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The coupling between molecular excitations and nanoparticles leads to promising applications. It is for example used to enhance the optical cross-section of molecules in surface enhanced Raman scattering, Purcell enhancement or plasmon enhanced dye lasers. In a coupled system new resonances emerge resulting from the original plasmon (ωpl) and exciton (ωex) resonances as $$\displaystyle \begin{aligned} \omega_{\pm} = \frac{1}{2}(\omega_{pl} + \omega_{ex}) \pm \sqrt{\frac{1}{4}(\omega_{pl}-\omega_{ex})^2+g^2}, {} \end{aligned} $$ (1) where g is the coupling parameter. Hence, the new resonances show a separation of Δ = ω+ − ω− from which the coupling strength can be deduced from the minimum distance between the two resonances, Ω = Δ(ω+ = ω−).

The equation of motion is introduced in Chap. 9 as $$ m\ddot{u}+c\dot{u}+ ku=-m{\ddot{u}}_g, $$ or $$ \ddot{u}+2h{\omega}_0\dot{u}+{\omega}_0^2u=-{\ddot{u}}_g $$ The damping term is recognized as the velocity proportional term in general. There is, however, more damping in the practice, which are listed in the following and will be explained in this chapter

The Rabi model (Rabi, Phys Rev 51:652–654, 1937) describes the simplest interaction between a cavity mode with a frequency ω and a two-level system with a resonance frequency ω 0. The model is characterized by the Hamiltonian (Rabi, Phys Rev 51:652–654, 1937; Schweber, Ann Phys 41:205–229, 1967) $$\displaystyle{ \hat{H}_{R} = \hslash \omega 1\!\!1 \hat{a}^{\dag }\hat{a} + \hslash g\sigma _{ 1}(\hat{a}^{\dag } +\hat{ a}) +\mu \sigma _{ 3}, }$$ (30.1) where \(\hat{a}\) and \(\hat{a}^{\dag }\) are the conventional boson annihilation and creation operators satisfying commutation relation \([\hat{a},\hat{a}^{\dag }] = 1\), g is a coupling constant, \(\mu = \hslash \omega _{0}/2\), \( 1\!\!1 \) is the unit matrix, σ j are the Pauli matrices in their standard representation, and we set the reduced Planck constant \(\hslash = 1\). In the Bargmann space of entire functions (Bargmann, Commun Pure Appl Math 14:187–214, 1961), the eigenfunctions of the Rabi model can be determined in terms of an entire function $$\displaystyle{ \upvarphi (z) =\sum _{ n=0}^{\infty }\phi _{ n}z^{n}. }$$ (30.2)

In the parallel robotic manipulators with uncoupled Schonflies motions presented in this chapter each independent velocity of the moving platform depends on one actuated joint velocity \( v_{i} = v_{i} (\dot{q}_{i} ) \), i = 1, 2, 3 and \( \omega_{\delta } = \omega_{\delta } (\dot{q}_{4} ) \). The Jacobian matrix in Eq. (1.18) is diagonal and the parallel robot has uncoupled motions.

In the parallel robotic manipulators with decoupled Schonflies motions presented in this chapter each translational velocity of the moving platform depends on one actuated joint velocity \( v_{i} = v_{i} (\dot{q}_{i} ) \), i = 1, 2, 3 and the rotational velocity on two actuated joint velocities \( \omega_{\delta } = \omega_{\delta } (\dot{q}_{3} ,\dot{q}_{4} ) \). The Jacobian matrix in Eq. (1.18) is triangular and the parallel robot has decoupled motions.

In the overactuated parallel robotic manipulator with coupled Schonflies motions presented in this section each operational velocity depends on maximum three independent actuated joint velocities \( v_{1} = v_{1} (\dot{q}_{1} ,\dot{q}_{2} ,\dot{q}_{3} ),\;v_{2} = v_{2} (\dot{q}_{1} ,\dot{q}_{2} ,\dot{q}_{3} ),\;v_{3} = v_{3} (\dot{q}_{4} ) \) and \( \omega_{\delta } = \omega_{\delta } (\dot{q}_{1} ,\dot{q}_{2} ,\dot{q}_{3} ) \). In these topologies the operational velocity v 3 is uncoupled, but the Jacobian matrix in Eq. ( 1.18) is not triangular and the parallel robot always has coupled motions.

Hyperdiamonds are covalently bonded carbon phases, more or less related to the diamond network, having a significant amount of sp3 carbon atoms and similar physical properties. Many of them have yet a hypothetical existence but a well-theorized description. Among these, the diamond D5 was studied in detail, as topology, at TOPO GROUP CLUJ, Romania. The theoretical instrument used was the Omega polynomial, also developed in Cluj. It was computed in several 3D network domains and analytical formulas have been derived, not only for D5 but also for the well-known diamond D6 and other known networks.

We give an overview of our recent works in which the a.c. nonlinear dielectric response of an archetypical glassformer (glycerol) was measured close to its glass transition temperature T g . The purpose was to investigate the prediction that the nonlinear susceptibility is directly related to the number of dynamically correlated molecules N { corr} (T). We explain that two nonlinear susceptibilities are available, namely χ3 (3) and χ3 (1), which correspond respectively to the nonlinear cubic response at the third harmonics and at the first harmonics. We describe how to measure these nonlinear responses, even if they yield signals much smaller than that of the linear response. We show that both \(\vert {\chi }_{3}^{(3)}(\omega,T)\vert\) and \(\vert {\chi }_{3}^{(1)}(\omega,T)\vert\) are peaked as a function of the angular frequency ω and mainly obeys critical scaling as a function of ωτα(T), where τα(T) is the relaxation time of the liquid. Both χ3 (3) and χ3 (1) decay with the same power-law of ω beyond the peak. The height of the peak increases as the temperature approaches T g : This yields an accurate determination of the temperature dependence of N { corr} (T), once the contribution of saturation of dipoles is disentangled from that of dynamical glassy correlations.

