The relaxation-time limit in the quantum hydrodynamic equations for semiconductors

The relaxation-time limit from the quantum hydrodynamic model to the quantum drift-diffusion equations in R³ is shown for solutions which are small perturbations of the steady state. The quantum hydrodynamic equations consist of the isentropic Euler equations for the particle density and current density including the quantum Bohm potential and a momentum relaxation term. The momentum equation is highly nonlinear and contains a dispersive term with third-order derivatives. The equations are self-consistently coupled to the Poisson equation for the electrostatic potential. The relaxation-time limit is performed both in the stationary and the transient model. The main assumptions are that the steady-state velocity is irrotational, that the variations of the doping profile and the velocity at infinity are sufficiently small and, in the transient case, that the initial data are sufficiently close to the steady state. As a by-product, the existence of global-in-time solutions to the quantum drift-diffusion model in R³ close to the steady-state is obtained.     

Sur Prolungamento Delle Funzioni Della Prima Classe

It is proved that a setH inR ⁿ has Lebesgue null measure if any function which is the restriction toH of a Baire function defined inR ⁿ is the restriction toH of a derivative of an interval function.     

Direct approach to quantum extensions of Fisher information

By manipulating classical Fisher information and employing various derivatives of density operators, and using entirely intuitive and direct methods, we introduce two families of quantum extensions of Fisher information that include those defined via the symmetric logarithmic derivative, via the right logarithmic derivative, via the Bogoliubov-Kubo-Mori derivative, as well as via the derivative in terms of commutators, as special cases. Some fundamental properties of these quantum extensions of Fisher information are investigated, a multi-parameter quantum Cramér-Rao inequality is established, and applications to characterizing quantum uncertainty are illustrated.      

Notes on the diamond-α dynamic derivative on time scales

Various dynamic derivative formulae have been proposed in the development of a time scales calculus, with the goal of unifying continuous and discrete analysis. Recent discussions of combined dynamic derivatives, in particular the α_? derivative defined as a linear combination of the Δ and the ∇ derivatives, have promised improved approximation formulae for computational applications. This paper presents an equivalent definition of the α_? functions without reference to the existing Δ and ∇ derivatives, examines the status of the α_? as a dynamic derivative and its properties relative to the Δ and ∇ derivatives, and compares data obtained using the various dynamic derivatives as approximation formulae in computational experiments. A α_? integral case is investigated.     

Fractional partial differential equations and modified Riemann-Liouville derivative new methods for solution

The paper deals with the solution of some fractional partial differential equations obtained by substituting modified Riemann-Liouville derivatives for the customary derivatives. This derivative is introduced to avoid using the so-called Caputo fractional derivative which, at the extreme, says that, if you want to get the first derivative of a function you must before have at hand its second derivative. Firstly, one gives a brief background on the fractional Taylor series of nondifferentiable functions and its consequence on the derivative chain rule. Then one considers linear fractional partial differential equations with constant coefficients, and one shows how, in some instances, one can obtain their solutions on by-passing the use of Fourier transform and/or Laplace transform. Later one develops a Lagrange method via characteristics for some linear fractional differential equations with nonconstant coefficients, and involving fractional derivatives of only one order. The key is the fractional Taylor series of non differentiable functionf(x + h) =E _α (h ^α D _x ^α )f(x).     

Derivatives of regularly varying functions in Rd and domains of attraction of stable distributions

In R² the integral of a regularly varying (RV) function f is regularly varying only if f is monotone. Generalization to R² of the one-dimensional result on regular variation of the derivative of an RV-function however is straightforward. Applications are given to limit theory for partial sums of i.i.d. positive random vectors in R²₊.     

Automorphisms of Ore extensions

Let R be a domain and σ an outer automorphism of R. For any automorphism g of the Ore extension R[t; σ], it is shown that either g(t) = at, g ^?1(t) = bt or g(t) = a, g ^?1(t) = b for some a, b ∈ R. As applications, we show first that R[t; σ] is essentially a quantum plane if R is a commutative domain and if R[t; σ] possesses an automorphism sending t into R. This shows an interesting analogy between the quantum plane and the Weyl algebra. We then determine all ring automorphisms of such R[t; σ].     

The thermodynamic limit of quantum Coulomb systems Part I. General theory

This article is the first in a series dealing with the thermodynamic properties of quantum Coulomb systems.In this first part, we consider a general real-valued function E defined on all bounded open sets of R³. Our aim is to give sufficient conditions such that E has a thermodynamic limit. This means that the limit E(Ωn)|Ωn|⁻¹ exists for all ‘regular enough’ sequence Ωn with growing volume, |Ωn|→∞, and is independent of the considered sequence.The sufficient conditions presented in our work all have a clear physical interpretation. In the next paper, we show that the free energies of many different quantum Coulomb systems satisfy these assumptions, hence have a thermodynamic limit.     

