The incompressible limit in nonlinear elasticity

The incompressible limit in nonlinear elasticity is shown to fall under the theory of singular limits of quasilinear symmetric hyperbolic systems developed by Klainerman and Majda. Specifically, initial-value problems for a family of hyperelastic materials with stored energy functions $$W\left( {\frac{{\partial x}}{{\partial X}}} \right) = W_\infty \left( {\frac{{\partial x}}{{\partial X}}} \right) + \lambda ^2 w\left( {\det \frac{{\partial x}}{{\partial X}}} \right)$$ are considered, whereX andx are reference and deformed coordinates respectively. Under the assumption that the elasticity tensor $$A_{kl}^{ij} \equiv \frac{{\partial ^2 W_\infty }}{{\partial \left( {\frac{{\partial x^i }}{{\partial X^k }}} \right)\partial \left( {\frac{{\partial x^j }}{{\partial X^l }}} \right)}}$$ is positive definite near the identity matrix and thatw″(1)>0, the following results are proven for appropriate initial data: i) existence of solutions of the corresponding evolution equations on a time interval independent of λ as λ→∞, and ii) convergence as λ → ∞ of the solutions to a solution of the incompressible elastodynamics equations.     

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.     

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 $$C = \frac{{IL_c ^2 }}{{\pi {\rm M}D\varepsilon ^{1/2} }}\ln \left[ {\frac{{\pi ^2 }}{2}\left( {\frac{{FDc_0 \varepsilon ^{3/2} }}{{I(1 - t_ + )L_c }} + \frac{1}{6}} \right)} \right]$$      

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 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.     

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 ^∞ \(\left( {\bar \Omega _{a/2} } \right)\) .     

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.     

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.     

Proofs of two conjectures related to the thermodynamic Bethe Ansatz

We prove that the solution to a pair of nonlinear integral equations arising in the thermodynamic Bethe Ansatz can be expressed in terms of the resolvent kernel of the linear integral operator with kernel $$\frac{{e^{ - (u(\theta ) + u(\theta \prime ))} }}{{\cosh ^{\frac{{\theta - \theta \prime }}{2}} }}$$ .     

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.     

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 $$\frac{{\Delta \tau }}{{\tau ^0 }} = \frac{{\tau ---\tau _0 }}{{\tau _0 }} \approx - \alpha F^2 .$$      

Orthogonal polynomial solutions of the Fokker-Planck equation

We have tabulated the form of the coefficientsg ₁(x) andg ₂(x) as well as the boundary values [a, b] of the Fokker-Planck equation $$\frac{{\partial P(x, t)}}{{\partial t}} = - \frac{\partial }{{\partial x}}[g_1 (x)P(x, t)] + \frac{{\partial ^2 }}{{\partial x^2 }}[g_2 (x)P(x, t)],a \leqslant x \leqslant b$$ for which the solution can be written as an eigenfunction expansion in the classical orthogonal polynomials. We also discuss the problem of finding solutions in terms of the discrete classical polynomials for the differential difference equations of stochastic processes.     

On concentration of positive bound states of nonlinear Schrödinger equations

We study the concentration behavior of positive bound states of the nonlinear Schrödinger equation $$ih\frac{{\partial \psi }}{{\partial t}} = \frac{{ - h^2 }}{{2m}}\Delta \psi + V\left( x \right)\psi - \gamma \left| \psi \right|^{p - 1} \psi .$$ Under certain condition ofV, we show that positive ground state solutions must concentrate at global minimum points ofV ash→0⁺; moreover, a point at which a sequence of positive bound states concentrates must be a critical point ofV. In cases thatV is radial, we prove that the positive radial solutions with least energy among all nontrivial radial solutions must concentrate at the origin ash→0⁺.     

Emission of neutrinos by alternating fields

It is supposed that the effective Lagrangian of interaction of a magnetic field with a neutrino can be written in the form $$L_{eff} = \frac{{G_{\mathbf{\gamma }} }}{{m_W^2 }} \frac{{\partial ^2 A^\mu }}{{\partial x^v \partial x_v }}[\bar \Psi _v {\mathbf{\gamma }}_\mu (1 + {\mathbf{\gamma }}^5 )\Psi _v ].$$ Formulas are obtained for the emission of neutrinos by alternating fields. In particular, neutrino synchrotron emission and neutrino emission in the case of collision of two classical charges are considered. Arguments are presented that this mechanism can make a contribution to the neutrino luminosity of stars.     

Eine neue Methode zur Berechnung der elektrischen Leitfähigkeit von Halbleitern in starken homogenen elektrischen Feldern

The Boltzmann equation for electrons in a semiconductor is assumed to be of the form $$\frac{{\partial f}}{{\partial t}} + F \cdot \frac{{\partial f}}{{\partial k}} = \frac{{h - f}}{{\tau _0 }} + \frac{1}{{\tau \left( k \right)}} \cdot \frac{1}{{4\pi }}\int {d\Omega 'w\left( \theta \right)\left( {f\left( {k,\vartheta '} \right) - f\left( {k,\vartheta '} \right)} \right)} $$ whereh is the Maxwell-Boltzmann distribution. The energy surface structure of the lattice electronsE(k) is assumed to be spheric. The stationary solutions for strong electric fields show a concentration of electrons into the field direction (field orientation), if the elastic collision frequency is not too large. This means, at least for large energies, that nearly all electrons are in a cone with small aperture around the field direction. Every transport problem whose collision operator can be reduced to the upper form at least for large energies, can be solved by a perturbation method whose zeroth order is the ideal field orientation. The conditions for a field orientation of the electron distribution to exist will be investigated.     

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.     

Equations of electrodynamics with noninvariant velocity of light

A five-dimensional spaceV ^{5(t, x, c)}, in which the velocity of light c is the fifth coordinate of the event point together with the radius vectorx and time t is is introduced. The equations rot $$rot E = - \frac{1}{c}\frac{{\partial H}}{{\partial t}}; div E = 4\pi \rho ;$$ $$rot H = - \frac{1}{c}\frac{{\partial E}}{{\partial t}} + 4\pi \rho \frac{{o - Nx\dot c/c}}{{c + (1 - N)\dot ct}}; div H = 0,$$ invariant relative to the symmetry group c ₁₆₀₀ ^tf of D'Alembert's equation are constructed in the space V⁵ (E andH are the electric and magnetic fields;ν= dx/dt, c=dc/dt,N and N is a numerical parameter). The transformation properties of the electromagnetic field and the electric charge density ρ are examined for the finite group of transformationsP ₁₀ ^N ×Δ ₁ ^N ·T ₁ ^N . A theorem for adding velocities is formulated. The limiting transition to a Minkowski space M⁴(t,x)?V⁵(t, x,c) is examined.     

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 $$\left\langle {\bar \psi \psi } \right\rangle \left( {Q^2 } \right) = \left( {{{\frac{1}{2}\log Q^2 } \mathord{\left/ {\vphantom {{\frac{1}{2}\log Q^2 } {\Lambda ^2 }}} \right. \kern-\nulldelimiterspace} {\Lambda ^2 }}} \right)^d \hat \mu ^3 ,d = {{12} \mathord{\left/ {\vphantom {{12} {\left( {33 - 2n_f } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {33 - 2n_f } \right)}}$$ .     

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 $$\parallel u(t)\parallel _\infty \leqq C(1 + |t|)^{ - 1/2} ,a.e. t \in \mathbb{R}.$$