arXiv Math @arxiv_math@qoto.org

In this paper, we have explored the impact of certain indices-dependent element-wise transformations on the null space of a matrix. We have found the conditions on this transformation that will preserve the rank and nullity of the original matrix. We have also found some transformations which give localized null vectors for the transformed matrix. Finally, some possible applications of these localized null vectors and eigenvalues are mentioned in different domains.

We consider the inverse eigenvalue problem of constructing a substochastic matrix from the given spectrum parameters with the corresponding eigenvector constraints. This substochastic inverse eigenvalue problem (SstIEP) with the specific eigenvector constraints is formulated into a nonconvex optimization problem (NcOP). The solvability for SstIEP with the specific eigenvector constraints is equivalent to identify the attainability of a zero optimal value for the formulated NcOP. When the optimal objective value is zero, the corresponding optimal solution to the formulated NcOP is just the substochastic matrix desired to be constructed. We develop the alternating minimization algorithm to solve the formulated NcOP, and its convergence is established by developing a novel method to obtain the boundedness of the optimal solution. Some numerical experiments are conducted to demonstrate the efficiency of the proposed method.

We give a full analytic solution to a particular case of the algebraic Riccati equation $XWW^*WX=W^*$ for any matrix $W$ (possibly non-square or non-symmetric) in using the Schur method, terms of the SVD decomposition of $W$. In particular, $(WX)^3=WX$ and $(XW)^3=XW$ for any solution $X$. We show that for $W=AB$, matrix $X=B^+ A^+$ is a solution of this equation if and only if the reverse order law holds, i.e., ${(AB)}^+=B^+ A^+$. For a Hermitian and invertible $W$ the maximal and stabilizing Hermitian solutions is shown to be equal to $W^+$. Equivalence to the equation $XWX=W^+$ is proven.

This paper conducts a geometric analysis of the Joint-Eigenspace Fourier transform of the symmetric space of the non-compact type. Our study shows how the Poisson transform builds up the well-known Helgason Fourier transform for an analysis of the complete duality of the underlying symetric space. Among other results, we establish an inversion formula, a Plancherel formula and (as our main result) a Paley-Wiener theorem for the Joint-Eigenspace Fourier transform on any noncompact symmetric space.

This article characterizes the associativity of two-place functions $T: [0,1]^2\rightarrow [0,1]$ defined by $T(x,y)=f^{(-1)}(F(f(x),f(y)))$ where $F:[0,1]^2\rightarrow[0,1]$ is a triangular norm (even a triangular subnorm), $f: [0,1]\rightarrow [0,1]$ is a strictly increasing function and $f^{(-1)}:[0,1]\rightarrow[0,1]$ is the pseudo-inverse of $f$. We prove that the associativity of functions $T$ only depends on the range of $f$, which is used to give a sufficient and necessary condition for the function $T$ being associative when the triangular norm $F$ is an ordinal sum of triangular norms and an ordinal sum of triangular subnorms in the sense of A. H. Clifford, respectively. These results finally are applied for describing classes of triangular norms generated by strictly increasing functions.

In this paper, we generalize the notion of joint eigenvalues and joint spectrum of matrices and operator tupples on a bi complex Hilbert space. We observe that unlike the spectrum of a bounded operator on a bi complex Hilbert space is bounded. But this is not the case here. Even the set of joint eigenvalues of matrix tuples is unbounded.

Cooperative perception, offering a wider field of view than standalone perception, is becoming increasingly crucial in autonomous driving. This perception is enabled through vehicle-to-vehicle (V2V) communication, allowing connected automated vehicles (CAVs) to exchange sensor data, such as light detection and ranging (LiDAR) point clouds, thereby enhancing the collective understanding of the environment. In this paper, we leverage an importance map to distill critical semantic information, introducing a cooperative perception semantic communication framework that employs intermediate fusion. To counter the challenges posed by time-varying multipath fading, our approach incorporates the use of orthogonal frequency-division multiplexing (OFDM) along with channel estimation and equalization strategies. Furthermore, recognizing the necessity for reliable transmission, especially in the low SNR scenarios, we introduce a novel semantic error detection method that is integrated with our semantic communication framework in the spirit of hybrid automatic repeated request (HARQ). Simulation results show that our model surpasses the traditional separate source-channel coding methods in perception performance, both with and without HARQ. Additionally, in terms of throughput, our proposed HARQ schemes demonstrate superior efficiency to the conventional coding approaches.

