A study of some exact solutions to Einstein's field equations which yield separable Hamilton-Jacobi equations

Exact solutions to Einstein's field equations, which give rise to a Stäckel-separable Hamilton-Jacobi equation of the form $$,y,z)\left[ {X(x)\left( {\frac{{\partial S}}{{\partial x}}} \right)^2 - 2\left( {\frac{{\partial S}}{{\partial x}}} \right)\left( {\frac{{\partial S}}{{\partial t}}} \right) - 2\left( {\frac{{\partial S}}{{\partial y}}} \right)\left( {\frac{{\partial S}}{{\partial t}}} \right) + Z(z)\left( {\frac{{\partial S}}{{\partial z}}} \right)^2 - 2\left( {\frac{{\partial S}}{{\partial z}}} \right)\left( {\frac{{\partial S}}{{\partial t}}} \right) - F(x,y,z)\left( {\frac{{\partial S}}{{\partial t}}} \right)^2 } \right] = \lambda $$ are considered. It is shown that there are no solutions for whichD is a function ofx orz, orx andz. The exact solutions are of Petrov typeN and are plane polarized waves without rotation. Some of the solutions are given explicitly, up to two arbitary functions. For these solutions the Hamilton-Jacobi equation is reduced to an uncoupled set of first-order ordinary differential equations.     

On bounded solutions of a classical yang-mills equation

We discuss bounded solutions of the equation $$r^2 \left( {\frac{{\partial ^2 u}}{{\partial r^2 }} + \frac{{\partial ^2 u}}{{\partial t^2 }}} \right) = u^3 - u$$ in the halfspacer>0. All solutions depending only ont/r are characterized topologically. Then we prove the existence of infinite dimensional manifolds oft-periodic as well as nonperiodic solutions which are small in a suitable norm.     

An analytical expression for the specific capacity of lithium ion cells

The concentration of lithium ions in the cathode of lithium ion cells has been obtained by solving the materials balance equation $$\frac{{\partial c}}{{\partial t}} = \varepsilon ^{1/2} D\frac{{\partial ^2 c}}{{\partial x^2 }} + \frac{{aj_n (1--t_ + )}}{\varepsilon }$$ by Laplace transform. On the assumption that the cell is fully discharged when there are zero lithium ions at the current collector of the cathode, the discharge timet _d is obtained as $$\tau = \frac{{r^2 }}{{\pi ^2 \varepsilon ^{1/2} }}\ln \left[ {\frac{{\pi ^2 }}{{r^2 }}\left( {\frac{{\varepsilon ^{1/2} }}{J} + \frac{{r^2 }}{6}} \right)} \right]$$ which, when substituted into the equationC=It _d /M, whereI is the discharge current andM is the mass of the separator and positive electrode, an analytical expression for the specific capacity of the lithium cell is given as $$C = \frac{{IL_c ^2 }}{{\pi {\rm M}D\varepsilon ^{1/2} }}\ln \left[ {\frac{{\pi ^2 }}{2}\left( {\frac{{FDc_0 \varepsilon ^{3/2} }}{{I(1 - t_ + )L_c }} + \frac{1}{6}} \right)} \right]$$      

Three-loop on-shell charge renormalization without integration:\Lambda _{QED}^{\overline {MS} } to four loops

