Quantum theta functions were introduced by the author in [Ma1]. They are certain elements in the function rings of quantum tori. By definition, they satisfy a version of the classical functional equations involving shifts by the multiplicative periods. This paper shows that for a certain subclass of period lattices (compatible with the quantization form), quantum thetas satisfy an analog of ano...

0. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 1. Notation and terminology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 2. Modular forms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 3. Discrim...

We study a model of proxy voting where the candidates, voters, and proxies are all located on real line, instead directly, each voter delegates its vote to closest proxy. The goal is find set that theta-representative, which entails for any anywhere favorite candidate within distance theta This property guarantees strong form representation as voters not required be fixed in advance, or even fi...

The $$(1 + (\lambda ,\lambda ))$$ genetic algorithm is a younger evolutionary trying to profit also from inferior solutions. Rigorous runtime analyses on unimodal fitness functions showed that it can indeed be faster than classical algorithms, though these simple problems the gains were only moderate. In this work, we conduct first analysis of multimodal problem class, jump benchmark. We show w...

We study exceptional sets of the local time continuous-time simple random walk in scaled-up (by N) versions $$D_N\subseteq {\mathbb {Z}}^2$$ bounded open domains $$D\subseteq {R}}^2$$ . Upon exit from $$D_N$$ , lands on a “boundary vertex” and then reenters through boundary edge next step. In parametrization by at we prove that, times corresponding to $$\theta $$ -multiple cover suitably define...

Motivated by the discovery that the eighth root of the theta series of the E 8 lattice and the 24th root of the theta series of the Leech lattice both have integer coefficients, we investigate the question of when an arbitrary element f ∈ Z[[x]]

Tke theta function of a sphere packing gives the number of centers at each distance from the origin. The theta functions of a number of important packings (An, D, , E,, the Leech lattice, and others) and tables of the first fifty or so of their coefficients are given in this paper.

It is shown that if $A \subseteq \mathbb{R}^3$ a Borel set of Hausdorff dimension $\dim A \in (3/2,5/2)$, then for a.e. $\theta [0,2\pi)$ the projection $\pi_{\theta}(A)$ $A$ onto 2-dimensional plane orthogonal to $\frac{1}{\sqrt{2}}(\cos \theta, \sin 1)$ satisfies \pi_{\theta}(A) \geq \max\left\{\frac{4\dim A}{9} + \frac{5}{6},\frac{2\dim A+1}{3} \right\}$. This improves bound Oberlin and Ober...

In this article, we introduce a new approach towards the statistical learning problem $\operatorname{argmin}_{\rho(\theta) \in \mathcal P_{\theta}} W_{Q}^2 (\rho_{\star},\rho(\theta))$ to approximate target quantum state $\rho_{\star}$ by set of parametrized states $\rho(\theta)$ in $L^2$-Wasserstein metric. We solve estimation considering Wasserstein natural gradient flows for density operator...