54,483 research outputs found

### Trace formulas for a class of compact complex surfaces

We give the trace formulas of weight $k$ for cocompact, torsion-free discrete
subgroups of $SU(2, 1)$ and prove the analogue of the Riemann hypothesis on
compact complex surfaces $M$ with $c_1^2(M)=3 c_2(M)$, where $c_i(M)$ is the
$i$-th Chern class of $M$, $c_2(M)$ is a multiple of three and $c_2(M)>0$.Comment: 63 page

### Equidistribution of expanding translates of curves in homogeneous spaces with the action of $(\mathrm{SO}(n,1))^k$

Given a homogeneous space $X = G/\Gamma$ with $G$ containing the group $H =
(\mathrm{SO}(n,1))^k$. Let $x\in X$ such that $Hx$ is dense in $X$. Given an
analytic curve $\phi: I=[a,b] \rightarrow H$, we will show that if $\phi$
satisfies certain geometric condition, then for a typical diagonal subgroup $A
=\{a(t): t \in \mathbb{R}\} \subset H$ the translates $\{a(t)\phi(I)x: t >0\}$
of the curve $\phi(I)x$ will tend to be equidistributed in $X$ as $t
\rightarrow +\infty$. The proof is based on the study of linear representations
of $\mathrm{SO}(n,1)$ and $H$.Comment: 19 page

### Dedekind $\eta$-function, Hauptmodul and invariant theory

We solve a long-standing open problem with its own long history dating back
to the celebrated works of Klein and Ramanujan. This problem concerns the
invariant decomposition formulas of the Hauptmodul for $\Gamma_0(p)$ under the
action of finite simple groups $PSL(2, p)$ with $p=5, 7, 13$. The cases of
$p=5$ and $7$ were solved by Klein and Ramanujan. Little was known about this
problem for $p=13$. Using our invariant theory for $PSL(2, 13)$, we solve this
problem. This leads to a new expression of the classical elliptic modular
function of Klein: $j$-function in terms of theta constants associated with
$\Gamma(13)$. Moreover, we find an exotic modular equation, i.e., it has the
same form as Ramanujan's modular equation of degree $13$, but with different
kinds of modular parametrizations, which gives the geometry of the classical
modular curve $X(13)$.Comment: 46 pages. arXiv admin note: substantial text overlap with
arXiv:1209.178

### Poincar\'{e} series and modular functions for U(n, 1)

We construct infinitely many nonholomorphic automorphic forms and modular
forms associated to a discrete subgroup of infinite covolume of $U(n, 1)$.Comment: 18 page

### Finite Heat conduction in 2D Lattices

This paper gives a 2D hamonic lattices model with missing bond defects, when
the capacity ratio of defects is enough large, the temperature gradient can be
formed and the finite heat conduction is found in the model. The defects in the
2D harmonic lattices impede the energy carriers free propagation, by another
words, the mean free paths of the energy carrier are relatively short. The
microscopic dynamics leads to the finite conduction in the model

### Proximal Gradient Method with Extrapolation and Line Search for a Class of Nonconvex and Nonsmooth Problems

In this paper, we consider a class of possibly nonconvex, nonsmooth and
non-Lipschitz optimization problems arising in many contemporary applications
such as machine learning, variable selection and image processing. To solve
this class of problems, we propose a proximal gradient method with
extrapolation and line search (PGels). This method is developed based on a
special potential function and successfully incorporates both extrapolation and
non-monotone line search, which are two simple and efficient accelerating
techniques for the proximal gradient method. Thanks to the line search, this
method allows more flexibilities in choosing the extrapolation parameters and
updates them adaptively at each iteration if a certain line search criterion is
not satisfied. Moreover, with proper choices of parameters, our PGels reduces
to many existing algorithms. We also show that, under some mild conditions, our
line search criterion is well defined and any cluster point of the sequence
generated by PGels is a stationary point of our problem. In addition, by
assuming the Kurdyka-{\L}ojasiewicz exponent of the objective in our problem,
we further analyze the local convergence rate of two special cases of PGels,
including the widely used non-monotone proximal gradient method as one case.
Finally, we conduct some numerical experiments for solving the $\ell_1$
regularized logistic regression problem and the $\ell_{1\text{-}2}$ regularized
least squares problem. Our numerical results illustrate the efficiency of PGels
and show the potential advantage of combining two accelerating techniques.Comment: This version addresses some typos in previous version and adds more
comparison

### Modular curves, invariant theory and $E_8$

The $E_8$ root lattice can be constructed from the modular curve $X(13)$ by
the invariant theory for the simple group $\text{PSL}(2, 13)$. This gives a
different construction of the $E_8$ root lattice. It also gives an explicit
construction of the modular curve $X(13)$.Comment: 39 pages. arXiv admin note: text overlap with arXiv:1511.0527

### Icosahedron, exceptional singularities and modular forms

We find that the equation of $E_8$-singularity possesses two distinct
symmetry groups and modular parametrizations. One is the classical icosahedral
equation with icosahedral symmetry, the associated modular forms are theta
constants of order five. The other is given by the group $\text{PSL}(2, 13)$,
the associated modular forms are theta constants of order $13$. As a
consequence, we show that $E_8$ is not uniquely determined by the icosahedron.
This solves a problem of Brieskorn in his ICM 1970 talk on the mysterious
relation between exotic spheres, the icosahedron and $E_8$. Simultaneously, it
gives a counterexample to Arnold's $A, D, E$ problem, and this also solves the
other related problem on the relation between simple Lie algebras and Platonic
solids. Moreover, we give modular parametrizations for the exceptional
singularities $Q_{18}$, $E_{20}$ and $x^7+x^2 y^3+z^2=0$ by theta constants of
order $13$, the second singularity provides a new analytic construction of
solutions for the Fermat-Catalan conjecture and gives an answer to a problem
dating back to the works of Klein.Comment: 41 page

### Complex version KdV equation and the periods solution

In this paper, the complex version KdV equation is discussed. The
corresponding coupled equations is a integrable system in the sense of the
bi-Hamiltonian structure, so the complex version KdV equation is integrable. A
new spectral form is given, the periodic solution of the complex version KdV
equation is obtained. It is showed that the periodic solution is the classical
solution

### Exact solutions of nonlinear PDE, nonlinear transformations and reduction nonlinear PDE to a quadrature

A method to construct the exact solution of the PDE is presents, which
combines the two kind methods(the nonlinear transformation and RQ(Reduction the
PDE to a Quadrature problem) method).The nonlinear diffusion equation is chosen
to illustrate the method and the exact solutions are obtained

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