Let X be a separable and metrizable space containing uncountably many pairwise disjoint copies of the compactum K. We discuss the question whether X must contain K × 2ω.

It is proved that C(K) has no equivalent uniformly Gâteaux differentiable norm (UGD) when K-is an uncountable separable scattered com¬pact space. This result is applied to obtain an example of scattered compact K such that K′″ = ∅ and C(K) has no UGD renorming.

We shall study the properties of typical n-dimensional subspaces of , or equivalently, -typical n-dimensional quotients of , where the meaning of what is typical and what is not is defined in terms of the Haar measure μn,N on the Grassmann manifold Gn,N of all n-dimensional subspaces of .

We prove a characterization of functions in B1/4(K)\C(K), where K is a compact metric space in terms of c0-spreading models, answering a Problem of R. Haydon, E. Odell and H. Rosenthal. Beginning with B1/4(K) we define a decreasing family (Vξ(K),║ · ║ξ)1≤ξ<ω1 of Banach spaces whose intersection is DBSC(K) and we prove an analogous stronger property for the functions in Vξ(K)\C(K). Defining the s-spreading model-index, we classify B1/4;(K) and we prove that s-SM[F]>ξ for every F∈ Vξ(K). Also we classify the separable Banach spaces by defining the c0-SM-index which measures the degree to which they have sequences with extending spreading models equivalent to the usual basis of c0. We give examples of Baire-1 functions and reflexive spaces with arbitrary large indices.

Let Ω be a regular domain in the extended complex plane, i.e., it is a bounded domain and its boundary consists of a finite number of disjoint analytic simple closed curves. Let dm(z) be the Lebesgue area measure on Ω and let ds = dm(z)/ω(z) be the Poincare metric on Ω, a Riemannian metric of negative constant curvature. It may be proved that Ω(z) ≈ Euclidean distance from z to the boundary of Ω (see [8]).

The impaction on symmetrical obstacles placed in uniform streams of aerosols is investigated. The governing equations of motion are nonlinear differential equations involving a parameter called the Stokes number. The study differentiates between the critical value of the Stokes number on the centre-line, kcr, below which no particles reach the stagnation point in finite time, and the critical value of the Stokes number on the obstacle, Kcr, below which no particles may be deposited on the obstacle in finite time. Based on the properties of the centre-line fluid velocity of the potential and viscous flows past a variety of symmetrically shaped obstacles, upper and lower bounds of Kcr and Kcr are established. Furthermore, using a numerical procedure for solving nonlinear differential equations with unknown parameters the critical values of Kcr and Kcr are obtained.

In this paper, we consider an extended model of a coupled nonlinear reaction-diffusion equations with Neumann-Neumann boundary conditions. We obtain upper linear growth bound for one of the components. We also find the corresponding bound for the case of Dirichlet-Dirichlet boundary conditions.

We establish a generalization of the Cesari-Kannan existence result for problems of the type Lx = N(x), x∈X where X is a separable Hilbert functional space, L is a selfadjoint linear differential operator with nontrivial finite dimensional kernel and N:X→X is a bounded continuous nonlinear operator. This generalization leads to new results when the dimension of the kernel of L is greater than one. Applications to systems of second order ordinary differential equations are given.

Our aim in this paper is to establish new comparison principles. The equation

is compared with the equation

The equations considered can be in canonical or noncanonical forms.

We shall give an explicit form of the Artin-Tschebotareff density theorem in function fields with several variable over finite fields. It may be an analogous prime number theorem in the higher dimensional case.

Let s, k and n be positive integers and define rs,k(n) to be the number of solutions of the diophantine equation

in positive integers xi. In 1922, using their circle method, Hardy and Littlewood [2] established the asymptotic formula

whenever s≥(k−2)2k−1 + 5. Here , the singular series, relates the local solubility of (1.1). For each k we define to be the smallest value of s0 such that for all s ≥ s0 we have (1.2), the asymptotic formula in Waring's problem. The main result of this memoir is the following theorem which improves upon bounds of previous authors when k≤9.

The construction of the fiber polytope ∑(P, Q) of a projection π:P→Q of polytopes is extended to flags of projections. While the faces of the fiber polytope are related to subdivisions of Q induced by the faces of P, those of an iterated fiber polytope are related to discrete homotopies between polyhedral subdivisions. In particular, in the case of projections

starting with an (n + 1)-simplex, vertices of the successive iterates correspond to, respectively, subsets, permutations and sequences of permutations of an n-set. The first iterate will always be combinatorially an n-cube, and, under certain conditions, the second will have the structure of the (n−1)-dimensional permutohedron.

Recently M. M. Kapranov [Kap] defined a poset KPAn−1, called the permuto-associahedron, which is a hybrid between the face poset of the permutohedron and the associahedron. Its faces are the partially parenthesized, ordered, partitions of the set {1, 2, …, n}, with a natural partial order.

Kapranov showed that KPAn−1, is the face poset of a regular CW-ball, and explored its connection with a category-theoretic result of MacLane, Drinfeld's work on the Knizhnik-Zamolodchikov equations, and a certain moduli space of curves. He also asked the question of whether this CW-ball can be realized as a convex polytope.

We show that indeed, the permuto-associahedron corresponds to the type An−1, in a family of convex polytopes KPW associated to the classical Coxeter groups, W = An−1, Bn, Dn. The embedding of these polytopes relies on the secondary polytope construction of the associahedron due to Gel'fand, Kapranov, and Zelevinsky. Our proofs yield integral coordinates, with all vertices on a sphere, and include a complete description of the facet-defining inequalities.

Also we show that for each W, the dual polytope KPW* is a refinement (as a CW-complex) of the Coxeter complex associated to W, and a coarsening of the barycentric subdivision of the Coxeter complex. In the case W = An−1, this gives a combinatorial proof of Kapranov's original sphericity result.

The main purposes of this paper are to investigate ℤ-injective rings with the representation extension property and its dual, to give a necessary and sufficient condition for a ℤ-injective ring to be an amalgamation base in the class of all rings and to determine structure of ℤ-injective Noetherian rings which are amalgamation bases. Further, in the class of all commutative rings, it is shown that a commutative ring has the representation extension property, if, and only if, it is an amalgamation base.

The last formula on page 65 of our paper [1] is incorrect; as it stands, the left hand side is bounded whereas the right hand side is not. The formula should read