SCIENTIA. Series A: Mathematical Science is an official journal of theUniversidad Técnica Federico Santa María, one of the leader Chilean institutions in science and education. It publishes original research papers in any branch of pure and applied mathematics. An international journal, its policy is to feature a wide variety of research areas, which in recent issues have included Complex Geometry, Complex and Real Analysis, Differential Equations, Dynamical Systems. The journal SCIENTIA is issued once a year by the university.The price is US$20 per volume, plus shipping. Subscription rates are available upon request. Scientia is also available on an exchange basis for journals of other universities or mathematical societies.
Departamento de Matemáticas. Universidad Técnica Federico Santa María
Generalization of the Laplace Transform of Powers of the sinc function
In this work, general expressions are found for the Laplace transform of \[L\left(\frac{\sin^m(at)}{t^n}\right) \quad \text{or} \quad L\left(\frac{\sin^n(t)\,\sin^m(at)}{t^p}\right).\] The results are expressed in terms of usual functions such as logarithms, arctangent, and rational functions. Thus, these expressions can be incorporated into the general literature to facilitate the calculation of Laplace transforms. In addition, the results of the article can be implemented in different mathematical software for a direct calculation of such transforms or to study the analytical properties of these transforms more directly.
Derechos de autor 2025 Scientia https://creativecommons.org/licenses/by-nc/4.0
A note on the Hurwitz-Lerch zeta function
In this work we derive a functional equation in terms of the Hurwitz-Lerch zeta function along with de nite integrals in terms of the incomplete gamma and Hurwitz-Lerch zeta functions. The method used in these derivations is contour integration. Special cases in terms of fundamental constants are produced.
In this work derivations of de nite integrals listed in Prudnikov volume I, Gradshteyn and Ryzhik and a few other tables are produced along with errata. Special cases of these integrals in terms of fundamental constants are also evaluated. The method used in these derivations is contour integration.
Derechos de autor 2025 Scientia https://creativecommons.org/licenses/by/4.0
We introduce the symmetric tensor power of graphs and explore the fundamental properties of this operation. A wide range of intriguing phenomena occur when one considers symmetric tensor powers of familiar graphs. We conclude with a host of open questions that we hope will spur future research.
A four parameter integral identity and a few consequences
The identity \[ \int_0^\infty \frac{e^{-ax} \, e^{-b\sqrt{x^2 + 2cx + a^2}}}{\sqrt{x^2 + 2cx + b^2}} \, dx = \int_0^\infty \frac{e^{-bx} \, e^{-a\sqrt{x^2 + 2cx + b^2}}}{\sqrt{x^2 + 2cx + a^2}} \, dx \] is derived, applied to the Struve function $H_0$, and used to deduce the reduction formula \[ \int_0^\infty \frac{F(\sqrt{x^2 + 2cx + b^2} + x)}{\sqrt{x^2 + 2cx + b^2}} \, dx = \int_0^\infty f(t)e^{-t} E_1[(c+b)t] \, dt, \] where $F$ is arbitrary.
Derechos de autor 2025 Scientia https://creativecommons.org/licenses/by/4.0/
On some improper integrals containing products between the tails of Maclaurin series for the sine and cosine functions
The values of twenty-six improper integrals containing products between the tails of Maclaurin series for the sine and cosine functions are evaluated using elementary methods. The values found extend work where the values for six improper integrals containing the squares of the tails of the Maclaurin series for the sine and cosine functions were recently reported.
Gain of regularity for a nonlinear dispersive equation
In this paper we study the gain of regularity of solutions of a dispersive evolution nonlinear equation. We consider the equation \[(1) =\begin{cases}\dfrac{\partial u}{\partial t} + \dfrac{\partial}{\partial x} \left[ f(u) \right] = \epsilon \, \dfrac{\partial}{\partial x} \left[ g\left( \dfrac{\partial u}{\partial x} \right) \right] - \delta \, \dfrac{\partial^3 u}{\partial x^3}, \\[6pt] u(x,0) = \varphi(x)\end{cases}\] where \( x \in \mathbb{R} \), \( t \in [0,T] \) and \( T \) is an arbitrary positive time. The flux \( f = f(u) \) and the (degenerate) viscosity \( g = g(\lambda) \) are smooth functions satisfying certain assumptions to be listed below. It is shown under certain additional conditions on \( f \) that \( C^\infty \)-solutions \( u(x,t) \) are obtained for all \( t > 0 \) if the initial data \( u(x,0) = \varphi(x) \) decays faster than polynomially on \( \mathbb{R}^+ = \{ x \in \mathbb{R} ; x > 0 \} \) and it has certain initial Sobolev regularity.
Abel summability and angular convergence
We construct an example of a series that is Abel summable but whose associated power series does not converge on angular sectors. We also give some related results.
