K-theory of twisted differential operators on flag varieties

Let be a semisimple Lie algebra overk, an algebraically closed field of characteristic zero, and let be a Cartan subalgebra inside a Borel subalgebra of . LetU be the enveloping algebra of . For letM() denote the corresponding Verma modúle and letU _u=U/AnnM(). LetW be the Weyl group and letW ⁰ be the stabiliser of inW. We prove the following theorem, which affirms a conjecture of T.J. Hodges.Oblatum 16-XII-1994     

Das Zeitverhalten von einfachen offenen exponentiellen Warteschlangensystemen mit unendlich vielen Warteplätzen

Zusammenfassung Die zeitabhängige (instationäre) Lösung für die Zustandswahrscheinlichkeiten und für einige Kenngrößen von Warteschlangensystemen mit einer Bedienungsstation, unendlich vielen Warteplätzen, exponentiellem Zu- und Abgang und beliebigem Anfangszustand wird bestimmt. Die ZustandswahrscheinlichkeitenP _v (), d. h. die Wahrscheinlichkeiten für Einheiten im System zur Zeit, ergeben sich als Integrale, in denen modifizierteSessel-Funktionen 1. Art auftreten. Der ErwartungswertL () und die VarianzV() der Zahl von Einheiten im System lassen sich als Integrale darstellen, in denen nur die ZustandswahrscheinlichkeitP ₀() auftritt.Für<1 und erreichen die Systeme einen stationären Zustand (für den die Lösung bekannt ist); für1 und giltP _v ()0 für alle, L(),V().Ist>1, dann wachsenL() undV() für große linear mit; ihre Asymptoten werden berechnet. Ist=1, dann wachsenL() und die Standardabweichung() für große mit ; einfache Näherungsformeln werden gefunden.

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Summary The time dependent solution is determined for the state probabilities and for some characteristic values of queuing systems with a single server, an infinite number of waiting places, exponentially distributed inter-arrival and service times, and any initial state. The state probabilitiesP _v (), i.e. the probabilities for units in the system at time, are given in the form of integrals in which modifiedBessel functions of the first kind occur. Integrating the state probalityP ₀() over leads to the meanL() and the varianceV() of the number of units in the system.For<1 and the systems tend to a steady state (for which the solution is known); for1 and we haveP _v ()0 for all, L(),V().If>1 asymptotic expansions for large are found givingL() andV() proportional to. If=1 simple approximate formulas for large are obtained givingL() and the standard deviation() proportional to .

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Vorgel. v.:J. Nitsche.     

On inertial wave and flow structure at low Rossby number

Flow disturbances produced by the slow relative motion of an impermeable body in a large, rapidly rotating vessel are studied as an asymptotic theory for an inviscid, incompressible fluid at a small Rossby number (u _c /L1). The axial distance between the vessel wallsH is assumed to be so much larger than the body scaleL that the reduced height H/L is of unit order or greater. This flow admits a columnar structure near the body and an outer nonlinear structure of the inertial-wave type far above the forcing region, at distances along the rotation-axis (z=0(L/)). The inner boundary condition for the outer problem is provided by transferring the impermeable surface condition through the columnar structure; the outer solution in turn determines the vorticity and the solution of the columnar inner region.For thin obstacles or shallow ground topography (1), the outer solution is governed by an equation comparable to a linear one for inertial waves. The linear solutions obtained for steady transverse motion in an infinite domain () shows that, in this case, surface (perturbation) velocities are orthogonal to those in an irrotational (non-rotating) flow over the same ground topography. In the far field, i.e., ( z/L l) disturbances are confined mainly behind a wedge-shaped caustic front downstream of the rotation axis , wherein their amplitudes are comparable to , in general accord with Lighthill's result from group-velocity consideration. The field behind the caustic supports, however, lee waves extending far downstream with diminishing strength. Their wave lengths belong to orderLz/x, and therefore these waves should appear to be densely packed in the wedge-shaped region. The question on tilting of the Taylor column is delineated; the structure of the caustic zone and its upstream flow behavior are also analyzed.

