A broken circuit model for chromatic homology theories

AU - Chandler, Alex AU - Sazdanovic, Radmila T2 - EUROPEAN JOURNAL OF COMBINATORICS AB - Using the tools of algebraic Morse theory, and the thin poset approach to constructing homology theories, we give the categorification of Whitney’s broken circuit theorem for the chromatic polynomial, and for Stanley’s chromatic symmetric function. DA - 2022/8// PY - 2022/8// DO - 10.1016/j.ejc.2022.103538 VL - 104 SP - SN - 1095-9971 ER - TY - JOUR TI - A high order compact time/space finite difference scheme for the 2D and 3D wave equation with a damping layer AU - Kahana, Adar AU - Smith, Fouche AU - Turkel, Eli AU - Tsynkov, Semyon T2 - JOURNAL OF COMPUTATIONAL PHYSICS AB - We consider fourth order accurate compact schemes, in both space and time, for the second order wave equation with a variable speed of sound. For unbounded domains we add a fourth order accurate sponge layer to damp the outgoing waves. We demonstrate that usually this is more efficient than lower order schemes despite being implicit and conditionally stable. Fast time marching of the implicit scheme is accomplished by iterative methods such as multi-grid. Computations confirm the design convergence rate for the in-homogeneous, variable wave speed equation. DA - 2022/7/1/ PY - 2022/7/1/ DO - 10.1016/j.jcp.2022.111161 VL - 460 SP - SN - 1090-2716 KW - Compact finite differences KW - Absorbing boundary layer KW - Unbounded domain KW - High-order accuracy KW - Wave equation ER - TY - JOUR TI - Influence of water table dynamics on spatial and temporal patterns of hydroclimate extremes over Lake Victoria Basin, East Africa: Comparison of wet and dry years AU - Anyah, Richard AU - Xia, Sun AU - Semazzi, Fredrick T2 - INTERNATIONAL JOURNAL OF CLIMATOLOGY AB - WRF model coupled to water table dynamics has been adapted to investigate the spatial and temporal evolution of wet and dry conditions over Lake Victoria Basin. Two 2-year long simulations were conducted using coupled model with water table and the uncoupled (without water table) for wet and dry periods. Influence of water table on land–atmosphere coupling and interconnections among precipitation, soil moisture, evapotranspiration, and surface energy fluxes were examined. Overall, the coupled model simulated significantly higher monthly rainfall amounts during both the short (March-May) and long (October-December) rains of the wet year, which was more consistent with observations particularly over the lake surface and immediate hinterlands. Simulated monthly rainfall differences between coupled and uncoupled were pronounced during the peak of long rains, exceeding 100mm over the lake surface. Toward the end of the rainfall season (May) the difference was minimal. During the short rains significant differences occurred mostly in November, especially over the eastern shores. But during the relatively dry year minimal differences were generally witnessed throughout the year except in May, east of Lake Victoria. The coupled model simulated stronger matching among rainfall, soil moisture, and evapotranspiration over areas with shallow water table, for example in Kisumu to the east and Bukoba to the west. In these areas coupled model simulated higher soil moisture corresponding to higher evapotranspiration and precipitation. These interconnections were more pronounced during short and long rains in 1997 (wet) compared to 2010 (dry). Wet conditions over the gulf of Kisumu corresponded with rise in water table especially during October–December 1997 consistent with ENSO-related flooding over the area. Hence, our study demonstrated that incorporating water table resulted in realistic interconnections between precipitation, soil moisture, ET, and surface energy fluxes, and could improve simulation and prediction of spatial-temporal evolution of wet and dry conditions over Lake Victoria Basin. DA - 2022/5/16/ PY - 2022/5/16/ DO - 10.1002/joc.7682 SP - SN - 1097-0088 KW - Lake Victoria Basin KW - water table dynamics KW - wet and dry conditions ER - TY - JOUR TI - Weak solutions in nonlinear poroelasticity with incompressible constituents AU - Bociu, Lorena AU - Muha, Boris AU - Webster, Justin T. T2 - NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS AB - We consider quasi-static nonlinear poroelastic systems with applications in biomechanics