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This is a sequel to the papers Oh and Wang (Real and Complex Submanifolds, Springer Proceedings in Mathematics and Statistics 106 (2014), 43–63, eds. by Y.-J. Suh and et al. for ICM-2014 satellite conference, Daejeon, Korea, August 2014; arXiv:1212.4817; Analysis of contact Cauchy–Riemann maps I: a priori$C^{k}$estimates and asymptotic convergence, submitted, preprint, 2012, arXiv:1212.5186v3). In Oh and Wang (Real and Complex Submanifolds, Springer Proceedings in Mathematics and Statistics 106 (2014), 43–63, eds. by Y.-J. Suh and et al. for ICM-2014 satellite conference, Daejeon, Korea, August 2014; arXiv:1212.4817), the authors introduced a canonical affine connection on $M$ associated to the contact triad $(M,\unicode[STIX]{x1D706},J)$. In Oh and Wang (Analysis of contact Cauchy–Riemann maps I: a priori$C^{k}$estimates and asymptotic convergence, submitted, preprint, 2012, arXiv:1212.5186v3), they used the connection to establish a priori$W^{k,p}$-coercive estimates for maps $w:\dot{\unicode[STIX]{x1D6F4}}\rightarrow M$ satisfying $\overline{\unicode[STIX]{x2202}}^{\unicode[STIX]{x1D70B}}w=0$, $d(w^{\ast }\unicode[STIX]{x1D706}\circ j)=0$without involving symplectization. We call such a pair $(w,j)$ a contact instanton. In this paper, we first prove a canonical neighborhood theorem of the locus $Q$ foliated by closed Reeb orbits of a Morse–Bott contact form. Then using a general framework of the three-interval method, we establish exponential decay estimates for contact instantons $(w,j)$ of the triad $(M,\unicode[STIX]{x1D706},J)$, with $\unicode[STIX]{x1D706}$ a Morse–Bott contact form and $J$ a CR-almost complex structure adapted to $Q$, under the condition that the asymptotic charge of $(w,j)$ at the associated puncture vanishes.We also apply the three-interval method to the symplectization case and provide an alternative approach via tensorial calculations to exponential decay estimates in the Morse–Bott case for the pseudoholomorphic curves on the symplectization of contact manifolds. This was previously established by Bourgeois (A Morse–Bott approach to contact homology, Ph.D. dissertation, Stanford University, 2002) (resp. by Bao (On J-holomorphic curves in almost complex manifolds with asymptotically cylindrical ends, Pacific J. Math. 278(2) (2015), 291–324)), by using special coordinates, for the cylindrical (resp. for the asymptotically cylindrical) ends. The exponential decay result for the Morse–Bott case is an essential ingredient in the setup of the moduli space of pseudoholomorphic curves which plays a central role in contact homology and symplectic field theory (SFT).

In this paper, we give a characterization of the simultaneous unitarizability of any finite set of SL(2, ℂ)-valued functions on and determine all possible ways of the unitarization. Such matrix functions can be regarded as images of the generators for the fundamental group of a surface in an -family, and the results of this paper have applications in the construction of constant mean curvature surfaces in space.

An algebraic group of type E7 over an algebraically closed field has an irreducible representation in a vector space of dimension 56 and is, in fact, the identity component of the automorphism group of a quartic form on the space. This paper describes the construction of the quartic form if the characteristic is ≠ 2, 3, taking into account a field of definition F. Certain F-forms of E7 appear in the automorphism groups of quartic forms over F, as well as forms of E6. Many of the results of the paper are known, but are perhaps not easily accessible in the literature.

