1. 1. This paper is the second in a series of papers discussing linear groups of prime degree, the first being (8). In this paper we discuss only linear groups of degree 7. Thus, G is a finite group with a faithful irreducible complex representation Xof degree 7 which is unimodular and primitive. The character of Xis x- The notation of (8) is used except here p= 7. Thus Pis a 7-Sylow group of G.In §§ 2 and 3 some general theorems about the 3-Sylow group and 5-Sylow group are given. In § 4 the statement of the results when Ghas a non-abelian 7-Sylow group is given. This corresponds to the case |P| =73 or |P|= 74. The proof is given in §§ 5 and 6. In a subsequent paper the results when Pis abelian will be given.

Two formulas for the multiplicity of the fiber cone $F\left( I \right)\,=\,\oplus _{n=0}^{\infty }{{I}^{n}}/\text{m}{{I}^{n}}$ of an $\text{m}$-primary ideal of a $d$-dimensional Cohen–Macaulay local ring $\left( R,m \right)$ are derived in terms of the mixed multiplicity ${{e}_{d-1}}\left( \text{m }|I \right)$, the multiplicity $e\left( I \right)$, and superficial elements. As a consequence, the Cohen–Macaulay property of $F\left( I \right)$ when $I$ has minimal mixed multiplicity or almost minimal mixed multiplicity is characterized in terms of the reduction number of $I$ and lengths of certain ideals. We also characterize the Cohen–Macaulay and Gorenstein properties of fiber cones of $\text{m}$-primary ideals with a $d$-generated minimal reduction $J$ satisfying $\ell \left( {{I}^{2}}/JI \right)\,=\,1$ or $l\left( Im \right)Jm) = 1$.

By a graph we mean a pair (X, R) where X is a non-void set and R ⊂ X × X. A mapping f: X → Y is called a compatible map (or morphism) from (X, R) into (Y, S) if 2f(R) ⊂ S, where 2f: X2 → Y2 is defined by 2f((x1, x2)) = (f(x1),f(x2)). The set of all compatible maps from (X, R) into itself forms a monoid (semigroup with a unit element) under composition, which is denoted by M(X, R). A graph (X1, R1) is said to be a full subgraph of (X, R) if X1 ⊂ X and R1 = R ∩ (X1 × X1). A graph (X, R) is said to be without loops if (x, x) ∉ R for all x ∈ X.

The Sturm-Liouville functions considered in this instalment are real (as are all other quantities discussed here) non-trivial solutions of the differential equation

1.1

Higher monotonicity properties, as defined in § 2, are investigated for a number of sequences (finite or infinite) associated with these functions. One such sequence, discussed in detail later, has the kth term

1.2

where the constant X > — 1 (to assure convergence of each integral), W(x) possesses higher monotonicity properties and, moreover, is such that, again, each integral converges, and X1, X2, … is a sequence (finite or infinite) of consecutive zeros of a solution of (1.1), which may or may not be linearly independent of y(x), in the interval of definition of the functions under consideration.

If G is a torsion–free group and F is a field, is the group ring F[G] a ring without zero divisors? This is true if G is an ordered group or various generalizations thereof - beyond this the question remains untouched. This paper proves a related result.

1.1. Throughout this note, will denote an associative ring but we shall not require to possess a unit.

If A and B are subsets of , then A + B will denote the set {x + y| x ∊ A, y ∊ B}. Aτ will denote the set {u ∊ | au = 0 for all a ∊ A} .

Elements a and b will be said to be orthogonal if ab = ba = 0.

If L is a Lie algebra with a representation Δ a→aΔ (a in L) (of finite degree), then by the trace form f = fΔ of Δ is meant the symmetric bilinear form on L obtained by taking the trace of the matrix products:

Then f is invariant, that is, f is symmetric and f(ab, c) — f(a, bc) for all a, b, c in L. By the Δ-radical L⊥ = L⊥ of L is meant the set of a in L such that f(a, b) = 0 for all b in L. Then L⊥ is an ideal and f induces a bilinear form , called a quotient trace form, on L/L⊥. Thus an algebra has a quotient trace form if and only if there exists a Lie algebra L with a representation Δ such that

We develop a method for deriving integral representations of certain orthogonal polynomials as moments. These moment representations are applied to find linear and multilinear generating functions for $q$-orthogonal polynomials. As a byproduct we establish new transformation formulas for combinations of basic hypergeometric functions, including a new representation of the $q$-exponential function $\text{ }{{\varepsilon }_{q}}$.

We present a new approach to singular reduction of Hamiltonian systems with symmetries. The tools we use are the category of differential spaces of Sikorski and the Stefan-Sussmann theorem. The former is applied to analyze the differential structure of the spaces involved and the latter is used to prove that some of these spaces are smooth manifolds.

Our main result is the identification of accessible sets of the generalized distribution spanned by the Hamiltonian vector fields of invariant functions with singular reduced spaces. We are also able to describe the differential structure of a singular reduced space corresponding to a coadjoint orbit which need not be locally closed.

