Searcharxiv⌕ Search

arXiv subjects

Juan D. Velez

Publications and source records attributed to Juan D. Velez.

13 recordsLinked to original sources
arXiv2026

Spectral Selection and Minimal Morse Structures on the Poincaré Dodecahedral Space

We study the long time behavior of the heat equation on the spherical Poincare dodecahedral space and introduce a spectral selection property P, asserting that for a dense open set of initial data, the solution eventually becomes a minimal Morse function. We first establish an obstruction principle. If the first positive eigenspace of the Laplace Beltrami operator contains a Morse function that is not minimal, then property P fails. Using an explicit representation theoretic description of the spherical first eigenspace, we show that the round metric on M violates property P. We then develop a perturbative spectral selection mechanism. Using conformal variations and a finite dimensional reduction of the first-order splitting of the lowest eigenvalue cluster, we construct metrics arbitrarily close to the spherical metric for which the first eigenvalue is simple and the corresponding eigenfunction is minimal Morse with exactly six critical points. As a consequence, these nearby metrics satisfy property P. This establishes both the failure and the restoration of minimal Morse selection on M, and provides a concrete spectral mechanism linking representation theory, eigenvalue splitting, and global Morse structure.

math.DG↗
arXiv2026

Inverse-Limit Formulas and Stable-Range Rigidity for Cyclotomic Sums

We study truncation compatible families F = (F_m)_{m>=1} over Q[z] through an inverse limit formalism, and we evaluate them at the punctured cyclotomic cosine points alpha_{k,n} = cos(2 pi k/n) with the specialization z equals n-1. For symmetric families of uniformly bounded total degree in x <= d, we prove a stable range rigidity theorem: for all n >= d+2, the cosine point evaluation factors through the finitely many punctured cosine power sums the finitely many power sums P1(n) through Pd(n). In the purely polynomial case this implies eventual polynomiality in n. We then extend the framework to include fixed product factors and package their cosine point contribution in multiplicative invariants MQ(n). In the stable range, the bounded degree symmetric part collapses as before; any remaining cyclotomic dependence occurs only through these explicit product terms. Finally, we show that coefficient extraction from such products produces further bounded degree symmetric families, and we apply this to complete symmetric functions h_r evaluated at cosine points.

math.CO↗
arXiv2017

On Preservation Properties and a Special Algebraic Characterization of Some Stronger Forms of the Noetherian Condition

We give an elementary proof prove of the preservation of the Noetherian condition for commutative rings with unity $R$ having at least one finitely generated ideal $I$ such that the quotient ring is again finitely generated, and $R$ is $I-$adically complete. Moreover, we offer as a direct corollary a new elementary proof of the fact that if a ring is Noetherian then the corresponding ring of formal power series in finitely many variables is Noetherian. In addition, we give a counterexample showing that the `completion' condition cannot be avoided on the former theorem. Lastly, we give an elementary characterization of Noetherian commutative rings that can be decomposed as a finite direct product of fields.

math.AC↗
arXiv2017

A General Version of the Nullstellensatz for Arbitrary Fields

We prove a general version of Bezout's form of the Nullstellensatz for arbitrary fields. The corresponding sufficient and necessary condition only involves the local existence of multi-valued roots for each of the polynomials belonging to the ideal in consideration. Finally, this version implies the standard Nullstellensatz when the coefficient field is algebraically closed.

math.AG↗
arXiv2017

Generalizations of the Direct Summand Theorem over UFD-s for some Bigenerated Extensions and an Asymptotic Version of Koh's Conjecture

This article deals with two different problems in commutative algebra. In the first part, we give a proof of generalized forms of the Direct Summand Theorem (DST (or DCS)) for module-finite extension rings of mixed characteristic $R\subset S$ satisfying the following hypotheses: The base ring $R$ is a Unique Factorization Domain of mixed characteristic zero. We assume that $S$ is generated by two elements which satisfy, either radical quadratic equations, or general quadratic equations under certain arithmetical restrictions. In the second part of this article, we discuss an asymptotic version of Koh's Conjecture. We give a model theoretical proof using "non-standard methods".

math.AC↗
arXiv2017

Containment-Division Rings and New Characterizations of Dedekind Domains

We introduce a new class of commutative rings with unity, namely, the Containment-Division Rings (CDR-s). We show that this notion has a very exceptional origin since it was essentially co-discovered with the qualitative help of a computer program (i.e. The Heterogeneous Tool Set (HETS)). Besides, we show that in a Noetherian setting, the CDR-s are just another way of describing Dedekind domains. Simultaneously, we see that for CDR-s, the Noetherian condition can be replaced by a weaker Divisor Chain Condition.

