Browsing by Author "Shaska, Tony"

Now showing 1 - 12 of 12
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    Curves of genus 2 with (n, n)--decomposable Jacobians
    (Journal of Symbolic Computation, 2001-05) Shaska, Tony
    Let 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.
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    Cyclic curves over the reals
    (eprint arXiv:1501.01559, 2014-11-15) Izquierdo, Milagros; Shaska, Tony
    In 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].
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    Design of a Compact Multi-Band (Cellular 5g/GNSS/V2x) Antenna and Rigorous Analysis of Antenna Performance on Glass Roofs for Vehicular Platforms
    (2023-01-01) Ibrahim, Ahmad Abu Elhassan Salih; Aloi, Daniel N; Kaur, Amanpreet; Qu, Guangzhi; Shaska, Tony
    The growing market competition between the automakers led to the implementation of more entertainment systems and extra features to satisfy the automotive customers. The entertainment systems depend on wireless services and communication which led to increasing numbers of antennas mounted on and inside the vehicles. This dissertation is focused on automotive antenna design and the effect of the vehicle environment on the antenna performance. For the first part of the dissertation, a compact multi-band monopole antenna is designed for vehicular roof top shark-fin applications. The proposed multi-band antenna covers 5G sub-6GHz, GNSS and V2X frequency bands starting at 617MHz to 5925MHz. The presented antenna is a three-dimensional monopole antenna with two branches to cover the required bands with compact size to fit inside a roof top shark-fin. The antenna is simulated, optimized and then a prototype is fabricated, and its radiation characteristics are measured when mounted on one-meter ground plane and on a vehicle's roof. For the second part of the dissertation, the analysis of a C-V2X quarter-wavelength monopole antenna performance when mounted on a vehicle's glass roof is presented. Antenna gain measurements performed on a full glass roof exhibited a performance degradation in a linear average gain of 8 dB compared to when the same antenna is mounted on a metallic ground plane. In addition, the antenna radiation pattern on the glass roof had deep nulls. The antenna was simulated using a full-wave, three-dimensional electromagnetic field solver on the full glass sample with low emissivity (low-E) coating on the edges of the full glass roof and the simulation results showed acceptable agreement with the measurements. Simulation shows that the C-V2X antenna performance on the full glass roof can be improved by moving the low-E coating from underneath the glass to top of the glass
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    Isogenous elliptic subcovers of genus two curves
    (2017-11-02) Beshaj, Lubjana; Elezi, Artur; Shaska, Tony
    We 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.
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    Mësimdhënia e matematikës nëpërmjet problemeve klasike
    (Albanian J. Math., 2016-12-03) Shaska, Tony; Shaska, Bedri
    In 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.
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    On generalized superelliptic Riemann surfaces
    (eprint arXiv:1609.09576, 2016-11-01) Hidalgo, Ruben; Quispe, Saul; Shaska, Tony
    A 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.
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    On the automorphism groups of some AG-codes based on C_{a,b} curves
    (2006-01-15) Shaska, Tony; Wang, Quang
    We 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.
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    Rational points in the moduli space of genus two
    (Contemporary Mathematics, 2018) Shaska, Tony; Beshaj, Lubjana; Hidalgo, Ruben; Kruk, Serge; Malmendier, Andreas; Quispe, Saul
    We build a database of genus 2 curves defined over the field of rationals.
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    Self-inversive polynomials, curves, and codes
    (American Mathematical Society, 2016-03-05) Shaska, Tony; Joyner, David
    We study connections between self-inversive and self-reciprocal polynomials, reduction theory of binary forms, minimal models of curves, and formally self-dual codes
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    Some open problems in computational algebraic geometry.
    (Albanian J. Math., 2007-12-15) Shaska, Tony
    The 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.
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    Special issue on algebra and computational algebraic geometry
    (Albanian J. Math., 2007-12) Elezi, Artur; Shaska, Tony
    Algebraic 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.
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    Thetanulls of cyclic curves of small genus
    (Albanian J. Math., 2007-12-15) Previato, Emma; Shaska, Tony; Wijesiri, Sujeeva
    We 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.