Constructions of Partial Difference Sets and Relative Difference Sets on p-Groups

By using some finite local rings, we construct some new partialdifference sets and relative difference sets on pgroups wherep is any prime. When p = 2, some of partial difference setsconstructed are reversible difference sets which include Dillon'sdifference sets.     

Coverings of Groups and Types

Assume that G is a group covered by countably many sets X_n,n < . It is proved that G is generated in two or three stepsby a small number of the sets X_n. These results are generalizedfor countable coverings of types and countable colourings ofgraphs.     

Asymptotic simplification for a reaction-diffusion problem with a nonlinear boundary condition

We study non-negative solutions of the porous medium equationwith a source and a nonlinear flux boundary condition, u_t =(u^m)_xx + u^p in (0, ), x (0, T); – (u^m)_x (0, t) = u^q (0,t) for t (0, T); u (x, 0) = u₀ (x) in (0, ), where m > 1,p, q > 0 are parameters. For every fixed m we prove thatthere are two critical curves in the (p, q-plane: (i) the criticalexistence curve, separating the region where every solutionis global from the region where there exist blowing-up solutions,and (ii) the Fujita curve, separating a region of parametersin which all solutions blow up from a region where both globalin time solutions and blowing-up solutions exist. In the caseof blow up we find the blow-up rates, the blow-up sets and theblow-up profiles, showing that there is a phenomenon of asymptoticsimplification. If 2q < p + m the asymptotics are governedby the source term. On the other hand, if 2q > p + m theevolution close to blow up is ruled by the boundary flux. If2q = p + m both terms are of the same order.     

Every Set has a Least Jump Enumeration

Given a computably enumerable set W, there is a Turing degreewhich is the least jump of any set in which W is computablyenumerable, namely 0'. Remarkably, this is not a phenomenonof computably enumerable sets. It is shown that for every subsetA of N, there is a Turing degree, c'_µ(A), which is theleast degree of the jumps of all sets X for which A is . In addition this result providesan isomorphism invariant method for assigning Turing degreesto certain torsion-free abelian groups.     

Riesz Transforms and Harmonic Lip1-Capacity in Cantor Sets

We estimate the L²-norm of the s-dimensional Riesz transformson some Cantor sets in R^d. Towards this end, we show that theRiesz transforms truncated at different scales behave in a quasiorthogonalway. As an application, we obtain some precise numerical estimatesfor the Lipschitz harmonic capacity of these sets. 2000 MathematicsSubject Classification 42B20, 42B25.     

On Sets Where Iterates of a Meromorphic Function Zip Towards Infinity

For a transcendental meromorphic function f, various propertiesof the set [formula] were obtained in [8] and [9]. Here we establish analogous propertiesfor the smaller sets [formula] introduced in [5], and [formula] We deduce a symmetry result for Julia sets J(f), and also indicatesome techniques for showing that certain invariant curves liein I'(f), Z(f) and J(f). 2000 Mathematics Subject Classification30D05, 37F10, 37F50.     

On Central Cantor Sets with Self-Arithmetic Difference of Positive Lebesgue Measure

We prove the existence of C^r - (but not C^r+1-) regular centralCantor sets with zero Lebesgue measure such that their selfarithmetic difference is a Cantor set with positive Lebesguemeasure. This is motivated by a conjecture in the field of bifurcationsof dynamical systems posed by Jacob Palis.     

Optimal Smoothing for Convex Polytopes

It is proved that, given a convex polytope P in Rⁿ, togetherwith a collection of compact convex subsets in the interiorof each facet of P, there exists a smooth convex body arbitrarilyclose to P that coincides with each facet precisely along theprescribed sets, and has positive curvature elsewhere. 2000Mathematics Subject Classification 53A07, 52B11, 53C45.     

