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The main object in this paper is a certain rational convex polytope whose lattice points give a polyhedral realization of a highest weight crystal basis. This is also identical to a Newton-Okounkov body of a flag variety, and it gives a toric degeneration. In this paper, we prove that a specific class of this polytope is given by Kiritchenko's Demazure operators on polytopes. This implies that polytopes in this class are all lattice polytopes. As an application, we give a sufficient condition for the corresponding toric variety to be Gorenstein Fano., Comment: 21 pages

In a number of papers by Edge, and in a related paper by Fano, several properties are discussed of the family of scrolls of degree 8, in the complex projective space, whose plane sections are projected bicanonical models of a genus 3 curve C. This beautiful subject is implicitely related to the moduli of semistable rank two vector bundles on C with bicanonical determinant. We revisit and reconstruct this matter in modern terms with a modular point of view. This paper is the refereed version of a contribution to a volume in honor of W.L Edge, to be published by European Journal of Mathematics., Comment: 13 pages

In \cite[Problem 72]{Fuchs60} Fuchs posed the problem of characterizing the groups which are the groups of units of commutative rings. In the following years, some partial answers have been given to this question in particular cases. In a previous paper \cite{DDcharp} we dealt with finite characteristic rings. In this paper we consider Fuchs' question for finite groups and we address this problem in two cases. Firstly, we study the case of torson-free rings and we obtain a complete classification of the finite groups of units which arise in this case. Secondly, we examine the case of characteristic zero rings obtaining, a pretty good description of the possible groups of units equipped with families of examples of both realizable and non-realizable groups. The main tools to deal with this general case are the Pearson and Schneider splitting of a ring \cite{PearsonSchneider70}, our previous results on finite characteristic rings \cite{DDcharp} and our classification of the groups of units of torsion-free rings. As a consequence of our results we completely answer Ditor's question \cite{ditor} on the possible cardinalities of the group of units of a ring., arXiv admin note: substantial text overlap with arXiv:1607.00965

The main inspiration for this paper is a paper by Elek where he introduces combinatorial cost for graph sequences. We show that having cost equal to 1 and hyperfiniteness are coarse invariants. We also show `cost-1' for box spaces behaves multiplicatively when taking subgroups. We show that graph sequences coming from Farber sequences of a group have property A if and only if the group is amenable. The same is true for hyperfiniteness. This generalises a theorem by Elek. Furthermore we optimise this result when Farber sequences are replaced by sofic approximations. In doing so we introduce a new concept: property almost-A., 20 pages. Comments welcome

Let R be an integral domain and I a nonzero ideal of R. A sub-ideal J of I is a t-reduction of I if (JI^{n})_{t}=(I^{n+1})_{t} for some positive integer n. An element x in R is t-integral over I if there is an equation x^{n} + a_{1}x^{n-1} +...+ a_{n-1}x + a_{n} = 0 with a_{i} in (I^{i})_{t} for I = 1,...,n. The set of all elements that are t-integral over I is called the t-integral closure of I. This paper investigates the t-reductions and t-integral closure of ideals. Our objective is to establish satisfactory t-analogues of well-known results, in the literature, on the integral closure of ideals and its correlation with reductions. Namely, Section 2 identifies basic properties of t-reductions of ideals and features explicit examples discriminating between the notions of reduction and t-reduction. Section 3 investigates the concept of t-integral closure of ideals, including its correlation with t-reductions. Section 4 studies the persistence and contraction of t-integral closure of ideals under ring homomorphisms. All along the paper, the main results are illustrated with original examples., 13 pages. Rocky Mountain Journal of Mathematics 2016

This paper focuses on integer linear programs where solutions are binary matrices, and the corresponding symmetry group is the set of all column permutations. Orbitopal fixing, as introduced by Kaibel et al., is a technique designed to break symmetries in the special case of partitioning (resp. packing) formulations involving matrices with exactly (resp. at most) one 1-entry in each row. The main result of this paper is to extend orbitopal fixing to the full orbitope, defined as the convex hull of binary matrices with lexicographically nonincreasing columns. We determine all the variables whose values are fixed in the intersection of an hypercube face with the full orbitope. Sub-symmetries arising in a given subset of matrices are also considered, thus leading to define the full sub-orbitope in the case of the sub-symmetric group. We propose a linear time orbitopal fixing algorithm handling both symmetries and sub-symmetries. We introduce a dynamic variant of this algorithm where the lexicographical order follows the branching decisions occurring along the B\&B search. Experimental results for the Unit Commitment Problem are presented. A comparison with state-of-the-art techniques is considered to show the effectiveness of the proposed variants of the algorithm.

In this paper, we study the partial bi-free $S$-transform of a pair $(a,b)$ of random variables, and the $S$-transform of the $2\times 2$ matrix-valued random variable $\left(\begin{matrix}a&0\\0&b\end{matrix}\right)$ associated with $(a,b)$ when restricted to upper triangular $2\times 2$ matrices. We first derive an explicit expression of bi-free multiplicative convolution (of probability measures on the bi-unit-sphere $\mathbb{T}^2$ of $\mathbb{C}^2$, or on $\mathbb{R}^2_+$ in $\mathbb{C}^2$) from a subordination equation for bi-free multiplicative convolution. We then show that, when $(a_1, b_1)$ and $(a_2,b_2)$ are bi-free, the $S$-transforms of $X_1=\left(\begin{matrix}a_1&0\\0&b_1\end{matrix}\right)$, $X_2=\left(\begin{matrix}a_2&0\\0&b_2\end{matrix}\right)$ satisfy Dykema's twisted multiplicative equation for free operator-valued random variables if and only if at least one of the two partial bi-free $S$-transforms of the pairs of random variables is the constant function 1 in a neighborhood of $(0,0)$. This is the case if and only if one of the two pairs, say $(a_1,b_1)$, has factoring two-band moments (that is, $��(a_1^mb_1^n)=��(a_1^m)��(b_1^n)$, for all $m,n=1, 2, \cdots$). We thus find tons of bi-free pairs of random variables to which the $S$-transforms of the corresponding matrix-value random variables do not satisfy Dykema's twisted multiplicative formula. Finally, if both $(a_1,b_1)$ and $(a_2,b_2)$ have factoring two-band moments, we prove that the $��$-transforms of $X_1$, $X_2$, and $X_1X_2$ satisfy a subordination equation., This is the final version of the paper, which will be published in Studia Mathematica

Let $$I_n(G)$$ denote the number of elements of order n in a finite group G. In 1979, Herzog (Proc Am Math Soc 77:313–314, 1979) conjectured that two finite simple groups containing the same number of involutions have the same order. In a 2018 paper (Arch Math 111:349–351, 2018), Zarrin disproved Herzog’s conjecture with a counterexample. Then he conjectured that “if S is a non-abelian simple group and G a group such that $$I_2(G)=I_2(S)$$ and $$I_p(G) =I_p(S)$$ for some odd prime divisor p, then $$|G|=|S|$$ ”. In this paper, we give more counterexamples to Herzog’s conjecture. Moreover, we disprove Zarrin’s conjecture.

