Differential Geometry from Differential Equations

We first show how, from the general 3rd order ODE of the form , one can construct a natural Lorentzian conformal metric on the four-dimensional space . When the function satisfies a special differential condition the conformal metric possesses a conformal Killing field, , which in turn, allows the conformal metric to be mapped into a three dimensional Lorentzian metric on the space ) or equivalently, on the space of solutions of the original differential equation. This construction is then generalized to the pair of differential equations, z _ss =S(z,z _s ,z _t ,z _st ,s,t) and z _tt =T(z,z _s ,z _t ,z _st ,s,t), with z _s and z _t the derivatives of z with respect to s and t. In this case, from S and T, one can again, in a natural manner, construct a Lorentzian conformal metric on the six dimensional space (z,z _s ,z _t ,z _st ,s,t). When the S and T satisfy differential conditions analogous to those of the 3^rd order ode, the 6-space then possesses a pair of conformal Killing fields, and which allows, via the mapping to the four-space of (z,z _s ,z _t ,z _st ) and a choice of conformal factor, the construction of a four-dimensional Lorentzian metric. In fact all four-dimensional Lorentzian metrics can be constructed in this manner. This construction, with further conditions on S and T, thus includes all (local) solutions of the Einstein equations. Received: 10 October 2000 / Accepted: 26 June 2001     

Lifting <Emphasis Type="Italic">q</Emphasis>-difference operators for Askey-Wilson polynomials and their weight function

We determine an explicit form of a q-difference operator that transforms the continuous q-Hermite polynomials H _n (x|q) of Rogers into the Askey-Wilson polynomials p _n (x; a, b, c, d|q) on the top level in the Askey q-scheme. This operator represents a special convolution-type product of four one-parameter q-difference operators of the form ɛ _q (c _q D _q ) (where c _q are some constants), defined as Exton’s q-exponential function ɛ _q (z) in terms of the Askey-Wilson divided q-difference operator D _q . We also determine another q-difference operator that lifts the orthogonality weight function for the continuous q-Hermite polynomialsH _n (x|q) up to the weight function, associated with the Askey-Wilson polynomials p _n (x; a, b, c, d|q).     

Finsler spaces with Riemannian geodesics

In Finsler spaces the intervalds=F(x ⁱ ,dx ⁱ ) is an arbitrary function of the coordinatesx ⁱ and coordinate incrementsdx ⁱ withF homogeneous of degree one in thedx ⁱ . It is shown that for Riemannian spacesds _R ²=g _ij dx ⁱ dx ⁱ which admit a non trivial covariantly constant tensorH _i .(x ^k ) there is an associated Finsler space with the same geodesic structure. The subset of such Finsler spaces withH _i .(x ^k ) a vector or second rank decomposable tensor is determined.     

Generalized holographic and Ricci dark energy models

In this paper, we consider generalized holographic and Ricci dark energy models where the energy densities are given as ρ _R =3c ² M _pl² Rf(H ²/R) and ρ _h =3c ² M _pl² H ² g(R/H ²), respectively; here f(x), g(y) are positive defined functions of the dimensionless variables H ²/R or R/H ². It is interesting that holographic and Ricci dark energy densities are recovered or recovered interchangeably when the function f(x)=g(y)≡1 or f(x)=Id and g(y)=Id are taken, respectively (for example f(x),g(x)=1−ε(1−x), ε=0or1, respectively). Also, when f(x)≡xg(1/x) is taken, the Ricci and holographic dark energy models are equivalent to a generalized one. When the simple forms f(x)=1−ε(1−x) and g(y)=1−η(1−y) are taken as examples, by using current cosmic observational data, generalized dark energy models are considered. As expected, in these cases, the results show that they are equivalent (ε=1−η=1.312), and Ricci-like dark energy is more favored relative to the holographic one where the Hubble horizon was taken as an IR cut-off. And the suggested combination of holographic and Ricci dark energy components would be 1.312R−0.312H ², which is 2.312H²+1.312[(H)\dot]2.312H^{2}+1.312\dot{H} in terms of H ² and [(H)\dot]\dot{H} .     

