The Partition Formalism and New Entropic-Information Inequalities for Real Numbers on an Example of Clebsch–Gordan Coefficients

We discuss the procedure of different partitions in the finite set of N integer numbers and construct generic formulas for a bijective map of real numbers s _y , where y = 1, 2,…, N, N = \( \underset{k=1}{\overset{n}{\varPi}}{X}_k, \) and X _k are positive integers, onto the set of numbers s(y(x ₁, x ₂,…, x _n )). We give the functions used to present the bijective map, namely, y(x ₁, x ₂, …, x _n ) and x _k (y) in an explicit form and call them the functions detecting the hidden correlations in the system. The idea to introduce and employ the notion of “hidden gates” for a single qudit is proposed. We obtain the entropic-information inequalities for an arbitrary finite set of real numbers and consider the inequalities for arbitrary Clebsch–Gordan coefficients as an example of the found relations for real numbers.     

On the intrinsic transverse momentum of partons

Using simple assumptions it is shown that the average transverse momentum of partons is determined by their structure functions. The x-dependence obtained this way agrees well with the experimental data. The size is controlled by the invariant mass of the nucleon core and the behaviour of the structure functions near x = 1. We obtain a lower limit for the intrinsic contribution to 〈k_⊥〉_μμ of ～ 400 MeV.     

The optical constants of di-iso-octyl phthalate and di-n-butyl phthalate

The optical constants-refractive index (n) and absorptive index (k) have been found for di-iso-octyl phthalate [C₆H₄(COO · C₈H₁₇)₂] and di-n-butyl phthalate [C₆H₄(COO · C₄H₉)₂] for a wave-length range extending from 2 to 16 μm. The absorptive index has been found by transmission measurements using a variable path length cell. Relative specular reflectances, at a near normal angle of incidence, were measured with distilled water as the reflectance standard. The reflectances and the measured k values were substituted into a form of the Fresnel equation to give the corresponding refractive index values. In the wavelength regions where the k value proved too great to be measured directly by the transmission technique the n and k values were found using a Kramers-Kronig dispersion analysis.     

Low activation energies in superionic glasses in the 4.2–77 K temperature region

The peaks in the ultrasonic attenuation observed in AgI-doped superionic glasses at temperatures higher than 77 K have always been explained in terms of classical, thermally activated, relaxation processes due to silver ions which are mobile in the glassy matrix. Here we complete an ultrasonic study with the loss characteristics and sound velocity in “binary” (Ag₂O·nB₂O₃) and “ternary” [(AgI)_x(Ag₂O·nB₂O₃) _{1 − x}] glasses in the 4.2–77 K temperature region. In such a way we obtain useful information about defects having low activation energy.     

Built-in-polarization and thermal conductivity of InxGa1-xN/GaN heterostructures

The effect of built-in-polarization (BIP) field on thermal properties of In_xGa_1−xN/GaN heterostructure has been investigated. The thermal conductivity k of In_xGa_1−xN alloy has been estimated using Callaway's formula including the BIP field for In content x = 0, 0.1, 0.3, 0.5 and 0.9. This study reports that irrespective of In content, the room temperature k of In_xGa_1−xN/GaN heterostructure is enhanced by BIP field. The result predicts the existence of a characteristic temperature T_p at which both thermal conductivities (including and excluding BIP field) show a crossover. This gives signature of pyroelectric nature of In_xGa_1−xN alloy which arises due to variation of polarization with temperature indicating that thermal conductivity measurement can reveal pyroelectric nature. The pyroelectric transition temperature of In_xGa_1−xN alloy has been predicted for various x. The composition dependent nature of room temperature k for x = 0.1 and 0.5 are in line with prior experimental studies. The result can be used to minimize the self heating effect in In_xGa_1−xN/GaN heterostructures.     

Characteristic electron energy loss spectra for diamond

The complete set of optical fundamental functions is determined for diamond in the range from 4 to 32 eV. The features of the bulk and surface characteristic energy loss spectra are elucidated and the functions n _eff(E) and ?_eff(E) are calculated. The energies of volume and surface plasmons are established.     

