N}_*}V   @VV{\\phi_*}V   \\\\\nH_*(M^2)  @>{j_*}>> H_*\\bigl(M^2, M^2-\\phi(M)\\bigr) \n@>{u''\\cap(\\ )}>{\\cong}>  H_{*-d}(M^2) @=  H_{*-d}(M^2)\n\\end{CD}\\end{equation*}\nThe commutativity implies that for $a,b\\in H_*(M)$, we have $\\phi_*\\phi_!\n=u''\\cap j_*(a\\times b)=j^*(u'')\\cap(a\\times b)=u\\cap(a\\times b)$. \n\nTo check the second formula, we first compute\n$\\phi_*\\phi_!\\phi_*([M])$. By the first formula,\n$\\phi_*\\phi_!\\phi_*([M])=u\\cap\\phi_*([M])=\\phi_*(\\phi^*(u)\\cap[M])$. Since\n$\\phi^*(u)$ is the Euler class $e_M$, this is equal to\n$\\phi_*(e_M\\cap[M])=\\chi(M)[(x_0,x_0)]$. Since $M$ is assumed to be\nconnected, $\\phi_*$ is an isomorphism in $H_0$. Hence\n$\\phi_!\\phi_*([M])=\\chi(M)[x_0]\\in H_0(M)$.\n\nFor the next formula, we examine the following commutative diagram. \n\\begin{equation*}\n\\begin{CD} \nH_*(M^2) @>{c_*}>> H_*(N,\\partial N) @>{u'\\cap(\\ )}>{\\cong}> H_{*-d}(N)\n@<{\\iota_*'}<{\\cong}< H_{*-d}(M) \\\\ \n@V{\\alpha\\cap(\\ )}VV\n@V{\\iota_N^*(\\alpha)\\cap(\\ )}VV @V{\\iota_N^*(\\alpha)\\cap(\\ )}VV\n@V{\\iota^*(\\alpha)}VV \\\\ \nH_{*-|\\alpha|}(M^2) @>{c_*}>>\nH_{*-|\\alpha|}(N,\\partial N) @>{u'\\cap(\\ )}>{\\cong}> H_{*-d-|\\alpha|}(N)\n@<{\\iota_*'}<{\\cong}< H_{*-d-|\\alpha|}(M)\n\\end{CD}\n\\end{equation*}\nwhere $\\iota': M\\rightarrow N$ is an inclusion map and\n$\\iota_*'=(\\pi_*)^{-1}$. The middle square commutes up to\n$(-1)^{|\\alpha|d}$. The commutativity of this diagram immediately\nimplies that $\\iota^*(\\alpha)\\cap\\phi_!(a\\times b)\n=(-1)^{|\\alpha|d}\\phi_!\\bigl(\\alpha\\cap(a\\times b)\\bigr)$.\n\nFor the last identity, we apply $\\phi_*$ on both sides and\ncompare. Since $a\\cdot b=\\phi^*(D(a)\\times D(b))\\cap[M]$, we have\n\\begin{align*} \n\\phi_*(a\\cdot b)&=\\bigl(D(a)\\times D(b)\\bigr)\\cap \\phi_*([M]) \\\\\n&=\\bigl(D(a)\\times D(b)\\bigr)\\cap \\bigl(u\\cap[M]\\bigr) \\\\\n&=(-1)^{d(|a|'-d)}u\\cap(a\\times b)=(-1)^{d(|a|'-d)}\\phi_*\\phi_!(a\\times b).\n\\end{align*} \nSince $\\phi_*$ is injective, we have $a\\cdot\nb=(-1)^{d(|a|'-d)}\\phi_!(a\\times b)$.\n\\end{proof} \n\nThese two intersection products, one defined using the Poincar\\'e\nduality, and the other using Pontrjagin Thom construction, differ only\nin signs. However, the formulas for graded commutativity take\ndifferent forms.\n\\begin{align}\na\\cdot b&=(-1)^{(d-|a|')(d-|b|')}b\\cdot a \\\\\n\\phi_!(a\\times b)&=(-1)^{|a|'|b|'+d}\\phi_!(b\\times a)\n\\end{align}\nThe sign $(-1)^d$ in the second formula above comes from the fact that\nthe Thom class $u\\in H^d(M\\times M)$ satisfies $T^*(u)=(-1)^du$, where\n$T$ is the switching map of factors.\n\nNext we turn to the loop product in $H_*(LM)$. We consider the\nfollowing diagram.\n\\begin{equation}\n\\begin{CD}\nLM\\times LM @<{j}<< LM\\times_M LM  @>{\\iota}>> LM  \\\\\n@V{p\\times p}VV    @V{q}VV        @.  \\\\\nM\\times M @<{\\phi}<<  M @.  \n\\end{CD}\n\\end{equation}\nwhere $LM\\times_M LM=(p\\times p)^{-1}(\\phi(M))$ consists of pairs of\nloops $(\\gamma,\\eta)$ with the same base point, and\n$\\iota(\\gamma,\\eta)=\\gamma\\cdot\\eta$ is the product of composable\nloops. Let $\\widetilde{N}=(p\\times p)^{-1}(N)$ and let $\\tilde{c}:\nLM\\times LM \\longrightarrow \\widetilde{N}\/\\partial\\widetilde{N}$ be the Thom collapse map. Let $\\tilde{\\pi}: \\widetilde{N} \\longrightarrow\nLM\\times_MLM$ be a projection map defined as follows. For\n$(\\gamma,\\eta)\\in\\widetilde{N}$, let their base points be $(x,y)\\in\nN$. Let $\\pi(x,y)=(z,z)\\in \\phi(M)$. Since $N\\cong D(\\nu)$ \nhas a bundle structure,\nlet $\\ell(t)=(\\ell_1(t),\\ell_2(t))$ be the straight ray in the fiber\nover $(z,z)$ from $(z,z)$ to $(x,y)$. Then let\n$\\tilde{\\pi}\\bigl((\\gamma,\\eta)\\bigr)=(\\ell_1\\cdot\\gamma\\cdot\\ell_1^{-1},\n\\ell_2\\cdot\\eta\\cdot\\ell_2^{-1})$. By considering $\\ell_{[t,1]}$, we\nsee that $\\tilde{\\pi}$ is a deformation retraction. \n\nIn fact, more is true. Stacey (\\cite{St}, Proposition 5.3) showed that when $L_{\\text{smooth}}M$ is the space of {\\it smooth} loops, $\\widetilde{N}$ has an actual structure of a tubular neighborhood of $LM\\times_MLM$ inside of $LM\\times LM$ equipped with a diffeomorphism  $p^*\\bigl(D(\\nu)\\bigr)\\cong \\widetilde{N}$. His proof only uses the smoothness of $M$ and exactly the same proof applies to the space $LM$ \nof {\\it continuous} loops and $\\widetilde{N}$ still has the structure of a tubular neighborhood and we again have a diffeomorphism $p^*\\bigl(D(\\nu)\\bigr)\\cong \\widetilde{N}$ between spaces of continuous loops. But we do not need this much here. \n\nLet $\\tilde{u}'=(p\\times p)^*(u')\\in\n\\tilde{H}^d(\\widetilde{N}\/\\partial\\widetilde{N})$, and $\\tilde{u}=(p\\times p)^*(u)\\in H^d(LM\\times LM)$ be pull-backs of Thom classes. Define the\ntransfer map $j_!$ by the following composition of maps.\n\\begin{equation} \nj_!: H_*(LM\\times LM) \\xrightarrow{\\tilde{c}_*} \n\\tilde{H}_*(\\widetilde{N}\/\\partial\\widetilde{N})\n\\xrightarrow[\\cong]{\\tilde{u}'\\cap(\\ )}  H_{*-d}(\\widetilde{N}) \n\\xrightarrow[\\cong]{\\tilde{\\pi}_*}\nH_{*-d}(LM\\times_MLM).\n\\end{equation}\nThe tubular neighborhood structure of $\\widetilde{N}$ implies that the middle map is a genuine Thom isomorphism. \n\n\\begin{definition}\\label{definition of loop product} \nLet $M$ be a closed oriented $d$-manifold. For $a,b\\in \\mathbb H_*(LM)$, their loop product, denoted by $a\\cdot b$, is defined by\n\\begin{equation}\\label{loop product} \na\\cdot b=(-1)^{d(|a|'-d)}\\iota_*j_!(a\\times b)\n=(-1)^{d|a|}\\iota_*j_!(a\\times b). \n\\end{equation}\n\\end{definition} \n\nThe sign $(-1)^{d(|a|'-d)}$ appears in \\cite{CJY} in the commutative\ndiagram (1-7). We include this sign explicitly in the definition of\nthe loop product for at least three reasons. The most trivial reason\nis that on the left hand side, the dot representing the loop product\nis between $a$ and $b$. On the right hand side, $j_!$ of degree $-d$\nrepresenting the loop product is in front of $a$. Switching $a$ and\n$j_!$ gives the sign $(-1)^{d|a|'}$. The other part of the sign\n$(-1)^d$ comes from our choice of orientation of $\\nu$ and ensures\nthat $s_*([M])\\in \\mathbb H_0(LM)$, with the $+$ sign, is the unit of the loop\nproduct. We quickly verify the correctness of the sign.\n\\begin{lemma} \nThe element $s_*([M])\\in\\mathbb H_0(LM)$ is the unit of the loop\nproduct. Namely for any $a\\in \\mathbb H_*(LM)$,\n\\begin{equation}\ns_*([M])\\cdot a=a\\cdot s_*([M])=a.\n\\end{equation}\n\\end{lemma}\n\\begin{proof} We consider the following diagram. \n\\begin{equation*}\n\\begin{CD}\n@. LM @= LM  \\\\\n@. @A{\\pi_2}AA @| @. \\\\\nM\\times M @<{1\\times p}<< M\\times LM  @<{j'}<< M\\times_M LM @= LM  \\\\\n@| @V{s\\times 1}VV @V{s\\times_M1}VV @|  \\\\\nM\\times M  @<{p\\times p}<< LM\\times LM  @<{j}<< LM\\times_M LM @>{\\iota}>> LM \n\\end{CD}\n\\end{equation*} \nIn the induced homology diagram with transfers $j_!$ and $j'_!$, the\nbottom middle square commutes because transfers are defined using Thom\nclasses pulled back from the same Thom class $u$ of the base\nmanifold. Thus,\n\\begin{equation*}\ns_*([M])\\cdot a=\\iota_*j_!(s\\times 1)_*([M]\\times a)=j'_!([M]\\times a). \n\\end{equation*}\nHere, since $[M]$ has degree $d$, the sign in \\eqref{loop product} is $+1$. Since $\\pi_2\\circ j'=1$, the identity on $LM$,\n\\begin{equation*}\nj'_!([M]\\times a)={\\pi_2}_*j'_*j'_!([M]\\times a)\n={\\pi_2}_*\\bigl((1\\times p)^*(u)\\cap([M]\\times a)\\bigr).\n\\end{equation*}\nDue to the way $\\nu$ is oriented, the Thom class $u$ is of the form\n$u=\\{M\\}\\times 1+\\dotsb+(-1)^d1\\times\\{M\\}$. Hence\n${\\pi_2}_*\\bigl((1\\times p)^*(u)\\cap([M]\\times\na)\\bigr)={\\pi_2}_*([x_0]\\times a+\\dotsm)=a$.\n\nThe other identity $a\\cdot s_*([M])=a$ can be proved similarly. This\ncompletes the proof.\n\\end{proof} \n\n\nThe second reason is that this choice of sign for the loop product is\nthe same sign appearing in the formula for the intersection product\ndefined in terms of the transfer map \\eqref{intersection\nproduct}. This makes the loop product compatible with the intersection\nproduct in the following sense.\n\n\\begin{proposition}\nBoth of the following maps are algebra maps preserving units between\nthe loop algebra $\\mathbb H_*(LM)$ and the intersection ring $\\mathbb H_*(M)$.\n\\begin{equation}\np_*: \\mathbb H_*(LM) \\longrightarrow \\mathbb H_*(M),\\qquad \ns_*: \\mathbb H_*(M) \\longrightarrow\n\\mathbb H_*(LM).\n\\end{equation}\n\\end{proposition} \n\\begin{proof} The proof is more or less straightforward, \nbut we discuss it briefly. We consider the following diagram.\n\\begin{equation*}\n\\begin{CD}\nLM\\times LM @<{j}<< LM\\times_M LM @>{\\iota}>> LM \\\\ @V{p\\times p}VV\n@V{p}VV @V{p}VV \\\\ M\\times M @<{\\phi}<< M @= M \\\\ @V{s\\times s}VV\n@V{s}VV @V{s}VV \\\\ LM\\times LM @<{j}<< LM\\times_M LM @>{\\iota}>> LM\n\\end{CD}\\end{equation*}\nSince the Thom classes for embeddings $j$ and $\\phi$ are compatible\nvia $(p\\times p)^*$, the induced homology diagram with transfers\n$j_!$ and $\\phi_!$ is commutative. Then by diagram chasing, we can\neasily check that $p_*$ and $s_*$ preserve products because of the\nsame signs appearing in \\eqref{intersection product} and \n\\eqref{loop product}. \n\\end{proof} \n\nThe third reason of the sign for the loop product is that it gives the\ncorrect graded commutativity, as given in \\cite{CS} proved in terms of chains. We discuss a homotopy theoretic proof of graded commutativity because \\cite{CJ} did not include it, and because the homotopy theoretic proof itself is not so trivial with careful treatment of transfers and signs. Contrast the present homotopy theoretic proof with the simple geometric proof given in \\cite{CS}. \n\n\\begin{proposition} \nFor $a,b\\in \\mathbb H_*(LM)$, the following graded commutativity relation holds\\textup{:}\n\\begin{equation} \na\\cdot b=(-1)^{(|a|'-d)(|b|'-d)}b\\cdot a=(-1)^{|a||b|}b\\cdot a. \n\\end{equation}\n\\end{proposition} \n\\begin{proof}\nWe consider the following commutative diagram, where $R_{\\frac12}$ is\nthe rotation of loops by $\\frac12$, that is,\n$R_{\\frac12}(\\gamma)(t)=\\gamma(t+\\frac12)$.\n\\begin{equation*}\n\\begin{CD}\nLM\\times LM @<{j}<< LM\\times_M LM @>{\\iota}>> LM \\\\\n@V{T}VV    @V{T}VV  @V{R_{\\frac12}}VV   \\\\\nLM\\times LM @<{j}<< LM\\times_M LM @>{\\iota}>> LM\n\\end{CD}\n\\end{equation*}\nSince $R_{\\frac12}$ is homotopic to the identity map, we have\n${R_{\\frac12}}_*=1$. Hence\n\\begin{equation*}\na\\cdot b=(-1)^{d(|a|'-d)}\\iota_*j_!(a\\times b)\n=(-1)^{d(|a|'-d)}\\iota_*T_*j_!(a\\times b).\n\\end{equation*} \nNext we show that the induced homology square with transfer $j_!$, we\nhave $T_*j_!=(-1)^dj_!T_*$. Since the left square in the above diagram\ncommutes on space level, we have that $T_*j_!$ and $j_!T_*$ coincides\nup to a sign. To determine this sign, we compose $j_*$ on the left of\nthese maps and compare. Since the homology square with induced\nhomology maps commute,\n\\begin{equation*}\nj_*T_*j_!(a\\times b)=T_*j_*j_!(a\\times\nb)=T_*\\bigl(\\tilde{u}\\cap(a\\times b)\\bigr).\n\\end{equation*}\nOn the other hand, \n\\begin{equation*}\nj_*j_!T_*(a\\times b)=\\tilde{u}\\cap T_*(a\\times b)\n=T_*\\bigl(T^*(\\tilde{u})\\cap(a\\times b)\\bigr). \n\\end{equation*}\nWe compare $T^*(\\tilde{u})$ and $\\tilde{u}$. Since $\\tilde{u}=(p\\times\np)^*(u)$, we have $T^*(\\tilde{u})=(p\\times p)^*T^*(u)$. Since $u$ is\ncharacterized by the property $u\\cap[M\\times M]=\\phi_*([M])$ and\n$T\\circ \\phi=\\phi$, we have\n\\begin{equation*}\n\\phi_*([M])=T_*\\phi_*([M])=T^*(u)\\cap T_*([M\\times\nM])=T^*(u)\\cap(-1)^d[M\\times M].\n\\end{equation*}\nThus $T^*(u)=(-1)^du$. Hence $T^*(\\tilde{u})=(-1)^d\\tilde{u}$. In view\nof the above two identities, this implies that\n$j_*T_*j_!=(-1)^dj_*j_!T_*$, or $T_*j_!=(-1)^dj_!T_*$. \n\nContinuing our computation,\n\\begin{equation*} \na\\cdot b=(-1)^{d|a|'}\\iota_*j_!T_*(a\\times b)\n=(-1)^{|a|'|b|'+d|\\alpha|}\\iota_*j_!(b\\times a)=(-1)^{(|a|'-d)(|b|'-d)}b\\cdot a.\n\\end{equation*}\nThis completes the homotopy theoretic proof of commutativity formula. \n\\end{proof}\n\n\\begin{remark}\nIf we let $\\mu=\\iota_*j_!:\\mathbb H_*(LM)\\otimes \\mathbb H_*(LM) \\longrightarrow \\mathbb H_*(LM)$, then using the method in \\cite{T2}, we can show that the associativity of $\\mu$ takes the form $\\mu\\circ(1\\otimes \\mu)=(-1)^d\\mu\\circ(\\mu\\otimes 1)$. With our choice of the sign for the loop product in Definition \\ref{definition of loop product}, we can get rid of the above sign and we have a usual associativity relation $(a\\cdot b)\\cdot c=a\\cdot(b\\cdot c)$ for the loop product without any signs for $a,b,c\\in\\mathbb H_*(LM)$. This is yet another reason of our choice of the sign in the definition of the loop product. \n\\end{remark}\n\n\nThe transfer map $j_!$ enjoys the following properties similar to\nthose satisfies by $\\phi_!$ as given in Proposition~\\ref{properties of\nphi}. The proof is similar, and we omit it.\n\n\\begin{proposition}\nFor $a,b\\in \\mathbb H_*(LM)$ and $\\alpha\\in H^*(LM\\times LM)$, the following\nformulas are valid.\n\\begin{align}\nj_*j_!(a\\times b)&=\\tilde{u}\\cap(a\\times b) \\label{j_!1}\\\\\nj_!\\bigl(\\alpha\\cap(a\\times b)\\bigr)&=\n(-1)^{d|\\alpha|}j^*(\\alpha)\\cap j_!(b\\times c) \\label{j_!2}\n\\end{align}\n\\end{proposition}\n\nThe second formula says that $j_!$ is a $H^*(LM\\times LM)$-module map. \n\n\n\n\n\n\n\n\n\n\n\\section{Cap products and extended BV algebra structure}\n\n\n\n\n\n\nWe examine compatibility of the cap product with the various\nstructures in the BV-algebra $\\mathbb{H}_*(LM)=H_{*+d}(LM)$.\n\nWe recall that a BV-algebra $A_*$ is an associative graded commutative\nalgebra equipped with a degree $1$ Lie bracket $\\{\\ ,\\ \\}$ and a\ndegree $1$ operator $\\Delta$ satisfying the following relations for\n$a,b,c\\in A_*$:\n\\begin{gather}\n\\Delta(a\\cdot b)=(\\Delta a)\\cdot b + (-1)^{|a|}a\\cdot \\Delta b +\n(-1)^{|a|}\\{a,b\\} \n\\tag{BV identity} \\\\ \n\\{a, b\\cdot c\\} =\\{a,b\\}\\cdot c + (-1)^{|b|(|a|+1)}b\\cdot\\{a,c\\} \n\\tag{Poisson identity} \\\\ \na\\cdot b=(-1)^{|a||b|}b\\cdot a,\\qquad \n\\{a,b\\}=-(-1)^{(|a|+1)(|b|+1)}\\{b,a\\}\n\\tag{Commutativity} \\\\\n\\{a,\\{b,c\\}\\}=\\{\\{a,b\\},c\\}+(-1)^{(|a|+1)(|b|+1)}\\{b,\\{a,c\\}\\}\n\\tag{Jacobi identity}\n\\end{gather} \nHere, degrees of elements are given by $\\Delta a\\in A_{|a|+1}, a\\cdot b\\in A_{|a|+|b|}$, and $\\{a,b\\}\\in A_{|a|+|b|+1}$. \nOne way to view these relations is to consider operators $D_a$ and\n$M_a$ acting on $A_*$ for each $a\\in A_*$ given by $D_a(b)=\\{a,b\\}$ and\n$M_a(b)=a\\cdot b$. Let $[x,y]=xy-(-1)^{|x||y|}yx$ be the graded\ncommutator of operators. Then the Poisson identity and the Jacobi\nidentity take the following forms:\n\\begin{equation}\n[D_a,M_b]=M_{\\{a,b\\}},\\qquad [D_a,D_b]=D_{\\{a,b\\}},\n\\end{equation}\nwhere degrees of operators are $|D_a|=|a|+1$ and $|M_b|=|b|$. \n\nOne nice context to understand BV identity is in the context of odd symplectic geometry (\\cite{G}, \\S2), where BV operator $\\Delta$ appears as a mixed second order odd differential operator, and BV identity can be simply understood as Leipnitz rule in differential calculus. This context actually arises in loop homology. In \\cite{T1}, we explicitly computed the BV structure of $\\mathbb H_*(LM)$ for the Lie group $\\text{SU}(n+1)$ and complex Stiefel manifolds. There, the BV operator $\\Delta$ is given by second order mixed odd differential operator as above, and $\\mathbb H_*(LM)$ is interpreted as the space of polynomial functions on the odd symplectic vector space. \n\nThe fact that the loop algebra $\\mathbb{H}_*(LM)$ is a\nBV-algebra was proved in \\cite{CS}. Note that the above BV relations\nare satisfied with respect to $\\mathbb{H}_*$-grading, rather than the\nusual homology grading. \nThe same is true for compatibility relations\nwith cap products. \n\nFirst we discuss cohomological $S^1$ action operator $\\Delta$ on $H^*(LM)$. Let $\\Delta:\nS^1\\times LM \\longrightarrow LM$ be the $S^1$ action map given by\n$\\Delta(t,\\gamma)=\\gamma_t$, where $\\gamma_t(s)=\\gamma(s+t)$ for\n$s,t\\in S^1=\\mathbb{R}\/\\mathbb{Z}$. The degree $-1$ operator\n$\\Delta: H^*(LM) \\longrightarrow H^{*-1}(LM)$ is defined by the\nfollowing formula for $\\alpha\\in H^*(LM)$:\n\\begin{equation}\n\\Delta^*(\\alpha)=1\\times\\alpha + \\{S^1\\}\\times \\Delta\\alpha\n\\end{equation}\nwhere $\\{S^1\\}$ is the fundamental cohomology class of $S^1$. The\nhomological $S^1$ action $\\Delta$ is not a derivation with respect to the loop product and the deviation from being a derivation is given \nby the loop bracket. However, the\ncohomology $S^1$-operator $\\Delta$ is a derivation with respect to the cup product.\n\n\\begin{proposition}\nThe cohomology $S^1$-operator $\\Delta$ satisfies $\\Delta^2=0$, and it acts as a derivation on the cohomology ring $H^*(LM)$. That is, for\n$\\alpha,\\beta\\in H^*(LM)$,\n\\begin{equation}\\label{delta and cup}\n\\Delta(\\alpha\\cup\\beta)=(\\Delta\\alpha)\\cup\\beta +\n(-1)^{|\\alpha|}\\alpha\\cup\\Delta\\beta.\n\\end{equation}\n\\end{proposition}\n\\begin{proof} The property $\\Delta^2=0$ is straightforward \nusing the following diagram\n\\begin{equation*}\n\\begin{CD}\nS^1\\times S^1 \\times LM @>{1\\times\\Delta}>> S^1\\times LM \\\\ @V{\\mu\\times 1}VV\n@V{\\Delta}VV \\\\ S^1\\times LM @>{\\Delta}>> LM\n\\end{CD}\n\\end{equation*}\nComparing both sides of $(1\\times\\Delta)^*\\Delta^*(\\alpha)=(\\mu\\times\n1)^*\\Delta^*(\\alpha)$, we obtain $\\Delta^2(\\alpha)=0$.\n\nFor the derivation property, we consider the following diagram. \n\\begin{equation*}\n\\begin{CD}\nS^1\\times LM @>{\\phi\\times\\phi}>>  (S^1\\times S^1)\\times (LM\\times LM) \n@>{1\\times T\\times 1}>>  (S^1\\times LM)\\times (S^1\\times LM)  \\\\\n@V{\\Delta}VV    @.   @V{\\Delta\\times\\Delta}VV    \\\\\nLM @>{\\phi}>> LM\\times LM @= LM\\times LM \n\\end{CD}\n\\end{equation*} \nOn the one hand, $\\Delta^*\\phi^*(\\alpha\\times\n\\beta)=\\Delta^*(\\alpha\\cup\\beta) =1\\times(\\alpha\\cup\\beta) +\n\\{S^1\\}\\times \\Delta(\\alpha\\cup\\beta)$. On the other hand,\n\\begin{equation*}\n(\\phi\\times\\phi)^*(1\\times T\\times 1)^*(\\Delta\\times\n\\Delta)^*(\\alpha\\times \\beta) =1\\times(\\alpha\\cup\\beta) +\n(-1)^{|\\alpha|}\\{S^1\\}\\times \\bigl(\\alpha\\cup\\Delta\\beta \n+ \\Delta\\alpha\\cup\\beta\\bigr).\n\\end{equation*}\nComparing the above two identities, we obtain the derivation formula.\n\\end{proof} \n\nWe can regard the cohomology ring $H^*(LM)$ together with cohomological $S^1$ action $\\Delta$ as a BV algebra with trivial bracket product. \n\nNow we show that the cap product is compatible with the loop product\nin the BV-algebra $\\mathbb H_*(LM)$. The following theorem describes the behavior of the cap product with those elements in the subalgebra of $H^*(LM)$ generated by $H^*(M)$ and $\\Delta\\bigl(H^*(M)\\bigr)$.\n\n\\begin{theorem} Let $\\alpha\\in H^*(M)$ and $b,c\\in \\mathbb H_*(LM)$. \nThe cap product with $p^*(\\alpha)$ behaves\nassociatively and graded commutatively with respect to the loop\nproduct. Namely\n\\begin{equation}\\label{cap and loop product} \np^*(\\alpha)\\cap(b\\cdot c)=(p^*(\\alpha)\\cap b)\\cdot c\n=(-1)^{|\\alpha||b|}b\\cdot(p^*(\\alpha)\\cap c).