We study estimates of the type

1

$${{\left\| \phi (D)\,-\,\phi ({{D}_{0}}) \right\|}_{E(\mathcal{M},\,\tau )}}\,\le \,C\,\cdot \,{{\left\| D\,-\,{{D}_{0}} \right\|}^{\alpha }},\,\,\,\,\,\,\,\alpha \,=\,\frac{1}{2},\,1$$

where $\phi (t)\,=\,t{{(1\,+\,{{t}^{2}})}^{-1/2}},\,{{D}_{0}}\,=\,D_{0}^{*}$ is an unbounded linear operator affiliated with a semifinite von Neumann algebra $M,D-{{D}_{0}}$ is a bounded self-adjoint linear operator from $\mathcal{M}$ and ${{(1+D_{0}^{2})}^{-1/2}}\in E(M,\tau )$, where $E(\mathcal{M},\tau )$ is a symmetric operator space associated with $\mathcal{M}$. In particular, we prove that $\phi \left( D \right)-\phi \left( {{D}_{0}} \right)$ belongs to the non-commutative ${{L}_{p}}$-space for some $p\in (1,\infty )$, provided ${{(1+D_{0}^{2})}^{-1/2}}$ belongs to the noncommutative weak ${{L}_{r}}$-space for some $r\in [1,p)$. In the case $\mathcal{M}\,=\,\mathcal{B}\left( \mathcal{H} \right)$ and $1\,\le \,p\,\le \,2$, we show that this result continues to hold under the weaker assumption ${{(1+D_{0}^{2})}^{-1/2}}\in {{C}_{p}}$. This may be regarded as an odd counterpart of A. Connes’ result for the case of even Fredholm modules.

In this paper the theory of Browder [2] and of Lions [3] on periodic solutions of non-linear evolution equations in Banach spaces is put in a more general framework so as to include the Navier-Stokes equations and their variants.

An abstract existence theorem is proved in § 1. Applications are given in § 2. The existence of periodic solutions of the Navier-Stokes equations without any restriction on the dimension of the space domain is established. Application of the abstract theorem to the following problem is given:

1. Let H be a Hilbert space and (., .)H the inner product in H. Let V and W be two reflexive separable Banach spaces with W ⊂ V ⊂ H. W is dense in V and V is dense in H.

We define below a notion for modules which is dual to that of faithful, and a notion of “fully divisible” which generalizes that of injectivity. We show that the bicommutator of a cofaithful, fully divisible left R-module is isomorphic to a subring of Qmax(R), the complete ring of left quotients of R.

In recent papers, Goldman [2] and Lambek [3] investigated rings of left quotients of a ring R constructed with respect to torsion radicals. It is known that every ring of left quotients of R is isomorphic to the bicommutator of an appropriate injective left R-module. We investigate below subrings of rings of quotients which are determined by radicals rather than torsion radicals, and show that any such ring can be constructed as the bicommutator of a fully divisible left R-module.

We consider a unitary representation of a discrete countable abelian group on a separable Hilbert space which is associated to a cyclic generalized frame multiresolution analysis. We extend Robertson’s theorem to apply to frames generated by the action of the group. Within this setup we use Stone’s theorem and the theory of projection valued measures to analyze wandering frame collections. This yields a functional analytic method of constructing a wavelet from a generalized frame multiresolution analysis in terms of the frame scaling vectors. We then explicitly apply our results to the action of the integers given by translations on ${{L}^{2}}(\mathbb{R})$.

We will say that a space X has property (*) if and only if each discrete subset of X is realcompact; i.e., the cardinality of each discrete subset of X is nonmeasurable. In [8], Shirota shows that a completely regular T1-space X is realcompact if and only if X has property (*) and X is complete with respect to some uniformity. In [7], Moran, using measure theoretic techniques, shows that any normal metacompact T1-space with property (*) is realcompact.

Let F be a skew field with a valuation (also called total) subring B, i.e. x in F\ B implies x-1 in B. Such rings are useful not only in the investigation and construction of division algebras (see for example [5],[6],[12]) but also in geometry ([15]).

Associated with B is an invariant subring R of F and a value group G. We investigate the relationship between properties like the distributivity of R and properties like being lattice ordered of G.

In solving the word problem for free lattices, Whitman [4] showed that free lattices admit canonical representations, that is, of all polynomials over the generating set representing an element of the lattice, the polynomial of shortest length is unique up to commutativity and associativity. These well-defined shortest polynomials have proved very important in analyzing the internal structure of free lattices in detail; see, e.g., [5].

Sorkin [3] proved that the free product of chains also admits canonical representations; these were exploited by Rolf [2]. In the above-mentioned paper, Sorkin also suggested that the free product of two copies of 22 does not admit canonical representations.

In this note, I shall establish necessary and sufficient conditions for the existence of critical lattices of an arbitrary point set, and I shall construct a non-trivial example of a point set without any critical lattice. In a previous paper, I proved that every star body of the finite type possesses at least one critical lattice.

Positive solutions are obtained for the boundary value problem

$$\left\{ _{u\left( 0 \right)\,=\,u\left( 1 \right)\,=\,0}^{-{{\left( {{\left| {{u}'} \right|}^{p-2}}{u}' \right)}^{\prime }}\,=\,\lambda f\left( t,\,u \right),\,t\,\in \,\left( 0,\,1 \right),\,p\,>\,1} \right.$$

Here $f(t,u)\ge -M$, ($M$ is a positive constant) for $(t,u)\in [0,1]\times (0,\infty )$. We will show the existence of two positive solutions by using degree theory together with the upper–lower solution method.

We give an explicit construction of polynomial (of arbitrary degree) $(\alpha ,\,\beta )$-metrics with scalar flag curvature and determine their scalar flag curvature. These Finsler metrics contain all nontrivial projectively flat $(\alpha ,\,\beta )$-metrics of constant flag curvature.

