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The article A “regular” pentagonal tiling of the plane by P. L. Bowers and K. Stephenson, Conform.Geom. Dyn. 1, 58–86, 1997, defines a conformal pentagonal tiling. This is a tiling of the planewith remarkable combinatorial and geometric properties. However, it doesn’t have finite localcomplexity in any usual sense, and therefore we cannot study it with the usual tiling theory. Theappeal of the tiling is that all the tiles are conformally regular pentagons. But conformal mapsare not allowable under finite local complexity. On the other hand, the tiling can be describedcompletely by its combinatorial data, which rather automatically has finite local complexity. Inthis paper we give a construction of the discrete hull just from the combinatorial data. The mainresult of this paper is that the discrete hull is a Cantor space.

In this paper, we prove the q -analogue of the fundamental theorem of Riordan arrays. In particular, by defining two new binary operations ∗q and ∗1/q , we obtain a q -analogue of the Riordan representation of the q -Pascal matrix. In addition, by aid of the q -Lagrange expansion formula we get q -Riordan representation for its inverse matrix. In this paper, we prove the q -analogue of the fundamental theorem of Riordan arrays. In particular, by defining two new binary operations ∗q and ∗1/q , we obtain a q -analogue of the Riordan representation of the q -Pascal matrix. In addition, by aid of the q -Lagrange expansion formula we get q -Riordan representation for its inverse matrix.

In this paper a new transformation of occupancy distributions, called merging, is introduced. In particular, it will be studied the effect of merging on a class of occupancy distributions that was recently introduced in Collet et al. (Probab Eng Inf Sci. 27:533–552 2013). These results have an interesting interpretation in the so-called entropy maximization inference. The last part of the paper is devoted to highlight the impact of our findings in this research area.

Let $ G $ be a connected standard simple algebraic group of type $ C $ or $ D $ over an algebraically closed field $ \Bbbk $ of positive characteristic $ p > 0 $, and $ \mathfrak{g}: = \mathrm{Lie}(G) $ be the Lie algebra of $ G $. Motivated by the variety of $ \mathbb{E}(r, \mathfrak{g}) $ of $ r $-dimensional elementary subalgebras of a restricted Lie algebra $ \mathfrak{g} $, in this paper we characterize the irreducible components of $ \mathbb{E}(\mathrm{rk}_{p}(\mathfrak{g})-1, \mathfrak{g}) $ where the $ p $-rank $ \mathrm{rk}_{p}(\mathfrak{g}) $ is defined to be the maximal dimension of an elementary subalgebra of $ \mathfrak{g} $.

The $ r $-Lah numbers generalize the Lah numbers to the $ r $-Stirling numbers in the same sense. The Stirling numbers and the central factorial numbers are one of the important tools in enumerative combinatorics. The $ r $-Lah number counts the number of partitions of a set with $ n+r $ elements into $ k+r $ ordered blocks such that $ r $ distinguished elements have to be in distinct ordered blocks. In this paper, the $ r $-central Lah numbers and the $ r $-central Lah-Bell numbers ($ r\in \mathbb{N} $) are introduced parallel to the $ r $-extended central factorial numbers of the second kind and $ r $-extended central Bell polynomials. In addition, some identities related to these numbers including the generating functions, explicit formulas, binomial convolutions are derived. Moreover, the $ r $-central Lah numbers and the $ r $-central Lah-Bell numbers are shown to be represented by Riemann integral, respectively.

In this paper, we introduce the concept of $ j $-hyperfilters, for all positive integers $ 1\leq j \leq n $ and $ n \geq 2 $, on (ordered) $ n $-ary semihypergroups and establish the relationships between $ j $-hyperfilters and completely prime $ j $-hyperideals of (ordered) $ n $-ary semihypergroups. Moreover, we investigate the properties of the relation $ \mathcal{N} $, which is generated by the same principal hyperfilters, on (ordered) $ n $-ary semihypergroups. As we have known from [21] that the relation $ \mathcal{N} $ is the least semilattice congruence on semihypergroups, we illustrate by counterexample that the similar result is not necessarily true on $ n $-ary semihypergroups where $ n\geq 3 $. However, we provide a sufficient condition that makes the previous conclusion true on $ n $-ary semihypergroups and ordered $ n $-ary semihypergroups where $ n\geq 3 $. Finally, we study the decomposition of prime hyperideals and completely prime hyperideals by means of their $ \mathcal{N} $-classes. As an application of the results, a related problem posed by Tang and Davvaz in [31] is solved.

The well-studied eccentric connectivity index directly consider the contribution of all edges in a graph. By considering the total eccentricity sum of all non-adjacent vertex, Hua et al. proposed a new topological index, namely, eccentric connectivity coindex of a connected graph. The eccentric connectivity coindex of a connected graph $ G $ is defined as \begin{document}$ \overline{\xi}^{c}(G) = \sum\limits_{uv\notin E(G)} (\varepsilon_{G}(u)+\varepsilon_{G}(v)). $\end{document} Where $ \varepsilon_{G}(u) $ (resp. $ \varepsilon_{G}(v) $) is the eccentricity of the vertex $ u $ (resp. $ v $). In this paper, some extremal problems on the $ \overline{\xi}^{c} $ of graphs with given parameters are considered. We present the sharp lower bounds on $ \overline{\xi}^{c} $ for general connecteds graphs. We determine the smallest eccentric connectivity coindex of cacti of given order and cycles. Also, we characterize the graph with minimum and maximum eccentric connectivity coindex among all the trees with given order and diameter. Additionally, we determine the smallest eccentric connectivity coindex of unicyclic graphs with given order and diameter and the corresponding extremal graph is characterized as well.

