arXiv subjects

Lubjana Beshaj

Publications and source records attributed to Lubjana Beshaj.

14 recordsLinked to original sources

arXiv2019

Solving machine learning optimization problems using quantum computers

Classical optimization algorithms in machine learning often take a long time to compute when applied to a multi-dimensional problem and require a huge amount of CPU and GPU resource. Quantum parallelism has a potential to speed up machine learning algorithms. We describe a generic mathematical model to leverage quantum parallelism to speed-up machine learning algorithms. We also apply quantum machine learning and quantum parallelism applied to a $3$-dimensional image that vary with time.

quant-ph↗

arXiv2019

Weighted greatest common divisors and weighted heights

We introduce the weighted greatest common divisor of a tuple of integers and explore some of it basic properties. Furthermore, for a set of heights $\mathfrak w=(q_0, \ldots , q_n)$, we use the concept of the weighted greatest common divisor to define a height $\mathfrak{h} (\mathfrak p)$ on weighted projective spaces $\mathbb{WP}_{\mathfrak w}^n (k)$. We prove some of the basic properties of this weighted height, including an analogue of the Northcott's theorem for heights on projective spaces.

math.NT↗

arXiv2019

Isogenous components of Jacobian surfaces

Let $\mathcal X$ be a genus 2 curve defined over a field $K$, $\mbox{char} K = p \geq 0$, and $\mbox{Jac} (\mathcal X, ι)$ its Jacobian, where $ι$ is the principal polarization of $\mbox{Jac} (\mathcal X)$ attached to $\mathcal X$. Assume that $\mbox{Jac} (\mathcal X)$ is $(n, n)$- geometrically reducible with $E_1$ and $E_2$ its elliptic components. We prove that there are only finitely many curves $\mathcal X$ (up to isomorphism) defined over $K$ such that $E_1$ and $E_2$ are $N$-isogenous for $n=2$ and $N=2,3, 5, 7$ with $\mbox{Aut} (\mbox{Jac} \mathcal X )\cong V_4$ or $n = 2$, $N = 3,5, 7$ with $\mbox{Aut} (\mbox{Jac} \mathcal X ) \cong D_4$. The same holds if $n=3$ and $N=5$. Furthermore, we determine the Kummer and the Shioda-Inose surfaces for the above $\mbox{Jac} \mathcal X$ and show how such results in positive characteristic $p>2$ suggest nice applications in cryptography.

math.AG↗

arXiv2019

Optimization problems with low SWaP tactical Computing

In a resource-constrained, contested environment, computing resources need to be aware of possible size, weight, and power (SWaP) restrictions. SWaP-aware computational efficiency depends upon optimization of computational resources and intelligent time versus efficiency tradeoffs in decision making. In this paper we address the complexity of various optimization strategies related to low SWaP computing. Due to these restrictions, only a small subset of less complicated and fast computable algorithms can be used for tactical, adaptive computing.

cs.AI↗

arXiv2018

On hyperelliptic curves of genus 3

We study the moduli space of genus 3 hyperelliptic curves via the weighted projective space of binary octavics. This enables us to create a database of all genus 3 hyperelliptic curves defined over $\mathbb Q$, of weighted moduli height $\mathcal h =1$.

math.AG↗

arXiv2018

The weighted moduli spaces of sextics

We use the weighted moduli height as defined in \cite{sh-h} to investigate the distribution of fine moduli points in the moduli space of genus two curves. We show that for any genus two curve with equation $y^2=f(x)$, its weighted moduli height $\mathfrak h (\mathfrak{p}) \leq 2^3 \sqrt{3 \cdot 5 \cdot 7} \, \cdot H(f)$, where $H(f)$ is the minimal naive height of the curve as defined in \cite{height}. Based on the weighted moduli height $\mathfrak h$ we create a database of genus two curves defined over $\mathbb Q$ with small $\mathfrak h$ and show that for small such height ($\mathfrak h < 5$) about 30% of points are fine moduli points.

math.AG↗

arXiv2016

On Prym varieties for the coverings of some singular plane curves

Let $k$ be a field of characteristic zero containing a primitive $n$-th root of unity. Let $C^0_n$ be a singular plane curve of degree $n$ over $k$ admitting an order $n$ automorphism, $n$ nodes as the singularities, and $C_n$ be its normalization. In this paper we study the factors of Prym variety $\mbox{Prym}(\widetilde{C}_n/C_n)$ associated to the double cover $\widetilde{C}_n$ of $C_n$ exactly ramified at the points obtained by the blow-up of the singularities. We provide explicit models of some algebraic curves related to the construction of $\mbox{Prym}(\widetilde{C}_n/C_n)$ as a Prym variety and determine the interesting simple factors other than elliptic curves or hyperelliptic curves with small genus which come up in $J_n$ so that the endomorphism rings contains the totally real field $\mathbb{Q}(ζ_n+ζ^{-1}_n)$.