and, in particular, tissue perfusion. The nonlinear permeability is taken to be dependent on solid dilation, and physical types of boundary conditions (Dirichlet, Neumann, and mixed) for the fluid pressure are considered. The system under consideration represents a nonlinear, implicit, degenerate evolution problem, which falls outside of the well-known implicit semigroup monotone theory. Previous literature related to proving existence of weak solutions for these systems is based on constructing solutions as limits of approximations, and energy estimates are obtained only for the constructed solutions. In comparison, in this treatment we provide for the first time a direct, fixed point strategy for proving the existence of weak solutions, which is made possible by a novel result on the uniqueness of weak solutions of the associated linear system (where the permeability is given as a function of space and time). The uniqueness proof for the associated linear problem is based on novel energy estimates for arbitrary weak solutions, rather than just for constructed solutions. The results of this work provide a foundation for addressing strong solutions, as well as uniqueness of weak solutions for nonlinear poroelastic systems. DA - 2022/10// PY - 2022/10// DO - 10.1016/j.nonrwa.2022.103563 VL - 67 SP - SN - 1878-5719 KW - Nonlinear poroelasticity KW - Implicit evolution equations KW - Quasilinear parabolic KW - Weak solutions KW - Energy methods KW - Incompressible constituents ER - TY - JOUR TI - Local Unitary Equivalence of Generic Multi-qubits Based on the CP Decomposition AU - Chang, Jingmei AU - Jing, Naihuan T2 - INTERNATIONAL JOURNAL OF THEORETICAL PHYSICS AB - The CANDECOMP/PARAFAC (CP) decomposition is a generalization of the spectral decomposition of matrices to higher-order tensors. In this paper we use the CP decomposition to study unitary equivalence of higher order tensors and construct several invariants of local unitary equivalence for general higher order tensors. Based on this new method, we study the coefficient tensors of 3-qubit states and obtain a necessary and sufficient criterion for local unitary equivalence of general tripartite states in terms of the CP decomposition. We also generalize this method to obtain some invariants of local unitary equivalence for general multi-partite qudits. DA - 2022/5/12/ PY - 2022/5/12/ DO - 10.1007/s10773-022-05106-w VL - 61 IS - 5 SP - SN - 1572-9575 KW - Local unitary equivalence KW - Tensors KW - CP decomposition ER - TY - JOUR TI - Consistent time‐homogeneous modeling of SPX and VIX derivatives AU - Papanicolaou, Andrew T2 - Mathematical Finance AB - Abstract This paper shows how to recover a stochastic volatility model (SVM) from a market model of the VIX futures term structure. Market models have more flexibility for fitting of curves than do SVMs, and therefore are better suited for pricing VIX futures and VIX derivatives. But the VIX itself is a derivative of the S&P500 (SPX) and it is common practice to price SPX derivatives using an SVM. Therefore, consistent modeling for both SPX and VIX should involve an SVM that can be obtained by inverting the market model. This paper's main result is a method for the recovery of a stochastic volatility function by solving an inverse problem where the input is the VIX function given by a market model. Analysis will show conditions necessary for there to be a unique solution to this inverse problem. The models are consistent if the recovered volatility function is non‐negative. Examples are presented to illustrate the theory, to highlight the issue of negativity in solutions, and to show the potential for inconsistency in non‐Markov settings. DA - 2022/5/14/ PY - 2022/5/14/ DO - 10.1111/mafi.12348 VL - 32 IS - 3 SP - 907-940 J2 - Mathematical Finance LA - en OP - SN - 0960-1627 1467-9965 UR - http://dx.doi.org/10.1111/mafi.12348 DB - Crossref KW - consistent pricing KW - market models KW - stochastic volatility KW - VIX futures ER - TY - JOUR TI - A new patch up technique for elliptic partial differential equation with irregularities AU - Singh, Swarn AU - Singh, Suruchi AU - Li, Zhilin T2 - JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS AB - This paper presents a new technique based on a collocation method using cubic splines for second order elliptic equation with irregularities in one dimension