We describe a new series of identities, which hold for certain general theta series, in two completely independent variables. We provide explicit examples of these identities involving the Dedekind eta function, Jacobi theta functions, and various theta functions of Ramanujan. Introduction Let z ∈ H = {x+ yi : x, y ∈ R, y > 0} and for each x ∈ R set q = exp(2πixz) and e(x) = exp(2πix). The Dede...

We reanalyse the computation of the cosmological constant Λ at two loops in recently proposed Superstring models without massless gravitini, both in the theta-function based formalism and by a detailed computation in the more explicit hyperelliptic description of the underlying genus two Riemann surface. Λ is expressed as the integral over the surface moduli of an amplitude which is zero if sus...

In this article, we further consider the above system. In particular, we give a sufficient condition under which the above system is Kato chaotic for $eta=0$ and a necessary condition for the above system to be Kato chaotic for $eta=0$. Moreover, it is deduced that for $eta=0$, if $Theta$ is P-chaotic then so is this system, where a continuous map $Theta$ from a compact metric space $Z$ to itse...

Abstract Given $$E_0, E_1, F_0, F_1, E$$ E 0 , 1 F rearrangement invariant function spaces, $$a_0$$ a , $$a_1$$ $$\mathrm {b}_0$$ b {b}_1$$ {b}$$ slowly varying f...

Let $A_\Phi$ be a matrix valued truncated Toeplitz operator -- the compression of multiplication to vector model space $H^2(E)\ominus \Theta H^2(E)$, where $\Theta$ is non constant inner function. Under supplementary assumptions, we find necessary and sufficient condition that product $A_\Phi A_\Psi$ itself operator.

In this paper, some results of Singh, Gopalakrishna and Kulkarni (1970s) have been extended to higher order derivatives. It has been shown that, if $sumlimits_{a}Theta(a, f)=2$ holds for a meromorphic function $f(z)$ of finite order, then for any positive integer $k,$ $T(r, f)sim T(r, f^{(k)}), rrightarrowinfty$ if $Theta(infty, f)=1$ and $T(r, f)sim (k+1)T(r, f^{(k)}), rrightarrowinfty$ if $Th...

Hippocampal theta has been related to locomotor speed, attention, anxiety, sensorimotor integration and memory among other emergent phenomena. One difficulty in understanding the function of theta is that the hippocampus (HPC) modulates voluntary behavior at the same time that it processes sensory input. Both functions are correlated with characteristic changes in theta indices. The current rev...

We report experimental measurements of particle dynamics on slowly sheared granular matter in a three-dimensional Couette cell. A closely packed ensemble of transparent spherical beads is confined by an external pressure and filled with fluid to match both the density and refractive index of the beads. This allows us to track tracer particles embedded in the system and obtain three-dimensional ...