We conducted a reproducibility study on Integrated Gradients (IG) based methods and the Important Direction Gradient Integration (IDGI) framework. IDGI eliminates the explanation noise in each step of the computation of IG-based methods that use the Riemann Integration for integrated gradient computation. We perform a rigorous theoretical analysis of IDGI and raise a few critical questions that we later address through our study. We also experimentally verify the authors' claims concerning the performance of IDGI over IG-based methods. Additionally, we varied the number of steps used in the Riemann approximation, an essential parameter in all IG methods, and analyzed the corresponding change in results. We also studied the numerical instability of the attribution methods to check the consistency of the saliency maps produced. We developed the complete code to implement IDGI over the baseline IG methods and evaluated them using three metrics since the available code was insufficient for this study.

Any choice of a spherical fusion category defines an invariant of oriented closed 3-manifolds, which is computed by choosing a triangulation of the manifold and considering a state sum model that assigns a 6j symbol to every tetrahedron in this triangulation. This approach has been generalized to oriented closed 3-manifolds with defect data by Meusburger. In a recent paper, she constructed a family of invariants for such manifolds parametrised by the choice of certain spherical fusion categories, bimodule categories, finite bimodule functors and module natural transformations. Meusburger defined generalised 6j symbols for these objects, and introduces a state sum model that assigns a generalised 6j symbol to every tetrahedron in the triangulation of a manifold with defect data, where the type of 6j symbol used depends on what defect data occur within the tetrahedron. The present work provides non-trivial examples of suitable bimodule categories, bimodule functors and module natural transformation, all over categories of $G$-graded vector spaces. Our main result is the description of module functors in terms of matrices, which allows us to classify these functors when $G$ is a finite cyclic group. Furthermore, we calculate the generalised 6j symbols for categories of $G$-graded vector spaces, (bi-)module categories over such categories and (bi-)module functors.

Breakthrough Sparsity Theorem, derived independently by Donoho and Elad \textit{[Proc. Natl. Acad. Sci. USA, 2003]}, Gribonval and Nielsen \textit{[IEEE Trans. Inform. Theory, 2003]} and Fuchs \textit{[IEEE Trans. Inform. Theory, 2004]} says that unique sparse solution to NP-Hard $\ell_0$-minimization problem can be obtained using unique solution of P-Type $\ell_1$-minimization problem. In this paper, we derive noncommutative version of their result using frames for Hilbert C*-modules.

Gentle algebras are a class of special biserial algebra whose representation theory has been thoroughly described. In this paper, we consider the infinite dimensional generalizations of gentle algebras, referred to as locally gentle algebras. We give combinatorial descriptions of the center, spectrum, and homological dimensions of a locally gentle algebra, including an explicit injective resolution. We classify when these algebras are Artin-Schelter Gorenstein, Artin-Schelter regular, and Cohen-Macaulay, and provide an analogue of Stanley's theorem for locally gentle algebras.

Let $R$ be a ring. In \cite{MD4} Mao and Ding defined an special class of $R$-modules that they called \( FP_n \)-projective $R$-modules. In this paper, we give some new characterizations of \( FP_n \)-projective $R$-modules and strong $n$-coherent rings. Some known results are extended and some new characterizations of the \( FP_n \)-injective global dimension in terms of \( FP_n \)-projective $R$-modules are obtained. Using the \( FP_n \)-projective dimension of an $R$-module defined by Ouyang, Duan and Li in \cite{Ouy} we introduce a slightly different \( FP_n \)-projective global dimension over the ring $R$ which measures how far away the ring is from being Noetherian. This dimension agrees with the $(n,0)$-projective global dimension of \cite{Ouy} when the ring in question is strong $n$-coherent.

This study investigates the iterative refinement method applied to the solution of linear discrete inverse problems by considering its application to the Tikhonov problem in mixed precision. Previous works on mixed precision iterative refinement methods for the solution of symmetric positive definite linear systems and least-squares problems have shown regularization to be a key requirement when computing low precision factorizations. For problems that are naturally severely ill-posed, we formulate the iterates of iterative refinement in mixed precision as a filtered solution using the preconditioned Landweber method with a Tikhonov-type preconditioner. Through numerical examples simulating various mixed precision choices, we showcase the filtering properties of the method and the achievement of comparable or superior accuracy compared to results computed in double precision as well as another approximate method.

