Nest Representations of Directed Graph Algebras

This paper is a comprehensive study of the nest representationsfor the free semigroupoid algebra L_G of a countable directedgraph G as well as its norm-closed counterpart, the tensor algebraT⁺(G). We prove that the finite-dimensional nest representations separatethe points in L_G, and a fortiori, in T⁺(G). The irreduciblefinite-dimensional representations separate the points in L_Gif and only if G is transitive in components (which is equivalentto being semisimple). Also the upper triangular nest representationsseparate points if and only if for every vertex x T(G) supportinga cycle, x also supports at least one loop edge. We also study faithful nest representations. We prove that L_G(or T⁺(G) admits a faithful irreducible representation if andonly if G is strongly transitive as a directed graph. More generally,we obtain a condition on G which is equivalent to the existenceof a faithful nest representation. We also give a conditionthat determines the existence of a faithful nest representationfor a maximal type N nest. 2000 Mathematics Subject Classification47L80, 47L55, 47L40.     

Two Models that show the Interpolation Theorem Fails in all L1(Q{alpha}) and L1, 1 (Q{alpha}, {alpha} = 0, 1, 2,...

The logics L¹(Q), L^1,1(Q) and L²(Q) are formed by adding quantifiersQ, Q^1,1 and Q² respectively to the first-order logic. In thispaper, for each ordinal (including = 0), we construct twoQ models to prove that the Interpolation Theorem fails in L(Q)and L^1,1 (Q).     

Isomorphisms between Semisimple Weighted Measure Algebras

It is shown that, if is an isomorphism between semisimple weightedmeasure algebras M(w₁) and M(w₂), then maps L¹(R⁺, w₁) ontoL¹(R⁺, w₂). This is used to describe all the automorphisms ofM(R⁺, w). A necessary and sufficient condition is given forM(w₁) and M(w₂) to be isomorphic.     

A Note on a Maximal Function Over Arbitrary Sets of Directions

Let M_S be the universal maximal operator over unit vectors ofarbitrary directions. This operator is not bounded in L²(R²).We consider a sequence of operators over sets of finite equidistributeddirections converging to M_S. We provide a new proof of N. Katz'sbound for such operators. As a corollary, we deduce that M_Sis bounded from some subsets of L² to L². These subsets arecomposed of positive functions whose Fourier transforms havea logarithmic decay or which are supported on a disc. 1991 MathematicsSubject Classification 42B25.     

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.     

Link Cobordism and the Intersection of Slice Discs

We work in the smooth category. An (oriented) (ordered) m-component n-(dimensional) link isa smooth oriented submanifold L = {K₁, ..., K_m} of Sⁿ⁺² whichis the ordered disjoint union of m manifolds, each PL-homeomorphicto the standard n-sphere. If m = 1, then L is called a knot. We say that m-component n-dimensional links L₀ and L₁ are (link-)concordantor (link-)cobordant if there is a smooth oriented submanifoldC = {C₁, ..., C_m} of Sⁿ⁺² x [0, 1] which meets the boundarytransversely in C, is PL-homeomorphic to L₀ x [0, 1], and meetsSⁿ⁺² x {l} in L_l (l = 0, 1). If m = 1, then we say that n-knotsL₀ and L_l are (knot-)concordant or (knot-)cobordant. Then wecall C a concordance-cylinder of the two n-knots L₀ and L_l. If an n-link L is concordant to the trivial link, then we callL a slice link. If an n-link L = {K₁, ..., K_m} Sⁿ⁺² = Bⁿ⁺³ Bⁿ⁺³ is slice,then there is a disjoint union of (n + 1)-discs in Bⁿ⁺³ such that is called a set of slice discs for L. If m = 1, then is called a slice disc for the knotL. 1991 Mathematics Subject Classification 57M25, 57Q45.     

Decay rates of solutions to thermoviscoelastic plates with memory

We consider the thermoelastic plate under the presence of along range memory. We find uniform rates of decay (in time)of the energy, provided that suitable assumptions on the relaxationfunctions are given. Namely, if the relaxation decays exponentiallythen the first order energy also decays exponentially. Whenthe relaxation g satisfies -c₁g(t)^1+1/_p g'(t) -c_og(t)^1+1/_p; and g,g^1-1/_p L¹ (R) withp > 2 then the energy decays as 1/(1+t)^p. A new Liapunov functionalis built for this problem.     

