It is known (Pursell and Shanks [9]) that an isomorphism between Lie algebras of infinitesimal automorphisms of C∞ structures with compact support on manifolds M and M′ yields an isomorphism between C∞ structures of M and M’.

Omori [5] proved that this is still true for some other structures on manifolds. More precisely, let M and M′ be Hausdorff and finite dimensional manifolds without boundary. Let α be one of the following structures:

In this paper it is shown that there is a unimodal Levy process with oscillating mode. After the author first constructed an example of such a self-decomposable process, Sato pointed out that it belongs to the class of semi-stable processes with β < 0. We prove that all non-symmetric semi-stable self-decomposable processes with β < 0 have oscillating modes.

Consider the function with variables z1…,zm

where υ and λj (j = 1,…,m) are complex numbers and.Г is a suitably chosen integral domain. In case m = 2, if we set [z1 z2]n as Г, it is the Selberg integral [22]. Our function can be regarded as an extention of it; so we may call (0.1) a Selberg type integral It is known that (0.1) satisfies a Gauss-Manin system, i.e. a system of rationally holonomic differential equations [3], [21].

Let Γ be a Fuchsian group (of finite type) acting on the upper half plane. To each parabolic cusp Ki (i = 1, …, h), corresponds a Eisenstein serie

In this paper, we improve a result of the third paper of this series, that is we show

THEOREM. Let K be a nilpotent extension of the rational number field Q with Galois group Γ, and G a Γ- stable finite subgroup of GLn(0K). Then G is of A-type.

Let be the space of cusp forms of weight k with respect to SL(2, Z). Let p be a prime number and let Tk(p) be the Hecke operator of degree p acting on as a linear endomorphism. Put Hk(X) = det (I – Tk(p)X + pk-lX2I), where I is the identity operator on . Hk(X) is a polynomial with coefficients of rational integers, which is called the Hecke polynomial.

Soit X un groupe abélien localement compact séparé et dénombrable á l’infini. On désignera par ξ la mesure de Haar sur X. Un noyau de convolution réel N (c’est-à-dire, une mesure de Radon réelle) sur X sera dit de type logarithmique s’il existe un semi-groupe (αt)t≥0 vaguement continu, markovien et récurrent des noyaux de convolution (non-négatifs) tel que limt → ∞ (vaguement) et, pour toute (pour la notation, voir § 2),

Our object in this paper is to prove the following result.

THEOREM. Let G be a finite group satisfying the following conditions:

• (*) G has subgroups L1, L2 such that L1 ≃ SL(2, q1), L2 ≃ SL(2, q2), [L1 L2] = 1, , where j is an involution, and |C(j): L1L2| = 2.

• (**) where n2 = 1,

In my recent paper [1], I associated a projectively invariant pseudo-distance dM to every affinely connected manifold M and proved the following

Theorem 1. Let M be a Riemannian manifold with metric and Ricci tensor RicM such that RicM ≤–c2. Let δM be the Riemannian distance defined by .

Let Ω be a locally compact separable metric space and let Ф be a positive symmetric kernel. Then the inner and outer capacities of subsets of Ω are defined by means of Ф-potentials of positive measures in the following manner. We define the capacity c(K) of a compact set K in a certain manner by means of Ф-potentials.

Let k be either a local or a global field, and K be a finite Galois extension of k with g = Gal (K/k). Let L be a Galois extension of K which is also Galois over k. Such an extension is called central if Gal(L/iT) lies inside the centre of Gal(L/K). Clearly L is abelian over K. Next set L* = L∩K · kab where kab is the maximal abelian extension of k in its algebraic closure. This is the genus field of L over K/k.

We prove that a finite-dimensional algebra $ \Lambda $ is $ \tau $ -tilting finite if and only if all the bricks over $ \Lambda $ are finitely generated. This is obtained as a consequence of the existence of proper locally maximal torsion classes for $ \tau $ -tilting infinite algebras.

Probability measure valued diffusion processes have been discussed by many authors, in connection with population genetics. Most papers studying probability measure valued diffusions are mainly concerned with the ones describing single locus models. In this paper, we will discuss a measure valued diffusion describing an n locus model. Random sampling, mutation and gene conversion, a kind of interaction between loci, which was introduced and investigated by T. Ohta in [5], [6], will be taken into consideration.

Theorem 5 of Azumaya’s recent article can be formulated in the following generalized form:

Let R be a ring. Let m be a finitely generated right-module of R such that mR = m. Assume that m = u1r + ubr + … + umr for every generating system of u1, u2, …, um of m and for every maximal right-ideal r of R. Then m = 0.

In the papers [11] and [18] Rohlin and Thom have introduced an equivalence relation into the set of compact orientable (not necessarily connected) differentiable manifolds, which, roughly speaking, is described in the following manner: two differentiable manifolds are equivalent (cobordantes), when they together form the boundary of a bounded differentiable manifold. The equivalence classes can be added and multiplied in a natural way and form a graded algebra Ω relative to the addition, the multiplication and the dimension of manifolds. The precise structures of the groups of cobordism Ωk of dimension k are not known thoroughly. Thom [18] has determined the free part of Ω and also calculated explicitly Ωk for 0 ≦ k ≦ 7.

For prime l we study the structure of the 2-part of the ideal class group Cl of ℚ(ζl). We prove that Cl ⊗ ℤ2) is a cyclic Galois module for all l < 10000 with one exception and compute the explicit structure in several cases.

Hartman proved that a diffeomorphism is topologically conjugate to a linear map on a neighbourhood of a hyperbolic fixed point ([3]). In this paper we study the topological conjugacy problem of a diffeomorphism on a neighbourhood of a hyperbolic set, and prove that for any hyperbolic set there is an arbitrarily slight extension to which a sub-shift of finite type is semi-conjugate.

One of the most important problems in the potential theory is the one of capacitability, that is, whether the inner capacity of an arbitrary borelian subset B is equal to the outer capacity of B. As for the capacities induced by the Newtonian potentials and other classical potentials, Choquet [5] has shown that every borelian and, more generally, every analytic set are capacitable. He goes on as follows : first he shows that, for the Newtonian capacity f, the inequality of strong subadditivity holds, that is,

and then, using this inequality, he shows that the outer capacity f* has the analogous property to one of the outer measure, more precisely, if an increasing sequence {An} of arbitrary subsets converges to A, then f*(A) = limf*(An)- This property plays an important role in his proof.

We establish the continuity of Hilbert–Kunz multiplicity and F-signature as functions from a Cohen–Macaulay local ring $(R,\mathfrak{m},k)$ of prime characteristic to the real numbers at reduced parameter elements with respect to the $\mathfrak{m}$-adic topology.

We [4], [5] examined some properties of balayaged measures in the theory of a Dirichlet space. In those papers, we showed that the singular measure of a Dirichlet space plays some important roles. In this paper, we shall precisely examine some properties of the singular measure of a Dirichlet space. Let X be a locally compact Hausdorff space in which there exists a positive Radon measure ξ which is everywhere dense in X.