arXiv subjects

Juan D. Vélez

Publications and source records attributed to Juan D. Vélez.

6 recordsLinked to original sources

arXiv2026

Galois-Orbit Structure, Ramanujan Sums, and Stable-Range Collapse for Cyclotomic Cosine Formulas

We study cyclotomic cosine evaluations beyond the fully symmetric setting by combining Galois-orbit structure with finite-frequency expansions. First, for truncation-compatible poly?nomial families over Q, we distinguish invariance of the evaluated value from invariance of the polynomial formula. Summability by a rational function of the level forces eventual rationality of the evaluated values. Conversely, every rational-valued sequence can be realized by a compatible family whose total degree is uniformly bounded by two, showing that value-level Galois invariance alone carries essentially no stable-range rigidity. We then introduce explicit orbit-structured pattern sums indexed by multiplicative configurations. For every fixed pattern, at each sufficiently large admissible level n, its cyclotomic cosine evaluation collapses to an affine function n times S0(r,h) minus 2 to the abs(h) power, where S0 is the zero-frequency coefficient of the associated Laurent polynomial. Consequently, every family uniformly generated from finitely many such patterns by a single polynomial recipe has eventually polynomial evaluation. Finally, decomposing the punctured index set into gcd-orbits yields an exact orbitwise formula in terms of Ramanujan sums. The semiprime and prime-power cases become transparent specializations of this general identity. These results identify both the obstruction to extending symmetric rigidity from value-level Galois invariance and a concrete non-symmetric class exhibiting uniform stable-range collapse

math.NT↗

arXiv2023

Discrete diffusion-type equation on regular graphs and its applications

We derive an explicit formula for the fundamental solution $K_{T_{q+1}}(x,x_{0};t)$ to the discrete-time diffusion equation on the $(q+1)$-regular tree $T_{q+1}$ in terms of the discrete $I$-Bessel function. We then use the formula to derive an explicit expression for the fundamental solution $K_{X}(x,x_{0};t)$ to the discrete-time diffusion equation on any $(q+1)$-regular graph $X$. Going further, we develop three applications. The first one is to derive a general trace formula that relates the spectral data on $X$ to its topological data. Though we emphasize the results in the case when $X$ is finite, our method also applies when $X$ has a countably infinite number of vertices. As a second application, we obtain a closed-form expression for the return time probability distribution of the uniform random walk on any $(q+1)$-regular graph. The expression is obtained by relating $K_{X}(x,x_{0};t)$ to the uniform random walk on a $(q+1)$-regular graph. We then show that if $\{X_{h}\}$ is a sequence of $(q+1)$-regular graphs whose number of vertices goes to infinity and which satisfies a certain natural geometric condition, then the limit of the return time probability distributions from $\{X_{h}\}$ is equal to the return time probability distribution on the tree $T_{q+1}$. As a third application, we derive formulas which express the number of distinct closed irreducible walks without tails on a finite graph $X$ in terms of moments of the spectrum of its adjacency matrix.

math.PR↗

arXiv2023

An integer factorization algorithm which uses diffusion as a computational engine

In this article we develop an algorithm which computes a divisor of an integer $N$, which is assumed to be neither prime nor the power of a prime. The algorithm uses discrete time heat diffusion on a finite graph. If $N$ has $m$ distinct prime factors, then the probability that our algorithm runs successfully is at least $p(m) = 1-(m+1)/2^{m}$. We compute the computational complexity of the algorithm in terms of classical, or digital, steps and in terms of diffusion steps, which is a concept that we define here. As we will discuss below, we assert that a diffusion step can and should be considered as being comparable to a quantum step for an algorithm which runs on a quantum computer. With this, we prove that our factorization algorithm uses at most $O((\log N)^{2})$ deterministic steps and at most $O((\log N)^{2})$ diffusion steps with an implied constant which is effective. By comparison, Shor's algorithm is known to use at most $O((\log N)^{2}\log (\log N) \log (\log \log N))$ quantum steps on a quantum computer. As an example of our algorithm, we simulate the diffusion computer algorithm on a desktop computer and obtain factorizations of $N=33$ and $N=1363$.

quant-ph↗

arXiv2022

On an approach for evaluating certain trigonometric character sums using the discrete time heat kernel

In this article we develop a general method by which one can explicitly evaluate certain sums of $n$-th powers of products of $d\geq 1$ elementary trigonometric functions evaluated at $\mathbf{m}=(m_1,\ldots,m_d)$-th roots of unity. Our approach is to first identify the individual terms in the expression under consideration as eigenvalues of a discrete Laplace operator associated to a graph whose vertices form a $d$-dimensional discrete torus $G_{\mathbf{m}}$ which depends on $\mathbf{m}$. The sums in question are then related to the $n$-th step of a Markov chain on $G_{\mathbf{m}}$. The Markov chain admits the interpretation as a particular random walk, also viewed as a discrete time and discrete space heat diffusion, so then the sum in question is related to special values of the associated heat kernel. Our evaluation follows by deriving a combinatorial expression for the heat kernel, which is obtained by periodizing the heat kernel on the infinite lattice $\mathbb{Z}^{d}$ which covers $G_{\mathbf{m}}$.

math.CO↗

arXiv2021

On the Topological Structure of the (Non-) Finitely-generated Locus of Frobenius Algebras emerging from Stanley-Reisner Rings

In this paper we study initial topological properties of the (non-)finitely-generated locus of Frobenius Algebra coming from Stanley-Reisner rings defined through face ideals. More specifically, we will give a partial answer to a conjecture made by M. Katzman about the openness of the finitely generated locus of such Frobenius algebras. This conjecture can be formulated in a precise manner: Let us define \begin{equation*} U=\{P\in \text{Spec}(R)~:~\mathcal{F}(E_{R_{p}})~\text{is a finitely generated}~R_{p}\text{-algebra}\} \end{equation*} where $\mathcal{F}$ denotes the Frobenius functor and $E_{R_{p}}$ the injective hull of the residual field of the local ring $(R_{p},pR_{p})$. Is the locus $U$ an open set in the Zariski Topology? In the case where $R$ is a ring of the form $R=K[[x_{1},\dots ,x_{n}]]/I,$ where $I\subset R$ is a face ideal, i.e., square-free monomial ideal, we show that the corresponding $U$ has non-empty interior. Even more, we prove that in general $U$ contains two different types of opens sets and that in specific situation its complement contains intersections of opens and closes sets in the Zariski topology. Furthermore, we explicitly verify in some non-trivial examples that $U$ is an non-trivial open set.

math.AC↗

arXiv2016

Limits of quotients of real polynomial functions of three variables

An algorithm for computing the limit of a quotient of bivariate real analytic functions has been developed by one of the authors in (Limits of quotients of bivariate real analytic functions, Journal of Symbolic Computation, 50, 2013, 197 207). In this paper we provide a theoretical method based on the work developed in that article to determine the existence of the limit of a quotient of two polynomial functions of three variables. An algorithm to compute such limits, in the case where the polynomials have rational coefficients, or more generally, coefficients in a real finite extension of the rationals, is also described.

math.AG↗