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Consider a graph G = ( V , E ) and a random initial coloring where each vertex is black independently with probability p b , and white with probability p w = 1 − p b . In each step, all vertices change their current color synchronously to the most frequent color in their neighborhood and in case of a tie, a vertex keeps its current color. This model is called the majority model. If in case of a tie a vertex always selects black color, it is called the biased majority model. We are interested in the behavior of these two processes, especially when the underlying graph is a two-dimensional torus (cellular automaton with (biased) majority rule). In the present paper, as our main result we prove that both majority and biased majority cellular automata exhibit a threshold behavior with two phase transitions. More precisely, we prove for a two-dimensional torus T n , n , there are two threshold values 0 ≤ p 1 , p 2 ≤ 1 such that p b ≪ p 1 , p 1 ≪ p b ≪ p 2 , and p 2 ≪ p b result in final complete occupancy by white, stable coexistence of both colors, and final complete occupancy by black, respectively in O ( n 2 ) number of steps. (For two functions f ( n ) and g ( n ) , we shortly write f ( n ) ≪ g ( n ) instead of f ( n ) ∈ o ( g ( n ) ) .) We finally argue that our proof techniques can be used to prove a similar threshold behavior for a larger class of models.

In this note, we give some new families of two-stage spaces for which the torus rank conjecture is affirmed., 5 pages

Combining McShane's identity on a hyperbolic punctured torus with Schmutz's work on the Markov Uniqueness Conjecture (MUC), we find that MUC is equivalent to the identity \begin{equation} \sum_{n=1}^\infty \, \left( 3- L_n \right) \, = \, 4 - \varphi - \sqrt 2 \end{equation} where $L_n$ is the $n$th Lagrange number and $\varphi=\frac{1+\sqrt5}2$ is the golden ratio., Comment: 5 pages, 2 figures, comments welcome!

We study various statistics regarding the distribution of the points \[\left\{\left(\frac{d}{q},\frac{\overline{d}}{q}\right) \in \mathbb{T}^2 : d \in (\mathbb{Z}/q\mathbb{Z})^{\times}\right\}\] as $q$ tends to infinity. Due to nontrivial bounds for Kloosterman sums, it is known that these points equidistribute on the torus. We prove refinements of this result, including bounds for the discrepancy, small scale equidistribution, bounds for the covering exponent associated to these points, sparse equidistribution, and mixing., Comment: 11 pages

The conditions for the unique solvability of the boundary-value problem for a functional differential equation with shifted and compressed arguments are expressed via the spectral radius formula for the corresponding class of functional operators. The use of this formula involves calculation of certain type limits, which, even in the simplest cases, exhibit an amazing “chaotic” dependence on the compression ratio. For example, it turns out that the spectral radius of the operator $$L_{2}(\mathbb{R}^{n})\ni u(x)\mapsto u(p^{-1}x+h)-u(p^{-1}x-h)\in L_{2}(\mathbb{R}^{n}),\quad p>1,\quad h\in\mathbb{R}^{n},$$ is equal to $$2p^{n/2}$$ for transcendental values of $$p$$ , and depends on the coefficients of the minimal polynomial for $$p$$ in the case where $$p$$ is an algebraic number. In this paper, we study this dependence. The starting point is the well-known statement that, given a velocity vector with rationally independent coordinates, the corresponding linear flow is minimal on the torus, i.e., the trajectory of each point is everywhere dense on the torus. We prove a version of this statement that helps to control the behavior of trajectories also in the case of rationally dependent velocities. Upper and lower bounds for the spectral radius are obtained for various cases of the coefficients of the minimal polynomial for $$p$$ . The main result of the paper is the exact formula of the spectral radius for rational (and roots of any degree of rational) values of $$p$$ .