A nondegenerate hypersurface in a pseudo-Euclidean space $ \mathbb{E}^{n+1}_{s} $ is called to have proper mean curvature vector if its mean curvature $ \vec{H} $ satisfies $ \Delta \vec{H} = \lambda \vec{H} $ for a constant $ \lambda $. In 2013, Arvanitoyeorgos and Kaimakamis conjectured [1]: any hypersurface satisfying $ \Delta \vec{H} = \lambda \vec{H} $ in pseudo-Euclidean space $ \mathbb E_{s}^{n+1} $ has constant mean curvature. This paper will give further support evidences to this conjecture by proving that a linear Weingarten hypersurface $ M^{n}_{r} $ in $ \mathbb E^{n+1}_{s} $ satisfying $ \Delta \vec{H} = \lambda \vec{H} $ has constant mean curvature if $ M^{n}_{r} $ has diagonalizable shape operator with less than seven distinct principal curvatures.

In this paper, we study the quotient and “blow-up” of graph-of-groups [Formula: see text] and of their automorphisms [Formula: see text]. We show that the existence of such a blow-up of any [Formula: see text], relative to a given family of “local” graph-of-groups isomorphisms [Formula: see text] depends crucially on the [Formula: see text]-conjugacy class of the correction term [Formula: see text] for any edge [Formula: see text] of [Formula: see text], where [Formula: see text]-conjugacy is a new but natural concept introduced here. As an application, we obtain a criterion as to whether a partial Dehn twist can be blown up relative to local Dehn twists, to give an actual Dehn twist. The results of this paper are also used crucially in the follow-up papers [Lustig and Ye, Normal form and parabolic dynamics for quadratically growing automorphisms of free groups, arXiv:1705.04110v2; Ye, Partial Dehn twists of free groups relative to local Dehn twists — A dichotomy, arXiv:1605.04479 ; When is a polynomially growing automorphism of [Formula: see text] geometric, arXiv:1605.07390 ].

In this paper we show by counter example that one of the main results in the paper "Edge Geodetic Covers in Graphsby Mariano and Canoy (International Mathematical Forum, 4, 2009, no. 46, 2301 - 2310) does not hold. Further, we partially characterize connected graphs G of order n for which its edge geodetic number g e(G) = n - 1.

In this paper, we study some aspects of the irreducibility of $\widetilde{M_g^{Pl}(G)}$ and its interrelation with the existence of "normal forms", i.e. non-singular plane equations (depending on a set of parameters) such that a specialization of the parameters gives a certain non-singular plane model associated to the elements of $\widetilde{M_g^{Pl}(G)}$. In particular, we introduce the concept of being equation strongly irreducible (ES-Irreducible) for which the locus $\widetilde{M_g^{Pl}(G)}$ is represented by a single "normal form". Henn, and Komiya-Kuribayashi, observed that $\widetilde{M_3^{Pl}(G)}$ is ES-Irreducible. In this paper we prove that this phenomena does not occur for any odd $d>4$. More precisely, let $\mathbb{Z}/m\mathbb{Z}$ be the cyclic group of order $m$, we prove that $\widetilde{M_g^{Pl}(\mathbb{Z}/(d-1)\mathbb{Z})}$ is not ES-Irreducible for any odd integer $d\geq5$, and the number of its irreducible components is at least two. Furthermore, we conclude the previous result when $d=6$ for the locus $\widetilde{M_{10}^{Pl}(\mathbb{Z}/3\mathbb{Z})}$. Lastly, we prove the analogy of these statements when $K$ is any algebraically closed field of positive characteristic $p$ such that $p>(d-1)(d-2)+1$., This paper is a recent version of chapter 1 of arXiv:1503.01149

As we proved earlier, for a triangulated category $\underline{C}$ endowed with a weight structure $w$ and a triangulated subcategory $\underline{D}$ of $\underline{C}$ (strongly) generated by cones of a set of morphisms $S$ in the heart $\underline{Hw}$ of $w$ there exists a weight structure $w'$ on the Verdier quotient $\underline{C}'=\underline{C}/\underline{D}$ such that the localization functor $\underline{C} \to \underline{C}'$ is weight-exact (i.e., "respects weights"). The goal of this paper is to find conditions ensuring that for any object of $\underline{C}'$ of non-negative (resp. non-positive) weights there exists its preimage in $\underline{C}$ satisfying the same condition; we call a certain stronger version of the latter assumption the left (resp., right) weight lifting property. We prove that these weight lifting properties are fulfilled whenever the set $S$ satisfies the corresponding (left or right) Ore conditions. Moreover, if $\underline{D}$ is generated by objects of $\underline{Hw}$ then any object of $\underline{Hw}'$ lifts to $\underline{Hw}$. We apply these results to obtain some new results on Tate motives and finite spectra (in the stable homotopy category). Our results are also applied to the study of the so-called Chow-weight homology in another paper., v3: 25 pages, A full revision has been made, an author added

Evolution in finite populations is often modelled using the classical Moran process. Over the last 10 years, this methodology has been extended to structured populations using evolutionary graph theory. An important question in any such population is whether a rare mutant has a higher or lower chance of fixating (the fixation probability) than the Moran probability, i.e. that from the original Moran model, which represents an unstructured population. As evolutionary graph theory has developed, different ways of considering the interactions between individuals through a graph and an associated matrix of weights have been considered, as have a number of important dynamics. In this paper, we revisit the original paper on evolutionary graph theory in light of these extensions to consider these developments in an integrated way. In particular, we find general criteria for when an evolutionary graph with general weights satisfies the Moran probability for the set of six common evolutionary dynamics.

