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We consider bounds on maximum nullity of a graph via transversal numbers of compatible collections of forts. Results include generalizations of theorems from symmetric to combinatorially symmetric matrices, special bases of matrix nullspaces derived from transversal sets, and examples of issues that arise when considering only minimal forts and how to avoid them. We also show an important difference between constructing symmetric and combinatorially symmetric matrices associated to a graph whose nullspaces are supported on collections of disjoint forts.

This paper is devoted to solving a multidimensional backward stochastic differential equation (BSDE) with a general random terminal time, which may take values in [0,+infinity]. The generator g satisfies a stochastic monotonicity condition in the first unknown variable y and a stochastic Lipschitz continuity condition in the second unknown variable z, and it can have a more general growth with respect to y than the classical one stated in (H5) of Briand et al. [2003]. Without imposing any restriction of finite moment on the stochastic coefficients, we establish a general existence and uniqueness result for the adapted solution of the previous BSDE in a proper weighted L2-space. This result is proved via some innovative ideas and delicate analytical techniques, and it unifies and strengthens many existing works on BSDEs with stochastic monotonicity generators, BSDEs with stochastic Lipschitz generators, and BSDEs with deterministic Lipschitz/monotonicity generators. Then, a continuous dependence property and a stability theorem for the weighted L2-solutions are given. We also derive the nonlinear Feynman-Kac formula for both parabolic and elliptic PDEs in this context.

We introduce a generalization of the concept of a chronological list of forces, called a relaxed chronology. This concept is used to introduce a new way of formulating the standard zero forcing process, which we refer to as parallel increasing path covers, or PIPs. The combinatorial properties of PIPs are utilized to identify bounds comparing standard zero forcing propagation time to positive semidefinite propagation time. A collection of paths within a set of PSD forcing trees, called a path bundle, is used to identify the PSD forcing analog of the reversal of a standard zero forcing process, as well as to draw a connection between PSD forcing and rigid-linkage forcing.

We prove that all tree-level amplitudes in pure (super-)gravity can be expressed as term-wise, gauge-invariant double-copies of those of pure (super-)Yang-Mills obtained via BCFW recursion. These representations are far from unique: varying the recursive scheme leads to a wide variety of distinct, but equally valid representations of gravitational amplitudes, all realized as double-copies., Comment: 5+3 pages; 3 tables

We investigate the sparsity of null vectors of real symmetric matrices whose off-diagonal pattern of zero and nonzero entries is described by the adjacencies of a graph. We use the definition of the spark of a matrix, the smallest number of nonzero coordinates of any null vector, to define the spark of a graph as the smallest possible spark of a corresponding matrix. We study connections of graph spark to well-known concepts including minimum rank, forts, orthogonal representations, Parter and Fiedler vertices, and vertex connectivity., Comment: 20 pages, 2 figures

Generating realistic human motion from given action descriptions has experienced significant advancements because of the emerging requirement of digital humans. While recent works have achieved impressive results in generating motion directly from textual action descriptions, they often support only a single modality of the control signal, which limits their application in the real digital human industry. This paper presents a Motion General-Purpose generaTor (MotionGPT) that can use multimodal control signals, e.g., text and single-frame poses, for generating consecutive human motions by treating multimodal signals as special input tokens in large language models (LLMs). Specifically, we first quantize multimodal control signals into discrete codes and then formulate them in a unified prompt instruction to ask the LLMs to generate the motion answer. Our MotionGPT demonstrates a unified human motion generation model with multimodal control signals by tuning a mere 0.4% of LLM parameters. To the best of our knowledge, MotionGPT is the first method to generate human motion by multimodal control signals, which we hope can shed light on this new direction. Visit our webpage at https://qiqiapink.github.io/MotionGPT/., Comment: 18 pages, 8 figures, accepted by AAAI 2024

Transformer is popular in recent 3D human pose estimation, which utilizes long-term modeling to lift 2D keypoints into the 3D space. However, current transformer-based methods do not fully exploit the prior knowledge of the human skeleton provided by the kinematic structure. In this paper, we propose a novel transformer-based model EvoPose to introduce the human body prior knowledge for 3D human pose estimation effectively. Specifically, a Structural Priors Representation (SPR) module represents human priors as structural features carrying rich body patterns, e.g. joint relationships. The structural features are interacted with 2D pose sequences and help the model to achieve more informative spatiotemporal features. Moreover, a Recursive Refinement (RR) module is applied to refine the 3D pose outputs by utilizing estimated results and further injects human priors simultaneously. Extensive experiments demonstrate the effectiveness of EvoPose which achieves a new state of the art on two most popular benchmarks, Human3.6M and MPI-INF-3DHP., Comment: 5 pages, 2 figures, 4 tables, published in the proceedings of IEEE ICASSP 2023

Many data insight questions can be viewed as searching in a large space of tables and finding important ones, where the notion of importance is defined in some adhoc user defined manner. This paper presents Holistic Cube Analysis (HoCA), a framework that augments the capabilities of relational queries for such problems. HoCA first augments the relational data model and introduces a new data type AbstractCube, defined as a function which maps a region-features pair to a relational table (a region is a tuple which specifies values of a set of dimensions). AbstractCube provides a logical form of data, and HoCA operators are cube-to-cube transformations. We describe two basic but fundamental HoCA operators, cube crawling and cube join (with many possible extensions). Cube crawling explores a region space, and outputs a cube that maps regions to signal vectors. Cube join, in turn, is critical for composition, allowing one to join information from different cubes for deeper analysis. Cube crawling introduces two novel programming features, (programmable) Region Analysis Models (RAMs) and Multi-Model Crawling. Crucially, RAM has a notion of population features, which allows one to go beyond only analyzing local features at a region, and program region-population analysis that compares region and population features, capturing a large class of importance notions. HoCA has a rich algorithmic space, such as optimizing crawling and join performance, and physical design of cubes. We have implemented and deployed HoCA at Google. Our early HoCA offering has attracted more than 30 teams building applications with it, across a diverse spectrum of fields including system monitoring, experimentation analysis, and business intelligence. For many applications, HoCA empowers novel and powerful analyses, such as instances of recurrent crawling, which are challenging to achieve otherwise., Comment: Establishing initial concepts of HoCA

We construct all six-particle helicity amplitude integrands in color-dressed, maximally supersymmetric Yang-Mills theory at two loops in a single prescriptive basis of master integrands., Comment: 5+2 pages; 15 figures. Updated for completeness and calrity

We outline the concrete steps involved in building prescriptive master integrand bases for scattering amplitudes beyond the planar limit. We highlight the role of contour choices in such bases, and illustrate the full process by constructing a complete, triangle power-counting basis at two loops for six particles. We show how collinear contour choices can be used to divide integrand bases into separately finite and divergent subspaces, and how double-poles can be used to further subdivide these spaces according to (transcendental) weight. Complete details of the basis constructed for six particles is provided in the ancillary files for this work's submission to the arXiv., Comment: 39 pages; 2 tables; dozens of figures. Complete prescriptive basis for six particles is provided in ancillary files

The tight upper bound $\operatorname{pt}_+(G) \leq \left\lceil \frac{\left\vert \operatorname{V}(G) \right\vert - \operatorname{Z}_+(G)}{2} \right\rceil$ is established for the positive semidefinite propagation time of a graph in terms of its positive semidefinite zero forcing number. To prove this bound, two methods of transforming one positive semidefinite zero forcing set into another and algorithms implementing these methods are presented. Consequences of the bound, including a tight Nordhaus-Gaddum sum upper bound on positive semidefinite propagation time, are established., Comment: 14 pages, 7 figures