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Paraskevopoulos1,2, F. Sebastiano1,2, C. G. Almudever3, and S. Feld1,2 +1Quantum and Computer Engineering Department, Delft University of Technology, 2628 CD Delft, The Netherlands +2QuTech, Delft University of Technology, 2628 CJ Delft, The Netherlands and +3Computer Engineering Department, Technical University of Valencia, Camino de Vera, s/n, 46022 Val`encia, Spain +In most qubit realizations, prototype devices are available and are already utilized in both industry and aca- +demic research. Despite being severely constrained, hardware- and algorithm-aware quantum circuit mapping +techniques have been developed for enabling successful algorithm executions during the NISQ era, targeting +mostly technologies with high qubit counts. Not so much attention has been paid to the implementation of com- +pilation methods for quantum processors based on spin-qubits due to the scarce availability of current experi- +mental devices and their small sizes. However, based on their high scalability potential and their rapid progress +it is timely to start exploring quantum circuit mapping solutions for these spin-qubit devices. In this work, +we discuss the unique mapping challenges of a scalable spin-qubit crossbar architecture with shared control +[1] and introduce SpinQ, the first native compilation framework for scalable spin-qubit architectures that maps +quantum algorithms on this crossbar architecture. At the core of SpinQ is the Integrated Strategy that addresses +the unique operational constraints of the crossbar while considering compilation (execution time) scalability, +having a O(n) computational complexity. To evaluate the performance of SpinQ on this novel architecture, we +compiled a broad set of well-defined quantum circuits and performed an in-depth analysis based on multiple +metrics such as gate overhead, depth overhead, and estimated success probability, which in turn allowed us to +create unique mapping and architectural insights. Finally, we propose novel mapping technique improvements +for the crossbar architecture that could increase algorithm success rates and potentially inspire further research +on quantum circuit mapping techniques for other scalable spin-qubit architectures. +I. +INTRODUCTION +The prospect of quantum computing advantage is steadily +becoming a reality [2–4]. The community is anticipating fur- +ther advances that will allow quantum computing systems to +become practical and to reach computational advantage [5]. +With such advancements, quantum computing systems are ex- +pected to solve a plethora of classically intractable problems. +Until then, current quantum systems belong to the so-called +Noisy Intermediate-Scale Quantum (NISQ) era [6], in which +devices can only handle small-sized quantum circuits. This is +due to limitations in the number of qubits and high operational +errors, the latter causing rapid quantum information deteriora- +tion. Combined with even more hardware constraints, such as +cross-talk and limited classical-control resources [7, 8], suc- +cessful quantum circuit execution is a difficult feat. Scientists, +both in academia and industry, face major engineering chal- +lenges in building both, hardware and corresponding system +software. +During the NISQ era, there have been significant efforts +[9–19] to extract the most out of these resource-constrained +and error-prone quantum computing systems. One of the ap- +proaches to do so is by developing hardware- and algorithm- +aware quantum circuit mapping techniques to maximize per- +formance. In general terms, mapping refers to the process of +modifying (potentially hardware-agnostic) quantum circuits +in such a way that they can be run on a given quantum com- +puting device by respecting all of its constraints while opti- +mizing performance (e.g., algorithm success rate). +So far, +several mapping techniques have been developed mostly for +superconducting and ion-trap qubit devices, as they are nowa- +days one of the most well-recognized and most-developed +qubit implementation technologies in terms of qubit counts +and availability to users. +However, spin-qubits emerge as +a promising technology for scaling up quantum computing +systems mainly due to their high integration potential [20– +25]. Therefore, the scientific community is envisioning two- +dimensional spin-qubit architectural proposals that could al- +leviate some of the major challenges towards scalability. Re- +cently, a crossbar array [26] has been experimentally demon- +strated showing great promise for architectures with shared +control. Such scalable architectural designs come with a new +set of hardware constraints for which novel quantum circuit +mapping techniques need to be developed. +In this paper, we present SpinQ, the first native compilation +framework focusing on scalable spin-qubit architectures. To +this purpose, we target the so-called crossbar architecture pro- +posed in [1]. By creating a deep understanding of its opera- +tional constraints, we draw a clear picture of unique mapping +challenges that arise in comparison to other qubit technolo- +gies. We discuss and implement possible mapping solutions +based on [27, 28] while improving those works from a scala- +bility standpoint. We emphasize the importance of performing +an extensive performance evaluation process of novel map- +ping techniques. Note, that this compilation framework will +not only allow quantum algorithm executions on scalable spin +qubit hardware but more importantly will also help to form +insights on the behaviour and performance of this new breed +of architectures and provide some design guidelines for future +developments. +The main contributions of this paper are: +1. An in-depth analysis of mapping challenges in order to +create novel mapping techniques for spin-qubit crossbar +architectures. +2. SpinQ – The first native compilation framework dedi- +cated to scalable spin-qubit architectures which utilizes +a more scalable compilation strategy compared to pre- +arXiv:2301.13241v1 [quant-ph] 30 Jan 2023 + +2 +vious proposals. +3. A thorough performance analysis of the main sources of +gate/depth overhead and estimated success probability +when mapping well-defined quantum algorithms on the +crossbar architecture. +4. Deriving algorithmic- and hardware-specific mapping +insights for the crossbar architecture and other spin- +qubit architectures. +The remainder of this paper is structured as follows: In Sec. +II the current progress and challenges of scalable spin-qubit +architectures are presented. In Sec. III the crossbar architec- +ture is introduced as a potential candidate in scaling quantum +devices in two dimensions, as well as its native operations. In +Sec. IV we comprehensively analyse the unique challenges +of mapping quantum algorithms on the crossbar architecture +which require novel mapping techniques. Then, in Sec.V we +introduce SpinQ – the first native compilation framework for +scalable spin-qubit architectures. In Sec. VII we thoroughly +analyze the performance of SpinQ when mapping a broad and +well-defined range of quantum algorithms on the crossbar ar- +chitecture after which we form architectural and mapping in- +sights. In Sec. VIII we discuss potential improvements of our +compilation strategy and we compare its computational com- +plexity to previous proposals. Finally, we conclude our work +in Sec. IX. +II. +SPIN QUBITS AS A SCALABLE PLATFORM +To fulfil the promise [6] of quantum computers being ma- +chines that solve some classically intractable problems, sub- +stantial system sizes have to be reached, i.e., a large number +of qubits [8, 29]. It still remains to be seen which qubit imple- +mentation technologies (e.g., superconducting, trapped ions, +quantum dots, photonics, defect-based on nitrogen-vacancy +diamond centres) will succeed in scaling up quantum comput- +ing systems with high-quality qubits [30, 31]. Spin qubits in +quantum dots are a promising technology for scalable quan- +tum computers due to the maturity of the semiconductor in- +dustry, the capability of high integration on a single die com- +pared to other qubit technologies, long coherence times, and +the ability to operate in super-kelvin temperatures [20–25]. +Despite the advantages just mentioned, there are still sev- +eral challenges today towards scaling spin-qubit devices in a +sustainable manner. One major challenge is the wiring scheme +between the quantum processor and the classical interface, the +so-called interconnect bottleneck [22]. Formally, the intercon- +nect bottleneck is described by Rent’s exponent [32], which is +a measure of optimization in the wiring scheme in both, clas- +sical and quantum processors. The existing scheme in most +quantum devices of having at least one control line per qubit +is not scalable in the long term. This is, mostly, due to the fact +that dilution refrigerators have an upper limit to I/O cable ca- +pacity and that more cables will progressively make it harder +to reach the desired milli-Kelvin temperature due to higher +heat dissipation. Therefore, qubit architectures and classical- +control electronics have to support multi-qubit shared-control +that requires a sub-linear number of control lines with an in- +creasing number of qubits. In other words, each control line +needs to address multiple qubits to effectively mitigate the in- +terconnect bottleneck when scaling up quantum hardware. +Going a step further, the inability to achieve a scalable +wiring scheme also originates from the low device unifor- +mity achieved by today’s fabrication tools. +In most cases, +this implies that qubits can not be made homogeneous enough +to control them effectively in a scalable architecture. +The +low uniformity results in resonance frequency deviations or +other control variations. This means that in an inhomoge- +neous device, a driving signal for a particular operation will +have to vary from one qubit to another to get the same out- +come [1, 22, 33]. This makes it difficult to successfully con- +trol many qubits with the same control line, thus contributing +to the wiring scheme challenge (i.e., the interconnect bottle- +neck). +There have been significant efforts [1, 22, 32, 34–38] to re- +duce the number of control lines reaching the qubits as de- +vices become ever denser. +Such efforts take advantage of +the miniaturization capabilities of spin qubits and the large- +scale integration of solid-state circuits to address the afore- +mentioned challenges. However, current experimental work +primarily has been focused on one-dimensional spin-qubit ar- +rays of small sizes [22], which are not easily scalable. Re- +cently, a 2×2 spin-qubit processor [39] and a 4×4 spin-qubit +device based on a crossbar architecture [1, 26] with shared +control has demonstrated the potential to scale spin-qubit de- +vices in two dimensions. +As the technology is advancing +and further reducing Rent’s exponent, there will be a need to +effectively map quantum algorithms on two-dimensional de- +vices such as the crossbar architecture which comes not only +with limited qubit connectivity but also with a new set of con- +straints. Therefore, there is an opportunity to explore its map- +ping challenges and propose novel solutions. +However, the sample space of these proposals is sparse and +lacks a detailed description of hardware constraints. In com- +bination with a lack of available devices for testing, leads to +a lack of a proper evaluation tool capable of benchmarking +various quantum algorithms. Therefore, mapping techniques +have not been studied as much as other qubit technologies +such as superconducting and ion traps. It also remains un- +clear whether existing techniques could be applicable. Then, +even if such techniques are realized they could be incompati- +ble with existing quantum compilation frameworks made for +other qubit technologies. This could be due to completely +different development requirements imposed by the particular +spin-qubit constraints and their scalability prospects. In other +words, a dedicated compilation framework for spin-qubit ar- +chitectures with a focus on scalability is still missing. All +these obstacles make it difficult to evaluate and compare var- +ious architectural proposals under relevant application cate- +gories. + +3 +CL0 +CL1 +CL2 +RL2 +RL1 +RL0 +1 +4 +2 +3 +5 +6 +8 +7 +QL-3 +QL-2 +QL-1 +QL0 +QL1 +QL2 +QL3 +FIG. 1: Schematic overview of the crossbar architecture and +operational control lines [1]. +III. +THE CROSSBAR ARCHITECTURE +The crossbar architecture for arranging spin qubits was in- +troduced in [1] as a scalable solution to the interconnect bot- +tleneck. Inspired by the crossbar architecture used in today’s +classical processors [1, 34], it adopts a similar characteristic, +namely shared control. This leads to a quadratic reduction +in control lines per qubit [28] and opens up the possibility +for high integration of up to 1, 000 qubits in a single pack- +age. Qubits are defined by electron (or hole) spin states in +Si-based quantum dots. In Figure 1, we illustrate a schematic +overview of the crossbar architecture in which each site (cir- +cles) represents a quantum dot, some of which are occupied +by spin-qubits (numbered, green circles). +Spin qubits are +usually sparsely initialized in a checker-board pattern to re- +duce potential cross-talk and to allow for long-range entan- +glement through shuttling qubits across the array [1]. Finally, +the crossbar architecture requires high uniformity in the fab- +rication of materials to minimize operational errors. Fortu- +nately, it is possible to mitigate such errors or even vanish +them by operating the crossbar at low magnetic fields and with +proper tuning (e.g., separated resonance frequencies between +columns). Furthermore, a crossbar module is envisioned to be +self-contained and to be duplicated in a network of modules. +This can provide the means to realize quantum error correc- +tion (QEC) in large-scale systems enabled by fast-shuttling, +low-error communication links. In this crossbar architecture, +three different kinds of shared control lines are used to per- +form operations on the qubits: vertical (column line, CL), +horizontal(row line, RL), and diagonal (qubit line, QL). No- +tably, each line affects all the sites that it is connected to. For +instance, in Fig. 1 line QL−2 affects the sites in which spin- +qubits 5 and 7 reside in. This imposes some restrictions in +the parallelization of instructions, which we will discuss in +Sec. IV. Below, we will abstractly describe the control prop- +erties for executing gates native to the crossbar architecture. +A more detailed explanation is provided in [28]. Two-qubit +gates. Two two-qubit gates CPHASE and +√ +SWAP are +supported by the crossbar, with the latter being chosen for this +work due to its higher operational fidelity and faster execu- +tion time according to [1]. A +√ +SWAP can be performed +when two qubits are vertically adjacent (i.e. same column) +and the horizontal barrier between them is lowered. Then the +QL lines going through the two qubits need to be in the same +voltage potential for a specific duration of time to complete +the +√ +SWAP. Qubit shuttling. In the crossbar architecture, +qubits can be moved around by performing shuttling opera- +tions. In this operation, the vertical and horizontal lines are +used as barrier gates. Lowering or raising these barriers can +create pathways from which qubits can move (shuttle) from +one site to another with the use of DC signals through the +diagonal lines. +Fig. +2 shows an example of shuttling, in +which spin-qubit 3 is moved one site to the left. Although +this architecture can support gate-based communication with +two subsequent +√ +SWAP gates as in superconducting qubits, +shuttling qubits is preferred due to higher operation fidelity +and shorter execution time. It should be noted that shuttling +horizontally, i.e., between columns, causes a Z-phase rota- +tion which should be mitigated by timing such operations well +([1]). In the crossbar architecture, single-qubit gate rotations +should be separated into two categories: Z-phase rotations and +X or Y rotations. Z-phase rotations. Z-phase qubit rotations +are controlled by a well-timed qubit shuttling to and from a +neighbouring column [1, 27, 28]. This is due to the differences +in Zeeman energies from column to column which imposes +an alternating magnetic field on qubits, thus rotating them in +the Z axis. When this shuttle is timed correctly, it rotates the +qubit in the correct Z state. The diagonal qubit line provides +the means to address multiple qubits, thus enabling parallel Z +phase shifts across the topology. X or Y rotations. As for +X or Y rotations, either all qubits belonging to red-coloured +columns or all qubits in blue-coloured columns are rotated +(see Fig. 1). This is called semi-global qubit rotation im- +plemented by electron-spin-resonance ([40]). Measurement. +The process of readout allows for local single qubit measure- +ments based on the Pauli Spin Blockade (PSB) process [41]. +With this process, the measurement outcome is determined by +whether a qubit shuttle towards a horizontally adjacent ancilla +qubit was successful or not. +IV. +QUANTUM CIRCUIT MAPPING CHALLENGES OF +THE CROSSBAR ARCHITECTURE +The quantum circuit mapping process plays an essential +role in the successful execution of algorithms on a quantum +computer. +It consists of a cascade of routines that trans- +form a (potentially hardware-agnostic) quantum circuit to a +hardware-compatible version. However, current NISQ quan- +tum processors are severely constrained and cannot run use- +ful applications successfully, yet. Examples of hardware con- +straints are low qubit connectivity, cross-talk, reduced prim- +itive gate set, low coherence time, fabrication imperfections, +and limited classical-control resources. Therefore, a mapping + +4 +process needs to consider such limitations and try to optimize +performance as much as possible to increase the algorithm’s +success rate. So far, there are a plethora of proposed solu- +tions which differ in strategy, methodology and performance +metrics to optimize [9–19, 42]. +Mapping techniques have been mostly developed for super- +conducting and ion-trap qubit devices. However, as of now, +there is not much focus on spin-qubit architectures and their +particular characteristics. Although spin-qubits are now in a +rather early development stage, their scalability potential is +undeniable and therefore it is timely to lay grounds for devel- +oping novel mapping techniques and inspire further research. +As previously mentioned, in this work we focus on the cross- +bar architecture that comes with a unique set of constraints +that affect the parallelization of quantum operations, the ap- +plication of X or Y rotations on single qubits, and the routing +of qubits (i.e. moving qubits around). +1. +Parallelization of quantum operations +CL0 +CL1 +CL2 +RL2 +RL1 +RL0 +1 +4 +2 +3 +5 +6 +8 +7 +QL-3 < QL-2 > QL-1 > QL0 +> +QL1 +QL2 +QL3 +FIG. 2: Shuttling example of qubit 3 moved one site to the +left, as the barrier CL0 between origin and destination site is +lowered and voltage of QL−1 is larger than > QL0. +Most of the operation parallelization restrictions come from +the fact that control lines are shared among multiple qubits, +while each line has a specific role and relation to one another. +It should also be noted that most operations must be imple- +mented with strict pulse durations and time intervals depend- +ing on the site that gets addressed [1] due to fabrication imper- +fections [28]. Although such pulse durations have to be care- +fully considered in the mapping process by providing recent +calibration data [13, 17, 18], in this work we consider an ideal +crossbar architecture, as such data are not available yet. De- +spite that, the mapping techniques proposed in this work are +compatible with such considerations and can be added once +calibration data are available. +To better illustrate what the conditions and constraints are +CL0 +CL1 +CL2 +RL2 +RL1 +RL0 +1 +4 +2 +3 +5 +6 +8 +7 +QL-3 < QL-2 > QL-1 > QL0 +>< +QL1 +< +QL2 +> +QL3 +FIG. 3: Parallelizing shuttles of qubit 3 and 6 is not allowed +due to violation of constraints. +when trying to parallelize quantum operations, let us consider +the following example in which two shuttles are performed in +parallel. As shown in Fig. 2, the following requirements must +be fulfilled to shuttle qubit 3 one site to the left: +1. The destination site must not be occupied by another +qubit. +2. The barrier between destination and origin sites must be +lowered. This is depicted as a dashed vertical CL0 line. +3. All barriers surrounding the origin and destination sites +must be raised. This is shown as solid red RL (RL0 and +RL1) and blue CL lines (CL1 and the always-raised +most-left CL line). +4. The voltage going through the QL line of the destination +site (QL−1) must be higher than the one going through +the origin site (QL0). This is shown as QL−1 > QL0 +in the top-right of Fig. 2. +5. To prevent other qubits in these two columns from shut- +tling, the voltage going through their QL lines must be +higher than their adjacent empty sites. This is depicted +as voltage level relations between QL lines. Note that +QLs with no voltage relations are irrelevant for this par- +ticular shuttle operation. +Now, we assume a shuttle of qubit 6 to the right (as de- +picted in Fig. +3) in parallel to the left shuttle of qubit 3. +This implies that all previously listed requirements (of qubit +3) need to be satisfied along with the new ones (of qubit 6). +However, the fourth requirement can not be satisfied as the +QL0 > QL1 relation we had before would have to be changed +to QL0 < QL1. If this change is allowed, we violate the fifth +requirement of the first shuttle and, as a consequence, qubit 1 +will shuttle to the right. Therefore, we can not shuttle qubits +3 and 6 at the same time. + +5 +Thus, we see that scheduling parallel gates in the crossbar +implies a strict simultaneous satisfaction of all signal require- +ments for each gate. Any violation of these conditions would +potentially result in shuttling of unwanted qubits, in unwanted +qubit interactions or unknown qubit states. As seen in the +previous example, performing quantum operations in parallel +without affecting other qubits and meeting all signal require- +ments is not always possible regardless of qubit distance. In +fact, it does not matter how far qubits are away from each +other, but whether control lines are shared between them or +not, and whether their operational requirements and relations +match or not. Unlike more popular qubit architectures based +on superconducting or ion traps, this form of operational con- +straint is unique. On one hand, sharing control lines tackles +the interconnect bottleneck, on the other hand, it intrinsically +constraints its parallelization capabilities. +Finally, in other qubit architectures, it is possible to per- +form different gate types in parallel. In the crossbar archi- +tecture, this is not always the case. For example, applying +single-qubit gates and shuttling operations at the same time is +not possible (see Fig. 4a), because the former CL lines need +to carry an alternating current (AC) signal while the latter re- +quire DC signals for raising or lowering the barriers. +2. +Application of X or Y rotations on single qubits +As established in Sec. III, X or Y qubit rotations are im- +plemented semi-globally, meaning that either all qubits in odd +or even column parities will be rotated. However, during an +arbitrary cycle of algorithm execution, not all qubits in odd +or even columns should be rotated. Therefore, to compen- +sate for unwanted X or Y rotations, one has to come up with +a specific rotation scheme such that only the targeted qubits +are rotated. In this work, we have implemented the scheme +introduced by [28]. We illustrate how it works in Fig. 4, in +which we are interested in rotating only qubit 5. This is an- +other unique characteristic of this architecture, as additional +gates are needed to perform single-qubit rotations on specific +qubits, which impose new challenges to the mapping process. +3. +Routing of Qubits +While we previously described the operational constraints +to parallelize various gates in the crossbar architecture, we +will now expand specifically on the qubit routing challenges. +Routing a qubit in the crossbar means that an electron +(or a hole) is physically ”pushed” to an empty site (i.e., an +empty quantum dot). This mechanism is similar to a Quan- +tum charged coupled device (QCCD) ion trap device when +ions are shuttled through a common channel from trap to trap, +assuming sufficient destination ion trap capacity [43]. The +QCCD architecture and the crossbar architecture fundamen- +tally differ in topology, but both require special algorithms or +additional routing routines to maintain control of qubit posi- +tions and avoid potential conflicts. +Focusing on the crossbar, shuttling a qubit does not only +depend on specific control signal requirements and available +empty sites but on the positions of other qubits as well. We +illustrate this fact with an example in Fig. 5, in which a verti- +cal shuttle operation of qubit 3 is indicated by a black arrow. +In this case, the horizontal barrier RL0 has to be lowered and +the QL lines have to be pulsed in certain voltage relations to +allow for correct shuttling. However, an unwanted interaction +between two other qubits in the same row (qubits 2 and 4, cir- +cled) is concurrently caused, regardless of the QL2 and QL3 +relation. Analogously, the same issue exists with a horizontal +shuttle when having two horizontally adjacent qubits in the +same columns where the shuttle takes place [27, 28]. Lastly, +there can be a blocked path conflict where there is no empty +site for a qubit to shuttle to. +Therefore, a dedicated qubit routing algorithm for the +crossbar architecture has to be developed to avoid collisions, +blocked paths, and unwanted interactions. Furthermore, even +if we had such a dedicated routing algorithm, the same con- +flicts have to be considered and prevented when scheduling +gates in parallel. +For that, control signals and qubit posi- +tions must be carefully monitored within the mapping pro- +cess. From the description above, it is clear that both, routing +and scheduling processes, need to jointly work in a strategy to +avoid conflicts and optimize for algorithm success rate. This +will be part of SpinQ, presented in the following section. +V. +SPINQ – THE FIRST NATIVE COMPILATION +FRAMEWORK FOR SCALABLE SPIN-QUBIT +ARCHITECTURES +In this work, we present the first native compilation frame- +work – SpinQ – dedicated to compiling and mapping quan- +tum circuits onto scalable spin-qubit architectures, such as the +previously described crossbar. We have based our mapping +techniques on previous works from [27, 28] while improving +them from a scalability standpoint. +Fig. 6 shows the schematic structure of our framework. As +input, SpinQ accepts QASM format files that describe quan- +tum circuits (used as benchmarks) in a device-independent +manner. To increase flexibility, custom operations and their +particular attributes can be defined in a hardware architec- +tural configuration file. It can include operational attributes +such as the duration of a gate, the mathematical description +of the unitary matrices, associated gate fidelities, and archi- +tectural constraints, among others. Moving on, the compiler +consists of a series of steps (called passes) to decompose +gates, route qubits, and schedule instructions. To address the +unique mapping constraints of the crossbar architecture, we +have conceptualized and developed the integrated strategy. +We did so not necessarily to maximize the performance of the +algorithms when being executed on the crossbar, but rather to +study how they behave on such architecture and focus on the +scalability potential of spin-qubit technologies (see also Sec. +VIII). The compiler’s output is a QASM file which is compat- +ible to be executed on the given crossbar architecture. Option- +ally, a verification step can take place to ensure the compiled + +6 +CL0 +CL1 +CL2 +RL2 +RL1 +RL0 +1 +4 +2 +3 +5 +6 +8 +7 +QL-3 QL-2 +QL-1 QL0 +QL1 +QL2 +QL3 +(a) Step 1 +CL0 +CL1 +CL2 +RL2 +RL1 +RL0 +1 +4 +2 +3 +5 +6 +8 +7 +QL-3 < QL-2 < QL-1 < QL0 +> +QL1 +QL2 +QL3 +(b) Step 2 +CL0 +CL1 +CL2 +RL2 +RL1 +RL0 +1 +4 +2 +3 +5 +6 +8 +7 +QL-3 QL-2 +QL-1 QL0 +QL1 +QL2 +QL3 +(c) Step 3 +CL0 +CL1 +CL2 +RL2 +RL1 +RL0 +1 +4 +2 +3 +5 +6 +8 +7 +QL-3 < QL-2 > QL-1 < QL0 +> +QL1 +QL2 +QL3 +(d) Step 4 +FIG. 4: Single-qubit gate on qubit 5: (a) Step 1: AC signals through the CL lines induce magnetic fields on qubits 1, 5, 6 and 2, +thus changing their state. The direction and frequency of these signals determine which columns (red or blue) and what rotation +(X or Y gate) will be applied to the corresponding qubits. (b) Step 2: The targeted qubit 5 is moved with a shuttle operation to +a different column parity. For this operation, the orthogonal lines (CL and RL) open and close as barriers and the diagonal lines +(QL) create potential gradients to allow for qubit 5 to move (shuttle). Note that QL needs to have voltage relations with their +neighbour QL lines. (c) Step 3: An inverse rotation is applied to qubits 1, 6 and 2 similarly to Step 1. (d) Step 4: Target qubit 5 +is moved with a shuttle operation to the initial position. +CL0 +CL1 +CL2 +RL2 +RL1 +RL0 +1 +4 +2 +3 +5 +6 +8 +7 +QL-3 +QL-2 QL-1 < QL0 +< +QL1 +QL2 +>< +QL3 +FIG. 5: Example of a conflict: the operational requirements +of shuttling qubit 3 downwards have lowered the RL0 barrier +thus causing an unwanted interaction between qubits 2 and 4 +circuit meets all operational constraints of the crossbar archi- +tecture without any conflicts. This step is implemented to be +able to check the compatibility of architectural proposals that +are not physically realized yet. Finally, several performance +metrics are extracted from the compiled circuit to evaluate +algorithm performance. In the next sections, we will further +discuss each of the elements of the compiler. +A. +Compilation steps +The compiler consists of the following steps: +Quantum Algorithms +(Benchmarks) +Compiler +Gate Decomposition +Initial Placement +Integrated Strategy +(Routing and Scheduling) +Architectural Configuration +Compiled Circuit +Metrics +Verification +Depth +Overhead +Gate Overhead +Estimated +Success +Probability +Compilation +time +Operational fidelities +Operational durations +Architectural constraints +Topology +FIG. 6: Overview of our SpinQ framework proposed in this +paper. +Decomposition of quantum gates. Inputted QASM quan- +tum circuits are transformed into a custom-made intermediate +representation (IR) data format. Quantum gates are then de- +composed into gates native to the architecture based on the +decomposition sequences specified in the architectural con- +figuration file. +Physical initialization of spin qubits. A checkerboard pat- +tern has been proposed [42] to allow space for qubits and an- +cilla qubits to move [27, 28]. The physical space achieved be- +tween the qubits not only facilitates shuttling qubits avoiding +possible conflicts but also reduces crosstalk and enables sur- +face code error correction [1]. As we will discuss later, main- +taining this placement throughout a circuit execution plays an +integral role in our compilation strategy. Having said that, +initializing qubits in alternative patterns and changing them +during execution is possible. This flexibility can be particu- +larly advantageous to highly specialized mapping techniques +for the crossbar as well as spin-qubit architectures in general. + +7 +Virtual-to-physical qubit initial placement. The current +version of SpinQ associates virtual qubits of an algorithm with +physical qubits (placed in the checkerboard pattern) in a one- +to-one manner by numbering the physical qubits from left to +right and from bottom to top (as shown in Fig. 1. In the re- +sults sections VII and VIII, we will provide insights on how +common initial placement algorithms can be adapted to im- +prove the performance of spin-qubit architectures (such as the +crossbar). +Integrated Strategy for Routing and Scheduling. As ex- +plained in Sec. IV, both routing and scheduling techniques +must avoid conflicts. To do that, a specific strategy needs +to be followed. There can be various strategies for various +goals with trade-offs between performance and compilation +time. The presented Integrated Strategy tilts towards mini- +mizing compilation time while having great prospects to be +competitive against other solutions that focus on algorithm +performance as will be discussed in Sec. VIII. +Firstly, in the Integrated Strategy, the checkerboard pat- +tern qubit placement [1], also known as ”idle-configuration” +in [28], should be maintained as much as possible. This pro- +vides at least two empty sites for every qubit to move towards +to, at the beginning of each cycle. To maintain the checker- +board pattern throughout circuit execution when routing for +two-qubit gates, a conflict-free shuttle-based SWAP technique +can be used ([27]) as shown in Fig. 7. Note that this move- +ment of qubits results in a gate overhead of 4 (i.e., 4 shuttle +operations), but a depth overhead of 2, as these two shuttle +pairs can always be executed in parallel. To bring the two +qubits to the appropriate sites and allow two-qubit interac- +tions, multiple shuttle-based SWAPs might be performed. For +that, we have implemented a shortest-path algorithm based +on the Manhattan distance. When one of the qubits is placed +in the desired position, the next step is a horizontal shuttle, +either to the left or to the right, after which the target and +control qubits are vertically adjacent for interaction, and the +checkerboard pattern is temporarily broken. Proceeding the +√ +SWAP, a shuttle instruction returns the qubit to the previ- +ous position and the checkerboard pattern gets restored. Note +that the aforementioned process can be successfully executed +only in that particular order, otherwise there can be a routing +conflict. +So far, we have only talked about a routing technique +for bringing together qubits for performing two-qubit gates. +However, qubit routing is also needed for X or Y rotations to +a specific qubit(s) and for shuttle-based Z rotations, as dis- +cussed in Sec. III. As a consequence, the ”idle configura- +tion” should be maintained when routing for these gates as +well. But once again, routing for single-qubit gates before +the scheduling stage can be problematic as it can cause con- +flicts. For that reason, the second consideration of the Inte- +grated Strategy is the integration of single-qubit gate routing +within the scheduling stage, hence the name ”integrated”. +Then the Integrated Strategy continues with two passes. In +the first pass, the scheduler tries to parallelize X or Y gates +in an ideal manner and Z gates individually, ignoring any po- +tential conflicts. This is no different than other single-qubit +gate scheduling processes proposed for other qubit architec- +tures. However, it differs on the second pass which integrates +the routing procedures for X, Y and Z gates. The second pass +iterates over each cycle produced by the first pass. For each +cycle, there are two possibilities: (a) if no conflicts are de- +tected when scheduling the necessary shuttle instructions re- +quired for each single-qubit gate, the new shuttle instructions +are inserted between the current cycle and the next. (b) if con- +flict(s) are detected, the subset of the problematic gate(s) is +removed and stored. Once the non-problematic gate subset +is scheduled according to case (a), the problematic subset is +recalled and iterated again. This time it constitutes a conflict- +free cycle and is scheduled according to case (a). This way +the second pass loops in total two times whenever there is a +detected conflict. +Overall, the current implementation does not parallelize +gates of different types in the same cycle, and thus each cycle +is dedicated to one instruction type. Fortunately, the strategy +described above and suggested extensions in Sec. VIII can be +adapted to a real setup. As explained in Sec. IV, a fabricated +crossbar device will most likely have material imperfections, +thus requiring pulse calibration per site. As pointed out by +[28, 44], pulsing control lines prematurely to account for ma- +terial variations could cause an unwanted interaction. Since, +however, the Integrated Strategy (or an extension thereof) ex- +clusively schedules gates of the same type in each cycle, fine- +tuning pulses within the cycle is possible before moving to the +next. +B. +Performance metrics +We will now introduce the metrics used in this work to eval- +uate the performance of SpinQ when mapping different algo- +rithms on the crossbar architecture. +Gate overhead. One commonly used metric to evaluate +the performance of a mapper and its underlying architecture +is gate overhead. We calculate it as the percentage relation of +additional gates inserted by the mapper to the number of gates +after decomposition. We do not count decomposition gate +overhead as it is always proportional to the number of gates. +Getting a clear view of the various sources of gate overhead +will help to form useful insights. Note that, unlike supercon- +ducting architectures where gate overhead results from rout- +ing instructions (i.e. SWAP gates) for performing two-qubit +gates, in the crossbar, it can be caused by single-qubit gates as +well. The main sources of gate overhead are the following: +• 4 additional shuttle instructions per shuttle-based +SWAP for two-qubit gates +• At least 3 additional instructions for each X or Y rota- +tion gate within the semi-global rotation scheme +• 2 additional shuttle instructions for each two-qubit gate +• 1 additional shuttle operation for each Z rotation gate +Depth overhead. Another commonly used metric to eval- +uate the performance of a mapper and its underlying architec- +ture is the depth overhead of a circuit. The depth of a circuit is + +8 +CL0 +CL1 +CL2 +RL2 +RL1 +RL0 +1 +4 +2 +3 +5 +6 +8 +7 +QL-3 < QL-2 > QL-1 > QL0 +< +QL1 +QL2 +QL3 +(a) Horizontal shuttling +CL0 +CL1 +CL2 +RL2 +RL1 +RL0 +1 +4 +2 +3 +5 +6 +8 +7 +QL-3 +QL-2 QL-1 < QL0 +> +QL1 +< +QL2 +> +QL3 +(b) Vertical shuttling +FIG. 7: Shuttle-based SWAP for two-qubit gate routing: With this technique, two diagonally neighbouring qubits exchange +their position by consecutively performing two horizontal and two vertical shuttles. +equal to the minimum number of time steps of a circuit when +executing gates in parallel [9, 10, 45–47]. We calculate depth +overhead as the percentage relation of additional depth pro- +duced by the mapper to the circuit depth after decomposition. +Note that the initial circuit depth is calculated after scheduling +the circuit only by its gate dependencies, meaning without any +architectural constraints. The main sources of depth overhead +are: +• At least 3 additional cycles for each X or Y rotation +gate due to the semi-global rotation scheme +• 2 additional cycles per shuttle-based SWAP for two- +qubit gates +• 2 additional cycles for each two-qubit gate +• 1 additional cycle for each Z rotation gate +Estimated Success Probability. A key metric to assess the +performance not only of the compiler but in general of a quan- +tum computing system is the algorithm success rate. From an +experimental point of view, the algorithm success rate is cal- +culated by executing the algorithm several times on a given +(real) quantum processor and creating the distribution of suc- +cessful executions, based on the expected measurement. An +alternative way to calculate the success rate without the need +for a real quantum processor is by using an approximation nu- +merical method. One of the most commonly used methods to +do so is considering the estimated success probability (ESP) +of an algorithm [48]: +ESP = +� +i +� +j +gate fidelityi,j +(1) +where i represents the ith time step and j the jth gate in the +ith time step. +This method is far more efficient compared to using a +Hamiltonian model. However, the accuracy of the estima- +tion can be low due to its simplicity. To expand it, we have +considered a per-type and per-location variability of gate fi- +delities, based on a normal distribution. This implies that, for +instance, a two-qubit gate ( +√ +SWAP) will have lower fidelity +than a single-qubit gate and that the actual fidelity will depend +on the exact location in the topology. These expansions con- +stitute a more realistic, i.e., closer to a real device, estimation +of circuit success probability: +ESP = +� +i +� +j +gate fidelityx,y +i,j +(2) +where i represents the ith time step, j the jth gate in the ith +time step and and x, y are the physical qubit(s) coordinates. +Compilation time. In this work, we are not only inter- +ested in building mapping techniques themselves, but also in +their scalability potential. This necessitates that our proposed +SpinQ strategy should remain efficient for a variety of quan- +tum circuit parameters (e.g., number of qubits or percentage +of two-qubit gates). By measuring the compilation time for +mapping quantum circuits, we get a reference of the scalabil- +ity of our implementations. +C. +Verification +A verification tool is important to this work due to the +lack of a working device for real-system testing. The tool +is searching for mismatches between all shuttling sequences +and the qubits position history stored during compilation. It +also checks for conflicts, architectural constraint violations +and state vector mismatches between and in each stage of the +mapper. The latter uses the Qiskit Aer library [49]. + +9 +VI. +EXPERIMENTAL METHODOLOGY +Benchmarks. We have generated 3, 630 random uniform +algorithms [50] containing X, Y, Z and +√ +SWAP gates (all +native to the crossbar architecture) to be used as benchmarks. +With this set, we can vary on demand the number of gates, +number of qubits, and percentage of two-qubit gates. +For +example, a random uniform benchmark with 50% of two- +qubit gates relative to single-qubit gates will have 33.33% of +X or Y gates, 33.33% of Z gates, and 33.33% of two-qubit +gates. Generating synthetic circuits provides a well-controlled +benchmark collection from which we can better understand +results and form insights. Moreover, we use real benchmarks +from the RevLib library in a [5 - 1400] gate range [51]. Quan- +tum circuits from this library are often used in related quantum +circuit compilation works [9, 11, 12] and it consists of quan- +tum algorithms with parameters ranging from 3 to 16 qubits, +18.75% to 100% of two-qubit gates and 5 to 512, 064 gates. +Finally, we also consider quantum circuits from the Qlib li- +brary [52] which contains real quantum algorithms in increas- +ing size. +Benchmarks characterization. When it comes to perfor- +mance evaluation, it is important to not only consider proper- +ties of the crossbar architecture but also the characteristics of +quantum circuits. The simplest and most commonly [14] used +parameters of quantum circuits are number of qubits, number +of gates, and absolute or relative (i.e., percentage) number of +two-qubit gates. However, only these three characteristics can +be misleading for two reasons. Firstly, two benchmarks, for +instance, could have the same parameter values but heavily +differ in the circuit’s structure [14]. When one of them has +all pairs of qubits interact with each other will require more +routing than the other which might have the same number of +interactions, but with only one pair of qubits interacting. The +structure of a quantum circuit is derived from its qubit inter- +action graph (QIG) which represents the number and distri- +bution of interactions (i.e., two-qubit gates) between virtual +qubits. Several internal circuit parameters can be extracted +from the QIG that better distil its properties [14]. Having said +that, we analyze QIGs visually only, as this is still an active +field of research [14]. Despite that, we can nonetheless make +concrete conclusions and form insights, making visual QIG +assessments a viable tool to characterize algorithms. The sec- +ond reason is that initial gates can be decomposed to natively +supported instructions for the underlying architecture. This +means that the number of gates and ratios (percentages) be- +tween each gate type can differ from the initial set to the actual +executable set, meaning that evaluations can become more ac- +curate when accounting for the decomposed set. +Experimental Setup. We run SpinQ on a laptop with an +Intel(R) Core(TM) i7-3610QM CPU @ 3.20GHz and 16GB +DDR3 memory. SpinQ is written in Python 3.9.6 version. +VII. +EVALUATION AND ANALYSIS +In this Section, we present an in-depth performance anal- +ysis of SpinQ when mapping a broad range of quantum al- +gorithms on the crossbar architecture. We then form architec- +tural and mapping insights for each performance metric. More +specifically, gate overhead and corresponding insights are pre- +sented in Sec. VII A and VII B, depth overhead in Sec. VII C +and VII D, and ESP in Sec. VII E and VII F. Finally, we show +results regarding compilation time of SpinQ in Sec. VII G to +asses its scalability capability. +A. +Gate Overhead +To start with, we analyse the gate overhead trend in a wide +range of quantum algorithms. In Fig. 8 we have mapped ran- +dom uniform circuits on the crossbar architecture. Focusing +on Fig. 8a, which reaches up to 25 qubits, we observe that as +we go from low to high number of qubits and from low to high +percentage of two-qubit gates, the gate overhead increases +(from blue to red color). More precisely, higher qubit counts +imply larger crossbar topologies, thus potentially longer rout- +ing distances, i.e., more shuttle-based SWAPs. Furthermore, +higher percentages of two-qubit gates potentially lead to more +routing of qubits. These observations verify that the main +source of gate overhead is indeed the routing of qubits for +two-qubit gates (see Sec. V A). We also notice that the num- +ber of gates has a small but noticeable influence on the gate +overhead. To further observe the trend when increasing the +number of qubits, we changed the range of qubits from [3 +– 25] to [25 – 99] in Fig. 8b. We see once more that the +gate overhead increases as we go from low to high number +of qubits and percentage of two-qubit gates. As expected, the +gate overhead, shown on the color bars, of the [25 – 99] qubit +range is on average 102.49% higher than that of the [3 – 25] +qubit range because of the increased routing distances. +So far, the above random algorithms were generated to have +control of different circuit parameters (i.e., number of qubits +and gates and two-qubit gate percentage) in a way to broadly +cover the parameter space and up to certain boundaries. How- +ever, they might not be representative of real algorithms from +a circuit structure point of view (e.g., how two-qubit gates +are distributed among qubits or the degree of operation par- +allelism). Therefore, we then mapped real algorithms from +the RevLib and Qlib libraries resulting in the gate overhead +shown in Fig. 9, Fig. 10, and Fig. 11. In Fig. 9 we can +observe that benchmarks “cluster” together in similar colours, +namely shades of blue, green, yellow and red. This implies +that similar benchmarks, meaning with similar parameters and +structure, have similar gate overhead. Note that whereas ran- +dom uniform algorithms have all the same circuit structure +because of the way they are generated, RevLib algorithms +present different structural parameters not only compared to +the randomly generated circuits but also between them. For +this reason, correlations such as the higher the number of +qubits and percentage of two-qubit gates gets, the higher the +gate overhead will be, are not as evident as before (i.e. for +random circuits). +To further analyse how structural circuit parameters impact +the gate overhead, we mapped algorithms with similar number +of gates, qubits, percentage of two-qubit gates and QIG from + +10 +Gates (before decomp.) +0 25005000750010000 +12500 +15000 +17500 +20000 +Qubits +5 +10 +15 +20 +25 +2-Q Gate Percentage (before decomp.) +0 +20 +40 +60 +80 +100 +MAX=1114.28, AVG=473.69, MED=423.23, MIN=124.53 +Gate Overhead [%] +200 +400 +600 +800 +1000 +(a) +Gates (before decomp.) +0 25005000750010000 +12500 +15000 +17500 +20000 +Qubits +30 +40 +50 +60 +70 +80 +90 +100 +2-Q Gate Percentage (before decomp.) +0 +20 +40 +60 +80 +100 +MAX=2416.29, AVG=959.18, MED=871.93, MIN=65.03 +Gate Overhead [%] +500 +1000 +1500 +2000 +(b) +FIG. 8: Resulting gate overhead when 3, 630 random uniform quantum algorithms are mapped onto the crossbar architecture. +The three axes correspond to benchmark characteristics, namely, the number of gates [50 - 20,000], number of qubits [3 - 99] +(split into two subfigures), and two-qubit gate percentage [0 – 100]. +the Qlib library onto the crossbar architecture (see Fig. 10). +With these simulations, we also want to perform a scalability +analysis of the algorithms which is not possible with RevLib +circuits. First, note that the Cuccaro Adder (top line in Fig. +10) has a small drop in the percentage of two-qubit gates that +goes from 71.43% to 66.75% when increasing in size (num- +ber of qubits) whereas the Vbe Adder (bottom line) main- +tains a lower percentage of 50% for the same increase in size. +One can immediately observe that the Cuccaro Adder shows a +higher gate overhead up to 284% due to the higher two-qubit +gate percentage compared to the 271% of Vbe Adder, match- +ing the conclusions made in Fig. 8. However, as we empha- +sized above, in the case of real algorithms comparisons can +only be properly made when looking not only at their circuit +parameters but also at their more structural ones such as the +QIG. +For this reason, in Fig. 11 we show the derived QIGs from +Vbe Adders’ 40-qubit circuit, Cuccaro Adders’ 38-qubit cir- +cuit and Cuccaro Multipliers’ 21-qubit circuit alongside their +gate overhead in relation to the number of qubits and percent- +age of two-qubit gates. In these QIGs, nodes correspond to +qubits and edges to qubit interactions, i.e., two-qubit gates. +The particular size selection of these QIGs was made to easily +show their structure. We immediately observe similarities in +the QIGs of the two Adders as the distribution of interactions +is almost identical. More specifically, we see 2 to 3 inter- +actions per qubit on average, with others close to their logical +qubit number. Therefore, we can conclude that the higher gate +overhead of Cuccaro Adder is due to the higher percentage of +two-qubit gates, compared to Vbe Adder. +However, note that the Cuccaro Multiplier has the highest +gate overhead of all three (309%) despite having a lower two- +qubit gate percentage than the Cuccaro Adder. The reason be- +hind this is the difference in its QIG, which is much more con- +nected implying a denser qubit interaction distribution com- +pared to the others. Because of this, more routing is needed to +connect (nearly) all qubits across the entire topology. +B. +Insights from gate overhead analysis +Accounting for the routing constraints, as discussed in Sec. +IV, mapping on the crossbar architecture is not a trivial task. +In fact, we have emphasized the importance of conceptu- +alizing and developing new routing techniques that specif- +ically can address the unique mapping challenges of spin- +qubit architectures. More specifically, with the adoption of +the checkerboard pattern combined with the shuttle-based +SWAPs, we can provide a scalable solution of qubit routing +for two-qubit gates. Additionally, the complexity only scales +with the number of two-qubit gates, therefore being a viable +solution for large-scale implementation. However, this tech- +nique makes two-qubit gate routing the highest source of gate +overhead and it can dramatically increase it with higher qubit +counts and a higher percentage of two-qubit gates (see Fig. +8 and 10). Moreover, in Fig. 11 we saw that gate overhead +can also be increased by a more connected QIG even if other +circuit parameter values are comparatively lower. This shows + +11 +Gates (before decomp.) +0 +200 +400 +600 +800 +1000 +1200 +1400 +Qubits +4 +6 +8 +10 +12 +14 +16 +2-Q Gate Percentage (before decomp.) +20 +30 +40 +50 +60 +70 +80 +90 +100 +MAX=306.6, AVG=210.59, MED=205.72, MIN=167.0 +Gate Overhead of Integrated Strategy [%] +180 +200 +220 +240 +260 +280 +300 +FIG. 9: Resulting gate overhead when mapping quantum +algorithms from the RevLib library onto the crossbar +architecture. The three axes correspond to benchmark +characteristics, namely, number of gates [5 - 1400], number +of qubits [3 - 16] and two-qubit gate percentage [18.75 - +100]. +the importance of basing circuit performance evaluation not +only on simple circuit parameters but also on other ‘hidden’ +structural characteristics such as the qubit interaction distribu- +tion.Having said that, the second biggest source of gate over- +head originates from X or Y qubit rotations, as it produces at +least 3 additional gates compared to 4 additional gates for each +shuttle-based SWAP. This is due to the unprecedented semi- +global rotation scheme which is the first time that single-qubit +gates require additional instructions (i.e., produce gate over- +head) compared to other qubit architectures. The previous +two facts inspire novel mapping techniques for the crossbar +architecture (and potentially for other spin-qubit architectures +with similar characteristics) that can increase performance, +namely: +1. Developing a routing solution dedicated to accounting +for potential conflicts and constraints can reduce the +gate overhead resulting from the shuttle-based SWAPs. +Such a generalized routing algorithm could also include +SWAP interactions (two consecutive +√ +SWAPs) and +CPHASE interactions. For instance, there can be sce- +narios that choosing a more noisy two-qubit interaction, +for the purpose of avoiding an upcoming conflict, that +could result in higher ESP. Additionally, such a heuris- +tic algorithm can allow multiple control or target qubits +([10]) to be shuttled around the topology allowing for +parallelization of many two-qubit gates while avoiding +high error variability in the topology [18]. However, +Gates (before decomp.) +0 +50 100 150 200 250 300 350 400 +Qubits +0 +20 +40 +60 +80 +100 +120 +2-Q Gate Percentage (before decomp.) +50 +55 +60 +65 +70 + + + +Gate Overhead [%] +200 +220 +240 +260 +280 +FIG. 10: Resulting gate overhead when mapping the Cuccaro +Adder (top line of data points) and the Vbe Adder (bottom) +quantum algorithms from the Qlib library onto the crossbar +architecture. The three axes correspond to benchmark +characteristics, namely, number of gates [4 - 385], number of +qubits [4 - 130] and two-qubit gate percentage [50 - 71.43]. +such a solution must be implemented with complexity +in mind such that it will not make it unviable on large +scale. +2. A more efficient routing algorithm for single-qubit +gates can significantly reduce the gate overhead, such +that a specific rotation scheme to rotate targeted qubits +is used less often. Such an algorithm can route qubits to +the appropriate odd or even columns before the execu- +tion of single-qubit gates without the need to apply any +scheme afterwards (see the example in Sec. IV). +3. Combining the previous two points, there can be a uni- +fied algorithm implementing both. +In such an algo- +rithm, upcoming routing for single-qubit gates is ac- +counted for when routing for two-qubit gates, and vice +versa. +4. Finally, an initial placement algorithm can take into ac- +count not only two-qubit gates but single-qubit gates as +well. Since the positions of qubits influence the gate +overhead resulting from single-qubit gates (due to the +semi-global rotation scheme), an extension of an initial +placement algorithm accounting for single-qubit gates +can reduce the gate overhead. +Last but not least, we have emphasized that to concretely +evaluate results, there has to be sufficient characterization of + +12 +Cuccaro Multiplier +Vbe Adder +Cuccaro Adder +FIG. 11: Resulitng gate overhead when the Vbe Adder, Cuccaro Adder and Cuccaro Multiplier from the Qlib library are +mapped onto the crossbar architecture alongside their Quantum Interaction Graphs (QIG) consisting of 40, 38 and 21 qubits, +respectively. The y-axis represents the two-qubit gate percentage and the x-axis the number of qubits. We see gate overhead to +be influenced not only by the number of qubits and two-qubit gate percentage but also by the qubit interaction distribution. +benchmarks, especially when evaluating novel architectures +and mapping techniques. In our analysis, we did not rely only +on simple benchmark parameters, such as the percentage of +two-qubit gates, but also on the internal structure of bench- +marks using the Quantum Interaction Graph (QIG). +C. +Depth Overhead +This time, we analyse the depth overhead when mapping +onto the crossbar the same random uniform benchmark set as +in Fig. 8. In Fig. 12, it can be observed that the trend (colours) +of the depth overhead changes for different ranges of number +of qubits as shown in the two subfigures. Knowing that the +main source of depth overhead originates from X or Y gates +(at least 3 additional cycles), we expect the depth overhead to +become higher in lower regions of two-qubit gate percentage. +That is observed in Fig. 12a, where the number of qubits goes +up to 25. However, moving on to Fig. 12b, we see that this +trend changes. Now, due to the higher number of qubits, rout- +ing distances have increased, thus routing for two-qubit gates +dominates the depth overhead. This is apparent by its increase +(from blue to red colour) as we go from lower qubit counts to +higher qubit counts, and as we go from low to higher percent- +age of two-qubit gates. Finally, this fact is also apparent in +the absolute values of depth overhead of the two subfigures. +Note also that the number of gates has a slight influence on +the depth overhead, but it is not as relevant as the other char- +acteristics discussed above. +Moving on, Fig. 13 shows the depth overhead of a Cuc- +caro Adder when scaling it up from 4 to 130 qubits. In the +range of 4 to 20 qubits, we observe an increase in depth over- +head as the percentage of two-qubit gates decreases, which +aligns with the remarks about the main source of depth over- +head (i.e., the X or Y gates). Then, for an increasing number +of qubits (from 20 qubits on) and at an almost constant two- +qubit gate percentage (67%), the depth overhead increases at +a slower rate. Here we conclude, once again, that two-qubit +gate routing starts to dominate the depth overhead as routing +distances become larger. +In most previous works, the amount of two-qubit gates is +the main circuit characteristic to anticipate how much qubit +routing will be needed for a specific quantum algorithm and +therefore the major and only source of gate/depth overhead. +However, in the crossbar architecture, and potentially in other +spin-qubit crossbar designs, single-qubit gates can also con- + +1334 +33 +36 +33.31 +28 +22 +1913 +Gates (before decomp.) +0 25005000750010000 +12500 +15000 +17500 +20000 +Qubits +5 +10 +15 +20 +25 +2-Q Gate Percentage (before decomp.) +0 +20 +40 +60 +80 +100 +MAX=6290.98, AVG=2217.92, MED=2132.74, MIN=262.0 +Depth Overhead [%] +1000 +2000 +3000 +4000 +5000 +6000 +(a) +Gates (before decomp.) +0 25005000750010000 +12500 +15000 +17500 +20000 +Qubits +30 +40 +50 +60 +70 +80 +90 +100 +2-Q Gate Percentage (before decomp.) +0 +20 +40 +60 +80 +100 +MAX=27786.21, AVG=12530.1, MED=11934.55, MIN=2250.0 +Depth Overhead [%] +5000 +10000 +15000 +20000 +25000 +(b) +FIG. 12: Resulting