arXiv subjects

V. I. Danilov

Publications and source records attributed to V. I. Danilov.

14 recordsLinked to original sources

arXiv2025

On the existence of stable contract systems

In 1962, Gale and Shapley \cite{GS} introduced the concept of stable marriages and proved their existence. Since then, the statement of the stability problem has been highly generalized. And a lot of proofs has emerged for the existence in these more general statements. It's time to review them and identify the similarities and differences. First, we will briefly discuss the classical case, because the existence proofs in the general case grew out of it. Or rather, from the idea of "deferred acceptance". When the best of the proposed contracts is temporarily retained until a better offer is received.

econ.TH↗

arXiv2020

$B_2$-crystals: axioms, structure, models

We present a list of ``local'' axioms and an explicit combinatorial construction for the regular $B_2$-crystals (crystal graphs of highest weight integrable modules over $U_q(sp_4)$). Also a new combinatorial model for these crystals is developed.

math.RT↗

arXiv2019

Cubillages of cyclic zonotopes

This paper (written in Russian) presents a survey of new and earlier results on fine zonotopal tilings (briefly, cubillages) of cyclic zonotopes. The combinatorial theory of these objects is of interest in its own right and also has a connection to higher Bruhat orders, triangulations of cyclic polytopes, and Tamari-Stasheff posets applied in the study of Kadomtsev--Petviashvily equations, and etc.

math.CO↗

arXiv2018

Cubillages on cyclic zonotopes, membranes, and higher separatation

We study certain structural properties of fine zonotopal tilings, or cubillages, on cyclic zonotopes $Z(n,d)$ of an arbitrary dimension $d$ and their relations to $(d-1)$-separated collections of subsets of a set $\{1,2,\ldots,n\}$. (Collections of this sort are well known as strongly separated ones when $d=2$, and as chord separated ones when $d=3$.)

math.CO↗

arXiv2018

On interrelations between strongly, weakly and chord separated set-systems (a geometric approach)

We consider three types of set-systems that have interesting applications in algebraic combinatorics and representation theory: maximal collections of the so-called strongly separated, weakly separated, and chord separated subsets of a set $[n]=\{1,2,\ldots,n\}$. These collections are known to admit nice geometric interpretations; namely, they are bijective, respectively, to rhombus tilings on the zonogon $Z(n,2)$, combined tilings on $Z(n,2)$, and fine zonotopal tilings (or `cubillages') on the 3-dimensional zonotope $Z(n,3)$. We describe interrelations between these three types of set-systems in $2^{[n]}$, by studying interrelations between their geometric models. In particular, we completely characterize the sets of rhombus and combined tilings properly embeddable in a fixed cubillage, explain that they form distributive lattices, give efficient methods of extending a given rhombus or combined tiling to a cubillage, and etc.

math.CO↗

arXiv2008

The crossing model for regular $A_n$-crystals

A regular $A_n$-crystal is an edge-colored directed graph, with $n$ colors, related to an irreducible highest weight integrable module over $U_q(sl_{n+1})$. Based on Stembridge's local axioms for regular simply-laced crystals and a structural characterization of regular $A_2$-crystals in \cite{DKK-07}, we present a new combinatorial construction, the so-called {\em crossing model}, and prove that this model generates precisely the set of regular $A_n$-crystals. Using the model, we obtain a series of results on the combinatorial structure of such crystals and properties of their subcrystals.

math.RT↗

arXiv2007

Measurable Systems and Behavioral Sciences

Individual choices often depend on the order in which the decisions are made. In this paper, we expose a general theory of measurable systems (an example of which is an individual's preferences) allowing for incompatible (non-commuting) measurements. The basic concepts are illustrated in an example of non-classical rational choice. We conclude with a discussion of some of the basic properties of non-classical systems in the context of social sciences. In particular, we argue that the distinctive feature of non-classical systems translates into a formulation of bounded rationality.

