Luis A Medina

University of Puerto Rico at Río Piedras | UPR-RP · Department of Mathematics

In this article we show that Walsh–Hadamard transformations of generalized p-ary functions whose components are symmetric, rotation symmetric or a combination or concatenation of them are C-finite sequences. This result generalized many of the known results for regular p-ary functions. We also present a study of the roots of the characteristic poly...

In this paper we define a new transform on (generalized) Boolean functions, which generalizes the Walsh-Hadamard, nega-Hadamard, 2k-Hadamard, consta-Hadamard and all HN-transforms. We describe the behavior of what we call the root-Hadamard transform for a generalized Boolean function f in terms of the binary components of f. Further, we define a no...

In this article we establish the asymptotic behavior of generating functions related to the exponential sum over finite fields of elementary symmetric functions and their perturbations. This asymptotic behavior allows us to calculate the probability generating function of the probability that the elementary symmetric polynomial of degree k and its...

In this article we establish the asymptotic behavior of generating functions related to the exponential sum over finite fields of elementary symmetric functions and their perturbations. This asymptotic behavior allows us to calculate the probability generating function of the probability that the the elementary symmetric polynomial of degree $k$ an...

Exponential sums have applications to a variety of scientific fields, including, but not limited to, cryptography, coding theory and information theory. Closed formulas for exponential sums of symmetric Boolean functions were found by Cai, Green and Thierauf in the late 1990's. Their closed formulas imply that these exponential sums are linear recu...

In this paper we define a new transform on (generalized) Boolean functions, which generalizes the Walsh-Hadamard, nega-Hadamard, $2^k$-Hadamard, consta-Hadamard and all $HN$-transforms. We describe the behavior of what we call the root- Hadamard transform for a generalized Boolean function $f$ in terms of the binary components of $f$. Further, we d...

Recently Schauz and Brink independently extended Chevalley's theorem to polynomials with restricted variables. In this note we give an improvement to Schauz-Brink's theorem via the ground field method. The improvement is significant in the cases where the degree of the polynomial is large compared to the weight of the degree of the polynomial.

Exponential sums of symmetric Boolean functions are linear recurrent with integer coefficients. This was first established by Cai, Green and Thierauf in the mid nineties. Consequences of this result has been used to study the asymptotic behavior of symmetric Boolean functions. Recently, Cusick extended it to rotation symmetric Boolean functions, wh...

In this paper we extend the covering method for computing the exact 2-divisibility of exponential sums of Boolean functions, improve results on the divisibility of the Hamming weight of deformations of Boolean functions, and provide criteria to obtain non-balanced functions. In particular, we present criteria to determine cosets of Reed-Muller code...

In this paper we provide new families of balanced symmetric functions over any finite field. We also generalize a conjecture of Cusick, Li, and Stǎnicǎ about the non-balancedness of elementary symmetric Boolean functions to any finite field and prove part of our conjecture.

In this article, we present a beautiful connection between Hadamard matrices and exponential sums of quadratic symmetric polynomials over Galois fields. This connection appears when the recursive nature of these sequences is analyzed. We calculate the spectrum for the Hadamard matrices that dominate these recurrences. The eigenvalues depend on the...

Exponential sums have applications to a variety of scientific fields, including, but not limited to, cryptography, coding theory and information theory. Closed formulas for exponential sums of symmetric Boolean functions were found by Cai, Green and Thierauf in the late 1990's. Their closed formulas imply that these exponential sums are linear recu...

For a prime p and an integer x, the p-adic valuation of x is denoted by . For a polynomial Q with integer coefficients, the sequence of valuations is shown to be either periodic or unbounded. The first case corresponds to the situation where Q has no roots in the ring of p-adic integers. In the periodic situation, the period length is determined.

Rotation symmetric Boolean functions are invariant under circular translation of indices. These functions have very rich cryptographic properties and have been used in different cryptosystems. Recently, Thomas Cusick proved that exponential sums of rotation symmetric Boolean functions satisfy homogeneous linear recurrences with integer coefficients...

Rotation symmetric Boolean functions are invariant under circular translation of indices. These functions have very rich cryptographic properties and have been used in different cryptosystems. Recently, Thomas Cusick proved that exponential sums of rotation symmetric Boolean functions satisfy homogeneous linear recurrences with integer coefficients...

This paper presents a study of perturbations of symmetric Boolean functions. In particular, it establishes a connection between exponential sums of these perturbations and Diophantine equations of the form Σ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l=0</sub> <sup xmlns:mml="http://www.w3.org/199...

This work presents a study of perturbations of symmetric Boolean functions. In particular, it establishes a connection between exponential sums of these perturbations and Diophantine equations of the form $$ \sum_{l=0}^n \binom{n}{l} x_l=0,$$ where $x_j$ belongs to some fixed bounded subset $\Gamma$ of $\mathbb{Z}$. The concepts of trivially balanc...

Following Boros-Moll, a sequence (a n ) is m-log-concave if \({\mathcal{L}^{j}(a_{n})\geqslant0}\) for all j = 0, 1, . . . , m. Here, \({\mathcal{L}}\) is the operator defined by \({\mathcal{L}(a_{n}) = a^{2}_{n}-a_{n-1}a_{n+1}}\). By a criterion of Craven-Csordas and McNamara-Sagan it is known that a sequence is ∞-log-concave if it satisfies the s...

This work brings techniques from the theory of recurrent integer sequences to the problem of balancedness of symmetric Boolean functions. In particular, the periodicity modulo $p$ ($p$ odd prime) of exponential sums of symmetric Boolean functions is considered. Periods modulo $p$, bounds for periods and relations between them are obtained for these...

