Grothendieck rings of a class of Hopf algebras of Kac-Paljutkin type

We construct the Grothendieck rings of a class of 2n²dimensional semisimple Hopf Algebras H_2n²,which can be viewed as a generalization of the 8 dimensional Kac-Paljutkin Hopf algebra H8.All irreducible H_2n²-modules are classified.Furthermore,we describe the Grothendieck rings r(H_2n²)by generators and relations explicitly.     

Oscillations of coefficients of symmetric square L-functions over primes

Let L(s, sym²f) be the symmetric-square L-function associated to a primitive holomorphic cusp form f for SL(2,?), with t_f(n,1) denoting the nth coefficient of the Dirichlet series for it. It is proved that, for N≥2 and any α ∈ ?, there exists an effective positive constant c such that ∑n≤NΛ(n)tf(n,1)e(nα)≪Nexp⁡(−clog⁡N), where Λ(n) is the von Mangoldt function, and the implied constant only depends on f. We also study the analogue of Vinogradov’s three primes theorem associated to the coefficients of Rankin-Selberg L-functions.     

Higher moment of coefficients of Dedekind zeta function

Let be a non-normal cubic extension over Q.We study the higher moment of the coefficients aK3(n)of Dedckind zeta function over sum of two squares∑n21+n22≤xa1K3(n21+n22),where 2≤l≤8 and n1,n2,l∈Z.     

Unit groups of quotient rings of complex quadratic rings

For a square-free integer d other than 0 and 1, let K=?(d), where ? is the set of rational numbers. Then K is called a quadratic field and it has degree 2 over ?. For several quadratic fields K=?(d), the ring R_dof integers of K is not a unique-factorization domain. For d<0, there exist only a finite number of complex quadratic fields, whose ring Rd of integers, called complex quadratic ring, is a unique-factorization domain, i.e., d = −1,−2,−3,−7,−11,−19,−43,−67,−163. Let ϑ denote a prime element of Rd, and let n be an arbitrary positive integer. The unit groups of Rd/〈vn〉 was determined by Cross in 1983 for the case d = −1. This paper completely determined the unit groups of Rd/〈vn〉 for the cases d = −2,−3.     

Minimizers of anisotropic Rudin-Osher-Fatemi models

We give the explicit formulas of the minimizers of the anisotropic Rudin-Osher-Fatemi models E1φ(u)=∫Ωφo(Du)dx+λ∫Ω|u−f|dx, u∈BV(Ω),E2φ(u)=∫Ωφo(Du)dx+λ∫Ω(u−f)2dx, u∈BV(Ω), where Ω⊂?2 is a domain, φo is an anisotropic norm on ?2, and f is a solution of the anisotropic 1-Laplacian equations.     

Transcendence of some multivariate power series

We prove some transcendence results for the sums of some multivariate serms of the form ∑j1,j2,...,jm=0 ^∞Cj1j2...jm（r1^j1r2^j2...rm^jm） for n = 1, 2, where Cj1j2...jm are some rational functions of j1 ＋ j2 ＋ ... ＋ jm.     

Sum-connectivity index of a graph

Let G be a simple connected graph, and let di be the degree of its i-th vertex. The sum-connectivity index of the graph G is defined as χ(G)=Σvivj∈E(G)? (di+dj)−1/2. We discuss the effect on χ(G) of inserting an edge into a graph. Moreover, we obtain the relations between sum-connectivity index and Randić index.     

Fourier transform of anisotropic mixed-norm Hardy spaces

Let a:=(a1,…,an)∈[1,∞)n,p:=(p1,…,pn)∈(0,1]n,Hpa(Rⁿ)be the anisotropic mixed-norm Hardy space associated with adefined via the radial maximal function,and let f belong to the Hardy space Hpa(Rⁿ).In this article,we show that the Fourier transform fcoincides with a continuous function g on?n in the sense of tempered distributions and,moreover,this continuous function g,multiplied by a step function associated with a,can be pointwisely controlled by a constant multiple of the Hardy space norm of f.These proofs are achieved via the known atomic characterization of Hpa(Rⁿ)and the establishment of two uniform estimates on anisotropic mixed-norm atoms.As applications,we also conclude a higher order convergence of the continuous function gat the origin.Finally,a variant of the Hardy-Littlewood inequality in the anisotropic mixed-norm Hardy space setting is also obtained.All these results are a natural generalization of the well-known corresponding conclusions of the classical Hardy spaces Hp(Rⁿ)with p∈0,1],and are even new for isotropic mixed-norm Hardy spaces on∈n.     

Subgroups and homology of extensions of centralizers of pro‐p groups

We study the growth of , where U is an open subgroup of and is a special class of pro‐p groups defined in 7 . Furthermore for non‐abelian we prove the core property: for pro‐p subgroups such that H is finitely generated and N is non‐trivial normal in G the index is always finite.     

On the strength of a weak variant of the axiom of counting

In this paper is used to denote Jensen's modification of Quine's ‘new foundations’ set theory () fortified with a type‐level pairing function but without the axiom of choice. The axiom is the variant of the axiom of counting which asserts that no finite set is smaller than its own set of singletons. This paper shows that proves the consistency of the simple theory of types with infinity (). This result implies that proves that consistency of , and that proves the consistency of .     

