Related papers: Fast (Multi-)Evaluation of Linearly Recurrent Sequ…

Let $A \in \{0,1\}^{n \times n}$ be a matrix with $z$ zeroes and $u$ ones and $x$ be an $n$-dimensional vector of formal variables over a semigroup $(S, \circ)$. How many semigroup operations are required to compute the linear operator…

In this paper, we consider a certain variation of the "isoperimetric problem" adopted for subsets of nonnegative integers. More specifically, we explore the sequence P(n) as described in OEIS A186053. We provide the first exact formulas for…

In this paper, we present a method for estimating the least common multiple of a large class of binary linear recurrence sequences. Let $P,Q,R_0$, and $R_1$ be fixed integers and let $\boldsymbol{R}=\left(R_n\right)_{n}$ be the recurrence…

In recent work, Rosenbaum and Wagner showed that isomorphism of explicitly listed $p$-groups of order $n$ could be tested in $n^{\frac{1}{2}\log_p n + O(p)}$ time, roughly a square root of the classical bound. The $O(p)$ term is entirely…

We reconsider the theory of Lagrange interpolation polynomials with multiple interpolation points and apply it to linear algebra. For instance, $A$ be a linear operator satisfying a degree $n$ polynomial equation $P(A)=0$. One can see that…

We give an algorithm to compute $N$ steps of a convolution quadrature approximation to a continuous temporal convolution using only $O(N \log N)$ multiplications and $O(\log N)$ active memory. The method does not require evaluations of the…

We present the Polar framework for fully automating the analysis of classical and probabilistic loops using algebraic reasoning. The central theme in Polar comes with handling algebraic recurrences that precisely capture the loop semantics.…

In this paper, we study the following problem of reconstructing a simple polygon: Given a cyclically ordered vertex sequence of an unknown simple polygon P of n vertices and, for each vertex v of P, the sequence of angles defined by all the…

The discrete acyclic convolution computes the 2n-1 sums sum_{i+j=k; (i,j) in [0,1,2,...,n-1]^2} (a_i b_j) in O(n log n) time. By using suitable offsets and setting some of the variables to zero, this method provides a tool to calculate all…

The approximate period recovery problem asks to compute all $\textit{approximate word-periods}$ of a given word $S$ of length $n$: all primitive words $P$ ($|P|=p$) which have a periodic extension at edit distance smaller than $\tau_p$ from…

This work investigates a new approach to find closed form analytical approximate solution of linear initial value problems. Classical Bernoulli polynomials have been used to derive a finite set of orthonormal polynomials and a finite…

We apply the method of spectral sequences to study classical problems in analysis. We illustrate the method by finding polynomial vector fields that commute with a given polynomial vector field and finding integrals of polynomial…

Pisot sequences (sequences $a_n$ with initial terms $a_0=x, a_1=y$, and defined for $n>1$ by $a_n= \lfloor a_{n-1}^2/a_{n-2} + \frac{1}{2} \rfloor$) often satisfy linear recurrences with constant coefficients that are valid for all $n \geq…

We introduce a subclass of linear recurrence sequences which we call poly-rational sequences because they are denoted by rational expressions closed under sum and product. We show that this class is robust by giving several…

In this note, we give an algorithm that computes the linearwidth of input $n$-vertex graphs in time $O^*(2^n)$, which improves a trivial $O^*(2^m)$-time algorithm, where $n$ and $m$ the number of vertices and edges, respectively.

A recurrence relation is said to have the Laurent property if all of its iterates are Laurent polynomials in the initial values with integer coefficients. We consider a family of nonlinear recurrences with the Laurent property, which were…

Statistical testing is classically used as an exploratory tool to search for association between a phenotype and many possible explanatory variables. This approach often leads to multiple testing under dependence. We assume a hierarchical…

We consider series expansions in bases of classical orthogonal polynomials. When such a series solves a linear differential equation with polynomial coefficients, its coefficients satisfy a linear recurrence equation. We interpret this…

Linear temporal logic (LTL) is a specification language for finite sequences (called traces) widely used in program verification, motion planning in robotics, process mining, and many other areas. We consider the problem of learning LTL…

Osborne's iteration is a method for balancing $n\times n$ matrices which is widely used in linear algebra packages, as balancing preserves eigenvalues and stabilizes their numeral computation. The iteration can be implemented in any norm…