On variational principles of relativistic theories involving torsion

We consider variational principles that are distinct from the formulation of Hehl et al. which reproduce their field equations in reductions of U₄. The results are at odds with conclusions which appeared in a recent letter by Safko et al.     

Integrals involving legendre polynomials that arise in the solution of radiation transfer

The integrals involving the product of the Legendre polynomials with the exponential integral function and the exponentials arise in the solution of equation of radiative transfer by a suitable expansion in the space variable in terms of the Legendre polynomials. The success of such methods of solution depends on the availability of rapidly converging analytic expressions for such integrals. In this work, analytic expressions are presented for some of such integrals.     

Integrals involving an exponential integral function and exponentials arising in the solution of radiation transfer

The solution of the equation of radiative transfer for a participating medium generally results in the evaluation of integrals involving a product of an exponential integral function or an exponential with a polynomial. Although expressions are available in the literature for the evaluation of such integrals, which appear to be structurally easy and simple to program, they are found to be not so accurate for certain parameters involved. To overcome such difficulties, alternative, computationally more accurate, analytic expressions are presented, and numerical techniques for their evaluation are discussed.     

Eigenvalues and eigenfunctions of a clover plate

We report a numerical study of the flexural modes of a plate using semi-classical analysis developed in the context of quantum systems. We first introduce the Clover billiard as a paradigm for a system inside which rays exhibit stable and chaotic trajectories. The resulting phase space explored by the ray trajectories is illustrated using the Poincare surface of section, and shows that it has both integrable and chaotic regions. Examples of the stable and the unstable periodic orbits in the geometry are presented. We numerically solve the biharmonic equation for the flexural vibrations of the Clover shaped plate with clamped boundary conditions. The first few hundred eigenvalues and the eigenfunctions are obtained using a boundary elements method. The Fourier transform of the eigenvalues show strong peaks which correspond to ray periodic orbits. However, the peaks corresponding to the shortest stable periodic orbits are not stronger than the peaks associated with unstable periodic orbits. We also perform statistics on the obtained eigenvalues and the eigenfunctions. The eigenvalue spacing distribution P(s) shows a strong peak and therefore deviates from both the Poisson and the Wigner distribution of random matrix theory at small spacings because of the C_4v symmetry of the Clover geometry. The density distribution of the eigenfunctions is observed to agree with the Porter-Thomas distribution of random matrix theory. Received 12 February 2001 and Received in final form 17 April 2001     

Analyticity properties of eigenfunctions and scattering matrix

For potentialsV=V(x)=O(|x|^–2–) for |x|,x³ we prove that if theS-matrix of (–, –+V) has an analytic extension to a regionO in the lower half-plane, then the family of generalized eigenfunctions of –+V has an analytic extension toO such that for |Imk|<b. Consequently, the resolvent (–+V–z ²)^–1 has an analytic continuation from ⁺ to {kOImk|<b} as an operator from _b ={f=e ^–b|x| g|gL ₂(³)} to _–b . Based on this, we define for potentialsW=o(e ^–2b|x|) resonances of (–+V, –+V+W) as poles of and identify these resonances with poles of the analytically continuedS-matrix of (–+V, –+V+W).The author would like to thank the Institute for Advanced Study for its hospitality and the National Science Foundation for financial support under Grant No. DMS-8610730(1)     

Spin Path Integrals and Generations

The spin of a free electron is stable but its position is not. Recent quantum information research by G. Svetlichny, J. Tolar, and G. Chadzitaskos have shown that the Feynman position path integral can be mathematically defined as a product of incompatible states; that is, as a product of mutually unbiased bases (MUBs). Since the more common use of MUBs is in finite dimensional Hilbert spaces, this raises the question “what happens when spin path integrals are computed over products of MUBs?” Such an assumption makes spin no longer stable. We show that the usual spin-1/2 is obtained in the long-time limit in three orthogonal solutions that we associate with the three elementary particle generations. We give applications to the masses of the elementary leptons.     

Hodge Integrals and Degenerate Contributions

Hodge integral techniques are used to compute the degree 1 degenerate contributions of curves of arbitrary genus in the Gromov–Witten theory of 3-folds. In the Calabi–Yau case, the contributions are compared to related M-theoretic calculations. In the Fano case, the contributions suggest new integrality conditions. Received: 6 April 1999 / Accepted: 14 July 1999     

Hodge Integrals and Integrable Hierarchies

We show that the generating series of some Hodge integrals involving one or two partitions are τ-functions of the KP hierarchy or the 2-Toda hierarchy, respectively. We also reformulate the results as a relationship between the relative invariants of some local Calabi-Yau geometries and integrable hierarchies and present some more examples using the topological vertex.     

Coherent-State Path Integrals and Brownian Motions

A stochastic approach to the rigorous foundation of the coherent-state (phase-space) path integral is given. Stochastic integrals and some generalizations of the Feynman–Kac theorem are used for this purpose. In this approach, quantum mechanics is described in terms of the Fock–Bargmann representation; a classical Hamiltonian is related to the corresponding quantum Hamiltonian on the Fock–Bargmann space, seen as a Hilbert subspace of L²(R²)L^{2}({\bf R}^{2}). The coherent-state path integral is realized as a conditional expectation of a stochastic process defined by the exponential of the Fisk–Stratonovich integral of the fundamental 1-form along a path of Brownian motion on the phase space R²{\bf R}^{2}.     

