This paper answers the following two questions: What are the easiest polynomial differential systems in R3 having an invariant hyperbolic, parabolic or elliptic cylinder?, and for such polynomial differential systems what are their phase portraits on such invariant cylinders?

This article focuses on the study of Lorentzian para-Sasakian manifolds Mn . It demonstrates that a W9-semisymmetric Lorentzian para-Sasakian manifold is a W9-flat manifold. Additionally, we explore Lorentzian para-Sasakian manifolds that satisfy the ζ-W9-flat condition, revealing that they represent a special type of η-Einstein manifold. Furthermore, it is shown that a W9-flat Lorentzian para-Sasakian manifold is a flat manifold. We also investigate Lorentzian para-Sasakian manifolds that meet W9-recurrent and ϕ-W9-semisymmetric conditions, presenting several significant results from this analysis. At last, we explore η-Ricci Solitons on Lorentzian para-Sasakian manifold satisfying W9(ζ, F1 ) · S = 0.

Let d ≥ 2. In this paper we prove that Id(ℓ, a) fills Rd face–to–face by translations. We prove that the symmetry group of Id(ℓ, a) contains the product of cyclic groups Cd × C2 as a subgroup. We compute the Lebesgue j–volume (i.e., the sum of the Lebesgue j–measures of the j–faces) of Id(ℓ, a), for 1 ≤ j < d. We compute the incidence numbers (as defined by Grünbaum) of the faces of Id(ℓ, a).

In this paper, we will prove a spectral theorem for self-adjoint compactoid operators. Also, we will study the condition on which the coefficient field must be imposed. In order to get the theorems, we will use the Fredholm theory for compactoid operators. Moreover, the property of maximal complete field is important for our study. These facts will allow us to find that the spectral theorem depends only on the residue class field, and is independent of the valuation group of the coefficient field. As a result, we can settle the problem of the spectral theorem in the case where the residue class field is formally real.

In this article we study the n-homogeneous polynomials P that are c-continuous on bounded subsets of l1 . We show that P can be decomposed in the form R + Q, where Q and R are n-homogeneous polynomials, with R weakly star continuous and Q (x) = 0 for all x ∈ ker u for u = (1, 1, . . . , 1, . . . ). We conclude that P = Σ un−j ⊗ Rj , where R is a weakly star continuous j-homogeneous polynomial for j = 0, 1, . . . , n.

In this paper, we are interested in the study of certain operators in non-Archimedean normed spaces of finite dimension. We introduce the notion of p-delta function, then we characterize the simple operators, the similarities and the expansions. We show if E has an orthogonal basis, then each injective operator on E is the composition of an isometry and an expansion.

We address a question by Henry Towsner about the possibility of representing linear operators between ultraproducts of Banach spaces by means of ultraproducts of nonlinear maps.We provide a bridge between these “accessible” operators and the theory of twisted sums through the so-called quasilinear maps. Thus, for many pairs of Banach spaces X and Y , there is an “accessible” operator XU → YU that is not the ultraproduct of a family of operators X → Y if and only if there is a short exact sequence of quasi-Banach spaces and operators 0 → Y → Z → X → 0 that does not split. We then adapt classical work by Ribe and Kalton–Peck to exhibit pretty concrete examples of accessible functionals and endomorphisms for the sequence spaces lp The paper is organized so that the main ideas are accessible to readers working on ultraproducts and requires only a rustic knowledge of Banach space theory.

We present a new characterization of Q-point ultrafilters and use it to optimize the result of Avilés, Martı́nez-Cervantes, and Rueda Zoca linking the existence of L-orthogonal sequences and L-orthogonal elements in Banach spaces via ultrafilter limits.

This is a survey of known results and still open problems on characteristic properties of classes of homothetic convex sets in the n-dimensional Euclidean space. These properties are formulated in terms of orthogonal projections, plane sections, homothety classes, Choquet simplices, and homothetic tilings and partitions.

A theorem of Arens and Calderon states that if A is a commutative Banach algebra with Jacobson radical Rad(A), and if a0 , . . . , an∈ A with a0 ∈ Rad(A) and a1 an invertible element of k A, then there exists y ∈ Rad(A) such that Σ ak yk = 0. In this paper, we give extensions of this result to commutative non-normed topological algebras, as this is vital for extending an embedding theorem of Allan in [2] regarding the embedding of the formal power series algebra C[[X]] into a commutative Banach algebra.