\n\\end{equation}\n\nThe cap product with \n$\\Delta\\bigl(p^*(\\alpha)\\bigr)$ is a derivation on the loop product. Namely, \n\\begin{equation}\\label{cap derivation} \n\\Delta\\bigl(p^*(\\alpha)\\bigr)\\cap(b\\cdot c)\n=\\bigl(\\Delta\\bigl(p^*(\\alpha)\\bigr)\\cap b\\bigr)\\cdot c +\n(-1)^{(|\\alpha|-1)|b|}b\\cdot\\bigl(\\Delta\\bigl(p^*(\\alpha)\\bigr)\\cap\nc\\bigr).\n\\end{equation}\n\\end{theorem} \n\\begin{proof}\nFor the first formula, we consider the following diagram, \nwhere $\\pi_i$ is the projection onto the $i$th factor for $i=1,2$.  \n\\begin{equation*}\n\\begin{CD} \nLM @<{\\pi_i}<<  LM\\times LM @<{j}<< LM\\times_M LM  @>{\\iota}>> LM  \\\\\n@V{p}VV  @V{p\\times p}VV  @V{q}VV   @V{p}VV \\\\\nM @<{\\pi_i}<< M\\times M  @<{\\phi}<< M @= M \n\\end{CD}\n\\end{equation*} \nSince $p^*(\\alpha)\\cap(b\\cdot c)\n=(-1)^{d|b|}\\iota_*\\bigl(\\iota^*p^*(\\alpha)\\cap j_!(b\\times c)\\bigr)$, \nwe need to understand $\\iota^*p^*(\\alpha)$. From the above commutative \ndiagram, we have $\\iota^*p^*(\\alpha)=j^*\\pi_i^*p^*(\\alpha)$, which is\nequal to either $j^*(p^*(\\alpha)\\times 1)$ or $j^*(1\\times\np^*(\\alpha))$. In the first case, continuting our computation using \n\\eqref{j_!2}, we have \n\\begin{align*}\np^*(\\alpha)\\cap(b\\cdot c)&=\n(-1)^{d|b|}\\iota_*\\bigl(j^*(p^*(\\alpha)\\times 1)\n\\cap j_!(b\\times c)\\bigr) \\\\\n&=(-1)^{d|b|+d|\\alpha|}\\iota_*j_!\n\\bigl((p^*(\\alpha)\\times 1)\\cap(b\\times c)\\bigr) \\\\\n&=(-1)^{d|b|+d|\\alpha|}\\iota_*j_!\n\\bigl((p^*(\\alpha)\\cap b)\\times c\\bigr)  \\\\\n&=(p^*(\\alpha)\\cap b)\\cdot c.\n\\end{align*}\nSimilarly, using $\\iota^*p^*(\\alpha)\n=j^*\\bigl(1\\times p^*(\\alpha)\\bigr)$, we get the other identity. This\nproves (1). \n\nFor (2), we first note that the element\n$\\Delta\\bigl(p^*(\\alpha)\\bigr)\\cap(b\\cdot c)$ is equal to \n\\begin{equation*}\n\\Delta\\bigl(p^*(\\alpha)\\bigr)\\cap (-1)^{d|b|}\\iota_*j_!(b\\times c)\n=(-1)^{d|b|}\\iota_*\\bigl(\\iota^*\\bigl(\\Delta(p^*(\\alpha))\\bigr)\n\\cap j_!(b\\times c)\\bigr).\n\\end{equation*}\nThus, we need to understand the element\n$\\iota^*\\bigl(\\Delta(p^*(\\alpha))\\bigr)$. We need some notations. Let\n$I=I_1\\cup I_2$, where $I_1=[0,\\frac12]$ and $I_2=[\\frac12,1]$, and\nset $S_i^1=I_i\/\\partial I_i$ for $i=1,2$. Let $r: S^1=I\/\\partial I \n\\longrightarrow I\/\\{0,\\frac12,1\\}=S^1_1\\vee S^1_2$ be an\nidentification map, and let $\\iota_i: S^1_i \\longrightarrow S^1_1\\vee\nS^1_2$ be the inclusion map into the $i$th wedge summand. We examine\nthe following diagram \n\\begin{equation*}\n\\begin{CD}\nS^1\\times(LM\\underset{M}{\\times}LM)  @>{r\\times 1}>> \n(S^1_1\\vee S^1_2)\\times\n(LM\\underset{M}{\\times}LM) @<<< \\{0\\}\\times (LM\\underset{M}{\\times}LM)  \\\\\n@V{1\\times\\iota}VV   @V{e'}VV   @V{\\iota}VV   \\\\\nS^1\\times LM  @>{e}>> M @<{p}<<  LM \n\\end{CD}\n\\end{equation*} \nwhere $e=p\\circ\\Delta$ is the evaluation map for $S^1\\times LM$, and\nthe other evaluation map $e'$ is given by \n\\begin{equation*}\ne'(t,\\gamma,\\eta)=\n\\begin{cases}\n\\gamma(2t)& 0\\le t\\le\\frac12,\\\\\n\\eta(2t-1)& \\frac12\\le t\\le 1.\n\\end{cases}\n\\end{equation*}\nFor $\\alpha\\in H^*(M)$, we let \n\\begin{equation*}\n{e'}^*(\\alpha)=1\\times \\iota^*p^*(\\alpha) \n+\\{s^1_1\\}\\times \\Delta_1(\\alpha) \n+\\{S^1_2\\}\\times \\Delta_2(\\alpha)\n\\end{equation*}\nfor some $\\Delta_i(\\alpha)\\in H^*(LM\\times_M LM)$ for $i=1,2$. The\nfirst term in the right hand side is identified using the right square\nof the above commutative diagram. Since $r^*(\\{S^1_i\\})=\\{S^1\\}$ for\n$i=1,2$,  \n\\begin{equation*}\n(r\\times 1)^*{e'}^*(\\alpha)=1\\times \\iota^*p^*(\\alpha) + \n\\{S^1\\}\\times \\bigl(\\Delta_1(\\alpha)+\\Delta_2(\\alpha)\\bigr). \n\\end{equation*}\nThe commutativity of the left square implies that this must be equal to \n\\begin{equation*}\n(1\\times\\iota)^*\\Delta^*p^*(\\alpha)\n=1\\times \\iota^*p^*(\\alpha)+\\{S^1\\}\\times \n\\iota^*\\Delta\\bigl(p^*(\\alpha)\\bigr). \n\\end{equation*}\nHence we have \n\\begin{equation*}\n\\iota^*\\bigl(\\Delta(p^*(\\alpha))\\bigr)\n=\\Delta_1(\\alpha)+\\Delta_2(\\alpha)\\in H_*(LM\\times_M LM). \n\\end{equation*}\nTo understand elements $\\Delta_i(\\alpha)$, we consider the following\ncommutative diagram, where $\\ell_1(t)=2t$ for $0\\le t\\le\\frac12$ and \n$\\ell_2(t)=2t-1$ for $\\frac12\\le t\\le 1$. \n\\begin{equation*}\n\\begin{CD}\nS^1_i\\times(LM\\times_M LM) @>{\\ell_i\\times j}>> S^1\\times(LM\\times LM)\n@>{1\\times\\pi_i}>> S^1\\times LM \\\\\n@V{\\iota_i\\times 1}VV @. @V{\\Delta}VV \\\\\n(S^1_1\\vee S^1_2)\\times(LM\\times_M LM) @>{e'}>> M  @<{p}<<  LM \n\\end{CD} \n\\end{equation*} \nOn the one hand, $(\\iota_1\\times 1)^*{e'}^*(\\alpha)\n=1\\times\\iota^*p^*(\\alpha)+ \\{S^1_1\\}\\times\\Delta_1(\\alpha)$. On the\nother hand, \n\\begin{equation*}\n(\\ell_1\\times j)^*(1\\times \\pi_1)^*\\Delta^*p^*(\\alpha)=\n1\\times j^*\\bigl(p^*(\\alpha)\\times 1\\bigr)\n+\\{S^1_1\\}\\times j^*\\bigl(\\Delta(p^*(\\alpha))\\times 1\\bigr).\n\\end{equation*}\nBy the commutativity of the diagram, we\nget $\\Delta_1(\\alpha)=j^*\\bigl(\\Delta(p^*(\\alpha))\\times\n1\\bigr)$. Similarly, $i=2$ case implies $\\Delta_2(\\alpha)\n=j^*\\bigl(1\\times\\Delta(p^*(\\alpha))\\bigr)$. Hence we finally obtain \n\\begin{equation*}\n\\iota^*\\bigl(\\Delta(p^*(\\alpha))\\bigr)\n=j^*\\bigl(\\Delta(p^*(\\alpha))\\times 1 + 1\\times\n\\Delta(p^*(\\alpha))\\bigr).\n\\end{equation*}\nWith this identification of $\\iota^*\\bigl(\\Delta(p^*(\\alpha))\\bigr)$\nas $j^*$ of some other element, we can continue our initial\ncomputation. \n\\begin{equation*}\n\\begin{split}\n\\Delta\\bigl(&p^*(\\alpha)\\bigr)\\cap(b\\cdot c)\n=(-1)^{d|b|}\\iota_*\\bigl(j^*\\bigl(\\Delta(p^*(\\alpha))\\times 1 + \n1\\times \\Delta\\bigl(p^*(\\alpha)\\bigr)\\bigr)\\cap j_!(b\\times c)\\bigr)\n\\\\\n&=(-1)^{d|b|+(|\\alpha|-1)d} \n\\iota_*j_!\\Bigl(\\bigl(\\Delta(p^*(\\alpha))\\times 1 \n+1\\times \\Delta\\bigl(p^*(\\alpha)\\bigr)\\bigr)\\cap(b\\times c)\\Bigr) \\\\\n&=(-1)^{d(|\\alpha|+|b|-1)}\\iota_*j_!\\Bigl(\n\\bigl(\\Delta(p^*(\\alpha))\\cap b\\bigr)\\times c +\n(-1)^{(|b|+d)(|\\alpha|-1)}b\\times \n\\bigl(\\Delta(p^*(\\alpha))\\cap c\\bigr)\\Bigr) \\\\\n&=\\bigl(\\Delta(p^*(\\alpha))\\cap b\\bigr)\\cdot c +\n(-1)^{(|\\alpha|-1)|b|}b\\cdot\\bigl(\\Delta(p^*(\\alpha))\\cap c\\bigr).\n\\end{split}\n\\end{equation*}\nThis completes the proof of the derivation property of the cap product\nwith respect to the loop product. \n\\end{proof} \n\nNext we describe the relation between the cap product and the BV\noperator in homology and cohomology. \n\n\\begin{proposition} For $\\alpha\\in H^*(LM)$ and $b\\in \\mathbb H_*(LM)$, the BV-operator $\\Delta$ satisfies \n\\begin{equation}\\label{delta and cap}\n\\Delta(\\alpha\\cap b)=(\\Delta\\alpha)\\cap b+(-1)^{|\\alpha|}\\alpha\\cap\n\\Delta b.\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nOn the one hand, the $S^1$-action map $\\Delta: S^1\\times LM\n\\longrightarrow LM$ satisfies \n\\begin{equation*}\n\\Delta_*\\bigl(\\Delta^*(\\alpha)\\cap([S^1]\\times b)\\bigr)\n=\\alpha\\cap\\Delta_*([S^1]\\times b)\n=\\alpha\\cap\\Delta b.\n\\end{equation*}\nOn the other hand, since $\\Delta^*(\\alpha)=1\\times\\alpha +\n\\{S^1\\}\\times \\Delta\\alpha$, we have \n\\begin{equation*}\n\\begin{split}\n\\Delta_*\\bigl(\\Delta^*(\\alpha)\\cap([S^1]\\times b)\\bigr)\n&=\\Delta_*\\bigl(\n(-1)^{|\\alpha|}[S^1]\\times(\\alpha\\cap b) +\n(-1)^{|\\alpha|-1}[pt]\\times(\\Delta\\alpha\\cap b)\\bigr)  \\\\\n&=(-1)^{|\\alpha|}\\Delta(\\alpha\\cap b) + \n(-1)^{|\\alpha|-1}\\Delta\\alpha\\cap b. \n\\end{split}\n\\end{equation*}\nComparing the above two formulas, we obtain \n$\\Delta(\\alpha\\cap b)=\\Delta\\alpha\\cap b + \n(-1)^{|\\alpha|}\\alpha\\cap\\Delta b$. \n\\end{proof} \nSince homology BV operator $\\Delta$ on $\\mathbb H_*(LM)$ acts trivially on $\\mathbb H_*(M)$, the\nfollowing corollary is immediate. \n\\begin{corollary}\nFor $\\alpha\\in H^*(M)$, the cap product of $\\Delta\\alpha$ with\n$\\mathbb H_*(M)\\subset \\mathbb H_*(LM)$ is trivial.\n\\end{corollary} \n\\begin{proof} For $b\\in\\mathbb H_*(M)$, the operator $\\Delta$ acts trivially on both $\\alpha\\cap b$ and $b$. Hence formula \\eqref{delta and cap} implies $(\\Delta\\alpha)\\cap b=0$. \n\\end{proof} \n\nNext, we discuss a behavior of the cap product with respect to the\nloop bracket.\n\n\\begin{theorem}  The cap product with $\\Delta\\bigl(p^*(\\alpha)\\bigr)$ \nis a derivation on the loop bracket. Namely, for $\\alpha\\in H^*(M)$\nand $b,c\\in \\mathbb H_*(LM)$,\n\\begin{equation}\\label{cap-loop bracket}\n\\Delta\\bigl(p^*(\\alpha)\\bigr)\\cap\\{b,c\\}\n=\\{\\Delta\\bigl(p^*(\\alpha)\\bigr)\\cap b, c\\} +\n(-1)^{(|\\alpha|-1)(|b|-1)}\\{b, \\Delta\\bigl(p^*(\\alpha)\\bigr)\\cap\nc\\}.\n\\end{equation}\n\\end{theorem} \n\\begin{proof}  Our proof is computational using previous results. \nWe use the BV identity as the definition of the loop bracket. Thus,\n\\begin{equation*}\n\\{b,c\\}=(-1)^{|b|}\\Delta(b\\cdot c)-(-1)^{|b|}(\\Delta b)\\cdot c \n-b\\cdot \\Delta c.\n\\end{equation*} \nWe compute the right hand side of \\eqref{cap-loop bracket}. \nFor simplicity, we write $\\Delta\\alpha$\nfor $\\Delta\\bigl(p^*(\\alpha)\\bigr)$.  Each term in the right hand side\nof \\eqref{cap-loop bracket} gives\n\\begin{gather*}\n\\!\\!\\!\\!\\!\\!\\!\\{\\Delta\\alpha\\cap b, c\\}\n=(-1)^{|b|-|\\alpha|+1}\\Delta\\bigl((\\Delta\\alpha\\cap b)\\cdot c\\bigr)\n-(-1)^{|b|}(\\Delta\\alpha\\cap \\Delta b)\\cdot c \n-(\\Delta\\alpha\\cap b)\\cdot\\Delta c, \\\\\n\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\{b,\\Delta\\alpha\\cap c\\}\n=(-1)^{|b|}\\Delta\\bigl(b\\cdot(\\Delta\\alpha\\cap c)\\bigr)\n-(-1)^{|b|}\\Delta b\\cdot(\\Delta\\alpha\\cap c) \n-(-1)^{|\\alpha|-1}b\\cdot(\\Delta\\alpha\\cap\\Delta c),\n\\end{gather*}\nHere we used \\eqref{delta and cap} for the second term in the first\nidentity and in the third term in the second identity. Combining these\nformulas, we get\n\\begin{multline*} \n\\{\\Delta\\alpha\\cap b, c\\}+ (-1)^{(|\\alpha|-1)(|b|+1)}\n\\{b,\\Delta\\alpha\\cap c\\}  \\\\\n=\\bigl[ \n(-1)^{|b|-|\\alpha|+1}\\Delta\\bigl((\\Delta\\alpha\\cap b)\\cdot c\\bigr)\n+(-1)^{|b|+(|\\alpha|-1)(|b|+1)}\n\\Delta\\bigl(b\\cdot(\\Delta\\alpha\\cap c)\\bigr)\\bigr]  \\\\\n-\\bigl[(-1)^{|b|}(\\Delta\\alpha\\cap\\Delta b)\\cdot c + \n(-1)^{(|\\alpha|-1)(|b|+1)+|b|}\\Delta b\\cdot(\\Delta\\alpha\\cap c)\\bigr] \\\\\n-\\bigl[(\\Delta\\alpha\\cap b)\\cdot \\Delta c +\n(-1)^{(|\\alpha|-1)(|b|+1)+|\\alpha|-1}b\\cdot(\\Delta\\alpha\\cap \\Delta c)\\bigr].\n\\end{multline*}\nUsing the derivation formula for $\\Delta\\alpha\\cap(\\ )$ with respect\nto the loop product \\eqref{cap derivation}, three pairs of terms above\nbecome\n\\begin{multline*}\n(-1)^{|b|-|\\alpha|+1}\\Delta\\bigl(\\Delta\\alpha\\cap(b\\cdot c)\\bigr)\n-(-1)^{|b|}\\Delta\\alpha\\cap(\\Delta b\\cdot c)\n-\\Delta\\alpha\\cap(b\\cdot\\Delta c) \\\\\n=\\Delta\\alpha\\cap\\bigl[(-1)^{|b|}\\Delta(b\\cdot c)\n-(-1)^{|b|}\\Delta b\\cdot c -b\\cdot\\Delta c\\bigr] = \\Delta\\alpha\\cap\\{b,c\\}.\n\\end{multline*}\nThis completes the proof of the derivation formula for the loop bracket. \n\\end{proof} \n\nRecall that in the BV algebra $\\mathbb H_*(LM)$, for every $a\\in \\mathbb H_*(LM)$ the\noperation $\\{a,\\ \\cdot \\ \\}$ of taking the loop bracket with $a$ is a\nderivation with respect to both the loop product and the loop bracket,\nin view of the Poisson identity and the Jacobi identity. Since we have\nproved that the cap product with $\\Delta p^*(\\alpha)$ for $\\alpha\\in\nH^*(M)$ is a derivation with respect to both the loop product and the\nloop bracket, we wonder if we can extend the BV structure in $\\mathbb H_*(LM)$ to a BV structure in $H^*(M)\\oplus \\mathbb H_*(LM)$. Indeed this is possible by extending the loop product and the loop bracket to elements in $H^*(M)$ as follows.\n\\begin{definition}\nFor $\\alpha,\\beta\\in H^*(M)$ and $b\\in \\mathbb H_*(LM)$, we define their loop product and loop bracket by\n\\begin{equation}\n\\begin{gathered}\n\\alpha\\cdot b=\\alpha\\cap b,\\qquad \n\\{\\alpha,b\\}=(-1)^{|\\alpha|}(\\Delta\\alpha)\\cap b,\\\\\n\\alpha\\cdot\\beta=\\alpha\\cup\\beta, \\qquad \\{\\alpha,\\beta\\}=0.\n\\end{gathered}\n\\end{equation}\nThis defines an associative graded commutative loop product by \\eqref{cap and loop product}, and a bracket product on $H^*(M)\\oplus\\mathbb H_*(LM)$. \n\\end{definition}\nNote that this loop product on $H^*(M)\\oplus\\mathbb H_*(LM)$ reduces to the ring structure on $H^*(M)\\oplus \\mathbb H_*(M)$ mentioned in the introduction. \n\nWith this definition, Poisson identities and Jacobi identities are\nstill valid in $H^*(M)\\oplus \\mathbb H_*(LM)$.\n\n\\begin{theorem} Let $\\alpha,\\beta\\in H^*(M)$, and let $b,c\\in \\mathbb H_*(LM)$. \n\n\\noindent\\textup{(I)} The following Poisson identities are valid in $H^*(M)\\oplus\\mathbb H_*(LM)$\\textup{:} \n\\begin{align}\n\\{\\alpha,\\beta\\cdot c\\}\n&=\\{\\alpha,\\beta\\}\\cdot c+ \n(-1)^{|\\beta|(|\\alpha|-1)}\\beta\\cdot\\{\\alpha,c\\} \\label{eq1}\\\\\n\\{\\alpha\\beta,c\\}&=\\alpha\\cdot\\{\\beta,c\\}\n+(-1)^{|\\alpha||\\beta|}\\beta\\cdot\\{\\alpha,c\\} \\label{eq2}\\\\\n\\{\\alpha,b\\cdot c\\}&=\\{\\alpha,b\\}\\cdot c\n+(-1)^{(|b|-d)(|\\alpha|-1)}b\\cdot\\{\\alpha,c\\} \\label{eq3}\\\\\n\\{\\alpha\\cdot b,c\\}&=\\alpha\\cdot\\{b,c\\}+\n(-1)^{|\\alpha|(|b|-d)}b\\cdot\\{\\alpha,c\\}. \\label{eq4} \n\\end{align}\n\n\\noindent\\textup{(II)} The following Jacobi identities are valid in $H^*(M)\\oplus\\mathbb H_*(LM)$\\textup{:}\n\\begin{align}\n\\{\\alpha,\\{\\beta,c\\}\\}&=\\{\\{\\alpha,\\beta\\},c\\}+\n(-1)^{(|\\alpha|-1)(|\\beta|-1)}\\{\\beta,\\{\\alpha,c\\}\\} \\label{eq5} \\\\\n\\{\\alpha,\\{b,c\\}\\}&=\\{\\{\\alpha,b\\},c\\}+\n(-1)^{(|\\alpha|-1)(|b|-d+1)} \\{b,\\{\\alpha,c\\}\\}.  \\label{eq6}\n\\end{align}\n\\end{theorem}\n\\begin{proof}  If we unravel definitions, we see that \\eqref{eq1} \nand \\eqref{eq5} are really the same as the graded commutativity of the\ncup product of the following form\n\\begin{align*}\n(\\Delta\\alpha)\\cap(b\\cap c)&=\n(-1)^{|\\beta|(|\\alpha|-1)}\\beta\\cap(\\Delta\\alpha\\cap c), \\\\\n(\\Delta\\alpha)\\cap(\\Delta\\beta\\cap c)&=(-1)^{(|\\alpha|-1)(|\\beta|-1)}\n(\\Delta\\beta)\\cap\\bigl((\\Delta\\alpha)\\cap c\\bigr).\n\\end{align*}\nthe identity \\eqref{eq2} is equivalent to the derivation formula \\eqref{delta and cup} of\nthe cohomology $S^1$ action operator with respect to the cup product.\n\\begin{equation*}\n\\Delta(\\alpha\\cup \\beta)=(\\Delta\\alpha)\\cup \\beta + (-1)^{|\\alpha|}\n\\alpha\\cup (\\Delta \\beta).\n\\end{equation*}\nThe identity \\eqref{eq3} says that $\\Delta\\alpha\\cap (\\ )$ is a\nderivation with respect to the loop product, and the identity\n\\eqref{eq6} says that $\\Delta\\alpha\\cap(\\ )$ is a derivation with\nrespect to the loop bracket. We have already verified both of these\ncases. Thus, what remains to be checked is formula \\eqref{eq4}, which\nsays\n\\begin{equation*}\n\\{\\alpha\\cap b,c\\}=\\alpha\\cap\\{b,c\\}+\n(-1)^{|\\alpha||b|+|\\alpha|}b\\cdot(\\Delta\\alpha\\cap c).\n\\end{equation*}\nUsing the BV identity, the derivation formula \\eqref{delta and cap} \nof the BV operator with\nrespect to the cap product, and properties of $\\alpha\\cap(\\ )$ and\n$\\Delta\\alpha\\cap(\\ )$, we can prove this identity as follows.\n\\begin{multline*}\n(-1)^{|b|-|\\alpha|}\\{\\alpha\\cap b,c\\}\n=\\Delta\\bigl((\\alpha\\cap b)\\cdot c\\bigr)-\\Delta(\\alpha\\cap b)\\cdot c\n-(-1)^{|b|-|\\alpha|}(\\alpha\\cap b)\\cdot\\Delta c \\\\\n=\\Delta\\bigl(\\alpha\\cap(b\\cdot c)\\bigr)\n-(\\Delta\\alpha\\cap b+(-1)^{|\\alpha|}\\alpha\\cap\\Delta b)\\cdot c\n-(-1)^{|b|-|\\alpha|}\\alpha\\cap(b\\cdot\\Delta c) \\\\\n=(\\Delta\\alpha)\\cap(b\\cdot c)\n-(\\Delta\\alpha\\cap b)\\cdot c\n+(-1)^{|\\alpha|}\\alpha\\cap\\Delta(b\\cdot c) \\\\\n-(-1)^{|\\alpha|}\\alpha\\cap(\\Delta b\\cdot c)\n-(-1)^{|b|-|\\alpha|}\\alpha\\cap(b\\cdot\\Delta c) \\\\\n=(-1)^{(|\\alpha|-1)|b|}b\\cdot(\\Delta\\alpha\\cap c)\n+(-1)^{|\\alpha|+|b|}\\alpha\\cap\\{b,c\\}.\n\\end{multline*}\nCanceling some signs, we get the desired formula. This completes the proof. \n\\end{proof}\n\nOther Poisson and Jacobi identities with cohomology elements in the second argument formally follow from above identities by making following definitions for $\\alpha\\in H^*(M)$ and $b\\in\\mathbb H_*(LM)$: \n\\begin{equation*}\nb\\cdot\\alpha=(-1)^{|\\alpha||b|}\\alpha\\cdot b,\\qquad \n\\{b,\\alpha\\}=-(-1)^{(|\\alpha|+1)(|b|+1)}\\{\\alpha,b\\}.\n\\end{equation*}\n\nFor $\\alpha\\in H^*(M)$ we showed that $\\Delta\\alpha\\cap(\\ )$ is a\nderivation for both the loop product and the loop bracket, and\n$\\alpha\\cap(\\ )$ is graded commutative and associative with respect to\nthe loop product.  What is the behavior\nof $\\alpha\\cap(\\ )$ is with respect to the loop bracket? \nFormula \\eqref{eq4} says that $\\alpha\\cap(\\ \\cdot\\ )$ on loop bracket  is not a derivation or graded commutativity: it is a Poisson identity!\n\nPoisson identities and Jacobi identities we have just proved in $A_*=H^*(M)\\oplus\\mathbb H_*(LM)$ show that $A_*$ is a Gerstenhaber algebra. In fact, $A_*$ can be formally turned into a BV algebra by defining a BV operator $\\boldsymbol\\Delta$ on $A_*$ to be trivial on $H^*(M)$ and to be the usual one on $\\mathbb H_*(LM)$ coming from the homological $S^1$ action. \n\n\\begin{corollary}\\label{BV structure on direct sum} \nThe direct sum $A_*=H^*(M)\\oplus\\mathbb H_*(LM)$ has the structure of a BV algebra. \n\\end{corollary}\n\\begin{proof} Since $\\mathbb H_*(LM)$ is a BV algebra and since we have already verified Poisson identities and Jacobi identities in $A_*$, we only have to verify BV identities in $A_*$. For $\\alpha,\\beta\\in H^*(M)$, an identity \n\\begin{equation*}\n\\boldsymbol\\Delta(\\alpha\\cup\\beta)=(\\boldsymbol\\Delta\\alpha)\\cup\\beta\n+(-1)^{|\\alpha|}\\alpha\\cup(\\boldsymbol\\Delta\\beta)\n+(-1)^{|\\alpha|}\\{\\alpha,\\beta\\}\n\\end{equation*}\nis trivially satisfied since all terms are zero by definition of BV operator $\\boldsymbol\\Delta$ and the loop bracket on $H^*(M)\\subset A_*$. \n\nNext, let $\\alpha\\in H^*(M)$ and $b\\in\\mathbb H_*(LM)$. Since the BV operator $\\boldsymbol\\Delta$ on $A_*$ acts trivially on $H^*(M)$, an identity \n\\begin{equation*}\n\\boldsymbol\\Delta(\\alpha\\cap b)=(\\boldsymbol\\Delta\\alpha)\\cap b\n+(-1)^{|\\alpha|}\\alpha\\cap(\\boldsymbol\\Delta b)\n+(-1)^{|\\alpha|}\\{\\alpha,b\\}\n\\end{equation*}\nis really a restatement of the derivative formula of the homology $S^1$ action operator $\\Delta$ on cap product: $\\Delta(\\alpha\\cap b)=(-1)^{|\\alpha|}\\alpha\\cap(\\Delta b)+(\\Delta\\alpha)\\cap b$ in formula \\eqref{delta and cap}. \n\\end{proof}\n\nIn connection with the above Corollary, we can ask whether $H^*(LM)\\oplus \\mathbb H_*(LM)$ has a structure of a BV algebra. Of course, $H^*(LM)$ together with the cohomological $S^1$ action operator $\\Delta$, which is a derivation, is a BV algebra with trivial bracket product. Thus, as a direct sum of BV algebras, $H^*(LM)\\oplus\\mathbb H_*(LM)$ is a BV algebra, although products between $H^*(LM)$ and $\\mathbb H_*(LM)$ are trivial. More meaningful question would be to ask whether the direct sum $H^*(LM)\\oplus\\mathbb H_*(LM)$ has a BV algebra structure extending the one on $A_*$ described in Corollary \\ref{BV structure on direct sum}. If we want to use the cap product as an extension of the loop product, the answer is no. This is because the cap product with an arbitrary element $\\alpha\\in H^*(LM)$ does not behave associatively with respect to the loop product in $\\mathbb H_*(LM)$: if $\\alpha$ is of the form $\\alpha=\\Delta\\beta$ for some $\\beta\\in H^*(M)$, then $\\alpha\\cap(\\ \\cdot\\ )$ acts as a derivation on loop product in $\\mathbb H_*(LM)$ due to \\eqref{cap derivation} and does not satisfy associativity.  \n\n\\begin{remark} In the course of our investigation, we noticed the \nfollowing curious identity, which is in some sense symmetric in three\nvariables, for $\\alpha\\in H^*(M)$ and $b,c\\in\\mathbb H_*(LM)$. \n\\begin{equation}\n\\begin{split}\n\\{\\alpha,b\\cdot c\\}+(-1)^{|b|}\\alpha\\cdot\\{b,c\\}\n&=\\{\\alpha,b\\}\\cdot c+(-1)^{|b|}\\{\\alpha\\cdot b,c\\} \\\\\n&=(-1)^{(|\\alpha|+1)|b|}\\bigl(b\\cdot\\{\\alpha,c\\}\n+(-1)^{|\\alpha|}\\{b,\\alpha\\cdot c\\}\\bigr).\n\\end{split}\n\\end{equation}\nThis identity is easily proved using Poisson identities. But we wonder\nthe meaning of this symmetry.\n\\end{remark} \n\n\n\n\n\n\n\n\n\n\\section{Cap products in terms of BV algebra structure} \n\n\n\n\n\n\nIn the previous section, we showed that the BV algebra structure in $\\mathbb H_*(LM)$ can be extended to the BV algebra structure in $H^*(M)\\oplus \\mathbb H_*(LM)$ by proving Poisson identities and Jacobi identities. This may be a bit surprising. But this turns out to be very natural through Poincar\\'e duality in the following way. For $a\\in \\mathbb H_*(M)$, we denote the element $s_*(a)\\in \\mathbb H_*(LM)$ by $a$, where $s:M\\to LM$ is the inclusion map.  \n\n\\begin{theorem} For $a\\in \\mathbb H_*(M)$, let $\\alpha=D(a)\\in H^*(M)$ be its Poincar\\'e dual. Then for any $b\\in \\mathbb H_*(LM)$, the following identities hold. \n\\begin{equation}\np^*(\\alpha)\\cap b=a\\cdot b,\\qquad \n(-1)^{|\\alpha|}\\Delta \\bigl(p^*(\\alpha)\\bigr)\\cap b=\\{a,b\\}.\n\\end{equation}\n\\end{theorem}\n\\begin{proof} Let $1=s_*([M])\\in \\mathbb H_0(LM)$ be the unit of the loop product. Since $p^*(\\alpha)\\cap b=p^*(\\alpha)\\cap(1\\cdot b)=\n\\bigl(p^*(\\alpha)\\cap1\\bigr)\\cdot b$ by \\eqref{cap and loop product}, and since \n\\begin{equation*}\np^*(\\alpha)\\cap1=p^*(\\alpha)\\cap s_*([M])=s_*\\bigl(s^*p^*(\\alpha)\\cap[M]\\bigr)\n=s_*(\\alpha\\cap[M])=a,\n\\end{equation*}\nwe have $p^*(\\alpha)\\cap b=a\\cdot b$. This proves the first identity. \n\nFor the second identity, in the BV identity\n\\begin{equation*}\n(-1)^{|a|}\\{a,b\\}=\\Delta(a\\cdot b)-(\\Delta a)\\cdot b\n-(-1)^{|a|}a\\cdot\\Delta b,\n\\end{equation*}\nthe first term in the right hand side gives \n\\begin{equation*}\n\\Delta(a\\cdot b)=\\Delta(p^*(\\alpha)\\cap\nb)=\\Delta\\bigl(p^*(\\alpha)\\bigr)\\cap b\n+(-1)^{|\\alpha|}p^*(\\alpha)\\cap\\Delta b\n\\end{equation*}\nin view of the first identity we just proved and the derivation\nproperty of the homological $A^1$ action operator on cap products. Here\n$p^*(\\alpha)\\cap\\Delta b=a\\cdot\\Delta b$. Since $a\\in \\mathbb H_*(M)$ is a\nhomology class of constant loops, we have $\\Delta a=0$. Thus,\n\\begin{equation*}\n(-1)^{|a|}\\{a,b\\}=\\Delta\\bigl(p^*(\\alpha)\\bigr)\\cap b\n+(-1)^{|\\alpha|}a\\cdot\\Delta b-(-1)^{|a|}a\\cdot\\Delta b\n=\\Delta\\bigl(p^*(\\alpha)\\bigr)\\cap b,\n\\end{equation*}\nsince $|\\alpha|=-|a|$. Thus, $\\{a,b\\}=(-1)^{|\\alpha|}\n\\Delta\\bigl(p^*(\\alpha)\\bigr)\\cap b$. This completes the proof. \n\\end{proof}\n\nIn view of this theorem, since $\\mathbb H_*(LM)$ is already a BV algebra, the\nPoisson identities and Jacobi identities we proved in section 4 may\nseem obvious. However, what we did in section 4 is that we gave a\n\\emph{new and elementary homotopy theoretic proof} of Poisson\nidentities and Jacobi identities using only basic properties of the\ncap product and the BV identity, when at least one of the elements is\nfrom $\\mathbb H_*(M)$.\n\nThe above theorem shows that loop products and loop brackets with elements in $\\mathbb H_*(M)$ can be written as cap products with cohomology elements in $LM$. Thus, compositions of loop products and loop brackets with elements in $\\mathbb H_*(M)$ corresponds to a cap product with the product of corresponding cohomology classes in $H^*(LM)$. Namely,\n\\begin{corollary}\nLet $a_0, a_1, \\dotsc, a_r\\in \\mathbb H_*(M)$, and let\n$\\alpha_0,\\alpha_1,\\dotsc\\alpha_r\\in H^*(M)$ be their Poincar\\'e\nduals. Then for $b\\in \\mathbb H_*(LM)$,\n\\begin{equation}\na_0\\cdot\\{a_1,\\{a_2,\\dotsc\\{a_r,b\\}\\dotsb\\}\\}\n=(-1)^{|a_1|+\\dotsb+|a_r|}\n\\bigl[\\alpha_0(\\Delta\\alpha_1)(\\Delta\\alpha_2)\n\\dotsm(\\Delta\\alpha_r)\\bigr]\\cap b.\n\\end{equation}\n\\end{corollary}\n\nIn section 2, we considered a problem of intersections of loops with\nsubmanifolds in certain configurations, and we saw that the homology\nclass of the intersections of interest can be given by a cap product\nwith cohomology cup products of the above form (Proposition \\ref{loop\nintersection}). The above corollary computes this homology class in\nterms of BV structure in $\\mathbb H_*(LM)$ using the homology classes of these\nsubmanifolds.\n\n\\begin{remark} \\label{exterior algebra}\nIn general, elements $\\alpha,\\Delta\\alpha$ for $\\alpha\\in H^*(M)$ do not generate the entire cohomology ring $H^*(LM)$. However, if $H^*(M;\\mathbb{Q})=\\Lambda_{\\mathbb{Q}}(\\alpha_1,\\alpha_2,\\dotsc\\alpha_r)$ is an exterior algebra, over $\\mathbb{Q}$, then using minimal models or spectral sequences, we have\n\\begin{equation}\nH^*(LM;\\mathbb{Q})=\\Lambda_{\\mathbb{Q}}(\\alpha_1,\\alpha_2,\\dotsc\\alpha_r)\\otimes\n\\mathbb{Q}[\\Delta\\alpha_1,\\Delta\\alpha_2,\\dotsc\\Delta\\alpha_r],\n\\end{equation}\nand thus we have the complete description of the cap products with any elements in $H^*(LM;\\mathbb{Q})$ in terms of the BV structure in $\\mathbb H_*(LM;\\mathbb{Q})$. \n\\end{remark}\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nNASA's \\textit{Kepler} mission has been continuously monitoring more than\n150\\,000 stars for the past 4 years, searching for transiting exoplanets\n\\citep{Borucki-Kepler}.\nThe unprecedented quality of the photometric light curves delivered by\n\\textit{Kepler} makes them also very well suited to study stellar variability in\ngeneral. Automated light curve classification techniques with the goal to\nrecognize \nand identify the many variable stars hidden in the \\textit{Kepler} database have\nbeen developed. The application of these methods to the\npublic \\textit{Kepler} Q1 data is described in \\cite{Debosscher:2011}. There,\nthe authors paid special attention to the detection of pulsating stars \nin eclipsing binary systems. These systems are relatively rare, and especially\ninteresting for asteroseismic studies \\citep[see\ne.g.][]{Maceroni:2009,Welsh:2011}. By modelling the orbital dynamics of the\nbinary, using photometric time series complemented with spectroscopic follow-up\nobservations, we can\nobtain accurate constraints on the masses and radii of the pulsating stars.\nThese\nconstraints are needed for asteroseismic modelling, and are difficult to obtain\notherwise. \n\nNumerous candidate pulsating binaries were identified in the\n\\textit{Kepler} data, and spectroscopic follow-up is ongoing. In this work, we\npresent the results obtained for KIC 11285625 (BD+48 2812), which turns out to be an\neclipsing binary system containing a $\\gamma$ Dor pulsator. The KIC (\\textit{Kepler} Input Catalog) lists the following properties for this target: $V = 10.143$ \\,mag,  $T_{\\rm eff}$ = 6882 K, $\\log g$ = 3.753,\n$R$ = 2.61 $\\unit{R_{\\sun}}$ and $[Fe\/H]$ = -0.127.\nSpectroscopic follow-up revealed it to be a double-lined binary (Section \\ref{rv}). Currently, only a few $\\gamma$ Dor pulsators in double-lined spectroscopic binaries are known, making their analysis very relevant for asteroseismology. \\cite{Maceroni:2013} studied a $\\gamma$ Dor pulsators in an eccentric binary system, observed by CoRoT. Here, we are dealing with a non-eccentric system with a longer orbital period. The longer time span and the higher photometric precision of the \\textit{Kepler} observations (almost a factor 6) allowed us to study the pulsation spectrum with significantly increased frequency resolution and down to lower amplitudes.\n\nA combined\nanalysis of the \\textit{Kepler} light curve and spectroscopic radial velocities\nallowed us to obtain a good binary model for KIC 11285625, resulting in accurate\nestimates of the masses and radii of both components (Section \\ref{binmod}). \nThis binary model was also\nused to disentangle the pulsations from the orbital variability in the\n\\textit{Kepler} light curve in an iterative way. In this paper, we describe\nprocedures to\nperform this task in an automated way. The low signal-to-noise composite\nspectra used for the determination of the radial velocities were used to obtain\nhigher signal-to-noise mean spectra of the components, by means of\nspectral disentangling. These spectra were then used to obtain fundamental\nparameters of the stars (Section \\ref{disentangling}). \nThe resulting pulsation signal of the primary is analysed in detail in Section \\ref{puls-spec}.\nThere we discuss the global characteristics of the frequency spectrum, we list\nthe dominant frequencies and their amplitudes detected by means of prewhitening,\nand search for signs of rotational splitting.  \n  \n\n\\section{\\textit{Kepler} data}\n\nKIC 11285625 has been almost continuously observed by \\textit{Kepler}; data are\navailable for observing quarters Q0-Q10. We only used long cadence data in this work (with a time resolution of 29.4 minutes), since short cadence data is only available for three quarters and is not needed for our purposes. During quarter Q4, one of the CCD modules failed, the reason why part of the Q4 data are missing. No Q8 data could be observed either, since\nthe target was positioned on the same broken CCD module during that quarter.\nThe \\textit{Kepler} spacecraft needs to make rolls every 3 months (for\ncontinuous illumination of its solar arrays), causing targets to fall on\ndifferent CCD modules depending on the observing quarter. \nGiven the different nature of the CCDs, and the different aperture masks used,\nthis caused some issues with the data reduction. The average flux level of the\nlight curve for KIC 11285625 varies significantly \nbetween quarters, and for some, instrumental trends are visible. The top\npanel of Fig. \\ref{LC-all-quarters} plots all the observed\ndatasets, showing the quarter-to-quarter\nvariations. Merging the quarters correctly is not\ntrivial, since the trends have to be removed for each quarter separately, and \nthe data have to be shifted so that all quarters are at the same average level\n(see below).\nOften, polynomials are used to remove the trends, but it is difficult to\ndetermine a reasonable order for the polynomial. This is especially the case\nwhen large amplitude variability, at time scales comparable to the total time\nspan of the data, is present in the light curve.\n\nFortunately, pixel target files are available for all observed quarters for\nKIC 11285625, allowing us to do the light curve extraction based on custom aperture masks. We can define a custom aperture mask, determining\nwhich pixels to include or not. It turned out that the standard aperture mask, used by\nthe data reduction pipeline, was not optimal for all quarters. \nThis is clearly visible in the top panel of Fig. \\ref{LC-all-quarters} for\nquarters Q3 and Q7. The clear upward trends and smaller variability amplitudes\ncompared to the other quarters are caused by a suboptimal aperture mask. The\ntrends can be explained by a small drift of the star on the CCD, changing the\namount of stellar flux included in the aperture during the quarter.\nChecking the target pixel files for those quarters revealed that pixels with\nsignificant flux contribution were not included in the mask. Fig. \\ref{mask}\nshows the \\textit{Kepler} aperture from the automated pipeline, and a single\ntarget pixel image obtained during quarter Q3. As can be seen, the\n\\textit{Kepler} aperture misses a pixel\nwith significant flux contribution (the lowest blue coloured pixel).   \nAdding this pixel to the aperture mask effectively removed the trend in the\nlight curve and increased the variability amplitude to the level of the other\nquarters.\n\nAn automated method was developed to optimize the aperture mask for each\nquarter, with the goal to maximize the signal-to-noise ratio (S\/N) in the\nFourier amplitude spectrum. \nThe standard mask provided with the target pixel files is used as a starting\npoint. The\nmethod then loops over each pixel in the images outside of the original mask\ndelivered by the \\textit{Kepler} pipeline. For each of those pixels, a new light\ncurve is constructed by adding the flux values of the pixel\nto the summed flux of the pixels within the original \\textit{Kepler} mask. The\namplitude spectrum of the resulting light curve is then computed and the S\/N of\nthe highest peak (in this case, the main pulsation frequency of the star) is\ndetermined.\nIf the addition of the pixel increases the S\/N (by a user specified amount), it\nwill be added to the final new light curve once each pixel has been analysed\nthis way. The method also avoids adding pixels containing significant flux of\nneighbouring contaminating targets, since these will normally decrease\nthe S\/N of the signal coming from the main target. It is also possible to detect\ncontaminating pixels within the original \\textit{Kepler} mask, using exactly the\nsame method, but now by excluding one pixel at a time from the original mask \nand checking the resulting S\/N of the new light curves.\n\nAfter determining a new optimal aperture mask for each quarter, the resulting\nlight curves still showed some small trends and offsets, but they were easily\ncorrected using second order polynomials.  \nSpecial care is needed however when shifting quarters to the same level, since\nthe average value of the light curve might be ill determined, especially\nwhen large amplitude non-sinusoidal variability is present at time scales\nsimilar\nto the duration of an observing quarter.\nIn our case, we want to make the average out-of-eclipse brightness match\nbetween different quarters, and not the global average of the quarters, since\nthe latter is shifted due to the presence of the eclipses. \nTherefore, we cut out the eclipses first and interpolated the data points in the\nresulting gaps using cubic splines. This was done only to determine the\npolynomial coefficients, which were then used to subtract \nthe trends from the original light curves (including the eclipses). We first\ntransformed the fluxes into magnitudes for each quarter separately, prior to\ntrend removal. The binary modelling code we describe further needs magnitudes\nas input, but the conversion to magnitude also\nresolves any potential quarter-to-quarter variability amplitude changes caused, e.g., by a difference in CCD gains (or any other instrumental effect changing the\nflux values in a linear way).        \n\nThe lower panel of Fig. \\ref{LC-all-quarters} shows the resulting light curve\nafter application of our detrending procedure using pixel target files.\n\n\n\n\\begin{figure*}\n \\centering\n  \n\\includegraphics[width=14cm,angle=270,scale=0.7]{kplr11285625-all-quarters.ps}\n\\caption{Combination of observing quarters Q1-Q10 for KIC 11285625. The top\npanel shows the `raw'\nSAP (simple aperture photometry) fluxes, as they were delivered, while the lower\npanel shows the resulting combined dataset after optimal mask selection and detrending.}\n\\label{LC-all-quarters}\n\\end{figure*}\n\n\n\n\\begin{figure*}\n \\centering\n   \\includegraphics[width=14cm,angle=0,scale=1.0]{aperture.ps}\n\\caption{\\textit{Kepler} aperture (left) and a single target pixel image (right)\nfor KIC 11285625, quarter Q3. Red colours in the pixel image indicate the lowest flux levels, white colours indicate the highest flux levels.}\n\\label{mask}\n\\end{figure*}\n\n\n\n\n\\section{Spectroscopic follow-up and RV determination}\n\\label{rv}\nSpectroscopic follow-up observations were obtained with the HERMES Echelle\nspectrograph at the Mercator telescope on La Palma (see \\cite{HERMES} for a detailed\ndescription of the instrument). \nIn total, 63 spectra with good orbital phase coverage were observed, with\nS\/N in \\textit{V} in the range 40-70. These spectra revealed the\ndouble-lined nature of the spectroscopic binary. \n\nRadial velocities were derived with the HERMES reduction pipeline, using the\ncross-correlation technique \nwith an F0 spectral mask. The choice for this mask was based on the F0 spectral\ntype given by SIMBAD and the effective temperature listed in the KIC (Kepler\nInput Catalogue). We also tried additional spectral masks, given the\ndouble-lined\nnature of the binary, but we did not obtain better results in terms of scatter\non the radial velocity points. Due to the numerous metal lines in the spectrum,\nthe cross-correlation technique worked very well, despite the relatively low S\/N\nspectra. The upper panel of  Fig. \\ref{rv-lc} shows the radial velocity measurements obtained for\nboth components of KIC 11285625.\nThe black circles correspond to the primary and the red circles to the\nsecondary. From the scatter on the radial velocity measurements, we conclude\nthat the primary component is pulsating. A Keplerian model was fitted for both\ncomponents (shown also in the upper panel of Fig. \\ref{rv-lc} ), resulting in the orbital parameters listed in Table \\ref{final-par}.\n\nUncertainties were estimated using a Monte-Carlo perturbation approach. \nAlthough the orbital period of the system can be a free parameter in the fitting\nprocedure for the Keplerian model, we fixed it to the much more accurate value obtained from the\n\\textit{Kepler} light curve, given its longer time span (see Section \\ref{binmod}). The quality\nof the fit obtained this way (as judged from the $\\chi^2$ values) is significantly better compared to the case where\nthe orbital period is left as a free parameter.\n\n\n\n\n\\section{Binary model}\n   \n\\label{binmod}\nWe used the combined \\textit{Kepler} Q1-Q10 data to obtain a binary model,\nwhich, combined with the results from the spectroscopic analysis, provided us\nwith accurate estimates of the main astrophysical properties\nof both components. Given that we are dealing with a detached binary with no or only\nlimited distortion of both components, we used JKTEBOP, written by\nJ. Southworth \\citep[see][]{Southworth:JKTEBOP1,Southworth:JKTEBOP2}.\nThis code is based on the EBOP code, originally developed by Paul B. Etzel\n\\citep[see][]{Etzel:EBOP,Popper:EBOP}. JKTEBOP has the advantage of being very\nstable, fast, and it is applicable to large datasets with thousands of\nmeasurements, such as the \\textit{Kepler} light curves. \nMoreover, it can easily be scripted (essential in our approach) and includes\nuseful error analysis options such as Monte Carlo and bootstrapping methods.\\\\\n\nSince we are dealing with a light curve containing both orbital variability\n(eclipses) and pulsations, we had to disentangle both phenomena in order to\nobtain a reliable binary model. \nIn the amplitude spectrum of the \\textit{Kepler} light curve (see Fig.