and two dimensions. The differential equation is first collocated at the two smooth sub domains divided by the interface. We extend the sub domains from the interior of the domain and then the scheme at the interface is developed by patching them up. The scheme obtained gives the second order accurate solution at the interface as well as at the regular points. Second order accuracy for the approximations of the first order and the second order derivative of the solution can also be seen from the experiments performed. Numerical experiments for 2D problems also demonstrate the second order accuracy of the present scheme for the solution u and the derivatives ux,uxx and the mixed derivative uxy. The approach to derive the interface relations, established in this paper for elliptic interface problems, can be helpful to derive high order accurate numerical methods. Numerical tests exhibit the super convergent properties of the scheme. DA - 2022/6// PY - 2022/6// DO - 10.1016/j.cam.2021.113975 VL - 407 SP - SN - 1879-1778 KW - Elliptic partial differential equation KW - Interface KW - Jump conditions KW - Cubic spline collocation KW - Irregularities KW - Patch up technique ER - TY - JOUR TI - Model-assisted deep learning of rare extreme events from partial observations AU - Asch, Anna AU - J. Brady, Ethan AU - Gallardo, Hugo AU - Hood, John AU - Chu, Bryan AU - Farazmand, Mohammad T2 - CHAOS AB - To predict rare extreme events using deep neural networks, one encounters the so-called small data problem because even long-term observations often contain few extreme events. Here, we investigate a model-assisted framework where the training data are obtained from numerical simulations, as opposed to observations, with adequate samples from extreme events. However, to ensure the trained networks are applicable in practice, the training is not performed on the full simulation data; instead, we only use a small subset of observable quantities, which can be measured in practice. We investigate the feasibility of this model-assisted framework on three different dynamical systems (Rössler attractor, FitzHugh-Nagumo model, and a turbulent fluid flow) and three different deep neural network architectures (feedforward, long short-term memory, and reservoir computing). In each case, we study the prediction accuracy, robustness to noise, reproducibility under repeated training, and sensitivity to the type of input data. In particular, we find long short-term memory networks to be most robust to noise and to yield relatively accurate predictions, while requiring minimal fine-tuning of the hyperparameters. DA - 2022/4// PY - 2022/4// DO - 10.1063/5.0077646 VL - 32 IS - 4 SP - SN - 1089-7682 UR - https://doi.org/10.1063/5.0077646 ER - TY - JOUR TI - Near-Optimal Variance-Based Uncertainty Relations AU - Xiao, Yunlong AU - Jing, Naihuan AU - Yu, Bing AU - Fei, Shao-Ming AU - Li-Jost, Xianqing T2 - FRONTIERS IN PHYSICS AB - Learning physical properties of a quantum system is essential for the developments of quantum technologies. However, Heisenberg's uncertainty principle constrains the potential knowledge one can simultaneously have about a system in quantum theory. Aside from its fundamental significance, the mathematical characterization of this restriction, known as `uncertainty relation', plays important roles in a wide range of applications, stimulating the formation of tighter uncertainty relations. In this work, we investigate the fundamental limitations of variance-based uncertainty relations, and introduce several `near optimal' bounds for incompatible observables. Our results consist of two morphologically distinct phases: lower bounds that illustrate the uncertainties about measurement outcomes, and the upper bound that indicates the potential knowledge we can gain. Combining them together leads to an \emph{uncertainty interval}, which captures the essence of uncertainties in quantum theory. Finally, we have detailed how to formulate lower bounds for product-form variance-based uncertainty relations by employing entropic uncertainty relations, and hence built a link between different forms of uncertainty relations. DA - 2022/4/1/ PY - 2022/4/1/ DO - 10.3389/fphy.2022.846330 VL - 10 SP - SN - 2296-424X KW -numbers:& nbsp;03.65.ta,& nbsp;& nbsp;03.67.a,& nbsp;& nbsp;42.50.lc

