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

Infinitely divisible probability measures on discrete spaces admitting a commutative convolution have been studied in various frameworks. For generalized convolutions related to Delphic structures an important contribution was made by Gilewski and Urbanik in [5]. In [11] Schwartz based his detailed analysis on convolutions arising from orthogonal series. Both of these approaches can be included into the framework of discrete hypergroups chosen f.e. by Gallardo and Gebuhrer in [4]. The main result common to these sources is the fact that, roughly speaking, all infinitely divisible probability measures are of Poisson type. Although the double coset spaces of Gelfand pairs are commutative hypergroups admitting an extended harmonic anafysis, the analytic methods developed in the theory of Gelfand pairs (see Dieudonné [3] and Heyer [7]) provide a more direct access to specific results like the characterization of divisible (idempotent, infinitely divisible) measures. For discrete Gelfand pairs (and their double coset spaces) Letac and his school have achieved remarkable results (see Letac [8], [9] and the references therein). The still unpublished thesis [1] of S. Ben Mansoor contains much information about divisible probability measures on cubes. In our exposition we reprove Ben Mansoor’s main theorem 3.5.4 with additional care, and discuss the special case of the ra-dimensional cube which was the basic object of study already in Letac, Takács [10]. It turns out that the main obstacle to be overcome in establishing the Poisson representation of infinitely divisible probability measures is the presence of idempotent factors, a problem that has been excluded f.e. in the work [4] of Gallardo and Gebuhrer.

To a certain Volevič system of singular partial differential equations, called a Fuchsian system, all the solutions of the homogeneous equation in a complex domain are constructed and parametrized in a good way, without any assumption on the characteristic exponents.

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].

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.

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.

The purpose of this paper is to give a result concerning the problem of geometric characterizations of the Euclidean n-space Cn and bounded domains. It is well known that a simply connected Riemann surface is biholomorphic to one of the Riemann sphere, the complex plane and the unit disc. And there are several results concerning the geometric characterization of these spaces. To show that some simply connected open Riemann surface is biholomorphic to the complex plane or the unit disc, it is sufficient to see that there exist non constant bounded sub-harmonic functions or not. But in the higher dimensional case, there is no uniformization theorem. By this reason to show that some complex manifold is biholomorphic to Cn or an open ball, we must construct a biholomorphic mapping directly.

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.

Let D be a division algebra of finite dimension n2 over it’s center F. Suppose D has an involution, τ, of the first kind, of symplectic type (e.g. [1], p. 169). By the theory of the pfaffian, τ symmetric elements have degree less than n/2 over F. On the other hand, Tamagawa has shown (unpublished) that involutions like τ are closely related to minimal symmetric idempotents in D ⊗ F D. This author began by examining and trying to generalize these relationships. But before any theory seemed possible for division algebras, a theory relating subfields and symmetric idempotents was required. This investigation gave rise to the results presented here, especially the main theorem in Section Two.

In this paper, we shall consider the logarithmic potential

where μ is a positive measure in the plane, P and Q are any points and PQ denotes the distance from P to Q. In general, consider the potential

of a positive measure μ taken with respect to a kernel K(P, Q) which is a continuous function in P and Q and may be + ∞ for P = Q. A kernel K (P, Q) is said to satisfy the balayage principle if, given any compact set F and any positive measure μ with compact support, there exists a positive measure μ′ supported by F such that K(P, μ′) = K(P, μ) on F with a possible exception of a set of k-capacity zero and K(P, μ′)≦K(P, μ) everywhere. A kernel K(P, Q) is said to satisfy the equilibrium principle if, given any compact set F there exists a positive measure λ supported by F such that K(P, λ) = V (a constant) on F with a possible exception of a set of K-capacity zero and K(p, λ)≦V everywhere.

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 cohomology theory of an associative algebra has been shown to be valuable in the study of the structure of algebras of finite cohomological dimension, especially those of dimension less than or equal to one over a field. M. Harada [9] has shown that every semi-primary hereditary algebra A (for example, A is finitely generated over a field R and has dimension < 1) is isomorphic to a generalized triangular matrix algebra. The concept of a central separable algebra over a commutative ring has been shown to be a useful generalization of the concept of a central simple algebra over a field, where a separable algebra is defined to be an algebra having cohomological dimension zero.

Consider a tower of principal fibrations

That is, Ei+1 is the pullback of Ei→Ri and the path fibration PRi → Ri. The question arises as to whether or not the tower can be shortened, that is, whether or not En+1→ B is fiber homotopically equivalent to a nice fibration E → B. If “nice” is taken to mean “principal” then sufficient conditions are known. They involve connectivity assumptions on the Ei. In this paper “nice” is taken to mean “D-relatively principal” for some space D.

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

This note is a correction of an error of the author’s paper mentioned in the title. (The reference [1]). The proof of the Prop. 6 of [1], Chap. II, p. 247, is an error. And the propositions and formulas in pp. 247-249 of [1] depending on this Prop. 6 must be corrected. All notations are referred to [1].

Coates and Wiles [1] and B. Perrin-Riou (see [2]) study the arithmetic of an elliptic curve E defined over a number field F with complex multiplication by an imaginary quadratic field K by using p-adic techniques, which combine the classical descent of Mordell and Weil with ideas of Iwasawa’s theory of Zp-extensions of number fields. In a special case they consider a non-cyclotomic Zp-extension F∞ defined via torsion points of E and a certain Iwasawa module attached to E/F, which can be interpreted as an abelian Galois group of an extension of F∞. We are interested in the corresponding non-abelian Galois group and we want to show that the whole situation is quite analogous to the case of the cyclotomic Zp-extension (which is generated by torsion points of Gm).

In the theory of integral quadratic forms or the related theory such as modular forms, it is important to give examples of forms with given discriminant for various computational purposes. However, little is known about this problem even in the case where the explicit formula for the class number has been given. Among the works that deal with satisfactorily many examples or give a constructive method of them, there are [1] for ternary forms, and [5] for quaternary forms with square discriminants. As for quaternary forms with prime discriminant, P. Ponomarev [8] gave an explicit way to construct such forms from ternary forms with discriminant 2q. It was based on the result of his previous work [7] and that of Y. Kitaoka [6], but was rather complicated.

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].