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This short paper is motivated by the paper of Bu et al. [C. Bu, C. Feng, P. Dong, A note on computational formulas for the Drazin inverse of certain block matrices, J. Appl. Math. Comput.(38) (2012) 631-640], where the authors gave additive formula for Drazin inverse for matrices under new conditions, and two representations under some specific conditions. Here is shown that the additive formula is not valid for all matrices which satisfy given conditions. Also, here is proved that the representations which were given in mentioned paper do not extend the results given by Hartwig et al. [R. Hartwig, X. Li, Y. Wei, Representations for the Drazin inverse of a 2 _ 2 block matrix, SIAM J. Matrix. Anal. Appl. (27)(2006) 757-771 ], in fact they are equivalent.

In this short paper we study the d-Hermite condition about stably free modules for skew $$\textit{PBW}$$ extensions. For this purpose, we estimate the stable rank of these non-commutative rings. In addition, and closely related with these questions, we will prove Kronecker’s theorem about the radical of finitely generated ideals for some particular types of skew $$\textit{PBW}$$ extensions.

In a recent paper, Sokół et al. [Applications of k-Fibonacci numbers for the starlike analytic functions, Hacet. J. Math. Stat. 44(1) (2015), 121{127] obtained an upper bound for the Fekete-Szegö functionalϕλwhenλ 2R of functions belong to the classSLkconnected withk-Fibonacci numbers. The main purpose of this paper is to obtain sharp bounds forϕλbothλ 2R andλ 2C.

In Sarkozy (2001) the third author of this paper presented two conjectures on the additive decomposability of the sequence of ”smooth” (or ”friable”) numbers. Elsholtz and Harper (2015) proved (by using sieve methods) the second (less demanding) conjecture. The goal of this paper is to extend and sharpen their result in three directions by using a different approach (based on the theory of S -unit equations).

A sequence $$(v_1,\ldots ,v_k)$$ of vertices in a graph G without isolated vertices is called a total dominating sequence if every vertex $$v_i$$ in the sequence totally dominates at least one vertex that was not totally dominated by $$\{v_1,\ldots , v_{i-1}\}$$ and $$\{v_1,\ldots ,v_k\}$$ is a total dominating set of G. The length of a shortest such sequence is the total domination number of G ( $$\gamma _{t}(G)$$ ), while the length of a longest such sequence is the Grundy total domination number of G ( $$\gamma _{gr}^t(G)$$ ). In this paper we study graphs with equal total and Grundy total domination numbers. We characterize bipartite graphs with both total and Grundy total dominations number equal to 4, and show that there is no connected chordal graph G with $$\gamma _{t}(G)=\gamma _{gr}^t(G)=4$$ . The main result of the paper is a characterization of bipartite graphs with $$\gamma _{t}(G)=\gamma _{gr}^t(G)=6$$ proved by establishing a surprising correspondence between the existence of such graphs and a classical but still open problem of the existence of certain finite projective planes.

Given a graph $G$ and a nondecreasing sequence $S=(s_1,\ldots,s_k)$ of positive integers, the mapping $c:V(G)\longrightarrow \{1,\ldots,k\}$ is called an $S$-packing coloring of $G$ if for any two distinct vertices $x$ and $y$ in $c^{-1}(i)$, the distance between $x$ and $y$ is greater than $s_i$. The smallest integer $k$ such that there exists a $(1,2,\ldots,k)$-packing coloring of a graph $G$ is called the packing chromatic number of $G$, denoted $\chi_{\rho}(G)$. The question of boundedness of the packing chromatic number in the class of subcubic (planar) graphs was investigated in several earlier papers; recently it was established that the invariant is unbounded in the class of all subcubic graphs. In this paper, we prove that the packing chromatic number of any 2-connected bipartite subcubic outerplanar graph is bounded by $7$. Furthermore, we prove that every subcubic triangle-free outerplanar graph has a $(1,2,2,2)$-packing coloring, and that there exists a subcubic outerplanar graph with a triangle that does not admit a $(1,2,2,2)$-packing coloring. In addition, there exists a subcubic triangle-free outerplanar graph that does not admit a $(1,2,2,3)$-packing coloring. A similar dichotomy is shown for bipartite outerplanar graphs: every such graph admits an $S$-packing coloring for $S=(1,3,\ldots,3)$, where $3$ appears $\Delta$ times ($\Delta$ being the maximum degree of vertices), and this property does not hold if one of the integers $3$ is replaced by $4$ in the sequence $S$., Comment: 24 pages