H−C Maps and elliptic SPDEs with polynomial and exponential perturbations of Nelson's Euclidean free field

Elliptic stochastic partial differential equations (SPDE) with polynomial and exponential perturbation terms defined in terms of Nelson's Euclidean free field on R^d are studied using results by S. Kusuoka and A.S. Üstünel and M. Zakai concerning transformation of measures on abstract Wiener space. SPDEs of this type arise, in particular, in (Euclidean) quantum field theory with interactions of the polynomial or exponential type. The probability laws of the solutions of such SPDEs are given by Girsanov probability measures, that are non-linearly transformed measures of the probability law of Nelson's free field defined on subspaces of Schwartz space of tempered distributions.     

Generalizing the column-row matrix decomposition to multi-way arrays

In this paper, we provide two generalizations of the CUR matrix decomposition Y=CUR (also known as pseudo-skeleton approximation method [1]) to the case of N-way arrays (tensors). These generalizations, which we called Fiber Sampling Tensor Decomposition types 1 and 2 (FSTD1 and FSTD2), provide explicit formulas for the parameters of a rank-(R,R,…,R) Tucker representation (the core tensor of size R×R×?×R and the matrix factors of sizes I_n×R, n=1,2,…N) based only on some selected entries of the original tensor. FSTD1 uses P^N-1(P?R)n-mode fibers of the original tensor while FSTD2 uses exactly R fibers in each mode as matrix factors, as suggested by the existence theorem provided in Oseledets et al. (2008) [2], with a core tensor defined in terms of the entries of a subtensor of size R×R×?×R. For N=2 our results are reduced to the already known CUR matrix decomposition where the core matrix is defined as the inverse of the intersection submatrix, i.e. U=W^-1. Additionally, we provide an adaptive type algorithm for the selection of proper fibers in the FSTD1 model which is useful for large scale applications. Several numerical results are presented showing the performance of our FSTD1 Adaptive Algorithm compared to two recently proposed approximation methods for 3-way tensors.     

Conditioning of the matrix-matrix exponentiation

If A has no eigenvalues on the closed negative real axis, and B is arbitrary square complex, the matrix-matrix exponentiation is defined as A ^B := e ^log(A)B . It arises, for instance, in Von Newmann’s quantum-mechanical entropy, which in turn finds applications in other areas of science and engineering. In this paper, we revisit this function and derive new related results. Particular emphasis is devoted to its Fréchet derivative and conditioning. We propose a new definition of bivariate matrix function and derive some general results on their Fréchet derivatives, which hold, not only to the matrix-matrix exponentiation but also to other known functions, such as means of two matrices, second order Fréchet derivatives and some iteration functions arising in matrix iterative methods. The numerical computation of the Fréchet derivative is discussed and an algorithm for computing the relative condition number of A ^B is proposed. Some numerical experiments are included.     

New kinematic parameters of the finite rotation of a rigid body

A new family of kinematic parameters for the orientation of a rigid body (global and local) is presented and described. All the kinematic parameters are obtained by mapping the variables onto a corresponding orientated subspace (hyperplane). In particular, a method of stereographically projecting a point belonging to a five-dimensional sphere S⁵ ⊂ R⁶ onto an orientated hyperplane R⁵ is demonstrated in the case of the classical direction cosines of the angles specifying the orientation of two systems of coordinates. A family of global kinematic parameters is described, obtained by mapping the Hopf five-dimensional kinematic parameters defined in the space R⁵ onto a four-dimensional orientated subspace R⁴. A correspondence between the five-dimensional and four-dimensional kinematic parameters defined in the corresponding spaces is established on the basis of a theorem on the homeomorphism of two topological spaces (a four-dimensional sphere S⁴ ⊂ R⁵ with one deleted point and an orientated hyperplane in R⁴). It is also shown to which global four-dimensional orientation parameters–quaternions defined in the space R⁴ the classical local parameters, that is, the three-dimensional Rodrigues and Gibbs finite rotation vectors, correspond. The kinematic differential rotational equations corresponding to the five-dimensional and four-dimensional orientation parameters are obtained by the projection method. All the rigid body kinematic orientation parameters enable one, using the kinematic equations corresponding to them, to solve the classical Darboux problem, that is, to determine the actual angular position of a body in a three-dimensional space using the known (measured) angular velocity of rotation of the object and its specified initial position.     