We consider volume preserving homeomorphisms of the torus $${\mathbb{T}^d}$$ (d ≥ 2) homotopic to the identity and prove that there exists a C0-open and dense subset of those homeomorphisms having stable rotation sets. Moreover, the rotation sets are polyhedrons with rational vertices and non-empty interior. Finally, we prove that the level sets in $${\mathbb{T}^d}$$ formed by points that realize the extremal vectors carry zero topological entropy. We obtain related results for continuous maps and general homeomorphisms homotopic to the identity.

In 1973, Brown, Erdős and Sós proved that if $\mathcal{H}$ is a 3-uniform hypergraph on $n$ vertices which contains no triangulation of the sphere, then $\mathcal{H}$ has $O(n^{5/2})$ edges, and this bound is the best possible up to a constant factor. Resolving a conjecture of Linial, also reiterated by Keevash, Long, Narayanan and Scott, we show that the same result holds for triangulations of the torus. Furthermore, we extend our result to every closed orientable surface $\mathcal{S}$.

Let $\Delta$ be a hyperbolic triangle with a fixed area $\varphi$. We prove that for all but countably many $\varphi$, generic choices of $\Delta$ have the property that the group generated by the $\pi$--rotations about the midpoints of the sides of the triangle admits no nontrivial relations. By contrast, we show for all $\varphi\in(0,\pi)\setminus\mathbb{Q}\pi$, a dense set of triangles does afford nontrivial relations, which in the generic case map to hyperbolic translations. To establish this fact, we study the deformation space $\mathfrak{C}_\theta$ of singular hyperbolic metrics on a torus with a single cone point of angle $\theta=2(\pi-\varphi)$, and answer an analogous question for the holonomy map $\rho_\xi$ of such a hyperbolic structure $\xi$. In an appendix by X.~Gao, concrete examples of $\theta$ and $\xi\in\mathfrak{C}_\theta$ are given where the image of each $\rho_\xi$ is finitely presented, non-free and torsion-free; in fact, those images will be isomorphic to the fundamental groups of closed hyperbolic 3--manifolds., Comment: 32 pages. To appear in Math. Z

We show that an Anosov map has a geodesic axis on the curve graph of the torus. The direct corollary of our result is the stable translation length of an Anosov map on the curve graph is always a positive integer. As the proof is constructive, we also provide an algorithm to calculate the exact translation length for any given Anosov map. The application of our result is threefold: (a) to determine which word realizes the minimal translation length on the curve graph within a specific class of words, (b) to establish the effective bound on the ratio of translation lengths of an Anosov map on the curve graph to that on Teichmuller space, and (c) to estimate the overall growth of the number of Anosov maps which have a sufficient number of Anosov maps with the same translation length .

The fractional matching preclusion number of a graph is the minimum number of edges whose deletion results in the remaining graph that has no fractional perfect matchings. For many networks, their optimal fractional matching preclusion sets are precisely those edges incident with a single vertex. The probability that all failures concentrate around a vertex is often small. To overcome the shortcoming, we consider the concept of conditional fractional matching preclusion, in which isolated vertices are not permitted in fault networks. We establish the conditional fractional matching preclusion numbers and all possible minimum conditional fractional matching preclusion sets for n -dimensional torus networks with n ≥ 3 .

Beineke, Harary, and Ringel discovered a formula for the minimum genus of a torus in which the n-dimensional hypercube graph can be embedded. We give a new proof of the formula by building this sur...

We consider an annular set of the form $$K=B\times \mathbb{T}^{\infty}$$ , where $$B$$ is a closed ball of the Banach space $$E$$ , $$\mathbb{T}^{\infty}$$ is the infinite-dimensional torus (the direct product of a countable number of circles with the topology of coordinatewise uniform convergence). For a certain class of smooth maps $$\Pi\colon K\to K$$ , we establish sufficient conditions for the existence and stability of an invariant toroidal manifold of the form $$A=\{(v, \varphi)\in K: v=h(\varphi)\in E,\,\varphi\in\mathbb{T}^{\infty}\},$$ where $$h(\varphi)$$ is a continuous function of the argument $$\varphi\in\mathbb{T}^{\infty}$$ . We also study the question of the $$C^m$$ -smoothness of this manifold for any natural $$m$$ .

