Given a fixed$p$≠2, we prove a simple and effective characterization of all radial multipliers of $ \mathcal{F}{L^p}\left( {{\mathbb{R}^d}} \right) $ , provided that the dimension$d$is sufficiently large. The method also yields new$L$^{$q$}space-time regularity results for solutions of the wave equation in high dimensions.

In this paper, we prove the DonohoStark uncertainty principle for locally compact quantum groups and characterize the minimizer which are bi-shifts of group-like projections. We also prove the HirschmanBeckner uncertainty principle for compact quantum groups and discrete quantum groups. Furthermore, we show Hardy's uncertainty principle for locally compact quantum groups in terms of bi-shifts of group-like projections.

No abstract uploaded!

In 1975, P. Erd\"{o}s proposed the problem of determining the maximum number $f(n)$ of edges in a graph with $n$ vertices in which any two cycles are of different lengths. In this paper, it is proved that $$f(n)\geq n+\frac{107}{3}t+\frac{7}{3}$$ for $t=1260r+169 \,\ (r\geq 1)$ and $n \geq \frac{2119}{4}t^{2}+87978t+\frac{15957}{4}$. Consequently, $\liminf\sb {n \to \infty} {f(n)-n \over \sqrt n} \geq \sqrt {2 + \frac{7654}{19071}},$ which is better than the previous bounds $\sqrt 2$ (Shi, 1988), $\sqrt {2.4}$ (Lai, 2003). The conjecture $\lim_{n \rightarrow \infty} {f(n)-n\over \sqrt n}=\sqrt {2.4}$ is not true.

We demonstrate that a supersymmetric and parity violating version of Vasiliev's higher spin gauge theory in AdS4 admits boundary conditions that preserve $\mathcal{N}=0,1,2,3,4$ or 6 supersymmetries. In particular, we argue that the Vasiliev theory with U(M) Chan–Paton and $\mathcal{N}=6$ boundary condition is holographically dual to the 2+1 dimensional U(N)_k × U(M)_{−k} ABJ theory in the limit of large N, k and finite M. In this system all bulk higher spin fields transform in the adjoint of the U(M) gauge group, whose bulk t'Hooft coupling is M/N. Analysis of boundary conditions in Vasiliev theory allows us to determine exact relations between the parity breaking phase of Vasiliev theory and the coefficients of two and three point functions in Chern–Simons vector models at large N. Our picture suggests that the supersymmetric Vasiliev theory can be obtained as a limit of type IIA string theory in AdS_4\times \mathbb {CP}^3, and that the non-Abelian Vasiliev theory at strong bulk 't Hooft coupling smoothly turn into a string field theory. The fundamental string is a singlet bound state of Vasiliev's higher spin particles held together by U(M) gauge interactions. This is illustrated by the thermal partition function of free ABJ theory on a two sphere at large M and N even in the analytically tractable free limit. In this system the traces or strings of the low temperature phase break up into their Vasiliev particulate constituents at a U(M) deconfinement phase transition of order unity. At a higher temperature of order $T=\sqrt{\frac{N}{M}}$ Vasiliev's higher spin fields themselves break up into more elementary constituents at a U(N) deconfinement temperature, in a process described in the bulk as black hole nucleation. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Higher spin theories and holography'.