In this expository paper, we consider the Hardy-Schr\"odinger operator $-\Delta -\gamma/|x|^2$ on a smooth domain \Omega of R^n with 0\in\bar{\Omega}, and describe how the location of the singularity 0, be it in the interior of \Omega or on its boundary, affects its analytical properties. We compare the two settings by considering the optimal Hardy, Sobolev, and the Caffarelli-Kohn-Nirenberg inequalities. The latter rewrites: $C(\int_{\Omega}\frac{u^{p}}{|x|^s}dx)^{\frac{2}{p}}\leq \int_{\Omega} |\nabla u|^2dx-\gamma \int_{\Omega}\frac{u^2}{|x|^2}dx$ for all $u\in H^1_0(\Omega)$, where \gamma, Comment: Expository paper. 48 pages

In this paper, we introduce and investigate an interesting subclass M§(∞;‚;-;') of analytic and bi-univalent functions of complex order in the open unit disk U: For func- tions belonging to the class M§(∞;‚;-;') we investigate the coe-cient estimates on the flrst two Taylor-Maclaurin coe-cients ja2j and ja3j: The results presented in this paper would generalize and improve some recent works of (1),(5),(9).

This paper studies the problem of finding optimal parameters for a Poincare section used for determining the type of behavior of a time series: a deterministic or stochastic one. To reach that goal optimization algorithms are coupled with the Poincare & Higuchi (P&H) method, which calculates the Higuchi dimension using points obtained by performing a Poincare section of a certain attractor. The P&H method generates distinctive patterns that can be used for determining if a given attractor is produced by a deterministic or a stochastic system, but this method is sensitive to the parameters of the Poincare section. Patterns generated by the P&H method can be characterized using numerical measures which in turn can be used for finding such parameters for the Poincare section for which the patterns produced by the P&H method are the most prominent. This paper studies several approaches to parameterization of the Poincare section. Proposed approaches are tested on twelve time series, six produced by deterministic chaotic systems and six generated randomly. The obtained results show, that finding good parameters of the Poincare section is important for determining the type of behavior of a time series. Among the tested methods the evolutionary algorithm was able to find the best Poincare sections for use with the P&H method.

In this paper, we consider the interval oscillation criteria for second-order damped differential equations with mixed nonlinearities $$\begin{aligned} \left( r(t)(x'(t))^\gamma \right) '+p(t)(x'(t))^\gamma +\sum ^n_{i=0}q_i(t)\left| x(g_i(t))\right| ^{\alpha _i}\text {sgn}\ x(g_i(t))=e(t), \end{aligned}$$ where $$\gamma $$ is a quotient of odd positive integers, $$\alpha _0=\gamma , \alpha _i>0, i=1,\ 2,\ldots ,n$$ with $$r,\ p,\ e$$ , and $$q_i\in C([t_0,\infty ),\mathbb {R}), r(t)>0, g_i:\ \mathbb {R}\rightarrow \mathbb {R}$$ are nondecreasing continuous functions on $$\mathbb {R}$$ and $$\lim _{t\rightarrow \infty }g_i(t)=\infty , i=0,\ 1,\ 2,\ldots ,n.$$ Our results in this paper extend and improve some known results. Some examples are given here to illustrate our main results.

The nonnegative rank of a matrix A is the smallest integer r such that A can be written as the sum of r rank-one nonnegative matrices. The nonnegative rank has received a lot of attention recently due to its application in optimization, probability and communication complexity. In this paper we study a class of atomic rank functions defined on a convex cone which generalize several notions of "positive" ranks such as nonnegative rank or cp-rank (for completely positive matrices). The main contribution of the paper is a new method to obtain lower bounds for such ranks which improve on previously known bounds. Additionally the bounds we propose can be computed by semidefinite programming. The idea of the lower bound relies on an atomic norm approach where the atoms are self-scaled according to the vector (or matrix, in the case of nonnegative rank) of interest. This results in a lower bound that is invariant under scaling and that is at least as good as other existing norm-based bounds. We mainly focus our attention on the two important cases of nonnegative rank and cp-rank where our bounds satisfying interesting properties: For the nonnegative rank we show that our lower bound can be interpreted as a non-combinatorial version of the fractional rectangle cover number, while the sum-of-squares relaxation is closely related to the Lov\'asz theta number of the rectangle graph of the matrix. We also prove that the lower bound inherits many of the structural properties satisfied by the nonnegative rank such as invariance under diagonal scaling, subadditivity, etc. We also apply our method to obtain lower bounds on the cp-rank for completely positive matrices. In this case we prove that our lower bound is always greater than or equal the plain rank lower bound, and we show that it has interesting connections with combinatorial lower bounds based on edge-clique cover number.

In this paper, we consider signal recovery via $l_{1}$-analysis optimisation. The signals we consider are not sparse in an orthonormal basis or incoherent dictionary, but sparse or nearly sparse in terms of some tight frame $D$. The analysis in this paper is based on the restricted isometry property adapted to a tight frame $D$ (abbreviated as $D$-RIP), which is a natural extension of the standard restricted isometry property. Assuming that the measurement matrix $A\in \mathbb{R}^{m\times n}$ satisfies $D$-RIP with constant ${\it\delta}_{tk}$ for integer $k$ and $t>1$, we show that the condition ${\it\delta}_{tk} guarantees stable recovery of signals through $l_{1}$-analysis. This condition is sharp in the sense explained in the paper. The results improve those of Li and Lin [‘Compressed sensing with coherent tight frames via $l_{q}$-minimization for $0’, Preprint, 2011, arXiv:1105.3299] and Baker [‘A note on sparsification by frames’, Preprint, 2013, arXiv:1308.5249].