Let Mn be an immersed umbilic-free hypersurface in the (n + 1)-dimensional unit sphere n+1, then Mn is associated with a so-called Möbius metric g, a Möbius second fundamental form B and a Möbius form Φ which are invariants of Mn under the Möbius transformation group of n+1. A classical theorem of Möbius geometry states that Mn (n ≥ 3) is in fact characterized by g and B up to Möbius equivalence. A Möbius isoparametric hypersurface is defined by satisfying two conditions: (1) Φ ≡ 0; (2) All the eigenvalues of B with respect to g are constants. Note that Euclidean isoparametric hyper-surfaces are automatically Möbius isoparametric, whereas the latter are Dupin hypersurfaces.In this paper, we prove that a Möbius isoparametric hypersurface in 4 is either of parallel Möbius second fundamental form or Möbius equivalent to a tube of constant radius over a standard Veronese embedding of ℝP2 into 4. The classification of hypersurfaces in n+1 (n ≥ 2) with parallel Möbius second fundamental form has been accomplished in our previous paper [6]. The present result is a counterpart of Pinkall’s classification for Dupin hypersurfaces in 4 up to Lie equivalence.

Let (G, ρ, X) be a prehomogeneous vector space with singular set S over an algebraic number field F. The main result of this paper is a proof for the convergence of the zeta fucntions Z(Φ, s) associated with (G, ρ, X) for large Re s under the assumption that S is a hypersurface. This condition is satisfied if G is reductive and (G, ρ, X) is regular. When the connected component of the stabilizer of a generic point x is semisimple and the group Πx of connected components of Gx is abelian, a clear estimate of the domain of convergence is given.Moreover when S is a hypersurface and the Hasse principle holds for G, it is shown that the zeta fucntions are sums of (usually infinite) Euler products, the local components of which are orbital local zeta functions. This result has been proved in a previous paper by the author under the more restrictive condition that (G, ρ, X) is irreducible, regular, and reduced, and the zeta function is absolutely convergent.

This paper is concerned with a finitely generated module M over a (commutative Noetherian) local ring R. In the case when R is a homomorphic image of a Gorenstein local ring, one can use the well-known associativity formula for multiplicities, together with local duality and Matlis duality, to produce analogous associativity formulae for the local cohomology modules of M with respect to the maximal ideal. The main purpose of this paper is to show that these formulae also hold in the case when R is universally catenary and such that all its formal fibres are Cohen–Macaulay.These formulae involve certain subsets of the spectrum of R called the pseudosupports of M; these pseudo-supports are closed in the Zariski topology when R is universally catenary and has the property that all its formal fibres are Cohen–Macaulay. However, examples are provided to show that, in general, these pseudo-supports need not be closed. We are able to conclude that the above-mentioned associativity formulae for local cohomology modules do not hold over all local rings.

It is proved that every holomorphically convex complex space endowed with an action of a compact Lie group K can be realized as an open K-stable subspace of a holomorphically convex space endowed with a holomorphic action of the complexified group K . Similar results are obtained for holomorphic if-bundles over such spaces. Let G be a real Lie group which acts by holomorphic transformations on a (reduced) complex space X. Suppose that the Lie algebra of the complexification G c of G (see [Ho, p. 204]) is the complexification of the Lie algebra of G. This holds for example in the case where G is simply connected. Then, by integrating the holomorphic vector fields given by the G-action, the complexification G c acts locally and holomorphically on X (see [K]). Adapting the terminology of Palais (see [P]), we say that a complex space X* which contains X as an open subset is a globalization of the complex G-space X whenever the local G-action on X extends to a global holomorphic action on X* and G c X = X*. The following results are proved in this paper. THEOREM 1. Let G be a compact Lie group and X a holomorphically convex complex G-space X. Then there exists a globalization X* of X satisfying the following conditions (i) X* is holomorphically convex (ii) Every G-equivariant holomorphic map ψ from X into a complex space Received August 7, 1995. Γhe first author would like to thank the Department of Mathematics of Brandeis University for its hospitality while a part of this paper was written. This project was partially supported by Sonderforschungbereich 237 "Unordnung und groβe Fluktuationeon" of the Deutsche Forschungsgemeinschaft. 2 The second author is supported by an Instituto Nazionale di Alt a Matematica grant which is gratefully acknowledged.

