The study of manifolds equipped with special linear connections has become an important topic in contemporary Riemannian geometry. In this work, we investigate Kenmotsu manifolds endowed with a class of semi-symmetric non-metric connections that generalize several previously known connections in the literature. Particular attention is devoted to the geometric behavior of symmetric and skew-symmetric parallel tensor fields associated with such connections. We further examine the conditions under which these manifolds satisfy Ricci-symmetric, weakly symmetric, and weakly Ricci-symmetric properties. In addition, the existence and structure of Ricci solitons related to these non-metric connections are studied within the setting of Kenmotsu geometry.

We consider a compact Riemannian manifold (M, g) of dimension N ≥ 3 and Σ a closed totally geodesic submanifold of dimension 1 ≤ k ≤ N -2, and h : M → R is a continuous function such that the linear operator -Δ g + h is coercive.We study existence of positive solutions u ∈ H 1 (M ) to the following nonlinear PDE with two Hardy-Sobolev critical exponents:where λ is a positive parameter, 0 < s 2 < s 1 < 2, the 2 * si := 2(N -si)N -2 (i = 1, 2) are two critical Hardy-Sobolev exponents and ρ Σ : M → R is the distance function to Σ.In this paper, we give sufficient condition depending on the local geometries of the submanifold Σ and the manifold M , for the existence of mountain pass solution to (0 1).

The main objective of this paper is to characterize a class of lattices, called splittable lattices, in the nilpotent semidirect product Lie group G = R n η R m .Here, η denotes the representation of the abelian Lie group R m in GL n (R) defined by η(t 1 , . . ., t m ) = exp ( m i=1 t i N i ) with (N i ) 1≤i≤m is a set of pairwise commuting nilpotent matrices in Q n×n .As an application we study the case of the nilpotent Lie group F n,m = R n η R m where N i = J i with J = n-1 k=1 E k+1,k where E k+1,k ∈ R n×n denotes the matrix given by the Kronecker delta, namely, E k+1,k i,j = δ i,k+1 δ j,k .

We introduce a hierarchical clustering framework for weighted projective spaces $\mathbb{P}_{\mathbf{q}}$ 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 $d_{F,\mathbb{Q}}([z], [w])$ for points of $\mathbb{P}_{\mathbf{q}}(\mathbb{Q})$. The norm carries a shape parameter $p$: the case $p=2$ is Riemannian and admits a closed-form distance, while $p\neq 2$ is genuinely Finsler, and the metric and clustering guarantees below hold for every $p\in[1,\infty)$. 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 $\mathbb{P}_{\mathbf{q}}$, 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 $\mathbf{q}$ 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.

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.

In this paper, we establish curvature estimates for a class of curvature equation F p (κ) = f (V, ν) for n 2 ≤ 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 Σ that satisfies the above equation.

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+1 4 ⌉.It follows that the Slater index of an indecomposable n-vertex tournament is at least ⌈ n+1 4 ⌉.This led A. Boussaïri 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 ⌈ v(T )+1 4 ⌉ 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.

The generalized forms of Pitt's inequality for the L p -Gabor transform on the groups of the form R n ; R n × 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.

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].