Let C1[0, T] denote (one-parameter) Wiener space; that is the space of continuous functions x on [0, T] such that x(0) = 0. In a recent expository essay [21], Nelson calls attention to some functions on Wiener space which were discussed in the book of Feynman and Hibbs [13, section 3-10] and in Feynman’s original paper [12, section 13]. These functions have the form

Let I be a Cohen-Macaulay ideal of grade g > 0 in a local Gorenstein ring (R, m) with residue class field k. An R-ideal J is said to be linked to I with respect to the regular sequence α = α1 …, αg ⊂ I ∩ J if J = (α): I and I = (α): J ([6]). In this paper we are concerned with the following question: how big is dimk ((α, mJ)/mJ)? Obviously this dimension is at most g, but it could be as small as 0. If it is g then the link from J to I is called a minimal link, which is in most respects the desired type of link. The only general result known in this direction is that if I is Gorenstein, then dimk {(α, mJ)/mJ) = g unless both I and J are complete intersections (see [1], Proposition 5.2). We are able to generalize this fact to the case where (R/I)p is Gorenstein for all prime ideals p in R/I with dim (R/J)p ≤ 4; however we have to assume that I is generically a complete intersection ideal, and that R is a complete intersection (Theorem 2.3). Without the assumption on R we prove that if I is generically a complete intersection, and if for a fixed integer r the type of (R/I)p is at most r for all prime ideals p in R/I with dim (R/I)P ≤ (r + 1)2, then dimk ((α, mJ/mJ)) ≥ g — r (Proposition 2.1). If r = 1, i.e. if R/I is Gorenstein in codimension 4, then this estimate shows the dimension is at least g — 1. Theorem 2.3 can also be interpreted to yield a strong upper bound for the codimension of the non-Gorenstein-locus of certain perfect ideals: Let R be a regular local ring. Let I be an R-ideal which is generically a complete intersection, and assume that I is in the even linkage class of a Gorenstein ideal (i.e., there exists a sequence of links I ~ I1 ~ I2 ~ … ~ I2n with I2n a Gorenstein ideal); then I is a Gorenstein ideal provided that {R/I)p is Gorenstein for all prime ideals p of R/I with dim (R/I)p ≤ 4 (Corollary 3.1).

An entire function F(z) = f(g(z)) is said to have f(z) and g(z) as left and right factors respe2tively, provided that f(z) is meromorphic and g(z) is entire (g may be meromorphic when f is rational). F(z) is said to be prime (pseudo-prime) if every factorization of the above form implies that one of the functions f and g is bilinear (a rational function). F is said to be E-prime (E-pseudo prime) if every factorization of the above form into entire factors implies that one of the functions f and g is linear (a polynomial). We recall here that an entire non-periodic function f is prime if and only if it is E-prime [5]. This fact will be useful in the sequel.

In this paper, we treat the problem of determining the rate of growth of meromorphic functions on the plane, which are solutions of nth order algebraic differential equations whose coefficients are arbitrary entire functions (i.e., equations of the form, Ω(z,y,dy/dz, • • •, dny/dzn) — 0, where Ω is a polynomial in y, dy/dz, • • •, dny/dzn whose coefficients are arbitrary entire functions of z.)

We shall present in this paper some ergodic properties of the stepping stone model. The model has been proposed by M. Kimura [2], to describe the evolution of a genetical population with mating and geographical structures. It has been investigated and developed by M. Kimura and G. H. Weiss [3], G. H. Weiss and M. Kimura [6], W. Fleming and C.-H. Su [1], S. Sawyer [5], and others.

A degeneration of K3 surfaces (over the complex number field) is a proper holomorphic map π: X→Δ from a three dimensional complex manifold to a disc, such that, for t ≠ 0, the fibres Xt = π-1(t) are smooth K3 surfaces (i.e. surfaces Xt with trivial canonical class KXt = 0 and dim H1(Xt, Oxt) = 0).

The existence and basic properties of what are now referred to as spinor exceptional integers for a genus of integral ternary quadratic forms were first observed in the 1950’s by Jones and Watson [7] and Kneser [8] in the context of indefinite forms. The study of these integers and their generalizations has been undertaken by a number of authors in recent years, and has contributed significantly to the understanding of representation properties unique to ternary forms. In this direction, the present author proved in a previous paper [4] that if c is a primitive spinor exceptional integer for a genus of integral ternary quadratic forms and f is some form in this genus, then a form in the spinor genus of f primitively represents c if and only if f primitively represents an integer of the type ct2, for some odd positive integer t, relatively prime to the discriminant d, which satisfies the condition that the Jacobi symbol (–cd/t) equals 1.

Recently a number of authors have studied the application of Grunsky’s coefficient inequalities to the study of the Bieberbach conjecture for the class of normalized regular univalent functions f(z) in the unit circle |z|< 1

Charzynski and Schiffer [2] applied this result to give an elementary proof of the inequality | a4 | Ȧ 4. One of the present authors [8] proved that if a2 is real non-negative then A natural first step in the study of the inequality for a coefficient is to prove local maximality for a2 near to 2.