math.AC↗
arXiv2017

Towards an Homological Generalization of the Direct Summand Theorem

We present a more general (parametric-) homological characterization of the Direct Summand Theorem. Specifically, we state two new conjectures: the Socle-Parameter conjecture (SPC) in its weak and strong forms. We give a proof for the week form by showing that it is equivalent to the Direct Summand Conjecture (DSC), now known to be true after the work of Y. André, based on Scholze's theory of perfectoids. Furthermore, we prove the SPC in its strong form for the case when the multiplicity of the parameters is smaller or equal than two. Finally, we present a new proof of the DSC in the equicharacteristic case, based on the techniques thus developed.

math.AC↗
arXiv2017

On Positive-Characteristic Semi-Parametric Local-Uniform Reductions of Varieties over Finitely Generated $\mathbb{Q}$-Algebras

We present a non-standard proof of the fact that the existence of a local (i.e. restricted to a point) characteristic-zero, semi-parametric lifting for a variety defined by the zero locus of polynomial equations over the integers is equivalent to the existence of a collection of local semi-parametric (positive-characteristic) reductions of such variety for almost all primes (i.e. outside a finite set), and such that there exists a global complexity bounding all the corresponding structures involved. Results of this kind are a fundamental tool for transferring theorems in commutative algebra from a characteristic-zero setting to a positive-characteristic one.

math.AC↗
arXiv2017

Global parameter test ideals

This paper shows the existence of ideals whose localizations and completions at prime ideals are parameter test ideals of the localized and completed rings. We do this for Cohen-Macaulay localizations (resp., completions) of non-local rings, for generalized Cohen-Macaulay rings, and for non-local rings with isolated non Cohen-Macaulay points, each being an isolated non $F$-rational point. The tools used to prove this results are constructive in nature and as a consequence our results yield algorithms for the computation of these global parameter test ideals. Finally, we illustrate the power of our methods by analyzing the HSL numbers of local cohomology modules with support at any prime ideal.

math.AC↗
arXiv2015

On the Classification of G-Graded Twisted Algebras over Finite Abelian Groups

Let G be a group and let W be an algebra over a field K. We will say that W is a G-graded twisted algebra if W can be written as a direct sum over the elements of G of one dimensional K-vector spaces. It is also assumed that W has no monomial which is a zero divisor. We also demand that W has a multiplicative identity element. We focus in the case where G is a finite abelian group and the field K is either the real numbers or the complex numbers. In this article, using methods of group cohomology, we classify all associative G-graded twisted algebras in the case G is a finite abelian group. On the other hand, by generalizing some of the arguments developed in (Velez et. al., 2014) we present a classification of all G-graded twisted algebras that satisfy certain symmetry condition.

math.RA↗
arXiv2013

On the classification of G-graded twisted algebras

Let G denote a group and let W be an algebra over a commutative ring R. We will say that W is a G-graded twisted algebra (not necessarily commutative, neither associative) if there exists a G-grading W=\bigoplus_{g \in G}W_{g} where each summand W_{g} is a free rank one R -module, and W has no monomial zero divisors (for each pair of nonzero elements w_{a},w_{b} en W_{a} and W_{b} their product is not zero, w_{a}w_{b}\neq 0). It is also assumed that W has an identity element. In this article, methods of group cohomology are used to study the general problem of classification under graded isomorphisms. We give a full description of these algebras in the associative cases, for complex and real algebras. In the nonassociative case, an analogous result is obtained under a symmetry condition of the corresponding associative function of the algebra, and when the group providing the grading is finite cyclic.

math.RA↗
arXiv2010

Limits of quotients of real analytic functions in two variables

Necessary and sufficient conditions for the existence of limits of the form {equation*} \lim_{(x,y)\rightarrow (a,b)}\frac{f(x,y)}{g(x,y)} {equation*} are given, under the hipothesis that $f$ and $g$ are real analytic functions near the point $(a,b)$, and $g$ has an isolated zero at $(a,b)$. An algorithm (implemented in MAPLE 12) is also provided. This algorithm determines the existence of the limit, and computes it in case it exists. It is shown to be more powerful than the one found in the latest versions of MAPLE. The main tools used throughout are Hensel's Lemma and the theory of Puiseux series.

math.AG↗