Higher Order Squeezed States of Anharmonic Oscillators

In this paper the N eigenstates of the Nth powers (N 3) of the annihilationoperator of an anharmonic oscillator are constructed and their completeness isinvestigated. We introduce a new kind of higher order squeezing, Mth-order SU(1,1) squeezing. The properties of Mth-order SU(1, 1) squeezing and anticorrelationof the N states are studied. The result show that these states may form a completeHilbert space; the Mth-order [M = (n + 1/2)N;n = 0,1, ...] SU(1, 1) squeezingeffects exist in all of the N states when N is even. There is anticorrelation in allof them.     

Stochastic homogenization of the laser intensity to improve the irradiation uniformity of capsules directly driven by thousands laser beams

Illumination uniformity of a spherical capsule directly driven by laser beams has been assessed numerically. Laser facilities characterized by N _D  = 12, 20, 24, 32, 48 and 60 directions of irradiation with associated a single laser beam or a bundle of N _B laser beams have been considered. The laser beam intensity profile is assumed super-Gaussian and the calculations take into account beam imperfections as power imbalance and pointing errors. The optimum laser intensity profile, which minimizes the root-mean-square deviation of the capsule illumination, depends on the values of the beam imperfections. Assuming that the N _B beams are statistically independents is found that they provide a stochastic homogenization of the laser intensity associated to the whole bundle, reducing the errors associated to the whole bundle by the factor \hbox{1/N_B^1/21/N_{B}^{1{/}2}} 1 / N B 1 / 2 , which in turn improves the illumination uniformity of the capsule. Moreover, it is found that the uniformity of the irradiation is almost the same for all facilities and only depends on the total number of laser beams N _tot  = N _D  × N _B .     

A Class of Integrable Flows on the Space of Symmetric Matrices

For a given skew symmetric real n × n matrix N, the bracket [X, Y] _N = XNY − YNX defines a Lie algebra structure on the space Sym(n, N) of symmetric n × n real matrices and hence a corresponding Lie-Poisson structure. The purpose of this paper is to investigate the geometry, integrability, and linearizability of the Hamiltonian system , or equivalently in Lax form, the equation on this space along with a detailed study of the Poisson geometry itself. If N has distinct eigenvalues, it is proved that this system is integrable on a generic symplectic leaf of the Lie-Poisson structure of Sym(n, N). This is established by finding another compatible Poisson structure. If N is invertible, several remarkable identifications can be implemented. First, (Sym(n, N), [·, ·]) is Lie algebra isomorphic with the symplectic Lie algebra associated to the symplectic form on given by N ⁻¹. In this case, the system is the reduction of the geodesic flow of the left invariant Frobenius metric on the underlying symplectic group Sp(n, N ⁻¹). Second, the trace of the product of matrices defines a non-invariant non-degenerate inner product on Sym(n, N) which identifies it with its dual. Therefore Sym(n, N) carries a natural Lie-Poisson structure as well as a compatible “frozen bracket” structure. The Poisson diffeomorphism from Sym(n, N) to maps our system to a Mischenko-Fomenko system, thereby providing another proof of its integrability if N is invertible with distinct eigenvalues. Third, there is a second ad-invariant inner product on Sym(n, N); using it to identify Sym(n, N) with itself and composing it with the dual of the Lie algebra isomorphism with , our system becomes a Mischenko- Fomenko system directly on Sym(n, N). If N is invertible and has distinct eigenvalues, it is shown that this geodesic flow on Sym(n, N) is linearized on the Prym subvariety of the Jacobian of the spectral curve associated to a Lax pair formulation with parameter of the system. If, on the other hand, N has nullity one and distinct eigenvalues, in spite of the fact that the system is completely integrable, it is shown that the flow does not linearize on the Jacobian of the spectral curve. Research partially supported by NSF grants CMS-0408542 and DMS-0604307. Research partially supported by the Swiss SCOPES grant IB7320-110721/1, 2005-2008, and MEdC Contract 2-CEx 06-11-22/25.07.2006. Research partially supported by the California Institute of Technology and NSF-ITR Grant ACI-0204932. Research partially supported by the Swiss NSF and the Swiss SCOPES grant IB7320-110721/1.     

Short Representations of SU(2,2/N) and Harmonic Superspace Analyticity

We consider the harmonic superspaces associated with SU(2,2/N) superconformal algebras. For arbitrary N, we show that massless representations, other than the chiral ones, correspond to [N/2] elementary ultrashort analytic superfields whose first component is a scalar in the k antisymmetric irrep of SU(N) (k=1...[N/2]) with top spin J _top=(N/2–k/2,0). For N=2n, we analyze UIR's obtained by tensoring the self-conjugate ultrashort multiplet J _top=(n/2,0) and show that N–1 different basic products give rise to all possible UIR's with residual shortening.     