UDC 512.5 In this paper, the normalizer N (σ) of the elementary net group E(σ) associated with a nonsplit maximal torus T (d) in the general linear group GL(n, k) over a field k of odd characteristic is computed. The nonsplit maximal torus T = T (d) is determined by the radical extension k( n √ d) of degree n of the ground field k (minisotropic torus). Bibliography: 18 titles. The problem of describing the overgroups of a split maximal torus in linear groups and in Chevalley groups is virtually solved. The fundamental contribution to the solution of this problem was made by the Leningrad–Petersburg algebraic school (Z. I. Borevich, N. A. Vavilov, and their pupils; for example, see [1, 2, 4–6]). The description of overgroups of a nonsplit torus is a significantly less investigated area. At the present time, a complete description of overgroups of a nonsplit torus has been obtained only for some special fields such as finite or local fields. For finite fields, this was carried out in papers by W. Kantor and G. Seitz (see [15, 17, 18]). Important results on overgroups of a nonsplit torus for local and global fields have been obtained by V. P. Platonov [16]. The case of the field of real numbers has been considered in [14]. For arbitrary fields, the study of overgroups of a nonsplit torus has been conducted in papers [3, 7–13]. The present paper is devoted to an investigation of intermediate subgroups of the general linear group that contain the nonsplit maximal torus related to a radical extension K = k( n √ d) of the ground field k, d ∈ k. The elementary net groups E(σ) associated with the torus T = T (d) and their normalizer N (σ) (which is an overgroup of the nonsplit torus) in the general linear group G =G L( n, k) are of primary importance in studies of the intermediate subgroups mentioned above (see [7–11]). In the present paper, we calculate the normalizer N (σ) of the elementary net group E(σ) associated with a nonsplit maximal torus T (d) in the general linear group GL(n, k )o ver afi eld k of odd characteristic. Moreover, the nonsplit maximal torus T = T (d) is determined by the radical degree-n extension k( n √ d) of the ground field k (minisotropic torus).

Let G be an edge-colored graph. A rainbow (heterochromatic, or multicolored) path of G is such a path in which no two edges have the same color. Let the color degree of a vertex v to be the number of different colors that are used on edges incident to v, and denote it by $$d^c(v)$$ . In a previous paper, we showed that if $$d^c(v)\ge k$$ (color degree condition) for every vertex v of G, then G has a rainbow path of length at least $$\lceil (k+1)/2\rceil $$ . Later, in another paper we first showed that if $$k\le 7$$ , G has a rainbow path of length at least $$k-1$$ , and then, based on this we used induction on k and showed that if $$k\ge 8$$ , then G has a rainbow path of length at least $$\lceil (3k)/5\rceil +1$$ . In 2010, Gyarfas and Mhalla showed that in any proper edge-colored complete graph $$K_n$$ , there is a rainbow path with no less than $$(2n+1)/3$$ vertices. In the present paper, by using a simpler approach we further improve the result by showing that if $$k\ge 8$$ , G has a rainbow path of length at least $$\lceil (2k)/3\rceil +1$$ .

It is well known that an integral domain D is a UFD if andonly if every nonzero prime ideal of D contains a nonzero principal prime.This is the so-called Kaplansky’s theorem. In this paper, we give this typeof characterizations of a graded PvMD (resp., G-GCD domain, GCDdomain, B´ezout domain, valuation domain, Krull domain, π-domain). 0. IntroductionThis is a continuation of our works on Kaplansky-type theorems [13, 20].It is well known that an integral domain D is a UFD if and only if everynonzero prime ideal of D contains a nonzero principal prime [19, Theorem 5].This is the so-called Kaplansky’s theorem. A generalization of this type oftheorems was first studied by Anderson and Zafrullah in [5], where they gaveseveral characterizations of this type for GCD domains, valuation domains, andPru¨fer domains. Also, characterizations for PvMDs and Krull domains weregiven in [15] and [6], respectively.Later, in [20], the second-named author gave a Kaplansky-type charac-terization of G-GCD domains and PvMDs and gave an ideal-wise version ofKaplansky-type theorems. This ideal-wise version is then used to give char-acterizations of UFDs, π-domains, and Krull domains. Let D be an integraldomain with quotient field K, X be an indeterminate over D, and D[X] bethe polynomial ring over D. A nonzero prime ideal Q of D[X] is called anupper to zero in D[X] if Q ∩ D = (0). Clearly, Q is an upper to zero in D[X]if and only if Q = fK[X] ∩ D[X] for some nonzero polynomial f ∈ D[X].For f ∈ D[X], let c(f) be the ideal of D generated by the coefficients of f.In [13], the first two authors of this paper studied an integral domain D suchthat every upper to zero in D[X] contains a prime (resp., primary) element

In [6], it is proved that the lengths of periods occurring inthe β-expansion of a rational series r noted by Per β (r) depend onlyon the denominator of the reduced form of r for quadratic Pisot unitseries. In this paper, we will show first that every rational r in theunit disk has strictly periodic β-expansion for Pisot or Salem unit basisunder some condition. Second, for this basis, if r = PQ is written inreduced form with |P| < |Q|, we will generalize the curious property“Per β ( PQ ) = Per β ( 1Q )”. 1. IntroductionThe notion of β-expansions of real numbers was introduced by A. R´enyi[11]. Since then, their arithmetic, diophantine and ergodic properties havebeen extensively studied by several researchers. In this paper, we consider ananalogue of this concept in algebraic function fields over finite fields. There arestriking analogies between these digit systems and the classical β-expansionsof real numbers. In order to pursue this analogy, we recall the definition of realβ-expansions and survey the problems corresponding to our results.Let β be a fixed real number greater than 1 and let x be a positive realnumber. A convergent seriesP

The problem of minimizing a polynomial over the standard simplex is one of the basic NP-hard nonlinear optimization problems --- it contains the maximum clique problem in graphs as a special case. It is known that the problem allows a polynomial-time approximation scheme (PTAS) for polynomials of fixed degree, which is based on polynomial evaluations at the points of a sequence of regular grids. In this paper, we provide an alternative proof of the PTAS property. The proof relies on the properties of Bernstein approximation on the simplex. We also refine a known error bound for the scheme for polynomials of degree three. The main contribution of the paper is to provide new insight into the PTAS by establishing precise links with Bernstein approximation and the multinomial distribution., 21 pages

In this paper we investigate the properties of (a, d)-vertex antimagic total labeling of a digraph D = (V, A). In this labeling, we assign to the vertices and arcs the consecutive integers from 1 to |V|+|A| and calculate the sum of labels at each vertex, i.e., the vertex label added to the labels on its out arcs. These sums form an arithmetical progression with initial term a and common difference d. We show the existence and non-existence of (a, d)-vertex antimagic total labeling for several class of digraphs, and show how to construct labelings for generalized de Bruijn digraphs. We conclude this paper with an open problem suitable for further research.

