On the ω-problem for some types of non-metrizable spaces

The ω-problem on a topological space X consists in finding out whether there exists a function whose oscillation is equal to a given upper semi-continuous (USC) function f:X→[0,∞] vanishing at isolated points of X. If such F exists, we call it an ω-primitive for f. Unlike the case of metrizable spaces, an ω-primitive need not exist if X is not metrizable. We study the ω-problem for f taking the value ∞ in the case of ordinal space, products of regular “constancy” spaces and the wedge sums of such spaces. Some open problems are formulated.     

The Reception of Frege''sBegriffsschrift

The topic of the paper is the public reception of Gottlob Frege's (1848–1925)Begriffsschriftright after its publication in 1879. According to a widespread conception, the reception of the book was “unfavorable” and even “tragic.” The aim of the paper is to correct this exaggerated and even false view. The arguments are based, above all, on the six journal reviews of Frege's book in 1879 and 1880, and on Leonhard Rabus's critical comment on it in his bookDie neuesten Bestrebungen auf dem Gebiete der Logik bei den Deutschen und die logische Frage(1880). The conclusion is that it is misleading to regard the reception of Frege's first work as unfair and hostile even though it is apparent thatsomereviewers of the book were rather poorly motivated to comment on theBegriffsschrift.Copyright 1998 Academic Press.Der Gegenstand dieses Beitrags ist die öffentliche Rezeption von Gottlob Freges (1848–1925)Begriffsschriftnach ihrer Publikation 1879. Nach einer weitverbreiteten Auffassung war die Rezeption “ungünstig,” sogar “tragisch.” Ziel des Beitrags ist es zu zeigen, dass solche Interpretationen überspannt, teilweise sogar falsch sind. Der Verfasser gründet seine Behauptung vor allem auf die Rezensionen, die in verschiedenen Zeitschriften in den Jahren 1879 und 1880 erschienen sind. Er benutzt auch den Kommentar von Leonhard RabusDie neuesten Bestrebungen auf dem Gebiete der Logik bei den Deutschen und Die logische Frage(1880). Die Untersuchung kommt zu dem Schluss, dass, obwohl offensichtlicheinigeRezensenten dem Buch eher ablehnend gegenüberstanden, es irreführend ist, von einer ungerechtfertigten oder abweisenden Rezeption der Fregeschen Begriffsschrift zu sprechenCopyright 1998 Academic Press.AMS subject classification: 00A30, 01A55, 03A05     

Univalent mappings associated with the Roper-Suffridge extension operator

The Roper-Suffridge extension operator provides a way of extending a (locally) univalent functionfεH(U) to a (locally) biholomorphic mappingF∈H(B_n). In this paper, we give a simplified proof of the Roper-Suffridge theorem: iff is convex, then so isF. We also show that iff∈S ^*, theF is starlike and that iff is a Bloch function inU, thenF is a Bloch mapping onB _n. Finally, we investigate some open problems. Partially supported by the Natural Sciences and Engineering Research Council of Canada under grant A9221.     

Newton's solution to the equiangular spiral problem and a new solution using only the equiangular property

This paper is a study of Proposition IX of Book I of Newton's Principia, the problem of determining the centripetal force for an equiangular spiral. In Newton's main proof of this proposition there is an error concerning his reason for the figure SPRQT being “given in kind,” and a very interesting technique of varying things in the neighborhood of a limit. This main proof utilized Newton's formula for the limit of SP²QT²/QR given in Corollary I to Proposition VI of the Principia. Newton also gave an alternate proof which utilized his formula for SY²PV given in Corollary III to Proposition VI. The “given” of Proposition IX was “a spiral PQS, cutting all the radii SP, SQ, &c., in a given angle.” Both the main proof and the alternate proof implicitly depend on the property of the equiangular spiral that the radius of curvature at any point is proportional to the pole distance SP. We here offer a new proof of Newton's proposition which does not depend on this implicit assumption.     

