Volume 75, Issue 5

The poset of copies of a relational structure 𝕏 is the partial order 〈ℙ(𝕏), ⊂〉, where P(X)={Y⊂X:Y≅X}. $\mathbb{P}(\mathbb{X})=\{Y \subset X: \mathbb{Y} \cong \mathbb{X}\}.$ Investigating the classification of structures related to isomorphism of the Boolean completions 𝔹 𝕏 = ro(sq(ℙ(𝕏))) we extend the results concerning linear orders to the class of structures definable in linear orders by first-order Σ 0 -formulas (monomorphic structures). So, BX≅BL $\mathbb{B}_{\mathbb{X}} \cong \mathbb{B}_{\mathbb{L}}$ holds for some linear order 𝕃, if 𝕏 is definable in a σ -scattered (in particular, countable) or additively indecomposable linear order. For example, BX≅ro(S), $\mathbb{B}_{\mathbb{X}} \cong \operatorname{ro}(\mathbb{S}),$ where 𝕊 is the Sacks forcing, whenever 𝕏 is a non-constant structure chainable by a real order type containing a perfect set.

A group G has the endomorphism kernel property (EKP) if every congruence relation θ on G is the kernel of an endomorphism on G . In this note we fully characterize extraspecial groups which have EKP and show two series of special groups which have EKP, these series are specific semidirect products of types Cp2 n⋊Cp and Cp×Cpn⋊Cp $\left(C_{p^2}{ }^n\right) \rtimes C_p \text { and }\left(C_p \times C_p\right)^n \rtimes C_p$ for all primes p and all n .

The Tribonacci sequence ( T n ) n > 0 is a generalization of the Fibonacci sequence whose first terms are 0, 1, 1 and each term afterwards is the sum of the three preceding terms. The present paper combines Baker’s theory of linear forms in logarithms with a variant of the Baker-Davenport reduction method to determine all sums of two Tribonacci numbers that are close to powers of 2, based on a notion of closeness introduced by Chern and Cui. Specifically, it determines all nonnegative integer solutions ( n , m , a ) to the Diophantine inequality ∣ T n + T m –2 a | < 2 a /2 . This work extends a previous result of Hasanalizade who found all sums of two Fibonacci numbers close to a power of 2.

Let k ≥ 3 be an integer. We prove that the set H $ \mathcal{H} $ of all nonzero hexagonal numbers is a k -additive uniqueness set for the set of multiplicative functions. That is, if a multiplicative function f k satisfies a multivariate Cauchy functional equation fk(x1+x2+⋯+xk)=fk(x1)+fk(x2)+⋯+fk(xk) $$ \begin{equation}f_k(x_1+x_2+\cdots+x_k)=f_k(x_1)+f_k(x_2)+\cdots+f_k(x_k) \end{equation} $$ for arbitrary x1,…,xk∈H $ x_1,\dots,x_k\in \mathcal{H} $ , then f k is the identity function f k ( n ) = n for all n∈N $ n\in\mathbb{N} $ . This extends the work of Kim et al. for k = 2.

Suppose that f ( x ) ∈ ℤ[ x ] is monic and irreducible over ℚ of degree N . We say that f ( x ) is monogenic if { 1 , θ, θ 2 , . . . , θ N− 1 } is a basis for the ring of integers of ℚ( θ ), where f ( θ ) = 0, and we say f ( x ) is cyclic if the Galois group of f ( x ) over ℚ is isomorphic to the cyclic group of order N . In this note, we prove that there do not exist any monogenic even cyclic sextic binomials or trinomials. Although the complete story on monogenic even cyclic sextic quadrinomials remains somewhat of a mystery, we nevertheless determine that the union of three particular infinite sets of cyclic sextic quadrinomials contains exactly four quadrinomials that are monogenic with distinct splitting fields. We also show that the situation can be quite different for quadrinomials whose Galois group is not cyclic.

In the paper, by establishing an integral representation of a specific Maclaurin power series and in light of two monotonicity rules, the authors present very elegant proofs for several basic properties, including the positivity, (absolute) monotonicity, logarithmic convexity, and inequalities, of the normalized remainders of the Maclaurin power series expansion of the exponential function.

The concept of degree independence, in the real number case, introduced in our earlier work is extended to non-archimedean fields. A sufficient condition for such independence is proved. As applications, sufficient conditions for degree independence of (i) elements in the field of formal Laurent series represented by Ruban continued fractions and of (ii) p -adic numbers are derived.

