We consider a symmetric positive definite weakly exchangeable infinite random matrix whose elements take a finite number of values and we prove that if the distribution of the matrix satisfies the Ghirlanda-Guerra identities then it is ultrametric with probability one.

Based on the Stein method and a general integration by parts framework we derive various bounds on the distance between probability measures. We show that this framework can be implemented on the Poisson space by covariance identities obtained from the Clark-Ocone representation formula and derivation operators. Our approach avoids the use of the inverse of the Ornstein-Uhlenbeck operator as in...

0; n < m for n and m non-negative integers. Binomial coefficients play an important role in many areas of mathematics, including number theory, statistics and probability. Reciprocal binomial coefficients are also prolific in the mathematical literature and many results on reciprocals of binomial coefficient identities may be seen in the papers of Mansour [1], Pla [2], Rockett [3], Sury [5], Su...

The Wnt and Notch signalling pathways represent two major channels of communication used by animal cells to control their identities and behaviour during development. A number of reports indicate that their activities are closely intertwined during embryonic development. Here, we review the evidence for this relationship and suggest that Wnt and Notch ('Wntch') signalling act as components of a...

We use supersymmetric Ward identities to relate multi-gluon helicity amplitudes involving a pair of massive quarks to amplitudes with massive scalars. This allows to use the recent results for scalar amplitudes with an arbitrary number of gluons obtained by on-shell recursion relations to obtain scattering amplitudes involving top quarks. [email protected] [email protected]...

Jarzynski’s identity relates the equilibrium free energy difference ∆F to the work W carried out on a system during a non-equilibrium transformation. In physics literature, the identity is usually written in the form: 〈e−βW 〉 = e−β∆F , where the average is said to be taken over all trajectories in the phase space. The identity in this form has been derived in different ways and published by man...