\n\\ref{puls-residuals}), the $\\gamma$ Dor type pulsations have numerous significant peaks in\nthe range 0-0.7 $\\unit{d^{-1}}$, with clear repeating patterns up to around 4 $\\unit{d^{-1}}$. In the same region of the amplitude spectrum, we\nalso find peaks \ncorresponding to the orbital variability: a comb-like pattern of harmonics of\nthe orbital frequency ($f_\\mathrm{orb}$, $2f_\\mathrm{orb}$, $3f_\\mathrm{orb}$,...). Moreover, the\nmain pulsation frequency of 0.567 $\\unit{d^{-1}}$ is close to $6f_\\mathrm{orb}$ (0.556\n$\\unit{d^{-1}}$ ), though the peaks are clearly separated, given an estimated\nfrequency resolution\nof $ 1\/T \\approx0.0012\\, d^{-1}$, with T the total time span \nof the combined \\textit{Kepler} Q1-Q10 data. This near coincidence complicated the disentangling\nof both types of variability in the light curve, unlike the case where the\norbital variability is well separated from the \npulsations in frequency domain (e.g. typical for a $\\delta$ Sct pulsator in a\nlong-period binary). We used an iterative procedure, consisting of an\nalternation\nof binary modelling with JKTEBOP, and prewhitening of \nthe remaining variability (pulsations) after removal of the binary model. This\nprocedure can be done in an automated way, and the number of iterations can be\nchosen. The idea is that we gradually improve both the binary model and the\nresidual pulsation spectrum at the same time. In each step of the procedure, we\nused the entire Q1-Q10 dataset without any rebinning. \nOur iterative method is similar to the one described in \\cite{Maceroni:2013} and\nconsists of the following steps: \n\\begin{enumerate}\n \\item Remove the eclipses from the original \\textit{Kepler} light curve and\ninterpolate the resulting gaps using cubic splines.\n \\item Derive a first estimate of the pulsation spectrum by means of iterative\nprewhitening of the light curve without eclipses.\n \\item Remove the pulsation model derived in the previous step from the original\n\\textit{Kepler} light curve.\n \\item Find the best fitting binary model to the residuals using JKTEBOP.\n \\item Remove this binary model from the original \\textit{Kepler} light curve\n(dividing by the model when working in flux, or subtracting the model when\nworking in magnitudes).\n \\item Perform frequency analysis on the residuals, which delivers a new\nestimate of the pulsation spectrum.  \n \\item Subtract the pulsation model obtained in the previous step from the\noriginal \\textit{Kepler} light curve. \n \\item Model the residuals (an improved estimate of the orbital variability)\nwith JKTEBOP and repeat the procedure starting from step five. \n\\end{enumerate}\n\nThe procedure is then stopped when convergence is obtained: the $\\chi^{2}$\nvalue of the binary model no longer decreases significantly.\nIn practice, convergence is obtained after just a few iterations, at least for\nKIC 11285625. The procedure can be run automatically,\nprovided that the user has a good initial guess for the orbital parameters.\nThe computation time is dominated by the prewhitening\nstep, which requires the repeated calculation of amplitude spectra. In our case,\nthe complete procedure required a few hours on a single desktop CPU. The top panel of Fig. \\ref{puls-residuals} shows part of the original \\textit{Kepler} data,\nwith the disentangled pulsation contribution overplotted in red, the lower\npanel shows the corresponding amplitude spectrum. In Fig.\n\\ref{rv-lc}, the Keplerian model fit to the HERMES RV data, the binary model\nfit to the \\textit{Kepler} data (pulsation part removed) and the residual light\ncurve are shown together, phased with the orbital period (using the\nzero-point $T_\\mathrm{0}$ = 2454953.751335 d, corresponding to a time of minimum\nof the primary eclipse). The slightly larger scatter in the residuals at the\ningress\nand egress of the primary eclipse is caused by the fact that the\nocculted surface of the pulsating primary is non-uniform and changing over\ntime, due to the non-radial pulsations.\n\nWe also investigated a different iterative approach, where we started by fitting\na binary model directly to the original \\textit{Kepler} light curve, instead of\nfirst removing the pulsations. This way, we do not remove information from the\nlight curve by cutting the eclipses, and we do not need to interpolate the data.\nAlthough the iterative procedure also converged quickly using this approach, the\nfinal binary model was not accurate. Removing the model from the original\n\\textit{Kepler} light curve introduced systematic offsets during the eclipses.\nThe reason is that the initial binary model obtained from the original light\ncurve is inaccurate due to the large amplitude and non-sinusoidal nature of the\npulsations, causing the mean light level between eclipses to be badly defined. \nThe approach starting from the light curve with the pulsations removed prior to\nbinary modelling provided much better results, as judged from the\n$\\chi^{2}$ values and visual inspection of the \nresiduals during the eclipses.\\\\\n\nA linear limb-darkening law was used for both stars, with coefficients\nobtained from \\cite{Prsa:2011}\\footnote{\\url{http:\/\/astro4.ast.villanova.edu\/aprsa\/?q=node\/8}}. \nThese coefficients have been computed for a grid of {$T_{\\rm eff}$},\n{$\\log g$} and [M\/H] values, taking into account \\textit{Kepler's} transmission,\nCCD quantum efficiency and optics. We estimated them using the KIC\n(Kepler Input Catalogue) parameters for a first\niteration, but later adjusted them using our obtained values for {$T_{\\rm eff}$},\n{$\\log g$} and [M\/H] from the combination of binary modelling and spectroscopic\nanalysis (see Section \\ref{disentangling}). We did not take gravity darkening and refection effects into account, given that both components are well separated, and are not significantly deformed by rapid rotation or binarity (the oblateness values returned by JKTEBOP are very small).\n\n\nDuring the iterative procedure, special care was paid to the following\nissues, since they all influence the quality of the final binary and\npulsation models:\n\n\n\\begin{itemize}\n \\item Removal of the pulsations from the original light curve: here, we first\ndetermined all the significant pulsation frequencies (or, more general:\nfrequencies most likely not caused by the orbital motion) from the light curve\nwith the\neclipses removed. Only frequencies with an amplitude signal-to-noise ratio (S\/N)\nabove four are considered. The noise level in the amplitude spectrum is\ndetermined from the region 20- 24 d$^{-1}$, where no significant peaks are\npresent. The often used procedure of computing the noise level in a region\naround the peak of interest would not provide reliable S\/N estimates in our\ncase, given the high density of significant peaks at low frequencies. \nWe also used false-alarm probabilities as an additional significance test,\nwith very similar results regarding the number of significant frequencies. \n\n\\item Initial parameters of the binary model: when running JKTEBOP, initial\nparameters have to be provided for the binary model. These are then refined\nusing non-linear optimization techniques (e.g. Levenberg-Marquardt). \nAlthough the optimization procedure is stable and converges fast, we have no\nguarantee that the global best solution is obtained. If the initial parameters\nare too far off from their true values, the procedure can end up in a local\nminimum. Therefore, we did some exploratory analysis first, to find a good set\nof initial \nparameters, aided by the constraints obtained from the radial velocity data. The\ninitial parameters were also refined: after completion of the first\niterative procedure, the final parameters were used as initial values for a new\nrun, etc. This also confirmed the stability of the solution, although it does\nnot guarantee that the overall best solution has been found.  \n\\end{itemize}\n\nError analysis of the final binary model was done using the Monte-Carlo method\nimplemented in JKTEBOP. Here, the input light curve (with the pulsations\nremoved) is perturbed by adding Gaussian noise with standard deviation estimated\nfrom the residuals, and the binary model is recomputed. This procedure is\nrepeated typically 10000 times, to obtain confidence intervals for the obtained\nparameter values.\nThe final parameters from the combined spectroscopic and photometric analysis,\nand their estimated uncertainties (1$\\sigma$), are listed in Table \\ref{final-par}.\n\nGiven the long time span and excellent time sampling of the light curve, we also checked for the presence of eclipse time variations (e.g. due to the presence of a third body).\nWe used two different methods to check for deviations of pure periodicity of the eclipse times. The first method consists of computing the amplitude spectrum of each observing quarter \nof the light curve separately, and comparing the peaks caused by the binary signal (the comb of harmonics of the orbital frequency). We could not detect any change in orbital frequency this way.\nMoreover, the orbital peaks in the amplitude spectrum of the entire light curve also do not show any broadening or significant sidelobes (indicative of frequency changes), compared to the amplitude spectrum of the purely periodic light curve of our best binary model computed with JKTEBOP. The second method consists of determining the times of minima for each eclipse individually by fitting a parabola to the bottom of each eclipse and determining the minimum. We then compared those times with the predicted values using our best value of the orbital period (O-C diagram). This was done both for the original light curve and the light curve with pulsations removed. In the first case, we found indications of periodic shifts of the eclipse times, but these are clearly linked to the pulsation signal in the light curve, which also affects the eclipses. No periodic shifts or trends could be found when analysing the light curve with pulsations removed, and we conclude that we do not detect eclipse time variations.\n\n\\begin{table}\n\\tiny\n\\renewcommand{\\tabcolsep}{0.4mm}\n\\center\n\\caption{Orbital and physical parameters for both components of KIC 11285625,\nobtained from the combined spectroscopic and photometric analysis.}\n \\begin{tabular}{lcc}\n \\hline\n &System&\\\\\n \\hline\n Orbital period  $P$ (days)&  10.790492 $\\pm$ 0.000003  &\\\\\n Eccentricity $e$ &$0.005\\pm0.003$ &\\\\\n Longitude of periastron $\\omega$ (\\degr)&90.103 $\\pm$ 0.006&\\\\\n Inclination $i$ (\\degr) & 85.32 $\\pm$ 0.02 &\\\\\n Semi-major axis $a$ ($\\unit{R_{\\sun}}$) &28.8 $\\pm$ 0.1&\\\\\n Light ratio $L_1\/L_2$ &0.38 $\\pm$ 0.01&\\\\\n System RV $\\gamma$ (km $\\unit{s^{-1}}$)&-11.7 $\\pm$ 0.2&\\\\\n $T_\\mathrm{0}$ (days)\\tablefootmark{a}&2454953.751335 $\\pm$ 0.000014 \\\\\n \\hline  \n &Primary&Secondary\\\\\n \\hline\n Mass $M$ ($\\unit{M_{\\sun}}$)&$1.543\\pm0.013$ &$1.200\\pm0.016$\\\\\n Radius $R$ ($\\unit{R_{\\sun}}$) & 2.123 $\\pm$ 0.010& 1.472 $\\pm$ 0.014\\\\\n log $g$ &3.973 $\\pm$ 0.006 & 4.18 $\\pm$ 0.01 \\\\\n\\hline\n\\end{tabular}\n\\tablefoottext{a}{Reference time of minimum of a primary eclipse.}\n\\label{final-par}\n\\end{table}\n\n\n\n\\begin{figure}\n \\centering\n   \\includegraphics[width=14cm,angle=0,scale=0.66]{rv-lc-model-plots-new.ps}\n\\caption{Upper panel: phased HERMES radial velocity data (open circles) and Keplerian model fits (lines) for both components (black: primary, red: secondary). Middle panel: phased \\textit{Kepler} data (pulsations removed) with the binary model overplotted in red.\nLower panel: phased residuals of the \\textit{Kepler} data, after removal of both the pulsations and the binary model. }\n\\label{rv-lc}\n\\end{figure}\n\n\n\\begin{figure*}\n \\centering\n  \n\\includegraphics[width=14cm,angle=270,scale=0.7]{kplr11285625-puls-ampspec-comparison.ps}  \n\\caption{Upper panel: resulting light curve after iterative removal of the\norbital variability (in red), with the original light curve shown for comparison\n(in black). Lower panel: Amplitude spectrum of the original\n\\textit{Kepler} light curve (in black), and the \namplitude spectrum of the pulsation `residuals' (in red) after iterative\nremoval of the orbital variability.}\n\\label{puls-residuals}\n\\end{figure*}\n\n\n\\section{Spectral disentangling}\n\\label{disentangling}\n\nTo derive the fundamental parameters  $T_{\\rm eff}$, $\\log g$, $[M\/H]$, etc. of the $\\gamma$ Dor pulsator from\nthe high\nresolution HERMES spectra, we first needed to separate the contributions\n of both stars in the measured composite spectra. Given the similar spectral types of the components, and the large number of metal lines in the spectra, \nwe could apply the technique of spectral disentangling to accomplish this\n\\citep{Simon:1994,Hadrava:1995}. Here,\nwe used the FDBinary code \\footnote{http:\/\/sail.zpf.fer.hr\/fdbinary\/}\n\\citep{Ilijic:2004} which is based on Hadrava's Fourier approach\n\\citep{Hadrava:1995}. The overall procedures used to determine the orbital\nparameters and reconstruct the spectra of the component stars from time series\nof observed composite spectra of a spectroscopic double-lined eclipsing binary\nhave been described extensively in \\cite{Hensberge:2000} and\n\\cite{Pavlovski:2005}. \n\nThe user has to provide good initial guesses and confidence intervals for the\norbital parameters, otherwise the method might not converge towards the correct\nsolution. Luckily, we had very good\ninitial parameter values available from the Keplerian orbital fit to the radial\nvelocity data, as was described in Section \\ref{rv}. The final orbital\nparameters\nturned out to be in excellent agreement with the initial values derived from\nspectroscopy. \n\nSpectral disentangling methods have the advantage that they enable us to\ndetermine\nthe orbital parameters of the system in an independent way (although good\ninitial estimates are necessary), and that the resulting component spectra have\na higher signal-to-noise ratio than the individual original composite spectra:\nS\/N $\\sim$ $\\sqrt{N}$, with N the number of composite spectra used for\ndisentangling. The increase in S\/N is illustrated in\nFig. \\ref{disentangled-primary}, where a single observed spectrum (corrected for\nDoppler shift) is compared to the disentangled spectrum for the primary\ncomponent. Renormalization of the disentangled spectra was done using\nthe light factors obtained from the binary modelling of the \\textit{Kepler}\nlight\ncurve. \n\n\\begin{figure}\n\n \\centering\n\n\\includegraphics[width=14cm,angle=270,scale=0.5]{disentangled-spectrum.ps}\n\\caption{Comparison of a single observed composite spectrum (black circles) with\nthe disentangled spectrum of the primary component (red lines), illustrating\nthe significant increase in S\/N that is obtained.}\n\\label{disentangled-primary}\n\\end{figure}\n\nFor the spectrum analysis of both components of KIC\\,11285625, we\nuse the GSSP code \\citep[Grid Search in Stellar Parameters,][]{Tkachenko2012} that finds the optimum\nvalues of $T_{\\rm eff}$, $\\log g$, $\\xi$, $[M\/H]$, and $v\\sin{i}$\\ from the minimum\nin $\\chi^2$ obtained from a comparison of the observed spectrum with\nthe synthetic ones computed from all possible combinations of the\nabove mentioned parameters. The errors of measurement (1$\\sigma$\nconfidence level) are calculated from the $\\chi^2$ statistics, using the\nprojections of the hypersurface of the $\\chi^2$ from all grid points\nof all parameters in question. In this way, the estimated error bars include\nany possible model-inherent correlations between the parameters but do not take\ninto account imperfections of the model (such as incorrect atomic data, non-LTE\neffects, etc.) and\/or continuum normalization. A\ndetailed description of the method and its application to the\nspectra of  \\textit{Kepler} $\\beta$\\,Cep and SPB candidate stars as well as\n$\\delta$\\,Sct and $\\gamma$\\,Dor candidate stars are given in\n\\citet{Lehmann2011} and \\citet{Tkachenko2012}, respectively.\n\nFor the calculation of synthetic spectra, we used the LTE-based\ncode SynthV \\citep{Tsymbal1996} which allows the computation of the\nspectra based on individual elemental abundances. The code uses\ncalculated atmosphere models which have been computed with the\nmost recent, parallelised version of the LLmodels program\n\\citep{Shulyak2004}. Both programs make use of the VALD database\n\\citep{Kupka2000} for a selection of atomic spectral lines.\nThe main limitation of the LLmodels code is that the models are\nwell suited for early and intermediate spectral type stars, but\nnot for very hot and cool stars where non-LTE effects or\nabsorption in molecular bands may become relevant, respectively.\n\n\n\\begin{table}\n\\caption{Fundamental parameters of both components of KIC\\,11285625.}\\label{Table: Fundamental parameters}\n\\begin{tabular}{lll}\n\\hline\\hline\n\\multicolumn{1}{c}{Parameter\\rule{0pt}{9pt}} & Primary & \\multicolumn{1}{c}{Secondary}\\\\\n\\hline\n$T_{\\rm eff}$\\,(K)\\rule{0pt}{11pt} &$6960\\pm 100$ & $7195\\pm200$\\\\\n$\\log g$\\,(fixed)\\rule{0pt}{11pt} & 3.97 & 4.18\\\\\n$\\xi$\\,(km\\,s$^{-1}$)\\rule{0pt}{11pt} & $0.95\\pm0.30$ & $0.09\\pm0.25$\\\\\n$v\\sin{i}$\\,(km\\,s$^{-1}$)\\rule{0pt}{11pt} & $14.2\\pm1.5$ & $8.4\\pm1.5$\\\\\n$[M\/H]$\\,(dex)\\rule{0pt}{11pt} & $-0.49\\pm0.15$ & $-0.37\\pm0.3$\\\\\n\\hline\n\\end{tabular}\n\\tablefoot{The temperature of the secondary is not reliable, as discussed in the text.}\n\\end{table}\n\nGiven that KIC\\,11285625 is an eclipsing, double-lined (SB2)\nspectroscopic binary for which unprecedented quality (\\textit{Kepler})\nphotometry is available, the masses and the radii of both components\nwere determined with very high precision. Having those two\nparameters, we evaluated surface gravities of the two stars with\nfar better precision than one would expect from the spectroscopic\nanalysis given that the S\/N of our spectra varies between 40 and 70,\ndepending on the weather conditions on the night when the\nobservations were taken. Thus, we fixed $\\log g$\\ for both\ncomponents to their photometric values (3.97 and 4.18 for the\nprimary and secondary, respectively) and adjusted the effective\ntemperature $T_{\\rm eff}$, micro-turbulent velocity $\\xi$, projected\nrotational velocity $v\\sin{i}$, and overall metallicity [M\/H] for both\nstars based on their disentangled spectra.\nGiven that the contribution of the primary component to the total\nlight of the system is significantly larger than that of the\nsecondary (72\\% compared to 28\\%) and that its decomposed spectrum\nis consequently better defined and is of higher quality than that of\nthe secondary, we were also able to evaluate individual abundances\nfor this star besides the fundamental atmospheric parameters.