In this note, we study the condition of being one-ended for a particular class of discrete groups, namely groups acting freely and co-compactly on some CAT(0) square complexes. In particular, we furnish an algorithm that associates to a one-ended squared group another square complex which is “smaller” and more tractable than the one given by the definition of the group, in particular with nicer local connectivity conditions.

In this paper, we present novel characterizations of weight functions for Hardy-type dynamic inequalities involving kernels on an arbitrary time scale 𝕋. We establish both necessary and sufficient conditions for a specific class of kernels. As a special case, when 𝕋 = ℕ, our results reduce to the classical discrete inequalities proved by Okpoti et al., while in the continuous case 𝕋 = ℝ, the resulting inequalities are essentially new.

In this paper, we first establish a novel integral identity involving functions of two variables by using the Riemann–Liouville fractional integrals. By taking the modulus of this newly derived identity, we obtain a new form of Newton-type inequality specifically for differentiable co-ordinated convex functions. Moreover, we give example using graph in order to show that our main results are correct. In addition, we derive several new inequalities by employing Hölder’s inequality. Furthermore, we present previously achieved results and new results by using special cases of the obtained theorems. These results not only extend existing inequalities in the literature but also offer new insights into the interplay between fractional calculus and convex analysis.

A normalized analytic function f defined on the open unit disc 𝔻 is called Ma-Minda starlike if the function zf′ / f is subordinate to the function φ. For 0 < λ ≤ 1, let 𝒰(λ) denote the class of functions f(z)=z+∑n=2∞anzn $f(z)=z+\sum\limits_{n=2}^{\infty} a_n z^n$ analytic in the unit disc 𝔻 satisfying the condition |( z/f ( z )) 2 f′ ( z )−1| < λ for z ∈ 𝔻. The class 𝒰 r (λ) consists of functions g defined by g ( z ) = (( z/f ( z )) − 1)/(− a 2 ), where f ∈ 𝒰(λ) and a 2 = f′′ (0)/2 ≠ 0. The radius of Janowski starlikeness and in particular, the radius of starlikeness of order γ are derived. Also, the sharp radii of Ma-Minda starlikeness associated with nephroid, lune, lemniscate of Bernoulli, cardioid, rational function, and so on, are estimated. The inclusion conditions based on λ are derived as specific cases so that functions in the class 𝒰 r (λ) are included in various Ma-Minda starlike classes.

This article aims to determine the sharp bounds for the logarithmic coefficients of inverse functions in several classes of univalent functions. We obtain the sharp bounds of logarithmic coefficients Γ 1 , Γ 2 and Γ 3 of the inverse functions for the classes of starlike and convex functions with respect to symmetric points respectively. In addition, we establish sharp inequalities for the second Hankel determinant H2,1Ff−1/2≤3/44 $\left|H_{2,1}\left(F_{f-1} / 2\right)\right| \leq 3 / 44$ and H2,1Ff−1/2≤23/3264 $\left|H_{2,1}\left(F_{f^{-1}} / 2\right)\right| \leq 23 / 3264$ of logarithmic coefficients of the inverse functions for starlike and convex functions associated with a lune region, where f − 1 is the inverse function of f .

In a recent paper, the first author obtained some q -congruences for truncated 4 ϕ 3 series from Singh’s quadratic transformation. In this paper, by applying Singh’s quadratic transformation again, we give some new q -congruences for truncated 3 ϕ 2 series. We also propose several related conjectures on q -congruences for further study.

In this paper, we consider a new type of dynamical systems called cycled dynamical systems (CDSs) based on the cell growth and division process. Firstly, the existence and uniqueness of cycled solutions of CDSs are studied. Secondly, we present the solutions of the CDSs through an appropriate cycled Cauchy matrix. Moreover, the stability of the linear homogeneous problems, linear perturbation problems and nonlinear problems is analyzed. Next, we investigate the controllability of linear and nonlinear CDSs. Finally, theoretical results are illustrated by providing examples.