In this paper, we study some spectral properties of the discrete Schrödinger operator = Δ + q defined on a locally finite connected graph with an automorphism group whose orbit space is a finite graph.The discrete Laplacian and its generalization have been explored from many different viewpoints (for instance, see [2] [4]). Our paper discusses the discrete analogue of the results on the bottom of the spectrum established by T. Kobayashi, K. Ono and T. Sunada [3] in the Riemannian-manifold-setting.

The aim of this paper is to extend and unify several results concerning complete ideals in 2-dimensional regular local rings by using the theory of joint reductions and mixed multiplicities. The theory of complete ideals in a 2-dimensional regular local ring was developed by Zariski in his 1938 paper [Z]. This theory is presented in a simpler and general form in [ZS, Appendix 5] and [H2].

In our paper [U2], we proved the irreducibility of the first differential equation y″ = 6y2 + x of Painlevé. In that paper we explained the origin of the problem and the importance of giving a rigorous proof. We can say that our method in [U2] is algebraic and finite dimensional in contrast to a prediction of Painlevé who expected a proof depending on the infinite dimensional differential Galois theory. Even nowadays the latter remains to be established. It seems that Painlevé needed an armament with the general theory (the infinite dimensional differential Galois theory) in the controversy with R. Liouville on the mathematical foundation of the proof of the irreducibility of the first differential equation (1902-03).

A Willmore surface $y:M\rightarrow S^{n+2}$ has a natural harmonic oriented conformal Gauss map $Gr_{y}:M\rightarrow SO^{+}(1,n+3)/SO(1,3)\times SO(n)$, which maps each point $p\in M$ to its oriented mean curvature 2-sphere at $p$. An easy observation shows that all conformal Gauss maps of Willmore surfaces satisfy a restricted nilpotency condition, which will be called “strongly conformally harmonic.” The goal of this paper is to characterize those strongly conformally harmonic maps from a Riemann surface $M$ to $SO^{+}(1,n+3)/SO^{+}(1,3)\times SO(n)$, which are the conformal Gauss maps of some Willmore surface in $S^{n+2}.$ It turns out that generically, the condition of being strongly conformally harmonic suffices to be associated with a Willmore surface. The exceptional case will also be discussed.

The first fundamental theorem of invariant theory for the orthosymplectic supergroup scheme $\text{OSp}(m|2n)$ states that there is a full functor from the Brauer category with parameter $m-2n$ to the category of tensor representations of $\text{OSp}(m|2n)$. This has recently been proved using algebraic supergeometry to relate the problem to the invariant theory of the general linear supergroup. In this work, we use the same circle of ideas to prove the second fundamental theorem for the orthosymplectic supergroup. Specifically, we give a linear description of the kernel of the surjective homomorphism from the Brauer algebra to endomorphisms of tensor space, which commute with the orthosymplectic supergroup. The main result has a clear and succinct formulation in terms of Brauer diagrams. Our proof includes, as special cases, new proofs of the corresponding second fundamental theorems for the classical orthogonal and symplectic groups, as well as their quantum analogues, which are independent of the Capelli identities. The results of this paper have led to the result that the map from the Brauer algebra ${\mathcal{B}}_{r}(m-2n)$ to endomorphisms of $V^{\otimes r}$ is an isomorphism if and only if $r.

In this paper, we will prove that any $\mathbb{A}^{3}$-form over a field $k$ of characteristic zero is trivial provided it has a locally nilpotent derivation satisfying certain properties. We will also show that the result of Kambayashi on the triviality of separable $\mathbb{A}^{2}$-forms over a field $k$ extends to $\mathbb{A}^{2}$-forms over any one-dimensional Noetherian domain containing $\mathbb{Q}$.

