Null cut loci of spacelike surfaces

The null cut locus of a spacelike submanifold of codimension 2 in a space-time is defined. In globally hyperbolic space-times, it is shown that the future (past) null cut locusc _n ⁺ (H) [c _n ^- (H)] of a compact, acausal, spacelike submanifoldH of codimension 2 is a closed subset of the space-time, and each pointx c _n ⁺ (H) is either a focal point ofH along some future-directed null geodesic meetingH orthogonally or there exist at least two null geodesics fromH tox, realizing the distance betweenH andx or both. Also, it can be shown that the assumptions of the Penrose's singularity theorem for open globally hyperbolic space-times may be weakened to the space-times which are conformal to an open subset of an open globally hyperbolic space-time.This study is based on Chapter 3 of the author's Ph.D. thesis.     

The global structure of simple space-times

According to a standard definition of Penrose, a space-time admitting well-defined future and past null infinitiesI ⁺ andI ^– is asymptotically simple if it has no closed timelike curves, and all its endless null geodesics originate fromI ^– and terminate atI ⁺. The global structure of such space-times has previously been successfully investigated only in the presence of additional constraints. The present paper deals with the general case. It is shown thatI ⁺ is diffeomorphic to the complement of a point in some contractible open 3-manifold, the strongly causal regionI ₀ ⁺ ofI ⁺ is diffeomorphic to , and every compact connected spacelike 2-surface inI ⁺ is contained inI ₀ ⁺ and is a strong deformation retract of bothI ₀ ⁺ andI ⁺. Moreover the space-time must be globally hyperbolic with Cauchy surfaces which, subject to the truth of the Poincaré conjecture, are diffeomorphic to ³.     

A relationship between future and past null infinity

By the addition of a single pointi at space-like infinity, null infinity,I =I ⁺ I ^-, may be regarded as the null cone with vertexi. We show that, given suitable regularity conditions ati, there exist natural Bondi scalings on the whole ofI which may be used to relate quantities onI ⁺ andI ^-. In particular, we show that, in terms of such a scaling, the mass aspect ₂ ^o onI satisfies lim _x°,xI+ ₂ ^° = lim _x°,xI- ₂ ^° where the limits are taken along the same generator ofI.     

Differing Averaged and Quenched Large Deviations for Random Walks in Random Environments in Dimensions Two and Three

We consider the quenched and the averaged (or annealed) large deviation rate functions I _q and I _a for space-time and (the usual) space-only RWRE on \mathbbZ^d{\mathbb{Z}^d} . By Jensen’s inequality, I _a  ≤ I _q . In the space-time case, when d ≥ 3 + 1, I _q and I _a are known to be equal on an open set containing the typical velocity ξ _o . When d = 1 + 1, we prove that I _q and I _a are equal only at ξ _o . Similarly, when d = 2 + 1, we show that I _a  < I _q on a punctured neighborhood of ξ _o . In the space-only case, we provide a class of non-nestling walks on \mathbbZ^d{\mathbb{Z}^d} with d = 2 or 3, and prove that I _q and I _a are not identically equal on any open set containing ξ _o whenever the walk is in that class. This is very different from the known results for non-nestling walks on \mathbbZ^d{\mathbb{Z}^d} with d ≥ 4.     

Optical absorption behavior of Tl<Superscript>+</Superscript> DOPED Rb(Br<Subscript>1–<Emphasis Type="Italic">x</Emphasis></Subscript>I<Subscript><Emphasis Type="Italic">x</Emphasis></Subscript>) system

The optical absorption behavior of Tl⁺ doped Rb(Br_1–x I _x ) mixed crystals (with x = 0.00, 0.05, and 0.10) grown under vacuum by slow cooling from the melt has been studied. Absorption spectra of the mixed crystals recorded at room temperature showed that the characteristic A-absorption band of Tl⁺ ions in the Rb(Br_1–x I _x ) system (0.1 mol. %) with x = 0.00 (i.e., RbBr:Tl⁺) broadened with the iodine content towards the low energy side. Changes in the absorption spectra of the mixed crystals are due to creation of some complex centers involving Tl⁺, Br^–, and I^- ions with energy levels inside the band gap while forming the mixed crystal. The absorption spectra of gamma-irradiated mixed crystals showed the F-band, which shifted towards the low energy side due to the existence of iodine ions in the mixed crystals.     

