A fixed point localization formula for the Fourier transform of regular semisimple coadjoint orbits

Let be a Lie group acting on an oriented manifold M, and let ω be an equivariantly closed form on M. If both and M are compact, then the integral is given by the fixed point integral localization formula (Theorem 7.11 in Berline et al. Heat Kernels and Dirac Operators, Springer, Berlin, 1992). Unfortunately, this formula fails when the acting Lie group is not compact: there simply may not be enough fixed points present. A proposed remedy is to modify the action of in such a way that all fixed points are accounted for.Let be a real semisimple Lie group, possibly noncompact. One of the most important examples of equivariantly closed forms is the symplectic volume form dβ of a coadjoint orbit Ω. Even if Ω is not compact, the integral exists as a distribution on the Lie algebra . This distribution is called the Fourier transform of the coadjoint orbit.In this article, we will apply the localization results described in [L1,L2] to get a geometric derivation of Harish-Chandra's formula (9) for the Fourier transforms of regular semisimple coadjoint orbits. Then, we will make an explicit computation for the coadjoint orbits of elements of which are dual to regular semisimple elements lying in a maximally split Cartan subalgebra of .     

Fixed points of dual quantum operations

Let ?A be a normal completely positive map on B(H) with Kraus operators . Denote M the subset of normal completely positive maps by . In this note, the relations between the fixed points of ?A and are investigated. We obtain that , where K(H) is the set of all compact operators on H and is the dual of ?A∈M. In addition, we show that the map is a bijection on M.     

Integrals of equivariant forms and a Gauss-Bonnet theorem for constructible sheaves

The Berline-Vergne integral localization formula for equivariantly closed forms ([N. Berline, M. Vergne, Classes caractéristiques équivariantes. Formules de localisation en cohomologie équivariante, C. R. Acad. Sci. Paris 295 (1982) 539-541], Theorem 7.11 in [N. Berline, E. Getzler, M. Vergne, Heat Kernels and Dirac Operators, Springer-Verlag, 1992]) is well-known and requires the acting Lie group to be compact. In this article, we extend this result to real reductive Lie groups G_R.As an application of this generalization, we prove an analogue of the Gauss-Bonnet theorem for constructible sheaves. If F is a G_R-equivariant sheaf on a complex projective manifold M, then the Euler characteristic of M with respect to F

     

Approximating Fourier transformation of orbital integrals

In this paper, we are concerned with orbital integrals on a class of real reductive Lie groups with non-compact Iwasawa K-component. The class contains all connected semisimple Lie groups with infinite center. We establish that any given orbital integral over general orbits with compactly supported continuous functions for a group G in is convergent. Moreover, it is essentially the limit of corresponding orbital integrals for its quotient groups in Harish-Chandra's class. Thus the study of orbital integrals for groups in class reduces to those of Harish-Chandra's class. The abstract theory for this limiting technique is developed in the general context of locally compact groups and linear functionals arising from orbital integrals. We point out that the abstract theory can be modified easily to include weighted orbital integrals as well. As an application of this limiting technique, we deduce the explicit Plancherel formula for any group in class .     

An answer to a question by Herings et al.

We answer an open question by Herings et al. [J.J. Herings, G. van der Laan, D. Talman, Z. Yang, A fixed point theorem for discontinuous functions, Operations Research Letters 36 (1) (2008) 89–93], by proving that their fixed point theorem for discontinuous functions works for mappings defined on convex compact subset of , and not only polytopes. This rests on a fixed point result of Toussaint [S. Toussaint, On the existence of equilibria in economies with infinitely many commodities and without ordered preferences, Journal of Economic Theory 33 (1) (1984) 98–115].     

Positive solutions for 2p-order and 2q-order nonlinear ordinary differential systems

In this paper, by using Krasnosel'skii fixed point theorem and under suitable conditions, we present the existence of single and multiple positive solutions to the following systems:

     

Existence of positive solutions for second-order boundary value problem with one parameter

In the case of not requiring f(t,u) to be nonnegative, by transforming the boundary value problem into the integral equation system, and applying the fixed point index theory, the author studies the following second-order boundary value problem with one parameter

     

Twin solutions to singular semipositone problems

In this paper, by a specially constructed cone and the fixed point index theory, we investigate the existence of multiple positive solutions for the following singular semipositone problem:

     

Positive solutions of a nonlinear four-point boundary value problems in Banach spaces

In this paper, by using the fixed points of strict-set-contractions, we study the existence of at least one or two positive solutions to the four-point boundary value problem

     

Symmetric solutions of a multi-point boundary value problem

We apply the fixed point theorem of Avery and Peterson to the nonlinear second-order multi-point boundary value problem

     

Positive solutions of some nonlocal fourth-order boundary value problem

By the use of the Krasnosel’skii’s fixed point theorem, the existence of one or two positive solutions for the nonlocal fourth-order boundary value problem

     

Positive periodic solutions of higher-dimensional nonlinear functional difference equations

By using a well-known fixed point index theorem, we study the existence, multiplicity and nonexistence of positive T-periodic solution(s) to the higher-dimensional nonlinear functional difference equations of the form

     

A general iterative method for nonexpansive mappings in Hilbert spaces

Let H be a real Hilbert space. Consider on H a nonexpansive mapping T with a fixed point, a contraction f with coefficient 0<α<1, and a strongly positive linear bounded operator A with coefficient . Let . It is proved that the sequence {xn} generated by the iterative method xn+1=(I−αnA)Txn+αnγf(xn) converges strongly to a fixed point which solves the variational inequality for x∈Fix(T).     

Positive solutions and eigenvalue intervals of nonlocal boundary value problems with delays

The paper is concerned with the delay differential equation u^″+λb(t)f(u(t−τ))=0 satisfying u(t)=0 for −τ?t?0 and , where denotes the Riemann-Stieltjes integral. By applying the fixed point theorem in cones, we show the relationship between the asymptotic behaviors of the quotient (at zero and infinity) and the open intervals (eigenvalue intervals) of the parameter λ such that the problem has zero, one and two positive solution(s). If g(t)=t, by using a property of the Riemann-Stieltjes integral, the above nonlocal boundary value problem educes a three-point boundary value problem with delay, for which some similar results are established.     

Existence results for singular three point boundary value problems on time scales

We prove the existence of a positive solution for the three point boundary value problem on time scale given by