The modulus of Rudin-Carleson extensions

We prove the followingTheorem. LetF be a closed subset of the unit circleT which has Lebesgue measure zero. Suppose thatp is a smooth positive function onT. GivenfC(F) which satisfies|f(s)|p(s) (sF) and a neighbourhoodU ofF there is an extension off in the disc algebra such that and .     

On some matrix analogs of the little Fermat theorem

Mathematical Notes - The rings over which every square matrix is representable as a sum of a nilpotent matrix and a q-potent matrix, where q is a positive integer power of a prime, are studied. As...     

Two applications of maschke's theorem

Let R ? G denote a crossed product of the finite group G over the ring R and let V be an R ? G-module. Maschke's theorem states that if 1/∣G∣ ε R and if V is completely reducible as an R-module, then V is also completely reducible as an R ? G -module. In this paper, we obtain two applications of this theorem, both under the assumption that R is semiprime with no ∣G∣ -torsion. The first concerns group actions and here we show that if G acts on R and if I is an essential right ideal of the fixed ring R^G , then IR is essential in Rs. This result, in turn, simplifies a number of proofs already in the literature. The second application here is a short proof of a theorem of Fisher and Montgomery which asserts that the crossed product R ? G is semiprime.     

On matrix analogs of Fermat’s little theorem

The theorem proved in this paper gives a congruence for the traces of powers of an algebraic integer for the case in which the exponent of the power is a prime power. The theorem implies a congruence in Gauss’ form for the traces of the sums of powers of algebraic integers, generalizing many familiar versions of Fermat’s little theorem. Applied to the traces of integer matrices, this gives a proof of Arnold’s conjecture about the congruence of the traces of powers of such matrices for the case in which the exponent of the power is a prime power.     

Two counterexamples related to the Kreiss matrix theorem

The modern version of the Kreiss matrix theorem states that A ⁿ esK (for alln0) ifA is ans ×s matrix satisfying a resolvent condition with respect to the unit disk with constantK. In this paper we show for any fixedK + 1 that the upper boundesK is sharp in the sense that a linear dependence on the dimensions is the best possible. An analogous result is obtained for the continuous version of the Kreiss matrix theorem, which states that exp(tA) esK (for allt0) ifA is ans ×s matrix satisfying a resolvent condition with respect to the left half plane with constantK.The research has been made possible by a fellowship of the Royal Netherlands Academy of Arts and Sciences (K.N.A.W.) and was carried out at the Department of Mathematics and Computer Science, University of Leiden, The Netherlands.     

Two variations of a theorem of Kronecker

We present two variations of Kronecker's classical result that every nonzero algebraic integer that lies with its conjugates in the closed unit disc is a root of unity. The first is an analogue for algebraic nonintegers, while the second is a several variable version of the result, valid over any field.     

Some analogs of the Carleson embedding theorem for certain Hilbetr spaces of analytic functions

The embedding theorems mentioned in the title are proved for Hilbert spaces of holomorphic and harmonic functions satisfying slow growth conditions near the boundary. Bibliography: 3 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 206, 1993. pp. 40–54. Translated by V. I. Vasyunin.     

Asymptotic analogs of the Floquet-Lyapunov theorem for some classes of periodic systems of ordinary differential equations

We prove asymptotic analogs of the Floquet-Lyapunov theorem and some reducibility theorems for various classes of linear and quasilinear systems of ordinary differential equations with periodic matrices with large and small amplitudes. We study such problems with the help of new versions of the splitting method in the theory of regular and singular perturbations, which complements the known results. We also adduce several examples.     

Two counterexamples to the spectral mapping theorem for semigroups of positive operators

LetX be anL ¹- orL ²-Banach lattice. Let (W(t)) _t0 be aC ₀-semigroup of positive operators onX with generator–T. Two examples ofC ₀-semigroups are presented for which the identity fails for almost allt>0.     

Two applications of a consistency theorem for systems of linear inequalities

A consistency criterion for systems of linear inequalities is applied to prove an existence theorem for non-negative matrices with restricted row sums, column sums and diagonal elements. As another application of the consistency criterion, a surprisingly short proof of a theorem of Hardy, Littlewood and Pólya is given.     

Derivation of the Cartwright-Levinson theorem from the Kolmogorov theorem

One says that an entire function f of finite exponential type belongs to the Cartwright class C, if Let N₊(r)(N_–(r)) denote the number of zeros of the function f in the disk ¦z¦R, such that Re z0 (Re z<0, respectively). We give a simple derivation of the folfollowing result of importance in the theory of entire functions, from a weak type Kolmogorov inequality.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 135, pp. 76–86, 1984.I am happy to note that this paper was written in connection with the 50th birthday of my friend V. P. Khavin.     

On the refinement of Cauchy's theorem and Pellet's theorem

Using the relationship of a polynomial and its associated polynomial, we derived a necessary and sufficient condition for determining all roots of a given polynomial on the circumference of a circle defined by its associated polynomial. By employing the technology of analytic inequality and the theory of distribution of zeros of meromorphic function, we refine two classical results of Cauchy and Pellet about bounds of modules of polynomial zeros. Sufficient conditions are obtained for the polynomial whose Cauchy's bound and Pellet's bounds are strict bounds. The characteristics is given for the polynomial whose Cauchy's bound or Pellet's bounds can be achieved by the modules of zeros of the polynomial.     

On Bayes' theorem and the inverse Bernoulli theorem

A recent assertion by S. M. Stigler that Thomas Bayes was perhaps anticipated in the discovery of the result that today bears his name is exposed to further scrutiny here. The distinction between Bayes' theorem and the inverse Bernoulli theorem is examined, and pertinent early writings on this matter are discussed. A careful examination of the difference between these two theorems leads to the conclusion that a result given by David Hartley in 1749 is more in line with the inverse Bernoulli theorem than with Bayes' result, and it is suggested that there is not sufficient evidence to remove Bayes from his place as originator of the method adopted.     

The Atiyah-Singer index theorem for families of Dirac operators: Two heat equation proofs

Summary The purpose of this paper is to give two heat equation proofs of the Index Theorem of Atiyah-Singer for a family of Dirac operators.     

Fermat's theorem as a corollary to the multinomial theorem

The multinomial theorem and a basic property of multinomial coefficients yield a way of counting the number of chains of coloured beads of a fixed length where each colour can be used repeatedly. This coupled with a simple counting argument forms a concise proof of Fermat's theorem.