In this paper, we introduce theta-functions of VOA-modules and show that the space spanned by them has a modular invariance property. If a VOA is a lattice VOA VL associated with an even lattice L (cf. [FLM]), then the above theta functions coincide to the ordinary theta functions θ(τ, ~v) = ∑ m∈L+β e πıτ+ for < β,L >⊆ Z, ~v ∈ CL and τ ∈ H.

Abstract This paper focuses on the most prominent theta roles used in the Sindhi language. The study attempts to answer to the research question, ‘How are theta roles prominently used in the Sindhi language?’ The data come from the young native Sindhi speakers. Each verb phrase in the data is examined with the help of Carnie’s (2007) ‘Theta Roles and Thematic Relations’ in order to find the pro...

Abstract This paper focuses on the most prominent theta roles used in the Sindhi language. The study attempts to answer to the research question, ‘How are theta roles prominently used in the Sindhi language?’ The data come from the young native Sindhi speakers. Each verb phrase in the data is examined with the help of Carnie’s (2007) ‘Theta Roles and Thematic Relations’ in order to find the pro...

It is explained how the Mordell integral ∫ R e −2πzx cosh(πx) dx unifies the mock theta functions, partial (or false) theta functions, and some of Zagier’s quantum modular forms. As an application, we exploit the connections between q-hypergeometric series and mock and partial theta functions to obtain finite evaluations of the Mordell integral for rational choices of τ and z. 1. The Mordell In...

The theta rhythm is one of the largest and most sinusoidal activity patterns in the brain. Here I survey progress in the field of theta rhythms research. I present arguments supporting the hypothesis that theta rhythms emerge owing to intrinsic cellular properties yet can be entrained by several theta oscillators throughout the brain. I review behavioral correlates of theta rhythms and consider...

The focus of this paper is on Ahlfors Q-regular compact sets $$E\subset \mathbb {R}^n$$ such that, for each $$Q-2<\alpha \le 0$$ , the weighted measure $$\mu _{\alpha }$$ given by integrating density $$\omega (x)=\text {dist}(x, E)^\alpha $$ yields a Muckenhoupt $$\mathcal {A}_p$$ -weight in ball B containing E. For E we show existence bounded linear trace operator acting from $$W^{1,p}(B,...

For a $d$-dimensional random vector $X$, let $p_{n, X}(\theta)$ be the probability that convex hull of $n$ independent copies $X$ contains given point $\theta$. We provide several sharp inequalities regarding and $N_X(\theta)$ denoting smallest for which X}(\theta)\ge1/2$. As main result, we derive totally general inequality $1/2 \le \alpha_X(\theta)N_X(\theta)\le 3d + 1$, where $\alpha_X(\thet...

is a modular form of weight n/2 on Γ0(N), where N is the level of Q, i.e. NQ−1 is integral and NQ−1 has even diagonal entries. This was proved by Schoeneberg [5] for even n and by Pfetzer [3] for odd n. Shimura [6] uses the Poisson summation formula to generalize their results for arbitrary n and he also computes the theta multiplier explicitly. Stark [8] gives a different proof by converting θ...

We show that for Lebesgue almost all $d$-tuples $(\theta\_1,\ldots,\theta\_d)$, with $|\theta\_j|>1$, any self-affine measure a homogeneous non-degenerate iterated function system ${Ax+a\_j}\_{j=1}^m$ in\~$\mathbb{R}^d$, where $A^{-1}$ is diagonal matrix the entries has power Fourier decay at infinity.