Continuous mappings on subspaces of products with the κ-box topology

Much of General Topology addresses this issue: Given a function fC(Y,Z) with YY^′ and ZZ^′, find , or at least , such that ; sometimes Z=Z^′ is demanded. In this spirit the authors prove several quite general theorems in the context Y^′=(X_I)_κ=∏_iIX_i in the κ-box topology (that is, with basic open sets of the form ∏_iIU_i with U_i open in X_i and with U_i≠X_i for <κ-many iI). A representative sample result, extending to the κ-box topology some results of Comfort and Negrepontis, of Noble and Ulmer, and of Hušek, is this. Theorem Let ωκα (that means: κ<α, and [β<α and λ<κ]β^λ<α) with α regular, be a set of non-empty spaces with each d(X_i)<α, π[Y]=X_J for each non-empty JI such that |J|<α, and the diagonal in Z be the intersection of <α-many regular-closed subsets of Z×Z. Then (a) Y is pseudo-(α,α)-compact, (b) for every fC(Y,Z) there is J[I]^<α such that f(x)=f(y) whenever x_J=y_J, and (c) every such f extends to .     

Inverse monoids and rational Schreier subsets of the free group

An inverse monoidM is an idempotent-pure image of the free inverse monoid on a setX if and only ifM has a presentation of the formM=Inv<X:e_o=f_i, i∈I>, wheree _i ,f _i are idempotents of the free inverse monoid: every inverse monoid is an idempotent-separating image of one of this type. IfR is anR-class of such an inverse monoid, thenR may be regarded as a Schreier subset of the free group onX. This paper is concerned with an examination of which Schreier subsets arise in this way. In particular, ifI is finite, thenR is a rational Schreier subset of the free group. Not every rational Schreier set arises in this way, but every positively labeled rational Schreier set does. Research supported by National Science Foundation grant #DMS8702019.     

ω* has (almost) no continuous images

We prove that the following statement follows from the Open Colouring Axiom (OCA): ifX is locally compactσ-compact but not compact and if its Čech-Stone remainderX* is a continuous image ofω*, thenX is the union ofω and a compact set. It follows that the remainders of familiar spaces like the real line or the sum of countably many Cantor sets need not be continuous images ofω*.     

Extremely weakly unconditionally convergent series

It is proved that if Σ _i=1 ^∞ X _i is a non-convergent series in a Banach spaceX such that Σ _i=1 ^∞ |f(X _i )|<∞ for all extreme pointsf of the unit ball ofX*, thenX contains a subspace isomorphic toc ₀, improving a result of Bessaga and Pelczynski. The proof uses Fonf’s result that Lindenstrauss-Phelps spaces contain isomorphs ofc ₀. Supported in part by NSF-MCS-8002393.     

On perfect images of ordinals

In the first section of this paper, the ordinal space [1, ω] is characterized. Then its quotients are completely described in easy terms. In the second section, the ideas of a well-ordered sum of an arbitrary collection of T₂spaces and a Chandelier space are introduced. The Chandelier spaces are obtained by taking compact ordinals and forming their interweaves and well ordered sums and one-point compactification of such spaces. Finally it is proved that the Hausdorff quotients of compact ordinals are precisely the Chandelier spaces. A problem of M. Hu ek on quotients of [1, ω] is thus solved.     

Hyperspaces of nowhere topologically complete spaces

It is proved that ifX is a connected locally continuumwise connected coanalytic nowhere topologically complete space, then the hyperspace 2 ^X of all nonempty compact subsets ofX is strongly universal in the class of all coanalytic spaces. Moreover, 2 ^X is homeomorphic to Π₂ ifX is a Baire space, and toQ∖Π₁ ifX contains a dense absoluteG _δ-setG ⊂X such that the intersectionG ∩U is connected for any open connectedU ⊂X. (Here Π₁, Π₁⊂X are the standard subsets of the Hilbert cubeQ absorbing for the classes of analytic and coanalytic spaces, respectively.) Similar results are obtained for higher projective classes. Translated fromMatematicheskie Zametki, Vol. 62, No. 1, pp. 35–51, July, 1997. Translated by O. V. Sipacheva     

On Fraïssé’s conjecture for linear orders of finite Hausdorff rank

We prove that the maximal order type of the wqo of linear orders of finite Hausdorff rank under embeddability is φ₂(0), the first fixed point of the ε-function. We then show that Fraïssé’s conjecture restricted to linear orders of finite Hausdorff rank is provable in  +“φ₂(0) is well-ordered” and, over , implies  +“φ₂(0) is well-ordered”.     