A generalization of Rayleigh's theorem for the infinite harmonic crystal

We show that if one replacesN masses in an infinite harmonic, perfect-dimensional crystal byN ₁ lighter andN ₂ heavier ones such thatN ₁+ N₂ =N is finite, then one introduces at most vN₁ (isolated) bound states. This can be considered as an extension of the results of Romerio and Wreszinski.Supported by the Netherlands Organization for the Advancement of Pure Research (Z.W.O.).     

Borel summability of the 1/N expansion for theN-vector [O(N) non-linear σ] models

We construct an analytic interpolation in 1/N for theN-vector [O(N) non-linear ] models withN-component fields on a lattice. This interpolation, valid at sufficiently high temperatures, extends over a large domain in the complex plane containing the half plane Re(1/N)>0. We use this result to show that the 1/N expansion of the free energy density and of the correlation functions is Borel summable in the thermodynamic limit and at high temperature.On leave from C.P.T., Ecole Polytechnique, F-91128 Palaiseau Cédex, France     

N=2 affine superalgebras and Hamiltonian reduction inN=2 superspace

We constructN=2 affine current algebras for the superalgebrassl(n/n-1)⁽¹⁾ in terms ofN=2 supercurrents subjected to nonlinear constraints and discuss the general procedure of the hamiltonian reduction inN=2 superspace at the classical level. We consider in detail the simplest case ofN=2sl(2/1)⁽¹⁾ and show howN=2 superconformal algebra inN=2 superspace follows via the hamiltonian reduction. Applying the hamiltonian reduction to the case ofN=2sl(3/2)⁽¹⁾, we find two new extendedN=2 superconformal algebras in a manifestly supersymmetricN=2 superfield form. Decoupling of four component currents of dimension 1/2 in them yields, respectively,u(2/1) andu(3) Knizhnik-Bershadsky superconformal algebras. We also discuss how theN=2 superfield formulations ofN=2W ₃ andN=2W ₃ ⁽²⁾ superconformal algebras come out in this framework, as well as some unusual extendedN=2 superconformal algebras containing constrainedN=2 stress tensor and/or spin 0 supercurrents.     

Large N expansion for strongly-coupled boson–fermion mixtures

We study a many-body mixture of an equal number of bosons and two-component fermions with a strong contact attraction. In this system bosons and fermions can be paired into composite fermions. We construct a large N extension where both bosons and fermions have the extra large N degrees of freedom and the boson–fermion interaction is extended to a four-point contact interaction which is invariant under the O(N) group transformation, so that the composite fermions become singlet in terms of the O(N) group. It is shown that such O(N) singlet fields have controllable quantum fluctuations suppressed by 1/N factors and yield a systematic 1/N-expansion in terms of composite fermions. We derive an effective action described by composite fermions up to the next-to-leading-order terms in the large N expansion, and show that there can be the BCS superfluidity of composite fermions at sufficiently low temperatures.     

Entropic repulsion of the lattice free field,II. The 0-boundary case

This paper is a continuation of [5]. We consider the Euclidean massless free field on a boxV _N of volumeN ^d with O-boundary condition; that is the centered Gaussian field with covariances given by the Green function of the simple random walk on ^d ,d3, killed as it exitsV _N . We show that the probability, that all the spins are positive in the boxV _N decays exponentially at a surface rateN ^d–1 . This is in contrast with the rateN ^d–2 logN for the infinite field of [5].     

Scaling of Linking and Writhing Numbers for Spherically Confined and Topologically Equilibrated Flexible Polymers

Scaling laws for Gauss linking number Ca and writhing number Wr for spherically confined flexible polymers with thermally fluctuating topology are analyzed. For ideal (phantom) polymers each of N segments of length unity confined to a spherical pore of radius R there are two scaling regimes: for sufficiently weak confinement (R≫N ^1/3) each chain has |Wr|≈N ^1/2, and each pair of chains has average |Ca|≈N/R ^3/2; alternately for sufficiently tight confinement (N ^1/3≫R), |Wr|≈|Ca|≈N/R ^3/2. Adding segment-segment avoidance modifies this result: for n chains with excluded volume interactions |Ca|≈(N/n)^1/2 f(φ) where f is a scaling function that depends approximately linearly on the segment concentration φ=nN/R ³. Scaling results for writhe are used to estimate the maximum writhe of a polymer; this is demonstrated to be realizable through a writhing instability that occurs for a polymer which is able to change knotting topology and which is subject to an applied torque. Finally, scaling results for linking are used to estimate bounds on the entanglement complexity of long chromosomal DNA molecules inside cells, and to show how “lengthwise” chromosome condensation can suppress DNA entanglement.     

