Related papers: Rank-Crank type PDE's for higher level Appell func…
A new approximation format for solutions of partial differential equations depending on infinitely many parameters is introduced. By combining low-rank tensor approximation in a selected subset of variables with a sparse polynomial…
Let $N_k(m,n)$ denote the number of partitions of $n$ with Garvan $k$-rank $m$. It is well-known that Andrews-Garvan-Dyson's crank and Dyson's rank are the $k$-rank for $k=1$ and $k=2$, respectively. In this paper, we prove that the…
We construct the first rigorously justified probabilistic algorithm for recovering the solution operator of a hyperbolic partial differential equation (PDE) in two variables from input-output training pairs. The primary challenge of…
Causal functions of sequences occur throughout computer science, from theory to hardware to machine learning. Mealy machines, synchronous digital circuits, signal flow graphs, and recurrent neural networks all have behaviour that can be…
Considering the L-function of exponential sums associated to a polynomial over a finite field F_q, Deligne proved that a reciprocal root's p-adic order is a rational number in the interval [0, 1]. Based on hypergeometric theory, in this…
We consider fractional differential equations of order $\alpha \in (0,1)$ for functions of one independent variable $t\in (0,\infty)$ with the Riemann-Liouville and Caputo-Dzhrbashyan fractional derivatives. A precise estimate for the order…
Fix an elliptic curve $E$ over a number field $F$ and an integer $n$ which is a power of $3$. We study the growth of the Mordell--Weil rank of $E$ after base change to the fields $K_d = F(\sqrt[2n]{d})$. If $E$ admits a $3$-isogeny, then we…
In this paper we discuss some results related to commuting ordinary differential operators of rank greater than one.
A systematic and unified approach to transformations and symmetries of general second order linear parabolic partial differential equations is presented. Equivalence group is used to derive the Appell type transformations, specifically…
Parameter-dependent models arise in many contexts such as uncertainty quantification, sensitivity analysis, inverse problems or optimization. Parametric or uncertainty analyses usually require the evaluation of an output of a model for many…
We combine the exact counting of all elliptic curves over $K = \mathbb{F}_q(t)$ with $\mathrm{char}(K) > 3$ by Bejleri, Satriano and the author, together with the torsion-free nature of most elliptic curves over global function fields…
This paper studies bilevel polynomial optimization in which lower-level constraint functions depend linearly on lower-level variables. We show that such bilevel program can be reformulated as a disjunctive program by using…
We consider self-adjoint extensions of differential operators of the type $ (-\frac{d^2}{dr^2} + \frac{l(l+1)}{r^2})^3 $ on the real semi-axis for l=1,2 with two kinds of boundary conditions: first that nullify the value of a function and…
We examine so-called rank function equations and their solutions consisting of non-nilpotent matrices. Secondly, we present some geometrical properties of the set of solutions to certain rank function equations in the nilpotent case.
In 2002 Watkins conjectured that given an elliptic curve defined over $\mathbb{Q}$, its Mordell-Weil rank is at most the $2$-adic valuation of its modular degree. We consider the analogous problem over function fields of positive…
We introduce the linear operators of fractional integration and fractional differentiation in the framework of the Riemann-Liouville fractional calculus. Particular attention is devoted to the technique of Laplace transforms for treating…
We introduce a generalization of the continued fraction and Chakravala algorithms for solving the Pell equation, utilizing the LLL-algorithm for rank 2 lattices.
In this paper, we study the Appell function $F_4$ from discrete point of view. In particular, we obtain regions of convergence, difference-differential equations, finite and infinite summation formulas and a list of recursion relations…
Low-rank plus diagonal (LRPD) decompositions provide a powerful structural model for large covariance matrices, simultaneously capturing global shared factors and localized corrections that arise in covariance estimation, factor analysis,…
Necessary conditions are obtained for certain types of rational delay differential equations to admit a non-rational meromorphic solution of hyper-order less than one. The equations obtained include delay Painlev\'e equations and equations…