Let G be a finite group. We prove that if every self-centralizing subgroup of G is nilpotent or subnormal or a TI-subgroup, then every subgroup of G is nilpotent or subnormal. Moreover, G has either a normal Sylow p-subgroup or a normal p-complement for each prime divisor p of |G|.

We consider an asymptotic analysis for series related to the work of Hardy and Littlewood (1923) on Diophantine approximation, as well as Davenport. In particular, we expand on ideas from some previous work on arithmetic series and the RH. To accomplish this, Mellin inversion is applied to certain infinite series over arithmetic functions to apply Cauchy’s residue theorem, and then the remainder of

Let T be a tree. Then a vertex of T with degree one is a leaf of T and a vertex of degree at least three is a branch vertex of T. The set of leaves of T is denoted by L(T) and the set of branch vertices of T is denoted by B(T). For two distinct vertices u, v of T, let PT[u, v] denote the unique path in T connecting u and v. Let T be a tree with B(T) ≠ ∅. For each leaf x of T, let yx denote the nearest

We present sufficient conditions for the existence of pth powers of a quasiho-mogeneous Toeplitz operator Teisθφ, where φ is a radial polynomial function and p, s are natural numbers. A large class of examples is provided to illustrate our results. To our best knowledge those examples are not covered by the current literature. The main tools in the proof of our results are the Mellin transform and

We study compressible isentropic Navier-Stokes-Poisson equations in ℝ3. With some appropriate assumptions on the density, velocity and potential, we show that the classical solution of the Cauchy problem for compressible unipolar isentropic Navier-Stokes-Poisson equations with attractive forcing will blow up in finite time. The proof is based on a contradiction argument, which relies on proving the

Let A be a finite-dimensional k-algebra and K/k be a finite separable field extension. We prove that A is derived equivalent to a hereditary algebra if and only if so is A ⊗kK.

We study the generalized k-connectivity κk(G) as introduced by Hager in 1985, as well as the more recently introduced generalized k-edge-connectivity λk(G). We determine the exact value of κk (G) and λk (G) for the line graphs and total graphs of trees, unicyclic graphs, and also for complete graphs for the case k = 3.

We show that for every p š (1, ∞) there exists a weight w such that the Lorentz Gamma space Γp,w is reflexive, its lower Boyd and Zippin indices are equal to zero and its upper Boyd and Zippin indices are equal to one. As a consequence, the Hardy-Littlewood maximal operator is unbounded on the constructed reflexive space Γp,w and on its associate space \(\Gamma _{p,w}^\prime\).

Suppose that f is a primitive Hecke eigenform or a Mass cusp form for Γ0(N) with normalized eigenvalues λf (n) and let X > 1 be a real number. We consider the sum \({{\cal S}_k}(X): = \sum\limits_{n < X} {\sum\limits_{n = {n_1},{n_2}, \ldots ,{n_k}} {{\lambda _f}({n_1}){\lambda _f}({n_2}) \ldots {\lambda _f}({n_k})}}\) and show that \({{\cal S}_k}(X){\ll _{f,\varepsilon }}{X^{1 - 3/(2(k + 3)) + \varepsilon}}\)

Let \({\Phi _n}(q)\) denote the nth cyclotomic polynomial in q. Recently, Guo, Schlosser and Zudilin proved that for any integer n > 1 with n ≡ 1 (mod 4), $$\sum\limits_{k = 0}^{n - 1} {{{({q^{- 1}};{q^2})_k^2{{({q^{- 2}};{q^4})}_k}} \over {({q^2};{q^2})_k^2{{({q^4};{q^4})}_k}}}} {q^{6k}} \equiv 0\,\,(\bmod \,{\Phi _n}{(q)^2}),$$ where (a; q)m = (1 − a)(1 − aq) … (1 − aqm−1). In this note, we give

Let μ be a finite positive measure on the unit disk and let j ⩾ 1 be an integer. D. Suárez (2015) gave some conditions for a generalized Toeplitz operator \(T_\mu ^{(j)}\) to be bounded or compact. We first give a necessary and sufficient condition for \(T_\mu ^{(j)}\) to be in the Schatten p-class for 1 ⩽ p < ∞ on the Bergman space A2, and then give a sufficient condition for \(T_\mu ^{(j)}\) to be

We show that there exist infinitely many consecutive square-free numbers of the form n2 + 1, n2 + 2. We also establish an asymptotic formula for the number of such square-free pairs when n does not exceed given sufficiently large positive number.