For a set of primes $\mathcal{P}$, let $\Psi(x, \mathcal{P})$ be the number of positive integers $n \leq x$ all of whose prime factors lie in $\mathcal{P}$. In this paper we classify the sets of primes $\mathcal{P}$ such that $\Psi(x, \mathcal{P})$ is within a constant factor of its expected value. This task was recently initiated by Granville, Koukoulopoulos and Matom\"aki and their main conjecture is proved in this paper. In particular our main theorem implies that, if not too many large primes are sieved out in the sense that \[ \sum_{\substack{p \in \mathcal{P} \\ x^{1/v} < p \leq x^{1/u}}} \frac{1}{p} \geq \frac{1 + \varepsilon}{u}, \] for some $\varepsilon > 0$ and $v \geq u \geq 1$, then \[ \Psi(x, \mathcal{P}) \gg_{\varepsilon, v} x \prod_{\substack{p \leq x\\ p \notin\mathcal{P}}} \left(1 - \frac{1}{p}\right). \], Comment: 23 pages

In this paper, we consider the boundary value problem $$y^{\prime\prime}_{j} + \lambda^{2}y_{j} = \sum^{n}_{k = 1} V_{jk} (x) y_{k}, \quad x \in{\mathbb R}_{+} : = ( 0, \infty),$$$$y^{\prime}_{j} (0) + (\alpha_{0} + \alpha_{1} \lambda+ \alpha_{2} \lambda^{2}) y_{j} (0) = 0, \quad j = 1, 2, \ldots, n,$$ where λ is the spectral parameter and $V (x) = \Vert V_{jk} (x)\Vert^{n}_{1}$ is a Hermitian matrix such that $$\int^{\infty}_{x} t\vert V (t)\vert dt < \infty, \quad x \in {\mathbb R}_{+}$$ and αi ∈ ℂ, i = 0, 1, 2, with α2 ≠ 0. In this paper, we investigate the eigenvalues and spectral singularities of L. In particular, we prove that L has a finite number of eigenvalues and spectral singularities with finite multiplicities, under the Naimark and Pavlov conditions.

This work is motivated by the papers [EG85] and [Ngu15] in which the following two problems are solved. Let $\mathcal{O}$ is a finitely generated $\mathbb{Z}$-algebra that is an integrally closed domain of characteristic zero, consider the following problems: (A) Fix $s$ that is integral over $\mathcal{O}$, describe all $t$ such that $\mathcal{O}[s]=\mathcal{O}[t]$. (B) Fix $s$ and $t$ that are integral over $\mathcal{O}$, describe all pairs $(m,n)\in\mathbb{N}^2$ such that $\mathcal{O}[s^m]=\mathcal{O}[t^n]$. In this paper, we solve these problems and provide a uniform bound for a certain "discriminant form equation" that is closely related to Problem (A) when $\mathcal{O}$ has characteristic $p>0$. While our general strategy roughly follows [EG85] and [Ngu15], many new delicate issues arise due to the presence of the Frobenius automorphisms $x\mapsto x^p$. Recent advances in unit equations over fields of positive characteristic together with classical results in characteristic zero play an important role in this paper., 27 pages, comments are welcome

In this paper, we consider the sum-product problem of obtaining lower bounds for the size of the set $$\frac{A+A}{A+A}:=\left \{ \frac{a+b}{c+d} : a,b,c,d \in A, c+d \neq 0 \right\},$$ for an arbitrary finite set $A$ of real numbers. The main result is the bound $$\left| \frac{A+A}{A+A} \right| \gg \frac{|A|^{2+\frac{2}{25}}}{|A:A|^{\frac{1}{25}}\log |A|},$$ where $A:A$ denotes the ratio set of $A$. This improves on a result of Balog and the author (arXiv:1402.5775), provided that the size of the ratio set is subquadratic in $|A|$. That is, we establish that the inequality $$\left| \frac{A+A}{A+A} \right| \ll |A|^{2} \Rightarrow |A:A| \gg \frac{ |A|^2}{\log^{25}|A|} . $$ This extremal result answers a question similar to some conjectures in a recent paper of the author and Zhelezov (arXiv:1410.1156)., In this version, Lemma 3.2 has been improved, and the new version of Lemma 3.2 is tight up to multiplicative constants. This results in a small improvement to the main result of the paper. To appear in JLMS. With thanks to Noga Alon, for providing the proof of the new and improved Lemma 3.2 via a private communication

We study biased Maker-Breaker positional games between two players, one of whom is playing randomly against an opponent with an optimal strategy. In this paper we consider the scenario when Maker plays randomly and Breaker is "clever", and determine the sharp threshold bias of classical graph games, such as connectivity, Hamiltonicity, and minimum degree-$k$. We treat the other case, that is when Breaker plays randomly, in a separate paper. The traditional, deterministic version of these games, with two optimal players playing, are known to obey the so-called probabilistic intuition. That is, the threshold bias of these games is asymptotically equal to the threshold bias of their random counterpart, where players just take edges uniformly at random. We find, that despite this remarkably precise agreement of the results of the deterministic and the random games, playing randomly against an optimal opponent is not a good idea: the threshold bias becomes significantly more tilted towards the random player. An important qualitative aspect of the probabilistic intuition carries through nevertheless: the bottleneck for Maker to occupy a connected graph is still the ability to avoid isolated vertices in her graph., This paper has been split in two parts, the second part can be found at http://arxiv.org/abs/1602.04628 . A full version from before the split is submission v2 of this arXiv entry, available at http://arxiv.org/abs/1507.06688v2