Using algebraic methods, we find the three-loop relation between the bare and physical couplings of one-flavourD-dimensional QED, in terms of Γ functions and a singleF ₃₂ series, whose expansion nearD=4 is obtained, by wreath-product transformations, to the order required for five-loop calculations. Taking the limitD→4, we find that the \(\overline {MS} \) coupling \(\bar \alpha (\mu )\) satisfies the boundary condition $$\begin{gathered} \frac{{\bar \alpha (m)}}{\pi } = \frac{\alpha }{\pi } + \frac{{15}}{{16}}\frac{{\alpha ^3 }}{{\pi ^3 }} + \left\{ {\frac{{11}}{{96}}\zeta (3) - \frac{1}{3}\pi ^2 \log 2} \right. \hfill \\ \left. { + \frac{{23}}{{72}}\pi ^2 - \frac{{4867}}{{5184}}} \right\}\frac{{\alpha ^4 }}{{\pi ^4 }} + \mathcal{O}(\alpha ^5 ), \hfill \\ \end{gathered} $$ wherem is the physical lepton mass and α is the physical fine structure constant. Combining this new result for the finite part of three-loop on-shell charge renormalization with the recently revised four-loop term in the \(\overline {MS} \) β-function, we obtain $$\begin{gathered} \Lambda _{QED}^{\overline {MS} } \approx \frac{{me^{3\pi /2\alpha } }}{{(3\pi /\alpha )^{9/8} }}\left( {1 - \frac{{175}}{{64}}\frac{\alpha }{\pi } + \left\{ { - \frac{{63}}{{64}}\zeta (3)} \right.} \right. \hfill \\ \left. { + \frac{1}{2}\pi ^2 \log 2 - \frac{{23}}{{48}}\pi ^2 + \frac{{492473}}{{73728}}} \right\}\left. {\frac{{\alpha ^2 }}{{\pi ^2 }}} \right), \hfill \\ \end{gathered} $$ at the four-loop level of one-flavour QED.     

Fractal wavelet dimensions and localization

In this paper we want to give a new definition of fractal dimensions as small scale behavior of theq-energy of wavelet transforms. This is a generalization of previous multi-fractal approaches. With this particular definition we will show that the 2-dimension (=correlation dimension) of the spectral measure determines the long time behavior of the time evolution generated by a bounded self-adjoint operator acting in some Hilbert space ?. It will be proved that for φ, ψ∈? we have $$\mathop {\lim \inf }\limits_{T \to \infty } \frac{{\log \int_0^T {d\omega \left| {\left\langle {\psi \left| {e^{ - iA\omega } } \right.\phi } \right\rangle } \right|^2 } }}{{\log T}} = - \kappa ^ + (2)$$ and that $$\mathop {\lim \sup }\limits_{T \to \infty } \frac{{\log \int_0^T {d\omega \left| {\left\langle {\psi \left| {e^{ - iA\omega } } \right.\phi } \right\rangle } \right|^2 } }}{{\log T}} = - \kappa ^ - (2),$$ wherek ^±(2) are the upper and lower correlation dimensions of the spectral measure associated with ψ and ?. A quantitative version of the RAGE theorem shall also be given.     

Collisional damping of transverse waves in a plasma

The effect of collisions on transverse waves in a homogeneous, field free plasma is investigated by means of Gross-Krook collision model. The dispersion relation is calculated by assuming the collision frequency to be small andKλ _D ?1. The damping rate ω _I is obtained as $$\omega _I = \frac{{\nu _{ei} }}{2}\frac{{\omega _p^2 }}{{\omega _0^2 }}\left[ {1 + \frac{{3K^2 \lambda _D^2 \omega _p^2 }}{{\omega _0^2 }} - \frac{{K^2 \lambda _D^2 \omega _p^4 }}{{\omega _0^4 }}} \right] + \frac{{\nu _{ee} }}{2}\frac{{\omega _p^2 }}{{\omega _0^2 }}\left( {\frac{{K^2 \lambda _D^2 \omega _p^2 }}{{\omega _0^2 }}} \right)$$ where ω ₀ ² =c ² K ²+ω ₂ ^p , andv _ei andv _ee are electron-ion and electron-electron collision frequency respectively.     

On relativistic deformable solids and the detection of gravitational waves

In this paper, the purpose of which is to complement a preceding work [1], it is shown, in agreement with the theory of relativistic deformable solids developed by A.C. Bringen and his coworkers, that the simplest conceivable dissipative constitutive equation — that of a socalled KelvinVoigt viscoelastic solid — yields a gravitational wave equation of propagation different from that of Weber: specifically, the following third order partial differential equation, $$\frac{{\partial ^2 \theta }}{{\partial t^2 }} - \left( {A + A'\frac{{\partial ^2 \theta }}{{\partial t}}} \right)\frac{{\partial ^2 \theta }}{{\partial x^2 }} = c^2 R_{1441'} $$ which can be solved by use of Fourier transform techniques, and where A and A′ are positive material coefficients.     