We combine the techniques of sequence spaces and general Orlicz functions that are broader than the classical cases of N-functions. We give three criteria for the weakly compact sets in general Orlicz sequence spaces. One criterion is related to elements of dual spaces. Under the restriction of \(\mathop {\lim }\limits_{u \to 0} M(u)/u = 0\), we propose two other modular types that are convenient to

We give a graph theoretic interpretation of r-Lah numbers, namely, we show that the r-Lah number \({\left\lfloor {\matrix{n \cr k \cr } } \right\rfloor _r}\) counting the number of r-partitions of an (n + r)-element set into k + r ordered blocks is just equal to the number of matchings consisting of n − k edges in the complete bipartite graph with partite sets of cardinality n and n + 2r − 1 (0 ⩽ k

It is well known that people can derive the radiation MHD model from an MHD-P1 approximate model. As pointed out by F. Xie and C. Klingenberg (2018), the uniform regularity estimates play an important role in the convergence from an MHD-P1 approximate model to the radiation MHD model. The aim of this paper is to prove the uniform regularity of strong solutions to an isentropic compressible MHD-P1 approximate

We show the existence of weak solutions in the extended sense of the Cauchy problem for the cubic fourth order nonlinear Schrödinger equation with the initial data u0 ∈ X, where \(X \in \{M_{2,q}^s(\mathbb{R}),\,{H^\sigma}(\mathbb{T}),\,{H^{{s_1}}}(\mathbb{R}) + {H^{{s_2}}}(\mathbb{T})\}\) and q ∈ [1, 2], s ⩾ 0, or σ ⩾ 0, or s2 ⩾ s1 ⩾ 0. Moreover, if M s2,q (ℝ) ↪ L3(ℝ), or if \(\sigma \geqslant {1

We give some characterizations for radial Minkowski additive operators and prove a new characterization of balls. Finally, we show the property of radial Minkowski homomorphism.

Let Dm be an elliptic curve over ℚ of the form y2 = x3 − m2x + m2, where m is an integer. In this paper we prove that the two points P−1 = (−m, m) and P0 = (0, m) on Dm can be extended to a basis for Dm(ℚ) under certain conditions described explicitly.

Tilting theory plays an important role in the representation theory of coalgebras. This paper seeks how to apply the theory of localization and colocalization to tilting torsion theory in the category of comodules. In order to better understand the process, we give the (co)localization for morphisms, (pre)covers and special precovers. For that reason, we investigate the (co)localization in tilting

A natural number n is said to be a (k, r)-integer if n = akb, where k > r > 1 and b is not divisible by the rth power of any prime. We study the distribution of such (k, r)-integers in the Piatetski-Shapiro sequence {⌊nc⌋} with c > 1. As a corollary, we also obtain similar results for semi-r-free integers.

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-quasinormal in K. We analyze how certain properties of SS-supplemented subgroups influence the structure of finite groups. Our results improve and generalize several recent results.

Let K be a field, and let G be a group. In the present paper, we investigate when the group ring K[G] has finite weak dimension and finite Gorenstein weak dimension. We give some analogous versions of Serre’s theorem for the weak dimension and the Gorenstein weak dimension.

We show that if the average number of (nonnormal) Sylow subgroups of a finite group is less than \({{29} \over 4}\) then G is solvable or G/F(G) ≌ A5. This generalizes an earlier result by the third author.

We study Morse-Bott functions with two critical values (equivalently, non-constant without saddles) on closed surfaces. We show that only four surfaces admit such functions (though in higher dimensions, we construct many such manifolds, e.g. as fiber bundles over already constructed manifolds with the same property). We study properties of such functions. Namely, their Reeb graphs are path or cycle

We study the multiplicative lattices L which satisfy the condition a = (a: (a: b))(a: b) for all a, b ∈ L. Call them sharp lattices. We prove that every totally ordered sharp lattice is isomorphic to the ideal lattice of a valuation domain with value group ℤ or ℝ. A sharp lattice L localized at its maximal elements are totally ordered sharp lattices. The converse is true if L has finite character.