Asymptotic symmetry and the global structure of future null infinity

Space-times for whichI ⁺ (future null infinity) is not necessarily homeomorphic toR×S ² are considered. It is shown that, depending on the global conformal structure ofI ⁺, a given space-time either (1) possesses an asymptotic symmetry group with a normal subgroup of supertranslations, similar in structure to the BMS group, or (2) possesses a simpler kind of asymptotic symmetry group, not involving supertranslations, or (3) has no asymptotic symmetry. The setting is Newman and Unti's approach to asymptotically flat space-times and use is made of the characterization of the asymptotic symmetry transformation as a conformal motion ofI ⁺ that preserves null angles.     

Einstein-Maxwell null fields with non-zero twist

It is shown that iff ^a is the propagation vector of a null Maxwell field in a space-time with metric⁰ g _ab , then it is also the propagation vector in the space-timeg _ab =⁰ g _ab +2fi _a i _b . This result, together with the Robinson metric for vacuum gravitational fields and Hughston's generalization to radiating fields, is used to set up equations for combined gravitational and electromagnetic null-fields with special reference to fields with non-zero twist.     

The Structure of Radiative Tunnel Recombination Sites in Emulsion Microcrystals of AgBr(I)

To identify the structure of emissive tunnel recombination sites in the emulsion microcrystals of silver bromide AgBr(I) with iodine contaminations and to determine the role of an emulsion medium in their formation, the temperature dependence of the luminescence spectra in the range from 77 to 120 K, the kinetics of the growth of the maximum luminescence intensity value at λ ≈ 560 nm, and the luminescence flash spectrum stimulated by the infrared light are investigated. Two types of the AgBr_{1 – x}(I _x ) (x = 0.03) microcrystals—namely, obtained in an aqueous solution and on a gelatin substrate—are used in the studies. It is established that the emissive tunnel recombination sites with a luminescence maximum at λ ≈ 560 nm in AgBr_{1 – x}(I _x ) (x = 0.03) are the {(I _a ^- I _a ^- )Ag _i ⁺ } donor–acceptor complexes with the I _a ^- iodine ions located in neighbor anionic sites of the AgBr(I) crystal lattice, next to which the Ag _i ⁺ interstitial silver ion is positioned. With an increase in the temperature, the {(I _a ^- I _a ^- )Ag _i ⁺ } sites undergo structural transformation into the {(I _a ^- I _a ^- )Ag_in⁺} sites, where n = 2, 3, …. Moreover, the {(I _a ^- I _a ^- )Ag _in ⁺ } sites (n = 2) after the capture of an electron and hole also provide the tunnel recombination with a luminescence maximum at λ ≈ 720 nm. The influence of an emulsion medium consists in that gelatin interacts with the surface electron-localization sites, i.e., the interstitial silver ions Ag _in ⁺ , n = 1, 2, and forms the complexes {Ag _in ⁰ G⁺} (n = 1, 2) with them. The latter are deeper electron traps with a small capture cross section as compared to the Ag _in ⁺ sites (n = 1, 2) and that manifest themselves in that the kinetics of the luminescence growth in AgBr(I) to a stationary level at λ ≈ 560 nm is characterized by the presence of “flash firing.” At the same time, the luminescence flash stimulated by IR light, for which the Ag _in ⁺ (n = 1, 2) electron-localization sites are responsible, is absent. It is supposed that the electrons localized on the {Ag _in ⁺ G⁺} complexes (n = 2) retain the capability for emissive tunnel recombination with holes localized on paired iodine sites with a luminescence maximum at λ ≈ 750 nm.     

Light-cone structure of vertex functions for composite fields

Vertex functions for composite fields are defined in a model field theory both on and off mass shell. Light-cone dominance at large momentum transfer is shown to hold, by the compositeness assumption, for the off-shell vertex function. On the other hand, it is in general untrue that the elastic form factor probes light-like distances between the constituents inside the nucleon. The relevant light-cone singularity (in the relative space-time separation x) is less important in this case than the large x₀ behaviour of the wave function at fixed x². It is found however that, under some conditions, the light-cone singularity determines the large x₀ behaviour of the wave function, and therefore the large q² behaviour of the form factor. For composite particles described by a Bethe-Salpeter equation, this result is equivalent to the known fact that at large q² the form factor depends on the binding interaction at small distances. A relation similar to that of Drell-Yan-West is finally established between the asymptotic behaviour of the elastic form factor and the threshold behaviour of the absorptive part of the vertex scaling function.     

