Related papers: Matrix Theory for Minimal Trellises
For a given tree tensor network $G$, we call a tuple of bond dimensions minimal if there exists a tensor $T$ that can be represented by this network but not on the same tree topology with strictly smaller bond dimensions. We establish…
Finite sample properties of random covariance-type matrices have been the subject of much research. In this paper we focus on the "lower tail" of such a matrix, and prove that it is subgaussian under a simple fourth moment assumption on the…
Minimal linear codes have interesting applications in secret sharing schemes and secure two-party computation. This paper uses characteristic functions of some subsets of $\mathbb{F}_q$ to construct minimal linear codes. By properties of…
Matrix-vector multiplication is one of the most fundamental computing primitives. Given a matrix $A\in\mathbb{F}^{N\times N}$ and a vector $b$, it is known that in the worst case $\Theta(N^2)$ operations over $\mathbb{F}$ are needed to…
We introduce a theoretical framework that connects multi-chart autoencoders in manifold learning with the classical theory of vector bundles and characteristic classes. Rather than viewing autoencoders as producing a single global Euclidean…
The main goal of this paper is to study the topological properties of tensors in tree-based Tucker format. These formats include the Tucker format and the Hierarchical Tucker format. A property of the so-called minimal subspaces is used for…
Integrality properties of partial sums over irreducible representations, along columns of character tables of finite groups, were recently derived using combinatorial topological string theories (CTST). These CTST were based on…
We make a systematic study of duality phenomena in tensor-triangular geometry, generalising and complementing previous results of Balmer--Dell'Ambrogio--Sanders and Dwyer--Greenlees--Iyengar. A key feature of our approach is the use of…
Let $V$ denote a vector space with finite positive dimension. We consider an ordered pair of linear transformations $A: V\to V$ and $A^*: V\to V$ that satisfy (i) and (ii) below: (i) There exists a basis for $V$ with respect to which the…
In this paper, we develop twisted $K$-theory for stacks, where the twisted class is given by an $S^1$-gerbe over the stack. General properties, including the Mayer-Vietoris property, Bott periodicity, and the product structure $K^i_\alpha…
Tracial Rokhlin property was introduced by Phillips to prove various structure theorems for crossed product. But it is defined for actions on simple C*-algebras only. This paper consists of two major parts: In section 2 and 3, we study the…
Minimal trap spaces (MTSs) capture subspaces in which the Boolean dynamics is trapped, whatever the update mode. They correspond to the attractors of the most permissive mode. Due to their versatility, the computation of MTSs has recently…
In this paper we define the adjoint Reidemeister torsion as a differential form on the character variety of a compact oriented 3-manifold with toral boundary, and prove it defines a regular volume form. Then we show that the torsion form…
Szalai et al. (SIAM J. on Sci. Comp. 28(4), 2006) gave a general construction for characteristic matrices for systems of linear delay-differential equations with periodic coefficients. First, we show that matrices constructed in this way…
Given n subspaces of a finite-dimensional vector space over a fixed finite field $\mathbb F$, we wish to find a linear layout $V_1,V_2,\ldots,V_n$ of the subspaces such that $\dim((V_1+V_2+\cdots+V_i) \cap (V_{i+1}+\cdots+V_n))\le k$ for…
This paper shows that an analytic space $X$ has a unique maximal model through which every proper surjective morphism from a non-singular analytic space to $X$ factors. This is called the {\sl geometric minimal model} of $X$ and…
A characterization of the tree $T^*$ such that $\mathrm{BP}(T^*)=\overleftrightarrow{\mathrm{DFUDS}(T)}$, the reversal of $\mathrm{DFUDS}(T)$ is given. An immediate consequence is a rigorous characterization of the tree $\hat{T}$ such that…
What are called secondary characteristic classes in Chern-Weil theory are a refinement of ordinary characteristic classes of principal bundles from cohomology to differential cohomology. We consider the problem of refining the construction…
A new framework is proposed to study rank-structured matrices arising from discretizations of 2D and 3D elliptic operators. In particular, we introduce the notion of a graph-induced rank structure (GIRS) which aims to capture the fine low…
Following Aguil\'{o}-Su\~{n}er-Torrens (2008), Koles\'{a}rov\'{a}-Mesiar-Mordelov\'{a}-Sempi (2006) and Mayor-Su\~{n}er-Torrens (2005), we continue to develop a theory of matrix representation for discrete copulas. To be more precise, we…