We construct a class of rotation surfaces with constant mean curvature in 4-dimensional Pseudo-Euclidean space E₂⁴.

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In this paper we will investigate the isomorphisms of A_2n⁽ⁿ⁾ which is defined in the introduction.

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Let X= {X_k(ω, t)}_t∈T, k∈N, be an independent sequence of Gaussian processes with mean 0 on a locally compact separable metric space (T, d), and assume that the covariance functions are non-negative and
??E[X_k(ω, s)X_k(ω, t)] = ulog⁺?? + O(1), s, t∈T,
for a positeve number u>0. Kahane [1985] defined a multiplicative chaos by X and gave sufficient conditions for the regularity or the singularity of a measure with respect to it in the case where (T, d) is compact and homogeneous in the sense of Coifman and Weiss.
The aim of this paper are to interprete Kahane's results from the viewpoint of the dimension of a measure, and extend them to the case where (T, d) is a locally compact metric space equipped with a majorizing measure and not necessarily homogeneous.

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A bounded linear operator T on a Hilbert space is called a θ-operator if T^* T and T+T^* commute. It is shown that θ-operators are dominant. We shall give useful construction of θ-operators. Using this we get an example of θ-operator which is not paranormal.

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Symmetric spin models were introduced by Jones to obtain invariants of links. In his paper he proposed to obtain the classification of models (X, w₊, w₋) with | X | = 4, 5, 6 and 7. In the present paper we complete this task by showing that the only spin models of these sizes are the Potts models, those coming from cyclic groups for 5≤n≤7, and for n=4 some other models obtained by the product construction of de la Harpe. Furthermore, we also classify the non-symmetric Jones-type spin models introduced by Kawagoe, Munemasa and Watatani, for n=4, 5.

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A real random variable X is sub-Gaussian iff there exist K>0 such that
E[exp(λX)]≤exp(??) for any λ ∈ R
J-P. Kahane [10] proved that a real random variable X is sub-Gaussian if and only if E[X]=0 and E[exp(εX²)] <∞ for some ε>0.
A probability measure μ on a Banach space B is said to be sub-Gaussian iff there exists C>0 such that
∫_Bexp(<y, x>)μ(dx)≤exp(??∫<y, x>²μ(dx))<∞ for any y ∈ B ^*. (1)
A Gaussian measure and the probability measure induced by a Rademacher series are typical examples of sub-Gaussian measures, and for these two probability measures, exp(ε||x||²) is integrable for some ε>0 ([3], [12]). We call this integrability the exponential square integrability. When B = L _P (p≥1), (1) is a sufficient condition for the exponential square integrability, but not necessary even if B is a Hilbert space ([4]).
For a sub-Gaussian measure μ, the L_p (μ) topologies (0 < p < ∞) and the L₀(μ) topology coincide on the family {<y, x> ; y ∈ B*}. To show that considerably many probability measures satisfy such a remarkable property, we shall propve the sub-Gaussian property of probability measures for two types. One is a probability measure identified with a positive generalized Wiener function (see H. Sugita [17]), and the other is a probability measure which is absolutely continuous with respect to the probability measure induced by a random Fourier series. The former is exponentially square integrable [17], and so is the latter under certain additional conditions.

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