Endodualisable and endoprimal finite double Stone algebras

The finite endodualisable double Stone algebras are characterised, and every finite endoprimal double Stone algebra is shown to be endodualisable. The authors wish to express their gratitude to B. A. Davey and T. Katriňák for their helpful remarks and to J. G. Pitkethly for her assistance with the pictures. A support by Slovak grants VEGA 1/4057/97, 1/3026/06 and APVV-51-009605 is acknowledged by the first author who also wishes to thank the Mathematical Institute of the University of Oxford and the School of Mathematical and Statistical Sciences of La Trobe University for their hospitality.     

Equations fonctionnelles et recherche de sous-groupes

Sans résumé

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Dedié à János Aczél à l'occasion de son 60ème anniversaire     

Clones on three elements preserving a binary relation

We describe the clones on 3 elements that can be expressed as Pol ρ for ρ a binary relation. We present the poset of these clones ordered by inclusion. This article is a shortened version of the author’s thesis, to give an idea of the whole work. Presented by R. P?schel. Received March 24, 2005; accepted in final form December 20, 2005.     

On factorizing meromorphic functions

Summary The paper determines all cases when a meromorphic functionF can be expressed both asf ⊗p andf ⊗q with the same meromorphicf and different polynomialsp andq. In all cases there are constantsk, β, a positive integerm, a root λ of unity of orderS and a polynomialr such thatp=(Lr) ^m+k,q=r ^m+k, whereLz=λz+β. We have eitherm=1,S arbitrary orm=2,S=2, which can occur even ifF andf are entire, or, in the remaining casesS=2, 3, 4 or 6,m dividesS andf(k+t ^m) is a doubly-periodic function.     

Ona-Wright convexity and the converse of Minkowski's inequality

Summary Leta (0, 1/2] be fixed. A functionf satisfying the inequalityf(ax + (1 – a)y) + f((1 – a)x + ay) f(x) + f(y), called herea-Wright convexity, appears in connection with the converse of Minkowski's inequality. We prove that every lower semicontinuousa-Wright convex function is Jensen convex and we pose an open problem. Moreover, using the fact that 1/2-Wright convexity coincides with Jensen convexity, we prove a converse of Minkowski's inequality without any regularity conditions.     

On the structure of varieties with equationally definable principal congruences IV

The notion of apseudo-interior algebra is introduced; it is a hybrid of a (topological) interior algebra and a residuated partially ordered monoid. The elementary arithmetic of pseudo-interior algebras is developed leading to a simple equational axiomatization. A notion ofopen filter analogous to the open filters of interior algebras is investigated. Pseudo-interior algebras represent, in algebraic form, the logic inherent in varieties with acommutative, regular ternary deductive (TD) term p(x, y, z), which is defined by the conditions: (1)p(x,y,z) z (mod(x, y)); (2) for fixed elementsa, b of an algebra A, {p(a, b, z):z A} is a transversal of the set of equivalence classes of (a, b); (3)p(a, b, z) andp(a,b,z) define the same transversal whenever(a,b)=(a,b); (4)(p(x, y, 1), 1)= (x, y) for some constant term 1. The TD term generalizes the (affine) ternary discriminator. Varieties with a commutative, regular TD term include most of the varieties of traditional algebraic logic as well as all double-pointed affine discriminator varieties andn-potent hoops (residuated commutative po-monoids in which the partial ordering is inverse divisibility). The main theorem:A variety has a commutative, regular TD term iff it is termwise definitionally equivalent to a pseudo-interior algebra with additional operations that are compatible with the open filters in a natural way.Presented by R. W. Quackenbush.The authors gratefully acknowledge the support of National Science Foundation Grants DMS-8703743 and DMS-8805870.     

The associativity equation revisited

Summary Consideration of the Associativity Equation,x (y z) = (x y) z, in the case where:I × I I (I a real interval) is continuous and satisfies a cancellation property on both sides, provides a complete characterization of real continuous cancellation semigroups, namely that they are topologically order-isomorphic to addition on some real interval: ( – ,b), ( – ,b], –, +), (a, + ), or [a, + ) — whereb = 0 or –1 anda = 0 or 1. The original proof, however, involves some awkward handling of cases and has defied streamlining for some time. A new proof is given following a simpler approach, devised by Páles and fine-tuned by Craigen.     

A lattice-theoretic approach to arbitrary real functions on frames

In this paper we model discontinuous extended real functions in pointfree topology following a lattice-theoretic approach, in such a way that, if L is a subfit frame, arbitrary extended real functions on L are the elements of the Dedekind-MacNeille completion of the poset of all extended semicontinuous functions on L. This approach mimicks the situation one has with a T₁-space X, where the lattice F?(X) of arbitrary extended real functions on X is the smallest complete lattice containing both extended upper and lower semicontinuous functions on X. Then, we identify real-valued functions by lattice-theoretic means. By construction, we obtain definitions of discontinuous functions that are conservative for T₁-spaces. We also analyze semicontinuity and introduce definitions which are conservative for T₀-spaces.     