physics.soc-ph↗

arXiv2007

The octahedron recurrence and RSK-correspondence

We start with an ``algebraic'' RSK-correspondence due to Noumi and Yamada. Given a matrix $X$, we consider a pyramidal array of solid minors of $X$. It turns out that this array satisfies an algebraic variant of octahedron recurrence. The main observation is that this array can also be constructed with the help of some square `genetic' array. Next we tropicalize this algebraic construction and consider $T$-{\em polarized} pyramidal arrays (that is arrays satisfying octahedral relations). As a result we get several bijections, viz: a) a linear bijection between non-negative arrays and supermodular functions; b) a piecewise linear bijection between supermodular functions and the so called infra-modular functions; c) a linear bijection between infra-modular functions and plane partitions. A composition of these bijections yields a bijection between non-negative arrays and plane partitions coinciding with the modified RSK-correspondence.

math.CO↗

arXiv2006

On the structure of regular $B_2$-type crystals

For simply-laced Kac-Moody algebras $\frak g$, Stembridge (2003) proposed a `local' axiomatization of crystal graphs of representations of $U_q(\frak g)$. In this paper we propose axioms for edge-2-colored graphs which characterize the crystals of integrable representations of $U_q(sp(4))$, regular crystal graphs of $B_2$-type. An edge-colored directed graph which obeys our Axioms (K0)--(K5) is called an R-{\em graph} (for brevity), and our main result is that the regular crystals of $B_2$-type are R-graphs and vice versa. We give a direct combinatorial construction for the crystals in question. On this way we introduce a new, so-called {\em crossing model}, which does not exploit Young tableaux. This combinatorial model consists of a two-component graph of a rather simple form and of a certain set of integer-valued functions on its vertices.

math.RT↗

arXiv2006

Non-Classical Expected Utility Theory

In this paper we extend Savage's theory of decision-making under uncertainty from a classical environment into a non-classical one. We formulate the corresponding axioms and provide representation theorems for qualitative measures and expected utility.

physics.soc-ph↗

arXiv2006

Combinatorics of $A_2$-crystals

We show that a connected regular $A_2$-crystal (the crystal graph of an irreducible representation of $sl_3$) can be produced from two half-grids by replicating them and glying together in a certain way. Also some extensions and related aspects are discussed.

math.RT↗

arXiv2005

Discrete strip-concave functions, Gelfand-Tsetlin patterns, and related polyhedra

Discrete strip-concave functions considered in this paper are, in fact, equivalent to an extension of Gelfand-Tsetlin patterns to the case when the pattern has a not necessarily triangular but convex configuration. They arise by releasing one of the three types of rhombus inequalities for discrete concave functions (or ``hives'') on a ``convex part'' of a triangular grid. The paper is devoted to a combinatorial study of certain polyhedra related to such functions or patterns, and results on faces, integer points and volumes of these polyhedra are presented. Also some relationships and applications are discussed. In particular, we characterize, in terms of valid inequalities, the polyhedral cone formed by the boundary values of discrete strip-concave functions on a grid having trapezoidal configuration. As a consequence of this result, necessary and sufficient conditions on a pair of vectors to be the shape and content of a semi-standard skew Young tableau are obtained.

math.CO↗

arXiv2003

Discrete convexity and unimodularity. I

In this paper we develop a theory of convexity for a free Abelian group M (the lattice of integer points), which we call theory of discrete convexity. We characterize those subsets X of the group M that could be call "convex". One property seems indisputable: X should coincide with the set of all integer points of its convex hull co(X) (in the ambient vector space V). However, this is a first approximation to a proper discrete convexity, because such non-intersecting sets need not be separated by a hyperplane. This issue is closely related to the question when the intersection of two integer polyhedra is an integer polyhedron. We show that unimodular systems (or more generally, pure systems) are in one-to-one correspondence with the classes of discrete convexity. For example, the well-known class of g-polymatroids corresponds to the class of discrete convexity associated to the unimodular system A_n:={\pm e_i, e_i-ej} in Z^n.

math.CO↗

arXiv2003

Mathematics of Plott choice functions

This paper is devoted to a study of mathematical structures arising from choice functions satisfying the path independence property (Plott functions). We broaden the notion of a choice function by allowing of empty choice. This enables us to define a lattice structure on the set of Plott functions. Moreover, this lattice is functorially dependent on its base. We introduce a natural convex structure on the set of linear orders (or words) and show that Plott functions are in one-to-one correspondence with convex subsets in this set of linear orders. That correspondence is compatible with both lattice structures. Keywords: Convex geometries, shuffle, linear orders, lattices, direct image, path independence, convex structure

math.CO↗