This paper investigates the existence, uniqueness, and multiplicity of radially symmetric k -admissible solutions for the k -Hessian equation subject to a homogeneous Dirichlet boundary condition: (P) S k ( D 2 u ) = λ k f ( − u ) in B , u = 0 on ∂ B , $$\left\{\begin{array}{ll} S_k(D^2 u)=\lambda^k f(-u)\qquad&\text{in}\,\,\ B, \\ u=0 &\text{on}\,\,\ \partial B, \end{array}\right. $$ where B = { x ∈ ℝ n : ∣ x ∣ < 1}, λ is a positive parameter, k ∈ {1, …, n }, f : [0, ∞) → [0, ∞) is continuous. In contrast to previous studies, our main results are established under the essential condition that the radially symmetric k -admissible solutions of (P) are generically non-convex. The main tool is the fixed point theorem in cones.

The main purpose of this paper is to study the existence of local and global strong solutions for a diffuse-interface model that describes the dynamics of incompressible two-phase viscous flows with surfactant in 3D whole space. We first establish the local existence of strong solutions by using the mollifier technique. Then, applying pure energy method and the standard continuity argument, one proves the global existence of strong solutions provided that initial data is sufficiently small.

The main aim of this article is to introduce the concept of generalized ( α, T )-contractive mappings in partial b -metric spaces involving a tri-simulation function T and obtain some new coincidence points and common fixed point results for a pair of self mappings utilizing such contractive condition. Our findings extend and unify several important results in the existing literature and also bring some new results as consequences. As an application of this study, we provide a unique solution of an integral equation.

In [Fong, C. K.—Tsui, S. K.: A note on positive operators , J. Operator Theory 5 (1) (1981), 73–76], the authors conjectured that T ∈ 𝔅( ℋ ) is self-adjoint if |T| ≤ |Re T |. After more than 40 years, the validity of the conjecture is still unknown. In this paper, we demonstrate, amongst other results, that the conjecture is true for binormal operators which are either injective, 2-quasinormal, or Im T is compact.

In this paper, Riemannian maps from a CR-submanifold of an almost Hermitian manifold to another almost Hermitian manifold are studied. Such Riemannian maps include the special class of CR-submersions, which are well known in the literature. First, the notion of Riemannian map from CR-submanifold to an almost Hermitian manifold is presented and the invariant of the image spaces of these maps is shown. The harmonicity of these maps is investigated. Also, the relation between the holomorphic sectional curvature of the target manifold and the holomorphic sectional curvature of the CR-submanifold is obtained depending on the second fundamental form of the submanifold and the second fundamental form of the Riemannian map. Finally, the Ricci curvature of the ambient manifold of the CR-submanifold in the direction of the invariant distribution is established in terms of the Ricci curvature of the image space, the Ricci curvature of the anti-invariant distribution, the mean curvature vector fields of the holomorphic distribution and the anti-invariant distribution.

Zero-dimensional structural numbers Z0ind  and Z0dim  $\mathcal{Z}_0^{\text {ind }} \text { and } \mathcal{Z}_0^{\text {dim }}$ w.r.t. dimensions ind and dim were introduced by Georgiou, Hattori, Megaritis, and Sereti. Somewhat similarly, we define structural numbers Sn 𝒜 for different subclasses 𝒜 of the class of hereditarily normal T 1 -spaces. In particular, if ℳ dim denotes the class of metrizable spaces Z with dim Z = 0 ; we show that: for any metrizable space X with dim X = n ≥ 0, we have 1≤SnMdimX≤n+1; $1 \leq \operatorname{Sn}^{\mathcal{M}_{\operatorname{dim}}} X \leq n+1 ;$ for any countable-dimensional metrizable space Y , we have 1≤SnMdimY≤ℵ0. $1 \leq \operatorname{Sn}^{\mathcal{M}_{\operatorname{dim}}} Y \leq \aleph_0 .$

A topological space X is Baire if the intersection of any sequence of open dense subsets of X is dense in X . In this paper, we establish that the property ( κ ) for a Tychonoff space X is equivalent to Baireness of B 1 ( X ), and hence, the Banakh property for C p ( X ) is equivalent to meagerness of B 1 ( X ). Thus, we obtain one characteristic of the Banakh property for C p ( X ) through the property of the space X .

In this paper, we analyze the existence and uniqueness of Stepanov-like weighted pseudo S -asymptotically ( ω, c )-periodic solutions for a class of fractional stochastic differential equations. We clarify the sufficient conditions ensuring the existence of mild solutions and explore their stability properties. The new existence and uniqueness results are provided and an illustrative example is presented to demonstrate the applicability of the established theoretical findings.