Let sl N ˆ ( C q ) be the core of extended affine Lie algebra of type A N − 1 coordinated by the rational quantum 2-torus C q . In this paper, we first prove that for any complex number l, the category of restricted sl N ˆ ( C q ) -modules of level l is canonically isomorphic to the category of twisted modules for the vertex algebra V C g ˆ ( l , 0 ) arising from a conformal Lie algebra C g , where C g ˆ is isomorphic to a toroidal Lie algebra. Then we prove that for any nonnegative integer l, the integrable restricted sl N ˆ ( C q ) -modules of level l are exactly the twisted modules for the quotient vertex algebra L C g ˆ ( l , 0 ) of V C g ˆ ( l , 0 ) . Finally, we classify irreducible graded twisted L C g ˆ ( l , 0 ) -modules.

We classify stably/retract rational norm one tori in dimension $p-1$ where $p$ is a prime number and in dimension up to ten with some minor exceptions., Comment: 12 pages. The statements of Theorem 1.9 and Theorem 1.11 are modified. arXiv admin note: substantial text overlap with arXiv:1210.4525, arXiv:1411.2790

Motion groups of links in the three sphere $\mathbb{S}^3$ are generalizations of the braid groups, which are motion groups of points in the disk $\mathbb{D}^2$. Representations of motion groups can be used to model statistics of extended objects such as closed strings in physics. Each $1$-extended $(3+1)$-topological quantum field theory (TQFT) will provide representations of motion groups, but it is difficult to compute such representations explicitly in general. In this paper, we compute representations of the motion groups of links in $\mathbb{S}^3$ with generalized axes from Dijkgraaf-Witten (DW) TQFTs inspired by dimension reduction. A succinct way to state our result is as a step toward a twisted generalization (Conjecture \ref{mainconjecture}) of a conjecture for DW theories of dimension reduction from $(3+1)$ to $(2+1)$: $\textrm{DW}^{3+1}_G \cong \oplus_{[g]\in [G]} \textrm{DW}^{2+1}_{C(g)}$, where the sum runs over all conjugacy classes $[g]\in [G]$ of $G$ and $C(g)$ the centralizer of any element $g\in [g]$. We prove a version of Conjecture \ref{mainconjecture} for the mapping class groups of closed manifolds and the case of torus links labeled by pure fluxes., Clarify the main conjecture as a twisted version of dimension reduction. To appear in Comm. Math Phys

We construct a minimal cubature formula of degree $$2 $$ for a torus in $${\mathbb R}^3 $$ .

An $$(n+1)$$ -toroid is a quotient of a tessellation of the n-dimensional Euclidean space with a lattice group. Toroids are generalisations of maps on the torus to higher dimensions and also provide examples of abstract polytopes. Equivelar toroids are those that are induced by regular tessellations. In this paper we present a classification of equivelar $$(n+1)$$ -toroids with at most n flag-orbits; in particular, we discuss a classification of 2-orbit toroids of arbitrary dimension.

The $\mathfrak {s}\mathfrak {l}_{3}$ colored Jones polynomial $J_{\lambda }^{\mathfrak {s}\mathfrak {l}_{3}}(L)$ is obtained by coloring the link components with two-row Young diagram λ. Although it is difficult to compute $J_{\lambda }^{\mathfrak {s}\mathfrak {l}_{3}}(L)$ in general, we can calculate it by using Kuperberg’s A2 skein relation. In this paper, we show some formulas for twisted two strands colored by one-row Young diagram in A2 web space and compute $J_{(n,0)}^{\mathfrak {s}\mathfrak {l}_{3}}(T(2,2m))$ for an oriented (2,2m)-torus link. These explicit formulas derives the $\mathfrak {s}\mathfrak {l}_{3}$ tail of T(2,2m). They also give explicit descriptions of the $\mathfrak {s}\mathfrak {l}_{3}$ false theta series with one-row coloring because the $\mathfrak {s}\mathfrak {l}_{2}$ tail of T(2,2m) is known as the false theta series.