depth overhead when 3, 630 random uniform quantum algorithms are mapped onto the crossbar +architecture. The three axes correspond to benchmark characteristics, namely, number of gates [50 - 20,000], number of qubits +[3 - 99] (split into two subfigures), and two-qubit gate percentage [0% – 100%]. +tribute to this overhead as discussed. It is then important to +have a closer look at the X or Y rotation gate percentage and +further analyse how it impacts the depth overhead. Addition- +ally, after the gate decomposition step, the percentages and +ratios between all gate types are changed. To illustrate this, +imagine a quantum circuit that originally consists of a low +number of CNOT gates and no Z gates. After the decompo- +sition to gates supported by the crossbar architecture, the per- +centage of Z rotation gates will increase, and consequently, +the two-qubit gate percentage will decrease, as CNOT gates +are decomposed as Ry( π +2 ), two +√ +SWAP, S, S†, Ry( −π +2 ). +Thus, it is relevant to consider this change in gate percentage +in our analysis as ultimately the executable circuit will only +consist of native gates. To summarize, as overhead comes +from mapping different types of gates on the crossbar, indi- +vidually distinguishing between them, in particular after de- +composition, can increase the accuracy of our evaluations. +To illustrate the previous point, in Fig. 14 we show the +depth overhead of the Cuccaro Adder (upper dots) and the +Vbe Adder (lower dots) with the same ranges as in Fig. 10. +Note that the y-axis corresponds to the percentage of X or Y +rotation gates after decomposition. From this new perspective, +we clearly see their difference in actual (i.e., executed by the +architecture) X or Y rotation gate percentage. On average +the depth overhead of the Vbe adder is 196% higher than the +Cuccaro Adder for the same range of qubits. As explained +before, the highest source of depth overhead comes from X +or Y rotations gates, which explains the large depth overhead +difference between those two algorithms. +D. +Insights from depth overhead analysis +From the previous analysis, we can observe that trends can +change based on the parameter ranges of benchmarks. This +is because different sources of depth overhead contribute with +different rates based on the number of qubits (i.e., crossbar +size). More specifically, the overhead contribution resulting +from mapping X/Y gates was higher up to a certain number +of qubits after which was exceeded by the contribution rate of +two-qubit gates. We saw that exceeding a threshold of more +than 20 qubits increases the depth overhead at a steadier pace, +which specifically favoured scalability for Cuccaro Adder in +Fig. 13 and 14. It is expected, however, that with different +algorithms, there will be different trends. With such observa- +tions, we stress the importance of distinguishing between all +gate types and especially after decomposition to better under- +stand the performance impact of mapping. With that knowl- +edge, we can create better mapping techniques and/or make +an informed selection of algorithms to execute. +As stated before, the fact that gate overhead can result from +mapping single-qubit gates is unprecedented. Furthermore, +we notice that mapping both, single- and two-qubit gates, re- +quires additional shuttles and they produce the highest gate +and depth overhead. Therefore, novel mapping techniques +minimizing all qubit movements (shuttles) can increase per- +formance substantially, such as the ones discussed in Sec. +VII B. From an architectural point of view, since the shuttle +operation is so relevant, there have to be as few operational + +14 +Gates (before decomp.) +0 +50 100 150 200 250 300 350 400 +Qubits +0 +20 +40 +60 +80 +100 +120 +2-Q Gate Percentage (before decomp.) +67 +68 +69 +70 +71 +MAX=586.97, AVG=563.11, MED=570.28, MIN=450.0 + + +Depth Overhead [%] +460 +480 +500 +520 +540 +560 +580 +FIG. 13: Resulting depth overhead when Cuccaro Adder +from the Qlib library is mapped onto the crossbar +architecture. The three axes correspond to benchmark +characteristics, namely, number of gates [4 - 385], number of +qubits [4 - 130] and two-qubit gate percentage [66.75 - +71.43]. +0 +20 +40 +60 +80 +100 +120 +Qubits +45.25 +45.50 +45.75 +46.00 +46.25 +46.50 +46.75 +X/Y Gate Percentage (after decomp.) +Depth Overhead [%] +450 +500 +550 +600 +650 +700 +750 +FIG. 14: Resulting depth overhead when Cuccaro Adder +(bottom line of data points) and Vbe Adder (top) from the +Qlib library are mapped onto the crossbar architecture. The +y-axis represents the X or Y gate percentage, and the x-axis +the number of qubits. +constraints as possible when mapping them. +E. +Estimated Success Probability +In this section, we will show how the success probability of +an algorithm drops after mapping it to the crossbar architec- +ture. Before we continue, we have to mention that even with +operational fidelities as high as 99.99% for single-qubit gates +and shuttles (as suggested in [1]) and 99.98% for +√ +SWAPs, +the ESP drops drastically to 0 in most algorithms with a high +number of gates. +For that reason, we just focused on the +Bernstein-Vazirani algorithm as it has got a low percentage of +two-qubit gates (usually there are only one or two CNOTs), +therefore the error is mostly introduced by single-qubit gates. +0 +100 +200 +300 +400 +500 +0 +20 +40 +60 +80 +100 +Estimated Success Probability (ESP) +0 +50 +100 +150 +200 +250 +ESP +Original ESP +Gates (before mapping) +Gates (after mapping) +FIG. 15: Estimated success probability (ESP) before and +after compilation of Bernstein-Vazirani algorithm from 2 to +129 qubits. +Fig. 15 shows the ESP of the Bernstein-Vazirani algorithm +when scaling it from 2 to 129 qubits. The red line “Origi- +nal ESP” refers to the ESP before mapping, and the blue line +”ESP” refers to ESP after mapping. We observe a sharp ESP +decrease approaching 10% for 267 gates after mapping with +a slope rate of −0.6 which is caused by the increased num- +ber of gates. For 529 gates after mapping we obtained a 0% +ESP. Another reason for the ESP decrease is the semi-global +single-qubit rotation; for each of the X or Y gates contained +in the circuit (after decomposition), all qubits in odd or even +columns are rotated (even the ones that are not targeted for +rotation). This is further explained in Sec. IV 2. + +15 +F. +Insights from Estimated Success Probability analysis +Our estimated success probability equation 2, although sim- +ple, is approximating a worse-case-scenario algorithm success +rate. +We observed a rapid decline in ESP in a minimally +connected algorithm (mostly X or Y rotation gates), even +though our equation did not include decoherence-induced er- +rors [28, 44]. The main reason for this decrease is the result- +ing overhead when implementing single-qubit gates on spe- +cific qubits given the semi-global rotation scheme. Note that +in this case, all qubits in either column parities are rotated thus +each contributing to this ESP drop. Therefore, it is essential +to determine which algorithms could take advantage of the +semi-global control and/or develop architecture-specific map- +ping techniques to minimize the need for a scheme. +On real NISQ quantum devices there are other sources of +noise noise that impact algorithm execution. Fortunately, it +is expected that processors will gradually become more ro- +bust with better fabrication tolerances and improved error- +mitigation and mapping techniques will be developed and ul- +timately quantum error correction protocols will be used. It +remains challenging, however, to accurately simulate errors in +large-scale devices to derive algorithm’s success probability. +G. +Compilation time +0 +2500 +5000 +7500 +10000 +12500 +15000 +17500 +20000 +Gates +0 +2 +4 +6 +8 +Seconds + Compilation Time [s] +qubits = 3 +qubits = 12 +qubits = 21 +qubits = 30 +qubits = 39 +qubits = 48 +qubits = 57 +qubits = 66 +qubits = 75 +qubits = 84 +qubits = 93 +FIG. 16: Compilation time when mapping random uniform +algorithms with 50% of two-qubit gates onto the crossbar +architecture. We observe a linear relation which makes +SpinQ suitable for scalable spin-qubit crossbar architectures. +Finally, we measure the compilation time of our solution +to evaluate its scalability. The compilation time of SpinQ In- +tegrated Strategy can be seen in Fig. 16 for a subset of the +random uniform circuits that have been used in Fig. 8 and 12. +This subset consists of circuits with only 50% of two-qubit +gates. With this subset we map the same number of gates for +each gate type, thus all internal SpinQ processes are weighted +equally. We observe a linear increase in compilation time in +relation to the number of gates for each qubit count. This im- +plies that our strategy is suited for scalable spin-qubit crossbar +architectures. Improvements can be directed towards reducing +the slopes for each qubit count. +VIII. +DISCUSSION AND FUTURE DIRECTIONS +TABLE I: Computational complexity comparison between +compilation strategies for the crossbar architecture [1]. With +n we denote the number of gates in a quantum circuit. +Strategy +Complexity +Backtrack [27] +O(n3) +Suffer a side effect [27] +O(n2log(n)) +Avoid the deadlock [27] +O(n) +Integrated (ours) +O(n) +Integrated strategy improvements. There can be a few +extensions to the Integrated Strategy that can provide better +performance (less overhead and higher ESP). These improve- +ments can be divided into two categories: a) improvements +that increase complexity marginally and b) improvements that +will increase complexity substantially. It is important to make +this differentiation because on large scale we have to consider +the trade-off between complexity (computation time as sizes +increase) and performance (less overhead and higher ESP). +Improvements in category (a) will involve a constraint and +conflict check for any shuttle-based type gate to enable com- +plete parallelization of all single-qubit gates within the second +pass. Note that, once again, each cycle remains dedicated to +one gate type, therefore, fine-tuning pulse durations in real +devices is still possible. +Moving on to the next category (b), it consists of all heuris- +tic mapping algorithms (routing and initial placement) dis- +cussed in Sections VII B, VII D and VII F, which can be ex- +tended to other scalable spin-qubit architectures. This will en- +able complete parallelization of two-qubit gates and less rout- +ing for both, one- and two-qubit gates. +Strategy Comparisons. As we discussed in Sec. IV, the +crossbar architecture comes with constraints that prevent full +parallelization of quantum instructions. The crossbar, how- +ever, may reach two types of conflicts (unwanted interactions +or blocked paths), even if all constraints are respected. For +that reason, there must be some kind of compilation strategy +between the scheduler and the router to prevent conflicts. In +this work, we have implemented the Integrated strategy which +is different from the three strategies suggested in [27]. Ta- +ble I compares the computational complexity of these three +strategies with our own. The backtrack strategy suggested in +[27] avoids conflicts by trying a different scheduling combi- + +16 +nation. If after repeating this process the scheduler has back- +tracked to the first instruction of the cycle (no more schedul- +ing combinations), a new routing path is given by the rout- +ing algorithm and the scheduling is repeated. This strategy +can be quite complex as the worst case scenario can un-route +and un-schedule all the gates going back to a completely un- +mapped circuit. An improved version of this strategy called +suffer a side effect, is a special case of the former and it +is only preferred whenever a corresponding conflict can be +corrected and if the correction is less costly than only fol- +lowing the ”backtracking” strategy. The final strategy, and +the one implemented in [27], is called avoid the deadlock. +This strategy, similar to our Integrated strategy, is trying to +avoid conflicts by parallelizing only X or Y gates. In this +way, +√ +SWAPs and shuttle operations can not cause a con- +flict. However, in this strategy there is no synergy between the +routing and scheduling stages as our Integrated strategy has, +therefore there is little flexibility for improvements and per- +formance can not be easily improved while keeping the same +complexity. Our strategy is able to maintain the same O(n) +complexity even after improvements. +General discussion. +When developing novel mapping +techniques for scalable quantum computing architectures such +as the si-spin crossbar two main factors have to be considered: +scalability and adaptability. As spin-qubit fabrication capa- +bilities are improving, new architectural designs with maybe +higher qubit counts will be explored. Therefore, from a com- +putation/compilation time point of view, mapping techniques +should be as scalable as the underlying technology. Practi- +cally, this implies that highly sophisticated and more complex +mapping techniques might be excellent for a particular archi- +tecture and up to a certain number of qubits, but could be +impractical for more qubits or even unusable for another ar- +chitecture. In addition, as we are slowly exiting the NISQ +era, quantum technologies will become more robust, espe- +cially with the use of quantum error correction techniques. By +that time, optimizing mapping techniques for specific hard- +ware and/or algorithm might not be as relevant as today, but +rather how fast and adaptable they are to a plethora of quan- +tum algorithms and increased number of qubits. +IX. +CONCLUSION +Different quantum circuit mapping techniques have been +developed to deal with the limitations that current quantum +hardware presents and are being consistently improved to ex- +pand its computational capabilities by getting better and better +algorithm success rates. The most advanced mapping meth- +ods focus on ion-trap and superconducting devices due to +their ‘maturity’ compared with other quantum technologies. +However, spin-qubit-based processors have a great potential +to rapidly scale and the first 2D crossbar architectures have +been recently demonstrated. In this work, we focused on the +quantum circuit mapping challenges of the newly emerging +spin qubit technology for which highly-specialized mapping +techniques are needed to take advantage of its operational +abilities. Specifically, we used the crossbar architecture as +a stepping stone to explore novel mapping solutions while +focusing on scalability. The crossbar architecture adopts a +shared-control scheme, thus making it a great candidate to +tackle the interconnect bottleneck. +On that note, we have +developed SpinQ, the first native compilation framework for +spin-qubit architecture which we used to analyze the perfor- +mance of synthetic and real quantum algorithms on the cross- +bar architecture. Through our analysis, we tried to inspire +novel algorithm- and hardware-specific mapping techniques +that can possibly increase the performance while taking into +account the compilation scalability. We also emphasized the +importance of characterizing benchmarks before and after de- +composition and by including their QIG structure to better +evaluate results. +X. +ACKNOWNLEDGEMENT +This work is part of the research program OTP with project +number 16278, which is (partly) financed by the Netherlands +Organisation for Scientific Research (NWO). This work has +also been partially supported by the Spanish Ministerio de +Ciencia e Innovaci´on, European ERDF under grant PID2021- +123627OB-C51 (CGA). We thank Menno Veldhorst and Hans +van Someren for their fruitful discussions. + +17 +[1] R. Li, L. Petit, D. P. Franke, J. P. Dehollain, J. Helsen, +M. Steudtner, N. K. Thomas, Z. R. Yoscovits, K. J. Singh, +S. Wehner, et al., A crossbar network for silicon quantum dot +qubits, Science advances 4, eaar3960 (2018). +[2] F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, +R. Barends, R. Biswas, S. Boixo, F. G. Brandao, D. A. 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Jha, Qlib: Quantum mod- +ule library, ACM Journal on Emerging Technologies in Com- +puting Systems (JETC) 11, 1 (2014). + diff --git a/2NFQT4oBgHgl3EQfFTWv/content/tmp_files/load_file.txt b/2NFQT4oBgHgl3EQfFTWv/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3594285ed904ae19a06db22f08ead92f66f86e40 --- /dev/null +++ b/2NFQT4oBgHgl3EQfFTWv/content/tmp_files/load_file.txt @@ -0,0 +1,1270 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf,len=1269 +page_content='SpinQ: Compilation strategies for scalable spin-qubit architectures N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Paraskevopoulos1,2, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Sebastiano1,2, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Almudever3, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Feld1,2 1Quantum and Computer Engineering Department, Delft University of Technology, 2628 CD Delft, The Netherlands 2QuTech, Delft University of Technology, 2628 CJ Delft, The Netherlands and 3Computer Engineering Department, Technical University of Valencia, Camino de Vera, s/n, 46022 Val`encia, Spain In most qubit realizations, prototype devices are available and are already utilized in both industry and aca- demic research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Despite being severely constrained, hardware- and algorithm-aware quantum circuit mapping techniques have been developed for enabling successful algorithm executions during the NISQ era, targeting mostly technologies with high qubit counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Not so much attention has been paid to the implementation of com- pilation methods for quantum processors based on spin-qubits due to the scarce availability of current experi- mental devices and their small sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' However, based on their high scalability potential and their rapid progress it is timely to start exploring quantum circuit mapping solutions for these spin-qubit devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' In this work, we discuss the unique mapping challenges of a scalable spin-qubit crossbar architecture with shared control [1] and introduce SpinQ, the first native compilation framework for scalable spin-qubit architectures that maps quantum algorithms on this crossbar architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' At the core of SpinQ is the Integrated Strategy that addresses the unique operational constraints of the crossbar while considering compilation (execution time) scalability, having a O(n) computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' To evaluate the performance of SpinQ on this novel architecture, we compiled a broad set of well-defined quantum circuits and performed an in-depth analysis based on multiple metrics such as gate overhead, depth overhead, and estimated success probability, which in turn allowed us to create unique mapping and architectural insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Finally, we propose novel mapping technique improvements for the crossbar architecture that could increase algorithm success rates and potentially inspire further research on quantum circuit mapping techniques for other scalable spin-qubit architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' INTRODUCTION The prospect of quantum computing advantage is steadily becoming a reality [2–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' The community is anticipating fur- ther advances that will allow quantum computing systems to become practical and to reach computational advantage [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' With such advancements, quantum computing systems are ex- pected to solve a plethora of classically intractable problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Until then, current quantum systems belong to the so-called Noisy Intermediate-Scale Quantum (NISQ) era [6], in which devices can only handle small-sized quantum circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' This is due to limitations in the number of qubits and high operational errors, the latter causing rapid quantum information deteriora- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Combined with even more hardware constraints, such as cross-talk and limited classical-control resources [7, 8], suc- cessful quantum circuit execution is a difficult feat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Scientists, both in academia and industry, face major engineering chal- lenges in building both, hardware and corresponding system software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' During the NISQ era, there have been significant efforts [9–19] to extract the most out of these resource-constrained and error-prone quantum computing systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' One of the ap- proaches to do so is by developing hardware- and algorithm- aware quantum circuit mapping techniques to maximize per- formance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' In general terms, mapping refers to the process of modifying (potentially hardware-agnostic) quantum circuits in such a way that they can be run on a given quantum com- puting device by respecting all of its constraints while opti- mizing performance (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=', algorithm success rate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' So far, several mapping techniques have been developed mostly for superconducting and ion-trap qubit devices, as they are nowa- days one of the most well-recognized and most-developed qubit implementation technologies in terms of qubit counts and availability to users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' However, spin-qubits emerge as a promising technology for scaling up quantum computing systems mainly due to their high integration potential [20– 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Therefore, the scientific community is envisioning two- dimensional spin-qubit architectural proposals that could al- leviate some of the major challenges towards scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Re- cently, a crossbar array [26] has been experimentally demon- strated showing great promise for architectures with shared control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Such scalable architectural designs come with a new set of hardware constraints for which novel quantum circuit mapping techniques need to be developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' In this paper, we present SpinQ, the first native compilation framework focusing on scalable spin-qubit architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' To this purpose, we target the so-called crossbar architecture pro- posed in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' By creating a deep understanding of its opera- tional constraints, we draw a clear picture of unique mapping challenges that arise in comparison to other qubit technolo- gies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' We discuss and implement possible mapping solutions based on [27, 28] while improving those works from a scala- bility standpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' We emphasize the importance of performing an extensive performance evaluation process of novel map- ping techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Note, that this compilation framework will not only allow quantum algorithm executions on scalable spin qubit hardware but more importantly will also help to form insights on the behaviour and performance of this new breed of architectures and provide some design guidelines for future developments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' The main contributions of this paper are: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' An in-depth analysis of mapping challenges in order to create novel mapping techniques for spin-qubit crossbar architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' SpinQ – The first native compilation framework dedi- cated to scalable spin-qubit architectures which utilizes a more scalable compilation strategy compared to pre- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='13241v1 [quant-ph] 30 Jan 2023 2 vious proposals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' A thorough performance analysis of the main sources of gate/depth overhead and estimated success probability when mapping well-defined quantum algorithms on the crossbar architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Deriving algorithmic- and hardware-specific mapping insights for the crossbar architecture and other spin- qubit architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' The remainder of this paper is structured as follows: In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' II the current progress and challenges of scalable spin-qubit architectures are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' III the crossbar architec- ture is introduced as a potential candidate in scaling quantum devices in two dimensions, as well as its native operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' IV we comprehensively analyse the unique challenges of mapping quantum algorithms on the crossbar architecture which require novel mapping techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Then, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='V we introduce SpinQ – the first native compilation framework for scalable spin-qubit architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' VII we thoroughly analyze the performance of SpinQ when mapping a broad and well-defined range of quantum algorithms on the crossbar ar- chitecture after which we form architectural and mapping in- sights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' VIII we discuss potential improvements of our compilation strategy and we compare its computational com- plexity to previous proposals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Finally, we conclude our work in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' SPIN QUBITS AS A SCALABLE PLATFORM To fulfil the promise [6] of quantum computers being ma- chines that solve some classically intractable problems, sub- stantial system sizes have to be reached, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=', a large number of qubits [8, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' It still remains to be seen which qubit imple- mentation technologies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=', superconducting, trapped ions, quantum dots, photonics, defect-based on nitrogen-vacancy diamond centres) will succeed in scaling up quantum comput- ing systems with high-quality qubits [30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Spin qubits in quantum dots are a promising technology for scalable quan- tum computers due to the maturity of the semiconductor in- dustry, the capability of high integration on a single die com- pared to other qubit technologies, long coherence times, and the ability to operate in super-kelvin temperatures [20–25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Despite the advantages just mentioned, there are still sev- eral challenges today towards scaling spin-qubit devices in a sustainable manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' One major challenge is the wiring scheme between the quantum processor and the classical interface, the so-called interconnect bottleneck [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Formally, the intercon- nect bottleneck is described by Rent’s exponent [32], which is a measure of optimization in the wiring scheme in both, clas- sical and quantum processors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' The existing scheme in most quantum devices of having at least one control line per qubit is not scalable in the long term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' This is, mostly, due to the fact that dilution refrigerators have an upper limit to I/O cable ca- pacity and that more cables will progressively make it harder to reach the desired milli-Kelvin temperature due to higher heat dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Therefore, qubit architectures and classical- control electronics have to support multi-qubit shared-control that requires a sub-linear number of control lines with an in- creasing number of qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' In other words, each control line needs to address multiple qubits to effectively mitigate the in- terconnect bottleneck when scaling up quantum hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Going a step further, the inability to achieve a scalable wiring scheme also originates from the low device unifor- mity achieved by today’s fabrication tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' In most cases, this implies that qubits can not be made homogeneous enough to control them effectively in a scalable architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' The low uniformity results in resonance frequency deviations or other control variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' This means that in an inhomoge- neous device, a driving signal for a particular operation will have to vary from one qubit to another to get the same out- come [1, 22, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' This makes it difficult to successfully con- trol many qubits with the same control line, thus contributing to the wiring scheme challenge (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=', the interconnect bottle- neck).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' There have been significant efforts [1, 22, 32, 34–38] to re- duce the number of control lines reaching the qubits as de- vices become ever denser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Such efforts take advantage of the miniaturization capabilities of spin qubits and the large- scale integration of solid-state circuits to address the afore- mentioned challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' However, current experimental work primarily has been focused on one-dimensional spin-qubit ar- rays of small sizes [22], which are not easily scalable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Re- cently, a 2×2 spin-qubit processor [39] and a 4×4 spin-qubit device based on a crossbar architecture [1, 26] with shared control has demonstrated the potential to scale spin-qubit de- vices in two dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' As the technology is advancing and further reducing Rent’s exponent, there will be a need to effectively map quantum algorithms on two-dimensional de- vices such as the crossbar architecture which comes not only with limited qubit connectivity but also with a new set of con- straints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Therefore, there is an opportunity to explore its map- ping challenges and propose novel solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' However, the sample space of these proposals is sparse and lacks a detailed description of hardware constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' In com- bination with a lack of available devices for testing, leads to a lack of a proper evaluation tool capable of benchmarking various quantum algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Therefore, mapping techniques have not been studied as much as other qubit technologies such as superconducting and ion traps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' It also remains un- clear whether existing techniques could be applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Then, even if such techniques are realized they could be incompati- ble with existing quantum compilation frameworks made for other qubit technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' This could be due to completely different development requirements imposed by the particular spin-qubit constraints and their scalability prospects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' In other words, a dedicated compilation framework for spin-qubit ar- chitectures with a focus on scalability is still missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' All these obstacles make it difficult to evaluate and compare var- ious architectural proposals under relevant application cate- gories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 3 CL0 CL1 CL2 RL2 RL1 RL0 1 4 2 3 5 6 8 7 QL-3 QL-2 QL-1 QL0 QL1 QL2 QL3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 1: Schematic overview of the crossbar architecture and operational control lines [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' THE CROSSBAR ARCHITECTURE The crossbar architecture for arranging spin qubits was in- troduced in [1] as a scalable solution to the interconnect bot- tleneck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Inspired by the crossbar architecture used in today’s classical processors [1, 34], it adopts a similar characteristic, namely shared control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' This leads to a quadratic reduction in control lines per qubit [28] and opens up the possibility for high integration of up to 1, 000 qubits in a single pack- age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Qubits are defined by electron (or hole) spin states in Si-based quantum dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' In Figure 1, we illustrate a schematic overview of the crossbar architecture in which each site (cir- cles) represents a quantum dot, some of which are occupied by spin-qubits (numbered, green circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Spin qubits are usually sparsely initialized in a checker-board pattern to re- duce potential cross-talk and to allow for long-range entan- glement through shuttling qubits across the array [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Finally, the crossbar architecture requires high uniformity in the fab- rication of materials to minimize operational errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Fortu- nately, it is possible to mitigate such errors or even vanish them by operating the crossbar at low magnetic fields and with proper tuning (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=', separated resonance frequencies between columns).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Furthermore, a crossbar module is envisioned to be self-contained and to be duplicated in a network of modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' This can provide the means to realize quantum error correc- tion (QEC) in large-scale systems enabled by fast-shuttling, low-error communication links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' In this crossbar architecture, three different kinds of shared control lines are used to per- form operations on the qubits: vertical (column line, CL), horizontal(row line, RL), and diagonal (qubit line, QL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' No- tably, each line affects all the sites that it is connected to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' For instance, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 1 line QL−2 affects the sites in which spin- qubits 5 and 7 reside in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' This imposes some restrictions in the parallelization of instructions, which we will discuss in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Below, we will abstractly describe the control prop- erties for executing gates native to the crossbar architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' A more detailed explanation is provided in [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Two-qubit gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Two two-qubit gates CPHASE and √ SWAP are supported by the crossbar, with the latter being chosen for this work due to its higher operational fidelity and faster execu- tion time according to [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' A √ SWAP can be performed when two qubits are vertically adjacent (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' same column) and the horizontal barrier between them is lowered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Then the QL lines going through the two qubits need to be in the same voltage potential for a specific duration of time to complete the √ SWAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Qubit shuttling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' In the crossbar architecture, qubits can be moved around by performing shuttling opera- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' In this operation, the vertical and horizontal lines are used as barrier gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Lowering or raising these barriers can create pathways from which qubits can move (shuttle) from one site to another with the use of DC signals through the diagonal lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 2 shows an example of shuttling, in which spin-qubit 3 is moved one site to the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Although this architecture can support gate-based communication with two subsequent √ SWAP gates as in superconducting qubits, shuttling qubits is preferred due to higher operation fidelity and shorter execution time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' It should be noted that shuttling horizontally, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=', between columns, causes a Z-phase rota- tion which should be mitigated by timing such operations well ([1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' In the crossbar architecture, single-qubit gate rotations should be separated into two categories: Z-phase rotations and X or Y rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Z-phase rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Z-phase qubit rotations are controlled by a well-timed qubit shuttling to and from a neighbouring column [1, 27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' This is due to the differences in Zeeman energies from column to column which imposes an alternating magnetic field on qubits, thus rotating them in the Z axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' When this shuttle is timed correctly, it rotates the qubit in the correct Z state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' The diagonal qubit line provides the means to address multiple qubits, thus enabling parallel Z phase shifts across the topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' X or Y rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' As for X or Y rotations, either all qubits belonging to red-coloured columns or all qubits in blue-coloured columns are rotated (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' This is called semi-global qubit rotation im- plemented by electron-spin-resonance ([40]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' The process of readout allows for local single qubit measure- ments based on the Pauli Spin Blockade (PSB) process [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' With this process, the measurement outcome is determined by whether a qubit shuttle towards a horizontally adjacent ancilla qubit was successful or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' QUANTUM CIRCUIT MAPPING CHALLENGES OF THE CROSSBAR ARCHITECTURE The quantum circuit mapping process plays an essential role in the successful execution of algorithms on a quantum computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' It consists of a cascade of routines that trans- form a (potentially hardware-agnostic) quantum circuit to a hardware-compatible version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' However, current NISQ quan- tum processors are severely constrained and cannot run use- ful applications successfully, yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Examples of hardware con- straints are low qubit connectivity, cross-talk, reduced prim- itive gate set, low coherence time, fabrication imperfections, and limited classical-control resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Therefore, a mapping 4 process needs to consider such limitations and try to optimize performance as much as possible to increase the algorithm’s success rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' So far, there are a plethora of proposed solu- tions which differ in strategy, methodology and performance metrics to optimize [9–19, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Mapping techniques have been mostly developed for super- conducting and ion-trap qubit devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' However, as of now, there is not much focus on spin-qubit architectures and their particular characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Although spin-qubits are now in a rather early development stage, their scalability potential is undeniable and therefore it is timely to lay grounds for devel- oping novel mapping techniques and inspire further research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' As previously mentioned, in this work we focus on the cross- bar architecture that comes with a unique set of constraints that affect the parallelization of quantum operations, the ap- plication of X or Y rotations on single qubits, and the routing of qubits (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' moving qubits around).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Parallelization of quantum operations CL0 CL1 CL2 RL2 RL1 RL0 1 4 2 3 5 6 8 7 QL-3 < QL-2 > QL-1 > QL0 > QL1 QL2 QL3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 2: Shuttling example of qubit 3 moved one site to the left, as the barrier CL0 between origin and destination site is lowered and voltage of QL−1 is larger than > QL0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Most of the operation parallelization restrictions come from the fact that control lines are shared among multiple qubits, while each line has a specific role and relation to one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' It should also be noted that most operations must be imple- mented with strict pulse durations and time intervals depend- ing on the site that gets addressed [1] due to fabrication imper- fections [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Although such pulse durations have to be care- fully considered in the mapping process by providing recent calibration data [13, 17, 18], in this work we consider an ideal crossbar architecture, as such data are not available yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' De- spite that, the mapping techniques proposed in this work are compatible with such considerations and can be added once calibration data are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' To better illustrate what the conditions and constraints are CL0 CL1 CL2 RL2 RL1 RL0 1 4 2 3 5 6 8 7 QL-3 < QL-2 > QL-1 > QL0 >< QL1 < QL2 > QL3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 3: Parallelizing shuttles of qubit 3 and 6 is not allowed due to violation of constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' when trying to parallelize quantum operations, let us consider the following example in which two shuttles are performed in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 2, the following requirements must be fulfilled to shuttle qubit 3 one site to the left: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' The destination site must not be occupied by another qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' The barrier between destination and origin sites must be lowered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' This is depicted as a dashed vertical CL0 line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' All barriers surrounding the origin and destination sites must be raised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' This is shown as solid red RL (RL0 and RL1) and blue CL lines (CL1 and the always-raised most-left CL line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' The voltage going through the QL line of the destination site (QL−1) must be higher than the one going through the origin site (QL0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' This is shown as QL−1 > QL0 in the top-right of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' To prevent other qubits in these two columns from shut- tling, the voltage going through their QL lines must be higher than their adjacent empty sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' This is depicted as voltage level relations between QL lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Note that QLs with no voltage relations are irrelevant for this par- ticular shuttle operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Now, we assume a shuttle of qubit 6 to the right (as de- picted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 3) in parallel to the left shuttle of qubit 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' This implies that all previously listed requirements (of qubit 3) need to be satisfied along with the new ones (of qubit 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' However, the fourth requirement can not be satisfied as the QL0 > QL1 relation we had before would have to be changed to QL0 < QL1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' If this change is allowed, we violate the fifth requirement of the first shuttle and, as a consequence, qubit 1 will shuttle to the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Therefore, we can not shuttle qubits 3 and 6 at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 5 Thus, we see that scheduling parallel gates in the crossbar implies a strict simultaneous satisfaction of all signal require- ments for each gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Any violation of these conditions would potentially result in shuttling of unwanted qubits, in unwanted qubit interactions or unknown qubit states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' As seen in the previous example, performing quantum operations in parallel without affecting other qubits and meeting all signal require- ments is not always possible regardless of qubit distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' In fact, it does not matter how far qubits are away from each other, but whether control lines are shared between them or not, and whether their operational requirements and relations match or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Unlike more popular qubit architectures based on superconducting or ion traps, this form of operational con- straint is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' On one hand, sharing control lines tackles the interconnect bottleneck, on the other hand, it intrinsically constraints its parallelization capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Finally, in other qubit architectures, it is possible to per- form different gate types in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' In the crossbar archi- tecture, this is not always the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' For example, applying single-qubit gates and shuttling operations at the same time is not possible (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 4a), because the former CL lines need to carry an alternating current (AC) signal while the latter re- quire DC signals for raising or lowering the barriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Application of X or Y rotations on single qubits As established in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' III, X or Y qubit rotations are im- plemented semi-globally, meaning that either all qubits in odd or even column parities will be rotated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' However, during an arbitrary cycle of algorithm execution, not all qubits in odd or even columns should be rotated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Therefore, to compen- sate for unwanted X or Y rotations, one has to come up with a specific rotation scheme such that only the targeted qubits are rotated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' In this work, we have implemented the scheme introduced by [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' We illustrate how it works in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 4, in which we are interested in rotating only qubit 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' This is an- other unique characteristic of this architecture, as additional gates are needed to perform single-qubit rotations on specific qubits, which impose new challenges to the mapping process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Routing of Qubits While we previously described the operational constraints to parallelize various gates in the crossbar architecture, we will now expand specifically on the qubit routing challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Routing a qubit in the crossbar means that an electron (or a hole) is physically ”pushed” to an empty site (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=', an empty quantum dot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' This mechanism is similar to a Quan- tum charged coupled device (QCCD) ion trap device when ions are shuttled through a common channel from trap to trap, assuming sufficient destination ion trap capacity [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' The QCCD architecture and the crossbar architecture fundamen- tally differ in topology, but both require special algorithms or additional routing routines to maintain control of qubit posi- tions and avoid potential conflicts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Focusing on the crossbar, shuttling a qubit does not only depend on specific control signal requirements and available empty sites but on the positions of other qubits as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' We illustrate this fact with an example in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 5, in which a verti- cal shuttle operation of qubit 3 is indicated by a black arrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' In this case, the horizontal barrier RL0 has to be lowered and the QL lines have to be pulsed in certain voltage relations to allow for correct shuttling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' However, an unwanted interaction between two other qubits in the same row (qubits 2 and 4, cir- cled) is concurrently caused, regardless of the QL2 and QL3 relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Analogously, the same issue exists with a horizontal shuttle when having two horizontally adjacent qubits in the same columns where the shuttle takes place [27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Lastly, there can be a blocked path conflict where there is no empty site for a qubit to shuttle to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Therefore, a dedicated qubit routing algorithm for the crossbar architecture has to be developed to avoid collisions, blocked paths, and unwanted interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Furthermore, even if we had such a dedicated routing algorithm, the same con- flicts have to be considered and prevented when scheduling gates in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' For that, control signals and qubit posi- tions must be carefully monitored within the mapping pro- cess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' From the description above, it is clear that both, routing and scheduling processes, need to jointly work in a strategy to avoid conflicts and optimize for algorithm success rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' This will be part of SpinQ, presented in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' SPINQ – THE FIRST NATIVE COMPILATION FRAMEWORK FOR SCALABLE SPIN-QUBIT ARCHITECTURES In this work, we present the first native compilation frame- work – SpinQ – dedicated to compiling and mapping quan- tum circuits onto scalable spin-qubit architectures, such as the previously described crossbar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' We have based our mapping techniques on previous works from [27, 28] while improving them from a scalability standpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 6 shows the schematic structure of our framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' As input, SpinQ accepts QASM format files that describe quan- tum circuits (used as benchmarks) in a device-independent manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' To increase flexibility, custom operations and their particular attributes can be defined in a hardware architec- tural configuration file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' It can include operational attributes such as the duration of a gate, the mathematical description of the unitary matrices, associated gate fidelities, and archi- tectural constraints, among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Moving on, the compiler consists of a series of steps (called passes) to decompose gates, route qubits, and schedule instructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' To address the unique mapping constraints of the crossbar architecture, we have conceptualized and developed the integrated strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' We did so not necessarily to maximize the performance of the algorithms when being executed on the crossbar, but rather to study how they behave on such architecture and focus on the scalability potential of spin-qubit technologies (see also Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' VIII).