Let $\mathbf{G}$ be one of the ind-groups $GL(\infty)$, $O(\infty)$, $Sp(\infty)$ and $\mathbf{P}\subset \mathbf{G}$ be a splitting parabolic ind-subgroup. The ind-variety $\mathbf{G}/\mathbf{P}$ has been identified with an ind-variety of generalized flags in the paper "Ind-varieties of generalized flags as homogeneous spaces for classical ind-groups" (Int. Math. Res. Not. 2004, no. 55, 2935--2953) by I. Dimitrov and I. Penkov. In the present paper we define a Schubert cell on $\mathbf{G}/\mathbf{P}$ as a $\mathbf{B}$-orbit on $\mathbf{G}/\mathbf{P}$, where $\mathbf{B}$ is any Borel ind-subgroup of $\mathbf{G}$ which intersects $\mathbf{P}$ in a maximal ind-torus. A significant difference with the finite-dimensional case is that in general $\mathbf{B}$ is not conjugate to an ind-subgroup of $\mathbf{P}$, whence $\mathbf{G}/\mathbf{P}$ admits many non-conjugate Schubert decompositions. We study the basic properties of the Schubert cells, proving in particular that they are usual finite-dimensional cells or are isomorphic to affine ind-spaces. We then define Schubert ind-varieties as closures of Schubert cells and study the smoothness of Schubert ind-varieties. Our approach to Schubert ind-varieties differs from an earlier approach by H. Salmasian in "Direct limits of Schubert varieties and global sections of line bundles" (J. Algebra 320 (2008), 3187--3198)., Keywords: Classical ind-group, Bruhat decomposition, Schubert decomposition, generalized flag, homogeneous ind-variety. [26 pages]

We consider changes in properties of a subgraph of an infinite graph resulting from the addition of open edges of Bernoulli percolation on the infinite graph to the subgraph. We give the triplet of an infinite graph, one of its subgraphs, and a property of the subgraphs. Then, in a manner similar to the way Hammersley's critical probability is defined, we can define two values associated with the triplet. We regard the two values as certain critical probabilities, and compare them with Hammersley's critical probability. In this paper, we focus on the following cases of a graph property: being a transient subgraph, having finitely many cut points or no cut points, being a recurrent subset, or being connected. Our results depend heavily on the choice of the triplet. Most results of this paper are announced in \cite{O16} without proofs. This paper gives full details of them., 23 pages, 3 figures, reorganized, final version, to appear in Indagationes Mathematicae

This paper is concerned with the performance of Orthogonal Matching Pursuit (OMP) algorithms applied to a dictionary $\mathcal{D}$ in a Hilbert space $\mathcal{H}$. Given an element $f\in \mathcal{H}$, OMP generates a sequence of approximations $f_n$, $n=1,2,\dots$, each of which is a linear combination of $n$ dictionary elements chosen by a greedy criterion. It is studied whether the approximations $f_n$ are in some sense comparable to {\em best $n$ term approximation} from the dictionary. One important result related to this question is a theorem of Zhang \cite{TZ} in the context of sparse recovery of finite dimensional signals. This theorem shows that OMP exactly recovers $n$-sparse signal, whenever the dictionary $\mathcal{D}$ satisfies a Restricted Isometry Property (RIP) of order $An$ for some constant $A$, and that the procedure is also stable in $\ell^2$ under measurement noise. The main contribution of the present paper is to give a structurally simpler proof of Zhang's theorem, formulated in the general context of $n$ term approximation from a dictionary in arbitrary Hilbert spaces $\mathcal{H}$. Namely, it is shown that OMP generates near best $n$ term approximations under a similar RIP condition., 12 pages

The iterative functional equation $ \alpha\rightarrow(\alpha\rightarrow \beta) = \alpha\rightarrow \beta $ and the law of importation $ (\alpha\wedge \beta)\rightarrow \gamma = \alpha\rightarrow (\beta\rightarrow \gamma) $ are two tautologies in classical logic. In fuzzy logics, they are two important properties, and are respectively formulated as $ I(\alpha, \beta) = I(\alpha, I(\alpha, \beta)) $ and $ I(T(\alpha, \beta), \gamma) = I(\alpha, I(\beta, \gamma)) $ where $ I $ is a fuzzy implication and $ T $ is a $ t $-norm. Over the past several years, solutions to these two functional equations involving different classes of fuzzy implications have been studied. However, there are no results about stability study of fuzzy functional equations involving fuzzy implication. This paper discusses fuzzy implications that do not strictly satisfying these equations, but approximately satisfy these equations. Then we establish the Hyers-Ulam stability of the iterative functional equation involving the $ (S, N) $-implication, where the $ (S, N) $-implication is a common class of fuzzy implications generated by a continuous $ t $-conorm $ S $ and a continuous fuzzy negation $ N $. Furthermore, given a fixed $ t $-norm (the minimum $ t $-norm or the product $ t $-norm) the Hyers-Ulam stability of the law of importation involving the $ (S, N) $-implication is studied.

