Mathematics and Statistics Faculty Scholarship
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Item Cyclic curves over the reals(eprint arXiv:1501.01559, 2014-11-15) Izquierdo, Milagros; Shaska, TonyIn this paper we study the automorphism groups of real curves admitting a regular meromorphic function f of degree p, so called real cyclic p-gonal curves. When p = 2 the automorphism groups of real hyperelliptic curves where given by Bujalance et al. in [4].Item Special issue on algebra and computational algebraic geometry(Albanian J. Math., 2007-12) Elezi, Artur; Shaska, TonyAlgebraic geometry is one of the main branches of modern mathematics with roots from classical Italian geometers. Its modern flavor started with Grothendieck and continued with many illustrious algebraic geometers of the second half of the 20-th century. During the last twenty years, the subject has changed drastically due to developments of new computational techniques and access to better computing power. Such changes have spurred a new direction of algebraic geometry, the so called computational algebraic geometry. While there is no universal agreement among mathematicians that what exactly is computational algebraic geometry, loosely stated it includes the areas of algebraic geometry where computer algebra can be used to obtain explicit results. It is obvious that such area will be of deep impact and importance in the future mathematics. Furthermore, such new developments have made possible applications of algebraic geometry in areas such as coding theory, computer security and cryptography, computer vision, mathematical biology, and many more.Item Thetanulls of cyclic curves of small genus(Albanian J. Math., 2007-12-15) Previato, Emma; Shaska, Tony; Wijesiri, SujeevaWe study relations among the classical thetanulls of cyclic curves, namely curves $\X$ (of genus $g(\X )>1$ ) with an automorphism $\s$ such that $\s$ generates a normal subgroup of the group $G$ of automorphisms, and $g \left( \X/ \<\s\> \right) =0$. Relations between thetanulls and branch points of the projection are the object of much classical work, especially for hyperelliptic curves, and of recent work, in the cyclic case. We determine the curves of genus 2 and 3 in the locus $\mathcal M_g (G, \textbf{C})$ for all $G$ that have a normal subgroup $\langle\s\rangle$ as above, and all possible signatures \textbf{C}, via relations among their thetanulls.Item Kalkulus I(AulonaPress, 2017-12-03) Shaska, TanushItem Rational points in the moduli space of genus two(Contemporary Mathematics, 2018) Shaska, Tony; Beshaj, Lubjana; Hidalgo, Ruben; Kruk, Serge; Malmendier, Andreas; Quispe, SaulWe build a database of genus 2 curves defined over the field of rationals.Item Some open problems in computational algebraic geometry.(Albanian J. Math., 2007-12-15) Shaska, TonyThe development of computational techniques in the last decade has made possible to attack some classical problems of algebraic geometry from a computational viewpoint. In this survey, we briefly describe some open problems of computational algebraic geometry which can be approached from such viewpoint. Some of the problems we discuss are the decomposition of Jacobians of genus two curves, automorphisms groups of algebraic curves and the corresponding loci in the moduli space of algebraic curves $\mathcal M_g$, inclusions among such loci, decomposition of Jacobians of algebraic curves with automorphisms, invariants of binary forms and the hyperelliptic moduli, theta functions of curves with automorphisms, etc. We decompose Jacobians of genus 3 curves with automorphisms and determine the inclusions among the loci for algebraic curves with automorphisms of genus 3 and 4.Item Curves of genus 2 with (n, n)--decomposable Jacobians(Journal of Symbolic Computation, 2001-05) Shaska, TonyLet C be a curve of genus 2 and ψ1: C − → E 1 a map of degree n, from C to an elliptic curveE1 , both curves defined over C. This map induces a degree n map φ1:P1 − → P 1 which we call a Frey–Kani covering. We determine all possible ramifications for φ1. If ψ1:C − → E 1 is maximal then there exists a maximal map ψ2: C − → E 2 , of degree n, to some elliptic curveE2 such that there is an isogeny of degree n2from the JacobianJC to E1 × E2. We say thatJC is (n, n)-decomposable. If the degree n is odd the pair (ψ2, E2) is canonically determined. For n = 3, 5, and 7, we give arithmetic examples of curves whose Jacobians are (n, n)-decomposable.Item On generalized superelliptic Riemann surfaces(eprint arXiv:1609.09576, 2016-11-01) Hidalgo, Ruben; Quispe, Saul; Shaska, TonyA closed Riemann surface X, of genus g ≥ 2, is called a generalized superelliptic curve of level n ≥ 2 if it admits an order n conformal automorphism τ so that X/hτihas genus zero and τ is central in Aut(X); the cyclic group H = hτiis called a generalized superelliptic group of level n for X. These Riemann surfaces are natural generalizations of hyperelliptic Riemann surfaces. We provide an algebraic curve description of these Riemann surfaces in terms of their groups of automorphisms. Also, we