Flattening and Subanalytic Sets in Rigid Analytic Geometry

Let K be an algebraically closed field endowed with a completenon-archimedean norm with valuation ring R. Let f: Y X be amap of K-affinoid varieties. In this paper we study the analyticstructure of the image f(Y) X; such an image is a typical exampleof a subanalytic set. We show that the subanalytic sets areprecisely the D-semianalytic sets, where D is the truncateddivision function first introduced by Denef and van den Dries.This result is most conveniently stated as a Quantifier Eliminationresult for the valuation ring R in an analytic expansion ofthe language of valued rings. To prove this we establish a Flattening Theorem for affinoidvarieties in the style of Hironaka, which allows a reductionto the study of subanalytic sets arising from flat maps, thatis, we show that a map of affinoid varieties can be renderedflat by using only finitely many local blowing ups. The caseof a flat map is then dealt with by a small extension of a resultof Raynaud and Gruson showing that the image of a flat map ofaffinoid varieties is open in the Grothendieck topology. Using Embedded Resolution of Singularities, we derive in thezero characteristic case, a Uniformization Theorem for subanalyticsets: a subanalytic set can be rendered semianalytic using onlyfinitely many local blowing ups with smooth centres. As a corollarywe obtain the fact that any subanalytic set in the plane R2is semianalytic. 2000 Mathematical Subject Classification: 32P05,32B20, 13C11, 12J25, 03C10.     

Long Arithmetic Progressions in Sum-Sets and the Number x-Sum-Free Sets

In this paper we obtain optimal bounds for the length of thelongest arithmetic progression in various kinds of sum-sets.As an application, we derive a sharp estimate for the numberof sets A of residues modulo a prime n such that no subsum ofA equals x modulo n, where x is a fixed residue modulo n. 2000Mathematics Subject Classification 05A16, 11B25, 11P32.     

How To Make Davies' Theorem Visible

We prove that for an arbitrary measurable set A R² and a -finiteBorel measure µ on the plane, there is a Borel set oflines L such that for each point in A, the set of directionsof those lines from L containing the point is a residual set,and, moreover, We show how this result may be used to characterise the sets of the planefrom which an invisible set is visible. We also characterisethe rectifiable sets C₁, C₂ for which there is a set which isvisible from C₁ and invisible from C₂.     

Intersecting Families of Separated Sets

A set A {1,2,...,n} is said to be k-separated if, when consideredon the circle, any two elements of A are separated by a gapof size at least k. A conjecture due to Holroyd and Johnson that an analogue ofthe Erds–Ko–Rado theorem holds for k-separated setsis proved. In particular, the result holds for the vertex-criticalsubgraph of the Kneser graph identified by Schrijver, the collectionof separated sets. A version of the Erds–Ko–Radotheorem for weighted k-separated sets is also given.     

Spectral Synthesis and Operator Synthesis for Compact Groups

Let G be a compact group and C(G) be the C^*-algebra of continuouscomplex-valued functions on G. The paper constructs an imbeddingof the Fourier algebra A(G) of G into the algebra V(G) = C(G)^hC(G)(Haagerup tensor product) and deduces results about parallelspectral synthesis, generalizing a result of Varopoulos. Itthen characterizes which diagonal sets in G x G are sets ofoperator synthesis with respect to the Haar measure, using thedefinition of operator synthesis due to Arveson. This resultis applied to obtain an analogue of a result of Froelich: atensor formula for the algebras associated with the pre-ordersdefined by closed unital subsemigroups of G.     

Singularities of Centre Symmetry Sets

The center symmetry set (CSS) of a smooth hypersurface S inan affine space Rⁿ is the envelope of lines joining pairs ofpoints where S has parallel tangent hyperplanes. The idea stemsfrom a definition of Janeczko, in an alternative version dueto Giblin and Holtom. For n = 2 the envelope is always real,while for n > 3 the existence of a real envelope dependson the geometry of the hypersurface. In this paper we make alocal study of the CSS, some results applying to n 5 and othersto the cases n = 2,3. The method is to construct a generatingfunction whose bifurcation set contains the CSS and possiblysome other redundant components. Focal sets of smooth hypersurfacesare a special case of the construction, but the CSS is an affineand not a euclidean invariant. Besides the familiar local formsof focal sets there are other local forms corresponding to boundarysingularities, and yet others which do not appear to have arisenelsewhere in a geometrical context. There are connections withFinsler geometry. This paper concentrates on the theory andthe proof of the local normal forms for the CSS. 2000 MathematicsSubject Classification 57R45, 58K40, 32S25, 58B20.     