The graph of irreducible parabolic subgroups is a combinatorial object associated to an Artin–Tits group A defined so as to coincide with the curve graph of the (n+1)‐times punctured disk when A is Artin's braid group on (n+1) strands. In this case, it is a hyperbolic graph, by the celebrated Masur–Minsky's theorem. Hyperbolicity of the graph of irreducible parabolic subgroups for more general Artin–Tits groups is an important open question. In this paper, we give a partial affirmative answer. For n⩾3, we show that the graph of irreducible parabolic subgroups associated to the Artin–Tits group of spherical type Bn is also isomorphic to the curve graph of the (n+1)‐times punctured disk; hence, it is hyperbolic. For n⩾2, we show that the graphs of irreducible parabolic subgroups associated to the Artin–Tits groups of euclidean type A∼n and C∼n are isomorphic to some subgraphs of the curve graph of the (n+2)‐times punctured disk which are not quasi‐isometrically embedded. We prove nonetheless that these graphs are hyperbolic.

The topic of this paper is the hat problem in which each of \(n\) players is uniformly and independently fitted with a blue or red hat. Then everybody can try to guess simultaneously his own hat color by looking at the hat colors of the other players. The team wins if at least one player guesses his hat color correctly, and no one guesses his hat color wrong; otherwise the team loses. The aim is to maximize the probability of winning. In this version every player can see everybody excluding himself. We consider such a problem on a graph, where vertices correspond to players, and a player can see each player to whom he is connected by an edge. The solution of the hat problem on a graph is known for trees and for cycles on four or at least nine vertices. In this paper first we give an upper bound on the maximum chance of success for graphs with neighborhood-dominated vertices. Next we solve the problem on unicyclic graphs containing a cycle on at least nine vertices. We prove that the maximum chance of success is one by two. Then we consider the hat problem on a graph with a universal vertex. We prove that there always exists an optimal strategy such that in every case some vertex guesses its color. Moreover, we prove that there exists a graph with a universal vertex for which there exists an optimal strategy such that in some case no vertex guesses its color. We also give some Nordhaus-Gaddum type inequalities.

We say that a set in a Euclidean space does not contain an angle α if the angle determined by any three points of the set is not equal to α. The goal of this paper is to construct compact sets of large Hausdorff dimension that do not contain a given angle α ∈ (0,π). We will construct such sets in ℝ n of Hausdorff dimension c(α)n with a positive c(α) depending only on α provided that α is different from π/3, π/2 and 2π/3. This improves on an earlier construction (due to several authors) that has dimension c(α) log n. The main result of the paper concerns the case of the angles π/3 and 2π/3. We present self-similar sets in ℝ n of Hausdorff dimension $c{{\sqrt[3]{n}} \mathord{\left/ {\vphantom {{\sqrt[3]{n}} {\log n}}} \right. \kern-\nulldelimiterspace} {\log n}}$ with the property that they do not contain the angles π/3 and 2π/3. The constructed sets avoid not only the given angle α but also a small neighbourhood of α.

In this paper, we continue the study of domination and total domination in cubic graphs. It is known [Henning M.A., Southey J., A note on graphs with disjoint dominating and total dominating sets, Ars Combin., 2008, 89, 159–162] that every cubic graph has a dominating set and a total dominating set which are disjoint. In this paper we show that every connected cubic graph on nvertices has a total dominating set whose complement contains a dominating set such that the cardinality of the total dominating set is at most (n+2)/2, and this bound is essentially best possible.

This paper deals with the estimation of the specific connectivity of a stationary random set in IRd. It turns out that the "natural" estimator is only asymptotically unbiased. The example of a boolean model of hypercubes illustrates the amplitude of the bias produced when the measurement field is relatively small with respect to the range of the random set. For that reason unbiased estimators are desired. Such an estimator can be found in the literature in the case where the measurement field is a right parallelotope. In this paper, this estimator is extended to apply to measurement fields of various shapes, and to possess a smaller variance. Finally an example from quantitative metallography (specific connectivity of a population of sintered bronze particles) is given.

This paper is mainly concerned with the limit distribution of (cos 2πn1x + ··· +c os 2πnN x)/ √ N on the unit interval when the increasing se- quence {nk} has bounded gaps, i.e., 1 nk+1 − nk = O(1). By Bobkov-Gotze (4), it was proved that the limiting variance must be less than 1/2 in this case. They proved that the centered Gaussian distribution with variance 1/4 together with mixtures of Gaussian distributions belonging to a huge class can be limit distri- butions. In this paper it is proved that any Gaussian distribution with variance less than 1/2 can be a limit distribution.

Let k, n ∈ N with k < n and let g k (R n ) denote the Grassmann manifold consisting of all k-dimensional linear subspaces in R n . In an earlier paper the authors showed that if the projections of a nonconvex closed set C C R n are convex and proper for projection directions from some nonempty open set P ⊂ g k (R n ), then C contains a closed copy of an (n―k―1)-manifold. In this paper we improve on that result by showing that that result remains valid under the weaker assumption that P is somewhere dense in g k (R n ).

