Let $z=x+iy \in \mathbb{H}:=\{z= x+ i y\in\mathbb{C}: y>0\}$ and $ \theta (s;z)=\sum_{(m,n)\in\mathbb{Z}^2 } e^{-s \frac{\pi }{y }|mz+n|^2}$ be the theta function associated with the lattice $\Lambda ={\mathbb Z}\oplus z{\mathbb Z}$. In this paper we consider the following pair of minimization problems \begin{equation}\aligned \min_{ \mathbb{H} } \theta (2;\frac{z+1}{2})+\rho\theta (1;z),\;\;\rho\in[0,\infty), \\ \min_{ \mathbb{H} } \theta (1; \frac{z+1}{2})+\rho\theta (2; z),\;\;\rho\in[0,\infty), \endaligned\end{equation} where the parameter $\rho\in[0,\infty)$ represents the competition of two intertwining lattices, the particular selection of the parameters $s=1,2$ is determined by the physical model, which can be generalized by our strategy and method proposed here. We find that as $\rho$ varies the optimal lattices admit a novel pattern: they move from rectangular (the ratio of long and short sides changes from $\sqrt3$ to 1 continuously), square, rhombus (the angle changes from $\pi/2$ to $\pi/3$ continuously) to hexagonal continuously; geometrically, up to an invariant group(a subgroup of the classical modular group), they move continuously on a special curve ; furthermore, there exists a closed interval of $\rho$ such that the optimal lattices is always a square lattice. This is the first, novel and also the complete result on the minimizer problem for theta functions with parameter $\rho$. This is in sharp contrast to optimal lattice shapes for single theta function ($\rho=\infty$ case), for which the hexagonal lattice prevails. As a consequence, we give a partial and positive answer to optimal lattice arrangements of vortices in competing systems of Bose-Einstein condensates as conjectured (and numerically and experimentally verified) by Mueller-Ho \cite{Mue2002}, this is the first progress on Mueller-Ho conjecture.

In 1962 Sprind\v{z}uk proved Mahler's conjecture in both the real and complex cases. Baker gave a generalized result by using a modified version of Sprind\v{z}uk's method. Later, Baker and Schmidt derived the Hausdorff dimensions of sets which are defiend in terms of approximation by algebraic numbers of bounded degrees by using Baker's theorem. In this article we will prove two analogue theorems in the fields of formal power series over finite finite fields.

Following the method developed by Waldspurger and Beuzart-Plessis in their proof of the local Gan-Gross-Prasad conjecture, we prove a local trace formula for the Ginzburg-Rallis model. By applying this trace formula, we prove the multiplicity one theorem for the Ginzburg-Rallis model over the tempered Vogan L-packets. In some cases, we also prove the epsilon dichotomy conjecture which gives a relation between the multiplicity and the exterior cube epsilon factor. This is a sequel work of [Wan15] in which we proved the geometric side of the trace formula.

The study of the present paper is inspired by Gangl, Kaneko and Zagier's result of the connection with double zeta values and modular forms. We introduce double Eisenstein series $E_{r,s}$ in positive characteristic with double zeta values $\zeta_A(r,s)$ as their constant term and compute the t-expansions of the double Eisenstein series. Moreover, we derive the shuffle relations of double Eisenstein series which match the shuffle relations of double zeta values in \cite{Huei2015}.

For each positive characteristic multiple zeta value defined by Thakur, the first and third authors constructed a t-module such that a certain coordinate of a logarithmic vector of a specified algebraic point is a rational multiple of that multiple zeta value. The main result in this paper gives explicit formulae for all of the coordinates of this logarithmic vector in terms of Taylor coefficients of t-motivic multiple zeta values and t-motivic Carlitz multiple star polylogarithms.