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Abstract: Abstract Metatrivial algebras (i.e., extensions of trivial algebras by trivial algebras) were studied by Agore and Militaru (J Algebra 426:1–31, 2015) for Leibniz algebras and by Militaru (Bull Malays Math Sci Soc 40:1639–1651, 2017) for associative algebras under names “metabelian" Leibniz and “metabelian" associative algebras respectively. We study metatrivial Poisson algebras. We first prove an analogy of Itô’s theorem for Poisson algebras. Then we construct a linear basis for a free metatrivial Poisson algebra. It turns out that such a linear basis depends on the characteristic of the underlying field. Finally, we elaborate the method of Gröbner–Shirshov basis for metatrivial Poisson algebras and show that the word problem for an arbitrary finitely presented metatrivial Poisson algebra is solvable. As a byproduct, we give a method of recognizing automorphisms of a finitely generated free metatrivial Poisson algebra. PubDate: 2023-11-08

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Abstract: Abstract We give two characterisations of when a map-germ admits a 1-parameter stable unfolding, one related to the \({\mathscr {K}}_e\) -codimension and another related to the normal form of a versal unfolding. We then prove that there are infinitely many finitely determined map-germs of multiplicity 4 from \({\mathbb {K}}^3\) to \({\mathbb {K}}^3\) which do not admit a 1-parameter stable unfolding. PubDate: 2023-11-02

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Abstract: Abstract We introduce and study the concept of essential Krull dimension as dual of small Krull dimension of a module, which are defined in the same vein as Krull dimension of a module. In particular, we aim to show that there are some natural relations between the essential Krull dimension and the Krull dimension of an arbitrary module A. We prove that if a module has finite Goldie dimension, then it has Krull dimension if, and only if, it has essential Krull dimension and show that \(\frac{A}{Soc(A) }\) has Krull dimension if, and only if, A has essential Krull dimension. We introduce and study the concepts of non-summand Krull dimension of an R-module over an arbitrary associative ring R as a generalization of essential Krull dimension and small Krull dimension and we prove that an R-module has non-summand Krull dimension if, and only if, it has Krull dimension or is semisimple. PubDate: 2023-10-25

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Abstract: Abstract We compute the cardinality of a set of Galois-invariant isomorphism classes of irreducible rank two \(\overline{{\mathbb {Q}}}_\ell \) -smooth sheaves on \(X_1-S_1\) , where \(X_1\) is a smooth projective absolutely irreducible curve of genus g over a finite field \({\mathbb {F}}_q\) and \(S_1\) is a reduced divisor, with pre-specified tamely ramified monodromy data at S, including precisely one point of principal unipotent monodromy, twisted by a tame character. Equivalently, we compute the number of the corresponding automorphic representations. The approach is based on using an explicit form of the trace formula for \({\text {GL}}(2)\) , extending the work “Counting local systems with tame ramification” to include a Steinberg (= special) component, twisted by a tame character, by employing a pseudo-coefficient thereof. PubDate: 2023-10-17

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Abstract: Abstract Recently, the notion of Lipschitz normally embedded sets introduced by Birbrair and Mostowski has been extensively studied. In the present paper we give a simple ‘effective’ criterion for a pure dimensional complex analytic germ not to be normally embedded. PubDate: 2023-10-13

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Abstract: Abstract This article establishes a method for deriving the spectral representation of an inherently Markov-switching bilinear \((MS{-}BL)\) process. The procedure is based on the application of the Riesz-Fisher theorem, which states that the spectral density can be obtained as the Fourier transform of the covariance function. We provide sufficient conditions for the second-order stationarity of \(MS{-}BL\) models, expressed in terms of the spectral radius of a specific matrix that involves the model’s coefficients. The exact form of the spectral density function demonstrates that it is impossible to distinguish between an \(MS{-}ARMA\) and an \(MS{-}BL\) model solely based on the second-order properties of the process. PubDate: 2023-10-13

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Abstract: Abstract In this work, we consider a fractional-order epidemiological model for computer viruses to study memory effects on population dynamics. This model is derived from a well-known integer-order epidemiological model and the Caputo fractional derivative. Our objective is to provide a rigorous mathematical analysis for dynamics of the fractional-order model. Here, positivity, linear invariant, asymptotic stability properties including local and global asymptotic stability, uniform and Mittag-Leffler stability are established. It is worth noting that the stability properties are investigated by a simple approach, which is based on stability theory for fractional-order dynamical systems and an appropriate linear Lyapunov function. As an important consequence, dynamical properties of the fractional-order model are determined fully. Additionally, a set of numerical experiments is conducted to support the theoretical findings. As we expect, the numerical results are consistent with the theoretical ones. PubDate: 2023-10-09

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Abstract: Abstract We consider planar polynomial differential systems of degree 1 and 2 on the cylinder and we study their limit cycles. We prove that such linear differential systems have at most one limit cycle and that such quadratic differential systems have at most two limit cycles. Moreover such upper bounds are reached. PubDate: 2023-09-27

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Abstract: Abstract In this article, we present a complete classification, with normal forms, of the real algebraic curves under blow-spherical homeomorphisms at infinity. PubDate: 2023-09-26

