I. Introduction. The purpose of this paper is to discuss functions defined on the continuous sample paths of Gaussian Markov processes which serve as Green’s functions for pairs of generalized Schroedinger equations. The results extend the author’s earlier paper [2] to a forward time version, and consider different boundary conditions.

We address two aspects of finitely generated modules of finite projective dimension over local rings and their connection in between: embeddability and grade of order ideals of minimal generators of syzygies. We provide a solution of the embeddability problem and prove important reductions and special cases of the order ideal conjecture. In particular, we derive that, in any local ring R of mixed characteristic p > 0, where p is a nonzero divisor, if I is an ideal of finite projective dimension over R and p 𝜖 I or p is a nonzero divisor on R/I, then every minimal generator of I is a nonzero divisor. Hence, if P is a prime ideal of finite projective dimension in a local ring R, then every minimal generator of P is a nonzero divisor in R.

In earlier work, the author introduced a method for constructing a Frobenius categorification of a cluster algebra with frozen variables by starting from the data of an internally Calabi–Yau algebra, which becomes the endomorphism algebra of a cluster-tilting object in the resulting category. In this paper, we construct appropriate internally Calabi–Yau algebras for cluster algebras with polarized principal coefficients (which differ from those with principal coefficients by the addition of more frozen variables) and obtain Frobenius categorifications in the acyclic case. Via partial stabilization, we then define extriangulated categories, in the sense of Nakaoka and Palu, categorifying acyclic principal coefficient cluster algebras, for which Frobenius categorifications do not exist in general. Many of the intermediate results used to obtain these categorifications remain valid without the acyclicity assumption, as we will indicate, and are interesting in their own right. Most notably, we provide a Frobenius version of Van den Bergh’s result that the Ginzburg dg-algebra of a quiver with potential is bimodule $3$-Calabi–Yau.

The well-known Thullen-Remmert-Stein’s theorem ([9], [7]) asserts that, for a domain D in CN and an n-dimensional irreducible analytic set S in D, a purely n-dimensional analytic set A in D — S has an essential singularity at any point in 5 if A has at least one essential singularity in S. In [1], E. Bishop generalized this to the case that A has the boundary of capacity zero in his sense. Afterwards, in [8], W. Rothstein obtained more precise informations on the essential singularities of A under the assumption dim A = 1. The main purpose in this paper is to generalize these Rothstein’s results to the case of arbitrary dimensional analytic sets.

In this paper we compute and continue meromorphically to the entire complex plane the zeta functions of twisted quaternionic Shimura varieties of dimension 3. The twist of the quaternionic Shimura varieties is done by a mod ℘ representation of the absolute Galois group.

Let X be a complex n-dimensional reduced analytic space with isolated singular point x0, and with a strongly plurisubharmonic function ρ : X → [0, ∞) such that ρ(x0) = 0. A smooth Kähler form on X \ {x0} is then defined by i∂∂ρ. The associated metric is assumed to have -curvature, to admit the Sobolev inequality and to have suitable volume growth near x0. Let E → X \ {x0} be a Hermitian-holomorphic vector bundle, and ξ a smooth (0, 1)-form with coefficients in E. The main result of this article states that if ξ and the curvature of E are both then the equation has a smooth solution on a punctured neighbourhood of x0. Applications of this theorem to problems of holomorphic extension, and in particular a result of Kohn-Rossi type for sections over a CR-hypersurface, are discussed in the final section.

In this article a new contribution to the following question is given: Let Ω ⊂ ⊂ Cn be a bounded pseudoconvex domain with C∞-smooth boundary, q ∈ ∂Ω a fixed point and H a k-dimensional affine complex plane such that q ∈ H and H intersects ∂Ω at q transversally. Let U be a suitably small neighborhood of q, and denote by r a C∞-defining function of Ω on U. Under which conditions on ∂Ω near q is it possible to find an exponent η>0 > 0 such that every holomorphic function f on Ω′ = H ∩Ω∩ U with

where dλ′ denotes the Lebesgue-measure on H, can be extended to a holomorphic function ^f on Ω ∩ U such that even

In this paper, the authors get the Lp estimates for the commutators generated by strongly singular integral operators and BMO functions and the corresponding multilinear operators by the scale changing method introduced by Carleson and Sjölin.