Omega Limit Sets in Positive Cones

The omega limit sets of a nonlinear operator T which is definedon a positive cone and satisfies certain ray-contractive typeconditions are discussed. Under the assumption that the restrictionof T to a compact subset is surjective, the following alternativesare proved: the omega limit set of a point in the cone eitherconsists of a fixed point or forms a 2-cycle. In addition, newproofs and extensions to relevant results are given. 2000 MathematicsSubject Classification 47H07, 47H09.     

Entropy of Absolute Convex Hulls in Hilbert Spaces

The metric entropy of absolute convex hulls of sets in Hilbertspaces is studied for the general case when the metric entropyof the sets is arbitrary. Under some regularity assumptions,the results are sharp. 2000 Mathematical Subject Classification41A46 (primary), 60G15 (secondary).     

Infinite Patterns that can be Avoided by Measure

A set A of real numbers is called universal (in measure) ifevery measurable set of positive measure necessarily containsan affine copy of A. All finite sets are universal, but no infiniteuniversal sets are known. Here we prove some results relatedto a conjecture of Erds that there is no infinite universalset. For every infinite set A, there is a set E of positivemeasure such that (x + tA)E fails for almost all (Lebesgue)pairs (x, t). Also, the exceptional set of pairs (x, t) (forwhich (x + tA)E) can be taken to project to a null set on thet-axis. Finally, if the set A contains large subsets whose minimumgap is large (in a scale-invariant way), then there is ER ofpositive measure which contains no affine copy of A. 1991 MathematicsSubject Classification 28A12.     

Extended conjugate points in the calculus of variations

This paper concerns a characterization of second-order optimalityconditions for the fixed-endpoint problem in the calculus ofvariations. The key new concept is a set S(x) with the propertythat S(x)=if and only if the second variation with respect tox, independently of non-singularity assumptions, is non-negativealong admissible variations. We show that, for this set of points,it may be much easier (and never more difficult) to prove itsnon-emptiness than directly finding variations that make thesecond variation negative. Earlier Loewen and Zheng, and Zeidan,introduced related sets C₁(x) and C₂(x), applicable to certainoptimal control problems, whose non-emptiness has been establishedmerely as a sufficient condition for the existence of negativesecond variations. These sets, when reduced to the problem weare considering, are related according to C₁(x) C₂(x) S(x).Contrary to the behaviour of S(x), verifying membership of C₁(x)or C₂(x) may be more difficult than verifying directly if thesecond-order condition holds. We provide several examples forwhich it is straightforward to prove that S(x) , but determiningthe sets C₁(x) or C₂(x) may be a very difficult or perhaps evena hopeless task.     

An Extension of the Ham Sandwich Theorem

It is shown that both the ‘ham sandwich theorem’and Richard Rado's theorem on general measure (see [6]), whichis known to be a measure theoretic equivalent of E. Helly'stheorem on convex sets, belong to the same family of resultsabout geometric, extremal properties of measures which are definedon Borel sets in Rⁿ.     

On the Completeness of Sets of Solutions to the Helmholtz Equation

Completeness in L²(D) is established for sets of functions formedfrom solutions to the two-dimensional Helmholtz equation ina domain D. Each function is a linear combination of a solution(found by separation of variables) and its normal derivativeon D, so the sets may be used to solve impedance-type boundaryvalue problems. Sets that contain either regular Bessel functionsor singular Hankel functions are considered. Methods of proofare employed that provide alternatives to the conventional potential-theoreticapproaches. In the majority of cases, the domain of interestis bounded and simply connected. One completeness result fora bounded, doubly-connected domain is proved. In some circumstances,one of the methods leads to a mild but inessential eigenvaluerestriction.