observe that the generalized superelliptic group H of level n is unique, with the exception of a very particular family of exceptional generalized superelliptic Riemann surfaces for n even. In particular, the uniqueness holds if either: (i) n is odd or (ii) the quotient X/H has all its cone points of order n. In the non-exceptional case, we use this uniqueness property to observe that the corresponding curves are definable over their fields of moduli if Aut(X)/H is neither trivial or cyclic.Item On the automorphism groups of some AG-codes based on C_{a,b} curves(2006-01-15) Shaska, Tony; Wang, QuangWe study Ca,b curves and their applications to coding theory. Recently, Joyner and Ksir have suggested a decoding algorithm based on the automorphisms of the code. We show how Ca,b curves can be used to construct MDS codes and focus on some Ca,b curves with extra automorphisms, namely y ³ = x⁴ + 1, y ³ = x⁴ −x, y ³ −y = x⁴. The automorphism groups of such codes are determined in most characteristics.Item Isogenous elliptic subcovers of genus two curves(2017-11-02) Beshaj, Lubjana; Elezi, Artur; Shaska, TonyWe prove that for $N=2,3, 5, 7$ there are only finitely many genus two curves $\X$ (up to isomorphism) defined over $\Q$ with $(2, 2)$-split Jacobian and $\Aut (\X)\iso V_4$, such that their elliptic subcovers are $N$-isogenous. Also, there are only finitely many genus two curves $\X$ (up to isomorphism) defined over $\Q$ with $(3, 3)$-split Jacobian such that their elliptic subcovers are $5$-isogenous.Item Self-inversive polynomials, curves, and codes(American Mathematical Society, 2016-03-05) Shaska, Tony; Joyner, DavidWe study connections between self-inversive and self-reciprocal polynomials, reduction theory of binary forms, minimal models of curves, and formally self-dual codesItem Mësimdhënia e matematikës nëpërmjet problemeve klasike(Albanian J. Math., 2016-12-03) Shaska, Tony; Shaska, BedriIn this paper we discuss how teaching of mathematics for middle school and high school students can be improved dramatically when motivation of concepts and ideas is done through the classical problems and the history of mathematics. This method improves intuition of students, awakens their curiosity, avoids memorizing useless formulas, and put concepts in a historical prospective. To illustrate we show how diagonalizing quadratic forms, elliptic integrals, discriminants of high degree polynomials, and geometric constructions can be introduced successfully in high school level.Item Combinatorial Algorithm for Quadratic Programs with Laplacian Structure(Utilitas Mathematica, 2016) Kruk, Serge; Nierman, Ryan; Shi, PeterAn algorithm is presented that uses a mostly combinatorial approach to solve a family of convex quadratic programs over box constraints. It is proved that for convex programs with the required structure, the algorithm converges in a finite number of iterations. Moreover, each iteration requires, at most, one function evaluation. On synthetic problems with thousands of variables, our implementation determines the optimal solution in seconds.Item Applications of Subset Selection Procedures and Bayesian Ranking Methods in Analysis of Traffic Fatality Data(WIREs Computational Statistics, 2016-11) McDonald, Gary C.Nonparametric and parametric subset selection procedures are used in the analysis of state motor vehicle traffic fatality rates (MVTFRs), for the years 1994 through 2012, to identify subsets of states that contain the ‘best’ (lowest MVTFR) and ‘worst’ (highest MVTFR) states with a prescribed probability. A new Bayesian model is developed and applied to the traffic fatality data and the results contrasted to those obtained with the subset selection procedures. All analyses are applied within the context of a two-way block design.Item Product measurability with applications to a stochastic contact problem with friction(2014-12) Kuttler, Kenneth; Shillor, MeirA new product measurability result for evolution equations with random inputs, when there is no uniqueness of the ω-wise problem, is established using results on measurable selection theorems for measurable multi- functions. The abstract result is applied to a general stochastic system of ODEs with delays and to a frictional contact problem in which the gap be- tween a viscoelastic body and the foundation and the motion of the foundation are random processes. The existence and uniqueness of a measurable solution for the problem with Lipschitz friction coefficient, and just existence for a discontinuous one, is obtained by using a sequence of approximate problems and then passing to the limit. The new result shows that the limit exists and is measurable. This new result opens the way to establish the existence of measurable solutions for various problems with random inputs in which the uniqueness of the solution is not known, which is the case in many problems involving frictional contact.Item A Model for Chagas Disease with Oral and Congenital Transmission(2013-06) Coffield, Daniel; Spagnuolo, Anna Maria; Shillor, Meir; Mema, Ensela; Pell, Bruce; Pruzinsky, Amanda; Zetye, AlexandraThis