Jay Jorgenson
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View article: The discrete analogue of the Gaussian
The discrete analogue of the Gaussian Open
This paper illustrates the utility of the heat kernel on as the discrete analogue of the Gaussian density function. The heat kernel on is the two-variable function where is a Bessel function, w…
View article: The discrete analogue of the Gaussian
The discrete analogue of the Gaussian Open
This paper illustrates the utility of the heat kernel on $\mathbb{Z}$ as the discrete analogue of the Gaussian density function. It is the two-variable function $K_{\mathbb{Z}}(t,x)=e^{-2t}I_{x}(2t)$ involving a Bessel function and variabl…
View article: Constructing heat kernels on infinite graphs
Constructing heat kernels on infinite graphs Open
Let $G$ be an infinite, edge- and vertex-weighted graph with certain reasonable restrictions. We construct the heat kernel of the associated Laplacian using an adaptation of the parametrix approach due to Minakshisundaram-Pleijel in the se…
View article: On the functional equation of twisted Ruelle zeta function and Fried's conjecture
On the functional equation of twisted Ruelle zeta function and Fried's conjecture Open
Let $M$ be a finite volume hyperbolic Riemann surface with arbitrary signature, and let $χ$ be an arbitrary $m$-dimensional multiplier system of weight $k$. Let $R(s,χ)$ be the associated Ruelle zeta function, and $φ(s,χ)$ the determinant …
View article: The resolvent kernel on the discrete circle and twisted cosecant sums
The resolvent kernel on the discrete circle and twisted cosecant sums Open
In this paper we present a general unifying principle for computing finite trigonometric sums of types that arise in physics and number theory. We obtain formulas that are more general than previous expressions and deduce linear recursions…
View article: The parametrix construction of the heat kernel on a graph \footnote{Keywords: heat kernels, graphs, parametrix, Bessel functions.}
The parametrix construction of the heat kernel on a graph \footnote{Keywords: heat kernels, graphs, parametrix, Bessel functions.} Open
In this paper we develop the parametrix approach for constructing the heat kernelon a graph G. In particular, we highlight two specific cases. First, we considerthe case when G is embedded in a Eulidean domain or manifold \Omega, andwe use…
View article: The parametrix construction of the heat kernel on a graph
The parametrix construction of the heat kernel on a graph Open
In this paper we develop the parametrix approach for constructing the heat kernel on a graph $G$. In particular, we highlight two specific cases. First, we consider the case when $G$ is embedded in a Eulidean domain or manifold $Ω$, and we…
View article: The resolvent kernel on the discrete circle and twisted cosecant sums
The resolvent kernel on the discrete circle and twisted cosecant sums Open
Let $X_m$ denote the discrete circle with $m$ vertices. For $x,y\in X_{m}$ and complex $s$, let $G_{X_m,χ_β}(x,y;s)$ be the resolvent kernel associated to the combinatorial Laplacian which acts on the space of functions on $X_{m}$ that are…
View article: Discrete diffusion-type equation on regular graphs and its applications
Discrete diffusion-type equation on regular graphs and its applications Open
We derive an explicit formula for the fundamental solution $K_{T_{q+1}}(x,x_{0};t)$ to the discrete-time diffusion equation on the $(q+1)$-regular tree $T_{q+1}$ in terms of the discrete $I$-Bessel function. We then use the formula to deri…
View article: On an approach for evaluating certain trigonometric character sums using the discrete time heat kernel
On an approach for evaluating certain trigonometric character sums using the discrete time heat kernel Open
In this article we develop a general method by which one can explicitly evaluate certain sums of $n$-th powers of products of $d\geq 1$ elementary trigonometric functions evaluated at $\mathbf{m}=(m_1,\ldots,m_d)$-th roots of unity. Our ap…
View article: An integer factorization algorithm which uses diffusion as a computational engine
An integer factorization algorithm which uses diffusion as a computational engine Open
In this article we develop an algorithm which computes a divisor of an integer $N$, which is assumed to be neither prime nor the power of a prime. The algorithm uses discrete time heat diffusion on a finite graph. If $N$ has $m$ distinct p…
View article: Kronecker limit functions and an extension of the Rohrlich-Jensen formula
Kronecker limit functions and an extension of the Rohrlich-Jensen formula Open
In 1984 Rohrlich proved a modular analogue of Jensen's formula. Under certain conditions, the Rohrlich-Jensen formula expresses an integral of the log-norm $\log \Vert f \Vert$ of a $\text{\rm PSL}(2,\ZZ)$ modular form $f$ in terms of the …
View article: Evaluating the Mahler measure of linear forms via Kronecker limit formulas on complex projective space
Evaluating the Mahler measure of linear forms via Kronecker limit formulas on complex projective space Open
In Cogdell et al., \it LMS Lecture Notes Series \bf 459, \rm 393--427 (2020), \rm the authors proved an analogue of Kronecker's limit formula associated to any divisor $\mathcal D$ which is smooth in codimension one on any smooth Kähler ma…