Canonical Projective Embeddings of the Deligne-Lusztig Curves Associated to 2 A 2 , 2 B 2 and 2 G 2 Daniel M. Kane May 15, 2015 Abstract The Deligne-Lusztig varieties associated to the Coxeter classes of the algebraic groups 2 A 2 , 2 B 2 and 2 G 2 are affine algebraic curves. We produce explicit projective models of the closures of these curves. Furthermore for d the Coxeter number of these groups, we find polynomials for each of these models that cut out the F q -points, the F q d -points and the F q d+1 - points, and demonstrate a relation satisfied by these polynomials. Introduction There are four kinds of finite groups of Lie type of rank 1. The associated Deligne-Lusztig varieties for the Coxeter classes of these groups all give affine algebraic curves. The completions of these curves have several applications in- cluding the representation theory of the associated group ([1, 5]), coding theory ([4]) and the construction of potentially interesting covers of P 1 ([3]). In this paper, we consider these curves associated to the groups G = 2 A 2 , 2 B 2 and 2 G 2 . The remaining curve is associated to G = A 1 and is P 1 , but we do not cover this case as it is easy and doesn’t follow many of the patterns found in the analysis of the other three cases. For each of these curves, we explicitly construct an embedding C ,→ P(W ) where W is a representation of G of dimension 3,5, or 14 respectively, and provide an explicit system of equations cutting out C. The curve associated to 2 A 2 is the Fermat curve. The curve associated to B 2 is also well-known though not immediately isomorphic to our embedding. Embeddings of the curve associated to 2 G 2 were not known until recently. In [6], they constructed an explicit curve with the correct genus, symmetry group and number of points. Later, in [2], Eid and Duursma use this description to independently arrive at the embedding we produce in this paper. ∗ University of California, San Diego, Department of Mathematics / Department of Computer Science and Engineering, 9500 Gilman Drive #0404, La Jolla, CA 92093 dakane@ucsd.edu

The normalized eigenvalues Ʌi(M, g) of the Laplace–Beltrami operator can be considered as functionals on the space of all Riemannian metrics g on a fixed surface M. In recent papers several explicit examples of extremal metrics were provided. These metrics are induced by minimal immersions of surfaces in 𝕊3 or 𝕊4. In this paper a family of extremal metrics induced by minimal immersions in 𝕊5 is investigated.

A subgroup H of a finite group G is said to be ss-supplemented in G if there exists a subgroup K of G such that G = HK and H ∩ K is s-permutable in K. In this paper, we first give an example to show that the conjecture in A.A. Heliel’s paper (2014) has negative solutions. Next, we prove that a finite group G is solvable if every subgroup of odd prime order of G is ss-supplemented in G, and that G is solvable if and only if every Sylow subgroup of odd order of G is ss-supplemented in G. These results improve and extend recent and classical results in the literature.

In this paper, we introduce the notion of $$\chi $$ -pure exact sequence. An exact sequence $$0\longrightarrow A\longrightarrow B\longrightarrow C\longrightarrow 0$$ is said to be $$\chi $$ -pure-exact if $$0\longrightarrow A\bigotimes M\longrightarrow B\bigotimes M\longrightarrow C\bigotimes M\longrightarrow 0$$ is again an exact sequence, where A, B, C are right R-modules and $$M\simeq R/I$$ is a left R-module for $$I\in \chi $$ , where $$\chi $$ denotes the collection of left ideals of R. In this paper, we establish several equivalent conditions for a pure exact sequence to be $$\chi $$ -pure exact. We further define a $$\chi $$ -pure injective module and study the topological aspects of this module and introduce a condition under which a $$\chi $$ -pure injective module coincides with an algebraically compact module.

It is well-known that the area of parabolic region between aparabola and any chord P 1 P 2 on the parabola is four thirds of the area oftriangle ∆P 1 P 2 P. Here we denote by P the point on the parabola wherethe tangent is parallel to the chord P 1 P 2 . In the previous works, thefirst and third authors of the present paper proved that this property isa characteristic one of parabolas. In this paper, with respect to triangles∆P 1 P 2 Q where Q is the intersection point of two tangents to X at P 1 and P 2 we establish some characterization theorems for parabolas. 1. IntroductionFor a smooth function f : I → Rdefined on an open interval, usually we saythat f is strictly convex if the graph of f has positive curvature κ with respectto the upward unit normal N. This condition is equivalent to f ′′ (x) > 0 on theinterval I.In this article, we study strictly locally convex plane curves. A regular planecurve X : I → R 2 defined on an open interval I, is called convex if, for all s ∈ Ithe trace X(I) of X lies entirely on one side of the closed half-plane determinedby the tangent line to X at s ([4]). A regular plane curve X : I → R

Let R be a commutative ring with identity and M be an R-module. Let Z(R) be the set of zero-divisors of R and Z(M) be the set of annihilators over M in R. The total graph of idealization, \(T\left( \Gamma (R(+)M)\right) \) was defined as the graph with all elements of the ring \(R(+)M =\lbrace (r,m)\mid r \in R, m \in M\rbrace \) as vertices where any two distinct vertices \((x,m), (y,n) \in R(+)M\) are adjacent if and only if \((x,m) + (y,n) \in Z(R(+)M)\), the set of zero-divisors of \(R(+)M\). In this paper we define the line graph of total graph of idealization, denoted by \(L\left( T(\Gamma (R(+)M))\right) \) as the graph with all the edges of \(T\left( \Gamma (R(+)M)\right) \) as vertices and any two distinct vertices are adjacent if and only if their corresponding edges share a common vertex in the graph \(T\left( \Gamma (R(+)M)\right) \). In this paper we discuss the diameter, girth and clique number of the graph \(L\left( T(\Gamma (R(+)M))\right) \). We also find condition for the graph \(L\left( T(\Gamma (R(+)M))\right) \) to be connected when \(Z(R(+)M)\) is not an ideal of the \(R(+)M\).

