相关论文: A counterexample to the "composition conjecture"

Given a finite tensor category $\ca$, an exact indecomposable $\ca$-module category $\Mo$, and a tensor subcategory $\Do\subseteq \ca^*_\Mo$, we describe a way to produce \textit{exact} commutative algebras in the center $Z(\ca)$, measuring…

The inverse of a large matrix can often be accurately approximated by a polynomial of degree significantly lower than the order of the matrix. The iteration polynomial generated by a run of the GMRES algorithm is a good candidate, and its…

This paper provides some counterexamples to Cantor's contributions to the foundations of Set Theory. The first counterexample forces Cantor's Diagonal Method (DM) to yield one of the numbers in the target list. To study this anomaly, and…

We develop the theory of mixed finite elements in terms of special inverse systems of complexes of differential forms, defined over cellular complexes. Inclusion of cells corresponds to pullback of forms. The theory covers for instance…

Using a new compactification (toroidal compactification) and desingularization, we obtain a complete characterization of monodromy at infinity for polynomial Newton system of arbitrary degree, in which we establish an equivalence between…

In this paper, I have proved that for a class of polynomial differential systems of degree n+1 ( where n is an arbitrary positive integer) the composition conjecture is true. I give the sufficient and necessary conditions for these…

We extend a conjecture of Kimberley-Robertson on the abelianizations of certain square complex groups.

In this paper we will investigate contramodules for algebraic groups. Namely, we give contra-analogs to two 20th century results about comodules. Firstly, we show that induction of contramodules over coordinate rings of algebraic groups is…

We classify fields having finitely many finite non-commutative (not necessarily central) division algebras over them. In the process, we introduce the notion of anti-closure of a field and also make comments on fields having a linear…

We introduce a handy construction of cluster algebras of type $\mathbb{A}_{\infty}$, we give a complete classification of the cluster algebras arising from the infinity-gon, and finally we construct the category of the diagonals of the…

The paper proposes a logical model of combinatorial problems, also it gives an example of a problem of the class NP that can not be solved in polynomial time on the dimension of the problem.

We give an explicit description of the category of central extensions of a group scheme by a sheaf of Abelian groups. Based on this, we describe a framework for computing with central extensions of finite commutative group schemes, torsors…

Machine learning algorithms generally suffer from a problem of explainability. Given a classification result from a model, it is typically hard to determine what caused the decision to be made, and to give an informative explanation. We…

We prove congruences, modulo a power of a prime p, for certain finite sums involving central binomial coefficients $\binom{2k}{k}$.

We give the counter-examples related to a Gaussian Brunn-Minkowski inequality and the (B) conjecture.

We show that there exists a stably free module over a polynomial ring which is not extended from the ground ring. This provides a counterexample to the Hermite ring conjecture.

Ultrafilters are useful mathematical objects having applications in nonstandard analysis, Ramsey theory, Boolean algebra, topology, and other areas of mathematics. In this note, we provide a categorical construction of ultrafilters in terms…

We present a theorem about irreducibility of a polynomial that is the resultant of two others polynomials. The proof of this fact is based on the field theory. We also consider the converse theorem and some examples.

The well-known Formanek's module finiteness theorem states that every unital prime PI-algebra (i.e. a central order in a matrix algebra by Posner's theorem) embeds into a finitely generated module over its center. An analogue of this…

The CMI Millennium "P vs NP Problem" can be resolved e.g. if one shows at least one counterexample to the conjecture "P is equal to NP". A certain class of problems being such counterexamples is formulated. This implies the rejection of the…