Light quark masses in quantum chromodynamics and chiral symmetry breaking

We derive model independent lower bounds for the sums of effective quark masses \(\bar m_u + \bar m_d \) and \(\bar m_u + \bar m_s \) . The bounds follow from the combination of the spectral representation properties of the hadronic axial currents two-point functions and their behavior in the deep euclidean region (known from a perturbative QCD calculation to two loops and the leading non-perturbative contribution). The bounds incorporate PCAC in the Nambu-Goldstone version. If we define the invariant masses \(\hat m\) by $$\bar m_i = \hat m_i \left( {{{\frac{1}{2}\log Q^2 } \mathord{\left/ {\vphantom {{\frac{1}{2}\log Q^2 } {\Lambda ^2 }}} \right. \kern-\nulldelimiterspace} {\Lambda ^2 }}} \right)^{{{\gamma _1 } \mathord{\left/ {\vphantom {{\gamma _1 } {\beta _1 }}} \right. \kern-\nulldelimiterspace} {\beta _1 }}} $$ and <F ²> is the vacuum expectation value of $$F^2 = \Sigma _a F_{(a)}^{\mu v} F_{\mu v(a)} $$ , we find, e.g., $$\hat m_u + \hat m_d \geqq \sqrt {\frac{{2\pi }}{3} \cdot \frac{{8f_\pi m_\pi ^2 }}{{3\left\langle {\alpha _s F^2 } \right\rangle ^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} }}} $$ ; with the value <α _u F ²?0.04GeV⁴, recently suggested by various analysis, this gives $$\hat m_u + \hat m_d \geqq 35MeV$$ . The corresponding bounds on \(\bar m_u + \bar m_s \) are obtained replacingm _π ² f _π bym _K ² f _K . The PCAC relation can be inverted, and we get upper bounds on the spontaneous masses, \(\hat \mu \) : $$\hat \mu \leqq 170MeV$$ where \(\hat \mu \) is defined by $$\left\langle {\bar \psi \psi } \right\rangle \left( {Q^2 } \right) = \left( {{{\frac{1}{2}\log Q^2 } \mathord{\left/ {\vphantom {{\frac{1}{2}\log Q^2 } {\Lambda ^2 }}} \right. \kern-\nulldelimiterspace} {\Lambda ^2 }}} \right)^d \hat \mu ^3 ,d = {{12} \mathord{\left/ {\vphantom {{12} {\left( {33 - 2n_f } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {33 - 2n_f } \right)}}$$ .     

Boundary regularity for some nonlinear elliptic degenerate equations

We consider the nonlinear elliptic degenerate equation (1) $$ - x^2 \left( {\frac{{\partial ^2 u}}{{\partial x^2 }} + \frac{{\partial ^2 u}}{{\partial y^2 }}} \right) + 2u = f(u)in\Omega _a ,$$ where $$\Omega _a = \left\{ {(x,y) \in \mathbb{R}^2 ,0< x< a,\left| y \right|< a} \right\}$$ for some constanta>0 andf is aC ^∞ functions on ? such thatf(0)=f′(0)=0. Our main result asserts that: ifu∈C \((\bar \Omega _a )\) satisfies (2) $$u(0,y) = 0for\left| y \right|< a,$$ thenx ^?2 u(x,y)∈C ^∞ \(\left( {\bar \Omega _{a/2} } \right)\) and in particularu∈C ^∞ \(\left( {\bar \Omega _{a/2} } \right)\) .     

Multiphonon thermal-field ionization of deep centers in semiconductors

The problem of thermal-field ionization of deep impurity centers in semiconductors is studied. It is shown that \(W_{ion} = W_0 e^{\alpha F^2 }\) , where F is the electric field strength. Also, the lifetime for multiphonon nonradiative capture is calculated as a function of F. It is shown that the relative change in lifetime is $$\frac{{\Delta \tau }}{{\tau ^0 }} = \frac{{\tau ---\tau _0 }}{{\tau _0 }} \approx - \alpha F^2 .$$      

Study of the mechanisms of pre-equilibrium nuclear reactions in the microscopic approach

