This corrigendum contains a corrected proof of Lemma 4 from the paper “On the embedding of Levi-flat hypersurfaces in the complex projective plane (and an appendix with L´aszl´o Lempert)” published in Rev. Roumaine Math. Pures Appl. 68 (2023), 95–114.
Let G be a finite group and let S(G) denote the solvable radical of G (the unique maximal subnormal solvable subgroup of G). In this article we present a chief factor characterization of S(G) that was suggested by the classic chief factor characterization of the Fitting subgroup of G (the unique maximal subnormal nilpotent subgroup of G). We close by extending these two results to a proper normal subgroup of G.
The Equivariant Nash conjecture states that any closed smooth C-manifold (C compact Lie) has an algebraic model. Now, let C be an abelian compact Lie group (respectively, an abelian compact affine Nash group) and M a closed smooth C-manifold (respectively, a closed smooth affine Nash C-manifold). With the aim to know whether the equivariant Nash conjecture is true for M, in this paper, we give some interesting necessary conditions. Moreover, if C is any compact affine Nash group, we find sufficient conditions so that a semialgebraic C-set or a Ck Nash C-manifold has an algebraic model.
Let F be a complete discretely valued field with residue field F, char F =/ 2. We express the u-invariant of F for one quadratic form, u(F), and a pair of quadratic forms, u2(F), in terms of the u-invariants of F for one quadratic form and a pair of quadratic forms, respectively. It is well known that u(F) = 2u(F). We give a new and simpler proof of this result that includes the case char F = 2. It is also known that u2(F) = 2u2(F) when F is a p-adic field. We extend this result to an arbitrary complete discretely valued field F with char F =/ 2. This paper gives a self-contained exposition of this material.
A family of polynomials linked to the set of the deltoid tangents and its associated algebraic hypersurfaces has been presented in recent years. In this paper, we study some related maximizing and free plane curves. We also analyse the bifurcations on polynomial Hamiltonian dynamical systems defined from such a family.
Mourtada and Plenat (2018) give minimal embedded toric resolutions of ADEsingularities in C3 by constructing regular refinements of their dual Newton polyhedrons with the elements of their embedded valuation sets derived from the jet schemes constructed by the first author in 2014. On the other hand, in the works by Aroca et al., the authors represent the Gro & uml;bner fan of a Newton non-degenerate variety and prove that a regular refinement of the Gro & uml;bner fan of such a singularity yields an embedded toric resolution. In this paper, we reconstruct embedded toric resolutions of ADE-singularities. We give the explicit constructions of their Gro & uml;bner fans and refine them using the concept of profile. This provides an alternative and computationally effective framework to the one based on jet schemes.
We consider coupled linear parabolic systems and we establish estimates in L^q-norm for the sources in terms of observations on the corresponding solutions on a part of the boundary. The main tool is a family of Carleman estimates in L^q-norm with boundary observations.
We consider the existence of solutions of the following weighted problem: where (Z L(sigma,xi)u :=g(integral(B) (sigma(x)|del u|(N) +xi(x)|u|(N))dx)[-div(sigma(x)|del u|(N-2)del u)+xi(x)u(N-1)] B is the unit ball of R-N, N > 2, sigma(x) = (log( e/|x| ))(N-1) the singular logarithm weight with the limiting exponent N-1 in the Trudinger-Moser embedding, xi(x) is a positive continuous function. The Kirchoff function g is positive and continuous on (0, +infinity). The nonlinearities are critical or subcritical growth in view of Trudinger-Moser inequalities of double exponential type. We prove the existence of positive solution by using Mountain Pass Theorem. In the critical case, the function of Euler-Lagrange does not fulfil the requirements of Palais-Smale conditions at all levels. We dodge this problem by using adapted test functions to identify this level of compactness.
