Norm inequalities for fractional powers of positive operators

It is shown that ifA, B andX are operators on a Hilbert space such thatA andB are positive andX belongs to a norm ideal associated with some unitarily invariant norm |·|, then for 0 r 1 we have |A ^r XB ^r | |X|^1-r |AXB| ^r . This is an extension of the classical Heinz-Kato inequality which was originally proved for the usual operator norm. Other related inequalities are also discussed.     

On the level order for Dirac operators

We start from the Dirac operator for the Coulomb potential and prove within first-order perturbation theory that degenerate levels split in a definite way depending on the sign of the laplacian of the perturbing potential.     

An inverse spectral theory of Gel'fand-Levitan-type for higher order differential operators

An inverse spectral theory is presented for certain linear ordinary differential operators of arbitrary even order n which generalizes the Gel'fand-Levitan theory for Sturm-Liouville operators. It is proved that the coefficients in these operators are uniquely determined by n–1 distinct spectral matrices. Our method of proof makes use of a transformation due to M.K. Fage which generalizes the Povzner-Levitan transformations for Sturm-Liouville operators     

On integrable systems with higher order corrections

By a canonical transformation equations with first-order corrections in the basic physical systems of dispersive nonlinear wave equations, for which the leading order equations are completely integrable, can be transformed into higher order integrable equations of the same order.     

Bell-type inequalities in orthomodular lattices. II. Inequalities of higher order

This paper is a continuation of the first part and it is devoted to the study of Bell-type inequalities of order at least 3 in orthomodular lattices. We give some necessary and sufficient conditions for the validity of Bell-type inequalities of order 3 and also, more generally, for those of ordern.     

More inequalities for critical exponents

A variety of rigorous inequalities for critical exponents is proved. Most notable is the low-temperature Josephson inequalitydv +2 2–. Others are 1 1 +v, 1 1 , 1,d 1 + 1/ (for d),dv, ₃ + (for d), ₄ , and _{2m 2m+2} (form 2). The hypotheses vary; all inequalities are true for the spin-1/2 Ising model with nearest-neighbor ferromagnetic pair interactions.NSF Predoctoral Fellow (1976–1979). Research supported in part by NSF Grant PHY 78-23952.     

Hardy and Lieb-Thirring Inequalities for Anyons

We consider the many-particle quantum mechanics of anyons, i.e. identical particles in two space dimensions with a continuous statistics parameter ${\alpha \in [0, 1]}$ α ∈ [ 0 , 1 ] ranging from bosons (α = 0) to fermions (α = 1). We prove a (magnetic) Hardy inequality for anyons, which in the case that α is an odd numerator fraction implies a local exclusion principle for the kinetic energy of such anyons. From this result, and motivated by Dyson and Lenard’s original approach to the stability of fermionic matter in three dimensions, we prove a Lieb-Thirring inequality for these types of anyons.     

Some critical exponent inequalities for percolation

For a large class of independent (site or bond, short- or long-range) percolation models, we show the following: (1) If the percolation densityP (p) is discontinuous atp _c , then the critical exponent (defined by the divergence of expected cluster size, nP _n (p) (P _c –P)^– asp p _c ) must satisfy 2. (2) or (defined analogously to, but asp p _c ) and [P _n (p _c ) (n ^–1–1/) asn ] must satisfy, 2(1 – 1/). These inequalities for improve the previously known bound 1(Aizenman and Newman), since 2 (Aizenman and Barsky). Additionally, result 1may be useful, in standardd-dimensional percolation, for proving rigorously (ind>2) that, as expected,P _x has no discontinuity atp _c .     

On higher order computations in strong coupling

An algorithm which simplifies manipulation of higher order graphs in strong coupling is presented.     

Partons and quark operators for higher twist

We compute the four-quark, twist-four contribution to the structure functions of leptoproduction in the parton model as a first step of a proposed approach towards a factorization algorithm for higher twist. Phenomenological implications for power corrections are also presented.     