In this paper, we formulate and prove a duality for cohomology of curves over perfect fields of positive characteristic with coefficients in Neron models of abelian varieties. This is a global function field version of the author's previous work on local duality and Grothendieck's duality conjecture. It generalizes the perfectness of the Cassels-Tate pairing in the finite base field case. The proof uses the local duality mentioned above, Artin-Milne's global finite flat duality, the non-degeneracy of the height pairing and finiteness of crystalline cohomology. All these ingredients are organized under the formalism of the rational etale site developed earlier., Comment: 79 pages. Accepted for publication in Nagoya Mathematical Journal

Let$k$be an imaginary quadratic field with$\operatorname{Cl}_{2}(k)\simeq V_{4}$. It is known that the length of the Hilbert$2$-class field tower is at least$2$. Gerth (On 2-class field towers for quadratic number fields with$2$-class group of type$(2,2)$, Glasgow Math. J.40(1) (1998), 63–69) calculated the density of$k$where the length of the tower is$1$; that is, the maximal unramified$2$-extension is a$V_{4}$-extension. In this paper, we shall extend this result for generalized quaternion, dihedral, and semidihedral extensions of small degrees.

In this paper, we show that the cohomology of a general stable bundle on a Hirzebruch surface is determined by the Euler characteristic provided that the first Chern class satisfies necessary intersection conditions. More generally, we compute the Betti numbers of a general stable bundle. We also show that a general stable bundle on a Hirzebruch surface has a special resolution generalizing the Gaeta resolution on the projective plane. As a consequence of these results, we classify Chern characters such that the general stable bundle is globally generated.

The main goal of this paper is to show that Castelnuovo–Enriques’ $P_{12}$ - theorem (a precise version of the rough classification of algebraic surfaces) also holds for algebraic surfaces $S$ defined over an algebraically closed field $k$ of positive characteristic ( $\text{char}(k)=p>0$ ). The result relies on a main theorem describing the growth of the plurigenera for properly elliptic or properly quasielliptic surfaces (surfaces with Kodaira dimension equal to 1). We also discuss the limit cases, i.e., the families of surfaces which show that the result of the main theorem is sharp.

Let $R$ be a commutative Noetherian ring of prime characteristic $p$. In this paper we give a short proof using filter regular sequences that the set of associated prime ideals of $H^t_I(R)$ is finite for any ideal $I$ and for any $t \ge 0$ when $R$ has finite $F$-representation type or finite singular locus. This extends a previous result by Takagi-Takahashi and gives affirmative answers for a problem of Huneke in many new classes of rings in positive characteristic. We also give a criterion about the singularities of $R$ (in any characteristic) to guarantee that the set of associated primes of $H^2_I(R)$ is always finite., Comment: We added Remark 1.1 about the work of Mel Hochster and Luis N\'u\~nez-Betancourt, who obtained the same result as our main Theorem. Also, Corollary 3.4 was added

In this paper, we give some rigidity results for both harmonic and pseudoharmonic maps from pseudo-Hermitian manifolds into Riemannian manifolds or Kähler manifolds. Some foliated results, pluriharmonicity and Siu–Sampson type results are established for both harmonic maps and pseudoharmonic maps.

Let $\xi$ be a stable Chern character on $\mathbb{P}^1 \times \mathbb{P}^1$, and let $M(\xi)$ be the moduli space of Gieseker semistable sheaves on $\mathbb{P}^1 \times \mathbb{P}^1$ with Chern character $\xi$. In this paper, we provide an approach to computing the effective cone of $M(\xi)$ after showing that it is a Mori dream space for all $\xi$. We find Brill-Noether divisors spanning extremal rays of the effective cone using resolutions of the general elements of $M(\xi)$ which are found using the machinery of exceptional bundles. We use this approach to provide many examples of extremal rays in these effective cones. In particular, we completely compute the effective cone of the first fifteen Hilbert schemes of points on $\mathbb{P}^1 \times \mathbb{P}^1$., Comment: 42 pages

In this paper, we show that for every nonnilpotent hyperbolic map $f$ on an infra-nilmanifold, the set $\operatorname{HPer}(f)$ is cofinite in $\mathbb{N}$. This is a generalization of a similar result for expanding maps in Lee and Zhao (J. Math. Soc. Japan 59(1) (2007), 179–184). Moreover, we prove that for every nilpotent map $f$ on an infra-nilmanifold, $\operatorname{HPer}(f)=\{1\}$.

