This paper contains in part the author’s Ph.D. dissertation written under the supervision of Professor F. Bagemihl at the University of Wisconsin-Milwaukee. This research was supported by grants from the National Science Foundation and the University of Wisconsin Graduate School.

A probability measure μ is called unimodal if there is a point α such that the distribution function of μ is convex on (− ∞, α) and concave on (α, ∞). The point α is called a mode of μ. When μ is unimodal, the mode of μ is not always unique; the set of modes is a one point set or a closed interval. If μ is a unimodal distribution with finite variance, Johnson and Rogers [6] give a bound

where m and v are mean and variance of μ (see also [11]).

Let dk(n) be the number of the factorizations of n into k positive numbers. It is known that the following asymptotic formula holds:

where r and q are co-prime integers with 0 < r < q, Pk is a polynomial of degree k − 1, φ(q) is the Euler function, and Δk(q; r) is the error term. (See Lavrik [3]).

We slightly modify the definitions of q-Hurwitz ζ-functions and q-L-series constructed by J. Satoh. By using these modified functions, we give some relations for the ordinary Dirichlet L-series. Especially we give an elementary proof of Katsurada’s formula on the values of Dirichlet L-series at positive integers.

The first order language ℒ that we consider has two nullary function symbols 0, 1, a unary function symbol –, a binary function symbol +, a unary relation symbol 0 <, and the binary relation symbol = (equality). Let ℒ′ be the language obtained from ℒ, by adding, for each integer n > 0, the unary relation symbol n| (read “n divides”).

The notion of a ray class field, which is fundamental in Takagi’s class field theory, has no immediate analogon in the function field case. The reason for this lies in the lacking of a distinguished maximal order. In this paper I overcome this difficulty by a relative version of the notion of ray class fields to be defined for every holomorphy ring of the field. The prototype for this new notion is M. Rosen’s definition of a Hilbert class field for function fields [6].

Let G be a finite reflection group acting in a complex vector space V = ℂr, whose coordinate ring will be denoted by S. Any element γ ∈ GL(V) which normalises G acts on the ring SG of G-invariants. We attach invariants of the coset Gγ to this action, and show that if G′ is a parabolic subgroup of G, also normalised by γ, the invariants attaching to G′γ are essentially the same as those of Gγ. Four applications are given. First, we give a generalisation of a result of Springer-Stembridge which relates the module structures of the coinvariant algebras of G and G′ and secondly, we give a general criterion for an element of Gγ to be regular (in Springer’s sense) in invariant-theoretic terms, and use it to prove that up to a central element, all reflection cosets contain a regular element. Third, we prove the existence in any well-generated group, of analogues of Coxeter elements of the real reflection groups. Finally, we apply the analysis to quotients of G which are themselves reflection groups.

Le but principal de ce mémoire est d’étudier les espaces fibres principaux différentiables et holomorphes de groupe abelien connexe ayant pour base une C-variété au sens de Wang [12]. Une C-variété X étant une variété complexe compacte simplement connexe et homogène, X est la base d’un espace fibre principal qui est un groupe de Lie où opère un sous-groupe fermé par les translations à droite comme groupe structural. Pour cette raison, on considère d’abord dans le § 1 les espaces fibres principaux différentiables de groupe abelien connexe A ayant pour base la base X d’un certain espace fibre principal differentiate Y de groupe B. Parmi ces espaces fibres principaux les plus simples sont ceux qui sont associés à Y par les homomorphismes du groupe B dans le groupe A.

This paper deals with cluster sets. While cluster sets can be considered in a more abstract setting, we shall limit ourselves to the study of functions f defined in the open unit disc D of the complex plane and taking their values on the Riemann sphere C̅. For p a point of the unit circle C, we denote by C(f,p) the cluster set of f at p, i.e., the set of all values w ∈ C̅ for which there is a sequence {zn}, zn ∈ D, such that zn → p and f(zn) → w. The point p is called a point of determination for f if C(f, p) is a singleton. In Section 1 we characterize the set of points of determination of a function f defined in D.

In our paper Cl] we showed that there exist Schottky groups whose singular sets have positive 1-dimensional measure. Since the example was very complicated, it is natural to seek for simpler examples. Further the problem how about the singular sets of more general groups occurs.

Let K be a commutative ring, A a K-algebra, and B a K-subalgebra of A. The object of this paper is to prove some results on higher derivations (in the sense of Jacobson [4]) of B into A. In § 1 we introduce a notion of equivalence among higher derivations. With this notion of equivalence, we prove in § 2 (Theorem 1) that the equivalence classes of higher K-derivations of B into A are in one-one correspondence with the isomorphism classes of certain filtered B ⊗ KA°-modules, where A° denotes the opposite algebra of A.

Let $f(x_{1},\ldots ,x_{n})$ be a regular indefinite integral quadratic form with $n\geqslant 9$, and let $t$ be an integer. Denote by $\mathbb{U}_{p}$ the set of $p$-adic units in $\mathbb{Z}_{p}$. It is established that $f(x_{1},\ldots ,x_{n})=t$ has solutions in primes if (i) there are positive real solutions, and (ii) there are local solutions in $\mathbb{U}_{p}$ for all prime $p$.

Let S be a simply connected domain in the w + u + iv plane and let ∂S denote its boundary which we assume passes through w= ∞. Suppose that the segment L= {u ≧ u0; v = 0} of the real axis lies in S and that w∞ is the point of ∂ S accessible along L. Let z = z(w) = x(w) + iy(w) map S in a (1 — 1) conformal way onto ∑ = {z = x + iy: — ∞ < x < + ∞ } so that . The inverse map is w = w(z) = u(z) + iv(z). S is said to possess a finite angular derivative at w∞ if z(w) — w approaches a finite limit (called the angular derivative) as w→w∞ in certain substrips of S.

A projective log variety (X, D) is called a log Fano manifold if X is smooth and if D is a reduced simple normal crossing divisor on Χ with − (KΧ + D) ample. The n-dimensional log Fano manifolds (X, D) with nonzero D are classified in this article when the log Fano index r of (X, D) satisfies either r ≥ n/2 with ρ(X) ≥ 2 or r ≥ n − 2. This result is a partial generalization of the classification of logarithmic Fano 3-folds by Maeda.

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.

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].

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.

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.