As in the previous chapter, we consider homogeneous inclusions and inhomogeneities in subvolumes \( {\Omega_r} \) of an infinitely extended homogeneous volume \( {\Omega_{{0}}} \) of a comparison medium or ‘matrix’ of stiffness \( {{L}_0} \); the total volume \( \Omega = {\Omega_{{0}}} + {\Omega_r} \). In Sects. 5.1.4 and in 5.2 and 5.3, we examine composite aggregates with dilute reinforcement, which may consist of many distinct inhomogeneities \( {{L}_r} \) in a matrix \( {{L}6557} \), as described in Sect. 4.4. Systems containing cracks are discussed in Sect. 5.4. Loads applied to both single and multiple inhomogeneity systems include displacement or traction fields acting at a remote boundary to generate uniform overall strain or stress, and piecewise uniform, physically based eigenstrains in both matrix and inhomogeneities. Those include thermal and moisture-induced strains, phase transformations, and inelastic strains. Low loading rates causing only small strains are assumed.

We investigate ion-orbital interaction in Car-Parinnello molecular dynamics (CPMD) analytically and numerically in order to probe the role of the fictitious orbital mass. We show analytically that this interaction can be described by linearly coupled oscillators when the system is sufficiently close to the ground state. This leads to ionic vibrational modes with frequency ωM that depends upon the ionic mass M and the orbital mass μ as \( {{{\omega }}_{\text{M}}}{ = }{{{\omega }}_{\text{0M}}}{{[1 - {\text C}(\mu /M)]}} \) in the limit of zero μ/M; ω0M is the Born-Oppenheimer ionic frequency and C depends upon the ion-orbital coupling force constants. This analysis provides new insight on the orbital mass dependence of the dynamics, and suggests a rigorous method of obtaining accurate ionic vibrational frequency using CPMD. We verify our analytical results with numerical simulations for N2, and discuss in detail the dynamical interaction between the ionic and the fictitious orbital modes in CPMD. Our results demonstrate that displacement from the ground state significantly affects ionic frequencies. In the linear regime this results in the linear dependence of ionic vibrational frequency upon μ/M. In the non-linear regime, even the ionic geometry deviates from the correct ground-state structure, highlighting the importance of staying close to the ground state in CPMD calculations.

Design of hypothetical crystal networks, consisting of most pentagon rings and generically called diamond D5, is presented. It is shown that the seed and repeat-units, as hydrogenated species, show good stability, compared with that of C60 fullerene, as calculated at DFT levels of theory. The topology of the network is described in terms of the net parameters and Omega polynomial.

It was shown that the covering of C60 “Buckminsterfullerene” is basically sumanenic, with the empty π-electron faces being only pentagons. Four series of cages, tessellated by sumanenic patterns S[r] = [r:(5,6) r/2], were generated by sequences of map operations, and their topology described. Among these cages, which all show all_R[5] 2-factors, those belonging to the series designed on the dual pair Dodecahedron/ Icosahedron by iterating the P 5 operation and closing by Le operation, show a unique term Omega signature, thus being classified as the C60 series. C60 itself also shows the unique signature and all the members of its family show large HOMO-LUMO gap values, larger than of the cages belonging to the other three series herein built up. Coverings are given in terms of circulenes/flowers and the relation with the Omega and Ring polynomials is evidenced. Analytical formulas for the net parameters and for the used polynomials are given.

A counting polynomial C(G,x) is a sequence description of a topological property so that the exponents express the extent of its partitions while the coefficients are related to the occurrence of these partitions. Basic definitions and properties of the Omega polynomial Ω(G,x) and Sadhana polynomial Sd(G,x) are presented. These polynomials for some infinite classes of fullerenes and nanotubes are also computed.

Planar parallel robotic manipulator (PPM) withuncoupled motionswith various degrees of overconstraint may be obtained by using simple and/or complex limbs. In these solutions, each operational velocity given by Eq. (1.18) depends on just one actuated joint velocity: \({\it v}_1={\it v}_1(\dot{q}_1), {\it v}_2={\it v}_2(\dot{q}_2) \)and\(\omega_\delta=\omega_\delta(\dot{q}_3) \)The Jacobian matrix in Eq. (1.18) is a diagonal matrix.

Maximally regular planar parallel robots are actuated by one rotating and two linear actuators and can have various degrees of overconstraint. In these solutions, the three operational velocities are equal to their corresponding actuated joint velocities: \({\it v}_1=\dot{q}_1,{\it v}_2=\dot{q}_2\)and\(\omega_\delta=\dot{q}_3.\)The Jacobian matrix in Eq. (1.18) is the identity matrix. We call planar Isoglide3-T2R1 the parallel mechanisms of this family.

The solutions of spatial parallel mechanism with planar motion of the moving platform have in their structure at least one spatial limb. In the general case, in a spatial parallel robotic manipulator (SPM) with coupled planar motions of the moving platform each operational velocity depends in the general case on three actuated joint velocities. In this section we focus on the solutions with decoupled rotation of the moving platform with \({\it v}_{\it 1}={\it v}_{\it 1}({\it\dot{q}_{\it1}},{\it\dot{q}_{\it 2}}), {\it v}_{\it 2}={\it v}_{\it2}({\it\dot{q}_{\it 1}},{\it\dot{q}_{\it 2}})\)and\(\omega_\delta=\omega_\delta({\it\dot{q}}_{\it 3})\)In these solutions, the Jacobian matrix in Eq. (6.1) is not triangular and the parallel robot is considered withcoupled motions. They have just a few partially decoupled motions.

