Quantum isometries and noncommutative spheres

We introduce and study two new examples of noncommutative spheres: the half-liberated sphere, and the free sphere. Together with the usual sphere, these two spheres have the property that the corresponding quantum isometry group is "easy", in the representation theory sense. We present as well some general comments on the axiomatization problem, and on the "untwisted" and "non-easy" case.Comment: 16 page

Quantum Symmetries and Strong Haagerup Inequalities

In this paper, we consider families of operators $\{x_r\}_{r \in \Lambda}$ in a tracial C$^\ast$-probability space $(\mathcal A, \phi)$, whose joint $\ast$-distribution is invariant under free complexification and the action of the hyperoctahedral quantum groups $\{H_n^+\}_{n \in \N}$. We prove a strong form of Haagerup's inequality for the non-self-adjoint operator algebra $\mathcal B$ generated by $\{x_r\}_{r \in \Lambda}$, which generalizes the strong Haagerup inequalities for $\ast$-free R-diagonal families obtained by Kemp-Speicher \cite{KeSp}. As an application of our result, we show that $\mathcal B$ always has the metric approximation property (MAP). We also apply our techniques to study the reduced C$^\ast$-algebra of the free unitary quantum group $U_n^+$. We show that the non-self-adjoint subalgebra $\mathcal B_n$ generated by the matrix elements of the fundamental corepresentation of $U_n^+$ has the MAP. Additionally, we prove a strong Haagerup inequality for $\mathcal B_n$, which improves on the estimates given by Vergnioux's property RD \cite{Ve}

Stability of the selfsimilar dynamics of a vortex filament

In this paper we continue our investigation about selfsimilar solutions of the vortex filament equation, also known as the binormal flow (BF) or the localized induction equation (LIE). Our main result is the stability of the selfsimilar dynamics of small pertubations of a given selfsimilar solution. The proof relies on finding precise asymptotics in space and time for the tangent and the normal vectors of the perturbations. A main ingredient in the proof is the control of the evolution of weighted norms for a cubic 1-D Schr\"odinger equation, connected to the binormal flow by Hasimoto's transform.Comment: revised version, 36 page

Quantum Isometries of the finite noncommutative geometry of the Standard Model

We compute the quantum isometry group of the finite noncommutative geometry F describing the internal degrees of freedom in the Standard Model of particle physics. We show that this provides genuine quantum symmetries of the spectral triple corresponding to M x F where M is a compact spin manifold. We also prove that the bosonic and fermionic part of the spectral action are preserved by these symmetries.Comment: 29 pages, no figures v3: minor change

Preliminary water splitting studies on Ag and graphite modified porous structures, as such or decorated with Pt particles

Porous materials were obtained using polyether foam modified with Ag nanowires and graphite paste. Some of those modified porous structures were decorated with Pt particles via double step chronoamperometry. All porous materials were investigated in terms of their electrocatalytic activity for the oxygen and hydrogen evolution reactions (OER and HER) in alkaline medium, after they were inserted into supports made from sintered graphite (for the OER experiments) or Ag wire (for the HER experiments). Electrochemical stability tests were also performed. The results of the OER and HER experiments show that the most catalytically active porous structure is the one modified with graphite paste and Pt particles. Stability tests data show that the porous electrode based on this structure is very stable

Spectral measures of small index principal graphs

The principal graph $X$ of a subfactor with finite Jones index is one of the important algebraic invariants of the subfactor. If $\Delta$ is the adjacency matrix of $X$ we consider the equation $\Delta=U+U^{-1}$. When $X$ has square norm $\leq 4$ the spectral measure of $U$ can be averaged by using the map $u\to u^{-1}$, and we get a probability measure $\epsilon$ on the unit circle which does not depend on $U$. We find explicit formulae for this measure $\epsilon$ for the principal graphs of subfactors with index $\le 4$, the (extended) Coxeter-Dynkin graphs of type $A$, $D$ and $E$. The moment generating function of $\epsilon$ is closely related to Jones' $\Theta$-series.Comment: 23 page

The K-theory of free quantum groups

In this paper we study the K -theory of free quantum groups in the sense of Wang and Van Daele, more precisely, of free products of free unitary and free orthogonal quantum groups. We show that these quantum groups are K -amenable and establish an analogue of the Pimsner–Voiculescu exact sequence. As a consequence, we obtain in particular an explicit computation of the K -theory of free quantum groups. Our approach relies on a generalization of methods from the Baum–Connes conjecture to the framework of discrete quantum groups. This is based on the categorical reformulation of the Baum–Connes conjecture developed by Meyer and Nest. As a main result we show that free quantum groups have a γ -element and that γ=1 . As an important ingredient in the proof we adapt the Dirac-dual Dirac method for groups acting on trees to the quantum case. We use this to extend some permanence properties of the Baum–Connes conjecture to our setting

Nondispersive solutions to the L2-critical half-wave equation

We consider the focusing $L^2$-critical half-wave equation in one space dimension $$i \partial_t u = D u - |u|^2 u,$$ where $D$ denotes the first-order fractional derivative. Standard arguments show that there is a critical threshold $M_* > 0$ such that all $H^{1/2}$ solutions with $\| u \|_{L^2} < M_*$ extend globally in time, while solutions with $\| u \|_{L^2} \geq M_*$ may develop singularities in finite time. In this paper, we first prove the existence of a family of traveling waves with subcritical arbitrarily small mass. We then give a second example of nondispersive dynamics and show the existence of finite-time blowup solutions with minimal mass $\| u_0 \|_{L^2} = M_*$. More precisely, we construct a family of minimal mass blowup solutions that are parametrized by the energy $E_0 >0$ and the linear momentum $P_0 \in \R$. In particular, our main result (and its proof) can be seen as a model scenario of minimal mass blowup for $L^2$-critical nonlinear PDE with nonlocal dispersion.Comment: 51 page

Dimensional distribution of PM2.5 and PM10 in the road proximity

The particulate matter (PM) is comprised of two kinds of particles, classified after their dimensions, the PM2.5 which encompasses particles with sizes smaller than 2.5 µm and the PM10 with particles ranging in size from 2.5 µm to 10 µm. As previous studies have shown, PMs have an undeniable influence, dependent on the exposure time, upon the health of the human cardiopulmonary system. In this study we focused on the dimensional distribution of PMs and the influence of altitude on their numbers. Our detailed investigation lead us to the conclusion that the particle number is increasing at higher altitudes and also that PM2.5, which represents a greater health risk factor, is much more abundant than PM10