20,625 research outputs found

### Numerical Solution of Quantum-Mechanical Pair Equations

We discuss and illustrate the numerical solution of the differential equation satisfied by the first‐order pair functions of Sinanoğlu. An expansion of the pair function in spherical harmonics and the use of finite difference methods convert the differential equation into a set of simultaneous equations. Large systems of such equations can be solved economically. The method is simple and straightforward, and we have applied it to the first‐order pair function for helium with 1 / r_(12) as the perturbation. The results are accurate and encouraging, and since the method is numerical they are indicative of its potential for obtaining atomic‐pair functions in general

### How many copies are needed for state discrimination?

Given a collection of states (rho_1, ..., rho_N) with pairwise fidelities
F(rho_i, rho_j) <= F < 1, we show the existence of a POVM that, given
rho_i^{otimes n}, will identify i with probability >= 1-epsilon, as long as
n>=2(log N/eps)/log (1/F). This improves on previous results which were either
dimension-dependent or required that i be drawn from a known distribution.Comment: 1 page, submitted to QCMC'06, answer is O(log # of states

### Numerical Solution of the (1s1s) and (1s2s) Hydrogenic Pair Equations

The pair functions which determine the exact first-order wave function for the ground state of the three-electron atom have been found with the matrix finite-difference method. The second- and third-order energies for the (1s1s)^1S, (1s2s)^3S, and (1s2s)^1S states of the two-electron atom are presented along with contour and perspective plots of the pair functions

### Spikes for the gierer-meinhardt system with many segments of different diffusivities

We rigorously prove results on spiky patterns for the
Gierer-Meinhardt system with a large number of jump
discontinuities in the diffusion coefficient of the inhibitor. Using
numerical computations in combination with a Turing-type instability
analysis, this system has been investigated by Benson, Maini and
Sherratt

### A flight-rated liquid-cooled garment for use within a full-pressure suit

A flight rated liquid cooled garment system for use inside a full pressure suit has been designed, fabricated, and tested. High temperature tests with this system have indicated that heat is absorbed at a rate decreasing from 224 kg-cal/hr to 143 kg-cal/hr over a 40-min period. The first 30 min are very comfortable; thereafter a gradual heat load builds that results in mild sweating at the end of the 40-min period. In flight tests during hot weather when this cooling system was worn under a regulation flight suit, the pilot reported that temperatures were comfortable and that the garment prevented sweating

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### A Higher-Order Energy Expansion to Two-Dimensional Singularly Neumann Problems

Of concern is the
following singularly perturbed semilinear elliptic problem
\begin{equation*}
\left\{ \begin{array}{c}
\mbox{${\epsilon}^2\Delta u -u+u^p =0$ in $\Omega$}\\
\mbox{$u>0$ in $\Omega$ and $\frac{\partial
u}{\partial \nu}=0$ on $\partial \Omega$},
\end{array}
\right.
\end{equation*}
where $\Omega$ is a bounded domain in ${\mathbf{R}}^N$ with smooth
boundary $\partial \Omega$, $\epsilon>0$ is a small constant and
$1< p<\left(\frac{N+2}{N-2}\right)_+$. Associated with the
above problem is the energy functional $J_{\epsilon}$ defined by
\begin{equation*}
J_{\epsilon}[u]:=\int_{\Omega}\left(\frac{\epsilon^2}{2}{|\nabla
u|}^2 +\frac{1}{2}u^2 -F(u)\right)dx
\end{equation*}
for $u\in H^1(\Omega)$, where $F(u)=\int_{0}^{u}s^p ds$.
Ni and Takagi (\cite{nt1}, \cite{nt2}) proved that for a single
boundary spike solution $u_{\epsilon}$, the following asymptotic
expansion holds:
\begin{equation*}
(1) \ \ \ \ \ \ \ \ J_{\epsilon}[u_{\epsilon}]=\epsilon^{N}
\left[\frac{1}{2}I[w]-c_1 \epsilon
H(P_{\epsilon})+o(\epsilon)\right],
\end{equation*}
where $I[w]$ is the energy of the ground state, $c_1 >0$ is a
generic constant, $P_{\epsilon}$ is the unique local maximum point
of $u_{\epsilon}$ and $H(P_{\epsilon})$ is the boundary mean
curvature function at $P_{\epsilon}\in \partial \Omega$. Later,
Wei and Winter (\cite{ww3}, \cite{ww4}) improved the result and
obtained a higher-order expansion of $J_{\epsilon}[u_{\epsilon}]$:
\begin{equation*}
(2) \ \ \ \ \ \ J_{\epsilon}[u_{\epsilon}]=\epsilon^{N}
\left[\frac{1}{2}I[\omega]-c_{1} \epsilon
H(P_{\epsilon})+\epsilon^2 [c_2(H(P_\epsilon))^2 +c_{3}
R(P_\epsilon)]+o(\epsilon^2)\right],
\end{equation*}
where $c_2$ and $c_3>0$ are generic constants and $R(P_\epsilon)$
is the scalar curvature at $P_\epsilon$. However, if $N=2$, the
scalar curvature is always zero. The expansion (2) is no longer sufficient to distinguish spike locations with same mean curvature.
In this paper, we consider
this case and assume that $2 \leq p <+\infty$. Without loss of generality, we may assume that the
boundary near P\in\partial\Om is represented by the graph $\{ x_2 = \rho_{P}
(x_1) \}$. Then we have the following higher order expansion of
$J_\epsilon[u_\epsilon]:$
\begin{equation*}
(3) \ \ \ \ \ J_\epsilon [u_\epsilon]
=\epsilon^N \left[\frac{1}{2}I[w]-c_1
\epsilon H({P_\epsilon})+c_2 \epsilon^2(H({P_\epsilon}))^2 ]
+\epsilon^3
[P(H({P_\epsilon}))+c_3S({P_\epsilon})]+o(\epsilon^3)\right],
\end{equation*}
where H(P_\ep)= \rho_{P_\ep}^{''} (0) is the curvature, $P(t)=A_1 t+A_2 t^2+A_3
t^3$ is a polynomial,
$c_1$, $c_2$, $c_3$ and $A_1$, $A_2$,$A_3$ are generic real
constants and S(P_\epsilon)= \rho_{P_\ep}^{(4)} (0). In
particular $c_3<0$. Some applications of this expansion are given

### A Resource Framework for Quantum Shannon Theory

Quantum Shannon theory is loosely defined as a collection of coding theorems,
such as classical and quantum source compression, noisy channel coding
theorems, entanglement distillation, etc., which characterize asymptotic
properties of quantum and classical channels and states. In this paper we
advocate a unified approach to an important class of problems in quantum
Shannon theory, consisting of those that are bipartite, unidirectional and
memoryless.
We formalize two principles that have long been tacitly understood. First, we
describe how the Church of the larger Hilbert space allows us to move flexibly
between states, channels, ensembles and their purifications. Second, we
introduce finite and asymptotic (quantum) information processing resources as
the basic objects of quantum Shannon theory and recast the protocols used in
direct coding theorems as inequalities between resources. We develop the rules
of a resource calculus which allows us to manipulate and combine resource
inequalities. This framework simplifies many coding theorem proofs and provides
structural insights into the logical dependencies among coding theorems.
We review the above-mentioned basic coding results and show how a subset of
them can be unified into a family of related resource inequalities. Finally, we
use this family to find optimal trade-off curves for all protocols involving
one noisy quantum resource and two noiseless ones.Comment: 60 page

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