We prove the holomorphic continuation of certain multi-variable multiple zeta-functions whose coefficients satisfy a suitable recurrence condition. In fact, we introduce more general vectorial zeta-functions and prove their holomorphic continuation. Moreover, we show a vectorial sum formula among those vectorial zeta-functions from which some generalizations of the classical sum formula can be deduced.

The Picard group Γ = PSL2 (Z(i)) is the group of linear transformations

with a, b, c, d Gaussian integers.

Γ is of interest both as an abstract group and in automorphic function theory [10]. In [10] Waldinger constructed a subgroup H of finite index which is a generalized free product, while in [1] Fine showed that T is a semidirect product with the subgroup H, contained as a subgroup of finite index in the normal factor.

The eigenfunctions of a boundary value problem are characterized by two quite distinct properties. They are solutions of ordinary differential equations, and they satisfy prescribed boundary conditions. It is a definite advantage to combine these two requirements into a single problem expressed by a unified formula. The use of integral equations is an example in point. The subject of this paper, namely the Schrödinger-Infeld Factorization Method, which is applicable to certain restricted. Sturm-Liouville problems, is based upon another combination of the two properties. The Factorization Method prescribes a manufacturing process.

Anderson, Clunie and Pommerenke defined and studied the family of Bloch functions on the unit disc (see [1]). This family strictly contains the space of bounded analytic functions. However, all Bloch functions are normal and therefore enjoy the “nice” properties of normal functions. The importance of the Bloch function concept is the combination of their richness as a family and their “nice” behavior.

Let A = [aij] be a matrix of 0's and 1's or a (0, 1)-matrix of size m by m′. The term rank of A is denned as the maximal number of 1's of A with no two of the 1's on the same row or colunn. A theorem due to D. König (3, Theorem 5.1, p. 55) asserts that the term rank of A is also equal to the minimal number of rows and columns of A that collectively contain all the 1's. The term rank of A will be denoted by ρ(A). Obviously it is invariant under arbitrary permutations of the rows and columns of A. We assume without loss of generality that all matrices considered have no rows or columns consisting entirely of 0's.

The similar sublattices of a planar lattice can be classified via its multiplier ring. The latter is the ring of rational integers in the generic case, and an order in an imaginary quadratic field otherwise. Several classes of examples are discussed, with special emphasis on concrete results. In particular, we derive Dirichlet series generating functions for the number of distinct similar sublattices of a given index, and relate them to zeta functions of orders in imaginary quadratic fields.

In this article we study exponential trichotomy for infinite dimensional discrete time dynamical systems. The goal of this article is to prove that finite time exponential trichotomy conditions allow us to derive exponential trichotomy for arbitrary times. We present an application to the case of pseudo orbits in some neighborhood of a normally hyperbolic set.

1. The theorem of Landau in question may be stated in the form that if the function F(Z) is regular for |Z| < 1 and does not take the values 0 and 1, while

F(Z) = a0 + a1Z + …

is its Taylor expansion about Z = 0, then |a1| has a bound depending only on a0. In fact |a1| has a bound depending only on |a0| and Hayman (1) gave the explicit bound

We consider the set (of homotopy classes) of co-H-structures on a Moore space M(G,n), where G is an abelian group and n is an integer ≥ 2. It is shown that for n > 2 the set has one element and for n = 2 the set is in one-one correspondence with Ext(G, G ⊗ G). We make a detailed investigation of the co-H-structures on M(G, 2) in the case G = ℤm, the integers mod m. We give a specific indexing of the co-H-structures on M(ℤm, 2) and of the homotopy classes of maps from M(ℤm, 2) to M(ℤn, 2) by means of certain standard homotopy elements. In terms of this indexing we completely determine the co-H-maps from M{ℤm, 2) to M(ℤn, 2) for each co-H-structure on M(ℤm, 2) and on M(ℤn, 2). This enables us to describe the action of the group of homotopy equivalences of M(ℤn, 2) on the set of co-H-structures of M(ℤm, 2). We prove that the action is transitive. From this it follows that if m is odd, all co-H-structures on M(ℤm, 2) are associative and commutative, and if m is even, all co-H-structures on M(ℤm, 2) are associative and non-commutative.

Letting $p$ vary over all primes and $E$ vary over all elliptic curves over the finite field ${{\mathbb{F}}_{p}}$, we study the frequency to which a given group $G$ arises as a group of points $E\left( {{\mathbb{F}}_{p}} \right)$. It is well known that the only permissible groups are of the form ${{G}_{m,\,k}}\,:=\,\mathbb{Z}\,/m\mathbb{Z}\,\times \,\mathbb{Z}/mk\mathbb{Z}$. Given such a candidate group, we let $M\left( {{G}_{m,\,k}} \right)$ be the frequency to which the group ${{G}_{m,\,k}}$ arises in this way. Previously, C.David and E. Smith determined an asymptotic formula for $M\left( {{G}_{m,\,k}} \right)$ assuming a conjecture about primes in short arithmetic progressions. In this paper, we prove several unconditional bounds for $M\left( {{G}_{m,\,k}} \right)$, pointwise and on average. In particular, we show that $M\left( {{G}_{m,\,k}} \right)$ is bounded above by a constant multiple of the expected quantity when $m\,\le \,{{k}^{A}}$ and that the conjectured asymptotic for $M\left( {{G}_{m,\,k}} \right)$ holds for almost all groups ${{G}_{m,\,k}}$ when $m\,\le \,{{k}^{1/4-\in }}$. We also apply our methods to study the frequency to which a given integer $N$ arises as a group order $\#E\left( {{\mathbb{F}}_{p}} \right)$.