Maximally regularspatial parallel robotic manipulators (SPMs) are actuated by one rotating and two linear actuators and can have various degrees of overconstraint. In these solutions, the three operational velocities are equal to their corresponding actuated joint velocities: \({\it v}_1={\it\dot{q}_1}, {\it v}_{\it 2}={\it\dot{q}_ 2} \)and\(\omega_\delta={\it\dot{q}}_3.\)The Jacobian matrix in Eq. (1.18) is the identity matrix. We call spatial Isoglide3-T2R1with planar motion of the moving platform the parallel mechanisms of this family in which at least one limb is a spatial kinematic chain.

In the general case, in a planar parallel robotic manipulator (PPM) with coupled motions each operational velocity depends in the general case on three actuated joint velocities: \({\it v}_{\it 1}={\it v}_{\it 1}({\it\dot{q}_{\it 1}},{\it\dot{q}_{\it 2}},{\it\dot{q}_{\it 3}}), {\it v}_{\it 2}={\it v}_{\it 2}({\it\dot{q}_{\it 1}},{\it\dot{q}_{\it 2}},{\it\dot{q}_{\it 3}})\) In some specific solutions, one or two operational velocities depend on just one or two actuator velocities. \(\omega_\delta=\omega_\delta({\it\dot{q}}_ {\it1},{\it\dot{q}}_{\it 2},{\it\dot{q}}_{\it 3})\) In some specific solutions, one or two operational velocities depend on just one or two actuator velocities. We note that, in this particular case, the Jacobian matrix in Eq. (6.1) is not triangular and the parallel robot always has coupled motions. They have just a few partially decoupled motions.

In this chapter, we shall consider the problem in which a prismatic bar occupying the region z>0 is loaded by transverse forces Fx, Fy in the negative x- and y-directions respectively on the end z =0, the sides of the bar being unloaded. Equilibrium considerations then show that there will be shear forces $$V_x = \int {\int_\Omega {\sigma _{zx} dxdy} } = F_x ;{\rm }V_y = \int {\int_\Omega {\sigma _{zy} dxdy} } = F_y$$ (17.1) and bending moments $$M_x = \int {\int_\Omega {\sigma _{zz} ydxdy} } = zF_x ;{\rm }M_y \equiv - \int {\int_\Omega {\sigma _{zz} xdxdy} } = - zF_x $$ (17.2) at any given cross-section Ω of the bar. In other words, the bar transmits constant shear forces, but the bending moments increase linearly with distance from the loaded end.

This paper is concerned with computable error estimates for approximations to a boundary-value problem $$\mathrm{curl}\ {\mu }^{-1}\mathrm{curl}\ u + {\kappa }^{2}u = j\quad \textrm{ in }\Omega ,$$ where μ > 0 and κ are bounded functions. We derive a posteriori error estimates valid for any conforming approximations of the considered problems. For this purpose, we apply a new approach that is based on certain transformations of the basic integral identity. The consistency of the derived a posteriori error estimates is proved and the corresponding computational strategies are discussed.

The theory of adiabatic invariants has a long history, and very important implications and applications in many different branches of physics, classically and quantally, but is rarely founded on rigorous results. Here we treat the general time-dependent one-dimensional linear (harmonic) oscillator, whose Newton equation \(\ddot q + \omega ^2 \left( t \right)q = 0\) cannot be solved in general. We follow the time-evolution of an initial ensemble of phase points with sharply defined energy E 0 at time t = 0 (microcanonical ensemble) and calculate rigorously the distribution of energy E1 after time t = T, which is fully (all moments, including the variance μ2) determined by the first moment E¯1. For example, \({{\mu ^2 = E_0^2 \left[ {\left( {{{\bar E_1 }/{E_0 }}} \right)^2 - \left( {{{\omega \left( T \right)}/{\omega \left( 0 \right)}}} \right)^2 } \right]}/{2,}}\), and all higher even moments are powers of μ2, whilst the odd ones vanish identically. This distribution function does not depend on any further details of the function E(t) and is in this sense universal, it is a normalized distribution function given by \(P(x) = \pi^{-1} (2\mu^2 - x^2)^{-\frac{1}{2}}\), where \(x = E_1 - \bar{E}_1.\ \bar{E}_1\) and μ2 can be calculated exactly in some cases. In ideal adiabaticity \(\bar{E}_1 = \omega(T)E_0/\omega(0)\), and the variance μ2 is zero, whilst for finite T we calculate \(\bar{E}_1\) , and μ2 for the general case using exact WKB-theory to all orders. We prove that if ω(t) is of class C m (all derivatives up to and including the order m are continuous) μ2∞T −2(m+1), whilst for class C∞ it is known to be exponential μ2 exp(-α T). Due to the positivity of μ2 we also see that the adiabatic invariant \(I = \bar{E}_1/\omega(T)\) at the average energy \(\bar{E}_1\) never decreases.