The mechanisms of pre-equilibrium nuclear reactions are investigated within the Statistical Multistep Direct Process (SMDP) + Statistical Multistep Compound Process (SMCP) formalism. It has been shown that from an analysis of linear part in such dependences as $$\ln \left[ {{{\frac{{d^2 \sigma }}{{d\varepsilon _b d\Omega _b }}} \mathord{\left/ {\vphantom {{\frac{{d^2 \sigma }}{{d\varepsilon _b d\Omega _b }}} {\varepsilon _b^{1/2} }}} \right. \kern-\nulldelimiterspace} {\varepsilon _b^{1/2} }}} \right]upon\varepsilon _b $$ and $$\ln \left[ {{{\frac{{d\sigma ^{SMDP \to SMCP} }}{{d\varepsilon _b }}} \mathord{\left/ {\vphantom {{\frac{{d\sigma ^{SMDP \to SMCP} }}{{d\varepsilon _b }}} {\frac{{d\sigma ^{SMDP} }}{{d\varepsilon _b }}}}} \right. \kern-\nulldelimiterspace} {\frac{{d\sigma ^{SMDP} }}{{d\varepsilon _b }}}}} \right]upon{{U_B } \mathord{\left/ {\vphantom {{U_B } {\left( {E_a - B_b } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {E_a - B_b } \right)}}$$ one can extract information about the type of mechanism (SMDP, SMCP, SMDP→SMCP) and the number of stages of the multistep emission of secondary particles. In the above approach, we have discussed the experimental data for a broad class of reactions in various entrance and exit channels.     

Secular-free solution up to third order of a model nonlinear equation for dispersive waves

The perturbation method of Lindstedt is applied to study the non linear effect of a nonlinear equation $$\nabla ^2 {\rm E} - \frac{1}{{c^2 }}\frac{{\partial ^2 {\rm E}}}{{\partial t^2 }} - \frac{{\omega _0^2 }}{{c^2 }}{\rm E} + \frac{{2v}}{{c^2 }}\frac{{\partial {\rm E}}}{{\partial t}} + E^2 \left[ {\frac{{\partial {\rm E}}}{{\partial t}} \times A} \right] = 0,$$ where (A. E)=0 andA,c, ω ₀ andν are constants in space and time. Amplitude dependent frequency shifts and the solution up to third order are derived.     

On the similarity solution of evolution equation u t =H(x,t, u,u x,u xx , ...)

Let $$\begin{gathered} u^* = u + \in \eta (x,{\text{ }}t,{\text{ }}u), \hfill \\ \hfill \\ \hfill \\ x^* = x + \in \xi (x, t, u{\text{),}} \hfill \\ \hfill \\ \hfill \\ {\text{t}}^{\text{*}} = {\text{ }}t + \in \tau {\text{(}}x,{\text{ }}t,{\text{ }}u), \hfill \\ \end{gathered}$$ be an infinitesimal invariant transformation of the evolution equation u _t =H(x,t,u,?u/?x,...,? ⁿ :u/?x ⁿ . In this paper we give an explicit expression for \(\eta ^{X^i }\) in the ‘determining equation’ $$\eta ^T = \sum\limits_{i = 1}^n {{\text{ }}\eta ^{X^i } {\text{ }}\frac{{\partial H}}{{\partial u_i }} + \eta \frac{{\partial H}}{{\partial u_{} }} + \xi \frac{{\partial H}}{{\partial x}} + \tau } \frac{{\partial H}}{{\partial t}},$$ where u _i =? ⁱ u/?x ⁱ . By using this expression we derive a set of equations with η, ξ, τ as unknown functions and discuss in detail the cases of heat and KdV equations.     