The goal of this paper is to characterize the operating functions on modulation spaces$M^{p,1}(\mathbb{R})$and Wiener amalgam spaces$W^{p,1}(\mathbb{R})$. This characterization gives an affirmative answer to the open problem proposed by Bhimani (Composition Operators on Wiener amalgam Spaces, arXiv: 1503.01606) and Bhimani and Ratnakumar (J. Funct. Anal.270(2016), pp. 621–648).

Let $K$ be a complete discrete valuation field of mixed characteristic $(0,p)$ with perfect residue field. Let $(\unicode[STIX]{x1D70B}_{n})_{n\geqslant 0}$ be a system of $p$-power roots of a uniformizer $\unicode[STIX]{x1D70B}=\unicode[STIX]{x1D70B}_{0}$ of $K$ with $\unicode[STIX]{x1D70B}_{n+1}^{p}=\unicode[STIX]{x1D70B}_{n}$, and define $G_{s}$ (resp. $G_{\infty }$) the absolute Galois group of $K(\unicode[STIX]{x1D70B}_{s})$ (resp. $K_{\infty }:=\bigcup _{n\geqslant 0}K(\unicode[STIX]{x1D70B}_{n})$). In this paper, we study $G_{s}$-equivariantness properties of $G_{\infty }$-equivariant homomorphisms between torsion crystalline representations.

Let u be a local homomorphism of noetherian local rings forming part of a commutative square vf=gu. We give some conditions on the square which imply that u is formally smooth. This result encapsulates a variety of (apparently unrelated) results in commutative algebra greatly improving some of them: Greco's theorem on descent of quasi-excellence property by finite surjective morphisms, Kunz's characterization of regular local rings in positive characteristic by means of the Frobenius homomorphism (and in fact the relative version obtained by Andre and Radu), etc. In the second part of the paper, we study a similar question for the complete intersection property instead of formal smoothness, giving also some applications., v3: Accepted Manuscript

In this paper we discuss a relationship between the spectral asymmetry and the surface symmetry. More precisely, we show that for every automorphism of a Hurwitz surface with the automorphism group $\text{PSL}(2,\mathbb{F}_{q})$, the $\unicode[STIX]{x1D702}$-invariant of the corresponding mapping torus vanishes if $q$ is sufficiently large.

We study the relationship between the reduction number of a primary ideal of a local ring relative to one of its minimal reductions and the multiplicity of the corresponding Sally module. This paper is focused on three goals: (i) to develop a change of rings technique for the Sally module of an ideal to allow extension of results from Cohen–Macaulay rings to more general rings; (ii) to use the fiber of the Sally modules of almost complete intersection ideals to connect its structure to the Cohen–Macaulayness of the special fiber ring; (iii) to extend some of the results of (i) to two-dimensional Buchsbaum rings. Along the way, we provide an explicit realization of the $S_{2}$-fication of arbitrary Buchsbaum rings.

In this paper, we exhibit explicit automorphisms of maximal Salem degree 22 on the supersingular K3 surface of Artin invariant one for all primes $p\equiv 3~\text{mod}\,4$ in a systematic way. Automorphisms of Salem degree 22 do not lift to any characteristic zero model.

In this paper, we investigate the nontrivial zeros of quadratic $L$-functions near the real axis. Assuming the generalized Riemann hypothesis, we give an asymptotic formula for the weighted pair correlation function of quadratic $L$-functions associated to the Kronecker symbols. From this formula, we obtain several results on the rate of simple zeros of quadratic $L$-functions and on the average distance of such nontrivial zeros.

The aim of this paper is to carry out an explicit construction of CAP representations of $\text{GL}(2)$ over a division quaternion algebra with discriminant two. We first construct cusp forms on such a group explicitly by lifting from Maass cusp forms for the congruence subgroup ${\rm\Gamma}_{0}(2)$. We show that this lifting is nonzero and Hecke-equivariant. This allows us to determine each local component of a cuspidal representation generated by such a lifting. We then show that our cuspidal representations provide examples of CAP (cuspidal representation associated to a parabolic subgroup) representations, and, in fact, counterexamples to the Ramanujan conjecture.