For each graph edge with length X k we consider 1D nonlinear hyperbolic system of equations \( \overrightarrow \nu _t + \overrightarrow F _{x_k } \left( {\overrightarrow \nu } \right) = \overrightarrow f ,\overrightarrow \nu = \left\{ {\nu _1 , \ldots ,\nu _1 } \right\},t \geqslant 0,0 \leqslant x_k \leqslant X_k ,k = 1, \ldots ,K \) (1) with initial conditions \( \overrightarrow \nu \left( {0,x_k } \right) = \overrightarrow \nu ^0 \left( {x_k } \right),k = 1, \ldots ,K \) and the next boundary conditions: for graph enters \( \left( {l^0 = 1, \ldots L^0 ,x_{k_ * } = 0} \right)\varphi _{li}^0 \left( {t,\overrightarrow \nu \left( {t,0} \right)} \right) = 0,i = 1, \ldots r_k^0 \leqslant I \) (2), for graph exits \( \left( {l^ * = 1, \ldots L^ * ,x_k = X_k } \right)\varphi _{li} \left( {t,\overrightarrow \nu \left( {t,X_k } \right)} \right) = 0,i = 1, \ldots ,r_k^ * \leqslant I \) (3) and for graph branchpoints \( l^ * = 1, \ldots L\psi _{lm} \left( {t,w_l ,\overrightarrow \nu _{l1} , \ldots \overrightarrow \nu _{lM_1 } } \right) = 0m = 1, \ldots M_l \) (4). Here K is the number of graph edges, LO - enters, LO - exits, L - branchpoints, M l - incoming and outgoing graph edges for the lth branchpoint, \( \overrightarrow \nu _{l1} , \ldots \overrightarrow \nu _{lM_l } \) - required vectors in the ends of edges adjoining to branchpoin l, W l - required vector for the branchpoint l. The matrix \( {{\partial \overrightarrow F } \mathord{\left/ {\vphantom {{\partial \overrightarrow F } {\partial \overrightarrow \nu }}} \right. \kern-\nulldelimiterspace} {\partial \overrightarrow \nu }} = A = \left\{ {a_{ij} } \right\}i,j = 1, \ldots ,I \) is Jacobi matrix and we can apply the identity \( A = \Omega ^{ - 1} \Lambda \Omega \), where \( \Lambda = \left\{ {\lambda _i } \right\} \) is the diagonal matrix of the matrix A eigenvalues, Ώ is the nonsingular matrix whose rows are linearly independent left-hand eigenvectors of the matrix A \( \left( {Det\Omega \ne 0} \right) \) and Ώ −1 is the matrix inverse to Ώ.

We are interested in the problem of data completion, which consists in recovering the data on an inaccessible part of the boundary of a solid using overspecified data measured on another part of it. This is an old problem mathematically known as the Cauchy problem. This kind of problem arises in many industrial, engineering or biomedical applications under various forms: identification of boundary conditions or expansion of measured surface fields inside a body. But also it can be the first step in general parameters identification problems and damage detection where only partial boundary data are under control. Hence, robust and efficient data completion method is an essential and basic tool in structural identification. In this paper we present a method for data completion based on the minimization of an energy-like error functional which depends on lacking data. We consider homogeneous elastic solid Ω with smooth boundaries, where Γc is the part of the boundary where the data Uc and Fc, respectively the displacement and the pressure fields, are known, and Γi is the part of the boundary where the data have to be recovered. Following, (σ, e and u) will denotes the stress, strain and displacement fields. The problem can be stated as follows, find (Ui, Fi) on Γi such that: $$ \left\{ \begin{gathered} \begin{array}{*{20}c} {div\left( {\sigma \left( u \right)} \right) = 0} & {in \Omega ,} & {\sigma \left( u \right).n = F_c ,u = U_c on \Gamma _c } \\ \end{array} \hfill \\ \begin{array}{*{20}c} {\sigma \left( u \right) = \mathbb{C}:\varepsilon \left( u \right)} & {in \Omega ,} & {\sigma \left( u \right).n = F_i ,u = U_i on \Gamma _i } \\ \end{array} \hfill \\ \end{gathered} \right. $$ (1) where ℂ is the fourth-order elasticity tensor. In the approach presented here, we consider, for a given pair (f, d), the following two mixed and well-posed problems, whose solutions are denoted by u1 and u2: $$ \begin{array}{*{20}c} {\left\{ {\begin{array}{*{20}c} \begin{gathered} div\left( {\sigma \left( {u_1 } \right)} \right) = 0 \hfill \\ \sigma \left( {u_1 } \right) = \mathbb{C}:\varepsilon \left( {u_1 } \right) \hfill \\ \end{gathered} & \begin{gathered} in \Omega ,u_1 = U_c \hfill \\ in \Omega ,\sigma \left( {u_1 } \right).n = f \hfill \\ \end{gathered} & \begin{gathered} on \Gamma _c \hfill \\ on \Gamma _i \hfill \\ \end{gathered} \\ \end{array} } \right.} & {\left\{ {\begin{array}{*{20}c} \begin{gathered} div\left( {\sigma \left( {u_2 } \right)} \right) = 0 \hfill \\ \sigma \left( {u_2 } \right) = \mathbb{C}:\varepsilon \left( {u_2 } \right) \hfill \\ \end{gathered} & \begin{gathered} in \Omega , \sigma \left( {u_2 } \right).n = F_c \hfill \\ in \Omega , u_2 = d \hfill \\ \end{gathered} & \begin{gathered} on \Gamma _c \hfill \\ on \Gamma _i \hfill \\ \end{gathered} \\ \end{array} } \right.} \\ \end{array} $$ The displacements fields u1 and u2 are equal when the pair (f, d) meets the real data (U i F i ) on the boundary Γi. Hence, we propose to solve the data completion problem via the minimization of the energy functional: $$ \left( {F_i ,U_i } \right) = \mathop {\arg \min }\limits_{\left( {f,d} \right)} \int\limits_\Omega {\left( {\sigma \left( {u_1 } \right) - \sigma \left( {u_2 } \right)} \right):\left( {\varepsilon \left( {u_1 } \right) - \varepsilon \left( {u_2 } \right)} \right)} $$ To explore the efficiency of this method, several numerical examples are presented for 2D and 3D situations. The results are in good agreement with the actual ones. The method turns out to be very efficient with respect to the precision of the solution but also with respect to the amount of computation needed. The formulation is very general and can be used with heterogeneous materials and other physical phenomena. Some variant of the problem (1) will also be addressed illustrating the flexibility of the approach and potential applications in experimental methods.