Random Matrix Theory and L-Functions at s= 1/2

Recent results of Katz and Sarnak [8, 9] suggest that the low-lying zeros of families of L-functions display the statistics of the eigenvalues of one of the compact groups of matrices U(N), O(N) or USp(2N). We here explore the link between the value distributions of the L-functions within these families at the central point s= 1/2 and those of the characteristic polynomials Z(U,θ) of matrices U with respect to averages over SO(2N) and USp(2N) at the corresponding point θ= 0, using techniques previously developed for U(N) in [10]. For any matrix size N we find exact expressions for the moments of Z(U,0) for each ensemble, and hence calculate the asymptotic (large N) value distributions for Z(U,0) and log Z(U,0). The asymptotic results for the integer moments agree precisely with the few corresponding values known for L-functions. The value distributions suggest consequences for the non-vanishing of L-functions at the central point. Received: 1 February 2000 / Accepted: 24 March 2000     

Moments and Distributions of Trajectories in Slow Random Monads

Given a finite set B (basin) with n>1 elements, which we call points, and a map M:B→B, we call such pairs (B,M) monads. Here we study a class of random monads, where the values of M(⋅) are independently distributed in B as follows: for all a,b∈B the probability of M(a)=a is s and the probability of M(a)=b, where a≠b, is (1−s)/(n−1). Here s is a parameter, 0≤s≤1. We fix a point ⊙∈B and consider the sequence M ^t (⊙), t=0,1,2,… . A point is called visited if it coincides with at least one term of this sequence. A visited point is called recurrent if it appears in this sequence at least twice; if a visited point appears in this sequence only once, it is called transient. We denote by Vis _n , Rec _n and Tra _n the numbers of visited, recurrent and transient points respectively. We prove that, when n tends to infinity, Vis _n and Tra _n converge in law to geometric distributions and Rec _n converges in law to a distribution concentrated at its lowest value, which is one. Now about moments. The case s=1 is trivial, so let 0≤s<1. For any natural number k there is a number such that the k-th moments of Vis _n , Rec _n and Tra _n do not exceed this number for all n. About Vis _n : for any natural k the k-th moment of Vis _n is an increasing function of n. So it has a limit when n→∞ and for all n it is less than this limit. About Rec _n : for any k the k-th moment of Rec _n tends to one when n tends to infinity. About Tra _n : for any k the k-th moment of Tra _n has a limit when n tends to infinity.     

Calculation of the Hyperfinestructure and gJ-Values of the 3d 4s 4p-Configuration of Scandium

Wave functions for the 3d 4s 4p, 3d² 4p and 4s² 4p configurations of ScI are calculated, taking into account departures from SL-coupling and configuration interaction and on fitting the radial integrals to the experimental fine structure energies. Using these wave functions g J -values are derived. The intermediate coupled hfs matrix elements of the 3d 4s 4p configuration are reduced to the unknown electron coupling constants a_s, a_p and a_d and calculated on estimating these constants from the spin-orbit coupling constants and fitting them to some experimental A-values. By this way the absolute phases of the experimental A((¹P)²D)- and A((¹P)²F)-values are obtained. Good agreement between calculated and most experimental data has been achieved.     

An exact formulation of the Blume-Emery-Griffiths model on a two-fold Cayley tree model

A two-fold Cayley tree graph with fully q-coordinated sites is constructed and the spin-1 Ising Blume-Emery-Griffiths model on the constructed graph is solved exactly using the exact recursion equations for the coordination number q = 3. The exact phase diagrams in (kT/J, K/J ) and (kT/J, D/J) planes are obtained for various values of constants D/J and K/J, respectively, and the tricritical behavior is found. It is observed that when the negative biquadratic exchange (K) and the positive crystal-field (D) interactions are large enough, the tricritical point disappears in the (kT/J, K/J) plane. On the other hand, the system always exhibits a tricritical behavior in the phase diagram of (kT/J, D/J) plane. Received 8 June 2001 and Received in final form 28 September 2001     

Solutions of Doubly and Higher Order Iterated Equations

Michael Fisher once studied the solution of the equation f(f(x))=x ²–2. We offer solutions to the general equation f(f(x))=h(x) in the form f(x)=g(ag ^–1(x)) where a is in general a complex number. This leads to solving duplication formulas for g(x). For the case h(x)=x ²–2, the solution is readily found, while the h(x)=x ²+2 case is challenging. The solution to these types of equations can be related to differential equations.     