Long-pin perturbations of the trivial solution for Hele-Shawcorner flows
We consider two-dimensional Hele-Shaw corner flows without effect of the surface tension and with an interface extending to the infinity along one of the walls. Explicit solutions that present a ”long-pin” deformations of the trivial solution are got. Making use of the Polubarinova-Galin approach we derive parametric equations for the moving interface in terms of univalent mappings of a canonical domain. For the right angle we repeat a result by Leo P. Kadanoff.
Fourth coefficient estimate in the class of univalent functions with quasiconformal extensions
We denote by \( S(k) \) the class of all univalent conformal maps \( f \) defined in the unit disk \( \Delta \) normalized by \[f(z) = z + \sum_{n=2}^{\infty} a_n z^n, \] such that all \( f \) admit \( k \)-quasiconformal homeomorphic extension to the whole Riemann sphere \( \hat{\mathbb{C}} \), and \( f(\infty) = \infty \). In our note we give a new estimate for \( |a_4| \) in \( S(k) \) making use of the Area Principle.
Some open problems related to 16b Hilbert problem.
This article contains a brief summary of the topics and concepts related to 16b Hilbert problem which refers to the existence of a bound on the number of limit cycles of a polynomial system in function of its degree. Both algebraic and analytic perspectives are described. Moreover, collateral open questions related to the mentioned problem are given. Finally, the bibliography, which does not intend to be exhaustive, contains more than one hundred of references.
On I-almost continuity and new descomposition of continuity
In this paper, I-almost continuous functions are introduced and their characterizations and properties are given. Relationships between I-almost continuity and other forms of continuity are discussed. Moreover, new decomposition of continuity with I-almost continuity as a factor is obtained.
Uniqueness Theorem for Hilbert Transform for Boehmians
In this paper a uniqueness theorem is proved for the Hilbert transform for the Boehmians of analytic function by using a relation between the Hilbert transform and the Fourier transform. Hilbert transform of Boehmians of analytic type is also discussed.
Some remarks, generalizations and misprints in the integrals in Gradshteyn and Ryzhik
This paper presents some remarks, generalizations and misprints to the classic table of integrals of Gradshteyn and Ryzhik. It is noted that the conditions for convergence in a number of integrals, that in the original source were obtained via Bessel transforms, are too restrictive. Also, a property of the parabolic cylinder function is used to simplify one of the integrals in Gradshteyn and Ryzhik into a result that enables the calculation of integrals for squares and higher powers of the error function. Subsequently, an indefinite integral for the normal distribution is specialized into several expressions that are reported in Gradshteyn and Ryzhik. The paper ends by listing integrals for which the aforementioned overly restrictive conditions emerge and/or for which generalizations or misprints are found
In this paper, we introduce a new class of algebras that related to distributive implication groupoids(DIG) and semigroups, call it a DIG-semigroup. We also define the concept of left(resp. right) deductive systems(LDS (resp. RDS) for short) of a DIG-semigroup and of unit divisors in DIG-semigroups. The notion of DIG-homomorphisms between DIG semigroups is introduced and investigate some of their properties and the quotient of DIG-semigroup via deductive systems is constructed.
Some Inequalities for Power Series with nonnegative coefficients via a divided difference reverse of Jensen inequality
Some inequalities for power series with nonnegative coefficients via a divided difference reverse of Jensen inequality are given. Applications for some fundamental functions defined by power series are also provided.
A table of definite integrals from the marriage of power and Fourier series
In this paper we show an unusual way to obtain values for definite integrals of the forms \(\int_{-\pi}^{\pi} f(x)dx\), \(\int_{-\pi}^{\pi} f(x)\cos(nx)dx\) and \(\int_{-\pi}^{\pi} f(x)\sin(nx)dx\) where \(n\) is a positive integer. Our method is indirect in that we do not start with the integral. The integral is obtained as the end result of a process that did not visualize the integral at the start. We begin with an analytic function, and obtain our integrals by comparing the coefficients of related power series and Fourier series. A table of 36 definite integrals results. Eleven of these can be reproduced by Mathematica or found in familiar tables, while 25 others cannot. These may be new integral evaluations.
One modulo \(N\) gracefulness of H-class of graphs
A function \(f\) is called a graceful labelling of a graph \(G\) with \(q\) edges if \(f\) is an injection from the vertices of \(G\) to the set \(\{0, 1, 2, \ldots, q\}\) such that, when each edge \(xy\) is assigned the label \(|f(x) - f(y)|\), the resulting edge labels are distinct. A graph \(G\) is said to be one modulo \(N\) graceful (where \(N\) is a positive integer) if there is a function \(\phi\) from the vertex set of \(G\) to \(\{0, 1, N, (N + 1), 2N, (2N + 1), \ldots, N(q - 1), N(q - 1) + 1\}\) in such a way that \( (i)\) \( \phi\) is \(1 - 1\) \( (ii) (\phi\) induces a bijection \(\phi^*\) from the edge set of \(G\) to \(\{1, N + 1, 2N + 1, \ldots, N(q - 1) + 1\}\) where \(\phi^*(uv)=|\phi(u) - \phi(v)|\). In this paper we prove that the H-graph, \(H \odot mK_1\) and \(H_{(n)}^{(a)}\) are one modulo \(N\) graceful for all positive integers \(N\).