View article: Spectral construction of non-holomorphic Eisenstein-type series and their Kronecker limit formulas
Spectral construction of non-holomorphic Eisenstein-type series and their Kronecker limit formulas Open
Let $X$ be a smooth, compact, projective Kähler variety and $D$ be a divisor of a holomorphic form $F$, and assume that $D$ is smooth up to codimension two. Let $ω$ be a Kähler form on $X$ and $K_{X}$ the corresponding heat kernel which is…
View article: An approach for computing generators of class fields of imaginary quadratic number fields using the Schwarzian derivative
An approach for computing generators of class fields of imaginary quadratic number fields using the Schwarzian derivative Open
Let be one of the distinct square-free integers such that the arithmetic group has genus one. We constructed canonical generators and for the associated function field (see Jorgenson, L. Smajlović, and H. Then [Exp. Math. 25 (2016), p…
View article: Transformation laws for generalized Dedekind sums associated to Fuchsian groups
Transformation laws for generalized Dedekind sums associated to Fuchsian groups Open
We establish transformation laws for generalized Dedekind sums associated to the Kronecker limit function of non-holomorphic Eisenstein series and their higher-order variants. These results apply to general Fuchsian groups of the first kin…
View article: Automorphic Forms and Related Topics
Automorphic Forms and Related Topics Open
The Langlands Programme predicts that a weight 2 newform f over a number eld K with integer Hecke eigenvalues generally should have an associated elliptic curve Ef over K. In [GMS14], we associated, building on works of Darmon [Dar01] and …
View article: Spectral asymptotics on sequences of elliptically degenerating Riemann surfaces
Spectral asymptotics on sequences of elliptically degenerating Riemann surfaces Open
In this article we study the spectral theory associated to families of hyperbolic Riemann surfaces obtained through elliptic degeneration, in particular the behavior of several spectral invariants. Some of these invariants, such as the Sel…
View article: Transformation laws for generalized Dedekind sums associated to Fuchsian groups
Transformation laws for generalized Dedekind sums associated to Fuchsian groups Open
We establish transformation laws for generalized Dedekind sums associated to the Kronecker limit function of non-holomorphic Eisenstein series and their higher-order variants. These results apply to general Fuchsian groups of the first kin…
View article: Superzeta functions, regularized products, and the Selberg zeta function on hyperbolic manifolds with cusps
Superzeta functions, regularized products, and the Selberg zeta function on hyperbolic manifolds with cusps Open
Let $Λ= \{λ_{k}\}$ denote a sequence of complex numbers and assume that that the counting function $#\{λ_{k} \in Λ: | λ_{k}| < T\} =O(T^{n})$ for some integer $n$. From Hadamard's theorem, we can construct an entire function $f$ of order a…
View article: ON THE DISTRIBUTION OF ZEROS OF THE DERIVATIVE OF SELBERG’S ZETA FUNCTION ASSOCIATED TO FINITE VOLUME RIEMANN SURFACES
ON THE DISTRIBUTION OF ZEROS OF THE DERIVATIVE OF SELBERG’S ZETA FUNCTION ASSOCIATED TO FINITE VOLUME RIEMANN SURFACES Open
We study the distribution of zeros of the derivative of the Selberg zeta function associated to a noncompact, finite volume hyperbolic Riemann surface $M$ . Actually, we study the zeros of $(Z_{M}H_{M})^{\prime }$ , where $Z_{M}$ is the Se…
View article: Modular Dedekind symbols associated to Fuchsian groups and higher-order\n Eisenstein series
Modular Dedekind symbols associated to Fuchsian groups and higher-order\n Eisenstein series Open
Let $E(z,s)$ be the non-holomorphic Eisenstein series for the modular group\n$SL(2,{\\mathbb Z})$. The classical Kronecker limit formula shows that the\nsecond term in the Laurent expansion at $s=1$ of $E(z,s)$ is essentially the\nlogarith…
View article: An evaluation of the central value of the automorphic scattering determinant
An evaluation of the central value of the automorphic scattering determinant Open
Let $M$ be a finite volume, non-compact hyperbolic Riemann surface, possibly with elliptic fixed points, and let $ϕ(s)$ denote the automorphic scattering determinant. From the known functional equation $ϕ(s)ϕ(1-s)=1$ one concludes that $ϕ(…
View article: Spectral asymptotics on sequences of elliptically degenerating Riemann\n surfaces
Spectral asymptotics on sequences of elliptically degenerating Riemann\n surfaces Open
This is the second in a series of two articles where we study various aspects\nof the spectral theory associated to families of hyperbolic Riemann surfaces\nobtained through elliptic degeneration. In the first article, we investigate\nthe …
View article: Kronecker’s Limit Formula, Holomorphic Modular Functions, and<i>q</i>-Expansions on Certain Arithmetic Groups
Kronecker’s Limit Formula, Holomorphic Modular Functions, and<i>q</i>-Expansions on Certain Arithmetic Groups Open
For any square-free integer $N$ such that the "moonshine group"\n$\\Gamma_0(N)^+$ has genus zero, the Monstrous Moonshine Conjectures relate the\nHauptmoduli of $\\Gamma_0(N)^+$ to certain McKay-Thompson series associated to\nthe represent…