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
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.
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.
Publicationes Mathematicae Debrecen, Jul 1, 2017
We study linear divisibility sequences of order 4, providing a characterization by means of their... more We study linear divisibility sequences of order 4, providing a characterization by means of their characteristic polynomials and finding their factorization as a product of linear divisibility sequences of order 2. Moreover, we show a new interesting connection between linear divisibility sequences and Salem numbers. Specifically, we generate linear divisibility sequences of order 4 by means of Salem numbers modulo 1.
arXiv (Cornell University), 2011
This paper is devoted to the study of eigen-sequences for some important operators acting on sequ... more This paper is devoted to the study of eigen-sequences for some important operators acting on sequences. Using functional equations involving generating functions, we completely solve the problem of characterizing the fixed sequences for the Generalized Binomial operator. We give some applications to integer sequences. In particular we show how we can generate fixed sequences for Generalized Binomial and their relation with the Worpitzky transform. We illustrate this fact with some interesting examples and identities, related to Fibonacci, Catalan, Motzkin and Euler numbers. Finally we find the eigen-sequences for the mutual compositions of the operators Interpolated Invert, Generalized Binomial and Revert.
Experimental Mathematics, Nov 6, 2018
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 the 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.
arXiv (Cornell University), Feb 5, 2013
In this paper, we present the problem of counting magic squares and we focus on the case of multi... more In this paper, we present the problem of counting magic squares and we focus on the case of multiplicative magic squares of order 4. We give the exact number of normal multiplicative magic squares of order 4 with an original and complete proof, pointing out the role of the action of the symmetric group. Moreover, we provide a new representation for magic squares of order 4. Such representation allows the construction of magic squares in a very simple way, using essentially only five particular 4 × 4 matrices .
arXiv (Cornell University), Dec 10, 2015
In this paper we generalize the study of Matiyasevich on integer points over conics, introducing ... more In this paper we generalize the study of Matiyasevich on integer points over conics, introducing the more general concept of radical points. With this generalization we are able to solve in positive integers some Diophantine equations, relating these solutions by means of particular linear recurrence sequences. We point out interesting relationships between these sequences and known sequences in OEIS. We finally show connections between these sequences and Chebyshev and Morgan-Voyce polynomials, finding new identities.
International Journal of Number Theory, Jul 15, 2014
In this paper, we study the representation of π as sum of arcotangents. In particular, we obtain ... more In this paper, we study the representation of π as sum of arcotangents. In particular, we obtain new identities by using linear recurrent sequences. Moreover, we provide a method in order to express π as sum of arcotangents involving the Golden mean, the Lucas numbers, and more in general any quadratic irrationality.
arXiv (Cornell University), 2010
In this paper we study the action of the Binomial and Invert (interpolated) operators on the set ... more In this paper we study the action of the Binomial and Invert (interpolated) operators on the set of linear recurrent sequences. We prove that these operators preserve this set, and we determine how they change the characteristic polynomials. We show that these operators, with the aid of two other elementary operators (essentially the left and right shifts), can transform any impulse sequence (a linear recurrent sequence starting from (0,. .. , 0, 1)) into any other impulse sequence, by two processes that we call construction and deconstruction. Finally, we give some applications to polynomial sequences and pyramidal numbers. We also find a new identity on Fibonacci numbers, and we prove that r-bonacci numbers are a Bell polynomial transform of the (r − 1)bonacci numbers.
Discrete Mathematics, Nov 1, 2014
This paper shows how the study of colored compositions of integers reveals some unexpected and or... more This paper shows how the study of colored compositions of integers reveals some unexpected and original connection with the Invert operator. The Invert operator becomes an important tool to solve the problem of directly counting the number of colored compositions for any coloration. The interesting consequences arising from this relationship also give an immediate and simple criterion to determine whether a sequence of integers counts the number of some colored compositions. Applications to Catalan and Fibonacci numbers naturally emerge, allowing to clearly answer to some open questions. Moreover, the definition of colored compositions with the "black tie" provides straightforward combinatorial proofs to a new identity involving multinomial coefficients and to a new closed formula for the Invert operator. Finally, colored compositions with the "black tie" give rise to a new combinatorial interpretation for the convolution operator, and to a new and easy method to count the number of parts of colored compositions.
Ricerche Di Matematica, Sep 5, 2017
We study some properties and perspectives of the Hurwitz series ring H R [[t]], for a commutative... more We study some properties and perspectives of the Hurwitz series ring H R [[t]], for a commutative ring with identity R. Specifically, we provide a closed form for the invertible elements by means of the complete ordinary Bell polynomials, we highlight some connections with well-known transforms of sequences, and we see that the Stirling transforms are automorphisms of H R [[t]]. Moreover, we focus the attention on some special subgroups studying their properties. Finally, we introduce a new transform of sequences that allows to see one of this subgroup as an ultrametric dynamic space.
arXiv (Cornell University), 2012
In this paper we present some results related to the problem of finding periodic representations ... more In this paper we present some results related to the problem of finding periodic representations for algebraic numbers. In particular, we analyze the problem for cubic irrationalities. We show an interesting relationship between the convergents of bifurcating continued fractions related to a couple of cubic irrationalities, and a particular generalization of the Rédei polynomials. Moreover, we give a method to construct a periodic bifurcating continued fraction for any cubic root paired with another determined cubic root.
arXiv (Cornell University), Sep 4, 2012
In this paper we study a general class of conics starting from a quotient field. We give a group ... more In this paper we study a general class of conics starting from a quotient field. We give a group structure over these conics generalizing the construction of a group over the Pell hyperbola. Furthermore, we generalize the definition of Rédei rational functions in order to use them for evaluating powers of points over these conics. Finally, we study rational approximations of irrational numbers over conics, obtaining a new result for the approximation of quadratic irrationalities.
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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.
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.