In this paper we study the approximation of vector valued continuous functions defined on a topological space and we apply this study to different problems. Thus we give a new proof of Machado's Theorem. Also we get a short proof of a Theorem of Katětov and we prove a generalization of Tietze's Extension Theorem for vector-valued continuous functions, thereby solving a question left open by Blair.

The purpose of this note is to prove the following. Suppose $\mathfrak{R}$ is a semiprime unity ring having an idempotent element $e$ ($e\neq 0,~e\neq 1$) which satisfies mild conditions. It is shown that every additive generalized Jordan derivation on $\mathfrak{R}$ is a generalized derivation.

We study cosine and sine Fourier transforms defined by F(t):= (2/π) and (t):= (2/π), where f is L1-integrable over[0, ∞]. We also assume than F are locally absolutely continuous over [0, ∞). In particular, this is the case if both f(x) and xf(x) are (L1-integrable over [0, ∞). Motivated by the inversion formulas, we consider the partial integras Sν (f, x):= and ν(f, x):= , the modified partial integrals uν (f, x):= sν(f, x) - F(ν)(sin νx)/x and ũν(f, x):= ν(f, x) + (ν) (cos νx)/x, where ν > 0. We give necessary and sufficient conditions for(L1 [0, ∞)-convergence of uν (f) and ũν (f) as well as for the L1 [0, X]-convergence of sν (f) and ν(f) to f as ν← ∞, where 0 < X < ∞ is fixed. On the other hand, in certain cases we conclude that sν(f) and ν(f) cannot belong to (L1 [0,∞). Conequently, it makes no sense to speak of their (L1 [0, ∞)-convergence as ν ← ∞.

As an intermediate tool, we use the Cesàro means of Fourier transforms. Then we prove Tauberian type results and apply Sidon type inequalities in order to obtain Tauberian conditions of Hardy-Karamata kind.

We extend these results to the complex Fourier transform defined by G(t):= , where g is L1- integrable over (−∞, ∞).

In this note, a one sided aspect of regular variation is considered, and some different results which can be obtained by bounding the ratio f(xλ)/f(x) above or below are given.

The purpose of this paper is to present a proof of the following theorem: Suppose Π is a set of dd primes, G is a finite Π-solvable group, and A is a nilpotent Π-subgroup of maximal order of G. Then G has a normal Π-complement, if and only if NG(ZJ(A)) has a normal Π-complement. (J(A) is the Thompson subgroup of A.)

In Russell (1973) a Riemann-type necessary and sufficient condition was given for the existence of (defined also in Russell (1975)) when f was bounded and g was k-convex in [a′, b′]. In this paper we present necessary and sufficient conditions for the existence of a particular Stieltjes-type integral without imposing a convexity condition upon g. These conditions are used to obtain an additivity result for the integral over adjoining intervals without any additional restrictions being imposed upon the functions involved.

A class (called c-radicals) is defined such that, given a c-radical P, there is in any class M′ a certain internal criterion that its upper radical UM′ = P, For P a non-c-radical (called a q-radical) there exists no smallest class M such that UM = P, and P is a q-radical if and only if for some M with P = UM there exists 0 ≠ R ∈ M such that when an image ¯ of R has a non-zero image in M there exists an infinite chain of epimorphisms ¯ → R1 → R2 → … with all R1 ∈ M and no Ri, the image of any Rj, with j > i. Several examples of such rings are constructed including a ring all of whose images are primitive. Thus all radicals contained in the Jacobson radical are q-radicals.

Let P be the probability distribution of a sample without replacement of size n from a finite population represented by the set N={1,2,…N}. For each r=0, 1, …, an approximation Pr is described such that the uniform norm ‖P − Pr‖ is of order (n2/N)r+1 if n2/N→0. The approximation Pr is a linear combination of uniform probability product-measures concentrated on certain subspaces of the sample space Nn.