The peculiar velocity reconstruction methods allow one to have a deeper insight into the distribution of dark matter: both to measure mean matter density and to obtain the primordial density fluctuations. We present here the Monge-Ampere-Kantorovitch method applied to mock catalogues mimicking in both redshift and distance catalogues. After having discussed the results obtained for a class of biases that may be corrected for, we focus on the systematics coming from the unknown distribution of unobserved mass and from the statistical relationship between mass and luminosity. We then show how to use these systematics to put constraints on the dark matter distribution. Finally a preliminary application to an extended version (c z < 3000 km/s) of the Neighbour Galaxy Catalogue is presented. We recover the peculiar velocities in our neighbourhood and present a preliminary measurement of the local Omega_M.

Let us consider the set of functions f (θ, φ), defined on the surface of a sphere in e3. The subset of square integrable uniform functions forms an infinite dimensional linear space, denoted by L2. This linear space becomes a unitary space if we introduce a scalar product defined by the relation $$\left. {f\left\langle {,g} \right.} \right\rangle = \int {\overline {f\left( {\theta ,\varphi } \right)} g\left( {\theta ,\varphi } \right)d\Omega ,} $$ (3.1.1) where dΩ=sinθ dθ dφ is the element of solid angle. The space L2 is complete, therefore it is also aHilbert space. To a rotation G of e3 there corresponds an operator T(G) acting in L2, which is unitary, linear, and generates a representation of SO (3), defined by the formula \(T\left( G \right)f\left( {\theta ,\varphi } \right) = f\left( {{G^{ - 1}}\left( {\theta ,\varphi } \right)} \right) = f\left( {\theta ',\varphi '} \right)\)

This paper adress to the equation \( (v - \omega {\partial _\omega })\bar \varphi (\omega ) = \frac{1}{{2\pi i}}P\int_0^\infty {\frac{{d\omega '}}{{\bar \varphi (\omega ) - \bar \varphi (\omega ')}}} \), in particular I will discuss solutions depending on the value of v. This equation is usefull for the study of the selfsimilar rolling-up of a semi-infinite vortex sheet.

The asymptotic interactions at large intermolecular distances are determined for two open-shell systems, H(X2S1/2) + O2 (X3∑{skg/−}) and \( O({}^3P_{j_O } ) + OH(X^2 \Pi _{\tilde \Omega } )\) for fixed values of intramolecular distances r. The electronic diabatic Hamiltonians are set up for two purposes: i) the direct diagonalization of the electronic Hamiltonian yielding two-dimensional potential energy surfaces (PES) which depend on the intermolecular distance R and the angle γ. between R and r, and ii) the incorporation of the diabatic electronic basis into the diabatic roronic basis which can be used in the construction of the roronic Hamiltonian in the total angular momentum representation. The former procedure allows one to compare the asymptotic PES with their ab initio counterparts, while the latter provides the input data for the calculation of low temperature capture rate constants within the statistical adiabatic channel model (SACM).

In the previous chapter we discussed some experiments which showed that particles like electrons, protons, neutrons, atoms, etc., exhibit wavelike properties. Indeed the wavelength is related to the momentum through the de Broglie relation: $$ \lambda = \frac{h}{p} $$ (1) or $$ p = \hbar k $$ (2) where k = 2π/λ and ħ = ħ/2π, h being the Planck’s constant. Thus we write $$ p = \hbar k $$ (3) where k denotes the wave vector. Further, as established by Einstein’s explanation of the photoelectric effect, the energy E of the particle is related to the frequency through the relation $$ E = \hbar \omega $$ (4)

We consider the linear elliptic system of two first-order equations $$ {{\partial }_{{\bar{z}}}}\omega + {{\mu }_{1}}\left( z \right){{\partial }_{z}}\omega + {{\mu }_{2}}\left( z \right)\overline {{{\partial }_{z}}\omega } = A\left( z \right)\omega + B\left( z \right)\bar{\omega } + F\left( z \right) $$ , where \( w\left( {z,\bar{z}} \right) = u + iv \) is an unknown complex-valued function, and the related integral operators and boundary value problems. We assume that \( A,B,F \in {{L}_{p}}\left( \Omega \right),p \leqslant 2 \), in contrast to the regular Vekua’s theory where p > 2. We prove that in this case the solutions of the system still preserve the properties, which correspond to the regular case with respect to: the structure of zeros, Liouville’s theorem, solvability of Riemann-Hilbert boundary value problems etc.

Let n ⩾ 2. Parametrize the hyperplanes in ℝ n by the unit normal vector and the distance to the origin: ξ W,d = {x ∈ ℝ n : (ω, x) = d}, where d ∈ ℝ and \( \omega \in \mathbb{S}^{n - 1} \). Assume that f ∈ L(ℝ n ). Then the Radon transform R f can be regarded as a function on \( \mathbb{S}^{n - 1} \times \mathbb{R} \) defined by the equality $$ Rf\left( {\omega ,d} \right) = \int\limits_{\xi w,d} {f\left( x \right)dm_{n - 1} \left( x \right)} , $$ (8.1) where dm n−1 is the (n− 1)-dimensional volume. By the Fubini theorem we see that the transform R is well defined for all \( \omega \in \mathbb{S}^{n - 1} \) and almost all d ∈ ℝ.

Throughout in this chapter we assume that n ⩾ 2. Let \( \mathcal{U} \) be a domain in ℝ n and let \( f \in L_{loc} \left( \mathcal{U} \right) \). For any \( x \in \mathcal{U} \) and almost all \( r \in \left( {0,dist\left( {x,\partial \mathcal{U}} \right)} \right) \) the spherical Radon transform of f is defined by $$ \mathcal{R}f\left( {x,r} \right) = \frac{1} {{\omega _{n - 1} }}\int\limits_{\mathbb{S}^{n - 1} } {f\left( {x + r\eta } \right)d\omega \left( \eta \right)} . $$ (1.1) (The reader should be warned that there does not seem to be a standard terminology in this area. Some authors use spherical Radon transform to refer to the transform \(\widehat f(\omega ,t)\) defined below in Section 1.2, which we have called the spherical Radon transform on spheres).

