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Using semi-tensor product of matrices, the structures of several kinds of symmetric games are investigated via the linear representation of symmetric group in the structure vector of games as its representation space. First of all, the…
This paper studies the complexity of computing a representation of a simple game as the intersection (union) of weighted majority games, as well as, the dimension or the codimension. We also present some examples with linear dimension and…
We study the space of functions computed by random-layered machines, including deep neural networks and Boolean circuits. Investigating the distribution of Boolean functions computed on the recurrent and layer-dependent architectures, we…
We examine a hierarchy of equivalence classes of quasi-random properties of Boolean Functions. In particular, we prove an equivalence between a number of properties including balanced influences, spectral discrepancy, local strong…
The relationship between quantum physics and discrete mathematics is reviewed in this article. The Boolean functions unitary representation is considered. The relationship between Zhegalkin polynomial, which defines the algebraic normal…
Many binary collective choice situations can be described as weighted simple voting games. We introduce weighted committee games to model decisions on an arbitrary number of alternatives in analogous fashion. We compare the effect of…
Arithmetic functions in Number Theory meet the Sprague-Grundy function from Combinatorial Game Theory. We study a variety of 2-player games induced by standard arithmetic functions, such as Euclidian division, divisors, remainders and…
We consider the volume of a Boolean expression of some congruent balls about a given system of centers in the $d$-dimensional Euclidean space. When the radius $r$ of the balls is large, this volume can be approximated by a polynomial of…
Boole polynomials play an important role in the area of number theory, algebra and umbral calculus. In this paper, we investigate some properties of Boole polynomials and consider Witt-type formulas for the Boole numbers and polynomials.…
We define Boolean algebras in the linear context and study its symmetric powers. We give explicit formulae for products in symmetric Boolean algebras of various dimensions. We formulate symmetric forms of the inclusion-exclusion principle.
The remoteness from a simple game to a weighted game can be measured by the concept of the dimension or the more general Boolean dimension. It is known that both notions can be exponential in the number of voters. For complete simple games…
We introduce a new class of games where each player's aim is to randomise her strategic choices in order to affect the other players' expectations aside from her own. The way each player intends to exert this influence is expressed through…
In this paper, we study various classes of partition functions such as those related to the parity of the number of parts, to differences of partition numbers, and to partitions with a repeated smallest part. We establish identities…
Scaled Boolean algebras are a category of mathematical objects that arose from attempts to understand why the conventional rules of probability should hold when probabilities are construed, not as frequencies or proportions or the like, but…
The concepts of Boolean metric space and convex combination are used to characterize polynomial maps in a class of commutative Von Neumann regular rings including Boolean rings and p-rings, that we have called CFG-rings. In those rings, the…
We analyze properties of apportionment functions in context of the problem of allocating seats in the European Parliament. Necessary and sufficient conditions for apportionment functions are investigated. Some exemplary families of…
Positional games are a mathematical class of two-player games comprising Tic-tac-toe and its generalizations. We propose a novel encoding of these games into Quantified Boolean Formulas (QBFs) such that a game instance admits a winning…
In this paper, we discover that the class of random polynomials arising from the equilibrium analysis of random asymmetric evolutionary games is \textit{exactly} the Kostlan-Shub-Smale system of random polynomials, revealing an intriguing…
An infinite family of Boolean polynomials which correspond to the discrete average maps, defined in [2], is constructed and their algebraic and combinatorial properties are investigated. They turn out to be balanced, and some recurrence…
Contextuality is arguably the fundamental property that makes quantum mechanics different from classical physics. It is responsible for quantum computational speedups in both magic-state-injection-based and measurement-based models of…