A dynamical approach to congruences: linking circle maps and aperiodic necklaces
ABSTRACT There are many possible proofs of Fermat’s little theorem. Among them we exemplify those... more ABSTRACT There are many possible proofs of Fermat’s little theorem. Among them we exemplify those using necklaces and dynamical systems. Both methods lead to a generalization. It is a congruence theorem which is already known from Gauss and Gegenbauer. A natural result from these proofs is a bijection between aperiodic necklaces and circle maps.
We investigate the computational complexity of the solution of y ′ = f (t, y) when f : R k+1 → R ... more We investigate the computational complexity of the solution of y ′ = f (t, y) when f : R k+1 → R k is primitively recursive computable. In particular, we show that there is some f and initial conditions with complexity on the level E n+1 , n ≥ 3, of Grzegorczyk's hierarchy, such that the complexity of the unique solution y does not collapse to the previous level E n . As a corollary, we present some connections between Shannon's General Purpose Analog Computer and Computable Analysis.
ABSTRACT In this paper we develop an appropriate theory of positive definite functions on the com... more ABSTRACT In this paper we develop an appropriate theory of positive definite functions on the complex plane from first principles and show some consequences of positive definiteness for meromorphic functions.
Positive definite matrices and integral equations on unbounded domains
Differential and Integral Equations, 2006
ABSTRACT The kernel of a continuous positive integral operator on an interval $I$ is a Moore matr... more ABSTRACT The kernel of a continuous positive integral operator on an interval $I$ is a Moore matrix on $I$. We show that, under minimal differentiability assumptions, this implies that the kernel satisfies a 2-parameter family of differential inequalities. These inequalities ensure that, for unbounded $I$, the corresponding integral operator is exceptionally well behaved: it is compact and thus the eigenfunctions for its discrete spectrum have the differentiability of the kernel and satisfy sharp Sobolev bounds, the symmetric mixed partial derivatives are again kernels of positive operators and the differentiated eigenfunction series converge uniformly and absolutely. Converse results are derived.
Mixed integral identities involving unit spheres and balls in complex context
International Journal of Mathematics, 2015
In this paper, we derive integral identities relating both unit spheres and unit balls of several... more In this paper, we derive integral identities relating both unit spheres and unit balls of several dimensions in the complex setting. More specifically, we find a chain of equations involving either balls or balls and spheres of different dimensions. In addition, as a result almost independent we prove a prototype of the Funk–Hecke formula for embedded subspheres within the unit sphere of [Formula: see text], allowing for a closed-form expression for the computation of the eigenvalues.
Janeiro 2006 -nº 150 51 algumas dificuldades e desta forma, talvez, consigam mais facilmente atin... more Janeiro 2006 -nº 150 51 algumas dificuldades e desta forma, talvez, consigam mais facilmente atingir os objectivos que o ciclo de ensino propõe para a Matemática.
A Dynamical Approach to Congruences: Linking Circle Maps and Aperiodic Necklaces
Springer Proceedings in Mathematics & Statistics, 2014
ABSTRACT There are many possible proofs of Fermat’s little theorem. Among them we exemplify those... more ABSTRACT There are many possible proofs of Fermat’s little theorem. Among them we exemplify those using necklaces and dynamical systems. Both methods lead to a generalization. It is a congruence theorem which is already known from Gauss and Gegenbauer. A natural result from these proofs is a bijection between aperiodic necklaces and circle maps.
The São Paulo Journal of Mathematical Sciences, 2008
It is well-known that stable Cantor sets are topologically conjugate to adding machines. In this ... more It is well-known that stable Cantor sets are topologically conjugate to adding machines. In this work we show are also conjugate to an algebraic object, the ring of P −adic integers with respect to group tramnslation. This ring is closely related to the field of p-adic numbers; connections and distintions are explored. The inverse limit construction provides a purely dynamical proof of an algebraic result: the classification of adding machines, or P −adic integers, up to group isomorphism.
IEEE International Conference on Acoustics Speech and Signal Processing, 2002
This paper shows how the sampling theorem relates with the variations along time of the second or... more This paper shows how the sampling theorem relates with the variations along time of the second order statistics of L 2 (R) nonstationary processes. As a consequence, and mainly due to the positive semidefiniteness of autocorrelation functions, it is possible to conclude if a nonstationary process is locally stationary (i.e., if its second order statistics vary slowly along time) by the direct observation of its 2-dimension power spectrum or its Wigner distribution. A simple example illustrates how two different strategies for the estimation of autocorrelation functions from a small number of data can lead to opposite results in terms of local stationarity.
Journal of Mathematical Analysis and Applications, 2004
We study positive integral operators K in L 2 (R) with continuous kernel k(x, y). We show that if... more We study positive integral operators K in L 2 (R) with continuous kernel k(x, y). We show that if k(x, x) ∈ L 1 (R) the operator is compact and Hilbert-Schmidt. If in addition k(x, x) → 0 as |x| → ∞, k is represented by an absolutely and uniformly convergent bilinear series of uniformly continuous eigenfunctions and K is trace class. Replacing the first assumption by the stronger k 1/2 (x, x) ∈ L 1 (R) then k ∈ L 1 (R 2 ) and the bilinear series converges also in L 1 . Sharp norm bounds are obtained and Mercer's theorem is derived as a special case.
