arXiv research

The paper extends von Neumann's theory to normed modules and shows how they can be represented.

The aim of this note is to analyse the structure of the $L^0$-normed $L^0$-modules over a metric measure space. These are a tool that has been introduced by N. Gigli to develop a differential calculus on spaces verifying the Riemannian Curvature Dimension condition. More precisely, we discuss under which conditions an …

Culler-Shalen theory uses the algebraic geometry of the SL(2,C)-character variety of a 3-manifold to construct essential surfaces in the manifold. There are module structures associated to the coordinate rings of the irreducible components of character varieties that are intimately related to essential surface construc…

Extends Hess-Schrader-Uhlenbrock inequality for 1-forms in tamed Dirichlet spaces.

The paper extends Johnson's characterization of amenable groups to homomorphisms and acyclicity in bounded cohomology.

We study the asymptotic behaviour of tame harmonic bundles. First of all, we prove a local freeness of the prolongation by an increasing order. Then we obtain the polarized mixed twistor structure. As one of the applications, we obtain the norm estimate of holomorphic or flat sections by weight filtrations of the monod…

DoRA improves adaptation efficiency for large models by factoring norms and fusing kernels.

We relate generalized Lebesgue decompositions of measures in terms of curve fragments (Alberti representations) and Weaver derivations. This correspondence leads to a geometric characterization of the local norm on the Weaver cotangent bundle of a metric measure space $(X,μ)$: the local norm of a form $df$ sees how fas…

We show that Residual Networks (ResNet) is equivalent to boosting feature representation, without any modification to the underlying ResNet training algorithm. A regret bound based on Online Gradient Boosting theory is proved and suggests that ResNet could achieve Online Gradient Boosting regret bounds through neural n…

In the early 2000's Cochran and Harvey introduced non-commutative Alexander polynomials for 3-manifolds. Their degrees give strong lower bounds on the Thurston norm. In this paper we make the case that the vanishing of a certain Novikov-Sikorav homology module is the correct notion of a monic non-commutative Alexander …

Recently, based on the idea of randomizing space theory, random convex analysis has been being developed in order to deal with the corresponding problems in random environments such as analysis of conditional convex risk measures and the related variational problems and optimization problems. Random convex analysis is …

Unified normative modeling for neuroimaging phenotypes using denoising diffusion models.

This work investigates implicit bias in multiclass separable data using a novel geometry-aware optimizer.

Logit dynamics formula reveals self-regulation in softmax policy gradient methods.

Pion optimizes LLMs by preserving weight matrix singular values.

Classifies modules of surface-knots in terms of their properties.

The purpose of this paper is to give a selective survey on recent progress in random metric theory and its applications to conditional risk measures. This paper includes eight sections. Section 1 is a longer introduction, which gives a brief introduction to random metric theory, risk measures and conditional risk measu…

Proves finiteness and holonomicity of skein modules for 3-manifolds.

Defines super projective modules and explores their properties.

Both the Dictionary Learning (DL) and Convolutional Neural Networks (CNN) are powerful image representation learning systems based on different mechanisms and principles, however whether we can seamlessly integrate them to improve the per-formance is noteworthy exploring. To address this issue, we propose a novel gener…

We introduce higher skein modules of links generalizing the Conway skein module. We show that these modules are closely connected to the HOMFLY polynomial.

A new method to derive presentations of skein modules is developed. For the case of homotopy skein modules it will be shown how the topology of a 3-manifold is reflected in the structure of the module. The freeness problem for q-homotopy skein modules is solved, and a natural skein module related to linking numbers is …

Sparse Singular Value Decomposition (SVD) models have been proposed for biclustering high dimensional gene expression data to identify block patterns with similar expressions. However, these models do not take into account prior group effects upon variable selection. To this end, we first propose group-sparse SVD model…

Neural Module Networks, originally proposed for the task of visual question answering, are a class of neural network architectures that involve human-specified neural modules, each designed for a specific form of reasoning. In current formulations of such networks only the parameters of the neural modules and/or the or…

Paper compares skein modules to Kauffman bracket modules.

Skein modules are the main objects of an algebraic topology based on knots (or position). In the same spirit as Leibniz we would call our approach "algebra situs." When looking at the panorama of skein modules we see, past the rolling hills of homologies and homotopies, distant mountains - the Kauffman bracket skein mo…

Generalized Steinberg module presentation for Gaussian and Eisenstein integers.

Enhanced Alexander module detects linking numbers in links.

We define 2-crossed module bundle 2-gerbes related to general Lie 2-crossed modules and discuss their properties. A 2-crossed module bundle 2-gerbe over a manifold is defined in terms of a so called 2-crossed module bundle gerbe, which is a crossed module bundle gerbe equipped with an extra sructure. It is shown that s…

Combinatorial approach to compute satellite knot invariants using graph theory.

Self-attention prefers sparse functions of input sequences, reducing sample complexity.

New sl(2) action defined on a mathematical module.

Introduces admissible skein modules for non-semisimple categories.

Studies modules over a category of Jacobi diagrams in handlebodies.

This paper generalizes L2 cohomology theory for complex manifolds.

Let $\{T_1, \ldots, T_n\}$ be a set of $n$ commuting bounded linear operators on a Hilbert space $\mathcal{H}$. Then the $n$-tuple $(T_1, \ldots, T_n)$ turns $\mathcal{H}$ into a module over $\mathbb{C}[z_1, \ldots, z_n]$ in the following sense: \[\mathbb{C}[z_1, \ldots, z_n] \times \mathcal{H} \raro \clh, \quad \quad …

Study Kauffman bracket skein modules of Seifert fibered spaces.

This paper presents a method to compute the {\it quasi-conformal parameterization} (QCMC) for a multiply-connected 2D domain or surface. QCMC computes a quasi-conformal map from a multiply-connected domain $S$ onto a punctured disk $D_S$ associated with a given Beltrami differential. The Beltrami differential, which me…

Formula for interleaving distance of rectangle persistence modules.

Introduces Floer lasagna modules using link Floer homology.

We show that the Kauffman bracket skein module of a cylinder over the torus embeds as a subalgebra of the noncommutative torus. Using this we derive nice formulas for the Jones-Wenzl idempotents and analyze the structure of the Kauffman bracket skein module of the unknot as a module over the Kauffman bracket skein modu…

New knot theory module shows torsion-ness in number theory.

Study quandle modules over geometric quandles and their relation to Lie-Yamaguti representations.

The multivariate Alexander module of a link L has several subsets that admit quandle operations defined using the module operations. One of them, the fundamental multivariate Alexander quandle, determines the link module sequence of L.

A complex vector space $V$ is a prehomogeneous $G$-module if $G$ acts rationally on $V$ with a Zariski-open orbit. The module is called etale if $\dim V=\dim G$. We study etale modules for reductive algebraic groups $G$ with one-dimensional center. For such $G$, even though every etale module is a regular prehomogeneou…

A commuting $n$-tuple $(T_1, \ldots, T_n)$ of bounded linear operators on a Hilbert space $\clh$ associate a Hilbert module $\mathcal{H}$ over $\mathbb{C}[z_1, \ldots, z_n]$ in the following sense: \[\mathbb{C}[z_1, \ldots, z_n] \times \mathcal{H} \rightarrow \mathcal{H}, \quad \quad (p, h) \mapsto p(T_1, \ldots, T_n)h…

Study on skein module dimensions at irreducible representations.

This document contains tables with the classification of prehomogeneous modules for reductive algebraic groups with up to two simple factors due to Sato, Kimura and many others, as well as corresponding tables of the étale modules appearing in this list, determined by the author. It is intended as a convenient referenc…