On data recovery with restraints on the spectrum range and the process range

The paper considers recovery of signals from incomplete observations and a problem of determination of the allowed quantity of missed observations, i.e. the problem of determination of the size of the uniqueness sets for a given data recovery procedures. The paper suggests a way to bypass solution of this uniqueness problem via imposing restrictions investigates possibility of data recovery for classes of finite sequences under a special discretization of the process range. It is shown that these sequences can be dense in the space of all sequences and that the uniqueness sets for them can be singletons. Some robustness with respect to rounding of input data can be achieved via including additional observations

On the Largest Prime factor of the k-generalized Lucas numbers

Let $(L_n^{(k)})_{n\geq 2-k}$ be the sequence of $k$--generalized Lucas numbers for some fixed integer $k\ge 2$ whose first $k$ terms are $0,\ldots,0,2,1$ and each term afterwards is the sum of the preceding $k$ terms. For an integer $m$, let $P(m)$ denote the largest prime factor of $m$, with $P(0)=P(\pm 1)=1$. We show that if $n \ge k + 1$, then $P (L_n^{(k)} ) > (1/86) \log \log n$. Furthermore, we determine all the $k$--generalized Lucas numbers $L_n^{(k)}$ whose largest prime factor is at most $7$

A Collaborative Framework for Non-Linear Integer Arithmetic Reasoning in Alt-Ergo

In this paper, we describe a collaborative framework for reasoning modulo simple properties of non-linear integer arithmetic. This framework relies on the AC(X) combination method and on interval calculus. The first component is used to handle equalities of linear integer arithmetic and associativity and commutativity properties of non-linear multiplication. The interval calculus component is used - in addition to standard linear operations over inequalities - to refine bounds of non-linear terms and to inform the SAT solver about judicious case-splits on bounded intervals. The framework has been implemented in the Alt-Ergo theorem prover. We show its effectiveness on a set of formulas generated from deductive program verification

Efficient Solution of Rational Conics

We present efficient algorithms for solving Legendre equations over Q (equivalently, for finding rational points on rational conics) and parametrizing all solutions. Unlike existing algorithms, no integer factorization is required, provided that the prime factors of the discriminant are known

The algorithmic resolution of diophantine equations: a computational cookbook

A coherent account of the computational methods used to solve diophantine equations