712 research outputs found
A New Two-Parameter Family of Potentials with a Tunable Ground State
In a previous paper we solved a countably infinite family of one-dimensional
Schr\"odinger equations by showing that they were supersymmetric partner
potentials of the standard quantum harmonic oscillator. In this work we extend
these results to find the complete set of real partner potentials of the
harmonic oscillator, showing that these depend upon two continuous parameters.
Their spectra are identical to that of the harmonic oscillator, except that the
ground state energy becomes a tunable parameter. We finally use these
potentials to analyse the physical problem of Bose-Einstein condensation in an
atomic gas trapped in a dimple potential.Comment: 15 pages, 5 figure
Non-equilibrium Berezinskii-Kosterlitz-Thouless Transition in a Driven Open Quantum System
The Berezinskii-Kosterlitz-Thouless mechanism, in which a phase transition is
mediated by the proliferation of topological defects, governs the critical
behaviour of a wide range of equilibrium two-dimensional systems with a
continuous symmetry, ranging from superconducting thin films to two-dimensional
Bose fluids, such as liquid helium and ultracold atoms. We show here that this
phenomenon is not restricted to thermal equilibrium, rather it survives more
generally in a dissipative highly non-equilibrium system driven into a
steady-state. By considering a light-matter superfluid of polaritons, in the
so-called optical parametric oscillator regime, we demonstrate that it indeed
undergoes a vortex binding-unbinding phase transition. Yet, the exponent of the
power-law decay of the first order correlation function in the (algebraically)
ordered phase can exceed the equilibrium upper limit -- a surprising
occurrence, which has also been observed in a recent experiment. Thus we
demonstrate that the ordered phase is somehow more robust against the quantum
fluctuations of driven systems than thermal ones in equilibrium.Comment: 11 pages, 9 figure
Monomial Testing and Applications
In this paper, we devise two algorithms for the problem of testing
-monomials of degree in any multivariate polynomial represented by a
circuit, regardless of the primality of . One is an time
randomized algorithm. The other is an time deterministic
algorithm for the same -monomial testing problem but requiring the
polynomials to be represented by tree-like circuits. Several applications of
-monomial testing are also given, including a deterministic
upper bound for the -set -packing problem.Comment: 17 pages, 4 figures, submitted FAW-AAIM 2013. arXiv admin note:
substantial text overlap with arXiv:1302.5898; and text overlap with
arXiv:1007.2675, arXiv:1007.2678, arXiv:1007.2673 by other author
Vertex Cover Kernelization Revisited: Upper and Lower Bounds for a Refined Parameter
An important result in the study of polynomial-time preprocessing shows that
there is an algorithm which given an instance (G,k) of Vertex Cover outputs an
equivalent instance (G',k') in polynomial time with the guarantee that G' has
at most 2k' vertices (and thus O((k')^2) edges) with k' <= k. Using the
terminology of parameterized complexity we say that k-Vertex Cover has a kernel
with 2k vertices. There is complexity-theoretic evidence that both 2k vertices
and Theta(k^2) edges are optimal for the kernel size. In this paper we consider
the Vertex Cover problem with a different parameter, the size fvs(G) of a
minimum feedback vertex set for G. This refined parameter is structurally
smaller than the parameter k associated to the vertex covering number vc(G)
since fvs(G) <= vc(G) and the difference can be arbitrarily large. We give a
kernel for Vertex Cover with a number of vertices that is cubic in fvs(G): an
instance (G,X,k) of Vertex Cover, where X is a feedback vertex set for G, can
be transformed in polynomial time into an equivalent instance (G',X',k') such
that |V(G')| <= 2k and |V(G')| <= O(|X'|^3). A similar result holds when the
feedback vertex set X is not given along with the input. In sharp contrast we
show that the Weighted Vertex Cover problem does not have a polynomial kernel
when parameterized by the cardinality of a given vertex cover of the graph
unless NP is in coNP/poly and the polynomial hierarchy collapses to the third
level.Comment: Published in "Theory of Computing Systems" as an Open Access
publicatio
Finding and counting vertex-colored subtrees
The problems studied in this article originate from the Graph Motif problem