Let be a smooth foliated manifold, and the Lie algebra of all leaf-tangent vector fields on M.

We establish certain deLeeuw type theorems for Littlewood-Paley functions. By these theorems, we know that the boundedness of a Littlewood-Paley function on ℝn is equivalent to the boundedness of its corresponding Littlewood-Paley function on the torus Tn.

To a pair $P$ and $Q$ of finite posets we attach the toric ring $K[P,Q]$ whose generators are in bijection to the isotone maps from $P$ to $Q$. This class of algebras, called isotonian, are natural generalizations of the so-called Hibi rings. We determine the Krull dimension of these algebras and for particular classes of posets $P$ and $Q$ we show that $K[P,Q]$ is normal and that their defining ideal admits a quadratic Gröbner basis.

Let $ZB$ be the center of a $p$-block $B$ of a finite group with defect group $D$. We show that the Loewy length $LL(ZB)$ of $ZB$ is bounded by $|D|/p+p-1$ provided $D$ is not cyclic. If $D$ is nonabelian, we prove the stronger bound $LL(ZB)<\min \{p^{d-1},4p^{d-2}\}$ where $|D|=p^{d}$. Conversely, we classify the blocks $B$ with $LL(ZB)\geqslant \min \{p^{d-1},4p^{d-2}\}$. This extends some results previously obtained by the present authors. Moreover, we characterize blocks with uniserial center.

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.

The purpose of this paper is to determine left-invariant vector fields on a Lie group G with a left-invariant Riemannian metric which induces C- flows on G.

A subgroup was defined by O. Ore to be quasinormal in a group if it permuted with all subgroups of the group, and he proved [5] that such a subgroup is subnormal (= subinvariant = accessible) in a finite group. Finite groups in which all subgroups are quasinormal were classified by K. Iwasawa [3], and more recently N. Ito and J. Szép [2] and the author [1] proved that a quasi-normal subgroup is an extension of a normal subgroup by a nilpotent group. Similar results were obtained by O. Kegel [4] and in [1] for subgroups which permute not necessarily with all subgroups but with those having some special property.

Let K be any field, and L a separable extension of K of finite degree. L has a structure of vector space over K, and we shall denote this space by V. The space of endomorphisms of V will be denoted by Let x be any element of L, and N(x) the norm of x relative to the extension L/K. N is then a function defined on V with values in K. We shall call N the norm form on V. The multiplicative groups of non-zero elements of K and L will be denoted by K* and L* respectively. Let H be any subgroup of if K*. Then the elements z of L* such that N(z)∈H form a subgroup of L*, which we shall denote by GH. On the other hand the elements s of such that N(sx) = Λ(s)N(x) with Λ(s)∈H for all X∈V, form obviously a subgroup of GL(V), which we shall denote by becomes an algebraic group if H=K* or {1}. In case will mean the group of linear transformations of V leaving semi-invariant the norm form of L/K and in case will mean the group of linear transformations of V leaving invariant the norm form of L/K.

We prove a multiplicity result of 2π-periodic solutions for certain weakly coupled system of ordinary differential equations with real parameters. The proofs are based on differential inequalities and coincidence degree.

An important problem in the theory of arithmetic automorphic functions is to construct, for a reductive algebraic group over Q which defines a bounded symmetric domain, a system of canonical models [2], [6], [18]. For the similitude group of a hermitian form over a quaternion algebra whose center is a totally real field, this is solved by Shimura [17], and for the similitude group of a hermitian form with respect to an involution of the second kind of a central division algebra over a CM-field, by Miyake [8], In this paper, we show that this also can be done for the special Clifford group of a quadratic form Q over a totally real algebraic number field. (We have to impose certain conditions on the signature of Q in order that G defines a bounded symmetric domain, see 1.1.)

Let A be a noetherian ring and let I be an ideal of A contained in the Jacobson radical of A: Rad (A). We assume that the form ring of A with respect to the ideal I: G = Gr (A, I), is a normal integral domain. Hence A is a normal integral domain and one can ask for the links between Cl(A) and Cl(G).