This paper surveys our results on integrable billiards. We consider various models of billiards, including Birkhoff, outer, magnetic, and Minkowski billiards. Also, we discuss wire billiards and billiards in cones. For four models of convex plane billiards, we also discuss an isoperimetric-type inequality for the Mather $\beta$-function. We conclude with a section of open questions on this subject.

We continue to explore the consequences of the recently discovered Minkowski space structure of the Higgs potential in the two-Higgs-doublet model. Here, we focus on the vacuum properties. The search for extrema of the Higgs potential is reformulated in terms of 3-quadrics in the 3+1-dimensional Minkowski space. We prove that 2HDM cannot have more than two local minima in the orbit space and that a twice-degenerate minimum can arise only via spontaneous violation of a discrete symmetry of the Higgs potential. Investigating topology of the 3-quadrics, we give concise criteria for existence of non-contractible paths in the Higgs orbit space. We also study explicit symmetries of the Higgs potential/lagrangian and their spontaneous violation from a wider perspective than usual.

As is well-known, a generalization of the classical concept of the factorial $n!$ for a real number $x\in {\mathbb R}$ is the value of Euler's gamma function $\Gamma(1+x)$. In this connection, the notion of a binomial coefficient naturally arose for admissible values of the real arguments. By elementary means, it is proved a number of properties of binomial coefficients $\binom{r}{\alpha}$ of real arguments $r,\,\alpha\in {\mathbb R}$ such as analogs of unimodality, symmetry, Pascal's triangle, etc. for classical binomial coefficients. The asymptotic behavior of such generalized binomial coefficients of a special form is established.

The Higgs potential of 2HDM keeps its generic form under the group of transformation GL(2,C), which is larger than the usually considered reparametrization group U(2). This reparametrization symmetry induces the Minkowski space structure in the orbit space of 2HDM. Exploiting this property, we present a geometric analysis of the number and properties of stationary points of the most general 2HDM potential. In particular, we prove that charge-breaking and neutral vacua never coexist in 2HDM and establish conditions when the most general explicitly CP-conserving Higgs potential has spontaneously CP-violating minima. Our analysis avoids manipulation with high-order algebraic equations.

This paper defines a new model which incorporates three key ingredients of a large class of wireless communication systems: (1) spatial interactions through interference, (2) dynamics of the queueing type, with users joining and leaving, and (3) carrier sensing and collision avoidance as used in, e.g., WiFi. In systems using (3), rather than directly accessing the shared resources upon arrival, a customer is considerate and waits to access them until nearby users in service have left. This new model can be seen as a missing piece of a larger puzzle that contains such dynamics as spatial birth-and-death processes, the Poisson-Hail model, and wireless dynamics as key other pieces. It is shown that, under natural assumptions, this model can be represented as a Markov process on the space of counting measures. The main results are then two-fold. The first is on the shape of the stability region and, more precisely, on the characterization of the critical value of the arrival rate that separates stability from instability. The second is of a more qualitative or perhaps even ethical nature. There is evidence that for natural values of the system parameters, the implementation of sensing and collision avoidance stabilizes a system that would be unstable if immediate access to the shared resources would be granted. In other words, for these parameters, renouncing greedy access makes sharing sustainable, whereas indulging in greedy access kills the system.

We consider some new estimates for general steady Navier-Stokes solutions in plane domains. According to our main result, if the domain is convex, then the difference between mean values of the velocity over two concentric circles is bounded (up to a constant factor) by the square-root of the Dirichlet integral in the annulus between the circles. The constant factor in this inequality is universal and does not depend on the ratio of the circle radii. Several applications of these formulas are discussed.

In the monograph, STRONG ARTIFICIAL INTELLIGENCE. On the Approaches to Superintelligence, published by Sberbank, provides a cross-disciplinary review of general artificial intelligence. As an anthropomorphic direction of research, it considers Brain Principles Programming, BPP) the formalization of universal mechanisms (principles) of the brain's work with information, which are implemented at all levels of the organization of nervous tissue. This monograph provides a formalization of these principles in terms of the category theory. However, this formalization is not enough to develop algorithms for working with information. In this paper, for the description and modeling of Brain Principles Programming, it is proposed to apply mathematical models and algorithms developed by us earlier that model cognitive functions, which are based on well-known physiological, psychological and other natural science theories. The paper uses mathematical models and algorithms of the following theories: this http URL's Theory of Functional Brain Systems, Eleonor Rosh's prototypical categorization theory, Bob Rehter's theory of causal models and natural classification. As a result, the formalization of the BPP is obtained and computer examples are given that demonstrate the algorithm's operation.