The Fixed Point of a Generalization of the Renormalization Group Maps for Self-Avoiding Paths on Gaskets

Let W(x,y) = ax ³+ bx ⁴+ f ₅ x ⁵+ f ₆ x ⁶+ (3 ax ²)² y+ g ₅ x ⁵ y + h ₃ x ³ y ² + h ₄ x ⁴ y ² + n ₃ x ³ y ³+a ₂₄ x ² y ⁴+a ₀₅ y ⁵+a ₁₅ xy ⁵+a ₀₆ y ⁶, and X = , , where the coefficients are non-negative constants, with a > 0, such that X ²(x,x ²)−Y(x,x ²) is a polynomial of x with non-negative coefficients. Examples of the 2 dimensional map Φ: (x,y)↦ (X(x,y),Y(x,y)) satisfying the conditions are the renormalization group (RG) maps (modulo change of variables) for the restricted self-avoiding paths on the 3 and 4 dimensional pre-gaskets. We prove that there exists a unique fixed point (x _f ,y _f ) of Φ in the invariant set . 2000 Mathematics Subject Classification Numbers: 82B28; 60G99; 81T17; 82C41.     

A new ℋ -space formalism

The equations governing geometrical objects in ? space are written in terms of operators adapted to families of left shear-freeσ ⁰= 0) cross sections of complexified null infinity (C I ⁺). The concept of ?-conformai weight (HCW) is introduced, and a derivative operatorI _a ^′ , which is closely connected with the covariant derivative but which (unlike the covariant derivative) maps objects having well-defined HCW to other such objects, is defined. A function ?, derived from the Gaussian curvature of left shear-free slicings ofC I ⁺ and having a well-defined HCW, is shown to contain all the curvature information for ? space.     

Asymptotics of Determinants of Bessel Operators

 For aL ^∞ (ℝ₊)∩L ¹ (ℝ₊) the truncated Bessel operator B _τ (a) is the integral operator acting on L ² [0,τ] with the kernel

Energy levels of the Schrödinger equation for some rational potentials in two-dimensional space using the inner product technique

The energy levels of a two-dimensional system are calculated for the rational potential,V(x, y; λ, g)=x ²+y ²+λ[x ²/(1+gx ²)+y ²/(1+gy ²)+a _xxx⁴+a _xyx² y ²+a _yyy⁴] using the inner product technique over a wide range of values of the perturbation parameters (g, λ) and for various eigenstates.     

Three classes of space-time with a uniformly accelerating and nonrotating gravitational source

Three classes of Petrov type-D nonrotating empty spaces, which we denote asA ₊ ^* ,A ₋ ^* , anda ₀ ^* , can be seen to represent, respectively, a gravitational field of a uniformly accelerating source which moves with velocity slower, faster than, or equal to that of light. The first two classes of space-time,A ₊ ^* andA ₋ ^* , approach to the last one,a ₀ ^* , as the acceleration becomes greater and greater. The inertial frame in a flat space-time, relative to which a faster-thanlight particle moving along a spacelike geodesic behaves as an ordinary particle, is given and discussed. The behaviors of Killing horizons in the classes,A ₊ ^* andA ₋ ^* are investigated. Comparisons are made between the analysis by Farhoosh and Zimmerman and ours. Our identification of the acceleration parameter seems to be more appropriate than that by the above authors.     