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Zusammenfassung Die Strömung um einen Körper, der sich langsam in einem grossen, rotierenden Behälter bewegt, wird untersucht mit einer asymptotischen Theorie für eine reibungsfreie, inkompressible Flüssigkeit bei einer kleinen Rossby-Zahl, d.h. u _c /L1. Der axiale AbstandH zwischen den Gefässwänden wird als soviel grösser als die Körperabmessung angenommen, dass für die reduzierte Höhe =H/L 0(1) gilt. Diese Strömung erlaubt eine säulenähnliche Struktur (Taylor column) in Körpernähe und eine äussere, nichtlineare Struktur vom Wellentyp für Höhenz=0(L/) über dem Köper. Die innere Randbedingung für das äussere Problem wird erhalten, indem die Bedingung an der Wand durch die Säule verschoben wird. Die äussere Lösung bestimmt ihrerseits die Rotation und damit die Lösung im inneren, säulenartigen Bereich.Wenn Körper oder Bodenform flach sind (1), so wird die Aussenlösung durch eine Gleichung bestimmt, welche vergleichbar ist mit einer linearen Gleichung für Trägheitswellen. Lineare Lösungen werden für gleichförmige Bewegung von der Achse weg in einem unbegrenzten Bereich ) gegeben. Sie zeigen, dass die Oberflächenstromlinien orthogonal zu den Stromlinien sind, die bei rotationsfreier (nichtdrehender) Strömung über der gleichen Bodenform entstehen. Im Fernfeld ( z/L l) sind Störungen hauptsächlich auf ein keilförmiges Gebiet stromabwärts von der Rotationsachse ( ) begrenzt. In diesem Gebiet geht ihre Amplitude mit , in Uebereinstimmung mit Lighthill's Resultat, das mit der Gruppengeschwindigkeit hergeleitet wurde. Im keiförmigen Gebiet bestehen drei Familien von Lee-Wellen, welche sich mit unveränderter Stärke weit stromabwärts erstrecken. Ihre Wellenlängen gehören zur OrdnungLx/z, weshalb die Wellen im keilförmigen Bereich dicht gepackt erscheinen. Die Frage der Neigung der Taylor Säule behandelt und die Struktur des Randes vom keilförmigen Bereich wird analysiert.

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This work is dedicated to Professor Nicholas Rott on the occasion of his sixtieth birthday.     

The Laplacian on $$ C(\overline \Omega) $$ with generalized Wentzell boundary conditions

In this note we prove that the Laplacian with generalized Wentzell boundary conditions on an open bounded regular domain in defined by generates an analytic semigroup of angle on for every > 0 and (for the definition of cf. (1.3)).Received: 13 July 2002     

On a Singular Semilinear Elliptic Boundary Value Problem and the Boundary Harnack Principle

On a bounded C ²-domain we consider the singular boundary-value problem 1/2u=f(u) in D, u _D =, where d3, f:(0,)(0,) is a locally Hölder continuous function such that f(u) as u0 at the rate u ^–, for some (0,1), and is a non-negative continuous function satisfying certain growth assumptions. We show existence of solutions bounded below by a positive harmonic function, which are smooth in D and continuous in . Such solutions are shown to satisfy a boundary Harnack principle.     

Double Extensions of Dynamical Systems and Construction of Mixing Filtrations. II. Quasihyperbolic Toral Automorphisms

Let T be an automorphism (an invertible measure preserving transformation) of a probability space and let U be a unitary operator on defined by . Let and be the generators of symmetric Markov transition semigroups acting on L₂. It is assumed that A_s and A_u satisfy the relations U_-1 A_s U =^-1 A_sand U^-1 A_u U A_u for some > 1. A nonnegative self-adjoint operator A on L₂ with the properties UA=AU and A_u+A_s A is called a T-invariant minorant for (A_u, A_s). Assuming that A_u and A_s commute, certain assumptions on a function f L₂ in terms of such a minorant are proposed under which the sequence (f T^k, k Z) satisfies the functional form of the central limit theorem and the law of the iterated logarithm. A special case of these assumptions was considered in an earlier paper by the author. Quasihyperbolic toral automorphisms are considered as an application. Bibliography: 9 titles.     

