Volume 75, Issue 3

Motivated by studies of oscillator networks, we study the spectrum of the join of several normal matrices with constant row sums. We apply our results to compute the characteristic polynomial of the join of several regular graphs. We then use this theorem to study several problems in spectral graph theory. In particular, we provide some simple constructions of Ramanujan graphs and give new proofs for some theorems in the classical book of Cvetković, Rowlinson, and Slobodan.

With the aim of studying subspaces in pointfree bitopology, we characterize extremal epimorphism in biframes and show that a smallest dense one always exists, providing an analogue of Isbell’s Density Theorem. Further we study the functoriality of assigning to each biframe its lattice of subbilocales and its smallest dense subbilocale.

It is proved that every positive integer can be expressed as a sum of at most n + 2 centered n -gonal numbers. Some specifying and generalizing results are added to this.

A new concept of the 𝔎-additive arithmetical function is introduced, generalizing the notion of the q -additive function. The properties of the 𝔎-additive arithmetical functions and their impact on the uniform distribution of sequences of the weighted sum-of-digits functions of elements of sequences generated by 𝔎-additive arithmetical functions are studied.

We study generalised class groups in dihedral extensions of number fields, focusing both on its p -adic and ℓ -adic properties along fake ℤ p -extensions, where p and ℓ are distinct odd primes. Our results generalise previous works in the (pro)-dihedral setting, with applications to the study of more general fake Iwasawa theories beyond the classical case of standard class groups.

In this paper, we find some new extensions of celebrated singular values inequalities, where convex and concave functions will be utilized repeatedly in this discussion. Applications of the obtained bounds will involve some new sharp bounds for unitarily invariant norms and the numerical radius.

In this paper, we study the class 𝒞 cos of normalized analytic functions f satisfying 1 + z f'' ( z ) / f' ( z ) ≺ cos( z ). We obtain the sharp coefficient bounds and Hankel determinants of second and third order for functions for 𝒞 cos . We also present the similar results for inverse and logarithm coefficients. These results improve the results recently obtained in [Marimuthu et al.: Coefficient estimates for starlike and convex functions associated with cosine function , Hacet. J. Math. Stat. 52 (2023), 596–618. Furthermore, our results provide examples of invariance of the coefficient bounds among the subclass of convex functions.

The purpose of this article is to find many solutions of the canonical q -difference equations for q -Appell functions Φ 2 , Φ 3 , Φ 4 , starting with the first four ones, previously published. Furthermore, q -Laplace integral expressions for two of these functions in the form q -confluent functions are used to find more solutions. In order to explain the discrepancies between meromorphic continuations of multiple q -hypergeometric functions and q -analogues of analytic continuations of corresponding multiple hypergeometric functions, the notion pseudo-meromorphic continuation is introduced to denote equivalence classes of solutions of other q -analogues of systems of partial differential equations. To this end, we introduce equivalence classes of (systems of) partial q -difference equations. Some q -integrals for corresponding q -Appell functions and the second q -Horn function are also proved. In the process, we find a meromorphic continuation of the second q -Horn function. In some formulas, abbreviated notations are used together with combinatorial reasonings. Many proofs use formulas from our first book, which are quoted in detail.

In this paper, a new technique, namely the Dzhumabaev parameterization method, is presented and applied to linear boundary value problem with parameter for delay differential equation to find an efficient algorithm for their approximate solutions. Effectiveness of this algorithm is tested by examples of second-order delay differential equations and linear boundary value problem with parameter for delay differential equations. Obtained results reveal that proposed algorithm of the Dzhumabaev parameterization method is highly efficient and straightforward to execute.

We consider the existence and uniqueness of solutions to the second order iterative boundary value problem u′′(t)=ft,u(t),u[2](t),a≤t≤b $$\begin{equation} u^{\prime \prime}(t)=f\left(t, u(t), u^{[2]}(t)\right), \quad a \leq t \leq b \end{equation}$$ where u [2]( t ) = u ( u ( t )), with solutions satisfying one of the following boundary conditions u ( a ) = a , u ( b ) = b or u ( a ) = b , u ( b ) = a . The Schauder fixed point theorem is used to establish our results.