In analogy with the complex analytic case, Mustaţă constructed (a family of) Bernstein–Sato polynomials for the structure sheaf${\mathcal{O}}_{X}$and a hypersurface$(f=0)$in$X$, where$X$is a regular variety over an$F$-finite field of positive characteristic (see Mustaţă,Bernstein–Sato polynomials in positive characteristic, J. Algebra321(1) (2009), 128–151). He shows that the suitably interpreted zeros of his Bernstein–Sato polynomials correspond to the$F$-jumping numbers of the test ideal filtration${\it\tau}(X,f^{t})$. In the present paper we generalize Mustaţă’s construction replacing${\mathcal{O}}_{X}$by an arbitrary$F$-regular Cartier module$M$on$X$and show an analogous correspondence of the zeros of our Bernstein–Sato polynomials with the jumping numbers of the associated filtration of test modules${\it\tau}(M,f^{t})$provided that$f$is a nonzero divisor on$M$.

Let I be an odd prime number and let K be an algebraic number field of degree I. Let M denote the genus field of K, i.e., the maximal extension of K which is a composite of an absolute abelian number field with K and is unramified at all the finite primes of K. In [4] Ishida has explicitly constructed M. Therefore it is of some interest to investigate unramified cyclic extensions of K of degree l, which are not contained in M. In the preceding paper [6] we have obtained some results about this problem in the case that K is a pure cubic field. The purpose of this paper is to extend those results.

Throughout the paper, a scheme means a noetherian scheme. By a curve C over a scheme S of genus g, we mean a proper and smooth S-scheme with irreducible curves of genus g as geometric fibres. In the previous paper [15], the author showed that the field of moduli for a non-hyperelliptic curve over a field coincides with the one for its canonically polarized jacobian variety, and in [16], he gave a partial result on the coincidence of the fields of rationality for a hyperelliptic curve and for its canonically polarized jacobian variety. In the present paper, we will discuss the isomorphy of the isomorphism schemes of two curves over a scheme and of their canonically polarized jacobian schemes, by using Oort-Steenbrink’s result [12].

In a previous paper [5], one of the present authors has worked out a theory of zeta functions of Selberg’s type for compact quotients of symmetric spaces of rank one. In the present paper, we consider the analogues of those results when G/K is a noncompact symmetric space of rank one and Γ is a discrete subgroup of G such that G/Γ is not compact but such that vol(G/Γ)Γ is a non-uniform lattice. Certain mild restrictions, which are fulfilled in many arithmetic cases, will be put on Γ, and we shall consider how one can define a zeta function ZΓ of Selberg’s type attached to the data (G, K, Γ).

This paper is a continuation of the two preceding papers [12], [13] where the classification of the de Jonquières type subgroups in the Cremona group of 3 variables is promised. However the classification of such subgroups is postponed until the article in preparation “On the maximal connected algebraic subgroups of the Cremona group II”. The purpose of this paper is to establish a general method to study algebraic subgroups in the Cremona group of n variables and to illustrate how it works and leads to the classification of Enriques (Theorem (2.25)) when applied to the 2 variable case. This method gives us also the classification of the maximal connected algebraic subgroups of the Cremona group of 3 variables.

In a recent paper [5], it was shown that the tensor product of a finite number of fields over a common subfield satisfies the property that each localization at a prime ideal is a primary ring (in the sense that a zero-divisor is in fact a nilpotent element).In the first section of this paper, we exploit the properties of associated primes and of flat extensions so as to generalize the above result to zero-dimensional algebras; a simple example shows that this is the best one can hope for. The converse situation is also investigated.

