Two classical upper bounds on the Shannon capacity of graphs are the θ-function due to Lovász and the minrank parameter due to Haemers. We provide several explicit constructions of n-vertex graphs with a constant θ-function and minrank at least n for a constant δ > 0 (over various prime order fields). This implies a limitation on the θ-function-based algorithmic approach to approximating the mi...

Cross Coherence time frequency transform and independent component analysis (ICA) method were used to analyse the electroencephalogram (EEG) signals in resting and action states during open and close eyes conditions. From the topographical scalp distributions of delta, theta, alpha, and beta power spectrum can clearly discriminate between the signal when the eyes were open or closed, but it was...

Some recent work of Gross and Prasad [14] suggests that the root numbers attached to certain symplectic representations of the Weil-Deligne group of a local field F control certain branching rules for representations of orthogonal groups over F . On a global level, this local phenomenon should have implications for the structure of the value at the center of symmetry for certain L-functions of ...

We show that, for the M\"obius function $\mu(n)$, we have $$ \sum_{x < n\leq x+x^{\theta}}\mu(n)=o(x^{\theta}) any $\theta>0.55$. This improves on a result of Ramachandra from 1976, which is valid $\theta>7/12$. Ramachandra's corresponded to Huxley's $7/12$ exponent prime number theorem in short intervals. The main new idea leading improvement using Ramar\'e's identity extract small factor $n$-...

An empirical exploration of five of Ramanujan’s intriguing false theta function identities leads to unexpected instances of Bailey’s transform which, in turn, lead to many new identities for both false and partial theta functions.

Given any number eld and Dirichlet character for that eld, we construct a theta function on a restricted product of p-adic half-planes. An explicit formula for the theta multiplier is obtained through local calculations.

In this paper we consider theta series with the highest order of vanishing at the cusp ∞. When the level is a power of 2, these theta series are the m-th powers of a certain theta function with characteristic, related to the lattice √ 2kZ. Instead if the level is a power of 3, these theta series are the m/2-th powers of a theta series associated to the 2−dimensional root lattice A2 with charact...

In this paper, we consider a problem in calculus of variations motivated by quantitative isoperimetric inequality the plane. More precisely, aim article is computation minimum variational $$\begin{aligned}\inf _{u\in {\mathcal {W}}} \frac{\displaystyle \int _{-\pi }^{\pi }[(u')^2-u^2]d\theta }{\displaystyle \left[ } |u| d\theta \right] ^2} \end{aligned}$$ where function $$u\in {W}}$$ $$H^1(-\pi...

1 Overview We will go over some of the classical valid inequalities for the stable set poly-hedron and then provide the proof of Lovász's Theta function and show SDP techniques can be used to optimize over TH(G) in polynomial time.