In this paper, we consider the nonlinear dynamic equation with variable delay (*) yΔ(t)+F(t,y(τ(t)))=0for t∈[t0,∞)T, $$\begin{equation} y^{\Delta}(t)+F\bigl(t,y(\tau(t))\bigr)=0\qquad\text{for}\ \,\,t\in[t_{0},\infty)_{\mathbb{T}}, \end{equation}$$ where 𝕋 is a time scale unbounded above, τ is an rd-continuous delay function and F is rd-continuous in its first component and continuous in its second component. We investigate the global attractivity of the trivial solution of ( ∗ ) by the well-known Lyapunov’s functional method. Our research significantly enhances and expands upon various established results in the literature, presents new results on time scales by defining a new companion function, and offers original perspectives for nonlinear delay dynamic equations on time scales. In addition, we present some illustrative examples on time scales to showcase the applicability of the new results.

In this paper, we investigate the existence, uniqueness and exponential stability of pseudo almost periodic (PAP-) mild solutions of the parabolic-elliptic (P-E) Keller-Segel system on a bounded domain Ω ∈ R n with smooth boundary. First, the well-posedness of the corresponding linear system is established by using the smoothing estimates of the Neumann heat semigroup on Ω. Then, the existence of PAP-mild solution of linear system is done by proving a Massera-type principle. Next, we obtain the well-posedness of such solutions for semilinear system by using the results of linear system and fixed point arguments. The exponential stability is proven by using again the estimates of the Neumann heat semigroup. Finally, we also consider the Keller-Segel system on the real hyperbolic manifolds.

In a recent study, Teymouri et al. [ Derivations into annihilators of the ideals of Banach algebras , Demonstr. Math. 52(1) (2019), 949–958] introduced the notions of AJ $\frac{A}{J}$ weak amenability and quotient ideal amenability for a Banach algebra A relative to a closed two-sided ideal J . They investigated the connection between the AJ $\frac{A}{J}$ -weak amenability of A and the weak amenability of AJ. $\frac{A}{J}.$ However, their primary theorem relied on a flawed conclusion, and Theorem 2.13 in their work includes an incorrect result. In this paper, we present counterexamples to highlight these issues, then refine and establish their main theorem under less restrictive assumptions. Additionally, we provide alternative results and extend the analysis of these concepts to specific classes of Banach algebras, particularly triangular Banach algebras.

This work delves into the characterization of mixed skew Lie and Jordan n -type ∗-derivations within the ∗-algebra 𝒜 , which features a non trivial projection with unit I . In particular, if 𝒜 be an ∗-algebra, then every unital non linear mixed skew Lie and Jordan n -type derivations are additive ∗-derivations. Additionally, we explore the relevance of these derivations in the context of prime ∗-algebras, Von-Neuman algebras, Standard operator algebras.

For any Tychonoff space X let D ( X ) be either the set C ( X ) of all continuous functions on X or the set C ∗ ( X ) of all bounded continuous functions on X . When D ( X ) is endowed with the pointwise convergence topology, we write D p ( X ). Let T : D p ( X ) → D p ( Y ) be a continuous linear surjection, where X is a metrizable space and Y is perfectly normal. We show that if X has some dimensional-like property 𝒫 , then so does Y . For example, 𝒫 could be one of the following properties: zero-dimensionality, countable-dimensionality or strong countable-dimensionality. This result remains true if T is a uniformly continuous and inversely bounded surjection. Also, we consider other properties 𝒫 : of being a scattered space, or a strongly σ -scattered space, or a Δ1-space. Our results strengthen and extend several results from the various recently published papers.

An updated version of the cumulative residual extropy (CRJ) with some dynamic features is introduced in this study as normalized dynamic survival extropy (NDSE). We observe that NDSE grabs attention in the reliability theory also, since NDSE of the relevant random variable in the age replacement model is always equal to the CRJ of that variable. This paper proposes an NDSE based non-parametric test that can be used to see if a data set follows an exponential trend. The proposed test makes itself unique due to its exact distribution of test statistics, scale invariant property, consistency, asymptotic normality, great power and simplicity in calculation. Being aware of the problem of censored observation, we explained how our test will be applied to it. We also provide an extensive power comparison with 5 different tests, each taking different alternatives, and a few real-life examples.

In the paper, in view of the monotonicity rule for the ratio between two Maclaurin power series and by virtue of establishment of a monotonicity result for a sequence involving the ratio between two Bernoulli numbers, the authors investigate the monotonicity of the ratio between two normalized remainders of the Maclaurin power series expansion for the square of the tangent function.