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' The compiler’s output is a QASM file which is compat- ible to be executed on the given crossbar architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Option- ally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' a verification step can take place to ensure the compiled ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='CL0 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='QL1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='QL2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='QL3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='(d) Step 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 4: Single-qubit gate on qubit 5: (a) Step 1: AC signals through the CL lines induce magnetic fields on qubits 1, 5, 6 and 2, thus changing their state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' The direction and frequency of these signals determine which columns (red or blue) and what rotation (X or Y gate) will be applied to the corresponding qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' (b) Step 2: The targeted qubit 5 is moved with a shuttle operation to a different column parity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' For this operation, the orthogonal lines (CL and RL) open and close as barriers and the diagonal lines (QL) create potential gradients to allow for qubit 5 to move (shuttle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Note that QL needs to have voltage relations with their neighbour QL lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' (c) Step 3: An inverse rotation is applied to qubits 1, 6 and 2 similarly to Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' (d) Step 4: Target qubit 5 is moved with a shuttle operation to the initial position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' CL0 CL1 CL2 RL2 RL1 RL0 1 4 2 3 5 6 8 7 QL-3 QL-2 QL-1 < QL0 < QL1 QL2 >< QL3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 5: Example of a conflict: the operational requirements of shuttling qubit 3 downwards have lowered the RL0 barrier thus causing an unwanted interaction between qubits 2 and 4 circuit meets all operational constraints of the crossbar archi- tecture without any conflicts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' This step is implemented to be able to check the compatibility of architectural proposals that are not physically realized yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Finally, several performance metrics are extracted from the compiled circuit to evaluate algorithm performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' In the next sections, we will further discuss each of the elements of the compiler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Compilation steps The compiler consists of the following steps: Quantum Algorithms (Benchmarks) Compiler Gate Decomposition Initial Placement Integrated Strategy (Routing and Scheduling) Architectural Configuration Compiled Circuit Metrics Verification Depth Overhead Gate Overhead Estimated Success Probability Compilation time Operational fidelities Operational durations Architectural constraints Topology FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 6: Overview of our SpinQ framework proposed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Decomposition of quantum gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Inputted QASM quan- tum circuits are transformed into a custom-made intermediate representation (IR) data format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Quantum gates are then de- composed into gates native to the architecture based on the decomposition sequences specified in the architectural con- figuration file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Physical initialization of spin qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' A checkerboard pat- tern has been proposed [42] to allow space for qubits and an- cilla qubits to move [27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' The physical space achieved be- tween the qubits not only facilitates shuttling qubits avoiding possible conflicts but also reduces crosstalk and enables sur- face code error correction [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' As we will discuss later, main- taining this placement throughout a circuit execution plays an integral role in our compilation strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Having said that, initializing qubits in alternative patterns and changing them during execution is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' This flexibility can be particu- larly advantageous to highly specialized mapping techniques for the crossbar as well as spin-qubit architectures in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 7 Virtual-to-physical qubit initial placement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' The current version of SpinQ associates virtual qubits of an algorithm with physical qubits (placed in the checkerboard pattern) in a one- to-one manner by numbering the physical qubits from left to right and from bottom to top (as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' In the re- sults sections VII and VIII, we will provide insights on how common initial placement algorithms can be adapted to im- prove the performance of spin-qubit architectures (such as the crossbar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Integrated Strategy for Routing and Scheduling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' As ex- plained in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' IV, both routing and scheduling techniques must avoid conflicts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' To do that, a specific strategy needs to be followed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' There can be various strategies for various goals with trade-offs between performance and compilation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' The presented Integrated Strategy tilts towards mini- mizing compilation time while having great prospects to be competitive against other solutions that focus on algorithm performance as will be discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Firstly, in the Integrated Strategy, the checkerboard pat- tern qubit placement [1], also known as ”idle-configuration” in [28], should be maintained as much as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' This pro- vides at least two empty sites for every qubit to move towards to, at the beginning of each cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' To maintain the checker- board pattern throughout circuit execution when routing for two-qubit gates, a conflict-free shuttle-based SWAP technique can be used ([27]) as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Note that this move- ment of qubits results in a gate overhead of 4 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=', 4 shuttle operations), but a depth overhead of 2, as these two shuttle pairs can always be executed in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' To bring the two qubits to the appropriate sites and allow two-qubit interac- tions, multiple shuttle-based SWAPs might be performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' For that, we have implemented a shortest-path algorithm based on the Manhattan distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' When one of the qubits is placed in the desired position, the next step is a horizontal shuttle, either to the left or to the right, after which the target and control qubits are vertically adjacent for interaction, and the checkerboard pattern is temporarily broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Proceeding the √ SWAP, a shuttle instruction returns the qubit to the previ- ous position and the checkerboard pattern gets restored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Note that the aforementioned process can be successfully executed only in that particular order, otherwise there can be a routing conflict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' So far, we have only talked about a routing technique for bringing together qubits for performing two-qubit gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' However, qubit routing is also needed for X or Y rotations to a specific qubit(s) and for shuttle-based Z rotations, as dis- cussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' As a consequence, the ”idle configura- tion” should be maintained when routing for these gates as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' But once again, routing for single-qubit gates before the scheduling stage can be problematic as it can cause con- flicts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' For that reason, the second consideration of the Inte- grated Strategy is the integration of single-qubit gate routing within the scheduling stage, hence the name ”integrated”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Then the Integrated Strategy continues with two passes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' In the first pass, the scheduler tries to parallelize X or Y gates in an ideal manner and Z gates individually, ignoring any po- tential conflicts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' This is no different than other single-qubit gate scheduling processes proposed for other qubit architec- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' However, it differs on the second pass which integrates the routing procedures for X, Y and Z gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' The second pass iterates over each cycle produced by the first pass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' For each cycle, there are two possibilities: (a) if no conflicts are de- tected when scheduling the necessary shuttle instructions re- quired for each single-qubit gate, the new shuttle instructions are inserted between the current cycle and the next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' (b) if con- flict(s) are detected, the subset of the problematic gate(s) is removed and stored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Once the non-problematic gate subset is scheduled according to case (a), the problematic subset is recalled and iterated again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' This time it constitutes a conflict- free cycle and is scheduled according to case (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' This way the second pass loops in total two times whenever there is a detected conflict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Overall, the current implementation does not parallelize gates of different types in the same cycle, and thus each cycle is dedicated to one instruction type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Fortunately, the strategy described above and suggested extensions in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' VIII can be adapted to a real setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' As explained in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' IV, a fabricated crossbar device will most likely have material imperfections, thus requiring pulse calibration per site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' As pointed out by [28, 44], pulsing control lines prematurely to account for ma- terial variations could cause an unwanted interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Since, however, the Integrated Strategy (or an extension thereof) ex- clusively schedules gates of the same type in each cycle, fine- tuning pulses within the cycle is possible before moving to the next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Performance metrics We will now introduce the metrics used in this work to eval- uate the performance of SpinQ when mapping different algo- rithms on the crossbar architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Gate overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' One commonly used metric to evaluate the performance of a mapper and its underlying architecture is gate overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' We calculate it as the percentage relation of additional gates inserted by the mapper to the number of gates after decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' We do not count decomposition gate overhead as it is always proportional to the number of gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Getting a clear view of the various sources of gate overhead will help to form useful insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Note that, unlike supercon- ducting architectures where gate overhead results from rout- ing instructions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' SWAP gates) for performing two-qubit gates, in the crossbar, it can be caused by single-qubit gates as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' The main sources of gate overhead are the following: 4 additional shuttle instructions per shuttle-based SWAP for two-qubit gates At least 3 additional instructions for each X or Y rota- tion gate within the semi-global rotation scheme 2 additional shuttle instructions for each two-qubit gate 1 additional shuttle operation for each Z rotation gate Depth overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Another commonly used metric to eval- uate the performance of a mapper and its underlying architec- ture is the depth overhead of a circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' The depth of a circuit is 8 CL0 CL1 CL2 RL2 RL1 RL0 1 4 2 3 5 6 8 7 QL-3 < QL-2 > QL-1 > QL0 < QL1 QL2 QL3 (a) Horizontal shuttling CL0 CL1 CL2 RL2 RL1 RL0 1 4 2 3 5 6 8 7 QL-3 QL-2 QL-1 < QL0 > QL1 < QL2 > QL3 (b) Vertical shuttling FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 7: Shuttle-based SWAP for two-qubit gate routing: With this technique, two diagonally neighbouring qubits exchange their position by consecutively performing two horizontal and two vertical shuttles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' equal to the minimum number of time steps of a circuit when executing gates in parallel [9, 10, 45–47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' We calculate depth overhead as the percentage relation of additional depth pro- duced by the mapper to the circuit depth after decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Note that the initial circuit depth is calculated after scheduling the circuit only by its gate dependencies, meaning without any architectural constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' The main sources of depth overhead are: At least 3 additional cycles for each X or Y rotation gate due to the semi-global rotation scheme 2 additional cycles per shuttle-based SWAP for two- qubit gates 2 additional cycles for each two-qubit gate 1 additional cycle for each Z rotation gate Estimated Success Probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' A key metric to assess the performance not only of the compiler but in general of a quan- tum computing system is the algorithm success rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' From an experimental point of view, the algorithm success rate is cal- culated by executing the algorithm several times on a given (real) quantum processor and creating the distribution of suc- cessful executions, based on the expected measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' An alternative way to calculate the success rate without the need for a real quantum processor is by using an approximation nu- merical method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' One of the most commonly used methods to do so is considering the estimated success probability (ESP) of an algorithm [48]: ESP = � i � j gate fidelityi,j (1) where i represents the ith time step and j the jth gate in the ith time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' This method is far more efficient compared to using a Hamiltonian model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' However, the accuracy of the estima- tion can be low due to its simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' To expand it, we have considered a per-type and per-location variability of gate fi- delities, based on a normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' This implies that, for instance, a two-qubit gate ( √ SWAP) will have lower fidelity than a single-qubit gate and that the actual fidelity will depend on the exact location in the topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' These expansions con- stitute a more realistic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=', closer to a real device, estimation of circuit success probability: ESP = � i � j gate fidelityx,y i,j (2) where i represents the ith time step, j the jth gate in the ith time step and and x, y are the physical qubit(s) coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Compilation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' In this work, we are not only inter- ested in building mapping techniques themselves, but also in their scalability potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' This necessitates that our proposed SpinQ strategy should remain efficient for a variety of quan- tum circuit parameters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=', number of qubits or percentage of two-qubit gates).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' By measuring the compilation time for mapping quantum circuits, we get a reference of the scalabil- ity of our implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Verification A verification tool is important to this work due to the lack of a working device for real-system testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' The tool is searching for mismatches between all shuttling sequences and the qubits position history stored during compilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' It also checks for conflicts, architectural constraint violations and state vector mismatches between and in each stage of the mapper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' The latter uses the Qiskit Aer library [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 9 VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' EXPERIMENTAL METHODOLOGY Benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' We have generated 3, 630 random uniform algorithms [50] containing X, Y, Z and √ SWAP gates (all native to the crossbar architecture) to be used as benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' With this set, we can vary on demand the number of gates, number of qubits, and percentage of two-qubit gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' For example, a random uniform benchmark with 50% of two- qubit gates relative to single-qubit gates will have 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='33% of X or Y gates, 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='33% of Z gates, and 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='33% of two-qubit gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Generating synthetic circuits provides a well-controlled benchmark collection from which we can better understand results and form insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Moreover, we use real benchmarks from the RevLib library in a [5 - 1400] gate range [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Quan- tum circuits from this library are often used in related quantum circuit compilation works [9, 11, 12] and it consists of quan- tum algorithms with parameters ranging from 3 to 16 qubits, 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='75% to 100% of two-qubit gates and 5 to 512, 064 gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Finally, we also consider quantum circuits from the Qlib li- brary [52] which contains real quantum algorithms in increas- ing size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Benchmarks characterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' When it comes to perfor- mance evaluation, it is important to not only consider proper- ties of the crossbar architecture but also the characteristics of quantum circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' The simplest and most commonly [14] used parameters of quantum circuits are number of qubits, number of gates, and absolute or relative (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=', percentage) number of two-qubit gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' However, only these three characteristics can be misleading for two reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Firstly, two benchmarks, for instance, could have the same parameter values but heavily differ in the circuit’s structure [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' When one of them has all pairs of qubits interact with each other will require more routing than the other which might have the same number of interactions, but with only one pair of qubits interacting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' The structure of a quantum circuit is derived from its qubit inter- action graph (QIG) which represents the number and distri- bution of interactions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=', two-qubit gates) between virtual qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Several internal circuit parameters can be extracted from the QIG that better distil its properties [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Having said that, we analyze QIGs visually only, as this is still an active field of research [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Despite that, we can nonetheless make concrete conclusions and form insights, making visual QIG assessments a viable tool to characterize algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' The sec- ond reason is that initial gates can be decomposed to natively supported instructions for the underlying architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' This means that the number of gates and ratios (percentages) be- tween each gate type can differ from the initial set to the actual executable set, meaning that evaluations can become more ac- curate when accounting for the decomposed set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Experimental Setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' We run SpinQ on a laptop with an Intel(R) Core(TM) i7-3610QM CPU @ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='20GHz and 16GB DDR3 memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' SpinQ is written in Python 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='6 version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' EVALUATION AND ANALYSIS In this Section, we present an in-depth performance anal- ysis of SpinQ when mapping a broad range of quantum al- gorithms on the crossbar architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' We then form architec- tural and mapping insights for each performance metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' More specifically, gate overhead and corresponding insights are pre- sented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' VII A and VII B, depth overhead in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' VII C and VII D, and ESP in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' VII E and VII F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Finally, we show results regarding compilation time of SpinQ in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' VII G to asses its scalability capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Gate Overhead To start with, we analyse the gate overhead trend in a wide range of quantum algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 8 we have mapped ran- dom uniform circuits on the crossbar architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Focusing on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 8a, which reaches up to 25 qubits, we observe that as we go from low to high number of qubits and from low to high percentage of two-qubit gates, the gate overhead increases (from blue to red color).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' More precisely, higher qubit counts imply larger crossbar topologies, thus potentially longer rout- ing distances, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=', more shuttle-based SWAPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Furthermore, higher percentages of two-qubit gates potentially lead to more routing of qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' These observations verify that the main source of gate overhead is indeed the routing of qubits for two-qubit gates (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' V A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' We also notice that the num- ber of gates has a small but noticeable influence on the gate overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' To further observe the trend when increasing the number of qubits, we changed the range of qubits from [3 – 25] to [25 – 99] in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 8b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' We see once more that the gate overhead increases as we go from low to high number of qubits and percentage of two-qubit gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' As expected, the gate overhead, shown on the color bars, of the [25 – 99] qubit range is on average 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='49% higher than that of the [3 – 25] qubit range because of the increased routing distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' So far, the above random algorithms were generated to have control of different circuit parameters (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=', number of qubits and gates and two-qubit gate percentage) in a way to broadly cover the parameter space and up to certain boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' How- ever, they might not be representative of real algorithms from a circuit structure point of view (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=', how two-qubit gates are distributed among qubits or the degree of operation par- allelism).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Therefore, we then mapped real algorithms from the RevLib and Qlib libraries resulting in the gate overhead shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 9, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 10, and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 9 we can observe that benchmarks “cluster” together in similar colours, namely shades of blue, green, yellow and red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' This implies that similar benchmarks, meaning with similar parameters and structure, have similar gate overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Note that whereas ran- dom uniform algorithms have all the same circuit structure because of the way they are generated, RevLib algorithms present different structural parameters not only compared to the randomly generated circuits but also between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' For this reason, correlations such as the higher the number of qubits and percentage of two-qubit gates gets, the higher the gate overhead will be, are not as evident as before (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' for random circuits).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' To further analyse how structural circuit parameters impact the gate overhead, we mapped algorithms with similar number of gates, qubits, percentage of two-qubit gates and QIG from 10 Gates (before decomp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=') 0 25005000750010000 12500 15000 17500 20000 Qubits 5 10 15 20 25 2-Q Gate Percentage (before decomp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=') 0 20 40 60 80 100 MAX=1114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='28, AVG=473.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='69, MED=423.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='23, MIN=124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='53 Gate Overhead [%] 200 400 600 800 1000 (a) Gates (before decomp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=') 0 25005000750010000 12500 15000 17500 20000 Qubits 30 40 50 60 70 80 90 100 2-Q Gate Percentage (before decomp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=') 0 20 40 60 80 100 MAX=2416.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='29, AVG=959.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='18, MED=871.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='93, MIN=65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='03 Gate Overhead [%] 500 1000 1500 2000 (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 8: Resulting gate overhead when 3, 630 random uniform quantum algorithms are mapped onto the crossbar architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' The three axes correspond to benchmark characteristics, namely, the number of gates [50 - 20,000], number of qubits [3 - 99] (split into two subfigures), and two-qubit gate percentage [0 – 100].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' the Qlib library onto the crossbar architecture (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' With these simulations, we also want to perform a scalability analysis of the algorithms which is not possible with RevLib circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' First, note that the Cuccaro Adder (top line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 10) has a small drop in the percentage of two-qubit gates that goes from 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='43% to 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='75% when increasing in size (num- ber of qubits) whereas the Vbe Adder (bottom line) main- tains a lower percentage of 50% for the same increase in size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' One can immediately observe that the Cuccaro Adder shows a higher gate overhead up to 284% due to the higher two-qubit gate percentage compared to the 271% of Vbe Adder, match- ing the conclusions made in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' However, as we empha- sized above, in the case of real algorithms comparisons can only be properly made when looking not only at their circuit parameters but also at their more structural ones such as the QIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' For this reason, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 11 we show the derived QIGs from Vbe Adders’ 40-qubit circuit, Cuccaro Adders’ 38-qubit cir- cuit and Cuccaro Multipliers’ 21-qubit circuit alongside their gate overhead in relation to the number of qubits and percent- age of two-qubit gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' In these QIGs, nodes correspond to qubits and edges to qubit interactions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=', two-qubit gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' The particular size selection of these QIGs was made to easily show their structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' We immediately observe similarities in the QIGs of the two Adders as the distribution of interactions is almost identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' More specifically, we see 2 to 3 inter- actions per qubit on average, with others close to their logical qubit number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Therefore, we can conclude that the higher gate overhead of Cuccaro Adder is due to the higher percentage of two-qubit gates, compared to Vbe Adder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' However, note that the Cuccaro Multiplier has the highest gate overhead of all three (309%) despite having a lower two- qubit gate percentage than the Cuccaro Adder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' The reason be- hind this is the difference in its QIG, which is much more con- nected implying a denser qubit interaction distribution com- pared to the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Because of this, more routing is needed to connect (nearly) all qubits across the entire topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Insights from gate overhead analysis Accounting for the routing constraints, as discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' IV, mapping on the crossbar architecture is not a trivial task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' In fact, we have emphasized the importance of conceptu- alizing and developing new routing techniques that specif- ically can address the unique mapping challenges of spin- qubit architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' More specifically, with the adoption of the checkerboard pattern combined with the shuttle-based SWAPs, we can provide a scalable solution of qubit routing for two-qubit gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Additionally, the complexity only scales with the number of two-qubit gates, therefore being a viable solution for large-scale implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' However, this tech- nique makes two-qubit gate routing the highest source of gate overhead and it can dramatically increase it with higher qubit counts and a higher percentage of two-qubit gates (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 8 and 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Moreover, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 11 we saw that gate overhead can also be increased by a more connected QIG even if other circuit parameter values are comparatively lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' This shows 11 Gates (before decomp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=') 0 200 400 600 800 1000 1200 1400 Qubits 4 6 8 10 12 14 16 2-Q Gate Percentage (before decomp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=') 20 30 40 50 60 70 80 90 100 MAX=306.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='6, AVG=210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='59, MED=205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='72, MIN=167.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='0 Gate Overhead of Integrated Strategy [%] 180 200 220 240 260 280 300 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 9: Resulting gate overhead when mapping quantum algorithms from the RevLib library onto the crossbar architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' The three axes correspond to benchmark characteristics, namely, number of gates [5 - 1400], number of qubits [3 - 16] and two-qubit gate percentage [18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='75 - 100].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' the importance of basing circuit performance evaluation not only on simple circuit parameters but also on other ‘hidden’ structural characteristics such as the qubit interaction distribu- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='Having said that, the second biggest source of gate over- head originates from X or Y qubit rotations, as it produces at least 3 additional gates compared to 4 additional gates for each shuttle-based SWAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' This is due to the unprecedented semi- global rotation scheme which is the first time that single-qubit gates require additional instructions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=', produce gate over- head) compared to other qubit architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' The previous two facts inspire novel mapping techniques for the crossbar architecture (and potentially for other spin-qubit architectures with similar characteristics) that can increase performance, namely: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Developing a routing solution dedicated to accounting for potential conflicts and constraints can reduce the gate overhead resulting from the shuttle-based SWAPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Such a generalized routing algorithm could also include SWAP interactions (two consecutive √ SWAPs) and CPHASE interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' For instance, there can be sce- narios that choosing a more noisy two-qubit interaction, for the purpose of avoiding an upcoming conflict, that could result in higher ESP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Additionally, such a heuris- tic algorithm can allow multiple control or target qubits ([10]) to be shuttled around the topology allowing for parallelization of many two-qubit gates while avoiding high error variability in the topology [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' However, Gates (before decomp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=') 0 50 100 150 200 250 300 350 400 Qubits 0 20 40 60 80 100 120 2-Q Gate Percentage (before decomp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=') 50 55 60 65 70 Gate Overhead [%] 200 220 240 260 280 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 10: Resulting gate overhead when mapping the Cuccaro Adder (top line of data points) and the Vbe Adder (bottom) quantum algorithms from the Qlib library onto the crossbar architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' The three axes correspond to benchmark characteristics, namely, number of gates [4 - 385], number of qubits [4 - 130] and two-qubit gate percentage [50 - 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' such a solution must be implemented with complexity in mind such that it will not make it unviable on large scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' A more efficient routing algorithm for single-qubit gates can significantly reduce the gate overhead, such that a specific rotation scheme to rotate targeted qubits is used less often.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Such an algorithm can route qubits to the appropriate odd or even columns before the execu- tion of single-qubit gates without the need to apply any scheme afterwards (see the example in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' IV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Combining the previous two points, there can be a uni- fied algorithm implementing both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' In such an algo- rithm, upcoming routing for single-qubit gates is ac- counted for when routing for two-qubit gates, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Finally, an initial placement algorithm can take into ac- count not only two-qubit gates but single-qubit gates as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Since the positions of qubits influence the gate overhead resulting from single-qubit gates (due to the semi-global rotation scheme), an extension of an initial placement algorithm accounting for single-qubit gates can reduce the gate overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Last but not least, we have emphasized that to concretely evaluate results, there has to be sufficient characterization of 12 Cuccaro Multiplier Vbe Adder Cuccaro Adder FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 11: Resulitng gate overhead when the Vbe Adder, Cuccaro Adder and Cuccaro Multiplier from the Qlib library are mapped onto the crossbar architecture alongside their Quantum Interaction Graphs (QIG) consisting of 40, 38 and 21 qubits, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' The y-axis represents the two-qubit gate percentage and the x-axis the number of qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' We see gate overhead to be influenced not only by the number of qubits and two-qubit gate percentage but also by the qubit interaction distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' benchmarks, especially when evaluating novel architectures and mapping techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' In our analysis, we did not rely only on simple benchmark parameters, such as the percentage of two-qubit gates, but also on the internal structure of bench- marks using the Quantum Interaction Graph (QIG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Depth Overhead This time, we analyse the depth overhead when mapping onto the crossbar the same random uniform benchmark set as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 12, it can be observed that the trend (colours) of the depth overhead changes for different ranges of number of qubits as shown in the two subfigures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Knowing that the main source of depth overhead originates from X or Y gates (at least 3 additional cycles), we expect the depth overhead to become higher in lower regions of two-qubit gate percentage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' That is observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 12a, where the number of qubits goes up to 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' However, moving on to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 12b, we see that this trend changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Now, due to the higher number of qubits, rout- ing distances have increased, thus routing for two-qubit gates dominates the depth overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' This is apparent by its increase (from blue to red colour) as we go from lower qubit counts to higher qubit counts, and as we go from low to higher percent- age of two-qubit gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Finally, this fact is also apparent in the absolute values of depth overhead of the two subfigures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Note also that the number of gates has a slight influence on the depth overhead, but it is not as relevant as the other char- acteristics discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Moving on, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 13 shows the depth overhead of a Cuc- caro Adder when scaling it up from 4 to 130 qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' In the range of 4 to 20 qubits, we observe an increase in depth over- head as the percentage of two-qubit gates decreases, which aligns with the remarks about the main source of depth over- head (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=', the X or Y gates).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Then, for an increasing number of qubits (from 20 qubits on) and at an almost constant two- qubit gate percentage (67%), the depth overhead increases at a slower rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Here we conclude, once again, that two-qubit gate routing starts to dominate the depth overhead as routing distances become larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' In most previous works, the amount of two-qubit gates is the main circuit characteristic to anticipate how much qubit routing will be needed for a specific quantum algorithm and therefore the major and only source of gate/depth overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' However, in the crossbar architecture, and potentially in other spin-qubit crossbar designs, single-qubit gates can also con- 1334 33 36 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='31 28 22 1913 Gates (before decomp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=') 0 25005000750010000 12500 15000 17500 20000 Qubits 5 10 15 20 25 2-Q Gate Percentage (before decomp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=') 0 20 40 60 80 100 MAX=6290.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='98, AVG=2217.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='92, MED=2132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='74, MIN=262.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='0 Depth Overhead [%] 1000 2000 3000 4000 5000 6000 (a) Gates (before decomp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=') 0 25005000750010000 12500 15000 17500 20000 Qubits 30 40 50 60 70 80 90 100 2-Q Gate Percentage (before decomp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=') 0 20 40 60 80 100 MAX=27786.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='21, AVG=12530.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='1, MED=11934.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='55, MIN=2250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='0 Depth Overhead [%] 5000 10000 15000 20000 25000 (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 12: Resulting depth overhead when 3, 630 random uniform quantum algorithms are mapped onto the crossbar architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' The three axes correspond to benchmark characteristics, namely, number of gates [50 - 20,000], number of qubits [3 - 99] (split into two subfigures), and two-qubit gate percentage [0% – 100%].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' tribute to this overhead as discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' It is then important to have a closer look at the X or Y rotation gate percentage and further analyse how it impacts the depth overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Addition- ally, after the gate decomposition step, the percentages and ratios between all gate types are changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' To illustrate this, imagine a quantum circuit that originally consists of a low number of CNOT gates and no Z gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' After the decompo- sition to gates supported by the crossbar architecture, the per- centage of Z rotation gates will increase, and consequently, the two-qubit gate percentage will decrease, as CNOT gates are decomposed as Ry( π 2 ), two √ SWAP, S, S†, Ry( −π 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Thus, it is relevant to consider this change in gate percentage in our analysis as ultimately the executable circuit will only consist of native gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' To summarize, as overhead comes from mapping different types of gates on the crossbar, indi- vidually distinguishing between them, in particular after de- composition, can increase the accuracy of our evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' To illustrate the previous point, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 14 we show the depth overhead of the Cuccaro Adder (upper dots) and the Vbe Adder (lower dots) with the same ranges as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Note that the y-axis corresponds to the percentage of X or Y rotation gates after decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' From this new perspective, we clearly see their difference in actual (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=', executed by the architecture) X or Y rotation gate percentage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' On average the depth overhead of the Vbe adder is 196% higher than the Cuccaro Adder for the same range of qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' As explained before, the highest source of depth overhead comes from X or Y rotations gates, which explains the large depth overhead difference between those two algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Insights from depth overhead analysis From the previous analysis, we can observe that trends can change based on the parameter ranges of benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' This is because different sources of depth overhead contribute with different rates based on the number of qubits (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=', crossbar size).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' More specifically, the overhead contribution resulting from mapping X/Y gates was higher up to a certain number of qubits after which was exceeded by the contribution rate of two-qubit gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' We saw that exceeding a threshold of more than 20 qubits increases the depth overhead at a steadier pace, which specifically favoured scalability for Cuccaro Adder in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 13 and 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' It is expected, however, that with different algorithms, there will be different trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' With such observa- tions, we stress the importance of distinguishing between all gate types and especially after decomposition to better under- stand the performance impact of mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' With that knowl- edge, we can create better mapping techniques and/or make an informed selection of algorithms to execute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' As stated before, the fact that gate overhead can result from mapping single-qubit gates is unprecedented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Furthermore, we notice that mapping both, single- and two-qubit gates, re- quires additional shuttles and they produce the highest gate and depth overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Therefore, novel mapping techniques minimizing all qubit movements (shuttles) can increase per- formance substantially, such as the ones discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' VII B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' From an architectural point of view, since the shuttle operation is so relevant, there have to be as few operational 14 Gates (before decomp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=') 0 50 100 150 200 250 300 350 400 Qubits 0 20 40 60 80 100 120 2-Q Gate Percentage (before decomp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=') 67 68 69 70 71 MAX=586.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='97, AVG=563.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='11, MED=570.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='28, MIN=450.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='0 Depth Overhead [%] 460 480 500 520 540 560 580 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 13: Resulting depth overhead when Cuccaro Adder from the Qlib library is mapped onto the crossbar architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' The three axes correspond to benchmark characteristics, namely, number of gates [4 - 385], number of qubits [4 - 130] and two-qubit gate percentage [66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='75 - 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 0 20 40 60 80 100 120 Qubits 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='25 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='50 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='75 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='00 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='25 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='50 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='75 X/Y Gate Percentage (after decomp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=') Depth Overhead [%] 450 500 550 600 650 700 750 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 14: Resulting depth overhead when Cuccaro Adder (bottom line of data points) and Vbe Adder (top) from the Qlib library are mapped onto the crossbar architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' The y-axis represents the X or Y gate percentage, and the x-axis the number of qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' constraints as possible when mapping them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Estimated Success Probability In this section, we will show how the success probability of an algorithm drops after mapping it to the crossbar architec- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Before we continue, we have to mention that even with operational fidelities as high as 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='99% for single-qubit gates and shuttles (as suggested in [1]) and 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='98% for √ SWAPs, the ESP drops drastically to 0 in most algorithms with a high number of gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' For that reason, we just focused on the Bernstein-Vazirani algorithm as it has got a low percentage of two-qubit gates (usually there are only one or two CNOTs), therefore the error is mostly introduced by single-qubit gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 0 100 200 300 400 500 0 20 40 60 80 100 Estimated Success Probability (ESP) 0 50 100 150 200 250 ESP Original ESP Gates (before mapping) Gates (after mapping) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 15: Estimated success probability (ESP) before and after compilation of Bernstein-Vazirani algorithm from 2 to 129 qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 15 shows the ESP of the Bernstein-Vazirani algorithm when scaling it from 2 to 129 qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' The red line “Origi- nal ESP” refers to the ESP before mapping, and the blue line ”ESP” refers to ESP after mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' We observe a sharp ESP decrease approaching 10% for 267 gates after mapping with a slope rate of −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content='6 which is caused by the increased num- ber of gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' For 529 gates after mapping we obtained a 0% ESP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Another reason for the ESP decrease is the semi-global single-qubit rotation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' for each of the X or Y gates contained in the circuit (after decomposition), all qubits in odd or even columns are rotated (even the ones that are not targeted for rotation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' This is further explained in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' IV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 15 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Insights from Estimated Success Probability analysis Our estimated success probability equation 2, although sim- ple, is approximating a worse-case-scenario algorithm success rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' We observed a rapid decline in ESP in a minimally connected algorithm (mostly X or Y rotation gates), even though our equation did not include decoherence-induced er- rors [28, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' The main reason for this decrease is the result- ing overhead when implementing single-qubit gates on spe- cific qubits given the semi-global rotation scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Note that in this case, all qubits in either column parities are rotated thus each contributing to this ESP drop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Therefore, it is essential to determine which algorithms could take advantage of the semi-global control and/or develop architecture-specific map- ping techniques to minimize the need for a scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' On real NISQ quantum devices there are other sources of noise noise that impact algorithm execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Fortunately, it is expected that processors will gradually become more ro- bust with better fabrication tolerances and improved error- mitigation and mapping techniques will be developed and ul- timately quantum error correction protocols will be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' It remains challenging, however, to accurately simulate errors in large-scale devices to derive algorithm’s success probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Compilation time 0 2500 5000 7500 10000 12500 15000 17500 20000 Gates 0 2 4 6 8 Seconds Compilation Time [s] qubits = 3 qubits = 12 qubits = 21 qubits = 30 qubits = 39 qubits = 48 qubits = 57 qubits = 66 qubits = 75 qubits = 84 qubits = 93 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 16: Compilation time when mapping random uniform algorithms with 50% of two-qubit gates onto the crossbar architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' We observe a linear relation which makes SpinQ suitable for scalable spin-qubit crossbar architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Finally, we measure the compilation time of our solution to evaluate its scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' The compilation time of SpinQ In- tegrated Strategy can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 16 for a subset of the random uniform circuits that have been used in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 8 and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' This subset consists of circuits with only 50% of two-qubit gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' With this subset we map the same number of gates for each gate type, thus all internal SpinQ processes are weighted equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' We observe a linear increase in compilation time in relation to the number of gates for each qubit count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' This im- plies that our strategy is suited for scalable spin-qubit crossbar architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Improvements can be directed towards reducing the slopes for each qubit count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' DISCUSSION AND FUTURE DIRECTIONS TABLE I: Computational complexity comparison between compilation strategies for the crossbar architecture [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' With n we denote the number of gates in a quantum circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Strategy Complexity Backtrack [27] O(n3) Suffer a side effect [27] O(n2log(n)) Avoid the deadlock [27] O(n) Integrated (ours) O(n) Integrated strategy improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' There can be a few extensions to the Integrated Strategy that can provide better performance (less overhead and higher ESP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' These improve- ments can be divided into two categories: a) improvements that increase complexity marginally and b) improvements that will increase complexity substantially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' It is important to make this differentiation because on large scale we have to consider the trade-off between complexity (computation time as sizes increase) and performance (less overhead and higher ESP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Improvements in category (a) will involve a constraint and conflict check for any shuttle-based type gate to enable com- plete parallelization of all single-qubit gates within the second pass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Note that, once again, each cycle remains dedicated to one gate type, therefore, fine-tuning pulse durations in real devices is still possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Moving on to the next category (b), it consists of all heuris- tic mapping algorithms (routing and initial placement) dis- cussed in Sections VII B, VII D and VII F, which can be ex- tended to other scalable spin-qubit architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' This will en- able complete parallelization of two-qubit gates and less rout- ing for both, one- and two-qubit gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Strategy Comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' As we discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' IV, the crossbar architecture comes with constraints that prevent full parallelization of quantum instructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' The crossbar, how- ever, may reach two types of conflicts (unwanted interactions or blocked paths), even if all constraints are respected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' For that reason, there must be some kind of compilation strategy between the scheduler and the router to prevent conflicts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' In this work, we have implemented the Integrated strategy which is different from the three strategies suggested in [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Ta- ble I compares the computational complexity of these three strategies with our own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' The backtrack strategy suggested in [27] avoids conflicts by trying a different scheduling combi- 16 nation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' If after repeating this process the scheduler has back- tracked to the first instruction of the cycle (no more schedul- ing combinations), a new routing path is given by the rout- ing algorithm and the scheduling is repeated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' This strategy can be quite complex as the worst case scenario can un-route and un-schedule all the gates going back to a completely un- mapped circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' An improved version of this strategy called suffer a side effect, is a special case of the former and it is only preferred whenever a corresponding conflict can be corrected and if the correction is less costly than only fol- lowing the ”backtracking” strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' The final strategy, and the one implemented in [27], is called avoid the deadlock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' This strategy, similar to our Integrated strategy, is trying to avoid conflicts by parallelizing only X or Y gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' In this way, √ SWAPs and shuttle operations can not cause a con- flict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' However, in this strategy there is no synergy between the routing and scheduling stages as our Integrated strategy has, therefore there is little flexibility for improvements and per- formance can not be easily improved while keeping the same complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Our strategy is able to maintain the same O(n) complexity even after improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' General discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' When developing novel mapping techniques for scalable quantum computing architectures such as the si-spin crossbar two main factors have to be considered: scalability and adaptability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' As spin-qubit fabrication capa- bilities are improving, new architectural designs with maybe higher qubit counts will be explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Therefore, from a com- putation/compilation time point of view, mapping techniques should be as scalable as the underlying technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Practi- cally, this implies that highly sophisticated and more complex mapping techniques might be excellent for a particular archi- tecture and up to a certain number of qubits, but could be impractical for more qubits or even unusable for another ar- chitecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' In addition, as we are slowly exiting the NISQ era, quantum technologies will become more robust, espe- cially with the use of quantum error correction techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' By that time, optimizing mapping techniques for specific hard- ware and/or algorithm might not be as relevant as today, but rather how fast and adaptable they are to a plethora of quan- tum algorithms and increased number of qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' CONCLUSION Different quantum circuit mapping techniques have been developed to deal with the limitations that current quantum hardware presents and are being consistently improved to ex- pand its computational capabilities by getting better and better algorithm success rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' The most advanced mapping meth- ods focus on ion-trap and superconducting devices due to their ‘maturity’ compared with other quantum technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' However, spin-qubit-based processors have a great potential to rapidly scale and the first 2D crossbar architectures have been recently demonstrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' In this work, we focused on the quantum circuit mapping challenges of the newly emerging spin qubit technology for which highly-specialized mapping techniques are needed to take advantage of its operational abilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Specifically, we used the crossbar architecture as a stepping stone to explore novel mapping solutions while focusing on scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' The crossbar architecture adopts a shared-control scheme, thus making it a great candidate to tackle the interconnect bottleneck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' On that note, we have developed SpinQ, the first native compilation framework for spin-qubit architecture which we used to analyze the perfor- mance of synthetic and real quantum algorithms on the cross- bar architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Through our analysis, we tried to inspire novel algorithm- and hardware-specific mapping techniques that can possibly increase the performance while taking into account the compilation scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' We also emphasized the importance of characterizing benchmarks before and after de- composition and by including their QIG structure to better evaluate results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' ACKNOWNLEDGEMENT This work is part of the research program OTP with project number 16278, which is (partly) financed by the Netherlands Organisation for Scientific Research (NWO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' This work has also been partially supported by the Spanish Ministerio de Ciencia e Innovaci´on, European ERDF under grant PID2021- 123627OB-C51 (CGA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' We thank Menno Veldhorst and Hans van Someren for their fruitful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' 17 [1] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Li, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} +page_content=' Petit, D.