The linearized theory of surface waves leads to several mixed boundary value problems which have been investigated by various methods. As the physical background of the theory has been repeatedly discussed, it will suffice to deal here mainly with the mathematical aspect of the question.

This paper analyzes the n-fold composition of a certain non-linear integral operator acting on a class of functions on [0,1 ]. The attracting orbit is identified and various properties of convergence to this orbit are derived. The results imply that the space-time scaling limit of a certain infinite system of interacting diffusions has universal behavior independent of model parameters.

Let D be an integral domain with 1 ≠ 0 . We consider “property SP” in D, which is that every ideal is a product of semiprime ideals. (A semiprime ideal is equal to its radical.) It is natural to consider property SP after studying Dedekind domains, which involve factoring ideals into prime ideals. We prove that a domain D with property SP is almost Dedekind, and we give an example of a nonnoetherian almost Dedekind domain with property SP.

If n is a non-negative integer, define pr(n) as the coefficient of xn in

;

otherwise define pr(n) as 0. In a recent paper (1) the author has proved that if r has any of the values 2, 4, 6, 8, 10, 14, 26 and p is a prime > 3 such that r(p + 1) ≡ 0 (mod 24), then

1, ,

where n is an arbitrary integer.

For X a metric continuum, 2X denotes the hyper space of all nonempty subcompacta, with the topology induced by the Hausdorff metric H, and C(X) ⊂ 2X the hyperspace of subcontinua. These hyperspaces are continua, in fact are arcwise-connected, since there exist order arcs between each hyperspace element and the element X. They also have trivial shape, i.e., maps of the hyperspaces into ANRs are homotopic to constant maps. For a detailed discussion of these and other general hyperspace properties, we refer the reader to Nadler's monograph [4].

The question of hyperspace contractibility was first considered by Wojdyslawski [8], who showed that 2X and C(X) are contractible if X is locally connected. Kelley [2] gave a more general condition (now called property K) which is sufficient, but not necessary, for hyperspace contractibility. The continuum X has property K if for every there exists δ > 0 such that, for every pair of points x, y with d(x, y) < δ and every subcontinuum M containing x, there exists a subcontinuum N containing y with .

For real or complex q, |q| < 1, let

(1.1)

for arbitrary λ and μ, so that

(1.2)

and

(1.3)

Define, as usual, a generalized q-hypergeometric function by (cf. [26, Chapter 3]; see also [18])

(1.4)

where, for convergence, |q| < 1 and |z| < ∞ when r is a positive integer, or |z| < 1 when r = 0, provided that no zeros appear in the denominator.

Let σ be a finite positive singular Borel measure defined on Euclidean space RN. For w ∈ RN and y > 0, its Poisson integral is defined by the formula

where CN is chosen so that

Since σ is singular, almost everywhere with respect to Lebesgue measure on RN. On the other hand, almost everywhere dσ. It follows that for all sufficiently small y,

is a non-empty open subset of RN. If σ has compact support then |Ey| → 0 as y → 0, where |Ey| denotes the Lebesgue measure of Ey. In this paper we give a lower bound on the rate at which |Ey| may go to zero. The lower bound depends on the smoothness of the measure; the smoother the measure, the more slowly |Ey| may approach 0.

Let K be a field and Mn﹛K) denote the vector space of n X n matrices over K. Marcus [4] posed the following general problem: Let W be a subspace of Mn(K) and S a subset of W. Describe the set L(S, W) of all linear transformations T on W such that T(S) is contained in S.

1. Definitions and notation. A generalized Steiner system (t-design, tactical configuration) with parameters t, λt, k, v is a system (T, B), where T is a set of v elements, B is a set of blocks each of which is a k-subset of T (but note that blocks bi and bj may be the same k-subset of T) and such that every set of t elements of T belongs to exactly λt of the blocks. If we put λt = u we denote by Su(t, k, v) the collection of all systems with these parameters. Thus Q ∈ Su(t, k, v) means Q = (T, B) is a system with the given parameters. If λt = u = 1, we write S(t, k, v) instead of S1(t, k, v) and refer to the system as a Steiner system. If t = 2, the system is called a balanced incomplete block design.

Set-valued mappings arise quite naturally in optimization and nonsmooth analysis. In optimization, typically one has a family of optimization problems that depend on some parameter. One can then associate to this family of problems the set-valued mappings that assign to the parameter the set of optimal solutions, the set of feasible solutions or the set of multipliers. Many of these set-valued mappings encountered in optimization have been shown to be “proto-differentiable” (see Rockafellar [16]) i.e., in some sense these set-valued mappings are “differentiable”. Using estimates provided by the proto-derivatives, see Proposition 2.1, one can then obtain information on how the sets depend on the parameter. The concept of proto-differentiation is described in Section 2.

One of the authors having just presented Cauchy's theorem on convex polyhedra to a class on “Convexity” at the University of Wisconsin, a discussion on Cauchy's lemma on convex polygons led to an exchange of letters which are here reproduced. The three letters are independently readable. Two new proofs of Cauchy's lemma are given, of which the second (§3) is very short.