Tomova (8) gave an upper bound for the distance of a bridge surface for a knot with two different bridge positions in a 3-manifold. In this paper, we show that the result of Tomova (8, Theorem 10.3) can be improved in the case when there are two different bridge spheres for a link in S 3 . Hempel (4) introduced the concept of distance of a Heegaard surface, and it is shown by many authors that it well represents various complexities of 3- manifolds. For example, Hartshorn (2) showed that the Euler characteristic of an incompressible surface in a 3-manifold bounds the distance of its Heegaard splittings, and Scharlemann and Tomova (7) showed that the Euler character- istic of any Heegaard splitting of a 3-manifold similarly bounds the distance of any non-isotopic Heegaard splitting. The above concept and results have been extended to bridge surfaces for knots and links in closed 3-manifolds, and have been studied by several authors. For example, Bachman and Schleimer (1) proved that Hartshorn's results can be extended to the distance of a bridge surface for a knot in a closed orientable 3-manifold, and also Tomova (8) proved that Scharlemann and Tomova's results can be extended to the distance of a bridge surface for a knot in a closed ori- entable 3-manifold. Moreover, Johnson and Tomova (6) proved that Tomova's result can be extended to a bridge surface for a tangle in a compact 3-manifold. Recently Jang (5) showed that for a link in a closed orientable 3-manifold, the result of Bachman and Schleimer (1) can be improved in the case when there exist essential meridional spheres. In this paper, we find a property of essential simple closed curves disjoint from the disk complex of a 3-ball containing trivial arcs (for detail, see Lemma 2.1). This allows us to improve the result of Tomova (8, Theorem 10.3) in the case when there are two different bridge spheres for a link in S 3 .

The aim of this paper is to define the notion of the Cohen- Macaulay cone of a Noetherian local domain R R and to present its applications to the theory of Hilbert-Kunz functions. It has been shown by the second author that with a mild condition on R R , the Grothendieck group G 0 ( R ) ¯ \overline {G_0(R)} of finitely generated R R -modules modulo numerical equivalence is a finitely generated torsion-free abelian group. The Cohen-Macaulay cone of R R is the cone in G 0 ( R ) ¯ R \overline {G_0(R)}_{\mathbb R} spanned by cycles represented by maximal Cohen-Macaulay modules. We study basic properties on the Cohen-Macaulay cone in this paper. As an application, various examples of Hilbert-Kunz functions in the polynomial type will be produced. Precisely, for any given integers ϵ i = 0 , ± 1 \epsilon _i = 0, \pm 1 ( d / 2 > i > d d/2 > i > d ), we shall construct a d d -dimensional Cohen-Macaulay local ring R R (of characteristic p p ) and a maximal primary ideal I I of R R such that the function ℓ R ( R / I [ p n ] ) \ell _R(R/I^{[p^n]}) is a polynomial in p n p^n of degree d d whose coefficient of ( p n ) i (p^n)^i is the product of ϵ i \epsilon _i and a positive rational number for d / 2 > i > d d/2 > i > d . The existence of such ring is proved by using Segre products to construct a Cohen-Macaulay ring such that the Chow group of the ring is of certain simplicity and that test modules exist for it.

Let $$r\ge 2$$ , $$\beta \ge 0$$ and $$2\le a\le b-\beta $$ be integers, and let $$G$$ be a graph of order $$n$$ with $$n\ge \frac{(a+b)(a+b+1+(r-2)(b-\beta ))}{a+\beta }$$ and $$\delta (G)\ge \frac{(r-1)(b-\beta )(b+1)}{a+\beta }$$ , and let $$g$$ and $$f$$ be two integer-valued functions defined on $$V(G)$$ satisfying $$a\le g(x)\le f(x)-\beta \le b-\beta $$ for each $$x\in V(G)$$ . In this paper, it is proved that $$G$$ is a fractional $$(g,f)$$ -deleted graph if $$G$$ satisfies $$\max \{d_G(x_1),d_G(x_2),\ldots ,d_G(x_r)\}\ge \frac{(b-\beta )n}{a+b}$$ for any independent subset $$\{x_1,x_2,\ldots ,x_r\}$$ in $$G$$ . Furthermore, it is shown that the result in this paper is best possible in some sense.

The paper considers upper bounds for the infinity norm of the inverse for matrices in two subclasses of the class of (nonsingular) H-matrices, both of which contain the class of Nekrasov matrices. The first one has been introduced recently and consists of the so-called S-Nekrasov matrices. For S-Nekrasov matrices, the known bounds are improved. The second subclass consists of the socalled QN- (quasi-Nekrasov) matrices, which are defined in the present paper. For QN-matrices, an upper bound on the infinity norm of the inverses is established. It is shown that in application to Nekrasov matrices the new bounds are generally better than the known ones. Bibliography: 15 titles.

In this paper we give a combinatorial characterization of tight fusion frame (TFF) sequences using Littlewood-Richardson skew tableaux. The equal rank case has been solved recently by Casazza, Fickus, Mixon, Wang, and Zhou. Our characterization does not have this limitation. We also develop some methods for generating TFF sequences. The basic technique is a majorization principle for TFF sequences combined with spatial and Naimark dualities. We use these methods and our characterization to give necessary and sufficient conditions which are satisfied by the first three highest ranks. We also give a combinatorial interpretation of spatial and Naimark dualities in terms of Littlewood-Richardson coefficients. We exhibit four classes of TFF sequences which have unique maximal elements with respect to majorization partial order. Finally, we give several examples illustrating our techniques including an example of tight fusion frame which can not be constructed by the existing spectral tetris techniques. We end the paper by giving a complete list of maximal TFF sequences in dimensions less than ten., 31 pages

For \(\alpha , \gamma \ge 0\) and \(\beta 0, \, \, \, z\in E \end{aligned}$$ for some \(\phi \in {\mathbb {R}}\). For \(f\in {{\mathcal {W}}_{\beta }(\alpha , \gamma )}\), we consider the integral transform $$\begin{aligned} V_{\lambda }(f)(z):=\int _{0}^{1}\lambda (t)\frac{f(tz)}{t}\mathrm{d}t, \end{aligned}$$ where \(\lambda \) is a non-negative real-valued integrable function satisfying the condition \(\int _{0}^{1}\lambda (t)\mathrm{d}t=1\). In a very recent paper, Ali et al. (J Math Anal Appl 385:808–822, 2012) discussed the starlikeness of the integral transform \(V_{\lambda }(f)\) when \(f\in {{\mathcal {W}}}_{\beta }(\alpha , \gamma )\). The aim of present paper is to find conditions on \(\lambda (t)\) such that \(V_{\lambda }(f)\) is starlike of order \(\delta \) (\(0\le \delta \le 1/2\)) when \(f\in {{\mathcal {W}}}_{\beta }(\alpha , \gamma )\). As applications, we study various choices of \(\lambda (t)\), related to classical integral transforms.