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQfFTWv/content/2301.13241v1.pdf'} diff --git a/2tAyT4oBgHgl3EQfovik/vector_store/index.pkl b/2tAyT4oBgHgl3EQfovik/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..a9e76cd42631ab8dc814f57386e1280cd3228c02 --- /dev/null +++ b/2tAyT4oBgHgl3EQfovik/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:0d1919dd2d47eb1aea95c922f1684f09e247bebb8204b2c0ec855ed938b39c0b +size 178800 diff --git a/2tE4T4oBgHgl3EQf0A0W/content/tmp_files/2301.05278v1.pdf.txt b/2tE4T4oBgHgl3EQf0A0W/content/tmp_files/2301.05278v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c4e8edc78d8832db82697b00b494eaaccde20bc4 --- /dev/null +++ b/2tE4T4oBgHgl3EQf0A0W/content/tmp_files/2301.05278v1.pdf.txt @@ -0,0 +1,1972 @@ +MIXED VOLUMES OF NORMAL COMPLEXES +LAUREN NOWAK, PATRICK O’MELVENY, AND DUSTIN ROSS +Abstract. Normal complexes are orthogonal truncations of polyhedral fans. In this paper, +we develop the study of mixed volumes for normal complexes. Our main result is a sufficiency +condition that ensures when the mixed volumes of normal complexes associated to a given fan +satisfy the Alexandrov–Fenchel inequalities. By specializing to Bergman fans of matroids, we +give a new proof of the Heron–Rota–Welsh Conjecture as a consequence of the Alexandrov– +Fenchel inequalities for normal complexes. +1. Introduction +The Alexandrov–Fenchel inequalities lie at the heart of convex geometry, asserting that, +for any convex bodies P♥, P♦, P3 . . . , Pd ∈ Rd, their mixed volumes satisfy +MVol(P♥, P♦, P3, . . . , Pd)2 ≥ MVol(P♥, P♥, P3, . . . , Pd) MVol(P♦, P♦, P3, . . . , Pd). +This paper is centered around developing an analogue of the Alexandrov–Fenchel inequalities +in a decidedly nonconvex setting. The geometric objects of interest to us are normal com- +plexes, which were recently introduced by A. Nathanson and the third author [NR21]. Given +a pure simplicial fan Σ, a normal complex associated to Σ is, roughly speaking, a polyhedral +complex obtained by truncating each cone of Σ with half-spaces perpendicular to the rays of +Σ. The choice of where to place the truncating half-spaces results in a family of normal com- +plexes associated to each fan Σ, and the question that motivates this work is: for a given fan +Σ, do the mixed volumes of the associated normal complexes satisfy the Alexandrov–Fenchel +inequalities? Our main result (Theorem 5.1) describes two readily verifiable conditions on +Σ that guarantee an affirmative answer to this question. +One of the motivations for studying mixed volumes of normal complexes is that, in the +special setting of tropical fans, they correspond to mixed degrees of divisors in associated +Chow rings. Thus, Alexandrov–Fenchel inequalities for normal complexes lead to nontrivial +numerical inequalities in these Chow rings. A class of tropical fans that have garnered a +great deal of attention in recent years are Bergman fans of matroids, and one application +of our main result (Theorem 6.2) is that normal complexes associated to Bergman fans of +matroids satisfy the Alexandrov–Fenchel inequalities. Translating these inequalities back to +matroid Chow rings, we obtain a volume-theoretic proof of the log-concavity of characteristic +polynomials of matroids, a result that was conjectured by Heron, Rota, and Welsh [Rot71, +Her72, Wel76] and first proved by Adiprasito, Huh, and Katz [AHK18]. +1 +arXiv:2301.05278v1 [math.CO] 12 Jan 2023 + +2 +L. NOWAK, P. O’MELVENY, AND D. ROSS +1.1. Overview of the paper. We begin in Section 2 by briefly recalling the construction +of normal complexes and their volumes. Normal complexes, denoted CΣ,∗(z), depend on +a marked simplicial d-fan Σ in a vector space NR with an inner product ∗ ∈ Inn(NR), as +well as a choice of pseudocubical truncating values z ∈ Cub(Σ, ∗) ⊆ RΣ(1). The volume of +CΣ,∗(z), denoted VolΣ,ω,∗(z), where ω is a weight function on the top-dimensional cones of +Σ, is defined as the weighted sum of the volumes of the maximal polytopes in CΣ,∗(z). We +recall the main result of [NR21], which asserts that, if (Σ, ω) is a tropical fan, then +(1.1) +VolΣ,ω,∗(z) = degΣ,ω(D(z)d) +where +D(z) = +� +ρ∈Σ(1) +zρXρ ∈ A1(Σ). +In Section 3, we introduce mixed volumes of normal complexes CΣ,∗(z1), . . . , CΣ,∗(zd), +denoted MVolΣ,ω,∗(z1, . . . , zd), which are weighted sums of mixed volumes of maximal poly- +topes. Analogous to mixed volumes in convex geometry, we show that mixed volumes of +normal complexes are characterized by being symmetric, multilinear, and normalized by vol- +ume (Proposition 3.1). Furthermore, we prove that mixed volumes are nonnegative on the +pseudocubical cone Cub(Σ, ∗) and positive on the cubical cone Cub(Σ, ∗) (Proposition 3.5). +For all tropical fans (Σ, ω), we leverage (1.1) to show (Theorem 3.6) that +(1.2) +MVolΣ,ω,∗(z1, . . . , zd) = degΣ,ω(D(z1) · · · D(zd)). +In Section 4, we develop the face structure of normal complexes, closely paralleling the +classical face structure of polytopes. In particular, the faces of a normal complex CΣ,∗(z) +are indexed by cones τ ∈ Σ, and each face is obtained as the intersection of CΣ,∗(z) with the +truncating hyperplanes indexed by the rays of τ. We describe how each face can, itself, be +viewed as a normal complex associated to the star fan Στ, and use this to define (mixed) +volumes of faces. +Our main result of this section (Proposition 4.13), shows how mixed +volumes of normal complexes can be computed in terms of mixed volumes of facets. +In Section 5, we introduce what it means for a triple (Σ, ω, ∗) to be AF—namely, that the +mixed volumes of cubical values satisfy the Alexandrov–Fenchel inequalities. Our main result +(Theorem 5.1), inspired by work of Cordero-Erausquin, Klartag, Merigot, and Santambrogio +[CEKMS19] and Br¨and´en and Leake [BL21], states that (Σ, ω, ∗) is AF if (i) all star fans Στ +of dimension at least three remain connected after removing the origin and (ii) the quadratic +volume polynomials associated to the two-dimensional star fans of Σ have exactly one positive +eigenvalue. In fact, under these conditions, we argue that the volume polynomial VolΣ,ω,∗(z) +is Cub(Σ, ∗)-Lorentzian, which then implies that (Σ, ω, ∗) is AF. +In Section 6, we briefly recall relevant notions regarding matroids and Bergman fans, and +then we use Theorem 5.1 to prove that Bergman fans of matroids are AF (Theorem 6.2). + +MIXED VOLUMES OF NORMAL COMPLEXES +3 +We conclude the paper by deducing the Heron–Rota–Welsh Conjecture as a consequence of +the Alexandrov–Fenchel inequalities for normal complexes. +1.2. Relation to other work. Since the original proof of the Heron–Rota–Welsh Conjec- +ture by Adiprasito, Huh, and Katz [AHK18], there have been a number of alternative proofs, +generalizations, and exciting related developments (an incomplete list includes [BHM+22, +BHM+20, BES20, ADH20, AP20, AP21, BH20, AGV21, ALGV19, ALGV18, CP21]). We +view the volume-theoretic approach in this paper as a new angle from which to view log- +concavity of characteristic polynomials of matroids, but we also want to acknowledge that +our methods share features of and are indebted to the approaches of several other teams +of mathematicians. In particular, our methods rely on the Chow-theoretic interpretation of +characteristic polynomials of matroids, proved by Huh and Katz [HK12], which was central +in the original proof of Adiprasito, Huh, and Katz [AHK18], as well as in the subsequent +proofs by Braden, Huh, Matherne, Proudfoot, and Wang [BHM+22] and Backman, Eur, and +Simpson [BES20]. In addition, our methods prove that volume polynomials are Lorentzian, +which is also a central feature in the methods of both Backman, Eur, and Simpson [BES20] +and Br¨and´en and Leake [BL21]. We note that, while the methods of [BES20] and [BL21] +seem to be tailored primarily for matroids, our methods readily extend to the more general +setting of tropical intersection theory (this extension will be spelled out in a forthcoming +work of the third author). By adding a new volume-theoretic approach to the Heron–Rota– +Welsh Conjecture to the literature, we hope that this paper will serve to welcome a new +batch of geometrically-minded folks into the fold of this flourishing area of research, opening +the door for further developments. +1.3. Acknowledgements. The authors would like to express their gratitude to Federico +Ardila, Matthias Beck, Emily Clader, Chris Eur, and Serkan Ho¸sten for sharing insights +related to this project. +This work was supported by a grant from the National Science +Foundation: DMS-2001439. +2. Background on normal complexes +In this section, we establish notation, conventions, and preliminary results regarding poly- +hedral fans and normal complexes. +2.1. Fan definitions and conventions. Let NR be a real vector spaces of dimension n. +Given a polyhedral fan Σ ⊆ NR, we denote the k-dimensional cones of Σ by Σ(k). Let ⪯ +denote the face containment relation among the cones of Σ, and for each cone σ ∈ Σ, let +σ(k) ⊆ Σ(k) denote the k-dimensional faces of σ. For any cone σ, let σ◦ denote the relative +interior of σ and denote the linear span of σ by Nσ,R ⊆ NR. + +4 +L. NOWAK, P. O’MELVENY, AND D. ROSS +We say that a fan Σ is pure if all of the maximal cones in Σ have the same dimension. +We say that Σ is marked if we have chosen a distinguished generating vector 0 ̸= uρ ∈ ρ for +each ray ρ ∈ Σ(1). Henceforth, we assume that all fans are pure, polyhedral, and marked, +and we use the term d-fan to refer to a pure, polyhedral, marked fan of dimension d. +We say that Σ is simplicial if dim(Nσ,R) = |σ(1)| for all σ ∈ Σ. The faces of a simplicial +cone σ are in bijective correspondence with the subsets of σ(1). For every face containment +τ ⪯ σ in a simplicial fan Σ, let σ \ τ denote the face of σ with rays σ(1) \ τ(1). Given two +faces τ, π ⪯ σ, denote by τ ∪ π the face of σ with rays τ(1) ∪ π(1). +Given a simplical d-fan Σ and a weight function ω : Σ(d) → R>0, we say that the pair +(Σ, ω) is a tropical fan if it satisfies the weighted balancing condition: +� +σ∈Σ(d) +τ≺σ +ω(σ)uσ\τ ∈ Nτ,R +for all +τ ∈ Σ(d − 1). +While the definition of tropical fans can be generalized to nonsimplicial fans, we will assume +throughout this paper that all tropical fans are simplicial. If ω(σ) = 1 for all σ ∈ Σ(d), we +say that Σ is balanced and we omit ω from the notation. +2.2. Chow rings and degree maps. Let MR denote the dual of NR and let ⟨−, −⟩ be the +duality pairing. Given a simplicial fan Σ ⊆ NR, the Chow ring of Σ is defined by +A•(Σ) ..= R +� +xρ | ρ ∈ Σ(1) +� +I + J +where +I ..= +� +xρ1 · · · xρk | R≥0{ρ1, . . . , ρk} /∈ Σ +� +and +J ..= +� � +ρ∈Σ(1) +⟨v, uρ⟩xρ +���� v ∈ MR +� +. +As both I and J are homogeneous, the Chow ring A•(Σ) is a graded ring, and we denote +by Ak(Σ) the subgroup of homogeneous elements of degree k. We denote the generators of +A•(Σ) by Xρ ..= [xρ] ∈ A1(Σ), and for any σ ∈ Σ(k), we define +Xσ ..= +� +ρ∈σ(1) +Xρ ∈ Ak(Σ). +If Σ is a simplicial d-fan, then every element of Ak(Σ) can be written as a linear combination +of Xσ with σ ∈ Σ(k) (see, for example, [AHK18, Proposition 5.5]). It follows that Ak(Σ) = 0 +for all k > d. If, in addition, (Σ, ω) is tropical, then there is a well-defined degree map +degΣ,ω : Ad(Σ) → R +such that degΣ,ω(Xσ) = ω(σ) for every σ ∈ Σ(d) (see, for example, [AHK18, Proposition 5.6]). + +MIXED VOLUMES OF NORMAL COMPLEXES +5 +2.3. Normal complexes. We now recall the construction of normal complexes from [NR21]. +In addition to a simplicial d-fan Σ ⊆ NR, the normal complex construction requires an +additional choice of an inner product ∗ ∈ Inn(NR) and a value z ∈ RΣ(1). Given such a ∗ +and z, we define a set of hyperplanes and half-spaces in NR associated to each ρ ∈ Σ by +Hρ,∗(z) ..= {v ∈ NR | v ∗ uρ = zρ} +and +H− +ρ,∗(z) ..= {v ∈ NR | v ∗ uρ ≤ zρ}. +We then define polytopes Pσ,∗(z), one for each σ ∈ Σ, by +Pσ,∗(z) ..= σ ∩ +� +ρ∈σ(1) +H− +ρ,∗(z). +Notice that Pσ,∗(z) is simply a truncation of the cone σ by hyperplanes that are normal to the +rays of σ—what it means to be normal is determined by ∗, and the locations of the normal +hyperplanes along the rays of the cone are determined by z. We would like to construct a +polytopal complex from these polytopes, but in general, they do not meet along faces. To +ensure that they meet along faces, we require a compatibility between z and ∗. +For each σ ∈ Σ, let wσ,∗(z) ∈ Nσ,R be the unique vector such that wσ,∗(z) ∗ uρ = zρ for all +ρ ∈ σ(1). That such a vector exists and is unique follows from the fact that the vectors uρ +with ρ ∈ σ(1) are linearly independent—this is equivalent to the simplicial hypothesis. We +then say that z is cubical (pseudocubical) with respect to (Σ, ∗) if +wσ,∗(z) ∈ σ◦ +(wσ,∗(z) ∈ σ) +for all +σ ∈ Σ. +In other words, the pseudocubical values are those values of z for which the truncating +hyperplanes intersect within each cone, and the cubical values are those for which they +intersect in the relative interior of each cone. The collection of cubical values are denoted +Cub(Σ, ∗) ⊆ RΣ(1) and the pseudocubical values are denoted Cub(Σ, ∗) ⊆ RΣ(1). +We now summarize key results from [NR21] that will be necessary for the developments +in this paper (see [NR21, Propositions 3.2, 3.3, and 3.7 ]). +Proposition 2.1. Let Σ ⊆ NR be a simplicial d-fan and let ∗ ∈ Inn(NR) be an inner product. +(1) The set Cub(Σ, ∗) ⊆ RΣ(1) is a polyhedral cone with Cub(Σ, ∗)◦ = Cub(Σ, ∗). +(2) For z ∈ Cub(Σ, ∗), the vertices of Pσ,∗(z) are {wτ,∗(z) | τ ⪯ σ}. +(3) For z ∈ Cub(Σ, ∗), the polytopes Pσ,∗(z) meet along faces. +For any polytope P, let �P denote the set of all faces of P. The third part of Proposition 2.1 +implies that +CΣ,∗(z) ..= +� +σ∈Σ(d) +� +Pσ,∗(z) +is a polytopal complex whenever z ∈ Cub(Σ, ∗), and this polytopal complex is called the +normal complex of Σ with respect to ∗ and z. + +6 +L. NOWAK, P. O’MELVENY, AND D. ROSS +Below, we depict a two-dimensional tropical fan and an associated normal complex. The +fan is comprised of nine two-dimensional cones glued along faces, and each of these nine +cones corresponds to a quadrilateral in the normal complex. +The next pair of images depict a three-dimensional fan comprised of two maximal cones +meeting along a two-dimensional face, and a corresponding normal complex. While this +fan is not tropical, the reader is welcome to view this image as just one small piece of a +three-dimensional tropical fan in some higher-dimensional vector space. +2.4. Volumes of normal complexes. Let Σ ⊆ NR be a simplicial d-fan, ∗ ∈ Inn(NR) +an inner product, and z ∈ Cub(Σ, ∗) a pseudocubical value. Informally, the volume of the +normal complex CΣ,∗(z) is the sum of the volumes of the polytopes Pσ,∗(z) with σ ∈ Σ(d); +however, some care is required in specifying what we mean by volume in each subspace Nσ,R. +For each cone σ ∈ Σ, define the discrete subgroup +Nσ ..= spanZ(uρ | ρ ∈ σ(1)) ⊆ NR, +and let Mσ denote its dual: Mσ ..= HomZ(Nσ, Z) ⊆ Mσ,R ..= HomR(Nσ,R, R). Using the inner +product ∗, we can identify Mσ,R with Nσ,R and thus, we can view Mσ as a lattice in Nσ,R. +For each σ ∈ Σ, let +Volσ : +� +polytopes in Nσ,R +� +→ R≥0 + +MIXED VOLUMES OF NORMAL COMPLEXES +7 +be the volume function determined by the property that a fundamental simplex of the lattice +Mσ ⊆ Nσ,R has unit volume. Define the volume of the normal complex CΣ,∗(z), denoted +VolΣ,∗(z) for brevity, as the sum of the volumes of the constituent d-dimensional polytopes: +VolΣ,∗(z) ..= +� +σ∈Σ(d) +Volσ(Pσ,∗(z)). +In slightly more generality, suppose that ω : Σ(d) → R>0 is a weight function on the maximal +cones of Σ. The volume of the normal complex CΣ,∗(z) weighted by ω is defined by +VolΣ,ω,∗(z) ..= +� +σ∈Σ(d) +ω(σ) Volσ(Pσ,∗(z)). +The main result of [NR21] is a Chow-theoretic interpretation of the weighted volumes of +normal complexes, valid whenever (Σ, ω) is tropical. +Theorem 2.2 ([NR21, Theorem 6.3]). Let (Σ, ω) be a tropical d-fan, ∗ ∈ Inn(NR) an inner +product, and z ∈ Cub(Σ, ∗) a pseudocubical value. Then +VolΣ,ω,∗(z) = degΣ,ω(D(z)d) +where +D(z) = +� +ρ∈Σ(1) +zρXρ ∈ A1(Σ). +3. Mixed Volumes of Normal Complexes +Our first aim in this paper is to enhance Theorem 2.2 to a statement about mixed volumes. +In order to do this, we briefly recall the classical theory of mixed volumes, for which we +recommend the comprehensive text by Schneider [Sch14] as a reference. +3.1. Mixed volumes of polytopes. Mixed volumes are the natural result of combining the +notion of volume with the operation of Minkowski addition. We start with a d-dimensional +real vector space V and a volume function Vol : {polytopes in V } → R≥0. The mixed +volume function +MVol : {polytopes in V }d → R≥0 +is the unique function determined by the following three properties. +• (Symmetry) For any permutation π ∈ Sd, +MVol(P1, . . . , Pd) = MVol(π(P1, . . . , Pd)). +• (Multilinearity) For any i = 1, . . . , d and λ ∈ R≥0, +MVol(P1, . . . , λPi + P ′ +i, . . . , Pd) = λ MVol(P1, . . . , Pi, . . . , Pd) ++ MVol(P1, . . . , P ′ +i, . . . , Pd), + +8 +L. NOWAK, P. O’MELVENY, AND D. ROSS +where the linear combination of polytopes is defined by +λPi + P ′ +i = {λv + w | v ∈ Pi, w ∈ P ′ +i}. +• (Normalization) For any polytope P, +MVol(P, . . . , P) = Vol(P). +That such a mixed volume function exists and is unique is due to Minkowski [Min03], who +proved that such a function exists more generally for convex bodies, not just for polytopes. +3.2. Mixed volumes of normal complexes. We now define a notion of mixed volumes +of normal complexes. Let Σ ⊆ NR be a simplicial d-fan and let ∗ ∈ Inn(NR) be an inner +product. Given pseudocubical values z1, . . . , zd ∈ Cub(Σ, ∗), we define the mixed volume +of the normal complexes CΣ,∗(z1), . . . , CΣ,∗(zd), denoted MVolΣ,∗(z1, . . . , zd) for brevity, +by +MVolΣ,∗(z1, . . . , zd) ..= +� +σ∈Σ(d) +MVolσ(Pσ,∗(z1), . . . , Pσ,∗(zd)). +In other words, the mixed volume is the sum of the mixed volumes of the polytopes associated +to the top-dimensional cones of Σ. More generally, if ω : Σ(d) → R>0 is a weight function, +then the mixed volume of the normal complexes CΣ,∗(z1), . . . , CΣ,∗(zd) weighted by +ω is defined by +MVolΣ,ω,∗(z1, . . . , zd) ..= +� +σ∈Σ(d) +ω(σ) MVolσ(Pσ,∗(z1), . . . , Pσ,∗(zd)). +In order to verify that this is a meaningful notion of mixed volumes for normal complexes, +we check that it is characterized by an analogue of the three characterizing properties of +mixed volumes of polytopes. +Proposition 3.1. Let Σ ⊆ NR be a simplicial d-fan, ∗ ∈ Inn(NR) an inner product, and +ω : Σ(d) → R>0 a weight function. +(1) For any z1, . . . , zd ∈ Cub(Σ, ∗) and π ∈ Sd, +MVolΣ,ω,∗(z1, . . . , zd) = MVolΣ,ω,∗(π(z1, . . . , zd)). +(2) For any i = 1, . . . , d, and for any z1, . . . , zi, z′ +i, . . . , zd ∈ Cub(Σ, ∗) and λ ∈ R≥0, +MVolΣ,ω,∗(z1, . . . , λzi + z′ +i, . . . , zd) = λ MVolΣ,ω,∗(z1, . . . , zi, . . . , zd) ++ MVolΣ,ω,∗(z1, . . . , z′ +i, . . . , zd). +(3) For any z ∈ Cub(Σ, ∗), +MVolΣ,ω,∗(z, . . . , z) = VolΣ,ω,∗(z). +Moreover, any function Cub(Σ, ∗)d → R≥0 satisfying Properties (1) – (3) must be MVolΣ,ω,∗. + +MIXED VOLUMES OF NORMAL COMPLEXES +9 +Proof. Given that +MVolΣ,ω,∗(z1, . . . , zd) = +� +σ∈Σ(d) +ω(σ) MVolσ(Pσ,∗(z1), . . . , Pσ,∗(zd)) +and the summands in the right-hand side are simply mixed volumes of polytopes, Proper- +ties (1) and (3) follow from the symmetry and normalization properties of mixed volumes in +the polytope setting. Moreover, once we prove that +(3.2) +Pσ,∗(λz + z′) = λPσ,∗(z) + Pσ,∗(z′) +for all z, z′ ∈ Cub(Σ, ∗) and λ ∈ R≥0, then Property (2) also follows from the multilinearity +property of mixed volumes in the polytope setting. Thus, it remains to prove (3.2), which +we accomplish by proving both inclusions. +First, suppose that v ∈ Pσ,∗(λz + z′). By Proposition 2.1, the vertices of Pσ,∗(λz + z′) are +{wτ,∗(λz + z′) | τ ⪯ σ}, so we can write v as a convex combination: +(3.3) +v = +� +τ⪯σ +aτ wτ,∗(λz + z′) +for some +aτ ∈ R≥0 +with +� +τ⪯σ +aτ = 1. +To prove that v ∈ λPσ,∗(z) + Pσ,∗(z′), our next step is to prove that the vertices are linear: +(3.4) +wτ,∗(λz + z′) = λwτ,∗(z) + wτ,∗(z′). +Since wτ,∗(λz + z′) is the unique vector in Nτ,R with wτ,∗(λz + z′) ∗ uρ = (λz + z′)ρ for +all ρ ∈ τ(1), proving (3.4) amounts to proving that λwτ,∗(z) + wτ,∗(z′) also satisfies these +equations. Using bilinearity of the inner product and the definition of the w vectors, we have +(λwτ,∗(z) + wτ,∗(z′)) ∗ uρ = λwτ,∗(z) ∗ uρ + wτ,∗(z′) ∗ uρ += λzρ + z′ +ρ += (λz + z′)ρ. +Therefore, (3.4) holds, and substituting (3.4) into (3.3) implies that +v = λ +� +τ⪯σ +aτwτ,∗(z) + +� +τ⪯σ +aτwτ,∗(z′) ∈ λPσ,∗(z) + Pσ,∗(z′). +To prove the other inclusion, suppose that v ∈ λPσ,∗(z) + Pσ,∗(z′). Then v = λw + w′ for +some w ∈ Pσ,∗(z) and w′ ∈ Pσ,∗(z′). This means that w, w′ ∈ σ and, in addition, w · uρ ≤ zρ +and w′ · uρ ≤ z′ +ρ for all ρ ∈ σ(1). Since σ is a cone, u = λw + w′ ∈ σ and, for every ρ ∈ σ(1), +we have +v ∗ uρ = (λw + w′) ∗ uρ += λw ∗ uρ + w′ ∗ uρ +≤ λzρ + z′ +ρ, + +10 +L. NOWAK, P. O’MELVENY, AND D. ROSS +from which we conclude that v ∈ Pσ,∗(λz + z′). +Finally, to prove the final assertion of the proposition, suppose that F : Cub(Σ, ∗)d → R≥0 +satisfies Properties (1) – (3). Our goal is to prove that F(z1, . . . , zd) = MVolΣ,ω,∗(z1, . . . , zd) +for any pseudocubical values z1, . . . , zd ∈ Cub(Σ, ∗). +Set z = λ1z1 + · · · + λdzd with +λ1, . . . , λd ∈ R≥0 arbitrary. Property (3) implies that +F(z, . . . , z) = VolΣ,ω,∗(z) = MVolΣ,ω,∗(z, . . . , z). +Using Properties (1) and (2) we can expand both the left- and right-hand sides of this +equation as polynomials in λ1, . . . , λd: +� +k1,...,kd +� +d +k1, . . . , kd +� +F(z1, . . . , z1 +� +�� +� +k1 +, . . . , zd, . . . , zd +� +�� +� +kd +)λk1 +1 · · · λkd +d += +� +k1,...,kd +� +d +k1, . . . , kd +� +MVolΣ,ω,∗(z1, . . . , z1 +� +�� +� +k1 +, . . . , zd, . . . , zd +� +�� +� +kd +)λk1 +1 · · · λkd +d +Equating the coefficients of λ1 · · · λd in these two polynomials leads to the desired conclusion: +F(z1, . . . , zd) = MVolΣ,ω,∗(z1, . . . , zd). +□ +Our methods for studying Alexandrov–Fenchel inequalities will also require the following +positivity result. +Proposition 3.5. Let Σ ⊆ NR be a simplicial d-fan, ∗ ∈ Inn(NR) an inner product, and +ω : Σ(d) → R>0 a weight function. Then +MVolΣ,ω,∗(z1, . . . , zd) ≥ 0 +for all +z1, . . . , zd ∈ Cub(Σ, ∗) +and +MVolΣ,ω,∗(z1, . . . , zd) > 0 +for all +z1, . . . , zd ∈ Cub(Σ, ∗). +Proof. The first statement follows from the definition of MVolΣ,ω,∗ and the nonnegativity +of mixed volumes of polytopes [Sch14, Theorem 5.1.7]. For the second statement, we first +observe that z ∈ Cub(Σ, ∗) implies that Pσ,∗(z) has dimension d for every σ ∈ Σ(d), which +follows from the fact that Pσ,∗(z) is combinatorially equivalent to a d-cube [NR21, Propo- +sition 3.8]. Thus, the second statement follows from the fact that mixed volumes of full- +dimensional polytopes are strictly positive [Sch14, Theorem 5.1.8]. +□ +3.3. Mixed volumes and mixed degrees. We now extend Theorem 2.2 to give a Chow- +theoretic interpretation of mixed volumes of normal complexes associated to tropical fans. +Theorem 3.6. Let (Σ, ω) be a tropical d-fan, let ∗ ∈ Inn(NR) be an inner product, and let +z1, . . . , zd ∈ Cub(Σ, ∗) be pseudocubical values. Then +MVolΣ,ω,∗(z1, . . . , zd) = degΣ,ω(D(z1) · · · D(zd)). + +MIXED VOLUMES OF NORMAL COMPLEXES +11 +Proof. By Proposition 3.1, it suffices to prove that the function +Cub(Σ, ∗)d → R≥0 +(z1, . . . , zd) �→ degΣ,ω(D(z1) · · · D(zd)) +is symmetric, multilinear, and normalized by VolΣ,ω,∗. Symmetry follows from the fact that +A•(Σ) is a commutative ring, multilinearity follows from the fact that degΣ,ω : Ad(Σ) → R +is a linear map, and normalization is the content of Theorem 2.2. +□ +4. Faces of Normal Complexes +In this section, we develop a face structure for normal complexes, analogous to the face +structure of polytopes. Parallel to the polytope case, we will see that each face is obtained +by intersecting the normal complex with supporting hyperplanes, that each face can, itself, +be viewed as a normal complex, and that a face of a face is, itself, a face. We then prove fun- +damental properties relating (mixed) volumes of normal complexes to the (mixed) volumes +of their facets, which perfectly parallel central results in the classical polytope setting. +4.1. Orthogonal decompositions. The face construction for normal complexes makes +heavy use of an orthogonal decomposition of NR associated to each cone τ ∈ Σ, which +we now describe. Associated to each τ ∈ Σ, we have already met the subspace Nτ,R ⊆ NR, +which is the linear span of τ, and we now introduce notation for the quotient space +N τ +R ..= NR/Nτ,R. +With the inner product ∗, we may identify N τ +R as the orthogonal complement of Nτ,R: +N τ +R = N ⊥ +τ,R = {v ∈ NR | v ∗ u = 0 for all u ∈ Nτ,R} ⊆ NR, +allowing us to decompose NR as an orthogonal sum NR = Nτ,R ⊕ N τ +R. +We denote the +orthogonal projections onto the factors of this decomposition by prτ and prτ. +As we will see below, given a normal complex CΣ,∗(z) and a cone τ ∈ Σ, we will associate +a face Fτ(CΣ,∗(z)), and this face will lie in the space N τ +R. +In order to help the reader +digest the construction of Fτ(CΣ,∗(z)) and its subsequent interpretation as a normal complex, +we henceforth make the convention that τ superscripts will be used exclusively for objects +associated to the vector space N τ +R. For example, Στ will denote a fan in N τ +R and ∗τ will denote +an inner product on N τ +R. +4.2. Faces of normal complexes. There are two primary steps in the face construction +for normal complexes. The first step is completely analogous to the polytope setting: we +intersect the normal complex with a collection of supporting hyperplanes to obtain a sub- +complex. However, in order to view this resulting subcomplex as a normal complex itself, + +12 +L. NOWAK, P. O’MELVENY, AND D. ROSS +the second step of the construction requires us to translate this polytopal subcomplex to the +origin, where we can then endow it with the structure of a normal complex inside N τ +R. +Let Σ ⊆ NR be a simplicial d-fan, ∗ ∈ Inn(NR) an inner product, and z ∈ Cub(Σ, ∗) a +pseudocubical value. For each cone τ ∈ Σ, define the neighborhood of τ in Σ by +NτΣ ..= {π | π ⪯ σ for some σ ∈ Σ with τ ⪯ σ}. +To illustrate this definition, we have darkened the neighborhood of the ray ρ in the following +two-dimensional fan. +ρ +Notice that NτΣ is, itself, a simplicial d-fan in NR whose cones are a subset of Σ, and the +maximal cones of NτΣ comprise all of the maximal cones of Σ that contain τ. Since every +maximal cone σ ∈ NτΣ(d) contains τ as a face, it follows from the definitions that each +hyperplane Hρ,∗(z) with ρ ∈ τ(1) is a supporting hyperplane of Pσ,∗(z): +Pσ,∗(z) ⊆ H− +ρ,∗(z) +for all +σ ∈ NτΣ(d) +and +ρ ∈ τ(1). +Thus, for each σ ∈ NτΣ(d), we obtain a face of Pσ,∗(z) by intersecting with all of these +hyperplanes: +Fτ(Pσ,∗(z)) ..= Pσ,∗(z) ∩ +� +ρ∈τ(1) +Hρ,∗(z). +The collection of these polytopes along with all of their faces forms a polytopal subcomplex +of CΣ,∗(z), which we denote +Fτ(CΣ,∗(z)) ..= +� +σ∈NτΣ(d) +� +Fτ(Pσ,∗(z)). +To illustrate how the polytopal subcomplex Fτ(CΣ,∗(z)) is constructed in a concrete exam- +ple, the following image depicts a two-dimensional normal complex where we have darkened +the collection of maximal polytopes associated to the neighborhood of a ray ρ. We have +also drawn in the hyperplane associated to ρ. The intersection of the hyperplane and the +darkened polytopes is Fρ(CΣ,∗(z)), which, in this example, is a polytopal complex comprised +of three line segments meeting at the point wρ,∗(z). + +MIXED VOLUMES OF NORMAL COMPLEXES +13 +Hρ,∗(z) +ρ +Fρ(CΣ,∗(z)) +One might be tempted to call Fτ(CΣ,∗(z)) a “face” of CΣ,∗(z); however, a drawback would +be that Fτ(CΣ,∗(z)) is not, itself, a normal complex (all normal complexes contain the origin, +for example, while Fτ(CΣ,∗(z)) generally does not). +Thus, our construction involves one +more step, which is to translate Fτ(CΣ,∗(z)) by the vector wτ,∗(z). Notice that, tracking +back through the definitions, there is an identification of affine subspaces +� +ρ∈τ(1) +Hρ,∗(z) = N τ +R + wτ,∗(z). +Since Fτ(CΣ,∗(z)) is, by definition, contained in the left-hand side, it follows that its trans- +lation by −wτ,∗(z) is a polytopal complex in N τ +R. We define the face of CΣ,∗(z) associated +to τ ∈ Σ to be this polytopal complex: +Fτ(CΣ,∗(z)) ..= Fτ(CΣ,∗(z)) − wτ,∗(z) ⊆ N τ +R. +The face associated to the ray ρ in our running example is depicted below inside N ρ +R. +N ρ +R = Hρ,∗(z) − wρ,∗(z) +ρ +F ρ(CΣ,∗(z)) = Fρ(CΣ,∗(z)) − wρ,∗(z) +The next pair of images depicts the subcomplex Fρ(CΣ,∗(z)) ⊆ CΣ,∗(z) and, after trans- +lating to the origin, the face Fρ(CΣ,∗(z)), where ρ is a ray of a three-dimensional fan. + +14 +L. NOWAK, P. O’MELVENY, AND D. ROSS +ρ +ρ +Fρ(CΣ,∗(z)) +F ρ(CΣ,∗(z)) +In the following subsections, it will also be useful to have notation for translates of the +polytopes Fτ(Pσ,∗(z)). We define +Fτ(Pσ,∗(z)) ..= Fτ(Pσ,∗(z)) − wτ,∗(z). +In terms of these translated polytopes, we can write the τ-face of CΣ,∗(z) as +Fτ(CΣ,∗(z)) = +� +σ∈NτΣ(d) +� +Fτ(Pσ,∗(z)). +4.3. Faces as normal complexes. Our aim in this subsection is to realize each face +Fτ(CΣ,∗(z)) as a normal complex. In order to do so, we require several ingredients; namely, +we require a marked, pure, simplicial fan Στ in N τ +R, an inner product ∗τ on N τ +R, and a +pseudocubical value zτ ∈ Cub(Στ, ∗τ). We now define each of these ingredients. +For each cone τ ∈ Σ, define the star of Σ at τ ∈ Σ to be the fan in N τ +R comprised of all +projections of cones in the neighborhood of τ: +Στ ..= {prτ(π) | π ∈ NτΣ}. +The star of a two-dimensional fan Σ at a ray ρ is depicted below. +In the image, there +are three two-dimensional cones in the neighborhood of ρ that are projected onto three +one-dimensional cones that comprise the maximal cones in the star fan Σρ. +ρ +Σ ⊆ NR +Σρ ⊆ N ρ +R +Henceforth, we use the shorthand πτ = prτ(π). + +MIXED VOLUMES OF NORMAL COMPLEXES +15 +Given any cone πτ ∈ Στ with π ∈ NτΣ, we can also view πτ as the projection of the larger +cone σ = π ∪ τ ∈ NτΣ. Note that σ is the unique maximal cone in NτΣ that projects onto +πτ, from which it follows that each cone in Στ is the projection of a distinguished cone in +NτΣ. In other words, there is a bijection +{σ ∈ NτΣ | τ ⪯ σ} → Στ +σ �→ στ. +From the assumptions that Σ is a simplicial d-fan, it follow that Στ is a simplicial fan in N τ +R +that is pure of dimension dτ = d − dim(τ). Moreover, the simplicial hypothesis on Σ implies +that each ray η ∈ Στ(1) is the projection of a unique ray ˆη ∈ NτΣ(1), and we can use this +to mark each ray η ∈ Στ(1) with the vector prτ(uˆη). +We now have a marked, pure, simplicial fan in N τ +R, so it remains to define an inner product +and pseudocubical value. The inner product ∗τ ∈ Inn(N τ +R) is simply defined as the restriction +of the inner product ∗ ∈ Inn(NR) to the subspace N τ +R. Lastly, given any z ∈ RΣ(1), we define +zτ ∈ RΣτ(1) by the rule +zτ +η = zˆη − wτ,∗(z) ∗ uˆη, +where, as before, ˆη ∈ NτΣ(1) is the unique ray with prτ(ˆη) = η. +We now have all the ingredients necessary to state and prove the following result, which +asserts that faces of normal complexes are, themselves, normal complexes. +Proposition 4.1. Let Σ ⊆ NR be a simplicial d-fan, ∗ ∈ Inn(NR) an inner product, and +τ ∈ Σ a cone. If z ∈ RΣ(1) is (pseudo)cubical with respect to (Σ, ∗), then zτ is (pseudo)cubical +with respect to (Στ, ∗τ) and +Fτ(CΣ,∗(z)) = CΣτ,∗τ(zτ). +We note that the first statement—that zτ is (pseudo)cubical—is necessary in order for +CΣτ,∗τ(zτ) to even be well-defined. Proposition 4.1 is a statement about normal complexes, +or equivalently, about the polytopes that comprise those complexes. +In order to prove +Proposition 4.1, we first prove the following key lemma, which concerns just the vertices of +the polytopes. +Lemma 4.2. Let Σ ⊆ NR be a simplicial d-fan, ∗ ∈ Inn(NR) an inner product, and τ ∈ Σ +a cone. For any σ ∈ Σ with τ ⪯ σ, we have +prτ(wσ,∗(z)) = wσ,∗(z) − wτ,∗(z) = wστ,∗τ(zτ). +Proof. We start by establishing the first equality. +To do so, we begin by arguing that +wσ,∗(z) − wτ,∗(z) ∈ N τ +R. +Since N τ +R = N ⊥ +τ,R, it suffices to prove that wσ,∗(z) − wτ,∗(z) is + +16 +L. NOWAK, P. O’MELVENY, AND D. ROSS +orthogonal to the basis {uρ | ρ ∈ τ(1)} ⊆ Nτ,R. By definition of the w vectors and the +assumption that τ ⪯ σ, we compute +(wσ,∗(z) − wτ,∗(z)) ∗ uρ = zρ − zρ = 0 +for all +ρ ∈ τ(1), +from which it follows that wσ,∗(z) − wτ,∗(z) ∈ N τ +R. Since NR = Nτ,R ⊕ N τ +R, the orthogonal +decomposition wσ,∗(z) = wτ,∗(z) + (wσ,∗(z) − wτ,∗(z)) then implies that +(4.3) +prτ(wσ,∗(z)) = wτ,∗(z) +and +prτ(wσ,∗(z)) = wσ,∗(z) − wτ,∗(z). +To prove that wσ,∗(z) − wτ,∗(z) = wστ,∗τ(zτ), we now argue that wσ,∗(z) − wτ,∗(z) is an +element of Nστ,R and is a solution of the equations defining wστ,∗τ(zτ): +(4.4) +v ∗τ uη = zτ +η +for all +η ∈ στ(1). +To check that wσ,∗(z) − wτ,∗(z) ∈ Nστ,R, we start by observing that we can write +wσ,∗(z) = +� +ρ∈σ(1) +aρ uρ +for some values aρ ∈ R, in which case +wσ,∗(z) − wτ,∗(z) = prτ(wσ,∗(z)) += +� +ρ∈σ(1)\τ(1) +aρ prτ(uρ) += +� +η∈στ(1) +aˆη uη, +where the first equality uses (4.3), the second uses that prτ vanishes on Nτ,R, and the third +uses that the rays of στ are in natural bijection with σ(1) \ τ(1). Lastly, we peel back the +definitions to check that wσ,∗(z) − wτ,∗(z) is a solution of Equations (4.4): +(wσ,∗(z) − wτ,∗(z)) ∗τ uη = (wσ,∗(z) − wτ,∗(z)) ∗ (uˆη − prτ(uˆη)) += wσ,∗(z) ∗ uˆη − wτ,∗(z) ∗ uˆη − +� +wσ,∗(z) − wτ,∗(z) +� +∗ prτ(uˆη) += zˆη − wτ,∗(z) ∗ uˆη += zτ +η, +where the first equality uses the orthogonal decomposition of uˆη and the fact that ∗τ is +just the restriction of ∗, the second equality uses linearity of the inner product, and the +third equality uses the definition of wσ,∗(z) along with the fact that the vectors prτ(uˆη) and +wσ,∗(z) − wτ,∗(z) = prτ(wσ,∗(z)) are in orthogonal subspaces. +□ + +MIXED VOLUMES OF NORMAL COMPLEXES +17 +Proof of Proposition 4.1. To prove the first statement in the cubical setting, assume that +z ∈ RΣ(1) is cubical. This means that, for every σ ∈ Σ, we can write +wσ,∗(z) = +� +ρ∈σ(1) +aρuρ +for some positive values aρ ∈ R>0. Consider any cone of Στ, which we can write as στ with +τ ⪯ σ. Applying the lemma, we then see that +wστ,∗τ(zτ) = prτ(wσ,∗(z)) += +� +ρ∈σ(1)\τ(1) +aρ prτ(uρ) += +� +η∈στ(1) +aˆη uη. +This shows that wστ,∗τ(zτ) can be written as a positive combination of the ray generators of +στ, proving that zτ ∈ Cub(Στ, ∗τ). The proof in the pseudocubical setting is identical but +with “positive” replaced by “nonnegative.” +To prove that +Fτ(CΣ,∗(z)) = CΣτ,∗τ(zτ), +it suffices to identify the maximal polytopes in these complexes. In other words, we must +prove that, for every σ ∈ NτΣ(d), we have +(4.5) +Fτ(Pσ,∗(z)) = Pστ,∗τ(zτ). +To prove (4.5), we analyze the vertices of these polytopes. +By Proposition 2.1, the vertices of Pσ,∗(z) are {wπ,∗(z) | π ⪯ σ}. Since +Fτ(Pσ,∗(z)) = Pσ,∗(z) ∩ +� +ρ∈τ(1) +Hρ,∗(z), +it follows that the vertices of Fτ(Pσ,∗(z)) are +{wπ,∗(z) | π ⪯ σ and wπ,∗(z) ∗ uρ = zρ for all ρ ∈ τ(1)}. +If a cone π ⪯ σ satisfies wπ,∗(z)∗uρ = zρ for all ρ ∈ τ(1), then the definition of the w-vectors +implies that wπ,∗(z) = wπ∪τ,∗(z), and it follows that the vertices of Fτ(Pσ,∗(z)) are +Vert +� +Fτ(Pσ,∗(z)) +� += {wπ,∗(z) | τ ⪯ π ⪯ σ}. +Upon translating by wτ,∗(z) to get from Fτ(Pσ,∗(z)) to F τ(Pσ,∗(z)), we see that +Vert +� +F τ(Pσ,∗(z)) +� += {wπ,∗(z) − wτ,∗(z) | τ ⪯ π ⪯ σ} += {wπτ,∗τ(zτ) | πτ ⪯ στ} += Vert +� +Pστ,∗τ(zτ)) +� +, + +18 +L. NOWAK, P. O’MELVENY, AND D. ROSS +where the second equality is an application of Lemma 4.2 and the third is an application +of Proposition 2.1. Having matched the vertices of the polytopes in (4.5), the equality of +polytopes then follows. +□ +The importance of Proposition 4.1 is that it allows us to endow each of the faces of a +normal complex with the structure of a normal complex, and in particular, it then allows +us to compute (mixed) volumes of faces. More specifically, if ω : Σ(d) → R>0 is a weight +function, then we obtain a weight function ωτ : Στ(dτ) → R>0 defined by ωτ(στ) = ω(σ) for +all σ ∈ Σ(d). The volume of the face Fτ(CΣ,∗(z)) weighted by ω is +VolΣτ,ωτ,∗τ(zτ). +Similarly, the mixed volume of the faces Fτ(CΣ,∗(z1)), . . . Fτ(CΣ,∗(zdτ)) weighted by ω is +MVolΣτ,ωτ,∗τ(zτ +1, . . . , zτ +dτ). +In the next two subsections, we use these concepts to prove fundamental results relating +(mixed) volumes of normal complexes to the (mixed) volumes of their facets. In making +arguments using mixed volumes, it will be useful to consider facets of facets; as such, the +next result—asserting that the face of a face of a normal complex is a face of the original +normal complex—will be useful. +Proposition 4.6. Let Σ ⊆ NR be a simplicial d-fan, ∗ ∈ Inn(NR) an inner product, and +z ∈ Cub(Σ, ∗) a pseudocubical value. If τ, π ∈ Σ with τ ⪯ π, then +Fπτ(Fτ(CΣ,∗(z))) = Fπ(CΣ,∗(z)). +Proof. By Proposition 4.1, the claim in this proposition is equivalent to +Fπτ(CΣτ,∗τ(zτ)) = CΣπ,∗π(zπ). +It suffices to match the maximal polytopes in these complexes, so we must prove: +(4.7) +Fπτ(Pστ,∗τ(zτ)) = Pσπ,∗π(zπ) +for all +σ ∈ Σ(d) +with +τ ⪯ σ. +The vertices of the polytope in the left-hand side of (4.7) are +{wµτ,∗τ(zτ) − wπτ,∗τ(zτ) | πτ ⪯ µτ ⪯ στ} +while the vertices in the right-hand side of (4.7) are +{wµπ,∗π(zπ) | µπ ⪯ σπ}. + +MIXED VOLUMES OF NORMAL COMPLEXES +19 +Notice that both sets of vertices are indexed by µ ∈ Σ with π ⪯ µ ⪯ σ, and we have +wµτ,∗τ(zτ) − wπτ,∗τ(zτ) = prπτ(wµτ,∗τ(zτ)) += prπτ(prτ(wµ,∗(z))) += prπ(wµ,∗(z)) += wµπ,∗π(zπ), +where the first, second, and fourth equalities are Lemma 4.2, while the second is the obser- +vation that the projection prπ can be broken up into two steps: prπ = prπτ ◦ prτ. Thus, the +vertices of the polytopes in (4.7) match up, and the proposition follows. +□ +4.4. Volumes and facets. This subsection is devoted to proving the following result, which +relates the volume of a normal complex to the volumes of its facets. +Proposition 4.8. Let Σ ⊆ NR be a simplicial d-fan with weight function ω : Σ(d) → R>0, +let ∗ ∈ Inn(NR) be an inner product, and let z ∈ Cub(Σ, ∗) be a pseudocubical value. Then +VolΣ,ω,∗(z) = +� +ρ∈Σ(1) +zρ VolΣρ,ωρ,∗ρ(zρ). +The sum in the right-hand side of the theorem corresponds to decomposing the normal +complex into pyramids over its facets, as depicted in the next image. +Proposition 4.8 follows from the following lemma relating the volume function Volσ on +Nσ,R to the volume function Volσρ on the hyperplane Nσρ,R ⊆ Nσ,R. +Lemma 4.9. Under the hypotheses of Proposition 4.8, let σ ∈ Σ(d) and ρ ∈ σ(1). For any +polytope P ⊆ Nσρ,R and a ∈ R≥0, we have +Volσ +� +conv(0, P + auρ) +� += a(uρ ∗ uρ) · Volσρ(P). +For intuition, we note that the polytope conv(0, P + auρ) appearing in the left-hand side +of Lemma 4.9 is obtained from the polytope P by first translating P along the ray ρ, which +is orthogonal to Nσρ,R, then taking the convex hull with the origin, the result of which can +be thought of as a pyramid with P as base and the origin as apex. The right-hand side can + +20 +L. NOWAK, P. O’MELVENY, AND D. ROSS +then be thought of as a “base-times-height” formula for the volume of this pyramid, where +the “height” of the vector auρ is a(uρ ∗ uρ). We now make this informal discussion precise. +Proof of Lemma 4.9. Let {vη | η ∈ σ(1)} ⊆ Mσ be the dual basis of {uη | η ∈ σ(1)} ⊆ Nσ, +defined uniquely by the equations +vη ∗ uµ = +� +� +� +1 +µ = η +0 +µ ̸= η. +Recall that each ray generator of σρ is of the form prρ(uη) for a unique η ∈ σ(1) \ {ρ}; we +claim that the dual vector of prρ(uη) in Mσρ is vη—in other words, the dual vector of prρ(uη) +is the same as the dual vector of uη. To verify this, note that, for any η, µ ∈ σ(1) \ {ρ}, we +have +prρ(uη) ∗ρ vµ = (uη − prρ(uη)) ∗ vµ += uη ∗ vµ += +� +� +� +1 +µ = η +0 +µ ̸= η, +where the first equality uses the decomposition of uη into its orthogonal components, along +with the fact that ∗ρ is just the restriction of ∗, and the second equality uses that prρ(uη) is +a multiple of uρ, along with uρ ∗ vµ = 0. +Using these dual bases, we defined simplices in each of vector spaces Nσ,R and Nσρ,R by +∆(σ) = conv(0, {vη | η ∈ σ(1)}) ⊆ Nσ,R +and +∆(σρ) = conv(0, {vη | η ∈ σ(1) \ {ρ}}) ⊆ Nσρ,R. +By our convention on how volumes are normalized in Nσ,R and Nσρ,R, along with our verifi- +cation above that {vη | η ∈ σ(1) \ {ρ}} is the dual basis of the ray generators of σρ, these +simplices have unit volume: +Volσ(∆(σ)) = Volσρ(∆(σρ)) = 1. +Notice that ∆(σρ) is a facet of ∆(σ) and we can write ∆(σ) = conv(vρ, ∆(σρ)). If we project +the vertex vρ of ∆(σ) onto the line spanned by ρ, we obtain a new simplex +∆1(σ) = conv(prρ(vρ), ∆(σρ)). +Since the projection prρ is parallel to the facet ∆(σρ), it follows that +Volσ(∆1(σ)) = Volσ(∆(σ)) = Volσρ(∆(σρ)). + +MIXED VOLUMES OF NORMAL COMPLEXES +21 +Now define a new simplex by sliding the vertex prρ(vρ) along ρ to the new vertex auρ: +∆2(σ) = conv(auρ, ∆(σρ)). +By the standard projection formula, we have prρ(vρ) = +uρ +uρ∗uρ, from which we see that ∆2(σ) +is obtained from ∆1(σ) by scaling the height of the vertex prρ(vρ) by a factor of a(uρ ∗ uρ). +It follows that the volume also scales by a(uρ ∗ uρ): +Volσ(∆2(σ)) = a(uρ ∗ uρ) · Volσ(∆1(σ)) = a(uρ ∗ uρ) · Volσρ(∆(σρ)). +More concisely, we have proved that +(4.10) +Volσ +� +conv(auρ, P) +� += a(uρ ∗ uρ) · Volσρ(P) +when P = ∆(σρ). +As a visual aid, we have depicted below the sequence of polytopes from the above discussion +in the specific setting of a two-dimensional cone σ, which we have visualized in R2 with the +usual dot product. +• +• +• +0 +σ +η +ρ +uη +uρ +Nσρ,R +• +vρ +•vη +• +0 +ρ +Nσρ,R +• +vρ +•vη +∆(σ) +∆(σρ) +• +0 +ρ +Nσρ,R +• +uρ +uρ∗uρ +•vη +∆1(σ) +∆(σρ) +• +0 +ρ +Nσρ,R +•auρ +•vη +∆2(σ) +∆(σρ) +We now extend (4.10) to any simplex P ⊆ Nσρ,R. To do so, first note that a simplex P can +be obtained from the specific simplex ∆(σρ) by a composition of a translation and a linear +transformation on Nσρ,R. Translating P within Nσρ,R does not affect the volume on either + +22 +L. NOWAK, P. O’MELVENY, AND D. ROSS +side of (4.10). Given a linear transformation T, on the other hand, we can extend it to a +linear transformation �T on Nσ,R by simply fixing the vector uρ, in which case we have +�T(conv(auρ, P)) = conv(auρ, T(P)). +Since det( �T) = det(T) and linear transformations scale volumes by the absolute values of +their determinants, we conclude that the equality in (4.10) is preserved upon taking linear +transforms of P: +Volσ +� +conv(auρ, T(P)) +� += Volσ +� �T(conv(auρ, P)) +� += | det( �T)| Volσ +� +conv(auρ, P) +� += | det(T)| · a(uρ ∗ uρ) · Volσρ(P) += a(uρ ∗ uρ) · Volσρ(T(P)). +Knowing that (4.10) holds for simplices, we extend it to arbitrary polytopes P ⊆ Nσρ,R +by triangulating P and applying (4.10) to each simplex in the triangulation. The lemma +then follows from (4.10) along with the observation that conv(auρ, P) is just a reflection of +conv(0, P + auρ), so has the same volume. +□ +We now use Lemma 4.9 to prove Proposition 4.8. +Proof of Proposition 4.8. For each top-dimensional cone σ ∈ Σ(d) and ρ ∈ σ(1), consider +the polytope face Fρ(Pσ,∗(z)) ⊆ Pσ,∗(z). By definition, we have +Fρ(Pσ,∗(z)) = Fρ(Pσ,∗(z)) + wρ,∗(z). +Noting that wρ,∗(z) = +zρ +uρ∗uρuρ, Lemma 4.9 computes the volume of the pyramid conv(0, Fρ(Pσ,∗(z))): +(4.11) +Volσ +� +conv(0, Fρ(Pσ,∗(z))) +� += zρ Volσρ(Fρ(Pσ,∗(z)) = zρ Volσρ(Pσρ,∗ρ(zρ)), +where the second equality is an application of (4.5). +Next, note that we can decompose each polytope Pσ,∗(z) into pyramids over the faces +Fρ(Pσ,∗(z)) with ρ ∈ σ(1), implying that +(4.12) +Volσ(Pσ,∗(z)) = +� +ρ∈σ(1) +Volσ +� +conv(0, Fρ(Pσ,∗(z)) +� +. + +MIXED VOLUMES OF NORMAL COMPLEXES +23 +We then compute: +VolΣ,ω,∗(z) = +� +σ∈Σ(d) +ω(σ) Volσ(Pσ,∗(z)) += +� +σ∈Σ(d) +ω(σ) +� +ρ∈σ(1) +Volσ +� +conv(0, Fρ(Pσ,∗(z)) +� += +� +σ∈Σ(d) +ω(σ) +� +ρ∈σ(1) +zρ Volσρ(Pσρ,∗ρ(zρ)) += +� +ρ∈Σ(1) +zρ +� +σρ∈Σρ(d−1) +ωρ(σρ) Volσρ(Pσρ,∗ρ(zρ)) += +� +ρ∈Σ(1) +zρ VolΣρ,ωρ,∗ρ(zρ), +where the first equality is the definition of VolΣ,ω,∗(z), the second and third are (4.12) and +(4.11), respectively, the fourth follows from the definition of ωρ and the fact that cones in +Σρ(d − 1) are in bijection with the cones in Σ(d) containing ρ via σρ ↔ σ, and the fifth is +the definition of VolΣρ,ωρ,∗ρ(zρ). +□ +4.5. Mixed volumes and facets. The aim of this subsection is to enhance Proposition 4.8 +to the following more general statement about mixed volumes. See [Sch14, Lemma 5.1.5] for +the analogous result in the classical setting of strongly isomorphic polytopes. +Proposition 4.13. Let Σ ⊆ NR be a simplicial d-fan with weight function ω : Σ(d) → R>0, +let ∗ ∈ Inn(NR) be an inner product, and let z1, . . . , zd ∈ Cub(Σ, ∗) be pseudocubical values. +Then +MVolΣ,ω,∗(z1, . . . , zd) = +� +ρ∈Σ(1) +z1,ρ MVolΣρ,ωρ,∗ρ(zρ +2, . . . , zρ +d). +Proof. We proceed by induction on d. +If d = 1, then mixed volumes are just volumes, +in which case Proposition 4.13 is a special case of Proposition 4.8. +Assume, now, that +Proposition 4.13 holds in dimension less than d > 1. Define +F(z1, . . . , zd) = +� +ρ∈Σ(1) +z1,ρ MVolΣρ,ωρ,∗ρ(zρ +2, . . . , zρ +d). +To prove that F = MVolΣ,∗,ω, Proposition 3.1 tells us that it suffices to prove that F is (1) +symmetric, (2) multilinear, and (3) normalized correctly with respect to volume; we check +these properties in reverse order. + +24 +L. NOWAK, P. O’MELVENY, AND D. ROSS +To check (3), we note that +F(z, . . . , z) = +� +ρ∈Σ(1) +zρ MVolΣρ,ωρ,∗ρ(zρ, . . . , zρ) += +� +ρ∈Σ(1) +zρ VolΣρ,ωρ,∗ρ(zρ) += VolΣ,ω,∗(z), +where the first equality is the definition of F, the second is Proposition 3.1 Part (3), and the +third is Proposition 4.8. +To check (2), there are two cases to consider: linearity in the first coordinate and linearity +in every other coordinate. Linearity in the first coordinate follows quickly from the definition +of F, while linearity in every other coordinate follows from Proposition 3.1 Part (2) applied +to (Σρ, ∗ρ, ωρ). +Finally, to check (1), we first note that Proposition 3.1 Part (1) applied to (Σρ, ∗ρ, ωρ) +implies that F is symmetric in the entries z2, . . . , zd. Thus, it remains to prove that F is +invariant under transposing z1 and z2. To do so, we first apply the induction hypothesis to +the mixed volumes appearing in the definition of F to obtain +(4.14) +F(z1, . . . , zd) = +� +ρ∈Σ(1) +z1,ρ +� +ηρ∈Σρ(1) +zρ +2,ηρ MVolΣρ,η,ωρ,η,∗ρ,η(zρ,η +3 , . . . , zρ,η +d ), +where, to avoid the proliferation of parentheses and superscripts, we have written, for exam- +ple, Σρ,η as short-hand for (Σρ)ηρ. Notice that the mixed volumes appearing in the right-hand +side of (4.14) are mixed volumes associated to faces of faces. Proposition 4.6 tells us that +the ηρ-face of the ρ-face of a normal complex is the same as the τ face of the original normal +complex, where τ ∈ Σ(2) is the 2-cone containing ρ and η as rays. Therefore, +MVolΣρ,η,ωρ,η,∗ρ,η(zρ,η +3 , . . . , zρ,η +d ) = MVolΣτ,ωτ,∗τ(zτ +3, . . . , zτ +d). +Keeping in mind that each 2-cone τ appears twice in (4.14), once for each ordering of the +rays, we have +F(z1, . . . , zd) = +� +τ∈Σ(2) +τ(1)={ρ,η} +(z1,ρzρ +2,ηρ + z1,ηzη +2,ρη) MVolΣτ,ωτ,∗τ(zτ +3, . . . , zτ +d). +Therefore, it remains to prove that z1,ρzρ +2,ηρ + z1,ηzη +2,ρη is invariant under transposing 1 and +2. Computing directly from the definition of zρ, we have +z1,ρzρ +2,ηρ + z1,ηzη +2,ρη = z1,ρ +� +z2,η − wρ,∗(z2) ∗ uη +� ++ z1,η +� +z2,ρ − wη,∗(z2) ∗ uρ +� +, +from which we see that it suffices to prove that both +z1,ρwρ,∗(z2) ∗ uη +and +z1,ηwη,∗(z2) ∗ uρ + +MIXED VOLUMES OF NORMAL COMPLEXES +25 +are invariant under transposing 1 and 2. This invariance follows from the computations +wρ,∗(z2) = +z2,ρ +uρ ∗ uρ +uρ +and +wη,∗(z2) = +z2,η +uη ∗ uη +uη. +□ +The following analytic consequence of Proposition 4.13 will be useful in our computations +in the next section. +Corollary 4.15. In addition to the hypotheses of Proposition 4.13, assume that Cub(Σ, ∗) +is nonempty. Then for any fixed z1, . . . , zk ∈ Cub(Σ, ∗), we have +∂ +∂zρ +MVolΣ,ω,∗(z1, . . . , zk, z, . . . , z +� �� � +d−k +) = (d − k) MVolΣρ,ωρ,∗ρ(zρ +1, . . . , zρ +k, zρ, . . . , zρ +� +�� +� +d−k−1 +). +Proof. The assumption that Cub(Σ, ∗) ̸= ∅ implies that MVolΣ,ω,∗(z1, . . . , zk, z, . . . , z) is a +degree d − k polynomial in R[zρ | ρ ∈ Σ(1)], so the derivatives are well-defined. Proposi- +tion 4.13 and symmetry of mixed volumes imply that +∂ +∂zi,ρ +MVolΣ,ω,∗(z1, . . . , zd) = MVolΣρ,ωρ,∗ρ(zρ +1, . . . , zρ +i−1, zρ +i+1, . . . , zρ +d). +Viewing MVolΣ,ω,∗(z1, . . . , zk, z, . . . , z) as the composition of MVolΣ,ω,∗(z1, . . . , zd) with the +specialization +zk+1 = · · · = zd = z, +the result then follows from the multivariable chain rule. +□ +5. Alexandrov–Fenchel inequalities +One of the most consequential properties of mixed volumes of polytopes (or, more gener- +ally, of mixed volumes of convex bodies) is the Alexandrov–Fenchel inequalities. Given +polytopes P1, . . . , Pd in a d-dimensional real vector space V with volume function Vol, the +Alexandrov–Fenchel inequalities state that +MVol(P1, P2, P3, . . . , Pd)2 ≥ MVol(P1, P1, P3, . . . , Pd) MVol(P2, P2, P3, . . . , Pd) +(see, for example, [Sch14, Theorem 7.3.1] for a proof and historical references). It is our aim +in this section to study Alexandrov–Fenchel inequalities in the setting of mixed volumes of +normal complexes. +Let Σ ⊆ NR be a simplicial d-fan, ω : Σ(d) → R>0 a weight function, and ∗ ∈ Inn(NR) +an inner product. We say that the triple (Σ, ω, ∗) is Alexandrov–Fenchel, or just AF for +short, if Cub(Σ, ∗) ̸= ∅ and +MVolΣ,ω,∗(z1, z2, z3, . . . , zd)2 ≥ MVolΣ,ω,∗(z1, z1, z3, . . . , zd) MVolΣ,ω,∗(z2, z2, z3, . . . , zd) +for all z1, . . . , zd ∈ Cub(Σ, ∗). In this section, we prove the following result, which provides +sufficient conditions for proving that a triple (Σ, ω, ∗) is AF. + +26 +L. NOWAK, P. O’MELVENY, AND D. ROSS +Theorem 5.1. Let Σ ⊆ NR be a simplicial d-fan, ω : Σ(d) → R>0 a weight function, and +∗ ∈ Inn(NR) an inner product such that Cub(Σ, ∗) ̸= ∅. The triple (Σ, ω, ∗) is AF if the +following two conditions are satisfied: +(i) Στ \ {0} is connected for any cone τ ∈ Σ(k) with k ≤ d − 3; +(ii) Hess +� +VolΣτ,ωτ,∗τ(z) +� +has exactly one positive eigenvalue for any τ ∈ Σ(d − 2). +Remark 5.2. Condition (i) in Theorem 5.1 can be thought of as requiring that the fan Σ +does not have any “pinch” points. For example, in dimension four, this condition rules out +fans that locally look like a pair of four-dimensional cones meeting along a ray, because the +star fan associated to that ray would comprise two three-dimensional cones that meet only +at the origin. +Remark 5.3. Condition (ii) of Theorem 5.1 concerns only the two-dimensional stars of Σ. +Since the volume polynomial of a two-dimensional fan is a quadratic form, the Hessians +appearing in Condition (ii) are constant matrices. Condition (ii) can be viewed as an ana- +logue of the Brunn–Minkowski inequality for polygons. For an example of a two-dimensional +(tropical) fan that does not satisfy Condition (ii), see [BH17]. +5.1. Proof of Theorem 5.1. Our proof of Theorem 5.1 is largely inspired by a proof +of the classical Alexandrov–Fenchel inequalities recently developed by Cordero-Erausquin, +Klartag, Merigot, and Santambrogio [CEKMS19]—for which the key geometric input is +Proposition 4.13. +While the arguments in [CEKMS19] can be employed in this setting +more-or-less verbatim, we present a more streamlined proof using ideas regarding Lorentzian +polynomials recently developed by Br¨and´en and Leake [BL21]. Before presenting a proof of +Theorem 5.1, we pause to introduce key ideas regarding Lorentzian polynomials. +5.1.1. Lorentzian polynomials on cones. One way to view the AF inequalities is as the non- +positivity of the 2 × 2 matrix +� +MVolΣ,ω,∗(z1, z1, z3, . . . , zd) +MVolΣ,ω,∗(z1, z2, z3, . . . , zd) +MVolΣ,ω,∗(z2, z1, z3, . . . , zd) +MVolΣ,ω,∗(z2, z2, z3, . . . , zd) +� +, +and this nonpositivity is equivalent to the matrix having exactly one positive eigenvalue. +Lorentzian polynomials are a clever tool for capturing the essence of this observation, and +are therefore a natural setting for understanding AF-type inequalities. +Our discussion of Lorentzian polynomials follows Br¨and´en and Leake [BL21]. Suppose +that C ⊆ Rn +>0 is a nonempty open convex cone, and let f ∈ R[x1, . . . , xn] be a homogeneous +polynomial of degree d. For each i = 1, . . . , n and v = (v1, . . . , vn) ∈ Rn, we use the following + +MIXED VOLUMES OF NORMAL COMPLEXES +27 +shorthand for partial and directional derivatives +∂i = ∂ +∂xi +and +∂v = +n +� +i=1 +vi∂i. +We say that f is C-Lorentzian if, for all v1, . . . , vd ∈ C, +(P) ∂v1 · · · ∂vdf > 0, and +(H) Hess(∂v3 · · · ∂vdf) has exactly one positive eigenvalue. +To relate Lorentzian polynomials back to AF-type inequalities, we recall the following key +observation (see [BH20, Proposition 4.4]). +Lemma 5.4. Let C ⊆ Rn +>0 be a nonempty open convex cone, and let f ∈ R[x1, . . . , xn] be +C-Lorentzian. Then for all v1, v2, v3 . . . , vd ∈ C, we have +� +∂v1∂v2∂v3 · · · ∂vdf +�2 ≥ +� +∂v1∂v1∂v3 · · · ∂vdf +�� +∂v2∂v2∂v3 · · · ∂vdf +� +. +Proof. Consider the symmetric 2 × 2 matrix +M = +� +∂v1∂v1∂v3 · · · ∂vdf +∂v1∂v2∂v3 · · · ∂vdf +∂v2∂v1∂v3 · · · ∂vdf +∂v2∂v2∂v3 · · · ∂vdf +� +. +By (P), the entries of M are positive, so the Peron–Frobenius Theorem implies that M has at +least one positive eigenvalue. On the other hand, M is a principal minor of Hess(∂v3 · · · ∂vdf), +which, by (H), has exactly one positive eigenvalue; thus, it follows from Cauchy’s Interlacing +Theorem that M has at most one positive eigenvalue. Therefore M has exactly one positive +eigenvalue, implying that the determinant of M is nonpositive, proving the lemma. +□ +The following result, proved by Br¨and´en and Leake [BL21], is particularly useful for the +study of Lorentzian polynomials on cones. We view this result as an effective implementation +of the key insights in [CEKMS19]; in essence, it eliminates the need for one of the induction +parameters in [CEKMS19] because that induction parameter is captured within the recursive +nature of Lorentzian polynomials. +Lemma 5.5 ([BL21], Proposition 2.4). Let C ⊆ Rn +>0 be a nonempty open convex cone, and +let f ∈ R[x1, . . . , xn] be a homogeneous polynomial of degree d. If +(1) ∂v1 · · · ∂vdf > 0 for all v1, . . . , vd ∈ C, +(2) Hess +� +∂v1 · · · ∂vd−2f +� +is irreducible1 and has nonnegative off-diagonal entries for all +v1, . . . , vd−2 ∈ C, and +(3) ∂if is C-Lorentzian for all i = 1, . . . , n, +then f is C-Lorentzian. +1An n×n matrix M is irreducible if the associated adjacency graph—the undirected graph on n labeled +vertices with an edge between the ith and jth vertex whenever the (i, j) entry of M is nonzero—is connected. + +28 +L. NOWAK, P. O’MELVENY, AND D. ROSS +5.1.2. Lorentzian volume polynomials. We now discuss how the above discussion of Lorentzian +polynomials on cones can be used to study mixed volumes of normal complexes. Let Σ ⊆ NR +be a simplicial d-fan, ω : Σ(d) → R>0 a weight function, and ∗ ∈ Inn(NR) an inner product. +We assume that Cub(Σ, ∗) ̸= ∅, in which case the function VolΣ,ω,∗ : Cub(Σ, ∗) → R is a +homogeneous polynomial of degree d in R[zρ | ρ ∈ Σ(1)]. By Proposition 3.1(3), we have +VolΣ,ω,∗(z) = MVolΣ,ω,∗(z, . . . , z). +It then follows from Proposition 3.1(1) and (2) (and the chain rule) that +(5.6) +∂z1 · · · ∂zk VolΣ,ω,∗(z) = +d! +d − k! MVolΣ,ω,∗(z1, . . . , zk, z, . . . , z +� �� � +d−k +) +for any z1, . . . , zk ∈ Cub(Σ, ∗). In particular, in order to prove that (Σ, ω, ∗) is AF, we now +see that it suffices (by Lemma 5.4) to prove that VolΣ,ω,∗ is Cub(Σ, ∗)-Lorentzian. Thus, +Theorem 5.1 is a consequence of the following stronger result. +Theorem 5.7. Let Σ ⊆ NR be a simplicial d-fan, ω : Σ(d) → R>0 a weight function, +and ∗ ∈ Inn(NR) an inner product such that Cub(Σ, ∗) ̸= ∅. Then VolΣ,ω,∗ is Cub(Σ, ∗)- +Lorentzian if the following two conditions are satisfied: +(i) Στ \ {0} is connected for any cone τ ∈ Σ(k) with k ≤ d − 3; +(ii) Hess +� +VolΣτ,ωτ,∗τ(z) +� +has exactly one positive eigenvalue for any τ ∈ Σ(d − 2). +Proof. We prove Theorem 5.7 by induction on d. +First consider the base case d = 2 (in which case Condition (i) is vacuous). Note that +VolΣ,ω,∗ satisfies (P) by (5.6) and the positivity of mixed volumes (Proposition 3.5), while +(H) for VolΣ,ω,∗ is equivalent to Condition (ii). Therefore, Theorem 5.7 holds when d = 2. +Now let d > 2 and assume (Σ, ω, ∗) satisfies Conditions (i) and (ii) in Theorem 5.7. +To prove that VolΣ,ω,∗ is Cub(Σ, ∗)-Lorentzian, we use Lemma 5.5. Translating the three +conditions of Lemma 5.5 using (5.6), we must prove that +(1) MVolΣ,ω,∗(z1, . . . , zd) > 0 for all z1, . . . , zd ∈ Cub(Σ, ∗), +(2) Hess +� +MVolΣ,ω,∗(z1, . . . , zd−2, z, z) +� +is irreducible and has nonnegative off-diagonal en- +tries for all z1, . . . , zd−2 ∈ Cub(Σ, ∗), and +(3) ∂ρ VolΣ,ω,∗(z) is Cub(Σ, ∗)-Lorentzian for all ρ ∈ Σ(1). +Note that (1) is just the positivity of mixed volumes (Proposition 3.5). To prove (3), note +that Proposition 3.1(3) and Corollary 4.15 (with k = 0) together imply that +∂ρ VolΣ,ω,∗(z) = d VolΣρ,ωρ,∗ρ(zρ). +Applying the induction hypothesis to (Σρ, ωρ, ∗ρ)—which we can do because any star fan +of Σρ is a star fan of Σ, so our assumption that (Σ, ω, ∗) satisfies the two conditions of + +MIXED VOLUMES OF NORMAL COMPLEXES +29 +Theorem 5.7 implies that (Σρ, ωρ, ∗ρ) also satisfies the two conditions of Theorem 5.7— +implies that ∂ρ VolΣ,ω,∗(z) is Lorentzian, verifying (3). +Finally, to prove (2), we use Corollary 4.15 to compute +∂ρ MVolΣ,ω,∗(z1, . . . , zd−2, z, z) = 2 MVolΣ,ω,∗(zρ +1, . . . , zρ +d−2, zρ). +If τ ∈ Σ(2) with rays ρ and η, then +zρ +ηρ = zη − wρ,∗(z) ∗ uη = zη − uρ ∗ uη +uρ ∗ uρ +zρ, +from which it follows that, +(5.8) +∂η∂ρ MVolΣ,ω,∗(z1, . . . , zd−2, z, z) = 2 MVolΣτ,ωτ,∗τ(zτ +1, . . . , zτ +d−2) +On the other hand, if ρ and η do not lie on a common cone τ ∈ Σ(2), then +(5.9) +∂η∂ρ MVolΣ,ω,∗(z1, . . . , zd−2, z, z) = 0. +The positivity of mixed volumes for cubical values, along with (5.8) and (5.9), then implies +that Hess +� +MVolΣ,ω,∗(z1, . . . , zd−2, z, z) +� +has nonnegative off-diagonal entries that are positive +whenever the row and column index are the rays of a cone τ ∈ Σ(2). The first condition in +Theorem 5.7 implies that we can travel from any ray of Σ to any other ray by passing only +through the relative interiors of one- and two-dimensional cones, which then implies that +Hess +� +MVolΣ,ω,∗(z1, . . . , zd−2, z, z) +� +is irreducible, concluding the proof. +□ +6. Application: the Heron–Rota–Welsh conjecture +As an application of our developments regarding mixed volumes of normal complexes, +we show in this section how Theorem 5.1 can be used to prove the Heron–Rota–Welsh +conjecture, which states that the coefficients of the characteristic polynomial of any matroid +are log-concave. The bridge between matroids and mixed volumes is the Bergman fan; we +begin this section by briefly recalling relevant notions regarding matroids and Bergman fans. +6.1. Matroids and Bergman fans. A (loopless) matroid M = (E, L) consists of a finite +set E, called the ground set, and a collection of subsets L ⊆ 2E, called flats, which satisfy +the following three conditions: +(F1) ∅ ∈ L, +(F2) if F1, F2 ∈ L, then F1 ∩ F2 ∈ L, and +(F3) if F ∈ L, then every element of E \ F is contained in exactly one flat that is minimal +among the flats that strictly contain F. + +30 +L. NOWAK, P. O’MELVENY, AND D. ROSS +We do not give a comprehensive overview of matroids; rather, we settle for a brief intro- +duction of key concepts. For a more complete treatment, see Oxley’s