The theory of the relationship between the symmetric group on a symbols, Σa, and the general linear group in n-dimensions, GL(n), was greatly developed by Weyl [4] who, in this connection, made use of tensor representations of GL(n). The set of mixed tensors

forms the basis of a representation of GL(n) if all the indices may take the values 1, 2, …, n, and if the linear transformation

is associated with every non-singular n × n matrix A. The representation is irreducible if the tensors are traceless and if the sets of covariant indices (α)a and contra variant indices (β)b themselves form the bases of irreducible representations (IRs) of Σa and Σb, respectively. These IRs of Σa and Σb may be specified by Young tableaux [μ]a and [v]b in the usual way [4].

The object of this paper is to determine the general solution of the functional equation

FE

where α is multiplicative. It turns out that non-trivial embeddings of the reals in the complex generate some interesting solutions.

In many applications, various special cases of (FE) have occurred ([1,3, 4, 6, 10, 11, 14]). The special case where f = g = h = k and α = the identity map is known as the fundamental equation of information, and has been extensively investigated by many authors ([5]). The case where f = g = h = k and α is multiplicative was treated in [13, 14]. The general solution of (FE) when α(1 – x) = (1 – x)β has been obtained in [9], except when β = 2.

The most important notion in topology is that of a homeomorphism f: X → Y from a topological space X onto a topological space Y. If a homeomorphism f: X → Y exists, then the topological spaces X and F are said to be homeomorphic (or topologically equivalent), in symbols,

X ≡ Y.

The relation ≡ among topological spaces is obviously reflexive, symmetric, and transitive; hence it is an equivalence relation. For an arbitrary family F of topological spaces, this equivalence relation ≡ divides /Mnto disjoint equivalence classes called the topology types of the family F. Then, the main problem in topology is the topological classification problem formulated as follows.

The topological classification problem: Given a family F of topological spaces, find an effective enumeration of the topology types of the family F and exhibit a representative space in each of these topology types.

Let A be a C*-algebra and a Hilbert space which is infinite dimensional and of Hilbert dimension ≧ dim π for all π ∈ Â. Suppose that the set Irr of all non-null *-representations π of A on , irreducible on the essential space , is given the relative strong topology as a subspace of Rep [2; 4; 6]. That is, the topology is that of simple convergence in with the strong topology. Finally, let ∼ denote equivalence of representations in Irr implemented by partial isometries in if and only if there exists a partial isometry with vv* ⊃ H(π1) and v*v ⊃ H(π2) satisfying π2(a) = v*π1(a)v for all a ∈ A.

There have been few published results concerning the relationship between the homology groups of branched and unbranched covering spaces of knots, despite the fact that these invariants are such powerful invariants for distinguishing knot types and have long been recognised as such [8]. It is well known that a simple relationship exists between these homology groups for cyclic covering spaces (see Example 3 in § 3), however for more complicated covering spaces, little has previously been known about the homology group, H1(M) of the branched covering space or about H1(U), U being the corresponding unbranched covering space, or about the relationship between these two groups.

We are concerned in this paper with category-theoretic aspects of homotopy theory. Originally, category theory developed as a simplifying language in the context of algebraic topology and yet one primary example: the category Π of spaces and homotopy classes of maps admits only limited use of the language owing to the very sparse occurrence of limits. Of course, full use has been made of them nevertheless: limits and colimits exist in the case of products and coproducts, and in almost no other case; yet, from this we obtain the theory of Samelson products, Whitehead products, and Hopf invariants which can all be expressed in Π see [8]. In addition, there are hosts of adjoint functors and yet the outcome is disappointing because the language applies only to special cases rather than to the situation as a whole.

We give an explicit treatment of cubic function fields of characteristic at least five. This includes an efficient technique for converting such a field into standard form, formulae for the field discriminant and the genus, simple necessary and sufficient criteria for non-singularity of the defining curve, and a characterization of all triangular integral bases. Our main result is a description of the signature of any rational place in a cubic extension that involves only the defining curve and the order of the base field. All these quantities only require simple polynomial arithmetic as well as a few square-free polynomial factorizations and, in some cases, square and cube root extraction modulo an irreducible polynomial. We also illustrate why and how signature computation plays an important role in computing the class number of the function field. This in turn has applications to the study of zeros of zeta functions of function fields.

Our work was motivated by attempts to find a criterion for the existence of a classical quotient ring, for a noetherian ring, in analogy with the various known criteria for the existence of an artinian classical quotient ring ([9], [10], [13], [2]).

We have restricted our attention to Krull symmetric noetherian rings R, and we make heavy use of the fact that all their Krull composition factors are non-singular (Proposition 7). The collection Kprime R of the associated primes of the Krull composition factors of R plays a central role, taking the place of the collection of the associated primes of R.

Let X be a Banach space and (fn)n be a bounded sequence in L1(X). We prove a complemented version of the celebrated Talagrand's dichotomy, i.e., we show that if (en)n denotes the unit vector basis of c0, there exists a sequence gn ∈ conv(fn, fn+1,...) such that for almost every ω, either the sequence (gn(ω) ⊗ en) is weakly Cauchy in or it is equivalent to the unit vector basis of ℓ1. We then get a criterion for a bounded sequence to contain a subsequence equivalent to a complemented copy of ℓ1 in L1(X). As an application, we show that for a Banach space X, the space L1(X) has Pełczyńiski's property (V*) if and only if X does.

