A finite dimensional power-associative algebra 𝒰 with a unity element 1 over a field J is called a nodal algebra by Schafer (7) if every element of 𝒰 has the form α1 + z where α is in J, z is nilpotent, and if 𝒰 does not have the form 𝒰 = ℐ1 + n with n a nil subalgebra of 𝒰. An algebra SI is called a non-commutative Jordan algebra if 𝒰 is flexible and 𝒰+ is a Jordan algebra. Some examples of nodal non-commutative Jordan algebras were given in (5) and it was proved in (6) that if 𝒰 is a simple nodal noncommutative Jordan algebra of characteristic not 2, then 𝒰+ is associative. In this paper we describe all simple nodal non-commutative Jordan algebras of characteristic not 2.

The relationship between compact and countably compact topological spaces has been studied by many topologists. In particular an important question is: “What conditions will make a countably compact space compact?” Conditions which are “covering axioms” have been extensively studied. The best results of this type appear in [19]. We wish to examine countably compact spaces which are separable or perfectly normal. Recall that a space is perfect if and only if every closed subset is a Gδ, and that a space is perfectly normal if and only if it is both perfect and normal. We show that the following statement follows from MA +┐ CH and thus is consistent with the usual axioms of set theory: Every countably compact perfectly normal space is compact. This result is Theorem 3 and can be understood without reading much of what goes before.

Let $R$ be a countable, principal ideal domain which is not a field and $A$ be a countable $R$-algebra which is free as an $R$-module. Then we will construct an ${{\aleph }_{1}}$-free $R$-module $G$ of rank ${{\aleph }_{1}}$ with endomorphism algebra $\text{En}{{\text{d}}_{R}}\,G=A$. Clearly the result does not hold for fields. Recall that an $R$-module is ${{\aleph }_{1}}$-free if all its countable submodules are free, a condition closely related to Pontryagin’s theorem. This result has many consequences, depending on the algebra $A$ in use. For instance, if we choose $A\,=\,R$, then clearly $G$ is an indecomposable ‘almost free’module. The existence of such modules was unknown for rings with only finitelymany primes like $R={{\mathbb{Z}}_{\left( p \right)}}$, the integers localized at some prime $p$. The result complements a classical realization theorem of Corner’s showing that any such algebra is an endomorphism algebra of some torsionfree, reduced $R$-module $G$ of countable rank. Its proof is based on new combinatorialalgebraic techniques related with what we call rigid tree-elements coming from a module generated over a forest of trees.

In this paper, we establish a universal volume comparison theorem for Finsler manifolds and give the Berger–Kazdan inequality and Santalá's formula in Finsler geometry. Based on these, we derive a Berger–Kazdan type comparison theorem and a Croke type isoperimetric inequality for Finsler manifolds.

A partially ordered set P has the fixed point property if every order-preserving map f : P → P has a fixed point, i.e. there exists x ∊ P such that f(x) = x. A. Tarski's classical result (see [4]), that every complete lattice has the fixed point property, is based on the following two properties of a complete lattice P:

• (A) For every order-preserving map f : P → P there exists x ∊ P such that x ≦ f(x).

• (B) Suprema of subsets of P exist; in particular, the supremum of the set {x|x ≦ f(x)} ⊂ P exists.

In a recent paper (7) Yood developed the beginnings of a theory of modular annihilator algebras. In this paper we extend his work on these algebras.

The definition of modular annihilator algebra is algebraic in nature (see §4) ; in fact the algebra need not be assumed even topological. However, a significant number of important normed algebras are modular annihilator algebras. A list of examples is given in §8.

The theory of modular annihilator algebras is related to the theory of certain important topological algebras. In §5 we consider the relationships between dual and annihilator algebras and modular annihilator algebras, and in §7, the relationship between completely continuous normed algebras and modular annihilator algebras.

A birth and death process is a stationary Markov process whose states are the nonnegative integers and the transition probabilities

(1.1)

satisfy

(1.2)

as t → 0. Here we assume βn > 0, δn + 1 > 0, n = 0, 1, …, but δ0 ≦ 0. Karlin and McGregor [10], [11], [12], showed that each birth and death process gives rise to two sets of orthogonal polynomials. The first is the set of birth and death process polynomials {Qn(x)} generated by

We wish to consider the class of (0,1)-matrices with prescribed row and column sums. Let R = (r1, r2, …, rm) and S= (s1, s2, …, sn) be vectors with nonnegative integral entries and r1 + r2 + … + rm = s1 + s2 + … + sn. We define the class to be the set of m × n (0, 1)-matrices with ith row sum ri and jth column sum sj for 1 ≦ i ≦ m and 1 ≦ j ≦ n.

Gale and Ryser independently found simple necessary and sufficient conditions for to be nonempty [9 , 14]. From R, we form an m × n (0, 1)-matrix Ā as follows. The ith row sum of Ā is ri and the 1‘s are as far to the left as possible. Let be the jth column sum of A. We define the sequence to be the conjugate of the sequence (ri).

We study the relationships between Gateaux, Fréchet and weak Hadamard differentiability of convex functions and of equivalent norms. As a consequence we provide related characterizations of infinite dimensional Banach spaces and of Banach spaces containing ł1. Explicit examples are given. Some renormings of WCG Asplund spaces are made in this vein.

