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We introduce a class of Gaussian processes with stationary increments which exhibit long-range dependence. The class includes fractional Brownian motion with Hurst parameter H > 1/2 as a typical example. We establish infinite and finite past prediction formulas for the processes in which the predictor coefficients are given explicitly in terms of the MA(∞) and AR(∞) coefficients.
For a prime p, we denote by Bn the cyclic group of order pn. Let ϕ be a faithful irreducible character of Bn, where p is an odd prime. We study the p-group G containing Bn such that the induced character ϕG is also irreducible. Set [NG(Bn):Bn] = pm and [G:Bn] = pM. The purpose of this paper is to determine the structure of G under the hypothesis [NG(Bn):Bn]2d ≤ pn, where d is the smallest integer not less than M/m.
The gradient flow associated to the Helfrich variational problem, called the Helfrich flow is considered. Here the n-dimensional Helfrich flow is investigated for any n, as a projected gradient flow. A result of local existence is proved. The uniqueness is shown for the cases (i) for the initial hypersurface with non-zero Gramian when n ≥ 2, (ii) for every initial curve when n = 1.
In this paper, we completely classify the rational solutions of the Sasano system of type A1(1), which is a degeneration of the Sasano system of type A4(2). These systems of differential equations are both expressed as coupled Painlevé II systems. The Sasano system of type A1(1) is a higher order version of the second Painlevé equation, PII, with the same affine Weyl group symmetry of type A1(1) as PII.
Necessary and sufficient conditions for positive Toeplitz operators on the Bergman space of a minimal bounded homogeneous domain to be bounded or compact are described in terms of the Berezin transform, the averaging function and the Carleson property
Let p be an odd prime and BP*(pt) ≅ $¥mathbb Z$(p)[v1,v2,…] the coefficient ring of the Brown-Peterson cohomology theory BP*(−) with |vi| = −2pi + 2. We study ABP*,*'(−) theory, which is the counter part in algebraic geometry of the BP*(−) theory. Let k be a field with k ⊂ $¥mathbb C$ and K*M(k) the Milnor K-theory. For a nonzero symbol a ∈ Kn+1M(k)/p, a norm variety Va is a smooth variety such that a|k(Va) = 0 ∈ Kn+1M(k(Va))/p and V a($¥mathbb C$) = vn. In particular, we compute ABP*,*'(Ma) for the Rost motive Ma which is a direct summand of the motive M(Va) of some norm variety Va.