Feynman’s operational calculi for noncommuting operators: The monogenic calculus

In recent papers the authors presented their approach to Feynman’s operational calculi for a system of not necessarily commuting bounded linear operators acting on a Banach space. The central objects of the theory are the disentangling algebra, a commutative Banach algebra, and the disentangling map which carries this commutative structure into the noncommutative algebra of operators. Under assumptions concerning the growth of disentangled exponential expressions, the associated functional calculus for the system of operators is a distribution with compact support which we view as the joint spectrum of the operators with respect to the disentangling map. In this paper, the functional calculus is represented in terms of a higher-dimensional analogue of the Riesz-Dunford calculus using Clifford analysis.     

Density Conditions for Panchromatic Colourings of Hypergraphs

Let be a hypergraph. A panchromatic t-colouring of is a t-colouring of its vertices such that each edge has at least one vertex of each colour; and is panchromatically t-choosable if, whenever each vertex is given a list of t colours, the vertices can be coloured from their lists in such a way that each edge receives at least t different colours. The Hall ratio of is . Among other results, it is proved here that if every edge has at least t vertices and whenever , then is panchromatically t-choosable, and this condition is sharp; the minimum such that every t-uniform hypergraph with is panchromatically t-choosable satisfies ; and except possibly when t = 3 or 5, a t-uniform hypergraph is panchromatically t-colourable if whenever , and this condition is sharp. This last result dualizes to a sharp sufficient condition for the chromatic index of a hypergraph to equal its maximum degree. Received November 10, 1998 RID="*" ID="*" This work was carried out while the first author was visiting Nottingham, funded by Visiting Fellowship Research Grant GR/L54585 from the Engineering and Physical Sciences Research Council. The work of this author was also partly supported by grants 96-01-01614 and 97-01-01075 of the Russian Foundation for Fundamental Research.     

Parabolic boundary integral operators symbolic representation and basic properties

In this paper we develop a theory of parabolic pseudodifferential operators in anisotropic spaces. We construct a symbolic calculus for a class of symbols globally defined on ⁿ⁺¹× ⁿ⁺¹, and then develop a periodisation procedure for the calculus of symbols on the cylinder ×. We show Gårding's inequality for suitable operators and precise estimates for the essential norm in anisotropic Sobolev spaces. These new mapping properties are needed in localization arguments for the analysis of numerical approximation methods.     

Dirichlet operators and the positive maximum principle

Let (A, D(A)) denote the infinitesimal generator of some strongly continuous sub-Markovian contraction semigroup onL ^p (m), p1 andm not necessarily -finite. We show under mild regularity conditions thatA is a Dirichlet operator in all spacesL ^q (m), qp. It turns out that, in the limitq,A satisfies the positive maximum principle. If the test functionsC _c D(A), then the positive maximum principle implies thatA is a pseudo-differential operator associated with a negative definite symbol, i.e., a Lévy-type operator. Conversely, we provide sufficient criteria for an operator (A, D(A)) onL ^p(m) satisfying the positive maximum principle to be a Dirichlet operator. If, in particular,A onL ² (m) is a symmetric integro-differential operator associated with a negative definite symbol, thenA extends to a generator of a regular (symmetric) Dirichlet form onL ² (m) with explicitly given Beurling-Deny formula.     

A Note on π-Engel Conditions

In this paper we show that the weakly -Engel conditions are closely related to the existance of normal -complements; while the -Engel conditions are closely related to the -nilpotent groups.AMS Subject Classification (2000): 20D20     

Some existence and nonexistence theorems for compact graphs of constant mean curvature with boundary in parallel planes

Given H≥0 and bounded convex curves α₁, ...,⇌_n, α in the plane z=0 bounding domains D₁, …, D_n, D, respectively, with if i ∈ j and with D_i ⊂ D, we obtain several results proving the existence of a constanth depending only on H and on the geometry of the curves α_i, α such that the Dirichlet problem for the constant mean curvature H equation: where may accept or not a solution.     

On hypersurfaces inR n+1 with integral bounds on curvature

We show that the L ^p norm of the second fundamental form of hypersurfaces in R ⁿ⁺¹ is very coercive in the GMT setting of Gauss graphs currents, when p exceeds the dimension n. A compactness result for immersed hypersurfaces and its application to a variational problem are provided.     

Note on hit-and-miss topologies

This is a continuation of [19]. We characterize first and second countability of the general hit-and-miss hyperspace topologyτ ₊ ^Δ for weakly-R ₀ base spaces. Further, metrizability ofτ ₊ ^Δ is characterized with no preliminary conditions on the base space and the generating family of closed sets and a new proof on uniformizability (i.e. complete regularity) ofτ ₊ ^Δ is given in this general setting, thus generalizing results of [3], [5] and [6].     

Exact Internal Controllability of Maxwell's Equations

In this paper we obtain two exact internal controllability results of Maxwell's equations in a general region by using multiplier techniques. The first one is exact controllability in a short time, in which we obtain the ``optimal" (observability) estimates when the location and the shape of the controller is fixed. What happens if we allow the controller to change? Under some conditions, we show that by doing that the system can be exactly controllable within any given time duration, which is our second exact controllability result. Accepted 30 September 1998     

An extension of Hawkes theorem on the Hausdorff dimension of a Galton–Watson tree

Let ? be the genealogical tree of a supercritical multitype Galton–Watson process, and let Λ be the limit set of ?, i.e., the set of all infinite self-avoiding paths (called ends) through ? that begin at a vertex of the first generation. The limit set Λ is endowed with the metric d(ζ, ξ) = 2 ⁻ⁿ where n = n(ζ, ξ) is the index of the first generation where ζ and ξ differ. To each end ζ is associated the infinite sequence Φ(ζ) of types of the vertices of ζ. Let Ω be the space of all such sequences. For any ergodic, shift-invariant probability measure μ on Ω, define Ω_μ to be the set of all μ-generic sequences, i.e., the set of all sequences ω such that each finite sequence v occurs in ω with limiting frequency μ(Ω(v)), where Ω(v) is the set of all ω′?Ω that begin with the word v. Then the Hausdorff dimension of Λ∩Φ⁻¹ (Ω_μ) in the metric d is