On Lévy (or stable) distributions and the Williams-Watts model of dielectric relaxation

This paper is concerned with the Lévy, or stable distribution function defined by the Fourier transform $$Q_\alpha \left( z \right) = \frac{1}{{2\pi }}\int {_{ - \infty }^\infty \exp \left( { - izu - \left| u \right|^\alpha } \right)du} with 0< \alpha \leqslant 2$$ Whenα=2 it becomes the Gauss distribution function and whenα=1, the Cauchy distribution. Whenα≠2 the distribution has a long inverse power tail $$Q_\alpha \left( z \right) \sim \frac{{\Gamma \left( {1 + \alpha } \right)\sin \tfrac{1}{2}\pi \alpha }}{{\pi \left| z \right|^{1 + \alpha } }}$$ In the regime of smallα, ifα¦logz¦?1, the distribution is mimicked by a log normal distribution. We have derived rapidly converging algorithms for the numerical calculation ofQ _α (z) for variousα in the range 0<α<1. The functionQ _α (z) appears naturally in the Williams-Watts model of dielectric relaxation. In that model one expresses the normalized dielectric parameter as $$ \in _n \left( \omega \right) \equiv \in '_n \left( \omega \right) - i \in ''_n \left( \omega \right) = - \int {_0^\infty e^{ - i\omega t} \left[ {{{d\phi \left( t \right)} \mathord{\left/ {\vphantom {{d\phi \left( t \right)} {dt}}} \right. \kern-\nulldelimiterspace} {dt}}} \right]} dt$$ with $$\phi \left( t \right) = \exp - \left( {{t \mathord{\left/ {\vphantom {t \tau }} \right. \kern-\nulldelimiterspace} \tau }} \right)^\alpha $$ It has been found empirically by various authors that observed dielectric parameters of a wide variety of materials of a broad range of frequencies are fitted remarkably accurately by using this form ofφ(t).ε″ _n (ω) is shown to be directly related toQ _α (z). It is also shown that if the Williams-Watts exponential is expressed as a weighted average of exponential relaxation functions $$\exp - \left( {{t \mathord{\left/ {\vphantom {t \tau }} \right. \kern-\nulldelimiterspace} \tau }} \right)^\alpha = \int {_0^\infty } g\left( {\lambda , \alpha } \right)e^{ - \lambda t} dt$$ the weight functiong(λ, α) is expressible as a stable distribution. Some suggestions are made about physical models that might lead to the Williams-Watts form ofφ(t).     

A measurement of the branching ratio∑ +→pγ/∑ +→pπ 0⋆

In an experiment performed in the CERN SPS hyperon beam we have obtained a value for the branching ratio $${{\Sigma ^ + \to p\gamma } \mathord{\left/ {\vphantom {{\Sigma ^ + \to p\gamma } {\Sigma ^ + \to p\pi }}} \right. \kern-\nulldelimiterspace} {\Sigma ^ + \to p\pi }}^0 of\left( {2.46_{ - 0.35}^{ + 0.30} } \right) \times 10^{ - 3} ,$$ corresponding to a branching ratio $${{\Sigma ^ + \to p\gamma } \mathord{\left/ {\vphantom {{\Sigma ^ + \to p\gamma } {\Sigma ^ + \to all}}} \right. \kern-\nulldelimiterspace} {\Sigma ^ + \to all}}of\left( {1.27_{ - 0.18}^{ + 0.16} } \right) \times 10^{ - 3} .$$ This result is discussed in the context of present understanding of hyperon radiative decays.     

Time decay of solutions to the cauchy problem for time-dependent Schrödinger-Hartree equations

We consider the time-dependent Schrödinger-Hartree equation (1) $$iu_t + \Delta u = \left( {\frac{1}{r}*|u|^2 } \right)u + \lambda \frac{u}{r},(t, x) \in \mathbb{R} \times \mathbb{R}^3 ,$$ (2) $$u(0,x) = \phi (x) \in \Sigma ^{2,2} ,x \in \mathbb{R}^3 ,$$ where λ≧0 and \(\Sigma ^{2,2} = \{ g \in L^2 ;\parallel g\parallel _{\Sigma ^{2,2} }^2 = \sum\limits_{|a| \leqq 2} {\parallel D^a g\parallel _2^2 + \sum\limits_{|\beta | \leqq 2} {\parallel x^\beta g\parallel _2^2< \infty } } \} \) . We show that there exists a unique global solutionu of (1) and (2) such that $$u \in C(\mathbb{R};H^{1,2} ) \cap L^\infty (\mathbb{R};H^{2,2} ) \cap L_{loc}^\infty (\mathbb{R};\Sigma ^{2,2} )$$ with $$u \in L^\infty (\mathbb{R};L^2 ).$$ Furthermore, we show thatu has the following estimates: $$\parallel u(t)\parallel _{2,2} \leqq C,a.c. t \in \mathbb{R},$$ and $$\parallel u(t)\parallel _\infty \leqq C(1 + |t|)^{ - 1/2} ,a.e. t \in \mathbb{R}.$$      