A dynamical approach to congruences: linking circle maps and aperiodic necklaces
ABSTRACT There are many possible proofs of Fermat’s little theorem. Among them we exemplify those... more ABSTRACT There are many possible proofs of Fermat’s little theorem. Among them we exemplify those using necklaces and dynamical systems. Both methods lead to a generalization. It is a congruence theorem which is already known from Gauss and Gegenbauer. A natural result from these proofs is a bijection between aperiodic necklaces and circle maps.
We investigate the computational complexity of the solution of y ′ = f (t, y) when f : R k+1 → R ... more We investigate the computational complexity of the solution of y ′ = f (t, y) when f : R k+1 → R k is primitively recursive computable. In particular, we show that there is some f and initial conditions with complexity on the level E n+1 , n ≥ 3, of Grzegorczyk's hierarchy, such that the complexity of the unique solution y does not collapse to the previous level E n . As a corollary, we present some connections between Shannon's General Purpose Analog Computer and Computable Analysis.
ABSTRACT In this paper we develop an appropriate theory of positive definite functions on the com... more ABSTRACT In this paper we develop an appropriate theory of positive definite functions on the complex plane from first principles and show some consequences of positive definiteness for meromorphic functions.
Positive definite matrices and integral equations on unbounded domains
Differential and Integral Equations, 2006
ABSTRACT The kernel of a continuous positive integral operator on an interval $I$ is a Moore matr... more ABSTRACT The kernel of a continuous positive integral operator on an interval $I$ is a Moore matrix on $I$. We show that, under minimal differentiability assumptions, this implies that the kernel satisfies a 2-parameter family of differential inequalities. These inequalities ensure that, for unbounded $I$, the corresponding integral operator is exceptionally well behaved: it is compact and thus the eigenfunctions for its discrete spectrum have the differentiability of the kernel and satisfy sharp Sobolev bounds, the symmetric mixed partial derivatives are again kernels of positive operators and the differentiated eigenfunction series converge uniformly and absolutely. Converse results are derived.
Mixed integral identities involving unit spheres and balls in complex context
International Journal of Mathematics, 2015
In this paper, we derive integral identities relating both unit spheres and unit balls of several... more In this paper, we derive integral identities relating both unit spheres and unit balls of several dimensions in the complex setting. More specifically, we find a chain of equations involving either balls or balls and spheres of different dimensions. In addition, as a result almost independent we prove a prototype of the Funk–Hecke formula for embedded subspheres within the unit sphere of [Formula: see text], allowing for a closed-form expression for the computation of the eigenvalues.
Janeiro 2006 -nº 150 51 algumas dificuldades e desta forma, talvez, consigam mais facilmente atin... more Janeiro 2006 -nº 150 51 algumas dificuldades e desta forma, talvez, consigam mais facilmente atingir os objectivos que o ciclo de ensino propõe para a Matemática.
A Dynamical Approach to Congruences: Linking Circle Maps and Aperiodic Necklaces
Springer Proceedings in Mathematics & Statistics, 2014
ABSTRACT There are many possible proofs of Fermat’s little theorem. Among them we exemplify those... more ABSTRACT There are many possible proofs of Fermat’s little theorem. Among them we exemplify those using necklaces and dynamical systems. Both methods lead to a generalization. It is a congruence theorem which is already known from Gauss and Gegenbauer. A natural result from these proofs is a bijection between aperiodic necklaces and circle maps.
The São Paulo Journal of Mathematical Sciences, 2008
It is well-known that stable Cantor sets are topologically conjugate to adding machines. In this ... more It is well-known that stable Cantor sets are topologically conjugate to adding machines. In this work we show are also conjugate to an algebraic object, the ring of P −adic integers with respect to group tramnslation. This ring is closely related to the field of p-adic numbers; connections and distintions are explored. The inverse limit construction provides a purely dynamical proof of an algebraic result: the classification of adding machines, or P −adic integers, up to group isomorphism.
IEEE International Conference on Acoustics Speech and Signal Processing, 2002
This paper shows how the sampling theorem relates with the variations along time of the second or... more This paper shows how the sampling theorem relates with the variations along time of the second order statistics of L 2 (R) nonstationary processes. As a consequence, and mainly due to the positive semidefiniteness of autocorrelation functions, it is possible to conclude if a nonstationary process is locally stationary (i.e., if its second order statistics vary slowly along time) by the direct observation of its 2-dimension power spectrum or its Wigner distribution. A simple example illustrates how two different strategies for the estimation of autocorrelation functions from a small number of data can lead to opposite results in terms of local stationarity.
Journal of Mathematical Analysis and Applications, 2004
We study positive integral operators K in L 2 (R) with continuous kernel k(x, y). We show that if... more We study positive integral operators K in L 2 (R) with continuous kernel k(x, y). We show that if k(x, x) ∈ L 1 (R) the operator is compact and Hilbert-Schmidt. If in addition k(x, x) → 0 as |x| → ∞, k is represented by an absolutely and uniformly convergent bilinear series of uniformly continuous eigenfunctions and K is trace class. Replacing the first assumption by the stronger k 1/2 (x, x) ∈ L 1 (R) then k ∈ L 1 (R 2 ) and the bilinear series converges also in L 1 . Sharp norm bounds are obtained and Mercer's theorem is derived as a special case.
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Papers by Jorge Buescu