Let A and B be two non-empty finite subsets of Z. We consider the dilated sumset A+k & centerdot;B = {a+kb : a is an element of A, b is an element of B}, where k is an element of Z(+). In 2010, Cilleruelo et al. proved that | A+ 3 & centerdot; A | >= 4| A| -| A | , where A is finite subset of Z. They proposed the following conjecture. If A is a non-empty finite subset of Z with arbitrarily large cardinality and k is an element of Z(+), then | A+k & centerdot;A| > (k+1)| A| -Gamma(k(2)+2k)/4). In 2009, Cilleruelo et al. proved this conjecture for k = p, with| A | > 3(k-1)(2)(k-1)!. In this article, we prove that this conjecture is true for smaller as well as larger value of | A | with | A| > 6 and k = 5.
The general sum-connectivity index and the general Sombor index have gained significant attention in recent years. In this study, we consider a more generalized form of these indices, denoted by KD alpha,beta For a graph G, this index is defined as KD alpha,beta(G) =Sigma(xy is an element of E(G))(d(G)(x)(beta)+ d(G)(y)(beta))(alpha), where alpha and /3 are nonzero real numbers. This paper establishes an upper bound on KD alpha,beta for chemical trees with given order n and branching vertices b >= 1, where alpha is an element of (0, 1] and /3 > 0. We also characterize the maximal chemical trees. Since the general sum-connectivity index is a special case of KD alpha,beta with /3 = 1, our results generalize the findings of the first Zagreb index in S. Noureen et al. [MATCH Commun. Math. Comput. Chem. 84 (2020), 513-534]. Similarly, since the general Sombor index corresponds to /3 = 2, our results extend the findings for the Sombor index in A. Ali et al. [AIMS Math. 8 (2023), 5369-5390].
We study the metaplectic representation of the Jacobi group (the semi-direct product of the Heisenberg group by SU(1, 1)) by using the complex Weyl correspondence. In particular, we give explicit formulas for the complex Weyl symbols of the metaplectic representation operators and we prove that the complex Weyl correspondence is a Stratonovich-Weyl correspondence for the metaplectic representation.
Let f and g be two distinct normalized primitive holomorphic cusp forms of even integral weights kappa(1) and kappa(2) for the full modular group Gamma = SL(2, Z), and let lambda(f) (n) and lambda(g)(n) denote the n-th normalized Fourier coefficients off and g, respectively. In this paper, we establish the asymptotic formulae for the summatory functions associated to lambda(2 & ell;)(f)(1)f (n)lambda(2 & ell;)(g)(2) (n), with & ell;(1), & ell;(2) >= 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.