\nTable~\\ref{Table: Fundamental parameters} lists the fundamental\nparameters of the two stars whereas Table~\\ref {Table: Individual abundances} summarizes the results\nof chemical composition analysis for the primary component. The overall metallicities of the two\nstars agree within the quoted errors, but the derived temperature for the\nsecondary is not reliable, since we find \nit to be about 200~K hotter than the primary. From the relative eclipse depths,\nwe estimate the temperature of the secondary to be about 6400~K. The cause of\nthis temperature discrepancy is the poor quality of the disentangled spectrum of\nthe secondary, given its smaller light contribution. Normalization errors in\nthe spectra can easily translate into temperature errors of several hundred\nKelvin. Note that the listed uncertainties for the spectroscopic temperatures do not take normalization errors into account.\n\n\nFigure~\\ref{Figure: HR diagram} shows the position of the primary component in\nthe $\\log(T_{\\rm eff})$-$\\log g$\\ diagram, with respect to the observational  $\\delta$~Sct\\  (solid\nlines) and $\\gamma$~Dor\\ (dashed lines) instability strips as given by\n\\cite{Rodriguez:2001} and \\cite{Handler:DSCUT}, respectively. The primary falls\ninto the $\\gamma$~Dor\\ instability strip, meaning that pure g-modes are expected to be\nexcited in its interior.\n\n\\begin{table}\n\\tabcolsep 2.2mm\\caption{Atmospheric chemical composition of the\nprimary component of KIC\\,11285625.}\n\\label{Table: Individual abundances}\n\\begin{tabular}{llllll}\n\\hline\\hline\n\\multicolumn{1}{c}{Element\\rule{0pt}{9pt}} & Value & \\multicolumn{1}{c}{Sun} & \\multicolumn{1}{c}{Element\\rule{0pt}{9pt}} & Value & \\multicolumn{1}{c}{Sun}\\\\\n\\hline\nFe\\rule{0pt}{11pt} & --0.58\\,(15) & --4.59 & Mg & --0.65\\,(23) & --4.51\\\\\nTi\\rule{0pt}{11pt} & --0.40\\,(20) & --7.14 & Ni & --0.53\\,(20) & --5.81\\\\\nCa\\rule{0pt}{11pt} & --0.41\\,(27) & --5.73 & Cr & --0.35\\,(27) & --6.40\\\\\nMn\\rule{0pt}{11pt} & --0.50\\,(35) & --6.65 & C & --0.28\\,(40) & --3.65\\\\\nSc\\rule{0pt}{11pt} & --0.31\\,(40) & --8.99 & Si & --0.68\\,(40) & --4.53\\\\\nY\\rule{0pt}{11pt} & --0.22\\,(45) & --9.83 &  &  & \\\\\n\\hline\n\\end{tabular}\n\\tablefoot{All values are in dex and on a\nrelative scale (compared to the Sun). Label ``Sun'' refers to the\nsolar composition given by \\citet{Grevesse2007}. Error bars\n(1-$\\sigma$ level) are given in parentheses in terms of last digits.}\n\\end{table}\n\n\\begin{figure}\n\\includegraphics[scale=0.9,clip=]{K11285625_Teff_logg.eps}\n\\caption{{\\small Location of the primary of KIC\\,11285625 in the\n$\\log(T_{\\rm eff})$-$\\log g$\\ diagram. The $\\gamma$~Dor\\ and the red edge of the $\\delta$~Sct\\\nobservational instability strips\nare represented by the dashed and solid lines, correspondingly. According to\nthe photometric $T_{\\rm eff}$\\ and $\\log g$, the secondary would be located in the lower\nright corner of the diagram, outside both instability strips. }} \\label{Figure:\nHR diagram}\n\\end{figure}\n\n\n\\section{Pulsation spectrum}\n\\label{puls-spec}\n\nFig. \\ref{as-puls-global} shows the amplitude spectrum of the \\textit{Kepler}\nlight curve with the binary model removed, in the region 0-2 d$^{-1}$, where\nmost of the dominant pulsation frequencies are found. Clearly visible\nare the three groups of peaks around 0.557, 1.124 and 1.684 {d$^{-1}$}. Some of the\nfrequencies found in the groups around 1.124 and\n1.684 {d$^{-1}$} are harmonics of frequencies present in the group around\n0.557 {d$^{-1}$}. These harmonics are caused by the\nnon-linear nature of the pulsations, and have been observed for many pulsators\nobserved by CoRoT and  \\textit{Kepler}, along almost the entire main-sequence\n\\citep{Degroote:2009,Poretti:2011,Breger:2011,Balona:2012}, and in $\\gamma$~Dor\\ stars in\nparticular \\citep{Tkachenko:2013}. Closer\ninspection of the\nmain frequency groups revealed that they consist of several closely spaced\npeaks,\nalmost\nequally spaced with a frequency of $\\sim$ 0.010 {d$^{-1}$}. A likely explanation is\namplitude modulation of the pulsation signal on a timescale of $\\sim$ 100\ndays. Mathematically, the effect of amplitude modulation can be\ndescribed as follows: imagine a simplified case where a single periodic\nnon-linear\npulsation signal is being modulated with a general periodic function. We can\nwrite this signal as a product of two sums of sines, where the number of terms\n(harmonics) in each sum depends on how non-linear (non-sinusoidal) the signals\nare: \n\\begin{equation}\n\\sum_{i=1}^{N_\\mathrm{mod}} a_\\mathrm{i} \\sin{[2\\pi f_\\mathrm{mod}\ni t + \\phi^\\mathrm{mod}_\\mathrm{i}]} \\sum_{j=1}^{N_\\mathrm{puls}} b_\\mathrm{j} \\sin{[2\\pi f_\\mathrm{puls}\nj t + \\phi^\\mathrm{puls}_\\mathrm{j}]} ,\n\\end{equation}\nwith $f_\\mathrm{mod}$ the modulation frequency and $f_\\mathrm{puls}$ the pulsation frequency.\nThis product can be rewritten as a sum, using Simpson's rule:\n\\begin{eqnarray}\n\\sum_{i=1}^{N_\\mathrm{mod}} \\sum_{j=1}^{N_\\mathrm{puls}} \\frac{a_\\mathrm{i} b_\\mathrm{j}}{2}(\\cos{[2\\pi\n(f_\\mathrm{puls} j-f_\\mathrm{mod} i)t+\\phi^\\mathrm{puls}_\\mathrm{j}-\\phi^\\mathrm{mod}_\\mathrm{i}]}-\\\\\n\\cos{[2\\pi (f_\\mathrm{puls} j+f_\\mathrm{mod} i)t+\\phi^\\mathrm{puls}_\\mathrm{j}+\\phi^\\mathrm{mod}_\\mathrm{i}]}). \\nonumber\n\\end{eqnarray}\nIn the Fourier transform of this signal, we will thus see peaks at\nfrequencies which are linear combinations of the pulsation frequency and the\nmodulation frequency, where the number of combinations depends on how\nnon-linear\nboth signals are. Most of the observed substructure in the amplitude spectrum\ncan be explained by amplitude modulation of a pulsation signal with\n$f_\\mathrm{mod}\\sim$ 0.010 {d$^{-1}$}. A more detailed description of amplitude\nand frequency modulation in light curves can be found in \\cite{Benko:2011}.\nSince the period of the amplitude modulation is close to the length of a\n\\textit{Kepler} observing quarter, we checked for a possible instrumental\norigin. A very strong argument against this, is the fact that the modulation is\nnot present in the eclipse signal of the light curve, but is only affecting the\npulsation peaks. \n\nTo study the pulsation signal in detail, we performed a complete frequency\nanalysis using an iterative prewhitening procedure. The Lomb-Scargle\nperiodogram was used in combination with false-alarm probabilities to detect the\nsignificant frequencies present in the light curve. Prewhitening of the\nfrequencies was performed using linear least-squares fitting with non-linear\nrefinement. In total, hundreds of formally significant frequencies were\ndetected. We stress here that formal significance does not imply that a\nfrequency has a physical interpretation, e.g. in the sense of all being\nindependent pulsation modes. In fact, most of them are not independent at all;\nmany combination\nfrequencies and harmonics are present and many peaks arise from the fact that\nwe are performing Fourier analysis on a signal that is not strictly periodic.\nTable \\ref{freqs} lists the 50\nmost significant frequencies, together with their amplitude, and estimated S\/N.\nThe noise level used to calculate the S\/N was estimated from the average\namplitude in the frequency range between 20 and 24 {d$^{-1}$}. The last column\nindicates possible combination frequencies and harmonics. The lowest order\ncombination is always listed, but for some frequencies, these can be\nwritten equivalently as a different higher order combination of more dominant\nfrequencies (they are listed in brackets). A complete list of frequencies\nis available online in electronic format at the CDS \\footnote{Centre de\nDonn\\'{e}es astronomiques de Strasbourg,\n  http:\/\/cdsweb.u-strasbg.fr\/}.\nClearly, many frequencies can\nbe explained by low-order linear combinations of the three most dominant\nfrequencies. However, given the amplitude modulation,\nthese dominant frequencies are likely not independent (corresponding to\ndifferent\npulsation modes). Instead, they are probably combination frequencies of the\nreal\npulsation frequency and (harmonics of) the modulation frequency $f_\\mathrm{mod}$\n$\\sim$ 0.010 {d$^{-1}$}. \nLooking back at the RV data for the pulsating component,\nwe expect the larger scatter in comparison to the secondary to be caused by the\npulsations. After subtraction of the Keplerian orbit fit, we performed\nfrequency analysis on the residuals, and found 2 significant\npeaks corresponding to frequencies detected in the \\textit{Kepler} data\n(f1 and f7 in Table \\ref{freqs}). The amplitude spectra of the RV residuals for\nboth components are shown in Fig. \\ref{ampspec-rv-res}. \n\nFrom spectroscopy, we found v sin i = 14.2 $\\pm$ 1.5 km\/s for the pulsating\nprimary component and v sin i = 8.4 $\\pm$ 1.5 km\/s for the secondary. Assuming that the rotation axes are perpendicular to the orbital plane, and using the orbital and stellar\nparameters listed in Table \\ref{final-par}, this corresponds to a rotation\nperiod of about 7.5 $\\pm$ 1.3 d for the primary and 8.8 $\\pm$ 1.9 d for the\nsecondary. These values suggest super-synchronous rotation, but on their own do\nnot provide sufficient evidence, given the uncertainties (1$\\sigma$), especially\nnot for the secondary.\n\nWe analysed the pulsation signal to\ndetect signs of rotational splitting of the pulsation frequencies in two\ndifferent ways:  by computing the autocorrelation function of the\namplitude spectrum and by using the list of detected significant frequencies.\nThe autocorrelation function between 0 and 0.7 {d$^{-1}$} is\nshown in Fig. \\ref{autocorrelation}. Clearly, the amplitude spectrum is\nself-similar for many different frequency shifts, as is evidenced by the\ncomplex structure of the autocorrelation function. Although the\nautocorrelation function is dominated by the highest peaks in the\namplitude spectrum and their higher harmonics, we can relate some smaller peaks\nto the properties of the system. For example, the peaks indicated with\nthe red arrows correspond to the orbital frequency and the rotational frequency of the primary as derived from spectroscopy (at 0.0926 and 0.1333 {d$^{-1}$} respectively). \n\nNext, we searched the list of significant frequencies for any possible frequency\ndifferences occurring several times (given the frequency resolution). This way,\nsplittings of lower amplitude peaks can be detected more easily, which is not the\ncase when using the autocorrelation function. Since the number of significant\nfrequencies is so large, several frequency differences occur multiple times\npurely by chance (this was checked by using a list of randomly generated\nfrequencies), so one must be careful when interpreting the results\n\\citep[see e.g.][]{Papics:2012}.\nIn our search for splittings, we used frequency lists of different lengths,\nranging from the first 50 to a maximum of several hundred significant\nfrequencies (down to a S\/N of 10). We used a rather strict cutoff value of\n0.0001 {d$^{-1}$} to accept frequency differences\nas being equal ( 0.1\/T, with T the total time span of the light curve). Next, we ordered all possible frequency differences in increasing\norder of occurrence. This always resulted in the same differences showing up in\nthe top of the lists. Table \\ref{freqdiffs} shows the most abundant differences detected in a conservative list of `only' 400 frequencies (down to a S\/N of about 50). \nClearly, many of these differences are related to the\namplitude modulation in the light curve (as discussed above), with values around\n0.010 {d$^{-1}$} (suspected modulation frequency) and 0.020 {d$^{-1}$} (twice the\nmodulation frequency) occurring often. Values around 0.567 {d$^{-1}$} are related to\nthe presence of higher harmonics of the pulsation frequencies. \nIn the autocorrelation function, clear peaks are present around the orbital\nfrequency, while the orbital frequency itself only\nshows up in the list of frequency differences when going down to S\/N values\naround 10. Most likely, this is caused by small residuals of the binary signal\n(eclipses) in the light curve. \nWe also find values very close to the rotation\nfrequency from spectroscopy and values in between the rotation frequency and the orbital frequency. They are also visible as the group of peaks in the\nautocorrelation function around 0.1 {d$^{-1}$}. We interpret these as the result of\nrotational splitting. The values are compatible with the spectroscopic results\nobtained for the primary, and hence provide stronger evidence for\nsuper-synchronous rotation. We cannot unambiguously identify one unique\nrotational splitting, but rather a range of possible values. The difference in\nmeasured splitting values across the spectrum suggests non-rigid rotation in\nthe interior of the primary \\citep[see e. g.][]{Aerts:2003, Aerts:book,Dziembowski:2008}.   \n                                               \n \nIn Fig. \\ref{as-puls-global}, the structures in the amplitude spectrum caused by\nrotational splitting are indicated by means of red arrows. Clearly, the pattern\nis repeated for the higher harmonics of the pulsation frequencies. \n\nFinally, the second-most dominant peak in\nthe amplitude spectrum almost coincides with the sixth harmonic of the orbital\nfrequency. This suggests tidally affected\npulsation, although this is not expected for non-eccentric systems.\nMoreover, the frequencies do not match perfectly, and the second-most dominant\npeak is likely not an independent pulsation mode, but caused by the amplitude\nmodulation of the main oscillation frequency.\n\n\\begin{figure*}\n \\centering\n   \\includegraphics[width=14cm,angle=270,scale=0.7]{as-puls-global.ps}\n\\caption{Part of the amplitude spectrum (below 2 {d$^{-1}$}) of the \\textit{Kepler}\nlight curve with the binary\nmodel removed. The red arrows indicate the splitting of groups of peaks caused\nby rotation with a  splitting value around 0.1 {d$^{-1}$}.}\n\\label{as-puls-global}\n\\end{figure*}\n\n\n\\begin{figure}\n \\centering\n   \\includegraphics[width=14cm,angle=270,scale=0.5]{autocorrelation.ps}\n\\caption{Autocorrelation of the amplitude spectrum, after removal of the binary\nmodel.}\n\\label{autocorrelation}\n\\end{figure}\n\n\n\n\n\\begin{table*}\n\\center\n\\caption{Dominant fifty frequencies with their amplitude and S\/N, detected in the\n\\textit{Kepler} light curve with the binary model removed, using the\nprewhitening technique as explained in the text.}\n \\begin{tabular}{c|c|c|c|c}\nNumber&Frequency (d$^{-1}$) & Amplitude (mmag)& S\/N&Combination\\\\\n\\hline\n 1       &0.5673 &17.487&720.1&-\\\\\n 2       &0.5575 &13.482&656.0&-\\\\\n 3       &0.5685 &8.378 &460.2&-\\\\\n 4       &1.1250 &7.468 &450.2&f1+f2\\\\\n 5       &0.5467 &6.559 &400.4&2f2-f3\\\\\n 6       &1.1359 &5.462 &355.8&f1+f3\\\\\n 7       &0.6594 &4.704 &323.4&-\\\\\n 8       &1.1141 &4.466 &311.7&f1+f5 (f1+2f2-f3)\\\\\n 9       &0.5563 &4.124 &296.7&f4-f3 (f1+f2-f3)\\\\\n 10      &0.4348 &3.192 &237.7&-\\\\\n 11      &0.4403 &3.288 &246.5&-\\\\\n 12      &1.6936 &2.842 &215.5&f3+f4 (f1+f2+f3)\\\\\n 13      &0.5666 &2.547 &195.9&-\\\\\n 14      &0.8985 &2.286 &178.8&-\\\\\n 15      &1.1041 &2.123 &168.1&f2+f5 (3f2-f3)\\\\\n 16      &0.3029 &2.080 &164.6&2f10-f13\\\\\n 17      &0.4483 &2.014 &161.4&-\\\\\n 18      &1.6925 &2.009 &163.2&f1+f4 (2f1+f2)\\\\\n 19      &1.6716 &1.969 &159.8&f4+f5 (f1+3f2-f3)\\\\\n 20      &0.5357 &1.963 &156.4&2f5-f2 (3f2-2f3)\\\\\n 21      &0.2365 &1.864 &150.7&2f1-f14\\\\\n 22      &1.1258 &1.847 &150.2&f2+f3\\\\\n 23      &0.4777 &1.831 &150.8&2f3-f7\\\\\n 24      &0.4337 &1.780 &148.4&2f5-f7 \\\\\n 25      &0.5784 &1.770 &147.7&f4-f5 (f1-f2+f3)\\\\\n 26      &1.2268 &1.766 &149.6&f1+f7\\\\\n 27      &1.6816 &1.738 &146.2&2f1+f5 (2f1+2f2-f3)\\\\\n 28      &0.4656 &1.712 &145.9&f4-f7 (f1+f2-f7)\\\\\n 29      &0.1272 &1.729 &149.5&f1-f11\\\\\n 30      &1.0023 &1.643 &142.2&f1+f10\\\\\n 31      &0.9979 &1.667 &146.7&f2+f11\\\\\n 32      &0.5582 &1.612 &143.8&f22-f1 (f2+f3-f1)\\\\\n 33      &0.6583 &1.539 &138.1&2f5-f10\\\\\n 34      &1.1346 &1.514 &135.7&2f1\\\\\n 35      &0.2477 &1.456 &131.7&2f7-2f20\\\\\n 36      &0.9804 &1.398 &127.6&f5+f24 (3f5-f7)\\\\\n 37      &1.2167 &1.399 &127.5&f2+f7\\\\\n 38      &1.7031 &1.388 &126.5&f1+f3\\\\\n 39      &0.4115 &1.363 &124.1&f7-f35 (2f20-f7)\\\\\n 40      &2.2609 &1.326 &122.2&f4+f6 (2f1+f2+f3)\\\\\n 41      &0.5257 &1.308 &120.3&2f5-f1\\\\\n 42      &0.5458 &1.267 &116.8&f8-f3 (f1+2f2-2f3)\\\\\n 43      &0.4200 &1.272 &117.2&2f7-f14\\\\\n 44      &0.0909 &1.240 &113.9&f7-f3\\\\\n 45      &0.5677 &1.220 &112.9&f1\\\\\n 46      &1.7043 &1.209 &113.8&f3+f6 (f1+2f3)\\\\\n 47      &0.6720 &1.201 &114.5&2f9-f11\\\\\n 48      &1.1368 &1.194 &114.4&2f3\\\\\n 49      &1.1470 &1.160 &113.0&2f6-f4 (f1-f2+2f3)\\\\\n 50      &0.2579 &1.159 &113.3&f4-2f24\\\\\n\\hline\n\n\n\\end{tabular}\n\\tablefoot{Typical uncertainties on the frequency values are\n$\\sim$ $10^{-3}$ d$^{-1}$ (using the Rayleigh criterion), and $\\sim 5 \\times\n10^{-3}$ mmag for the amplitudes. The last column indicates possible\ncombination frequencies and harmonics.}\n\\label{freqs}\n\\end{table*}\n\n\n\\begin{table*}\n\\center\n\\caption{Frequency differences and corresponding periods (in order of decreasing occurrence), detected in the list of the first 400 frequencies obtained using iterative prewhitening.}\n \\begin{tabular}{c|c|c}\nFrequency difference (d$^{-1}$)&  Period (d)&Remarks\\\\\n\\hline\n       0.5674  &      1.7624 & harmonics of pulsation frequencies\\\\ \n       0.0108  &    92.5926 & amplitude modulation\\\\\n       0.5675  &    1.7621   & harmonics of pulsation frequencies\\\\\n       0.1244   &     8.0386  & rotational splitting? \\\\ \n       0.5575   &     1.7937 &  harmonics of pulsation frequencies  \\\\\n       0.1133   &     8.8261  &  rotational splitting? \\\\\n       0.5685   &     1.7590   & harmonics of pulsation frequencies  \\\\ \n       0.5672   &     1.7630  & harmonics of pulsation frequencies \\\\\n       0.1010   &     9.9010   & rotational splitting? \\\\\n       0.0210  &     47.6190  &  amplitude modulation\\\\ \n       0.5577  &      1.7931 & harmonics of pulsation frequencies  \\\\\n       0.3132  &      3.1928  &  \\\\\n       0.2492  &      4.0128   &  \\\\ \n       0.0110  &     90.9091  &  amplitude modulation \\\\\n       0.0284  &     35.2113   &  \\\\\n       0.2267  &      4.4111    &  \\\\ \n       0.3318  &      3.0139   &  \\\\\n       0.0109  &     91.7431    & amplitude modulation \\\\\n       0.0070  &    142.8571  & \\\\ \n       0.3353  &      2.9824 &  \\\\\n       ...         &     ... &\\\\\n\\hline                       \n\\end{tabular}\n\\label{freqdiffs}\n\\end{table*}\n\n\n\n\\begin{figure}\n \\centering\n   \\includegraphics[width=14cm,angle=270,scale=0.5]{ampspec-rv-res.ps}\n\\caption{Amplitude spectrum of the radial velocity residuals for both\ncomponents, after subtraction of the best Keplerian orbit fit. The two highest\npeaks for the primary (indicated) correspond to frequencies detected in the\n\\textit{Kepler} light curve.}\n\\label{ampspec-rv-res}\n\\end{figure}\n\n\n\n\n\n\\section{Conclusions}\n\nWe have obtained accurate system parameters and astrophysical\nproperties for KIC 11285625, a double-lined eclipsing binary\nsystem with a $\\gamma$ Dor pulsator discovered by the \\textit{Kepler} space\nmission. The excellent \\textit{Kepler} data with a total time span of almost\n1000 days have been analysed together with high resolution HERMES spectra.