Given two disjoint vertex-sets, S = { s 1 , … , s k } and T = { t 1 , … , t k } in a graph, a paired many-to-many k -disjoint path cover joining S and T is a set of pairwise vertex-disjoint paths { P 1 , … , P k } that altogether cover every vertex of the graph, in which each path P i runs from s i to t i . In this paper, we propose a family of graphs, called torus-like graphs,that include torus networks, and reveal that a torus-like graph, if built from lower dimensional torus-like graphs that have good Hamiltonian and disjoint-path-cover properties, retain such good properties. As a result, every m -dimensional nonbipartite torus, m ≥ 2 , with at most f vertex and/or edge faults has a paired many-to-many k -disjoint path cover joining arbitrary disjoint sets S and T of size k each, subject to k ≥ 2 and f + 2 k ≤ 2 m . The bound 2 m on f + 2 k is nearly optimal.

In this paper we study the extreme and the periodic $L_2$ discrepancy of plane point sets. The extreme discrepancy is based on arbitrary rectangles as test sets whereas the periodic discrepancy uses "periodic intervals", which can be seen as intervals on the torus. The periodic $L_2$ discrepancy is, up to a multiplicative factor, also known as diaphony. The main results are exact formulas for these kinds of discrepancies for the Hammersley point set and for rational lattices. In order to value the obtained results we also prove a general lower bound on the extreme $L_2$ discrepancy for arbitrary point sets in dimension $d$, which is of order of magnitude $(\log N)^{(d-1)/2}$, like the standard and periodic $L_2$ discrepancies, respectively. Our results confirm that the extreme and periodic $L_2$ discrepancies of the Hammersley point set are of best possible asymptotic order of magnitude. This is in contrast to the standard $L_2$ discrepancy of the Hammersley point set. Furthermore our exact formulas show that also the $L_2$ discrepancies of the Fibonacci lattice are of the optimal order. We also prove that the extreme $L_2$ discrepancy is always dominated by the standard $L_2$ discrepancy, a result that was already conjectured by Morokoff and Caflisch when they introduced the notion of extreme $L_2$ discrepancy in the year 1994.

In this paper, we will prove that a closed Möbius minimal and Möbius isotropic hypersurface without umbilic points in the unit sphere $ \mathbb{S}^{5} $ is Möbius equivalent to either the torus $ \mathbb{S}^{2}(\frac{1}{\sqrt{2}})\times\mathbb{S}^{2}(\frac{1}{\sqrt{2}})\rightarrow \mathbb{S}^{5} $ or the Cartan minimal hypersurface in $ \mathbb{S}^{5} $ with four distinct principal curvatures.

We consider a variant of Cops and Robbers wherein each edge traversed by the robber is deleted from the graph. The focus is on determining the minimum number of cops needed to capture a robber on a graph $G$, called the {\em bridge-burning cop number} of $G$ and denoted $c_b(G)$. We determine $c_b(G)$ exactly for several elementary classes of graphs and give a polynomial-time algorithm to compute $c_b(T)$ when $T$ is a tree. We also study two-dimensional square grids and tori, as well as hypercubes, and we give bounds on the capture time of a graph (the minimum number of rounds needed for a single cop to capture a robber on $G$, provided that $c_b(G) = 1$).

