Abstract: We present a recently developed approach to calculate electron-hole excitations and the optical spectra of condensed matter from first principles The key concept is to describe the excitations of the electronic system by the corresponding one- and two-particle Green's function The method combines three computational techniques First, the electronic ground state is treated within density-functional theory Second, the single-particle spectrum of the electrons and holes is obtained within the $\mathrm{GW}$ approximation to the electron self-energy operator Finally, the electron-hole interaction is calculated and a Bethe-Salpeter equation is solved, yielding the coupled electron-hole excitations The resulting solutions allow the calculation of the entire optical spectrum This holds both for bound excitonic states below the band gap, as well as for the resonant spectrum above the band gap We discuss a number of technical developments needed for the application of the method to real systems To illustrate the approach, we discuss the excitations and optical spectra of spatially isolated systems (atoms, molecules, and semiconductor clusters) and of extended, periodic crystals (semiconductors and insulators)

Abstract: An efficient method for the calculation of Breit-Pauli spin-orbit matrix elements for internally contracted multireference configuration interaction wavefunctions is presented. Instead of taking all two-electron contributions of the wavefunction explicitly into account, the most important two-electron contributions of the spin-orbit operator are incorporated by means of an effective one-electron Fock operator. As a further refinement, explicit two-electron contributions can be reinstated for the dominant all-internal parts of the wavefunctions.

Abstract: The existing methods to estimate the magnitude of spin–orbit coupling for arbitrary molecules and multiconfigurational wave functions are reviewed. The form-factor method is extended from the original singlet–triplet formulation into arbitrary multiplicities. A simplified version of the mean-field method (the partial two-electron method, P2E) is formulated and tested versus the full two-electron operator on a set of representative molecules. The change of the one and two-electron spin–orbit coupling down the Periodic Table is investigated, and it is shown that the computationally much less demanding P2E method has an accuracy comparable to that of the full two-electron method.

Abstract: The aim of this paper is to present the Ordered Weighted Geometric (OWG) operator. The OWG operator is based on the geometric mean and the OWA operator. It is a fuzzy majority guided aggregation operator proposed to aggregate information given on a ratio scale. Therefore, it allows us to incorporate the concept of fuzzy majority in problems where the information is provided using a ratio scale. Its properties are studied and an application for multicriteria decision making problems with multiplicative preference relations is presented [1].

Abstract: We consider the following eigenvalue optimization problem: Given a bounded domain Ω⊂ℝ and numbers α > 0, A∈[ 0, |Ω|], find a subset D⊂Ω of area A for which the first Dirichlet eigenvalue of the operator −Δ+αχ D is as small as possible. We prove existence of solutions and investigate their qualitative properties. For example, we show that for some symmetric domains (thin annuli and dumbbells with narrow handle) optimal solutions must possess fewer symmetries than Ω on the other hand, for convex Ω reflection symmetries are preserved. Also, we present numerical results and formulate some conjectures suggested by them.

Abstract: We analyze further the IR singularities that appear in noncommutative field theories on R^d. We argue that all IR singularities in nonplanar one loop diagrams may be interpreted as arising from the tree level exchanges of new light degrees of freedom, one coupling to each relevant operator. These exchanges are reminiscent of closed string exchanges in the double twist diagrams in open string theory. Some of these degrees of freedom are required to have propagators that are inverse linear or logarithmic. We suggest that these can be interpreted as free propagators in one or two extra dimensions respectively. We also calculate some of the IR singular terms appearing at two loops in noncommutative scalar field theories and find a complicated momentum dependence which is more difficult to interpret.

Abstract: We present an approach and numerical results for scaling up fine grid information to coarse scales in an approximation to a nonlinear parabolic system governing two-phase flow in porous media The technique allows upscaling of the usual parameters porosity and relative and absolute permeabilities, and also the location of wells and capillary pressure Some of these are critical nonlinear terms that need to be resolved on the fine scale, or serious errors will result Upscaling is achieved by explicitly decomposing the differential system into a coarse-grid-scale operator coupled to a subgrid-scale operator, which we localize by imposing a closure assumption We approximate the coarse-grid-scale operator with a mixed finite element method that has a second order accurate velocity coupled implicitly to the subgrid scale The subgrid-scale operator is approximated locally by a first order accurate mixed method A numerical Greens influence function technique allows us to solve these subgrid problems independently of the coarse-grid approximation No explicit macroscopic coefficients nor pseudo-functions result The method is easily seen to be optimally convergent in the case of a single linear parabolic equation