The literature on subjective probabilities contains a number of functions that have been proposed as ‘scoring functions’. The principal requirement is that, with several events that may occur in the future and to which subjective ‘probabilities’ are assigned, the expected score given by these ‘probabilities’ will be extremised if the values assigned equal the ‘true probabilities’ of the various outcomes. This article discusses the question of what other scoring functions might be used (beyond those so far proposed).

For a normalized analytic function $f(z)=z+\sum _{n=2}^{\infty }a_{n}z^{n}$ in the unit disk $\mathbb{D}:=\{z\in \mathbb{C}:|z|<1\}$, the estimate of the integral means

$$\begin{eqnarray}L_{1}(r,f):=\frac{r^{2}}{2{\it\pi}}\int _{-{\it\pi}}^{{\it\pi}}\frac{d{\it\theta}}{|f(re^{i{\it\theta}})|^{2}}\end{eqnarray}$$

$f(z)$

$\mathbb{D}\setminus \{0\}$

${\rm\Delta}(r,f)$

$\mathbb{D}_{r}:=\{z\in \mathbb{C}:|z|<r\}$

$f$

$0<r\leq 1$

$L_{1}(r,f)$

${\rm\Delta}(r,z/f)$

$r$

$f$

$m$

We prove general Dwork-type congruences for constant terms attached to tuples of Laurent polynomials. We apply this result to establishing arithmetic and p-adic analytic properties of functions originating from polynomial solutions modulo $p^s$ of hypergeometric and Knizhnik–Zamolodchikov (KZ) equations, solutions which come as coefficients of master polynomials and whose coefficients are integers. As an application, we show that the simplest example of a p-adic KZ connection has an invariant line subbundle while its complex analog has no nontrivial subbundles due to the irreducibility of its monodromy representation.

A variety B is called solutionally complete if any system of algebraic equations over an algebra A in B has a solution in A provided it is solvable in B and has at most one solution in any extension of A in B. B is called solutionally compatible if every solvable system of equations over an algebra in B is also solvable over any extension of that algebra. It is shown that solutional compatibility is equivalent with the amalgamation property and that a weaker form of the strong amalgamation property is sufficient but not necessary for equational completeness.

An approximation method based on a certain Sonnenschein matrix is studied. Results are obtained for approximation in an interval and in the complex plane. A connection between convergence of the approximation process and regularity of the matrix is also discussed.

Let R be the incidence algebra of a finite partially ordered set T over a commutative noetherian ring A. Then the spectrum of R is homeomorphic to the product (Spec A) x T, where Spec A has the usual Zariski topology and T has the order topology. An explicit construction is given for the structure sheaf of R over its spectrum.

The author investigates M(n)= min¦Σηkk−1¦ where the minimum is over all sets of signs η1 = ±1 and shows M(n) < n½−(1−e)log2n.

Let K and L be lattices, and let ϕ be a homomorphism of K into L.Then ϕ induces a natural 0-preserving join-homomorphism of Con K into Con L.

Extending a result of Huhn, the authors proved that if D and E are finite distributive lattices and ψ is a 0-preserving join-homomorphism from D into E, then D and E can be represented as the congruence lattices of the finite lattices K and L, respectively, such that ψ is the natural 0-preserving join-homomorphism induced by a suitable homomorphism ϕ: K → L. Let m and n denote the number of join-irreducible elements of D and E, respectively, and let k = max (m, n). The lattice L constructed was of size O(22(n+m)) and of breadth n+m.

We prove that K and L can be constructed as ‘small’ lattices of size O(k5) and of breadth three.

A transcendence theorem is proved for functions satisfying functional equations of the shape P(z, f(z), f(zp)) = 0, where P is a polynomial and p ≥ 2 is an integer.

Let f be a transcendental entire function and denote the n-th iterate fn. For n ≥ 2, we give an explict estimate of the number of periodic points of f with period n, that is, fix-points of fn which are not fix-points of fk for 1 ≤ k <n.

A mathematical theory of the separating flow past a cascade of aerofoils is developed. The flow is assumed to be inviscid, incompressible and twodimensional. The wakes are represented by regions of stationary fluid, which could, in the general case, maintain a pressure gradient, although much of the theory is developed for the case of constant wake pressure.