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Numerical solvers of Partial Differential Equations (PDEs) are of fundamental significance to science and engineering. To date, the historical reliance on legacy techniques has circumscribed possible integration of big data knowledge and…
We investigate a relaxation of the notion of treewidth-fragility, namely tree-independence-number-fragility. In particular, we obtain polynomial-time approximation schemes for independent packing problems on fractionally…
We study the form of possible algebraic relations between functions satisfying linear differential equations. In particular , if f and g satisfy linear differential equations and are algebraically dependent, we give conditions on the…
We characterize unicyclic graphs that are singular using the support of the null space of their pendant trees. From this, we obtain closed formulas for the independence and matching numbers of a unicyclic graph, based on the support of its…
The problem of algebraic dependence of solutions to (non-linear) first order autonomous equations over an algebraically closed field of characteristic zero is given a `complete' answer, obtained independently of model theoretic results on…
An independent set in a graph is a set of pairwise non-adjacent vertices. Let $\alpha(G)$ denote the cardinality of a maximum independent set in the graph $G = (V, E)$. Gutman and Harary defined the independence polynomial of $G$ \[ I(G;x)…
We develop new polynomial methods for studying systems of word equations. We use them to improve some earlier results and to analyze how sizes of systems of word equations satisfying certain independence properties depend on the lengths of…
In this note we develop a coalgebraic approach to the study of solutions of linear difference equations over modules and rings. Some known results about linearly recursive sequences over base fields are generalized to linearly (bi)recursive…
We present the first formulation of the optimal polynomial approximation of the solution of linear non-autonomous systems of ODEs in the framework of the so-called $\star$-product. This product is the basis of new approaches for the…
For a tuple $(\theta_1,..,\theta_M)$ of complex number, buliding on the approximation techniques in earlier papers of this series, this paper engages in deducing lower estimates on the transcendence degree of the field generated by…
We present a successive constraint approach that makes it possible to cheaply solve large-scale linear matrix inequalities for a large number of parameter values. The efficiency of our method is made possible by an offline/online…
This paper is devoted to the study of independent spaces of q-polymatroids. With the aid of an auxiliary q-matroid it is shown that the collection of independent spaces satisfies the same properties as for q-matroids. However, in contrast…
We propose a class of Pad\'e interpolation problems whose solutions are expressible in terms of determinants of hypergeometric series.
Let E/Q be an elliptic curve with a fixed modular parametrization F : X_0(N) --> E and let P_1,...,P_r be Heegner points on E attached to the rings of integers of distinct quadratic imaginary field k_1,...,k_r. We prove that if the odd…
We address the problem of the existence of a Lagrangian for a given system of linear PDEs with constant coefficients. As a subtask, this involves bringing the system into a pre-Lagrangian form, wherein the number of equations matches the…
A set of real $n$th roots that is pairwise linearly independent over the rationals must also be linearly independent. We show how this result may be extended to more general fields.
This paper investigates the independence polynomials arising from iterated strong products of cycle graphs, examining their algebraic symmetries and combinatorial structures. Leveraging modular arithmetic and Galois theory, we establish…
Given r elements in an integral domain over an algebraically closed field such that any two are linearly independent, we show that there is an integer e between 1 and r! such the eth powers of these elements are linearly independent.
This paper presents a novel parallel-in-time algorithm able to compute time-periodic solutions of problems where the period is not given. Exploiting the idea of the multiple shooting method, the proposed approach calculates the initial…
For the challenging task of modeling multivariate time series, we propose a new class of models that use dependent Mat\'ern processes to capture the underlying structure of data, explain their interdependencies, and predict their unknown…