Let M be a real analytic manifold with a complex neighborhood X. Let P be a microdifferential operator defined in a neighborhood U in T * X ofq˙ ∈ T M * X\M. We assume that the characteristic variety of P satisfies $$Char(P) \subset \{ q \in U;{p_1}(q) \cdot {P_2}q = 0\} $$ with homogeneous holomorphic functions p 1 andp 2 on U. We assume that $${p_1}and{p_2}arerealveluedonT_M^*X,$$ (1) $$d{p_1} \wedge d{p_2} \wedge {\omega _X}(q) \ne 0if{p_1}(q) = {p_2}(q) = 0,$$ (2) $$\{ {p_1},{p_2}\} (q) = 0if{p_1}(q) = {p_2}(q) = 0.$$ (3)

Electron scattering is known to be one of the most powerful means to study both the structure of nuclei and the internal structure of the nucleon, especially the less known strange and axial form factors. In particular, inclusive (e,e′) processes at or near quasielastic peak kinematics have attracted attention in the last two decades and several experiments have been performed with the aim of disentangling the longitudinal and trans-verse contributions to the quasielastic cross section. These are linked to the hadronic tensor $$ {W^{\mu \nu }} = \mathop {\bar \sum }\limits_i \mathop \sum \limits_f \langle f\mid \mathop {{J^\mu }}\limits^ \wedge \mid i\rangle *\langle f\mid \mathop {{J^v}}\limits^ \wedge \mid i\rangle \delta ({E_i} + \omega -{E_f}) $$ (1) via the relations $$ {R^L}(q,\omega ) = {\left( {{{{q^2}} \over {{Q^2}}}} \right)^2}\left[ {{W^{00}} -{\omega \over q}({W^{03}} + {W^{30}}) + {{{\omega ^2}} \over {{q^2}}}{W^{33}}} \right] $$ (2) $$ {R^T}(q,\omega ) = {W^{11}} + {W^{22}}, $$ (3) where Q µ = (ω, q) is the four-momentum carried by the virtual photon, Ĵ µ is the nuclear many-body current operator and the nuclear states |i〉 and |f〉 are exact eigenstates of the nuclear Hamiltonian with definite fourmomentum. The general form (1) includes all possible final states that can be reached through the action of the current operator Ĵ µ on the exact ground state; here we focus on the one-particle one-hole (1p–1h) excitations.

We investigate the discrete real and complex solutions of the modal equation of an asymmetric pseudochiral slab waveguide, in which both the film and the substrate are pseudochiral media. When the pseudochiral parameter exceeds a certain transition value, power leakage occurs since one of the characteristic waves ceases to be internally reflected at the film-substrate interface. The analysis includes a parametric study of the effect of the media on the propagation characteristics of multilayered omega structures. The dependence of the propagation characteristics on the frequency is also addressed.

By extrapolating to O/H = N/H = 0 the empirical correlations Y -O/H and Y -N/H defined by a relatively large sample of ~ 45 Blue Compact Dwarfs (BCDs), we have obtained a primordial 4Helium mass fraction Y p = 0.2443 ± 0.0015 with dY/dZ = 2.4 ± 1.0. This result is in excellent agreement with the average Y p = 0.2452 ± 0.0015 determined in the two most metal-deficient BCDs known, I Zw 18 (Z ⊙/50) and SBS 0335–052 (Z⊙/41), where the correction for He production is smallest. The quoted error (lσ) of ≲ 1% is statistical and does not include systematic effects. We examine various systematic effects including collisional excitation of hydrogen lines, ionization structure and temperature fluctuation effects, and underlying stellar He i absorption, and conclude that combining all systematic effects, our Y p may be underestimated by ~ 2–4%. Taken at face value, our Y p implies a baryon-to-photon number ratio \(\eta = \left( {4.7_{ - 0.8}^{ + 1.0}} \right) \times {10^{ - 10}}\) and a baryon mass fraction \(\Omega bh_{100}^2 = 0.017 \pm 0.005\left( {2\sigma } \right)\) , consistent with the values obtained from deuterium and Cosmic Microwave Background measurements. Correcting Y p upward by 2–4% would make the agreement even better.

Twenty-five years ago in [Gir12, p.269] the general formula for the Stieltjes transform of limit normalized spectral function µ(u) of eigenvalues of the large order ACE-Gram matrices $$ \int_0^\infty {(1} + tu{)^{ - 1}}\;{\rm{d}}\mu {\rm{(}}u{\rm{) = }}\int_0^1 x {d_x}{G_t}(x),\;t > 0, $$ was found, where G t (x) satisfies the so-called Stochastic canonical equation for ACE-Gram matrices (Gram matrices with asymptotically constant entries): $$ {G_t}\;(x) = \;P\;\left\{ {\left. {\frac{1}{{1\; + \;t{\theta _1}\;{\rm{\{ }}\omega {\rm{, (1 + }}t{\theta _2}{\rm{\{ (*), }}{G_t}{\rm{(*)\} }}{{\rm{)}}^{ - 1}}{\rm{\} }}}}\;{\rm{ < }}x} \right\}} \right., $$ the random functional θ 1{ω, ✱} is given on the set of bounded random real continuous functions, the random functional θ 2{ω, G t (✱)} is given on the set of distribution functions G t (x) and these functionals are mutually independent.