KW - uncertainty relation KW - variance-based KW - uncertainty interval ER - TY - JOUR TI - Detection of Multipartite Entanglement Based on Heisenberg-Weyl Representation of Density Matrices AU - Zhao, Hui AU - Yang, Yu AU - Jing, Naihuan AU - Wang, Zhi-Xi AU - Fei, Shao-Ming T2 - INTERNATIONAL JOURNAL OF THEORETICAL PHYSICS AB - We study entanglement and genuine entanglement of tripartite and four-partite quantum states by using Heisenberg-Weyl (HW) representation of density matrices. Based on the correlation tensors in HW representation, we present criteria to detect entanglement and genuine tripartite and four-partite entanglement. Detailed examples show that our method can detect more entangled states than previous criteria. DA - 2022/5/10/ PY - 2022/5/10/ DO - 10.1007/s10773-022-05123-9 VL - 61 IS - 5 SP - SN - 1572-9575 KW - Heisenberg-Weyl representation KW - Genuine entanglement KW - Correlation tensor ER - TY - JOUR TI - Tukey Depths and Hamilton-Jacobi Differential Equations AU - Molina-Fructuoso, Martin AU - Murray, Ryan T2 - SIAM JOURNAL ON MATHEMATICS OF DATA SCIENCE AB - Widespread application of modern machine learning has increased the need for robust statistical algorithms. This work studies one such fundamental statistical concept known as the Tukey depth. We study the problem in the continuum (population) limit. In particular, we formally derive the associated necessary conditions, which take the form of a first-order partial differential equation which is necessarily satisfied at points where the Tukey depth is smooth. We discuss the interpretation of this formal necessary condition in terms of the viscosity solution of a Hamilton--Jacobi equation, but with a nonclassical Hamiltonian with discontinuous dependence on the gradient at zero. We prove that this equation possesses a unique viscosity solution and that this solution always bounds the Tukey depth from below. In certain cases we prove that the Tukey depth is equal to the viscosity solution, and we give some illustrations of standard numerical methods from the optimal control community which deal directly with the partial differential equation. We conclude by outlining several promising research directions both in terms of new numerical algorithms and theoretical challenges. DA - 2022/// PY - 2022/// DO - 10.1137/21M1411998 VL - 4 IS - 2 SP - 604-633 SN - 2577-0187 UR - https://doi.org/10.1137/21M1411998 KW - statistical depths KW - robust statistics KW - Hamilton-Jacobi equations KW - viscosity solutions ER - TY - JOUR TI - Shape-morphing reduced-order models for nonlinear Schrodinger equations AU - Anderson, William AU - Farazmand, Mohammad T2 - NONLINEAR DYNAMICS AB - We consider reduced-order modeling of nonlinear dispersive waves described by a class of nonlinear Schrödinger (NLS) equations. We compare two nonlinear reduced-order modeling methods: (i) The reduced Lagrangian approach which relies on the variational formulation of NLS and (ii) the recently developed method of reduced-order nonlinear solutions (RONS). First, we prove the surprising result that, although the two methods are seemingly quite different, they can be obtained from the real and imaginary parts of a single complex-valued master equation. Furthermore, for the NLS equation in a stationary frame, we show that the reduced Lagrangian method fails to predict the correct group velocity of the waves, whereas RONS predicts the correct group velocity. Finally, for the modified NLS equation, where the reduced Lagrangian approach is inapplicable, the RONS reduced-order model accurately approximates the true solutions. DA - 2022/4/25/ PY - 2022/4/25/ DO - 10.1007/s11071-022-07448-w VL - 4 IS - 4 SP - SN - 1573-269X UR - https://doi.org/10.1007/s11071-022-07448-w KW - Model order reduction KW - Partial differential equations KW - Nonlinear Schrodinger equation KW - Variational methods ER - TY - JOUR TI - A multilevel approach to stochastic trace estimation AU - Hallman, Eric AU - Troester, Devon T2 - LINEAR ALGEBRA AND ITS APPLICATIONS AB - This article presents a randomized matrix-free method for approximating the trace of f(A), where A is a large symmetric matrix and f is a function analytic in a closed interval containing the eigenvalues of A. Our method uses a combination of stochastic trace estimation (i.e., Hutchinson's method), Chebyshev approximation, and multilevel Monte Carlo techniques. We