Simultaneous solution of Kompaneets equation and radiative transfer equation in the photon energy range 1-125 keV

Radiative transfer equation in plane parallel geometry and Kompaneets equation is solved simultaneously to obtain theoretical spectrum of 1-125 keV photon energy range. Diffuse radiation field are calculated using time-independent radiative transfer equation in plane parallel geometry, which is developed using discrete space theory (DST) of radiative transfer in a homogeneous medium for different optical depths. We assumed free-free emission and absorption and emission due to electron gas to be operating in the medium. The three terms n, n² and (∂n/∂x_k) where n is photon phase density and x_k=(hν/kT_e), in Kompaneets equation and those due to free-free emission are utilized to calculate the change in the photon phase density in a hot electron gas. Two types of incident radiation are considered: (1) isotropic radiation with the modified black body radiation I^MB[1] and (2) anisotropic radiation which is angle dependent. The emergent radiation at τ=0 and reflected radiation τ=τ_max are calculated by using the diffuse radiation from the medium. The emergent and reflected radiation contain the free-free emission and emission from the hot electron gas. Kompaneets equation gives the changes in photon phase densities in different types of media. Although the initial spectrum is angle dependent, the Kompaneets equation gives a spectrum which is angle independent after several Compton scattering times.     

Effects of complex parameters on classical trajectories of Hamiltonian systems

Anderson et al have shown that for complex energies, the classical trajectories of real quartic potentials are closed and periodic only on a discrete set of eigencurves. Moreover, recently it was revealed that when time is complex t \((t=t_{r}\mathrm {e}^{i\theta _{\tau }}),\) certain real Hermitian systems possess close periodic trajectories only for a discrete set of values of ?? _τ . On the other hand, it is generally true that even for real energies, classical trajectories of non-PT symmetric Hamiltonians with complex parameters are mostly non-periodic and open. In this paper, we show that for given real energy, the classical trajectories of complex quartic Hamiltonians H=p ²+a x ⁴+b x ^k (where a is real, b is complex and k=1 or 2) are closed and periodic only for a discrete set of parameter curves in the complex b-plane. It was further found that the given complex parameter b, the classical trajectories are periodic for a discrete set of real energies (i.e., classical energy gets discretized or quantized by imposing the condition that trajectories are periodic and closed). Moreover, we show that for real and positive energies (continuous), the classical trajectories of complex Hamiltonian H = p ² + μx ⁴, μ = μ _r e ^i?? ) are periodic when ??=4 tan^?1[(n/(2m+n))] for ? n and \(m\in \mathbb {Z}\) .     

Gaussian Fluctuations of Eigenvalues in Wigner Random Matrices

We study the fluctuations of eigenvalues from a class of Wigner random matrices that generalize the Gaussian orthogonal ensemble. We begin by considering an n×n matrix from the Gaussian orthogonal ensemble (GOE) or Gaussian symplectic ensemble (GSE) and let x _k denote eigenvalue number k. Under the condition that both k and n?k tend to infinity as n→∞, we show that x _k is normally distributed in the limit. We also consider the joint limit distribution of eigenvalues $(x_{k_{1}},\ldots,x_{k_{m}})$ from the GOE or GSE where k ₁, n?k _m and k _i+1?k _i , 1≤i≤m?1, tend to infinity with n. The result in each case is an m-dimensional normal distribution. Using a recent universality result by Tao and Vu, we extend our results to a class of Wigner real symmetric matrices with non-Gaussian entries that have an exponentially decaying distribution and whose first four moments match the Gaussian moments.     

Integrable Structure of Ginibre’s Ensemble of Real Random Matrices and a Pfaffian Integration Theorem

In the recent publication (E. Kanzieper and G. Akemann in Phys. Rev. Lett. 95:230201, 2005), an exact solution was reported for the probability p _n,k to find exactly k real eigenvalues in the spectrum of an n×n real asymmetric matrix drawn at random from Ginibre’s Orthogonal Ensemble (GinOE). In the present paper, we offer a detailed derivation of the above result by concentrating on the proof of the Pfaffian integration theorem, the key ingredient of our analysis of the statistics of real eigenvalues in the GinOE. We also initiate a study of the correlations of complex eigenvalues and derive a formula for the joint probability density function of all complex eigenvalues of a GinOE matrix restricted to have exactly k real eigenvalues. In the particular case of k=0, all correlation functions of complex eigenvalues are determined.     