All ten symmetry classes of plane partitions that fit in a given box are known to be enumerated by simple product formulas, but there is still no unified proof for all of them. Progress towards this goal can be made by establishing identities connecting the various symmetry classes. We present in this paper four such identities, involving all ten symmetry classes. We discuss their proofs and generalizations. The main result of this paper is to give a generalization of one of them, in the style of the identity presented in "A factorization theorem for rhombus tilings," M. Ciucu and C. Krattenthaler, arXiv:1403.3323., 14 pages

In this paper, we present several new constructions of differentially 4-uniform permutations over $\F_{2^{2m}}$ by modifying the values of the inverse function on some subsets of $\F_{2^{2m}}$. The resulted differentially 4-uniform permutations have high nonlinearities and algebraic degrees, which provide more choices for the design of cryptographic substitution boxes., 15 pages. This paper has been accepted for publication in SCIENCE CHINA Mathematics

The representation theory of the symmetric group has been intensively studied for over 100 years and is one of the gems of modern mathematics. The full transformation monoid $\mathfrak T_n$ (the monoid of all self-maps of an $n$-element set) is the monoid analogue of the symmetric group. The investigation of its representation theory was begun by Hewitt and Zuckerman in 1957. Its character table was computed by Putcha in 1996 and its representation type was determined in a series of papers by Ponizovski{\u��}, Putcha and Ringel between 1987 and 2000. From their work, one can deduce that the global dimension of $\mathbb C\mathfrak T_n$ is $n-1$ for $n=1,2,3,4$. We prove in this paper that the global dimension is $n-1$ for all $n\geq 1$ and, moreover, we provide an explicit minimal projective resolution of the trivial module of length $n-1$. In an appendix with V.~Mazorchuk we compute the indecomposable tilting modules of $\mathbb C\mathfrak T_n$ with respect to Putcha's quasi-hereditary structure and the Ringel dual (up to Morita equivalence)., Explicit minimal projective resolutions of the exterior powers of the standard module are computed allowing a simpler proof of the main result. Made minor corrections and added an appendix with Volodymyr Mazorchuk computing the characteristic tilting module and the Ringel dual

[EN] The aim of this paper is to prove the following result: let π be a set of odd primes. If the finite group G = AB is a product of two π-decomposable subgroups A = Oπ(A)×Oπ (A) and B = Oπ(B)×Oπ (B), then Oπ(A)Oπ(B)=Oπ(B)Oπ(A) and this is a Hall π-subgroup of G., The second and third authors were supported by Proyecto MTM2010-19938-C03-02, Ministerio de Economia y Competitividad, Spain. The first author would like to thank the Universitat de Valencia and the Universitat Politecnica de Valencia for their warm hospitality during the preparation of this paper and the RFBR Project 13-01-00469 for its support. The authors are also grateful to A. S. Kondratiev for helpful comments and suggestions during his visit to Valencia.

Figueroa-Centeno et al. [4] introduced the following product of digraphs let D be a digraph and let Γ be a family of digraphs such that V (F) = V for every F∈Γ . Consider any function h:E(D)→Γ . Then the product D⊗hΓ is the digraph with vertex set V(D)×V and ((a,x),(b,y))∈E(D⊗hΓ) if and only if (a,b)∈E(D) and (x,y)∈E(h(a,b)) . In this paper, we deal with the undirected version of the ⊗h -product, which is a generalization of the classical direct product of graphs and, motivated by the ⊗h -product, we also recover a generalization of the classical lexicographic product of graphs, namely the ∘h -product, that was introduced by Sabidussi in 1961. We provide two characterizations for the connectivity of G⊗hΓ that generalize the existing one for the direct product. For G∘hΓ , we provide exact formulas for the connectivity and the edge-connectivity, under the assumption that V (F) = V , for all F∈Γ . We also introduce some miscellaneous results about other invariants in terms of the factors of both, the ⊗h -product and the ∘h -product. Some of them are easily obtained from the corresponding product of two graphs, but many others generalize the existing ones for the direct and the lexicographic product, respectively. We end up the paper by presenting some structural properties. An interesting result in this direction is a characterization for the existence of a nontrivial decomposition of a given graph G in terms of ⊗h -product. The research conducted in this document by the first author has been supported by the Spanish Research Council under project MTM2011-28800-C02-01 and by the Catalan Research Council under grant 2009SGR1387.

For a finite set of polynomials F={fj}, B. Reznick [in Algorithmic and quantitative real algebraic geometry (Piscataway, NJ, 2001), 101–121, DIMACS Ser. Discrete Math. Theoret. Comput. Sci., 60, Amer. Math. Soc., Providence, RI, 2003; MR1995017] introduced the ticket T(F)={d∈N:{fdj} is linearly dependent}. Motivated by results in the above paper, the author of the current paper studies the "degree'' of linear dependence of the set of powers of polynomials within its ticket, i.e., dimspan{fdj} for d∈T(F). This seems to be a rather difficult question for an arbitrary set of polynomials. The author therefore focuses on the special case when the polynomials are linear forms and proves various interesting results. The interested reader should consult the paper for the details.

In a graph G, the distance between two vertices is the length of the shortest path between them. The maximum distance between a vertex to any other vertex is considered as the eccentricity of the vertex. In this paper, we introduce the first general Zagreb eccentricity index and found upper and lower bounds on this index in terms of order, size and diameter. Moreover, we characterize the extremal graphs in the class of trees, trees with pendant vertices and bipartite graphs. Results on some famous topological indices can be presented as the corollaries of our main results.

