This paper presents a necessary and sufficient condition for a topological vector group to be locally compact. We also introduce several sufficient conditions that ensure the local compactness of topological vector groups. Furthermore, we establish a sufficient condition for a topological vector group to be first countable.
This paper investigates the Maxwell-Boltzmann system in the framework of a Bianchi type III space-time. We conduct a rigorous mathematical analysis of the system, emphasizing its structural properties, functional setting, and energy estimates. By employing a combination of the Faedo-Galerkin method and the standard iteration technique, we establish the local-in-time existence and uniqueness of a solution with good regularity. Our approach carefully integrates the geometric constraints imposed by the Bianchi type III background, ensuring a well-posed formulation of the problem. These results contribute to a broader study of the relativistic kinetic theory in anisotropic cosmological models.
We develop topological partitions for m-to-1 local homeomorphisms on compact metric spaces-maps that arise naturally in non-invertible dynamical systems, such as expanding and covering maps. These partitions enable a symbolic representation of the dynamics via the zip shift, an extended bilateral shift in the non-invertible setting. Inspired by Smale's horseshoe construction, this approach generalizes topological partitions to a broader class of systems and opens new directions for studying their topological and ergodic properties.
In this paper, we establish curvature estimates for a class of curvature equation Fp(,c) = f(V, v) for n2 <= p <= n-1 in the warped product manifolds M . Additionally, by imposing some constraints on the right-hand side function, we also obtain an existence result for the starshaped hypersurface Sigma that satisfies the above equation.
This paper is concerned with the nonlinear strongly damped wave equations involving the fractional Laplacian and regional fractional Laplacian with various boundary conditions. We first prove the existence and uniqueness of weak solutions using the compactness method and weak convergence techniques in Orlicz spaces. Then we study the existence and regularity of global attractors of associated semigroups. The main novelty of the obtained results here is to improve and extend the previous results in [6, 7, A.N. Carvalho and J.W. Cholewa] and [24, J. Shomberg].
The generalized forms of Pitt's inequality for the Lp-Gabor transform on the groups of the form Rn; Rn x K, K being a Lie group of type I, in particular, a connected nilpotent Lie group; Heisenberg motion group and diamond Lie groups have been established.
The Slater index (resp. decomposability index) of a tournament is the minimum number of arcs that must be reversed in that tournament in order to make it a total order (resp. indecomposable (under modular decomposition)). The first author [H. Belkhechine, Decomposability index of tournaments, Discrete Math. 340 (2017) 2986-2994] showed that for every integer n >= 5, the decomposability index of the n-vertex total order equals [n+14 1. It follows that the Slater index of an indecomposable n-vertex tournament is at least [n+14 1. This led A. Boussairi to ask the following question during the thesis defense of the second author on July 2, 2021: what are the indecomposable tournaments T whose Slater index is minimum over all indecomposable tournaments with the same vertex set as T? These tournaments are then the indecomposable tournaments T obtained from a total order by reversing exactly 1v(T )+1 4 1 arcs, where v(T) is the number of vertices of T. In this paper, we characterize such tournaments by means of so-called irreducible pairings. 2020 Mathematics Subject Classification. 05A18, 05C09, 05C20, 05C35, 05C75, 06A05.
We introduce a hierarchical clustering framework for weighted projective spaces ℙ_𝐪 built on Finsler geometry. From an optimization-based Finsler norm that quotients out the weighted scaling action, we construct a scaling-invariant distance d_F([z], [w]) and a rational analogue d_F,ℚ([z], [w]) for points of ℙ_𝐪(ℚ). The norm carries a shape parameter p: the case p=2 is Riemannian and admits a closed-form distance, while p≠ 2 is genuinely Finsler, and the metric and clustering guarantees below hold for every p∈[1,∞). Whereas earlier work measured proximity in these spaces through non-metric dissimilarities, we prove that d_F satisfies the triangle inequality and is therefore a genuine metric; this is what equips the induced clustering with its theoretical guarantees, including monotone dendrograms and Gromov–Hausdorff stability under perturbation of the data. The metric respects the intrinsic scaling symmetry and weighted topology of ℙ_𝐪, avoiding the distortions of a flat-space embedding. We develop the framework's arithmetic applications – clustering rational points in the moduli space of genus two curves and analyzing rational functions in arithmetic dynamics – and indicate prospective extensions to quantum state spaces, where the weights 𝐪 model anisotropic noise. More broadly, the construction offers a rigorous metric foundation for graded neural networks and related machine-learning techniques on graded algebraic varieties.