We prove that for normal operators N1, \({N_2} \in {\cal L}({\cal H})\), the generalized commutator [N1, N2; X] approaches zero when [N1, N2; [N1, N2; X]] tends to zero in the norm of the Schatten-von Neumann class \({{\cal C}_p}\) with p > 1 and X varies in a bounded set of such a class.

A perturbation of the de Rham complex was introduced by Witten for an exact 1-form Θ and later extended by Novikov for a closed 1-form on a Riemannian manifold M. We use invariance theory to show that the perturbed index density is independent of Θ; this result was established previously by J. A. Álvarez López, Y. A. Kordyukov and E. Leichtnam (2020) using other methods. We also show the higher order

Let K be a number field defined by an irreducible polynomial F(X) ∈ ℤ[X] and ℤK its ring of integers. For every prime integer p, we give sufficient and necessary conditions on F(X) that guarantee the existence of exactly r prime ideals of ℤK lying above p, where \(\overline F (X)\) factors into powers of r monic irreducible polynomials in \({\mathbb{F}_p}[X]\). The given result presents a weaker condition

Let p be an odd prime. By using the elementary methods we prove that: (1) if 2 ∤ x, p = ±3 (mod 8), the Diophantine equation (2x − 1)(py − 1) = 2z2 has no positive integer solution except when p = 3 or p is of the form \(p = 2a_0^2 + 1\), where a0 > 1 is an odd positive integer. (2) if 2 ∤ x, 2 ∣; y, y ≠ 2, 4, then the Diophantine equation (2x − 1)(py − 1) = 2z2 has no positive integer solution.

We investigate signed graphs with just 2 or 3 distinct eigenvalues, mostly in the context of vertex-deleted subgraphs, the join of two signed graphs or association schemes.

Let r(G1, G2) be the Ramsey number of the two graphs G1 and G2. For n1 ⩾ n2 ⩾ 1 let S(n1, n2) be the double star given by \(V\left( {S\left( {{n_1},{n_2}} \right)} \right) = \left\{ {{v_0},{v_1}, \ldots ,{v_{{n_1}}},{w_0},{w_1}, \ldots ,{w_{{n_2}}}} \right\}\) and \(E\left( {S\left( {{n_1},{n_2}} \right)} \right) = \left\{ {{v_0}{v_1}, \ldots ,{v_0}{v_{{n_1}}},{v_0}{w_0},{w_0}{w_1}, \ldots ,{w_0}{w_{{n_2}}}}

We give a novel upper bound on graph energy in terms of the vertex cover number, and present a complete characterization of the graphs whose energy equals twice their matching number.

The main purpose of this paper is to consider a new definition of Hom-left-symmetric bialgebra. The coboundary Hom-left-symmetric bialgebra is also studied. In particular, we give a necessary and sufficient condition that s-matrix is a solution of the Hom-S-equation by a cocycle condition.

Let G be a finite group. If G has two rows which differ in only one entry in the character table, we call G an RD1-group. We investigate the character tables of RD1-groups and get some necessary and sufficient conditions about RD1-groups.

Let \(\mathscr{C}\) be a triangulated category and \(\mathscr{K}\) be a cluster tilting subcategory of \(\mathscr{C}\). Koenig and Zhu showed that the quotient category \(\mathscr{C}\) / \(\mathscr{K}\) is Gorenstein of Gorenstein dimension at most one. But this is not always true when \(\mathscr{C}\) becomes an exact category. The notion of an extriangulated category was introduced by Nakaoka and

We study Hom-Lie superalgebras of Heisenberg type. For 3-dimensional Heisenberg Hom-Lie superalgebras we describe their Hom-Lie super structures, compute the cohomology spaces and characterize their infinitesimal deformations.

We provide combinatorial interpretations for three new classes of partitions, the so-called chromatic partitions. Using only combinatorial arguments, we show that these partition identities resemble well-know ordinary partition identities.