We introduce a new method for expanding an abelian category and study it using recollements. In particular, we give a criterion for the existence of cotilting objects. We show, using techniques from noncommutative algebraic geometry, that our construction encompasses the category of coherent sheaves on Geigle–Lenzing weighted projective lines. We apply our construction to some concrete examples and obtain new weighted projective varieties, and analyze the endomorphism algebras of their tilting bundles.

E. Cartan has proved that the prolonged system of a Pfaffian system in involution is also in involution. But he has treated only the case of a Pfaffian system of some special type. On the other hand in his book [3] he has reduced the solution of any differential system to the solution of a Pfaffian system of the type mentioned above, and this reduction is precisely the method of the prolongation of a differential system. Therefore it seems to be disirable to establish the above mentioned theorem in the case of an arbitrary differential system.

In this note we give a lattice path proof of the ribbon determinant formula for Schur functions ((1) below) which was originally formulated and proved in [2].

This paper is a continuation to the previous one [3], We shall show that, for the inhomogeneous linear group ISL(n + 1, R) (resp. ISp(n, R)), the coadjoint invariant polynomial algebra is generated by one (resp. n) algebraically independent element. We shall state our results more precisely.

In [15] Titchmarsh posed and solved under the generalized Riemann Hypothesis, the problem of an asymptotic behavior of the number of the solutions of the equation 1 = p — n1n2 for a prime p ≦ x and natural numbers n1 and n2. When we put then the above problem is to get an asymptotic law for the sum

Let I’ be a maximal order over a complete discrete rank one valuation ring R in a central simple algebra over the quotient field of R. The purpose of this paper is to determine necessary and sufficient conditions for I’ to be equivalent to a crossed product over a tamely ramified extension of R.

It is a classical result that every central simple algebra over a field k is equivalent to a crossed product over a Galois extension of k. Furthermore, it has been proved by Auslander and Goldman in [2] that every central separable algebra over a local ring is equivalent to a crossed product over an unramified extension.

Most of the problems in differential geometry can be reduced to problems in differential equations and differential inequalities on Riemannian manifolds. Our main purpose of this paper is to study such differential inequalities on complete Riemannian manifolds. In [5], H. Omori proved a very important theorem. S. Y. Cheng and S. T. Yau gave a simplification and a generalization of it which was called the generalized maximum principle in [2] and [7], and many interesting applications in differential geometry in [2], [3], [7], and [8].

The aim of this paper is to establish the principle of limiting absorption for uniformly propagative systems Λ(x, Dx) = E(x)-1AjDj, Dj = — i∂/∂xj, with perturbations of long-range class, where the perturbation of long-range class, roughly speaking, means that E(x) approaches to E0, E0 being the N × N identity matrix, as | x | → ∞ with order O(| x |-δ), 0 < δ ≦ 1. (The more precise assumptions will be stated below and we require some additional assumptions on the derivatives of E(x).) The spectral and scattering problem for uniformly propagative systems was first formulated by Wilcox [10]. Since then, the principle of limiting absorption has been proved by many authors ([5], [7], [8], [11] etc.). The perturbations discussed in their works belong to the short-range class with δ > 1.

In D. Aldous gave a sufficient condition for the tightness of a sequence (Xn)n≥0 of right continuous (with left limits) processes taking their values in a separable complete metric space S. As already noted by Aldous this condition is far from being necessary when the processes Xn are not continuous. More precisely the Aldous-condition implies the left-quasi-continuity of all the weak limits of the sequence (Xn)n≥0.

Soit V une variété complexe satisfaisant aux conditions suivantes:

• 1) un groupe de Lie semi-simple connexe G opère sur V de manière transitive et holomorphe;

• 2) la variété V admet une mesure invariante par G.

On dira par abus de langage que la variété V est semi-simple. On se propose d’étudier dans ce travail des variétés complexes compactes semi-simples. Les variétés de ce genre font l’objet de deux travaux de Wang [5], [6]. Dans le mémoire [5] il a étudié le cas simplement connexe et dans [6] le cas parallélisable. Le résultat obtenu dans ce travail montre que toute variété complexe compacte semi-simple est en un certain sens une combinaison de deux types de variétés étudiées par Wang. On établit en effet le théorème suivant.

We shall work over a fixed algebraically closed field k of arbitrary characteristic. By an algebraic variety over k we shall mean a reduced algebraic scheme over k. Fix a positive integer n and e = (e0, el,…, en) a system of n + 1 weights (i.e. n + 1 positive integers e0, el,…, en). If k[T0, Tl,…, Tn] is the polynomial k-algebra in n + 1 variables, graded by the conditions deg(Ti) = ei i = 0, 1,…, n, denote by Pn(e) = Proj(k[T0, T1,…, Tn]) the n-dimensional weighted projective space over k of weights e. We refer the reader to [3] for the basic properties of weighted projective spaces.