Topological graph indices have been of great interest in the research of several properties of chemical substances as it is possible to obtain these properties only by using mathematical calculations. The irregularity indices are the ones to determine the degree of irregularity of a graph. Albertson and Bell indices are two of them. Edge and vertex deletion and addition are important and useful methods in calculating several properties of a given graph. In this paper, the effects of adding a new edge or a new vertex to a graph on the Albertson and Bell indices are determined.

This paper delves into the study of critical sets of Latin squares having a given isotopism in their autotopism group. Particularly, we prove that the sizes of these critical sets only depend on both the main class of the Latin square and the cycle structure of the isotopism under consideration. Keeping then in mind that the autotopism group of a Latin square acts faithfully on the set of entries of the latter, we enumerate all the critical sets based on autotopisms of Latin squares of order up to five. Junta de Andalucía FQM-016

In the paper, the authors find and apply necessary and sufficient conditions for a bivariate mean of two positive numbers with three parameters to be Schur convex or Schur harmonically convex respectively.

This paper is mainly concerned with the existence of multiple solutions for the following boundary value problems of fractional differential equations with generalized Caputo derivatives: \begin{document}$ \hskip 3mm \left\{ \begin{array}{lll} ^{C}_{0}D^{\alpha}_{g}x(t)+f(t, x) = 0, \ 0 where $ 2 < \alpha < 3 $, $ 1 < \nu < 2 $, $ \alpha-\nu-1 > 0 $, $ f\in C([0, 1]\times \mathbb{R}^{+}, \mathbb{R}^{+}) $, $ g' > 0 $, $ h\in C([0, 1], \mathbb{R}^{+}) $, $ \mathbb{R}^{+} = [0, +\infty) $. Applying the fixed point theorem on cone, the existence of multiple solutions for considered system is obtained. The results generalize and improve existing conclusions. Meanwhile, the Ulam stability for considered system is also considered. Finally, three examples are worked out to illustrate the main results.

Algebras of distributions of binary isolating and semi-isolating formulas are derived objects for given theory and reflect binary formula relations between realizations of 1-types. These algebras are associated with the following natural classification questions: 1) for a given class of theories, determine which algebras correspond to the theories from this class and classify these algebras; 2) to classify theories from a given class depending on the algebras defined by these theories of isolating and semi-isolating formulas. Here the description of a finite algebra of binary isolating formulas unambiguously entails a description of the algebra of binary semi-isolating formulas, which makes it possible to track the behavior of all binary formula relations of a given theory. The paper describes algebras of binary formulae for root products. The Cayley tables are given for the obtained algebras. Based on these tables, theorems describing all algebras of binary formulae distributions for the root multiplication theory of regular polygons on an edge are formulated. It is shown that they are completely described by two algebras.

In this paper, by employing the AGW criterion and determining the number of solutions to some equations over finite fields, we further investigate nine classes of permutation polynomials over $ \mathbb{F}_{p^n} $ with the form $ (x^{p^m}-x+\delta)^{s_1}+(x^{p^m}-x+\delta)^{s_2}+x $ and propose five classes of complete permutation polynomials over $ \mathbb{F}_{p^{2m}} $ with the form $ ax^{p^m}+bx+h(x^{p^m}-x) $.

Let O be a maximal order in the quaternion algebra B_p over Q ramified at p and infinity. The paper is about the computational problem: Construct a supersingular elliptic curve E over F_p such that End(E) = O. We present an algorithm that solves this problem by taking gcds of the reductions modulo p of Hilbert class polynomials. New theoretical results are required to determine the complexity of our algorithm. Our main result is that, under certain conditions on a rank three sublattice O^T of O, the order O is effectively characterized by the three successive minima and two other short vectors of O^T. The desired conditions turn out to hold whenever the j-invariant j(E), of the elliptic curve with End(E) = O, lies in F_p. We can then prove that our algorithm terminates with running time O(p^{1+\epsilon}) under the aforementioned conditions. As a further application we present an algorithm to simultaneously match all maximal order types with their associated j-invariants. Our algorithm has running time O(p^{2.5+\epsilon}) operations and is more efficient than Cervino's algorithm for the same problem., Comment: Full version of paper published by the LMS Journal of Computation and Mathematics

We present an $O^*(n^3)$ randomized algorithm for estimating the volume of a well-rounded convex body given by a membership oracle, improving on the previous best complexity of $O^*(n^4)$. The new algorithmic ingredient is an accelerated cooling schedule where the rate of cooling increases with the temperature. Previously, the known approach for potentially achieving this asymptotic complexity relied on a positive resolution of the KLS hyperplane conjecture, a central open problem in convex geometry. We also obtain an $O^*(n^3)$ randomized algorithm for integrating a standard Gaussian distribution over an arbitrary convex set containing the unit ball. Both the volume and Gaussian volume algorithms use an improved algorithm for sampling a Gaussian distribution restricted to a convex body. In this latter setting, as we show, the KLS conjecture holds and for a spherical Gaussian distribution with variance $\sigma^2$, the sampling complexity is $O^*(\max\{n^3, \sigma^2n^2\})$ for the first sample and $O^*(\max\{n^2, \sigma^2n^2\})$ for every subsequent sample., Comment: This paper is a combination of two previously published conference papers: "A Cubic Algorithm for Computing Gaussian Volume" (SODA 2014, arXiv:1306.5829) and "Bypassing KLS: Gaussian Cooling and an $O^*(n^3)$ Volume Algorithm" (STOC 2015). Additionally, this version has a major simplification to the main proof in the latter conference paper. (Lemma 3.2 in this version) 36 pages

