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Abstract: Abstract Can one tell if an ideal is radical just by looking at the degrees of the generators' In general, this is hopeless. However, there are special collections of degrees in multigraded polynomial rings, with the property that any multigraded ideal generated by elements of those degrees is radical. We call such a collection of degrees a radical support. In this paper, we give a combinatorial characterization of radical supports. Our characterization is in terms of properties of cycles in an associated labelled graph. We also show that the notion of radical support is closely related to that of Cartwright-Sturmfels ideals. In fact, any ideal generated by multigraded generators whose multidegrees form a radical support is a Cartwright-Sturmfels ideal. Conversely, a collection of degrees such that any multigraded ideal generated by elements of those degrees is Cartwright-Sturmfels is a radical support. PubDate: 2022-08-05

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Abstract: Abstract We survey the computation of polytope volumes by the algorithms of Normaliz to which the Lawrence algorithm has recently been added. It has enabled us to master volume computations for polytopes from social choice in dimension 119. This challenge required a sophisticated implementation of the Lawrence algorithm. PubDate: 2022-07-25

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Abstract: Abstract In this article, we obtain some new modular equations involving \(k(q) = \chi (q) \chi (-q^2)\) which is a special case of general continued fraction in Ramanujan’s lost notebook. Also, we obtain 2 and 4-dissection of k(q) and record some applications to partitions. PubDate: 2022-07-25

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Abstract: Abstract In this paper we consider the Hermitian curve \(y^{q_0} + y = x^{q_0+1}\) over the field \({\mathbb {F}}_q (q:=q_0^2)\) . The automorphism group of this curve is known to be the projective unitary group \(\text {PGU}(3,q)=\mathrm{{ ^2A_2}}(q)\) with \(q_0^3(q_0^3+1)(q_0^2-1)\) elements. We follow the construction done for the Suzuki code in Eid et al. (Designs Codes Cryptogr 81(3):413–425, 2016, https://doi.org/10.1007/s10623-015-0164-5). We construct algebraic geometry codes over \({\mathbb {F}}_{q^3}\) from an \(\mathrm{{ ^2A_2}}(q)\) -invariant divisor D, give an explicit basis for the Riemann–Roch space \(L(\ell D)\) for \(0 < \ell \le q_0^3-1\) . These families of codes have good parameters and information rate close to one. In addition, they are explicitly constructed. The dual codes of these families are of the same kind if \(q_0^3-2g+1 \le \ell \le q_0^3-1\) . PubDate: 2022-07-13

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Abstract: Abstract In this paper, we study the number of limit cycles that can bifurcate from a period annulus in discontinuous planar piecewise linear Hamiltonian differential system with three zones separated by two parallel straight lines. More precisely, we consider the case where the period annulus, bounded by a heteroclinic orbit or homoclinic loop, is obtained from a real center of the central subsystem, i.e. the system defined between the two parallel lines, and two real saddles of the others subsystems. Denoting by H(n) the number of limit cycles that can bifurcate from this period annulus by polynomial perturbations of degree n, we prove that if the period annulus is bounded by a heteroclinic orbit then \(H(1)\ge 2\) , \(H(2)\ge 3\) and \(H(3)\ge 5\) . Now, if the period annulus is bounded by a homoclinic loop then \(H(1)\ge 3\) , \(H(2)\ge 4\) and \(H(3)\ge 7\) . For this, we study the number of zeros of a Melnikov function for piecewise Hamiltonian system. PubDate: 2022-07-11

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Abstract: Abstract We consider q-matroids and their associated classical matroids derived from Gabidulin rank-metric codes. We express the generalized rank weights of a Gabidulin rank-metric code in terms of Betti numbers of the dual classical matroid associated to the q-matroid corresponding to the code. In our main result, we show how these Betti numbers and their elongations determine the generalized weight polynomials for q-matroids, in particular, for the Gabidulin rank-metric codes. In addition, we demonstrate how the weight distribution and higher weight spectra of such codes can be determined directly from the associated q-matroids by using Möbius functions of its lattice of q-flats. PubDate: 2022-07-07

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Abstract: Abstract In this work a linear complementary dual code associated to the Haar wavelet transform over the finite field \({\mathbb {Z}}_p\) for certain values for p is given. Examples are presented illustrating the results. PubDate: 2022-06-28

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Abstract: Abstract In coding theory, super- \(\lambda\) graphs were used to build linear codes. Thus, in order to see whether the zero-divisor graphs might be useful into this context, it is natural to study when zero-divisor graphs of some non elementary ring constructions are super- \(\lambda\) graphs. In this paper, using the finite direct product of finite fields, the ring of the residues, and the trivial extension of rings by a module, we show that there are various classes of rings whose zero-divisor graphs are super- \(\lambda\) . We apply these results to determine parameters of some linear codes associated to zero-divisor graphs. PubDate: 2022-06-23

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Abstract: Abstract We give a new family of planar polynomial differential systems of degree seven with a non elementary point at the origin. We show the integrability of this family by transforming it into a Riccati equation. We determine sufficient conditions for the coexistence of algebraic and non-algebraic limit cycle surrounding this non elementary point. Moreover these limits cycles are explicitly given. An example is given and its phase portrait is drawn as an illustration of our result. PubDate: 2022-06-21