Abstract: We apply a versatile numerical technique to establishing the existence of cavity solitons (CS) in a semiconductor microresonator with bulk GaAs or multiple quantum well GaAs/AlGaAs as its active layer. Based on a Newton method, our approach implies the evaluation of the linearized operator describing deviations from the exact stationary state. The eigenvalues of this operator determine the dynamical stability of the CS. A typical eigenspectrum contains a zero eigenvalue with which a ``neutral mode'' of the CS is associated. Such neutral modes are characteristic of models with translational symmetry. All other eigenvalues typically have negative real parts large enough to cause any excitations to die out in a few medium response times. The neutral mode thus dominates the response to external random or deterministic perturbations, and its excitation induces a simple translation of the CS, which are thus stable and robust. We show how to relate the speed with which a CS moves under external perturbations to the projection of the perturbations on to the neutral mode, and give some examples, including weak gradients on the driving field and interaction with other CS. Finally, we show that the separatrix between two stable coexisting solutions: the homogeneous solution and the CS is the intervening unstable CS solution. Our results are important with a view to future applications of CS to optical information processing.

Abstract: We discuss the existence of large isolated (non-unit) eigenvalues of the Perron-Frobenius operator for expanding interval maps. Corresponding to these eigenvalues (or `resonances') are distributions which approach the invariant density (or equilibrium distribution) at a rate slower than that prescribed by the minimal expansion rate. We consider the transitional behaviour of the eigenfunctions as the eigenvalues cross this `minimal expansion rate' threshold, and suggest dynamical implications of the existence and form of these eigenfunctions. A systematic means of constructing maps which possess such isolated eigenvalues is presented.

Abstract: For the integrable case of the discrete self-trapping (DST) model we construct a Backlund transformation. The dual Lax matrix and the corresponding dual Backlund transformation are also found and studied. The quantum analogue of the Backlund transformation (Q -operator) is constructed as the trace of a monodromy matrix with an infinite-dimensional auxiliary space. We present the Q -operator as an explicit integral operator as well as describing its action on the monomial basis. As a result we obtain a family of integral equations for multivariable polynomial eigenfunctions of the quantum integrable DST model. These eigenfunctions are special functions of the Heun class which is beyond the hypergeometric class. The integral equations found are new and they shall provide a basis for efficient analytical and numerical studies of such complicated functions.

Abstract: We compute the two point correlation function of a dimension 4 operator in a nonconformal cascading N=1 SUSY gauge theory using the supergravity dual found by Klebanov and Strassler[hep-th/0007191]. The two point function has a logarithmic correction to conformal behavior which is related to the scale dependence of the effective number of colors. The nonsingular behavior of this correlator suggests that the theory remains a local 4-dimensional quantum field theory at all scales. We also compute the spectrum of low-lying glueball modes corresponding to the above operator.

Abstract: The Roper-Suffridge extension operator provides a way of extending a (locally) univalent functionfeH(U) to a (locally) biholomorphic mappingF∈H(Bn). In this paper, we give a simplified proof of the Roper-Suffridge theorem: iff is convex, then so isF. We also show that iff∈S *, theF is starlike and that iff is a Bloch function inU, thenF is a Bloch mapping onB n. Finally, we investigate some open problems.

Abstract: I show, on a simple example, that the two approaches lead to exactly the same density operator for the whole spin system, with the same equations of motion for the matrix elements (ignoring relaxation and diffusion). Hence, the two approaches are equivalent in all respects.

Abstract: A dynamic integro-differential operator of variable order is suggested for a more adequate description of processes, which involve state dependent measures of elastic and inelastic material features. For any negative constant order this operator coincides with the well-known operator of fractional integration. The suggested operator is especially effective in cases with strong dependence of the behavior of the material on its present state-i.e., with pronounced nonlinearity. Its efficiency is demonstrated for cases of viscoelastic and elastoplastic spherical indentation into such materials (aluminum, vinyl) and into an elastic material (steel) used as a reference. Peculiarities in the behavior of the order function are observed in these applications, demonstrating the physicality of this function which characterizes the material state. Mathematical generalization of the fractional-order integration-differentiation in the sense of yariability of the operator order, as well as definitions and techniques, are discussed.

Abstract: The present application discloses an axial operator that is configured for use with a door assembly. The axial operator comprises a rotatable operator output member that rotates about an operator axis, the operator output member being constructed and arranged to be operatively connected within the door assembly such that the operator output axis extends generally vertically. An electric motor has a rotatable motor output member that rotates about the operator axis. A reduction transmission is connected between the motor output member and the operator output member. The reduction transmission comprises (a) an orbit gear, (b) a planet gear carrier, and (c) a planet gear.

Abstract: We present a family of bidirectional beam propagation methods based on the application of Pade approximants to the square-root expressions appearing in the transition operator. We show that the numerical instability associated with the use of real Pade primes can be removed by using branch-cut rotation of the square-root operator.