book [Oxl11]. +The closure of a set S ⊆ E, denoted cl(S), is the smallest flat containing S. +A set +I ⊆ E is called independent if cl(I1) ⊊ cl(I2) for any I1 ⊊ I2 ⊆ I. The rank of a set +S ⊆ E, denoted rk(S), is the maximum size of an independent subset of S, and the rank +of M, denoted rk(M) is defined to be the rank of E. While we have chosen to characterize +matroids in terms of their flats, we note that matroids can also be characterized in terms of +their independent sets or their rank function. +A flag of flats (of length k) in M is a chain of the form +F = (F1 ⊊ · · · ⊊ Fk) +with +F1, . . . , Fk ∈ L. +It can be checked from the matroid axioms that every maximal flag has one flat of each rank +0, . . . , rk(M). We let ∆M denote the set of flags of flats, which naturally has the structure of +a simplicial complex of dimension rk(M) + 1. Since every maximal flag contains ∅ and E, we +often restrict our attention to studying proper flats. We use the notation L∗ = L \ {∅, E} +for the set of proper flats and ∆∗ +M for the set of flags of proper flats, which is a simplicial +complex of dimension rk(M) − 1. +Given a matroid M, consider the vector space RE with basis {ve | e ∈ E}. For each subset +S ⊆ E, define +vS = +� +e∈S +ve ∈ RE. +Set NR = RE/RvE and denote the image of vS in the quotient space NR by uS. For each +flag F = (F1 ⊊ · · · ⊊ Fk) ∈ ∆∗ +M, define a polyhedral cone +σF = R≥0{uF1, . . . , uFk} ⊆ NR. +The Bergman fan of M, denoted ΣM, is the polyhedral fan +ΣM = {σF | F ∈ ∆∗ +M}. +Note that ΣM is simplicial, pure of dimension d = rk(M) − 1, and marked by the vectors uF. +Consider a cone σF ∈ ΣM(d − 1) corresponding to a flag +F = (F1 ⊊ · · · ⊊ Fk−1 ⊊ Fk+1 ⊊ · · · ⊊ Fd) +with +rk(Fi) = i. +The d-cones containing σF are indexed by flats F with Fk−1 ⊊ F ⊊ Fk+1. If there are ℓ such +flats, then (F3) implies that +� +F ∈L +Fk−1⊊F ⊊Fk+1 +uF = (ℓ − 1)uFk−1 + uFk+1. + +MIXED VOLUMES OF NORMAL COMPLEXES +31 +Since the right-hand side lies in NσF,R, this observation implies that ΣM is balanced (tropical +with weights all equal to 1). +In order to check that Bergman fans are AF, we require a working understanding of the +star fans of Bergman fans. Consider a cone σF associated to a flat F = (F1 ⊊ · · · ⊊ Fk). Set +F0 = ∅ and Fk+1 = E, and for each j = 0, . . . , k consider the matroid minor M[Fj, Fj+1], +which is the matroid on ground set Fj+1 \ Fj with flats of the form F \ Fj where F is a flat +of M satisfying Fj ⊆ F ⊆ Fj+1. Notice that the star fan ΣσF +M lives in the quotient space +N σF +R += +NR +R{uF1, . . . , uFk} = +RE +R{vF1, . . . , vFk+1} = +k +� +j=0 +RFk+1\Fk +RvFk+1\Fk +, +and one checks that this natural isomorphism of vector spaces identifies the star of ΣM at +σF as the product of the Bergman fans of the associated matroid minors: +(6.1) +ΣσF +M = +k +� +j=0 +ΣM[Fj,Fj+1]. +6.2. Bergman fans are AF. We are now ready to use Theorem 5.1 to prove that Bergman +fans of matroids are AF. +Theorem 6.2. Let M be a matroid of rank d+1 and let ΣM ⊆ NR be the associated Bergman +fan. If ∗ ∈ Inn(NR) is any inner product with Cub(ΣM, ∗) ̸= ∅, then (ΣM, ∗) is AF. +Remark 6.3. We are assuming the weight function ω is equal to 1 because, as noted in the +previous subsection, ΣM is balanced. Thus, we omit ω from the notation in this section. +To prove Theorem 6.2, we verify the two conditions of Theorem 5.1. We accomplish this +through the following three lemmas. The first lemma verifies that Bergman fans satisfy (a +slight strengthening of) Condition (i) of Theorem 6.2. +Lemma 6.4. ΣσF +M \ {0} is connected for any cone σF ∈ ΣM(k) with k ≤ d − 2. +Proof. We begin by arguing that ΣM \{0} is connected for any matroid of rank at least 3. It +suffices to prove that, for any two rays ρF, ρF ′ ∈ ΣM(1) associated to flats F, F ′ ∈ L∗, there +are sequences ρ1, . . . , ρℓ ∈ ΣM(1) and τ1, . . . , τℓ+1 ∈ ΣM(2) such that +ρF ≺ τ1 ≻ ρ1 ≺ · · · ≻ ρℓ ≺ τℓ+1 ≻ ρF ′. +If F ∩ F ′ = G ̸= ∅, then G ∈ L∗ by (F2) and the following is such a sequence +ρF ≺ τG⊊F ≻ ρG ≺ τG⊊F ′ ≻ ρF ′. +If, on the other hand, F ∩ F ′ = ∅, choose rank-one flats G ⊆ F and G′ ⊆ F ′. By (F3), there +is exactly one rank-two flat H that contains G and G′, so we can construct a sequence +(ρF ≺ τG⊊F ≻)ρG ≺ τG⊊H ≻ ρH ≺ τG′⊊H ≻ ρG′(≺ τG′⊊F ′ ≻ ρF ′), + +32 +L. NOWAK, P. O’MELVENY, AND D. ROSS +where the parenthetical pieces should be omitted if G = F or G′ = F ′. +Now consider any star fan ΣσF +M where F = (F1 ⊊ · · · ⊊ Fk) with k ≤ d − 2. Notice that +such a star fan has dimension at least two, and we can write it as a product of Bergman fans +on matroid minors +ΣσF +M = +k +� +j=0 +ΣM[Fj,Fj+1]. +Consider two rays ρ, ρ′ ∈ ΣσF +M (1). If the two rays happen to come from different factors in +the product, then we can connect them through the sequence +ρ ≺ ρ × ρ′ ≻ ρ′. +If, on the other hand, they lie in the same factor, there are two cases to consider. If the +matroid minor of the factor that the rays lie in has rank at least 3, then the rays can be +connected via the argument above. If, on the other hand, the matroid minor has rank 2, +then one of the other matroid minors must also have rank at least 2. Choosing any ray ρ′′ in +the Bergman fan of the second matroid minor, we can connect ρ and ρ′ through the sequence +ρ ≺ ρ × ρ′′ ≻ ρ′′ ≺ ρ′ × ρ′′ ≻ ρ′. +□ +In order to verify Condition (ii) of Theorem 5.1, there are two cases to consider, depending +on whether the two-dimensional star fan in question is, itself, a Bergman fan, or whether it +is the product of two one-dimensional Bergman fans. In both cases, we use the fact that, +in order to prove that the Hessian of a quadratic form f ∈ R[x1, . . . , xn] has exactly one +eigenvalue, it suffices (by Sylvester’s Law of Inertia) to find an invertible change of variables +y1(x), . . . , yn(x) such that +f = +n +� +i=1 +aiyi(x)2 +with exactly one positive ai. We now consider the two cases in the following two lemmas. +Lemma 6.5. If M is a rank-three matroid, then the Hessian of degΣM(D(z)2) has exactly +one positive eigenvalue. +Proof. For a flat F ∈ L∗, we use the shorthand XF = XρF and zF = zρF . In order to compute +degΣM(D(z)2), we must compute degΣM(XFXG) for any two flats F, G ∈ L∗. If F ⊊ G, then +the degree is one, by definition of the degree function, and if F and G are incomparable, +then the degree is zero. +Thus, it remains to compute the degree of the squared terms. +Using the definition of A•(ΣM) and the flat axioms, the reader is encouraged to verify that +degΣM(X2 +F) = 1 − |{G ∈ L∗ | F ⊊ G}| if rk(F) = 1 and degΣM(X2 +G) = −1 if rk(G) = 2. It + +MIXED VOLUMES OF NORMAL COMPLEXES +33 +follows that +degΣM(D(z)2) = 2 +� +F,G∈L∗ +F ⊊G +zFzG + +� +F ∈L∗ +rk(F )=1 +z2 +F − +� +F,G∈L∗ +F ⊊G +z2 +F − +� +G∈L∗ +rk(G)=2 +z2 +G. +By creatively organizing the terms, we can rewrite this as +degΣM(D(z)2) = +� � +F ∈L∗ +rk(F )=1 +zF +�2 +− +� +G∈L∗ +rk(G)=2 +� +zG − +� +F ∈L∗ +F ⊊G +zF +�2 +, +where the only key matroid assertion used in the equivalence of these two formulas is that +there exists a unique rank-two flat containing any two distinct rank-one flats. Sylvester’s +Law of Inertia implies that the Hessian of this quadratic form has exactly one positive +eigenvalue. +□ +Lemma 6.6. If M and M′ are rank-two matroids, then the Hessian of degΣM×ΣM′(D(z)2) has +exactly one positive eigenvalue. +Proof. By definition of A•(ΣM × ΣM′), the reader is encouraged to verify that +degΣM×ΣM′(XρXη) = +� +� +� +0 +ρ, η ∈ ΣM(1) or ρ, η ∈ ΣM′(1), +1 +ρ ∈ ΣM(1) and η ∈ ΣM′(1). +Therefore, +degΣM×ΣM′(D(z)2) = +� +ρ∈ΣM(1), η∈ΣM′(1) +2zρzη, +which can be rewritten as +degΣM×ΣM′(D(z)2) = 1 +2 +� +� +ρ∈ΣM(1) +zρ + +� +η∈ΣM′(1) +zρ +�2 +− 1 +2 +� +� +ρ∈ΣM(1) +zρ − +� +η∈ΣM′(1) +zρ +�2 +. +Sylvester’s Law of Inertia implies that the Hessian of this quadratic form has exactly one +positive eigenvalue. +□ +We now have all the ingredients we need to prove Theorem 6.2. +Proof of Theorem 6.2. We prove that Bergman fans satisfy the two conditions of Theo- +rem 5.1. That Bergman fans satisfy Condition (i) is the content of Lemma 6.4. To prove +Condition (ii), we first note that, since Bergman fans are balanced, their star fans are also +balanced, so Theorem 2.2 implies that the volume polynomials in Condition (ii) are inde- +pendent of ∗ and are equal to +degΣσF +M (D(z)2). +By the product decomposition of star fans given in (6.1), ΣσF +M is either a two-dimensional +Bergman fan or a product of two one-dimensional Bergman fans; in the former case, the + +34 +L. NOWAK, P. O’MELVENY, AND D. ROSS +Hessian of the volume polynomial has exactly one positive eigenvalue by Lemma 6.5, and in +the latter case, by Lemma 6.6. +□ +6.3. Revisiting the Heron–Rota–Welsh Conjecture. The characteristic polynomial +of a matroid M = (E, L) can be defined by +χM(λ) = +� +S⊆E +(−1)|S|λrk(M)−rk(S). +It can be checked that χM(λ) has a root at λ = 1 for any positive-rank matroid, and the +reduced characteristic polynomial is defined by +χM(λ) = χM(λ) +λ − 1 . +We use the notation µa(M) and µa(M) for the (unsigned) coefficients of these polynomials: +χM(λ) = +rk(M) +� +a=0 +(−1)aµa(M)λrk(M)−a +and +χM(λ) = +rk(M)−1 +� +a=0 +(−1)aµa(M)λrk(M)−1−a. +The Heron–Rota–Welsh Conjecture, developed in [Rot71, Her72, Wel76], asserts that the +sequence of nonnegative integers µ0(M), . . . , µrk(M)(M) is unimodal and log-concave: +0 ≤ µ0(M) ≤ · · · ≤ µk(M) ≥ · · · ≥ µrk(M)(M) ≥ 0 +for some +k ∈ {0, . . . , rk(M)} +and +µk(M)2 ≥ µk−1(M)µk+1(M) +for every +k ∈ {1, . . . , rk(M) − 1}. +The Heron–Rota–Welsh Conjecture was first proved by Adiprasito, Huh, and Katz [AHK18]. +Our aim here is to show how this result also follows from the developments in this paper. +It is elementary to check that the unimodality and log-concavity of the coefficients of the +characteristic polynomial is implied by the analogous properties for the coefficients of the +reduced characteristic polynomial. The bridge from characteristic polynomials to the content +of this paper, then, is a result of Huh and Katz [HK12, Proposition 5.2] (see also [AHK18, +Proposition 9.5] and [DR22, Proposition 3.11]), which asserts that +µa(M) = degΣM(αd−aβa) +where rk(M) = d + 1 and α, β ∈ A1(ΣM) are defined by +α = +� +e0∈F +XF +and +β = +� +e0 /∈F +XF +for some e0 ∈ E (these Chow classes are independent of the choice of e0). + +MIXED VOLUMES OF NORMAL COMPLEXES +35 +Choose any e0 ∈ E, and let ∗ ∈ Inn(NR) be the inner product with orthonormal basis +{ue | e ̸= e0} ⊆ NR = RE/RuE. For two flats F1, F2 ∈ L∗, we compute +uF1 ∗ uF2 = +� +� +� +� +� +� +� +� +� +|F1 ∩ F2| +e0 /∈ F1 and e0 /∈ F2, +−|F1 ∩ F c +2| +e0 /∈ F1 and e0 ∈ F2, +|F c +1 ∩ F c +2| +e0 ∈ F1 and e0 ∈ F2. +Define zα, zβ ∈ RΣM(1) = RL∗ by +zα +F = +� +� +� +1 +e0 ∈ F, +0 +e0 /∈ F, +and +zβ +F = +� +� +� +1 +e0 /∈ F, +0 +e0 ∈ F, +so that D(zα) = α and D(zβ) = β in A1(ΣM). The following lemma allows us to connect +characteristic polynomials to mixed volumes of normal complexes. +Lemma 6.7. zα, zβ ∈ Cub(ΣM, ∗). +Proof. We must argue that wσ,∗(zα), wσ,∗(zβ) ∈ σ for every cone σ ∈ ΣM. Consider a flag +F = (F1 ⊊ · · · ⊊ Fk) corresponding to a cone σF ∈ ΣM. It suffices to prove that +(6.8) +wσF,∗(zα) = +� +� +� +1 +|F c +k|uFk +e0 ∈ Fk +0 +e0 /∈ Fk, +and +(6.9) +wσF,∗(zβ) = +� +� +� +1 +|F1|uF1 +e0 /∈ F1 +0 +e0 ∈ F1, +We verify (6.8); the verification (6.9) is similar. +To verify (6.8), first suppose that e0 ∈ Fk. Then for any j = 1, . . . , k, it follows from the +definition of ∗ that +uFk ∗ uFj = +� +� +� +|F c +k| +e0 ∈ Fj, +0 +e0 /∈ Fj. +Using this, we verify that +1 +|F c +k|uFk satisfies the defining equations of wσF,∗(zα): +1 +|F c +k|uFk ∗ uFj = zα +Fj +for all +j = 1, . . . , k. +Now suppose that e0 /∈ Fk. Then e0 /∈ Fj for any j = 1, . . . , k, so zα +Fj = 0. Thus, the defining +equation for wσF,∗(zα) become +wσF,∗(zα) ∗ uFj = 0 +for all +j = 1, . . . , k, +showing that wσF,∗(zα) = 0. +□ + +36 +L. NOWAK, P. O’MELVENY, AND D. ROSS +It follows from Theorem 3.6 that the coefficients of the reduced characteristic polynomial +have a volume-theoretic interpretation: +µa(M) = MVolΣM,∗(zα, . . . , zα +� +�� +� +d−a +, zβ, . . . , zβ +� +�� +� +a +). +By [NR21, Proposition 7.4], we know that Cub(ΣM, ∗) ̸= ∅, and since the cubical cone +is the interior of the pseudocubical cone, we may approximate zα, zβ ∈ Cub(ΣM, ∗) with +zα +t , zβ +t ∈ Cub(ΣM, ∗) such that +lim +t→0 zα +t = zα +and +lim +t→0 zβ +t = zβ. +Define +µa +t (M) = MVolΣM,∗(zα +t , . . . , zα +t +� +�� +� +d−a +, zβ +t , . . . , zβ +t +� +�� +� +a +). +By Theorem 6.2, we know that (ΣM, ∗) is AF, and the AF inequalities applied to the mixed +volumes µa +t (M) imply that the sequence µ0 +t(M), . . . , µd +t (M) is log-concave. Since mixed vol- +umes of cubical values are positive (Proposition 3.5), and since all log-concave sequences of +positive values are unimodal, we see that the sequence µ0 +t(M), . . . , µd +t (M) is also unimodal. +Since both unimodality and log-concavity are preserved under limits, we conclude that +µ0(M), . . . , µd(M) +is unimodal and log-concave, verifying the Heron–Rota–Welsh Conjecture. +References +[ADH20] +F. +Ardila, +G. +Denham, +and +J. +Huh. +Lagrangian +geometry +of +matroids. +Preprint: +arXiv:2004.13116, 2020. +[AGV21] +N. Anari, S. O. Gharan, and C. Vinzant. Log-concave polynomials, I: entropy and a determinis- +tic approximation algorithm for counting bases of matroids. Duke Math. J., 170(16):3459–3504, +2021. +[AHK18] +K. Adiprasito, J. Huh, and E. Katz. Hodge theory for combinatorial geometries. Ann. of Math. +(2), 188(2):381–452, 2018. +[ALGV18] +N. Anari, K. Liu, S. O. Gharan, and C. Vinzant. 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Lorentzian polynomials. Ann. of Math. (2), 192(3):821–891, 2020. +[BHM+20] +T. Braden, J. Huh, J. P. Matherne, N. Proudfoot, and B. Wang. Singular hodge theory for +combinatorial geometries. Preprint: arXiv:2010.06088, 2020. +[BHM+22] +T. Braden, J. Huh, J. P. Matherne, N. Proudfoot, and B. Wang. A semi-small decomposition +of the Chow ring of a matroid. Adv. Math., 409(part A):Paper No. 108646, 49, 2022. +[BL21] +P. Br¨and´en and J. Leake. Lorentzian polynomials on cones and the Heron-Rota-Welsh conjec- +ture. Preprint: arXiv:2110.08647, 2021. +[CEKMS19] D. Cordero-Erausquin, B. Klartag, Q. Merigot, and F. Santambrogio. One more proof of the +Alexandrov-Fenchel inequality. C. R. Math. Acad. Sci. Paris, 357(8):676–680, 2019. +[CP21] +S. H. Chan and I. Pak. Log-concave poset inequalities. Preprint: arXiv:2110.10740, 2021. +[DR22] +J. Dastidar and D. Ross. Matroid psi classes. Selecta Math. (N.S.), 28(3):Paper No. 55, 38, +2022. +[Her72] +A. P. Heron. Matroid polynomials. In Combinatorics (Proc. Conf. Combinatorial Math., Math. +Inst., Oxford, 1972), pages 164–202, 1972. +[HK12] +J. Huh and E. Katz. Log-concavity of characteristic polynomials and the Bergman fan of ma- +troids. Math. Ann., 354(3):1103–1116, 2012. +[Min03] +H. Minkowski. Volumen und Oberfl¨ache. Math. Ann., 57(4):447–495, 1903. +[NR21] +R. Nathanson and D. Ross. Tropical fans and normal complexes. Preprint: arXiv:2110.08647, +2021. +[Oxl11] +J. Oxley. Matroid theory, volume 21 of Oxford Graduate Texts in Mathematics. Oxford Univer- +sity Press, Oxford, second edition, 2011. +[Rot71] +G.-C. Rota. Combinatorial theory, old and new. In Actes du Congr`es International des +Math´ematiciens (Nice, 1970), Tome 3, pages 229–233. 1971. +[Sch14] +R. Schneider. Convex bodies: the Brunn-Minkowski theory, volume 151 of Encyclopedia of Math- +ematics and its Applications. Cambridge University Press, Cambridge, expanded edition, 2014. +[Wel76] +D. J. A. Welsh. Matroid theory. Academic Press [Harcourt Brace Jovanovich, Publishers], +London-New York, 1976. L. M. S. Monographs, No. 8. +Department of Mathematics, University of Washington +Email address: lnowak@uw.edu +Department of Mathematics, San Francisco State University +Email address: pomelveny@mail.sfsu.edu +Department of Mathematics, San Francisco State University +Email address: rossd@sfsu.edu + diff --git a/2tE4T4oBgHgl3EQf0A0W/content/tmp_files/load_file.txt b/2tE4T4oBgHgl3EQf0A0W/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..23596f4b8d9ecd6b3225a6bf6b70aa45bfb32497 --- /dev/null +++ b/2tE4T4oBgHgl3EQf0A0W/content/tmp_files/load_file.txt @@ -0,0 +1,1523 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf,len=1522 +page_content='MIXED VOLUMES OF NORMAL COMPLEXES LAUREN NOWAK, PATRICK O’MELVENY, AND DUSTIN ROSS Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Normal complexes are orthogonal truncations of polyhedral fans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' In this paper, we develop the study of mixed volumes for normal complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Our main result is a sufficiency condition that ensures when the mixed volumes of normal complexes associated to a given fan satisfy the Alexandrov–Fenchel inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' By specializing to Bergman fans of matroids, we give a new proof of the Heron–Rota–Welsh Conjecture as a consequence of the Alexandrov– Fenchel inequalities for normal complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Introduction The Alexandrov–Fenchel inequalities lie at the heart of convex geometry, asserting that, for any convex bodies P♥, P♦, P3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , Pd ∈ Rd, their mixed volumes satisfy MVol(P♥, P♦, P3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , Pd)2 ≥ MVol(P♥, P♥, P3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , Pd) MVol(P♦, P♦, P3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , Pd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' This paper is centered around developing an analogue of the Alexandrov–Fenchel inequalities in a decidedly nonconvex setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' The geometric objects of interest to us are normal com- plexes, which were recently introduced by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Nathanson and the third author [NR21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Given a pure simplicial fan Σ, a normal complex associated to Σ is, roughly speaking, a polyhedral complex obtained by truncating each cone of Σ with half-spaces perpendicular to the rays of Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' The choice of where to place the truncating half-spaces results in a family of normal com- plexes associated to each fan Σ, and the question that motivates this work is: for a given fan Σ, do the mixed volumes of the associated normal complexes satisfy the Alexandrov–Fenchel inequalities?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Our main result (Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='1) describes two readily verifiable conditions on Σ that guarantee an affirmative answer to this question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' One of the motivations for studying mixed volumes of normal complexes is that, in the special setting of tropical fans, they correspond to mixed degrees of divisors in associated Chow rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Thus, Alexandrov–Fenchel inequalities for normal complexes lead to nontrivial numerical inequalities in these Chow rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' A class of tropical fans that have garnered a great deal of attention in recent years are Bergman fans of matroids, and one application of our main result (Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='2) is that normal complexes associated to Bergman fans of matroids satisfy the Alexandrov–Fenchel inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Translating these inequalities back to matroid Chow rings, we obtain a volume-theoretic proof of the log-concavity of characteristic polynomials of matroids, a result that was conjectured by Heron, Rota, and Welsh [Rot71, Her72, Wel76] and first proved by Adiprasito, Huh, and Katz [AHK18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='05278v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='CO] 12 Jan 2023 2 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' NOWAK, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' O’MELVENY, AND D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' ROSS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Overview of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' We begin in Section 2 by briefly recalling the construction of normal complexes and their volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Normal complexes, denoted CΣ,∗(z), depend on a marked simplicial d-fan Σ in a vector space NR with an inner product ∗ ∈ Inn(NR), as well as a choice of pseudocubical truncating values z ∈ Cub(Σ, ∗) ⊆ RΣ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' The volume of CΣ,∗(z), denoted VolΣ,ω,∗(z), where ω is a weight function on the top-dimensional cones of Σ, is defined as the weighted sum of the volumes of the maximal polytopes in CΣ,∗(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' We recall the main result of [NR21], which asserts that, if (Σ, ω) is a tropical fan, then (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='1) VolΣ,ω,∗(z) = degΣ,ω(D(z)d) where D(z) = � ρ∈Σ(1) zρXρ ∈ A1(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' In Section 3, we introduce mixed volumes of normal complexes CΣ,∗(z1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , CΣ,∗(zd), denoted MVolΣ,ω,∗(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zd), which are weighted sums of mixed volumes of maximal poly- topes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Analogous to mixed volumes in convex geometry, we show that mixed volumes of normal complexes are characterized by being symmetric, multilinear, and normalized by vol- ume (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Furthermore, we prove that mixed volumes are nonnegative on the pseudocubical cone Cub(Σ, ∗) and positive on the cubical cone Cub(Σ, ∗) (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' For all tropical fans (Σ, ω), we leverage (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='1) to show (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='6) that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='2) MVolΣ,ω,∗(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zd) = degΣ,ω(D(z1) · · · D(zd)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' In Section 4, we develop the face structure of normal complexes, closely paralleling the classical face structure of polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' In particular, the faces of a normal complex CΣ,∗(z) are indexed by cones τ ∈ Σ, and each face is obtained as the intersection of CΣ,∗(z) with the truncating hyperplanes indexed by the rays of τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' We describe how each face can, itself, be viewed as a normal complex associated to the star fan Στ, and use this to define (mixed) volumes of faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Our main result of this section (Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='13), shows how mixed volumes of normal complexes can be computed in terms of mixed volumes of facets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' In Section 5, we introduce what it means for a triple (Σ, ω, ∗) to be AF—namely, that the mixed volumes of cubical values satisfy the Alexandrov–Fenchel inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Our main result (Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='1), inspired by work of Cordero-Erausquin, Klartag, Merigot, and Santambrogio [CEKMS19] and Br¨and´en and Leake [BL21], states that (Σ, ω, ∗) is AF if (i) all star fans Στ of dimension at least three remain connected after removing the origin and (ii) the quadratic volume polynomials associated to the two-dimensional star fans of Σ have exactly one positive eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' In fact, under these conditions, we argue that the volume polynomial VolΣ,ω,∗(z) is Cub(Σ, ∗)-Lorentzian, which then implies that (Σ, ω, ∗) is AF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' In Section 6, we briefly recall relevant notions regarding matroids and Bergman fans, and then we use Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='1 to prove that Bergman fans of matroids are AF (Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' MIXED VOLUMES OF NORMAL COMPLEXES 3 We conclude the paper by deducing the Heron–Rota–Welsh Conjecture as a consequence of the Alexandrov–Fenchel inequalities for normal complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Relation to other work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Since the original proof of the Heron–Rota–Welsh Conjec- ture by Adiprasito, Huh, and Katz [AHK18], there have been a number of alternative proofs, generalizations, and exciting related developments (an incomplete list includes [BHM+22, BHM+20, BES20, ADH20, AP20, AP21, BH20, AGV21, ALGV19, ALGV18, CP21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' We view the volume-theoretic approach in this paper as a new angle from which to view log- concavity of characteristic polynomials of matroids, but we also want to acknowledge that our methods share features of and are indebted to the approaches of several other teams of mathematicians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' In particular, our methods rely on the Chow-theoretic interpretation of characteristic polynomials of matroids, proved by Huh and Katz [HK12], which was central in the original proof of Adiprasito, Huh, and Katz [AHK18], as well as in the subsequent proofs by Braden, Huh, Matherne, Proudfoot, and Wang [BHM+22] and Backman, Eur, and Simpson [BES20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' In addition, our methods prove that volume polynomials are Lorentzian, which is also a central feature in the methods of both Backman, Eur, and Simpson [BES20] and Br¨and´en and Leake [BL21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' We note that, while the methods of [BES20] and [BL21] seem to be tailored primarily for matroids, our methods readily extend to the more general setting of tropical intersection theory (this extension will be spelled out in a forthcoming work of the third author).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' By adding a new volume-theoretic approach to the Heron–Rota– Welsh Conjecture to the literature, we hope that this paper will serve to welcome a new batch of geometrically-minded folks into the fold of this flourishing area of research, opening the door for further developments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' The authors would like to express their gratitude to Federico Ardila, Matthias Beck, Emily Clader, Chris Eur, and Serkan Ho¸sten for sharing insights related to this project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' This work was supported by a grant from the National Science Foundation: DMS-2001439.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Background on normal complexes In this section, we establish notation, conventions, and preliminary results regarding poly- hedral fans and normal complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Fan definitions and conventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Let NR be a real vector spaces of dimension n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Given a polyhedral fan Σ ⊆ NR, we denote the k-dimensional cones of Σ by Σ(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Let ⪯ denote the face containment relation among the cones of Σ, and for each cone σ ∈ Σ, let σ(k) ⊆ Σ(k) denote the k-dimensional faces of σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' For any cone σ, let σ◦ denote the relative interior of σ and denote the linear span of σ by Nσ,R ⊆ NR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' 4 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' NOWAK, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' O’MELVENY, AND D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' ROSS We say that a fan Σ is pure if all of the maximal cones in Σ have the same dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' We say that Σ is marked if we have chosen a distinguished generating vector 0 ̸= uρ ∈ ρ for each ray ρ ∈ Σ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Henceforth, we assume that all fans are pure, polyhedral, and marked, and we use the term d-fan to refer to a pure, polyhedral, marked fan of dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' We say that Σ is simplicial if dim(Nσ,R) = |σ(1)| for all σ ∈ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' The faces of a simplicial cone σ are in bijective correspondence with the subsets of σ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' For every face containment τ ⪯ σ in a simplicial fan Σ, let σ \\ τ denote the face of σ with rays σ(1) \\ τ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Given two faces τ, π ⪯ σ, denote by τ ∪ π the face of σ with rays τ(1) ∪ π(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Given a simplical d-fan Σ and a weight function ω : Σ(d) → R>0, we say that the pair (Σ, ω) is a tropical fan if it satisfies the weighted balancing condition: � σ∈Σ(d) τ≺σ ω(σ)uσ\\τ ∈ Nτ,R for all τ ∈ Σ(d − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' While the definition of tropical fans can be generalized to nonsimplicial fans, we will assume throughout this paper that all tropical fans are simplicial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' If ω(σ) = 1 for all σ ∈ Σ(d), we say that Σ is balanced and we omit ω from the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Chow rings and degree maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Let MR denote the dual of NR and let ⟨−, −⟩ be the duality pairing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Given a simplicial fan Σ ⊆ NR, the Chow ring of Σ is defined by A•(Σ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='.= R � xρ | ρ ∈ Σ(1) � I + J where I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='.= � xρ1 · · · xρk | R≥0{ρ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , ρk} /∈ Σ � and J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='.= � � ρ∈Σ(1) ⟨v, uρ⟩xρ ���� v ∈ MR � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' As both I and J are homogeneous, the Chow ring A•(Σ) is a graded ring, and we denote by Ak(Σ) the subgroup of homogeneous elements of degree k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' We denote the generators of A•(Σ) by Xρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='.= [xρ] ∈ A1(Σ), and for any σ ∈ Σ(k), we define Xσ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='.= � ρ∈σ(1) Xρ ∈ Ak(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' If Σ is a simplicial d-fan, then every element of Ak(Σ) can be written as a linear combination of Xσ with σ ∈ Σ(k) (see, for example, [AHK18, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' It follows that Ak(Σ) = 0 for all k > d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' If, in addition, (Σ, ω) is tropical, then there is a well-defined degree map degΣ,ω : Ad(Σ) → R such that degΣ,ω(Xσ) = ω(σ) for every σ ∈ Σ(d) (see, for example, [AHK18, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' MIXED VOLUMES OF NORMAL COMPLEXES 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Normal complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' We now recall the construction of normal complexes from [NR21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' In addition to a simplicial d-fan Σ ⊆ NR, the normal complex construction requires an additional choice of an inner product ∗ ∈ Inn(NR) and a value z ∈ RΣ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Given such a ∗ and z, we define a set of hyperplanes and half-spaces in NR associated to each ρ ∈ Σ by Hρ,∗(z) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='.= {v ∈ NR | v ∗ uρ = zρ} and H− ρ,∗(z) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='.= {v ∈ NR | v ∗ uρ ≤ zρ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' We then define polytopes Pσ,∗(z), one for each σ ∈ Σ, by Pσ,∗(z) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='.= σ ∩ � ρ∈σ(1) H− ρ,∗(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Notice that Pσ,∗(z) is simply a truncation of the cone σ by hyperplanes that are normal to the rays of σ—what it means to be normal is determined by ∗, and the locations of the normal hyperplanes along the rays of the cone are determined by z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' We would like to construct a polytopal complex from these polytopes, but in general, they do not meet along faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' To ensure that they meet along faces, we require a compatibility between z and ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' For each σ ∈ Σ, let wσ,∗(z) ∈ Nσ,R be the unique vector such that wσ,∗(z) ∗ uρ = zρ for all ρ ∈ σ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' That such a vector exists and is unique follows from the fact that the vectors uρ with ρ ∈ σ(1) are linearly independent—this is equivalent to the simplicial hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' We then say that z is cubical (pseudocubical) with respect to (Σ, ∗) if wσ,∗(z) ∈ σ◦ (wσ,∗(z) ∈ σ) for all σ ∈ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' In other words, the pseudocubical values are those values of z for which the truncating hyperplanes intersect within each cone, and the cubical values are those for which they intersect in the relative interior of each cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' The collection of cubical values are denoted Cub(Σ, ∗) ⊆ RΣ(1) and the pseudocubical values are denoted Cub(Σ, ∗) ⊆ RΣ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' We now summarize key results from [NR21] that will be necessary for the developments in this paper (see [NR21, Propositions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='3, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='7 ]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Let Σ ⊆ NR be a simplicial d-fan and let ∗ ∈ Inn(NR) be an inner product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' (1) The set Cub(Σ, ∗) ⊆ RΣ(1) is a polyhedral cone with Cub(Σ, ∗)◦ = Cub(Σ, ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' (2) For z ∈ Cub(Σ, ∗), the vertices of Pσ,∗(z) are {wτ,∗(z) | τ ⪯ σ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' (3) For z ∈ Cub(Σ, ∗), the polytopes Pσ,∗(z) meet along faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' For any polytope P, let �P denote the set of all faces of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' The third part of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='1 implies that CΣ,∗(z) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='.= � σ∈Σ(d) � Pσ,∗(z) is a polytopal complex whenever z ∈ Cub(Σ, ∗), and this polytopal complex is called the normal complex of Σ with respect to ∗ and z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' 6 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' NOWAK, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' O’MELVENY, AND D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' ROSS Below, we depict a two-dimensional tropical fan and an associated normal complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' The fan is comprised of nine two-dimensional cones glued along faces, and each of these nine cones corresponds to a quadrilateral in the normal complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' The next pair of images depict a three-dimensional fan comprised of two maximal cones meeting along a two-dimensional face, and a corresponding normal complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' While this fan is not tropical, the reader is welcome to view this image as just one small piece of a three-dimensional tropical fan in some higher-dimensional vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Volumes of normal complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Let Σ ⊆ NR be a simplicial d-fan, ∗ ∈ Inn(NR) an inner product, and z ∈ Cub(Σ, ∗) a pseudocubical value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Informally, the volume of the normal complex CΣ,∗(z) is the sum of the volumes of the polytopes Pσ,∗(z) with σ ∈ Σ(d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' however, some care is required in specifying what we mean by volume in each subspace Nσ,R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' For each cone σ ∈ Σ, define the discrete subgroup Nσ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='.= spanZ(uρ | ρ ∈ σ(1)) ⊆ NR, and let Mσ denote its dual: Mσ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='.= HomZ(Nσ, Z) ⊆ Mσ,R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='.= HomR(Nσ,R, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Using the inner product ∗, we can identify Mσ,R with Nσ,R and thus, we can view Mσ as a lattice in Nσ,R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' For each σ ∈ Σ, let Volσ : � polytopes in Nσ,R � → R≥0 MIXED VOLUMES OF NORMAL COMPLEXES 7 be the volume function determined by the property that a fundamental simplex of the lattice Mσ ⊆ Nσ,R has unit volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Define the volume of the normal complex CΣ,∗(z), denoted VolΣ,∗(z) for brevity, as the sum of the volumes of the constituent d-dimensional polytopes: VolΣ,∗(z) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='.= � σ∈Σ(d) Volσ(Pσ,∗(z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' In slightly more generality, suppose that ω : Σ(d) → R>0 is a weight function on the maximal cones of Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' The volume of the normal complex CΣ,∗(z) weighted by ω is defined by VolΣ,ω,∗(z) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='.= � σ∈Σ(d) ω(σ) Volσ(Pσ,∗(z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' The main result of [NR21] is a Chow-theoretic interpretation of the weighted volumes of normal complexes, valid whenever (Σ, ω) is tropical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='2 ([NR21, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Let (Σ, ω) be a tropical d-fan, ∗ ∈ Inn(NR) an inner product, and z ∈ Cub(Σ, ∗) a pseudocubical value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Then VolΣ,ω,∗(z) = degΣ,ω(D(z)d) where D(z) = � ρ∈Σ(1) zρXρ ∈ A1(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Mixed Volumes of Normal Complexes Our first aim in this paper is to enhance Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='2 to a statement about mixed volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' In order to do this, we briefly recall the classical theory of mixed volumes, for which we recommend the comprehensive text by Schneider [Sch14] as a reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Mixed volumes of polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Mixed volumes are the natural result of combining the notion of volume with the operation of Minkowski addition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' We start with a d-dimensional real vector space V and a volume function Vol : {polytopes in V } → R≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' The mixed volume function MVol : {polytopes in V }d → R≥0 is the unique function determined by the following three properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' (Symmetry) For any permutation π ∈ Sd, MVol(P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , Pd) = MVol(π(P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , Pd)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' (Multilinearity) For any i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , d and λ ∈ R≥0, MVol(P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , λPi + P ′ i, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , Pd) = λ MVol(P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , Pi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , Pd) + MVol(P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , P ′ i, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , Pd), 8 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' NOWAK, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' O’MELVENY, AND D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' ROSS where the linear combination of polytopes is defined by λPi + P ′ i = {λv + w | v ∈ Pi, w ∈ P ′ i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' (Normalization) For any polytope P, MVol(P, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , P) = Vol(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' That such a mixed volume function exists and is unique is due to Minkowski [Min03], who proved that such a function exists more generally for convex bodies, not just for polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Mixed volumes of normal complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' We now define a notion of mixed volumes of normal complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Let Σ ⊆ NR be a simplicial d-fan and let ∗ ∈ Inn(NR) be an inner product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Given pseudocubical values z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zd ∈ Cub(Σ, ∗), we define the mixed volume of the normal complexes CΣ,∗(z1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , CΣ,∗(zd), denoted MVolΣ,∗(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zd) for brevity, by MVolΣ,∗(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zd) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='.= � σ∈Σ(d) MVolσ(Pσ,∗(z1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , Pσ,∗(zd)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' In other words, the mixed volume is the sum of the mixed volumes of the polytopes associated to the top-dimensional cones of Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' More generally, if ω : Σ(d) → R>0 is a weight function, then the mixed volume of the normal complexes CΣ,∗(z1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , CΣ,∗(zd) weighted by ω is defined by MVolΣ,ω,∗(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zd) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='.= � σ∈Σ(d) ω(σ) MVolσ(Pσ,∗(z1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , Pσ,∗(zd)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' In order to verify that this is a meaningful notion of mixed volumes for normal complexes, we check that it is characterized by an analogue of the three characterizing properties of mixed volumes of polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Let Σ ⊆ NR be a simplicial d-fan, ∗ ∈ Inn(NR) an inner product, and ω : Σ(d) → R>0 a weight function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' (1) For any z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zd ∈ Cub(Σ, ∗) and π ∈ Sd, MVolΣ,ω,∗(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zd) = MVolΣ,ω,∗(π(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zd)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' (2) For any i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , d, and for any z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zi, z′ i, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zd ∈ Cub(Σ, ∗) and λ ∈ R≥0, MVolΣ,ω,∗(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , λzi + z′ i, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zd) = λ MVolΣ,ω,∗(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zd) + MVolΣ,ω,∗(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , z′ i, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' (3) For any z ∈ Cub(Σ, ∗), MVolΣ,ω,∗(z, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , z) = VolΣ,ω,∗(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Moreover, any function Cub(Σ, ∗)d → R≥0 satisfying Properties (1) – (3) must be MVolΣ,ω,∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' MIXED VOLUMES OF NORMAL COMPLEXES 9 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Given that MVolΣ,ω,∗(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zd) = � σ∈Σ(d) ω(σ) MVolσ(Pσ,∗(z1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , Pσ,∗(zd)) and the summands in the right-hand side are simply mixed volumes of polytopes, Proper- ties (1) and (3) follow from the symmetry and normalization properties of mixed volumes in the polytope setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Moreover, once we prove that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='2) Pσ,∗(λz + z′) = λPσ,∗(z) + Pσ,∗(z′) for all z, z′ ∈ Cub(Σ, ∗) and λ ∈ R≥0, then Property (2) also follows from the multilinearity property of mixed volumes in the polytope setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Thus, it remains to prove (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='2), which we accomplish by proving both inclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' First, suppose that v ∈ Pσ,∗(λz + z′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='1, the vertices of Pσ,∗(λz + z′) are {wτ,∗(λz + z′) | τ ⪯ σ}, so we can write v as a convex combination: (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='3) v = � τ⪯σ aτ wτ,∗(λz + z′) for some aτ ∈ R≥0 with � τ⪯σ aτ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' To prove that v ∈ λPσ,∗(z) + Pσ,∗(z′), our next step is to prove that the vertices are linear: (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='4) wτ,∗(λz + z′) = λwτ,∗(z) + wτ,∗(z′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Since wτ,∗(λz + z′) is the unique vector in Nτ,R with wτ,∗(λz + z′) ∗ uρ = (λz + z′)ρ for all ρ ∈ τ(1), proving (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='4) amounts to proving that λwτ,∗(z) + wτ,∗(z′) also satisfies these equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Using bilinearity of the inner product and the definition of the w vectors, we have (λwτ,∗(z) + wτ,∗(z′)) ∗ uρ = λwτ,∗(z) ∗ uρ + wτ,∗(z′) ∗ uρ = λzρ + z′ ρ = (λz + z′)ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Therefore, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='4) holds, and substituting (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='4) into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='3) implies that v = λ � τ⪯σ aτwτ,∗(z) + � τ⪯σ aτwτ,∗(z′) ∈ λPσ,∗(z) + Pσ,∗(z′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' To prove the other inclusion, suppose that v ∈ λPσ,∗(z) + Pσ,∗(z′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Then v = λw + w′ for some w ∈ Pσ,∗(z) and w′ ∈ Pσ,∗(z′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' This means that w, w′ ∈ σ and, in addition, w · uρ ≤ zρ and w′ · uρ ≤ z′ ρ for all ρ ∈ σ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Since σ is a cone, u = λw + w′ ∈ σ and, for every ρ ∈ σ(1), we have v ∗ uρ = (λw + w′) ∗ uρ = λw ∗ uρ + w′ ∗ uρ ≤ λzρ + z′ ρ, 10 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' NOWAK, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' O’MELVENY, AND D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' ROSS from which we conclude that v ∈ Pσ,∗(λz + z′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Finally, to prove the final assertion of the proposition, suppose that F : Cub(Σ, ∗)d → R≥0 satisfies Properties (1) – (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Our goal is to prove that F(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zd) = MVolΣ,ω,∗(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zd) for any pseudocubical values z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zd ∈ Cub(Σ, ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Set z = λ1z1 + · · · + λdzd with λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , λd ∈ R≥0 arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Property (3) implies that F(z, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , z) = VolΣ,ω,∗(z) = MVolΣ,ω,∗(z, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Using Properties (1) and (2) we can expand both the left- and right-hand sides of this equation as polynomials in λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , λd: � k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=',kd � d k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , kd � F(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , z1 � �� � k1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zd, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zd � �� � kd )λk1 1 · · · λkd d = � k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=',kd � d k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , kd � MVolΣ,ω,∗(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , z1 � �� � k1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zd, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zd � �� � kd )λk1 1 · · · λkd d Equating the coefficients of λ1 · · · λd in these two polynomials leads to the desired conclusion: F(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zd) = MVolΣ,ω,∗(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' □ Our methods for studying Alexandrov–Fenchel inequalities will also require the following positivity result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Let Σ ⊆ NR be a simplicial d-fan, ∗ ∈ Inn(NR) an inner product, and ω : Σ(d) → R>0 a weight function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Then MVolΣ,ω,∗(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zd) ≥ 0 for all z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zd ∈ Cub(Σ, ∗) and MVolΣ,ω,∗(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zd) > 0 for all z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zd ∈ Cub(Σ, ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' The first statement follows from the definition of MVolΣ,ω,∗ and the nonnegativity of mixed volumes of polytopes [Sch14, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' For the second statement, we first observe that z ∈ Cub(Σ, ∗) implies that Pσ,∗(z) has dimension d for every σ ∈ Σ(d), which follows from the fact that Pσ,∗(z) is combinatorially equivalent to a d-cube [NR21, Propo- sition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Thus, the second statement follows from the fact that mixed volumes of full- dimensional polytopes are strictly positive [Sch14, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Mixed volumes and mixed degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' We now extend Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='2 to give a Chow- theoretic interpretation of mixed volumes of normal complexes associated to tropical fans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Let (Σ, ω) be a tropical d-fan, let ∗ ∈ Inn(NR) be an inner product, and let z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zd ∈ Cub(Σ, ∗) be pseudocubical values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Then MVolΣ,ω,∗(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zd) = degΣ,ω(D(z1) · · · D(zd)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' MIXED VOLUMES OF NORMAL COMPLEXES 11 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='1, it suffices to prove that the function Cub(Σ, ∗)d → R≥0 (z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zd) �→ degΣ,ω(D(z1) · · · D(zd)) is symmetric, multilinear, and normalized by VolΣ,ω,∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Symmetry follows from the fact that A•(Σ) is a commutative ring, multilinearity follows from the fact that degΣ,ω : Ad(Σ) → R is a linear map, and normalization is the content of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Faces of Normal Complexes In this section, we develop a face structure for normal complexes, analogous to the face structure of polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Parallel to the polytope case, we will see that each face is obtained by intersecting the normal complex with supporting hyperplanes, that each face can, itself, be viewed as a normal complex, and that a face of a face is, itself, a face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' We then prove fun- damental properties relating (mixed) volumes of normal complexes to the (mixed) volumes of their facets, which perfectly parallel central results in the classical polytope setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Orthogonal decompositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' The face construction for normal complexes makes heavy use of an orthogonal decomposition of NR associated to each cone τ ∈ Σ, which we now describe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Associated to each τ ∈ Σ, we have already met the subspace Nτ,R ⊆ NR, which is the linear span of τ, and we now introduce notation for the quotient space N τ R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='.= NR/Nτ,R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' With the inner product ∗, we may identify N τ R as the orthogonal complement of Nτ,R: N τ R = N ⊥ τ,R = {v ∈ NR | v ∗ u = 0 for all u ∈ Nτ,R} ⊆ NR, allowing us to decompose NR as an orthogonal sum NR = Nτ,R ⊕ N τ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' We denote the orthogonal projections onto the factors of this decomposition by prτ and prτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' As we will see below, given a normal complex CΣ,∗(z) and a cone τ ∈ Σ, we will associate a face Fτ(CΣ,∗(z)), and this face will lie in the space N τ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' In order to help the reader digest the construction of Fτ(CΣ,∗(z)) and its subsequent interpretation as a normal complex, we henceforth make the convention that τ superscripts will be used exclusively for objects associated to the vector space N τ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' For example, Στ will denote a fan in N τ R and ∗τ will denote an inner product on N τ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Faces of normal complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' There are two primary steps in the face construction for normal complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' The first step is completely analogous to the polytope setting: we intersect the normal complex with a collection of supporting hyperplanes to obtain a sub- complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' However, in order to view this resulting subcomplex as a normal complex itself, 12 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' NOWAK, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' O’MELVENY, AND D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' ROSS the second step of the construction requires us to translate this polytopal subcomplex to the origin, where we can then endow it with the structure of a normal complex inside N τ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Let Σ ⊆ NR be a simplicial d-fan, ∗ ∈ Inn(NR) an inner product, and z ∈ Cub(Σ, ∗) a pseudocubical value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' For each cone τ ∈ Σ, define the neighborhood of τ in Σ by NτΣ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='.= {π | π ⪯ σ for some σ ∈ Σ with τ ⪯ σ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' To illustrate this definition, we have darkened the neighborhood of the ray ρ in the following two-dimensional fan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' ρ Notice that NτΣ is, itself, a simplicial d-fan in NR whose cones are a subset of Σ, and the maximal cones of NτΣ comprise all of the maximal cones of Σ that contain τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Since every maximal cone σ ∈ NτΣ(d) contains τ as a face, it follows from the definitions that each hyperplane Hρ,∗(z) with ρ ∈ τ(1) is a supporting hyperplane of Pσ,∗(z): Pσ,∗(z) ⊆ H− ρ,∗(z) for all σ ∈ NτΣ(d) and ρ ∈ τ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Thus, for each σ ∈ NτΣ(d), we obtain a face of Pσ,∗(z) by intersecting with all of these hyperplanes: Fτ(Pσ,∗(z)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='.= Pσ,∗(z) ∩ � ρ∈τ(1) Hρ,∗(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' The collection of these polytopes along with all of their faces forms a polytopal subcomplex of CΣ,∗(z), which we denote Fτ(CΣ,∗(z)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='.= � σ∈NτΣ(d) � Fτ(Pσ,∗(z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' To illustrate how the polytopal subcomplex Fτ(CΣ,∗(z)) is constructed in a concrete exam- ple, the following image depicts a two-dimensional normal complex where we have darkened the collection of maximal polytopes associated to the neighborhood of a ray ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' We have also drawn in the hyperplane associated to ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' The intersection of the hyperplane and the darkened polytopes is Fρ(CΣ,∗(z)), which, in this example, is a polytopal complex comprised of three line segments meeting at the point wρ,∗(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' MIXED VOLUMES OF NORMAL COMPLEXES 13 Hρ,∗(z) ρ Fρ(CΣ,∗(z)) One might be tempted to call Fτ(CΣ,∗(z)) a “face” of CΣ,∗(z);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' however, a drawback would be that Fτ(CΣ,∗(z)) is not, itself, a normal complex (all normal complexes contain the origin, for example, while Fτ(CΣ,∗(z)) generally does not).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Thus, our construction involves one more step, which is to translate Fτ(CΣ,∗(z)) by the vector wτ,∗(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Notice that, tracking back through the definitions, there is an identification of affine subspaces � ρ∈τ(1) Hρ,∗(z) = N τ R + wτ,∗(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Since Fτ(CΣ,∗(z)) is, by definition, contained in the left-hand side, it follows that its trans- lation by −wτ,∗(z) is a polytopal complex in N τ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' We define the face of CΣ,∗(z) associated to τ ∈ Σ to be this polytopal complex: Fτ(CΣ,∗(z)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='.= Fτ(CΣ,∗(z)) − wτ,∗(z) ⊆ N τ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' The face associated to the ray ρ in our running example is depicted below inside N ρ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' N ρ R = Hρ,∗(z) − wρ,∗(z) ρ F ρ(CΣ,∗(z)) = Fρ(CΣ,∗(z)) − wρ,∗(z) The next pair of images depicts the subcomplex Fρ(CΣ,∗(z)) ⊆ CΣ,∗(z) and, after trans- lating to the origin, the face Fρ(CΣ,∗(z)), where ρ is a ray of a three-dimensional fan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' 14 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' NOWAK, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' O’MELVENY, AND D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' ROSS ρ ρ Fρ(CΣ,∗(z)) F ρ(CΣ,∗(z)) In the following subsections, it will also be useful to have notation for translates of the polytopes Fτ(Pσ,∗(z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' We define Fτ(Pσ,∗(z)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='.= Fτ(Pσ,∗(z)) − wτ,∗(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' In terms of these translated polytopes, we can write the τ-face of CΣ,∗(z) as Fτ(CΣ,∗(z)) = � σ∈NτΣ(d) � Fτ(Pσ,∗(z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Faces as normal complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Our aim in this subsection is to realize each face Fτ(CΣ,∗(z)) as a normal complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' In order to do so, we require several ingredients;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' namely, we require a marked, pure, simplicial fan Στ in N τ R, an inner product ∗τ on N τ R, and a pseudocubical value zτ ∈ Cub(Στ, ∗τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' We now define each of these ingredients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' For each cone τ ∈ Σ, define the star of Σ at τ ∈ Σ to be the fan in N τ R comprised of all projections of cones in the neighborhood of τ: Στ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='.