Für ein eigentliches konvexes Polyeder P des k-dimensionalen euklidischen Raumes gilt die isoperimetrische Ungleichung

(1)

wobei V das Volumen und F die Oberfläche von P bedeuten; die Grôsse wk bezeichnet das Volumen der k-dimensionalen Einheitskugel, also

(2)

The Laplacians form a class of the most important differential operators in white noise analysis. The goal of this paper is to give their characterizations. Our main tools are the Fock expansions of operators in terms of integral kernel operators and rotation-invariance. In Section 1, the fundamental setting of white noise analysis is introduced briefly. In Section 2, integral kernel operators and the Fock expansions of operators are given. The characterization theorems for number operator, Gross-Laplacian and Euler operator are given in Sections 3, 4 and 5 respectively.

We relate the analytic spread of a module expressed as the direct sum of two submodules with the analytic spread of its components. We also study a class of submodules whose integral closure can be expressed in terms of the integral closure of its row ideals, and therefore can be obtained by means of a simple computer algebra procedure. In particular, we analyze a class of modules, not necessarily of maximal rank, whose integral closure is determined by the family of Newton polyhedra of their row ideals.

In [M] Murthy asked if every subvariety V of kn (k-any field) with a trivial conormal bundle is a (scheme theoretic) complete intersection.

In [B] we were able to prove that all such subvarieties of kn are at least set theoretic complete intersections. Mohan Kumar has shown in [MK] that if one assumes moreover that n ≥ 2 dim V + 2 then V is a complete intersection. The aim of this paper is to prove the following result which extends the Mohan Kumar Theorem in the nonsingular (and connected) case and moreover sheds some light on his bound for n.

Cobordism invariants have been applied to both real and complex categories. For example, the index of 4k-dimensional manifolds was treated by Hirzebruch for the generalization of the Riemann-Roch theorem [3]. He also considered, in relation with this, virtual genus or virtual characteristics. But many invariants, such as virtual characteristics, have their origin in complex analytic category. In view of this, we consider certain complex analytic cobordism, i.e., quasilinear cobordism among quasilinear subvarieties in complex manifolds (see for the definition of quasilinear structure [4].) Quasilinear body has very simple type of singularities, as well as its quasilinear boundaries. Therefore, the theory of quasilinear cobordism can be reduced, through σ-processes, to that of non-singular cobordism theory.

Let X be a compact space and f be a continuous map from X into itself. The topological entropy of f, h(f), was defined by Adler, Konheim and McAndrew [1]. After that Bowen [4] defined the topological entropy for uniformly continuous maps of metric spaces, and proved that the two entropies coincide when the spaces are compact. The definition of Bowen is useful in calculating entropy of continuous maps.

In the previous paper [7], we have studied the relationship between the Riemannian connection of an n-dimensional Riemannian space M imbedded into the (n + k)-dimensional Euclidean space Rn+k and the canonical connection in the bundle Pn, k = O(n + k)/{l} Х O(k) over the Grassmann manifold Mn, k = O(n + k)/O(n) Х O(k).

In the first half of the present paper, the relationship between the canonical connections in bundles and the invariant Riemannian connection on Mn, k will be discussed. We obtain the holonomy groups of these canonical connections.

The study of finiteness for Riemannian manifolds, which has been done originally by J. Cheeger [5] and A. Weinstein [13], is to investigate what bounds on the sizes of geometrical quantities imply finiteness of topological types, —e.g. homotopy types, homeomorphism or diffeomorphism classes-— of manifolds admitting metrics which satisfy the bounds. For a Riemannian manifold M we denote by RM and KM respectively the curvature tensor and the sectional curvature, by Vol (M) the volume, and by diam(M) the diameter.

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 $r<(m+1)(n+1)$.

This is a continuation of the previous paper [8] concerning the relation between the arithmetic of imaginary quadratic fields and cusp forms of weight one on a certain congruence subgroup. Let K be an imaginary quadratic field, say K = with a prime number q ≡ − 1 mod 8, and let h be the class number of K. By the classical theory of complex multiplication, the Hubert class field L of K can be generated by any one of the class invariants over K, which is necessarily an algebraic integer, and a defining equation of which is denoted by

Φ(x) = 0.