任意光场湮没算符高次幂本征态的内部结构

任意光场湮没算符高次幂的本征态都具有类似于奇偶相干态的内部奇偶结构.根据这一结果,对于单模光场,指出了怎么由aM本征态来产生a2M的本征态.     

A toy model for higher spin Dirac operators

This paper deals with the higher spin Dirac operator Q_2,1 acting on functions taking values in an irreducible representation space for so(m) with highest weight $ (\tfrac{5} {2},\tfrac{3} {2},\tfrac{1} {2},...,\tfrac{1} {2}) $ (\tfrac{5} {2},\tfrac{3} {2},\tfrac{1} {2},...,\tfrac{1} {2}) . This operator acts as a toy model for generalizations of the classical Rarita—Schwinger equations in Clifford analysis. Polynomial null solutions for this operator are studied in particular.     

Thermodynamics of higher dimensional black holes with higher order thermal fluctuations

In this paper, we consider higher order corrections of the entropy, which coming from thermal fluctuations, and find their effect on the thermodynamics of higher dimensional charged black holes. Leading order thermal fluctuation is logarithmic term in the entropy while higher order correction is proportional to the inverse of original entropy. We calculate some thermodynamics quantities and obtain the effect of logarithmic and higher order corrections of entropy on them. Validity of the first law of thermodynamics investigated and Van der Waals equation of state of dual picture studied. We find that five-dimensional black hole behaves as Van der Waals, but higher dimensional case have not such behavior. We find that thermal fluctuations are important in stability of black hole hence affect unstable/stable black hole phase transition.     

Ginzburg-Landau theory for higher order phase transition

Ginzburg-Landau theory for studying phase transitions of higher order has been derived using coarse graining and lattice formulation within Ehrenfest thermodynamics. Our developed Hamiltonian leads directly to the functional of the system. We studied the evolution of the order parameter using our developed model equations for third and fourth order phase transitions. The periodic nature of the system can be likened to spatially varying periodic soliton/antisoliton lattice of holes in condensate. This is different from what one observes for any conventional solitary wave in the second order phase regime.     

Geometric critical exponent inequalities for general random cluster models

A set of new critical exponent inequalities,d(1 –1 /)2 –, dv(1 – 1/), andd> 1, is proved for a general class of random cluster models, which includes (independent or dependent) percolations, lattice animals (with any interactions), and various stochastic cluster growth models. The inequalities imply that the critical phenomena in the models are inevitably not mean-field-like in the dimensions one, two, and three.The present work was reported at the 56th Statistical Mechanics Meeting (Rutgers, December 1986).     

Strict inequalities for some critical exponents in two-dimensional percolation

For 2D percolation we slightly improve a result of Chayes and Chayes to the effect that the critical exponent for the percolation probability isstrictly less than 1. The same argument is applied to prove that ifL():={(x, y):x=r cos, y=r sin for some r0, or} and():=limpp _c [log(p–p _c )]^–1 log P_cr {itO is connected to by an occupied path inL()}, then() is strictly decreasing in on [0, 2]. Similarly, lim_n [–logn]^–1 logP _cr {itO is connected by an occupied path inL()() to the exterior of [–n, n]×[–n, n] is strictly decreasing in on [0, 2].     

Inequalities for higher order Ising spins and for continuum fluids

The recently derived Fortuin, Kasteleyn and Ginibre (FKG) inequalities for lattice gasses are investigated for higher order Ising spin systems and multi-component lattice gasses. Conditions are given for the validity of the FKG inequalities for higher order spin systems with Hamiltonians of the form used recently as models for various physical systems, e.g.He ³–He ⁴ mixtures. We also investigate various inequalities for binary lattice gases and show how these can be carried over to continuum systems.Supported in part by U.S.A.F.O.S.R. # F 44620-71-C-0013.N.S.F. Graduate Trainee.