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Lee-Peng Teo

Publications and source records attributed to Lee-Peng Teo.

At least 19 recordsLinked to original sources
arXiv2025

Superior Highly Composite Numbers and the Explicit Upper Bound of Generalized Divisor Functions

For $k\geq 2$, we give a detailed exposition of the superior $k$-highly composite numbers. We then consider the function \[f_k(n)=\frac{\log d_k(n)\log\log n}{\log k\log n},\quad n\geq 3\] which has a maximum value $λ(k)$ at a superior $k$-highly composite number. We develop an efficient algorithm to compute $λ(k)$ and the positive integer $N_{\max}(k)$ where $f_k$ achieves the value $λ(k)$. The results for $2\leq k\leq 100$ are tabled.

math.NT↗
arXiv2025

Generalized Curvatures of Curves in $\mathbb{R}^n$

For a curve $\boldsymbolγ:I\to\mathbb{R}^n$ of order $n-1$, we prove that the generalized curvatures $κ_1, \ldots, κ_{n-1}$ can be expressed in terms of the leading principal minors of the matrix $\mathbf{A}(t)^T\mathbf{A}(t)$, where $\mathbf{A}(t)$ is the $n\times n$ matrix whose $i$-th column is $\boldsymbolγ^{(i)}(t)$. This gives an efficient algorithm to calculate the curvatures.

math.DG↗
arXiv2025

Explicit Bound of $\pmb{|ζ\left(1+it\right)|}$

In this work, we show that for all $t\geq e$, \[|ζ(1+it)|\leq 0.6443 \log t. \] The equality is achieved when $t=17.7477$. We also use the Riemann-Siegel formula and numerical computations to show that \[|ζ(1+it)|\leq\frac{1}{2}\log t+0.6633\hspace{1cm}\text{when}\;t\geq e.\]When $t\geq 100$, the bound $\frac{1}{2}\log t+0.6633$ is better than the bound $0.6443\log t$.

math.NT↗
arXiv2024

The Brioschi Formula for the Gaussian Curvature

The Brioschi formula expresses the Gaussian curvature $K$ in terms of the functions $E, F$ and $G$ in local coordinates of a surface $S$. This implies the Gauss' theorema egregium, which says that the Gaussian curvature just depends on angles, distances, and their rates of change. In most of the textbooks, the Gauss' theorema egregium was proved as a corollary to the derivation of the Gauss equations, a set of equations expressing $EK, FK$ and $GK$ in terms of the Christoffel symbols. The Christoffel symbols can be expressed in terms of $E$, $F$ and $G$. In principle, one can derive the Brioschi formula from the Gauss equations after some tedious calculations. In this note, we give a direct elementary proof of the Brioschi formula without using Christoffel symbols. The key to the proof are properties of matrices and determinants.

math.DG↗
arXiv2024

Local Index Theorem for Cofinite Hyperbolic Riemann Surfaces

We discuss the local index theorem for cofinite Riemann surfaces in a pedagogical way, from a more computational perspective. Given a cofinite Riemann surface $X$, let $Δ_n$ be the $n$-Laplacian and let $N_n$ be the Gram matrix of a basis of holomorphic $n$-differentials on $X$. The local index theorem says that on the Teichmüller space $T(X)$, the second variation of $\log\detΔ_n-\log \det N_n$ can be written as a sum of three symplectic forms $ω_{\text{WP}}$, $ω_{\text{TZ}}^{\text{cusp}}$ and $ω_{\text{TZ}}^{\text{ell}}$. These are the symplectic forms for the three Kähler metrics on $T(X)$ -- the Weil-Petersson metric, the parabolic Takhtajan-Zograf (TZ) metric and the elliptic Takhtajan-Zograf metric. Using Ahlfors' variational formulas and projection formulas, we derive explicitly integral formulas for the variations of $\log\detΔ_n$ and $\log \det N_n$. The integrals are regular integrals that allow explicit computations. In the spirit of the Selberg trace formula, we identify the identity, hyperbolic, parabolic and elliptic contributions to the second variations of $\log\detΔ_n$ and $\log \det N_n$. We showed that the Weil-Petersson term comes from the identity contribution, while the parabolic TZ metric and elliptic TZ metric terms come from parabolic and elliptic contributions respectively. The hyperbolic contributions are cancelled. As a byproduct, we obtain alternative integral formulas for the parabolic TZ metric and the elliptic TZ metric.

math.DG↗
arXiv2023

Mathematical Analysis Volume I

This is the first volume of a textbook for a two-semester course in mathematical analysis. This first volume is about analysis of functions of a single variable. The topics covered include completeness axiom, Archimedean property, sequentially compact subsets of $\mathbb{R}$, limits of functions, continuous functions, intermediate value theorem, extreme value theorem, differentiation, mean value theorem, l'Hopital's rule, Riemann integrals, improper integrals, elementary transcendental functions, sequences and series of numbers, infinite products, sequences and series of functions, uniform convergence, power series, Taylor series and Taylor polynomials. At the end of the book, we include some classical examples such as the irrationality of the number $e$, the existence of a non-analytic infinitely differentiable function, the existence of a nowhere differentiable continuous function. The book is concluded with the proof of the Weierstrass approximation theorem.