The exponential interaction in Rn

We prove that the Schwinger functions for the ultraviolet cut-off exponential interaction with euclidean measure exp {;?λ∫Λ:e^αξk(x):dx} dμ₀(ξ/ ∫ exp{?λ∫Λ:e^αξk(x):dx} dμ₀(ξ), λ > 0, converge as the ultraviolet cut-off is removed. The limits are the free Schwinger functions in the case of space-time dimension n ? 3. In the case n = 2 this holds for |α| sufficiently big, whereas for |α| < 2 √π, one has the well-known nontrivial Schwinger functions of the exponential interaction.     

Moderate Deviations for Random Field Curie-Weiss Models

The random field Curie-Weiss model is derived from the classical Curie-Weiss model by replacing the deterministic global magnetic field by random local magnetic fields. This opens up a new and interestingly rich phase structure. In this setting, we derive moderate deviations principles for the random total magnetization S _n , which is the partial sum of (dependent) spins. A typical result is that under appropriate assumptions on the distribution of the local external fields there exist a real number m, a positive real number ??, and a positive integer k such that (S _n ?nm)/n ^?? satisfies a moderate deviations principle with speed n ^1?2k(1???) and rate function ??x ^2k /(2k)!, where 1?1/(2(2k?1))<??<1.     

Boundary Structure and Module Decomposition of the Bosonic Z2 Orbifold Models with R=1/2k

The Z₂ bosonic orbifold models with compactification radius R²=1/2k are examined in the presence of boundaries. Demanding the extended algebra characters to have definite conformal dimension and to consist of an integer sum of Virasoro characters, one arrives at the right splitting of the partition function. This is used to derive a free field representation of a complete, consistent set of boundary states, compatible with the modular transformations of the characters. Finally the modules of the extended symmetry algebra that correspond to the finitely many characters are identified inside the direct sum of Fock modules that constitute the space of states of the theory.     

On the photogalvanic effect in asymmetric quantum wells of different shapes

In this work we studied the charge carriers' behaviour in quantum structures where the symmetry with respect to space coordinates and time-reversal symmetry are broken simultaneously. As the models of such structures we considered finite triangular as well as finite semi-parabolic quantum wells placed in external magnetic field. We have shown by numerical analysis that the energy spectra of charge carriers in such structures are anisotropic with respect to in-plane (transverse) motion ?n(+kx)≠?n(−kx). This leads to the anisotropy of charge carrier's in-plane momentum transfer which can be very naturally explained by introducing the concept of charge carriers ‘renormalized’ effective masses. The anisotropy of momentum transfer leads to interesting photo-galvanic effect, the anisotropy of photo-conductivity σ(+kx)≠σ(−kx) and as it follows from our calculations, the effect though not very great, could be measurable for the magnetic field of about few T.     

Remarks concerning the connection between properties of the 4-point-function and the Wilson-Zimmermann expansion

This paper concerns an investigation of the Wilson-Zimmermann (or “short distance”) expansion forA(x)A(y) withx →y whereA(x) is a real scalar field fulfilling Wightman's axioms. If one assumes that such an expansion exists, where the terms of the expansion are operators relatively local toA(x), then the singularities arising in the 4-point-function forx ₃ →x ₄ must control the singularities of then-point functions (n=4, 5, 6, ...) arising forx _j →x _x+1,j=1,2,...,n?1. A similar consequence can be drawn if the terms of the expansion are assumed to exist only as bilinear-forms (Section 2). For certain classes of fields one can show that this condition necessary for the short distance expansion is indeed fulfilled (Section 3). The result of the last section is that the above mentioned condition is also sufficient for the Wilson-Zimmermann expansion, interpreted as an expansion into bilinear forms, and also as an operator expansion in a somewhat modified sense.     

Effects of germanium incorporation on optical performances of silicon germanium passive devices for group-IV photonic integrated circuits

Optical interconnect in integrated optoelectronic circuits is one of the promising next-generation technologies for replacing metalized interconnect. Efforts have been made to use silicon (Si)-compatible materials such as germanium (Ge) and Ge-buffered III–V compound semiconductors, along with Si, as optical sources for Si and group-IV integrated optoelectronic systems. This opens the possibility that higher fraction of Ge with its high refractive index (n) can be incorporated in Si waveguide for optical interconnect and the graftability between Si and group-IV or III–V materials would be improved in silicon photonics. In this work, advantageous features of nano-structured silicon germanium (Si_1−xGe_x) optical waveguide with different Ge fraction (x) were evaluated by both optical simulations and theoretical calculations, which are mainly found in the enhanced optical confinement and better interfacing capability. Along with the SiGe waveguide, performance of Si_1−xGe_x microring resonator under material loss in the effect of extinction coefficient (k) has been investigated to suggest the necessity of optimizing the Ge content in Si_1−xGe_x passive devices. While carrying out the establish design criteria, n and k have been modelled in closed-form functions of Ge fraction at 1550 nm. Furthermore, by examining high-resolution transmission electron microscopy (HR-TEM) images, process compatibility of Ge with either group-IV alloys or III–V compound semiconductors is confirmed for the monolithically integrated photonic circuits.     