Distance signless Laplacian eigenvalues of graphs

Suppose that the vertex set of a graph G is V(G)=\{v\mathrm{1},v\mathrm{2},\mathrm{...},vn\}. The transmission Tr(vi) (or D_i) of vertex vi is defined to be the sum of distances from vi to all other vertices. Let Tr(G) be the n\times n diagonal matrix with its (i, i)-entry equal to {{\displaystyle Tr}}_G(vi). The distance signless Laplacian spectral radius of a connected graph G is the spectral radius of the distance signless Laplacian matrix of G, defined as L(G)=Tr(G)+D(G), where D(G) is the distance matrix of G. In this paper, we give a lower bound on the distance signless Laplacian spectral radius of graphs and characterize graphs for which these bounds are best possible. We obtain a lower bound on the second largest distance signless Laplacian eigenvalue of graphs. Moreover, we present lower bounds on the spread of distance signless Laplacian matrix of graphs and trees, and characterize extremal graphs.     

The size-Ramsey number of powers of paths

Given graphs and and a positive integer , say that is -Ramsey for , denoted , if every -coloring of the edges of contains a monochromatic copy of . The size-Ramsey number of a graph is defined to be . Answering a question of Conlon, we prove that, for every fixed , we have , where is the th power of the -vertex path (ie, the graph with vertex set and all edges such that the distance between and in is at most ). Our proof is probabilistic, but can also be made constructive.     

Continuity properties for commutators of multilinear type operators on product of certain Hardy spaces

Similar to the property of a linear Calderdn-Zygmund operator, a linear fractional type operator Is associated with a BMO function b fails to satisfy the continuity from the Hardy space Hp into Lp for p ≤ 1. Thus, an alternative result was given by Y. Ding, S. Lu and P. Zhang, they proved that [b,Iα] is continuous from an atomic Hardy space Hp b into Lp, where Hp b is a subspace of the Hardy space Hp for n/（n ＋ 1） 〈 p ≤ 1. In this paper, we study the commutators of multilinear fractional type operators on product of certain Hardy spaces. The endpoint （Hp1 b1 ×... × HP2, Lp） boundedness for multilinear fractional type operators is obtained. We also give the boundedness for the commutators of multilinear Calderdn-Zygmund operators and multilinear fractional type operators on product of certain Hardy spaces when b ∈ （Lipβ）m（Rn）.     

Dimension of divergence sets for dispersive equation

Consider the generalized dispersive equation defined by{iδtu+Ф(√-△)u=0,(x,t)∈R^n×R,u(x,0)=f(x),F∈F(R^n),(*)whereФ(√-△)is a pseudo-differential operator with symbolФ(|ζ|).In the present paper,assuming thatФsatisfies suitable growth conditions and the initial data in H^s(R^n),we bound the Hausdorff dimension of the sets on which the pointwise convergence of solutions to the dispersive equations(*)fails.These upper bounds of Hausdorff dimension shall be obtained via the Kolmogorov-Seliverstov-Plessner method.     

Limits of Near‐Coloring of Sparse Graphs

Let be nonnegative integers. A graph G is ‐colorable if its vertex set can be partitioned into sets such that the graph induced by has maximum degree at most d for , while the graph induced by is an edgeless graph for . In this article, we give two real‐valued functions and such that any graph with maximum average degree at most is ‐colorable, and there exist non‐‐colorable graphs with average degree at most . Both these functions converge (from below) to when d tends to infinity. This implies that allowing a color to be d‐improper (i.e., of type ) even for a large degree d increases the maximum average degree that guarantees the existence of a valid coloring only by 1. Using a color of type (even with a very large degree d) is somehow less powerful than using two colors of type (two stable sets).     

On the Location of Maxima of Solutions of Schrödinger's Equation

We prove an inequality with applications to solutions of the Schrödinger equation. There is a universal constant c > 0 such that if is simply connected, vanishes on the boundary ∂Ω, and |u| assumes a maximum in , then (1) It was conjectured by Pólya and Szegő (and proven, independently, by Makai and Hayman) that a membrane vibrating at frequency λ contains a disk of size . Our inequality implies a refined result: the point on the membrane that achieves the maximal amplitude is at distance from the boundary. We also give an extension to higher dimensions (generalizing results of Lieb and of Georgiev and Mukherjee): if u solves on with Dirichlet boundary conditions, then the ball B with radius centered at the point in which |u| assumes a maximum is almost fully contained in Ω in the sense that © 2018 Wiley Periodicals, Inc.     

S-semiembedded subgroups of finite groups

A subgroup H of a finite group G is said to be s-semipermutable in G if it is permutable with every Sylow p-subgroup of G with (p, |H|) = 1. We say that a subgroup H of a finite group G is S-semiembedded in G if there exists an s-permutable subgroup T of G such that TH is s-permutable in G and T∩H≤Hs¯G, where Hs¯G is an s-semipermutable subgroup of G contained in H. In this paper, we investigate the influence of S-semiembedded subgroups on the structure of finite groups.