introduced by Lacroix et al. in the context of biological networks. The problem
is to decide if a vertex-colored graph has a connected subgraph whose colors
equal a given multiset of colors . It is a graph pattern-matching problem
variant, where the structure of the occurrence of the pattern is not of
interest but the only requirement is the connectedness. Using an algebraic
framework recently introduced by Koutis et al., we obtain new FPT algorithms
for Graph Motif and variants, with improved running times. We also obtain
results on the counting versions of this problem, proving that the counting
problem is FPT if M is a set, but becomes W[1]-hard if M is a multiset with two
colors. Finally, we present an experimental evaluation of this approach on real
datasets, showing that its performance compares favorably with existing
software.Comment: Conference version in International Symposium on Mathematical
Foundations of Computer Science (MFCS), Brno : Czech Republic (2010) Journal
Version in Algorithmic
Growing spherulitic calcite grains in saline, hyperalkaline lakes: experimental evaluation of the effects of Mg-clays and organic acids
The origin of spherical-radial calcite bodies â spherulites â in sublacustrine, hyperalkaline and saline systems is unclear, and therefore their palaeoenvironmental significance as allochems is disputed. Here, we experimentally investigate two hypotheses concerning the origin of spherulites. The first is that spherulites precipitate from solutions super-saturated with respect to magnesium-silicate clays, such as stevensite. The second is that spherulite precipitation happens in the presence of dissolved, organic acid molecules. In both cases, experiments were performed under sterile conditions using large batches of a synthetic and cell-free solution replicating waters found in hyperalkaline, saline lakes (such as Mono Lake, California). Our experimental results show that a highly alkaline and highly saline solution supersaturated with respect to calcite (control solution) will precipitate euhedral to subhedral rhombic and trigonal bladed calcite crystals. The same solution supersaturated with respect to stevensite precipitates sheet-like stevensite crystals rather than a gel, and calcite precipitation is reduced by ~ 50% compared to the control solution, producing a mixture of patchy prismatic subhedral to euhedral, and minor needle-like, calcite crystals. Enhanced magnesium concentration in solution is the likely the cause of decreased volumes of calcite precipitation, as this raised equilibrium ion activity ratio in the solution. On the other hand, when alginic acid was present then the result was widespread development of micron-size calcium carbonate spherulite bodies. With further growth time, but falling supersaturation, these spherules fused into botryoidal-topped crusts made of micron-size fibro-radial calcite crystals. We conclude that the simplest tested mechanism to deposit significant spherical-radial calcite bodies is to begin with a strongly supersaturated solution that contains specific but environmentally-common organic acids. Furthermore, we found that this morphology is not a universal consequence of having organic acids dissolved in the solution, but rather spherulite development requires specific binding behaviour. Finally, we found that the location of calcite precipitation was altered from the air:water interface to the surface of the glassware when organic acids were present, implying that attached calcite precipitates reflect precipitation via metalâorganic intermediaries, rather than direct forcing via gas exchange
Statistical Inference in a Directed Network Model with Covariates
Networks are often characterized by node heterogeneity for which nodes
exhibit different degrees of interaction and link homophily for which nodes
sharing common features tend to associate with each other. In this paper, we
propose a new directed network model to capture the former via node-specific
parametrization and the latter by incorporating covariates. In particular, this
model quantifies the extent of heterogeneity in terms of outgoingness and
incomingness of each node by different parameters, thus allowing the number of
heterogeneity parameters to be twice the number of nodes. We study the maximum
likelihood estimation of the model and establish the uniform consistency and
asymptotic normality of the resulting estimators. Numerical studies demonstrate
our theoretical findings and a data analysis confirms the usefulness of our
model.Comment: 29 pages. minor revisio
- âŠ