At optical frequencies the dielectric response of metals is dominated by the plasma like behaviour of the electron gas: $$ \varepsilon \left( \omega \right) = 1 - \frac{{\omega _p^2}}{{\omega \left( {\omega + i\gamma } \right)}}.$$ There is a characteristic plasma frequency which is the natural frequency of oscillation of the electron gas $$ \omega _p^2 = \frac{{n{e^2}}}{{{\varepsilon _0}{m_e}}}.$$ Dissipation is introduced through the damping factor, γ, which in turn can be related to the conductivity of the metal if we assume that the same form persists to low frequencies, $$ \gamma = {\sigma ^{ - 1}}{\varepsilon _0}.$$ It is customary to ignore the dependence of e on wave vector, q, and this is a good approximation for many purposes. However in some circumstances, for example where we consider the response of nanostructures to light, we may have to worry about the short wavelength, large q behaviour of e. One obvious cut-off length is the separation between electrons in the metal. Another might be the inelastic scattering length for electrons which is typically a few nanometres. The very short wavelength response of metals at optical frequencies has been studied in the electron microscope where losses at large momentum transfers can be measured.

In general, there are two kinds of secondary currents. One is called the ‘secondary currents of Prandtl’s first kind’. The other is called the ‘secondary currents of Prandtl’s second kind’. The vorticity equation is obtained from the equations of motion in the vertical(y) and spanwise(z) directions, as follows: $$V\frac{{\partial \Omega }}{{\partial y}} + W\frac{{\partial \Omega }}{{\partial z}} = \frac{{{\partial ^2}}}{{\partial y\partial z}}\left( {\overline {{v^2}} - \overline {{w^2}} } \right) + \left( {\frac{{{\partial ^2}}}{{\partial {z^2}}} - \frac{{{\partial ^2}}}{{\partial {y^2}}}} \right)\overline {vw} + v{\nabla ^2}\Omega $$ (1) $$\Omega = \frac{{\partial W}}{{\partial y}} - \frac{{\partial V}}{{\partial z}}$$ (2) in which, V and W are the time averaged vertical and spanwise velocities, respectively. v and w are the turbulence fluctuations of the vertical and spanwise velocities, respectively. The first term and second term on the left-hand side are the advection terms. The first term and second term on the right-hand side are the generation term and Reynolds stress term, respectively. Nezu & Nakagawa(1984) have implied experimentally and Demuren & Rodi(1984) implied numerically that the generation term and the Reynolds-stress term are the predominant ones. However, they could not measure the Reynolds stress term of \(\overline {vw} \) . Therefore, the Reynolds-stress term was evaluated as (advection term)-(generation term)-(viscous term).

In this chapter, we consider degenerate linear random functionals in Theorem 13.1, i.e., functionals whose Laplace transforms are equal to $$ \begin{array}{*{20}{c}} {E exp \{ - s{\theta _{1\alpha }}\{ {\eta _\alpha }(*,*,t),z\} \} = \exp \left\{ {\;\left. { - s\;\int_0^\gamma {{\sigma ^2}} (\upsilon ,\;z)\;[E\eta \;(\omega ,\;\upsilon ,\;t)]\,d\upsilon } \right\},} \right.}\\ {E exp \{ - s{\theta _{2\alpha }}\{ {G_\alpha }(*,*,t),z\} \} = \exp \left\{ {\;\left. { - s\;\int_0^\gamma {{\sigma ^2}} (\upsilon ,\;z)\;[\int_0^1 {y\;{{\rm{d}}_y}G} \;(y,\;\upsilon ,\;t)]\,d\upsilon } \right\}.} \right.} \end{array} $$

The geodynamo process of magnetic field generation occurs in the outer core of the Earth and is described by 3D MHD-equations. The focus of this paper is on a numerical solution and thus the equations will be considered, without the loss of generality, in a simple Boussinesq approximation. Denoting B the magnetic field, v the velocity, p the pressure and T the temperature, the dimensionless equations read $$ \frac{{\partial B}} {{\partial t}}{\text{ = }}\nabla {\text{ }} \times {\text{ (v }} \times {\text{ B) + }}\nabla ^2 B, $$ (1) $$ R_0 (\frac{{\partial v}} {{\partial t}}{\text{ + v}} \cdot \nabla v){\text{ = }}\nabla p + F + E\nabla ^2 {\text{v,}} $$ (2) $$ \frac{{\partial T}} {{\partial t}}{\text{ + v}} \cdot \nabla {\text{ (}}T{\text{ + }}T_0 {\text{) = }}q\nabla ^2 T{\text{,}} $$ (3) $$ \nabla \cdot {\text{B = 0,}} $$ (4) $$ \nabla \cdot {\text{v = 0,}} $$ (5) where F is the sum of Archimedean, Corioliss and Lorenz forces $$ {\text{F = }}qR_a Tr1_r {\text{ - (v}} \times {\text{1}}_z {\text{) + }}\Lambda {\text{(}}\nabla \times {\text{B)}} \times {\text{B,}} $$ and where the following dimensionless numbers are introduced $$ \begin{gathered} E = \frac{v} {{2\Omega L^2 }}{\text{ [Ekman number], R}}_0 = \frac{\eta } {{2\Omega L^2 }}{\text{ [Rossby number],}} \hfill \\ \Lambda {\text{ = }}\frac{{B_0^2 }} {{2\Omega \eta \mu \rho }}{\text{ [The Elsasser number], q = }}\frac{\kappa } {\eta }{\text{ [Roberts number],}} \hfill \\ R_a = \frac{{\alpha g_0 \Theta L}} {{2\Omega \kappa }}[Raleigh number]. \hfill \\ \end{gathered} $$

Illuminating in a quasi-cross-stream plane of an open channel flow of water at low Re and injecting fluorescene dye from a bed slit, mushroom patterns reliably inferred to result from counter-rotating streamwise vortices are found to be typically oriented at some 30° from the cross-stream vertical. PIV in the cross-stream plane shows that the averaged field, and two-point correlation, of the quantity \({Q_x} = \left( {\omega _x^2 - S_x^2} \right)/2,\left( {{S_x} = \frac{{\partial v}}{{\partial z}} + \frac{{\partial w}}{{\partial y}}} \right)\) exhibit preferred directions in the cross-stream plane at about 45 degrees to the vertical. Spatio-temporal measurements of streamwise vorticity provide direct experimental evidence, possibly the first, for the staggered arrangement of streamwise vortices previously educed from DNS data.

