The proof of Proposition 2 and that of Corollary to Proposition 3 in my previous paper “On the structure of complete local rings” are not correct Here we want to correct them.

We have introduced in our former work [1] and [2] the comprehension operator “{·)”, which maps every binary relation to a binary relation. The definition of this operator

is remarkable in that it can be defined in any formal system having the membership relation ∈, which is hereafter called the universal system and is denoted by U. In this work, we would like to point out that the axiom of choice, in the strong sense as well as in the weak sense, can be formulated in an extremely simple style by making use of the comprehension operator.

Usually the group structure in a homotopy group is defined directly and explicitly. But the algebraic approach to the topology, now common, seems to raise the following question : is that the only group sturcture which is natural from the algebraic topological point of view? On the other hand, several algebraists have begun to feel a necessity to construct a “homotopy or cohomotopy theory of groups,” and it may be allowed to say that one of the first steps to the problem is the axiomatization of homotopy groups. Our first question is of course a special case of the latter problem.

The problem to determine the Gevrey index of solutions of a given hypoelliptic partial differential equation seems to be not yet well investigated. In this paper, we shall show the Gevrey indices of solutions of the equations of Grushin type, [6], are determined by a rather simple application of a straightforward extension of the results given in [7], [8] and [13]. For simplicity to construct left parametrices in the operator valued sense, we shall consider the equations under the stronger condition than that of [6] (cf. Condition 1 of Section 3). Typical examples of Grushin type are given by which will be discussed in Section 4. We remark that our approach may be compared with the one to a similar problem discussed in [17] by using suitable L2-estimates constructed in [16].

In [Mo.l], S. Momm studied the boundary behaviour of extremal plurisubharmonic functions by using the pluricomplex Green function gΩ of a bounded convex domain Ω in Cn to exhaust the domain by a family of sublevel sets. Let Ω be a bounded convex domain in Cn containing the origin in its interior. The pluricomplex green function of Ω with a pole is defined by

(1.1)

Let R be a commutative Noetherian ring. We prove that if R is either an equidimensional finitely generated algebra over a perfect field, or an equidimensional equicharacteristic complete local ring with a perfect residue field, then the annihilator of the singularity category of R coincides with the Jacobian ideal of R up to radical. We establish a relationship between the annihilator of the singularity category of R and the cohomological annihilator of R under some mild assumptions. Finally, we give an upper bound for the dimension of the singularity category of an equicharacteristic excellent local ring with isolated singularity. This extends a result of Dao and Takahashi to non-Cohen–Macaulay rings.

Let G and H be finite groups. If a group G̅ has an invariant subgroup H̅, which is isomorphic with H, such that the factor group G̅/H̅ is isomorphic with G. then we say that G̅ is an extension of H by G. Now let G be the Galois group of a normal extension K over an algebraic number field k of finite degree. The imbedding problem concerns us with the question, under what conditions K can be imbedded in a normal extension L over k such that the Galois group of L over k is isomorphic with G̅ and K corresponds to H̅. Brauer connected this problem with the structure of algebras over k, whose splitting fields are isomorphic with K.

The present paper is a continuation of [6] and [7] in which the principle of limiting absorption has been verified for symmetric systems of first order with long-range perturbations, but the operators considered there are of constant rank. On the other hand, operators with non-constant rank are also important in application as well as from a purely theoretical point of view. In this paper, we consider the linearized equation of magnetogasdynamics with long-range perturbation as an important example of such operators.

Let k be an algebraically closed field of characteristic p>0. Let K/k be a function field of one variable and L/K be an unramified separable abelian extension of degree pr over K. The galois automorphisms ε1, …, εpr of L/K are naturally extended to automorphisms η(ε1), … , η(εpr) of the jacobian variety JL of L/k. If we take a svstem of p-adic coordinates on JL, we get a representation {Mp(η(εv))} of the galois group G(L/K) of L/K over p-adic integers.

