arXiv subjects
Publications and source records attributed to Ronald Orozco López.
In this paper, we find solutions to first-order linear proportional difference equations via the $(s,t)$-integration factor method. The $(s,t)$-integration factor involves the $(s,t)$-Pantograph function, which is a generalization of the partial Theta function. Other equations are solved including the $(s,t)$-analog of the Bernoulli equation.
In this paper, we study the existence of solutions of the functional difference equations with proportional delay on deformed generalized Fibonacci polynomials via successive approximation method and Bell polynomials. First, we introduce the deformed generalized Fibonacci polynomials and show that the $q$-numbers can be viewed as "bifurcation" of deformed $(s,t)$-numbers. These deformations are closely related to proportional delay. Second, a differential and integral calculus on deformed generalized Fibonacci polynomials is introduced. The main reason for introducing this calculation is to have a framework for solving proportional functional equations and thus obtain the Pell calculus, Jacobsthal calculus, Chebysheff calculus, and Mersenne calculus, among others. We study the convergence of $(s,t)$-exponential type series and its dependence on the deformation parameter. We define the deformed $(s,t)$-exponential functions and we give its analytic and algebraic properties. In addition, we study the $(1,u)$-deformed $(s,t)$-exponential function and use it to prove the existence of functional difference equations with proportional delay. The solution is not unique when it is related to $q$-periodic functions.
In this paper a Ward-Fontené differential universal algebra is constructed. In this algebra it is possible to obtain a product $ψ$-rule and a general $ψ$-rule of Leibniz for any calculus on sequences. In particular, the simplicial polytopic calculus and the calculus on Bell numbers are introduced.
In this paper we use Faa di Bruno's formula to associate Bell polynomial values to differential equations of the form $y^{\prime}=f(y)$. That is, we use partial Bell polynomials to represent the solution of such an equation and use the solution to compute special values of partial Bell polynomials.
In this paper special values of Bell polynomials are given by using the power series solution of the equation $y^{(k)}=e^{ay}$. In addition, complete and partial exponential autonomous functions, exponential autonomous polynomials, autonomous polynomials and $(k,a)$-autonomous coefficients are defined. Finally, we show the relationship between various numbers counting combinatorial objects and binomial coefficients, Stirling numbers of the second kind and autonomous coefficients.
We construct a ring of flows where we can decompose autonomous nonlinear dynamical systems into smaller parts, then solve each part and finally put everything together to obtain the exact solution of these systems.
In this article, the ring of flows of autonomous differential equations of order one on integral domains is constructed. First, we build the autonomous ring $\Opa(\hurw_{R}[[x]])$ and then its structure is studied. Next, we build the ring of formal exponential generating series of the ring $\Opa(\hurw_ {R} [[x]]) $, where it is possible to find solutions of differential equations of order one when the vector field of the system can be decomposed in sum or product of functions.
In this paper, autonomous differential equations type delta of order one on integral domains are defined. For this we will use the autonomous ring defined on the Hurwitz expansion ring of exponential generating functions with coefficients in an integral domain. We will also use delta operators, which behave like derivatives when acting on polynomials, along with Umbral calculus. As a particular example of a delta operator we have the forward difference operator that defines the difference equations. Then the delta-type equations generalize to the ordinary equations and to the difference equations.
In this paper the circulant Hadamard conjecture is proved.
In this paper the relationship between $S$-subgroups in $\Z_{2}^{n}$ and binary codes is shown. If the codes used are both $P(T)$-codes and $G$-codes, then the $S$-subgroup is free. The codes constructed are cyclic, decimated or symmetric and the $S$-subgroups obtained are free under the action the cyclic permutation subgroup, invariants under the action the decimated permutation subgroup and symmetric under the action of symmetric permutation subgroup, respectively. Also it is shows that there is no codes generating whole $\Z_{2}^{n}$ in any $\G_{n}(a)$-complete $S$-set of the $S$-ring $\mathfrak{S}(\Z_{2}^{n},S_{n})$.
In this paper a variety of issues are discussed, Schur ring, $S$-sets, circulant orbits, decimation operator and Hadamard matrices and their relation between them is shown. Firstly we define the complete $S$-sets. Next, we study the structure of Schur ring with circulant basic sets over $\Z_{2}^{n}$ and we define the free and non-free circulant $S$-sets, the symmetric, non-symmetric and antisymmetric circulant $S$-sets. We prove that all this $S$-sets are invariants under decimation. Finally, we prove that if a Hadamard matrix exist then this is contained in a complete $S$-set. Also, we prove that can't exist circulant and with one core Hadamard matrices with some particular structure. These theorems include a result known on symmetric circulant Hadamard matrices of order $4n$ only when $n$ is an odd number.
In this paper three Schur ring are discussed, namenly: Hamming, circulant orbists and decimated circulant orbits Schur ring. By using autocorrelation function and the run structure of binary sequences we proof the relation between this Schur ring and combinatorial structures such as Hadamard matrices, periodic compatible binary sequences and perfect binary sequences. Cai proved for binary sequences that the autocorrelation function is in fact completely determined by its run structure. Also, he characterised the structure of the circulant Hadamard matrices. We characterise a more general structure, called periodic compatible binary sequences ($PComS$ for brevety), which generalises Hadamard matrices, periodic complementary binary sequences and binary sequences with $2$-level autocorrelation. Families of periodic compatibles binary sequences are presented. Also, we compute a bounds on familias $PComS$ in Hamming Schur ring. The results obtained are applied to families of $PComS$ such as circulant, with one and two circulant cores, Goethals-Seidel type and partial Hadamard matrices and perfect binary sequences.
Turyn prove that if a circulant Hadamard matrix of order $n$ exists then $n$ must be of the form $n=4m^{2}$ for some odd integer $m$. In this paper we use the structure constant of Schur ring of $\Z_{2}^{4m^{2}}$ to prove that there is no circulant Hadamard matrix in $\Z_{2}^{4m^{2}}$ except possibly for sequences with Hamming weight $a+b$, with $\frac{m^{2}-m}{2}\leq a\leq\frac{3m^{2}-m}{2}$ and $b=2m^{2}-m-a$ and with $\frac{m^{2}+m}{2}\leq 2m^{2}-a\leq\frac{3m^{2}+m}{2}$ and $b=m+a$.