Laser-Induced Flourescence Excitation Spectroscopy and Photophysics of Naphthalene Bichromophoric Molecules in Supersonic Jets

We present studies of interchromophore interactions under supersonic jet conditions in a large number of dinaphthyl bichromophoric molecules by measuring their laser-induced fluorescence (LIF) excitation spectra. The molecules are composed of two naphthalene chromophores connected by an n-methylene bridge. The length of the bridge was varied as a function of the number of methylene units (n = 0, 1, 2, 4, 6), of the general type NnN(i,j), were N denotes naphthalene moiety, n the number of methylene units in the bridge, and (i,j) are the or positions of the bridge at each of the chromophores. We obtained high-quality LIF spectra of these bichromophoric olecules. In the molecules N1N(2,2), N1N(1,2), N2N(2,2), and N2N(1,2), the spectrum is characterized by an intense 0–0 region, with series of low-frequency progressions. These progressions are assigned as vibrational modes of the bridge. The appearance of several series of progressions is explained either by the excitation of different chromophores (in the mixed molecules) or by the excitation of different populated conformers. The spectrum of N4N(1,1) is different in several aspects from these spectra. The origin is shifted farther to the red, to 31,402 cm^–1. Low-frequency progressions or other transitions are not observed near the origin, but typical intrachromophore naphthalene vibrations are intense. The spectra of N6N(1,1) and N6N(2,2) are also characterized by intense intrachromophore vibrations, however, the spectrum of N6N(2,2) is very complicated due to many populated conformations, while that of N6N(1,1) is more simple.     

The N = 2 Supersymmetric Unconstrained Matrix GNLS Hierarchies

The generalization of the N = 2 supersymmetric chiral matrix (k | n,m)-GNLS hierarchy (Lett. Math. Phys. 45 (1998) 63, solv-int/9711009) to the case when matrix entries are bosonic and fermionic unconstrained N = 2 superfields is proposed. This is done by exhibiting the corresponding matrix Lax-pair representation in terms of N = 2 unconstrained superfields. It is demonstrated that when matrix entries are chiral and antichiral N = 2 superfields, it reproduces the N = 2 chiral matrix (k | n,m)-GNLS hierarchy, while in the scalar case, k = 1, it is equivalent to the N = 2 supersymmetric multicomponent hierarchy (J. Phys. A 29 (1996) 1281, hep-th/9510185). The simplest example—the N = 2 unconstrained (1 | 1,0)-GNLS hierarchy—and its reduction to the N = 2 supersymmetric = 1 KdV hierarchy are discussed in more detail, and its rich symmetry structure is uncovered.     

Finite- N effects for ideal polymer chains near a flat impenetrable wall

This paper addresses the statistical mechanics of ideal polymer chains next to a hard wall. The principal quantity of interest, from which all monomer densities can be calculated, is the partition function, G _N(z) , for a chain of N discrete monomers with one end fixed a distance z from the wall. It is well accepted that in the limit of infinite N , G _N(z) satisfies the diffusion equation with the Dirichlet boundary condition, G _N(0) = 0 , unless the wall possesses a sufficient attraction, in which case the Robin boundary condition, G _N(0) = - G _N ^′(0) , applies with a positive coefficient, . Here we investigate the leading N ^-1/2 correction, G _N(z) . Prior to the adsorption threshold, G _N(z) is found to involve two distinct parts: a Gaussian correction (for z aN ^1/2 with a model-dependent amplitude, A , and a proximal-layer correction (for z a described by a model-dependent function, B(z) .     

Complete integrability of relativistic Calogero-Moser systems and elliptic function identities

Poincaré-invariant generalizations of the Galilei-invariant Calogero-MoserN-particle systems are studied. A quantization of the classical integralsS ₁, ...,S _N is presented such that the operators ₁, ..., _N mutually commute. As a corollary it follows thatS ₁, ...,S _N Poisson commute. These results hinge on functional equations satisfied by the Weierstrass - and -functions. A generalized Cauchy identity involving the -function leads to anN×N matrixL whose symmetric functions are proportional toS ₁, ...,S _N .     