Spectroscopic investigations on NO+(x1∑+,a3∑+,A1Ⅱ) ion using multi-reference configuration interaction method and correlation-consistent sextuple b

Three low-lying electronic states (x¹∑⁺,a³∑⁺,and A¹Ⅱ) of NO⁺ ion are studied using the complete active space self-consistent-field (CASSCF) method followed by highly accurate valence internally contracted multi-reference configuration interaction (MRCI) approach in combination of the correlation-consistent sextuple basis set augmented with diffuse functions, aug-cc-pV6Z. The potential energy curves (PECs) of the NO⁺(x¹∑⁺,a³∑⁺,A¹Ⅱ) are calculated. Based on the PECs, the spectroscopic parameters R_e, D_e, ω_e, ω_eχ_e, α _e, B_e, and D₀ are reproduced, which are in excellent agreement with the available measurements. By numerically solving the radial Schrödinger equation of nuclear motion using the Numerov method, the first 20 vibrational levels, inertial rotation and centrifugal distortion constants of NO⁺(x¹∑⁺,a³∑⁺,A¹Ⅱ) ion are derived when the rotational quantum number J is equal to zero (J = 0) for the first time, which accord well with the available measurements. Finally, the analytical potential energy functions of these states are fitted, which are used to accurately derive the first 20 classical turning points when J = 0. These results are compared in detail with those of previous investigations reported in the literature.     

FTIR spectrum of vinyl fluoride near 3.6 μm: rovibrational analysis of the ν4+ν7 band and modelling Coriolis resonances in a seven-level polyad

ABSTRACT

The Fourier transform infrared (FTIR) spectrum of vinyl fluoride, H₂C=CHF, has been widely investigated in the region of the ν₄+ν₇ combination band around 2800 cm^?1 at a resolution of 0.005 cm^?1. This vibration of A' symmetry gives rise to an a/b-hybrid band with a predominant a-type component. The rovibrational structure is strongly perturbed and the analysis has been rather complicated since this combination band is involved at least in a seven-level interacting polyad, including the ν₈+2ν₁₀, 2ν₈+ν₁₀, 2ν₇+ν₉, ν₇+ν₈+ν₁₂, ν₅+ν₉+ν₁₀ and ν₇+ν₁₀+ν₁₂ vibrational states. The study has been further complicated by the absence of transitions coming from the perturbers that were considered as dark states. The spectral analysis resulted in the identification of 936 transitions with J" ≤ 46 and K_a" ≤ 11, all belonging to the a-type component. Most of the assigned data have been fitted using the Watson's A-reduction Hamiltonian in the I^r representation and proper Coriolis perturbation operators. The model employed includes seven different resonances within a complex polyad resonant system and a set of spectroscopic constants for the ν₄+ν₇ combination band, for the dark states, and Coriolis coupling coefficients have been determined.     

Dependence of electromagnetic form factors of hadrons on light-cone frames

A constituent quark model is developed for an arbitrary light-cone direction so that the light-front time is x_LF⁺ = ω · x with a constant lightlike four-vector ω. Form factors are obtained from free one-body electromagnetic current matrix elements. They are found to be ω-independent for spin-0 mesons, nucleons and the Λ-hyperon, while there is an ω-dependence for spin-1 systems like the deuteron.     

Universality Limits of a Reproducing Kernel for a Half-Line Schrödinger Operator and Clock Behavior of Eigenvalues

We extend some recent results of Lubinsky, Levin, Simon, and Totik from measures with compact support to spectral measures of Schrödinger operators on the half-line. In particular, we define a reproducing kernel S _L for Schrödinger operators and we use it to study the fine spacing of eigenvalues in a box of the half-line Schrödinger operator with perturbed periodic potential. We show that if solutions u(ξ, x) are bounded in x by ${e^{\epsilon x}}We extend some recent results of Lubinsky, Levin, Simon, and Totik from measures with compact support to spectral measures of Schr?dinger operators on the half-line. In particular, we define a reproducing kernel S _L for Schr?dinger operators and we use it to study the fine spacing of eigenvalues in a box of the half-line Schr?dinger operator with perturbed periodic potential. We show that if solutions u(ξ, x) are bounded in x by e^ex{e^{\epsilon x}} uniformly for ξ near the spectrum in an average sense and the spectral measure is positive and absolutely continuous in a bounded interval I in the interior of the spectrum with x₀ ? I{\xi_0\in I}, then uniformly in I,

\fracSL(x0 + a/L, x0 + b/L)SL(x0, x0)? \fracsin(pr(x0)(a - b))pr(x0)(a - b),\frac{S_L(\xi_0 + a/L, \xi_0 + b/L)}{S_L(\xi_0, \xi_0)}\rightarrow \frac{\sin(\pi\rho(\xi_0)(a - b))}{\pi\rho(\xi_0)(a - b)},