Volume 67, Issue 2

In this paper, we study the zero-divisor graphs of a subclass of dismantlable lattices. These graphs are characterized in terms of the non-ancestor graphs of rooted trees.

Let R be a ring with identity and M be a unitary left R -module. The intersection graph of submodules of M , denoted by G ( M ), is defined to be a graph whose vertices are in one to one correspondence with all non-trivial submodules of M and two distinct vertices are adjacent if and only if the corresponding submodules of M have non-zero intersection. In this paper, we consider the intersection graph of submodules of a module. We determine the structure of modules whose clique numbers are finite. We show that if 1 < ω ( G ( M )) < ∞, then M is a direct sum of a finite module and a cyclic module, where ω ( G ( M )) denotes the clique number of G ( M ). We prove that if ω ( G ( M )) is not finite, then M contains an infinite clique. Among other results, it is shown that a Noetherian R -module whose intersection of all non-trivial submodules is non-zero, is Artinian.

The parities of ideal class numbers of compositum of quadratic function fields are studied. Especially, the parities of ideal class numbers of Fq(t)(P,Q) and Fq(t)(P,Q,R) $\mathbb{F}_q(t)(\sqrt P,\sqrt Q) \text{ and } \mathbb{F}_q(t)(\sqrt P,\sqrt Q,\sqrt R)$ are completely determined in detail, where P , Q , R are monic irreducible polynomials of even degrees.

A generalized prime system 𝓟 is a sequence of positive reals p 1 , p 2 , p 3 , … satisfying 1 < p 1 ≤ p 2 ≤ ⋯ ≤ p n ≤ ⋯ and for which p n → ∞ as n → ∞. The { p n } are called generalized primes (or Beurling primes) with the products p1a1⋅ p2a2… pkak ${p^{a_{1}}_{1}}\cdot {p^{a_{2}}_{2}}\dots {p^{a_{k}}_{k}}$ (where k ∈ ℕ and a 1 , a 2 , ⋯, a k ∈ ℕ ∪ {0}) forming the generalized integers (or Beurling integers). In this article we generalise Balanzario’s result [BALANZARIO, E.: An example in Beurling’s theory of primes , Acta Arith. 87 (1998), 121–139] by adapting his method to show that for any 0 < α < 1 there is a continuous g-prime system for which Π P(x)=li(x)+O(xe− (log⁡ x)α ), $$ \Pi_{\mathcal {P}}(x)={\rm li}(x)+ O(x\text{e}^{-(\log x)^\alpha}), $$ (0.1) and NP(x)=ρ x+Ω ± (xe− c(log⁡ x)β ), $$ \mathcal {N}_{\mathcal {P}}(x)= \rho x + \Omega_{\pm}(x\text{e}^{-c(\log x)^\beta}), $$ (0.2) We use the method developed by Diamond, Montgomery and Vorhauer [DIAMOND, H.—MONTGOMERY, H.—VORHAUER, U.: Beurling primes with large oscillation , Math. Ann. 334 (2006), 1–36] and Zhang [ZHANG, W.: Beurling primes with RH and Beurling primes with large oscillation , Math. Ann. 337 (2007), 671–704] to prove (by using some measure theoretical results) that there is a discrete system of Beurling primes satisfying (0.1) and (0.2) which is similar to the continuous system. Finding discrete example is typically more challenging since one cannot control the various growth rates (of π 𝓟 ( x ), 𝓝 𝓟 ( x ) and ζ 𝓟 ( s )) so easily.

Based on the multiplication theorem for Bernoulli polynomials we define the matrix A . We prove that ±det( A ) = 2 m -2 hp− $h_p^-$ .

Let { n k } be a sequence of non-zero real numbers. We prove that the law of the iterated logarithm for discrepancies of the sequence { n k x } is permutational invariant if | n k +1/ n k | → ∞ is satisfied.

We evaluate a class of sums of triple aerated Fibonomial coefficients with a generalized Fibonacci number as coefficient. The technique is to rewrite everything in terms of a variable q and then to use Rothe’s identity from classical q -calculus.

We present new proof of non-bijective Wigner theorem on symmetries of quantum systems using only basic linear algebra. It is based on showing that any non-zero Jordan ∗-homomorphism between matrix algebras preserving rank-one projections is implemented by either a unitary or an anitiunitary map. As a new application we extend hitherto known results on preservers of quantum relative entropy to infinite quantum systems.

A group G is said to be a V -group if every subnormal subgroup of G has only finitely many conjugates. It is proved here that if G is a group admitting an ascending normal series whose factors have finite rank, and all proper subgroups of G have the V -property, then G itself is a V -group, provided that G belongs to a suitable class of generalized soluble groups, containing in particular all locally (soluble-by-finite) groups. On the other hand, an example shows that there exist periodic metabelian minimal non- V groups.

For a natural number n and the prime p , let L be an almost simple group with the socle PSL (2, p n ) such that p does not divide [ L : PSL (2, p n )]. In this paper, we prove that L is uniquely determined by the first column of its character table. In particular, this implies that L is uniquely determined by the structure of its complex group algebra.