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Abstract: Abstract We study multi-parameter deformations of isolated singularity function-germs on either a subanalytic set or a complex analytic space. We prove that if such a deformation has no coalescing of singular points, then it has weak constant topological type. This extends some classical results due to Lê and Ramanujam (Am J Math 98:67–78, 1976) and Parusiński (Bull Lond Math Soc 31(6):686–692, 1999), as well as a recent result due to Jesus-Almeida and the first author (Int Math Res Notices 2023(6):4869–4886, 2023). It also provides a sufficient condition for a one-parameter family of complex isolated singularity surfaces in \({\mathbb {C}}^3\) to have weak constant topological type. On the other hand, for complex isolated singularity families defined on an isolated determinantal singularity, we prove that \(\mu\) -constancy implies weak constant topological type. PubDate: 2023-09-25

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Abstract: Abstract The aim of this paper is to study the algebraic structure of the space \(R(\Gamma _{n,m})\) of representations of the torus knot groups, \(\Gamma _{n,m}=\left\langle x,y:x^{n}=y^{m}\right\rangle\) , into the linear special group \(SL(2,{\mathbb {C}})\) . PubDate: 2023-09-22

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Abstract: Abstract In this paper we introduce some subclasses of normalized univalent functions, more exactly subclasses of \(S^\star\) , which we will call convex functions of degree \(n\) . The main tool used is the idea behind Alexander’s duality theorem between \(S^\star\) and \(K\) . Also, we present a necessary and sufficient condition for convexity of degree \(n\) in two complex variables. PubDate: 2023-09-08 DOI: 10.1007/s40863-023-00376-6

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Abstract: Abstract Continuous distributions can be used to characterize risk exposure successfully. It is preferable to use a numerical value, or at the very least, a limited selection of numbers, to show the degree of exposure to a particular threat. These risk exposure figures, often known as major risk indicators, are indisputably a specific model’s output. The risk exposure in the reinsurance revenues data is defined in this study using five important indicators. We create a new XGamma extension specifically for this use. The maximum-likelihood method, maximum product spacing, and least square estimation were used to estimate the parameters. Under a certain set of circumstances and controls, a Monte Carlo simulation study is carried out. Five crucial risk indicators, including value-at-risk, tail-value-at-risk, tail variance, tail mean-variance, and mean excess loss function, were also used to explain the risk exposure in the reinsurance revenue data. These statistical measurements were created for the new model that was provided. PubDate: 2023-09-06 DOI: 10.1007/s40863-023-00373-9

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Abstract: Abstract In this note we show how conjectures and current problems on determinants and eigenvalues of highly structured tridiagonal matrices can be solved using very classical results. PubDate: 2023-08-10 DOI: 10.1007/s40863-023-00372-w

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Abstract: Abstract We study the topology of real analytic maps in a neighborhood of a (possibly non-isolated) critical point. We prove fibration theorems à la Milnor for real analytic maps with non-isolated critical values. Here we study the situation for maps with arbitrary critical set. We use the concept of d-regularity introduced in an earlier paper for maps with an isolated critical value. We prove that this is the key point for the existence of a Milnor fibration on the sphere in the general setting. Plenty of examples are discussed along the text, particularly the interesting family of functions \((f,g):{\mathbb {R}}^n \rightarrow {\mathbb {R}}^2\) of the type $$\begin{aligned} (f,g) = \left( \sum _{i=1}^n a_i x_i^p, \sum _{i=1}^n b_i x_i^q \right) , \end{aligned}$$ where \(a_i, b_i \in {\mathbb {R}}\) are constants in generic position and \(p,q \ge 2\) are integers. PubDate: 2023-08-03 DOI: 10.1007/s40863-023-00370-y

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Abstract: Abstract In this paper, under certain conditions on the mean and scalar curvatures, we prove that there are no strongly stable linear Weingarten closed two-sided hypersurfaces immersed in a certain region determined by a geodesic sphere of the \((n+1)\) -dimensional real projective space \(\mathbb{R}\mathbb{P}^{n+1}\) . We also provide a rigidity result for these hypersurfaces. PubDate: 2023-07-31 DOI: 10.1007/s40863-023-00371-x

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Abstract: Abstract A smooth map of a closed n-dimensional manifold into \(\textbf{R}^p\) with \(1 \le p \le n\) is a special generic map if it has only definite folds as its singularities. We show that for \(1 \le p < n\) and \(n \ge 6\) , a homotopy n-sphere admits a special generic map into \(\textbf{R}^p\) with standard properties if and only if its Gromoll filtration is equal to p. PubDate: 2023-07-03 DOI: 10.1007/s40863-023-00369-5

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Abstract: Abstract We consider a finite-dimensional Jordan superalgebra \(\mathcal {A}\) over a field of characteristic zero \(\mathbb {F}\) such that \(\mathcal {N}\) is the solvable radical of \(\mathcal {A}\) . We proved that if \(\mathcal {N}\,^2=0\) and \(\mathcal {A}/\mathcal {N}\) is isomorphic to simple Jordan superalgebra of Grassmann Poisson bracket \(\mathfrak {K}\textrm{an}(2)\) , then an analogous to Wedderburn Principal Theorem (WPT) holds. PubDate: 2023-06-19 DOI: 10.1007/s40863-023-00367-7

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Abstract: Abstract We prove that Thompson’s group F has a generating set with two elements such that every two powers of them generate a finite index subgroup of F. PubDate: 2023-06-19 DOI: 10.1007/s40863-023-00360-0

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