establish general bounds on the approximation error of this method by extending an existing error bound for Hutchinson's method to multilevel trace estimators. Numerical experiments are conducted for common applications such as estimating the log-determinant, nuclear norm, and Estrada index. We find that using multilevel techniques can substantially reduce the variance of existing single-level estimators. DA - 2022/4/1/ PY - 2022/4/1/ DO - 10.1016/j.laa.2021.12.010 VL - 638 SP - 125-149 SN - 1873-1856 KW - Spectral function KW - Trace estimation KW - Chebyshev approximation KW - Hutchinson's trace estimator KW - Multilevel Monte Carlo ER - TY - JOUR TI - On Weissler's Conjecture on the Hamming Cube I AU - Ivanisvili, P. AU - Nazarov, F. T2 - INTERNATIONAL MATHEMATICS RESEARCH NOTICES AB - Abstract Let $1\leq p \leq q <\infty $ and let $w \in \mathbb{C}$. Weissler conjectured that the Hermite operator $e^{w\Delta }$ is bounded as an operator from $L^{p}$ to $L^{q}$ on the Hamming cube $\{-1,1\}^{n}$ with the norm bound independent of $n$ if and only if $$\begin{align*} |p-2-e^{2w}(q-2)|\leq p-|e^{2w}|q. \end{align*}$$It was proved in [ 1], [ 2], and [ 17] in all cases except $2<p\leq q <3$ and $3/2<p\leq q <2$, which stood open until now. The goal of this paper is to give a full proof of Weissler’s conjecture in the case $p=q$. Several applications will be presented. DA - 2022/4/25/ PY - 2022/4/25/ DO - 10.1093/imrn/rnaa363 VL - 2022 IS - 9 SP - 6991-7020 SN - 1687-0247 ER - TY - JOUR TI - The Green polynomials via vertex operators AU - Jing, Naihuan AU - Liu, Ning T2 - JOURNAL OF PURE AND APPLIED ALGEBRA AB - An iterative formula for the Green polynomial is given using the vertex operator realization of the Hall-Littlewood function. Based on this, (1) a general combinatorial formula of the Green polynomial is given; (2) several compact formulas are given for Green's polynomials associated with upper partitions of length $\leq 3$ and the diagonal lengths $\leq 3$; (3) a Murnaghan-Nakayama type formula for the Green polynomial is obtained; and (4) an iterative formula is derived for the bitrace of the finite general linear group $G$ and the Iwahori-Hecke algebra of type $A$ on the permutation module of $G$ by its Borel subgroup. DA - 2022/8// PY - 2022/8// DO - 10.1016/j.jpaa.2022.107032 VL - 226 IS - 8 SP - SN - 1873-1376 KW - Green's polynomials KW - Vertex operators KW - Hall-Littlewood functions KW - Hecke algebra ER - TY - JOUR TI - A new parameter free partially penalized immersed finite element and the optimal convergence analysis AU - Ji, Haifeng AU - Wang, Feng AU - Chen, Jinru AU - Li, Zhilin T2 - NUMERISCHE MATHEMATIK AB - This paper presents a new parameter free partially penalized immersed finite element method and convergence analysis for solving second order elliptic interface problems. A lifting operator is introduced on interface edges to ensure the coercivity of the method without requiring an ad-hoc stabilization parameter. The optimal approximation capabilities of the immersed finite element space is proved via a novel new approach that is much simpler than that in the literature. A new trace inequality which is necessary to prove the optimal convergence of immersed finite element methods is established on interface elements. Optimal error estimates are derived rigorously with the constant independent of the interface location relative to the mesh. The new method and analysis have also been extended to variable coefficients and three-dimensional problems. Numerical examples are also provided to confirm the theoretical analysis and efficiency of the new method. DA - 2022/4// PY - 2022/4// DO - 10.1007/s00211-022-01276-1 VL - 150 IS - 4 SP - 1035-1086 SN - 0945-3245 KW - 65N15 KW - 65N30 KW - 35R05 ER - TY - JOUR TI - From Graph Cuts to Isoperimetric Inequalities: Convergence Rates of Cheeger Cuts on Data Clouds AU - Trillos, Nicolas Garcia AU - Murray, Ryan AU - Thorpe, Matthew T2 - ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS AB - Abstract In this work we study statistical properties of graph-based clustering algorithms that rely on the optimization of balanced graph cuts, the main example being the optimization of Cheeger cuts. We consider proximity graphs built from data sampled from an underlying distribution supported on a generic smooth compact manifold $${\mathcal {M}}$$