= {prτ(π) | π ∈ NτΣ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' The star of a two-dimensional fan Σ at a ray ρ is depicted below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' In the image, there are three two-dimensional cones in the neighborhood of ρ that are projected onto three one-dimensional cones that comprise the maximal cones in the star fan Σρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' ρ Σ ⊆ NR Σρ ⊆ N ρ R Henceforth, we use the shorthand πτ = prτ(π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' MIXED VOLUMES OF NORMAL COMPLEXES 15 Given any cone πτ ∈ Στ with π ∈ NτΣ, we can also view πτ as the projection of the larger cone σ = π ∪ τ ∈ NτΣ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Note that σ is the unique maximal cone in NτΣ that projects onto πτ, from which it follows that each cone in Στ is the projection of a distinguished cone in NτΣ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' In other words, there is a bijection {σ ∈ NτΣ | τ ⪯ σ} → Στ σ �→ στ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' From the assumptions that Σ is a simplicial d-fan, it follow that Στ is a simplicial fan in N τ R that is pure of dimension dτ = d − dim(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Moreover, the simplicial hypothesis on Σ implies that each ray η ∈ Στ(1) is the projection of a unique ray ˆη ∈ NτΣ(1), and we can use this to mark each ray η ∈ Στ(1) with the vector prτ(uˆη).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' We now have a marked, pure, simplicial fan in N τ R, so it remains to define an inner product and pseudocubical value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' The inner product ∗τ ∈ Inn(N τ R) is simply defined as the restriction of the inner product ∗ ∈ Inn(NR) to the subspace N τ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Lastly, given any z ∈ RΣ(1), we define zτ ∈ RΣτ(1) by the rule zτ η = zˆη − wτ,∗(z) ∗ uˆη, where, as before, ˆη ∈ NτΣ(1) is the unique ray with prτ(ˆη) = η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' We now have all the ingredients necessary to state and prove the following result, which asserts that faces of normal complexes are, themselves, normal complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Let Σ ⊆ NR be a simplicial d-fan, ∗ ∈ Inn(NR) an inner product, and τ ∈ Σ a cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' If z ∈ RΣ(1) is (pseudo)cubical with respect to (Σ, ∗), then zτ is (pseudo)cubical with respect to (Στ, ∗τ) and Fτ(CΣ,∗(z)) = CΣτ,∗τ(zτ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' We note that the first statement—that zτ is (pseudo)cubical—is necessary in order for CΣτ,∗τ(zτ) to even be well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='1 is a statement about normal complexes, or equivalently, about the polytopes that comprise those complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' In order to prove Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='1, we first prove the following key lemma, which concerns just the vertices of the polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Let Σ ⊆ NR be a simplicial d-fan, ∗ ∈ Inn(NR) an inner product, and τ ∈ Σ a cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' For any σ ∈ Σ with τ ⪯ σ, we have prτ(wσ,∗(z)) = wσ,∗(z) − wτ,∗(z) = wστ,∗τ(zτ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' We start by establishing the first equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' To do so, we begin by arguing that wσ,∗(z) − wτ,∗(z) ∈ N τ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Since N τ R = N ⊥ τ,R, it suffices to prove that wσ,∗(z) − wτ,∗(z) is 16 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' NOWAK, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' O’MELVENY, AND D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' ROSS orthogonal to the basis {uρ | ρ ∈ τ(1)} ⊆ Nτ,R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' By definition of the w vectors and the assumption that τ ⪯ σ, we compute (wσ,∗(z) − wτ,∗(z)) ∗ uρ = zρ − zρ = 0 for all ρ ∈ τ(1), from which it follows that wσ,∗(z) − wτ,∗(z) ∈ N τ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Since NR = Nτ,R ⊕ N τ R, the orthogonal decomposition wσ,∗(z) = wτ,∗(z) + (wσ,∗(z) − wτ,∗(z)) then implies that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='3) prτ(wσ,∗(z)) = wτ,∗(z) and prτ(wσ,∗(z)) = wσ,∗(z) − wτ,∗(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' To prove that wσ,∗(z) − wτ,∗(z) = wστ,∗τ(zτ), we now argue that wσ,∗(z) − wτ,∗(z) is an element of Nστ,R and is a solution of the equations defining wστ,∗τ(zτ): (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='4) v ∗τ uη = zτ η for all η ∈ στ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' To check that wσ,∗(z) − wτ,∗(z) ∈ Nστ,R, we start by observing that we can write wσ,∗(z) = � ρ∈σ(1) aρ uρ for some values aρ ∈ R, in which case wσ,∗(z) − wτ,∗(z) = prτ(wσ,∗(z)) = � ρ∈σ(1)\\τ(1) aρ prτ(uρ) = � η∈στ(1) aˆη uη, where the first equality uses (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='3), the second uses that prτ vanishes on Nτ,R, and the third uses that the rays of στ are in natural bijection with σ(1) \\ τ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Lastly, we peel back the definitions to check that wσ,∗(z) − wτ,∗(z) is a solution of Equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='4): (wσ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='∗(z) − wτ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='∗(z)) ∗τ uη = (wσ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='∗(z) − wτ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='∗(z)) ∗ (uˆη − prτ(uˆη)) = wσ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='∗(z) ∗ uˆη − wτ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='∗(z) ∗ uˆη − � wσ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='∗(z) − wτ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='∗(z) � ∗ prτ(uˆη) = zˆη − wτ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='∗(z) ∗ uˆη = zτ η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' where the first equality uses the orthogonal decomposition of uˆη and the fact that ∗τ is just the restriction of ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' the second equality uses linearity of the inner product,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' and the third equality uses the definition of wσ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='∗(z) along with the fact that the vectors prτ(uˆη) and wσ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='∗(z) − wτ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='∗(z) = prτ(wσ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='∗(z)) are in orthogonal subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' □ MIXED VOLUMES OF NORMAL COMPLEXES 17 Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' To prove the first statement in the cubical setting, assume that z ∈ RΣ(1) is cubical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' This means that, for every σ ∈ Σ, we can write wσ,∗(z) = � ρ∈σ(1) aρuρ for some positive values aρ ∈ R>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Consider any cone of Στ, which we can write as στ with τ ⪯ σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Applying the lemma, we then see that wστ,∗τ(zτ) = prτ(wσ,∗(z)) = � ρ∈σ(1)\\τ(1) aρ prτ(uρ) = � η∈στ(1) aˆη uη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' This shows that wστ,∗τ(zτ) can be written as a positive combination of the ray generators of στ, proving that zτ ∈ Cub(Στ, ∗τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' The proof in the pseudocubical setting is identical but with “positive” replaced by “nonnegative.” To prove that Fτ(CΣ,∗(z)) = CΣτ,∗τ(zτ), it suffices to identify the maximal polytopes in these complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' In other words, we must prove that, for every σ ∈ NτΣ(d), we have (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='5) Fτ(Pσ,∗(z)) = Pστ,∗τ(zτ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' To prove (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='5), we analyze the vertices of these polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='1, the vertices of Pσ,∗(z) are {wπ,∗(z) | π ⪯ σ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Since Fτ(Pσ,∗(z)) = Pσ,∗(z) ∩ � ρ∈τ(1) Hρ,∗(z), it follows that the vertices of Fτ(Pσ,∗(z)) are {wπ,∗(z) | π ⪯ σ and wπ,∗(z) ∗ uρ = zρ for all ρ ∈ τ(1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' If a cone π ⪯ σ satisfies wπ,∗(z)∗uρ = zρ for all ρ ∈ τ(1), then the definition of the w-vectors implies that wπ,∗(z) = wπ∪τ,∗(z), and it follows that the vertices of Fτ(Pσ,∗(z)) are Vert � Fτ(Pσ,∗(z)) � = {wπ,∗(z) | τ ⪯ π ⪯ σ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Upon translating by wτ,∗(z) to get from Fτ(Pσ,∗(z)) to F τ(Pσ,∗(z)), we see that Vert � F τ(Pσ,∗(z)) � = {wπ,∗(z) − wτ,∗(z) | τ ⪯ π ⪯ σ} = {wπτ,∗τ(zτ) | πτ ⪯ στ} = Vert � Pστ,∗τ(zτ)) � , 18 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' NOWAK, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' O’MELVENY, AND D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' ROSS where the second equality is an application of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='2 and the third is an application of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Having matched the vertices of the polytopes in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='5), the equality of polytopes then follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' □ The importance of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='1 is that it allows us to endow each of the faces of a normal complex with the structure of a normal complex, and in particular, it then allows us to compute (mixed) volumes of faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' More specifically, if ω : Σ(d) → R>0 is a weight function, then we obtain a weight function ωτ : Στ(dτ) → R>0 defined by ωτ(στ) = ω(σ) for all σ ∈ Σ(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' The volume of the face Fτ(CΣ,∗(z)) weighted by ω is VolΣτ,ωτ,∗τ(zτ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Similarly, the mixed volume of the faces Fτ(CΣ,∗(z1)), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Fτ(CΣ,∗(zdτ)) weighted by ω is MVolΣτ,ωτ,∗τ(zτ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zτ dτ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' In the next two subsections, we use these concepts to prove fundamental results relating (mixed) volumes of normal complexes to the (mixed) volumes of their facets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' In making arguments using mixed volumes, it will be useful to consider facets of facets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' as such, the next result—asserting that the face of a face of a normal complex is a face of the original normal complex—will be useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Let Σ ⊆ NR be a simplicial d-fan, ∗ ∈ Inn(NR) an inner product, and z ∈ Cub(Σ, ∗) a pseudocubical value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' If τ, π ∈ Σ with τ ⪯ π, then Fπτ(Fτ(CΣ,∗(z))) = Fπ(CΣ,∗(z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='1, the claim in this proposition is equivalent to Fπτ(CΣτ,∗τ(zτ)) = CΣπ,∗π(zπ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' It suffices to match the maximal polytopes in these complexes, so we must prove: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='7) Fπτ(Pστ,∗τ(zτ)) = Pσπ,∗π(zπ) for all σ ∈ Σ(d) with τ ⪯ σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' The vertices of the polytope in the left-hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='7) are {wµτ,∗τ(zτ) − wπτ,∗τ(zτ) | πτ ⪯ µτ ⪯ στ} while the vertices in the right-hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='7) are {wµπ,∗π(zπ) | µπ ⪯ σπ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' MIXED VOLUMES OF NORMAL COMPLEXES 19 Notice that both sets of vertices are indexed by µ ∈ Σ with π ⪯ µ ⪯ σ, and we have wµτ,∗τ(zτ) − wπτ,∗τ(zτ) = prπτ(wµτ,∗τ(zτ)) = prπτ(prτ(wµ,∗(z))) = prπ(wµ,∗(z)) = wµπ,∗π(zπ), where the first, second, and fourth equalities are Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='2, while the second is the obser- vation that the projection prπ can be broken up into two steps: prπ = prπτ ◦ prτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Thus, the vertices of the polytopes in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='7) match up, and the proposition follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Volumes and facets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' This subsection is devoted to proving the following result, which relates the volume of a normal complex to the volumes of its facets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Let Σ ⊆ NR be a simplicial d-fan with weight function ω : Σ(d) → R>0, let ∗ ∈ Inn(NR) be an inner product, and let z ∈ Cub(Σ, ∗) be a pseudocubical value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Then VolΣ,ω,∗(z) = � ρ∈Σ(1) zρ VolΣρ,ωρ,∗ρ(zρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' The sum in the right-hand side of the theorem corresponds to decomposing the normal complex into pyramids over its facets, as depicted in the next image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='8 follows from the following lemma relating the volume function Volσ on Nσ,R to the volume function Volσρ on the hyperplane Nσρ,R ⊆ Nσ,R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Under the hypotheses of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='8, let σ ∈ Σ(d) and ρ ∈ σ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' For any polytope P ⊆ Nσρ,R and a ∈ R≥0, we have Volσ � conv(0, P + auρ) � = a(uρ ∗ uρ) · Volσρ(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' For intuition, we note that the polytope conv(0, P + auρ) appearing in the left-hand side of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='9 is obtained from the polytope P by first translating P along the ray ρ, which is orthogonal to Nσρ,R, then taking the convex hull with the origin, the result of which can be thought of as a pyramid with P as base and the origin as apex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' The right-hand side can 20 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' NOWAK, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' O’MELVENY, AND D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' ROSS then be thought of as a “base-times-height” formula for the volume of this pyramid, where the “height” of the vector auρ is a(uρ ∗ uρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' We now make this informal discussion precise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Let {vη | η ∈ σ(1)} ⊆ Mσ be the dual basis of {uη | η ∈ σ(1)} ⊆ Nσ, defined uniquely by the equations vη ∗ uµ = � � � 1 µ = η 0 µ ̸= η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Recall that each ray generator of σρ is of the form prρ(uη) for a unique η ∈ σ(1) \\ {ρ};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' we claim that the dual vector of prρ(uη) in Mσρ is vη—in other words, the dual vector of prρ(uη) is the same as the dual vector of uη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' To verify this, note that, for any η, µ ∈ σ(1) \\ {ρ}, we have prρ(uη) ∗ρ vµ = (uη − prρ(uη)) ∗ vµ = uη ∗ vµ = � � � 1 µ = η 0 µ ̸= η, where the first equality uses the decomposition of uη into its orthogonal components, along with the fact that ∗ρ is just the restriction of ∗, and the second equality uses that prρ(uη) is a multiple of uρ, along with uρ ∗ vµ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Using these dual bases, we defined simplices in each of vector spaces Nσ,R and Nσρ,R by ∆(σ) = conv(0, {vη | η ∈ σ(1)}) ⊆ Nσ,R and ∆(σρ) = conv(0, {vη | η ∈ σ(1) \\ {ρ}}) ⊆ Nσρ,R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' By our convention on how volumes are normalized in Nσ,R and Nσρ,R, along with our verifi- cation above that {vη | η ∈ σ(1) \\ {ρ}} is the dual basis of the ray generators of σρ, these simplices have unit volume: Volσ(∆(σ)) = Volσρ(∆(σρ)) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Notice that ∆(σρ) is a facet of ∆(σ) and we can write ∆(σ) = conv(vρ, ∆(σρ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' If we project the vertex vρ of ∆(σ) onto the line spanned by ρ, we obtain a new simplex ∆1(σ) = conv(prρ(vρ), ∆(σρ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Since the projection prρ is parallel to the facet ∆(σρ), it follows that Volσ(∆1(σ)) = Volσ(∆(σ)) = Volσρ(∆(σρ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' MIXED VOLUMES OF NORMAL COMPLEXES 21 Now define a new simplex by sliding the vertex prρ(vρ) along ρ to the new vertex auρ: ∆2(σ) = conv(auρ, ∆(σρ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' By the standard projection formula, we have prρ(vρ) = uρ uρ∗uρ, from which we see that ∆2(σ) is obtained from ∆1(σ) by scaling the height of the vertex prρ(vρ) by a factor of a(uρ ∗ uρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' It follows that the volume also scales by a(uρ ∗ uρ): Volσ(∆2(σ)) = a(uρ ∗ uρ) · Volσ(∆1(σ)) = a(uρ ∗ uρ) · Volσρ(∆(σρ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' More concisely, we have proved that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='10) Volσ � conv(auρ, P) � = a(uρ ∗ uρ) · Volσρ(P) when P = ∆(σρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' As a visual aid, we have depicted below the sequence of polytopes from the above discussion in the specific setting of a two-dimensional cone σ, which we have visualized in R2 with the usual dot product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' 0 σ η ρ uη uρ Nσρ,R vρ vη 0 ρ Nσρ,R vρ vη ∆(σ) ∆(σρ) 0 ρ Nσρ,R uρ uρ∗uρ vη ∆1(σ) ∆(σρ) 0 ρ Nσρ,R auρ vη ∆2(σ) ∆(σρ) We now extend (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='10) to any simplex P ⊆ Nσρ,R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' To do so, first note that a simplex P can be obtained from the specific simplex ∆(σρ) by a composition of a translation and a linear transformation on Nσρ,R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Translating P within Nσρ,R does not affect the volume on either 22 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' NOWAK, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' O’MELVENY, AND D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' ROSS side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Given a linear transformation T, on the other hand, we can extend it to a linear transformation �T on Nσ,R by simply fixing the vector uρ, in which case we have �T(conv(auρ, P)) = conv(auρ, T(P)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Since det( �T) = det(T) and linear transformations scale volumes by the absolute values of their determinants, we conclude that the equality in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='10) is preserved upon taking linear transforms of P: Volσ � conv(auρ, T(P)) � = Volσ � �T(conv(auρ, P)) � = | det( �T)| Volσ � conv(auρ, P) � = | det(T)| · a(uρ ∗ uρ) · Volσρ(P) = a(uρ ∗ uρ) · Volσρ(T(P)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Knowing that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='10) holds for simplices, we extend it to arbitrary polytopes P ⊆ Nσρ,R by triangulating P and applying (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='10) to each simplex in the triangulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' The lemma then follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='10) along with the observation that conv(auρ, P) is just a reflection of conv(0, P + auρ), so has the same volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' □ We now use Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='9 to prove Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' For each top-dimensional cone σ ∈ Σ(d) and ρ ∈ σ(1), consider the polytope face Fρ(Pσ,∗(z)) ⊆ Pσ,∗(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' By definition, we have Fρ(Pσ,∗(z)) = Fρ(Pσ,∗(z)) + wρ,∗(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Noting that wρ,∗(z) = zρ uρ∗uρuρ, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='9 computes the volume of the pyramid conv(0, Fρ(Pσ,∗(z))): (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='11) Volσ � conv(0, Fρ(Pσ,∗(z))) � = zρ Volσρ(Fρ(Pσ,∗(z)) = zρ Volσρ(Pσρ,∗ρ(zρ)), where the second equality is an application of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Next, note that we can decompose each polytope Pσ,∗(z) into pyramids over the faces Fρ(Pσ,∗(z)) with ρ ∈ σ(1), implying that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='12) Volσ(Pσ,∗(z)) = � ρ∈σ(1) Volσ � conv(0, Fρ(Pσ,∗(z)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' MIXED VOLUMES OF NORMAL COMPLEXES 23 We then compute: VolΣ,ω,∗(z) = � σ∈Σ(d) ω(σ) Volσ(Pσ,∗(z)) = � σ∈Σ(d) ω(σ) � ρ∈σ(1) Volσ � conv(0, Fρ(Pσ,∗(z)) � = � σ∈Σ(d) ω(σ) � ρ∈σ(1) zρ Volσρ(Pσρ,∗ρ(zρ)) = � ρ∈Σ(1) zρ � σρ∈Σρ(d−1) ωρ(σρ) Volσρ(Pσρ,∗ρ(zρ)) = � ρ∈Σ(1) zρ VolΣρ,ωρ,∗ρ(zρ), where the first equality is the definition of VolΣ,ω,∗(z), the second and third are (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='12) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='11), respectively, the fourth follows from the definition of ωρ and the fact that cones in Σρ(d − 1) are in bijection with the cones in Σ(d) containing ρ via σρ ↔ σ, and the fifth is the definition of VolΣρ,ωρ,∗ρ(zρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Mixed volumes and facets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' The aim of this subsection is to enhance Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='8 to the following more general statement about mixed volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' See [Sch14, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='5] for the analogous result in the classical setting of strongly isomorphic polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Let Σ ⊆ NR be a simplicial d-fan with weight function ω : Σ(d) → R>0, let ∗ ∈ Inn(NR) be an inner product, and let z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zd ∈ Cub(Σ, ∗) be pseudocubical values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Then MVolΣ,ω,∗(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zd) = � ρ∈Σ(1) z1,ρ MVolΣρ,ωρ,∗ρ(zρ 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zρ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' We proceed by induction on d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' If d = 1, then mixed volumes are just volumes, in which case Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='13 is a special case of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Assume, now, that Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='13 holds in dimension less than d > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Define F(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zd) = � ρ∈Σ(1) z1,ρ MVolΣρ,ωρ,∗ρ(zρ 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zρ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' To prove that F = MVolΣ,∗,ω, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='1 tells us that it suffices to prove that F is (1) symmetric, (2) multilinear, and (3) normalized correctly with respect to volume;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' we check these properties in reverse order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' 24 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' NOWAK, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' O’MELVENY, AND D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' ROSS To check (3), we note that F(z, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , z) = � ρ∈Σ(1) zρ MVolΣρ,ωρ,∗ρ(zρ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zρ) = � ρ∈Σ(1) zρ VolΣρ,ωρ,∗ρ(zρ) = VolΣ,ω,∗(z), where the first equality is the definition of F, the second is Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='1 Part (3), and the third is Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' To check (2), there are two cases to consider: linearity in the first coordinate and linearity in every other coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Linearity in the first coordinate follows quickly from the definition of F, while linearity in every other coordinate follows from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='1 Part (2) applied to (Σρ, ∗ρ, ωρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Finally, to check (1), we first note that Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='1 Part (1) applied to (Σρ, ∗ρ, ωρ) implies that F is symmetric in the entries z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Thus, it remains to prove that F is invariant under transposing z1 and z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' To do so, we first apply the induction hypothesis to the mixed volumes appearing in the definition of F to obtain (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='14) F(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zd) = � ρ∈Σ(1) z1,ρ � ηρ∈Σρ(1) zρ 2,ηρ MVolΣρ,η,ωρ,η,∗ρ,η(zρ,η 3 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zρ,η d ), where, to avoid the proliferation of parentheses and superscripts, we have written, for exam- ple, Σρ,η as short-hand for (Σρ)ηρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Notice that the mixed volumes appearing in the right-hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='14) are mixed volumes associated to faces of faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='6 tells us that the ηρ-face of the ρ-face of a normal complex is the same as the τ face of the original normal complex, where τ ∈ Σ(2) is the 2-cone containing ρ and η as rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Therefore, MVolΣρ,η,ωρ,η,∗ρ,η(zρ,η 3 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zρ,η d ) = MVolΣτ,ωτ,∗τ(zτ 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zτ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Keeping in mind that each 2-cone τ appears twice in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='14), once for each ordering of the rays, we have F(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zd) = � τ∈Σ(2) τ(1)={ρ,η} (z1,ρzρ 2,ηρ + z1,ηzη 2,ρη) MVolΣτ,ωτ,∗τ(zτ 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zτ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Therefore, it remains to prove that z1,ρzρ 2,ηρ + z1,ηzη 2,ρη is invariant under transposing 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Computing directly from the definition of zρ, we have z1,ρzρ 2,ηρ + z1,ηzη 2,ρη = z1,ρ � z2,η − wρ,∗(z2) ∗ uη � + z1,η � z2,ρ − wη,∗(z2) ∗ uρ � , from which we see that it suffices to prove that both z1,ρwρ,∗(z2) ∗ uη and z1,ηwη,∗(z2) ∗ uρ MIXED VOLUMES OF NORMAL COMPLEXES 25 are invariant under transposing 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' This invariance follows from the computations wρ,∗(z2) = z2,ρ uρ ∗ uρ uρ and wη,∗(z2) = z2,η uη ∗ uη uη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' □ The following analytic consequence of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='13 will be useful in our computations in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' In addition to the hypotheses of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='13, assume that Cub(Σ, ∗) is nonempty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Then for any fixed z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zk ∈ Cub(Σ, ∗), we have ∂ ∂zρ MVolΣ,ω,∗(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zk, z, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , z � �� � d−k ) = (d − k) MVolΣρ,ωρ,∗ρ(zρ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zρ k, zρ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zρ � �� � d−k−1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' The assumption that Cub(Σ, ∗) ̸= ∅ implies that MVolΣ,ω,∗(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zk, z, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , z) is a degree d − k polynomial in R[zρ | ρ ∈ Σ(1)], so the derivatives are well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Proposi- tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='13 and symmetry of mixed volumes imply that ∂ ∂zi,ρ MVolΣ,ω,∗(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zd) = MVolΣρ,ωρ,∗ρ(zρ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zρ i−1, zρ i+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zρ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Viewing MVolΣ,ω,∗(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zk, z, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , z) as the composition of MVolΣ,ω,∗(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zd) with the specialization zk+1 = · · · = zd = z, the result then follows from the multivariable chain rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Alexandrov–Fenchel inequalities One of the most consequential properties of mixed volumes of polytopes (or, more gener- ally, of mixed volumes of convex bodies) is the Alexandrov–Fenchel inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Given polytopes P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , Pd in a d-dimensional real vector space V with volume function Vol, the Alexandrov–Fenchel inequalities state that MVol(P1, P2, P3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , Pd)2 ≥ MVol(P1, P1, P3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , Pd) MVol(P2, P2, P3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , Pd) (see, for example, [Sch14, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='1] for a proof and historical references).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' It is our aim in this section to study Alexandrov–Fenchel inequalities in the setting of mixed volumes of normal complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Let Σ ⊆ NR be a simplicial d-fan, ω : Σ(d) → R>0 a weight function, and ∗ ∈ Inn(NR) an inner product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' We say that the triple (Σ, ω, ∗) is Alexandrov–Fenchel, or just AF for short, if Cub(Σ, ∗) ̸= ∅ and MVolΣ,ω,∗(z1, z2, z3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zd)2 ≥ MVolΣ,ω,∗(z1, z1, z3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zd) MVolΣ,ω,∗(z2, z2, z3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zd) for all z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zd ∈ Cub(Σ, ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' In this section, we prove the following result, which provides sufficient conditions for proving that a triple (Σ, ω, ∗) is AF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' 26 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' NOWAK, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' O’MELVENY, AND D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' ROSS Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Let Σ ⊆ NR be a simplicial d-fan, ω : Σ(d) → R>0 a weight function, and ∗ ∈ Inn(NR) an inner product such that Cub(Σ, ∗) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' The triple (Σ, ω, ∗) is AF if the following two conditions are satisfied: (i) Στ \\ {0} is connected for any cone τ ∈ Σ(k) with k ≤ d − 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' (ii) Hess � VolΣτ,ωτ,∗τ(z) � has exactly one positive eigenvalue for any τ ∈ Σ(d − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Condition (i) in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='1 can be thought of as requiring that the fan Σ does not have any “pinch” points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' For example, in dimension four, this condition rules out fans that locally look like a pair of four-dimensional cones meeting along a ray, because the star fan associated to that ray would comprise two three-dimensional cones that meet only at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Condition (ii) of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='1 concerns only the two-dimensional stars of Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Since the volume polynomial of a two-dimensional fan is a quadratic form, the Hessians appearing in Condition (ii) are constant matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Condition (ii) can be viewed as an ana- logue of the Brunn–Minkowski inequality for polygons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' For an example of a two-dimensional (tropical) fan that does not satisfy Condition (ii), see [BH17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Our proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='1 is largely inspired by a proof of the classical Alexandrov–Fenchel inequalities recently developed by Cordero-Erausquin, Klartag, Merigot, and Santambrogio [CEKMS19]—for which the key geometric input is Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' While the arguments in [CEKMS19] can be employed in this setting more-or-less verbatim, we present a more streamlined proof using ideas regarding Lorentzian polynomials recently developed by Br¨and´en and Leake [BL21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Before presenting a proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='1, we pause to introduce key ideas regarding Lorentzian polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Lorentzian polynomials on cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' One way to view the AF inequalities is as the non- positivity of the 2 × 2 matrix � MVolΣ,ω,∗(z1, z1, z3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zd) MVolΣ,ω,∗(z1, z2, z3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zd) MVolΣ,ω,∗(z2, z1, z3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zd) MVolΣ,ω,∗(z2, z2, z3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zd) � , and this nonpositivity is equivalent to the matrix having exactly one positive eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Lorentzian polynomials are a clever tool for capturing the essence of this observation, and are therefore a natural setting for understanding AF-type inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Our discussion of Lorentzian polynomials follows Br¨and´en and Leake [BL21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Suppose that C ⊆ Rn >0 is a nonempty open convex cone, and let f ∈ R[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , xn] be a homogeneous polynomial of degree d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' For each i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , n and v = (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , vn) ∈ Rn, we use the following MIXED VOLUMES OF NORMAL COMPLEXES 27 shorthand for partial and directional derivatives ∂i = ∂ ∂xi and ∂v = n � i=1 vi∂i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' We say that f is C-Lorentzian if, for all v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , vd ∈ C, (P) ∂v1 · · · ∂vdf > 0, and (H) Hess(∂v3 · · · ∂vdf) has exactly one positive eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' To relate Lorentzian polynomials back to AF-type inequalities, we recall the following key observation (see [BH20, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Let C ⊆ Rn >0 be a nonempty open convex cone, and let f ∈ R[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , xn] be C-Lorentzian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Then for all v1, v2, v3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , vd ∈ C, we have � ∂v1∂v2∂v3 · · · ∂vdf �2 ≥ � ∂v1∂v1∂v3 · · · ∂vdf �� ∂v2∂v2∂v3 · · · ∂vdf � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Consider the symmetric 2 × 2 matrix M = � ∂v1∂v1∂v3 · · · ∂vdf ∂v1∂v2∂v3 · · · ∂vdf ∂v2∂v1∂v3 · · · ∂vdf ∂v2∂v2∂v3 · · · ∂vdf � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' By (P), the entries of M are positive, so the Peron–Frobenius Theorem implies that M has at least one positive eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' On the other hand, M is a principal minor of Hess(∂v3 · · · ∂vdf), which, by (H), has exactly one positive eigenvalue;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' thus, it follows from Cauchy’s Interlacing Theorem that M has at most one positive eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Therefore M has exactly one positive eigenvalue, implying that the determinant of M is nonpositive, proving the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' □ The following result, proved by Br¨and´en and Leake [BL21], is particularly useful for the study of Lorentzian polynomials on cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' We view this result as an effective implementation of the key insights in [CEKMS19];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' in essence, it eliminates the need for one of the induction parameters in [CEKMS19] because that induction parameter is captured within the recursive nature of Lorentzian polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='5 ([BL21], Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Let C ⊆ Rn >0 be a nonempty open convex cone, and let f ∈ R[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , xn] be a homogeneous polynomial of degree d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' If (1) ∂v1 · · · ∂vdf > 0 for all v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , vd ∈ C, (2) Hess � ∂v1 · · · ∂vd−2f � is irreducible1 and has nonnegative off-diagonal entries for all v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , vd−2 ∈ C, and (3) ∂if is C-Lorentzian for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , n, then f is C-Lorentzian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' 1An n×n matrix M is irreducible if the associated adjacency graph—the undirected graph on n labeled vertices with an edge between the ith and jth vertex whenever the (i, j) entry of M is nonzero—is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' 28 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' NOWAK, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' O’MELVENY, AND D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' ROSS 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Lorentzian volume polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' We now discuss how the above discussion of Lorentzian polynomials on cones can be used to study mixed volumes of normal complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Let Σ ⊆ NR be a simplicial d-fan, ω : Σ(d) → R>0 a weight function, and ∗ ∈ Inn(NR) an inner product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' We assume that Cub(Σ, ∗) ̸= ∅, in which case the function VolΣ,ω,∗ : Cub(Σ, ∗) → R is a homogeneous polynomial of degree d in R[zρ | ρ ∈ Σ(1)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='1(3), we have VolΣ,ω,∗(z) = MVolΣ,ω,∗(z, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' It then follows from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='1(1) and (2) (and the chain rule) that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='6) ∂z1 · · · ∂zk VolΣ,ω,∗(z) = d!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' d − k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' MVolΣ,ω,∗(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zk, z, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , z � �� � d−k ) for any z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zk ∈ Cub(Σ, ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' In particular, in order to prove that (Σ, ω, ∗) is AF, we now see that it suffices (by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='4) to prove that VolΣ,ω,∗ is Cub(Σ, ∗)-Lorentzian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Thus, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='1 is a consequence of the following stronger result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Let Σ ⊆ NR be a simplicial d-fan, ω : Σ(d) → R>0 a weight function, and ∗ ∈ Inn(NR) an inner product such that Cub(Σ, ∗) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Then VolΣ,ω,∗ is Cub(Σ, ∗)- Lorentzian if the following two conditions are satisfied: (i) Στ \\ {0} is connected for any cone τ ∈ Σ(k) with k ≤ d − 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' (ii) Hess � VolΣτ,ωτ,∗τ(z) � has exactly one positive eigenvalue for any τ ∈ Σ(d − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' We prove Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='7 by induction on d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' First consider the base case d = 2 (in which case Condition (i) is vacuous).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Note that VolΣ,ω,∗ satisfies (P) by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='6) and the positivity of mixed volumes (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='5), while (H) for VolΣ,ω,∗ is equivalent to Condition (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Therefore, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='7 holds when d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Now let d > 2 and assume (Σ, ω, ∗) satisfies Conditions (i) and (ii) in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' To prove that VolΣ,ω,∗ is Cub(Σ, ∗)-Lorentzian, we use Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Translating the three conditions of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='5 using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='6), we must prove that (1) MVolΣ,ω,∗(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zd) > 0 for all z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zd ∈ Cub(Σ, ∗), (2) Hess � MVolΣ,ω,∗(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zd−2, z, z) � is irreducible and has nonnegative off-diagonal en- tries for all z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zd−2 ∈ Cub(Σ, ∗), and (3) ∂ρ VolΣ,ω,∗(z) is Cub(Σ, ∗)-Lorentzian for all ρ ∈ Σ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Note that (1) is just the positivity of mixed volumes (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' To prove (3), note that Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='1(3) and Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='15 (with k = 0) together imply that ∂ρ VolΣ,ω,∗(z) = d VolΣρ,ωρ,∗ρ(zρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Applying the induction hypothesis to (Σρ, ωρ, ∗ρ)—which we can do because any star fan of Σρ is a star fan of Σ, so our assumption that (Σ, ω, ∗) satisfies the two conditions of MIXED VOLUMES OF NORMAL COMPLEXES 29 Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='7 implies that (Σρ, ωρ, ∗ρ) also satisfies the two conditions of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='7— implies that ∂ρ VolΣ,ω,∗(z) is Lorentzian, verifying (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Finally, to prove (2), we use Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='15 to compute ∂ρ MVolΣ,ω,∗(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zd−2, z, z) = 2 MVolΣ,ω,∗(zρ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zρ d−2, zρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' If τ ∈ Σ(2) with rays ρ and η, then zρ ηρ = zη − wρ,∗(z) ∗ uη = zη − uρ ∗ uη uρ ∗ uρ zρ, from which it follows that, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='8) ∂η∂ρ MVolΣ,ω,∗(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zd−2, z, z) = 2 MVolΣτ,ωτ,∗τ(zτ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zτ d−2) On the other hand, if ρ and η do not lie on a common cone τ ∈ Σ(2), then (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='9) ∂η∂ρ MVolΣ,ω,∗(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zd−2, z, z) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' The positivity of mixed volumes for cubical values, along with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='8) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='9), then implies that Hess � MVolΣ,ω,∗(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zd−2, z, z) � has nonnegative off-diagonal entries that are positive whenever the row and column index are the rays of a cone τ ∈ Σ(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' The first condition in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='7 implies that we can travel from any ray of Σ to any other ray by passing only through the relative interiors of one- and two-dimensional cones, which then implies that Hess � MVolΣ,ω,∗(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zd−2, z, z) � is irreducible, concluding the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Application: the Heron–Rota–Welsh conjecture As an application of our developments regarding mixed volumes of normal complexes, we show in this section how Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='1 can be used to prove the Heron–Rota–Welsh conjecture, which states that the coefficients of the characteristic polynomial of any matroid are log-concave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' The bridge between matroids and mixed volumes is the Bergman fan;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' we begin this section by briefly recalling relevant notions regarding matroids and Bergman fans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Matroids and Bergman fans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' A (loopless) matroid M = (E, L) consists of a finite set E, called the ground set, and a collection of subsets L ⊆ 2E, called flats, which satisfy the following three conditions: (F1) ∅ ∈ L, (F2) if F1, F2 ∈ L, then F1 ∩ F2 ∈ L, and (F3) if F ∈ L, then every element of E \\ F is contained in exactly one flat that is minimal among the flats that strictly contain F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' 30 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' NOWAK, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' O’MELVENY, AND D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' ROSS We do not give a comprehensive overview of matroids;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' rather, we settle for a brief intro- duction of key concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' For a more complete treatment, see Oxley’s book [Oxl11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' The closure of a set S ⊆ E, denoted cl(S), is the smallest flat containing S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' A set I ⊆ E is called independent if cl(I1) ⊊ cl(I2) for any I1 ⊊ I2 ⊆ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' The rank of a set S ⊆ E, denoted rk(S), is the maximum size of an independent subset of S, and the rank of M, denoted rk(M) is defined to be the rank of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' While we have chosen to characterize matroids in terms of their flats, we note that matroids can also be characterized in terms of their independent sets or their rank function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' A flag of flats (of length k) in M is a chain of the form F = (F1 ⊊ · · · ⊊ Fk) with F1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , Fk ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' It can be checked from the matroid axioms that every maximal flag has one flat of each rank 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , rk(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' We let ∆M denote the set of flags of flats, which naturally has the structure of a simplicial complex of dimension rk(M) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Since every maximal flag contains ∅ and E, we often restrict our attention to studying proper flats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' We use the notation L∗ = L \\ {∅, E} for the set of proper flats and ∆∗ M for the set of flags of proper flats, which is a simplicial complex of dimension rk(M) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Given a matroid M, consider the vector space RE with basis {ve | e ∈ E}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' For each subset S ⊆ E, define vS = � e∈S ve ∈ RE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Set NR = RE/RvE and denote the image of vS in the quotient space NR by uS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' For each flag F = (F1 ⊊ · · · ⊊ Fk) ∈ ∆∗ M, define a polyhedral cone σF = R≥0{uF1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , uFk} ⊆ NR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' The Bergman fan of M, denoted ΣM, is the polyhedral fan ΣM = {σF | F ∈ ∆∗ M}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Note that ΣM is simplicial, pure of dimension d = rk(M) − 1, and marked by the vectors uF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Consider a cone σF ∈ ΣM(d − 1) corresponding to a flag F = (F1 ⊊ · · · ⊊ Fk−1 ⊊ Fk+1 ⊊ · · · ⊊ Fd) with rk(Fi) = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' The d-cones containing σF are indexed by flats F with Fk−1 ⊊ F ⊊ Fk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' If there are ℓ such flats, then (F3) implies that � F ∈L Fk−1⊊F ⊊Fk+1 uF = (ℓ − 1)uFk−1 + uFk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' MIXED VOLUMES OF NORMAL COMPLEXES 31 Since the right-hand side lies in NσF,R, this observation implies that ΣM is balanced (tropical with weights all equal to 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' In order to check that Bergman fans are AF, we require a working understanding of the star fans of Bergman fans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Consider a cone σF associated to a flat F = (F1 ⊊ · · · ⊊ Fk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Set F0 = ∅ and Fk+1 = E, and for each j = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , k consider the matroid minor M[Fj, Fj+1], which is the matroid on ground set Fj+1 \\ Fj with flats of the form F \\ Fj where F is a flat of M satisfying Fj ⊆ F ⊆ Fj+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Notice that the star fan ΣσF M lives in the quotient space N σF R = NR R{uF1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , uFk} = RE R{vF1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , vFk+1} = k � j=0 RFk+1\\Fk RvFk+1\\Fk , and one checks that this natural isomorphism of vector spaces identifies the star of ΣM at σF as the product of the Bergman fans of the associated matroid minors: (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='1) ΣσF M = k � j=0 ΣM[Fj,Fj+1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Bergman fans are AF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' We are now ready to use Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='1 to prove that Bergman fans of matroids are AF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Let M be a matroid of rank d+1 and let ΣM ⊆ NR be the associated Bergman fan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' If ∗ ∈ Inn(NR) is any inner product with Cub(ΣM, ∗) ̸= ∅, then (ΣM, ∗) is AF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' We are assuming the weight function ω is equal to 1 because, as noted in the previous subsection, ΣM is balanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Thus, we omit ω from the notation in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' To prove Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='2, we verify the two conditions of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' We accomplish this through the following three lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' The first lemma verifies that Bergman fans satisfy (a slight strengthening of) Condition (i) of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' ΣσF M \\ {0} is connected for any cone σF ∈ ΣM(k) with k ≤ d − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' We begin by arguing that ΣM \\{0} is connected for any matroid of rank at least 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' It suffices to prove that, for any two rays ρF, ρF ′ ∈ ΣM(1) associated to flats F, F ′ ∈ L∗, there are sequences ρ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , ρℓ ∈ ΣM(1) and τ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , τℓ+1 ∈ ΣM(2) such that ρF ≺ τ1 ≻ ρ1 ≺ · · · ≻ ρℓ ≺ τℓ+1 ≻ ρF ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' If F ∩ F ′ = G ̸= ∅, then G ∈ L∗ by (F2) and the following is such a sequence ρF ≺ τG⊊F ≻ ρG ≺ τG⊊F ′ ≻ ρF ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' If, on the other hand, F ∩ F ′ = ∅, choose rank-one flats G ⊆ F and G′ ⊆ F ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' By (F3), there is exactly one rank-two flat H that contains G and G′, so we can construct a sequence (ρF ≺ τG⊊F ≻)ρG ≺ τG⊊H ≻ ρH ≺ τG′⊊H ≻ ρG′(≺ τG′⊊F ′ ≻ ρF ′), 32 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' NOWAK, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' O’MELVENY, AND D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' ROSS where the parenthetical pieces should be omitted if G = F or G′ = F ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Now consider any star fan ΣσF M where F = (F1 ⊊ · · · ⊊ Fk) with k ≤ d − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Notice that such a star fan has dimension at least two, and we can write it as a product of Bergman fans on matroid minors ΣσF M = k � j=0 ΣM[Fj,Fj+1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Consider two rays ρ, ρ′ ∈ ΣσF M (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' If the two rays happen to come from different factors in the product, then we can connect them through the sequence ρ ≺ ρ × ρ′ ≻ ρ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' If, on the other hand, they lie in the same factor, there are two cases to consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' If the matroid minor of the factor that the rays lie in has rank at least 3, then the rays can be connected via the argument above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' If, on the other hand, the matroid minor has rank 2, then one of the other matroid minors must also have rank at least 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Choosing any ray ρ′′ in the Bergman fan of the second matroid minor, we can connect ρ and ρ′ through the sequence ρ ≺ ρ × ρ′′ ≻ ρ′′ ≺ ρ′ × ρ′′ ≻ ρ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' □ In order to verify Condition (ii) of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='1, there are two cases to consider, depending on whether the two-dimensional star fan in question is, itself, a Bergman fan, or whether it is the product of two one-dimensional Bergman fans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' In both cases, we use the fact that, in order to prove that the Hessian of a quadratic form f ∈ R[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , xn] has exactly one eigenvalue, it suffices (by Sylvester’s Law of Inertia) to find an invertible change of variables y1(x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , yn(x) such that f = n � i=1 aiyi(x)2 with exactly one positive ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' We now consider the two cases in the following two lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' If M is a rank-three matroid, then the Hessian of degΣM(D(z)2) has exactly one positive eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' For a flat F ∈ L∗, we use the shorthand XF = XρF and zF = zρF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' In order to compute degΣM(D(z)2), we must compute degΣM(XFXG) for any two flats F, G ∈ L∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' If F ⊊ G, then the degree is one, by definition of the degree function, and if F and G are incomparable, then the degree is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Thus, it remains to compute the degree of the squared terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Using the definition of A•(ΣM) and the flat axioms, the reader is encouraged to verify that degΣM(X2 F) = 1 − |{G ∈ L∗ | F ⊊ G}| if rk(F) = 1 and degΣM(X2 G) = −1 if rk(G) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' It MIXED VOLUMES OF NORMAL COMPLEXES 33 follows that degΣM(D(z)2) = 2 � F,G∈L∗ F ⊊G zFzG + � F ∈L∗ rk(F )=1 z2 F − � F,G∈L∗ F ⊊G z2 F − � G∈L∗ rk(G)=2 z2 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' By creatively organizing the terms, we can rewrite this as degΣM(D(z)2) = � � F ∈L∗ rk(F )=1 zF �2 − � G∈L∗ rk(G)=2 � zG − � F ∈L∗ F ⊊G zF �2 , where the only key matroid assertion used in the equivalence of these two formulas is that there exists a unique rank-two flat containing any two distinct rank-one flats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Sylvester’s Law of Inertia implies that the Hessian of this quadratic form has exactly one positive eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' □ Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' If M and M′ are rank-two matroids, then the Hessian of degΣM×ΣM′(D(z)2) has exactly one positive eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' By definition of A•(ΣM × ΣM′), the reader is encouraged to verify that degΣM×ΣM′(XρXη) = � � � 0 ρ, η ∈ ΣM(1) or ρ, η ∈ ΣM′(1), 1 ρ ∈ ΣM(1) and η ∈ ΣM′(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Therefore, degΣM×ΣM′(D(z)2) = � ρ∈ΣM(1), η∈ΣM′(1) 2zρzη, which can be rewritten as degΣM×ΣM′(D(z)2) = 1 2 � � ρ∈ΣM(1) zρ + � η∈ΣM′(1) zρ �2 − 1 2 � � ρ∈ΣM(1) zρ − � η∈ΣM′(1) zρ �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Sylvester’s Law of Inertia implies that the Hessian of this quadratic form has exactly one positive eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' □ We now have all the ingredients we need to prove Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' We prove that Bergman fans satisfy the two conditions of Theo- rem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' That Bergman fans satisfy Condition (i) is the content of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' To prove Condition (ii), we first note that, since Bergman fans are balanced, their star fans are also balanced, so Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='2 implies that the volume polynomials in Condition (ii) are inde- pendent of ∗ and are equal to degΣσF M (D(z)2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' By the product decomposition of star fans given in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='1), ΣσF M is either a two-dimensional Bergman fan or a product of two one-dimensional Bergman fans;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' in the former case, the 34 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' NOWAK, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' O’MELVENY, AND D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' ROSS Hessian of the volume polynomial has exactly one positive eigenvalue by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='5, and in the latter case, by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Revisiting the Heron–Rota–Welsh Conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' The characteristic polynomial of a matroid M = (E, L) can be defined by χM(λ) = � S⊆E (−1)|S|λrk(M)−rk(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' It can be checked that χM(λ) has a root at λ = 1 for any positive-rank matroid, and the reduced characteristic polynomial is defined by χM(λ) = χM(λ) λ − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' We use the notation µa(M) and µa(M) for the (unsigned) coefficients of these polynomials: χM(λ) = rk(M) � a=0 (−1)aµa(M)λrk(M)−a and χM(λ) = rk(M)−1 � a=0 (−1)aµa(M)λrk(M)−1−a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' The Heron–Rota–Welsh Conjecture, developed in [Rot71, Her72, Wel76], asserts that the sequence of nonnegative integers µ0(M), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , µrk(M)(M) is unimodal and log-concave: 0 ≤ µ0(M) ≤ · · · ≤ µk(M) ≥ · · · ≥ µrk(M)(M) ≥ 0 for some k ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , rk(M)} and µk(M)2 ≥ µk−1(M)µk+1(M) for every k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , rk(M) − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' The Heron–Rota–Welsh Conjecture was first proved by Adiprasito, Huh, and Katz [AHK18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Our aim here is to show how this result also follows from the developments in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' It is elementary to check that the unimodality and log-concavity of the coefficients of the characteristic polynomial is implied by the analogous properties for the coefficients of the reduced characteristic polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' The bridge from characteristic polynomials to the content of this paper, then, is a result of Huh and Katz [HK12, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='2] (see also [AHK18, Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='5] and [DR22, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='11]), which asserts that µa(M) = degΣM(αd−aβa) where rk(M) = d + 1 and α, β ∈ A1(ΣM) are defined by α = � e0∈F XF and β = � e0 /∈F XF for some e0 ∈ E (these Chow classes are independent of the choice of e0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' MIXED VOLUMES OF NORMAL COMPLEXES 35 Choose any e0 ∈ E, and let ∗ ∈ Inn(NR) be the inner product with orthonormal basis {ue | e ̸= e0} ⊆ NR = RE/RuE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' For two flats F1, F2 ∈ L∗, we compute uF1 ∗ uF2 = � � � � � � � � � |F1 ∩ F2| e0 /∈ F1 and e0 /∈ F2, −|F1 ∩ F c 2| e0 /∈ F1 and e0 ∈ F2, |F c 1 ∩ F c 2| e0 ∈ F1 and e0 ∈ F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Define zα, zβ ∈ RΣM(1) = RL∗ by zα F = � � � 1 e0 ∈ F, 0 e0 /∈ F, and zβ F = � � � 1 e0 /∈ F, 0 e0 ∈ F, so that D(zα) = α and D(zβ) = β in A1(ΣM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' The following lemma allows us to connect characteristic polynomials to mixed volumes of normal complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' zα, zβ ∈ Cub(ΣM, ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' We must argue that wσ,∗(zα), wσ,∗(zβ) ∈ σ for every cone σ ∈ ΣM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Consider a flag F = (F1 ⊊ · · · ⊊ Fk) corresponding to a cone σF ∈ ΣM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' It suffices to prove that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='8) wσF,∗(zα) = � � � 1 |F c k|uFk e0 ∈ Fk 0 e0 /∈ Fk, and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='9) wσF,∗(zβ) = � � � 1 |F1|uF1 e0 /∈ F1 0 e0 ∈ F1, We verify (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='8);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' the verification (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='9) is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' To verify (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='8), first suppose that e0 ∈ Fk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Then for any j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , k, it follows from the definition of ∗ that uFk ∗ uFj = � � � |F c k| e0 ∈ Fj, 0 e0 /∈ Fj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Using this, we verify that 1 |F c k|uFk satisfies the defining equations of wσF,∗(zα): 1 |F c k|uFk ∗ uFj = zα Fj for all j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Now suppose that e0 /∈ Fk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Then e0 /∈ Fj for any j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , k, so zα Fj = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Thus, the defining equation for wσF,∗(zα) become wσF,∗(zα) ∗ uFj = 0 for all j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , k, showing that wσF,∗(zα) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' □ 36 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' NOWAK, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' O’MELVENY, AND D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' ROSS It follows from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='6 that the coefficients of the reduced characteristic polynomial have a volume-theoretic interpretation: µa(M) = MVolΣM,∗(zα, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zα � �� � d−a , zβ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zβ � �� � a ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' By [NR21, Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='4], we know that Cub(ΣM, ∗) ̸= ∅, and since the cubical cone is the interior of the pseudocubical cone, we may approximate zα, zβ ∈ Cub(ΣM, ∗) with zα t , zβ t ∈ Cub(ΣM, ∗) such that lim t→0 zα t = zα and lim t→0 zβ t = zβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Define µa t (M) = MVolΣM,∗(zα t , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zα t � �� � d−a , zβ t , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , zβ t � �� � a ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' By Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='2, we know that (ΣM, ∗) is AF, and the AF inequalities applied to the mixed volumes µa t (M) imply that the sequence µ0 t(M), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , µd t (M) is log-concave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Since mixed vol- umes of cubical values are positive (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='5), and since all log-concave sequences of positive values are unimodal, we see that the sequence µ0 t(M), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , µd t (M) is also unimodal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Since both unimodality and log-concavity are preserved under limits, we conclude that µ0(M), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' , µd(M) is unimodal and log-concave, verifying the Heron–Rota–Welsh Conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' References [ADH20] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Ardila, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Denham, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Huh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Lagrangian geometry of matroids.