Effective-mass eigenfunctions in superlattices and their role in well-capture

The rate of capture into the wells of a superlattice is calculated using the effective-mass eigenfunctions and the superlattice folded spectrum of polar optical phonons. The dependence on well-width is followed for the specific case of 100Å barriers. A general trend of diminishing rate with diminishing well-width is obtained, but superimposed on this is a resonance structure (periodic in well-width) associated with ‘nearly’ bound states. The results are significantly different from other published work, principally because the latter has been hitherto on the bulk-phonon spectrum, and on single wells.     

Relation between the parabolic and spherical eigenfunctions of hydrogen

It is shown that the transformation coefficients relating the eigenfunctions of the Kepler problem in parabolic and spherical coordinates respectively are the normalized Clebsch-Gordan coefficients.     

同时测量扭转角度和扭转方向的侧漏光子晶体光纤扭转传感器

基于所研制的侧漏型光子晶体光纤,提出并构建了出一种同时检测扭转角度 和扭转方向的高灵敏度Sagnac干涉仪型光纤扭转传感器.顺时针扭转时, 传感器传输谱向短波长方向偏移;逆时针扭转,向长波长方向偏移. 对传感器扭转特性的实验研究结果表明,构成Sagnac干涉仪的侧漏型光子晶体光纤的长度, 对扭转敏感系数和扭转角度测量范围起着决定性作用.当光纤长度较短时, 扭转传感器具有较大的扭转灵敏度,但扭转角度测量范围较小;光纤长度增加时,扭转灵敏度减小, 扭转角度测量范围增大.当构成Sagnac干涉仪的侧漏型光子晶体光纤长度为14.85 cm时, 传感器的扭转敏感系数可达到0.9354 nm/(°),扭转角度测量范围为-90°—90°; 光纤长度为32 cm时,最大扭转敏感系数降为0.2132 nm/(°), 扭转角度测量范围扩展至-180°—180°. 采用二维测量矩阵法可以有效排除温度对扭转角度的测量的影响. 关键词： 光纤传感器 侧漏型光子晶体光纤 扭转传感器 Sagnac干涉仪     

Feynman Integrals and Motives of Configuration Spaces

We formulate the problem of renormalization of Feynman integrals and its relation to periods of motives in configuration space instead of momentum space. The algebro-geometric setting is provided by the wonderful compactifications [`(Conf)]_G(X){\overline{Conf}_\Gamma(X)} of arrangements of subvarieties associated to the subgraphs of a Feynman graph Γ, with X a (quasi)projective variety. The motive and the class in the Grothendieck ring are computed explicitly for these wonderful compactifications, in terms of the motive of X and the combinatorics of the Feynman graph, using recent results of Li Li. The pullback to the wonderful compactification of the form defined by the unrenormalized Feynman amplitude has singularities along a hypersurface, whose real locus is contained in the exceptional divisors of the iterated blowup that gives the wonderful compactification. A regularization of the Feynman integrals can be obtained by modifying the cycle of integration, by replacing the divergent locus with a Leray coboundary. The ambiguities are then defined by Poincaré residues. While these residues give periods associated to the cohomology of the exceptional divisors and their intersections, the regularized integrals give rise to periods of the hypersurface complement in the wonderful compactification.     

First Integrals of Shear-Free Fluids and Complexity

A single master equation governs the behaviour of shear-free neutral perfect fluid distributions arising in gravity theories. In this paper, we study the integrability of yxx=f(x)y2, find new solutions, and generate a new first integral. The first integral is subject to an integrability condition which is an integral equation which restricts the function f(x). We find that the integrability condition can be written as a third order differential equation whose solution can be expressed in terms of elementary functions and elliptic integrals. The solution of the integrability condition is generally given parametrically. A particular form of which corresponds to repeated roots of a cubic equation is given explicitly, which is a new result. Our investigation demonstrates that complexity of a self-gravitating shear-free fluid is related to the existence of a first integral, and this may be extendable to general matter distributions.     

Path Integrals and Statistics of Identical Particles

We summarize the essential ingredients, which enabled us to derive the path-integral for a system of harmonically interacting spin-polarized identical particles in a parabolic confining potential, including both the statistics (Bose–Einstein or Fermi–Dirac) and the harmonic interaction between the particles. This quadratic model, giving rise to repetitive Gaussian integrals, allows to derive an analytical expression for the generating function of the partition function. The calculation of this generating function circumvents the constraints on the summation over the cycles of the permutation group. Moreover, it allows one to calculate the canonical partition function recursively for the system with harmonic two-body interactions. Also, static one-point and two-point correlation functions can be obtained using the same technique, which make the model a powerful trial system for further variational treatments of realistic interactions.     

Matrix Integrals and the Geometry of Spinors

Abstract

We obtain the collection of symmetric and symplectic matrix integrals and the collection of Pfaffian tau-functions, recently described by Peng and Adler and van Moerbeke, as specific elements in the Spin-group orbit of the vacuum vector of a fermionic Fock space. This fermionic Fock space is the same space as one constructs to obtain the KP and 1-Toda lattice hierarchy.     

States on Pseudo Effect Algebras and Integrals

We show that every state on an interval pseudo effect algebra E satisfying an appropriate version of the Riesz Decomposition Property (RDP for short) is an integral through a regular Borel probability measure defined on the Borel σ-algebra of a Choquet simplex K. In particular, if E satisfies the strongest type of RDP, the representing Borel probability measure can be uniquely chosen to have its support in the set of the extreme points of K.