相关论文: Characterization and enumeration on Lam\'e equatio…

The number of Lame equations with finite (ordinary or projective) monodromy has been conjectured by S. R. Dahmen, and a few proofs have been proposed. It is known that Lame equations with unitary monodromy are corresponding to spherical…

In this paper, we study the algebraic form of the symmetric generalized Lam\'e equations which have finite projective monodromy groups. In particular, we consider equations with $3$ regular singular points on a flat torus $T$ which takes…

Motivated by the finite-gap structure of the classical Lam\'{e} equation (1.2) and its central role in mathematical physics, generalized Lam\'{e}-type equations (1.12) are investigated. For the fundamental case $n=1$, a monodromy…

In this paper, the second in a series, we continue to study the generalized Lam\'{e} equation with the Treibich-Verdier potential \begin{equation*} y^{\prime \prime }(z)=\bigg[ \sum_{k=0}^{3}n_{k}(n_{k}+1)\wp(z+\tfrac{…

In this paper, we prove Dahmen and Beukers' conjecture that the number of integral Lam\'{e} equations with index $n$ modulo scalar equivalence with the monodromy group dihedral $D_{N}$ of order $2N$ is given by \[L_{n}(N)=\frac{1}{2}\left(…

We obtain an explicit formula for the number of Lam\'e equations (modulo scalar equivalence) with index $n$ and projective monodromy group of order $2N$, for given $n \in \Z$ and $N \in \N$. This is done by performing the combinatorics of…

We explain how algebraic geometry comes into play in the study of non-linear mean field (singular Liouville) equations $$ \triangle u + e^u = 4\pi \sum_{i = 1}^N \ell_i \delta_{p_i} $$ on a flat torus $E = \Bbb C/\Lambda$, where $N, \ell_1,…

In this paper, the third in a series, we continue to study the generalized Lam\'{e} equation H$(n_0,n_1,n_2,n_3;B)$ with the Darboux-Treibich-Verdier potential \begin{equation*} y^{\prime \prime }(z)=\bigg[…

If $A$ and $B$ are $n$- and $m$-representation finite $k$-algebras, then their tensor product $\Lambda = A\otimes_k B$ is not in general $(n+m)$-representation finite. However, we prove that if $A$ and $B$ are acyclic and satisfy the weaker…

We show that for every finite colouring of the natural numbers there exists $a,b >1$ such that the triple $\{a,b,a^b\}$ is monochromatic. We go on to show the partition regularity of a much richer class of patterns involving exponentiation.…

We introduce a sequent calculus with a simple restriction of Lambek's product rules that precisely captures the classical Tamari order, i.e., the partial order on fully-bracketed words (equivalently, binary trees) induced by a…

Slattery has generalized Brauer's theory of p-blocks of finite groups to pi-blocks of pi-separable groups where pi is a set of primes. In this setting we show that the order of a defect group of a pi-block B is bounded in terms of the…

In this paper it is stressed that there is no {\em physical} reason for symmetries to be linear and that Lie group theory is therefore too restrictive. We illustrate this with some simple examples. Then we give a readable review on the…

This text gives a rough, but linear summary covering some key definitions, notations, and propositions from Lambda Calculus: Its Syntax and Semantics, the classical monograph by Barendregt. First, we define a theory of untyped extensional…

For any $\Lambda>0$, let $\mathcal{M}_{n,\Lambda}$ denote the space containing all locally Lipschitz minimal graphs of dimension $n$ and of arbitrary codimension $m$ in Euclidean space $\mathbb{R}^{n+m}$ with uniformly bounded 2-dilation…

We develop a theory connecting the following three areas: (a) the mean field equation (MFE) $\triangle u + e^u = \rho\, \delta_0$, $\rho \in \mathbb R_{>0}$ on flat tori $E_\tau = \mathbb C/(\mathbb Z + \mathbb Z\tau)$, (b) the classical…

Let K be an abstract elementary classes which has arbitrarily large models and satisfies the amalgamation and joint embedding properties. Theorem 1. Suppose K is \chi-tame. If K is categorical in some \lambda^+ >LS(K) then it is categorical…

We define a Weil-\'etale complex with compact support for duals (in the sense of the Bloch dualizing cycles complex $\mathbb{Z}^c$) of a large class of $\mathbb{Z}$-constructible sheaves on an integral $1$-dimensional proper arithmetic…

We introduce a sequent calculus with a simple restriction of Lambek's product rules that precisely captures the classical Tamari order, i.e., the partial order on fully-bracketed words (equivalently, binary trees) induced by a…

We generalize several classical theorems in extremal combinatorics by replacing a global constraint with an inequality which holds for all objects in a given class. In particular we obtain generalizations of Tur\'an's theorem, the…