Let X be a Banach space and B a bounded subset of X. For each x ∈ X, define R(x) = sup{‖x – y‖ : y ∈ B}. If C is a nonempty subset of X, we call the number R = inƒ{R(x) : x ∈ C} the Chebyshev radius of B in C and the set the Chebyshev center of B in C. It is well known that if C is weakly compact and convex, then and if, in addition, X is uniformly convex, then the Chebyshev center is unique; see e.g., [9].

Let {Bα : α ∈ ∧} be a decreasing net of bounded subsets of X. For each x ∈ X and each α ∈ ∧, define

In a previous paper (1) the notion of the α-width ∈A(α) of a (0, 1)-matrix A was introduced, and a formula for the minimal α-width taken over the class of all (0, 1)-matrices having the same row and column sums as A, was obtained. The main tool in arriving at this formula was a block decomposition theorem (1, Theorem 2.1; repeated below as Theorem 2.1) that established the existence, in the class generated by A, of certain matrices having a simple block structure. The block decomposition theorem does not itself directly involve the notion of minimal α-width, but rather centres around a related class concept, that of multiplicity. We review both of these notions in § 2, together with some other pertinent definitions and results.

This paper is a continuation of [1] where we began the study of intertwining analytic Toeplitz operators. Recall that X intertwines two operators A and B if XA = BX. Let H2 be the Hilbert space of analytic functions in the open unit disk D for which the functions fr(θ) = f(reiθ) are bounded in the L2 norm, and H∞ be the set of bounded functions in H2. For φ ∊ Hφ, Tφ (or Tφ(z)) is the analytic Toeplitz operator defined on H2 by the relation (Tφf)(z) = φ(z)f(z). For φ ∊ H∞, we shall denote {φ(z): |z| < 1} by Range (φ) or φ(D). Then where and σ(Tφ) = Closure(φ(D)) [1]. If φ ∊ H∞ maps D into D, then we define the composition operator Cφ on H2 by the relation (Cφf) (z) = f(φ(z)). J. Ryff has shown [11, Theorem 1] that Cφ, is a bounded linear operator on H2.

We show that every finite, finitely related algebra in a congruence distributive variety has a near unanimity term operation. As a consequence we solve the near unanimity problem for relational structures: it is decidable whether a given finite set of relations on a finite set admits a compatible near unanimity operation. This consequence also implies that it is decidable whether a given finite constraint language defines a constraint satisfaction problem of bounded strict width.

Let C be an arbitrary Grothendieck site. The purpose of this note is to show that, with the closed model structure on the category S Pre(C) of simplicial presheaves in hand, it is a relatively simple matter to show that the category S Pre(C)stab of presheaves of spectra (of simplicial sets) satisfies the axioms for a closed model category, giving rise to a stable homotopy theory for simplicial presheaves. The proof is modelled on the corresponding result for simplicial sets which is given in [1], and makes direct use of their Theorem A.7.

This result gives a precise description of the associated stable homotopy category Ho(S Pre(C))stab, according to well known results of Quillen [6]. One will recall, however, that it is preferable to have several different descriptions of the stable homotopy category, for the construction of smash products and the like.

A graph G is an ordered pair G = (V, E) where E is a set of 2-element subsets of V. The set V is the set of vertices, and E is the set of edges. The vertices x and y are joined by an edge if {x, y} is an edge. If X is a set (of colors) and χ : V →X, then we say that χ is an X-coloring of G if whenever two vertices x and y are joined by an edge, then χ(x) ≠ x(y). A graph is X-colorable if there is an χ-coloring of it. We will identify the natural number n with the set {0, 1, …, n – 1}, and often refer to n-colorings and to graphs being n-colorable.

The problem of finding a lattice with a given abstract group of automorphisms has been solved by Garrett Birkhoff who proved that for any group of order g there exists a distributive lattice with at most elements. That this number can be somewhat reduced by modifications of Birkhoff's original procedure has already been shown by the author; it turns out, however, that it remains rather high for finite groups of relatively low order.

Denote by C0 the collection of complex-valued functions which are continuous and compactly supported on (0, ∞). The transformations of the title are defined on C0 by

1.1

and

1.2

respectively, where Yv(x) is the Bessel function of the second kind, and Hv(x) is the Struve function; see [1; 7.5.4(55)]. The two transformations are studied briefly in [6; §8.4]; tables of transform pairs are given in [2; Chapters IX and XI], where it is also stated that, for – , each of the transformations is the inverse of the other.

This paper is a direct continuation of the article “Proximal analysis and boundaries of closed sets in Banach space, Part I: Theory”, by the same authors. It is devoted to a detailed analysis of applications of the theory presented in the first part and of its limitations.

Theorem 2.1 has important consequences for geometry of Banach spaces. We start the presentation with a discussion of density and existence of R-proper points (Definition 1.3) for closed sets in Banach spaces. Our considerations will be based on the “lim inf” inclusions proven in the first part of our paper.

We consider the product of two power series with positive coefficients: .

What conditions will ensure that the coefficients wn shall be either (i) monotonic, or (ii) logarithmically convex? By the latter, we mean that wn2 ≤ Wn-1Wn+1 for n = 1, 2, … In investigating this question, which was suggested by a special example, we have found it convenient to express the conditions in terms of the ratios of un, vn to certain binomial coefficients, rather than in terms of un, vn themselves.

Bambah (1) has recently determined the most economical covering of three dimensional space by equal spheres whose centres form a lattice, the density of this covering being

1.1.