Amorphe Potenzen kompakter Räume

A set is amorphous, if it is not a union of two disjoint infinite subsets. The following variants of the Tychonoff product theorem are investigated in the hierarchy of weak choice principles. TA1: An amorphous power of a compactT ₂ space is compact. TA2: An amorphous power of a compactT ₂ space which as a set is wellorderable is compact. In ZF⁰TA1 is equivalent to the assertion, that amorphous sets are finite. RT is Ramsey's theorem, that every finite colouring of the set ofn-element subsets of an infinite set has an infinite homogeneous subset and PW is Rubin's axiom, that the power set of an ordinal is wellorderable. In ZF⁰RT+PW implies TA2. Since RT+PW is compatible with the existence of infinite amorphous sets, TA2 does not imply TA1 in ZF⁰. But TA2 cannot be proved in ZF⁰ alone. As an application, we prove a theorem of Stone, using a weak wellordering axiomD ₃ (a set is wellorderable, if each of its infinite subsets is structured) together with RT.

Diese Arbeit ist Teil der Habilitationsschrift des Verfassers im Fachgebiet Mathematische Analysis an der Technischen Universität Wien.     

Supernumary polylogarithmic ladders and related functional equations

Summary The nature of the polylogarithmic ladder is briefly reviewed, and its close relationship to the associated cyclotomic equation explained. Generic results for the base determined by the family of equationsu ^p +u ^q = 1 are developed, and many new supernumary ladders, existing for particular values ofp andq, are discussed in relation to theirad hoc cyclotomic equations. Results for ordersn from 6 through 9, for which no relevant functional equations are known, are reviewed; and new results for the base , where ³ + = 1, are developed through the sixth order.Special results for the exponentp from 4 through 6 are determined whenever a new cyclotomic equation can be constructed. Only the equationu ⁵+u ³ = 1 has so far resisted this process. The need for the constraint (p,q) = 1 is briefly considered if redundant formulas are to be avoided.The equationu ^6m+1 +u ^6r–1 = 1 is discussed and some valid results deduced. This equation is divisible byu ² –u + 1, and the quotient polynomial is useful for constructing cyclotomic equations. The casem = 1,r = 2 is the first example encountered for which no valid ladders have yet been found.New functional equations to give the supernumary -ladders of index 24 are developed, but their construction runs into difficulty at the third order, apparently requiring the introduction of an adjoint set of variables that blocks the extension to the fourth order.A demonstration, based on the indices of existing accessible and supernumary ladders, indicates that functional equations based on arguments ±z ^m (1–z) ^r (1 +z) ^s are not capable of extension to the sixth order.There are some miscellaneous supernumary ladders that seem incapable, at this time, of analytic proof, and these are briefly discussed. In conclusion, applications of ladders are considered, and attention drawn to the existence of ladders with the base on the unit circle giving rise to Clausenfunction formulas which may play an important role inK-theory.     

Series like Taylor's series

A theorem and some examples are given concerning the convergence, in a space of generalized functions, of power series whose terms contain successive derivatives of a given function. One example is the Euler-Maclaurin sum formula, where there are some novelties.     

Holomorphic solutions of an inhomogeneous Cauchy equation

Summary We consider the functional equation(x + y) – (x) – (y) = f(x)f(y)h(x + y) and we find all its homomorphic solutionsf, h, defined in a neighbourhood of the origin.     

An Invariant Harnack Inequality for a Class of Hypoelliptic Ultraparabolic Equations

We show an invariant Harnack inequality for a class of hypoelliptic ultraparabolic operators with underlying homogeneous Lie group structures. As a byproduct we prove a Liouville type theorem for the related stationary operators. We also introduce a notion of link of homogeneous Lie Groups that allows us to show that our results apply to wide classes of operators.     

Linearisierung formal-biholomorpher Abbildungen und Iterationsprobleme

Ohne Zusammenfassung

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Further results on supernumary polylogarithmic ladders

Summary With the help of the PARI computer program a number of matters left unresolved from previous work have now been settled. It will be recalled that a ladder is a rational sum of polylogarithms, with predetermined coefficients, of powers of a given algebraic base. The simplest bases considered are the roots in (0, 1) ofu ^p +u ^q = 1 for various integersp andq.. They possess a number of generic results, together with some additional equations, termed supernumary for certain specific values ofp andq. In particular, ladders of the base (see [1]) have been extended to the sixth order, and involve a new index, 60, found by the PARI program. The base from (p, q) = (11, 7) has an additional index 20, and this combines with earlier results to produce a valid ladder. The apparent barren feature of certain equations is now explained in terms of a need to work with a sufficient number of results. It is confirmed that the equation with (p, q) = (5, 3) indeed does not possess any supernumary results.A complete investigation of the smallest Salem number of degree four is given: it possesses results to the 8th order. An introduction is given to similar studies for the smallest known Salem number, which has now been shown to extend to the 16th order.Some ladder results for combined bases are found, with one such formula deducible from a three-variable dilogarithmic functional equation. Formulas of a new type are developed in which summation over conjugate roots enables ladders to be extended fromn = 2 to 3.     

Packing random intervals

Summary Letn random intervalsI ₁, ...,I _n be chosen by selecting endpoints independently from the uniform distribution on [0.1]. Apacking is a pairwise disjoint subset of the intervals; itswasted space is the Lebesgue measure of the points of [0,1] not covered by the packing. In any set of intervals the packing with least wasted space is computationally easy to find; but its expected wasted space in the random case is not obvious. We show that with high probability for largen, this best packing has wasted space . It turns out that if the endpoints 0 and 1 are identified, so that the problem is now one of packing random arcs in a unit-circumference circle, then optimal wasted space is reduced toO(1/n). Interestingly, there is a striking difference between thesizes of the best packings: about logn intervals in the unit interval case, but usually only one or two arcs in the circle case.