This paper is a contribution to applied stability theory. Our purpose is to investigate the complexity of lattices by determining the stability of their first order theories.

Stability measures the complexity of a theory T by counting the number of different “kinds” of elements in models of T. The notion of ω-stability was introduced by Morley [26] in 1965 and generalized by Shelah [31] in 1969. Shelah classified all first order theories according to their stability properties.

Stability and -categoricity are closely related (see [26] and [1]). In fact, the notions of stable, superstable and ω-stable can be regarded as successive approximations of -categorical. -categoricity is a very strong property while stability, superstability and ω-stability facilitate the classification of more “complex” theories.

In this paper we determine the limiting distributional behavior for sums of infinitesimal conditionally free random variables. We show that the weak convergence of classical convolution and that of conditionally free convolution are equivalent for measures in an infinitesimal triangular array, where the measures may have unbounded support. Moreover, we use these limit theorems to study the conditionally free infinite divisibility. These results are obtained by complex analytic methods without reference to the combinatorics of $\text{c}$-free convolution.

Throughout this paper T will denote a bounded linear operator which is defined on a Banach space and whose spectrum lies on a rectifiable Jordan curve J .

The operators having some growth conditions on their resolvents have been the subject of discussion for a long time. Many sufficient conditions have been found to ensure that such operators have invariant subspaces [2 ; 3 ; 7 ; 8 ; 12 ; 13; 14; 21; 27; 28; 29], are S-operators [14], are quasidecomposable [9], are decomposable [4 ; 11], are spectral [7 ; 10 ; 15 ; 17], are similar to normal operators [16 ; 23 ; 25 ; 26], or are normal [15 ; 18 ; 22]. In this line we are going to show that many such operators are decomposable.

Our first result is a decomposition theorem for free groups relative to a set of elements. This enables us to formulate several algebraic conditions, some necessary and some sufficient, for various surfaces in the boundary of a 3-dimensional handlebody to be incompressible. Moreover, we show that there exists an algorithm to determine whether or not these algebraic conditions are met.

Many of our algebraic ideas are similar to those of Shenitzer [3]. Conversations with Professor Roger Lyndon were helpful in the initial development of these results, and he reviewed an earlier version of this paper, suggesting Theorem 1 (iii) and its proof. Our notation and techniques are standard (cf. [1], [2]). A set X of elements in a finitely generated free group F is a basis if it is a minimal generating set, and X±l denotes the set of all elements in X, together wTith their inverses.

We shall consider a multiparameter eigenvalue problem of the form

(1.1)

where λ ∈ Rk while Tn and Vn(λ) are self-adjoint linear operators on a Hilbert space Hn. If λ = (λ1, …, λk) ∈ Rk and satisfy (1.1) then we call λ an eigenvalue, x an eigenvector and (λ, x) an eigenpair. While our main thrust is towTards the general case of several parameters λn, the method ultimately involves reduction to a sequence of one parameter problems. Our chief contributions are (i) to generalise the conditions under which this reduction is possible, and (ii) to develop methods for the one parameter problem particularly suited to the multiparameter application. For example, we give rather general results on the magnitude and direction of the movement of non-linear eigenvalues under perturbation.

Let

(W): W0, W1,…, Wn,…

be a linear integral recurring sequence of order r ≥ 2; that is, a particular solution of the recurrence

(1.1) ,

where P1, P2,… , Pr ≠ 0 are integers, and the initial values W0, W1, … , Wr−1 are integers.

If there is a nonsingular symmetric bilinear form defined on a distributive algebra A with basis elements C1, C2, … , Cr over a field F such that and , then the so called Casimir operator [1,2]

Introduction. The concept of a reducible integer was introduced recently [3] : if P(m) denotes the greatest prime factor of m then n is said to be reducible if P ( 1 + n2) < 2n. The reason for the term is that reducibility is a condition necessary and sufficient for the existence of a relation of the form where the ƒi are integers and the ni positive integers less than n. J. C. P. Miller pointed out to us the regularity of the distribution of the reducible integers (less than 600).

If a1, a2, … , ak are relatively prime positive integers then the equation

always has solutions in non-negative xi for n sufficiently large. Sylvester called the number of non-negative solutions of (1) the denumerant of the equation. We shall denote the denumerant by n(a1, … , ak).

If $f,\,g:\,{{\mathbb{R}}^{n}}\,\to \,{{\mathbb{R}}_{\ge 0}}$ are non-negative measurable functions, then the Prékopa–Leindler inequality asserts that the integral of the Asplund sum (provided that it is measurable) is greater than or equal to the 0-mean of the integrals of $f$ and $g$. In this paper we prove that under the sole assumption that $f$ and $g$ have a common projection onto a hyperplane, the Prékopa–Leindler inequality admits a linear refinement. Moreover, the same inequality can be obtained when assuming that both projections (not necessarily equal as functions) have the same integral. An analogous approach may be also carried out for the so-called Borell-Brascamp-Lieb inequality.

We study Riemannian manifolds with boundary under a lower weighted Ricci curvature bound. We consider a curvature condition in which the weighted Ricci curvature is bounded from below by the density function. Under the curvature condition and a suitable condition for the weighted mean curvature for the boundary, we obtain various comparison geometric results.