Namely, we prove that the random planar map converges in the scaling limit to Liouville quantum gravity under a discrete conformal embedding which we call the Cardy embedding.
Namely, we prove that this problem is both NP-hard and coNP-hard, and is in PSPACE.
It is as though the sciences had done what the humanities never could have achieved, namely, to prove demonstrably the validity of this concern.
Namely, we prove that the scaled l2 distance between any low energy local minimizers of the TAP free energy and the mean of the Gibbs measure vanishes in the large size limit.
Namely, we prove that this property is equivalent to the Poincaré inequality, and that it is also equivalent to the solvability of −Δu=f in L∞ for all f∈L∞.
We obtain this as a consequence of a new property of almost transitive spaces with a Schauder basis, namely we prove that in such spaces the unit vector basis of ℓ22 belongs to the two-dimensional asymptotic structure and we obtain some information about the asymptotic structure in higher dimensions.
In this paper we give a very natural joint generalisation of Wigner's and Molnár's theorems, namely, we prove a characterisation of all (not necessarily bijective) transformations on the Grassmann space which fix the quantity TrPQ (i.e. the sum of the squares of cosines of principal angles) for every pair of rank-n projections P and Q.
Namely, we prove that Γ is removable for Lu=Ψ u) if and only if its Orlicz capacity is zero, that is, the integral ∫B dx Ψ(∫Γ |x−y|2−d ν dy)) is equal to 0 or ∞ for every measure ν concentrated on Γ, where B stands for any ball containing Γ.
Namely, we prove the following.
Namely, we prove that every multifunction with closed values, defined on a closed set and contractive with respect to any one of these measures, has the origin as a fixed point.
Namely, we prove that there are constants and such that for any th Lebesgue integrable or continuous function defined on the sphere, where is the th modulus of smoothness of.
