Average electron momenta in many-electron atoms

In many-electron atoms, the average electron momentum p represents the mean momentum of a single electron when all the electron motions are averaged. If any two electrons are considered simultaneously, however, the average momentum p splits into two different momenta, low momentum p_< and high momentum p_>. For the 102 atoms He through Lr in their ground states, the momenta p_< and p_> are systematically examined at the Hartree–Fock limit level. It is also shown that the sum p_>+p_< and the difference p_>–p_< of the two momenta constitute upper and lower bounds to the electron-pair relative momentum p₁₂=|p₁–p₂| and to the electron-pair center-of-mass momentum P=|p₁+p₂|/2. The tightness of the bounds is discussed for the 102 atoms.     

Proof of some conjectures on the mean-value of Titchmarsh series — III

With some applications in view, the following problem is solved in some special case which is not too special. LetF(s) =Σ _n ^∞ =1a_n λ _n ^−s be a generalized Dirichlet series with 1 =λ ₁ <λ ₂ < …,λ _n≤Dn, andλ _n+1 -λ _n ≥D ^{− 1} λ _n+1 ^{− a} where α>0 andD(≥ 1) are constants. Then subject to analytic continuation and some growth conditions, a lower bound is obtained for . These results will be applied in other papers to appear later.     

Effective permittivity of liquid crystal nanodispersions and composites with different pore structure

The volume filling fraction dependence of the effective permittivity of the nematic liquid crystal 4-n-pentyl-4’-cyanobiphenyl embedded in different porous membranes and dispersed with aerosil nanoparticles was determined using broadband dielectric spectroscopy in the frequency range from 10⁶ to 10⁹ Hz. The experimental data were analyzed and compared with some existing theories based on the effective medium approximation and their modifications. The obtained effective permittivities as a function of the volume filling fraction lie between the lower limits of the Wiener and Hashin–Shtrikman bounds. The observed shift of the experimental points reflects the changes in the structure of the investigated composites.     

A priori bounds and multiplicity for fully nonlinear equations with quadratic growth in the gradient

We consider fully nonlinear uniformly elliptic equations with quadratic growth in the gradient, such as

$?F(x,u,Du,D^{2}u)=\lambda c(x)u+〈M(x)Du,Du〉+h(x)$

in a bounded domain with a Dirichlet boundary condition; here $\lambda \in \mathbb{R}$, $c,h\in L^p(\mathrm{\Omega })$, $p>n\ge 1$, $c?0$ and the matrix M satisfies $0<{\mu }_{1}I\le M\le {\mu }_{2}I$. Recently this problem was studied in the “coercive” case $\lambda c\le 0$, where uniqueness of solutions can be expected; and it was conjectured that the solution set is more complex for noncoercive equations. This conjecture was verified in 2015 by Arcoya, de Coster, Jeanjean and Tanaka for equations in divergence form, by exploiting the integral formulation of the problem. Here we show that similar phenomena occur for general, even fully nonlinear, equations in nondivergence form. We use different techniques based on the maximum principle.We develop a new method to obtain the crucial uniform a priori bounds, which permit to us to use degree theory. This method is based on basic regularity estimates such as half-Harnack inequalities, and on a Vázquez type strong maximum principle for our kind of equations.     

Analysis of risk bounds in partially specified additive factor models

The study of worst case scenarios for risk measures (e.g. the Value at Risk) when the underlying risk vector (or portfolio of risks) is not completely specified is a central topic in the literature on robust risk measurement. In this paper we discuss partially specified factor models as introduced in Bernard et al. (2017) in more detail for the class of additive factor models which admit more explicit results. These results allow to describe in more detail the reduction of risk bounds obtainable by this method in dependence on the degree of positive resp. negative dependence induced by the systematic risk factors. The insight may help in applications of this reduction method to get a better qualitative impression on the range of influence of the partially specified factor structure.     

Global existence and bounds for blow‐up time in a class of nonlinear pseudo‐parabolic equations with a memory term

In this article, we consider the initial boundary value problem for a class of nonlinear pseudo‐parabolic equations with a memory term: Under suitable assumptions, we obtain the local and global existence of the solution by Galerkin method. We prove finite‐time blow‐up of the solution for initial data at arbitrary energy level and obtain upper bounds for blow‐up time by using the concavity method. In addition, by means of differential inequality technique, we obtain a lower bound for blow‐up time of the solution if blow‐up occurs.     

An optimal perturbation bound

Cai and Zhang establish separate perturbation bounds for distances with spectral and Frobenius norms (Cai T, Zhang A. Rate‐optimal perturbation bounds for singular subspaces with applications to high‐dimensional statistics. The Annals of Statistics. 2018; Vol. 46, No. 1: 60?89). We extend their theorem to each unitarily invariant norm. It turns out that our estimation is optimal as well.     

A sharp‐interface immersed boundary framework for simulations of high‐speed inviscid compressible flows

A new finite‐volume flow solver based on the hybrid Cartesian immersed boundary (IB) framework is developed for the solution of high‐speed inviscid compressible flows. The IB method adopts a sharp‐interface approach, wherein the boundary conditions are enforced on the body geometry itself. A key component of the present solver is a novel reconstruction approach, in conjunction with inverse distance weighting, to compute the solutions in the vicinity of the solid‐fluid interface. We show that proposed reconstruction leads to second‐order spatial accuracy while also ensuring that the discrete conservation errors diminish linearly with grid refinement. Investigations of supersonic and hypersonic inviscid flows over different geometries are carried out for an extensive validation of the proposed flow solver. Studies on cylinder lift‐off and shape optimisation in supersonic flows further demonstrate the efficacy of the flow solver for computations with moving and shape‐changing geometries. These studies conclusively highlight the capability of the proposed IB methodology as a promising alternative for robust and accurate computations of compressible fluid flows on nonconformal Cartesian meshes.     

Parametric error bounds for convex approximations of two-stage mixed-integer recourse models with a random second-stage cost vector

We consider two-stage recourse models with integer restrictions in the second stage. These models are typically non-convex and hence, hard to solve. There exist convex approximations of these models with accompanying error bounds. However, it is unclear how these error bounds depend on the distributions of the second-stage cost vector q. In this paper, we derive parametric error bounds whose dependence on the distribution of q is explicit: they scale linearly in the expected value of the ${?}_1$-norm of q.