Weak and Partial Symmetries of Nonlinear PDE in Two Independent Variables

Abstract

Nonclassical infinitesimal weak symmetries introduced by Olver and Rosenau and partial symmetries introduced by the author are analyzed. For a family of nonlinear heat equations of the form u _t = (k(u) u _x)_x + q(u), pairs of functions (k(u), q(u)) are pointed out such that the corresponding equations admit nontrivial two-dimensional modules of partial symmetries. These modules yield explicit solutions that look like u(t, x) = F (θ(t) x + φ(t)) or u(t, x) = G(f(x) + g(t)).     

The Thermodynamic Evolution of the Cosmological Event Horizon

By manipulating the integral expression for the proper radius R _e of the cosmological event horizon (CEH) in a Friedmann-Robertson-Walker (FRW) universe we obtain an analytical expression for the change δR _e in response to a uniform fluctuation δρ in the average cosmic background density ρ. We stipulate that the fluctuation arises within a vanishing interval of proper time, during which the CEH is approximately stationary, and evolves subsequently such that δρ/ρ is constant. The respective variations 2πR _e δR _e and δE _e in the horizon entropy S _e and enclosed energy E _e should be therefore related through the cosmological Clausius relation. In that manner we find that the temperature T _e of the CEH at an arbitrary time in a flat FRW universe is E _e /S _e , which recovers asymptotically the usual static de Sitter temperature. Furthermore it is proven that during radiation-dominance and in late times the CEH conforms to the fully dynamical First Law T _e dS _e =PdV _e −dE _e , where V _e is the enclosed volume and P is the average cosmic pressure.     

Quantum BRST charge for quadratically nonlinear Lie algebras

We consider the construction of a nilpotent BRST charge for extensions of the Virasoro algebra of the form {T _a ,T _b }=f _ab ^c T _c +V _ab ^cd T _c T _d , (classical algebras in terms of Poisson brackets) and [T _a ,T _b ]=h _ab I+f _ab ^c T _c +V _ab ^cd (T _c T _d )(quantum algebras in terms of commutator brackets; normal ordering of the product (T _c T _d ) is understood). In both cases we assume that the set of generators {T _a } splits into a set {H _i } generating an ordinary Lie algebra and remaining generators {S }, such that only theV _ij are nonvanishing. In the classical case a nilpotent BRST charge can always be constructed; for the quantum case we derive a condition which is necessary and sufficient for the existence of a nilpotent BRST charge. Non-trivial examples are the spin-3 algebra with central chargec=100 and theso(N)-extended superconformal algebras with levelS=–2(N–3).     

Classification of bicovariant differential calculi on quantum groups of type A,B, C and D

Under the assumptions thatq is not a root of unity and that the differentialsdu _j ⁱ of the matrix entries span the left module of first order forms, we classify bicovariant differential calculi on quantum groupsA _n–1 ,B _n ,C _n andD _n . We prove that apart one dimensional differential calculi and from finitely many values ofq, there are precisely2n such calculi on the quantum groupA _n–1 =SL _q (n) forn3. All these calculi have the dimensionn ². For the quantum groupsB _n ,C _n andD _n we show that except for finitely manyq there exist precisely twoN ²-dimensional bicovariant calculi forN3, whereN=2n+1 forB _n andN=2n forC _n ,D _n . The structure of these calculi is explicitly described and the corresponding ad-invariant right ideals of ker are determined. In the limitq1 two of the 2n calculi forA _n–1 and one of the two calculi forB _n ,C _n andD _n contain the ordinary classical differential calculus on the corresponding Lie group as a quotient.     

