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 B_n the cyclic group of order pⁿ. Let ϕ be a faithful irreducible character of B_n, where p is an odd prime. We study the p-group G containing B_n such that the induced character ϕ^G is also irreducible. Set [N_G(B_n):B_n] = p^m and [G:B_n] = p^M. The purpose of this paper is to determine the structure of G under the hypothesis [N_G(B_n):B_n]^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, P_II, with the same affine Weyl group symmetry of type A₁⁽¹⁾ as P_II.

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 |v_i| = −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 ∈ K_n+1^M(k)/p, a norm variety V_a is a smooth variety such that a|_k(Va) = 0 ∈ K_n+1^M(k(V_a))/p and V _a($¥mathbb C$) = v_n. In particular, we compute ABP^*,*'(M_a) for the Rost motive M_a which is a direct summand of the motive M(V_a) of some norm variety V_a.