A Spectral Calculus for Lorentz Invariant Measures on Minkowski Space

This paper presents a spectral calculus for computing the spectra of causal Lorentz invariant Borel complex measures on Minkowski space, thereby enabling one to compute their densities with respect to Lebesque measure. The spectra of certain elementary convolutions involving Feynman propagators of scalar particles are computed. It is proved that the convolution of arbitrary causal Lorentz invariant Borel complex measures exists and the product of such measures exists in a wide class of cases. Techniques for their computation in terms of their spectral representation are presented.

On approximations of the point measures associated with the Brownian web by means of the fractional step method and the discretization of the initial interval

UDC 519.21 We establish the rate of weak convergence in the fractional step method for the Arratia flow in terms of the Wasserstein distance between the images of the Lebesque measure under the action of the flow. We introduce finite-dimensional densities that describe sequences of collisions in the Arratia flow and derive an explicit expression for them. With the initial interval discretized, we also discuss the convergence of the corresponding approximations of the point measure associated with the Arratia flow in terms of such densities.

On the instability of the Millionshchikov linear systems depending on a real parameter

The present article considers one-parameter families of second-order linear differential systems with a coefficient matrix depending on the real parameter, which is a diagonal matrix at each odd time interval of unit length. The Cauchy matrix is the rotation matrix at each odd time interval, whereas the angle is the sum of a parameter value and some real number. Earlier, it has been has proved that the upper Lyapunov exponent of each such a system, which is considered to be the function of parameter, is positive on the set of the positive Lebesque measure if the diagonal part of the coefficient matrix is independent on a parameter and separated from zero. The proof of this result essentially uses a complex matrix of special type. In recent article, the author has given another way to prove this theorem based on implementing the Parseval equality for trygonometric sums. Besides, the author considers the special case of the above systems. Now the diagonal part of the coefficient matrix is time-independent and is sufficiently big, whereas the rotation angle is defined by a maximum degree of two that divides the number of the corresponding time interval. For such a system, in the case of a continious coefficient dependence on a parameter it is proved that such a value exists, at which the corresponding system is unstable.

Lebesque measure zero subsets of the real line and an infinite game

Let denote the ideal of Lebesgue measure zero subsets of the real line. Then add() denotes the minimal cardinality of a subset of whose union is not an element of . In [1] Bartoszynski gave an elegant combinatorial characterization of add(), namely: add() is the least cardinal number κ for which the following assertion fails:For every family of at mostκ functions from ω to ω there is a function F from ω to the finite subsets of ω such that:1. For each m, F(m) has at most m + 1 elements, and2. for each f inthere are only finitely many m such that f(m) is not an element of F(m).The symbol A(κ) will denote the assertion above about κ. In the course of his proof, Bartoszynski also shows that the cardinality restriction in 1 is not sharp. Indeed, let (Rm: m < ω) be any sequence of integers such that for each m Rm, ≤ Rm+1, and such that limm→∞Rm = ∞. Then the truth of the assertion A(κ) is preserved if in 1 we say instead that1′. For each m, F(m) has at most Rm elements.We shall use this observation later on. We now define three more statements, denoted B(κ), C(κ) and D(κ), about cardinal number κ.

Compactness of Invariant Densities for Families of Expanding, Piecewise Monotonic Transformations

Let I = [0,1] and let be the space of all integrable functions on I, where m denotes Lebesque measure on I. Let ∥ ∥1 be the ℒ-1-norm and let be a measurable, nonsingular transformation on I. Let denote the space of densities. The probability measure μ is invariant under τ if for all measurable sets A, The measure μ is absolutely continuous if there exists an such that for any measurable set A We refer to ƒ* as the invariant density of τ (with respect to m). It is well-known that ƒ * is a fixed point of the Frobenius-Perron operator defined by

A Practical Two-Dimensional Ergodic Theorem

Let τ:[0, 1] → [0, 1] be defined by τ(x) = 2x on [0, 1/2,] and τ(JC) = 2(1 - x) on [5, 1], and let T:[0, 1] x [0, 1] → [0, 1] x [0, 1] be defined by T(x,y) = (τ(x), τ(y))- Letwhere p is a prime > 2, and a and M are integers. Consider T restricted to θM x θN, 1 < M < N. Let X = ((2a)/(pM), (2b)/(pN)) ∈ θM x θN and let per(X) denote the length of the period of X.Then,where m is Lebesque measure on [0, 1], and C is independent of p, N, M, a and b. Thus, as p → or as N - M and M→ →,

On uniformly distributed orbits of certain horocycle flows

Let(where t ε ℝ) and let μ be the G-invariant probability measure on G/Γ. We show that if x is a non-periodic point of the flow given by the (ut)-action on G/Γ then the (ut)-orbit of x is uniformly distributed with respect to μ; that is, if Ω is an open subset whose boundary has zero measure, and l is the Lebesque measure on ℝ then, as T→∞, converges to μ(Ω).

Projections on Bergman Spaces Over Plane Domains

Let D be a bounded plane domain and let Lp(D) stand for the usual Lebesgue spaces of functions with domain D, relative to the area Lebesque measure dσ(z) = dxdy. The class of all holomorphic functions in D will be denoted by H(D) and we write Bp(D) = Lp(D) ∩ H(D). Bp(D) is called the Bergman p-space and its norm is given byLet be the Bergman kernel of D and consider the Bergman projection(1.1)It is well known that P is not bounded on Lp(D), p = 1, ∞, and moreover, it can be shown that there are no bounded projections of L∞(Δ) onto B∞(Δ).