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.     

Witt Groups of the Punctured Spectrum of a 3-Dimensional Regular Local Ring and a Purity Theorem

Let A be a regular local ring with quotient field K. Assumethat 2 is invertible in A. Let W(A)W(K) be the homomorphisminduced by the inclusion AK, where W( ) denotes the Witt groupof quadratic forms. If dim A4, it is known that this map isinjective [6, 7]. A natural question is to characterize theimage of W(A) in W(K). Let Spec¹(A) be the set of prime idealsof A of height 1. For PSpec¹(A), let _P be a parameter of thediscrete valuation ring A_P and k(P) = A_P/PA_P. For this choiceof a parameter _P, one has the second residue homomorphism _P:W(K)W(k(P))[9, p. 209]. Though the homomorphism _P depends on the choiceof the parameter _P, its kernel and cokernel do not. We havea homomorphism A part of the so-called Gersten conjecture is the followingquestion on ‘purity’. Is the sequence exact? This question has an affirmative answer for dim(A)2 [1;3, p. 277]. There have been speculations by Pardon and Barge-Sansuc-Vogelon the question of purity. However, in the literature, thereis no proof for purity even for dim(A) = 3. One of the consequencesof the main result of this paper is an affirmative answer tothe purity question for dim(A) = 3. We briefly outline our main result.     

Cupping the Recursively Enumerable Degrees by D.R.E. Degrees

We prove that there are two incomplete d.r.e. degrees (the Turingdegrees of differences of two recursively enumerable sets) suchthat every non-zero recursively enumerable degree cups at leastone of them to 0', the greatest recursively enumerable (Turing)degree. 1991 Mathematics Subject Classification: primary 03D25,03D30; secondary 03D35.     

Isolation in the CEA hierarchy

Examining various kinds of isolation phenomena in the Turing degrees, I show that there are, for every n>0, (n+1)-c.e. sets isolated in the n-CEA degrees by n-c.e. sets below them. For n1 such phenomena arise below any computably enumerable degree, and conjecture that this result holds densely in the c.e. degrees as well. Surprisingly, such isolation pairs also exist where the top set has high degree and the isolating set is low, although the complete situation for jump classes remains unknown.     

Rado Partition Theorem for Random Subsets of Integers

For an l x k matrix A = (a_ij) of integers, denote by L(A) thesystem of homogenous linear equations a_i1x₁ + ... + a_ikx_k =0, 1 i l. We say that A is density regular if every subsetof N with positive density, contains a solution to L(A). Fora density regular l x k matrix A, an integer r and a set ofintegers F, we write if for any partition F = F₁ ... F_r there exists i {1, 2,..., r} and a column vector x such that Ax = 0 and all entriesof x belong to F_i. Let [n]_N be a random N-element subset of{1, 2, ..., n} chosen uniformly from among all such subsets.In this paper we determine for every density regular matrixA a parameter = (A) such that lim_n P([n]_N (A)_r)=0 if N =O(n) and 1 if N = (n). 1991 Mathematics Subject Classification:05D10, 11B25, 60C05     

Discontinuous Homomorphisms from Non-Commutative Banach Algebras

In the 1970s, a question of Kaplansky about discontinuous homomorphismsfrom certain commutative Banach algebras was resolved. Let Abe the commutative C*-algebra C(), where is an infinite compactspace. Then, if the continuum hypothesis (CH) be assumed, thereis a discontinuous homomorphism from C() into a Banach algebra[2, 7]. In fact, let A be a commutative Banach algebra. Then(with (CH)) there is a discontinuous homomorphism from A intoa Banach algebra whenever the character space _A of A is infinite[3, Theorem 3] and also whenever there is a non-maximal, primeideal P in A such that |A/P|=2^₀ [4, 8]. (It is an open questionwhether or not every infinite-dimensional, commutative Banachalgebra A satisfies this latter condition.) 1991 MathematicsSubject Classification 46H40.     

A Criterion for the Existence of Transversals of Set Systems

Nash-Williams [6] formulated a condition that is necessary andsufficient for a countable family A=(A_i)_iI of sets to have atransversal. In [7] he proved that his criterion applies alsowhen we allow the set I to be arbitrary and require only that_iJA_i=Ø for any uncountable JI. In this paper, we formulateanother criterion of a similar nature, and prove that it isequivalent to the criterion of Nash-Williams for any familyu. We also present a self-contained proof that if _iJA_i=Øfor any uncountable JI, then our condition is necessary andsufficient for the family u to have a transversal.     