\nThe individual composite spectra could not be used to derive fundamental\nparameters such as\n{$T_{\\rm eff}$} and {$\\log g$}, given their insufficient S\/N. This was achieved after using\nthe spectral disentangling technique for both components. \n\nAn iterative automated method was developed to separate the orbital variability\nin the \\textit{Kepler} light curve from the variability due to the pulsations of\nthe primary. The fact that the orbital frequency and its overtones are located \nin the same frequency range as the pulsation frequencies, made a simple\nseparation technique (such as a filter in the frequency domain) insufficient.\nWe plan to develop this technique further and apply it to other binary systems\nin the \\textit{Kepler} database.\n\nAfter removal of the best binary model, we studied the residual pulsation\nsignal in detail, and found indications for rotational splitting of the pulsation\nfrequencies, compatible with super-synchronous and non-rigid internal rotation.\nA detailed asteroseismic analysis of the $\\gamma$ Dor pulsator and comparison\nwith theoretical models can now be attempted on the basis of this observational work, which\nconstitutes an excellent starting point for stellar modelling of a $\\gamma$~Dor\\\nstar. A concrete interpretation of the detected amplitude modulation must await a much longer \\textit{ Kepler} light curve, given the relatively long modulation period.\n\n\\begin{acknowledgements}\nThe research leading to these results has received funding from the European\nResearch Council under the European Community's Seventh Framework Programme\n(FP7\/2007--2013)\/ERC grant agreement n$^\\circ$227224 (PROSPERITY), from the\nResearch Council of K.U.Leuven (GOA\/2008\/04), and from the Belgian federal\nscience\npolicy office (C90309: CoRoT Data Exploitation); A. Tkachenko and P. Degroote\nare postdoctoral fellows of the Fund for Scientific Research (FWO), Flanders,\nBelgium. \nFunding for the \\textit{Kepler} Discovery mission is provided by NASA's Science\nMission Directorate.\nSome of the data presented in this paper were obtained from the\nMultimission Archive at the Space Telescope Science Institute (MAST). STScI is\noperated by the Association of Universities for Research in Astronomy, Inc.,\nunder NASA contract NAS5-26555. Support for MAST for non-HST data is provided by\nthe NASA Office of Space Science via grant NNX09AF08G and by other grants and\ncontracts.\nThis research has made use of the SIMBAD database,\noperated at CDS, Strasbourg, France.\nWe would like to express our special thanks to the numerous people who helped\nmake the \\textit{Kepler} mission possible.\n\\end{acknowledgements}\n\n\\bibliographystyle{aa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and preliminary results}\n\nIt is well known from elementary calculus that an arbitrary\npolynomial $f$ with complex coefficients (complex polynomial) of\ndegree $n \\in \\mathbb{N}$\n$$f(z)= a_0 z^{n}+ a_1 z^{n-1}+ \\dots + a_{n-1} z + a_n, \\ a_0\\neq\n0,\\eqno(1)$$\nhaving a root $\\lambda \\in \\mathbb{C}$ of multiplicity $\\mu, \\ 1\\le\n\\mu\\le n$,  shares it with each of its derivatives  up to order $\\mu-1$,\nbut $f^{(\\mu)}(\\lambda)\\neq 0$. When $\\lambda$ is a unique root of\n$f$, it has the  form  $f(z)=a(z-\\lambda)^n$, $\\mu=n$ and $\\lambda$ is\nthe same root of each derivative of $f$ up to order $n-1$. We will\ncall such a polynomial as a trivial polynomial.  Obviously, as it\nfollows from fundamental theorem of algebra, $f$ has at least two\ndistinct roots, i.e. a polynomial of degree $n$ is non-trivial, if\nand only if its maximum multiplicity of roots $r$ does not exceed\n$n-1$.\n\n\nIn 2001 Casas- Alvero \\cite{CA} conjectured that an arbitrary polynomial $f$  degree $n\n\\ge 1$ with complex coefficients  is of the form   $f(z)=\na(z-b)^n, a, b \\in \\mathbb{C}$,    if   and \\ only   if $f$  shares a root with each of its derivatives $f^{(1)}, f^{(2)},\n\\dots, f^{(n-1)}.$\n\nWe will call a possible non-trivial polynomial,  which has a common root with each of its non-constant derivatives   as the CA-polynomial. The conjecture says that there  exist no CA-polynomials.  The problem is still open. However, it is proved for small degrees, for infinitely many degrees, for instance, for all powers $n$,   when $n$ is a prime   (see in \\cite{Drai}, \\cite{Graf}, \\cite{Pols} ).  We observe that such a kind of  CA-polynomial of degree $n \\ge 2$  cannot have all distinct roots since at least one root is common   with its  first derivative.  Therefore it has a multiplicity at least 2 and a maximum of possible distinct  roots is $n-1$.\n\n\n\n Our main goal here is to derive necessary and sufficient conditions for an arbitrary polynomial (1) to be trivial.  For example, solving a simple differential equation of the first order, we easily prove that a polynomial is trivial, if and only if it is divisible by its first derivative.  In the sequel we establish other criteria, which will guarantee that an arbitrary polynomial has a unique joint root.\n\n Without loss of generality one can assume in the sequel that $f$ is a monic polynomial of degree $n$, i.e. $a_0=1$ in (1).  Generally, it has $k$ distinct roots $\\lambda_j$  of multiplicities $r_j, \\   j= 1,  \\dots,  k, 1\\le k\\le n$ such that\n %\n$$ r_1+ r_2+ \\dots r_{k} = n\\eqno(2).$$\n %\n By $r$ we will denote the maximum of multiplicities (2), $r= \\hbox{max}_{1\\le j\\le k}  (r_j)$,  $r_0= \\hbox{min}_{1\\le j\\le k}  (r_j)$ and by  $  \\xi^{(m)}_\\nu, \\  \\nu = 1,\\dots, n-m$  zeros of $m$-th derivative $f^{(m)}, \\ m=1,\\dots, n-1.$\n For further needs we specify zeros of the $n-1$-th and $n-2$-th derivatives, denoting them by $\\xi^{(n-1)}_1=z_{n-1}$ and $\\xi^{(n-2)}_2=z_{n-2}$, respectively. It is easy to find another zero of the $n-2$-th derivative, which is equal to $\\xi^{(n-2)}_1= 2z_{n-1} -z_{n-2} $.  When  zeros $ z_{n-1},\\  z_{n-2}$ are real we write, correspondingly,   $ x_{n-1},\\  x_{n-2}$.    The value $z_{n-1}$ is called the centroid. It is a center of gravity of roots and by Gauss-Lucas theorem it is contained in the convex hull of all non-constant polynomial derivatives (see details in \\cite{Rah}).\n\n The paper is structured as follows: In Section 2 we study properties of the Abel-Goncharov interpolation polynomials, including integral and series representations and upper bounds.  Section 3 deals with the Sz.-Nagy type identities and Obreshkov-Chebotarev type inequalities for roots of polynomials and   their derivatives. As applications  new criteria are found for an arbitrary polynomial with only real roots to be trivial.   Section 4 is devoted to the Laguerre type inequalities for polynomials with only real roots to localize their zeros. The final Section 5 contains applications of these results towards  solution of the  Casas-Alvero  conjecture and its particular cases.\n\n\n \\section{Abel-Goncharov polynomials, their upper bounds and  integral and genetic sum's representations}\n\nWe begin, choosing a sequence of complex numbers (repeated terms are permitted)\n$z_0, z_1, z_2, \\dots, z_{n-1}, n \\in \\mathbb{N}$, where $z_0 \\in \\{\\lambda_1, \\lambda_2, \\dots, \\lambda_k\\},\\\nz_m \\in  \\{\\xi^{(m)}_1,\\  \\xi^{(m)}_2, \\dots, \\xi^{(m)}_{n-m}\\}, \\  m=1,2,\\dots, n-1$, satisfying conditions $f^{(m)}(z_m) =0, \\\nm =0, 1,\\dots,  \\ n-1$ and, clearly $f^{(n)} (z)= n!$.     Then we represent  $f(z)$ in the form\n\n$$\n    f(z)= z^n + P_{n-1} (z),\\eqno(3)\n$$\nwhere $P_{n-1} (z)$ is a polynomial of degree at most $n-1$. To determine  $P_{n-1} (z)$  we  differentiate the latter equality $m$ times,  and we  calculate the corresponding derivatives in $z_m$ to obtain\n$$\n P_{n-1}^{(m)} (z_m) =  -  \\frac{n!}{ (n-m)!}  z_m ^{n- m}  ,   \\quad  m= 0,1, \\dots, n-1.\\eqno(4)\n$$\nBut this is the known Abel-Goncharov interpolation problem (see \\cite{Evgrafov}) and the polynomial $P_{n-1}(z)$ can be uniquely determined via the linear system (4) of $n$ equations with $n$ unknowns and triangular\nmatrix with non-zero determinant.   So, following \\cite{Evgrafov}, we derive\n\n$$\n     P_{n-1} (z) = -  \\sum_{k=0}^{n-1}  \\frac{n!}{ (n- k)!}  z_k ^{n- k} G_k(z) , \\eqno(5)\n$$\nwhere $G_k(z),  k=0, 1,\\dots, n-1$ is the system of the Abel-Goncharov polynomials\n \\cite{Evgrafov}, \\cite{Levinson1},  \\cite{Levinson2}.    On the other hand it is known that\n\n$$\n     G_n(z)= z^n  -  \\sum_{k=0}^{n-1}  \\frac{n!}{ (n- k)!}  z_k ^{n- k}  G_k(z).\\eqno(6)\n$$\nThus comparing with (3),  we  find that\n$$\nG_n(z)\\equiv  G_n\\left(z, z_0, z_1, z_2,\\dots, z_{n-1}\\right)=f(z),\n$$\nand\n$$\n G_n\\left(\\lambda_j, z_0, z_1, z_2,\\dots, z_{n-1}\\right) = f(\\lambda_j)= 0,   \\quad\nj= 1,2, \\dots,  k.\n$$\nPlainly, one can make a relationship of possible CA-polynomials with the corresponding Abel-Goncharov polynomials, fixing a sequence $\\{z_m\\}_0^{n-1}$ such that\n$$z_m \\in \\{\\lambda_1, \\lambda_2, \\dots, \\lambda_k\\}, \\  m=0,1,\\dots,\\ n-1.$$\n\nFurther, It is known \\cite{Evgrafov} that the Abel-Goncharov polynomial can\nbe represented as a multiple integral in the complex plane\n$$\nG_n(z)=   n! \\int_{z_0}^z  \\int_{z_1}^{s_{1}} \\dots\n\\int_{z_{n-1}}^{s_{n-1}} d s_{n} \\dots d s_{1}.\\eqno (6)\n$$\nMoreover, making  simple changes  of variables in (6),  it can be verified that $G_n(z)$ is\n a homogeneous function of degree $n$ (cf. \\cite{Levinson1}). Therefore\n$$G_n(\\alpha z)= G_n\\left(\\alpha z, \\alpha z_0, \\alpha z_1,\\dots, \\alpha z_{n-1}\\right)\n = \\alpha^n G_n(z), \\ \\alpha \\neq 0.\\eqno(7)$$\nThe following Goncharov upper bound holds for  $G_n$ (see \\cite{Gon}, \\cite{Evgrafov}, \\cite{Levinson1}, \\cite{Ibra})\n$$\n\\left| G_n(z) \\right|  \\le  \\left( |z-z_0| + \\sum_{s=0}^{n-2} \\left\n|z_{s+1}- z_s\\right| \\right)^n.\\eqno(8)\n$$\n  Let us represent the Abel-Goncharov polynomials $G_n(z)$ in a different  way.  To do this, we will use the following representation of the Gauss hypergeometric function given by relation (2.2.6.1) in \\cite{Prud}, namely\n$$\\int_a^b (z-a)^{\\alpha-1} (b-z)^{\\beta-1}(z + c)^\\gamma dz =  (b-a)^{\\alpha+\\beta-1} (a+c)^\\gamma\nB(\\alpha,\\beta) {}_2F_1 \\left(\\alpha, -\\gamma; \\alpha +\\beta;\n\\frac{a-b}{ a+c} \\right),\\eqno(9)$$ where $\\alpha, \\beta, \\gamma $\nare positive integers,  $a, b, c \\in \\mathbb{C}$ and\n$B(\\alpha,\\beta)$ is the Euler beta-function. So, our goal  will be a\nrepresentation of the Abel-Goncharov polynomials  in terms of the\nso-called genetic sums considered, for instance, in \\cite{Apteka}.\nMoreover, this will drive us to a  sharper upper bound for these  polynomials, improving  the Goncharov  bound (8).\n Indeed,   $G_1(z)= z-z_0$. When $n \\ge 2$, we employ  multiple integral (6),  and appealing to\n representation (9), we obtain recursively\n$$\nG_n(z)=   n! \\int_{z_0}^z  \\int_{z_1}^{s_{1}} \\dots \\int_{z_{n-2}}^{s_{n-2}} \\  (s_{n-1} - z_{n-1})\n d s_{n-1} \\dots d s_{1}\n$$\n$$=   n! (z_{n-2} - z_{n-1})  \\int_{z_0}^z  \\int_{z_1}^{s_{1}} \\dots \\int_{z_{n-3}}^{s_{n-3}} \\  (s_{n-2} - z_{n-2})\n {}_2F_1 \\left(1, -1 ;  \\ 2; \\   \\frac{ z_{n-2} - s_{n-2}}{ z_{n-2} -  z_{n-1} } \\right) d s_{n-2} \\dots d s_{1} $$\n$$=  n!   \\sum_{j_1=0} ^ 1\\frac{(-1)_{j_1}(-1)^{j_1} }{(2)_{j_1}}  (z_{n-2} - z_{n-1})^{1-j_1} \\int_{z_0}^z\n\\int_{z_1}^{s_{1}} \\dots \\int_{z_{n-3}}^{s_{n-3}} \\  (s_{n-2} - z_{n-2})^{1+j_1}  d s_{n-2} \\dots d s_{1} .$$\nHence, employing  properties of the Pochhammer symbol and repeating this process, we find\n$$\nG_n(z)=  n!   \\sum_{j_1=0} ^ 1\\frac{(z_{n-2} - z_{n-1})^{1-j_1}  }{(2)_{j_1} (1-j_1)!}   \\int_{z_0}^z  \\int_{z_1}^{s_{1}}\n\\dots \\int_{z_{n-3}}^{s_{n-3}} \\  (s_{n-2} - z_{n-2})^{1+j_1}  d s_{n-2} \\dots d s_{1} $$\n$$= n!   \\sum_{j_1=0} ^ 1  \\sum_{j_2=0} ^{1+j_1} \\frac{(z_{n-2} - z_{n-1})^{1-j_1} (z_{n-3} - z_{n-2})^{1+j_1-j_2}  }\n{(2)_{j_2} (1-j_1)!(1+j_1 -j_2)!}   \\int_{z_0}^z  \\int_{z_1}^{s_{1}} \\dots \\int_{z_{n-4}}^{s_{n-4}} \\\n(s_{n-3} - z_{n-3})^{1+j_2}  d s_{n-3} \\dots d s_{1} .$$\nContinuing to calculate  iterated integrals with the use of (9), we\narrive  finally  at the following genetic sum's representation of\nthe Abel-Goncharov polynomials ($j_0 = j_n=0,\\   z_{-1}\\equiv z $)\n$$\nG_n(z)=   n!   \\sum_{j_1=0} ^ 1  \\sum_{j_2=0} ^{1+j_1} \\dots  \\sum_{j_{n-1}=0} ^{1+j_{n-2}}  \\\n \\prod_{s=0}^{n-1} \\frac{ (z_{n-2-s}- z_{n-1-s} )^ {1+ j_s- j_{s+1}} }{ (1+ j_s- j_{s+1})!}.\\eqno(10) $$\nAnalogously, we derive the genetic  sum's representation for  the $m$-th derivative\n$G_{n}^{(m)}(z)$, namely  ($j_0  =0 $)\n $$\nG_n^{(m)} (z)=   n!   \\sum_{j_1=0} ^ 1  \\sum_{j_2=0} ^{1+j_1} \\dots  \\sum_{j_{n-1-m }=0} ^{1 +j_{n-2-m}}\n  \\frac{ (z - z_{m} )^ {1 + j_{n-1-m}} }{ (1 + j_{n-1-m})!} \\\n \\prod_{s=0}^{n-2-m} \\frac{ (z_{n-2-s}- z_{n-1-s} )^ {1 + j_s- j_{s+1}} }{ (1+ j_s- j_{s+1} )!},\\eqno(11) $$\nwhere $m= 0,1,\\dots, n-1$.\n\nMeanwhile,  the Taylor expansions of $ G_n^{(m)} (z)$ in the neighborhood of points  $z_m$ give the formulas\n$$G_n^{(m)} (z)  = \\frac{n!}{(n- m)!} (z-z_m)^{n-m}  + \\frac {G_n^{(n-1)} (z_m)}{(n-m-1)!} (z-z_m)^{n-m- 1} + \\dots\n+  G_n^{(1+m)} (z_m)  (z-z_m),\\eqno(12)$$\nwhere $m= 0,1,\\dots, n-1$.  Thus comparing coefficients in front of $(z-z_m)^s, \\  s= 1, \\dots , n- m-1$ in (11) and (12), we find the values of derivatives $G_n^{(s+ m)} (z_m)$ in terms  $z_m, z_{m+1}, \\dots, z_{n-1}$. Precisely, we obtain   ($j_0  =0 $)\n$$G_n^{(s+m)} (z_m)   =   n!   \\sum_{j_1=0} ^ 1  \\sum_{j_2=0} ^{1+j_1} \\dots\n\\sum_{j_{n-2-m }=0} ^{1 +j_{n-3-m}}   \\frac{ (z_m - z_{m+1} )^ {2 + j_{n-2-m}-s} }{ (2 + j_{n- 2-m}-s)!} \\\n \\prod_{l=0}^{n-3-m} \\frac{ (z_{n-2-l}- z_{n-1-l} )^ {1 + j_l- j_{l+1}} }{ (1+ j_l- j_{l+1} )!},\\eqno(13)$$\nwhere $s=1,2,\\dots, n-m, \\  m=0,1,\\dots, n-1$.\n\nFinally, in this section,  we will establish a  sharper  upper bound for the\nAbel-Goncharov polynomials.    We have\n\n{\\bf Theorem 1}. {\\it Let $z, z_0, z_1, z_2,\\dots, z_{n-1} \\in\n\\mathbb{C},\\ n \\ge 1$. The following  upper bound holds for the\nAbel-Goncharov polynomials\n$$| G_n\\left(z, z_0, z_1, z_2,\\dots, z_{n-1}\\right)| \\le\n\\sum_{k_0=0}^1  \\sum_{k_1=0} ^{2-k_0} \\dots  \\sum_{k_{n-2}=0} ^{n-1\n- k_0-k_1-\\dots- k_{n-3}}  \\ {n! \\choose k_0! k_1! \\dots k_{n-2}! \\\n(n - k_0-k_1-\\dots- k_{n-2})! }$$\n$$\\times \\prod_{s=0}^{n-1} |z_{n-2-s}- z_{n-1-s}|^ {k_s},\\eqno(14) $$\nwhere $ z_{-1}\\equiv z$ and\n$${n! \\choose l_0! l_1! \\dots l_m!} = \\frac{n!}{l_0! l_1! \\dots\nl_m!}, \\ l_0+l_1+\\dots +l_m= n$$ are multinomial coefficients. This\nbound is sharper than the Goncharov upper bound $(8)$.}\n\n\\begin{proof} In fact, making simple substitutions $k_s= 1+j_s- j_{s+1}, \\ s=0,1,\\dots, n-1,  j_0=j_n=0$\nand writing identity (10) for the Abel-Goncharov polynomials (6), we\nestimate their  absolute value, coming out immediately with inequality\n(14). Furthermore, appealing to the multinomial theorem, we estimate\nthe right-hand side of (14) in the following way\n$$\\sum_{k_0=0}^1  \\sum_{k_1=0} ^{2-k_0} \\dots  \\sum_{k_{n-2}=0} ^{n-\n1- k_0-k_1-\\dots- k_{n-3}}  \\ {n! \\choose k_0! k_1! \\dots k_{n-2}\\\n(n - k_0-k_1-\\dots- k_{n-2})! } \\prod_{s=0}^{n-1} |z_{n-2-s}-\nz_{n-1-s}|^ {k_s}$$$$\\le \\sum_{l_0+l_1+\\dots+ l_{n-1} = n} \\ {n!\n\\choose  l_0! l_1!\\dots l_{n-1}!} \\prod_{s=0}^{n-1} |z_{n-2-s}-\nz_{n-1-s}|^{l_s}$$\n$$=  \\left(\\sum_{m=0}^{n-1} \\left |z_{m-1}- z_{m}\\right|\\right)^n,$$\nwhere the summation now  is taken over all combinations of\nnonnegative integer indices $l_0$ through $l_{n-1}$ such that the\nsum of all $l_j$ is $n$. Thus it  yields (8) and completes the\nproof.\n\n\\end{proof}\n\n\n\n\\section{Sz.-Nagy type identities for roots of polynomials and their derivatives}\n\nIn this section we prove  Sz.-Nagy type identities \\cite{Rah} for zeros of monic polynomials with complex coefficients and their derivatives.  All   notations of  roots and their multiplicities given in Section 1 are involved.\n\n    We begin with\n\n{\\bf  Lemma 1.}  {\\it Let $f$ be  a monic polynomial of degree $n \\ge 2$ with complex coefficients,  $m= 0,1,\\dots, n-2$ and  $z \\in \\mathbb{C}$. Then the following Sz.-Nagy type identities, which are related to the roots of $f$ and its $m$-th derivative,  hold }\n\n$$z_{n-1} - z  = {1\\over n} \\sum_{j=1}^k r_{j}(\\lambda_j- z) =   {1\\over n-m} \\sum_{j=1}^{n-m}  (\\xi^{(m)}_j - z),\\eqno(15)$$\n\n$$   (z_{n-1} - z_{n-2})^2={1\\over n(n-1)} \\left[ \\sum_{j=1}^k r_{j}(\\lambda_j- z)^2-  n (z_{n-1} -  z)^2\\right]\n=  {1\\over (n-m)(n-m -1)}$$$$\\times  \\left[ \\sum_{j=1}^{n-m}\n(\\xi^{(m)}_j - z)^2 -  (n-m) (z_{n-1} -  z)^2\\right],\\eqno(16)$$\n\n$$   (z_{n-1} - z_{n-2})^2={1\\over n^2(n-1)} \\sum_{1\\le j <  s\\le k} r_{j}r_{s}(\\lambda_j- \\lambda_s)^2\n=  {1\\over (n-m)^2(n-m -1)} \\sum_{1\\le j < s \\le n-m} (\\xi^{(m)}_j -\n\\xi^{(m)}_s)^2.\\eqno(17)$$\n\n\n\n\\begin{proof}  In fact,  the first Vi\\'{e}te  formula (see \\cite{Rah}) says that the coefficient $a_1$ ($a_0=1$) in (1) is equal to\n$$- a_1= r_{1}\\lambda_1  +  r_{2} \\lambda_{2} +   \\dots +  r_{k}\\lambda_{k}.$$\nOn the other hand, differentiating (1) $n-1$ times, we find $z_{n-1}\n= - a_1\/ n$.  Thus minding (2) we prove the first equality in (15). The second\nequality can be done similarly, using the properties of centroid,\nwhich is differentiation invariant, see, for instance, in\n\\cite{Rah}.  In order to establish the first equality in (16), we\ncall formula (11) to find\n$$\\frac{f^{(n-2)} (z)}{(n-2)!} =  \\frac{n(n-1)}{2} (z - z_{n-2}) (z + z_{n-2}- 2 z_{n-1}).\\eqno(18)$$\nMoreover,  as a consequence of the second Vi\\'{e}te formula,  the\ncoefficient $a_2$ in (1), which  equals\n$$a_2=   \\frac{f^{(n-2)} ( z)}{(n-2)!} -  \\frac{n(n-1)}{2} z^2 + n(n-1) z_{n-1}z\\eqno(19)$$\ncan be expressed as follows\n$$a_2=   {1\\over 2} \\left(\\sum_{j=1}^k r_{j} \\lambda_j\\right)^2 -  {1\\over 2} \\sum_{j=1} ^k r_j\\lambda_j^2.\\eqno(20) $$\nHence letting $z=z_{n-2}$ in (18),  and taking into account (15) with $z=0$,\nwe deduce\n\n$$2a_2=   n^2z^2_{n-1}  -   \\sum_{j=1} ^k r_{j} \\lambda_j^2 =  2 n(n-1) z_{n-1}z_{n-2}  -  n(n-1) z_{n-2}^2 . $$\nTherefore,   using again (15) and (2), we easily come out with the\nfirst equality in (16). The second one can be prove in the same\nmanner, involving roots of derivatives. Finally, we prove the first\nequality in (17). Concerning the second equality,  see Lemma 6.1.5\nin \\cite{Rah}. Indeed, calling the first equality in (16),  letting\n$z= z_{n-1}$ and employing (15), we derive\n\n$$  n^2(n-1)(z_{n-1} - z_{n-2})^2= n \\sum_{j=1}^k r_{j}\\lambda^2_j +\n\\left(\\sum_{s=1}^k r_{s}\\lambda_s\\right)^2 - 2\\left(\\sum_{s=1}^k\nr_{s}\\lambda_s\\right)\\left(\\sum_{j=1}^k r_{j}\\lambda_j\\right)$$\n$$= n \\sum_{j=1}^k r_{j}\\lambda^2_j - \\sum_{s=1}^k r^2_{j}\\lambda^2_j -\n2\\sum_{1\\le j <  s\\le k} r_{j}r_{s}\\lambda_j\\lambda_s=  \\sum_{1\\le j\n<  s\\le k} r_{j}r_{s}(\\lambda_j- \\lambda_s)^2.$$\n\n\\end{proof}\n\nThe following result gives an identity, which is associated with  zeros of a monic\npolynomial and common zeros  of its derivatives. Precisely, we have\n\n\n{\\bf Lemma 2.