This paper studies the asymptotic behavior of the exact constants of the Nikolskii inequalities for the space $$\Pi _n^d$$ of spherical polynomials of degree at most n on the unit sphere $$\mathbb{S}{^d} \subset {^{d + 1}}$$ as n → ∞. It is shown that for 0 < p < ∞, $$\mathop {\lim }\limits_{x \to \infty } \sup \left\{ {\frac{{{{\left\| P \right\|}_{{L^\infty }({\mathbb{S}^d})}}}}{{{n^{\tfrac{d}{p}}}{{\left\| P \right\|}_{{L^p}({\mathbb{S}^d})}}}}:P \in \Pi _n^d} \right\} = \sup \left\{ {\frac{{{{\left\| f \right\|}_{{L^\infty }({\mathbb{R}^d})}}}}{{{{\left\| f \right\|}_{{L^p}({\mathbb{R}^d})}}}}:f \in \varepsilon _p^d} \right\},$$ where $$\varepsilon _p^d$$ denotes the space of all entire functions of spherical exponential type at most 1 whose restrictions to ℝd belong to the space Lp(ℝd), and it is agreed that 0/0 = 0. It is also proved that for 0 < p < q < ∞, $$\liminf _{n \rightarrow \infty} \sup \left\{\frac{\|P\|_{L^{q}\left(\mathbb{S}^{d}\right)}}{n^{d(1 / p-1 / q)}\|P\|_{L^{p}\left(\mathbb{S}^{d}\right)}}: P \in \Pi_{n}^{d}\right\} \geq \sup \left\{\frac{\|f\|_{L^{q}\left(\mathbb{R}^{d}\right)}}{\|f\|_{L^{p}\left(\mathbb{R}^{d}\right)}}: f \in \mathcal{E}_{p}^{d}\right\}.$$ These results extend the recent results of Levin and Lubinsky for trigonometric polynomials on the unit circle. The paper also determines the exact value of the Nikolskii constant for nonnegative functions with p = 1 and q = ∞: $$\lim _{n \rightarrow \infty} \sup _{0 \leq P \in \Pi_{n}^{d}} \frac{\|P\|_{L^{\infty}\left(\mathbb{S}^{d}\right)}}{\|P\|_{L^{1}\left(\mathbb{S}^{d}\right)}}=\sup _{0 \leq f \in \mathcal{E}_{1}^{d}} \frac{\|f\|_{L^{\infty}\left(\mathbb{R}^{d}\right)}}{\|f\|_{L^{1} \mathbb{R}^{d}}}=\frac{1}{4^{d} \pi^{d / 2} \Gamma(d / 2+1)}.$$

In this paper, we continue to investigate the exceptional sets for sums of five and six almost equal cubes of primes. We would also like to establish that almost all natural numbers n, subjected to certain congruence conditions, can be written as n = p 1 3 + ⋯ + p s 3 {n=p_{1}^{3}+\cdots+p_{s}^{3}} ( s = 5 , 6 {s=5,6} ) with | p j - ( n / s ) 1 / 3 | ≤ n θ s / 3 + ε {|p_{j}-(n/s)^{1/3}|\leq n^{\theta_{s}/3+\varepsilon}} ( 1 ≤ j ≤ s {1\leq j\leq s} ), where θ s {\theta_{s}} is as small as possible. The main result of this paper is to improve θ 6 = 5 / 6 + ε {\theta_{6}=5/6+\varepsilon} , which is proven in [M. Wang, Exceptional sets for sums of five and six almost equal prime cubes, Acta Math. Hungar. 156 2018, 2, 424–434], to θ 6 = 9 / 11 + ε {\theta_{6}=9/11+\varepsilon} , as well as prove θ 5 = 8 / 9 + ε {\theta_{5}=8/9+\varepsilon} in another way.

Diaconis and Griffiths (2014) study the multivariate Krawtchouk polynomials orthogonal on the multinomial distribution. In this paper we derive the reproducing kernel orthogonal polynomials Q n ( x , y ; N , p ) on the multinomial distribution which are sums of products of orthonormal polynomials in x and y of fixed total degree n = 0 , 1 , … , N . The Poisson kernel ∑ n = 0 N ρ n Q n ( x , y ; N , p ) arises naturally from a probabilistic argument. An application to a multinomial goodness of fit test is developed, where the chi-squared test statistic is decomposed into orthogonal components which test the order of fit. A new duplication formula for the reproducing kernel polynomials in terms of the 1-dimensional Krawtchouk polynomials is derived. The duplication formula allows a Lancaster characterization of all reversible Markov chains with a multinomial stationary distribution whose eigenvectors are multivariate Krawtchouk polynomials and where eigenvalues are repeated within the same total degree. The χ 2 cutoff time, and total variation cutoff time is investigated in such chains. Emphasis throughout the paper is on a probabilistic understanding of the polynomials and their applications, particularly to Markov chains.

In this paper we make a comparison between certain probabilistic and deterministic point sets and show that some deterministic constructions (especially spherical t -designs) are better or as good as probabilistic ones like the jittered sampling model. We find asymptotic equalities for the discrete Riesz s -energy of sequences of well separated t -designs on the unit sphere S d ⊂ R d + 1 , d ≥ 2 . The case d = 2 was studied in Hesse (2009) and Hesse and Leopardi (2008). In Bondarenko et al., (2015) it was established that for d ≥ 2 , there exists a constant c d , such that for every N > c d t d there exists a well-separated spherical t -design on S d with N points. This paper gives results, based on recent developments that there exists a sequence of well separated spherical t -designs such that t and N are related by N ≍ t d .

In this paper we study the so called optimal polynomial meshes for domains in K ⊂ R d , d ≥ 2 . These meshes are discrete point sets Y n of cardinality c n d which have the property that ‖ p ‖ K ≤ A ‖ p ‖ Y n for every polynomial p of degree at most n with a constant A > 1 independent of n . It was conjectured earlier that optimal polynomial meshes exist in every convex domain. This statement was previously shown to hold for polytopes and C 2 like domains. In this paper we give a complete affirmative answer to the above conjecture when d = 2 .

In this paper we extend the idea of weighted density of Balcerzak et al. (2015) by using a modulus function and introduce the idea f -density of weight g of subsets of ω ≔ { 0 , 1 , … } (at the same time extending the notion of f -density (Aizpuru et al., 2014)), which we name d g f where g : ω → [ 0 , ∞ ) satisfies g ( n ) → ∞ and n ∕ g ( n ) ↛ 0 and f is a modulus function. The aim of this paper is to show that we can get new ideals Z g ( f ) consisting of sets A ⊂ ω for which d g f ( A ) = 0 different from all the previously constructed ideals Z g of Balcerzak et al. (2015) and moreover they retain all the nice properties of the ideals Z g .