We study the simplicial volume of manifolds obtained from Davis' reflection group trick, the goal being characterizing those having positive simplicial volume. In particular, we focus on checking whether manifolds in this class with nonzero Euler characteristic have positive simplicial volume (Gromov asked whether this holds in general for aspherical manifolds). This leads to a combinatorial problem about triangulations of spheres: we define a partial order on the set of triangulations -- the relation being the existence of a nonzero-degree simplicial map between two triangulations -- and the problem is to find the minimal elements of a specific subposet. We solve explicitly the case of triangulations of the two-dimensional sphere, and then perform an extensive analysis, with the help of computer searches, of the three-dimensional case. Moreover, we present a connection of this problem with the theory of graph minors.

We survey a number of incompleteness results in operator algebras stemming from the recent undecidability result in quantum complexity theory known as $\operatorname{MIP}^*=\operatorname{RE}$, the most prominent of which is the Gödelian refutation of the Connes Embedding Problem. We also discuss the very recent use of $\operatorname{MIP}^*=\operatorname{RE}$ in refuting the Aldous-Lyons conjecture in probability theory.

Decarbonizing global energy systems requires extensive integration of renewable energy into the electric grid. However, the intermittency and variable nature of wind and other non-dispatchable renewable resources make integration a great challenge. Hybridizing renewables with energy storage to form integrated energy systems (IESs) helps mitigate these concerns by improving reliability and resiliency. This paper systematically studies the limitations of the prevailing price-taker assumption for techno-economic analysis (TEA) and optimization of hybrid energy systems. As an illustrative case study, we retrofit an existing wind farm in the RTS-GMLC test system (which loosely mimics the Southwest U.S.) with battery energy storage to form an IES. We show that the standard price-taker model overestimates the electricity revenue and the net present value (NPV) of the IES up to 178% and 50%, respectively, compared to our more rigorous multiscale optimization. These differences arise because introducing storage creates a more flexible resource that impacts the larger wholesale electricity market. Moreover, this work highlights the impact of the IES has on the market via various strategic bidding, and underscores the importance of moving beyond price-taker for optimal storage sizing and TEA of IESs. We conclude by discussing opportunities to generalize the proposed framework to other IESs, and highlight emerging research questions regarding the complex interactions between IESs and markets.

While much attention of neural network methods is devoted to high-dimensional PDE problems, in this work we consider methods designed to work for elliptic problems on domains $Ω\subset \mathbb{R} ^d, $ $d=1,2,3$ in association with more standard finite elements. We suggest to connect finite elements and neural network approximations through training, i.e., using finite element spaces to compute the integrals appearing in the loss functionals. This approach, retains the simplicity of classical neural network methods for PDEs, uses well established finite element tools (and software) to compute the integrals involved and it gains in efficiency and accuracy. We demonstrate that the proposed methods are stable and furthermore, we establish that the resulting approximations converge to the solutions of the PDE. Numerical results indicating the efficiency and robustness of the proposed algorithms are presented.

Driven by applications in the natural, social and computer sciences several algorithms have been proposed to enumerate all sets $X$ of vertices of a graph $G$ that induce a connected subgraph. Our algorithm AllMetricSets enumerates all $X$'s that induce (more exquisite) metric subgraphs. Here "metric" means that any distinct $s,t\in X$ are joined by a globally shortest $s-t$ path.

Convex programming with linear constraints plays an important role in the operation of a number of everyday systems. However, absent any additional protections, revealing or acting on the solutions to such problems may reveal information about their constraints, which can be sensitive. Therefore, in this paper, we introduce a method for solving convex programs while keeping linear constraints private. First, we prove that this method is differentially private and always generates a feasible optimization problem (i.e., one whose solution exists). Then we show that the solution to the privatized problem also satisfies the original, non-private constraints. Next, we bound the expected loss in performance from privacy, which is measured by comparing the cost with privacy to that without privacy. Simulation results apply this framework to constrained policy synthesis in a Markov decision process, and they show that a typical privacy implementation induces only an approximately $9\%$ loss in solution quality.

In his paper on the Mordell-Lang conjecture, Hrushovski employed techniques from model theory to prove the function field version of the conjecture. In doing so he was able to answer a related question of Voloch, which we refer to henceforth as Hrushovski's theorem. In this paper we shall give an alternative proof of said theorem in the characteristic $p$ setting, but using purely algebro-geometric ideas.