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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.
2025
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