Convergence of Finite-Difference Schemes for Elliptic Equations with Variable Coefficients

We study the convergence of finite-difference schemes for second-orderelliptic equations with variable coefficients. We prove thatthe convergence rate in the discrete W₂¹ norm is of the orderh^{s –1} if the solution of the boundary value problem belongsto the Sobolev space W₂^s (1 < s 3).     

A Version of the Poincare-Birkhoff-Witt Theorem

In this note I shall prove that if L is a finite-dimensionalLie algebra over a field F of characteristic zero which is generatedas an algebra by a set of elements {e₁, e₂,...,e_k}, then theuniversal enveloping algebra U(L) of L is linearly generatedby monomials spanned by the elements {e_i} of an a priori boundedwidth. As an application, a criterion of Kostant for a leftideal of U(L) to be of finite codimension is proved by purelyalgebraic means.     

The Fast Numerical Solution of Very Large Elliptic Difference Schemes

Let L_kv_k = g_k be a system of difference equations discretizingan elliptic boundary value problem. Assume the system to be"very large", that means that the number of unknowns exceedsthe capacity of storage. We present a method for solving theproblem with much less storage requirement. For two-dimensionalproblems the size of the needed storage decreases from O(h^–2)to (or even O(h^–5/4)). The computational work increasesonly by a factor about six. The technique can be generalizedto nonlinear problems. The algorithm is also useful for computerswith a small number of parallel processors.     

The Effect of Sidewalls on Overstable Convection in a Rotating Fluid

A simple two-dimensional model is used to demonstrate some interestingeffects which arise when Chandrasekhar's (1962) theory of overstableconvection in an infinite rotating fluid layer is modified totake account of lateral walls. The aim of the investigationis to determine how sidewalls aligned with the convective rollsaffect the critical Rayleigh number and frequency of oscillationand also how the overstable eigensolutions are related to thepreviously determined stationary solutions of the equations(Daniels, 1977). For containers of large aspect ratio, L, thecritical Rayleigh number for overstability is R_o+O(L^–1)(where R_o is the value for the infinite layer) and in the neighbourhoodof this single perturbed value it is found that there is aninfinite spectrum of overstable eigenvalues with frequencieswhich differ by O(L^–1). The O(L^–1) correction toR_o is determined analytically for the case of small Prandtlnumber and rapid rotation.     

Convex Bodies with Homothetic Sections

We prove that if K is a convex body in Eⁿ⁺¹, n2, and p₀ is apoint of K with the property that all n-sections of K throughp₀ are homothetic, then K is a Euclidean ball.     

Harmonic morphisms on heaven spaces

We prove that any (real or complex) analytic horizontally conformalsubmersion from a three-dimensional conformal manifold (M³,c_M) to a two-dimensional conformal manifold (N², c_N) can be,locally, ‘extended’ to a unique harmonic morphismfrom the (eaven)-space (H⁴, g) of (M³, c_N) to (N², c_N). Moreover,any positive harmonic morphism with two-dimensional fibres from(H⁴, g) is obtained in this way.     

Sharp Results Concerning the Expression of Functions as Sums of Finite Differences