The Busy Period of the M/GI/∞ Queue

An equation for the distribution Z() of the duration T of the busy period in a stationary M/GI/ service system is constructed from first principles. Two scenarios are examined, being distinguished by the half-plane Re()>₀ for some ₀0 in which the generic service time random variable S, always assumed to have a finite mean E(S), has an analytic Laplace–Stieltjes transform E(e^–S ). If ₀<0 then E(e^–T ) is analytic in a half-plane (₁,), where ₀₁<0 and ₁ is determined by the distribution of S; then for any 0<s<|₁|.When ₀=0, E(e^–T ) is analytic in (0,), and now more is known about T. Inequalities on the tail () are used to show that for any 1, E(T ) is finite if and only if E(S ) is finite. It follows that the point process consisting of the starting epochs of busy periods is long range dependent if and only if E(S ²)=, in which case it has Hurst index equal to [frac12](3–), where is the moment index of S.If also the tail (x)=Pr{Sx} of the service time distribution satisfies the subexponential density condition ₀ ^x (x–u) (u)du/ (x)2E(S) as x, then (x)/ (x)e^E(S), where is the arrival rate.     

The Reynolds number and large time behaviour for weak solutions of the Navier-Stokes equations

We consider time-dependent perturbationsu of R. Finn's stationary PR-solution of the Navier-Stokes equations, which converges to a constant vectorv as ¦x¦ .We investigate the large time behaviour ofu in the case, that the (energetic) Reynolds numberR _e of has the critical valueR _e =1. We show that perturbationsu with small initial kinetic energy and small driving force become smooth after finite time.     

On singular perturbation of a semilinear hyperbolic equation

We consider a hyperbolic version of Eells-Sampson's equation: . This equation is semilinear with respect to space derivative and time derivative. Letu (x) be the solution with initial data u(0) and (0), and putv (t,x)=u (t,x). We show that when the resistance ,V (t,x) converges to a solution of the original parabolic Eells-Sampson's equation: . Note thatv _t(0)= (0) diverges when . We show that this phenomena occurs in more general situations.This article was processed by the author using the Springer-Verlag Pjourlg macro package.     

Gauge Theory Solitons on the Noncommutative Cylinder

The solution-generating technique originally suggested for gauge theories on the noncommutative plane is generalized to the noncommutative cylinder. For this, we construct partial isometry operators and a complete set of orthogonal projection operators in the algebra _C of the cylinder, and an isomorphism between the free module _C and its direct sum _{C C} with the Fock module on the cylinder. We explicitly construct the gauge theory soliton and evaluate the spectrum of perturbations about this soliton.     

Acceleration of Superlinearly Convergent Sequences

A method analogous to the Aitken extrapolation is proposed to accelerate the convergence of sequences of real or complex numbers with asymptotic behavior where e _i is the error of the ith element of the sequence, d _j1 for all 1jr, and d_j > 1. The R-order of the resulting sequences is computed using methods which are also of independent interest.     

Besov Regularity of Stochastic Integrals with Respect to the Fractional Brownian Motion with Parameter H > 1/2

Let {B _t ,t[0,1]} be a fractional Brownian motion with Hurst parameter H > 1/2. Using the techniques of the Malliavin calculus we show that the trajectories of the indefinite divergence integral ^t ₀ u _s B _s belong to the Besov space _p,q for all , provided the integrand u belongs to the space . Moreover, if u is bounded and belongs to for some even integer p2 and for some large enough, then the trajectories of the indefinite divergence integral ^t ₀ u _s B _s belong to the Besov space _p, ^H .     

Maximum Norm Resolvent Estimates for Elliptic Finite Element Operators

We study a finite element approximation A _h, based on simplicial Lagrange elements, of a second order elliptic operator A under homogeneous Dirichlet boundary conditions in two and three dimensions, where h is thought of as a meshsize. The main result of the paper is a new resolvent estimate for the operator A _h in the L -norm. This estimate is uniform with respect to h for the case with at least quadratic elements. In the case with linear elements, the estimate contains on the right a factor proportional to (log log ), where = 1 or = in two or three dimensions, respectively.This revised version was published online in October 2005 with corrections to the Cover Date.     