The modified Korteweg-de Vries equation on the quarter plane with t-periodic data

We study the modified Korteweg-de Vries equation posed on the quarter plane with asymptotically t-periodic Dirichlet boundary datum u(0,t) in the sense that u(0,t) tends to a periodic function g?₀ (t) with period τ as t → ∞. We consider the perturbative expansion of the solution in a small ε > 0. Here we show that if the unknown boundary data u_x(0,t) and u_xx(0,t) are asymptotically t-periodic with period τ which tend to the functions g?₁ (t) and g?₂ (t) as t → ∞, respectively, then the periodic functions g?₁ (t) and g?₂ (t) can be uniquely determined in terms of the function g?₀ (t). Furthermore, we characterize the Fourier coefficients of g?₁ (t) and g?₂ (t) to all orders in the perturbative expansion by solving an infinite system of algebraic equations. As an illustrative example, we consider the case of a sine-wave as Dirichlet datum and we explicitly determine the coefficients for large t up to the third order in the perturbative expansion.     

Inverse resonance scattering for Jacobi operators

The Jacobi operator (Jf) _n = a _n−1 f _n−1 +a _n f _n+1 + b _n f _n on ℤ with real finitely supported sequences (a _n − 1) _n∈ℤ and (b _n ) _n∈ℤ is considered. The inverse problem for two mappings (including their characterization): (a _n , b _n , n ∈ ℤ) → {the zeros of the reflection coefficient} and (a _n , b _n , n ∈ ℤ) → {the eigenvalues and the resonances} is solved. All Jacobi operators with the same eigenvalues and resonances are also described.     

Orientational order parameter studies on 3.Om and 3O.Om liquid crystals

Orientational order parameter S is evaluated in the nematic phase of six liquid crystal compounds, N-(p-n-propyl benzylidene)-p-n-alkoxy anilines, 3.Om and N-(p-n-propyloxy benzylidene)-p-n-alkoxy anilines, 3O.Om compounds with m = 6, 7 and 8, using different methods. The techniques employed are S from birefringence δn, Haller's approximation from (1?T/T_c) ^β, effective geometry parameter α_g and Vuks’ scaling factor S_C. The values of S obtained using the above methods are compared with one another and with the results on a number of liquid crystals; the liquid crystals favor isotropic Vuks’ method.     

Symmetry Analysis of the Multidimensional Polywave Equation

Abstract

We present symmetry classification of the polywave equation □ ^l u = F (u). It is established that the equation in question is invariant under the conformal group C(1, n) iff F (u) = λe ^u, n + 1 = 2l or F (u) = λu ^{(n+1+2l)/(n+1?2l)}, n + 1 6= 2l. Symmetry reduction for the biwave equation □ ² u = λu ^k is carried out. Some exact solutions are obtained.     

Dispersive Estimates for Schrödinger Equations with Threshold Resonance and Eigenvalue

Let H=−Δ+V(x) be a three dimensional Schrödinger operator. We study the time decay in L^p spaces of scattering solutions e^−itHP_cu, where P_c is the orthogonal projection onto the continuous spectral subspace of L²(R³) for H. Under suitable decay assumptions on V(x) it is shown that they satisfy the so-called L^p-L^q estimates ||e^−itHP_cu||_p≤(4π|t|)^{−3(1/2−1/p)}||u||_q for all 1≤q≤2≤p≤∞ with 1/p+1/q=1 if H has no threshold resonance and eigenvalue; and for all 3/2<q≤2≤p<3 if otherwise.     

Nonlinear optical effects for plasma diagnostics

Experimental investigations show that the 1/f noise intensityC and the contact resistanceR can be used to analyse contacts. The simply prepared contacts are fritted by discharging a capacitor, resulting in a multi-spot contact. A model relatesC andR to a number of contact spotsk with radiusa. More impulse-frittings at increasing energies decreaseC andR, thus enhancing the values ofk anda. From experimentalC vsR plots two regions are determined for GaAs: A fritting (a=constant) and A+B fritting (a∝k). Calculated values ofk are in good agreement with the number of peaks or pits formed by etching the semiconductor surface. From experimentalC vsW orR vsW curves, withW the cumulative impulse-fritting energy, the conclusion can be made thatka ³ is proportional toW.