Quite some years ago a number of mathematicians were interested in obtaining explicit expressions for the bounds in Schottky's and Landau's theorems, specifically numerically évaluable bounds of a particular form. The best bounds of this type in Schottky's theorem were given by the author [3]. For Landau's theorem the chosen form is as follows. Let F(Z) be regular in |Z| < 1, omit the values 0 and 1 and have Taylor expansion about Z = 0

Then

Using the same method employed for Schottky's theorem the author showed that one can take K = 5.94. By a slight modification of the author's method Lai [6] gave the further value K = 4.76.

Let V be an n-dimensional vector space over the field F. Let ꕕm V be the rath tensor power of V. If ᓂ ∈ Sm, the symmetric group, there exists a linear operator P (ᓂ1) on ꕕm V such that

for all x1, … , xm ∈ V. (Here, x1 ꕕ … ꕕ xm denotes the decomposable tensor product of the indicated vectors.) If c is any function of Sm taking its values in F, we define

Until recently none of the numerous papers on the distribution of quadratic and higher power residues was concerned with questions of the following sort: Let k and m be positive integers. According to a theorem of Brauer (1), for every sufficiently large prime p there exist m consecutive positive integers r, r + 1 , . . . , r + m — 1, each of which is a kth power residue of p. Let r(k, m, p) denote the least such r.

The measure (x1x2x3)2adm(x) on the unit sphere in R3 is invariant under sign-changes and permutations of the coordinates; here dm denotes the rotation-invariant surface measure. The more general measure

corresponds to the measure

on the triangle

(where ). Appell ([1] Chap. VI) constructed a basis of polynomials of degree n in v1, v2 orthogonal to all polynomials of lower degree, and a biorthogonal set for the case γ = 0. Later Fackerell and Littler [6] found a biorthogonal set for Appell's polynomials for γ ≠ 0. Meanwhile Pronol [10] had constructed an orthogonal basis in terms of Jacobi polynomials.

This paper describes a systematic procedure which yields in a finite number of steps a solution to the following problem. Let G be a group generated by a finite set of generators g1, g2, g3, . . . , gr and defined by a finite set of relations R1 = R2 = . . . = Rk = I, where I is the unit element of G and R1R2, . . . , Rk are words in the gi and gi-1. Let H be a subgroup of G, known to be of finite index, and generated by a finite set of words, W1, W2, . . . , Wt. Let W be any word in G. Our problem is the following. Can we find a new set of generators for H, together with a set of representatives h1 = 1, h2, . . . , hu of the right cosets of H (i.e. G = H1 + Hh2 + . . . + Hhu) such that W can be expressed in the form W = Uhp, where U is a word in .

Le théoréme qui suit résoud une question qui m'avait été posée indépendamment par M. H. Mirkil et Professeur E. Farah, dans le cas particulier de p = 2 :

Théorème 1. Soit M un espace localement compact muni d'une mesure bornée μ, et soit 1 ≤ p < + ∞. Soit H un sous-espace vectoriel de L∞(μ), fermé dans LP(μ). Alors H est de dimension finie.

A number of combinatorial problems may be regarded as particular instances of the following rather general situation. Given a set X composed of n elements x1, x2, ..., xn, and m subsets X1 X2, … , Xm of X, find a minimal system of representatives for X1, X2, … , Xm. That is, single out a subset X* of X such that Xi ∩ X* is non-empty for i = 1,2, … ,m, and no subset of X containing fewer elements than X* has this property. To illustrate, each of the following can be thought of in these terms.

In a recent paper, Kaplansky [2] proved the following theorem: Let R be a ring with centre Z, and such that xn(x) ∈ Z for every x∈ R. If R, in addition, is semi-simple then it is also commutative

The existence of non-commutative rings in which every element is nilpotent rules out the possibility of extending this result to all rings. One might hope, however, that if R is such that xn(x) ∈ Z for all x ∈ R and the nilpotent elements of R are reasonably “well-behaved,” then Kaplansky's theorem should be true without the restriction of semi-simplicity.

We say that the restricted cancellation law for ideals (RCL) holds in the commutative ring R if from the equation AB = AC, where A, B, and C are ideals of R and AB ≠ (0), it follows that B = C. RCL is a weakened form of the cancellation law for ideals (CL) : If A, B, and C are ideals of R such that AB = AC and A ≠ (0), then B = C. A ring in which CL holds is an integral domain and in an integral domain, RCL is equivalent to CL.

It is well known (1, p. 162) that the lattice of subalgebras of a finite Boolean algebra is dually isomorphic to a finite partition lattice. In this paper we study the lattice of subalgebras of an arbitrary Boolean algebra. One of our main results is that the lattice of subalgebras characterizes the Boolean algebra. In order to prove this result we introduce some notions which enable us to give a characterization and representation of the lattices of subalgebras of a Boolean algebra in terms of a closure operator on the lattice of partitions of the Boolean space associated with the Boolean algebra. Our theory then has some analogy to that of the lattice theory of topological vector spaces. Of some interest is the problem of classification of Boolean algebras in terms of the properties of their lattice of subalgebras, and we obtain some results in this direction.

Let t be a Dedekind ring whose quotient field we denote by K. If is a prime ideal of t, let o denote the ring of all -regular elements of K. If is a torsion free i-module, let K denote the smallest K-module into which can be embedded and let o denote the o-submodule of K generated by .

