Related papers: On some impossible disentanglement puzzles
We study problems related to indecomposability of modules over certain local finite dimensional trivial extension algebras. We do this by purely combinatorial methods. We introduce the concepts of graph of cyclic modules, of combinatorial…
We prove that certain quiver varieties are irreducible and therefore are isomorphic to Hilbert schemes of points of the total spaces of the bundles $\mathcal O_{\mathbb P^1}(-n)$ for $n \ge 1$.
Motivated by the problem of finding finite versions of classical incompleteness theorems, we present some conjectures that go beyond ${\bf NP\neq co NP}$. These conjectures formally connect computational complexity with the difficulty of…
In the present paper, mappings satisfying one modular inequality with respect to cylinders in a space, are considered. Distorting of modulus is majorized by an integral which depends from some locally integrable function. The result on…
In this paper, we proposed an interesting problem that might be classified into enumerative combinatorics. Featuring a distinctive two-fold dependence upon the sequences' terms, our problem can be really difficult, which calls for novel…
We consider cylindrical algebraic decomposition (CAD) and the key concept of delineability which underpins CAD theory. We introduce the novel concept of projective delineability which is easier to guarantee computationally. We prove results…
This work gathers new results concerning the semi-geostrophic equations: existence and stability of measure valued solutions, existence and uniqueness of solutions under certain continuity conditions for the density, convergence to the…
Several results about the union-closed sets conjecture are presented.
Multiparameter persistence is a natural extension of the well-known persistent homology, which has attracted a lot of interest. However, there are major theoretical obstacles preventing the full development of this promising theory. In this…
We consider apictorial edge-matching puzzles, in which the goal is to arrange a collection of puzzle pieces with colored edges so that the colors match along the edges of adjacent pieces. We devise an algebraic representation for this…
Will the cosmological multiverse, when described mathematically, have easily stated properties that are impossible to prove or disprove using mathematical physics? We explore this question by constructing lattice multiverses which exhibit…
This article presents a general solution to the problem of computational complexity. First, it gives a historical introduction to the problem since the revival of the foundational problems of mathematics at the end of the 19th century.…
In the present popular-science paper, we find out which rectangles can be dissected into squares. The proof is based on a physical interpretation in terms of electrical networks. Only a secondary school background is assumed in the paper.
In this paper, we pose many challenging conjectures on congruences involving binomial coefficients and Ap\'ery-like numbers.
We collect here various conjectures on congruences made by the author in a series of papers, some of which involve binary quadratic forms and other advanced theories. Part A consists of 100 unsolved conjectures of the author while…
The main result of this paper is a probabilistic construction of finite rigid structures. It yields a finitely axiomatizable class of finite rigid structures where no L^omega_{infty, omega} formula with counting quantifiers defines a linear…
We give a proof of impossibility of probabilistic exact $1\to 2$ cloning of any three different states of a qubit. The simplicity of the proof is due to the use of a surprising result of remote state preparation [M.-Yong Ye, Y.-Sheng Zhang…
Unrefinable partitions are a subset of partitions into distinct parts which satisfy an additional unrefinability property. More precisely, being an unrefinable partition means that none of the parts can be written as the sum of smaller…
I shall describe a general model-theoretic task to construct expansions of pseudofinite structures and discuss several examples of particular relevance to computational complexity. Then I will present one specific situation where finding a…
Understanding the Impossibility of a Tie in Hex via Fixed Point Theorems, the Hex Theorem, and Their Equivalence.