Logarithmic correlation functions in two-dimensional turbulence

We consider the correlation functions of two-dimensional turbulence in the presence and absence of a three-dimensional perturbation, by means of conformal field theory. In the presence of three-dimensional perturbation, we show that in the strong coupling limit of a small scale random force, there is some logarithmic factor in the correlation functions of velocity stream functions. We show that the logarithmic conformal field theory c_8,1 describes the 2D-turbulence both in the absence and in the presence of the perturbation. We obtain the energy spectrum E(k) ∼ k^−5.125 ln(k) for perturbed 2D-turbulence and E(k) ∼ k⁻⁵ ln(k) for unperturbed turbulence. Recent numerical simulation and experimental results confirm our prediction.     

A-dependence of shadowing and the small-x EMC data

We examine the gluon-interaction model for shadowing in the light of the new EMC data on nuclear structure functions. In the model, x_n and x_A characterise the points where shadowing begins abd becomes total. Good agreement with the data occurs when x_n varies with A but x_A does not - conversely to the original formulation of the model. The A-dependence of x_n can be described by the same dynamics that generates the A-dependence of the EMC effect at large x.     

Transverse instability of a plane front of fast impact ionization waves

The transverse instability of a plane front of fast impact ionization waves in p ⁺-n-n ⁺ semiconductor structures with a finite concentration of donors N in the n layer has been theoretically analyzed. It is assumed that the high velocity u of impact ionization waves is ensured owing to the avalanche multiplication of the uniform background of electrons and holes whose concentration ??_b ahead of the front is high enough for the continuum approximation to be applicable. The problem of the calculation of the growth rate s of a small harmonic perturbation with wavenumber k is reduced to the eigenvalue problem for a specific homogeneous Volterra equation of the second kind containing the sum of double and triple integrals of an unknown eigenfunction. This problem has been solved by the method of successive approximations. It has been shown that the function s(k) for small k values increases monotonically in agreement with the analytical theory reported in Thermal Engineering 58 (13), 1119 (2011), reaches a maximum s _M at k = k _M, then decreases, and becomes negative at k > k ₀₁. This behavior of the function s(k) for short-wavelength perturbations is due to a decrease in the distortion of the field owing to a finite thickness of the space charge region of the front and ??smearing?? of perturbation of concentrations owing to the transverse transport of charge carriers. The similarity laws for perturbations with k ? k _M have been established: at fixed ??_b values and the maximum field strength on the front E _0M, the growth rate s depends only on the ratio k/N and the boundary wavenumber k ₀₁ ?? N. The parameters s _M, k _M, and k ₀₁, which determine the perturbation growth dynamics and the upper boundary of the instability region for impact ionization waves, have been presented as functions of E _0M. These dependences indicate that the model of a plane impact ionization wave is insufficient for describing the operation of avalanche voltage sharpers and that fronts of fast streamers in the continuum approximation should be stable with respect to transverse perturbations in agreement with the previously reported numerical simulation results. The results have been confirmed by the numerical simulation of the evolution of small harmonic perturbations of the steady-state plane impact ionization wave.     

Constraints fulfilled by density moments and density correlations

The first set of equations we are going to consider follow from the conservation of additive quantum numbers. These constraints assume a simpler form than the corresponding relations for inclusive cross sections and inclusive correlations and can all be fulfilled consistently with any collection of higher order density correlation functions G_ms(x,k_⊥1,ν₁,…, x_m, k_⊥m,ν_m), m ≥ 2, identically zero. We will also derive a set of constraints from the requirements that the probability of any particular class of final states must be a number greater than or equal to zero, and the number of particles of certain type which appear in any region of momentum space must also be positive of zero. These relations can be fulfilled with any finite sequence of density correlations G_ms(x₁, k_⊥1,ν₁,…, x_m, k_⊥m, ν_m), m ≥ 3, identically equal to zero, or with all correlations of order m ≥ 2 identically zero.