A connected graph \(G\) is optimal-\(\kappa \) if the connectivity \(\kappa (G)=\delta (G)\), where \(\delta (G)\) is the minimum degree of \(G\). It is super-\(\kappa \) if every minimum vertex cut isolates a vertex. An optimal-\(\kappa \) graph \(G\) is \(m\)-optimal-\(\kappa \) if for any vertex set \(S\subseteq V(G)\) with \(|S|\le m\), \(G-S\) is still optimal-\(\kappa \). The maximum integer of such \(m\), denoted by \(O_{\kappa }(G)\), is the vertex fault tolerance of \(G\) with respect to the property of optimal-\(\kappa \). The concept of vertex fault tolerance with respect to the property of super-\(\kappa \), denoted by \(S_{\kappa }(G)\), is defined in a similar way. In a previous paper, we have proved that \(\min \{\kappa _1(G)-\delta (G),\delta (G)-1\}\le O_\kappa (G)\le \delta (G)-1\) and \(\min \{\kappa _1(G)-\delta (G)-1,\delta (G)-1\}\le S_\kappa (G)\le \delta (G)-1\). We also have \(S_\kappa (G)\le O_\kappa (G)\le \delta (G)-1\). In this paper, we study the realizability problems concerning the above three bounds. By construction, we proved that for any non-negative integers \(a,b,c\) with \(a\le b\le c\), (i) there exists a graph \(G\) such that \(\kappa _1(G)-\delta (G)=a\), \(O_{\kappa }(G)=b\), and \(\delta (G)-1=c\); (ii) there exists a graph \(G\) with \(\kappa _1(G)-\delta (G)-1=a\), \(S_{\kappa }(G)=b\), and \(\delta (G)-1=c\); and (iii) there exists a graph \(G\) such that \(S_{\kappa }(G)=a\), \(O_{\kappa }(G)=b\), and \(\delta (G)-1=c\).

In this paper, concentrating on the Hou Zhentings paper "The criterion for uniqueness of a Q- process", we review a number of associated results of the existence and uniqueness for Q -processes, especially Hou-Conditions ( H -conditions) at the construction of Q -processes. We successively summarize the existence and uniqueness for honest Q -processes, revisable Q -processes, stochastically monotone Q -processes, dual Q -processes and Q -processes in abstract space, give some associated results for birth-death processes, and point out three unsolved open problems. We wish to help Chinese young probability scholars to understand the outstanding contributions of Chinese probability seniors especially Mr. Hou.

This is the capstone of a series of papers studying the units of a compatible nearring \(R\) satisfying the descending chain condition on right ideals using a faithful compatible module \(G\) of \(R\). The approach is an inductive type one. Letting \(H\) be a direct sum of isomorphic minimal \(R\)-ideals of \(G\), assume we know the unit group of \(R/Ann_R(G/H)\). The unit group of \(R\) is then the extension group of the unit group of \(1 + Ann_R(G/H)\) by the unit group of \(R/Ann_R(G/H)\). Previous papers in this series have been devoted to the determination of the unit group of \(1 + Ann_R(G/H)\) which has now been completed. This leaves the study of this extension group as the final step to complete, and that is the purpose of this paper. In its most general form, we will see that our extension group is a more general type of a wreath product we shall call a Linz wreath product in honor of its place of birth with an amalgamation. We then proceed to doing a number of examples where we explicitly determine these unit groups.

In this paper we complete the study of the following elliptic equation driven by a general non-local integrodifferential operator $$\mathcal {L}_K$$ $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} \mathcal {L}_K u+\lambda u+|u|^{2^*-2}u+f(x, u)=0 &{} \hbox {in } \Omega \\ u=0 &{} \hbox {in } {\mathbb {R}}^n{\setminus } \Omega , \end{array}\right. \end{aligned}$$ that was started by Servadei and Valdinoci (Commun Pure Appl Anal 12(6):2445–2464, 2013). Here $$s\in (0,1),\, \Omega $$ is an open bounded set of $${\mathbb {R}}^n,\, n>2s$$ , with continuous boundary, $$\lambda $$ is a positive real parameter, $$2^*=2n/(n-2s)$$ is a fractional critical Sobolev exponent and $$f$$ is a lower order perturbation of the critical power $$|u|^{2^*-2}u$$ , while $$\mathcal {L}_K$$ is the integrodifferential operator defined as $$\begin{aligned} \mathcal {L}_Ku(x)= \int _{{\mathbb {R}}^n}\left( u(x+y)+u(x-y)-2u(x)\right) K(y)\,dy, \quad x\in {\mathbb {R}}^n. \end{aligned}$$ Under suitable growth condition on $$f$$ , we show that this problem admits non-trivial solutions for any positive parameter $$\lambda $$ . This existence theorem extends some results obtained in [15, 19, 20]. In the model case, that is when $$K(x)=|x|^{-(n+2s)}$$ (this gives rise to the fractional Laplace operator $$-(-\Delta )^s$$ ), the existence result proved along the paper may be read as the non-local fractional counterpart of the one obtained in [12] (see also [9]) in the framework of the classical Laplace equation with critical nonlinearities.

Balas introduced disjunctive cuts in the 1970s for mixed-integer linear programs. Several recent papers have attempted to extend this work to mixed-integer conic programs. In this paper we study the structure of the convex hull of a two-term disjunction applied to the second-order cone and develop a methodology to derive closed-form expressions for convex inequalities describing the resulting convex hull. Our approach is based on first characterizing the structure of undominated valid linear inequalities for the disjunction and then using conic duality to derive a family of convex, possibly nonlinear, valid inequalities that correspond to these linear inequalities. We identify and study the cases where these valid inequalities can equivalently be expressed in conic quadratic form and where a single inequality from this family is sufficient to describe the convex hull. In particular, our results on two-term disjunctions on the second-order cone generalize related results on split cuts by Modaresi, Kilinc, and Vielma, and by Andersen and Jensen.