In this paper we improve the bounds for the Carath\'eodory number, especially on algebraic varieties and with small gaps (not all monomials are present). We provide explicit lower and upper bounds on algebraic varieties, $\mathbb{R}^n$, and $[0,1]^n$. We also treat moment problems with small gaps. We find that for every $\varepsilon>0$ and $d\in\mathbb{N}$ there is a $n\in\mathbb{N}$ such that we can construct a moment functional $L:\mathbb{R}[x_1,\dots,x_n]_{\leq d}\rightarrow\mathbb{R}$ which needs at least $(1-\varepsilon)\cdot\left(\begin{smallmatrix} n+d\\ n\end{smallmatrix}\right)$ atoms $l_{x_i}$. Consequences and results for the Hankel matrix and flat extension are gained. We find that there are moment functionals $L:\mathbb{R}[x_1,\dots,x_n]_{\leq 2d}\rightarrow\mathbb{R}$ which need to be extended to the worst case degree $4d$, $\tilde{L}:\mathbb{R}[x_1,\dots,x_n]_{\leq 4d}\rightarrow\mathbb{R}$, in order to have a flat extension., Comment: The first version contained the Carath\'eodory numbers from Hilbert functions and the shape reconstruction from derivatives of moments. The second part part extended and the paper is split into two papers: "Carath\'eodory numbers from Hilbert functions" and "Shape and Gaussian Mixture Reconstruction from Derivatives of Moments"

In part I of this paper we studied additive decomposability of the set F y of the y -smooth numbers and the multiplicative decomposability of the shifted set G y = F y + { 1 } . In this paper, focusing on the case of ’large’ functions y , we continue the study of these problems. Further, we also investigate a problem related to the m-decomposability of k -term sumsets, for arbitrary k .

Given an infinite family of extended real-valued functions fi, i∈I, and a family H of nonempty finite subsets of I, the H-partial robust sum of fi, i∈I, is the supremum, for J∈H, of the finite sums ∑j∈Jfj. These infinite sums arise in a natural way in location problems as well as in functional approximation problems, and include as particular cases the well-known sup function and the so-called robust sum function, corresponding to the set H of all nonempty finite subsets of I, whose unconstrained minimization was analyzed in previous papers of three of the authors (https://doi.org/10.1007/s11228-019-00515-2 and https://doi.org/10.1007/s00245-019-09596-9). In this paper, we provide ordinary and stable zero duality gap and strong duality theorems for the minimization of a given H-partial robust sum under constraints, as well as closedness and convex criteria for the formulas on the subdifferential of the sup-function. This research was supported by the National Foundation for Science & Technology Development (NAFOSTED), Vietnam, Project 101.01-2018.310, and by Ministerio de Ciencia, Innovación y Universidades (MCIU), Agencia Estatal de Investigación (AEI), and European Regional Development Fund (ERDF), Project PGC2018-097960-B-C22.

The weighted T-free 2-matching problem is the following problem: given an undirected graph G, a weight function on its edge set, and a set T of triangles in G, find a maximum weight 2-matching containing no triangle in T. When T is the set of all triangles in G, this problem is known as the weighted triangle-free 2-matching problem, which is a long-standing open problem. A main contribution of this paper is to give the first polynomial-time algorithm for the weighted T-free 2-matching problem under the assumption that T is a set of edge-disjoint triangles. In our algorithm, a key ingredient is to give an extended formulation representing the solution set, that is, we introduce new variables and represent the convex hull of the feasible solutions as a projection of another polytope in a higher dimensional space. Although our extended formulation has exponentially many inequalities, we show that the separation problem can be solved in polynomial time, which leads to a polynomial-time algorithm for the weighted T-free 2-matching problem., A preliminary version of this paper appears in [Kobayashi, Y.: Weighted triangle-free 2-matching problem with edge-disjoint forbidden triangles. In: Integer Programming and Combinatorial Optimization, pp. 280–293. Springer, New York (2020).].

Variants of the coordinate descent approach for minimizing a nonlinear function are distinguished in part by the order in which coordinates are considered for relaxation. Three common orderings are cyclic (CCD), in which we cycle through the components of $x$ in order; randomized (RCD), in which the component to update is selected randomly and independently at each iteration; and random-permutations cyclic (RPCD), which differs from CCD only in that a random permutation is applied to the variables at the start of each cycle. Known convergence guarantees are weaker for CCD and RPCD than for RCD, though in most practical cases, computational performance is similar among all these variants. There is a certain type of quadratic function for which CCD is significantly slower than for RCD; a recent paper by Sun & Ye (2016, Worst-case complexity of cyclic coordinate descent: $O(n^2)$ gap with randomized version. Technical Report. Stanford, CA: Department of Management Science and Engineering, Stanford University. arXiv:1604.07130) has explored the poor behavior of CCD on functions of this type. The RPCD approach performs well on these functions, even better than RCD in a certain regime. This paper explains the good behavior of RPCD with a tight analysis.