Solutions for a Cosmic String

For a conformally flat metric ds² = a²(η)(dη² – dx² – y² – dz²) Vilenkin obtained the equation $$\frac{\partial }{{\partial _{\eta } }}\left[ {\frac{{a^2 \left( {\eta } \right)\dot y}}{{\sqrt {1 + y'^2 - \dot y^2 } }}} \right] = \frac{\partial }{{\partial _x }}\left[ {\frac{{a^2 \left( {\eta } \right)\dot y'}}{{\sqrt {1 + y'^2 - \dot y^2 } }}} \right]$$ for a cosmic string and gave some particular solutionsboth for a = const and a const. The present workcompletely solves the equation for a = const and extendthe work of Vilenkin for a ≠ const.     

Absence of long-range order in one-dimensional spin systems

For a one-dimensional Ising model with interaction energy $$E\left\{ \mu \right\} = - \sum\limits_{1 \leqslant i< j \leqslant N} {J(j - i)} \mu _\iota \mu _j \left[ {J(k) \geqslant 0,\mu _\iota = \pm 1} \right]$$ it is proved that there is no long-range order at any temperature when $$S_N = \sum\limits_{k = 1}^N {kJ\left( k \right) = o} \left( {[\log N]^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} } \right)$$ The same result is shown to hold for the corresponding plane rotator model when $$S_N = o\left( {\left[ {{{\log N} \mathord{\left/ {\vphantom {{\log N} {\log \log N}}} \right. \kern-\nulldelimiterspace} {\log \log N}}} \right]} \right)$$      

On the absence of breakdown of symmetry for the plane rotator model with long-range random interaction

We study the plane rotator model with hamiltonian $$ - \frac{1}{2}\sum\limits_{x \ne y} {J_{xy} \frac{{\cos (\theta _x - \theta _y )}}{{\left| {\left. {x - y} \right|} \right.^{3 + \in } }}}$$ whereJ _xy for different pair (x, y) are independent symmetric random variables. It is proved that for almost allJ, all the Gibbs statesP(J) are rotation invariant.     

More corrections to the sixth-order anomalous magnetic moment of the muon

The contribution to the sixth-order muon anomaly from second-order electron vacuum polarization is determined analytically to orderm _e/m _μ. The result, including the contributions from graphs containing proper and improper fourth-order electron vacuum polarization subgraphs, is $$\begin{gathered} \left( {\frac{\alpha }{\pi }} \right)^3 \left\{ {\frac{2}{9}\log ^2 } \right.\frac{{m_\mu }}{{m_e }} + \left[ {\frac{{31}}{{27}}} \right. + \frac{{\pi ^2 }}{9} - \frac{2}{3}\pi ^2 \log 2 \hfill \\ \left. { + \zeta \left( 3 \right)} \right]\log \frac{{m_\mu }}{{m_e }} + \left[ {\frac{{1075}}{{216}}} \right. - \frac{{25}}{{18}}\pi ^2 + \frac{{5\pi ^2 }}{3}\log 2 \hfill \\ \left. { - 3\zeta \left( 3 \right) + \frac{{11}}{{216}}\pi ^4 - \frac{2}{9}\pi ^2 \log ^2 2 - \frac{1}{9}log^4 2 - \frac{8}{3}a_4 } \right] \hfill \\ + \left[ {\frac{{3199}}{{1080}}\pi ^2 - \frac{{16}}{9}\pi ^2 \log 2 - \frac{{13}}{8}\pi ^3 } \right]\left. {\frac{{m_e }}{{m_\mu }}} \right\} \hfill \\ \end{gathered} $$ . To obtain the total sixth-order contribution toa _μ?a _e, one must add the light-by-light contribution to the above expression.