Positive-energy irreps of the quantum anti de Sitter algebra

We obtain positive-energy irreducible representations of theq-deformed anti de Sitter algebraU _q (so(3, 2)) by deformation of the classical ones. When the deformation parameterq isN-th root of unity, all these irreducible representations become unitary and finite-dimensional. Generically, their dimensions are smaller than those of the corresponding finite-dimensional non-unitary representations ofso(3, 2). We discuss in detail the singleton representations, i.e. the Di and Rac. WhenN is odd, the Di has dimension 1/2(N ²–1) and the Rac has dimension 1/2(N ²+1), while ifN is even, both the Di and Rac have dimension 1/2N ². These dimensions are classical only forN=3 when the Di and Rac are deformations of the two fundamental non-unitary representations ofso(3, 2).Presented at the 4th Colloquium Quantum groups and integrable systems, Prague, 22–24 June 1995.On leave from Bulgarian Acad. Sci., Institute of Nuclear Research and Nuclear Energy, 72 Tsarigradsko Chaussee, 1784 Sofia, Bulgaria.On leave from Pennsylvania State University (Fulbright scholar).     

Scaling contributions to the free energy in the 1/n expansion of O(n) nonlinear sigma models in d-dimensions

Within the 1/N expansion of O(N) nonlinear σ models for d≤4 it is possible to separate consistently the spin-wave and the massive-mode contributions to the scaling part of the free energy near criticality, and to evaluate them to O(1/N). For critical dimensions d=2+2/n the Abe-Hikami anomaly is recovered, while for d=2 the removal of the spin-wave term is justified.     

Suppression of Noise Effects in Color Correction by Spectral Sensitivities of Image Sensors

The accurate transformation of device responses to tristimulus values is important at an early stage in color management and this transformation is usually called color correction. In this paper it is shown that the influence of noise on color correction can be suppressed if the square of the singular values by the singular value decomposition (SVD) of a sensor matrix represented by surface reflectance, SLVA^1/2, is larger than the noise variance, where S is the M × N matrix of M channel spectral sensitivities, L is the N × N diagonal matrix for an recording illuminant, V is the N × N matrix which is composed of eigenvectors of an autocorrelation matrix for spectral reflectance of samples, A is the N × N diagonal matrix with eigenvalues of the autocorrelation matrix along the diagonal and N is the number of the dimension, respectively. A method to suppress the influence of noise on color correction by spectral sensitivities of sensors is presented for the first time.     

Analysis of R. P. Smith's and D. C. Parris-R. B. Mclellan's carbon diffusivity data

A phenomenological model is proposed to account for the variation of carbon diffusivityD(N₁) with composition in austenite. This model is based on Parris-McLellan's statement that the increase ofD(N₁) with carbon concentrationN ₁ is due to the increase in chemical driving force. The gradient of the carbon activity in Wagner's and in Zupp-Stevenson's interpretation is presumed to be the intrinsic driving force. The predictions of the model are compared with carbonD(N₁) values both with [1] and without [5] a chemical concentration gradient. TheD(0) values and the carbon-carbon interaction coefficients ₁ are calculated by the use of the least squares method from the experimental data. The application of Zupp-Stevenson's definition of the activity coefficient ₁ results in a better fit of the theory to the experimental data.     

Nonlocal quark model beyond mean field

The nonlocal chiral quark model is extended beyond mean field using a strict 1/N _c expansion scheme. The nonlocal interaction has the advantage that all diagrams are finite and leads to unique evaluation of the 1/N _c corrections. Parameters of the nonlocal model are refitted making use of the physical values of the pion mass and the weak pion decay constant. The size of the 1/N _c correction to the quark condensate is carefully studied in nonlocal and local Nambu-Jona-Lasinio models. It is found that even the sign of the corrections can be different.     

One Dimensional Behavior of Singular N Dimensional Solutions of Semilinear Heat Equations

We consider u(x,t) a solution of u _t =Δu+|u| ^{p − 1} u that blows up at time T, where u:ℝ ^N ×[0, T)→ℝ, p>1, (N−2)p<N+2 and either u(0)≥ 0 or (3N−4)p<3N+8. We are concerned with the behavior of the solution near a non isolated blow-up point, as T−t→ 0. Under a non-degeneracy condition and assuming that the blow-up set is locally continuous and N−1 dimensional, we escape logarithmic scales of the variable T−t and give a sharper expansion of the solution with the much smaller error term (T−t)^{1, 1/2−η} for any η>0. In particular, if in addition p>3, then the solution is very close to a superposition of one dimensional solutions as functions of the distance to the blow-up set. Finally, we prove that the mere hypothesis that the blow-up set is continuous implies that it is C ^{1, 1/2−η} for any η>0. Received: 20 June 2001 / Accepted: 6 October 2001