The relationship between the frame bounds of frames (Gabor) for the space $L^2(\mathbb{R})$ with several generators from the Weyl-Heisenberg group and the scalars linked to the sum of frames is examined in this paper. We give sufficient conditions for the finite sum of frames of the space $L^2(\mathbb{R})$ from the Weyl-Heisenberg group, with explicit frame bounds, in terms of frame bounds and scalars involved in the finite sum of frames, to be a frame for $L^2(\mathbb{R})$. It is shown that if a series of square roots of upper frame bounds of countably infinite frames from the Weyl-Heisenberg group is convergent and some lower frame bound majorizes the sum of all other frame bounds, then the infinite sum of frames for $L^2(\mathbb{R})$ space turns out to be a frame for the space $L^2(\mathbb{R})$. We show that the sum of frames from the Weyl-Heisenberg group and its dual frame always constitutes a frame. We provide sufficient conditions for the sum of images of frames under bounded linear operators acting on $L^2(\mathbb{R})$ in terms of lower bounds of their Hilbert adjoint operator to be a frame. The finite sum of frames where frames are perturbed by bounded sequences of scalars is also discussed. As an application of the results, we show that the frame bounds of sums of frames can increase the rate of approximation in the frame algorithm. Our results are true for all types of frames.
In this paper, we survey valuable results to analyze discrete models frequently appearing in social and natural sciences. We review some well-known results and tools to analyze these systems, trying to make them as practical as possible so that not only mathematicians but also physicists, economists or biologists can use them to explore their models. Applying these methods requires some basic knowledge of computational tools and basic programming. The reviewed topics vary from the local stability of equilibrium points to the characterization of topological and physically observable chaos.
In this paper, we survey valuable results to analyze discrete models frequently appearing in social and natural sciences. We review some well-known results and tools to analyze these systems, trying to make them as practical as possible so that not only mathematicians but also physicists, economists or biologists can use them to explore their models. Applying these methods requires some basic knowledge of computational tools and basic programming. The reviewed topics vary from the local stability of equilibrium points to the characterization of topological and physically observable chaos.
In this paper, having analyzed the previously obtained results devoted to the root vectors series expansion in the Abel-Lidskii sense, we come to the conclusion that the concept can be formulated in the classical terms of the spectral theorem. Though, the spectral theorem for a sectorial operator has not been formulated even in the m-sectorial case, we can consider from this point of view a most simplified case related to the sectorial operator with a discrete spectrum. Thus, in accordance with the terms of the spectral theorem, we naturally arrive at the functional calculus for sectorial operators which is the main focus of this paper. Due to the functional calculus methods, we construct the operator class with the asymptotics more subtle then one of the power type.
Fix x is an element of N, a multiprojective space Y and a very ample line bundle L on Y. We say that (Y, L) satisfies +/- x*-non-defectivity if the s-secant variety of (Y, L) has the expected dimension if either (dim Y + 1)(s + x) <= h(0)(L) or (dim Y + 1)(s-x) >= h(0)(L). Natural examples arise when L is the Segre line bundle and all factors have the same dimension (Abo-Ottaviani-Peterson). We take integers r > 0, t >= 2 and set X := Y x P (R). Let L[t] be the line bundle on X coming from L and O(P)r(t). Under certain assumptions on x, dim Y, h(0)(L), r and t we prove that L[t] is not secant defective. Two of the main results are for r <= 2. In particular we extend a recent result by Ballico, Bernardi and Manduz on the non-defectivity of Segre-Veronese embeddings of multidegree (t1,. .. , tk) of (P-2)k, k >= 3, to the case in which ti = 1 for y > 0 integers i: we require y >= 9.
Let G be a permutation group on a set c with no fixed points in c, and let m be a positive integer. If for each subset T of c the size |Tg-T| is bounded, for g is an element of G, the movement of g is defined as move(g) := max |Tg-T| over all subsets T of c, and move(G) is defined as the maximum of move(g) over all non-identity elements of g is an element of G. Suppose that G is not a 2-group. It was shown by Praeger that |c|<= 2mp p-1 + t-1, where t is the number of G-orbits on c and p is the least odd prime dividing |G|. In this paper, we classify all permutation groups with maximum possible degree |c| = 2mp p-1 + t-1 for t = 2, in which every non-identity element has constant movement m. 2000 Mathematics Subject Classification. 20B05
This paper complements the description of finite-dimensional Jordan symplectic metric superalgebras on algebraically closed fields of characteristic zero. We discuss the graduation of the metric and the symplectic structures and use a new type of generalized double extension by two-dimensional Jordan superalgebras.
In this paper, we study a new class of operators, so called A(n, m)-iso-contra-expansive operators. These new families of operators are considered as a generalization that combines the m-expansive operators as well as mcontractive operators and the classes of (A, m)-expansive and (A, m)-contractive operators and we recover the notion of n-quasi-(A, m)-isometric operators. Some spectral properties of these kind of operators are provided, we derive also a conditions to have the single-valued extension property (SVEP) and we finish by an application of Toeplitz operators on Bergman spaces.
In this paper, we reconsider and slightly generalize various classes of Weyl almost automorphic functions ([29], [33]). More precisely, we consider here various classes of metrically Weyl almost automorphic functions of the form F : ]1n x X- Y and metrically Weyl almost automorphic sequences of the form F : Zn x X- Y, where X and Y are complex Banach spaces. The main structural characterizations for the introduced classes of metrically Weyl almost automorphic functions and sequences are established. In addition to the above, we provide several illustrative examples, useful remarks and applications of the theoretical results.