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Abstract: Abstract Determining if an integer is a quadratic residue with a composite modulo that has a known prime factorization is usually a matter of applying the Chinese Remainder Theorem. Conversely, in this paper, we report a direct criterion for the solubility of a quadratic congruence. In this paper we take the modulo to be restricted in such way that to the modulo in which the number of quadratic residues is odd, in other words, it is decomposed only into prime factors congruent to 3 modulo 4 and factors of two limited to third power. This condition has a complexity of \(O((\log n )^3)\) bit operations when given the number of quadratic residue modulo n. PubDate: 2022-06-20

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Abstract: Abstract In this work we present a class of locally recoverable codes, i.e. codes where an erasure at a position P of a codeword may be recovered from the knowledge of the entries in the positions of a recovery set \(R_P\) . The codes in the class that we define have availability, meaning that for each position P there are several distinct recovery sets. Also, the entry at position P may be recovered even in the presence of erasures in some of the positions of the recovery sets, and the number of supported erasures may vary among the various recovery sets. PubDate: 2022-06-09

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Abstract: Abstract This article is focused on modular equation of degree 5 of composite degrees and further as an application of this, we evaluate theta function identities. PubDate: 2022-06-06

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Abstract: Abstract We compute all the simply connected homogeneous and infinitesimally homogeneous surfaces admitting one or more invariant affine connections. We find exactly six non equivalent simply connected homogeneous surfaces admitting more than one invariant connections and four classes of simply connected homogeneous surfaces admitting exactly one invariant connection. PubDate: 2022-05-09 DOI: 10.1007/s40863-022-00306-y

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Abstract: Abstract This article gives a survey of the e-value, a statistical significance measure a.k.a. the evidence rendered by observational data, X, in support of a statistical hypothesis, H, or, the other way around, the epistemic value of H given X. The e-value and the accompanying FBST, the Full Bayesian Significance Test, constitute the core of a research program that was started at IME-USP, is being developed by over 20 researchers worldwide, and has, so far, been referenced by over 200 publications. The e-value and the FBST comply with the best principles of Bayesian inference, including the likelihood principle, complete invariance, asymptotic consistency, etc. Furthermore, they exhibit powerful logic or algebraic properties in situations where one needs to compare or compose distinct hypotheses that can be formulated either in the same or in different statistical models. Moreover, they effortlessly accommodate the case of sharp or precise hypotheses, a situation where alternative methods often require ad hoc and convoluted procedures. Finally, the FBST has outstanding robustness and reliability characteristics, outperforming traditional tests of hypotheses in many practical applications of statistical modeling and operations research. PubDate: 2022-05-01 DOI: 10.1007/s40863-020-00171-7

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Abstract: Abstract In this paper we survey the estimation of nonparametric regression models using wavelets, under different conditions on the innovations, on the predictor variables, on the function spaces involved and on the regularity conditions imposed. We begin with the seminal works of Donoho and co-authors, in the regular fixed design and independent and identically Gaussian noise and move towards non-regular designs, random designs and correlated errors. PubDate: 2022-05-01 DOI: 10.1007/s40863-021-00240-5

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Abstract: Abstract The purpose of this manuscript associated to the Golden Jubilee of the IME-USP is to present selected material from the author’s scientific contribution dealing with nonlinear phenomena of dispersive type. That study has been a source for modern research in the dynamic of traveling wave solutions of different type and it has been disseminated through publication of scientific articles and/or books. Here, we provide the reader with information about current research in stability theory for nonlinear dispersive equations and possible developments. Also, I hope it inspires future developments in this important and fascinating subject. In this manuscript we consider the following topics: stability theory of solitary waves and the applicability of the concentration–compactness principle, the existence and orbital (in)stability of periodic traveling wave solutions for nonlinear dispersive models, nonlinear Schrödinger and Korteweg–de Vries models on star-shaped metric graphs. The use of tools of the theory of spaces of Hilbert, the spectral theory for unbounded self-adjoint operators, Sturm–Liouville’s theory, variational methods, analytic perturbation theory of operators, and the extension theory of symmetric operators are pieces fundamental in our study. The methods presented in this manuscript have prospect for the study of the dynamic of solutions for nonlinear evolution equations around of different traveling waves profiles which may appear in non-standard environments such as star-shaped metric graphs. PubDate: 2022-05-01 DOI: 10.1007/s40863-020-00195-z

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Abstract: Abstract In this paper we present a generalization of the theory of isoparametric families of hypersurfaces in a Riemannian manifold of constant curvature, in that we consider families of submanifolds of codimension greater than one. Our starting point is a definition of isoparametric family of submanifolds which coincides with the classical definition formulated by Cartan (Ann Mat 17:177–191, 1938) in the case of codimension 1. That definition is given in Sect. 1, where we also show that the mean curvature vector of the submanifolds that belong to an isoparametric family has constant length (cf. Proposition 1.4). In Sect. 2, we study the isoparametric families whose orthogonal distribution is integrable. In this case we prove that each leaf of the isoparametric family has flat normal bundle, the integral manifolds of the orthogonal distribution are totally geodesic and that the mean curvature vector of each submanifold in the isoparametric family is parallel with respect to the normal connection (Proposition 2.3). In Sect. 3 we study the parallel submanifolds. In this way we obtain an effective procedure to construct normal isoparametric families (i.e. whose orthogonal distributions are integrable) which, however, is not explored in the present paper (cf. Proposition 3.5). In Sect. 4 we show that the leaves of an isoparametric family of submanifolds in a Riemannian manifold of constant curvature have constant principal curvatures. Finally, in Sect. 5, we obtain a relation between these principal curvatures which generalizes the fundamental formula of Cartan to higher codimensions. PubDate: 2022-04-12 DOI: 10.1007/s40863-022-00295-y