Abstract: We consider the parabolic Anderson problem $\partial_t u=\kappa\Delta u+\xi u$ on $(0,\infty)\times \Z^d$ with random i.i.d. potential $\xi=(\xi(z))_{z\in\Z^d}$ and the initial condition $u(0,\cdot)\equiv1$. Our main assumption is that $\esssup\xi(0)=0$. Depending on the thickness of the distribution $\prob(\xi(0)\in\cdot)$ close to its essential supremum, we identify both the asymptotics of the moments of $u(t,0)$ and the almost-sure asymptotics of $u(t,0)$ as $t\to\infty$ in terms of variational problems. As a by-product, we establish Lifshitz tails for the random Schrodinger operator $-\kappa\Delta-\xi$ at the bottom of its spectrum. In our class of $\xi$ distributions, the Lifshitz exponent ranges from $d/2$ to $\infty$; the power law is typically accompanied by lower-order corrections.

Abstract: In this paper we prove, under convexity assumptions for the data, the existence of classical solutions for a Bernoulli-type free boundary problem, with the p-Laplacian as the governing operator. The method employed here originates from a pioneering work of A. Beurling where he proves the existence for the harmonic case in the plane, though with no geometrical restrictions.

Abstract: It is shown that any elloptic or parabolic operator in nondivergence form with measurable coefficients has a global fundamental solution verifying certain pointwise bounds.

Abstract: It is shown that any elliptic or parabolic operator in nondiver- gence form with measurable coeficients has a global fundamental solution verifying certain pointwise bounds.

Abstract: By using a rather simple algebraic approach, we revisit and further bridge between two most commonly used quantum dissipation theories, the Bloch–Redfield theory and a class of Fokker–Planck equations. The nature of the common approximation scheme involving in these two theories is analyzed in detail. While the Bloch–Redfield theory satisfies the detailed-balance relation, we also construct a class of Fokker–Planck equations that satisfy the detailed-balance relation up to the second moments in phase-space. Developed is also a generalized Fokker–Planck equation that preserves the general positivity of the reduced density operator. Both T1-relaxation and pure-T2 dephasing are considered, and their temperature dependence is shown to be very different. Provided is also an analogy between the quantum pure-T2 dephasing and the classical heat transport.

Abstract: We propose an efficient stochastic method to implement numerically the Bogolubov approach to study finite-temperature Bose-Einstein condensates. Our method is based on the Wigner representation of the density matrix describing the non-condensed modes and a Brownian motion simulation to sample the Wigner distribution at thermal equilibrium. Allowing it to sample any density operator Gaussian in the field variables, our method is very general and it applies both to the Bogolubov and to the Hartree-Fock Bogolubov approach, in the equilibrium case as well as in the time-dependent case. We think that our method can be useful to study trapped Bose-Einstein condensates in two or three spatial dimensions without rotational symmetry properties, as in the case of condensates with vortices, where the traditional Bogolubov approach is difficult to implement numerically due to the need to diagonalize very big matrices.

Abstract: We consider entanglement for quantum states defined in vector spaces over the real numbers. Such real entanglement is different from entanglement in standard quantum mechanics over the complex numbers. The differences provide insight into the nature of entanglement in standard quantum theory. Wootters [Phys. Rev. Lett. 80, 2245 (1998)] has given an explicit formula for the entanglement of formation of two qubits in terms of what he calls the concurrence of the joint density operator. We give a contrasting formula for the entanglement of formation of an arbitrary state of two ``rebits,'' a rebit being a system whose Hilbert space is a 2-dimensional real vector space.

Abstract: We introduce the theory of operator monotone functions and employ it to derive a new inequality relating the quantum relative entropy and the quantum conditional entropy. We present applications of this new inequality and in particular we prove a new lower bound on the relative entropy of entanglement and other properties of entanglement measures.

Abstract: A simple derivation of the spin probability current density from the expectation value of the spin operator is given. The properties of the spin probability current density are then examined in detail. We show that the spin probability current is solenoidal, virtual, and gives null contribution to the momentum of the particle. Expressions of the spin probability current density are derived for the Gaussian wave packet and the s states of the hydrogen atom.

Abstract: In this paper we develop explicit formulas for induced convolution operator norms and their bounds. These results generalize established induced operator norms for linear dynamical systems with various classes of input–output signal pairs.

Abstract: We consider a stationary Schrodinger–Poisson problem modeling a self-consistent transport in a quantum coupler. The Schrodinger equation is set on a bounded domain with transparent boundary conditions describing incoming scattering states of the Schrodinger operator. The coupling with the Poisson equation is done thanks to a nonlinear limiting absorption procedure. The charge density of the limit potential is shown to be equal to the sum of the scattering state and bound state densities.