Classes of analytic functions for which both f and f ' are univalent in the open unit disc E = {z : z < 1} was investigated earlier by Silverman in 1987. However, the application of Gaussian hypergeometric functions on the classes of analytic functions for which both f and f ' are univalent in the open unit disc E is not being studied in the literature. By exploring this, we investigate the necessary and sufficient conditions and inclusion relations for certain function involving Gaussian hypergeometric functions to be in few subclasses of analytic functions for which both f and f ' are univalent in the open unit disc E in this article. Further, we consider an integral operator related to Gaussian hypergeometric functions and several mapping properties are discussed. We also pointed out certain corollaries and consequences of the main results.
Any analytic self-map f of the open unit disk D has a distinguished (so-called Denjoy-Wolff) fixed point zeta(0 )is an element of D at which | f '(zeta(0)) < 1. Any other fixed point zeta of f (if exists) is unimodular and f '(zeta) > 1. C. Cowen and Ch. Pommerenke (1982) established sharp estimates for the series Sigma (1)/(f '(zeta )-1) (over all "non-distinguished" fixed points) in terms of f (zeta(0 )) and f '(zeta(0 )) for the elliptic (| zeta(0)| < 1) and hyperbolic (| zeta(0)| = 1 and f '(zeta(0)) < 1) cases. The parabolic case (| zeta(0)| = 1 and f '(zeta(0)) < 1) was settled by the authors and M. Elin. In this paper, we propose a somewhat different approach that allows to cover all three cases in a unified way. The same approach also applies to a greater class of holomorphic pseudo-contractions and a closely related class of infinitesimal generators, for which similar inequalities for derivatives at fixed points come up with no extra efforts.
This paper aims to discuss a stabilization problem for quasi-linear systems and to study the asymptotic behavior of a distributed system on an evolution domain with a p-Laplace operator in a containing structure of a nanolayer. The epi-convergence method is considered to find the limit problem with interface conditions. This approach consists of studying the stability of the approximate problem associated with our initial problem, then studying the limit behavior in order to determine the stability of the limit problem. The obtained results are numerically tested.
We study the continuity and the boundedness of bilinear mappings between topological vector spaces. We investigate and characterize the compacity of these mappings. As an application, we prove bilinear versions of the Banach-Steinhaus and closed graph theorems in the framework of topological vector spaces.
Let K-3 be a non-normal cubic extension over Q. And let tau(K)(k)(3) (n) denote the k-dimensional divisor function in the number field K-3/Q. In this paper, we investigate the asymptotic behaviour of higher power moments of tau(K)(k)(3) (n) over a certain sparse sequence of positive integers. In a more explicit manner, we consider the asymptotic formula of the following type Sigma(2)(n=a)1+a(2)2+a(2)3+a(2)4+a(2)5+a(2)6 <= x (a1, a2, a3, a4, a5, a6)is an element of Z(6) (tau(K)(k)(3) (n))(& ell;), where k >= 2, & ell; >= 2 are any given positive integers. Furthermore, as an application, we also establish the asymptotic formula of the variance of (tau(K)(k)(3) (n))(& ell;). These results generalize the recent works in this direction.
In this note, we extend results by Denef and Loughran, Skorobogatov, and Smeets concerning the arithmetical surjectivity conjecture of Colliot-Th´el`ene. The question is about giving necessary and sufficient birational conditions for morphisms of varieties to be surjective on local points for almost all localizations of the base field.
This is mainly a small exposition on extensions of valuation rings
We consider ideals in a polynomial ring generated by collections of power sum polynomials, and obtain conditions under which these define complete intersection rings, normal domains, and unique factorization domains. We also settle a key case of a conjecture of Conca, Krattenthaler, and Watanabe, and prove other results in that direction.
We survey old and new results on the existence of moduli spaces of semistable coherent sheaves both in algebraic and in complex geometry.
We compute the Du Bois complexes of abstract cones over singular varieties, and use this to describe the local cohomological dimension and the non-positive K-groups of such cones.
In this paper, we characterize the compact orbifolds, quotients X = D/Γ of a bounded symmetric domain D of tube type by the action of a discontinuous group Γ, as those projective orbifolds with ample canonical divisor possessing a slope zero tensor of “orbifold type”.
We investigate the emptiness of adjoint linear systems associated to successive multiples of a given positive divisor with real coefficients