We prove that a pair of proximal Klee-Phelps convex groupoids A (∘), B (∘) in a finite-dimensional normed linear space E are normed proximal, i.e., A (∘) δ B (∘) if and only if the groupoids are normed proximal. In addition, we prove that the groupoid neighbourhood N z (∘) ⊆ S z (∘) is convex in E if and only if N z (∘) = S z (∘).

The aim of present article is to introduce and study new subclasses in conic regions. These classes unifies several known classes studied by various well-known authors. Many interesting properties including sufficiency criteria, arc length problem, distortion results are investigated for these newly defined subclasses.

We introduce and characterize interpolation sets in a weak sense for the Lipschitz class in the unit disc of the complex plane. Interpolation sets in the classical sense and in a strong sense for this space have already been examined.

Given a regular plane condenser ( E , F ), let α and β be triangular point tables; α ∈ E , β ∈ F . In the present paper, a criterium relating to the extremal distribution of the tables α and β is proved. The results are provided in terms of generalized Padé approximant, associated with the point tables α and β .

The common exponential functions whose exponents are the scalar products 〈λ, x 〉, where x is a real variable and λ is an integer, admit two generalizations to any higher dimension, the symmetric and the antisymmetric ones [KLIMYK, A.—PATERA, J.: (Anti)symmetric multivariate exponential functions and corresponding Fourier transforms , J. Phys. A: Math. Theor. 40 (2007), 10473–10489]. Restriction in the paper to the three variables only allows us to work out many specific properties of the symmetric and antisymmetric functions useful in applications. Such are (i) the orthogonalities, both the continuous one and the discrete one on the 3 D lattice of any density; (ii) corresponding discrete and continuous Fourier transforms; (iii) generating functions for the related polynomials in three variables, and others. Rapidly increasing precision of the interpolation with increasing density of the 3 D lattice is shown in an example.

In the paper, we investigate a class of four-point integral boundary value problems for the nonlinear coupled system involving higher-order Caputo fractional derivatives and Riemann-Stieltjes integral boundary conditions. By employing Guo-Krasnoselskii fixed point theorem, some sufficient conditions are obtained to guarantee the existence of at least one or two positive solutions for this system. Meanwhile, the eigenvalue intervals of existence for positive solutions are also given. As applications, some examples are provided to illustrate the validity of our main results.

In this paper, six functionals fixed point theorem is used to investigate the existence of at least three positive solutions for a nonlinear n th-order m -point p -Laplacian impulsive boundary value problem. As an application, we give an example to demonstrate our result.

The paper deals with the boundary value problem for differential equation with ϕ -Laplacian and state-dependent impulses of the form ϕ (z′(t))′=f(t,z(t),z′(t)) for a.e. t∈ [0,T]⊂ R,Δ z′(t)=M(z(t),z′(t− )),t=γ (z(t)),z(0)=z(T)=0. $$\begin{array}{} \left(\phi(z'(t))\right)' = f(t,z(t),z'(t))\qquad \text{ for a.e. } t\in [0,T]\subset\mathbb R,\\ \Delta z'(t) = M(z(t),z'(t-)),\qquad t=\gamma (z(t)),\\ z(0) = z(T) = 0. \end{array} $$ Here, T > 0, ϕ : ℝ → ℝ is an increasing homeomorphism, ϕ (ℝ) = ℝ, ϕ (0) = 0, f : [0, T ] × ℝ 2 → ℝ satisfies Carathéodory conditions, M : ℝ → ℝ is continuous and γ : ℝ → (0, T ) is continuous, Δ z ′( t ) = z ′( t +) − z ′( t −). Sufficient conditions for the existence of at least one solution to this problem having no pulsation behaviour are provided.

New oscillation criteria for certain third order nonlinear dynamic equations with a nonlinear damping term are established. Examples to illustrate the results are included.

A result for the existence of homoclinic orbits is obtained for ( q , p )-Laplacian systems.

Semi-equivelar maps are generalizations of maps on the surfaces of Archimedean solids to surfaces other than the 2-sphere. The well known 11 types of normal tilings of the plane suggest the possible types of semi-equivelar maps on the torus and the Klein bottle. In this article we classify (up to isomorphism) semi-equivelar maps on the torus and the Klein bottle with few vertices.

Following Friedlander & Iwaniec [FRIEDLANDER, J. B.—IWANIEC, H.: Summation formulae for coefficients of L-functions, Canad. J. Math. 57 (2005), 494—505], the objective of this note are the coefficients a n of the Dirichlet series for L ( s , χ 1 ) L ( s , χ 2 ) L ( s , χ 3 ) where χ 1 , χ 2 , χ 3 are primitive Dirichlet characters with modules D 1 , D 2 , D 3 . For ∑ n≤ xan $\sum\limits_{n\le x}a_n$ , with x large, sharp asymptotics are established which are uniform in D 1 , D 2 , D 3 . To this end, the modern method for the estimation of exponential sums, due to [HUXLEY, M. N.: Area, Lattice Points, and Exponential Sums, LMS Monographs, New Ser. 13 , University Press, Oxford, 1996] and others, is applied with gain.