An extension of the recursively enumerable Turing degrees

Consider the countable semilattice _T consisting of the recursivelyenumerable Turing degrees. Although _T is known to be structurallyrich, a major source of frustration is that no specific, naturaldegrees in _T have been discovered, except the bottom and topdegrees, 0 and 0'. In order to overcome this difficulty, weembed _T into a larger degree structure which is better behaved.Namely, consider the countable distributive lattice _w consistingof the weak degrees (also known as Muchnik degrees) of massproblems associated with non-empty ₀¹ subsets of 2. It is knownthat _w contains a bottom degree 0 and a top degree 1 and isstructurally rich. Moreover, _w contains many specific, naturaldegrees other than 0 and 1. In particular, we show that in _wone has 0 < d < r₁ < f(r₂, 1) < 1. Here, d is theweak degree of the diagonally non-recursive functions, and r_nis the weak degree of the n-random reals. It is known that r₁can be characterized as the maximum weak degree of a ₀¹ subsetof 2 of positive measure. We now show thatf(r₂, 1) can be characterizedas the maximum weak degree of a ₀¹ subset of 2, the Turing upwardclosure of which is of positive measure. We exhibit a naturalembedding of _T into _w which is one-to-one, preserves the semilatticestructure of _T, carries 0 to 0, and carries 0' to 1. Identifying_T with its image in _w, we show that all of the degrees in _Texcept 0 and 1 are incomparable with the specific degrees d,r₁, andf(r₂, 1) in _w.     

Beardon's Diophantine Equations and Non-Free Mobius Groups

Let µ be a real number. The Möbius group G_µis the matrix group generated by It is known that G_µ is free if |µ| 2 (see [1])or if µ is transcendental (see [3, 8]). Moreover, thereis a set of irrational algebraic numbers µ which is densein (–2, 2) and for which G_µ is non-free [2, p. 528].We may assume that µ > 0, and in this paper we considerrational µ in (0, 2). The following problem is difficult. Let G^nf denote the set of all rational numbers µ in (0,2) for which G_µ is non-free. In 1969 Lyndon and Ullman[8] proved that G^nf contains the elements of the forms p/(p²+ 1) and 1/(p + 1), where p = 1, 2, ..., and that if µ₀ G^nf, then µ₀/p G^nf for p = 1, 2, .... In 1993 Beardon[2] studied problem (P) by means of the words of the form A^rB^s A^t and A^r B^s A^t B^u A^v, and he obtained a sufficient conditionfor solvability of (P), included implicitly in [2, pp. 530–531],by means of the following Diophantine equations: 1991 Mathematics SubjectClassification 20E05, 20H20, 11D09.     

On the Discrepancy for Cartesian Products

Let B₂ denote the family of all circular discs in the plane.It is proved that the discrepancy for the family {B₁ x B₂ :B₁, B₂ B₂} in R⁴ is O(n^1/4+) for an arbitrarily small constant > 0, that is, it is essentially the same as that for thefamily B₂ itself. The result is established for the combinatorialdiscrepancy, and consequently it holds for the discrepancy withrespect to the Lebesgue measure as well. This answers a questionof Beck and Chen. More generally, we prove an upper bound forthe discrepancy for a family {^k_i=1A_i:A_iA_i, i = 1, 2, ..., k},where each A_i is a family in R^d_i, each of whose sets is describedby a bounded number of polynomial inequalities of bounded degree.The resulting discrepancy bound is determined by the ‘worst’of the families A_i, and it depends on the existence of certaindecompositions into constant-complexity cells for arrangementsof surfaces bounding the sets of A_i. The proof uses Beck's partialcoloring method and decomposition techniques developed for therange-searching problem in computational geometry.     

Euler Characteristics of Coxeter Groups, PL-Triangulations of Closed Manifolds, and Cohomology of Subgroups of Artin Groups

The motivation for the theory of Euler characteristics of groups,which was introduced by C. T. C. Wall [21], was topology, butit has interesting connections to other branches of mathematicssuch as group theory and number theory. This paper investigatesEuler characteristics of Coxeter groups and their applications.In his paper [20], J.-P. Serre obtained several fundamentalresults concerning the Euler characteristics of Coxeter groups.In particular, he obtained a recursive formula for the Eulercharacteristic of a Coxeter group, as well as its relation tothe Poincaré series (see 3). Later, I. M. Chiswell obtainedin [10] a formula expressing the Euler characteristic of a Coxetergroup in terms of orders of finite parabolic subgroups (Theorem1). These formulae enable us to compute Euler characteristicsof arbitrary Coxeter groups. On the other hand, the Euler characteristics of Coxeter groupsW happen to be intimately related to their associated complexesF_W, which are defined by means of the posets of nontrivial parabolicsubgroups of finite order (see 2.1 for the precise definition).In particular, it follows from the recent result of M. W. Davis[13] that if F_W is a product of a simplex and a generalizedhomology 2n-sphere, then the Euler characteristic of W is zero(Corollary 3.1). The first objective of this paper is to generalizethe previously mentioned result to the case when F_W is a PL-triangulationof a closed 2n-manifold which is not necessarily a homology2n-sphere. In other words (as given below in Theorem 3), ifW is a Coxeter group such that F_W is a PL-triangulation of aclosed 2n-manifold, then the Euler characteristic of W is equalto 1–(F_W)/2.     