Our aim in this paper is to establish Trudinger’s inequality on Musielak-Orlicz-Morrey spaces LΦ,κ(G) under conditions on Φ which are essentially weaker than those considered in a former paper. As an application and example, we show Trudinger’s inequality for double phase functionals Φ(x, t) = tp(x) + a(x)tq(x), where p(·) and q(·) satisfy log-Hölder conditions and a(·) is nonnegative, bounded and

We provide a construction of monomial ideals in R = K[x, y] such that μ(I2) < μ(I), where μ denotes the least number of generators. This construction generalizes the main result of S. Eliahou, J. Herzog, M. Mohammadi Saem (2018). Working in the ring R, we generalize the definition of a Freiman ideal which was introduced in J. Herzog, G. Zhu (2019) and then we give a complete characterization of such

In 1998, Michael Hirschhorn discovered the 5-dissection formulas of the Rogers-Ramanujan continued fraction R(q) and its reciprocal. We obtain the 5-dissections for functions R(q)R(q2)2 and R(q)2/R(q2), which are essentially Ramanujan’s parameter and its companion. Additionally, 5-dissections of the reciprocals of these two functions are derived. These 5-dissection formulas imply that the coefficients

We identify new classes of structured matrices whose numerical range is of the elliptical type, that is, an elliptical disk or the convex hull of elliptical disks.

Let χ be a nonprincipal Dirichlet character modulo a prime number p ≽ 3 and let \({\mathfrak{a}_{\cal X}}: = {1 \over 2}\left( {1{ - _{\cal X}}\left( { - 1} \right)} \right)\). Define the mean value $${{\cal M}_p}\left( { - s,{\cal X}} \right)\,\,: = {2 \over {p - 1}}\,\sum\limits_{\matrix{ {\psi \,\left( {\bmod \,p} \right)} \cr {\psi \left( { - 1} \right) = - 1} \cr } } {L\,\left( {1,\psi } \right)L\left(

By considering arbitrary mappings ω from a Banach algebra A into the set of all nonempty, compact subsets of the complex plane such that for all a ∈ A, the set ω(a) lies between the boundary and connected hull of the exponential spectrum of a, we create a general framework in which to generalize a number of results involving spectra such as the exponential and singular spectra. In particular, we discover

We study the inverse problem of the determination of a group algebra from the knowledge of its Wedderburn decomposition. We show that a certain class of matrix rings always occur as summands of finite group algebras.

The aim of this paper is to characterize the Lp — Lq boundedness of two classes of integral operators from Lp(\({\cal U}\), dVα) to Lq(\({\cal U}\), dVβ) in terms of the parameters a, b, c, p, q and α, β, where \({\cal U}\) is the Siegel upper half-space. The results in the presented paper generalize a corresponding result given in C. Liu, Y. Liu, P. Hu, L. Zhou (2019).

We construct a family of modular forms from harmonic Maass Jacobi forms by considering their Taylor expansion and using the method of holomorphic projection. As an application we present a certain type Hurwitz class relations which can be viewed as a generalization of Mertens’ result in M. H. Mertens (2016).

We study algebraic properties of two Toeplitz operators on the weighted Bergman space on the unit disk with harmonic symbols. In particular the product property and commutative property are discussed. Further we apply our results to solve a compactness problem of the product of two Hankel operators on the weighted Bergman space on the unit bidisk.

Let N be a sufficiently large integer. We prove that almost all sufficiently large even integers n ∈ [N − 6U, N + 6U] can be represented as $$\left\{ {\matrix{ {n = p_1^2 + p_2^3 + p_3^3+ p_4^3 + p_5^3 + p_6^3,} \hfill \;\;\; {} \hfill \cr {\left| {p_1^2 - {N \over 6}} \right| \leqslant U,\,\,\,\,\,\left| {p_i^3 - {N \over 6}} \right| \leqslant U,\,\,} \hfill \;\;\; {i = 2,3,4,5,6,} \hfill \cr } }

We explore the connection between atomicity in Prüfer domains and their corresponding class groups. We observe that a class group of infinite order is necessary for non-Noetherian almost Dedekind and Prüfer domains of finite character to be atomic. We construct a non-Noetherian almost Dedekind domain and exhibit a generating set for the ideal class semigroup.