math.HO↗
arXiv2023

Mathematical Analysis Volume II

This is the second volume of a textbook for a two-semester course in mathematical analysis. This second volume is about analysis of multi-variable functions. The topics covered include Euclidean spaces, convergence of sequences, open sets and closed sets, limits and continuity, uniform continuity, connectedness, compactness, intermediate value theorem, extreme value theorem, partial derivatives, differentiability, chain rule, mean value theorem, first and second order approximations, local extrema, inverse function theorem, implicit function theorem, constrained extrema problems and Lagrange multipliers, Riemann integrals of functions of several variables, Jordan measurable sets, iterated integrals, Fubini's theorem, change of variables theorem, Fourier series and its convergence, Fourier transforms.

math.HO↗
arXiv2023

Rademacher's Formula for the Partition Function

For a positive integer $n$, let $p(n)$ be the number of ways to express $n$ as a sum of positive integers. In this note, we revisit the derivation of the Rademacher's convergent series for $p(n)$ in a pedagogical way, with all the details given. We also derive the leading asymptotic behavior of $p(n)$ when $n$ approaches infinity. Some numerical results are tabled.

math.NT↗
arXiv2023

An Elementary Proof of the Transformation Formula for the Dedekind Eta Function

The Dedekind eta function $η(τ)$ is defined by \[η(τ)=e^{πiτ/12}\prod_{n=1}^{\infty}\left(1-e^{2πi nτ}\right),\quad\text{when}\;\text{Im}\,τ>0.\] It plays an important role in number theory, especially in the theory of modular forms. Its 24$^{\text{th}}$ power, $η(τ)^{24}$, is a modular form of weight 12 for the modular group $\text{PSL}\,(2,\mathbb{Z})$. Up to a constant, $η(τ)^{24}$ is equal to the modular discriminant $Δ(τ)$. In this note, we give an elementary proof of the transformation formula for the Dedekind eta function under the action of the modular group $\text{PSL}\,(2,\mathbb{Z})$. We start by giving a proof of the transformation formula \[η\left(-\frac{1}τ\right)=(-iτ)^{1/2}η(τ)\]using the Jacobi triple product identity and the Poisson summation formula. Both of these formulas have elementary proofs. After we establish some identities for the Dedekind sum, the transformation formula for $η(τ)$ under the transformation \[τ\mapsto \frac{aτ+b}{cτ+d}\]induced by a general element of the modular group $\text{PSL}\,(2,\mathbb{Z})$ is derived by induction.

math.NT↗
arXiv2021

Resolvent Trace Formula and Determinants of $n$ Laplacians on Orbifold Riemann Surfaces

For $n$ a nonnegative integer, we consider the $n$-Laplacian $Δ_n$ acting on the space of $n$-differentials on a confinite Riemann surface $X$ which has ramification points. The trace formula for the resolvent kernel is developed along the line à la Selberg. Using the trace formula, we compute the regularized determinant of $Δ_n+s(s+2n-1)$, from which we deduce the regularized determinant of $Δ_n$, denoted by $\det\!'Δ_n$. Taking into account the contribution from the absolutely continuous spectrum, $\det\!'Δ_n$ is equal to a constant $\mathcal{C}_n$ times $Z(n)$ when $n\geq 2$. Here $Z(s)$ is the Selberg zeta function of $X$. When $n=0$ or $n=1$, $Z(n)$ is replaced by the leading coefficient of the Taylor expansion of $Z(s)$ around $s=0$ and $s=1$ respectively. The constants $\mathcal{C}_n$ are calculated explicitly. They depend on the genus, the number of cusps, as well as the ramification indices, but is independent of the moduli parameters.

math.CV↗
arXiv2019

Ruelle zeta function for cofinite hyperbolic Riemann surfaces with ramification points

We consider the Ruelle zeta function $R(s)$ of a genus $g$ hyperbolic Riemann surface with $n$ punctures and $v$ ramification points. $R(s)$ is equal to $Z(s)/Z(s+1)$, where $Z(s)$ is the Selberg zeta function. The main result of this work is the leading behavior of $R(s)$ at $s=0$. If $n_0$ is the order of the determinant of the scattering matrix $φ(s)$ at $s=0$, we find that \begin{align*} \lim_{s\rightarrow 0}\frac{R(s)}{s^{2g-2+n-n_0}}=(-1)^{\frac{A}{2}+1}(2π)^{2g-2+n }\tildeφ(0)^{-1} \prod_{j=1}^v m_j, \end{align*}which says that $R(s)$ has order $2g-2+n-n_0$ at $s=0$, and its leading coefficient can be expressed in terms of $m_1$, $m_2$, $\ldots$, $m_v$, the ramification indices at the ramification points, and $\tildeφ(0)$, the leading coefficient of $φ(s)$ at $s=0$. The constant $A$ is an even integer, equal to twice the multiplicity of the eigenvalue $-1$ in the scattering matrix $Φ(s)$ at $s=1/2$, and $(-1)^{\frac{A}{2}}=φ\left(\frac{1}{2}\right)$. We also consider the order of the Ruelle zeta function at other integers.

