Advances in Pure and Applied Mathematics

APAM - ISSN 1869-6090 - © ISTE Ltd

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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.

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.

The generalized forms of Pitt’s inequality for the $$$L^p$$$-Gabor transform on the groups of the form $$$ℝ^n; ℝ^n \times K, K$$$ 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 $$$n \geq 5$$$, the decomposability index of the $$$n$$$-vertex total order equals $$$\left\lceil \frac{n+1}{4} \right\rceil$$$. It follows that the Slater index of an indecomposable $$$n$$$-vertex tournament is at least $$$\left\lceil \frac{n+1}{4} \right\rceil$$$. 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 $$$\left\lceil \frac{v(T)+1}{4} \right\rceil$$$ 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.

In this paper, we establish curvature estimates for a class of curvature equation $$$\mathcal{F}_{p}(\kappa)=f(V,\nu) for \frac{n}{2} \leq p \leq n-1$$$ in the warped product manifolds $$$\bar{M}$$$. 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.

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 $$$x\in \mathbb{N}$$$, a multiprojective space $$$Y$$$ and a very ample line bundle $$$L$$$ on $$$Y$$$ . We say that $$$(Y,L)$$$ satisfies $$$\pm{x}\star$$$-non-defectivity if the $$$s$$$-secant variety of $$$(Y, L)$$$ has the expected dimension if either $$$(\dim Y+1)(s+x)\le h^0(L)$$$ or $$$(\dim Y +1)(s-x)\ge h^0(L)$$$. Natural examples arise when $$$L$$$ is the Segre line bundle and all factors have the same dimension (Abo - Ottaviani - Peterson). We take integers $$$r > 0$$$, $$$t\ge 2$$$ and set $$$X:= Y\times \mathbb{P}^r$$$. Let $$$L[t]$$$ be the line bundle on $$$X$$$ coming from $$$L$$$ and $$$\mathcal{O}_{\mathbb{P}^r}(t)$$$. 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 $$$r\le 2$$$. In particular we extend a recent result by Ballico, Bernardi and Mańduz on the non-defectivity of Segre-Veronese embeddings of multidegree $$$(t_1,\dots ,t_k)$$$ of $$$(\mathbb{P}^2)^k$$$, $$$k\ge 3$$$, to the case in which $$$t_i=1$$$ for $$$y > 0$$$ integers $$$i:$$$ we require $$$y\ge 9$$$.

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.