We characterize diametrically maximal and constant width sets in $C(K)$, where $K$ is any compact Hausdorff space. These results are applied to prove that the sum of two diametrically maximal sets needs not be diametrically maximal, thus solving a question raised in a paper by Groemer. A characterization of diametrically maximal sets in $\ell _{1}^{3}$ is also given, providing a negative answer to Groemer's problem in finite dimensional spaces. We characterize constant width sets in ${{c}_{0}}\left( I \right)$, for every $I$, and then we establish the connections between the Jung constant of a Banach space and the existence of constant width sets with empty interior. Porosity properties of families of sets of constant width and rotundity properties of diametrically maximal sets are also investigated. Finally, we present some results concerning non-reflexive and Hilbert spaces.

We study the asymptotic behaviour of the Bloch–Kato–Shafarevich–Tate group of a modular form $f$ over the cyclotomic ${{\mathbb{Z}}_{p}}$-extension of $\mathbb{Q}$ under the assumption that $f$ is non-ordinary at $p$. In particular, we give upper bounds of these groups in terms of Iwasawa invariants of Selmer groups defined using $p$-adic Hodge Theory. These bounds have the same form as the formulae of Kobayashi, Kurihara, and Sprung for supersingular elliptic curves.

The purpose of this note is to provide a simple proof of the sharp polynomial upper bound for the resonance counting function of a Schrödinger operator in odd dimensions. At the same time we generalize the result to the class of superexponentially decreasing potentials.

Among the classical field theories, general relativity theory occupies a somewhat peculiar place. Unlike those of most other field theories, the field equations in relativity theory are non-linear. This implies that many facts, well known in linear theories, have no analogues in general relativity theory, and conversely. The equations of motion of the sources of the gravitational field are contained in the field equations, a fact which does not apply for the motion of an electron in the electromagnetic field. Conversely, it is difficult to define the notion of a wave (familiar in electrodynamics) in relativity theory; for, the linear principle of superposition is crucial for the existence of waves (at least in the sense that the notion of a wave is normally used).

A contractive $n$-tuple $A\,=\,({{A}_{1}},...,{{A}_{n}})$ has a minimal joint isometric dilation $S\,=\,({{S}_{1}},...,{{S}_{n}})$ where the ${{S}_{i}}$’s are isometries with pairwise orthogonal ranges. This determines a representation of the Cuntz-Toeplitz algebra. When $A$ acts on a finite dimensional space, the wot-closed nonself-adjoint algebra $\mathfrak{S}$ generated by $S$ is completely described in terms of the properties of $A$. This provides complete unitary invariants for the corresponding representations. In addition, we show that the algebra $\mathfrak{S}$ is always hyper-reflexive. In the last section, we describe similarity invariants. In particular, an $n$-tuple $B$ of $d\,\times \,d$ matrices is similar to an irreducible $n$-tuple $A$ if and only if a certain finite set of polynomials vanish on $B$.

In what follows, a theorem on blocking sets is generalized to higher dimensions. The result is then used to study maximal partial spreads of odd-dimensional projective spaces.

Notation. The number of elements in a set X is denoted by |X|. Those elements in a set A which are not in the set Bare denoted by A — B. In a projective space Σ = PG(n, q) of dimension n over the field GF(q) of order q, ┌d(Ωd, Λd, etc.) will mean a subspace of dimension d. A hyperplane of Σ is a subspace of dimension n — 1, that is, of co-dimension one.

A blocking set in a projective plane π is a subset S of the points of π such that each line of π contains at least one point in S and at least one point not in S. The following result is shown in [1], [2].

We adopt the following definition of a completely regular semigroup S: for every element a of S, there exists a unique element a-1 of S such that

Soit A un anneau intègre de corps des fractions K, on s'interesse ici aux polynômes de K[X] qui prennent sur A leurs valeurs dans A. Ils forment bien évidemment un sous anneau de K[X] qui contient A[X], nous le notons As.

Cet article reprend, dans un langage moderne, les idées de deux publications assez anciennes, dues à Polya [3] et Ostrowsky [4], en 1919. Les auteurs montraient que si A est principal, As est un A-module libre qui admet sur A une base formée de polynômes de degré croissant ; si A est plus généralement l'anneau des entiers d'un corps de nombres ils mettaient en évidence une suite d'idéaux fractionnaires de A. En exposant le cas de l'anneau A principal, pour un anneau de valisation discrète dans le premier paragraphe, puis en généralisant aux anneaux de Dedekind dans notre second paragraphe, nous pouvons énoncer de façon plus explicite, que As est un A-module projectif, que l'on peut décomposer canoniquement en somme d'idéaux fractionnaires de A.

In this work type II Hermite-Padé approximants for a vector of Cauchy transforms of smooth Jacobi-type densities are considered. It is assumed that densities are supported on mutually disjoint intervals (an Angelesco system with complex weights). The formulae of strong asymptotics are derived for any ray sequence of multi-indices.

1. Introduction. It is the object of this paper to describe in more detail than has hitherto been done the general methods by which a square may be dissected into smaller unequal non-overlapping squares. Examples of such dissections are given

A semigroup (R, ⋅) is said to be a unique addition ring (UA-ring) if there exists a unique binary operation + making (R, ⋅, + ) into a ring. All our results can be presented in this semigroup theoretic setting. However, we prefer the following equivalent ring theoretic formulation: a ring R is a UA-ring if and only if any semigroup isomorphism α: (R, ⋅) ≅ (S, ⋅) with another ring S is always a ring isomorphism.

UA-rings have been studied in (8; 4) and are also touched on in (1; 2; 6; 7). In this note we generalize Rickart's methods to much wider classes of rings. In particular, we show that, for a ring R with a 1 and n ≧ 2, the (n × n) matrix ring over R and its subring of lower triangular matrices are UA-rings.