} {\\it Let $f$ be a monic polynomial of exact degree $n\\ge 2$, having $k$ distinct  roots of multiplicities $(2)$. Let  $z_{n-1}=\\lambda_1$ be a common root of $f$ of multiplicity $r_1$ with  the unique root of its $n-1$-th\nderivative. Let also $z_m= \\xi_{n-m}^{(m)}= \\lambda_{k_m}$ be a common root of $f$ of multiplicity $r_{k_m}$ and\nits $m$-th derivative, $m \\in \\{1,2,\\dots,  n-2\\}$. Then,  involving other roots of $f^{(m)}$,\nthe following identity holds}\n\n$$\\left[\\frac{n-m-2}{(n-m)^2} + \\frac{r_{k_m}+r_1-n}{n (n-1)}\\right]\\sum_{s=1}^{n-m-1} (z_m- \\xi^{(m)}_s)^2\n+ \\frac{n-m-2}{(n-m)^2}\\sum_{1\\le s < t \\le n-m-1} (\\xi^{(m)}_s- \\xi^{(m)}_t)^2\n$$$$= \\frac{(n-m)^2 r_{k_m}- (n-r_1)(n-m+2)}{n (n-1)} (z_m- z_{n-1})^2$$$$+ \\\n\\frac{2}{n (n-1)}\\sum_{j\\neq 1, k_m} r_j \\sum_{1 \\le s < t \\le\nn-m-1} (\\lambda_j- \\xi^{(m)}_s)(\\lambda_j- \\xi^{(m)}_t).\\eqno(21)$$\n\n\\begin{proof}  We begin, appealing to  (15) and letting  $z=0$.  We get\n$$\\sum_{s=1}^{n-m} \\xi^{(m)}_s= (n-m) z_{n-1}, \\quad   \\xi_{n-m}^{(m)}=z_m.\\eqno(22) $$\nHence via identities  (17) with $z=z_m$ we write the chain of equalities\n$$\\sum_{1\\le s < t \\le n-m} (\\xi^{(m)}_s- \\xi^{(m)}_t)^2=\n\\frac{(n-m-1)(n-m)^2}{n (n-1)} r_{k_m}(z_m- z_{n-1})^2 +\n\\frac{(n-m-1)(n-m)^2}{n (n-1)}\\sum_{j\\neq 1, k_m} r_j (\\lambda_j -\nz_{n-1})^2$$\n$$=\\frac{(n-m-1)(n-m)^2}{n (n-1)} r_{k_m}(z_m- z_{n-1})^2 +\n\\frac{n-m-1}{n (n-1)}\\sum_{j\\neq 1, k_m} r_j \\left(\\lambda_j - z_m+\n\\sum_{s=1}^{n-m-1} (\\lambda_j- \\xi^{(m)}_s) \\right)^2$$\n$$=\\frac{(n-m-1)(n-m)^2}{n (n-1)} r_{k_m}(z_m- z_{n-1})^2 +\n\\frac{n-m-1}{n (n-1)}\\left[\\sum_{j\\neq 1, k_m} r_j (\\lambda_j -\nz_m)^2 +  \\sum_{j\\neq 1, k_m} r_j \\left(\\sum_{s=1}^{n-m-1}\n(\\lambda_j- \\xi^{(m)}_s) \\right)^2\\right.$$\n$$\\left.+ 2 \\sum_{j\\neq 1, k_m} r_j \\sum_{s=1}^{n-m-1} (\\lambda_j -\nz_m)(\\lambda_j- \\xi^{(m)}_s)\\right]= \\frac{(n-m-1)(n-m)^2}{n (n-1)}\nr_{k_m}(z_m- z_{n-1})^2$$\n$$+ \\frac{n-m-1}{n (n-1)}\\left[(2(n-m)-1) \\sum_{j\\neq 1, k_m} r_j (\\lambda_j -\nz_m)^2 +  \\sum_{j\\neq 1, k_m} r_j \\left(\\sum_{s=1}^{n-m-1}\n(\\lambda_j- \\xi^{(m)}_s) \\right)^2\\right.$$\n$$\\left.+ 2 \\sum_{j\\neq 1, k_m} r_j \\sum_{s=1}^{n-m-1} (\\lambda_j -\nz_m)(z_m - \\xi^{(m)}_s)\\right]= \\frac{(n-m-1)(n-m)^2}{n (n-1)}\nr_{k_m}(z_m- z_{n-1})^2$$\n$$+ \\frac{n-m-1}{n (n-1)}\\left[(2(n-m)-1) \\sum_{j\\neq 1, k_m} r_j (\\lambda_j -\nz_m)^2 +  \\sum_{j\\neq 1, k_m} r_j \\left(\\sum_{s=1}^{n-m-1}\n(\\lambda_j- \\xi^{(m)}_s) \\right)^2\\right.$$\n$$\\left. - 2 (n-m)(n-r_1)(z_m-z_{n-1})^2 \\right]= \\frac{(n-m-1)}{n (n-1)}\n\\left((n-m)^2 r_{k_m}- n+r_1\\right) (z_m- z_{n-1})^2$$\n$$+ (n-m-1)(3(n-m)-2)(z_{n-1}- z_{n-2})^2  +  \\frac{(n-m-1)(n-r_1)}{n (n-1)}\\sum_{s=1}^{n-m-1}\n(z_{n-1}- \\xi^{(m)}_s)^2$$$$-    \\frac{r_{k_m}(n-m-1)}{n\n(n-1)}\\sum_{s=1}^{n-m-1} (z_{m}- \\xi^{(m)}_s)^2+ 2 \\ \\frac{n-m-1}{n\n(n-1)}\\sum_{j\\neq 1, k_m} r_j \\sum_{1 \\le s < t \\le n-m-1}\n(\\lambda_j- \\xi^{(m)}_s)(\\lambda_j- \\xi^{(m)}_t).$$ Applying again\n (17), (22), we split the right-hand side of the latter\nequality in (17) in two parts, selecting the root $z_m$. Thus in the\nsame manner after straightforward calculations it becomes\n$$\\left[\\frac{n-m-2}{(n-m)^2} + \\frac{r_{k_m}+r_1-n}{n (n-1)}\\right]\\sum_{s=1}^{n-m-1} (z_m- \\xi^{(m)}_s)^2\n+ \\frac{n-m-2}{(n-m)^2}\\sum_{1\\le s < t \\le n-m-1} (\\xi^{(m)}_s-\n\\xi^{(m)}_t)^2\n$$$$= \\frac{(n-m)^2 r_{k_m}- (n-r_1)(n-m+2)}{n (n-1)} (z_m- z_{n-1})^2+ \\\n\\frac{2}{n (n-1)}\\sum_{j\\neq 1, k_m} r_j \\sum_{1 \\le s < t \\le\nn-m-1} (\\lambda_j- \\xi^{(m)}_s)(\\lambda_j- \\xi^{(m)}_t),$$\ncompleting the proof of Lemma 2.\n\n\\end{proof}\n\n{\\bf Remark 1}.  It is easy to verify identity (21) for the least case $m=n-2$, when double sums are\nempty and $\\xi^{(n-2)}_1= 2z_{n-1} -z_{n-2} $ (see above).\n\n{\\bf  Corollary 1.}  {\\it A polynomial with only real roots of degree $n\\ge 2$ is trivial, if and only if its  $n-2$-th derivative has a double root}.\n\n\\begin{proof}  Indeed, necessity is obvious.  To prove sufficiency we see that since the $n-2$-th derivative  has a double real root $x_{n-2}$,  it  is equal to the root $x_{n-1}$ of the $n-1$-th derivative. Therefore letting in (16)\n$z= x_{n-1}$, we find that its left-hand side becomes zero and,\ncorrespondingly, all squares in the right-hand side are zeros. This\ngives a conclusion that all roots are equal to $x_{n-1}$.\n\n\\end{proof}\n\n{\\bf  Corollary 2.}  {\\it Let $f$ be an arbitrary  polynomial of\ndegree $n \\ge 3$ with at least two distinct roots, whose $n-2$-th\nderivative has a double root. Then it contains at least one  complex\nroot}.\n\n\\begin{proof} In fact, if all roots are real it is trivial via Corollary 1.\n\n\\end{proof}\n\n Evidently, each derivative up to $f^{(r-1)}$ of a  polynomial $f$ with only real roots,  where $r$ is the maximum of multiplicities of roots  shares a root with $f$. Moreover, since  via the Rolle theorem all roots of $f^{(m)}, \\  m=r,r+1,\\dots, n-1$ are simple,  we have that a possible common root with $f$ is simple too (we note,  that a number of common roots does not exceed $k-2$, because  minimal and  maximal  roots cannot be zeros of $f^{(m)},\\ m\\ge r$).  This circumstance gives an immediate\n\n {\\bf Corollary 3.} {\\it There exists no non-trivial polynomial with only real roots, having two distinct zeros  and sharing a root  with at least one of its derivatives, whose order exceeds $r-1,\\  r= \\hbox{max}_{1\\le j\\le k}  (r_j)$. }\n\n\\begin{proof} Indeed, in the case of existence of such a polynomial,  these two distinct roots cannot be within zeros  of any derivative $f^{(m)},\\  m > r$ owing to the Rolle theorem.  Moreover, if any of two roots  is in common with roots of  $f^{(r)}$, its multiplicity is greater than $r$, which is impossible.\n\n\\end{proof}\n\nWe extend Corollary 3 on three distinct real roots.  Precisely, it drives to\n\n {\\bf Corollary 4.} {\\it There exists no non-trivial polynomial $f$ of degree $n \\ge 3$ with only real roots, having three distinct zeros  and sharing a root  with its $n-2$-th and $n-1$-th derivatives. }\n\n\\begin{proof}  Let such a polynomial exist. Calling its roots  $\\lambda_1= x_{n-1}, \\   \\lambda_2= x_{n-2}$ and $\\lambda_3$ of multiplicities  $r_1, \\ r_2, \\ r_3$, respectively. Hence employing identities (16), we write for this case\n$$  (n^2- n- r_2) (x_{n-1} - x_{n-2})^2=  r_{3}(\\lambda_3 - x_{n-1}) ^2.$$\nIn the meantime, making square of both sides of the first equality in (15) for this case after simple modifications , we obtain\n$$ r_2^2(x_{n-1} - x_{n-2})^2=  r_{3}^2 (\\lambda_3 - x_{n-1}) ^2.$$\nHence, comparing with the  previous equality, we come out with the relation\n$$ (n^2- n- r_2) r_3=  r_2^2.$$\nBut $n=r_1+r_2+r_3,\\ r_j \\ge 1, j=1,2,3.$  Consequently,\n$$r_2^2 \\ge  n(n-1)- r_2 >  (n-1)^2- r_2 \\ge (r_1+r_2)^2-r_2 \\ge r_2^2+ r_2+ r_1^2 > r_2^2,$$\nwhich is impossible.\n\\end{proof}\n\n{\\bf Remark 2.}  If we omit the condition for $f$ to have a common root with the $n-2$-th derivative in Corollary 4, it becomes false. In fact,  this circumstance can be shown  by the counterexample $f(x)= x^3-x.$\n\nThe following result deals with the case of 4 distinct roots.  We have,\n\n{\\bf Corollary 5.} {\\it There exists no non-trivial polynomial $f$ of degree $n \\ge 4$ with only real roots, having four  distinct zeros  and sharing a root  with its $n-2$-th and $n-1$-th derivatives. }\n\n\\begin{proof}  Similarly to the previous corollary, we assume the existence of such a polynomial and call  its  roots\n$\\lambda_1= x_{n-1}, \\   \\lambda_2= x_{n-2}$ and $\\lambda_3, \\lambda_4$ of multiplicities $r_j, \\  j=1,2,3,4$, respectively.\nHence the first  identity  in (16)  yields\n$$  (n^2- n- r_2) (x_{n-1} - x_{n-2})^2=  r_{3}(\\lambda_3 - x_{n-1}) ^2+   r_{4}(\\lambda_4 - x_{n-1}) ^2 .\\eqno(23)$$\nMeanwhile, using  the first equality in (15) for this case, we derive in a similar manner\n$$ r_2^2(x_{n-1} - x_{n-2})^2=  r_{3}^2 (\\lambda_3 - x_{n-1}) ^2+ r_{4}^2 (\\lambda_4 - x_{n-1}) ^2+\n2 r_{3}r_4 (\\lambda_3 - x_{n-1}) (\\lambda_4 - x_{n-1}).$$\nThus,  after straightforward calculations,  we come out with the quadratic equation\n$$Ay^2+ By+ C=0$$\nin variable $y=  (\\lambda_3 - x_{n-1}) \/  (\\lambda_4 - x_{n-1})$ with coefficients $A= r_3r_2^2- r_3^2(n^2-n-r_2),\\\nB= - 2 r_3r_4 (n^2-n-r_2),\\ C= r_4r_2^2- r_4^2(n^2-n-r_2).$   But, it is easy to verify that $B^2-4AC >0.$ Therefore the quadratic\nequation has two distinct real roots.  Writing $\\lambda_3 - x_{n-1}= y (\\lambda_4 - x_{n-1})$ and substituting into (23), we obtain\n$$ (n^2- n- r_2) (x_{n-1} - x_{n-2})^2= ( r_{3}y^2+ r_4) (\\lambda_4 - x_{n-1})^2.$$\nAt the same time, since $y\\neq 0$,  we have $  \\lambda_4 - x_{n-1}=\ny^{-1}  (\\lambda_3 - x_{n-1})$ and\n$$y^2 (n^2- n- r_2) (x_{n-1} - x_{n-2})^2= ( r_{3}y^2+ r_4) (\\lambda_3 - x_{n-1})^2.$$\nHence,\n$$\\lambda_4= x_{n-1}  \\pm \\sqrt{\\frac{n^2- n- r_2}{ r_{3}y^2+ r_4}} \\  |x_{n-1} - x_{n-2}|,$$\n$$\\lambda_3= x_{n-1}  \\pm |y| \\sqrt{\\frac{n^2- n- r_2}{ r_{3}y^2+ r_4}}\\   |x_{n-1} - x_{n-2}|.$$\nConsequently,\n$$\\lambda_4- \\lambda_3 =  \\sqrt{\\frac{n^2- n- r_2}{ r_{3}y^2+ r_4}} \\  |x_{n-1} - x_{n-2}| ( 1-|y|)=\n - \\sqrt{\\frac{n^2- n- r_2}{ r_{3}y^2+ r_4}} \\  |x_{n-1} - x_{n-2}| ( 1+|y|)$$\n $$=  \\sqrt{\\frac{n^2- n- r_2}{ r_{3}y^2+ r_4}} \\  |x_{n-1} - x_{n-2}| ( 1+|y|)=\n \\sqrt{\\frac{n^2- n- r_2}{ r_{3}y^2+ r_4}} \\  |x_{n-1} - x_{n-2}| (|y|-1),$$\nwhich are possible only in the case $x_{n-1}=x_{n-2},$\\  $\\lambda_3=\\lambda_4$. Thus we get a contradiction with Corollary 1 and complete the proof.\n\\end{proof}\n\nIn the same manner we prove\n\n{\\bf Corollary 6.} {\\it There exists no non-trivial polynomial $f$ of degree $n \\ge 5$ with only real roots, having five   distinct zeros  and sharing  roots  with its $n-2$-th and $n-1$-th derivatives. }\n\n\\begin{proof}  Assuming its existence, it has  the roots  $\\lambda_1= x_{n-1}, \\   \\lambda_2= x_{n-2}$,  $\\lambda_3= 2x_{n-1}- x_{n-2}, \\  \\lambda_4$ and $\\lambda_5$ of multiplicities $r_j, \\  j=1,2,3,4, 5$, respectively. Hence\n$$  (n^2- n- r_2-r_3) (x_{n-1} - x_{n-2})^2=  r_{4}(\\lambda_4 - x_{n-1}) ^2+   r_{5}(\\lambda_5 - x_{n-1}) ^2 .$$\nTherefore  using similar ideas as in the proof of Corollary 5, we come out again to the contradiction.\n\n\\end{proof}\n\nFor general number of distinct zeros we establish the following\n\n{\\bf Corollary 7.} {\\it There exists no non-trivial polynomial $f$ of degree $n$ with only real roots, having $k \\ge 2$   distinct zeros of multiplicities $(2)$ $r_j,\\  j=1,\\dots, k$ and  among them all roots of $f^{(m)}$ for some  $m$, satisfying   the relations\n$$ r \\le m <  {1\\over 2}\\left(1-{1\\over r_0}\\right)(n-1),\\eqno(24)$$\nwhere $r,\\  r_0$ are maximum and minimum multiplicities of roots of $f$.}\n\n\\begin{proof}  In fact, as a consequence of (16) we have the identity \n$${(n-m)(n-m -1)\\over n(n-1)}  \\sum_{j=1}^k r_{j}(\\lambda_j- x_{n-1})^2 =  \n \\sum_{j=1}^{n-m} (\\xi^{(m)}_j - x_{n-1})^2\\eqno(25)$$\nfor some $m$,  satisfying condition (24).  Hence, since $m\\ge r$,  it has $n-m \\le k-2$ and $\\xi^{(m)}_j= \\lambda_{m_j}, \\  m_j \\in \\{1,\\dots, k\\}, \\  j= 1,\\dots, n-m$  are simple roots of $f^{(m)}$.  Thus  we find \n$$ \\sum_{j=1}^{n-m} \\left[ r_{m_j} {(n-m)(n-m -1)\\over n(n-1)} -1\\right]  (\\lambda_{m_j}- x_{n-1})^2 + \n{(n-m)(n-m -1)\\over n(n-1)}   \\sum_{j=n-m+1}^k r_{m_j}(\\lambda_{m_j} - x_{n-1})^2 =  0.$$\nBut, owing to condition (24) \n$$r_{m_j} {(n-m)(n-m -1)\\over n(n-1)} -1 \\ge  r_{0} {(n-m)(n-m -1)\\over n(n-1)} -1 \\ge 0, \\ j=  1,\\dots, n-m.$$\nIndeed, we have from the latter inequality \n$$m \\le n- {1\\over 2} - \\sqrt{\\frac{n^2-n}{r_0} + {1\\over 4}}$$\nand, in turn,\n$$ n- {1\\over 2} - \\sqrt{\\frac{n^2-n}{r_0} + {1\\over 4}}=  \\frac{2 (1-r_0^{-1}) (n^2-n)}\n{ 2n- 1  + \\sqrt{4(n^2-n)r^{-1}_0 + 1}}\\ge \\frac{ (1-r_0^{-1}) (n^2-n)}\n{ 2n- 1} >  {1\\over 2}\\left(1-{1\\over r_0}\\right)(n-1).$$\nTherefore  $\\lambda_j= x_{n-1}, \\ j=1,\\dots, k$ and this contradicts to the fact that all roots are distinct.  \n\\end{proof}\n\n\n\nFinally, in this section, we will employ identities (17) to prove an analog of the Obreshkov- Chebotarev theorem for multiple roots (see \\cite{Rah}, Theorem 6.4.3), involving estimates for smallest and largest of distances between consecutive zeros of polynomials and their derivatives.  Namely, it has\n\n{\\bf Theorem 2.} {\\it  Let $f$ be a polynomial of degree $n >  2$ with only real zeros. Denote the largest and the smallest of the distances between consecutive zeros of $f$ by $\\Delta$ and $\\delta$, respectively.  Denoting the corresponding quantities associated with $f^{(m)}, \\ m=1,2,\\dots,\\  n-2$ by  $\\Delta^{(m)}$ and $\\delta^{(m)}$, the following inequalities take place\n\n$$\\delta^{(m)}  \\le \\Delta \\  {rk \\over n} \\  \\sqrt{  \\frac{k^2-1}{ (n-m+1)(n-1)}},\\eqno(26)$$\n$$\\delta \\  {r_0 k \\over n} \\  \\sqrt{  \\frac{k^2-1}{ (n-m+1)(n-1)}}\\le  \\Delta^{(m)} ,\\eqno(27)$$\n$$\\delta \\  {r_0 k \\over 2 n} \\  \\sqrt{  \\frac{k^2-1}{ 3 (n-1)}}\\le |x_{n-1} - x_{n-2}| \\le\n \\Delta \\  {rk \\over 2 n} \\  \\sqrt{  \\frac{k^2-1}{ 3(n-1)}},\\eqno(28)$$\nwhere $r_0,\\  r$ are minimum and maximum multiplicities of roots of $f$, respectively,  and $k \\ge 2$ is a number of distinct roots.}\n\n\\begin{proof}  Following similar ideas as in the proof of Theorem 6.4.3 in \\cite{Rah},  we assume distinct roots of $f$ in the increasing order and roots of its  $m$-th derivative in the non-decreasing order, and taking the second identity in (17), we deduce\n$$ {[\\delta^{(m)}]^2 \\over (n-m)^2(n-m -1)} \\sum_{1\\le j < s \\le n-m} (s-j)^2 \\le {[\\Delta r]^2  \\over n^2(n-1)} \\sum_{1\\le j < s \\le k} (s-j)^2.$$\nHence, minding the value of the sum\n$$ \\sum_{1\\le j < s \\le q} (s-t)^2 = {1\\over 12} q^2(q^2-1),$$\nafter simple manipulations we arrive at the inequality (26).  In the same manner (cf. \\cite{Rah}) we establish inequalities (27), (28), basing Sz.-Nagy type identities (17). \n\\end{proof}\n\n\n\n\\section{Laguerre's  type inequalities }\n\nIn 1880 Laguerre proved his famous theorem for polynomials with only\nreal roots, which provides their localization with  upper and lower\nbounds (see details in \\cite{Rah}).  Precisely,  we have the following\nLaguerre inequalities\n$$x_{n-1}-  (n-1)\\left|x_{n-1}- x_{n-2}\\right| \\le w_j \\le  x_{n-1} + (n-1)\\left|x_{n-1}- x_{n-2}\\right|, \\\n j=1,\\dots, n,$$\nwhere $w_j$ are  roots of the polynomial $f$ of degree $n$ and $x_{n-1}, \\  x_{n-2}$ are roots of  $f^{(n-1)},\\ f^{(n-2)}$, respectively.   First  we prove an analog of the Laguerre inequalities  for multiple roots.\n\n{\\bf  Lemma 3.}  {\\it Let $f$ be a polynomial with only real roots of degree  $n \\in \\mathbb{N}$,  having $k$  distinct roots\n $\\lambda_j, \\ j=1,\\ \\dots, k$ of multiplicities $(2)$ and $x_{n-1}, \\  x_{n-2}$ be roots of  $f^{(n-1)},\\ f^{(n-2)}$, respectively.\n  Then the following Laguerre   type inequalities  hold}\n$$x_{n-1}-  \\sqrt{\\frac{(n-r_j)(n-m-1)}{r_j-m} }\\left|x_{n-1}- x_{n-2}\\right| \\le \\lambda_j \\le  x_{n-1} +\n\\sqrt{\\frac{(n-r_j)(n-m-1)}{r_j-m} }\\left|x_{n-1}- x_{n-2}\\right|,\\eqno(29)$$\nwhere $ j=1,\\dots, k, \\  m= 0,1,\\dots, r_j-1.$\n\n\\begin{proof} In fact, appealing to the  Sz.-Nagy type identities (15), (16) and the Cauchy -Schwarz inequality, we find\n$$   (x_{n-1} - x_{n-2})^2={1\\over (n-m)(n-m-1)} \\left[ \\sum_{s=1}^{n-m} (\\xi^{(m)}_s-  \\lambda_j)^2-  (n-m) (x_{n-1} -\n\\lambda_j)^2\\right]$$$$ \\ge  {1\\over (n-m)(n-m-1)} \\left[\n\\frac{1}{n-r_j} \\left( \\sum_{s=1}^{n-m} (\\xi^{(m)}_s-\n\\lambda_j)\\right)^2-  (n-m) (x_{n-1} - \\lambda_j)^2\\right]$$$$=\n\\frac{r_j-m} {(n-r_j)(n-m-1)} \\left(x_{n-1}- \\lambda_j\\right)^2, \\\nm= 0,1,\\dots, r_j-1,$$ which yields (29).\n\n\\end{proof}\n\nAs a corollary we improve the Laguerre inequality (28) for multiple roots.\n\n\n{\\bf Corollary 8.} {\\it  Let $f$ be a polynomial with only real roots of degree  $n \\in \\mathbb{N}$.\nThen the multiple zero $\\lambda_j$ of multiplicity $r_j\\ge 1, \\   j=1,\\dots, k$ lies in the interval}\n$$\\left[ x_{n-1}-  \\sqrt{\\left(\\frac{n}{r_j}-1\\right)(n-1) }\\left|x_{n-1}- x_{n-2}\\right|, \\quad   x_{n-1} +\n\\sqrt{\\left(\\frac{n}{r_j}-1\\right)(n-1) }\\left|x_{n-1}- x_{n-2}\\right| \\right].\\eqno(30)$$\n\n\n\\begin{proof}  Indeed, the fraction $\\frac{(n-r_j)(n-m-1)}{r_j-m}$ attains its minimum value, letting $m=0$ in (29).\n\\end{proof}\n\n{\\bf Remark 3.}  When all roots are simple, the latter interval\ncoincides with the one generated by (28).\n\nA localization of roots of the  $m$-th derivative $f^{(m)}, \\ m=0,1,\\dots,  n-2$ is given by\n\n{\\bf  Lemma 4.}  {\\it Roots of the $m$-th derivative $f^{(m)}, \\ m=0,1,\\dots,  n-2$ satisfy the following Laguerre   type inequalities}\n$$x_{n-1}-  (n-m-1)\\left| x_{n-1}- x_{n-2}\\right| \\le \\xi^{(m)}_\\nu \\le\n x_{n-1} +  (n-m-1)\\left| x_{n-1}- x_{n-2}\\right|,\\eqno(31)$$\nwhere   $\\nu=1,\\dots, n-m.$\n\n\\begin{proof} Similarly to the proof of Lemma 3, we employ  the  Sz.-Nagy type identities (15), (16) and the Cauchy -Schwarz inequality to deduce\n$$   (x_{n-1} - x_{n-2})^2={1\\over (n-m)(n-m-1)} \\left[ \\sum_{s=1}^{n-m} (\\xi^{(m)}_s-  \\xi^{(m)}_\\nu)^2-  (n-m) (x_{n-1} -\n\\xi^{(m)}_\\nu)^2\\right]$$$$ \\ge  {1\\over (n-m)(n-m-1)} \\left[\n\\frac{1}{n-m-1} \\left( \\sum_{s=1}^{n-m} (\\xi^{(m)}_s-\n\\xi^{(m)}_\\nu)\\right)^2-  (n-m) (x_{n-1} -\n\\xi^{(m)}_\\nu)^2\\right]$$$$= \\frac{1}{(n-m-1)^2} \\left(x_{n-1}-\n\\xi^{(m)}_\\nu\\right)^2, \\  m= 0,1,\\dots, n-2.$$ Thus we come out\nwith (31) and complete the proof.\n\n\\end{proof}\n\nWhen $x_{n-1}=\\lambda_1$ be in common with $f$ of multiplicity $r_1$, we have\n\n{\\bf  Lemma 5.}  {\\it Let $f$ be a polynomial with only real roots\nof degree $n \\ge 2$ and $x_{n-1}=\\lambda_1$ be a  common zero with\n$f$ of multiplicity $r_1$, having $k \\ge 2$  distinct roots $\\lambda_j$ of multiplicities\n$r_j, j=1,\\dots, k$. Then the following Laguerre type inequalities\nhold}\n$$x_{n-1}-   \\sqrt{\\left({1\\over r_s}- {1\\over n-r_1}\\right) (n^2-n)}\\left| x_{n-1}- x_{n-2}\\right| \\le \\lambda_s \\le  x_{n-1}\n $$$$+  \\sqrt{\\left({1\\over r_s}- {1\\over n-r_1}\\right) (n^2-n)}\\left| x_{n-1}- x_{n-2}\\right|,\\eqno(32)$$\nwhere   $s=2,\\dots, k.$\n\n\\begin{proof} In the same manner  we involve   the  first Sz.-Nagy type identity in  (15) with $z=  \\lambda_s$,  which can be written in the form\n$$(n-r_1) (x_{n-1} - \\lambda_s)  =  \\sum_{j=2}^k r_{j}(\\lambda_j- \\lambda_s).$$\nHence squaring  both sides of the latter equality and appealing to the Cauchy -Schwarz inequality, we derive by virtue of  (16)\n$$(n-r_1)^2 (x_{n-1} - \\lambda_s)^2  = \\left( \\sum_{j=2}^k r_{j}(\\lambda_j- \\lambda_s)\\right)^2$$\n$$\\le (n-r_1-r_s)\\sum_{j=2}^k r_{j}(\\lambda_j- \\lambda_s)^2=  (n-r_1-r_s)\\left[ (n^2-n) ( x_{n-1}- x_{n-2})^2 +\n(n-r_1)(x_{n-1}- \\lambda_s)^2\\right].$$\nThus after simple calculations we easily arrive at (32).\n\n\\end{proof}\n\n\n{\\bf Remark 4}. Inequalities (27) are sharper than the corresponding relations, generated by interval (30).\n\n\nThe following result gives a Laguerre type localization for common roots of a possible CA-polynomial with only real roots and its $m$-th derivative.\n\n\n{\\bf Lemma 6.