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Satisfiability of word equations is an important problem in the intersection of formal languages and algebra: Given two sequences consisting of letters and variables we are to decide whether there is a substitution for the variables that…
It is well-known that for any non-constant polynomial $P$ with integer coefficients the sequence $(P(n))_{ n\in \mathbb N}$ has the property that there are infinitely many prime numbers dividing at least one term of this sequence.…
Formulas are presented for the recursive generation of four-body integrals in which the integrand consists of arbitrary integer powers (>= -1) of all the interparticle distances r_ij, multiplied by an exponential containing an arbitrary…
Recurrence equations have played a central role in static cost analysis, where they can be viewed as abstractions of programs and used to infer resource usage information without actually running the programs with concrete data. Such…
Linear algebraic expressions are the essence of many computationally intensive problems, including scientific simulations and machine learning applications. However, translating high-level formulations of these expressions to efficient…
Linear recurrence equations with constant coefficients define the power series coefficients of rational functions. However, one usually prefers to have an explicit formula for the sequence of coefficients, provided that such a formula is…
Certain completely logarithmic formula for a set of reversely iterated integrals (energies) is proved in this paper. Namely, in this case we have that integral powers of $\ln T$ are contained on input as well as on output of corresponding…
Zeckendorf's theorem states that every positive integer can be written uniquely as the sum of non-consecutive shifted Fibonacci numbers $\{F_n\}$, where we take $F_1=1$ and $F_2=2$. This has been generalized for any Positive Linear…
We study the decidability of the Skolem Problem, the Positivity Problem, and the Ultimate Positivity Problem for linear recurrences with real number initial values and real number coefficients in the bit-model of real computation. We show…
Starting from a result of Stewart, Tijdeman and Ruzsa on iterated difference sequences, we introduce the notion of iterated compositions of linear operations. We prove a general result on the stability of such compositions (with bounded…
The need for analytic continuation arises frequently in the context of inverse problems. Notwithstanding the uniqueness theorems, such problems are notoriously ill-posed without additional regularizing constraints. We consider several…
Let $(X,T,\mu,d)$ be a metric measure-preserving system for which $3$-fold correlations decay exponentially for Lipschitz continuous observables. Suppose that $(M_k)$ is a sequence satisfying some weak decay conditions and suppose there…
We introduce a permutation analogue of the celebrated Szemeredi Regularity Lemma, and derive a number of consequences. This tool allows us to provide a structural description of permutations which avoid a specified pattern, a result that…
To reliably model real robot characteristics, interval linear systems of equations allow to describe families of problems that consider sets of values. This allows to easily account for typical complexities such as sets of joint states and…
We propose a recursive algorithm for identifying all finite sequences of positive integers whose product equals their sum. Our method uses solutions of strictly shorter length that are iteratively extended in pursuit of a valid solution.…
We propose a novel combination of optimization tools with learning theory bounds in order to analyze the sample complexity of optimal kernel sum classifiers. This contrasts the typical learning theoretic results which hold for all…
In many instances in first order logic or computable algebra, classical theorems show that many problems are undecidable for general structures, but become decidable if some rigidity is imposed on the structure. For example, the set of…
Sums-of-squares formulas over the integers have been studied extensively using their equivalence to consistently signed intercalate matrices. This representation, combined with combinatorial arguments, has been used to produce…
Training in machine learning generally consists in finding one model, whose parameters minimize a data-dependent loss. Yet, empirical work shows that ensemble learning, an approach in which multiple models are sampled, can improve…
Consider a real matrix $\Theta$ consisting of rows $(\theta_{i,1},\ldots,\theta_{i,n})$, for $1\leq i\leq m$. The problem of making the system linear forms $x_{1}\theta_{i,1}+\cdots+x_{n}\theta_{i,n}-y_{i}$ for integers $x_{j},y_{i}$ small…