math.NT↗
arXiv2017

Liouville action and Holography on quasi-Fuchsian deformation spaces

We study the Liouville action for quasi-Fuchsian groups with parabolic and elliptic elements. In particular, when the group is Fuchsian, the contribution of elliptic elements to the classical Liouville action is derived in terms of the Bloch-Wigner functions. We prove the first and second variation formulas for the classical Liouville action on the quasi-Fuchsian deformation space. We prove an equality expressing the holography principle, which relates the Liouville action and the renormalized volume for quasi-Fuchsian groups with parabolic and elliptic elements. We also construct the potential functions of the Kähler forms corresponding to the Takhtajan-Zograf metrics associated to the elliptic elements in the quasi-Fuchsian groups.

math-ph↗
arXiv2017

Alternating Double Euler Sums, Hypergeometric Identities and a Theorem of Zagier

In this work, we derive relations between generating functions of double stuffle relations and double shuffle relations to express the alternating double Euler sums $ζ\left(\overline{r}, s\right)$, $ζ\left(r, \overline{s}\right)$ and $ζ\left(\overline{r}, \overline{s}\right)$ with $r+s$ odd in terms of zeta values. We also give a direct proof of a hypergeometric identity which is a limiting case of a basic hypergeometric identity of Andrews. Finally, we gave another proof for the formula of Zagier on the multiple zeta values $ζ(2,\ldots,2,3,2,\ldots,2)$.

math.CV↗
arXiv2015

Potentials and Chern forms for Weil-Petersson and Takhtajan-Zograf metrics on moduli spaces

For the TZ metric on the moduli space $\mathscr{M}_{0,n}$ of $n$-pointed rational curves, we construct a Kähler potential in terms of the Fourier coefficients of the Klein's Hauptmodul. We define the space $\mathfrak{S}_{g,n}$ as holomorphic fibration $\mathfrak{S}_{g,n}\rightarrow\mathfrak{S}_{g}$ over the Schottky space $\mathfrak{S}_{g}$ of compact Riemann surfaces of genus $g$, where the fibers are configuration spaces of $n$ points. For the tautological line bundles $\mathscr{L}_{i}$ over $\mathfrak{S}_{g,n}$ we define Hermitian metrics $h_{i}$ in terms of Fourier coefficients of a covering map $J$ of the Schottky domain. We define the regularized classical Liouville action $S$ and show that $\exp\{S/π\}$ is a Hermitian metric in the line bundle $\mathscr{L}=\otimes_{i=1}^{n}\mathscr{L}_{i}$ over $\mathfrak{S}_{g,n}$. We explicitly compute the Chern forms of these Hermitian line bundles $$c_{1}(\mathscr{L}_{i},h_{i})=\frac{4}{3}ω_{\mathrm{TZ},i},\quad c_{1}(\mathscr{L},\exp\{S/π\})=\frac{1}{π^{2}}ω_{\mathrm{WP}}.$$ We prove that a smooth real-valued function $-\mathscr{S}=-S+π\sum_{i=1}^{n}\log h_{i}$ on $\mathfrak{S}_{g,n}$, a potential for this special difference of WP and TZ metrics, coincides with the renormalized hyperbolic volume of a corresponding Schottky $3$-manifold. We extend these results to the quasi-Fuchsian groups of type $(g,n)$.

math.AG↗
arXiv2015

Bubble-wall Casimir interaction in fermionic environments

We consider the Casimir interaction, mediated by massless fermions, between a spherical defect and a flat potential barrier, assuming hard (bag-type) boundary conditions at both the barrier and the surface of the sphere. The computation of the quantum interaction energy is carried out using the multiple scattering approach, adapted here to the setup in question. We find an exact integral formula for the energy, from which we extract both the large and short distance asymptotic behaviour. At large distance the fermionic contribution is found to scale as $L^{-3}$, in contrast to that of electromagnetic vacuum fluctuations that, assuming perfectly conducting boundaries, scales as $L^{-4}$. At short distance, we compute the leading and sub-leading contribution to the vacuum energy. The leading one coincides with what it is expected from the proximity force approximation, while the sub-leading term gives, contrary to the electromagnetic case, a positive correction to the proximity force result.

hep-th↗
arXiv2010

The Multicomponent KP Hierarchy: Differential Fay Identities and Lax Equations

In this article, we show that four sets of differential Fay identities of an $N$-component KP hierarchy derived from the bilinear relation satisfied by the tau function of the hierarchy are sufficient to derive the auxiliary linear equations for the wave functions. From this, we derive the Lax representation for the $N$-component KP hierarchy, which are equations satisfied by some pseudodifferential operators with matrix coefficients. Besides the Lax equations with respect to the time variables proposed in \cite{2}, we also obtain a set of equations relating different charge sectors, which can be considered as a generalization of the modified KP hierarchy proposed in \cite{3}.

math-ph↗