В настоящей заметке мы доказываем теорему редукции для подгрупп полной линейной группы ${\operatorname{GL}}(n,T)$ над телом $T$, порожденных парой микровесовых торов одного и того же типа. Оказывается, что любая пара торов вычета $m$ сопряжена такой же паре в ${\operatorname{GL}}(3m,T)$. При этом пары, которые не могут быть вложены далее в ${\operatorname{GL}}(3m-1,T)$, образуют единственную ${\operatorname{GL}}(3m,T)$-орбиту.В случае $m=1$ нам остаётся проанализировать ${\operatorname{GL}}(2,T)$, что было сделано два десятилетия назад вторым автором, Коэном, Кюйперсом и Стерком. Для следующего значения $m=2$ это означает, что единственными случаями, которые должны быть рассмотрены, являются группы${\operatorname{GL}}(4,T)$ и ${\operatorname{GL}}(5,T)$. В этих случаях задача может быть полностью решена (прямыми, но достаточно длинными) матричными вычислениями, которые осуществлены в готовящейся статье авторов.

A set of vertices W of a graph G is a resolving set if every vertex of G is uniquely determined by its vector of distances to W. In this paper, the Maker–Breaker resolving game is introduced. The game is played on a graph G by Resolver and Spoiler who alternately select a vertex of G not yet chosen. Resolver wins if at some point the vertices chosen by him form a resolving set of G, whereas Spoiler wins if the Resolver cannot form a resolving set of G. The outcome of the game is denoted by o(G), and $$R_{\mathrm{MB}}(G)$$ (resp. $$S_{\mathrm{MB}}(G)$$ ) denotes the minimum number of moves of Resolver (resp. Spoiler) to win when Resolver has the first move. The corresponding invariants for the game when Spoiler has the first move are denoted by $$R'_{\mathrm{MB}}(G)$$ and $$S'_\mathrm{MB}(G)$$ . Invariants $$R_{\mathrm{MB}}(G)$$ , $$R'_{\mathrm{MB}}(G)$$ , $$S_\mathrm{MB}(G)$$ , and $$S'_{\mathrm{MB}}(G)$$ are compared among themselves and with the metric dimension $$\mathrm{dim}(G)$$ . A large class of graphs G is constructed for which $$R_{\mathrm{MB}}(G) > \mathrm{dim}(G)$$ holds. The effect of twin equivalence classes and pairing resolving sets on the Maker–Breaker resolving game is described. As an application, o(G), as well as $$R_{\mathrm{MB}}(G)$$ and $$R'_{\mathrm{MB}}(G)$$ (or $$S_{\mathrm{MB}}(G)$$ and $$S'_{\mathrm{MB}}(G)$$ ), is determined for several graph classes, including trees, complete multi-partite graphs, grid graphs, and torus grid graphs.

Any knot $K$ in genus-$1$ $1$-bridge position can be moved by isotopy to lie in a union of $n$ parallel tori tubed by $n-1$ tubes so that $K$ intersects each tube in two spanning arcs, which we call a leveling of the position. The minimal $n$ for which this is possible is an invariant of the position, called the level number. In this work, we describe the leveling by the braid group on two points in the torus, which yields a numerical invariant of the position, called the $(1, 1)$-length. We show that the $(1, 1)$-length equals the level number. We then find braid descriptions for $(1,1)$-positions of all $2$-bridge knots providing upper bounds for their level numbers, and also show that the $(-2, 3, 7)$-pretzel knot has level number two., Comment: 24 pages, 17 figures

Let [Formula: see text] be the special linear group over integers and [Formula: see text] [Formula: see text], or [Formula: see text] products of spheres and tori. We prove that any group action of [Formula: see text] on [Formula: see text] by diffeomorphims or piecewise linear homeomorphisms is trivial if [Formula: see text] This confirms a conjecture on Zimmer’s program for these manifolds.