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content=' Department of Mathematics, University of Washington Email address: lnowak@uw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='edu Department of Mathematics, San Francisco State University Email address: pomelveny@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='sfsu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='edu Department of Mathematics, San Francisco State University Email address: rossd@sfsu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQf0A0W/content/2301.05278v1.pdf'} +page_content='edu' metadata={'source': 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a/6dE0T4oBgHgl3EQffAAW/content/tmp_files/2301.02397v1.pdf.txt b/6dE0T4oBgHgl3EQffAAW/content/tmp_files/2301.02397v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..2eb7ed1a7518b1f48cf9ec245514e192802f9dfa --- /dev/null +++ b/6dE0T4oBgHgl3EQffAAW/content/tmp_files/2301.02397v1.pdf.txt @@ -0,0 +1,3265 @@ +arXiv:2301.02397v1 [math.AP] 6 Jan 2023 +Fine boundary regularity for fully nonlinear mixed local-nonlocal +problems +MITESH MODASIYA AND ABHROJYOTI SEN +Abstract. We consider Dirichlet problems for fully nonlinear mixed local-nonlocal non-translation +invariant operators. For a bounded C2 domain Ω ⊂ Rd, let u ∈ C(Rd) be a viscosity solution of +such Dirichlet problem. We obtain global Lipschitz regularity and fine boundary regularity for u by +constructing appropriate sub and supersolutions coupled with a weak version of Harnack inequality. +We apply these results to obtain Hölder regularity of Du up to the boundary. +1. Introduction and main results +In this article, for a bounded C2 domain Ω ⊂ Rd we establish the boundary regularity of the +solution u to the in-equations +Lu + C0|Du| ≥ −K +in Ω, +Lu − C0|Du| ≤ K +in Ω, +u = 0 +in Ωc, +(1.1) +where C0, K ≥ 0 and L is a fully nonlinear integro-differential operator of the form +Lu(x) := L[x, u] = sup +θ∈Θ +inf +ν∈Γ +� +Tr aθν(x)D2u(x) + Iθν[x, u] +� +, +(1.2) +for some index sets Θ, Γ. The coefficient aθν : Ω → Rd×d is a matrix valued function and Iθν is a +nonlocal operator defined as +Iθνu(x) := Iθν[x, u] = +ˆ +Rd(u(x + y) − u(x) − 1B1(y)Du(x) · y)Nθν(x, y) dy. +(1.3) +The above in-equations (1.1) are motivated by Hamilton-Jacobi equations of the form +Iu(x) := sup +θ∈Θ +inf +ν∈Γ {Lθνu(x) + fθν(x)} = 0, +where +Lθνu(x) = Tr aθν(x)D2u(x) + Iθν[x, u] + bθν(x) · Du(x), +(1.4) +bθν(·) and fθν(·) are bounded functions on Ω. These linear operators (1.4) are extended generator +for a wide class of d-dimensional Feller processes (more precisely, jump diffusions) and the nonlinear +operator Iu(·) has its connection to the stochastic control problems and differential games (see [12,13] +and the references therein). The first term in (1.4) represents the diffusion, the second term represents +the jump part of a Feller process, and the third represents the drift. We refer to [1, 14, 15] and the +references therein for more on the connections between the operators of the form (1.4) and stochastic +differential equations. For a precise application of these type of operators in finance and biological +models, we refer to [20,23,24] and the references therein. +Department +of +Mathematics, +Indian +Institute +of +Science +Education +and +Research, +Dr. +Homi +Bhabha +Road, +Pune +411008, +India. +Email: +mitesh.modasiya@students.iiserpune.ac.in; +abhrojyoti.sen@acads.iiserpune.ac.in +2020 Mathematics Subject Classification. Primary: 35D40, 47G20, 35J60, 35B65 . +Key words and phrases. Operators of mixed order, viscosity solution, fine boundary regularity, fully nonlinear +integro-PDEs, Harnack inequality, gradient estimate. +1 + +2 +BOUNDARY REGULARITY +We set the following assumptions on the coefficient aθν(·) and the kernel Nθν(x, y), throughout +this article. +Assumption 1.1. +(a) aθν(·) are uniformly continuous and bounded in ¯Ω, uniformly in θ, ν for θ ∈ Θ, ν ∈ Γ. Further- +more, aθν(·) satisfies the uniform ellipticity condition λI ≤ aθν(·) ≤ ΛI for some 0 < λ ≤ Λ +where I denotes the d × d identity matrix. +(b) For each θ ∈ Θ, ν ∈ Γ, Nθν : Ω × Rd is a measurable function and for some α ∈ (0, 2) there +exists a kernel k that is measurable in Rd \{0} such that for any θ ∈ Θ, ν ∈ Γ, x ∈ Ω, we have +0 ≤ Nθν(x, y) ≤ k(y) +and +ˆ +Rd(1 ∧ |y|α)k(y)dy < +∞, +where we denote p ∧ q := min{p, q} for p, q ∈ R. +Let us comment briefly on Assumption 1.1. The uniform continuity of aθν(·) is required for the +stability of viscosity sub or supersolutions under appropriate limits and useful in Lemma A.1 which +is a key step for proving interior C1,γ regularity (cf. Lemma 2.1). The Assumption 1.1(b) includes a +large class of kernels. We mention some of them below. +Example 1.1. Consider the following kernels Nθν(x, y) : +(i) Nθν(x, y) = +1 +|y|d+σ for σ ∈ (0, 2). Clearly we can take k(y) = +1 +|y|d+σ and +´ +Rd(1 ∧ |y|α)k(y)dy is +finite for α ∈ [1 + σ/2, 2). +(ii) Nθν(x, y) = �∞ +i=1 +ai +|y|d+σi for σi ∈ (0, 2), σ0 = supi σi < 2 and �∞ +i=1 ai = 1. Similarly taking +Nθν(x, y) = k(y) we can see +´ +Rd(1 ∧ |y|α)k(y) < +∞ for α ∈ [1 + σ0/2, 2). +(iii) Nθν(x, y) = + + + +(1−log |y|)β +|y|d+σ +for 0 < |y| ≤ 1 +(1+log |y|)−β +|y|d+σ +for |y| ≥ 1, +where σ ∈ (0, 2). +(a) For 2(2 − σ) > β ≥ 0, taking Nθν(x, y) = k(y) we have +´ +Rd(1 ∧ |y|α)k(y)dy < +∞ for +α ∈ [1 + σ +2 + β +4 , 2). +(b) For −σ < β < 0, taking Nθν(x, y) = k(y) we have +´ +Rd(1 ∧ |y|α)k(y)dy < +∞ for +α ∈ [1 + σ +2 , 2). +Proof of (a): +ˆ +Rd(1 ∧ |y|α)k(y)dy = +ˆ +|y|≤1 +|y|α(1 − log |y|)β +|y|d+σ +dy + +ˆ +|y|>1 +(1 + log |y|)−β +|y|d+σ +dy := I1 + I2. +Using (1 − log |y|) ≤ +1 +√ +|y| + 1 and the convexity of ξ(t) = tp for p ≥ 1 we get +(1 − log |y|)β ≤ C +� +1 +|y|β/2 + 1 +� +. +Therefore +I1 ≤ +ˆ +|y|≤1 +Cdy +|y|β/2+d+σ−α + +ˆ +|y|≤1 +Cdy +|y|d+σ−α < +∞ for α ∈ [1 + σ/2 + β/4, 2), +and +I2 ≤ +ˆ +|y|>1 +dy +|y|d+σ < +∞. + +BOUNDARY REGULARITY +3 +Proof of (b): Since β < 0 in this case, we have (1−log |y|)β ≤ 1 and I1 < +∞ for α ∈ [1+ σ +2 , 2). +To estimate I2, observe (1 + log |y|)−β ≤ (1 + |y|)−β and +I2 ≤ C +ˆ +|y|>1 +(1 + |y|−β) +|y|d+σ +dy < +∞ since σ > −β. +(iv) Nθν(x, y) = +Ψ(1/|y|2) +|y|d+σ(x,y), where σ : Rd × Rd → R satisfying +0 < σ− := +inf +(x,y)∈Rd×Rd σ(x, y) ≤ +sup +(x,y)∈Rd×Rd σ(x, y) := σ+ < 2. +and Ψ is a Bernstein function (for several examples of such functions, see [50]) vanishing at +zero. Furthermore, Ψ is non-decreasing, concave and satisfies a weak upper scaling property +i.e, there exists µ ≥ 0 and c ∈ (0, 1] such that +Ψ(λx) ≤ cλµΨ(x) for x ≥ s0 > 0, λ ≥ 1. +For µ < 2(2 − σ+), we can take +k(y) = + + + +Ψ(1) +|y|d+2µ+σ+ , +if 0 < |y| ≤ 1, +Ψ(1) +|y|d+σ− , +if |y| > 1 +and +´ +Rd(1 ∧ |y|α)k(y)dy < +∞ for α ∈ [1 + µ + σ+/2, 2). +The main purpose of this article is to establish a global Lipschitz regularity and boundary regularity +of the solutions satisfying (1.1) under the Assumption 1.1. On the topic of regularity theory for +linear elliptic equations, Hölder estimate plays a key role and it can be obtained by using Harnack +inequality. +The pioneering contributions are by DeGiorgi-Nash-Moser [29, 42, 45] who proved Cα +regularity for solutions to the second order elliptic equations in divergence form with measurable +coefficients under the assumption of uniform ellipticity. For equations of non-divergence form, the +corresponding regularity theory was established by Krylov and Safonov [41]. +We refer [16] for a +comprehensive overview on the regularity theory for fully nonlinear elliptic equations. In [40], Krylov +studied the boundary regularity for local second order elliptic equations in non-divergence form with +bounded measurable coefficients. He obtained the Hölder regularity of u +δ up to the boundary where +δ denotes the distance function, i.e, δ(x) = dist(x, Ωc). +Turning our attention towards the nonlocal equations, first Hölder estimates and Harnack inequal- +ities for s-harmonic functions are proved by Bass and Kassmann [3–5], however their approach was +purely probabilistic. In the realm of analytic setup, Silvestre [52] proved Hölder continuity of u satis- +fying (1.3) with some structural assumptions on the operator and kernel related to the assumptions of +Bass and Kassmann. Analogous to the local case [40], in the nonlocal setting, for a bounded domain +Ω ⊂ Rd with C1,1 boundary the first result concerning boundary regularity of u solving the Dirichlet +problem for (−∆)s with bounded right hand side is obtained by Ros-Oton and Serra [46] where they +established a Hölder regularity of u/δs up to the boundary. This result is proved by using a method +of Krylov (see [33]). The idea is to obtain a bound for u with respect to a constant multiple of δs +and this controls the oscillation of u/δs near the boundary ∂Ω. The Hölder regularity of u/δs, (i) for +more general nonlocal linear operators with C1,α domain is established in [48], (ii) for smooth domain +with smooth right hand side is established in [30,31], (iii) for kernel with variable order see [34] and +(iv) for Dirichlet problem for fractional p-Laplacian, see [32]. +In a seminal paper, Caffarelli and Silvestre [17] studied the regularity theory for fully nonlinear +integro-differential equations of the form : supθ∈Θ infν∈Γ I[x, u] where I[x, u] is given by (1.3). By +obtaining a nonlocal ABP estimate, they established the Hölder regularity and Harnack inequality +when Nθν(y) (Nθν(y) denotes the x-independent form of Nθν(x, y)) is positive, symmetric and com- +parable with the kernel of the fractional Laplacian. From a large amount of literature that extend +the work of Caffarelli and Silvestre [17], we mention [36] where the authors considered integro-PDEs + +4 +BOUNDARY REGULARITY +with regularly varying kernel, [9,19,37] where regularity results are obtained for symmetric and non- +symmetric stable-like operators and [35] for kernels with variable order. Also a recent paper [38] +studies Hölder regularity and a scale invariant Harnack inequality under some weak scaling condition +on the kernel. Boundary regularity results for fully nonlinear integro-differential equations are ob- +tained by Ros-Oton and Serra in [47]. They considered a restricted class of kernels L∗ where Nθν(x, y) +is x-independent and of the following form +Nθν(y) := µ(y/|y|) +|y|d+2s +with µ ∈ L∞(Sd−1), +satisfying µ(θ) = µ(−θ) and λ ≤ µ ≤ Λ where 0 < λ ≤ Λ are the ellipticity constants. An interesting +feature of L∗ is +L(xd)s ++ = 0 in {xd > 0} for all L ∈ L∗ +which is useful to construct barriers in their case. Note that our operators do not enjoy such property +for having different orders. Furthermore, with Assumption 1.1 the nonlocal part (1.3) is not scale +invariant in our case, that is one may not find any 0 ≤ β ≤ 2 such that Iθν[x, u(r·)] = rβIθν[rx, u(·)] +for any 0 < r < 1. +Recently, the mathematical study of mixed local-nonlocal integro-differential equations have been +received a considerable attention, for instance see [2,6–8,10,26]. The regularity results and Harnack +inequality for mixed fractional p-Laplace equations are recently obtained in [27,28]. The interior Cα +regularity theory for HJBI-type integro-PDEs has been studied by Mou [43]. He obtained Hölder +regularity for viscosity solutions under uniform ellipticity condition and a slightly weaker condition +on kernels in compared to the Assumption 1.1 (b), that is +´ +Rd(1 ∧ |y|2)k(y)dy < +∞. More recently +global Lipschitz regularity (compare it with Biagi et. al. [7]) and fine boundary regularity have been +obtained for linear mixed local-nonlocal operators in [11]. Since the nonlocal operator applied on the +distance function becomes singular near the boundary for certain range of order of the kernel, one +of the main challenges was to construct appropriate sub and supersolutions and prove an oscillation +lemma following [46]. To do such analysis, along with several careful estimates, the authors borrowed +a Harnack inequality from [25]. Note that for fully nonlinear mixed operators of the form (1.2) no +such Harnack inequality is available in the literature. +In this current contribution, we continue the study started in [11] to obtain the boundary regularity +for fully nonlinear integro-differential problems of the form (1.1). Below, we present our first result +that is the Lipschitz regularity of u satisfying (1.1) up to the boundary. Note that (1.5) can be +achieved under some weaker assumptions on the domain and kernel. For this result, we only assume +∂Ω to be C1,1 and +´ +Rd(1 ∧ |y|2)k(y)dy < +∞. +Theorem 1.1. Let Ω be a bounded C1,1 domain in Rd and u be a continuous function which solves the +in-equations (1.1) in viscosity sense. Then u is in C0,1(Rd) and there exists a constant C, depending +only on d, Ω, λ, Λ, C0, +´ +Rd(1 ∧ |y|2)k(y)dy, such that +∥u∥C0,1(Rd) ≤ CK. +(1.5) +To prove Theorem 1.1, the first step is to show that the distance function δ(x) = dist(x, Ωc) can be +used as a barrier to u in Ω. Once this is done, we can complete the proof by considering different cases +depending on the distance between any two points in Ω or their distance from ∂Ω and combining +|u| ≤ Cδ with an interior C1,γ-estimate for scaled operators (cf. Lemma 2.1). +Next we show the fine boundary regularity, that is the Hölder regularity of u/δ up to the boundary. +Theorem 1.2. Suppose that Assumption 1.1 holds. +Let Ω be a bounded C2 domain and u be a +viscosity solution to the in-equations (1.1). Then there exists κ ∈ (0, ˆα) such that +∥u/δ∥Cκ(Ω) ≤ C1K, +(1.6) + +BOUNDARY REGULARITY +5 +for some constant C1, where κ, C1 depend on d, Ω, C0, Λ, λ, α and +´ +Rd(1 ∧ |y|α)k(y)dy. Here ˆα is +given by +ˆα = +� +1 +if α ∈ (0, 1] +2−α +2 +if α ∈ (1, 2). +To prove Theorem 1.2, following [46] we prove an oscillation lemma (cf. Proposition 4.1). For +this, first we need to construct sub and supersolutions carefully since Iθνδ becomes singular near +the boundary ∂Ω for α ∈ (1, 2). Then we shall use a “weak version” of Harnack inequality (cf. +Theorem 4.1). This weak version of Harnack inequality is new and needed to be developed due to +the unavailability of classical Harnack inequality. Also, we must point out that one needs to bypass +the use of comparison principle [10, Theorem 5.1] in such analysis, since the mentioned theorem is +for translation invariant linear operators. For non-translation invariant operators, such comparison +principle is unavailable, see Remark 2.1 for details. +Now applying (1.6), we prove the Hölder regularity of Du up to the boundary. +Theorem 1.3. Suppose that Assumption 1.1 holds and Ω be a bounded C2 domain. Then for any +viscosity solution u to the in-equations (1.1) we have +||Du||Cη(Ω) ≤ CK, +for some η ∈ (0, 1) and C, depending only on d, Ω, C0, Λ, λ, α and +´ +Rd(1 ∧ |y|α)k(y)dy. +The interior C1,η-regularity for fully nonlinear integro-differential equations is studied in [17] by +introducing a new ellipticity class where the kernels are C1 away from the origin. Kriventsov [39] +extended this result without the additional assumption on kernels (sometimes referred as rough ker- +nels). Also see [49] for its parabolic version. For HJBI-type integro-PDEs, interior C1,η-regularity is +established by Mou and Zhang [44] and for mixed local nonlocal fractional p-Laplacian, see [22]. The +C1,η-regularity up to the boundary for linear mixed local-nonlocal operators is recently obtained in +[11]. +The rest of the article is organized as follows. In Section 2, we introduce the necessary preliminaries +and collect all the auxiliary results which will be used throughout the article. In Section 3 we prove +Theorem 1.1. Theorem 1.2 is proved in Section 4. In Section 5 we prove Theorem 1.3. Lastly, in +Appendix A, following an approximation and scaling argument, we give a proof of C1,γ regularity for +a scaled operator i.e, Lemma 2.1. +2. Notation and preliminary results +This section sets the notation which we use throughout the paper and collects the necessary results. +2.1. Notations and Definitions. We start by setting the notations. We use Br(x) to denote an +open ball of radius r > 0 centred at a point x ∈ Rd and for x = 0, we denote Br := Br(0). For any +subset U ⊆ Rd and for α ∈ (0, 1), we denote Cα(U) as the space of all bounded, α-Hölder continuous +functions equipped with the norm +||f||Cα(U) := sup +x∈U +|f(x)| + sup +x,y∈U +|f(x) − f(y)| +|x − y|α +. +Note that for α = 1, C0,1(U) denotes the space of all Lipschitz continuous functions on U. The space +of all bounded functions with bounded α-Hölder continuous derivatives is denoted by C1,α(U) with +the norm +||f||C1,α(U) := sup +x∈U +|f(x)| + ||Df||Cα(U). +We use USC(Rd), LSC(Rd), C(Rd), Cb(Rd), Md to denote the space of upper semicontinuous, +lower semicontinuous, continuous functions, bounded continuous functions on Rd and d×d symmetric +matrices respectively. + +6 +BOUNDARY REGULARITY +Now let us introduce the scaled operators. For 0 < s ≤ 1, we define scaled version of (1.2) as +following. +Ls[x, u] = sup +θ∈Θ +inf +ν∈Γ +� +Tr aθν(sx)D2u(x) + Is +θν[x, u] +� +, +where +Is +θν[x, u] = +ˆ +Rd(u(x + y) − u(x) − 1B 1 +s (y)∇u(x) · y)sd+2Nθν(sx, sy)dy. +Next we define extremal Pucci operators for second order term and the nonlocal term. +P+u(x) := sup +� +Tr(AD2u(x)), A ∈ Md, λI ≤ A ≤ ΛI +� +, +P−u(x) := inf +� +Tr(AD2u(x)), A ∈ Md, λI ≤ A ≤ ΛI +� +, +and +P+ +k,su(x) := +ˆ +Rd(u(x + y) − u(x) − 1B 1 +s (y)∇u(x) · y)+sd+2k(sy)dy, +P− +k,su(x) := − +ˆ +Rd(u(x + y) − u(x) − 1B 1 +s (y)∇u(x) · y)−sd+2k(sy)dy. +Denote P+ +k,1 = P+ +k and P− +k,1 = P− +k . +We recall the definition of viscosity sub and supersolution. First of all, we say a function ϕ touches +from above (below) at x if, for a small r > 0, +ϕ(x) = u(x) and u(y) ≤ (≥)ϕ(y) for all y ∈ Br(x). +Definition 2.1. A function u ∈ USC(Rd) ∩ L∞(Rd) (resp. u ∈ LSC(Rd) ∩ L∞(Rd)) is said to be a +viscosity subsolution (resp. supersolution) to (1.1) if whenever ϕ touches u from above (resp. below) +for some bounded test function ϕ ∈ C2(Br(x)) ∩ C(Rd), then +v = +� +ϕ +in Br(x) +u +in Bc +r(x) +satisfies Lv(x) + C0|Dv(x)| ≥ −K (resp. Lv(x) − C0|Dv(x)| ≤ K). +2.2. Auxiliary lemmas. We collect some preliminary results here. The first result is the interior +C1,γ regularity for the scaled operator Ls. +Lemma 2.1. Let 0 < s ≤ 1 and u ∈ L∞(Rd) ∩ C(Rd) solves the in-equations +Ls[x, u] + C0s|Du(x)| ≥ −K in B2, +Ls[x, u] − C0s|Du(x)| ≤ K in B2, +(2.1) +in the viscosity sense. Then there exist constants 0 < γ < 1 and C > 0 independent of s, such that +||u||C1,γ(B1) ≤ C +� +||u||L∞(Rd) + K +� +, +where γ and C depend only on d, λ, Λ, C0 and +´ +Rd(1 ∧ |y|2)k(y)dy. +Proof. The proof essentially uses the approximation arguments for nonlocal equations [18] and we +postpone it to Appendix A. +□ +Now we present a maximum principle type result similar to [10, Theorem 5.2]. We report the proof +here for reader’s convenience. +Lemma 2.2. Let u be a bounded function on Rd which is in USC(Ω) and satisfies P+u + P+ +k u + +C0|Du| ≥ 0 in Ω. Then we have supΩ u ≤ supΩc u. + +BOUNDARY REGULARITY +7 +Proof. From [43, Lemma 5.5] we can find a non-negative function χ ∈ C2(¯Ω) ∩ Cb(Rd) satisfying +P+χ + P+ +k χ + C0|Dχ| ≤ −1 +in Ω. +Note that, since χ ∈ C2(¯Ω), the above inequality holds in the classical sense. For ε > 0, we let φM +to be +φM(x) = M + εχ. +Then P+φM(x0) + P+ +k φM + C0|DφM| ≤ −ε in Ω. +Let M0 be the smallest value of M for which φM ≥ u in Rd. We show that M0 ≤ supΩc u. Suppose, +to the contrary, that M0 > supΩc u. Then there must be a point x0 ∈ Ω for which u(x0) = φM0(x0). +Otherwise using the upper semicontinuity of u, we get a M1 < M0 such that φM1 ≥ u in Rd, which +contradicts the minimality of M0. +Now φM0 would touch u from above at x0 and thus, by the +definition of the viscosity subsolution, we would have that P+φM0(x0) + P+ +k φM0 + C0|DφM0| ≥ 0. +This leads to a contradiction. Therefore, M0 ≤ supΩc u which implies that for every x ∈ Rd +u ≤ φM0 ≤ M0 + ε sup +Rd χ ≤ sup +Ωc u + ε sup +Rd χ. +The result follows by taking ε → 0. +Remark 2.1. Although we have the above maximum principle, one can not simply compare two +viscosity sub and supersolutions for the operator (1.2). More precisely, if u, v are bounded functions +and u ∈ USC(Rd), v ∈ LSC(Rd) satisfy +Lu + C|Du| ≥ f and Lv + C|Dv| ≤ g in Ω +in viscosity sense for two continuous functions f and g, and for some C ≥ 0, then L(u−v)+C|D(u− +v)| ≥ f − g may not always holds true in Ω. However, if one of them is C2, then we have +P+(u − v) + P+ +k (u − v) + C|D(u − v)| ≥ f − g in Ω. +Indeed, without loss of generality, let us assume v ∈ C2(Ω) and ϕ be a C2 test function that touches +u − v at x ∈ Ω from above then clearly ϕ + v touches u at x from above. By definition of viscosity +subsolution we have L(ϕ + v)(x) + C|D(ϕ + v)(x)| ≥ f(x), which implies +P+ϕ(x) + P+ +k ϕ(x) + Lv(x) + C|Dϕ(x)| + C|Dv(x)| ≥ f(x) +and hence we obtain +P+ϕ(x) + P+ +k ϕ(x) + C|Dϕ(x)| ≥ f(x) − g(x). +□ +3. Global Lipschitz regularity +In this section we establish the Lipschitz regularity of the solution u up to the boundary. We start +by showing that the distance function δ(x) is a barrier to u. +Lemma 3.1. Let Ω be a bounded C1,1 domain in Rd and u be a continuous function which solves (1.1) +in the viscosity sense. Then there exists a constant C which depends only on d, λ, Λ, C0, diam(Ω), +radius of exterior sphere and +´ +Rd(1 ∧ |y|2)k(y)dy, such that +|u(x)| ≤ CKδ(x) +for all x ∈ Ω. +(3.1) +Proof. First we show that +|u(x)| ≤ κ K +x ∈ Rd, +(3.2) +for some constant κ. From [43, Lemma 5.5], there exists a non-negative function χ ∈ C2(¯Ω)∩Cb(Rd), +with infRd χ > 0, satisfying +P+χ + P+ +k χ + C0|Dχ| ≤ −1 +in Ω. +We define ψ = Kχ which gives that infRd ψ ≥ 0 and +P+ψ + P+ +k ψ + C0|Dψ| ≤ −K +in Ω. + +8 +BOUNDARY REGULARITY +Then by using Remark 2.1, we get +P+(u − ψ) + P+ +k (u − ψ) + C0|D(u − ψ)| ≥ 0. +Now applying Lemma 2.2 on u − ψ we obtain +sup +Ω +(u − ψ) ≤ sup +Ωc (u − ψ) ≤ 0. +Note that in the second inequality above we used u = 0 in Ωc. This proves that u ≤ ψ in Rd. Similar +calculation using −u will also give us −u ≤ ψ in Rd. Thus +|u| ≤ sup +Rd |χ| K +in Rd, +which gives (3.2). +Now we shall prove (3.1). Since ∂Ω is C1,1, Ω satisfies a uniform exterior sphere condition from +outside. Let r◦ be a radius satisfying uniform exterior condition. From [43, Lemma 5.4] there exists +a bounded, Lipschitz continuous function ϕ, Lipschitz constant being r−1 +◦ , satisfying +ϕ = 0 +in +¯Br◦, +ϕ > 0 +in +¯Bc +r◦, +ϕ ≥ ε +in +Bc +(1+δ)r◦, +P+ϕ + P+ +k ϕ + C0|Dϕ| ≤ −1 +in +B(1+δ)r◦ \ ¯Br◦, +for some constants ε, δ, dependent on C0, d, λ, Λ, d and +´ +Rd(1 ∧ |y|2)k(y)dy. Furthermore, ϕ is C2 +in B(1+δ)r◦ \ ¯Br◦. For any point y ∈ ∂Ω, we can find another point z ∈ Ωc such that Br◦(z) ⊂ Ωc +touches ∂Ω at y. Let w(x) = ε−1κKϕ(x − z). Also P+(w) + P+ +k (w) + C0|Dw| ≤ −K. Then by using +Remark 2.1 we have +P+(u − w) + P+ +k (u − w) + C0|D(u − w)| ≥ 0 +in B(1+δ)r◦(z) ∩ Ω. +Since, by (3.2) u − w ≤ 0 in (B(1+δ)r◦(z) ∩ Ω)c, applying Lemma 2.2 on u − w, it follows that +u(x) ≤ w(x) in Rd. Repeating a similar calculation for −u, we can conclude that |u(x)| ≤ w(x) in +Rd. Since this relation holds for any y ∈ ∂Ω, taking x ∈ Ω with dist(x, ∂Ω) < r◦, one can find y ∈ ∂Ω +satisfying dist(x, ∂Ω) = |x − y| < r◦. Then using the previous estimate we would obtain +|u(x)| ≤ ε−1κKϕ(x − z) ≤ ε−1κK(ϕ(x − z) − ϕ(y − z)) ≤ ε−1κK r−1 +◦ +dist(x, ∂Ω), +which gives us (3.1). +□ +Now we are ready to prove that u ∈ C0,1(Rd). +Proof of Theorem 1.1. Let x0 ∈ Ω and s ∈ (0, 1] be such that 2s = dist(x0, ∂Ω) ∧ 1. Without loss of +any generality, we assume x0 = 0. Define v(x) = u(sx) in Rd. Using Lemma 3.1 we already have +|u(x)| ≤ C1Kδ(x), from that one can deduce +|v(x)| ≤ C1 Ks(1 + |x|) +for all x ∈ Rd, +(3.3) +for some constant C1 independent of s. We recall the scaled operator +Is +θν[x, f] := +ˆ +Rd(f(x + y) − f(x) − 1B 1 +s (y)∇f(x) · y)sd+2Nθν(sx, sy)dy. +To compute Ls[x, v] + C0s|Dv(x)| in B2, first we observe that D2v(x) = s2D2u(sx) and Dv(x) = +sDu(sx). Also +Is +θν[x, v] = s2 +ˆ +Rd(v(x + y) − v(x) − 1B 1 +s (y)∇v(x) · y)Nθν(sx, sy)sddy += s2 +ˆ +Rd(u(sx + sy) − u(sx) − 1B1(sy)∇u(sx) · sy)Nθν(sx, sy)sddy = s2Iθν[sx, u]. + +BOUNDARY REGULARITY +9 +Thus, it follows from (1.1) that +Ls[x, v] + C0s|Dv(x)| ≥ −Ks2 +in +B2, +Ls[x, v] − C0s|Dv(x)| ≤ Ks2 +in +B2. +(3.4) +Now consider a smooth cut-off function ϕ, 0 ≤ ϕ ≤ 1, satisfying +ϕ = +� +1 +in B3/2, +0 +in Bc +2. +Let w = ϕv. +Clearly, ((ϕ − 1)v)(y) = 0 for all y ∈ B3/2, which gives D((ϕ − 1)v) = 0 and +D2((ϕ − 1)v) = 0 in x ∈ B3/2. Since w = v + (ϕ − 1)v, from (3.4) we obtain +Ls[x, w] + C0s|Dw(x)| ≥ −Ks2 − | sup +θ∈Θ +inf +ν∈Γ Is +θν[x, (ϕ − 1)v)]| +in +B1, +Ls[x, w] − C0s|Dw(x)| ≤ Ks2 + | sup +θ∈Θ +inf +ν∈Γ Is +θν[x, (ϕ − 1)v)]| +in +B1. +(3.5) +Again, since (ϕ − 1)v = 0 in B3/2, we have in B1 that +|Is +θν[x, (ϕ − 1)v]| = +��� +ˆ +|y|≥1/2 +((ϕ − 1)v)(x + y) − ((ϕ − 1)v)(x))sd+2Nθν(sx, sy)dy +��� +≤ +ˆ +|y|≥1/2 +|v(x + y)|sd+2Nθν(sx, sy)dy + |v(x)| +ˆ +|y|≥1/2 +sd+2Nθν(sx, sy)dy +:= I1 + I2. +Since x ∈ B1, using sd+2Nθν(sx, sy) ≤ sd+2k(sy) and (3.3) we can have the following estimate, +I2 ≤ 2C1Ks +ˆ +Rd(1 ∧ |y|2)dy. +Now write +I1 = +ˆ +1/2≤|y|≤1/s +|v(x + y)|sd+2Nθν(sx, sy)dy + +ˆ +|y|≥1/s +|v(x + y)|sd+2Nθν(sx, sy)dy += Is,1 + Is,2 . +Let us first estimate Is,1. Since x ∈ B1 and |y| ≥ 1 +2 we have 1 + |x + y| ≤ 5|y|. By using this estimate +and (3.3) we obtain +Is,1 = sd+2 +ˆ +1 +2≤|y|≤ 1 +s +|v(x + y)|Nθν(sx, sy)dy +≤ 5C1K +ˆ +1 +2≤|y|≤ 1 +s +|sy|sd+2k(sy)dy ≤ 5C1Ks +ˆ +s +2 ≤|z|≤1 +|sz|k(z)dz +≤ C2s +ˆ +s +2 ≤|z|≤1 +|z|2k(z)dz ≤ C2s +ˆ +Rd(1 ∧ |y|2)k(z)dz ≤ C3s, +for some constants C3. For Is,2, a change of variable and (3.2) gives +Is,2 ≤ κs2K +ˆ +s|y|>1 +sdk(ry)dy = κs2K +ˆ +|y|>1 +k(y)dy +≤ κs2K +ˆ +Rd(1 ∧ |y|2)k(y)dy ≤ C4s2K +for some constant C4. Therefore, putting the estimates of I1 and I2 in (3.5) we obtain +Ls[x, w] + C0s|Dw(x)| ≥ −C5Ks +in +B1, +Ls[x, w] − C0s|Dw(x)| ≥ C5Ks +in +B1, +(3.6) + +10 +BOUNDARY REGULARITY +for some constant C5. Now applying Lemma 2.1, from (3.6) we have +∥v∥C1(B 1 +2 ) ≤ C6 +� +∥v∥L∞(B2) + sK +� +(3.7) +for some constant C6. From (3.3) and (3.7) we then obtain +sup +y∈Bs/2(x),y̸=x +|u(x) − u(y)| +|x − y| +≤ C7K, +(3.8) +for some constant C7. +Now we can complete the proof. Note that if |x − y| ≥ 1 +8, then +|u(x) − u(y)| +|x − y| +≤ 2κK, +by (3.2). So we consider |x − y| < 1 +8. If |x − y| ≥ 8−1(δ(x) ∨ δ(y)), then using Lemma 3.1 we get +|u(x) − u(y)| +|x − y| +≤ 4CK(δ(x) + δ(y))(δ(x) ∨ δ(y))−1 ≤ 8CK. +Now let |x − y| < 8−1 min{δ(x) ∨ δ(y), 1}. Then either y ∈ B δ(x)∧1 +8 +(x) or x ∈ B δ(y)∧1 +8 +(y). Without +loss of generality, we suppose y ∈ B δ(x)∧1 +8 +(x). From (3.8) we get +|u(x) − u(y)| +|x − y| +≤ C7K. +This completes the proof. +□ +4. Fine boundary regularity +Aim of this section is to prove Theorem 1.2. Since u is Lipschitz, (1.1) can be written as +|Lu| ≤ CK in Ω, +and u = 0 in Ωc. +We start by constructing subsolutions which will be useful later on to prove oscillation lemma. +Lemma 4.1. There exists a constant ˜κ, which depends only on d, λ, Λ, +´ +Rd(1∧ |y|2)k(y)dy, such that +for any r ∈ (0, 1], we have a bounded radial function φr satisfying + + + + + + + + + +P−φr + P− +k φr ≥ 0 +in B4r \ ¯Br, +0 ≤ φr ≤ ˜κr +in Br, +φr ≥ 1 +˜κ(4r − |x|) +in B4r \ Br, +φr ≤ 0 +in Rd \ B4r. +Moreover, φr ∈ C2(B4r \ ¯Br). +Proof. We use the same subsolution constructed in [11] and show that it is indeed a subsolution +with respect to minimal Pucci operators. Fix r ∈ (0, 1] and define vr(x) = e−ηq(x) − e−η(4r)2, where +q(x) = |x|2 ∧ 2(4r)2 and η > 0. Clearly, 1 ≥ vr(0) ≥ vr(x) for all x ∈ Rd. Thus using the fact that +1 − e−ξ ≤ ξ for all ξ ≥ 0 we have +vr(x) ≤ 1 − e−η(4r)2 ≤ η(4r)2, +(4.1) +Again for x ∈ B4r \ Br, we have that +vr(x) = e−η(4r)2(eη((4r)2−q(x)) − 1) ≥ ηe−η(4r)2((4r)2 − |x|2) += ηe−η(4r)2(4r + |x|)(4r − |x|) ≥ 5ηre−η(4r)2(4r − |x|). +(4.2) + +BOUNDARY REGULARITY +11 +Fix x ∈ B4r \ ¯Br. We start by estimating the local minimal Pucci operator P− of v. Using rotational +symmetry we may always assume x = (l, 0, · · · , 0) Then +∂ivr(x) = −2ηe−η|x|2xi = +� +−2ηe−η|x|2l +i = 1, +0 +i ̸= 1 +and +∂ijvr(x) = +� +4η2x2 +i e−η|x|2 − 2ηe−η|x|2 +i = j, +4η2xixje−η|x|2 +i ̸= j. += + + + + + +4η2l2e−η|x|2 − 2ηe−η|x|2 +i = j = 1, +−2ηe−η|x|2 +i = j ̸= 1, +0 +i ̸= j. +By the above calculation, for any x ∈ B4r \ ¯Br, choosing η > 1 +r2 we have +P−vr(x) = λ4η2l2e−η|x|2 − λ2ηe−η|x|2 − Λ(d − 1)2ηe−η|x|2 +≥ λ4η2l2e−η|x|2 − dΛ2ηe−η|x|2. +Now to determine nonlocal minimal Pucci operator, using the convexity of exponential map we get, +e−η|x+y|2 − e−η|x|2 + 2η1{|y|≤1}y · xe−η|x|2 +≥ −ηe−η|x|2 � +|x + y|2 − |x|2 − 21{|y|≤1}y · x +� +. +Since P− +k vr = P− +k (vr + e−η(4r)2) and using above inequality we obtain +P− +k (e−ηq(·))(x) = − +ˆ +Rd +� +e−ηq(x+y) − e−ηq(x) − 1B1(y)∇e−ηq(x) · y +�− +k(y)dy +≥ −ηe−η|x|2 ˆ +|y|≤r +� +|x + y|2 − |x|2 − 2y · x +� +k(y)dy +− +ˆ +r<|y|≤1 +���e−η(|x|2+2(4r)2) − e−η|x|2 + 2ηy · xe−η|x|2��� k(y)dy +− +ˆ +|y|>1 +���e−η(|x|2+2(4r)2) − e−η|x|2��� k(y)dy +≥ −ηe−η|x|2 +�ˆ +|y|1 +����� +1 − e−2η(4r)2 +η +����� k(y)dy +≥ −ηe−η|x|2 +�ˆ +|y|1 +2(4r)2k(y)dy +� +≥ −ηe−η|x|243 +ˆ +Rd(1 ∧ |y|2)k(y)dy, +where in the second line we used |x + y|2 ∧ 2(4r)2 ≤ |x|2 + 2(4r)2. Combining the above estimates +we see that, for x ∈ B4r \ ¯Br, +P −vr(x) + P − +k vr(x) ≥ ηe−η|x|2� +4ηλ|x|2 − 2dΛ − 43 +ˆ +Rd(1 ∧ |y|2)k(y)dy +� +≥ ηe−η|x|2� +4ηλr2 − 2dΛ − 43 +ˆ +Rd(1 ∧ |y|2)k(y)dy +� +. + +12 +BOUNDARY REGULARITY +Thus, finally letting η = +1 +λr2(2dΛ + 43 ´ +Rd(1 ∧ |y|2)k(y)dy), we obtain +P−vr + P−vr > 0 +in B4r \ ¯Br. +Note that the final choice of η is admissible since +1 +λr2(2dΛ + 43 ´ +Rd(1 ∧ |y|2)k(y)dy) > +1 +r2. Now set +φr = rvr and the result follows from (4.1)-(4.2). +□ +Next we prove a weak version of Harnack inequality. +Theorem 4.1. Let s ∈ (0, 1], α′ = 1∧(2−α) and u be a continuous non-negative function satisfying +P−u + P− +k,su ≤ C0s1+α′, +P+u + P+ +k,su ≥ −C0s1+α′ +in B2. +Furthermore if supRd |u| ≤ M0 and |u(x)| ≤ M0s(1 + |x|) for all x ∈ Rd, then +u(x) ≤ C(u(0) + (M0 ∨ C0)s1+α′) +for every x ∈ B 1 +2 and for some constant C which only depends on λ, Λ, d, +´ +Rd(1 ∧ |y|α)k(y)dy. +Proof. Dividing by u(0)+(M0 ∨C0)s1+α′, it can be easily seen that supRd |u| ≤ s−(1+α′) and |u(x)| ≤ +s−α′(1 + |x|) for all x ∈ Rd and u satisfies +P−u + P− +k,su ≤ 1, +P+u + P+ +k,su ≥ −1. +Fix ε > 0 from [43, Corollary 3.14] and let γ = d +ε. Let +t0 := min +� +t : u(x) ≤ ht(x) := t(1 − |x|)−γ for all x ∈ B1 +� +. +Clearly this set is nonempty since u(0) ≤ 1, thus t0 exist. Let x0 ∈ B1 be such that u(x0) = ht0(x0). +Let η = 1 − |x0| be the distance of x0 from ∂B1. For r = η +2 and x ∈ Br(x0), we can write +Br(x0) = +� +u(x) ≤ u(x0) +2 +� +∪ +� +u(x) > u(x0) +2 +� +:= A + ˜A. +The goal is to estimate |Br(x0)| in terms of |A| and | ˜A|. Proceeding this way, we show that t0 < C +for some universal C which, in turn, implies that u(x) < C(1 − |x|)−γ. This would prove our result. +Next, Using [43, Corollary 3.14] we obtain +| ˜A ∩ B1| ≤ C +���� +2 +u(x0) +���� +ε +≤ Ct−ε +0 ηd , +whereas |Br| = ωd(η/2)d. In particular, +�� ˜A ∩ Br(x0) +�� ≤ Ct−ε +0 |Br|. +(4.3) +So if t0 is large, ˜A can cover only a small portion of Br(x0). We shall show that for some δ > 0, +independent of t0 we have +|A ∩ Br(x0)| ≤ (1 − δ)|Br|, +which will provide an upper bound on t0 completing the proof. We start by estimating |A ∩ Bθr(x0)| +for θ > 0 small. For every x ∈ Bθr(x0) we have +u(x) ≤ ht0(x) ≤ t0 +�2η − θη +2 +�−γ +≤ u(x0) +� +1 − θ +2 +�−γ +, +with +� +1 − θ +2 +� +close to 1. Define +v(x) := +� +1 − θ +2 +�−γ +u(x0) − u(x). +Then we get v ≥ 0 in Bθr(x0) and also P−v + P− +k,sv ≤ 1 as P+u + P+ +k,su ≥ −1. + +BOUNDARY REGULARITY +13 +We would like to apply [43, Corollary 3.14] to v, but v need not be non-negative in the whole of +Rd. Thus we consider the positive part of v, i.e, w = v+ and find an upper bound of P−w + P− +k,sw. +Since v− is C2 in B θr +4 (x0), we have +P−w(x) + P− +k,sw(x) ≤ [P−v(x) + P− +k,sv(x)] + [P+v−(x) + P+ +k,sv−(x)] ≤ 1 + P+v−(x) + P+ +k,sv−(x). +(4.4) +Also, using v−(x) = Dv−(x) = D2v−(x) = 0 for all x ∈ B θr +4 (x0), we get +P+v−(x) + P+ +k,sv−(x) = +ˆ +Rd∩{v(x+y)≤0} +v−(x + y)sd+2k(sy)dy. +(4.5) +Now plugging (4.5) into (4.4), for all x ∈ B θr +4 (x0) we obtain +P−w(x) + P− +k,sw(x) ≤ 1 + +ˆ +Rd\B θr +2 +(x−x0) +� +u(x + y) − +� +1 − θ +2 +�−γ +u(x0) +�+ +sd+2k(sy)dy +≤ 1 + +ˆ +Rd\B θr +2 +(x−x0) +|u(x + y)| sd+2k(sy)dy + +ˆ +Rd\B θr +2 +(x−x0) +����� +� +1 − θ +2 +�−γ +u(x0) +����� sd+2k(sy)dy +≤ 1 + +ˆ +Rd\B θr +4 +|u(x + y)| sd+2k(sy)dy + +ˆ +Rd\B θr +4 +����� +� +1 − θ +2 +�−γ +u(x0) +����� sd+2k(sy)dy := 1 + I1 + I2. +Estimate of I1: Let us write +I1 = +ˆ +θr +4 ≤|y|≤ 1 +s +|u(x + y)| sd+2k(sy)dy + +ˆ +|y|≥ 1 +s +|u(x + y)| sd+2k(sy)dy := I11 + I12. +Simply using change of variable and supRd |u| ≤ s−(1+α′), we obtain +I12 ≤ +ˆ +|z|≥1 +k(z)dz. +Now we estimate I11 using |u(x)| ≤ s−α′(1 + |x|) for all x ∈ Rd. +I11 ≤ +ˆ +θr +4 ≤|y|≤ 1 +s +(1 + |x + y|) sd+2−α′k(sy)dy +≤ 5 +4 +ˆ +θr +4 ≤|y|≤ 1 +s +sd+2−α′k(sy)dy + +ˆ +θr +4 ≤|y|≤ 1 +s +sd+2−α′|y|k(sy)dy . +We consider two cases. First consider the case α′ = 1 so α ≤ 1. This implies +I11 ≤ 5 +4 +ˆ +θrs +4 ≤|z|≤1 +sk(z)dz + +ˆ +θrs +4 ≤|z|≤1 +|z|k(z)dz ≤ 6(θr)−1 +ˆ +Rd(1 ∧ |z|α)k(z)dz. +Now consider the case α′ = 2 − α, and hence α > 1. In this case +I11 ≤ 5 +4 +ˆ +θr +4 ≤|y|≤ 1 +s +sαsdk(sy)dy + +ˆ +θr +4 ≤|y|≤ 1 +s +sα−1|sy|sdk(sy)dy += 5 · 4α−1(θr)−α +ˆ +θrs +4 ≤|z|≤1 +�θr +4 s +�α +k(z)dz + +�θr +4 +�1−α ˆ +θrs +4 ≤|z|≤1 +�θr +4 s +�α−1 +|z|k(z)dz +≤ C(θr)−2 +ˆ +Rd(1 ∧ |z|α)k(z)dz . + +14 +BOUNDARY REGULARITY +Combining the estimates of I11 and I12, we get +I1 ≤ C(θr)−2 +ˆ +Rd(1 ∧ |z|α)k(z)dz. +Estimate of I2: If α′ = 1, then α ≤ 1 and using |u(x0)| ≤ s−α′(1 + |x0|) we have +I2 := +ˆ +Rd\B θr +4 +����� +� +1 − θ +2 +�−γ +u(x0) +����� sd+2k(sy)dy ≤ C +ˆ +Rd\B θr +4 +sd+2−α′k(sy)dy += C +ˆ +Rd\B θrs +4 +sk(z)dz ≤ C +�ˆ +θrs +4 ≤|z|≤1 +sk(z)dz + +ˆ +|z|≥1 +sk(z)dz +� +≤ C +� +4 +θr +ˆ +θrs +4 ≤|z|≤1 +|z|αk(z)dz + +ˆ +|z|>1 +k(z)dz +� +≤ C(θr)−1 +ˆ +Rd(1 ∧ |y|α)k(z)dz. +If α′ = 2 − α then α > 1. In that case, using similar calculation as above we have +I2 := +ˆ +Rd\B θr +4 +����� +� +1 − θ +2 +�−γ +u(x0) +����� sd+2k(sy)dy ≤ C +ˆ +Rd\B θr +4 +sd+αk(sy)dy += C +ˆ +Rd\B θrs +4 +sαk(z)dz ≤ C(θr)−α +ˆ +Rd(1 ∧ |y|α)k(z)dz. +Since α ∈ (0, 2), combining the above estimates we obtain +P−w + P− +k,sw ≤ +C +(θr)2 +in B θr +4 (x0) . +Now using [43, Corollary 3.14] for w we get +|A ∩ B θr +8 (x0)| = +���� +� +w ≥ u(x0)((1 − θ/2)−γ − 1/2) +� +∩ B θr +8 (x0) +���� +≤ C(θr)d +� +inf +B θr +8 +(x0) w + θr +8 · +C +(θr)2 +�ε +· +� +u(x0)((1 − θ/2)−γ − 1/2) +�−ε +≤ C(θr)d� � +(1 − θ +2)−γ − 1 +2 +� ++ C +8 (θr)−1t−1 +0 (2r)d�ε +≤ C(θr)d �� +(1 − θ/2)−γ − 1 +�ε + C0(θr)−εt−ε +0 rdε� +. +Now let us choose θ > 0 small enough (independent of t0) to satisfy +C(θr)d � +(1 − θ/2)−γ − 1 +�ε ≤ 1 +4|B θr +8 (x0)| . +With this choice of θ if t0 becomes large, then we also have +C(θr)dθ−εr(n−1)εt−ε +0 +≤ 1 +4|B θr +8 (x0)| , +and hence +|A ∩ B θr +8 (x0)| ≤ 1 +2|B θr +8 (x0)| . +This estimate of course implies that +| ˜A ∩ B θr +8 (x0)| ≥ C2|Br|, +but this is contradicting (4.3). Therefore t0 cannot be large and this completes the proof. +□ + +BOUNDARY REGULARITY +15 +Corollary 4.1. Let u satisfies the conditions of Theorem 4.1, then the following holds. +sup +B 1 +4 +u ≤ C +� +inf +B 1 +4 +u + (M0 ∨ C0)s1+α′ +� +. +Proof. Take any point x0 ∈ B 1 +4 such that u(x0) = infB 1 +4 u(x). Clearly B 1 +4 ⊂ B 1 +2(x0). Defining +˜u(x) := u(x + x0) and applying Theorem 4.1 on ˜u we find +˜u(x) ≤ C +� +˜u(0) + (M0 ∨ C0)s1+α′� +in B 1 +2 . +This implies +sup +B 1 +4 +u(x) ≤ +sup +B 1 +2 (x0) +u(x) ≤ C +� +inf +B 1 +4 +u(x) + (M0 ∨ C0)s1+α′ +� +. +This proves the claim. +□ +Now we will give some auxiliary lemmas which will be used to construct appropriate supersolutions +that are crucial to prove the oscillation estimate. +Lemma 4.2. Let Ω be a bounded C2 domain in Rd, then for any 0 < ǫ < 1, we have the following +estimate +��Iθν(δ1+ǫ) +�� ≤ C +� +1 + 1(1,2)(α)δ1−α� +in Ω, +(4.6) +where C > 0 depends only on d, Ω and +´ +Rd(1 ∧ |y|α)k(y)dy. +Proof. Since δ ∈ C0,1(Rd)∩C2(¯Ω) [21, Theorem 5.4.3], using the Lipschtiz continuity of δ1+ǫ near the +origin and boundedness away from the origin we can easily obtain the estimate (4.6) for α ∈ (0, 1]. +Next consider the case α ∈ (1, 2). For any x ∈ Ω we have +��Iθν(δ1+ǫ)(x) +�� ≤ +ˆ +Rd +��δ1+ǫ(x + y) − δ1+ǫ(x) − 1B1(y)y · ∇δ1+ǫ(x) +�� k(y)dy += +ˆ +|y|< δ(x) +2 ++ +ˆ +δ(x) +2 ≤|y|≤1 ++ +ˆ +|y|>1 +:= I1 + I2 + I3 . +Since |y| ≤ δ(x) +2 +and δ(x) < 1, we have the following estimate on I1. +��δ1+ǫ(x + y) − δ1+ǫ(x) − 1B1(y)y · ∇δ1+ǫ(x) +�� ≤ ||δ1+ǫ||C2(B δ(x) +2 +(x))|y|2 +≤ 4C +||δ||C2(¯Ω) +δ(x)1−ǫ |y|2 ≤ 4C +||δ||C2(¯Ω)δ(x)2−α +δ(x)1−ǫ +|y|α. +This implies +I1 ≤ 4C||δ||C2(¯Ω)δ(x)1+ǫ−α +ˆ +Rd |y|αk(y)dy ≤ 4C0C||δ||C2(¯Ω)δ(x)1+ǫ−α. +(4.7) +Again for I2 we have +I2 ≤ C +ˆ +δ(x) +2 ≤|y|≤1 +|y|k(y)dy ≤ +�Cδ(x) +2 +�1−α ˆ +δ(x) +2 ≤|y|≤1 +|y|αk(y)dy +≤ +�Cδ(x) +2 +�1−α ˆ +Rd (1 ∧ |y|α) k(y)dy. +Finally, +I3 = +ˆ +|y|>1 +|δ1+ǫ(x + y) − δ1+ǫ(x)|k(y)dy ≤ 2(diam Ω)1+ǫ +ˆ +Rd(1 ∧ |y|α)k(y)dy. +(4.8) +Combining (4.7)-(4.8) we obtain (4.6). +□ + +16 +BOUNDARY REGULARITY +Next we obtain an estimate on minimal Pucci operator P− applied on δ1+ǫ. +Lemma 4.3. Let Ω be a bounded C2 domain in Rd, then for any 0 < ǫ < 1, we have the following +estimate +P− � +δ1+ǫ� +≥ C1 · ǫδǫ−1 − C2 in Ω, +where C1, C2 depends only on d, Ω, λ, Λ. +Proof. Since ∂Ω is C2, we have δ1+ǫ ∈ C2(Ω) and for any x ∈ Ω +∂2 +∂xi∂xj +δ1+ǫ(x) = (1 + ǫ) +� +δǫ(x) +∂2 +∂xi∂xj +δ(x) + ǫδǫ−1(x)∂δ(x) +∂xi +· ∂δ(x) +∂xj +� +:= A + B +where A, B are two d × d matrices given by +A := (ai,j)1≤i,j≤d = (1 + ǫ)δǫ(x) +∂2 +∂xi∂xj +δ(x) +and +B := (bi,j)1≤i,j≤d = (1 + ǫ)ǫδǫ−1(x)∂δ(x) +∂xi +· ∂δ(x) +∂xj +. +Note that B is a positive definite matrix and ||A|| ≤ d2(1 + ǫ)(diam Ω)ǫ||δ||C2(¯Ω). Therefore we have +P−(δ1+ǫ(x)) = P−(A + B) ≥ P−(B) + P−(A) +≥ P−(B) − d2Λ(1 + ǫ)(diam Ω)ǫ||δ||C2(¯Ω) +≥ ǫ(1 + ǫ)δǫ−1(x)λ|Dδ(x)|2 − d2Λ(1 + ǫ)(diam Ω)ǫ||δ||C2(¯Ω) +≥ C1 · ǫδǫ−1(x) − C2. +□ +Next we obtain an estimate on Lδ in Ω. +Lemma 4.4. Let Ω be a bounded C2 domain in Rd. Then we have the following estimate +|Lδ| ≤ C(1 + 1(1,2)δ1−α) in Ω, +(4.9) +where constant C depends only on d, Ω, λ, Λ and +´ +Rd(1 ∧ |y|α)k(y)dy. +Proof. First of all, for all x ∈ Ω we have +|Lδ(x)| ≤ sup +θ,ν +| Tr(aθν(x)D2δ(x))| + sup +θ,ν +|Iθνδ(x)| ≤ κ + sup +θ,ν +|Iθνδ(x)|, +(4.10) +for some constant κ, depending on Ω and uniform bound of aθν. For α ∈ (0, 1], (4.9) follows from the +same arguments of Lemma 4.2. For α ∈ (1, 2), it is enough to obtain the estimate (4.9) for all x ∈ Ω +such that δ(x) < 1. We follow the similar calculation as in Lemma 4.2 and get +|Iθνδ(x)| ≤ +ˆ +Rd |δ(x + y) − δ(x) − 1B1(y)y · ∇δ(x)|k(y)dy += +ˆ +|y|≤ δ(x) +2 ++ +ˆ +δ(x) +2 <|y|<1 ++ +ˆ +|y|>1 +and +|Iθνδ(x)| ≤ κ1 +ˆ +Rd(1 ∧ |y|α)k(y)dyδ1−α(x) +for some constant κ1. Inserting these estimates in (4.10) we obtain +|Lδ(x)| ≤ κ2δ1−α(x) +for some constant κ2 and (4.9) follows. +□ + +BOUNDARY REGULARITY +17 +Let us now define the sets that we use for our oscillation estimates. We borrow the notations of +[46]. +Definition 4.1. Let κ ∈ (0, 1 +16) be a fixed small constant and let κ′ = 1/2 + 2κ. Given a point +x0 ∈ ∂Ω and R > 0, we define +DR = DR(x0) = BR(x0) ∩ Ω, +and +D+ +κ′R = D+ +κ′R(x0) = Bκ′R(x0) ∩ {x ∈ Ω : (x − x0) · n(x0) ≥ 2κR} , +where n(x0) is the unit inward normal at x0. For any bounded C1,1-domain, we know that there +exists ρ > 0, depending on Ω, such that the following inclusions hold for each x0 ∈ ∂Ω and R ≤ ρ: +BκR(y) ⊂ DR(x0) +for all y ∈ D+ +κ′R(x0), +(4.11) +and +B4κR(y∗ + 4κRn(y∗)) ⊂ DR(x0), +and +BκR(y∗ + 4κRn(y∗)) ⊂ D+ +κ′R(x0) +(4.12) +for all y ∈ DR/2, where y∗ ∈ ∂Ω is the unique boundary point satisfying |y − y∗| = dist(y, ∂Ω). Note +that, since R ≤ ρ, y ∈ DR/2 is close enough to ∂Ω and hence the point y∗ + 4κR n(y∗) belongs to the +line joining y and y∗. +Remark 4.1. In the remaining part of this section, we fix ρ > 0 to be a small constant depending +only on Ω, so that (4.11)-(4.12) hold whenever R ≤ ρ and x0 ∈ ∂Ω. Also, every point on ∂Ω can be +touched from both inside and outside Ω by balls of radius ρ. We also fix σ > 0 small enough so that +for 0 < r ≤ ρ and x0 ∈ ∂Ω we have +Bηr(x0) ∩ Ω ⊂ B(1+σ)r(z) \ ¯Br(z) +for +η = σ/8, σ ∈ (0, γ), +for any x′ ∈ ∂Ω ∩ Bηr(x0), where Br(z) is a ball contained in Rd \ Ω that touches ∂Ω at point x′. +In the following lemma, using Lemma 4.2 and Lemma 4.3 we construct supersolutions. We denote +Ωρ := {x ∈ Ω| dist(x, Ωc) < ρ}. +Lemma 4.5. Let Ω be a bounded C2 domain in Rd and α ∈ (1, 2), then there exist ρ1 > 0 and a C2 +function φ1 satisfying + + + + + +P+φ1(x) + P+ +k φ1(x) ≤ −Cδ− α +2 (x) +in +Ωρ1, +C−1δ(x) ≤ φ1(x) ≤ Cδ(x) +in +Ω, +φ1(x) = 0 +in +Rd \ Ω, +where the constants ρ1 and C depend only on d, α, Ω, λ, Λ and +´ +Rd(1 ∧ |y|α)k(y)dy. +Proof. Let ǫ = 2−α +2 +and c = +1 +(diam Ω)2 , and define +φ1(x) = δ(x) − cδ1+ǫ(x). +Since both δ and δ1+ǫ are in C2(Ω), we have P+φ1(x) ≤ P+δ(x) − cP−δ1+ǫ(x). Then by Lemma 4.3 +and supθν | Tr(aθν(x)D2δ(x))| ≤ ˜C, we get for all x ∈ Ωρ +P+φ1(x) ≤ P+δ(x) − cP−δ1+ǫ(x) ≤ C − c(C1 · ǫδǫ−1(x)). +Similarly for all x ∈ Ωρ, using Lemma 4.2 and Lemma 4.4 we get +P+ +k φ1(x) ≤ |P+ +k δ(x)| + c|P− +k δ1+ǫ(x)| ≤ C2δ1−α(x). +Combining the above inequalities we have +P+φ1(x) + P+ +k φ1(x) ≤ C − cC1ǫδǫ−1(x) + C2δ1−α(x) +≤ −δǫ−1(x) +� C1(2 − α) +2(diam Ω)2 − Cδ +α +2 (x) − C2δ +2−α +2 (x) +� +, + +18 +BOUNDARY REGULARITY +for all x ∈ Ωρ. Now choose 0 < ρ1 ≤ ρ < 1 such that +� C1(2 − α) +2(diam Ω)2 − Cρ +α +2 +1 − C2ρ +2−α +2 +1 +� +≥ C1(2 − α) +4(diam Ω)2 . +Thus for all x ∈ Ωρ1, we have +P+φ1(x) + P+ +k φ1(x) ≤ − C1(2 − α) +4(diam Ω)2 δ− α +2 (x). +Finally the construction of φ1 immediately gives us that +C−1δ(x) ≤ φ1(x) ≤ Cδ(x) +in +Ω, +and φ1 = 0 in Ωc. This completes the proof of the lemma. +□ +As mentioned earlier, the key step of proving Theorem 1.2 is to obtain the oscillation lemma +Proposition 4.1. For this we next prove two preparatory lemmas. In the first lemma we obtain a +lower bound of infD R +2 +u +δ whereas the second lemma controls supD+ +κ′R +u +δ by using that lower bound. +Lemma 4.6. Let α ∈ (0, 2) and Ω be a bounded C2 domain in Rd. Also, let u be such that u ≥ 0 in +Rd, and |Lu| ≤ C2(1 + 1(1,2)(α)δ1−α) in DR, for some constant C2. If ˆα is given by +ˆα = +� +1 +if α ∈ (0, 1], +2−α +2 +if α ∈ (1, 2), +then there exists a positive constant C depending only on d, Ω, Λ, λ, α, +´ +Rd(1 ∧ |y|α)k(y)dy, such that +inf +D+ +κ′R +u +δ ≤ C +� +inf +D R +2 +u +δ + C2Rˆα +� +(4.13) +for all R ≤ ρ0, where the constant ρ0 depends only on d, Ω, λ, Λ, α and +´ +Rd(1 ∧ |y|α)k(y)dy. +Proof. Suppose R ≤ ηρ, where ρ is given by Remark 4.1 and η ≤ 1 be some constant that will be +chosen later. Define m = infD+ +κ′ R +u/δ ≥ 0. Let us first observe that by (4.11) we have, +u ≥ mδ ≥ m (κR) +in D+ +κ′R. +(4.14) +Moreover by (4.12), for any y ∈ DR/2, we have either y ∈ D+ +κ′R or δ(y) < 4κR. If y ∈ D+ +κ′R, then by +the definition of m we get m ≤ u(y)/δ(y). +Next we consider δ(y) < 4κR. Let y∗ be the nearest point to y on ∂Ω, i.e, dist(y, ∂Ω) = |y − y∗| +and define ˜y = y∗ + 4κR n(y∗). Again by (4.12), we have +B4κR(˜y) ⊂ DR and BκR(˜y) ⊂ D+ +κ′R. +Denoting r = κR and using the subsolution constructed in Lemma 4.1, define ˜φr(x) := 1 +˜κφr(x − ˜y). +We will consider two cases. +Case 1: α ∈ (0, 1]. Take r′ = R +η . Since r′ ≤ ρ, points of ∂Ω can be touched by exterior ball of radius +r′. In particular, for y∗ ∈ ∂Ω, we can find a point z ∈ Ωc such that ¯Br′(z) ⊂ Ωc touches ∂Ω at y∗. +Now from [43, Lemma 5.4] there exists a bounded, Lipschitz continuous function ϕr′, with Lipschitz +constant 1 +r′ , that satisfies + + + + + +ϕr′ = 0, +in +¯Br′, +ϕr′ > 0, +in +¯Bc +r′, +P+ϕr′ + P+ +k ϕr′ ≤ − +1 +(r′)2 , +in +B(1+σ)r′ \ ¯Br′, + +BOUNDARY REGULARITY +19 +for some constant σ, independent of r′. Without any loss of any generality we may assume σ ≤ γ +(see Remark 4.1). Then setting η = σ +8 and using Remark 4.1, we have +DR ⊂ B(1+σ)r′(z) \ Br′(z) +and by (4.12) we have +B4r(˜y) \ Br(˜y) ⊂ DR ⊂ B(1+σ)r′(z) \ Br′(z). +We show that v(x) = m˜φr(x) − C2(r′)2ϕr′(x − z) is an appropriate subsolution. Since both ˜φr and +ϕr′ are C2 functions in B4r(˜y) \ ¯Br(˜y), we conclude that v is C2 function in B4r(˜y) \ ¯Br(˜y). For +x ∈ B4r(˜y) \ ¯Br(˜y), +P−v(x) + P− +k v(x) ≥ m +� +P− ˜φr(x) + P− +k ˜φr(x) +� +− C2(r′)2 � +P+ϕr′(x − z) + P+ +k ϕr′(x − z) +� +≥ C2. +Therefore by Remark 2.1 we have +P+(v − u) + P+ +k (v − u) ≥ 0 in B4r(˜y) \ ¯Br(˜y). +Furthermore, using (4.14) and u ≥ 0 in Rd we obtain u(x) ≥ m˜φr(x) − C2(r′)2ϕr′(x − z) in +� +B4r(˜y) \ ¯Br(˜y) +�c . Hence an application of maximum principle (cf. +Lemma 2.2) gives u ≥ v in +Rd. Now for y ∈ DR/2, using the Lipschitz continuity of ϕr′ we get +m˜φr(y) ≤ u(y) + C2(r′)2 [ϕr′(y − z) − ϕr′(y∗ − z)] ≤ u(y) + C2r′ · δ(y) +and as y lies on the line segment joining y∗ to ˜y we get +u(y) +δ(y) + C2r′ ≥ +m +(˜κ)2 . +This gives +inf +D+ +κ′R +u +δ ≤ C +� +inf +DR/2 +u +δ + C2 +R +η +� +and finally choosing ρ0 = ηρ we have (4.13). +Case 2: α ∈ (1, 2). Let ρ1 as in Lemma 4.5 and consider R ≤ ρ1 < 1. Here we aim to construct +an appropriate subsolution using ˜φr(x) and supersolution constructed in Lemma 4.5. Since δ(x) ≤ 1 +in DR, we have |Lu(x)| ≤ C2(1 + δ1−α(x)) ≤ 2C2δ1−α(x) in DR. Also by Lemma 4.5, we have a +bounded function φ1 which is C2 in Ωρ1 ⊃ DR and satisfies +P+φ1(x) + P+ +k φ1(x) ≤ −Cδ− α +2 (x) = −C +1 +δ +2−α +2 (x) +δ1−α(x) ≤ −C +Rˆα δ1−α(x), +for all x ∈ DR. Now we define the subsolutions as +v(x) = m˜φr(x) − µ Rˆαφ1(x), +where the constant µ is chosen suitably so that P−v(x) + P− +k v(x) ≥ 2C2δ1−α(x) in B4r(˜y) \ ¯Br(˜y) +(i.e. µ = 2C2 +C ). Also u ≥ v in (B4r(˜y) \ ¯Br(˜y))c. Using the same calculation as previous case for v − u +and maximum principle Lemma 2.2 we derive that u ≥ v in Rd. Again, repeating the arguments of +Case 1 we get +inf +D+ +κ′R +u +δ ≤ C +� +inf +D R +2 +u +δ + 2C2Rˆα +� +. +Choosing ρ0 = ηρ ∧ ρ1 completes the proof. +□ +Lemma 4.7. Let α′ = 1 ∧ (2 − α) and Ω be a bounded C2 domain in Rd. Also, let u be a bounded +continuous function such that u ≥ 0 and u ≤ M0δ(x) in Rd, and |Lu| ≤ C2(1 + 1(1,2)(α)δ1−α) in + +20 +BOUNDARY REGULARITY +DR, for some constant C2. Then, there exists a positive constant C, depending only on d, λ, Λ, Ω and +´ +Rd(1 ∧ |y|α)k(y)dy, such that +sup +D+ +κ′R +u +δ ≤ C +� +inf +D+ +κ′R +u +δ + (M0 ∨ C2)Rα′ +� +(4.15) +for all R ≤ ρ, where constant ρ is given by Remark 4.1. +Proof. We will use the weak Harnack inequality proved in Theorem 4.1 to show (4.15). Let R ≤ ρ. +Then for each y ∈ D+ +κ′R, we have BκR(y) ⊂ DR. Hence we have |Lu| ≤ C2(1 + 1(1,2)(α)δ1−α(x)) in +BκR(y). Without loss of generality, we may assume y = 0. Let s = κR and define v(x) = u(sx) for +all x ∈ Rd. Then, it can be easily seen that +s2L[sx, u] = Ls[x, v] := sup +θ∈Θ +inf +ν∈Γ +� +Tr aθν(sx)D2v(x) + Is +θν[x, v] +� +for all x ∈ B2. +This gives +|Ls[x, v]| ≤ C2s2(1 + 1(1,2)(α)δ1−α(sx)) +≤ C2 +� +s2 + 1(1,2)(α)s2 (κR)1−α� +≤ C2s1+α′ , +in B2 where α′ = 1 ∧ (2 − α). In second line, we used that for each x ∈ BκR, |sx| < κR and hence +δ(sx) > κR +2 = s +2. From u ≤ M0δ(x) we have v(y) ≤ M0 diam Ω and v(y) ≤ M0s(1 + |y|) in whole Rd. +Hence by Corollary 4.1, we obtain +sup +B 1 +4 +v ≤ C +� +inf +B 1 +4 +v + (M0 ∨ C2)s1+α′ +� +, +where constant C does not depend on s, M0, C2. This of course, implies +sup +B κR +64 +(y) +u ≤ C +� +inf +B κR +64 +(y) u + (M0 ∨ C2)R1+α′ +� +, +for all y ∈ D+ +κ′R. Now cover D+ +κ′R by a finite number of balls BκR/64(yi), independent of R, to obtain +sup +D+ +κ′R +u ≤ C +� +inf +D+ +κ′R +u + (M0 ∨ C2)R1+α′ +� +. +Then (4.15) follows since κR/2 ≤ δ ≤ 3κR/2 in D+ +κ′R. +□ +Now we are ready to prove the oscillation lemma. +Proposition 4.1. Let u be a bounded continuous function such that |Lu| ≤ K in Ω, for some constant +K, and u = 0 in Ωc. Given any x0 ∈ ∂Ω, let DR be as in the Definition 4.1. Then for some τ ∈ (0, ˆα) +there exists C, dependent on Ω, d, λ, Λ, α and +´ +Rd(1 ∧ |y|α)k(y)dy but not on x0, such that +sup +DR +u +δ − inf +DR +u +δ ≤ CKRτ +(4.16) +for all R ≤ ρ0, where ρ0 > 0 is a constant depending only on Ω, d, λ, Λ, α and +´ +Rd(1 ∧ |y|α)k(y)dy. +Proof. For the proof we follow a standard method, similar to [46], with the help of Lemmas 4.4, 4.6, +and 4.7. Fix x0 ∈ ∂Ω and consider ρ0 > 0 to be chosen later. With no loss of generality, we assume +x0 = 0. In view of (3.2), we only consider the case K > 0. By considering u/K instead of u, we +may assume that K = 1, that is, |Lu| ≤ 1 in Ω. From Theorem 1.1 we note that ||u||C0,1(Rd) ≤ C1. +Below, we consider two cases. + +BOUNDARY REGULARITY +21 +Case 1: For α ∈ (0, 1], Iθνu is classically defined and |Iθνu| ≤ ˜C in Ω for all θ and ν. Consequently, +one can combine the nonlocal term on the rhs and only deal with local nonlinear operator ˜L[x, u] := +supθ∈Θ infν∈Γ +� +Tr aθν(x)D2u(x) +� +. In this case the proof is simpler and can be done following the +same method as for the local case. However, the method we use below would also work with an +appropriate modification. +Case 2: Now we deal with the case α ∈ (1, 2). We show that there exists K > 0, ρ1 ∈ (0, ρ0) and +τ ∈ (0, 1), dependent only on Ω, d, λ, Λ, α and +´ +Rd(1 ∧ |y|α)k(y)dy, and monotone sequences {Mk} +and {mk} such that, for all k ≥ 0, +Mk − mk = +1 +4kτ , +−1 ≤ mk ≤ mk+1 < Mk+1 ≤ Mk ≤ 1, +(4.17) +and +mk ≤ K−1 u +δ ≤ Mk +in +DRk, +where +Rk = ρ1 +4k . +(4.18) +Note that (4.18) is equivalent to the following +mkδ ≤ K−1u ≤ Mkδ, +in +BRk, +where +Rk = ρ1 +4k . +(4.19) +Next we construct monotone sequences {Mk} and {mk} by induction. +The existence of M0 and m0 such that (4.17) and (4.19) hold for k = 0 is guaranteed by Lemma 3.1. +Assume that we have the sequences up to Mk and mk. We want to show the existence of Mk+1 and +mk+1 such that (4.17)-(4.19) hold. We set +uk = 1 +Ku − mkδ. +Note that to apply Lemma 4.7 we need uk to be nonnegative in Rd. Therefore we shall work with +u+ +k , the positive part of uk. Let uk = u+ +k − u− +k and by the induction hypothesis, +u+ +k = uk +and +u− +k = 0 +in +BRk. +(4.20) +We need to find a lower bound on uk. Since uk ≥ 0 in BRk and uk is Lipschitz in Rd, we get for +x ∈ Bc +Rk that +uk(x) = uk(Rkxu) + uk(x) − uk(Rkxu) ≥ −CL|x − Rkxu|, +(4.21) +where zu = +1 +|z|z for z ̸= 0 and CL denotes a Lipschitz constant of uk which can be chosen independent +of k. Using Lemma 3.1 we also have |uk| ≤ K−1 + diam(Ω) = C1 for all x ∈ Rd. Thus using (4.20) +and (4.21) we calculate L[x, u− +k ] in D Rk +2 . Let x ∈ DRk/2. By (4.20), D2u− +k (x) = 0. Then +0 ≤ Iθν[x, u− +k ] = +ˆ +x+y̸∈BRk +u− +k (x + y)Nθν(x, y)dy +≤ +ˆ +� +|y|≥ Rk +2 ,x+y̸=0 +� u− +k (x + y)k(y)dy +≤ CL +ˆ +� Rk +2 ≤|y|≤1, x+y̸=0 +� +���(x + y) − Rk(x + y)u +���k(y)dy + C1 +ˆ +|y|≥1 +k(y)dy +≤ CL +ˆ +Rk +2 ≤|y|≤1 +(|x| + Rk) k(y) dy + CL +ˆ +Rk +2 ≤|y|≤1 +|y|k(y) dy + C1 +ˆ +Rd(1 ∧ |y|α)k(y) dy +≤ κ3 +�ˆ +Rd(1 ∧ |y|α)k(y) dy +� � +R1−α +k ++ 1 +� +≤ κ4R1−α +k +, +(4.22) +for some constants κ3, κ4, independent of k. + +22 +BOUNDARY REGULARITY +Now we write u+ +k = K−1u − mkδ + u− +k . Since δ is C2 and u− +k = 0 in D Rk +2 , first note that +Lu+ +k ≤ K−1 − (P− + P− +k )(mkδ) + (P+ + P+ +k )(u− +k ), +Lu+ +k ≥ −K−1 − (P+ + P+ +k )(mkδ) + (P− + P− +k )(u− +k ). +Using Lemma 4.4 and (4.22) in the above estimate we have +|Lu+ +k | ≤ K−1 + mkCδ1−α + κ4(Rk)1−α in D Rk +2 . +(4.23) +Since ρ1 ≥ Rk ≥ δ in DRk, for α > 1, we have R1−α +k +≤ δ1−α, and hence, from (4.23), we have +|Lu+ +k | ≤ +� +K−1[(ρ1)]α−1 + C + κ4 +� +δ1−α(x) := κ5δ1−α(x) +in +DRk/2. +Now we are in a position to apply Lemmas 4.6 and 4.7. Recalling that +u+ +k = uk = K−1u − mkδ +in +DRk, +and using Lemma 3.1 we also have |u+ +k | ≤ |uk| ≤ (K−1 + 1)δ(x) = C1δ(x) for all x ∈ Rd. We get +from Lemmas 4.6 and 4.7 that +sup +D+ +κ′Rk/2 +� +K−1 u +δ − mk +� +≤ C +� +inf +D+ +κ′Rk/2 +� +K−1 u +δ − mk +� ++ (κ5 ∨ C1)Rˆα +k +� +≤ C +� +inf +DRk/4 +� +K−1 u +δ − mk +� ++ (κ5 ∨ C1)Rˆα +k +� +. +(4.24) +Repeating a similar argument for the function ˜uk = Mkδ − K−1u, we find +sup +D+ +κ′Rk/2 +� +Mk − K−1 u +δ +� +≤ C +� +inf +DRk/4 +� +Mk − K−1 u +δ +� ++ (κ5 ∨ C1)Rˆα +k +� +. +(4.25) +Combining (4.24) and (4.25) we obtain +Mk − mk ≤ C +� +inf +D+ +Rk/4 +� +Mk − K−1 u +δ +� ++ inf +D+ +Rk/4 +� +K−1 u +δ − mk +� ++ (κ5 ∨ C1)Rˆα +k +� += C +� +inf +DRk+1 +K−1 u +δ − sup +DRk+1 +K−1 u +δ + Mk − mk + (κ5 ∨ C1)Rˆα +k +� +. +(4.26) +Putting Mk − mk = +1 +4τk in (4.26), we have +sup +DRk+1 +K−1 u +δ − +inf +DRk+1 +K−1 u +δ ≤ +�C − 1 +C +1 +4τk + (κ5 ∨ C1)Rˆα +k +� += +1 +4τk +�C − 1 +C ++ (κ5 ∨ C1)Rˆα +k4τk� +. +(4.27) +Since Rk = ρ1 +4k for ρ1 ∈ (0, ρ0), we can choose ρ0 and τ small so that +�C − 1 +C ++ (κ5 ∨ C1)Rˆα +k 4τk� +≤ 1 +4τ . +Putting in (4.27) we obtain +sup +DRk+1 +K−1 u +δ − +inf +DRk+1 +K−1 u +δ ≤ +1 +4τ(k+1) . +Thus we find mk+1 and Mk+1 such that (4.17) and (4.18) hold. +It is easy to prove (4.16) from +(4.17)-(4.18). +□ +Next we establish Hölder regularity of u/δ up to the boundary, that is Theorem 1.2. + +BOUNDARY REGULARITY +23 +Proof of Theorem 1.2. Replacing u by +u +CK we may assume that |Lu| ≤ 1 in Ω. Let v = u/δ. From +Lemma 3.1 we then have +∥v∥L∞(Ω) ≤ C, +for some constant C and from Theorem 1.1 we have +∥u∥C0,1(Rd) ≤ C. +(4.28) +Also from Proposition 4.1 for each x0 ∈ ∂Ω and for all r > 0 we have +sup +Dr(x0) +v − +inf +Dr(x0) v ≤ Crτ. +(4.29) +where Dr(x0) = Br(x0) ∩ Ω as before. To complete the proof we shall show that +sup +x,y∈Ω,x̸=y +|v(x) − v(y)| +|x − y|κ +≤ C, +(4.30) +for some κ > 0. +Let r = |x − y| and there exists x0, y0 ∈ ∂Ω such that δ(x) = |x − x0| and +δ(y) = |y − y0|. If r ≥ 1 +8, then +|v(x) − v(y)| +|x − y|κ +≤ 2 · 8κ||v||L∞(Ω). +If r < 1 +8 and r ≥ 1 +8(δ(x) ∨ δ(y))p for some p > 2 then clearly y ∈ Bκr1/p(x0) for some κ > 0. Now +using (4.29) we obtain +|v(x) − v(y)| ≤ +sup +Dκr1/p(x0) +v − +inf +Dκr1/p(x0) v ≤ Cκrτ/p. +If r < 1 +8 and r < 1 +8(δ(x) ∨ δ(y))p, then r < 1 +8(δ(x) ∨ δ(y)) and this implies y ∈ B 1 +8(δ(x)∨δ(y))(x) or +x ∈ B 1 +8(δ(x)∨δ(y))(y). Without loss of any generality assume δ(x) ≥ δ(y) and y ∈ B δ(x) +8 (x). Using +(4.28) and the Lipschitz continuity of δ, we get +|v(x) − v(y)| = +���� +u(x) +δ(x) − u(y) +δ(y) +���� ≤ M(K, diam Ω)r +δ(x) · δ(y) +. +Also we have (8r)1/pδ(y) < δ(x) · δ(y). This implies +|v(x) − v(y)| ≤ M(K, diam Ω)r +δ(x) · δ(y) +< M(K, diam Ω) +81/p +· r1−1/p +δ(y) . +Now if r < 1 +8(δ(y))p then one obtains +|v(x) − v(y)| < M(K, diam Ω) +81/p +· r1−1/p +δ(y) +≤ Cr1−2/p. +On the other hand, if r ≥ 1 +8(δ(y))p, since δ(y) > +1 +64δ(x) we have r ≥ 1 +8 · +� 1 +64 +�p (δ(x))p and this case +can be treated as previous. Therefore choosing κ = (1 − 2 +p) ∧ τ +p we conclude (4.30). This completes +the proof. +□ +5. Global Hölder regularity of the gradient +In this section we prove the Hölder regularity of Du up to the boundary. First, let us recall +L[x, u] = sup +θ∈Θ +inf +ν∈γ +� +Tr aθν(x)D2u(x) + Iθν[x, u] +� +. +We denote v = u +δ . Following [11], next we obtain the in-equations satisfied by v. + +24 +BOUNDARY REGULARITY +Lemma 5.1. Let Ω be bounded C2 domain in Rd. If |Lu| ≤ K in Ω and u = 0 in Ωc, then we have +Lv + 2K0d2 |Dδ| +δ +|Dv| ≥ 1 +δ +� +− K − |v|(P + + P + +k )δ − sup +θ,ν +Zθν[v, δ] +� +, +Lv − 2K0d2 |Dδ| +δ +|Dv| ≤ 1 +δ +� +K − |v|(P − + P − +k )δ − inf +θ,ν Zθν[v, δ] +� +(5.1) +for some K0, where +Zθν[v, δ](x) = +ˆ +Rd(v(y) − v(x))(δ(y) − δ(x))Nθν(x, y − x)dy. +Proof. First note that, since u ∈ C1(Ω) by Lemma 2.1, we have v ∈ C1(Ω). Therefore, Zθν[v, δ] is +continuous in Ω. Consider a test function ψ ∈ C2(Ω) that touches v from above at x ∈ Ω. Define +ψr(z) = +� +ψ(z) +in Br(x), +v(z) +in Bc +r(x). +By our assertion, we have ψr ≥ v for all r small. To verify the first inequality in (5.1) we must show +that +L[x, ψr] + 2k0d2 |Dδ(x)| +δ(x) +· |Dψr(x)| ≥ +1 +δ(x)[−K − |v(x)|(P+ + P+ +k )δ(x) − sup +θ,ν +Zθν[v, δ](x)], +(5.2) +for some r small. We define +˜ψr(z) = +� +δ(z)ψ(z) +in Br(x), +u(z) +in Bc +r(x). +Then, ˜ψr ≥ u for all r small. Since |Lu| ≤ K and δψr = ˜ψr, we obtain at a point x +−K ≤L[x, ˜ψr] += sup +θ∈Θ +inf +ν∈γ +� +δ(x) +� +Tr aθν(x)D2ψr(x) + Iθνψr(x) +� ++ ψr(x) +� +Tr aθν(x)D2δ(x) + Iθνδ(x) +� ++ Tr +� � +aθν(x) + aT +θν(x) +� +· (Dδ(x) ⊗ Dψr(x)) +� ++ Zθν[ψr, δ](x) +� +≤ δ(x)L[x, ψr] + sup +θ,ν +� +|ψr(x)| +� +Tr aθν(x)D2δ(x) + Iθνδ(x) +� ++ Tr +�� +aθν(x) + aT +θν(x) +� +· (Dδ(x) ⊗ Dψr(x)) +� ++ Zθν[ψr, δ](x) +� +≤ δ(x)L[x, ψr] + |v(x)| +� +P+ + P+ +k +� +δ(x) + 2K0d2|Dδ(x)| · |Dψr(x)| + sup +θ,ν +Zθν[ψr, δ](x), +for all r small and some constant K0, where Dδ(x)⊗Dψr(x) := +� +∂δ +∂xi · ∂ψr +∂xj +� +i,j . Rearranging the terms +we have +− K − |v(x)| +� +P+ + P+ +k +� +δ(x) − sup +θ,ν +Zθν[ψr, δ](x) ≤ δ(x)L[x, ψr] + 2K0d2|Dδ(x)| · |Dψr(x)|. +(5.3) +Let r1 ≤ r. Since ψr is decreasing with r, we get from (5.3) that +δ(x)L[x, ψr] + 2K0d2|Dδ(x)| · |Dψr(x)| ≥ δ(x)L[x, ψr1] + 2K0d2|Dδ(x)| · |Dψr1(x)| +≥ lim +r1→0 +� +−K − |v(x)| +� +P+ + P+ +k +� +δ(x) − sup +θ,ν +Zθν[ψr1, δ](x) +� += +� +−K − |v(x)| +� +P+ + P+ +k +� +δ(x) − sup +θ,ν +Zθν[v, δ](x) +� +, + +BOUNDARY REGULARITY +25 +by dominated convergence theorem. This gives (5.2). Similarly we can verify the second inequality +of (5.1). +□ +Next we obtain a the following estimate on v, away from the boundary. Denote Ωσ = {x ∈ Ω : +dist(x, Ωc) ≥ σ}. +Lemma 5.2. Let Ω be bounded C2 domain in Rd. If |Lu| ≤ K in Ω and u = 0 in Ωc, then for some +constant C it holds that +∥Dv∥L∞(Ωσ) ≤ CKσκ−1 +for all σ ∈ (0, 1). +(5.4) +Furthermore, there exists η ∈ (0, 1) such that for any x ∈ Ωσ and 0 < |x − y| ≤ σ/8 we have +|Dv(y) − Dv(x)| +|x − y|η +≤ CKσκ−1−η, +for all σ ∈ (0, 1). +Proof. Using Lemma 5.1 we have +Lv + 2K0d2 |Dδ| +δ +|Dv| ≥ 1 +δ +� +− K − |v|(P+ + P+ +k )δ − sup +θ,ν +Zθν[v, δ] +� +, +Lv − 2K0d2 |Dδ| +δ +|Dv| ≤ 1 +δ +� +K − |v|(P− + P− +k )δ − inf +θ,ν Zθν[v, δ] +� +(5.5) +in Ω. Fix a point x0 ∈ Ωσ and define +w(x) = v(x) − v(x0). +From (5.5) we then obtain +Lw + 2K0d2 |Dδ| +δ +|Dw| ≥ +� +− 1 +δ K − ℓ1 +� +, +Lw − 2K0d2 |Dδ| +δ +|Dw| ≤ +�1 +δ K + ℓ2 +� +(5.6) +in Ω, where +ℓ1(x) = +1 +δ(x) +� +|w(x)|(P+ + P+ +k )δ(x) + sup +θ,ν +Zθν[w, δ](x) + |v(x0)|(P+ + P+ +k )δ(x) +� +And +ℓ2(x) = +1 +δ(x) +� +|w(x)|(P− + P− +k )δ(x) − inf +θ,ν Zθν[w, δ](x) − |v(x0)|(P− + P− +k )δ(x) +� +. +We set r = σ +2 and claim that +∥ℓi∥L∞(Br(x0)) ≤ κ1σκ−2, +for all σ ∈ (0, 1) and i = 1, 2, +(5.7) +for some constant κ1. Let us denote by +ξ± +1 = |w(x)|(P± + P± +k )δ +δ +, +ξ2 = 1 +δ sup +θ,ν +Zθν[w, δ], +ξ± +3 = |v(x0)|(P± + P± +k )δ +δ +, +ξ4 = 1 +δ inf +θ,ν Zθν[v, δ]. +Recall that κ ∈ (0, ˆα). Since +∥P±δ∥L∞(Ω) < ∞ +and +∥P± +k δ∥L∞(Ωσ) ≲ +� +1 + 1(1,2)(α)δ1−α� +(cf Lemma 4.4 ), and +∥v∥L∞(Rd) < ∞, +∥w∥L∞(Br(x0)) ≲ rκ, +it follows that +∥ξ± +3 ∥L∞(Br(x0)) ≲ +� +1 +σ +if α ∈ (0, 1], +1 +σα +if α ∈ (1, 2) +� +≲ σκ−2, + +26 +BOUNDARY REGULARITY +and +∥ξ± +1 ∥L∞(Br(x0)) ≲ +� +σκ +δ2 +if α ∈ (0, 1], +σκ +δα +if α ∈ (1, 2) +� +≲ σκ−2. +Next we estimate ξ2 and ξ4. Let x ∈ Br(x0). Denote by ˆr = δ(x)/4. Note that +δ(x) ≥ δ(x0) − |x − x0| ≥ 2r − r = r ⇒ ˆr ≥ r/4. +Since u ∈ C1(Ω) by Lemma 2.1 and |u| ≤ Cδ in Rd by Lemma 3.1. Thus we have +|Dv| ≤ +���� +Du +δ +���� + +���� +uDδ +δ2 +���� ≲ +1 +δ(x) +in Bˆr(x). +(5.8) +Now we calculate +|Zθν[w, δ](x)| ≤ +ˆ +Rd |δ(x) − δ(y)||v(x) − v(y)|k(y − x)dy = +ˆ +Bˆr(x) ++ +ˆ +B1(x)\Bˆr(x) ++ +ˆ +Bc +1(x) += I1 + I2 + I3. +To estimate I1, first we consider α ≤ 1. Since δ is Lipschitz continuous and v bounded on Rd, I1 can +be written as +I1 = +ˆ +Bˆr(x) +|δ(x) − δ(y)| +|x − y| +|v(x) − v(y)| · |x − y|k(y − x)dy +≲ +ˆ +Bˆr(x) +|x − y|αk(y − x)dy ≤ +ˆ +Rd(1 ∧ |z|α)k(z)dz. +For α ∈ (1, 2), using the Lipschitz continuity of δ and (5.8) we get +I1 = +ˆ +Bˆr(x) +|δ(x) − δ(y)| +|x − y| +· |v(x) − v(y)| +|x − y| +· |x − y|α|x − y|2−αk(y − x)dy +≲ ˆr2−α +δ(x) +ˆ +Bˆr(x) +|x − y|αk(y − x)dy ≲ δ(x)1−α +ˆ +Rd(1 ∧ |z|α)k(z)dz ≲ σκ−1. +Bounds on I2 can be computed as follows: for α ≤ 1, we write +I2 = +ˆ +B1(x)\Bˆr(x) +|δ(x) − δ(y)||v(x) − v(y)|k(y − x)dy ≲ +ˆ +B1(x)\Bˆr(x) +|x − y|αk(y − x)dy +≲ +ˆ +Rd(1 ∧ |z|α)k(z)dz. +In the second line of the above inequality we used +|δ(x) − δ(y)| ≲ |x − y| and ||v||L∞(Rd) < ∞. +For α ∈ (1, 2) we can compute I2 as +ˆ +B1(x)\Bˆr(x) +|δ(x) − δ(y)||v(x) − v(y)|k(y − x)dy ≲ +ˆ +B1(x)\Bˆr(x) +|x − y|1−α · |x − y|αk(y − x)dy +≲ δ(x)1−α +ˆ +Rd(1 ∧ |z|α)k(z)dz ≲ σκ−1. +Moreover, since δ and v are bounded in Rd, we get I3 ≤ κ3. Combining the above estimates we obtain +∥ξi∥L∞Br(x0) ≲ σκ−2 for i = 2, 4. +Thus the claim (5.7) is established. +Let us now define ζ(z) = w( r +2z + x0). Letting b(z) = +Dδ( r +2 z+x0) +2δ( r +2z+x0) it follows from (5.6) that +˜Lrζ + K0d2rb(z) · |Dζ| ≥ −r2 +4 +�1 +δ K + l1 +� �r +2z + x0 +� +(5.9) + +BOUNDARY REGULARITY +27 +˜Lrζ − K0d2rb(z) · |Dζ| ≤ r2 +4 +�1 +δ K + l2 +� �r +2z + x0 +� +in B2(0), where +˜Lr[x, u] := sup +θ∈Θ +inf +ν∈Γ +� +Tr +� +aθν +�r +2x + x0 +� +D2u(x) +� ++ ˜Ir +θν[x, u] +� +and ˜Ir +θν is given by +˜Ir +θν[x, f] = +ˆ +Rd +� +f(x + y) − f(x) − 1B 1 +r (y)∇f(x) · y +� �r +2 +�d+2 +Nθν +�r +2x + x0, ry +� +dy. +Consider a cut-off function ϕ satisfying ϕ = 1 in B3/2 and ϕ = 0 in Bc +2. Defining ˜ζ = ζϕ we get +from (5.9) that +˜Lr[z, ˜ζ] + K0d2rb(z).