All rings in this paper are assumed to be commutative with identity, and they will generally also be Noetherian.In several recent papers the asymptotic theory of ideals in Noetherian rings has been introduced and developed. In this new theory the roles played in the standard theory by associated primes, R-sequences, classical grade, and Cohen-Macaulay rings are played by, respectively, asymptotic prime divisors, asymptotic sequences, asymptotic grade, and locally quasi-unmixed Noetherian rings. And up to the present time it has been shown that quite a few results from the standard theory have a valid analogue in the asymptotic theory, and a number of interesting and useful new results concerning the asymptotic prime divisors of an ideal in a Noetherian ring have also been proved. In fact the analogy between the two theories is so good that a very useful (but not completely valid) working guide is: results from the standard theory should have a valid analogue in the asymptotic theory. And, although asymptotic sequences are coarser than R-sequences (for example, they behave nicely when passing to R/z with z a minimal prime ideal in R), the converse of this working guide has also proved useful.

Let R be a field and S a separable algebraic closure of R with galois group R. In [8] Harrison succeeded in describing R/′R in terms of R only. More precisely, he constructed a certain complex (R, Q/Z) and proved Homc, where Homc denotes continuous homomorphisms and H2 stands for the second cohomology group of the complex . In this paper, which is mainly expository in nature, we reexamine Harrison’s proof and show how [8] connects with Kummer theory and the theory of galois algebras [16]. We emphasize that most of the ideas on which this paper is based originate in [8].

The classification of Riemann surfaces has largely reached its completion. The purpose of the present paper is to lay the foundation for a new intriguing field in the classification theory: Riemannian spaces with Euclidean metrics. The paper is self-contained, both for the Riemann surface expert and the reader whose main interest is with higher dimensions. The significance of locally Euclidean spaces lies, first of all, in that their function-theoretic nature differs for dimensions n>2 and n=2. The existence or nonexistence of Green's functions and positive or bounded harmonic functions in Rn, punctured Rn, and in the punctured flat torus offer simple examples. A striking phenomenon is that, despite such differences, the basic inclusion relations remain valid. Moreover, capacities and null-classes can be defined for components of point sets in Rn.

The present paper is devoted to the study of differential geometry of Kaehlerian homogeneous spaces. In section 1 we deal with the canonical decomposition of a simply connected complete Kaehlerian space and that of its largest connected group of automorphisms. We know that a simply connected complete Riemannian space V is the product of Riemannian spaces V0, V1, …, Vn, where V0 is a Euclidean space and V1, …,Vn are not locally flat and their homogeneous holonomy groups are irreducible [2]. Moreover, if V is homogeneous, so are all Vk [10]. We shall show that if V is Kaehlerian space with real analytic metric (resp. Kaehlerian homogeneous space), each factor Vk is also Kaehlerian (resp. Kaehlerian homogeneous) and that V is the product of V0, V1, …, Vn as Kaehlerian space. We call this decomposition the de Rham decomposition of the Kaehlerian space V. Although this result is supposedly known, there is no published proof as yet. Using this decomposition theorem we shall show that the largest connected group of automorphisms of a simply connected complete Kaehlerian space with real analytic metric is the direct product of those of the factors of the de Rham decomposition. In the Riemannian case this result has be been established in [3] by one of the authors of the present paper.

It is the purpose of the present paper to give a natural method of realizing bounded symmetric domains as matrix-spaces. Our method yields, as special cases, the well-known bounded models of irreducible bounded symmetric domains of classical type (I)-(IV), as were already described in the original paper of E. Cartan [1] (see §3; we follow in this paper the classification table in [14], not in [1]). A direct application of this method will be to determine the canonical bounded models of the irreducible bounded symmetric domains of exceptional type; it will be published in another paper (see [6], [7] for the summary of the results).

The theory of stochastic differential equations in a differentiate manifold has been established by many authors from different view-points, especially by R Lévy [2], F. Perrin [1], A. Kolmogoroff [1] [2] and K. Yosida [1] [2]. It is the purpose of the present paper to discuss it by making use of stochastic integrals.

In [5], Ikeda-Nagao-Nakayama gave a characterization of algebras of cohomological dimension ≦n In a subsequent paper [4] Eilenberg gave an alternative treatment of the same question. The present paper is devoted to the discussion of a number of questions suggested by the results of [4] and [5]. Among others it is shown that the conditions employed in stating the main results in [4] and [5] are equivalent, so that the main results of these two papers are in accord. Further, the cohomological dimension of a residue-algebra is studied in terms of that of the original algebra and the (module-) dimension of the associated ideal. The terminology and notation employed here are that of [3].