We study the Kronecker sequence $\{n\alpha\}_{n\leq N}$ on the torus ${\mathbb T}^d$ when $\alpha$ is uniformly distributed on ${\mathbb T}^d.$ We show that the discrepancy of the number of visits of this sequence to a random box, normalized by $\ln^d N$, converges as $N\to\infty$ to a Cauchy distribution. The key ingredient of the proof is a Poisson limit theorem for the Cartan action on the space of $d+1$ dimensional lattices., Comment: 56 pages. This is a revised and expanded version of the prior submissions

We characterize global Denjoy–Carleman hypoellipticity for the family of systems acting on the torus $${\mathbb {T}}_{t}^{N} \times {\mathbb {T}}_{x}$$ given by $$L_{j} = \frac{\partial }{\partial t_{j}} + \left( a_{j}(t_{j}) + ib_{j}(t_{j}) \right) \frac{\partial }{\partial x} + \lambda _j$$ , where $$a_j, b_j$$ are real-valued $$2\pi$$ -periodic elements of the classes and $$j = 1, 2,\ldots , N$$ .

Given two $n \times n $ matrices $A = (a_{ij})$ and $B=(b_{ij}) $ with entries in $B(H)$ for some Hilbert space $H$, their block Schur product is the $n \times n$ matrix $ A\square B := (a_{ij}b_{ij})$. Given two continuous functions $f$ and $g$ on the torus with Fourier coefficients $(f_n)$ and $(g_n)$ their convolution product $f \star g$ has Fourier coefficients $(f_n g_n)$. Based on this, the Schur product on scalar matrices is also known as the Hadamard product. We show that for a C*-algebra $\mathcal{A} $, and a discrete group $G$ with an action $\alpha _g$ of $G$ on $\mathcal{A} $ by *-automorphisms, the reduced crossed product C*-algebra $\mathrm {C}^*_r(\mathcal{A} , \alpha , G)$ possesses a natural generalization of the convolution product, which we suggest should be named the Hadamard product. We show that this product has a natural Stinespring representation and we lift some known results on block Schur products to this setting, but we also show that the block Schur product is a special case of the Hadamard product in a crossed product algebra.

We study the abelian period sets of Sturmian words, which are codings of irrational rotations on a one-dimensional torus. The main result states that the minimum abelian period of a factor of a Sturmian word of angle $\alpha$ with continued fraction expansion $[0; a_1, a_2, \ldots]$ is either $tq_k$ with $1 \leq t \leq a_{k+1}$ (a multiple of a denominator $q_k$ of a convergent of $\alpha$) or $q_{k,\ell}$ (a denominator $q_{k,\ell}$ of a semiconvergent of $\alpha$). This result generalizes a result of Fici et. al stating that the abelian period set of the Fibonacci word is the set of Fibonacci numbers. A characterization of the Fibonacci word in terms of its abelian period set is obtained as a corollary., Comment: 27 pages, 3 figures

This paper studies the compression of a kth-order slant Toeplitz operator on the Hardy space $${H^2}\left({\mathbb{T}{^n}} \right)$$ for integers k ⩾ 2 and n ⩾ 1. It also provides a characterization of the compression of a kth-order slant Toeplitz operator on $${H^2}\left({\mathbb{T}{^n}} \right)$$ . Finally, the paper highlights certain properties, namely isometry, eigenvalues, eigenvectors, spectrum and spectral radius of the compression of kth-order slant Toeplitz operator on the Hardy space $${H^2}\left({\mathbb{T}{^n}} \right)$$ of n-dimensional torus $$\mathbb{T}{^n}$$ .

The local structure of the derived tilings $$ \mathcal{T} $$ of the two-dimensional torus 𝕋2 is studied. The polygonal types of stars in these tilings are classified. It is proved that in the nondegenerate case, the tilings $$ \mathcal{T} $$ contain stars of seven different types, and all the types are represented by stars with interior vertices from the crown Cr of the tiling $$ \mathcal{T} $$ . Also the maximum principle is established, based on which the layer-by-bayer growth algorithm for the derived tilings $$ \mathcal{T} $$ is constructed.

