We describe the exponent of a group-theoretical fusion category C = C(G,ω, F, α) associated to a finite group G in terms of group cohomology. We show that the exponent of C divides both e(ω) expG and (expG), where e(ω) is the cohomological order of the 3-cocycle ω. In particular exp C divides (dim C).

S (in alphabetic order by speaker surname) Speaker: Boris Adamczewski (Institu Camille Jordan & CNRS) Title: Combinatorics on words and Diophantine approximation Abstract: A very fruitful interplay between combinatorics on words and Diophantine approximation comes up with the use of numeration systems. Finite and infinite words occur naturally in Number Theory when one considers the expansion o...

For any fixed power exponent, it is shown that the first digits of powerful integer powers follow a generalized Benford law (GBL) with size-dependent exponent that converges asymptotically to a GBL with the inverse double power exponent. In particular, asymptotically as the power goes to infinity these sequences obey Benford's law. Moreover, the existence of a one-parametric size-dependent expo...

We review and discuss some different techniques for describing local dispersion properties in fluids. A recent Lagrangian diagnostics, based on the Finite Scale Lyapunov Exponent (FSLE), is presented and compared to the Finite Time Lyapunov Exponent (FTLE), and to the Okubo-Weiss (OW) and Hua-Klein (HK) criteria. We show that the OW and HK are a limiting case of the FTLE, and that the FSLE is t...

An analog of the Falconer distance problem in vector spaces over finite fields asks for the threshold α > 0 such that |∆(E)| & q whenever |E| & q, where E ⊂ Fq , the d-dimensional vector space over a finite field with q elements (not necessarily prime). Here ∆(E) = {(x1 − y1) 2 + · · ·+ (xd − yd) 2 : x, y ∈ E}. The second listed author and Misha Rudnev ([4]) established the threshold d+1 2 , an...

We say that a space X admits a homology exponent if there exists an exponent for the torsion subgroup of H * (X; Z). Our main result states if an H-space of finite type admits a homology exponent, then either it is, up to 2-completion, a product of spaces of the form BZ/2 r , S 1 , CP ∞ , and K(Z, 3), or it has infinitely many non-trivial homotopy groups and k-invariants. We then show with the ...

Numerical studies of the Anderson transition are based on finite-size scaling analysis of the smallest positive Lyapunov exponent. We prove numerically that the same scaling holds also for higher Lyapunov exponents. This scaling supports the hypothesis of the one-parameter scaling of the conductance distribution. From collected numerical data for quasi one dimensional systems up to system size ...

To study dissipative quantum mechanics we adopt the Caldeira-Leggett model where environmental harmonic oscillators are coupled to the target variable. After integrating out the environmental degrees of freedom, effective interactions of infinitely long range appear. As the simplest example we take 2-state model for the target variable, and then we investigate the 1-dimensional Ising model with...

We provide a comprehensive view of various phase transitions in random K-satisfiability problems solved by stochastic-local-search algorithms. In particular, we focus on the finite-size scaling (FSS) exponent, which is mathematically important and practically useful in analyzing finite systems. Using the FSS theory of nonequilibrium absorbing phase transitions, we show that the density of unsat...