} {\\it     Let $f$ be a CA-polynomial of degree $n \\ge\n2$ with only real distinct zeros of multiplicities $(2)$,   including common\nroots $x_{n-1}=\\lambda_1$ of its $n-1$-th derivative and $x_m$ of\nits $m$-th derivative,   $m=  r, r+1, \\dots, n-2$, where  $r=\n\\hbox{max}_{1\\le j\\le k} (r_j)$. Then the following Laguerre type\ninequality holds\n$$ \\frac{n-r_1- r_{k_m}}{(n-r_1)^2}\n\\left(n^2-r_1+ (n-r_1)(n-m) (n-m-2)\\right) (x_{n-1}-  x_{n-2})^2 \\ge  (x_{n-1}- x_{m})^2,\\eqno(33)$$\nwhere $x_{n-2}$ is a root of $f^{(n-2)}$ and $r_{k_m}$ is the multiplicity of  $x_m$ as a root of $f$}.\n\n\\begin{proof}   Appealing  again to  Sz.-Nagy's type  identities (15), (16) with $z=x_m$, inequality (31)  and the Cauchy-Schwarz inequality, we  find\n$$   (x_{n-1} - x_{n-2})^2={1\\over n(n-1)} \\left[ \\sum_{j=2}^k r_{j}(\\lambda_j- x_m)^2-  (n-r_1) (x_{n-1} -\nx_m)^2\\right]$$$$ \\ge {1\\over n(n-1)} \\left[ \\sum_{j=2}^k\nr_{j}(\\lambda_j- x_m)^2- (n-r_1)(n-m-1)^2(x_{n-1}-  x_{n-2})^2\\right]$$\n$$ \\ge {1\\over n(n-1)} \\left[ {1\\over n-r_1- r_{j_m}} \\left(\\sum_{j=2}^k\nr_{j}(\\lambda_j- x_m)\\right)^2 -  (n-r_1)(n-m-1)^2(x_{n-1}-  x_{n-2})^2\\right]$$\n$$= {n-r_1\\over n(n-1)} \\left[ {n-r_1\\over n-r_1- r_{j_m}} (x_{n-1}- x_m)^2 - (n-m-1)^2(x_{n-1}-  x_{n-2})^2\\right].$$\nHence, making straightforward calculations,  we  derive (33), completing the proof of Lemma 6.\n\n\\end{proof}\n\n\nLet us denote by $d,\\ d^{(m)},\\  D, D^{(m)}$ the following values\n$$   d=  \\hbox{min}_{2\\le j\\le k}  |\\lambda_j-   x_{n-1}|, \\quad\n  d^{(m)}=  \\hbox{min}_{1\\le j\\le n-m}   |\\xi^{(m)}_j -   x_{n-1}|,\\eqno(34)$$\n$$   D=  \\hbox{max}_{2\\le j\\le k}  |\\lambda_j-   x_{n-1}|, \\quad\n  D^{(m)}=  \\hbox{max}_{1\\le j\\le n-m}   |\\xi^{(m)}_j -   x_{n-1}|,\\eqno(35)$$\nand by\n$$\\hbox{span} (f) = \\lambda^*- \\lambda_*,$$\nwhere\n$$  \\lambda^*= \\hbox{max}_{1\\le j\\le k} (\\lambda_j), \\quad     \\lambda_*= \\hbox{min}_{1\\le j\\le k} (\\lambda_j)$$\nare roots of $f$ of multiplicities $r^*, \\ r_*$, respectively.   It has the properties  $ D^{(m+1)}  \\le  D^{(m)}\\le D$ and\n  (cf. \\cite{Rah})  $\\hbox{span}(f^{(m+1)}) \\le \\hbox{span}(f^{(m)}) \\le \\hbox{span}(f)$, where $\\hbox{span}(f^{(m)})$\n   is the span of the $m$-th derivative.  Moreover, the strict inequalities $ D^{(m)}  <  D$,\\    $\\hbox{span}(f^{(m)}) < \\hbox{span}(f)$   hold when $m$ is sufficiently large.\n\n  {\\bf Lemma 7}. {\\it Let $x_{n-1}=\\lambda_1, \\  x_{n-2}=\\lambda_2$ be  common roots of $f$  with its $n-1$-th, $n-2$-th derivatives, respectively,  of multiplicities $r_1, r_2$ as roots of $f$,  and the maximum distance $D$ (see $(35)$) be attained at the root $\\lambda_{s_0}, \\ s_0 \\in \\{ 3,\\dots, k\\},\\ k \\ge 3$   of $f$ of multiplicity $r_{s_0}$.   Then the following inequalities hold }\n  %\n    $$\\sqrt{\\frac{n^2-n- r_2}{n-r_1-r_2}}\\  \\left|x_{n-1} - x_{n-2}\\right|   \\le D \\le \\sqrt{\\frac{n^2-n-r_2}{r_{s_0}}}\n     \\left|x_{n-1} - x_{n-2}\\right|,\\eqno(36)$$\n %\n  $${1\\over 2} \\sqrt{ {r_{s_0} \\over 3(n-r_1) }\\left(5 + \\frac{ r_2} {n^2-n-r_2}\\right)}\\hbox{span}(f) \\le D \\le \\sqrt{{1\\over n-r_1} \\left[ n-r_1- {r_{s_0} \\over 4}\\left(5 + \\frac{ r_2} {n^2-n-r_2}\\right)\\right] } \\hbox{span}(f).\\eqno(37)$$\n \n\\begin{proof}  In order to establish (36), we employ identities (16) and under condition of the lemma we write \n$$ (n^2-n- r_2)(x_{n-1} -   x_{n-2})^2 = \\sum_{j=3}^k r_{j}(\\lambda_j- x_{n-1})^2 \\le (n-r_1-r_2) D^2.$$\nSince $n > r_1+r_2$ and $x_{n-2}\\neq \\lambda_{s_0}$ (otherwise $f$ is trivial, because equalities $x_{n-2}= \\lambda_{s_0}= \\lambda^*$ or  $x_{n-2}= \\lambda_{s_0}= \\lambda_*$ mean that the maximum multiplicity $r > n-2$,  and we appeal to Corollary 3),  we come up with the lower  bound (36) for $D$.  The lower bound comes immediately from the estimate \n$$(n^2-n- r_2)(x_{n-1} -   x_{n-2})^2 = \\sum_{j=3}^k r_{j}(\\lambda_j- x_{n-1})^2    \\ge  r_{s_0} D^2.$$\nNow,  since $2D \\ge  \\hbox{span}(f)$, we find from (36)\n$$\\hbox{span}(f) \\le 2 \\sqrt{\\frac{n^2-n-r_2}{r_{s_0}}} \\  \\left|x_{n-1} - x_{n-2}\\right|.$$\nHence,  since  $D = \\hbox{max} \\left(|\\lambda^*- x_{n-1}|, \\ |\\lambda_*- x_{n-1}|\\right)$, the\n$n-2$-th derivative has roots $x_{n-2}$ and $2x_{n-1}- x_{n-2}$ and\n$\\hbox{span}(f)= D + \\Lambda$, where $\\Lambda = \\hbox{min}\n\\left(|\\lambda^*- x_{n-1}|, \\ |\\lambda_*- x_{n-1}|\\right)$, we\nappeal to the first equality in (16), letting $z= \\lambda_{s_0}$ and writing it in the form\n$$ (n-r_1)(x_{n-1} -  \\lambda_{s_0})^2 = \\sum_{j=2}^k r_{j}(\\lambda_j- \\lambda_{s_0})^2 - n(n-1)(x_{n-1} -\nx_{n-2})^2.$$\nTherefore,\n$$(n-r_1)D^2 \\le \\left[ n-r_1- {5\\over 4} r_{s_0} - \\frac{r_{s_0} r_2} {4(n^2-n-r_2)}\\right]  [\\hbox{span}(f)]^2$$ \nand we establish the upper bound (37) for $D$. On the other hand $\\hbox{span}(f)= D+ \\Lambda$. So,\n$$D^2 \\le \\left(1 -   {r_{s_0} \\over 4(n-r_1) }\\left(5 + \\frac{ r_2} {n^2-n-r_2}\\right)\\ \\right) \\left(D^2 + \\Lambda^2 +\n2D\\Lambda\\right)$$ and we easily come out with the lower bound (37)\nfor $D$, completing the proof of Lemma 7.\n\n\\end{proof}\n\n{\\bf Lemma 8}. {\\it Let $x_{n-1}=\\lambda_1, x_{n-2}=\\lambda_2$ be\ncommon roots of $f$ with its $n-1$-th, $n-2$-th   derivatives of\nmultiplicities $r_1, r_2,\\ r_1+r_2 < n$, respectively. Then  we\n have the following lower bound for $\\hbox{span}(f)$}\n  %\n  $$\\hbox{span}(f)\\ge \\sqrt{\\frac{n^2-r_1}{n-r_1-r_2}}\\ |x_{n-1}-x_{n-2}|.\\eqno(38)$$\n\n\\begin{proof} Indeed,  identities  (16) with $z=x_{n-2}$ yield\n$$(n^2-r_1)(x_{n-1}-x_{n-2})^2=\\sum_{j=3}^k r_{j}(\\lambda_j- x_{n-2})^2$$\nand we derive\n$$(n^2-r_1)(x_{n-1}-x_{n-2})^2\\le (n-r_1-r_2)[\\hbox{span}(f)]^2,$$\nwhich implies (38).\n\\end{proof}\n\nNext, we establish an analog of Lemma 5 for roots of derivatives.   Precisely, it has\n\n {\\bf Lemma 9}. {\\it Let  $x_{n-1},  \\  x_{n-2}$ be roots of the $n-1$-, $n-2$-th   derivatives of $f$, respectively.  Then\n %\n $$D^{(m)} \\ge \\sqrt{n-m-1}\\  |x_{n-1}-x_{n-2}|,\\eqno(39)$$\n where $m \\in \\{r, r+1,\\dots, n-2 \\}, \\   r= \\hbox{max}_{1\\le j\\le k} (r_j).$  Besides, if $x_{n-1}$ is a root of $f^{(m)}$, then\n we have a stronger inequality $$D^{(m)} \\ge \\sqrt{n-m}\\  |x_{n-1}-x_{n-2}|.\\eqno(40)$$\nMoreover,\n  %\n  $$2\\  D^{(m)} \\ge \\hbox{span}(f^{(m)}) \\ge   \\frac{n-m}{n-m-1}\\   D^{(m)}.\\eqno(41)$$\n  and if $x_{n-1}$ is a  root of  $f^{(m)}$,  it becomes\n   $$2\\  D^{(m)} \\ge \\hbox{span}(f^{(m)}) \\ge  \\sqrt{ \\frac{(n-m)(n-m-1)+1}{(n-m-1)(n-m-2)}}\\   D^{(m)},\\eqno(42)$$\nwhere  $m \\in \\{r, r+1,\\dots, n-3 \\}.$}\n\\begin{proof}  In fact,  since (see (16))\n$$(n-m)(n-m-1)(x_{n-1} - x_{n-2})^2= \\sum_{j=1}^{n-m} (\\xi^{(m)}_{j} -  x_{n-1})^2 \\le (n-m) \\left[ D^{(m)}\\right]^2,$$\nwe get (39). Analogously, we immediately come out with (40), when $x_{n-1}$ is a root of $f^{(m)}$, because one element of the sum of squares is zero.  In order to prove (41), we appeal again to (16), letting  $z= \\xi^{(m)}_{s_0}, \\ s_0 \\in \\{1,2, \\dots,  n-m\\}$,  $m \\in \\{r, r+1,\\dots, n-2 \\}, \\   r= \\hbox{max}_{1\\le j\\le k} (r_j)$, which is a root of  $f^{(m)}$, where the maximum $D^{(m)}$ is attained.  Hence owing to Laguerre type inequality (31)\n$$(n-m)\\left[D^{(m)}\\right]^2\\le (n- m-1) [\\hbox{span}(f^{(m)})]^2 - \\frac{n-m}{n-m-1} \\left[D^{(m)}\\right]^2,$$\nwhich drives to  the lower bound for $ \\hbox{span}(f^{(m)})$ in   (41).   The upper bound is straightforward\nsince $x_{n-1}$ belongs to the smallest interval containing roots of $f^{(m)}$.  In the same manner we establish (42),  since in this case\n$$(n-m-1)\\left[D^{(m)}\\right]^2\\le (n- m-2) [\\hbox{span}(f^{(m)})]^2 - \\frac{n-m}{n-m-1} \\left[D^{(m)}\\right]^2.$$\n\n\\end{proof}\n\n{\\bf Remark 5}.  The case $m=n-2$ gives  equalities in (39), (41).  Letting the same value of $m$ in (40), we easily get\na contradiction, which means that the only trivial polynomial is within polynomials with only real roots,\nwhose derivatives $f^{(n-2)}, \\  f^{(n-1)}$ have a common root (see Corollary 1).\n\n\n\n\\section{Applications to the Casas- Alvero conjecture}\n\nIn this final section we will discuss properties of possible CA-polynomials,  which share roots with each of their non-constant derivatives. We will investigate particular cases of the Casas-Alvero conjecture, especially for polynomials with only real roots, showing when it holds true or, possibly, is false. \n\nWe begin with \n\n\n{\\bf Proposition 1}. {\\it The Casas-Alvero conjecture holds true, if and only if  it is true for common roots $\\{z_\\nu \\}_0^{n-1}$ lying in the unit circle.}\n\n\\begin{proof}   The necessity is trivial. Let's l prove the sufficiency. Let the conjecture be true for common roots $\\{z_\\nu \\}_0^{n-1}$ of a complex polynomial $f$ and its non-constant derivatives, which lie in the unit circle.  Associating with $f$ an Abel-Goncharov polynomial $G_n$ (6),    one can choose an arbitrary $\\alpha  >0$ such that $\\ |z_\\nu| < \\alpha^{-1}, \\   \\nu=0, 1, \\dots , n-1. $  Hence owing to  (7)\n$$ f (\\alpha z_\\nu) = G_n\\left(\\alpha z_0, \\alpha z_\\nu, \\alpha z_1,\\dots, \\alpha z_{n-1}\\right)\n= \\alpha^n G_n(  z_\\nu) = \\alpha^n f (  z_\\nu)=  0,\\  \\nu=0, 1,\\dots , n-1,$$\nand\n$$f^{(\\nu)} _n(\\alpha z ) =  n! {d^{\\nu}\\over d z^{\\nu}}\n\\int_{\\alpha z_0}^{\\alpha z}  \\int_{\\alpha z_1}^{s_{1}} \\dots \\int_{\\alpha z_{n-1}}^{s_{n-1}} d s_{n} \\dots d s_{1}\n=n!  \\alpha ^\\nu \\int_{\\alpha z_\\nu}^{\\alpha z}  \\int_{\\alpha z_{\\nu+1}}^{s_{\\nu+ 1}} \\dots \\int_{\\alpha z_{n-1}}^{s_{n-1}} d s_{n} \\dots d s_{\\nu+1},$$\nwe find $f^{(\\nu)} _n(\\alpha z_\\nu ) =0$.   Hence  $\\alpha z_\\nu, \\  \\nu=0, 1,\\dots , n-1$ are common\nroots of  $\\nu$-th derivatives  $f^{(\\nu)}$ and $f$,  lying in the unit circle.   Consequently,  since via assumption the\nCasas-Alvero conjecture is true when common roots are inside the unit circle, we have that  $f$ is trivial  and $z_0=z_1=\\dots = z_{n-1} = a$ is a unique joint root of $f$ of the multiplicity $n$.  Proposition 1  is proved.\n\\end{proof}\n\n\n\nThe following lemma will be useful in the sequel.\n\n{\\bf Lemma 10}. {\\it Let  $f$ be a  CA-polynomial with only real roots of degree $n \\ge 2$ and  $\\{x_\\nu\n\\}_{0}^{n-1}$ be a sequence of common  roots of $f$ and the corresponding derivatives $f^{(\\nu)}$.  \nLet  $f^{(s+\\nu)} (x_\\nu) \\ge 0, \\  s =1,2,\\dots, n-\\nu-1$ and $\\nu=0,1,\\dots, n-1$. Then $x_\\nu$ is a\nmaximal root of the derivative  $f^{(\\nu)}$.}\n\n\\begin{proof}  In fact,  the proof is an immediate consequence of  expansion (12), where we let $G_n(x)=f(x)$.  Indeed,  $f^{(\\nu)} (x_\\nu) =0,   \\nu=0,1,\\dots, n-1$ and when $ x > x_\\nu$   we have from (12) $f^{(\\nu)} (x) > 0,   \\nu=0,1,\\dots, n-1$.   So,  this  means that there is no roots, which are bigger than $x_\\nu$. This completes  the proof of Lemma 10.\n\\end{proof}\n\n\n{\\bf Proposition  2}. {\\it  Under conditions of Lemma 10  the Casas-Alvero conjecture holds true for polynomials with only real roots.}\n\n\\begin{proof}  We will show that under conditions of Lemma 10 there exists no CA-polynomial $f$ with only real roots.  Indeed, assuming its existence,  we find via conditions of  the lemma that the root $x_0$ is a maximal zero  of $f(x)$. This means that $x_0 \\ge x_1$. On the other hand, classical Rolle's theorem states that between zeros $x_0, \\  x_1$\nin the case $x_0 > x_1$ there exists at least one zero of the derivative $f^{(1)} (x)$, say $\\xi_1^{(1)}$,\n which is bigger than $x_1$.  But this is impossible because $x_1$ is a maximal zero of the first derivative.\n  Thus $x_0=x_1\\ge x_2$.   Then   between $x_1$ and $x_2$ in the case $x_1 > x_2$ there exists a zero $\\xi_2^{(1)}$\n   of the first derivative such that $x_1>  \\xi_2^{(1)}  > x_2$.  Hence between $x_1$ and   $\\xi_2^{(1)}$ there exists at least one     zero of the second derivative, which is bigger than $x_2$. But this is impossible, since $x_2$ is a maximal zero of $f^{(2)} (x)$.      Therefore   $x_0=x_1=x_2$.   Continuing this process we observe that the sequence  $\\{x_\\nu \\}_0^{n-1}$ is stationary and $f$ has a unique joint root, which contradicts the definition of the CA-polynomial. \n\\end{proof}\n\n{\\bf Corollary 9}. {\\it   There exists no CA-polynomial $f$ with only real roots, having non-increasing sequence $\\{x_\\nu \\}_0^{n-1}$ of roots in common with $f$ and its non-constant derivatives.}\n\n\\begin{proof}    Obviously,  via (13) $f^{(s+\\nu)} (x_\\nu) \\ge 0, \\  s=1,2,\\dots, n-\\nu-1$ and conditions of Lemma 10 are satisfied.\n\\end{proof}\n\n\n{\\bf Corollary 10}. {\\it   There exists no CA-polynomial $f$ with only real roots, such that each $x_\\nu$ in the  sequence $\\{x_\\nu \\}_0^{n-1}$  is a maximal root of the derivative  $f^{(\\nu)} (x),  \\  \\nu=0,1,\\dots, n-1$.}\n\n\\begin{proof}    The proof is similar to the proof of Proposition  2.\n\\end{proof}\n\nAn immediate consequence of Corollaries 3,4,5 is \n\n{\\bf Corollary 11}. {\\it   The CA-polynomial, if any,   with only real roots has at least 5 distinct zeros. }\n\n\n\n Let us  denote by $l(m)$ the  number of distinct roots of the $m$-th derivative $f^{(m)},\\ m=0, 1,\\dots, n-2$,   which are in common with $f$ and different from $\\lambda_1=x_{n-1}$, which is a common root with $f^{(n-1)}$,   i.e.  the $m$-th derivative $f^{(m)}$ has $l(m)$ common roots with $f$\n\n$$\\lambda_{j_1}, \\dots,  \\lambda_{j_{l(m)}}  \\subseteq  \\{ \\lambda_2, \\lambda_3, \\dots, \\  \\lambda_k\\} $$\nof multiplicities\n$$r_{j_1}, \\dots,  r_{j_{l(m)}}  \\subseteq  \\{ r_2, r_3, \\dots, \\  r_k\\} .$$\nFor instance,  $l(0)= k-1, \\   l(1)= k-1-s$, where $s$ is a number\nof simple roots of $f$. So, we see that $n-m \\ge l(m) \\ge 0$ and\nsince $f$ is a  CA-polynomial,  $l(m)=0$ if and only if\n$x_{n-1}=\\lambda_1$ is the only common root of $f$ with $f^{(m)}$.\n\n\n\n{\\bf Lemma  11}. {\\it There exists no  CA-polynomial with only real roots, having the property\n$l(m)= l(m+1)=0$ for some $m \\in \\{r, r+1, \\dots,n-2\\},$ where $r= \\hbox{max}_{1\\le j\\le k} (r_j)$.}\n\n\\begin{proof} In fact, as we saw above, since all roots are real,  it follows that all roots of $f^{(m)}, \\  m \\ge r$ are simple, which contradicts equalities $l(m)= l(m+1)=0$.   Indeed, the latter equalities yield that $x_{n-1}$ is a multiple root of $f^{(m)}$. Therefore $r\\ge r_1 >  m+1\\ge r+1$, which is impossible.\n\\end{proof}\n\nFurther, as in Lemma 7 we involve the root  $\\lambda_{s_0}$ of multiplicity $r_{s_0}$,  and $D= |\\lambda_{s_0}- x_{n-1}|$ (see (35)). Thus $\\lambda_{s_0}= \\lambda_*$ or $\\lambda_{s_0}= \\lambda^*$ and, correspondingly,  $r_{s_0}=r_*$ or $r_{s_0}=r^*$. Hence, calling  Sz.-Nagy identities (15), we  let  $z= x_{n-1}$ and assume without loss of generality that $\\lambda_{s_0}= \\lambda^*$. Then  we obtain for  $m \\ge r$\n$$r_*(x_{n-1}- \\lambda_*) = r^*D + \\sum_{j=2, \\ r_j\\neq r_*,\\  r^*}^{k} r_j(\\lambda_j -  x_{n-1}) \\ge  r^*D - D^{(m)}\n \\sum_{s=1}^{l(m)} r_{j_s} -   D^{(m+1)} \\sum_{s=1}^{l(m+1)} r_{l_s} $$\n$$- \\left(n-r_1-  r^*- r_*-  \\sum_{s=1}^{l(m)} r_{j_s}-  \\sum_{s=1}^{l(m+1)} r_{l_s}\\right)D.$$\nBut $x_{n-1}- \\lambda_*= \\hbox{span}(f)- D$.  Therefore,\n$$r_*  \\hbox{span}(f) + \\left(n-r_1- 2(  r^*+ r_*)\\right) D \\ge (D- D^{(m)})  \\sum_{s=1}^{l(m)} r_{j_s} +\n(D- D^{(m+1)})  \\sum_{s=1}^{l(m+1)} r_{l_s}.$$\nThe right-hand side of the latter inequality is, obviously,  greater or equal to $r_0 \\left(l(m)+ l(m+1)\\right) (D- D^{(m)}) $, where $1 \\le r_0= \\hbox{min}_{1\\le j\\le k} (r_j)$.  Moreover,  since $ \\hbox{span}(f) \\le 2D$,  the left-hand side does not exceed $\\left(n-r_1\\right) D-  r^* \\hbox{span}(f)$. Thus  we come out with the inequality\n$$r_0 \\left(l(m)+ l(m+1)\\right) (D- D^{(m)}) \\le \\left(n-r_1\\right) D-  r^* \\hbox{span}(f)$$\nor since $D- D^{(m)} > 0$ \\ ($m \\ge r$), it becomes\n$$l(m)+ l(m+1)\\le \\frac{\\left(n-r_1\\right) D-  r^* \\hbox{span}(f)}{r_0(D- D^{(m)}) }.\\eqno(43)$$\nMeanwhile, appealing to (16), we get similarly \n$$n(n-1) ( x_{n-1}- x_{n-2})^2 = r^*D^2 + r_* (\\lambda_*- x_{n-1})^2 +  \\sum_{j=2, \\ r_j\\neq r_*,\\  r^*}^{k}\n r_{j}(\\lambda_j-  x_{n-1})^2 $$$$\n\\le  r^*D^2 + r_*\\left(\\hbox{span}(f)- D\\right)^2+  \\left[D^{(m)}\\right]^2 \\sum_{s=1}^{l(m)} r_{j_s} +   \\left[D^{(m+1)}\\right]^2 \\sum_{s=1}^{l(m+1)} r_{l_s}$$$$+ \\left(n-r_1-  r^*- r_*-  \\sum_{s=1}^{l(m)} r_{j_s}-\n \\sum_{s=1}^{l(m+1)} r_{l_s}\\right)D^2.$$\n %\n Therefore, analogously to (43), we arrive at the inequality\n %\n $$l(m)+ l(m+1)\\le   \\frac{(n-r_1)D^2 + r_* \\left[\\hbox{span}(f)\\right]^2 - n(n-1) ( x_{n-1}- x_{n-2})^2 -2D r_*\\  \\hbox{span}(f)}{r_0(D^2- \\left[D^{(m)}\\right]^2) }.$$\n{\\bf Proposition 3}. {\\it There exists no CA- polynomial with only real roots of degree $n$ such that }\n$$  \\hbox{span}(f) >  \\left(r^*\\right)^{-1} \\left[ (n-r_1-r_0)D + r_0 D^{(m)}\\right],\\   m\\ge r.\\eqno(44)$$\n\\begin{proof}  Under condition (44), the right-hand side of (43) is less than one.  Thus $l(m)= l(m+1)=0$ and Lemma 11 completes the proof.\n\\end{proof}\nLet $m=n-2$.   Then since $l(n-1)=0$, inequality (43) becomes\n$$l(n-2) \\le  \\frac{ (n-r_1)D -  r^* \\hbox{span}(f)}{r_0(D-   \\left|x_{n-1}- x_{n-2}\\right|)}.\\eqno(45)$$\n{\\bf Proposition 4}. {\\it There exists no CA- polynomial with only real roots of degree $n$ such that }\n$$D <  \\left[r^* \\  \\sqrt{ \\frac{n^2-r_1}{n-r_1-r_2}} -r_0\\right] \\frac{ \\left|x_{n-1}- x_{n-2}\\right|}{n-r_1-r_0}.\\eqno(46)$$\n\n\\begin{proof}  Indeed, employing the lower bound (38) for $ \\hbox{span}(f)$, we find that under condition (46) the right-hand side of (45) is strictly less than one.  Consequently,  $l(n-2)=0$ and owing to Corollary 1 $f$ is trivial. If the maximum of multiplicities $r > n-2$,    $f$ has at most 2 distinct zeros and  it is trivial via Corollary 3.\n\\end{proof}\n\nFinally, we prove \n\n{\\bf Proposition 5}. {\\it Let  CA- polynomial with only real roots exist.  Then it has the property\n$$  \\frac{d}{D} \\le \\sqrt{\\frac{2(n-m-1)}{2(k-1)-1}},\\eqno(47)$$\nwhere $d, D$ are defined by $(34), (35)$, respectively,  and $m,\\ m+1$ belong to the interval $\\left[r, \\   {1\\over 2}\\left(1-{1\\over r_0}\\right)(n-1) \\right)$.}\n\\begin{proof}  Since $m,\\ m+1$ are chosen from  the interval $\\left[r, \\   {1\\over 2}\\left(1-{1\\over r_0}\\right)(n-1) \\right)$,  condition (24) holds for these values. Hence assuming the existence of the CA-polynomial, we return to the Sz.-Nagy type identity (25) to  have the estimate \n$$0 \\ge  l(m)  \\left( r_{0} {(n-m)(n-m -1)\\over n(n-1)} -1\\right) d^2 +  \\left(k-1-l(m)\\right) d^2- (n-m-l(m)) D^2$$\n$$\\ge  (k-1) d^2- (n-m) D^2  + l(m) (D^2-d^2).$$ \nWriting the same inequality for $m+1$\n$$0 \\ge  (k-1) d^2- (n-m-1) D^2  + l(m+1) (D^2-d^2)$$ \nand adding two inequalities, we find\n$$0 \\ge  2 (k-1) d^2- (2(n-m) -1) D^2  +(l(m)+  l(m+1)) (D^2-d^2),$$\nwhich means\n$$l(m)+  l(m+1) \\le \\frac{(2(n-m) -1) D^2 - 2 (k-1) d^2}{D^2-d^2}.$$\nSo, for the existence of the CA-polynomial it is necessary that the right-hand side of the latter inequality is more or equal to 1.  Thus we come out with condition (47) and complete the proof. \n\\end{proof}\n\n\n\n\n{\\bf  Acknowledgment}.   The present investigation was supported, in\npart, by the \"Centro de Matem{\\'a}tica\" of the University of Porto.\n\n\n\n\n\\bibliographystyle{amsplain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}