This work brings techniques from the theory of recurrent integer sequences to the problem of balancedness of symmetric Boolean functions. In particular, the periodicity modulo $p$ ($p$ odd prime) of exponential sums of symmetric Boolean functions is considered. Periods modulo $p$, bounds for periods and relations between them are obtained for these...

For a prime $p$ and an integer $x$, the $p$-adic valuation of $x$ is denoted by $\nu_{p}(x)$. For a polynomial $Q$ with integer coefficients, the sequence of valuations $\nu_{p}(Q(n))$ is shown to be either periodic or unbounded. The first case corresponds to the situation where $Q$ has no roots in the ring of $p$-adic integers. In the periodic sit...

In this work, the p-adic valuation of Eulerian numbers is explored. A tree whose nodes contain information about the p-adic valuation of these numbers is constructed, and this tree, along with some classical results for Bernoulli numbers, is used to compute the exact p divisibility for the Eulerian numbers when the first variable lies in a congruen...

In this paper we compute the exact 2-divisibility of exponential sums associated to elementary symmetric Boolean functions. Our computation gives an affirmative answer to most of the open boundary cases of Cusick-Li-Stǎnicǎ’s conjecture. As a byproduct, we prove that the 2-divisibility of these families satisfies a linear recurrence. In particular,...

In this paper we consider perturbations of symmetric Boolean functions \({{\sigma_{n,k_1}} +\ldots+{\sigma_{n,k_s}}}\) in n-variable and degree k s . We compute the asymptotic behavior of Boolean functions of the type $${\sigma_{n,k_1}} +\ldots+{\sigma_{n,k_s}} +F(X_1, . . . , X_j)$$for j fixed. In particular, we characterize all the Boolean functi...

In this paper we give an improvement of the degree of the homogeneous linear recurrence with integer coefficients that exponential sums of symmetric Boolean functions satisfy. This improvement is tight. We also compute the asymptotic behavior of symmetric Boolean functions and provide a formula that allows us to determine if a symmetric boolean fun...

Boolean functions are one of the most studied objects in math-ematics. In this paper, we use the covering method to compute the exact 2-divisibility of exponential sums of boolean functions with prescribed leading monomials. Our results generalize those of [4] and [8] for the binary field. As an application of our findings, we provide families of b...

A new iterative method for high-precision numerical integration of rational functions on the real line is presented. The algorithm transforms the rational integrand into a new rational function preserving the integral on the line. The coefficients of the new function are explicit polynomials in the original ones. These transformations depend on the...

Let p > 2 be a prime. The p-adic valuation of Stirling numbers of the second kind is analyzed. Two types of tree diagrams that encode this information are introduced. Conditions that describe the infinite branching of these trees, similar to the case p = 2, are presented.

The evaluation of iterated primitives of powers of logarithms is expressed in closed form. The expressions contain polynomials with coefficients given in terms of the harmonic numbers and their generalizations. The logconcavity of these polynomials is established. Comment: 11 pages

We show that the $p$-adic valuation of the sequence of Fibonacci numbers is a $p$-regular sequence for every prime $p$. For $p \neq 2, 5$, we determine that the rank of this sequence is $\alpha(p) + 1$, where $\alpha(m)$ is the restricted period length of the Fibonacci sequence modulo $m$.

In a recent article in American Scientist, Theodore Hill described a coin-tossing game whose pay-off is the number of heads over the total number of throws. Suppose that at a given point during the game you have 5 heads and 3 tails, should you stop and get 5/8, or should you keep playing, hoping to get a better score? This is still an open problem....

We present a systematic study of integrals of the form $$I_{Q}=\int_{0}^{1}Q(x)\log\log\frac{1}{x}dx,$$ where Q is a rational function.

The table of Gradshteyn and Ryzhik contains some integrals that can be expressed in terms of the incomplete beta function. We describe some elementary properties of this function and use them to check some of the formulas in the mentioned table.

The sequence {xn} defined by xn=(n+xn−1)/(1−nxn−1), with x1=1, appeared in the context of some arctangent sums. We establish the fact that xn≠0 for n⩾4 and conjecture that xn is not an integer for n⩾5. This conjecture is given a combinatorial interpretation in terms of Stirling numbers via the elementary symmetric functions. The problem features li...

Many integrals in the classical table by Gradshteyn and Ryzhik can be evaluated in terms of the digamma function (= the logarithmic derivative of the gamma function). Some of them are presented here.

Let t[n] be a sequence that satisfies a first order homogeneous recurrence t[n] = Q[n]*t[n-1], where Q is a polynomial with integer coefficients. The asymptotic behavior of the p-adic valuation of t[n] is described under the assumption that all the roots of Q in Z/pZ have nonvanishing derivative.

Let t[n] be a sequence that satisfies a first order homogeneous recurrence t[n] = Q[n]*t[n-1], where Q is a polynomial with integer coefficients. The asymptotic behavior of the p-adic valuation of t[n] is described under the assumption that all the roots of Q in Z/pZ have nonvanishing derivative.

We present evalauations and provide proofs of definite integrals involving the function x^p cos^n x. These formulae are generalizations of 3.761.11 and 3.822.1, among others, in the classical table of integrals by I. S. Gradshteyn and I. M. Ryzhik.

The table of Gradshteyn and Ryzhik contains many entries where the integrand is a combination of a rational function and a logarithmic function. The proofs presented here, complete the evaluation of all entries in Section 4.231 and 4.291.