For every polytope $\mathcal{P}$ there is the universal regular polytope of the same rank as $\mathcal{P}$ corresponding to the Coxeter group $\mathcal{C}\,=\,\left[ \infty ,\,.\,.\,.\,,\,\infty \right]$. For a given automorphism $d$ of $\mathcal{C}$, using monodromy groups, we construct a combinatorial structure ${{P}^{d}}$. When ${{P}^{d}}$ is a polytope isomorphic to $\mathcal{P}$ we say that $\mathcal{P}$ is self-invariant with respect to $d$, or $d$-invariant. We develop algebraic tools for investigating these operations on polytopes, and in particular give a criterion on the existence of a $d$-automorphism of a given order. As an application, we analyze properties of self-dual edge-transitive polyhedra and polyhedra with two flag-orbits. We investigate properties of medials of such polyhedra. Furthermore, we give an example of a self-dual equivelar polyhedron which contains no polarity (duality of order 2). We also extend the concept of Petrie dual to higher dimensions, and we show how it can be dealt with using self-invariance.

In this work we study the global geometry of planar quadratic Hamiltonian systems with a center and we sum up the dynamics of these systems in geometrical terms. For this we use the algebro-geometric concept of multiplicity of intersection Ip(P,Q) of two complex projective curves P(x, y, z) = 0, Q(x,y,z) = 0 at a point p of the plane. This is a convenient concept when studying polynomial systems and it could be applied for the analysis of other classes of nonlinear systems.

In this paper, we construct abelian varieties associated to Gauss’ and Appell–Lauricella hypergeometric series. Abelian varieties of this kind and the algebraic curves we define to construct them were considered by several authors in settings ranging from monodromy groups (Deligne, Mostow), exceptional sets (Cohen, Wolfart, Wüstholz), modular embeddings (Cohen, Wolfart) to CM-type (Cohen, Shiga, Wolfart) and modularity (Darmon). Our contribution is to provide a complete, explicit and self-contained geometric construction.

Let H be a subgroup of a group G (all groups considered throughout this article are finite); then H will be called primitive if the subgroup

is distinct from H. Such subgroups, which are also called meet-irreducible, arise naturally in connection with minimal permutation representations of groups and in other contexts; for example, every subgroup of a group G can be written as an intersection of primitive subgroups of G, and the set of all primitive subgroups of G is characterized by its minimality with respect to this property. While maximal subgroups are always primitive, most groups contain non-maximal subgroups which are primitive (see remark at end of article). Note that a subgroup H of an abelian group G is primitive if, and only if, G/H is cyclic of prime-power order.

Jategaonkar (5) has constructed a class of rings which can be used to provide counterexamples to problems concerning unique factorization in non-commutative domains, the left-right symmetry of the global dimension for a right- Noetherian ring and the transhnite powers of the Jacobson radical of a right- Noetherian ring. These rings have the following property:

(W) Every non-empty family of right ideals of the ring R contains exactly one maximal element.

In the present paper we wish to consider rings, with unit element, which satisfy property (W). This property means that the right ideals are inverse well-ordered by inclusion, and it is our aim to describe these rings by their order type. Rings of this kind appear as a generalization of discrete valuation rings in R; see (1; 2).

In the following, R will always denote a ring with unit element satisfying (W).

The martingale convergence theorem was first proved by Doob (3) who considered a sequence of real-valued random variables. Since various collections of real-valued random variables can be regarded as vector lattices, it seems of interest to prove the martingale convergence theorem in an arbitrary vector lattice. In doing so we use the concept of order convergence that is related to convergence almost everywhere, the type of convergence used in Doob's theorem.

In [11] G. G. Kasparov defined the “operator K-functor” KK(A, B) associated with the graded C*-algebras A and B. If the algebras A and B are trivially graded and A is nuclear he proves six term exact sequence theorems. He asks whether this extends to the graded case.

Here we prove such “six-term exact sequence” results in the graded case. Our proof does not use nuclearity of the algebra A. This condition is replaced by a completely positive lifting condition (Theorem 1.1).

Using our result we may extend the results by M. Pimsner and D. Voiculescu on the K groups of crossed products by free groups to KK groups [15]. We give however a different way of computing these groups using the equivariant KK-theory developed by G. G. Kasparov in [12]. This method also allows us to compute the KK groups of crossed products by PSL2(Z).

Explicit evaluations of the symmetric Euler integral $\int _{0}^{1}\,{{u}^{\alpha }}{{(1-u)}^{\alpha }}f(u)\,du$ are obtained for some particular functions $f$. These evaluations are related to duplication formulae for Appell’s hypergeometric function ${{F}_{1}}$ which give reductions of ${{F}_{1}}(\alpha ,\beta ,\beta ,2\alpha ,y,z)$ in terms of more elementary functions for arbitrary $\beta $ with $z=y/(y-1)$ and for $\beta =\alpha +\frac{1}{2}$ with arbitrary $y,z$. These duplication formulae generalize the evaluations of some symmetric Euler integrals implied by the following result: if a standard Brownian bridge is sampled at time 0, time 1, and at $n$ independent randomtimes with uniformdistribution on $[0,1]$, then the broken line approximation to the bridge obtained from these $n+2$ values has a total variation whose mean square is $n(n+1)/(2n+1)$.

An arithmetic function ƒ(n) is said to be additive if it satisfies ƒ(ab) = ƒ(a) + ƒ(b) whenever a and b are coprime integers. For such a function we define

A standard form of the Turán-Kubilius inequality states that

(1)

holds for some absolute constant c1, uniformly for all complex-valued additive arithmetic functions ƒ (n), and real x ≧ 2. An inequality of this type was first established by Turán [11], [12] subject to some side conditions upon the size of │ƒ(pm)│. For the general inequality we refer to [10].

This inequality, and more recently its dual, have been applied many times to the study of arithmetic functions. For an overview of some applications we refer to [2]; a complete catalogue of the applications of the inequality (1) would already be very large. For some applications of the dual of (1) see [3], [4], and [1].