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Carlos A. Cadavid

Publications and source records attributed to Carlos A. Cadavid.

9 recordsLinked to original sources
arXiv2026

Eventual Morse Minimality of Heat Flow on Lens Spaces

We prove that heat flow generically produces Morse-minimal functions on round lens spaces. The result gives a rigorous explanation of a phenomenon first suggested by computational experiments: after high-frequency modes have been suppressed by diffusion, generic heat evolutions on lens spaces tend to settle into Morse functions with the smallest possible number of critical points. Precisely, for every lens space \(L(p,q)\) with \(p\geq2\), \(1\leq q\leq p/2\), and \((p,q)=1\), there is an open dense set of initial data \(f\in L^2(L(p,q),\mathbb R)\) such that the heat evolution \(e^{tΔ}f\), where \(Δ\) is taken in the non-positive sign convention, is, for all sufficiently large \(t\), a Morse function with exactly four critical points, of indices \(0,1,2,3\). The proof analyzes the asymptotic spectral expansion of the heat flow. In the most delicate case \(1<q<p/2\), the leading term is Morse--Bott with two critical circles, and the higher heat modes break these circles through a resonant Fourier mechanism. An arithmetic estimate shows that the fundamental reduced frequency dominates all higher multiples, and generic nonvanishing of the corresponding resonant coefficients gives the minimal critical-point count.

math.DG↗
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

Diffusion Computation versus Quantum Computation: A Comparative Model for Order Finding and Factoring

We study a hybrid computational model for integer factorization in which the only non-classical resource is access to an \emph{iterated diffusion process} on a finite graph. Concretely, a \emph{diffusion step} is defined to be one application of a symmetric stochastic matrix (the half-lazy walk operator) to an $\ell^{1}$--normalized state vector, followed by an optional readout of selected coordinates. Let $N\ge 3$ be an odd integer which is neither prime nor a prime power, and let $b\in(\mathbb{Z}/N\mathbb{Z})^\ast$ have odd multiplicative order $r={\rm ord}_N(b)$. We construct, without knowing $r$ in advance, a weighted Cayley graph whose vertex set is the cyclic subgroup $\langle b\rangle$ and whose edges correspond to the powers $b^{\pm 2^t}$ for $t\le \lfloor \log_2 N\rfloor+1$. Using an explicit spectral decomposition together with an elementary doubling lemma, we show that $r$ can be recovered from a single heat-kernel value after at most $O((\log_2 N)^2)$ diffusion steps, with an effective bound. We then combine this order-finding model with the standard reduction from factoring to order finding (in the spirit of Shor's framework) to obtain a randomized factorization procedure whose success probability depends only on the number $m$ of distinct prime factors of $N$. Our comparison with Shor's algorithm is \emph{conceptual and model-based}. We replace unitary $\ell^2$ evolution by Markovian $\ell^1$ evolution, and we report complexity in two cost measures: digital steps and diffusion steps. Finally, we include illustrative examples and discussion of practical implementations.

math.SP↗
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↗
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↗
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↗