This paper studies four families of polyhedral norms parametrized by a single parameter. The first two families consist of the CVaR norm (which is equivalent to the D-norm, or the largest-$$k$$k norm) and its dual norm, while the second two families consist of the convex combination of the $$\ell _1$$l1- and $$\ell _\infty $$l?-norms, referred to as the deltoidal norm, and its dual norm. These families contain the $$\ell _1$$l1- and $$\ell _\infty $$l?-norms as special limiting cases. These norms can be represented using linear programming (LP) and the size of LP formulations is independent of the norm parameters. The purpose of this paper is to establish a relation of the considered LP-representable norms to the $$\ell _p$$lp-norm and to demonstrate their potential in optimization. On the basis of the ratio of the tight lower and upper bounds of the ratio of two norms, we show that in each dual pair, the primal and dual norms can equivalently well approximate the $$\ell _p$$lp- and $$\ell _q$$lq-norms, respectively, for $$p,q\in [1,\infty ]$$p,q?[1,?] satisfying $$1/p+1/q=1$$1/p+1/q=1. In addition, the deltoidal norm and its dual norm are shown to have better proximity to the $$\ell _p$$lp-norm than the CVaR norm and its dual. Numerical examples demonstrate that LP solutions with optimized parameters attain better approximation of the $$\ell _{2}$$l2-norm than the $$\ell _1$$l1- and $$\ell _\infty $$l?-norms do.

We study the hit problem, set up by F. Peterson, of finding a minimal set of generators for the polynomial algebra $P_k := \mathbb F_2[x_1,x_2,...,x_k]$ as a module over the mod-2 Steenrod algebra, $\mathcal{A}$. In this paper, we study a minimal set of generators for $\mathcal A$-module $P_k$ in some so-call generic degrees and apply these results to explicitly determine the hit problem for $k=4$., Comment: 68 pages, Quy Nhon University preprint, Viet Nam, 2011. A shorter version of this paper was published in Advances in Mathematics

The aim of this series of papers is to study cyclic extensions of the field of rational numbers of odd prime power degree. This paper, which is the third of this series, concentrates on the situation where some of the ramified primes are (partially) decomposed.

Let $G$ be a directed graph. A \textit{$2$-directed block} in $G$ is a maximal vertex set $C^{2d}\subseteq V$ with $|C^{2d}|\geq 2$ such that for each pair of distinct vertices $x,y \in C^{2d}$, there exist two vertex-disjoint paths from $x$ to $y$ and two vertex-disjoint paths from $y$ to $x$ in $G$. In contrast to the $2$-vertex-connected components of $G$, the subgraphs induced by the $2$-directed blocks may consist of few or no edges. In this paper we present two algorithms for computing the $2$-directed blocks of $G$ in $O(\min\lbrace m,(t_{sap}+t_{sb})n\rbrace n)$ time, where $t_{sap}$ is the number of the strong articulation points of $G$ and $t_{sb}$ is the number of the strong bridges of $G$. Furthermore, we study two related concepts: the $2$-strong blocks and the $2$-edge blocks of $G$. We give two algorithms for computing the $2$-strong blocks of $G$ in $O( \min \lbrace m,t_{sap} n\rbrace n)$ time and we show that the $2$-edge blocks of $G$ can be computed in $O(\min \lbrace m, t_{sb} n \rbrace n)$ time. In this paper we also study some optimization problems related to the strong articulation points and the $2$-blocks of a directed graph. Given a strongly connected graph $G=(V,E)$, find a minimum cardinality set $E^{*}\subseteq E$ such that $G^{*}=(V,E^{*})$ is strongly connected and the strong articulation points of $G$ coincide with the strong articulation points of $G^{*}$. This problem is called minimum strongly connected spanning subgraph with the same strong articulation points. We show that there is a linear time $17/3$ approximation algorithm for this NP-hard problem. We also consider the problem of finding a minimum strongly connected spanning subgraph with the same $2$-blocks in a strongly connected graph $G$. We present approximation algorithms for three versions of this problem, depending on the type of $2$-blocks.

In this paper we introduce and study the new properties (W �), (UWa), (UW E) and (UW �). The main goal of this paper is to study relationship between these new properties and other Weyl type theorems. Moreover, we reconsider several earlier results obtained respec- tively in (11), (18), (14), (1) and (13) for which we give stronger versions. by σa(T) the approximate point spectrum of T. If the range R(T) of T is closed and α(T) < ∞ (resp. β(T) < ∞), then T is called an upper semi-Fredholm (resp. a lower semi-Fredholm) operator. If T ∈ L(X) is either upper or lower semi Fredholm, then T is called a semi-Fredholm operator, and the index of T is defined by ind(T) = α(T) − β(T). If both of α(T) and β(T) are finite, then T is called a Fredholm operator. An operator T ∈ L(X) is called a Weyl operator if it is a Fredholm operator of index zero. The Weyl spectrum σW(T) of T is defined by σW(T) = {λ ∈ C | T − λI is not a Weyl operator}. For a bounded linear operator T and a nonnegative integer n, define T(n) to be the restriction of T to R(T n ), viewed as a map from R(T n ) into R(T n ) (in particular T(0) = T). If for some integer n the range space R(T n ) is closed and T(n) is an upper (resp. a lower) semi-Fredholm operator, then T is called an upper (resp. a lower) semi-B-Fredholm operator. A semi-B-Fredholm operator T is an upper or a lower semi-B-Fredholm operator, and in this case the index of T is defined as the index of the semi-Fredholm operator T(n), see (12). Moreover if T(n) is a Fredholm operator, then T is called a B-Fredholm operator, see