Let * denote convolution and let _x denote the Dirac measureat a point x. A function in L²(R)) is called a difference oforder 1 if it is of the form g-_x * g for some x R and g L²(R)).Also, a difference of order 2 is a function of the form for some x R and g L²(R)). In fact,the concept of a ‘difference of order s’ may bedefined in a similar manner for each s 0. If f denotes the Fouriertransform of f, it is known that a function f in L²(R)) is afinite sum of differences of order s if and only if , and the vector space of all suchfunctions is denoted by D_s (L²(R)). Every function in D_s (L²(R))is a sum of int(2s) + 1 differences of order s, where int(t)denotes the integer part of t. Thus, every function in D₁ (L²(R))is a sum of three first order differences, but it was provedin 1994 that there is a function in D₁ (L(R)) which is neverthe sum of two first order differences. This complemented, forthe group R, the corresponding result for first order differencesobtained by Meisters and Schmidt in 1972 for the circle group.The results show that there is a function in L₂ R such that,for each s 1/2, this function is a sum of int (2s) + 1 differencesof order s but it is never the sum of int (2s) differences oforder s. The proof depends upon extending to higher dimensionsthe following result in two dimensions obtained by Schmidt in1972 in connection with Heilbronn's problem: if x₁, x__n arepoints in the unit square, Following on from the work of Meisters and Schmidt, this workfurther develops a connection between certain estimates in combinatorialgeometry and some questions of sharpness in harmonic analysis.2000 Mathematics Subject Classification 42A38 (primary), 52A40(secondary).     

An Application of the Fefferman-Stein Inequality to the Study of the Tent Spaces

The purpose of this note is to show how a result on tent spacesproved by Coifman, Meyer and Stein, namely the L^p-norm relationshipbetween the functionals A_q and C_q appearing in the definitionof such spaces, can be derived easily from the Fefferman-SteinL^p-inequality for the sharp maximal function.     

On the L-Function of the Curves y2 = x5 + A

Let C_A/Q be the curve y² = x⁵ + A, and let L(s, J_A) denote theL-series of its Jacobian. Under the assumption that the signin the functional equation for L(s, J_A) is +1, the criticalvalue L(1, J_A) is evaluated in terms of the value of a thetaseries for depending on Aat a complex multiplication point coming from Q(₅).     

Rigidity of Continuous Coboundaries

We consider the functional equation F_oT–F=f, where T isa measure-preserving transformation and f is a continuous function.We show that if there is an L function F which satisfies thisequation, then F is constrained to satisfy a number of regularityconditions, and, in particular, if T is a one-sided Bernoullishift, then we show that there is a continuous function F satisfyingthis equation. We show that this is not the case for the two-sidedshift. 1991 Mathematics Subject Classification 28D05, 58F11.     

Isomorphisms of Group Algebras

Let G₁ and G₂ be locally compact groups. If T is an algebraisomorphism of L¹(G₁) onto L¹(G₂) with ||T|| (1+3), then G₁and G₂ are isomorphic. This improves on earlier results, and,in a certain sense, is best possible. However, the main conjecturethat the groups are isomorphic if ||T|| < 2 remains unsolvedexcept for abelian groups and for connected groups. Similarresults are given for the measure algebra M(G) and for the algebraC(G) of continuous functions when the group G is compact.     

A Class of Infinite Dimensional Simple Lie Algebras

Let A be an abelian group, F be a field of characteristic 0,and , ß be linearly independent additive maps fromA to F, and let ker()\{0}. Then there is a Lie algebra L = L(A,, ß, ) = _xA Fe_x under the product [e_x, e_y]]=(x–y)e_x+y+(ß) (x, y) e_x+y–. If, further, ß() = 1, and ß(A) = Z, thereis a subalgebra L⁺:=L(A⁺, , ß, ) = _xA⁺ Fe_x, whereA⁺ = {xA|ß(x)0}. The necessary and sufficient conditionsare given for L' = [L, L] and L⁺ to be simple, and all semi-simpleelements in L' and L⁺ are determined. It is shown that L' andL⁺ cannot be isomorphic to any other known Lie algebras andL' is not isomorphic to any L⁺, and all isomorphisms betweentwo L' and all isomorphisms between two L⁺ are explicitly described.     

Minimal Determinants and Lattice Inequalities

Some results of P. McMullen on determinants of sublattices ofZ^d induced by rational subspaces are generalized to arbitrarylattices. As an application, we obtain an equality for the minimaldeterminants introduced by J. M. Wills, namely D_t(L) = D_d(L)D_d–1((L*).Using an inequality of Lagarias, Lenstra and Schnorr, we generalizetwo isoperimetric inequalities withlattice constraints by Bokowski,Hadwiger and Wills, and Hadwiger, respectively, to arbitrarylattices.