work presents a new mathematical model for the domestic transmission of Chagas disease, a parasitic disease affecting humans and other mammals throughout Central and South America. The model takes into account congenital transmission in both humans and domestic mammals as well as oral transmission in domestic mammals. The model has time-dependent coefficients to account for seasonality and consists of four nonlinear differential equations, one of which has a delay, for the populations of vectors, infected vectors, infected humans, and infected mammals in the domestic setting. Computer simulations show that congenital transmission has a modest effect on infection while oral transmission in domestic mammals substantially contributes to the spread of the disease. In particular, oral transmission provides an alternative to vector biting as an infection route for the domestic mammals, who are key to the infection cycle. This may lead to high infection rates in domestic mammals even when the vectors have a low preference for biting them, and ultimately results in high infection levels in humans.Item Dynamic contact of two GAO beams(2012-11) Ahn, Jeongho; Kuttler, Kenneth; Shillor, MeirThe dynamic contact of two nonlinear Gao beams that are connected with a joint is modeled, analyzed, and numerically simulated. Contact is modeled with either (i) the normal compliance condition, or (ii) the unilateral Signorini condition. The model is in the form of a variational equality in case (i) and a variational inequality in case (ii). The existence of the unique variational solution is established for the problem with normal compliance and the existence of a weak solution is proved in case (ii). The solution in the second case is obtained as a limit of the solutions of the first case when the normal compliance stiffness tends to infinity. A numerical algorithm for the problem is constructed using finite elements and a mixed time discretization. Simulation results, based on the implementation of the algorithm, of the two cases when the horizontal traction vanishes or when it is sufficiently large to cause buckling, are presented. The spectrum of the vibrations, using the FFT, shows a rather noisy system.Item Regularity result for the problem of vibrations of a nonlinear beam(2008-02) M'Bengue, M.F.; Shillor, MeirA model for the dynamics of the Gaonon linear beam, which allows for buckling, is studied. Existence and uniqueness of the local weak solution was established in Andrews et al. (2008). In this work the further regularity in time of the weak solution is shown using recent results for evolution problems. Moreover, the weak solution is shown to be global, existing on each finite time interval.Item A frictional contact problem for an electro-viscoelastic body(2007-12) Lerguet, Zhor; Shillor, Meir; Sofonea, MirceaA mathematical model which describes the quasistatic frictional contact between a piezoelectric body and a deformable conductive foundation is studied. A nonlinear electro-viscoelastic constitutive law is used to model the piezoelectric material. Contact is described with the normal compliance condition, a version of Coulomb’s law of dry friction, and a regularized electrical conductivity condition. A variational formulation of the model, in the form of a coupled system for the displacements and the electric potential, is derived. The existence of a unique weak solution of the model is established under a smallness assumption on the surface conductance. The proof is based on arguments of evolutionary variational inequalities and fixed points of operators.Item A one-dimensional spot welding model(2006-11) Andrews, K.T.; Guessous, L.; Nassar, S.; Putta, S.V.; Shillor, MeirA one-dimensional model is proposed for the simulations of resistance spot welding, which is a common industrial method used to join metallic plates by electrical heating. The model consists of the Stefan problem, in enthalpy form, coupled with the equation of charge conservation for the electrical potential. The temperature dependence of the density, thermal conductivity, specific heat, and electrical conductivity are taken into ac- count, since the process generally involves a large temperature range, on the order of 1000 K. The model is general enough to allow for the welding of plates of different thicknesses or dissimilar materials and to account for variations in the Joule heating through the material thickness due to the dependence of electrical resistivity on the temperature. A novel feature in the model is the inclusion of the effects of interface resistance between the plates which is also assumed to be temperature dependent. In addition to construct- ing the model, a finite difference scheme for its numerical approximations is described, and representative computer simulations are depicted. These describe welding processes involving different interface resistances, different thicknesses, different materials, and different voltage forms. The differences in the process due to AC or DC currents are depicted as well.