Quaestiones Mathematicae

New congruences modulo 9 for the coefficients of Gordon-Mcintosh's mock theta function ξ (q)

Olivia X.M. Yao — Fri, 21 Jun 2024 00:00:00 +0000

In recent years, congruence properties for the coefficients of mock theta functions have been studied by mathematicians. Recently, Silva and Sellers proved some congruences modulo 3 and 9 for the coefficients of the third order mock theta function ξ(q) given by Gordon and McIntosh. Motivated by their work, we prove new congruences modulo 9 for the coefficients of ξ(q) based on the formula for the number of representations of an integer as sums of seven squares in this paper. Those congruences involve infinitely many primes.

Pointwise multipliers between spaces of analytic functions

Daniel Girela, Noel Merchan — Fri, 21 Jun 2024 00:00:00 +0000

A Banach space X of analytic function in D, the unit disc in C, is said to be admissible if it contains the polynomials and convergence in X implies uniform convergence in compact subsets of D.
If X and Y are two admissible Banach spaces of analytic functions in D and g is a holomorphic function in D, g is said to be a multiplier from X to Y if g · f ∈ Y for every f ∈ X. The space of all multipliers from X to Y is denoted M(X; Y ), and M (X) will stand for M (X;X).
The closed graph theorem shows that if g ∈ M (X; Y ) then the multiplication operator M_g, defined by M_g(f) = g · f, is a bounded operator from X into Y .
It is known that M (X) ⊂ H^∞and that if g ∈ M (X), then ∥g∥_H^∞≤ ∥M_g∥. Clearly, this implies that M (X; Y ) ⊂ H^∞if Y ⊂ X. If Y ⊄ X, the inclusion M (X; Y ) ⊂ H^∞may not be true.
In this paper we start presenting a number of conditions on the spaces X and Y which imply that the inclusion M (X; Y ) ⊂ H^∞holds. Next, we concentrate our attention on multipliers acting an BMOA and some related spaces, namely, the Q_s-spaces (0 < s < ∞ ).

Certain results on gradient almost η - Ricci-Bourguignon Soliton

Santu Dey — Fri, 21 Jun 2024 00:00:00 +0000

The present research article deals with the study of almost η -Ricci- Bourguignon soliton and gradient almost η-Ricci-Bourguignon soliton on almost Kenmotsu manifolds. It is shown that if the metric of a Kenmotsu manifold M²ⁿ⁺¹admits a gradient almost η-Ricci-Bourguignon soliton, then it is η-Einstein. More-over, if the manifold is complete and ξ leaves the scalar curvature invariant, then it is locally isometric to Hyperbolic space H²ⁿ⁺¹(-1). Next, we demonstrate that if a (κ; μ) almost Kenmotsu manifold admits an almost η-Ricci-Bourguignon soliton, then the manifold is η -Einstein. Besides, we explore the condition for non-normal almost Kenmotsu manifolds satisfying gradient almost η-Ricci-Bourguignon soliton. In addition, we have also investigated an almost η-Ricci-Bourguignon soliton on (κ; μ) ' -almost Kenmotsu manifold.

On binary matrix properties

Michael Hoefnagel, Pierre-Alain Jacqmin, Zurab Janelidze, Emil van der Walt — Fri, 21 Jun 2024 00:00:00 +0000

The so-called matrix properties of finitely complete categories are a special type of exactness properties that can be encoded as extended matrices of integers. The relation of entailment of matrix properties gives an interesting preorder on the set of such matrices, which can be investigated independently of the categorytheoretic considerations. In this paper, we conduct such investigation and obtain several results dealing with binary matrices, i.e., when the only integer entries in the matrix are 0 or 1. Central to these results is a complete description of the preorder in the case of a special type of binary matrices, which we call diagonal matrices, and which include matrices that define Mal'tsev, majority and arithmetical categories.

On the existence of block-transitive Steiner designs with k divides v

Zheng Huang, Weijun Liu, Lihua Feng — Fri, 21 Jun 2024 00:00:00 +0000

In this paper, we prove that there is no block-transitive 4-(ν; k; 1) designs and 6-(ν; k; 1) designs with k divides ν.

Upper bounds for the numerical radii of powers of Hilbert space operators

Mohammed Al-Dolat, Fuad Kittaneh — Fri, 21 Jun 2024 00:00:00 +0000

We present several upper bounds for the numerical radii of 2X2 operator matrices. We employ these bounds to improve on some known numerical radius inequalities for powers of Hilbert space operators. In particular, we show that if T is a bounded linear operator on a complex Hilbert space, then

w^2r(T) ≤ 1 + ∝ / 4 |||T|^2r+ |T*|^2r|| + 1 + ∝ / 2 w^r(T²) for every r ≥ 1 and ∝ ∈ [0.1] .

This substantially improves on the existing inequality w^2r(T) 1/2 |||T|^2r+ |T*|^2r||. Here w(.) and ||.|| denote the numerical radius and the usual operator norm, respectively.

