Let f and g be two distinct normalized primitive holomorphic cusp forms of even integral weights κ1 and κ2 for the full modular group Γ = SL(2, Z), and let λf (n) and λg(n) denote the n-th normalized Fourier coefficients of f and g, respectively. In this paper, we establish the asymptotic formulae for the summatory functions associated to λ2ℓ1 f (n)λ2ℓ2 g (n), with ℓ1, ℓ2 ≥ 2 being any fixed integers, over two sparse sequences of positive integers, which generalizes the existing results in the literature in this direction.

This paper investigates critical points of the energy functional in sub-Finsler manifolds. Critical points of the energy functional correspond to sub-Finsler geodesics, i.e., horizontal curves that are extremal for the energy, emphasizing their importance in sub-Finsler geometry. Assuming that the sub-Finsler metric is bumpy (in the Morse–Bott sense), the energy functional on the free loop space has the Morse--Bott property, and we establish a direct link between critical points and closed geodesics. We shed light on the existence of at least one closed geodesic on compact sub-Finsler manifolds.

In this paper, we study the relationship between the $k$-th derivative of an entire function and its shift sharing two distinct pairs of small functions. For such a function $f$, a nonzero complex value $c$ and a positive integer $k$, we establish a linear relationship between $f(z+c)$ and $f^{(k)}(z)$ which generalizes a result due to Huang, Deng and Fang [Open Mathematics, 19 (2021), 144--156]. The examples are also given in support of our result.

The purpose of this paper is to investigate an ℓ-ring structure of algebraic integers from an arithmetic point of view. We endow the algebra D of hyperbolic numbers with its standard f-algebra structure [7]. We introduce the ring of hyperbolic integers Zh as a sub f-ring of the ring ZD of integers of D. Next, we prove that Zh is the unique, up to ring isomorphism, Archimedean f-ring of quadratic integers. Our study focuses on arithmetic properties of Zh related to its latticeordered structure. We show that many basic properties of the ring of integers Z such as primes, unique factorization theorem and the notions of floor and ceiling functions can be extended to Zh. A surprising fact is that prime numbers seen as hyperbolic integers are semiprimes. We also obtain some properties of hyperbolic Gaussian integers. As an application, we discuss the Dirichlet divisor problem using hyperbolic intervals.

The recently introduced {k}-packing function problem is considered in this paper. Relationship between cases when k = 1, k ≥ 2 and linear programming relaxation are introduced with sufficient conditions for optimality. For arbitrary simple connected graph

Let Γ(G) denote the set of commutators of a group G, G′ = ⟨Γ(G)⟩ and c(G) (or cw(G)) the minimal number such that every element of G′ can be expressed as a product of at most c(G) commutators. Recently we proved that if G is a finite p-group of maximal cl

We study cyclic quotient singularities given by automorphisms of maximum order on K3 surfaces. In particular, we describe fixed loci of such automorphisms, and provide types of these singularities

"The general Randi´c index of a graph G is defined as Rα(G) = P uv∈E(G)(dudv)α, where du and dv denote the degrees of the vertices u and v, respectively, α is a real number, and E(G) is the edge set of G. The minimum number of edges of a graph G whose removal makes G as acyclic is known as the cyclomatic number and it is usually denoted by ν. A graph with the maximum degree at most 4 is known as a chemical graph. For ν = 0, 1, 2 and α > 1, the problem of finding graph(s) with the minimum general Randi´c index Rα among all n-vertex chemical graphs with the cyclomatic number ν has already been solved. In this paper, this problem is solved for the case when ν ≥ 3, n ≥ 5(ν − 1), and 1 < α < α0, where α0 ≈ 11.4496 is the unique positive root of the equation 4(8α − 6α) + 4α − 9α = 0."

"Let Mn be an n-dimensional (n ≥ 3) complete smooth connected and oriented hypersurface in a real space form Mn+1(c) (c = 0, 1, −1) with constant quasiGauss-Kronecker curvature and two distinct principal curvatures. Denoting by H the mean curvature, |A| 2 the squared norm of the second fundamental form and Kq the quasi-Gauss-Kronecker curvature of Mn , we obtain some characterizations of S k (a)×Rn−k or S k (a)×S n−k ( √ 1 − a 2) or S k (a)×Hn−k (− √ 1 + a 2) in terms of H, |A| 2 and Kq, where 1 ≤ k ≤ n−1 and S k (a) is the k-dimensional sphere with radius a"