The purpose of this paper is to investigate the invariance of the arithmetic mean with respect to two weighted Bajraktarevic means, i.e., to solve the functional equation $$\begin{aligned} \left( \frac{f}{g}\right) ^{\!\!-1}\!\!\left( \frac{tf(x)+sf(y)}{tg(x)+sg(y)}\right) +\left( \frac{h}{k}\right) ^{\!\!-1}\!\!\left( \frac{sh(x)+th(y)}{sk(x)+tk(y)}\right) =x+y \qquad (x,y\in I), \end{aligned}$$ where $$f,g,h,k:I\rightarrow \mathbb {R}$$ are unknown continuous functions such that g, k are nowhere zero on I, the ratio functions f / g, h / k are strictly monotone on I, and $$t,s\in \mathbb {R}_+$$ are constants different from each other. By the main result of this paper, the solutions of the above invariance equation can be expressed either in terms of hyperbolic functions or in terms of trigonometric functions and an additional weight function. For the necessity part of this result, we will assume that $$f,g,h,k:I\rightarrow \mathbb {R}$$ are four times continuously differentiable.

In 1933, Lehmer exhibited the polynomial $$\begin{aligned} L(z)=z^{10} + z^9 - z^7 - z^6 - z^5 - z^4 - z^3 + z + 1 \end{aligned}$$ with Mahler measure $$\mu _0>1$$ . Then he asked if $$\mu _0$$ is the smallest Mahler measure, not 1. This question became known as the Lehmer conjecture and it was apparently solved in the positive, while this paper was in preparation [19]. In this paper we consider those polynomials of the form $$\chi _A$$ , that is, Coxeter polynomials of a finite dimensional algebra A (for instance $$L(z)=\chi _{\mathbb {E}_{10}}$$ ). A polynomial in $$\mathbb {Z}[T]$$ which is either cyclotomic or with Mahler measure $$\ge \mu _0$$ is called a Lehmer polynomial. We give some necessary conditions for a polynomial to be Lehmer. We show that A being a tree algebra is a sufficient condition for $$\chi _A$$ to be Lehmer.

In this paper we consider the following question. When does there exist a square root of a probability measure supported on R+? This question is naturally related to subnormality of weighted shifts. The main result of this paper is that if μ is a finitely atomic probability measure having at most 4 atoms, then μ has a square root, i.e., there exists a measure ν such that μ=ν*ν (* means the convolution) if and only if the Aluthge transform of a subnormal weighted shift with Berger measure μ is subnormal. As an application of them, we give non-trivial, large classes of probability measures having a square root. We also prove that there are 6 and 7-atomic probability measures which don't have any square root. Our results have a connection to the following long-open problem in Operator Theory: characterize the subnormal operators having a square root.

In this paper, the Taylor sequence space t p θ of non-absolute type is introduced which consists of all sequences whose Taylor transforms of order θ , ( 0 θ 1 ) , are in the space l p . It is shown that the space t p θ is a normed space which includes the space l p where 1 ≤ p ≤ ∞ . Moreover, in this paper a general upper estimate is obtained for the norm of Hausdorff matrices and Norlund matrices as operators from l p into the Taylor sequence space t p θ . Finally, the results are extended to the matrix domain of an arbitrary invertible matrix E in the sequence space l p .

Let N be a module in category $$\sigma [M]$$ . N is called generalized $$\delta$$ –M-small (briefly G– $$\delta$$ –M) if, $$N \subseteq \delta (L)$$ for some $$L \in \sigma [M]$$ . In this paper we characterize G– $$\delta$$ –M modules and get some suitable results related to this kind of modules. We will show that a module $$N \in \sigma [M]$$ is G– $$\delta$$ –M if and only if $$N \subseteq \delta (\hat{N})$$ , where $$\hat{N}$$ is the M-injective envelope of N in $$\sigma [M]$$ . Also we prove that if there is no non-zero G– $$\delta$$ –M module, then M is cosemisimple and by giving an example we show that the converse need not be true. The relation between G– $$\delta$$ –M modules and some other classes of modules would be investigated in this paper. The torsion theory cogenerated by this class of modules will be introduced and studied in this paper. For a module $$N \in \sigma [M]$$ we show that $$N = \mathrm{{Re}}_{GD[M]}(N)$$ iff for every nonzero homomorphism $$f : N \longrightarrow K$$ in $$\sigma [M],$$ $$\mathrm{{Im}}(f) \not \subseteq \delta (K)$$ iff $$\Delta _{\delta }(N,A) = 0,$$ for all $$A \in \sigma [M]$$ .

This paper is concerned with so-called generic properties of general linear conic programs. Many results have been obtained on this subject during the last two decades. For example, it is known that uniqueness, strict complementarity, and nondegeneracy of optimal solutions hold for almost all problem instances. Strong duality holds generically in a stronger sense, i.e., it holds for a generic subset of problem instances. In this paper, we survey known results and present new ones. In particular we give an easy proof of the fact that Slater’s condition holds generically in linear conic programming. We further discuss the problem of stability of uniqueness, nondegeneracy, and strict complementarity. We also comment on the fact that in general, a conic program cannot be treated as a smooth problem and that techniques from nonsmooth geometric measure theory are needed.

Let k be a nonzero complex number. In this paper we consider k - circulant matrices with arithmetic sequence and investigate the eigenvalues, determinants and Euclidean norms of such matrices. Also, for k=1 , the inverses of such ( invertible ) matrices are obtained (in a way different from the way presented in the paper: M. Bahsi and S. Solak , On the Circulant Matrices with Arithmetic Sequence , Int. J. Contemp . Math. Sci. 5 (25) (2010), 1213-1222, and the Moore-Penrose inverses of such (singular) matrices are derived.

