This article is devoted to the study of continuous colorings of the n-element subsets of a Polish space. The homogeneity number hm(c) of an n-coloring c : [X]n → 2 is the least size of a family of c-homogeneous sets that covers X. An n-coloring is uncountably homogeneous if hm(c) > א0. Answering a question of B. Miller, we show that for every n > 1 there is a finite family B of continuous n-col...

We prove that there is only one imprimitive homogeneous simple unstable 3-graph with finite classes such that all predicates are realised in the union of two classes. 1 Definitions and context. Definition 1.1. An n-graph is a structure (M,R1, . . . , Rn) in which each Ri is binary, irreflexive and symmetric; also, for all distinct x, y ∈ M exactly one of the Ri holds and n ≥ 2. We assume that a...

Let N , N1 and N2 be point processes such that N1 is obtained from N by homogeneous independent thinning and N2 = N −N1. We give a new elementary proof that N1 and N2 are independent if and only if N is a Poisson point process. We also present an application of this result to test if a homogeneous point process is a Poisson point process.

Koszul property was generalized to homogeneous algebras of degree N > 2 in [5], and related to N -complexes in [7]. We show that if the N -homogeneous algebra A is generalized Koszul, AS-Gorenstein and of finite global dimension, then one can apply the Van den Bergh duality theorem [23] to A, i.e., there is a Poincaré duality between Hochschild homology and cohomology of A, as for N = 2. Mathem...

In this paper we study a two-phase free boundary problem for a semilinear elliptic equation on a bounded domain $Dsubset mathbb{R}^{n}$ with smooth boundary‎. ‎We give some results on the growth of solutions and characterize the free boundary points in terms of homogeneous harmonic polynomials using a fundamental result of Caffarelli and Friedman regarding the representation of functions whose ...

In this paper, we prove superpolynomial lower bounds for the class of homogeneous depth 4 arithmetic circuits. We give an explicit polynomial in VNP of degree n in n variables such that any homogeneous depth 4 arithmetic circuit computing it must have size n . Our results extend the works of Nisan-Wigderson [NW95] (which showed superpolynomial lower bounds for homogeneous depth 3 circuits), Gup...

We consider self-similar statistical ensembles with the phase space whose volume is invariant under the deformation that squeezes (expands) the coordinate and expands (squeezes) the momentum. Related probability distribution function is shown to possess a discrete symmetry with respect to manifold action of the Jackson derivative to be a homogeneous function with a self-similarity degree q fixe...

We obtain expressions for the vacuum expectations of the energy-momentum tensor of the scalar field with an arbitrary coupling to the curvature in an N-dimensional homogeneous isotropic space for the vacuum determined by diagonalization of the Hamiltonian. We generalize the n-wave procedure to N-dimensional homogeneous isotropic space-time. Using the dimensional regularization, we investigate t...