Related papers: A method for obtaining the algebraic generating fu…
The gap probability generating function has as its coefficients the probability of an interval containing exactly $k$ eigenvalues. For scaled random matrices with orthogonal symmetry, and the interval at the hard or soft spectrum edge, the…
We present an algorithm for effectively generating binary sequences which would be rated by people as highly likely to have been generated by a random process, such as flipping a fair coin.
Let G be the space of generating functions of a periodic infinite order linear recurrence. In this paper we provide an explicit procedure for computing a basis of G.
The FLAME methodology makes it possible to derive provably correct algorithms from a formal description of a linear algebra problem. So far, the methodology has been successfully used to automate the derivation of direct algorithms such as…
A well-known problem in Algebraic Combinatorics, is the enumeration of circulant graphs. The failure of Adam's Conjecture for such graphs with order containing a repeated prime, led researchers to investigate the problem using two different…
Choosing the most adequate kernel is crucial in many Machine Learning applications. Gaussian Process is a state-of-the-art technique for regression and classification that heavily relies on a kernel function. However, in the Gaussian…
Using geometric considerations, we provide a clear derivation of the integral representation for the error function, known as the Craig formula. We calculate the corresponding power series expansion and prove the convergence. The same…
Generating high-quality geometry problems is both an important and challenging task in education. Compared to math word problems, geometry problems further emphasize multi-modal formats and the translation between informal and formal…
This paper describes a new method for Symbolic Regression that allows to find mathematical expressions from a dataset. This method has a strong mathematical basis. As opposed to other methods such as Genetic Programming, this method is…
Holonomic functions play an essential role in Computer Algebra since they allow the application of many symbolic algorithms. Among all algorithmic attempts to find formulas for power series, the holonomic property remains the most important…
Let $F(t,u)\equiv F(u)$ be a formal power series in $t$ with polynomial coefficients in $u$. Let $F\_1, ..., F\_k$ be $k$ formal power series in $t$, independent of $u$. Assume all these series are characterized by a polynomial equation $$…
The number of linear independent algebraic relations among elementary symmetric polynomial functions over finite fields is computed. An algorithm able to find all such relations is described. It is proved that the basis of the ideal of…
In this paper we propose an idea of constructing a macro--scale matrix system given a micro--scale matrix linear system. Then the macro--scale system is solved at cheaper computing costs. The method uses the idea of the generalized…
Homogeneous generative meta-programming (HGMP) enables the generation of program fragments at compile-time or run-time. We present the first foundational calculus which can model powerful HGMP languages such as Template Haskell. The…
Mixed-integer linear programming (MILP) is widely employed for modeling combinatorial optimization problems. In practice, similar MILP instances with only coefficient variations are routinely solved, and machine learning (ML) algorithms are…
We propose a class of generating functions denoted by $\textrm{RGF}_p(x)$, which is related to the Sylvester denumerant for the quotients of numerical semigroups. Using MacMahon's partition analysis, we can obtain $\textrm{RGF}_p(x)$ by…
We have presented a multivariate polynomial function termed as factor elimination function,by which, we can generate prime numbers. This function's mapping behavior can explain the irregularities in the occurrence of prime numbers on the…
Explaining neural network models is important for increasing their trustworthiness in real-world applications. Most existing methods generate post-hoc explanations for neural network models by identifying individual feature attributions or…
Synthesis from examples enables non-expert users to generate programs by specifying examples of their behavior. A domain-specific form of such synthesis has been recently deployed in a widely used spreadsheet software product. In this paper…
New methods for $D$-decomposition analysis are presented. They are based on topology of real algebraic varieties and computational real algebraic geometry. The estimate of number of root invariant regions for polynomial parametric families…