The spectrum $\omega(G)$ of a finite group $G$ is the set of element orders of $G$. Finite groups $G$ and $H$ are isospectral if their spectra coincide. Suppose that $L$ is a simple classical group of sufficiently large dimension (the lower bound varies for different types of groups but is at most 62) defined over a finite field of characteristic $p$. It is proved that a finite group $G$ isospectral to $L$ cannot have a nonabelian composition factor which is a group of Lie type defined over a field of characteristic distinct from $p$. Together with a series of previous results this implies that every finite group $G$ isospectral to $L$ is `close' to $L$. Namely, if $L$ is a linear or unitary group, then $L\leqslant G\leqslant\operatorname{Aut}(L)$, in particular, there are only finitely many such groups $G$ for given $L$. If $L$ is a symplectic or orthogonal group, then $G$ has a unique nonabelian composition factor $S$ and, for given $L$, there are at most 3 variants for $S$ (including $S\simeq L$).

The Erdős--Gallai Theorem states that for $k \geq 3$, any $n$-vertex graph with no cycle of length at least $k$ has at most $\frac{1}{2}(k-1)(n-1)$ edges. A stronger version of the Erdős--Gallai Theorem was given by Kopylov: If $G$ is a 2-connected $n$-vertex graph with no cycle of length at least $k$, then $e(G) \leq \max\{h(n,k,2),h(n,k,\lfloor \frac{k-1}{2}\rfloor)\}$, where $h(n,k,a) := {k - a \choose 2} + a(n - k + a)$. Furthermore, Kopylov presented the two possible extremal graphs, one with $h(n,k,2)$ edges and one with $h(n,k,\lfloor \frac{k-1}{2}\rfloor)$ edges. In this paper, we complete a stability theorem which strengthens Kopylov's result. In particular, we show that for $k \geq 3$ odd and all $n \geq k$, every $n$-vertex $2$-connected graph $G$ with no cycle of length at least $k$ is a subgraph of one of the two extremal graphs or $e(G) \leq \max\{h(n,k,3),h(n,k,\frac{k-3}{2})\}$. The upper bound for $e(G)$ here is tight.

For a graph $H$, a graph $G$ is $H$-saturated if $G$ does not contain $H$ as a subgraph but for any $e \in E(\overline{G})$, $G+e$ contains $H$. In this note, we prove a sharp lower bound for the number of paths and walks on length $2$ in $n$-vertex $K_{r+1}$-saturated graphs. We then use this bound to give a lower bound on the spectral radii of such graphs which is asymptotically tight for each fixed $r$ and $n\to\infty$.

Optimal page replacement is an important problem in efficient buffer management. The range of replacement strategies known in the literature varies from simple but efficient FIFO-based algorithms to more accurate but potentially costly methods tailored to specific data access patterns. The principal issue in adopting a pattern-specific replacement logic in a DB buffer manager is to guarantee non-degradation in general high-load regimes. In this paper, we propose a new family of page replacement algorithms for DB buffer manager which demonstrate a superior performance wrt competitors on custom data access patterns and imply a low computational overhead on TPC-C. We provide theoretical foundations and an extensive experimental study on the proposed algorithms which covers synthetic benchmarks and an implementation in an open-source DB kernel evaluated on TPC-C.

Driven by the need for numerical solutions to the Einstein field equations, we derive a first-order reduction of the second-order $f(T)$-teleparallel gravity equations in the pure-tetrad formulation (no spin connection). We then restrict our attention to the teleparallel equivalent of general relativity (TEGR) and propose a 3+1 decomposition of these equations suitable for computational implementation. We demonstrate that in vacuum (matter-free spacetime) the obtained system of first-order equations is equivalent to the tetrad reformulation of general relativity by Estabrook, Robinson, Wahlquist, and Buchman and Bardeen, and therefore also admits a symmetric hyperbolic formulation. However, the question of hyperbolicity of the 3+1 TEGR equations for arbitrary spacetimes remains unaddressed so far. Furthermore, the structure of the 3+1 equations resembles a lot of similarities with the equations of relativistic electrodynamics and the recently proposed dGREM tetrad-reformulation of general relativity.

We present and discuss some open problems formulated by participants of the International Workshop "Knots, Braids, and Auto\-mor\-phism Groups" held in Novosibirsk, 2014. Problems are related to palindromic and commutator widths of groups; properties of Brunnian braids and two-colored braids, corresponding to an amalgamation of groups; extreme properties of hyperbolic 3-orbifold groups, relations between inner and quasi-inner automorphisms of groups; and Staic's construction of symmetric cohomology of groups.