Quantum Integrable and Nonintegrable Nonlinear Schrödinger Models for Realizable Bose–Einstein Condensation in d+1 Dimensions (d = 1, 2, 3)

We evaluate finite-temperature equilibrium correlators for thermal time ordered Bose fields to good approximations by new methods of functional integration in d=1,2,3 dimensions and with the trap potentials V(r)0. As in the translationally invariant cases, asymptotic behaviors fall as to longer-range condensate values for and only for d=3 in agreement with experimental observations; but there are generally significant corrections also depending on due to the presence of the traps. For d=1, we regain the exact translationally invariant results as the trap frequencies 0. In analyzing the attractive cases, we investigate the time-dependent c-number Gross–Pitaevskii (GP) equation with the trap potential for a generalized nonlinearity –2c||²ⁿ and c<0. For n=1, the stationary form of the GP equation appears in the steepest-descent approximation of the functional integrals. We show that collapse in the sense of Zakharov can occur for c=0 and nd2 and a functional E _NLS[]0 even when V(r)0. The singularities typically arise as -functions centered on the trap origin r=0.     

L p -estimates for the solutions of second order elliptic equations

Summary We investigate the homogeneous Dirichlet problem in H^2,p for a second order elliptic partial differential equation in nondivergence form Lu=f in the case in which the leading coefficients of L belong to H^1,n(), Rⁿ. We prove that if p belongs to a suitable neighbourhood of 2, then the above problem, has a unique solution u satisfying D²u_p Cf_p; furthermore, if f H^k,p, k=1,2, ..., and the coefficients of L satisfy some natural conditions, then the solution satisfies .Lavoro eseguito nell'ambito del gruppi 40% e 60% del M.P.I.     

Designs and partial geometries over finite fields

Let be a finite field, and let (, B) be a nontrivial 2-(n, k, 1)-design over . Then each point induces a (k–1)-spread S on /. (, B) is said to be a geometric design if S is a geometric spread on / for each . In this paper, we prove that there are no geometric designs over any finite field .Research partially supported by NSF grant DMS-8703229.     

The Eigenvalues of Very Sparse Random Symmetric Matrices

Let W _n be an n × n random symmetric sparse matrix with independent identically distributed entries such that the values 1 and 0 are taken with probabilities p/n and 1-p/n, respectively; here is independent of n. We show that the limit of the expected spectral distribution functions of W _n has a discrete part. Moreover, the set of positive probability points is dense in (- +). In particular, the points , and 0 belong to this set.     

Brownian processes in infinite dimension

LetE be a locally convex space endowed with a centered gaussian measure . We construct a continuousE-valued brownian motionW _t with covariance . The main goal is to solve the SDE of Langevin type dX _t= dW _t–AX _t wherea andA are unbounded operators of the Cameron-Martin space of (E, ). It appears as the unique linear measurable extension of the solution of the classical Cauchy problemv(t)= u–Av(t).     

The local stability of positive solutions to the Hammerstein equation with a nonmonotonic nemytskii operator

LetE be a Banach lattice which consists of some functionsu: over a fixed domain . This article is concerned with the local stability of a non-zero positive solutionu ^*E to the Hammerstein equationu(x)= x, wherek0, andf:×[0, )[0, ) is not necessarily increasing in the second variable. It is assumed thatf(x, 0)=0 and forx,u>0. Under some mild additional hypotheses onE, k, andf it is proved that the spectral radius of the Fréchet derivative atu ^* of the Hammerstein operator is less than one. Also the impact of local stability on secondary bifurcations is investigated. The proof of the main result is based on the spectral theory for completely continuous and irreducible positive operators on Banach lattices.     

On convergence of types and processes in Euclidean space

Let be an Euclidean space; Y _n , Z, U random vectors in ; h _n , g _n affine transformations and let þ be a subgroup of the group G of all the in vertible affine transformations, closed relative to G. Suppose that g_n and where Z is nonsingular. The behaviour of _n = h _n g _n ^–1 as n is discussed first. The results are used then to prove that if for all t(0, ), where h _n þ and Z ₁ is nonsingular and nonsymmetric with respect to þ then H, for all t(0,) and is a continuous homomorphism of the multiplicative group of (0, ) into þ. The explicit forms of the possible are shown.