Let $H_1$, $H_2$, $H_3$ be real Hilbert spaces, let $A:H_1\rightarrow H_3$, $B:H_2\rightarrow H_3$ be two bounded linear operators. The general split common fixed-point problem under consideration in this paper is to $$\text{find}\ \ x \in \cap_{i=1}^p F(U_i),\ \ y \in \cap_{j=1}^r F(T_j)\ \ \text{such that}\ \ Ax = By,\eqno{(1)}$$ where $p$, $r\geq 1$ are integers, $U_i:H_1\rightarrow H_1$ $(1\leq i\leq p)$ and $T_j:H_2\rightarrow H_2$ $(1\leq j\leq r)$ are quasi-nonexpansive mappings with nonempty common fixed-point sets $\cap_{i=1}^pF(U_i)=\cap_{i=1}^p\{x\in H_1:U_ix=x\}$ and $\cap_{j=1}^rF(T_j)=\cap_{j=1}^r\{x\in H_2:T_jx=x\}$. Note that, the above problem (1) allows asymmetric and partial relations between the variables $x$ and $y$. If $H_2=H_3$ and $B=I$, then the general split common fixed-point problem (1) reduces to the general split common fixed-point problem proposed by Censor and Segal $\cite{C}$. In this paper, we introduce simultaneous parallel and cyclic algorithms for the general split common fixed-point problems (1). We introduce a way of selecting the stepsizes such that the implementation of our algorithms does not need any prior information about the operator norms. We prove the weak convergence of the proposed algorithms and apply the proposed algorithms to the multiple-set split feasibility problems. Our results improve and extend the corresponding results announced by many others.

Suppose $R$ is an associative ring with identity 1. In this paper, We give the necessary and sufficient conditions for the existence and explicit representations of the group inverses of the following two classes block matrices. (i) $M=\left(\begin{array}{cc}AX+YBA (ii) $M=\left(\begin{array}{cc}A&B\\C&D\end{array}\right)$, where $A^{\sharp}$ exists, $A^{\pi}B=0$ and $(D-CA^{\sharp}B)^{\sharp}$ exists.\\ The paper's results generalize some relative results of Li et al. (J. Harbin Engineering Univ., 34, 658-661, 2013), Cao et al. (Appl. Math. Comput., 217, 10271-10277, 2011), Bu et al. (Appl. Math. Comput., 215, 132-139, 2009), Deng et al. (Linear Algebra Appl, 435, 2766-2783, 2011), and Liu et al. (J. Appl. Math. http://dx.doi.org/10.1155 /2013/247028). Some examples are given to illustrate our results.

The (un)weighted stability for some matrices on a graph is one of essential hypotheses in time-frequency analysis and applied harmonic analysis. In the first part of this paper, we show that for a localized matrix in a Beurling algebra, its weighted stabilities for different exponents and Muckenhoupt weights are equivalent to each other, and reciprocal of its optimal lower stability bound for one exponent and weight is controlled by a polynomial of reciprocal of its optimal lower stability bound for another exponent and weight. Banach algebras of matrices with certain off-diagonal decay is of great importance in many mathematical and engineering fields, and its inverse-closed property can be informally interpreted as localization preservation. Let $${{\mathcal {B}}}(\ell ^p_w)$$ be the Banach algebra of bounded linear operators on the weighted sequence space $$\ell ^p_w$$ on a graph. In the second part of this paper, we prove that Beurling algebras of localized matrices on a connected simple graph are inverse-closed in $${{\mathcal {B}}}(\ell ^p_w)$$ for all $$1\le p

In the present paper, we expand the domain of work on theconcept of semiderivations in 3-prime near-rings through the study ofstructure and commutativity of near-rings admitting semiderivations sat-isfying certain differential identities. Moreover, several examples havebeen provided at places which show that the assumptions in the hypothe-ses of various theorems are not altogether superfluous. 1. IntroductionThroughout this paper, N is a zero-symmetric left near ring. A near ringN is called zero symmetric if 0x= 0 for all x∈ N (recall that in a left nearring x0 = 0 for all x∈ N). N is called 3-prime if xNy = {0} implies x= 0or y = 0. The symbol Z(N) will represent the multiplicative center of N,that is, Z(N) = {x∈ N | xy= yxfor all y∈ N}.For any x,y∈ N; as usual[x,y] = xy−yxand x◦y= xy+yxwill denote the well-known Lie product andJordan product, respectively. Recall that N is called 2-torsion free if 2x= 0implies x= 0 for all x∈ N. For terminologies concerning near-rings we referto G. Pilz [7].An additive mapping d: N → N is said to be a derivation if d(xy) = xd(y)+d(x)yforall x,y∈ N, orequivalently, asnotedin [8], that d(xy) = d(x)y+xd(y)for all x,y ∈ N. An additive mapping d: N → N is called semiderivation ifthere exists a function g : N → N such that d(xy) = xd(y) + d(x)g(y) =g(x)d(y)+d(x)yand d(g(x)) = g(d(x)) for all x,y∈ N.Obviously, any deriva-tion is a semiderivation, but the converse is not true in general (see [6]). Therehas been a greatdeal of workconcerning derivations in near-rings(see [1, 2, 4, 5]where further references can be found). In this paper, we study the commuta-tivity of addition and multiplication of near-rings. Two well-known results forderivations in near-rings have been generalized for semiderivation. In fact, ourresults generalize some theorems obtained by the authors together with Rajiin [1].

