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 this paper we attempt to set up a notion of homomorphism for continuous pseudogroups and show that the kernel exists (as a continuous pseudogroup) in the transitive case. This paper is really an extension of the paper by Kuranishi and Rodrigues [11] which essentially examines the question of the existence (as a continuous pseudogroup) of an image of a homomorphism. A certain amount of overlap in definitions and statements of results was unavoidable, especially in sections 2 and 3, but for many proofs and constructions the reader is referred to that paper. For the basic notions of the theory of continuous pseudogroups as used in section 4, see Kuranishi [9] and for the terminology of the Cartan-Kãhler theory used in section 5, see Kuranishi [6] Fuller expositions may be found in Cartan [I] Kàhler [4], Kumpera [5], Kuranishi [7], and Schouten and v.d. Kulk [13].

Let k be a field complete with respect to a non-trivial, non-archimedean valuation and let g be a positive integer. Consider the following question : if Γ is a multiplicative subgroup of Gg = (k*)g satisfying certain “Riemann conditions”, can one construct in a natural way an abelian variety defined over k having Gg/Γ as its set of k-rational points? This problem was first considered by Morikawa [3]. J. Tate provided a complete solution for g = 1 (cf. for example [6]). J. McCabe [2] gave a partial solution for g > 1. He showed how to attach to Γ a graded ring R of theta functions such that A = Proj. R is g-dimensional abelian variety over k.

This paper is concerned with the p-Ginzburg–Landau (p-GL) type model with $p\neq 2$ . First, we obtain global energy estimates and energy concentration properties by the singularity analysis. Next, we give a decay rate of $1-|u_\varepsilon |$ in the domain away from the singularities when $\varepsilon \to 0$ , where $u_\varepsilon $ is a minimizer of p-GL functional with $p \in (1,2)$ . Finally, we obtain a Liouville theorem for the finite energy solutions of the p-GL equation on $\mathbb {R}^2$ .

We give a notion of ordinary Enriques surfaces and their canonical lifts in any positive characteristic, and we prove Torelli-type results for this class of Enriques surfaces.

In their renowned paper (2011, Inventiones Mathematicae 184, 591–627), I. Vollaard and T. Wedhorn defined a stratification on the special fiber of the unitary unramified PEL Rapoport–Zink space with signature $(1,n-1)$ . They constructed an isomorphism between the closure of a stratum, called a closed Bruhat–Tits stratum, and a Deligne–Lusztig variety which is not of classical type. In this paper, we describe the $\ell $ -adic cohomology groups over $\overline {{\mathbb Q}_{\ell }}$ of these Deligne–Lusztig varieties, where $\ell \not = p$ . The computations involve the spectral sequence associated with the Ekedahl–Oort stratification of a closed Bruhat–Tits stratum, which translates into a stratification by Coxeter varieties whose cohomology is known. Eventually, we find out that the irreducible representations of the finite unitary group which appear inside the cohomology contribute to only two different unipotent Harish-Chandra series, one of them belonging to the principal series.

Let $A$ be an expansive dilation on $\mathbb{R}^{n}$ and $\unicode[STIX]{x1D711}:\mathbb{R}^{n}\times [0,\infty )\rightarrow [0,\infty )$ an anisotropic growth function. In this article, the authors introduce the anisotropic weak Musielak–Orlicz Hardy space $\mathit{WH}_{A}^{\unicode[STIX]{x1D711}}(\mathbb{R}^{n})$ via the nontangential grand maximal function and then obtain its Littlewood–Paley characterizations in terms of the anisotropic Lusin-area function, $g$-function or $g_{\unicode[STIX]{x1D706}}^{\ast }$-function, respectively. All these characterizations for anisotropic weak Hardy spaces $\mathit{WH}_{A}^{p}(\mathbb{R}^{n})$ (namely, $\unicode[STIX]{x1D711}(x,t):=t^{p}$ for all $t\in [0,\infty )$ and $x\in \mathbb{R}^{n}$ with $p\in (0,1]$) are new. Moreover, the range of $\unicode[STIX]{x1D706}$ in the anisotropic $g_{\unicode[STIX]{x1D706}}^{\ast }$-function characterization of $\mathit{WH}_{A}^{\unicode[STIX]{x1D711}}(\mathbb{R}^{n})$ coincides with the best known range of the $g_{\unicode[STIX]{x1D706}}^{\ast }$-function characterization of classical Hardy space $H^{p}(\mathbb{R}^{n})$ or its weighted variants, where $p\in (0,1]$.

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.

In the present paper by an algebraic system (algebra) A we shall mean a system with a set F of operations fλ: (x1,…, xn) ∈ A × · · · × A → fλ(x1,…, xn) ∈ A. A polynomial p(x1, …, xr) is a function of variables x1,…, xr which is either one of the xi, or (recursively) a result of some operation fλ(p1,…, pn) performed on other polynomials pi. An algebra A may satisfy a set R of identities p(x1,…, xr) = q(x1,…, xs), and then A shall be called an (F, R)-algebra.

Let p denote an odd prime integer and let q = pf where f is a positive integer. Let ℋ denote the projective symplectic group PSp(4,q), the Dickson group G2(q), or the Steinberg “triality twisted” group over a field Fq of q elements. Then ℋ is simple and the Sylow 2-subgroups of ℋ have centers of order 2 so that involutions which centralize a Sylow 2-subgroup of ℋ form a single conjugacy class of ℋ.

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)

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.

Several years ago, K. Oikawa, my colleague, investigated the properties of Schottky coverings of closed Riemann surfaces, leaving an interesting problem as open [3]:

Does exist a Schottky covering between the basic surface and a given planar covering?

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.

A holomorphic n × n-matric automorphic factor with respect to a lattice L in Cg means a system of holomorphic n × n-matrices {μα(z) | α ∈ L} such that

The purpose of this paper is to discuss some ergodic properties of the stepping stone model proposed by Kimura, M. [4] and developed by Weiss, G.H. and Kimura, M. [12]. Our model to be discussed in this paper involves selection in addition to mutation and migration which are dealt with in [4], [12]. Because of the additional factor selection, the stochastic process describing our model becomes complicated and presents particularly interesting profound structure of the random phenomena in question.

Let X = (Xt, 0 ≤ t < ∞) be a Lévy process on the Euclidean space Rd, that is, a process on Rd with stationary independent increments which has right continuous paths with left limits. We denote by Px the probability measure such that PX(X0 = x) = 1 and by Ex the expectation relative to Px. The process is characterized by the exponent Ψ through

.

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.

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

All rings in this paper are assumed to be commutative with identity and the terminology is standard.