Let D be the unit disk, C the unit circle, and f a continuous function from D into the Riemann sphere W. We say that f is normal if f is uniformly continuous with respect to the non-Euclidean hyperbolic metric in D and the chordal metric in W.

The notion of fields of moduli introduced first by Matsusaka [8] has been developed by Shimura [12] exclusively in the area of polarized abelian varieties. Later Koizumi [7] gave an axiomatic treatment for the notion.

Let M > 1 be a positive number. Let be a family of holomorphic functions f in some domain D ⊂ ℂ for which there exists an integer k = k(f) > 2 such that |(fk)′(ζ)| ≤ Mk for every periodic point ζ of period k of f in D. We show first that is quasinormal of order at most one in D. This strengthens a result of W. Bergweiler. Secondly, for the case M = 1, we prove that is normal in D if there exists a positive number K < 3 such that | f(η)| ≤ K for each f ∈ and every fixed point η of f in D. This improves a result of M. Esséen and S. J. Wu. We also construct an example which shows that the condition |f’(η)| ≤ K < 3 can not be replaced by | f′(η) | < 3.

To any f ∈ ℂ[x1, … ,xn] \ ℂ with f(0) = 0 one can associate the motivic zeta function. Another interesting singularity invariant of f-1{0} is the zeta function on the level of Hodge polynomials, which is actually just a specialization of the motivic one. In this paper we generalize for the Hodge zeta function the result of Veys which provided for n = 2 a complete geometric determination of the poles. More precisely we give in arbitrary dimension a complete geometric determination of the poles of order n − 1 and n. We also show how to obtain the same results for the motivic zeta function.

In this note, we prove the semiampleness conjecture for Kawamata log terminal Calabi–Yau (CY) surface pairs over an excellent base ring. As applications, we deduce that generalized abundance and Serrano’s conjecture hold for surfaces. Finally, we study the semiampleness conjecture for CY threefolds over a mixed characteristic DVR.

The purpose of this paper is to characterize by means of simple quadratic forms the set of rational primes that are decomposed completely in a non-abelian central extension which is abelian over a quadratic field. More precisely, let L = Q be a bicyclic biquadratic field, and let K = Q. Denote by the ray class field mod m of K in narrow sense for a large rational integer m. Let be the maximal abelian extension over Q contained in and be the maximal extension contained in such that Gal(/L) is contained in the center of Gal(/Q). Then we shall show in Theorem 2.1 that any rational prime p not dividing d1d2m is decomposed completely in /Q if and only if p is representable by rational integers x and y such that x ≡ 1 and y ≡ 0 mod m as follows

where a, b, c are rational integers such that b2 − 4ac is equal to the discriminant of K and (a) is a norm of a representative of the ray class group of K mod m.

Moreover is decomposed completely in if and only if .

The Poisson process , as is well-known, is a temporally and spatially homogeneous Markoff process satisfying

(1)

(2)

where k is a non-negative integer and λ is a positive constant. In this note we consider the random variable Lm(ω) which denotes the length of t-interval such that X(t, w) = m (m = 0, 1, 2,…) and some of other properties concerning them.

By a global field we mean a field which is either an algebraic number field, or an algebraic function field in one variable over a finite constant field. The purpose of the present note is to show that the Galois cohomology group of the ring of integers of a global field is isomorphic to that of the ring of integers of its adele ring and is reduced to asking for that of the ring of local integers.

In this paper, we prove that the set of equivalence classes of dormant opers of rank $p-1$ over a projective smooth curve of genus ${\geqslant}2$ over an algebraically closed field of characteristic $p>0$ is of cardinality one.

For complex numbers q and u, Carlitz defined the q-Bernoulli numbers {βk(q)} and the q-Euler numbers {Hk(u, q)} associated to u by

and

Let K be an extension of a field k, and p denotes the characteristic. It was proved by M. Gerstenhaber ([1]) that if K is separable over k, then it is rigid and it was conjectured in [1] that, if K is not separable over k, then it is not rigid. We studied in [4] the above conjecture in certain special case. In this note we shall extend the results of [4] to inseparable algebraic extension fields.