The integrals in Gradshteyn and Ryzhik Part 28: The confluent hypergeometric function and Whittaker functions
The table of Gradshteyn and Ryzhik contains many entries where the integrand contains the confluent hypergeometric function \(_{1}F_{1}(a;c;z)\) and a closely associated function studied by Whittaker. A selection of these entries are evaluated.
The integrals in Gradshteyn and Ryzhik Part 27: More logarithmic examples
The table of Gradshteyn and Ryzhik contains many entries where the integrand is a combination of an elementary function and the logarithmic of another function of the same type. This paper presents proofs of some of these. A sample of examples where the elementary function is replaced by an algebraic function is also discussed.
The integrals in Gradshteyn and Ryzhik Part 26: The exponential integral
The table of Gradshteyn and Ryzhik contains many entries where the evaluation is given in terms of the exponential integral. A selection of these formulas are established.
In this paper, we discuss spaces with \(\aleph_0\)-\(sn\)-networks and give some characterizations of \(\aleph_0\)-\(sn\)-metric spaces. For a space \(X\) we prove that \(X\) has a point-countable \(\aleph_0\)-\(sn\)-network if and only if \(X\) is an sequentially quotient, countable-to-one image of a metric space. and \(X\) is an \(\aleph_0\)-\(sn\)-metric space if and only if \(X\) is a sequentially quotient, \(\sigma\), countable-to-one image of a metric space
\(\aleph_0\)-\(sn\)-weakly first-countable
Some integrals related to the Basel problem
We evaluate several arctangent and logarithmic integrals depending on a parameter. This provides a closed form summation of certain series and also integral and series representation of some classical constants.
On two families of stochastic predation models with Allee effect
Two classes of stochastic models of predator-prey interaction with Allee effect on prey are presented in this paper. Both families are introduced as generalizations of Leslie type predation models with Holling type functional response. The stochastic perturbations are considered of polynomial kind in order to model environmental random noise affecting both species. The only difference between both families of models is the way the Allee effect is modeled, namely, by means of a generic term whose conveniently chosen properties give rise to a behavior that is representative of either additive or multiplicative Allee effect, respectively. We show that both classes of models are well-posed in the sense that any positive solution starting in the open first quadrant remains in that region and, hence, it never becomes negative. In the particular case when the random noise is absent, we find that all the solutions are bounded for both sets of models. We also prove that the stochastic solutions exist, are unique in pathwise sense and have bounded moments. Moreover, we find that, as a consequence of our modeling approach, for sufficiently low population densities, the solutions of the random systems behave like the associated deterministic trajectories and, in doing so, the overall dynamics is determined by the features of the Allee effect.
This paper is devoted to investigate scatteredness on generalized topological spaces. The concept of \(g\)-scattered spaces is introduced. Their characterizations, properties and mapping theorems are obtained.
Evaluation of entries in Gradshteyn and Ryzhik employing the method of brackets
The method of brackets was created for the evaluation of definite integrals appearing in the resolution of Feynman diagrams. This method consists of a small number of heuristic rules and it is quite easy to apply. The first of these is Ramanujan’s Master Theorem, one of his favorite tools to evaluate integrals. The current work illustrates its applicability by evaluating a variety of entries from the classical table of integrals by I. S. Gradshteyn and I. M. Ryzhik.
Certain Inequalities Related to the Chebyshev’s Functional Involving Erdélyi-Kober Operators
The object of the present investigation is to obtain certain Chebyshev type integral inequalities involving the Erd´elyi-Kober fractional integral operators. Some consequent results and special cases of the main results are also pointed out.
Coexistence of limit cycles and invariant curves for Extended Kukles systems
We work with a certain class of extended Kukles system of arbitrary degree n with at least three invariant straight lines. We show that for a certain values of the parameters, the system has an lower bound of limit cycles. By writing the system as a perturbation of a Hamiltonian system, we show that the first Poincar´e-Melnikov integral of the system is a polynomial whose coefficients are the Lyapunov quantities. The maximum number of simples zero of this polynomial gives the maximum number of the global limit cycles; the multiplicity of the origin as a root the polynomial gives the maximum weakness that the weak focus at the origin. On the other hand, we also work with a certain extended Kukles system of order four with a invariant circumference. We show that for certain values of the parameters the system has an lower bound of limit cycles at the origin.
Poisson’s formula with principal value integrals and some special Gradshteyn and Ryzhik integrals
Poisson’s integral formula for holomorphic functions on the right half plane can be used to quickly evaluate certain integrals from Gradshteyn and Ryzhik’s table. In addition we prove a version of Poisson’s formula for principal value integrals and use it in several interesting cases.
Companions of Hermite-Hadamard Inequality for Convex Functions (I)
Companions of Hermite-Hadamard inequalities for convex functions defined on the positive axis in the case when the integral has the weight \(\frac{1}{t^3}\), \(t > 0\) are given. Applications for special means are provided as well.