Related papers: Efficient generation of projective modules: a moti…

This work presents a generative modeling approach based on successive subspace learning (SSL). Unlike most generative models in the literature, our method does not utilize neural networks to analyze the underlying source distribution and…

We prove a recognition principle for motivic infinite P1-loop spaces over a perfect field. This is achieved by developing a theory of framed motivic spaces, which is a motivic analogue of the theory of E-infinity-spaces. A framed motivic…

We study the problem of 3D object generation. We propose a novel framework, namely 3D Generative Adversarial Network (3D-GAN), which generates 3D objects from a probabilistic space by leveraging recent advances in volumetric convolutional…

Let $X=Spec{A}$ denote a regular affine scheme, over a field $k$, with $1/2\in k$ and $\dim X=d$. Let $P$ denote a projective $A$-module of rank $n\geq 2$. Let $\pi_0\left({\mathcal LO}(P)\right)$ denote the (Nori) Homotopy Obstruction set,…

We give a moduli-theoretic treatment of the existence and properties of moduli spaces of semistable quiver representations, avoiding methods from geometric invariant theory. Using the existence criteria of Alper--Halpern-Leistner--Heinloth,…

We show that for polytopes P_1, P_2, ..., P_r \subset \R^d, each having n_i \ge d+1 vertices, the Minkowski sum P_1 + P_2 + ... + P_r cannot achieve the maximum of \prod_i n_i vertices if r \ge d. This complements a recent result of Fukuda…

Let $S_g$ be the closed oriented surface of genus g and let $\text{Mod}(S_g)$ be the mapping class group. When the genus is at least 3, $\text{Mod}(S_g)$ can be generated by torsion elements. We prove the follow results. For $g \geq 4$,…

A commutative ring R has finite rank r, if each ideal of R is generated at most by r elements. A commutative ring R has the r-generator property, if each finitely generated ideal of R can be generated by r elements. Such rings are closely…

For $k$ a perfect field of characteristic $p>0$ and $G/k$ a split reductive group with $p$ a non-torsion prime for $G,$ we compute the mod $p$ motivic cohomology of the geometric classifying space $BG_{(r)}$, where $G_{(r)}$ is the $r$th…

We formalize an abstraction of Grothendieck's philosophy of motives and construct a category of derived motivic spectra in the Segal category $\mathbb{R} \underline{\text{Hom}} ((\text{dSt}_k)^{\text{op}}_{/F}, \text{Top})$ ($\text{dSt}_k$…

In the last decade, interest in projective MV-algebras has grown greatly; see [1], [5] e [6]. In this paper we establish a necessary and sufficient condition for n elements of the free n-generator MV-algebra to generate a projective…

Let $S_g$ be the closed oriented surface of genus g and let $\text{Mod}^{\pm}(S_g)$ be the extended mapping class group of $S_g$. When the genus is at least 5, we prove that $\text{Mod}^{\pm}(S_g)$ can be generated by two torsion elements.…

We call a monoidal category ${\mathcal C}$ a Serre category if for any $C$, $D \in {\mathcal C}$ such that $C\ot D$ is semisimple, $C$ and $D$ are semisimple objects in ${\mathcal C}$. Let $H$ be an involutory Hopf algebra, $M$, $N$ two…

Given a reduced effective divisor D on a smooth variety X, we describe the generating function for the classes of the Hodge ideals of D in the Grothendieck group of coherent sheaves on X in terms of the motivic Chern class of the complement…

In this article, the projectivity of finitely generated flat modules of a commutative ring are studied from a topological point of view. Then various interesting results are obtained. For instance, it is shown that if a ring has either a…

To assess the knowledge proficiency of a learner, multiple choice question is an efficient and widespread form in standard tests. However, the composition of the multiple choice question, especially the construction of distractors is quite…

A few pages in Siegel describe how, starting with a fundamental polygon for a compact Riemann surface, one can construct a symplectic basis of its homology. This note retells that construction, specializing to the case where the surface is…

We prove that for any fixed d the generating function of the projection of the set of integer points in a rational d-dimensional polytope can be computed in polynomial time. As a corollary, we deduce that various interesting sets of lattice…

An object in motivic homotopy theory is called cellular if it can be built out of motivic spheres using homotopy colimit constructions. We explore some examples and consequences of cellularity. We explain why the algebraic K-theory and…

For modules over group rings we introduce the following numerical parameter. We say that a module A over a ring R has finite r-generator property if each f.g. (finitely generated) R-submodule of A can be generated exactly by r elements and…