Existence results for fractional dirichlet problems of p-Laplacian with instantaneous and non-instantaneous impulses on time scales

Xing Hu, Yongkun Li — Fri, 21 Jun 2024 00:00:00 +0000

In this paper, by means of a relation between Riemann-Liouville fractional derivative and Caputo fractional derivative on time scales and with the help of variational methods and critical point theory, we obtain the existence results for a class of fractional Dirichlet problems of p-Laplacian with instantaneous and non-instantaneous impulses on time scales.

Optimal control and feedback control for nonlinear impulsive evolutionary equations

Bin Yin, Biao Zeng — Fri, 21 Jun 2024 00:00:00 +0000

The aim of this paper is to study the existence, optimal control and feed-back control for a class of nonlinear impulsive evolution equations involving weakly continuous operators. We first obtain the existence result of the evolution equation without impulse in any subinterval by Rothe method, and construct a solution of impulsive evolution equation by using this result. Moreover, we obtain the existence of optimal control pairs for the optimal control problem and feasible pairs for the feed-back control system by establishing several sufficient assumptions. An application on a quasistatic frictional contact problem is provided.

Form of the solutions of difference equations via lie symmetry analysis and Fibonacci numbers

Melih Göcen, Mensah Folly-Gbetoula — Fri, 21 Jun 2024 00:00:00 +0000

In this paper, we give the form of the solutions of the following rational difference equation

^Xn + 1 = ^Xn - 3^Xn -1 / ^an^Xn - 3 + bn^Xn - 1

where a_n and b_n are sequences of real numbers, by using Lie symmetry analysis and associated with Fibonacci numbers, respectively. We find an interesting relation between the exact solution of Equation (1) and the classical Fibonacci sequence.

Well-posedness of fractional differential equations with finite delay in complex Banach spaces

Shangquan Bu, Gang Cai — Fri, 21 Jun 2024 00:00:00 +0000

We study the well-posedness of the fractional differential equations with finite delay: D^∝u(t) + BD^βu(t) = Au(t) + Fu_t + f(t); (0 ≤ t ≤ 2π ) on Lebesgue-Bochner spaces L^p(T;X) and periodic Besov spaces B^s_p;_q(T;X), where A and B are closed linear operators in a complex Banach space X satisfying D(A) ⊏ D(B), 0 ≤ β < ∝ , ∝ ≥ 1 and F is a bounded linear operator from L^p([-2π; 0];X) (resp. B^s_p;_q([- 2π; 0];X)) into X. We give necessary and sufficient conditions for the L^p-well- posedness and the B^s_p;_q-well-posedness of above equations by using known operator- valued Fourier multiplier theorems.

A system of two elliptic equations with nonlinear convection and coupled reaction terms

Dragos-Patru Covei — Fri, 21 Jun 2024 00:00:00 +0000

The present paper is concerned with the existence of solutions for a class of elliptic systems involving nonlinearities of the Keller- Osserman type and combined with the convection terms. Firstly, we establish a result involving sub and super-solution for a class of elliptic systems whose nonlinearity may depend of the gradient of the solution. This result permits the study of the existence of blow-up solution for a large class of systems.

Lie derivable maps at unit operator on nest algebras

Kaipeng Li, Lei Liu — Fri, 21 Jun 2024 00:00:00 +0000

Let N be a nontrivial nest on a separable Hilbert space and algN be the associated nest algebra. In this note, we prove that, if N contains an atom of infinite rank or N is uncountable, then the unit operator I is a Lie all-derivable point of algN.

On property-(R₁) and relative Chebyshev centers In Banach spaces-II

Syamantak Das, Tanmoy Paul — Fri, 21 Jun 2024 00:00:00 +0000

We continue to study (strong) property-(R₁) in Banach spaces. As discussed by Pai & Nowroji in [On restricted centers of sets, J. Approx. Theory, 66(2), 170{189 (1991)], this study corresponds to a triplet (X; V;F), where X is a Banach space, V is a closed convex set, and F is a subfamily of closed, bounded subsets of X. It is observed that if X is a Lindenstrauss space then (X;B_X;K(X)) has strong property-(R₁), where K(X) represents the compact subsets of X. It is established that for any F ∈ K(X), Cent_BX(F) ̸= ∅. This extends the well-known fact that a compact subset of a Lindenstrauss space X admits a nonempty Chebyshev center in X. We extend our observation that Cent_BX is Lipschitz continuous in K(X) if X is a Lindenstrauss space. If Y is a subspace of a Banach space X and F represents the set of all finite subsets of B_X then we observe that B_Y exhibits the condition for simultaneously strongly proximinal (viz. property-(P₁)) in X for F ∈ F if (X; Y;F(X)) satisfies strong property-(R₁), where F(X) represents the set of all infite subsets of X. It is demonstrated that if P is a bi-contractive projection in ℓ∞ , then (ℓ∞;Range(P);K(ℓ∞)) exhibits the strong property-(R1), where K(ℓ∞) represents the set of all compact subsets of ℓ∞. Furthermore, stability results for these properties are derived in continuous function spaces, which are then studied for various sums in Banach spaces.