In this paper, we obtain a class of irreducible Virasoro modules by taking tensor products of the irreducible Virasoro modules $\Omega(\lambda,b)$ defined in [LZ], with irreducible highest weight modules $V(\theta,h)$ or with irreducible Virasoro modules Ind$_{\theta}(N)$ defined in [MZ2]. We determine the necessary and sufficient conditions for two such irreducible tensor products to be isomorphic. Then we prove that the tensor product of $\Omega(\lambda,b)$ with a classical Whittaker module is isomorphic to the module $\mathrm{Ind}_{\theta,\lambda}(\mathbb{C_\mathbf{m}})$ defined in [MW]. As a by-product we obtain the necessary and sufficient conditions for the module $\mathrm{Ind}_{\theta, \lambda}(\mathbb{C_\mathbf{m}})$ to be irreducible. We also generalize the module $\mathrm{Ind}_{\theta, \lambda}(\mathbb{C_\mathbf{m}})$ to $\mathrm{Ind}_{\theta,\lambda}(\mathcal{B}^{(n)}_{\mathbf{s}})$ for any non-negative integer $ n$ and use the above results to completely determine when the modules $\mathrm{Ind}_{\theta,\lambda}(\mathcal{B}^{(n)}_{\mathbf{s}})$ are irreducible. The submodules of $\mathrm{Ind}_{\theta,\lambda}(\mathcal{B}^{(n)}_{\mathbf{s}})$ are studied and an open problem in [GLZ] is solved. Feigin-Fuchs' Theorem on singular vectors of Verma modules over the Virasoro algebra is crucial to our proofs in this paper., Comment: 17 Pages

In this paper, we propose a new incremental gradient methodfor solving a regularized minimization problem whose objective is thesum of msmooth functions and a (possibly nonsmooth) convex function.This method uses an adaptive stepsize. Recently proposed incrementalgradient methods for a regularized minimization problem need O(mn)storage, where n is the number of variables. This is the drawback ofthem. But, the proposed new incremental gradient method requires onlyO(n) storage. 1. IntroductionIn this paper, we consider the regularized minimization problem whose formis(1) min x∈ℜ n F λ (x) := f(x) +λP(x),where λ > 0, P : ℜ n → (−∞,∞] is a proper, convex, lower semicontinuous(lsc) function [20], and(2) f(x) :=X mi=1 f i (x),where each function f i is real-valued and smooth (i.e., continuously differen-tiable) on an open subset of ℜ n containing domP = {x | P(x) < ∞}.The minimization problem (1) we consider arises in many applications suchas (supervised) learning [7, 12, 27], regression [17, 23], neural network training[11, 21, 29], and data mining/classification [5, 15, 22, 28]. For the l

In a series of papers recently "checkerboard discrepancy" has been introduced, where a black-and-white checkerboard background induces a coloring on any curve, and thus a discrepancy, i.e., the difference of the length of the curve colored white and the length colored black. Mainly straight lines and circles have been studied and the general situation is that, no matter what the background coloring, there is always a curve in the family studied whose discrepancy is at least of the order of the square root of the length of the curve. In this paper we generalize the shape of the background, keeping the lattice structure. Our background now consists of lattice copies of any bounded fundamental domain of the lattice, and not necessarily of squares, as was the case in the previous papers. As the decay properties of the Fourier Transform of the indicator function of the square were strongly used before, we now have to use a quite different proof, in which the tiling and spectral properties of the fundamental domain play a role., Comment: 10 pages, 3 figures

This paper gives simple proofs of Sylvester (` = 2) and Frobenius (` = 3) inequalities. Moreover, a new sufficient condition for the equality of the Frobenius inequality is provided. In addition, an extension for ` > 3 matrices and an application to generalized inverses are provided., This paper has been partially supported by Ministerio de Economia y Competitividad (Grant DGI MTM2013-43678P and Red de Excelencia MTM2015-68805-REDT). The author thanks the referees for their valuable suggestions.

The aim of this paper is to continue our investigations in line of our recent paper, Savas [24] and [26]. We introduce the notion of AI-double statistical convergence which includes the new summabilitymethods studied in [24] and [23] as special cases and make certain observations on this new and moregeneral summability method. © 2016, University of Nis. All rights reserved.

The standard quadratic optimization problem (StQP), i.e. the problem of minimizing a quadratic form $$\mathbf{x}^TQ\mathbf{x}$$ on the standard simplex $$\{\mathbf{x}\ge \mathbf{0}: \mathbf{x}^T\mathbf{e}=1\}$$ , is studied. The StQP arises in numerous applications, and it is known to be NP-hard. Chen, Peng and Zhang showed that almost certainly the StQP with a large random matrix $$Q=Q^T$$ , $$Q_{i,j},\, (i\le j)$$ being independent and identically concave-distributed, attains its minimum at a point $$\mathbf{x}$$ with support size $$|\{j: x_j>0\}|$$ bounded in probability. Later Chen and Peng proved that for $$Q=(M+M^T)/2$$ , with $$M_{i,j}$$ i.i.d. normal, the likely support size is at most 2. In this paper we show that the likely support size is poly-logarithmic in n, the problem size, for a considerably broader class of the distributions. Unlike the cited papers, the mild constraints are put on the asymptotic behavior of the distribution at a single left endpoint of its support, rather than on the distribution’s shape elsewhere. It also covers the distributions with the left endpoint $$-\infty $$ , provided that the distribution of $$Q_{i,j},\, (i\le j)$$ (of $$M_{i,j}$$ , if $$Q=(M+M^T)/2$$ resp.) has a (super/sub) exponentially narrow left tail.

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.

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.