We introduce an operation on modules over an F-finite ring of characteristic p. We call this operation tight interior. While it exists more generally, in some cases this operation is equivalent to the Matlis dual of tight closure. Moreover, the interior of the ring itself is simply the big test ideal. We directly prove, without appeal to tight closure, results analogous to persistence, colon capturing, and working modulo minimal primes, and we begin to develop a theory dual to phantom homology. Using our dual notion of persistence, we obtain new and interesting transformation rules for tight interiors, and so in particular for the test ideal. Using our theory of phantom homology, we prove a vanishing theorem for maps of Ext. We also compare our theory with Blickle’s notion of Cartier modules, and in the process we prove new existence results for Blickle’s test submodule. Finally, we apply the theory we developed to the study of test ideals in nonnormal rings, proving that the finitistic test ideal coincides with the big test ideal in some cases.

Stark [9] has explicitly evaluated some symplectic Gauss sums when the “denominator” matrix has odd prime level. This result is useful in computing the exact tranformation formulas of multivariable theta functions (see Stark [10], Friedberg [3] and Styer [11]). It is particularly useful when considering theta functions with quadratic forms having an odd number of variables, often a troublesome case (see Eichler [2] and Andrianov-Maloletkin [1]).

For a closed and bounded set E in the complex plane, let A(E) denote the collection of all functions continuous on E and analytic on E°, its interior; let R(E) denote the collection of all functions which are uniform limits on E of rational functions with poles outside E. Then let A denote the collection of all closed, bounded sets for which A(E) = R(E). The purpose of this paper is to formulate a condition on a set, which is essentially of a geometric nature, in order that the set belong to A. Then using approximation techniques, we shall construct a meromorphic function having a certain boundary behavior on a perfect set; this answers a question raised in [1].

The purpose of this paper is to study the distribution of Fatou points of annular functions introduced by Bagemihl and Erdös [1]. Recall that a function f(z), regular in the open unit disk D: | z | < 1, is referred to as an annular function if there exists a sequence {Jn} of closed Jordan curves, converging out to the unit circle C: | z | = 1, such that the minimum modulus of f(z) on Jn increases to infinity. If the Jn can be taken as circles concentric with C, f(z) will be called strongly annular.

As a continuation of the previous paper [7], we shall consider in this paper the problem of prediction given bounded intervals and obtain integral representations of predictors and prediction errors. For that purpose we shall introduce innovation processes well matched with bounded intervals. We follow the notation and terminology in [6].

Main purpose of the present paper is to exhibit a class of truth-value evaluations of the primitive logic LO and its sentential part LOS.

In this article, we study short exact sequences of finitary 2-representations of a weakly fiat 2-category. We provide a correspondence between such short exact sequences with fixed middle term and coidempotent subcoalgebras of a coalgebra 1-morphism defining this middle term. We additionally relate these to recollements of the underlying abelian 2-representations.

In Part II, we shall be concerned with applications of classical invariant theory, to statistic physics and to theta functions. Main theorem in Chapter 2 is stated as follows:

For a partition function

satisfying γl ≥ 0 (l ≥ 1) and α > 0, the 2n-apolar of ξ(s)

has the expansion

such that βn,1 ≥ 0 (l ≥ 2). This means, for a given partition function ξ(s) with nonnegative relative probabilities, we construct a sequence of partition functions A2n (ξ(s), ξ(s))n≥1 with the same properties, which may be considered a sequence of symbolical higher derivative of ξ(s). The main theorem in Chapter 3 is stated as follows: For given theta functions φ1(z) and φ2(z) of level n1 and n2 respectively, in g variables z = (z1, z2,…, zg), then r = (r1, r2,…, rg-apolar

is a theta function of level n1 + n2, and

We show an explicit functional equation of the standard L-function associated with an automorphic form on a definite orthogonal group over a totally real algebraic number field. This is a completion and a generalization of our previous paper, in which we constructed standard L-functions by using Rankin-Selberg convolution and the theory of Shintani functions under certain technical conditions. In this article we remove these conditions. Furthermore we show that the L-function of f has a pole at s = m/2 if and only if f is a constant function.

Dans cet article, on donne la classification des orbites des préhomogènes de type parabolique (g0, g1) correspondants aux paraboliques maximaux, notés (Δ, λ0), suivants: (A2n-1, αn), (Bn, α1) ou (Dn, α1), (Cn, αn), (D2n, α2n) ou (D2n, α2n-1), (E7, α7), (F4, α1), (E6, α2), (E7, α1), (E8, α8) lorsque F est un corps local ou global de caractéristique zéro, à l’aide de formes quadratiques, ceci lorsque le sous-espace radiciel est de dimension un.