In this note we summarize some necessary and sufficient conditions for subspaces invariant with respect to the backward shift to contain smooth functions. We also discuss smoothness of moduli of functions in such subspaces.

An extremal plunsubharmonic function that generalizes Green's function of one complex variable is introduced and studied. It is used to define a biholomorphically invariant pseudodistance.

The wave functions of a quantum isotropic harmonic oscillator in N-space modified by barriers at the coordinate hyperplanes can be expressed in terms of certain generalized spherical harmonics. These are associated with a product-type weight function on the sphere. Their analysis is carried out by means of differential-difference operators. The symmetries of this system involve the Weyl group o...

Divergence is a discrepancy measure between two objects, such as functions, vectors, matrices, and so forth. In particular, divergences defined on probability distributions are widely employed in probabilistic forecasting. As the dissimilarity measure, the divergence should satisfy some conditions. In this paper, we consider two conditions: The first one is the scale-invariance property and the...

Let V be a real finite dimensional representation of a compact Lie group G. It is well-known that the algebra R[V ] of G-invariant polynomials on V is finitely generated, say by σ1, . . . , σp. Schwarz [38] proved that each G-invariant C-function f on V has the form f = F (σ1, . . . , σp) for a Cfunction F on R. We investigate this representation within the framework of Denjoy–Carleman classes....

The main goal of this paper is to answer question 1.10 and settle conjecture 1.11 of BenjaminiLyons-Schramm [BLS99] relating harmonic Dirichlet functions on a graph to those on the infinite clusters in the uniqueness phase of Bernoulli percolation. We extend the result to more general invariant percolations, including the Random-Cluster model. We prove the existence of the nonuniqueness phase f...

It is shown that there are no nilpotent invariants in N = 4 analytic superspace for n ≤ 4 points. It is argued that there is (at least) one such invariant for n = 5 points which is not invariant under U (1) Y. The consequences of these results are that the n = 2 and 3 point correlation functions of the N = 4 gauge-invariant operators which correspond to KK multiplets in AdS supergravity are giv...

This paper explores the existence of affine invariant descent directions for unconstrained minimization. While there may exist several affine invariant descent directions for smooth functions f at a given point, it is shown that for quadratic functions there exists exactly one invariant descent direction in the strictly convex case and generally none in the nondegenerate indefinite case. These ...

The crossed channels of generalized reaction γN → γN have been considered. The transformation coefficients from the independent helicity amplitudes to the invariant functions are calculated. The explicit expressions for invariant functions have been obtained with the subject to the contribution of Born diagrams in s-, u-, and t-channel and six resonances in sand u-channel. It has been shown tha...

Momentum space Ward identities are derived for the amputated n-point Green’s functions in 3 + 1 dimensional non-relativistic conformal field theory. For n = 4 and 6 the implications for scattering amplitudes (i.e. on-shell amputated Green’s functions) are considered. Any scale invariant 2-to-2 scattering amplitude is also conformally invariant. However, conformal invariance imposes constraints ...