A Necessary and Sufficient Condition for a Linear Differential System to be Strongly Monotone

In order to present the results of this note, we begin withsome definitions. Consider a differential system [formula] where IR is an open interval, and f(t, x), (t, x)IxRⁿ, is acontinuous vector function with continuous first derivativesf_r/x_s, r, s=1, 2, ..., n. Let D_xf(t, x), (t, x)IxRⁿ, denote the Jacobi matrix of f(t,x), with respect to the variables x₁, ..., x_n. Let x(t, t₀,x₀), tI(t₀, x₀) denote the maximal solution of the system (1)through the point (t₀, x₀)IxRⁿ. For two vectors x, yRⁿ, we use the notations x>y and x>>yaccording to the following definitions: [formula] An nxn matrix A=(a_rs) is called reducible if n2 and there existsa partition [formula] (p1, q1, p+q=n) such that [formula] The matrix A is called irreducible if n=1, or if n2 and A isnot reducible. The system (1) is called strongly monotone if for any t₀I, x₁,x₂Rⁿ [formula] holds for all t>t₀ as long as both solutions x(t, t₀, x_i),i=1, 2, are defined. The system is called cooperative if forall (t, x)IxRⁿ the off-diagonal elements of the nxn matrix D_xf(t,x) are nonnegative. 1991 Mathematics Subject Classification34A30, 34C99.     

Degrees of Autostability Relative to Strong Constructivizations of Graphs

We show that each computably enumerable Turing degree is a degree of autostability relative to strong constructivizations for a decidable directed graph. We construct a decidable undirected graph whose autostability spectrum relative to strong constructivizations is equal to the set of all PA-degrees.     

Beurling and Lipschitz Algebras

It is well known that there exist infinite closed subsets Eof T such that A(E) = C(E) (see, for example, [3]). Such setsare called Helson sets. Let E be a closed subset of T, let 0< < 1, and let A(E) be the restriction of the Beurlingalgebra A(T). Then A(E) lipE. We shall show that A(E) = lipEif and only if E is finite. This answers a question raised byPedersen [5], where partial results were obtained. 1991 MathematicsSubject Classification 46J10.     

A Strengthening of Resolution of Singularities in Characteristic Zero

Let X be a closed subscheme embedded in a scheme W, smooth overa field k of characteristic zero, and let I (X) be the sheafof ideals defining X. Assume that the set of regular pointsof X is dense in X. We prove that there exists a proper, birationalmorphism, : W_r W, obtained as a composition of monoidal transformations,so that if X_r W_r denotes the strict transform of X W then: (1) the morphism : W_r W is an embedded desingularization ofX (as in Hironaka's Theorem); (2) the total transform of I (X) in factors as a product of an invertible sheaf of ideals L supportedon the exceptional locus, and the sheaf of ideals defining thestrict transform of X (that is, . Thus (2) asserts that we can obtain, in a simple manner, theequations defining the desingularization of X. 2000 MathematicalSubject Classification: 14E15.     

Sandwiching C0-Semigroups

Let T = {T(t)}_t0 be a C₀-semigroup on a Banach space X. Thefollowing results are proved. (i) If X is separable, there exist separable Hilbert spacesX₀ and X₁, continuous dense embeddings j₀:X₀ X and j₁:X X₁,and C₀-semigroups T₀ and T₁ on X₀ and X₁ respectively, suchthat j₀ T₀(t) = T(t) j₀ and T₁(t) j₁ = j₁ T(t) for all t 0. (ii) If T is -reflexive, there exist reflexive Banach spacesX₀ and X₁ , continuous dense embeddings j:D(A²) X₀, j₀:X₀ X, j₁:X X₁, and C₀-semigroups T₀ and T₁ on X₀ and X₁ respectively,such that T₀(t) j = j T(t), j₀ T₀(t) = T(t) j₀ and T(t) j₁ = j₁ T(t) for all t 0, and such that (A₀) = (A) = (A₁),where A_k is the generator of T_k, k = 0, Ø, 1.     

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.     

Automorphisms of Relatively Free Groups in the Variety N2A {wedge} AN2 {wedge} Nc

For positive integers n and c, with n 2, let G_{n, c} be a relativelyfree group of finite rank n in the variety N₂A AN₂ N_c. Itis shown that the subgroup of the automorphism group Aut(G_n,c) of G_{n, c} generated by the tame automorphisms and an explicitlydescribed finite set of IA-automorphisms of G_{n, c} has finiteindex in Aut(G_{n, c}). Furthermore, it is proved that there areno non-trivial elements of G_{n, c} fixed by every tame automorphismof G_{n, c}.     

On the Discreteness and Convergence in n-Dimensional Mobius Groups

Throughout this paper, we adopt the same notations as in [1,6, 8] such as the Möbius group M(Rⁿ), the Clifford algebraC_n–1, the Clifford matrix group SL(2, _n), the Cliffordnorm of ||A||=(|a|²+|b|²+|c|²+|d|²) (1) and the Clifford metric of SL(2, _n) or of the Möbius groupM(Rⁿ) d(A₁,A₂)=||A₁–A₂||(|a₁–a₂|²+|b₁–b₂|²+|c₁–c₂|²+|d₁–d₂|²)(2) where |·| is the norm of a Clifford number and represents f_i M(), i = 1,2, and so on. In addition, we adopt some notions in [6, 12]:the elementary group, the uniformly bounded torsion, and soon. For example, the definition of the uniformly bounded torsionis as follows.