In the game of cops and robber, the cops try to capture a robber moving on the vertices of the graph. The minimum number of cops required to win on a given graph G is called the cop number of G. The biggest open conjecture in this area is the one of Meyniel, which asserts that for some absolute constant C, the cop number of every connected graph G is at most C|VG|. In this paper, we show that Meyniel's conjecture holds asymptotically almost surely for the binomial random graph Gn,p, which improves upon existing results showing that asymptotically almost surely the cop number of Gn,p is Onlogn provided that pni¾?2+elogn for some e>0. We do this by first showing that the conjecture holds for a general class of graphs with some specific expansion-type properties. This will also be used in a separate paper on random d-regular graphs, where we show that the conjecture holds asymptotically almost surely when d=dni¾?3. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 396-421, 2016

In this paper, we classified finite p-groups Gsuch thatC G (x)/hxiis cyclic for all non-central elements x∈G. This solved a problem pro-posed By Y. Berkovoch. 1. IntroductionIf G is a finite group, then 1 ≤ hxi ≤ C G (x) ≤ G for all elements x ∈ G.In particular, G is abelian if and only if |G : C G (x)| = 1 for all x ∈ G.Moreover, K. Ishikawa [5] classified finite p-groups with |G : C G (x)| ≤ p 2 .Along another line, X. H. Li and J. Q. Zhang [6] classified finite p-groups with|C G (x) : hxi| ≤ p k , where k = 1,2 and p > 2. Y. Berkovich [1] proposed thefollowing:Problem 116(ii). Classify the p-groups G such that C G (H)/H is cyclic forall noncentral cyclic H < G.In other words, Problem 116(ii) requires to classify finite p-groups G suchthat C G (x)/hxi is cyclic for all non-central elements x ∈ G.For convenience, the groups in Problem 116(ii) are called P-groups. LetS = {G G is a P-group}. In this paper, S is determined, and hence theProblem 116(ii) is solved.2. PreliminariesAssume G is a finite p-group. Let r(G) = max{log

The commuting graph of a finite non-commutative semigroup S, denoted G(S), is a simple graph whose vertices are the non-central elements of S and two distinct vertices x, y are adjacent if xy = yx. Let I(X) be the symmetric inverse semigroup of partial injective transformations on a finite set X. The semigroup I(X) has the symmetric group Sym(X) of permutations on X as its group of units. In 1989, Burns and Goldsmith determined the clique number of the commuting graph of Sym(X). In 2008, Iranmanesh and Jafarzadeh found an upper bound of the diameter of G(Sym(X)), and in 2011, Dolžan and Oblak claimed that this upper bound is in fact the exact value. The goal of this paper is to begin the study of the commuting graph of the symmetric inverse semigroup I(X). We calculate the clique number of G(I(X)), the diameters of the commuting graphs of the proper ideals of I(X), and the diameter of G(I(X)) when |X| is even or a power of an odd prime. We show that when |X| is odd and divisible by at least two primes, then the diameter of G(I(X)) is either 4 or 5. In the process, we obtain several results about semigroups, such as a description of all commutative subsemigroups of I(X) of maximum order, and analogous results for commutative inverse and commutative nilpotent subsemigroups of I(X). The paper closes with a number of problems for experts in combinatorics and in group or semigroup theory.

Lyubeznik introduced the concept of size of a monomial ideal and showed that the size of a monomial ideal increased by \(1\) is a lower bound for its depth. Associated with this, Herzog, Popescu and Vladoiu showed that the size increased by \(1\) is simultaneously a lower bound for its Stanley depth. It turns out that the splitting-variables technique in their paper is particularly suited for comparing the Stanley depth and size of squarefree monomial ideals. In this paper, we generalize this technique and investigate the quotient \(I/J\) when \(J\subsetneq I\) are general monomial ideals. As an application, we consider lower bounds for Stanley depth of quotients of monomial ideals under various conditions. Through Alexander duality, we also consider upper bounds for the Stanley support-regularity of edge ideals of clutters. Meanwhile, recently, Katthan and Seyed Fakhari established another lower bound for depth and Stanley depth of \(I/J\) in terms of the lcm number. Using the splitting-variables technique, we provide an alternate proof of their result.

Let f be a transcendental meromorphic function of nite order in the plane such that f (m) has nitely many zeros for some positive integer m ≥ 2: Suppose that f (k) and f share a CM, where k ≥ 1 is a positive integer, a 0 is a nite complex value. Then f is an entire function such that f (k) − a = c(f − a); where c 0 is a nonzero constant. The results in this paper are concerning a conjecture of Bruck (5). An example is provided to show that the results in this paper, in a sense, are the best possible.

We study the braid indices of the Kanenobu knots. It is known that the Kanenobu knots have the same HOMFLYPT polynomial and the same Khovanov-Rozansky homology. The MFW inequality is known for giving a lower bound of the braid index of a link by applying the HOMFLYPT polynomial. Therefore, it is not easy to determine the braid indices of the Kanenobu knots. In our previous paper, we gave upper bounds and sharper lower bounds of the braid indices of the Kanenobu knots by applying the 2-cable version of the zeroth coefficient HOMFLYPT polynomial. In this paper, we give sharper upper bounds of the braid indices of the Kanenobu knots.

In this paper we are concerned with the boundedness of Hormander–Mihlin multipliers of order $$\,r\, (1\le r1.$$ On the other hand the boundedness extends to $$\,\mathcal H_{2\pi }$$ if $$\,r>1,$$ but not if $$\,r=1.$$ Generalizing this result in the present paper we show that the scale of Hardy spaces $$\,\mathcal H^p_{2\pi }\, (01\,$$ we give a sharp bound $$\,p_rp_r\,$$ then the Hormander–Mihlin condition of order $$\,r\,$$ is sufficient on $$\mathcal H^p_{2\pi }$$ .

The power graph $${\fancyscript{P}}(G)$$ of a group $$G$$ is the graph which has a vertex set of the group elements and two elements are adjacent if one is a power of the other. Chakrabarty, Ghosh, and Sen proved the main properties of the undirected power graph of a finite group. The aim of this paper is to generalize some results of their work and presenting some counterexamples for one of the problems raised by these authors. It is also proved that the power graph of a $$p$$ -group is $$2$$ -connected if and only if the group is a cyclic or generalized quaternion group and if $$G$$ is a nilpotent group which is not of prime power order then the power graph $${\fancyscript{P}}(G)$$ is $$2$$ -connected. We also prove that the number of edges of the power graph of the simple groups is less than or equal to the number of edges in the power graph of the cyclic group of the same order. This partially answers to a question in an earlier paper. Finally, we give a complete classification of groups in which the power graph is a union of complete graphs sharing a common vertex.