Aronhold’s classical result states that a plane quartic can be recovered by the configuration of any Aronhold systems of bitangents, i.e., special 7-tuples of bitangents such that the six points at which any sub-triple of bitangents touches the quartic do not lie on the same conic in the projective plane. Lehavi (cf. [L05]) proved that a smooth plane quartic can be explicitly reconstructed from its 28 bitangents; this result improved Aronhold’s method of recovering the curve. In a 2011 paper [PSV11] Plaumann, Sturmfels and Vinzant introduced an 8 × 8 symmetric matrix that parametrizes the bitangents of a nonsingular plane quartic. The starting point of their construction is Hesse’s result for which every smooth quartic curve has exactly 36 equivalence classes of linear symmetric determinantal representations. In this paper we tackle the inverse problem, i.e., the construction of the bitangent matrix starting from the 28 bitangents of the plane quartic, and we provide a Sage script intended for computing the bitangent matrix of a given curve.

We investigate the ratio asymptotic behavior of the sequence $(Q_{n})_{n=0}^{\infty}$ of multiple orthogonal polynomials associated with a Nikishin system of $p\geq 1$ measures that are compactly supported on the star-like set of $p+1$ rays $S_{+}=\{z\in\mathbb{C}: z^{p+1}\geq 0\}$. The main algebraic property of these polynomials is that they satisfy a three-term recurrence relation of the form $zQ_{n}(z)=Q_{n+1}(z)+a_{n} Q_{n-p}(z)$ with $a_{n}>0$ for all $n\geq p$. Under a Rakhmanov-type condition on the measures generating the Nikishin system, we prove that the sequence of ratios $Q_{n+1}(z)/Q_{n}(z)$ and the sequence $a_{n}$ of recurrence coefficients are limit periodic with period $p(p+1)$. Our results complement some results obtained by the first author and Mi\~{n}a-D\'{i}az in a recent paper in which algebraic properties and weak asymptotics of these polynomials were investigated. Our results also extend some results obtained by the first author in the case $p=2$., Comment: Minor corrections were done for this version. A continuation of this work is in arXiv: 1907.03002. This paper has 37 pages, 1 table, no figures

The purpose of this paper has twofold. The first is to establish a second main theorem with truncated counting functions for algebraically nondegenerate meromorphic mappings into an arbitrary projective variety intersecting a family of hypersurfaces in subgeneral position. In our result, the truncation level of the counting functions is estimated explicitly. Our result is an extension of the classical second main theorem of H. Cartan, also is a generalization of the recent second main theorem of M. Ru and improves some recent results. The second purpose of this paper is to give another proof for the second main theorem for the special case where the projective variety is a projective space, by which the truncation level of the counting functions is estimated better than that of the general case., This paper is accepted for publication in Transactions of the American Mathematical Society

Multiple zeta values (MZVs) are generalizations of Riemann zeta values at positive integers to multiple variable setting. These values can be further generalized to level N multiple polylog values by evaluating multiple polylogs at Nth roots of unity. In this paper, we consider another level N generalization by restricting the indices in the iterated sums defining MZVs to congruence classes modulo N, which we call the MZVs at level N. The goals of this paper are twofold. First, we shall lay down the theoretical foundations of these values such as their regularizations and double shuffle relations. Second, we will generalize the bracket functions related to multiple divisor sums defined by Bachmann and Kuhn to arbitrary level N and study their relations to MZVs at level N. The brackets are all q-series and similar to MZVs, they have both weight and depth filtrations. But unlike that of MZVs, the product of brackets usually has mixed weights; however, after projecting to the highest weight we can obtain an algebra homomorphism from brackets to MZVs. Moreover, the image of the derivation \({\mathfrak{D}=q\frac{d}{dq}}\) on brackets vanishes on the MZV side, which gives rise to many nontrivial \({\mathbb{Q}}\)-linear relations.

This paper presents sufficient conditions for Hamiltonian paths and cycles in graphs. Letting $\lambda\left( G\right) $ denote the spectral radius of the adjacency matrix of a graph $G,$ the main results of the paper are: (1) Let $k\geq1,$ $n\geq k^{3}/2+k+4,$ and let $G$ be a graph of order $n$, with minimum degree $\delta\left( G\right) \geq k.$ If \[ \lambda\left( G\right) \geq n-k-1, \] then $G$ has a Hamiltonian cycle, unless $G=K_{1}\vee(K_{n-k-1}+K_{k})$ or $G=K_{k}\vee(K_{n-2k}+\overline{K}_{k})$. (2) Let $k\geq1,$ $n\geq k^{3}/2+k^{2}/2+k+5,$ and let $G$ be a graph of order $n$, with minimum degree $\delta\left( G\right) \geq k.$ If \[ \lambda\left( G\right) \geq n-k-2, \] then $G$ has a Hamiltonian path, unless $G=K_{k}\vee(K_{n-2k-1}+\overline {K}_{k+1})$ or $G=K_{n-k-1}+K_{k+1}$ In addition, it is shown that in the above statements, the bounds on $n$ are tight within an additive term not exceeding $2$., Comment: 18 pages. This version gives tighter bounds

This paper deals with the characteristic roots of different types of lattice matrices and proves that a matrix and its transpose have the same characteristic roots. Also this paper introduces the concept of similar lattice matrices and proves that similar lattice matrices have the same characteristic roots.