|D˜ζ(z)| ≥ −r2 +4 +�K +δ + |l1| +� �r +2z + x0 +� +− +����sup +θ∈Θ +inf +ν∈Γ +˜Ir +θν[z, (ϕ − 1)ζ] +���� +˜Lr[z, ˜ζ] − K0d2rb(z).|D˜ζ(z)| ≤ r2 +4 +�K +δ + |l1| +� �r +2z + x0 +� +− +����sup +θ∈Θ +inf +ν∈Γ +˜Ir +θν[z, (ϕ − 1)ζ] +���� +in B1. Since +∥rb∥L∞(B1(0)) ≤ κ3 +for all σ ∈ (0, 1), +applying Lemma 2.1 we obtain, for some η ∈ (0, 1), +∥Dζ∥Cη(B1/2(0)) ≤ κ6 +� +∥˜ζ∥L∞(Rd) + κ4σ + κ5σκ� +, +(5.10) +for some constant κ6 independent of σ ∈ (0, 1), where we used +���˜Ir +θν[z, (ϕ − 1)ζ] +��� ≲ σ +(cf. the proof of Theorem 1.1) and |l1|(r +2 · +x0) ≲ σκ−2. +Since v is in Cκ(Rd), it follows that +∥˜ζ∥L∞(Rd) = ∥˜ζ∥L∞(B2) ≤ ∥ζ∥L∞(B2) ≲ rκ. +Putting these estimates in (5.10) and calculating the gradient at z = 0 we obtain +|Dv(x0)| ≲ σκ−1, +for all σ ∈ (0, 1). This proves the Hölder estimate (5.4). +For the second part, compute the Hölder ratio with Dζ(0) − Dζ(z) where z = +2 +r(y − x0) for +|x0 − y| ≤ σ/8. This completes the proof. +□ +Now we can complete the proof of Theorem 1.3. If u is solution of the in-equation (1.1) then using +Theorem 1.1 we have |Lu| ≤ CK. Now the proof can be obtained by following the same lines as in +[11, Theorem 1.3]. We present it here for the sake of completeness. +Proof of Theorem 1.3. Since u = vδ it follows that +Du = vDδ + δDv. +Since δ ∈ C2(¯Ω), it follows from Theorem 1.2 that vDδ ∈ Cκ(¯Ω). Thus, we only need to concentrate +on ϑ = δDv. Consider η from Lemma 5.2 and with no loss of generality, we may fix η ∈ (0, κ). +For |x − y| ≥ 1 +8(δ(x) ∨ δ(y)) it follows from (5.4) that +|ϑ(x) − ϑ(y)| +|x − y|η +≤ CK(δκ(x) + δκ(y))(δ(x) ∨ δ(y))−η ≤ 2CK. + +28 +BOUNDARY REGULARITY +So consider the case |x − y| < +1 +8(δ(x) ∨ δ(y)). +Without loss of generality, we may assume that +|x − y| < 1 +8δ(x). Then +9 +8δ(x) ≥ |x − y| + δ(x) ≥ δ(y) ≥ δ(x) − |x − y| ≥ 7 +8δ(x). +By Lemma 5.2, it follows +|ϑ(x) − ϑ(y)| +|x − y|η +≤ |Dv(x)||δ(x) − δ(y)| +|x − y|η ++ δ(y)|Dv(x) − Dv(y)| +|x − y|η +≲ δ(x)κ−1(δ(x))1−η + δ(y)[δ(x)]κ−1−η +≤ CK. +This completes the proof. +□ +A. Appendix +In this section we aim to present a proof of Lemma 2.1. For this purpose, we first introduce the +scaled operator. Let x0 ∈ Ω and r > 0, we define the doubly scaled operator as +Lr,s(x0)[x, u] = sup +θ∈Θ +inf +ν∈Γ +� +Tr aθν(sr(x − x0) + sx0)D2u(x) + Ir,s +θν (x0)[x, u] +� +(A.1) +where +Ir,s +θν (x0)[x, u] = +ˆ +Rd(u(x + y) − u(x) − 1B 1 +sr (y)∇u(x) · y)rd+2(sd+2Nθν(rs(x − x0) + sx0, sry)dy. +Further, we define +L0,s(x0)[x, u] := sup +θ∈Θ +inf +ν∈Γ +� +Tr aθν(sx0)D2u(x) +� +. +(A.2) +Now we give the definition of weak convergence of operators. +Definition A.1. Let Ω ⊂ Rd be open and 0 < r < 1. A sequence of operators Lm is said to converge +weakly to L in Ω, if for any test function ϕ ∈ L∞(Rd) ∩ C2(Br(x0)) for some Br(x0) ⊂ Ω, we have +Lm[x, ϕ] → L[x, ϕ] +uniformly in B r +2 (x0) as m → ∞. +The next lemma is a slightly modified version of [44, Lemma 4.1] which can be proved by similar +arguments. +Lemma A.1. For any x0 ∈ B1, r > 0 and 0 < s < 1, Let Lr,s(x0) and L0,s(x0) is given by (A.1) +and (A.2) respectively where the Assumption 1.1 are satisfied by the corresponding coefficients with +Ω = B2. Moreover, for given M, ε > 0 and a modulus of continuity ρ, there exists r0, η > 0 independent +of x0 and s such that if +(i) r < r0, L0,s(x0)[x, u] = 0 in B1, +(ii) +Lr,s(x0)[x, u] + C0rs|Du(x)| ≥ −η in B1 +Lr,s(x0)[x, u] − C0rs|Du(x)| ≤ η in B1, +u = v in ∂B1. +(iii) |u(x)| + |v(x)| ≤ M in Rd and |u(x) − u(y)| + |v(x) − v(y)| ≤ ρ(|x − y|) for all x, y ∈ B1, +then we have +|u − v| ≤ ε +in B1. + +BOUNDARY REGULARITY +29 +It is worth mentioning that in [44], the authors have set a uniform continuity assumption on the +nonlocal kernels Nθν(x, y) ( for the precise assumption, see Assumption (C) of [44, p. 391] ) which +is a standard assumption to make for the stability property of viscosity solutions. Namely, if we +have a sequence of integro-differential operators Lm converging weakly to L in Ω and a sequence +of subsolutions (or supersolutions) in Ω converging locally uniformly on any compact subset of Ω, +then the limit is also a subsolution (or supersolution) with respect to L. However in the case of +the operator Lr,s defined in (A.1), the nonlocal term Ir,s +θν can be treated as a lower order term that +converges to zero as r → 0 without any kind of continuity assumptions on nonlocal kernels Nθν. +Now we give the proof of Lemma 2.1. +Proof of Lemma 2.1. We will closely follow the proof of [44, Theorem 4.1]. Fix any x0 ∈ B1, let +Lrk,s(x0) and L0,s(x0) is given by (A.1) and (A.2) respectively. Then by [44, Lemma 3.1] as rk → 0, +we have +Lrk,s(x0) → L0,s(x0), +in the sense of Definition A.1. By interior regularity [16, Corollary 5.7], L0,s(x0) has C1,β estimate +for an universal constant β > 0. Now without loss of any generality we may assume that x0 = 0. Also +dividing u by ||u||L∞(Rd) + K in (2.1) we may assume that K = 1 and ||u||L∞(Rd) ≤ 1. +Using the Hölder regularity [44, Lemma 2.1], we have u ∈ Cβ(B1). Following [18, Theorem 52], +we will show that there exists δ, µ ∈ (0, 1 +4), independent of s and a sequence of linear functions +lk(x) = ak + bkx such that + + + + + + + + + + + + + +(i) +sup +B2δνk +|u − lk| ≤ µk(1+γ) , +(ii) |ak − ak−1| ≤ µ(k−1)(1+γ) , +(iii) µk−1|bk − bk−1| ≤ Cµ(k−1)(1+γ) , +(iv) |u − lk| ≤ µ−k(γ′−γ)δ−(1+γ′)|x|1+γ′ for x ∈ Bc +2δµk , +(A.3) +where 0 < γ < γ′ < β do not depend on s. We plan to proceed by induction, when k = 0, since +||u||L∞(Rd) ≤ 1, (A.3) holds with l−1 = l0 = 0. Assume (A.3) holds for some k and we shall show +(A.3) for k + 1. +Let ξ : Rd → [0, 1] be a continuous function such that +ξ(x) = +� +1 for x ∈ B3, +0 for x ∈ Bc +4. +Let us define +wk(x) = (u − ξlk)(δµkx) +µk(1+γ) +. +We claim that there exists a universal constant C > 0, such that for all k, we have +Lrk,s[x, wk] − C0rks|Dwk(x)| ≤ Cδ2µk(1−γ) ≤ Cδ2, +Lrk,s[x, wk] + C0rks|Dwk(x)| ≥ −Cδ2µk(1−γ) ≥ −Cδ2, +(A.4) +in B2 in viscosity sense. Let φ ∈ C2(B2) ∩ C(Rd) which touches wk from below at x′ in B2. Let +ψ(x) := µk(1+γ)φ +� x +δµk +� ++ ξlk(x). +Then ψ ∈ C2(B2δµk) ∩ C(Rd) is bounded and touches u from below at δµkx′. Taking rk = δµk, we +have +Irk,s +θν [x′, φ] = δ2µk(1−γ)Is +θν[rkx′, ψ − ξlk]. + +30 +BOUNDARY REGULARITY +Thus we get +Lrk,s[x′, φ] − C0rks|Dφ(x′)| += δ2µk(1−γ)� +sup +θ∈Θ +inf +ν∈Γ +� +Tr aθν(srkx′)D2ψ(rkx′) + Is +θν[rkx′, ψ − ξlk] +� +− sC0|Dψ(rkx′) − bk| +� +≤ δ2µk(1−γ)� +Ls[rkx′, ψ] − sC0|Dψ(rkx′)| + sup +θ∈Θ +inf +ν∈Γ{−Is +θν[rkx′, ξlk]} + sC0|bk| +� +≤ Cδ2µk(1−γ) ≤ Cδ2. +In the second last inequality we use that +Ls[x, u] − C0s|Du(x)| ≤ 1, +and |ak|, |bk| are uniformly bounded and for all x′ ∈ B2, sup +θ∈Θ +inf +ν∈Γ{−Is +θν[rkx′, ξlk]} is bounded inde- +pendent of s and k . Thus we have proved +Lrk,s[x, wk] − C0rks|Dwk(x)| ≤ Cδ2 in B2, +in viscosity sense. Similarly the other inequality in (A.4) can be proven. +Define w′ +k(x) := max {min {wk(x), 1} , −1} . We see that w′ +k is uniformly bounded independent of +k. We claim that in B 3 +2 +Lrk,s[x, w′ +k] − C0rks|Dw′ +k(x)| ≤ Cδ2 + ω1(δ), +Lrk,s[x, w′ +k] + C0rks|Dw′ +k(x)| ≥ −Cδ2 − ω1(δ) +(A.5) +Now take any bounded φ ∈ C2(B2) ∩ C(Rd) that touches w′ +k from below at x′ in B3/2. By the +definition of w′ +k, in B2 we have |wk| = |w′ +k| ≤ 1 and φ touches wk from below at x′. Hence +sup +θ∈Θ +inf +ν∈Γ +� +Tr aθν(srkx′)D2φ(x′) ++ +ˆ +B1/2 +(φ(x′ + z) + φ(x′) − 1B 1 +rs (z)Dφ(x′) · z)(rks)d+2Nθν(rksx, srkz)dz +− +ˆ +Rd\B1/2 +(wk(x′ + z) − w′ +k(x′ + z) ++ w′ +k(x′ + z) − φ(x′) − 1B 1 +rs (z)Dφ(x′) · z))(rks)d+2Nθν(rksx, srkz)dz +� +− C0rks|Dφ(x′)| ≤ Cδ2 +Therefore by Definition 2.1 of viscosity supersolution and using the bounds on the kernel we get the +following estimate: +Lrk,s[x, w′ +k] − C0rks|Dw′ +k(x)| ≤ +ˆ +Rd\B1/2 +��wk(x′ + z) − w′ +k(x′ + z) +�� (rks)d+2k(rksz)dz + Cδ2. +in the viscosity sense. By the inductive assumptions, we have ak and bk uniformly bounded. Since +||u||L∞(Rd) ≤ 1 and ξlk is uniformly bounded, |wk| ≤ Cµ−k(1+γ) in Rd. Using (iv) from (A.3) we have +|wk(x)| = (u − ξlk)(rkx) +µk(1+γ) +≤ +� 1 +rk +�1+γ′ +|rkx|1+γ′ = |x|1+γ′, +for any x ∈ Bc +2 ∩ B 2 +rk . Again for any x ∈ Bc +2/rk, we find +|wk(x)| ≤ Cµ−k(1+γ′) · µ−k(γ−γ′) ≤ Cµ−k(1+γ′) ≤ C δ1+γ′ +2 +|x|1+γ′ ≤ C|x|1+γ′. + +BOUNDARY REGULARITY +31 +Now, since w′ +k is uniformly bounded, we have for x ∈ Bc +2, +|wk| + |w′ +k − wk| ≤ C min{|x|1+γ′, µ−k(1+γ)}. +(A.6) +For x′ ∈ B3/2, using (A.6) we have the following estimate. +ˆ +Rd +��wk(x′ + z) − w′ +k(x′ + z) +�� (rks)d+2k(rksz)dz +≤ +ˆ +{z:|x′+z|≥2}∩B1/rk +��wk − w′ +k +�� (x′ + z)(rks)d+2k(rksz)dz + δ2µk(1−γ) +ˆ +Bc +1 +rk +(rks)d+2k(rksz) +(δµ−k)2 +dz +≤ C +� ˆ +Bc +1/2∩B +1 +√rk +|z|2(rks)d+2k(rksz)dz + r +(1−γ′) +2 +k +ˆ +Bc +1 +√rk +∩B 1 +rk +|z|2(rks)d+2k(rksz)dz ++ δ2µk(1−γ) +ˆ +Bcs +s2k(z)dz +� +≤ C +� ˆ +B√rk +|y|2k(y)dy + (r +(1−γ′) +2 +k ++ δ2µk(1−γ)) +ˆ +Rd(1 ∧ |y|2)k(y)dy +� +. +Hence, +ˆ +Rd +��wk(x′ + z) − w′ +k(x′ + z) +�� krk,s(z)dz ≤ ˜C +�ˆ +B√ +δ +|y|2k(y)dy + δ +1−γ′ +2 ++ δ2 +� += ω1(δ) +where ω1(δ) → 0 as δ → 0. Therefore we proved Lrk,s[x, w′ +k] − C0rks|Dw′ +k(x)| ≤ Cδ2 + ω1(δ). The +other inequality of (A.5) can be proved in a similar manner. +Since w′ +k satisfies the equation (A.5), by [44, Lemma 2.1] we have ||w′ +k||Cβ(B1) ≤ M1 for some M1 +independent of k, s. Now we consider the a function h which solves +L0,s(x0)[x, h] = 0 +in B1 +h = w′ +k +on ∂B1. +Existence of such h can be seen from [51, Theorem 1]. Moreover, using [51, Theorem 2] we have +||h||Cα(B1) ≤ M2 where α < β +2 and M2 is independent of k, s. Now for any 0 < ε < 1, let r0 := r0(ε) +and η := η(ε) as given in Lemma A.1. Also for x ∈ B1 and δ := δ(ε) ≤ r0, we have +Lrk,s[x, w′ +k] + C0rks|Dw′ +k(x)| ≥ −η, +Lrk,s[x, w′ +k] − C0rks|Dw′ +k(x)| ≤ η. +Therefore by Lemma A.1, we conclude |w′ +k −h| ≤ ε in B1. Again by using [16, Corollary 5.7], we have +h ∈ C1,β(B1/2) and we can take a linear part l(x) := a + bx of h at the origin. By C1,β estimate of +L0,s(x0) and |w′ +k| ≤ 1 in B1 we obtain that the coefficients of l, i.e, a, b are bounded independent of +k, s. Further for x ∈ B1/2, we have +|h(x) − l(x)| ≤ C1|x|1+β, +where C1 is independent of k, s. Hence using the previous estimate we get +|w′ +k(x) − l(x)| ≤ ǫ + C1|x|1+β in B1/2. +Again using (A.6) and |wk| ≤ 1 in B2 we have +|wk(x) − l(x)| ≤ 1 + |a| + |b| ≤ C2 in B1, +|wk(x) − ξ(δµkx)l(x)| ≤ C|x|1+γ′ + C3|x| in Bc +1. + +32 +BOUNDARY REGULARITY +Next defining +lk+1(x) := lk(x) + µk(1+γ)l +� +δ−1µ−kx +� +, +wk+1(x) := (u − ξlk+1)(δµk+1x) +µ(k+1)(1+γ) +, +and following the proof of [44, Theorem 4.1] we conclude that (A.3) holds for k + 1. This completes +the proof. +□ +Acknowledgement. We thank Anup Biswas for several helpful discussions during the prepara- +tion of this article. +Mitesh Modasiya is partially supported by CSIR PhD fellowship (File no. +09/936(0200)/2018-EMR-I). +References +[1] D. Applebaum: Lévy processes and stochastic calculus. Second edition. Cambridge Studies in Advanced Mathe- +matics, 116. 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J. 55 (2006), no. 3, 1155–1174. + diff --git a/6dE0T4oBgHgl3EQffAAW/content/tmp_files/load_file.txt b/6dE0T4oBgHgl3EQffAAW/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e5f339fe719ff7427aca15c31eb3fd5c3ab0ae6c --- /dev/null +++ b/6dE0T4oBgHgl3EQffAAW/content/tmp_files/load_file.txt @@ -0,0 +1,1403 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf,len=1402 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='02397v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='AP] 6 Jan 2023 Fine boundary regularity for fully nonlinear mixed local-nonlocal problems MITESH MODASIYA AND ABHROJYOTI SEN Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' We consider Dirichlet problems for fully nonlinear mixed local-nonlocal non-translation invariant operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' For a bounded C2 domain Ω ⊂ Rd, let u ∈ C(Rd) be a viscosity solution of such Dirichlet problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' We obtain global Lipschitz regularity and fine boundary regularity for u by constructing appropriate sub and supersolutions coupled with a weak version of Harnack inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' We apply these results to obtain Hölder regularity of Du up to the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Introduction and main results In this article, for a bounded C2 domain Ω ⊂ Rd we establish the boundary regularity of the solution u to the in-equations Lu + C0|Du| ≥ −K in Ω, Lu − C0|Du| ≤ K in Ω, u = 0 in Ωc, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='1) where C0, K ≥ 0 and L is a fully nonlinear integro-differential operator of the form Lu(x) := L[x, u] = sup θ∈Θ inf ν∈Γ � Tr aθν(x)D2u(x) + Iθν[x, u] � , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='2) for some index sets Θ, Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' The coefficient aθν : Ω → Rd×d is a matrix valued function and Iθν is a nonlocal operator defined as Iθνu(x) := Iθν[x, u] = ˆ Rd(u(x + y) − u(x) − 1B1(y)Du(x) · y)Nθν(x, y) dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='3) The above in-equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='1) are motivated by Hamilton-Jacobi equations of the form Iu(x) := sup θ∈Θ inf ν∈Γ {Lθνu(x) + fθν(x)} = 0, where Lθνu(x) = Tr aθν(x)D2u(x) + Iθν[x, u] + bθν(x) · Du(x), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='4) bθν(·) and fθν(·) are bounded functions on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' These linear operators (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='4) are extended generator for a wide class of d-dimensional Feller processes (more precisely, jump diffusions) and the nonlinear operator Iu(·) has its connection to the stochastic control problems and differential games (see [12,13] and the references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' The first term in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='4) represents the diffusion, the second term represents the jump part of a Feller process, and the third represents the drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' We refer to [1, 14, 15] and the references therein for more on the connections between the operators of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='4) and stochastic differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' For a precise application of these type of operators in finance and biological models, we refer to [20,23,24] and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Department of Mathematics, Indian Institute of Science Education and Research, Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Homi Bhabha Road, Pune 411008, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Email: mitesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='modasiya@students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='iiserpune.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='in;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' abhrojyoti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='sen@acads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='iiserpune.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='in 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Primary: 35D40, 47G20, 35J60, 35B65 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Operators of mixed order, viscosity solution, fine boundary regularity, fully nonlinear integro-PDEs, Harnack inequality, gradient estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' 1 2 BOUNDARY REGULARITY We set the following assumptions on the coefficient aθν(·) and the kernel Nθν(x, y), throughout this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' (a) aθν(·) are uniformly continuous and bounded in ¯Ω, uniformly in θ, ν for θ ∈ Θ, ν ∈ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Further- more, aθν(·) satisfies the uniform ellipticity condition λI ≤ aθν(·) ≤ ΛI for some 0 < λ ≤ Λ where I denotes the d × d identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' (b) For each θ ∈ Θ, ν ∈ Γ, Nθν : Ω × Rd is a measurable function and for some α ∈ (0, 2) there exists a kernel k that is measurable in Rd \\{0} such that for any θ ∈ Θ, ν ∈ Γ, x ∈ Ω, we have 0 ≤ Nθν(x, y) ≤ k(y) and ˆ Rd(1 ∧ |y|α)k(y)dy < +∞, where we denote p ∧ q := min{p, q} for p, q ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Let us comment briefly on Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' The uniform continuity of aθν(·) is required for the stability of viscosity sub or supersolutions under appropriate limits and useful in Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='1 which is a key step for proving interior C1,γ regularity (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' The Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='1(b) includes a large class of kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' We mention some of them below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Consider the following kernels Nθν(x, y) : (i) Nθν(x, y) = 1 |y|d+σ for σ ∈ (0, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Clearly we can take k(y) = 1 |y|d+σ and ´ Rd(1 ∧ |y|α)k(y)dy is finite for α ∈ [1 + σ/2, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' (ii) Nθν(x, y) = �∞ i=1 ai |y|d+σi for σi ∈ (0, 2), σ0 = supi σi < 2 and �∞ i=1 ai = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Similarly taking Nθν(x, y) = k(y) we can see ´ Rd(1 ∧ |y|α)k(y) < +∞ for α ∈ [1 + σ0/2, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' (iii) Nθν(x, y) = \uf8f1 \uf8f2 \uf8f3 (1−log |y|)β |y|d+σ for 0 < |y| ≤ 1 (1+log |y|)−β |y|d+σ for |y| ≥ 1, where σ ∈ (0, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' (a) For 2(2 − σ) > β ≥ 0, taking Nθν(x, y) = k(y) we have ´ Rd(1 ∧ |y|α)k(y)dy < +∞ for α ∈ [1 + σ 2 + β 4 , 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' (b) For −σ < β < 0, taking Nθν(x, y) = k(y) we have ´ Rd(1 ∧ |y|α)k(y)dy < +∞ for α ∈ [1 + σ 2 , 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Proof of (a): ˆ Rd(1 ∧ |y|α)k(y)dy = ˆ |y|≤1 |y|α(1 − log |y|)β |y|d+σ dy + ˆ |y|>1 (1 + log |y|)−β |y|d+σ dy := I1 + I2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Using (1 − log |y|) ≤ 1 √ |y| + 1 and the convexity of ξ(t) = tp for p ≥ 1 we get (1 − log |y|)β ≤ C � 1 |y|β/2 + 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Therefore I1 ≤ ˆ |y|≤1 Cdy |y|β/2+d+σ−α + ˆ |y|≤1 Cdy |y|d+σ−α < +∞ for α ∈ [1 + σ/2 + β/4, 2), and I2 ≤ ˆ |y|>1 dy |y|d+σ < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' BOUNDARY REGULARITY 3 Proof of (b): Since β < 0 in this case, we have (1−log |y|)β ≤ 1 and I1 < +∞ for α ∈ [1+ σ 2 , 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' To estimate I2, observe (1 + log |y|)−β ≤ (1 + |y|)−β and I2 ≤ C ˆ |y|>1 (1 + |y|−β) |y|d+σ dy < +∞ since σ > −β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' (iv) Nθν(x, y) = Ψ(1/|y|2) |y|d+σ(x,y), where σ : Rd × Rd → R satisfying 0 < σ− := inf (x,y)∈Rd×Rd σ(x, y) ≤ sup (x,y)∈Rd×Rd σ(x, y) := σ+ < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' and Ψ is a Bernstein function (for several examples of such functions, see [50]) vanishing at zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Furthermore, Ψ is non-decreasing, concave and satisfies a weak upper scaling property i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='e, there exists µ ≥ 0 and c ∈ (0, 1] such that Ψ(λx) ≤ cλµΨ(x) for x ≥ s0 > 0, λ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' For µ < 2(2 − σ+), we can take k(y) = \uf8f1 \uf8f2 \uf8f3 Ψ(1) |y|d+2µ+σ+ , if 0 < |y| ≤ 1, Ψ(1) |y|d+σ− , if |y| > 1 and ´ Rd(1 ∧ |y|α)k(y)dy < +∞ for α ∈ [1 + µ + σ+/2, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' The main purpose of this article is to establish a global Lipschitz regularity and boundary regularity of the solutions satisfying (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='1) under the Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' On the topic of regularity theory for linear elliptic equations, Hölder estimate plays a key role and it can be obtained by using Harnack inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' The pioneering contributions are by DeGiorgi-Nash-Moser [29, 42, 45] who proved Cα regularity for solutions to the second order elliptic equations in divergence form with measurable coefficients under the assumption of uniform ellipticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' For equations of non-divergence form, the corresponding regularity theory was established by Krylov and Safonov [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' We refer [16] for a comprehensive overview on the regularity theory for fully nonlinear elliptic equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' In [40], Krylov studied the boundary regularity for local second order elliptic equations in non-divergence form with bounded measurable coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' He obtained the Hölder regularity of u δ up to the boundary where δ denotes the distance function, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='e, δ(x) = dist(x, Ωc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Turning our attention towards the nonlocal equations, first Hölder estimates and Harnack inequal- ities for s-harmonic functions are proved by Bass and Kassmann [3–5], however their approach was purely probabilistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' In the realm of analytic setup, Silvestre [52] proved Hölder continuity of u satis- fying (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='3) with some structural assumptions on the operator and kernel related to the assumptions of Bass and Kassmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Analogous to the local case [40], in the nonlocal setting, for a bounded domain Ω ⊂ Rd with C1,1 boundary the first result concerning boundary regularity of u solving the Dirichlet problem for (−∆)s with bounded right hand side is obtained by Ros-Oton and Serra [46] where they established a Hölder regularity of u/δs up to the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' This result is proved by using a method of Krylov (see [33]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' The idea is to obtain a bound for u with respect to a constant multiple of δs and this controls the oscillation of u/δs near the boundary ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' The Hölder regularity of u/δs, (i) for more general nonlocal linear operators with C1,α domain is established in [48], (ii) for smooth domain with smooth right hand side is established in [30,31], (iii) for kernel with variable order see [34] and (iv) for Dirichlet problem for fractional p-Laplacian, see [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' In a seminal paper, Caffarelli and Silvestre [17] studied the regularity theory for fully nonlinear integro-differential equations of the form : supθ∈Θ infν∈Γ I[x, u] where I[x, u] is given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' By obtaining a nonlocal ABP estimate, they established the Hölder regularity and Harnack inequality when Nθν(y) (Nθν(y) denotes the x-independent form of Nθν(x, y)) is positive, symmetric and com- parable with the kernel of the fractional Laplacian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' From a large amount of literature that extend the work of Caffarelli and Silvestre [17], we mention [36] where the authors considered integro-PDEs 4 BOUNDARY REGULARITY with regularly varying kernel, [9,19,37] where regularity results are obtained for symmetric and non- symmetric stable-like operators and [35] for kernels with variable order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Also a recent paper [38] studies Hölder regularity and a scale invariant Harnack inequality under some weak scaling condition on the kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Boundary regularity results for fully nonlinear integro-differential equations are ob- tained by Ros-Oton and Serra in [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' They considered a restricted class of kernels L∗ where Nθν(x, y) is x-independent and of the following form Nθν(y) := µ(y/|y|) |y|d+2s with µ ∈ L∞(Sd−1), satisfying µ(θ) = µ(−θ) and λ ≤ µ ≤ Λ where 0 < λ ≤ Λ are the ellipticity constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' An interesting feature of L∗ is L(xd)s + = 0 in {xd > 0} for all L ∈ L∗ which is useful to construct barriers in their case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Note that our operators do not enjoy such property for having different orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Furthermore, with Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='1 the nonlocal part (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='3) is not scale invariant in our case, that is one may not find any 0 ≤ β ≤ 2 such that Iθν[x, u(r·)] = rβIθν[rx, u(·)] for any 0 < r < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Recently, the mathematical study of mixed local-nonlocal integro-differential equations have been received a considerable attention, for instance see [2,6–8,10,26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' The regularity results and Harnack inequality for mixed fractional p-Laplace equations are recently obtained in [27,28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' The interior Cα regularity theory for HJBI-type integro-PDEs has been studied by Mou [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' He obtained Hölder regularity for viscosity solutions under uniform ellipticity condition and a slightly weaker condition on kernels in compared to the Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='1 (b), that is ´ Rd(1 ∧ |y|2)k(y)dy < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' More recently global Lipschitz regularity (compare it with Biagi et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' [7]) and fine boundary regularity have been obtained for linear mixed local-nonlocal operators in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Since the nonlocal operator applied on the distance function becomes singular near the boundary for certain range of order of the kernel, one of the main challenges was to construct appropriate sub and supersolutions and prove an oscillation lemma following [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' To do such analysis, along with several careful estimates, the authors borrowed a Harnack inequality from [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Note that for fully nonlinear mixed operators of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='2) no such Harnack inequality is available in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' In this current contribution, we continue the study started in [11] to obtain the boundary regularity for fully nonlinear integro-differential problems of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Below, we present our first result that is the Lipschitz regularity of u satisfying (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='1) up to the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Note that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='5) can be achieved under some weaker assumptions on the domain and kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' For this result, we only assume ∂Ω to be C1,1 and ´ Rd(1 ∧ |y|2)k(y)dy < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Let Ω be a bounded C1,1 domain in Rd and u be a continuous function which solves the in-equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='1) in viscosity sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Then u is in C0,1(Rd) and there exists a constant C, depending only on d, Ω, λ, Λ, C0, ´ Rd(1 ∧ |y|2)k(y)dy, such that ∥u∥C0,1(Rd) ≤ CK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='5) To prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='1, the first step is to show that the distance function δ(x) = dist(x, Ωc) can be used as a barrier to u in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Once this is done, we can complete the proof by considering different cases depending on the distance between any two points in Ω or their distance from ∂Ω and combining |u| ≤ Cδ with an interior C1,γ-estimate for scaled operators (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Next we show the fine boundary regularity, that is the Hölder regularity of u/δ up to the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Suppose that Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Let Ω be a bounded C2 domain and u be a viscosity solution to the in-equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Then there exists κ ∈ (0, ˆα) such that ∥u/δ∥Cκ(Ω) ≤ C1K, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='6) BOUNDARY REGULARITY 5 for some constant C1, where κ, C1 depend on d, Ω, C0, Λ, λ, α and ´ Rd(1 ∧ |y|α)k(y)dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Here ˆα is given by ˆα = � 1 if α ∈ (0, 1] 2−α 2 if α ∈ (1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' To prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='2, following [46] we prove an oscillation lemma (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' For this, first we need to construct sub and supersolutions carefully since Iθνδ becomes singular near the boundary ∂Ω for α ∈ (1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Then we shall use a “weak version” of Harnack inequality (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' This weak version of Harnack inequality is new and needed to be developed due to the unavailability of classical Harnack inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Also, we must point out that one needs to bypass the use of comparison principle [10, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='1] in such analysis, since the mentioned theorem is for translation invariant linear operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' For non-translation invariant operators, such comparison principle is unavailable, see Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='1 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Now applying (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='6), we prove the Hölder regularity of Du up to the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Suppose that Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='1 holds and Ω be a bounded C2 domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Then for any viscosity solution u to the in-equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='1) we have ||Du||Cη(Ω) ≤ CK, for some η ∈ (0, 1) and C, depending only on d, Ω, C0, Λ, λ, α and ´ Rd(1 ∧ |y|α)k(y)dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' The interior C1,η-regularity for fully nonlinear integro-differential equations is studied in [17] by introducing a new ellipticity class where the kernels are C1 away from the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Kriventsov [39] extended this result without the additional assumption on kernels (sometimes referred as rough ker- nels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Also see [49] for its parabolic version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' For HJBI-type integro-PDEs, interior C1,η-regularity is established by Mou and Zhang [44] and for mixed local nonlocal fractional p-Laplacian, see [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' The C1,η-regularity up to the boundary for linear mixed local-nonlocal operators is recently obtained in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' The rest of the article is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' In Section 2, we introduce the necessary preliminaries and collect all the auxiliary results which will be used throughout the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' In Section 3 we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='2 is proved in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' In Section 5 we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Lastly, in Appendix A, following an approximation and scaling argument, we give a proof of C1,γ regularity for a scaled operator i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='e, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Notation and preliminary results This section sets the notation which we use throughout the paper and collects the necessary results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Notations and Definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' We start by setting the notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' We use Br(x) to denote an open ball of radius r > 0 centred at a point x ∈ Rd and for x = 0, we denote Br := Br(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' For any subset U ⊆ Rd and for α ∈ (0, 1), we denote Cα(U) as the space of all bounded, α-Hölder continuous functions equipped with the norm ||f||Cα(U) := sup x∈U |f(x)| + sup x,y∈U |f(x) − f(y)| |x − y|α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Note that for α = 1, C0,1(U) denotes the space of all Lipschitz continuous functions on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' The space of all bounded functions with bounded α-Hölder continuous derivatives is denoted by C1,α(U) with the norm ||f||C1,α(U) := sup x∈U |f(x)| + ||Df||Cα(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' We use USC(Rd), LSC(Rd), C(Rd), Cb(Rd), Md to denote the space of upper semicontinuous, lower semicontinuous, continuous functions, bounded continuous functions on Rd and d×d symmetric matrices respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' 6 BOUNDARY REGULARITY Now let us introduce the scaled operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' For 0 < s ≤ 1, we define scaled version of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='2) as following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Ls[x, u] = sup θ∈Θ inf ν∈Γ � Tr aθν(sx)D2u(x) + Is θν[x, u] � , where Is θν[x, u] = ˆ Rd(u(x + y) − u(x) − 1B 1 s (y)∇u(x) · y)sd+2Nθν(sx, sy)dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Next we define extremal Pucci operators for second order term and the nonlocal term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' P+u(x) := sup � Tr(AD2u(x)), A ∈ Md, λI ≤ A ≤ ΛI � , P−u(x) := inf � Tr(AD2u(x)), A ∈ Md, λI ≤ A ≤ ΛI � , and P+ k,su(x) := ˆ Rd(u(x + y) − u(x) − 1B 1 s (y)∇u(x) · y)+sd+2k(sy)dy, P− k,su(x) := − ˆ Rd(u(x + y) − u(x) − 1B 1 s (y)∇u(x) · y)−sd+2k(sy)dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Denote P+ k,1 = P+ k and P− k,1 = P− k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' We recall the definition of viscosity sub and supersolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' First of all, we say a function ϕ touches from above (below) at x if, for a small r > 0, ϕ(x) = u(x) and u(y) ≤ (≥)ϕ(y) for all y ∈ Br(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' A function u ∈ USC(Rd) ∩ L∞(Rd) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' u ∈ LSC(Rd) ∩ L∞(Rd)) is said to be a viscosity subsolution (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' supersolution) to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='1) if whenever ϕ touches u from above (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' below) for some bounded test function ϕ ∈ C2(Br(x)) ∩ C(Rd), then v = � ϕ in Br(x) u in Bc r(x) satisfies Lv(x) + C0|Dv(x)| ≥ −K (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Lv(x) − C0|Dv(x)| ≤ K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Auxiliary lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' We collect some preliminary results here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' The first result is the interior C1,γ regularity for the scaled operator Ls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Let 0 < s ≤ 1 and u ∈ L∞(Rd) ∩ C(Rd) solves the in-equations Ls[x, u] + C0s|Du(x)| ≥ −K in B2, Ls[x, u] − C0s|Du(x)| ≤ K in B2, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='1) in the viscosity sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Then there exist constants 0 < γ < 1 and C > 0 independent of s, such that ||u||C1,γ(B1) ≤ C � ||u||L∞(Rd) + K � , where γ and C depend only on d, λ, Λ, C0 and ´ Rd(1 ∧ |y|2)k(y)dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' The proof essentially uses the approximation arguments for nonlocal equations [18] and we postpone it to Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' □ Now we present a maximum principle type result similar to [10, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' We report the proof here for reader’s convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Let u be a bounded function on Rd which is in USC(Ω) and satisfies P+u + P+ k u + C0|Du| ≥ 0 in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Then we have supΩ u ≤ supΩc u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' BOUNDARY REGULARITY 7 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' From [43, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='5] we can find a non-negative function χ ∈ C2(¯Ω) ∩ Cb(Rd) satisfying P+χ + P+ k χ + C0|Dχ| ≤ −1 in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Note that, since χ ∈ C2(¯Ω), the above inequality holds in the classical sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' For ε > 0, we let φM to be φM(x) = M + εχ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Then P+φM(x0) + P+ k φM + C0|DφM| ≤ −ε in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Let M0 be the smallest value of M for which φM ≥ u in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' We show that M0 ≤ supΩc u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Suppose, to the contrary, that M0 > supΩc u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Then there must be a point x0 ∈ Ω for which u(x0) = φM0(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Otherwise using the upper semicontinuity of u, we get a M1 < M0 such that φM1 ≥ u in Rd, which contradicts the minimality of M0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Now φM0 would touch u from above at x0 and thus, by the definition of the viscosity subsolution, we would have that P+φM0(x0) + P+ k φM0 + C0|DφM0| ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' This leads to a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Therefore, M0 ≤ supΩc u which implies that for every x ∈ Rd u ≤ φM0 ≤ M0 + ε sup Rd χ ≤ sup Ωc u + ε sup Rd χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' The result follows by taking ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Although we have the above maximum principle, one can not simply compare two viscosity sub and supersolutions for the operator (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' More precisely, if u, v are bounded functions and u ∈ USC(Rd), v ∈ LSC(Rd) satisfy Lu + C|Du| ≥ f and Lv + C|Dv| ≤ g in Ω in viscosity sense for two continuous functions f and g, and for some C ≥ 0, then L(u−v)+C|D(u− v)| ≥ f − g may not always holds true in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' However, if one of them is C2, then we have P+(u − v) + P+ k (u − v) + C|D(u − v)| ≥ f − g in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Indeed, without loss of generality, let us assume v ∈ C2(Ω) and ϕ be a C2 test function that touches u − v at x ∈ Ω from above then clearly ϕ + v touches u at x from above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' By definition of viscosity subsolution we have L(ϕ + v)(x) + C|D(ϕ + v)(x)| ≥ f(x), which implies P+ϕ(x) + P+ k ϕ(x) + Lv(x) + C|Dϕ(x)| + C|Dv(x)| ≥ f(x) and hence we obtain P+ϕ(x) + P+ k ϕ(x) + C|Dϕ(x)| ≥ f(x) − g(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Global Lipschitz regularity In this section we establish the Lipschitz regularity of the solution u up to the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' We start by showing that the distance function δ(x) is a barrier to u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Let Ω be a bounded C1,1 domain in Rd and u be a continuous function which solves (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='1) in the viscosity sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Then there exists a constant C which depends only on d, λ, Λ, C0, diam(Ω), radius of exterior sphere and ´ Rd(1 ∧ |y|2)k(y)dy, such that |u(x)| ≤ CKδ(x) for all x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='1) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' First we show that |u(x)| ≤ κ K x ∈ Rd, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='2) for some constant κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' From [43, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='5], there exists a non-negative function χ ∈ C2(¯Ω)∩Cb(Rd), with infRd χ > 0, satisfying P+χ + P+ k χ + C0|Dχ| ≤ −1 in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' We define ψ = Kχ which gives that infRd ψ ≥ 0 and P+ψ + P+ k ψ + C0|Dψ| ≤ −K in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' 8 BOUNDARY REGULARITY Then by using Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='1, we get P+(u − ψ) + P+ k (u − ψ) + C0|D(u − ψ)| ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Now applying Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='2 on u − ψ we obtain sup Ω (u − ψ) ≤ sup Ωc (u − ψ) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Note that in the second inequality above we used u = 0 in Ωc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' This proves that u ≤ ψ in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Similar calculation using −u will also give us −u ≤ ψ in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Thus |u| ≤ sup Rd |χ| K in Rd, which gives (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Now we shall prove (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Since ∂Ω is C1,1, Ω satisfies a uniform exterior sphere condition from outside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Let r◦ be a radius satisfying uniform exterior condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' From [43, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='4] there exists a bounded, Lipschitz continuous function ϕ, Lipschitz constant being r−1 , satisfying ϕ = 0 in ¯Br◦, ϕ > 0 in ¯Bc r◦, ϕ ≥ ε in Bc (1+δ)r◦, P+ϕ + P+ k ϕ + C0|Dϕ| ≤ −1 in B(1+δ)r◦ \\ ¯Br◦, for some constants ε, δ, dependent on C0, d, λ, Λ, d and ´ Rd(1 ∧ |y|2)k(y)dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Furthermore, ϕ is C2 in B(1+δ)r◦ \\ ¯Br◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' For any point y ∈ ∂Ω, we can find another point z ∈ Ωc such that Br◦(z) ⊂ Ωc touches ∂Ω at y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Let w(x) = ε−1κKϕ(x − z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Also P+(w) + P+ k (w) + C0|Dw| ≤ −K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Then by using Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='1 we have P+(u − w) + P+ k (u − w) + C0|D(u − w)| ≥ 0 in B(1+δ)r◦(z) ∩ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Since, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='2) u − w ≤ 0 in (B(1+δ)r◦(z) ∩ Ω)c, applying Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='2 on u − w, it follows that u(x) ≤ w(x) in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Repeating a similar calculation for −u, we can conclude that |u(x)| ≤ w(x) in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Since this relation holds for any y ∈ ∂Ω, taking x ∈ Ω with dist(x, ∂Ω) < r◦, one can find y ∈ ∂Ω satisfying dist(x, ∂Ω) = |x − y| < r◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Then using the previous estimate we would obtain |u(x)| ≤ ε−1κKϕ(x − z) ≤ ε−1κK(ϕ(x − z) − ϕ(y − z)) ≤ ε−1κK r−1 dist(x, ∂Ω), which gives us (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' □ Now we are ready to prove that u ∈ C0,1(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Let x0 ∈ Ω and s ∈ (0, 1] be such that 2s = dist(x0, ∂Ω) ∧ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Without loss of any generality, we assume x0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Define v(x) = u(sx) in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='1 we already have |u(x)| ≤ C1Kδ(x), from that one can deduce |v(x)| ≤ C1 Ks(1 + |x|) for all x ∈ Rd, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='3) for some constant C1 independent of s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' We recall the scaled operator Is θν[x, f] := ˆ Rd(f(x + y) − f(x) − 1B 1 s (y)∇f(x) · y)sd+2Nθν(sx, sy)dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' To compute Ls[x, v] + C0s|Dv(x)| in B2, first we observe that D2v(x) = s2D2u(sx) and Dv(x) = sDu(sx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Also Is θν[x, v] = s2 ˆ Rd(v(x + y) − v(x) − 1B 1 s (y)∇v(x) · y)Nθν(sx, sy)sddy = s2 ˆ Rd(u(sx + sy) − u(sx) − 1B1(sy)∇u(sx) · sy)Nθν(sx, sy)sddy = s2Iθν[sx, u].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' BOUNDARY REGULARITY 9 Thus, it follows from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='1) that Ls[x, v] + C0s|Dv(x)| ≥ −Ks2 in B2, Ls[x, v] − C0s|Dv(x)| ≤ Ks2 in B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='4) Now consider a smooth cut-off function ϕ, 0 ≤ ϕ ≤ 1, satisfying ϕ = � 1 in B3/2, 0 in Bc 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Let w = ϕv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Clearly, ((ϕ − 1)v)(y) = 0 for all y ∈ B3/2, which gives D((ϕ − 1)v) = 0 and D2((ϕ − 1)v) = 0 in x ∈ B3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Since w = v + (ϕ − 1)v, from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='4) we obtain Ls[x, w] + C0s|Dw(x)| ≥ −Ks2 − | sup θ∈Θ inf ν∈Γ Is θν[x, (ϕ − 1)v)]| in B1, Ls[x, w] − C0s|Dw(x)| ≤ Ks2 + | sup θ∈Θ inf ν∈Γ Is θν[x, (ϕ − 1)v)]| in B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='5) Again, since (ϕ − 1)v = 0 in B3/2, we have in B1 that |Is θν[x, (ϕ − 1)v]| = ��� ˆ |y|≥1/2 ((ϕ − 1)v)(x + y) − ((ϕ − 1)v)(x))sd+2Nθν(sx, sy)dy ��� ≤ ˆ |y|≥1/2 |v(x + y)|sd+2Nθν(sx, sy)dy + |v(x)| ˆ |y|≥1/2 sd+2Nθν(sx, sy)dy := I1 + I2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Since x ∈ B1, using sd+2Nθν(sx, sy) ≤ sd+2k(sy) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='3) we can have the following estimate, I2 ≤ 2C1Ks ˆ Rd(1 ∧ |y|2)dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Now write I1 = ˆ 1/2≤|y|≤1/s |v(x + y)|sd+2Nθν(sx, sy)dy + ˆ |y|≥1/s |v(x + y)|sd+2Nθν(sx, sy)dy = Is,1 + Is,2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Let us first estimate Is,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Since x ∈ B1 and |y| ≥ 1 2 we have 1 + |x + y| ≤ 5|y|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' By using this estimate and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='3) we obtain Is,1 = sd+2 ˆ 1 2≤|y|≤ 1 s |v(x + y)|Nθν(sx, sy)dy ≤ 5C1K ˆ 1 2≤|y|≤ 1 s |sy|sd+2k(sy)dy ≤ 5C1Ks ˆ s 2 ≤|z|≤1 |sz|k(z)dz ≤ C2s ˆ s 2 ≤|z|≤1 |z|2k(z)dz ≤ C2s ˆ Rd(1 ∧ |y|2)k(z)dz ≤ C3s, for some constants C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' For Is,2, a change of variable and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='2) gives Is,2 ≤ κs2K ˆ s|y|>1 sdk(ry)dy = κs2K ˆ |y|>1 k(y)dy ≤ κs2K ˆ Rd(1 ∧ |y|2)k(y)dy ≤ C4s2K for some constant C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Therefore, putting the estimates of I1 and I2 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='5) we obtain Ls[x, w] + C0s|Dw(x)| ≥ −C5Ks in B1, Ls[x, w] − C0s|Dw(x)| ≥ C5Ks in B1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='6) 10 BOUNDARY REGULARITY for some constant C5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Now applying Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='1, from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='6) we have ∥v∥C1(B 1 2 ) ≤ C6 � ∥v∥L∞(B2) + sK � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='7) for some constant C6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='3) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='7) we then obtain sup y∈Bs/2(x),y̸=x |u(x) − u(y)| |x − y| ≤ C7K, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='8) for some constant C7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Now we can complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Note that if |x − y| ≥ 1 8, then |u(x) − u(y)| |x − y| ≤ 2κK, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' So we consider |x − y| < 1 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' If |x − y| ≥ 8−1(δ(x) ∨ δ(y)), then using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='1 we get |u(x) − u(y)| |x − y| ≤ 4CK(δ(x) + δ(y))(δ(x) ∨ δ(y))−1 ≤ 8CK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Now let |x − y| < 8−1 min{δ(x) ∨ δ(y), 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Then either y ∈ B δ(x)∧1 8 (x) or x ∈ B δ(y)∧1 8 (y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Without loss of generality, we suppose y ∈ B δ(x)∧1 8 (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='8) we get |u(x) − u(y)| |x − y| ≤ C7K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Fine boundary regularity Aim of this section is to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Since u is Lipschitz, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='1) can be written as |Lu| ≤ CK in Ω, and u = 0 in Ωc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' We start by constructing subsolutions which will be useful later on to prove oscillation lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' There exists a constant ˜κ, which depends only on d, λ, Λ, ´ Rd(1∧ |y|2)k(y)dy, such that for any r ∈ (0, 1], we have a bounded radial function φr satisfying \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 P−φr + P− k φr ≥ 0 in B4r \\ ¯Br, 0 ≤ φr ≤ ˜κr in Br, φr ≥ 1 ˜κ(4r − |x|) in B4r \\ Br, φr ≤ 0 in Rd \\ B4r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Moreover, φr ∈ C2(B4r \\ ¯Br).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' We use the same subsolution constructed in [11] and show that it is indeed a subsolution with respect to minimal Pucci operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Fix r ∈ (0, 1] and define vr(x) = e−ηq(x) − e−η(4r)2, where q(x) = |x|2 ∧ 2(4r)2 and η > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Clearly, 1 ≥ vr(0) ≥ vr(x) for all x ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Thus using the fact that 1 − e−ξ ≤ ξ for all ξ ≥ 0 we have vr(x) ≤ 1 − e−η(4r)2 ≤ η(4r)2, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='1) Again for x ∈ B4r \\ Br, we have that vr(x) = e−η(4r)2(eη((4r)2−q(x)) − 1) ≥ ηe−η(4r)2((4r)2 − |x|2) = ηe−η(4r)2(4r + |x|)(4r − |x|) ≥ 5ηre−η(4r)2(4r − |x|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='2) BOUNDARY REGULARITY 11 Fix x ∈ B4r \\ ¯Br.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' We start by estimating the local minimal Pucci operator P− of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Using rotational symmetry we may always assume x = (l, 0, · · · , 0) Then ∂ivr(x) = −2ηe−η|x|2xi = � −2ηe−η|x|2l i = 1, 0 i ̸= 1 and ∂ijvr(x) = � 4η2x2 i e−η|x|2 − 2ηe−η|x|2 i = j, 4η2xixje−η|x|2 i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 4η2l2e−η|x|2 − 2ηe−η|x|2 i = j = 1, −2ηe−η|x|2 i = j ̸= 1, 0 i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' By the above calculation, for any x ∈ B4r \\ ¯Br, choosing η > 1 r2 we have P−vr(x) = λ4η2l2e−η|x|2 − λ2ηe−η|x|2 − Λ(d − 1)2ηe−η|x|2 ≥ λ4η2l2e−η|x|2 − dΛ2ηe−η|x|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' Now to determine nonlocal minimal Pucci operator, using the convexity of exponential map we get, e−η|x+y|2 − e−η|x|2 + 2η1{|y|≤1}y · xe−η|x|2 ≥ −ηe−η|x|2 � |x + y|2 − |x|2 − 21{|y|≤1}y · x � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='Since P− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='k vr = P− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='k (vr + e−η(4r)2) and using above inequality we obtain ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='P− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='k (e−ηq(·))(x) = − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='ˆ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='Rd ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='e−ηq(x+y) − e−ηq(x) − 1B1(y)∇e−ηq(x) · y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='�− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='k(y)dy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='≥ −ηe−η|x|2 ˆ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='|y|≤r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='|x + y|2 − |x|2 − 2y · x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='k(y)dy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='ˆ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='r<|y|≤1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='���e−η(|x|2+2(4r)2) − e−η|x|2 + 2ηy · xe−η|x|2��� k(y)dy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='ˆ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='|y|>1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='���e−η(|x|2+2(4r)2) − e−η|x|2��� k(y)dy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='≥ −ηe−η|x|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='�ˆ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQffAAW/content/2301.02397v1.pdf'} +page_content='|y|