Let r(k) be the smallest upper bound for the radius of k congruent balls in the 3-dimensional cubical flat torus. It will be proved that $$r(1)={1/2}$$ , $$r(2)={\sqrt{3}/4}$$ , $$r(3)={\sqrt{2}/4}$$ and $$r(4)={\sqrt{2}/4}.$$

Let ( x n ) n = 1 ∞ be a sequence on the torus T (normalized to length 1). A sequence ( x n ) is said to have Poissonian pair correlation if, for all s > 0 , lim N → ∞ ⁡ 1 N # { 1 ≤ m ≠ n ≤ N : | x m − x n | ≤ s N } = 2 s . It is known that this implies uniform distribution of the sequence ( x n ) . Hinrichs, Kaltenbock, Larcher, Stockinger & Ullrich extended this result to higher dimensions and showed that sequences ( x n ) in [ 0 , 1 ] d that satisfy, for all s > 0 , lim N → ∞ ⁡ 1 N # { 1 ≤ m ≠ n ≤ N : ‖ x m − x n ‖ ∞ ≤ s N 1 / d } = ( 2 s ) d are also uniformly distributed. We prove the same result for the extension by the Euclidean norm: if a sequence ( x n ) in T d satisfies, for all s > 0 , lim N → ∞ ⁡ 1 N # { 1 ≤ m ≠ n ≤ N : ‖ x m − x n ‖ 2 ≤ s N 1 / d } = ω d s d , where ω d is the volume of the unit ball, then ( x n ) is uniformly distributed. Our approach shows that Poissonian Pair Correlation implies an exponential sum estimate that resembles and implies the Weyl criterion.

We study bond percolation on the hypercube $\{0,1\}^m$ in the slightly subcritical regime where $p = p_c (1-\varepsilon_m)$ and $\varepsilon_m = o(1)$ but $\varepsilon_m \gg 2^{-m/3}$ and study the clusters of largest volume and diameter. We establish that with high probability the largest component has cardinality $\Theta\left(\varepsilon_m^{-2} \log(\varepsilon_m^3 2^m)\right)$, that the maximal diameter of all clusters is $(1+o(1)) \varepsilon_m^{-1} \log(\varepsilon_m^3 2^m)$, and that the maximal mixing time of all clusters is $\Theta\left(\varepsilon_m^{-3} \log^2(\varepsilon_m^3 2^m)\right)$. These results hold in different levels of generality, and in particular, some of the estimates hold for various classes of graphs such as high-dimensional tori, expanders of high degree and girth, products of complete graphs, and infinite lattices in high dimensions., Comment: 38 pages

In the present paper, we investigate the complexity of infinite family of graphs $H_n=H_n(G_1,\,G_2,\ldots,G_m)$ obtained as a circulant foliation over a graph $H$ on $m$ vertices with fibers $G_{1},\,G_{2},\ldots,G_{m}.$ Each fiber $G_{i}=C_{n}(s_{i,1},\,s_{i,2},\ldots,s_{i,k_{i}})$ of this foliation is the circulant graph on $n$ vertices with jumps $s_{i,1},\,s_{i,2},\ldots,s_{i,k_{i}}.$ This family includes the family of generalized Petersen graphs, $I$-graphs, sandwiches of circulant graphs, discrete torus graphs and others. We obtain a closed formula for the number $\tau(n)$ of spanning trees in $H_{n}$ in terms of Chebyshev polynomials, investigate some arithmetical properties of this function and find its asymptotics as $n\to\infty.$, Comment: 14 pages