This paper concerns the worst-case complexity of cyclic coordinate descent (C-CD) for minimizing a convex quadratic function, which is equivalent to Gauss-Seidel method and can be transformed to Kaczmarz method and projection onto convex sets (POCS). We observe that the known provable complexity of C-CD can be $O(n^2)$ times slower than randomized coordinate descent (R-CD), but no example was rigorously proven to exhibit such a large gap. In this paper we show that the gap indeed exists. We prove that there exists an example for which C-CD takes at least $O(n^4 \kappa_{\text{CD}} \log\frac{1}{\epsilon})$ operations, where $\kappa_{\text{CD}}$ is related to Demmel's condition number and it determines the convergence rate of R-CD. It implies that in the worst case C-CD can indeed be $O(n^2)$ times slower than R-CD, which has complexity $O( n^2 \kappa_{\text{CD}} \log\frac{1}{\epsilon})$. Note that for this example, the gap exists for any fixed update order, not just a particular order. Based on the example, we establish several almost tight complexity bounds of C-CD for quadratic problems. One difficulty with the analysis is that the spectral radius of a non-symmetric iteration matrix does not necessarily constitute a \textit{lower bound} for the convergence rate. An immediate consequence is that for Gauss-Seidel method, Kaczmarz method and POCS, there is also an $O(n^2) $ gap between the cyclic versions and randomized versions (for solving linear systems). We also show that the classical convergence rate of POCS by Smith, Solmon and Wager [1] is always worse and sometimes can be infinitely times worse than our bound., Comment: 47 pages. Add a few tables to summarize the main convergence rates; add comparison with classical POCS bound; add discussions on another example

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.

We show that (with one possible exception) there exist strongly dense free subgroups in any semisimple algebraic group over a large enough field. These are nonabelian free subgroups all of whose subgroups are either cyclic or Zariski dense. As a consequence, we get new generating results for finite simple groups of Lie type and a strengthening of a theorem of Borel related to the Hausdorff-Banach-Tarski paradox. In a sequel to this paper, we use this result to also establish uniform expansion properties for random Cayley graphs over finite simple groups of Lie type., 27 pages, no figures, submitted, Israel J. Math. It turns out that there is one specific family of algebraic group - namely Sp(4) in characteristic 3 - which our methods are unable to resolve, and we have adjusted the paper to exclude this case from the results

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

Let f be a continuous function defined on Ω:=[0,1]N which depends on only l coordinate variables, \(f(x_{1},\ldots,x_{N})=g(x_{i_{1}},\ldots,x_{i_{\ell}})\). We assume that we are given m and are allowed to ask for the values of f at m points in Ω. If g is in Lip1 and the coordinates i1,…,il are known to us, then by asking for the values of f at m=Ll uniformly spaced points, we could recover f to the accuracy |g|Lip1L−1 in the norm of C(Ω). This paper studies whether we can obtain similar results when the coordinates i1,…,il are not known to us. A prototypical result of this paper is that by asking for C(l)Ll(log 2N) adaptively chosen point values of f, we can recover f in the uniform norm to accuracy |g|Lip1L−1 when g∈Lip1. Similar results are proven for more general smoothness conditions on g. Results are also proven under the assumption that f can be approximated to some tolerance e (which is not known) by functions of l variables.

In a paper Cheung, Cucker and Peña (in press) [5] that can be seen as the first part of this one, we extended the well-known condition numbers for polyhedral conic systems C(A) Renegar (1994, 1995) [7–9] and C(A) Cheung and Cucker (2001) [3] to versions C¯(A) and C¯(A) that are finite for all input matrices A∈Rn×m. In this paper we compare C¯(A) and C¯(A) with other condition measures for the same problem that are also always finite.

Let $${S =\{x\in \mathbb{R}^n: g_1(x)\geq 0, \ldots, g_m(x)\geq 0\}}$$be a semialgebraic set defined by multivariate polynomials g i (x). Assume S is convex, compact and has nonempty interior. Let $${S_i =\{x\in \mathbb{R}^n:\, g_i(x)\geq 0\}}$$, and ∂ S (resp. ∂ S i ) be the boundary of S (resp. S i ). This paper, as does the subject of semidefinite programming (SDP), concerns linear matrix inequalities (LMIs). The set S is said to have an LMI representation if it equals the set of solutions to some LMI and it is known that some convex S may not be LMI representable (Helton and Vinnikov in Commun Pure Appl Math 60(5):654–674, 2007). A question arising from Nesterov and Nemirovski (SIAM studies in applied mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1994), see Helton and Vinnikov in Commun Pure Appl Math 60(5):654–674, 2007 and Nemirovski in Plenary lecture, International Congress of Mathematicians (ICM), Madrid, Spain, 2006, is: given a subset S of $${\mathbb{R}^n}$$, does there exist an LMI representable set Ŝ in some higher dimensional space $${ \mathbb{R}^{n+N}}$$whose projection down onto $${\mathbb{R}^n}$$equals S. Such S is called semidefinite representable or SDP representable. This paper addresses the SDP representability problem. The following are the main contributions of this paper: (i) assume g i (x) are all concave on S. If the positive definite Lagrange Hessian condition holds, i.e., the Hessian of the Lagrange function for optimization problem of minimizing any nonzero linear function l T x on S is positive definite at the minimizer, then S is SDP representable. (ii) If each g i (x) is either sos-concave ( − ∇2 g i (x) = W(x)T W(x) for some possibly nonsquare matrix polynomial W(x)) or strictly quasi-concave on S, then S is SDP representable. (iii) If each S i is either sos-convex or poscurv-convex (S i is compact convex, whose boundary has positive curvature and is nonsingular, i.e., ∇g i (x) ≠ 0 on ∂ S i ∩ S), then S is SDP representable. This also holds for S i for which ∂ S i ∩ S extends smoothly to the boundary of a poscurv-convex set containing S. (iv) We give the complexity of Schmudgen and Putinar’s matrix Positivstellensatz, which are critical to the proofs of (i)–(iii).