Improved Approximation Algorithms for Computing k Disjoint Paths Subject to Two Constraints

For a given graph $G$ with positive integral cost and delay on edges, distinct vertices $s$ and $t$, cost bound $C\in Z^{+}$ and delay bound $D\in Z^{+}$, the $k$ bi-constraint path ($k$BCP) problem is to compute $k$ disjoint $st$-paths subject to $C$ and $D$. This problem is known NP-hard, even when $k=1$ \cite{garey1979computers}. This paper first gives a simple approximation algorithm with factor-$(2,2)$, i.e. the algorithm computes a solution with delay and cost bounded by $2*D$ and $2*C$ respectively. Later, a novel improved approximation algorithm with ratio $(1+\beta,\,\max\{2,\,1+\ln\frac{1}{\beta}\})$ is developed by constructing interesting auxiliary graphs and employing the cycle cancellation method. As a consequence, we can obtain a factor-$(1.369,\,2)$ approximation algorithm by setting $1+\ln\frac{1}{\beta}=2$ and a factor-$(1.567,\,1.567)$ algorithm by setting $1+\beta=1+\ln\frac{1}{\beta}$. Besides, by setting $\beta=0$, an approximation algorithm with ratio $(1,\, O(\ln n))$, i.e. an algorithm with only a single factor ratio $O(\ln n)$ on cost, can be immediately obtained. To the best of our knowledge, this is the first non-trivial approximation algorithm for the $k$BCP problem that strictly obeys the delay constraint.Comment: 12 page

Multiplicativity of maximal output purities of Gaussian channels under Gaussian inputs

We address the question of the multiplicativity of the maximal p-norm output purities of bosonic Gaussian channels under Gaussian inputs. We focus on general Gaussian channels resulting from the reduction of unitary dynamics in larger Hilbert spaces. It is shown that the maximal output purity of tensor products of single-mode channels under Gaussian inputs is multiplicative for any p>1 for products of arbitrary identical channels as well as for a large class of products of different channels. In the case of p=2 multiplicativity is shown to be true for arbitrary products of generic channels acting on any number of modes.Comment: 9 page

Majorization criterion for distillability of a bipartite quantum state

Bipartite quantum states are classified into three categories: separable states, bound entangled states, and free entangled states. It is of great importance to characterize these families of states for the development of quantum information science. In this paper, I show that the separable states and the bound entangled states have a common spectral property. More precisely, I prove that for undistillable -- separable and bound entangled -- states, the eigenvalue vector of the global system is majorized by that of the local system. This result constitutes a new sufficient condition for distillability of bipartite quantum states. This is achieved by proving that if a bipartite quantum state satisfies the reduction criterion for distillability, then it satisfies the majorization criterion for separability.Comment: 4 pages, no figures, REVTEX. A new lemma (Lemma 2) added. To appear in Physical Review Letter

CHIANTI - an Atomic Database for Emission Lines. Paper VI: Proton Rates and Other Improvements

The CHIANTI atomic database contains atomic energy levels, wavelengths, radiative transition probabilities and electron excitation data for a large number of ions of astrophysical interest. Version 4 has been released, and proton excitation data is now included, principally for ground configuration levels that are close in energy. The fitting procedure for excitation data, both electrons and protons, has been extended to allow 9 point spline fits in addition to the previous 5 point spline fits. This allows higher quality fits to data from close-coupling calculations where resonances can lead to significant structure in the Maxwellian-averaged collision strengths. The effects of photoexcitation and stimulated emission by a blackbody radiation field in a spherical geometry on the level balance equations of the CHIANTI ions can now be studied following modifications to the CHIANTI software. With the addition of H I, He I and N I, the first neutral species have been added to CHIANTI. Many updates to existing ion data-sets are described, while several new ions have been added to the database, including Ar IV, Fe VI and Ni XXI. The two-photon continuum is now included in the spectral synthesis routines, and a new code for calculating the relativistic free-free continuum has been added. The treatment of the free-bound continuum has also been updated.Comment: CHIANTI is available at http://wwwsolar.nrl.navy.mil/chianti.htm

Spectral Conditions on the State of a Composite Quantum System Implying its Separability

For any unitarily invariant convex function F on the states of a composite quantum system which isolates the trace there is a critical constant C such that F(w)<= C for a state w implies that w is not entangled; and for any possible D > C there are entangled states v with F(v)=D. Upper- and lower bounds on C are given. The critical values of some F's for qubit/qubit and qubit/qutrit bipartite systems are computed. Simple conditions on the spectrum of a state guaranteeing separability are obtained. It is shown that the thermal equilbrium states specified by any Hamiltonian of an arbitrary compositum are separable if the temperature is high enough.Comment: Corrects 1. of Lemma 2, and the (under)statement of Proposition 7 of the earlier version

Subsidence Affects Due to Gypsum Solution in Syracuse, New York

Two apartment buildings owned by Syracuse University in the Slocum Heights area underwent extensive differential settlement manifested by foundation cracks. Examination of rock exposures in the nearby area revealed the presence of gypsum beds. Onsite boring and investigation revealed that regrading and installation of french drains and storm sewers influenced the groundwater regime resulting in solutioning of the gypsum and subsequent collapse of underground cavities. Remedial measures included alteration of drainage system and underpinning of apartment structures

Effectiveness of a Slurry Cutoff Wall at Loeffel Site

In recent years slurry cutoff walls have been successfully employed to mitigate seepage and isolate liquid waste and leachate in the groundwater environment. However , a success of the slurry cutoff wall depends on the hydrological condition of the site. A post construction and pre-construction groundwater budget analysis can demonstrate the effectiveness of a slurry cutoff wall. In this paper, a detailed groundwater budget analysis of Loeffel site in the Southwestern Rensselaer County of New York is discussed. The analysis shows that the use of a slurry cutoff wall effectively mitigates the release of contaminated groundwater from the site

Effectiveness of a Slurry Cutoff Wall at Loeffel Site

In recent years slurry cutoff walls have been successfully employed to mitigate seepage and isolate liquid waste and leachate in the groundwater environment. However , a success of the slurry cutoff wall depends on the hydrological condition of the site. A post construction and pre-construction groundwater budget analysis can demonstrate the effectiveness of a slurry cutoff wall. In this paper, a detailed groundwater budget analysis of Loeffel site in the Southwestern Rensselaer County of New York is discussed. The analysis shows that the use of a slurry cutoff wall effectively mitigates the release of contaminated groundwater from the site

Structural and Dielectric Studies of Fe-Li Substituted Hexagonal Barium Titanate

ABSTRACT: We have studied the influence of a new lead-free Fe/Li incorporated H-Bt prepared by the solid state reaction method and compared it with pure hexagonal barium titanate. The hexagonal phase was stabilised at room temperature due to substitution with Fe-Li. Lattice constants were refined as a = 5.720 A.U., c = 13.96 A.U. Both XRD and IR imply that Li enters octahedral as well as Ba sites. Dielectric measurements show an increase from 800 to 10 5 indicating sample to be potential colossal dielectric constant (CDC) material. Abnormal grain growth (AGG) was restricted in the substituted sample and normal grain growth was observed