Given a free ideal J of subsets of a set X, we consider games where player ONE plays an increasing sequence of elements of the σ-completion of J, and player TWO tries to cover the union of this sequence by playing one set at a time from J. We describe various conditions under which player TWO has a winning strategy that uses only information about the most recent k moves of ONE, and apply some of these results to the Banach-Mazurgame.

Given a finite positive measure on the Borel subsets of the complex plane with compact support containing infinitely many points, we deduce some formulas for the sequence of monic orthogonal polynomials associated to a rational modification of the measure. These expressions depend on so called functions of the second kind. Some examples for particular Jordan curves are given

The purpose of this paper is to discuss the existence, structure, and properties of certain projective and injective Hopf algebras in the category of Hopf algebras that support the structure one expects on the homology of an infinite loop space. As an auxiliary project, we show that these projective and injective Hopf algebras can be realized as the homology of infinite loop spaces associated to spectra obtained from Brown-Gitler spectra by Spanier-Whitehead duality and Brown-Comenetz duality, respectively. We concentrate mainly on indecomposable projectives and injectives, and we work only at the prime 2.

A fraction-dense (semi-prime) commutative ring A with 1 is one for which the classical quotient ring is rigid in its maximal quotient ring. The fractiondense ƒ- rings are characterized as those for which the space of minimal prime ideals is compact and extremally disconnected. For archimedean lattice-ordered groups with this property it is shown that the Dedekind and order completion coincide. Fractiondense spaces are defined as those for which C(X) is fraction-dense. If X is compact, then this notion is equivalent to the coincidence of the absolute of X and its quasi-F cover. R-embeddings of Tychonoff spaces are re-introduced and examined in the context of fraction-density.

In the present paper we completely describe the isomorphisms between projective elementary groups PSLn and PSLm (n ≥ 2, m ≥ 2) over division rings. It was found that such groups can be isomorphic only if n = m; the division rings are isomorphic or anti-isomorphic, except for the following groups:

PSL(2,F7) and PSL(3,F2); PSL(2, F4) and PSL(2,F5).

The idea is based on a deepening of the classical Hua's approach. This problem has been solved independently by H. Ren, Z. Wan and X. Wu using a different way

A strongly maximal triangular AF algebra which is defined by a realvalued cocycle is said to be analytic. Formulas for generic cocycles are given separately for both the integer-valued case and the real-valued coboundary case, and also for certain nest algebras. In the case of an integer-valued cocycle, there is an associated partial homeomorphismof the maximal ideal space of the diagonal. If the partial homeomorphism extends to a homeomorphism, then the algebra embeds in a crossed product. This occurs for a large class of subalgebras of UHF algebras, but an example shows that this does not always occur. An example is given of a triangular AF algebra which is analytic via a coboundary but is not a nest algebra; also, it is shown that a nest algebra need not be analytic

Continuous-trace C*-algebras A with spectrum T can be characterized as those algebras which are locally Monta equivalent to C0(T). The Dixmier-Douady class δ(A) is an element of the Čech cohomology group Ȟ3(T, ℤ) and is the obstruction to building a global equivalence from the local equivalences. Here we shall be concerned with systems (A, G, α) which are locally Monta equivalent to their spectral system (C0(T),G, τ), in which G acts on the spectrum T of A via the action induced by α. Such systems include locally unitary actions as well as N-principal systems. Our new Dixmier-Douady class δ (A, G, α) will be the obstruction to piecing the local equivalences together to form a Monta equivalence of (A, G, α) with its spectral system. Our first main theorem is that two systems (A, G, α) and (B, G, β) are Monta equivalent if and only if δ (A, G, α) = δ (B, G, β). In our second main theorem, we give a detailed formula for δ (A ⋊α G) when (A, G, α) is N-principal.

We display a number with a surprising continued fraction expansion and show that we may explain that expansion as a specialisation of the continued fraction expansion of a formal series: A series ΣchX-h has a continued fraction expansion with partial quotients polynomials in X of positive degree (other, perhaps than the 0-th partial quotient). Simple arguments, let alone examples, demonstrate that it is noteworthy if those partial quotients happen to have rational integer coefficients only. In that special case one may replace the variable X by an integer ≥ 2; that is: one may 'specialise' and thereby proceed to obtain the regular continued fraction expansion of values of the series. And that is significant because, generally, it is difficult to obtain the explicit continued fraction expansion of a number presented in different shape. Our example leads to a series with a specialisable continued fraction expansion and, a little surprisingly, our arguments suggest that the phenomenon of specialisability for series of the kind appearing here may be reserved to just the special subclass of series we happen to have stumbled upon.

In this paper we characterize the bounded harmonic functions ƒ and g on the unit disk for which the Toeplitz operators Tƒ and Tg defined on the Bergman space of the unit disk are essentially commuting.

Consider the second order nonlinear differential equation

y" + a(t)f(y) = 0

where a(t) ∈ C[0,∞),f(y) GC1 (-∞, ∞),ƒ'(y) ≥ 0 and yf(y) > 0 for y ≠ 0. Furthermore, f(y) also satisfies either a superlinear or a sublinear condition, which covers the prototype nonlinear function f(y) = |γ|γ sgny with γ > 1 and 0 < γ < 1 known as the Emden-Fowler case. The coefficient a(t) is allowed to be negative for arbitrarily large values of t. Oscillation criteria involving integral averages of a(t) due to Wintner, Hartman, and recently Butler, Erbe and Mingarelli for the linear equation are shown to remain valid for the general equation, subject to certain nonlinear conditions on f(y). In particular, these results are therefore valid for the Emden-Fowler equation.

In this paper, we are concerned with the basic problem defined in [9]. Formulas for δV(0)and δ∞V(0),respectively the generalized and asymptotic gradient of the value function at zero, corresponding to an L2 -additive perturbation of dynamics are given. Under the normality condition, δV(0)turns out to be a compact subset of L2, formed entirely of arcs, and V is locally finite and Lipschitz at 0. Moreover, estimations of the generalized directional derivative and Dini's derivative of V at 0 are derived. Supplementary conditions imply that Dini's derivative of V at 0 exists, and V is actually strictly differentiate at this point.