Homogeneous random fields and statistical mechanics

We illustrate the connection between homogeneous perturbations of homogeneous Gaussian random fields over Rⁿ or Zⁿ, with values in R^m, and classical as well as quantum statistical mechanics. In particular we construct homogeneous non-Gaussian random fields as weak limits of perturbed Gaussian random fields and study the infinite volume limit of correlation functions for a classical continuous gas of particles with inner degrees of freedom. We also exhibit the relation between quantum statistical mechanics of lattice systems (anharmonic crystals) at temperature β^?1 and homogeneous random fields over Zⁿ × S_β, where S_β is the circle of length β, which then provides a connection also with classical statistical mechanics. We obtain the infinite volume limit of real and imaginary times Green's functions and establish its properties. We also give similar results for the Gibbs state of the correspondent classical lattice systems and show that it is the limit as h → 0 of the quantum statistical Gibbs state.     

Rings whose multiplicative endomorphisms are power functions

Let R be a commutative ring. For any positive integer m, the power function f:R→R defined by f(x):=x ^m is easily seen to be an endomorphism of the multiplicative semigroup (R,?). In this note, we characterize the commutative rings R with identity for which every multiplicative endomorphism of (R,?) is equal to a power function. Specifically, we show that every endomorphism of (R,?) is a power function if and only if R is a finite field.     

Extremum problems for the cone volume functional of convex polytopes

Lutwak, Yang and Zhang defined the cone volume functional U over convex polytopes in Rn containing the origin in their interiors, and conjectured that the greatest lower bound on the ratio of this centro-affine invariant U to volume V is attained by parallelotopes. In this paper, we give affirmative answers to the conjecture in R² and R³. Some new sharp inequalities characterizing parallelotopes in Rn are established. Moreover, a simplified proof for the conjecture restricted to the class of origin-symmetric convex polytopes in Rn is provided.     

Solution to n-dimensional Sturm-Liouville-like equations using multi-dimensional Schwarzian

In the present work we produce the solution to the n-dimensional Sturm-Liouville-like equations in Rn. To make it, we define the multi-dimensional Schwarzian derivative of a real-valued function of n variables and show that its basic properties related to its invariance under the action of a group of multi-dimensional Möbius transformations defined in Rn correspond to a straightforward generalization of those of the one-dimensional Schwarzian.     

Manifolds whose higher odd order curvature operators have constant eigenvalues at the basepoint

LetR ^k be the higher-order curvature operator of a Riemannian manifoldM. IfR ^k has constant eigenvalues at the basepointP ofM for all oddk, then all the odd covariant derivatives of the curvature tensor vanish atP. If the metric is real analytic, the geodesic involution is a local isometry atP.     

Essential self-adjointness of Dirichlet operators on a path space with Gibbs measures via an SPDE approach

The main objective of this paper is to prove the essential self-adjointness of Dirichlet operators in L²(μ) where μ is a Gibbs measure on an infinite volume path space C(R,Rd). This operator can be regarded as a perturbation of the Ornstein-Uhlenbeck operator by a nonlinearity and corresponds to a parabolic stochastic partial differential equation (= SPDE, in abbreviation) on R. In view of quantum field theory, the solution of this SPDE is called a P₁(?)-time evolution.     

Interpolation series for fractional derivatives and iterates of functions and operators

For a class of complex valued functions on the real line a fractional derivative is defined which is an entire function of exponential type of the order. It is shown that these derivatives can be found by a Newton interpolation series. For a class of linear operators, a fractional derivative for their resolvents also is defined. These fractional derivatives and the fractional iterates of these operators are related and both can be found by a Newton interpolation series on the nth-order iterates of the operators.     

On subharmonic functions dominated by certain functions

Given two kinds of functionsf(X) andh(y) defined on them-dimensional Euclidean spaceR ^m (m≧1) and the set of positive real numbers respectively, we give an estimation of growth of subharmonic functionsu(P) defined onR ^m+n (n≧1) such that $$u(P) \leqq f\left( X \right)h\left( {\left\| Y \right\|} \right)$$ for anyP=(X, Y),X ∈R ^m, Y ∈R ⁿ, where ‖Y ‖ denotes the usual norm ofY. Using an obtained result, we give a sharpened form of an ordinary Phragmén-Lindelöf theorem with respect to the generalized cylinderD ×R ⁿ, with a bounded domainD inR ^m.