It is argued from the pressure dependence of the transport properties that the Mott-Hubbard transition in the AMO3 perovskites is first-order with a dynamic phase segregation appearing in the metallic phase on the approach to the transition from the itinerant-electron side. A transition with decreasing bandwidth from a strongly enhanced Pauli paramagnetism toward a Curie-Wiess law is observed in the metallic CuO3 array of La1-xNdxCuO3,0 ≤ x ≤ 0.6, and in the metallic phase of the compounds LnNiO3, Ln = La, Pr, Nd, Sm0.5Nd0.5. The LnNiO3 family undergoes an antiferromagnetic-insulator to metal transition at a temperature Tt that is sensitive to 18O/16O isotope exchange and disappears in LaNiO3 and PrNiO3 under 15 kbar hydrostatic pressure. We suggest a bandwidth of the form $$ W = W_b exp( - \lambda \varepsilon _{sc} /h\omega _O ) $$ where λ ∼ ɛSC/Wb, Wb is the tight-binding bandwidth, and a pressure-sensitive ω−1 0 is the period of the locally cooperative oxygen displacements that define strong-correlation fluctuations stabilized by an energy ɛsc. LaMnO3 exhibits an insulator-conductive electronic transition at a cooperative Jahn-Teller orbital ordering below TJT; the magnetic susceptibility obeys a Curie-Weiss law in which ?, but not C, changes discontinuously at TJT. We propose a double-exchange coupling involving vibrons.

Loeb measures have been applied in various fields of real analysis. In his fundamental paper [9] Peter Loeb has given the first applications to probability theory. Also developed at that time (and published later in [10]) was an application constructing representing measures in potential theory. (See Section 3.12.2.) The next convincing example of the usefulness of Loeb measures is Bob Anderson’s [2] construction of Brownian motion from a hyperfinite model of tossing an unbiased coin. Let us briefly sketch Anderson’s approach: Fix an unlimited positive integer H} and put T := { 1,...,H} This set T is infinite, but * finite, and can be interpreted as a “time line”, which is closely related to the continuous time line [0,1], because each real number between 0 and 1 is infinitely close to some \( \frac{k} {H} \) with k ∈T. Let { -1, l} T be the set of all internal H-tuples of the numbers -1 and 1. As noted in Section 3.12, this set can be interpreted as the set of all outcomes of tossing a coin H-times. Anderson defines an internal process A : -1,1 T × T → *ℝ by setting $$ A\left( {\omega ,t} \right): = \sum\limits_{s < t} {\omega \left( s \right)} \frac{1} {{\sqrt H }}; $$ A (ωt) can be understood as the profit (or loss) at time t during the game ω ∈ if the gamblers are playing for the infinitesimal stake of \( \frac{1} {{\sqrt H }} \) (dollar, mark, euro, lira pound sterling, it always remains an infinitestimal amount of money). for example, if ω (1) = -1 and ω (t)= 1 for each t ∈ T with t ω 1, then \( A\left( {\omega ,t} \right) = \left( {t - 2} \right)\frac{1} {{\sqrt H }} \) for each t ∈ T. Therefore A (ωt) may be unlimited.

When solving problems of the classical calculus of variations and examining the solution of the Euler equation obtained via second-order conditions, there arises the problem of verifying the positive semi-definitness of the integral quadxatic form. This form looks as follows: $$U(x) = \int_0^1 {\left\langle {A(t)\dot x(t),\dot x(t)} \right\rangle \quad + \left\langle {B(t)x(t),x(t)} \right\rangle \quad + 2\left\langle {C(t)\dot x(t),x(t)} \right\rangle dt\quad + \left\langle {\Omega (x(0),x(1)),(x(0),x(1))} \right\rangle } $$ (1.1)

We are going, in this chapter, to give all the local models, up to isomorphism, of Pfaffian systems of constant rank and class on ℝ n with 3 ≤ n ≤ 5. Let us point out some conventions and notations used in the preceding chapters. Let (S) be a Pfaffian system defined by the equations $$\left\{ \begin{gathered} {\omega _1} = 0 \hfill \\ \vdots \hfill \\ {\omega _r} = 0 \hfill \\ \end{gathered} \right. $$ .

We study solutions to the 3D Euler vorticity equation of the form \(\omega = \tilde \omega (x,t)\left( {\frac{{\partial t}}{{\partial {x_2}}},\frac{{\partial t}}{{\partial {x_1}}},0} \right) \) in a neighborhood U. When the curvesf (x 1 x 2 t) = const are circles then these solutions are the well known axisymmetric 3D flow without swirl, and for this case there is no vortex stretching. If we assumef(x 1,x2, t) = const to be a set of curves that contain a simple hyperbolic saddle then vortex stretching may take place. We show that the angle of the saddle can not close faster than a double exponential in time and there is no breakdown. Similar results are obtain in two dimensional models.

White noise analysis was initiated by T. Hida in 1975. This is an infinite dimensional stochastic analysis, the basic idea of which is to view Wiener functionals as functionals of white noise. More precisely, let Ω denote the space of all continuous functions f on ∝, null at 0, equipped with the topology of uniform convergence on bounded sets. Then Ω is a Prechet space. Let B(Ω) denote the Borel σ-field on Ω and ℙ the standard Wiener measure on (Ω,B(Ω)). Put \( Wt(\omega ) = \omega (t), t \in , \omega \in \Omega . \)