As is well known, this problem may be interpreted in terms of the inhomogeneous minimum of a positive definite quadratic form.

We study closed extensions $\underset{\scriptscriptstyle-}{A}$ of an elliptic differential operator $A$ on a manifold with conical singularities, acting as an unbounded operator on a weighted ${{L}_{p}}$-space. Under suitable conditions we show that the resolvent ${{\left( \lambda -\underset{\scriptscriptstyle-}{A} \right)}^{-1}}$ exists in a sector of the complex plane and decays like $1/\left| \lambda \right|$ as $\left| \lambda \right|\to \infty $. Moreover, we determine the structure of the resolvent with enough precision to guarantee existence and boundedness of imaginary powers of $\underset{\scriptscriptstyle-}{A}$.

As an application we treat the Laplace–Beltrami operator for a metric with straight conical degeneracy and describe domains yielding maximal regularity for the Cauchy problem $\dot{u}\,-\,\Delta u\,=\,f,$$u\left( 0 \right)\,=\,0$.

The notion of decompositon class in a semisimple Lie algebra is a common generalization of nilpotent orbits and the set of regular semisimple elements.We prove that the closure of a decomposition class has many properties in common with nilpotent varieties, e.g., its normalization has rational singularities.

The famous Grothendieck simultaneous resolution is related to the decomposition class of regular semisimple elements. We study the properties of the analogous commutative diagrams associated to an arbitrary decomposition class.

It is shown that simple stably projectionless ${{\text{C}}^{*}}$-algebras which are inductive limits of certain specified building blocks with trivial $\text{K}$-theory are classified by their cone of positive traces with distinguished subset. This is the first example of an isomorphism theorem verifying the conjecture of Elliott for a subclass of the stably projectionless algebras.

We study the algebraic properties of Generalized Laguerre Polynomials for negative integral values of the parameter. For integers $r,n\ge 0$, we conjecture that $L_{n}^{(-1-n-r)}(x)=\Sigma _{j=0}^{n}\left( _{n-j}^{n-j+r} \right){{x}^{j}}/j!$ is a $\mathbb{Q}$-irreducible polynomial whose Galois group contains the alternating group on $n$ letters. That this is so for $r=n$ was conjectured in the 1950's by Grosswald and proven recently by Filaseta and Trifonov. It follows fromrecent work of Hajir and Wong that the conjecture is true when $r$ is large with respect to $n\ge 5$. Here we verify it in three situations: (i) when $n$ is large with respect to $r$, (ii) when $r\le 8$, and (iii) when $n\le 4$. The main tool is the theory of $p$-adic Newton Polygons.

Throughout this paper, Z is the ring of integers, ƒ*(t) (ƒ(t)) is an integer monic (co-monic) polynomial in the indeterminate t (i.e., each coefficient of ƒ* (ƒ) is in Z and its highest (lowest) coefficient is 1 (5, p. 121, Definition) and M* (M) is the multiplicative semigroup of all integer monic (co-monic) polynomials ƒ* (ƒ) having no constant term. In (3, Theorem 2), Herstein proved that if R is a division ring with centre C such that

1

then R = C. In this paper we seek a generalization of Herstein's result to semi-simple rings. We also study the following condition:

(1)*

Our results are quite complete for a semi-simple ring R in which there exists a bound for the codegree ofƒ (ƒ*) (i.e., the degree of the lowest monomial of ƒ(ƒ*)) appearing in the left-hand side of (1) ((1)*).

A finite automaton may be thought of as a possible abstraction of a digital computer. Imagine a tape or sequence of letters from some alphabet being fed into a device with a finite number of internal states. When the device is in a particular state and receives an input letter, the system passes to another internal state and prints a letter on an output tape. This operation is continued until the entire input sequence has been processed.

Both Harary (6) and Ginsburg (4) have focused attention on the previously unsolved problem of counting the number of equivalence classes of finite automata. In the present paper, this problem is solved completely by proving a variety of theorems about the enumeration of functions.

It is well known [13] that the irreducible tensor representations (IRs) of the unitary, orthogonal, and symplectic groups in an n-dimensional space may be specified by means of Young tableaux associated with partitions (σ)s = (σ1, σ2, …, σp) with σ1 + σ2 + … + σp = s. Formulae for the dimensions of the corresponding representations have been established [1; 8; 9; 13] in terms of the row lengths of these tableaux. It has been shown [12] for the unitary group, U(n), that the formula may be written as a quotient whose numerator is a polynomial in n containing s factors, and whose denominator is a number independent of n, which likewise may be expressed as a product of s factors. This formula is valid for all n.

Throughout this paper R denotes the set of all real numbers, m(K) the Lebesgue measure of K ⊆ R, H a Hilbert space, L(H) the set of all linear continuous mappings of H into H, endowed with the usual structure of a Banach space.

We consider the mapping F of the set R into L(H) such that

holds for all x, y ∈ R. In (2) we have solved this equation under the assumption that H is of finite dimension. In this paper we prove that a weak measurability of F implies its weak continuity in the case of separable Hilbert space. In Theorem 2 we prove that every weakly continuous solution of (1) in the set of normal transformations has the form F(x) = cos (xN), where the normal transformation N does not depend on x.

Coercive closed forms on L2-spaces are studied whose associated L2-semigroups are positivity preserving. Earlier work by other authors is extended by further developing the potential theory of such forms and completed by proving an analytic characterization of those of these forms which have a probabilistic counterpart, i.e., are associated with (special standard) Markov processes. Examples with finite and infinite dimensional state spaces are discussed.