Papers by Umberto Cerruti
Springer eBooks, 2002
Abstract. This paper explores the possibility of using the evolution
of a population of finite s... more Abstract. This paper explores the possibility of using the evolution
of a population of finite state machines (FSMs) as a measure of the
‘randomness’ of a given binary sequence. An FSM with binary input and
output alphabet can be seen as a predictor of a binary sequence. For
any finite binary sequence, there exists an FSM able to perfectly predict
the string but such a predictor, in general, has a large number of states.
In this paper, we address the problem of finding the best predictor for
a given sequence. This is an optimization problem over the space of all
possible FSMs with a fixed number of states evaluated on the sequence
considered. For this optimization an evolutionary algorithm is used: the
better the FSMs found are, the less ‘random’ the given sequence will be.
arXiv (Cornell University), Mar 9, 2020
We discuss the use of matrices for providing sequences of rationals that approximate algebraic ir... more We discuss the use of matrices for providing sequences of rationals that approximate algebraic irrationalities. In particular, we study the regular representation of algebraic extensions, proving that ratios between two entries of the matrix of the regular representation converge to specific algebraic irrationalities. As an interesting special case, we focus on cubic irrationalities giving a generalization of the Khovanskii matrices for approximating cubic irrationalities. We discuss the quality of such approximations considering both rate of convergence and size of denominators. Moreover, we briefly perform a numerical comparison with well-known iterative methods (such as Newton and Halley ones), showing that the approximations provided by regular representations appear more accurate for the same size of the denominator.
American Mathematical Monthly, 2017
arXiv (Cornell University), Dec 15, 2012
In this paper we study the action of a generalization of the Binomial interpolated operator on th... more In this paper we study the action of a generalization of the Binomial interpolated operator on the set of linear recurrent sequences. We find how the zeros of characteristic polynomials are changed and we prove that a subset of these operators form a group, with respect to a well-defined composition law. Furthermore, we study a vast class of linear recurrent sequences fixed by these operators and many other interesting properties. Finally, we apply all the results to integer sequences, finding many relations and formulas involving Catalan numbers, Fibonacci numbers, Lucas numbers and triangular numbers.
arXiv (Cornell University), Mar 19, 2011
In this paper, we define a new product over R ∞ , which allows us to obtain a group isomorphic to... more In this paper, we define a new product over R ∞ , which allows us to obtain a group isomorphic to R * with the usual product. This operation unexpectedly offers an interpretation of the Rédei rational functions, making more clear some of their properties, and leads to another product, which generates a group structure over the Pell hyperbola. Finally, we join together these results, in order to evaluate solutions of Pell equation in an original way.
Journal of Algebra, Jul 1, 1995
Vector Linear Recurrence Sequences in Commutative Rings
Springer eBooks, 1996
Let R be a commutative ring with identity. Let X be a vector sequence in \(\mathfrak{M}: = {R^t}\... more Let R be a commutative ring with identity. Let X be a vector sequence in \(\mathfrak{M}: = {R^t}\), such that X (m) ∑ h k =1 X (m−h) G h , with G h € Mat(t,R). The main result of this paper is to show that X can be computed as a linear recurrence sequence (in \(\mathfrak{M}\)) with scalar coefficients.
Counting the Number of Solutions of Congruences
Springer eBooks, 1993
arXiv (Cornell University), Apr 15, 2020
Given a commutative ring R with identity, let H R be the set of sequences of elements in R. We in... more Given a commutative ring R with identity, let H R be the set of sequences of elements in R. We investigate a novel isomorphism between (H R , +) and (H R , *), where + is the componentwise sum, * is the convolution product (or Cauchy product) andH R the set of sequences starting with 1 R. We also define a recursive transform over H R that, together to the isomorphism, allows to highlight new relations among some well studied integer sequences. Moreover, these connections allow to introduce a family of polynomials connected to the D'Arcais numbers and the Ramanujan tau function. In this way, we also deduce relations involving the Bell polynomials, the divisor function and the Ramanujan tau function. Finally, we highlight a connection between Cauchy and Dirichlet products. 
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Papers by Umberto Cerruti
of a population of finite state machines (FSMs) as a measure of the
‘randomness’ of a given binary sequence. An FSM with binary input and
output alphabet can be seen as a predictor of a binary sequence. For
any finite binary sequence, there exists an FSM able to perfectly predict
the string but such a predictor, in general, has a large number of states.
In this paper, we address the problem of finding the best predictor for
a given sequence. This is an optimization problem over the space of all
possible FSMs with a fixed number of states evaluated on the sequence
considered. For this optimization an evolutionary algorithm is used: the
better the FSMs found are, the less ‘random’ the given sequence will be.