We shall describe how to construct a fundamental solution for the Pell equation x2 — my2 = 1 over finite fields of characteristic p ≠ 2. Especially, a complete description of the structure of these fundamental solutions will be given using Chebyshev polynomials. Furthermore, we shall describe the structure of the solutions of the general Pell equation x2 — my2 = n.

An m × n matrix R with nonnegative entries is called row stochastic if the sum of entries on every row of R is 1. Let Mm,n be the set of all m × n real matrices. For A, B ∈ Mm,n, we say that A is row Hadamard majorized by B (denoted by A ≺ RHB) if there exists an m × n row stochastic matrix R such that A = R ο B, where X ο Y is the Hadamard product (entrywise product) of matrices X, Y ∈ Mm,n. In this

In this article we introduce a higher dimensional analogue of Engel structure, motivated by the Cartan prolongation of contact manifolds. We study the stability of such structure, generalizing the Gray-type stability for Engel manifolds.

for each squarefree monomial ideal I ⊂ S = k[x1, …, xn], we associate a simple finite graph GI by using the first linear syzygies of I. The nodes of GI are the generators of I, and two vertices ui and uj are adjacent if there exist variables x, y such that xui = yuj. In the cases, where GI is a cycle or a tree, we show that I has a linear resolution if and only if I has linear quotients and if and

In Theorem 6 of [1], if R is a von Neumann regular ring, then every 2-prime ideal of R is a prime ideal. But the converse of this implication does not hold. Thus, we correct Theorem 6 of [1] as follows:

Let R be a Noetherian ring, and I and J be two ideals of R. Let S be a Serre subcategory of the category of R-modules satisfying the condition CI and M be a ZD-module. As a generalization of the S-depth(I, M) and depth(I, J, M), the S-depth of (I, J) on M is defined as $$S{\rm{ - depth}}\left( {I,J,M} \right) = \inf \left\{ {S{\rm{ - depth}}\left( {\mathfrak{a},M} \right): \mathfrak{a}\in \widetilde{W}\left(

We characterize generalized Douglas-Weyl Randers metrics in terms of their Zermelo navigation data. Then, we study the Randers metrics induced by some important classes of almost contact metrics. Furthermore, we construct a family of generalized Douglas-Weyl Randers metrics which are not R-quadratic. We show that the Randers metric induced by a Kenmotsu manifold is a Douglas metric which is not of

We construct some nondecreasing quantities associated to the first eigenvalue of $$-\Delta_{\varphi}+cR(c\geqslant\frac{1}{2}(n-2)/(n-1))$$ along the Yamabe flow, where Δφ is the Witten-Laplacian operator with a C2 function φ. We also prove a monotonic result on the first eigenvalue of $$-\Delta_{\varphi}+\frac{1}{4}(n/(n-1))R$$ along the Yamabe flow. Moreover, we establish some nondecreasing quantities

Let $H$ denote a finite index subgroup of the modular group $\Gamma$ and let $\rho$ denote a finite-dimensional complex representation of $H.$ Let $M(\rho)$ denote the collection of holomorphic vector-valued modular forms for $\rho$ and let $M(H)$ denote the collection of modular forms on $H$. Then $M(\rho)$ is a $\textbf{Z}$-graded $M(H)$-module. It has been proven that $M(\rho)$ may not be projective

Let $$\mathbb{k}=\mathbb{Q}(\sqrt{2},\sqrt{d})$$ be an imaginary bicyclic biquadratic number field, where d is an odd negative square-free integer and $$\mathbb{k}_{2}^{(2)}$$ its second Hilbert 2-class field. Denote by $$G=\text{Gal}(\mathbb{k}_{2}^{(2)}/\mathbb{k})$$ the Galois group of $$\mathbb{k}_{2}^{(2)}/\mathbb{k}$$ . The purpose of this note is to investigate the Hilbert 2-class field tower

Let $\A$ be a finitary hereditary abelian category. We give a Hall algebra presentation of Kashaev's theorem on the relation between Drinfeld double and Heisenberg double. As applications, we obtain realizations of the Drinfeld double Hall algebra of $\A$ via its derived Hall algebra and Bridgeland Hall algebra of $m$-cyclic complexes.