math.AG↗

arXiv2015

The case for superelliptic curves

There is a natural question to ask whether the rich mathematical theory of the hyperelliptic curves can be extended to all superelliptic curves. Moreover, one wonders if all of the applications of hyperelliptic curves such as cryptography, mathematical physics, quantum computation, diophantine geometry, etc can carry over to the superelliptic curves. In this short paper we make the case that the superelliptic curves are exactly the curves that one should study.

math.AG↗

arXiv2015

Reduction theory of binary forms

In these lectures we give an introduction to the reduction theory of binary forms starting with quadratic forms with real coefficients, Hermitian forms, and then define the Julia quadratic for any degree $n$ binary form. A survey of a reduction algorithm over $\mathbb Z$ is described based on recent work of Cremona and Stoll.

math.NT↗

arXiv2014

Equations for superelliptic curves over their minimal field of definition

Let $\mathcal X_g$ be a genus $g\geq 2$ superelliptic curve, $F$ its field of moduli, and $K$ the minimal field of definition. In this short note we construct an equation of the curve $\mathcal X_g$ over its minimal field of definition $K$ when $\mathcal X_g$ has extra automorphisms. We make use of the dihedral invariants of superelliptic curves as defined by Shaska in [6] and results on the automorphism groups of superelliptic curves as in [10].

math.NT↗

arXiv2014

On Jacobians of curves with superelliptic components

We investigate the decomposition of Jacobians of superelliptic curves based on their automorphisms. For curve with equation $y^n=f(x^m)$ we provide an necessary and sufficient condition in terms of $m$ and $n$ for the decomposition of the Jacobian induced by the automorphisms of the curve. Moreover, we generalize a construction in \cite{Ya} of a family of non-hyperelliptic curves $\mathcal X_{r,s} $ and determine arithmetic conditions on $r$ and $s$ that the Jacobians $\mbox{Jac} (\mathcal X_{r, s})$ decomposes.

math.AG↗

arXiv2012

The arithmetic of genus two curves

Genus 2 curves have been an object of much mathematical interest since eighteenth century and continued interest to date. They have become an important tool in many algorithms in cryptographic applications, such as factoring large numbers, hyperelliptic curve cryptography, etc. Choosing genus 2 curves suitable for such applications is an important step of such algorithms. In existing algorithms often such curves are chosen using equations of moduli spaces of curves with decomposable Jacobians or Humbert surfaces. In these lectures we will cover basic properties of genus 2 curves, moduli spaces of (n,n)-decomposable Jacobians and Humbert surfaces, modular polynomials of genus 2, Kummer surfaces, theta-functions and the arithmetic on the Jacobians of genus 2, and their applications to cryptography. The lectures are intended for graduate students in algebra, cryptography, and related areas.

math.AG↗

arXiv2012

Singular locus on the space of genus 2 curves with decomposable Jacobians

We study the singular locus on the algebraic surface $§_n$ of genus 2 curves with a $(n, n)$-split Jacobian. Such surface was computed by Shaska in \cite{deg3} for $n=3$, and Shaska at al. in \cite{deg5} for $n=5$. We show that the singular locus for $n=2$ is exactly th locus of the curves of automorphism group $D_4$ or $D_6$. For $n=3$ we use a birational parametrization of the surface $§_3$ discovered in \cite{deg3} to show that the singular locus is a 0-dimensional subvariety consisting exactly of three genus 2 curves (up to isomorphism) which have automorphism group $D_4$ or $D_6$. We further show that the birational parametrization used in $§_3$ would work for all $n \geq 7$ if $§_n$ is a rational surface.

math.AG↗

arXiv2012

On superelliptic curves of level $n$ and their quotients, I

We study families of superelliptic curves with fixed automorphism groups. Such families are parametrized with invariants expressed in terms of the coefficients of the curves. Algebraic relations among such invariants determine the lattice of inclusions among the loci of superelliptic curves and their field of moduli. We give a Maple package of how to compute the normal form of an superelliptic curve and its invariants. A complete list of all superelliptic curves of genus $g \leq 10$ defined over any field of characteristic $\neq 2$ is given in a subsequent paper.

math.AG↗