Representations of braid group $B_n$ on $n \geq 2$ strands by automorphisms of a free group of rank $n$ go back to Artin (1925). In 1991 Kauffman introduced a theory of virtual braids and virtual knots and links. The virtual braid group $VB_n$ on $n \geq 2$ strands is an extension of the classical braid group $B_n$ by the symmetric group $S_n$. In this paper we consider flat virtual braid groups $FVB_n$ on $n\geq 2$ strands and construct a family of representations of $FVB_n$ by automorphisms of free groups of rank $2n$. It has been established that these representations do not preserve the forbidden relations between classical and virtual generators. We investigated some algebraic properties of the constructed representations. In particular, we established conditions of faithfulness in case $n=2$ and proved that the kernel contains a free group of rank two for $n\ge3$.

There are only 10 Euclidean forms, that is flat closed three dimensional manifolds: six are orientable $\mathcal{G}_1,\dots,\mathcal{G}_6$ and four are non-orientable $\mathcal{B}_1,\dots,\mathcal{B}_4$. In the present paper we investigate the manifold $\mathcal{G}_6$, also known as Hantzsche-Wendt manifold; this is the unique Euclidean $3$-form with finite first homology group $H_1(\mathcal{G}_6) = \mathbb{Z}^2_4$. The aim of this paper is to describe all types of $n$-fold coverings over $\mathcal{G}_{6}$ and calculate the numbers of non-equivalent coverings of each type. We classify subgroups in the fundamental group $\pi_1(\mathcal{G}_{6})$ up to isomorphism. Given index $n$, we calculate the numbers of subgroups and the numbers of conjugacy classes of subgroups for each isomorphism type and provide the Dirichlet generating series for the above sequences.

We describe the classification of orthogonal arrays OA$(2048,14,2,7)$, or, equivalently, completely regular $\{14;2\}$-codes in the $14$-cube ($30848$ equivalence classes). In particular, we find that there is exactly one almost-OA$(2048,14,2,7{+}1)$, up to equivalence. As derived objects, OA$(1024,13,2,6)$ ($202917$ classes) and completely regular $\{12,2;2,12\}$- and $\{14, 12, 2; 2, 12, 14\}$-codes in the $13$- and $14$-cubes, respectively, are also classified. Keywords: binary orthogonal array, completely regular code, binary 1-perfect code.

We solve the problem of existence of perfect codes in the Doob graph. It is shown that 1-perfect codes in the Doob graph D(m,n) exist if and only if 6m+3n+1 is a power of 2; that is, if the size of a 1-ball divides the number of vertices. Keywords: perfect codes, distance-regular graphs, Doob graphs, Eisenstein-Jacobi integers.

Word-representable graphs, which are the same as semi-transitively orientable graphs, generalize several fundamental classes of graphs. In this paper we propose a novel approach to study word-representability of graphs using a technique of homomorphisms. As a proof of concept, we apply our method to show word-representability of the simplified graph of overlapping permutations that we introduce in this paper. For another application, we obtain results on word-representability of certain subgraphs of simplified de Bruijn graphs that were introduced recently by Petyuk and studied in the context of word-representability.

A graph $G=(V,E)$ is a \emph{word-representable graph} if there exists a word $W$ over the alphabet $V$ such that letters $x$ and $y$ alternate in $W$ if and only if $(x,y)\in E$ for each $x\neq y$. In this paper we give an effective characterization of word-representable graphs in terms of orientations. Namely, we show that a graph is word-representable if and only if it admits a \emph{semi-transitive orientation} defined in the paper. This allows us to prove a number of results about word-representable graphs, in particular showing that the recognition problem is in NP, and that word-representable graphs include all 3-colorable graphs. We also explore bounds on the size of the word representing the graph. The representation number of $G$ is the minimum $k$ such that $G$ is a representable by a word, where each letter occurs $k$ times; such a $k$ exists for any word-representable graph. We show that the representation number of a word-representable graph on $n$ vertices is at most $2n$, while there exist graphs for which it is $n/2$.

We assume that current state of the Universe can be described by the Inert Doublet Model, containing two scalar doublets, one of which is responsible for EWSB and masses of particles and the second one having no couplings to fermions and being responsible for dark matter. We consider possible evolutions of the Universe to this state during cooling down of the Universe after inflation. We found that in the past Universe could pass through phase states having no DM candidate. In the evolution via such states in addition to a possible EWSB phase transition (2-nd order) the Universe sustained one 1-st order phase transition or two phase transitions of the 2-nd order.