The Arnoux-Ito theory of geometric substitutions allows to construct sequences of generalized exchanged tilings of the d-dimensional torus. These tilings consist of parallelepipeds of d + 1 type, and the action of a certain toric shift on the tiling reduces to exchanging of the d + 1 central parallelepipeds. Moreover, the set of vertices of all parallelepipeds of the tiling is a fragment of the orbit of zero point under considered toric shift. The considered tilings are actively used in various problems of number theory, combinatorics, and the theory of dynamical systems. In this paper, we study the local structure of toric tilings obtained using geometric substitutions. The n-corona of the parallelepiped is a set of all parallelepipeds located at a distance of not greater than n from a given parallelepiped in the natural metric of the tiling. The problem is to describe all possible types of n-coronas. With each parallelepiped in the tiling we can naturally assigned a number — its number in the orbit of the corresponding central parallelepiped with respect to the toric shift. It is proved that the set of all parallelepipeds numbers splits into a finite number of half-intervals defining possible types of n-coronas. Moreover, it is proved that the boundaries of the corresponding half-open intervals are determined by the numbers of the parallelepipeds in the n-corona of the set of d + 1 central parallelepiped. It is shown that this result can be considered as some multi-dimensional generalization of the famous three lengths theorem. Earlier, a similar description was obtained for 1-coronas of the toric tilings obtained using one specific geometric substitution: the Rauzy substitution. In addition, similar results were previously obtained for some quasiperiodic plane tilings. In conclusion, some directions for further research are formulated.

International audience; We analyze the stability of a differential equation with two delays originating from a model for a population divided into two subpopulations, immature and mature, and we apply this analysis to a model for platelet production. The dynamics of mature individuals is described by the following nonlinear differential equation with two delays: x'(t) = −γx(t) + g(x(t − τ1)) − g(x(t − τ1 − τ2))e(−γ τ2). The method of D-decomposition is used to compute the stability regions for a given equilibrium. The centre manifold theory is used to investigate the steady-state bifurcation and the Hopf bifurcation. Similarly, analysis of the centre manifold associated with a double bifurcation is used to identify a set of parameters such that the solution is a torus in the pseudo-phase space. Finally, the results of the local stability analysis are used to study the impact of an increase of the death rate γ or of a decrease of the survival time τ2 of platelets on the onset of oscillations. We show that the stability is lost through a small decrease of survival time (from 8.4 to 7 days), or through an important increase of the death rate (from 0.05 to 0.625 1/day).; Nous analysons la stabilité d’une équation différentielle à deux retards issue de la dynamique de populations structurées en deux catégories, immatures et matures, puis nous appliquons cette analyse à un modèle pour la production des plaquettes. La dynamique du nombre d’individus matures est alors décrite par une équation à deux retards non-linéaire de la forme x'(t) = −γx(t) + g(x(t − τ1)) − g(x(t − τ1 − τ2))e(−γ τ2) . La méthode de la D-décomposition est utilisée pour procéder numériquement à une exploration exhaustive des configurations que peut prendre la région de stabilité dans les plans (A,B) et (τ,B). L’analyse de la variété centre est utilisée pour étudier la bifurcation "steady-state" et les bifurcations de Hopf. De manière similaire, l'analyse de la variété centre associée aux doubles bifurcations permet d'identifier un jeu de paramètres auquel correspond une solution prenant la forme d’un tore dans l’espace de pseudo-phase. Enfin, les résultats de l’analyse de stabilité locale est utilisé pour étudier l’impact d’une augmentation du taux de mort des plaquettes γ ainsi que celui d’une diminution du temps de survie des plaquettes τ2 sur l’apparition d’oscillations. Nous montrons alors que la stabilité peut être perdue à la suite d’une diminution légère du temps de survie (de 8.4 jours à 7 jours), ou par une forte augmentation du taux de mort (de 0.05 à 0.6/jour).

We show that in 5-connected planar and projective planar triangulations on n vertices, the number of Hamiltonian cycles grows exponentially with n. The result is best possible in the sense that 4-connected triangulations on n vertices on any fixed surface may have only polynomially many cycles. Also, there is an infinite class of 5-connected graphs (not on a fixed surface) which have only polynomially many cycles. The result also extends to 5-connected triangulations of the torus if a long standing conjecture of Nash-Williams holds. For any fixed surface, we show that every 5-connected triangulation of large face-width on that surface contains exponentially many cycles.