Let S denote the integral closure of a complete discrete rank one valuation ring R in a finite Galois extension of the quotient field of R, G the Galois group of the quotient field extension, and f an element of Z2(G,U(S)) where U(S) denotes the multiplicative group of units of S. A crossed product Δ(f, S, G) whose radical is generated as a left ideal by the prime element II of S is an hereditary order according to the Corollary to Thm. 2. 2 of [2], and we call such a crossed product a II-principal hereditary order. In previous papers the author has studied II-principal hereditary orders Δ(f, S, G) for tamely and wildly ramified extensions S of R (see [10] and [11]). The purpose of this paper is to study II-principal hereditary orders Δ(f, S, G) with no restriction on the extension S of R.

We consider in this paper eleven conditions on a commutative ringR. The first of these is thatRcontains an identity. It is well known that each of the other properties is a consequence of the first condition. This paper considers other relations which exist between these properties. A complete diagram of all simple implications which exist between the eleven properties, together with proof of these implications, is given in section 3. Examples illustrating simple implications which do not hold are presented in section 4.

Throughout this paper all rings are assumed to have unit elements. A ring Λ is said to be semi-primary if its Jacobson radical N is nilpotent and Г = Λ/N satisfies the minimum condition. The main objective of this paper isTHEOREM I. Let A be a semi-primary algebra over a field K. Let N be the radical of Λ and Г = Λ/N. IfThendim Λ = gl.dim Λ.

In the previous paper [6], we studied the liftings of tensor fields to tangent bundles of higher order. The purpose of the present paper is to generalize the results of [6] to the tangent bundles of pr-velocities in a manifold M— notions due to C. Ehresmann [1] (see also [2]). In §1, we explain the pr-velocities in a manifold and define the (Λ)-lifting of differentiable functions for any multi-index λ -(λ1, λ2,…,λp) of non-negative integers λi satisfying ΣΛi≤r. In § 2, we construct ‹Λ›-lifts of any vector fields and ‹Λ›-lifts of 1-forms. The ‹Λ›-lift is a little bit different from the ‹Λ›-lift of vector fields in [6].

The concept of semi-local rings was introduced by C. Chevalley [1], which the writer has generalized in a recent paper [7] by removing the chain condition. The present paper aims mainly at the study of completions of semi-local rings. First in § 1 we investigate semi-local rings which are subdirect sums of semi-local rings, and we see in § 2 that a Noetherian semi-local ringRis complete if (and only if)R/pis complete for every minimal prime divisorpof zero ideal, together with some other properties. Further we consider in § 3 subrings of the completion of a semi-local ring. § 4 gives some supplementary remarks to [7], Chapter II, Proposition 8.

In this paper, coupled systemsof Korteweg-de Vries type are considered, where u = u(x, t), v = v(x, t) are real-valued functions and where x, t∈R. Here, subscripts connote partial differentiation andare quadratic polynomials in the variables u and v. Attention is given to the pure initial-value problem in which u(x, t) and v(x, t) are both specified at t = 0, namely,for x ∈ ℝ. Under suitable conditions on P and Q, global well-posedness of this problem is established for initial data in the L2-based Sobolev spaces Hs(ℝ) × Hs(ℝ) for any s > ‒3/4.

Let A be a Noetherian local ring with the maximal ideal m, and let I be an m-primary ideal in A. This paper examines the equality on Hilbert coefficients of I first presented by Elias and Valla, but without assuming that A is a Cohen–Macaulay local ring. That equality is related to the Buchsbaumness of the associated graded ring of I.

Let p and ℓ be two distinct primes. The aim of this paper is to show how, under a certain congruence hypothesis, the mod ℓ cohomology complex of the Lubin-Tate tower, together with a natural Lefschetz operator, provides a geometric interpretation of Vignéras’s local Langlands correspondence modulo ℓ for unipotent representations.