In a series of four papers we prove the following relaxation of the Loebl-Komlos-Sos Conjecture: For every $\alpha>0$ there exists a number $k_0$ such that for every $k>k_0$ every $n$-vertex graph $G$ with at least $(\frac12+\alpha)n$ vertices of degree at least $(1+\alpha)k$ contains each tree $T$ of order $k$ as a subgraph. The method to prove our result follows a strategy similar to approaches that employ the Szemer\'edi regularity lemma: we decompose the graph $G$, find a suitable combinatorial structure inside the decomposition, and then embed the tree $T$ into $G$ using this structure. Since for sparse graphs $G$, the decomposition given by the regularity lemma is not helpful, we use a more general decomposition technique. We show that each graph can be decomposed into vertices of huge degree, regular pairs (in the sense of the regularity lemma), and two other objects each exhibiting certain expansion properties. In this paper, we introduce this novel decomposition technique. In the three follow-up papers, we find a combinatorial structure suitable inside the decomposition, which we then use for embedding the tree., Comment: 41 pages, 6 figures; further referees' comments incorporated, the most substantial of which being a newly written Section 3.8

In this paper we investigate the existence and multiplicity of classical homoclinic solutions for the following second order Hamiltonian systems $$ (\mbox{HS}) \ddot u-L(t)u+\nabla W(t,u)=0, $$ where $L\in C(\mathbb R,\mathbb R^{n^2})$ is a symmetric and positive definite matrix for all $t\in \mathbb R$, $W\in C^1(\mathbb R\times\mathbb R^n,\mathbb R)$ and $\nabla W(t,u)$ is the gradient of $W$ at $u$. The novelty of this paper is that, assuming that $L$ is bounded in the sense that there are constants $0\lt \tau_1\lt \tau_2$ such that $\tau_1 |u|^2\leq (L(t)u,u)\leq \tau_2 |u|^2$ for all $(t,u)\in \mathbb R\times \mathbb R^n$ and $W(t,u)$ is of subquadratic growth at infinity, we are able to establish two new criteria to guarantee the existence and multiplicity of classical homoclinic solutions for (HS), respectively. Recent results in the literature are extended and significantly improved.

DAHA-Jones polynomials of torus knots T(r, s) are studied systematically for reduced root systems and in the case of $$C^\vee C_1$$ . We prove the polynomiality and evaluation conjectures from the author’s previous paper on torus knots and extend the theory by the color exchange and further symmetries. The DAHA-Jones polynomials for $$C^\vee C_1$$ depend on five parameters. Their surprising connection to the DAHA-superpolynomials (type A) for the knots $$T(2p+1,2)$$ is obtained, a remarkable combination of the color exchange conditions and the author’s duality conjecture (justified by Gorsky and Negut). The uncolored DAHA-superpolynomials of torus knots are expected to coincide with the Khovanov–Rozansky stable polynomials and the superpolynomials defined via rational DAHA and/or in terms of certain Hilbert schemes. We end the paper with certain arithmetic counterparts of DAHA-Jones polynomials for the absolute Galois group in the case of $$C^\vee C_1$$ , developing the author’s previous results for $$A_1$$ .

We investigate weighted Sobolev spaces on metric measure spaces $(X,d,m)$. Denoting by $\rho$ the weight function, we compare the space $W^{1,p}(X,d,\rho m)$ (which always concides with the closure $H^{1,p}(X,d,\rho m)$ of Lipschitz functions) with the weighted Sobolev spaces $W^{1,p}_\rho(X,d,m)$ and $H^{1,p}_\rho(X,d,m)$ defined as in the Euclidean theory of weighted Sobolev spaces. Under mild assumptions on the metric measure structure and on the weight we show that $W^{1,p}(X,d,\rho m)=H^{1,p}_\rho(X,d, m)$. We also adapt results by Muckenhoupt and recent work by Zhikov to the metric measure setting, considering appropriate conditions on $\rho$ that ensure the equality $W^{1,p}_\rho(X,d,m)=H^{1,p}_\rho(X,d,m)$., Comment: 26 pages. Removed Proposition 5.2 which was incorrect in previous versions of the paper; this does not affect any other part of the paper

Let $ G $ be a simple graph with edge set $ E(G) $. The modified Sombor index is defined as $ ^{m}SO(G) = \sum\limits_{uv\in E(G)}\frac{1}{\sqrt{d_{u}^{2}~~+~~d_{v}^{2}}} $, where $ d_{u} $ (resp. $ d_{v} $) denotes the degree of vertex $ u $ (resp. $ v $). In this paper, we determine some bounds for the modified Sombor indices of graphs with given some parameters (e.g., maximum degree $ \Delta $, minimum degree $ \delta $, diameter $ d $, girth $ g $) and the Nordhaus-Gaddum-type results. We also obtain the relationship between modified Sombor index and some other indices. At last, we obtain some bounds for the modified spectral radius and energy.

In this paper we prove a Dwork-type supercongruence: for any prime $ p\geq3 $ and integer $ r\geq 1 $, \begin{document}$ \begin{align*} \sum\limits_{k = 0}^{p^r-1}\frac{3k+1}{16^k}{\binom{2k}{k}}^3\equiv p\sum\limits_{k = 0}^{p^{r-1}-1}\frac{3k+1}{16^k}{\binom{2k}{k}}^3\pmod{p^{3r+1-\delta_{p, 3}}}, \end{align*} $\end{document} which extends a result of Guo and Zudilin.