On bases,finite dimensional decompositions and weaker structures in Banach spaces

This is an investigation of the connections between bases and weaker structures in Banach spaces and their duals. It is proved, e.g., thatX has a basis ifX* does, and that ifX has a basis, thenX* has a basis provided thatX* is separable and satisfies Grothendieck’s approximation property; analogous results are obtained concerning π-structures and finite dimensional Schauder decompositions. The basic results are then applied to show that every separableℒ _p space has a basis. The second and third named authors have been supported by the NSF Grant GP 12997.     

On furstenberg’s characterization of harmonic functions on symmetric spaces

We provide a necessary and sufficient condition on a radial probability measureμ on a symmetric space for whichf =f *μ, f bounded, implies thatf is harmonic. In particular, we obtain a short and elementary proof of a theorem of Furstenberg which says that iff is a bounded function on a symmetric space which satisfiesf =f *μ for some radialabsolutely continuous probability measureμ, thenf is harmonic.     

Irregularity of certain algebraic fiber spaces

LetX, Y be smooth complex projective varieties, andf: X →Y be a fiber space whose general fiber is a curve of genusg. Denote byq _f the relative irregularity off. It is proved thatq _f ≤5g+1 / 6, iff is not generically trivial; moreover, if either a)f is non-constant and the general fiber is either hyperelliptic or bielliptic or b)q(Y)=0, thenq _f ≤g+1 / 2, and the bound is best possible. A classification of fiber surfaces of genus 3 withq _f =2 is also given in this note. Project supported by China Postdoctoral Science Foundation     

Hilbert’s irreducibility theorem for prime degree and general polynomials

Letf (X, t)εℚ[X, t] be an irreducible polynomial. Hilbert’s irreducibility theorem asserts that there are infinitely manyt ₀εℤ such thatf (X, t ₀) is still irreducible. We say thatf (X, t) isgeneral if the Galois group off (X, t) over ℚ(t) is the symmetric group in its natural action. We show that if the degree off with respect toX is a prime ≠ 5 or iff is general of degree ≠ 5, thenf (X, t ₀) is irreducible for all but finitely manyt ₀εℤ unless the curve given byf (X, t)=0 has infinitely many points (x ₀,t ₀) withx ₀εℚ,t ₀εℤ. The proof makes use of Siegel’s theorem about integral points on algebraic curves, and classical results about finite groups, going back to Burnside, Schur, Wielandt, and others. Supported by the DFG.     

C(K) quotients of separable ℒ∞ spaces

It is shown that ifX is an ℒ⫗ space with separable dual, thenX has a quotient isomorphic toC(ω)^α if, and only if, there is anɛ >0, such that theɛ-Szlenk index ofX is at leastα. It was previously known, that ifX is a separable ℒ⫗ space with a non-separable dual, thenX has a quotient isomorphic toC[0, 1]. Research partially supported by NSF Grant MCS-74-07509-A03.     

Quelques théorèmes sur la structure complexe

We prove the following result: If the function Max (log|ω -f ₁(z)|, ..., log|ω -f _k(z)|) is plurisubharmonic in the open setD×ℂ (D open of ℂ ⁿ ), thenf ₁,...,f _k are analytic functions iff ₁,...,f _k are continuous functions onD(k≥2). We prove also some other results.     

On the Convergence and Iterates of q-Bernstein Polynomials

The convergence properties of q-Bernstein polynomials are investigated. When q1 is fixed the generalized Bernstein polynomials _nf of f, a one parameter family of Bernstein polynomials, converge to f as n→∞ if f is a polynomial. It is proved that, if the parameter 0<q<1 is fixed, then _nf→f if and only if f is linear. The iterates of _nf are also considered. It is shown that _n^Mf converges to the linear interpolating polynomial for f at the endpoints of [0,1], for any fixed q>0, as the number of iterates M→∞. Moreover, the iterates of the Boolean sum of _nf converge to the interpolating polynomial for f at n+1 geometrically spaced nodes on [0,1].     

Geometrical implications of the existence of very smooth bump functions in Banach spaces

We show that ifX is a Banach space and if there is a non-zero real-valuedC ^∞-smooth function onX with bounded support, then eitherX contains an isomorphic copy ofc ₀(N), or there is an integerk greater than or equal to 1 such thatX is of exact cotype 2k and, in this case,X contains an isomorphic copy ofl ^2k(N). We also show that ifX is a Banach space such that there is onX a non-zero real-valuedC ⁴-smooth function with bounded support and ifX is of cotypeq forq<4, thenX is isomorphic to a Hilbert space.