60 research outputs found
On the Threshold of Intractability
We study the computational complexity of the graph modification problems
Threshold Editing and Chain Editing, adding and deleting as few edges as
possible to transform the input into a threshold (or chain) graph. In this
article, we show that both problems are NP-complete, resolving a conjecture by
Natanzon, Shamir, and Sharan (Discrete Applied Mathematics, 113(1):109--128,
2001). On the positive side, we show the problem admits a quadratic vertex
kernel. Furthermore, we give a subexponential time parameterized algorithm
solving Threshold Editing in time,
making it one of relatively few natural problems in this complexity class on
general graphs. These results are of broader interest to the field of social
network analysis, where recent work of Brandes (ISAAC, 2014) posits that the
minimum edit distance to a threshold graph gives a good measure of consistency
for node centralities. Finally, we show that all our positive results extend to
the related problem of Chain Editing, as well as the completion and deletion
variants of both problems
Shape Invariance and Its Connection to Potential Algebra
Exactly solvable potentials of nonrelativistic quantum mechanics are known to
be shape invariant. For these potentials, eigenvalues and eigenvectors can be
derived using well known methods of supersymmetric quantum mechanics. The
majority of these potentials have also been shown to possess a potential
algebra, and hence are also solvable by group theoretical techniques. In this
paper, for a subset of solvable problems, we establish a connection between the
two methods and show that they are indeed equivalent.Comment: Latex File, 10 pages, One figure available on request. Appeared in
the proceedings of the workshop on "Supersymmetric Quantum Mechanics and
Integrable Models" held at University of Illinois, June 12-14, 1997; Ed. H.
Aratyn et a
Connection Between Type A and E Factorizations and Construction of Satellite Algebras
Recently, we introduced a new class of symmetry algebras, called satellite
algebras, which connect with one another wavefunctions belonging to different
potentials of a given family, and corresponding to different energy
eigenvalues. Here the role of the factorization method in the construction of
such algebras is investigated. A general procedure for determining an so(2,2)
or so(2,1) satellite algebra for all the Hamiltonians that admit a type E
factorization is proposed. Such a procedure is based on the known relationship
between type A and E factorizations, combined with an algebraization similar to
that used in the construction of potential algebras. It is illustrated with the
examples of the generalized Morse potential, the Rosen-Morse potential, the
Kepler problem in a space of constant negative curvature, and, in each case,
the conserved quantity is identified. It should be stressed that the method
proposed is fairly general since the other factorization types may be
considered as limiting cases of type A or E factorizations.Comment: 20 pages, LaTeX, no figure, to be published in J. Phys.
Exactly solvable models of supersymmetric quantum mechanics and connection to spectrum generating algebra
For nonrelativistic Hamiltonians which are shape invariant, analytic
expressions for the eigenvalues and eigenvectors can be derived using the well
known method of supersymmetric quantum mechanics. Most of these Hamiltonians
also possess spectrum generating algebras and are hence solvable by an
independent group theoretic method. In this paper, we demonstrate the
equivalence of the two methods of solution by developing an algebraic framework
for shape invariant Hamiltonians with a general change of parameters, which
involves nonlinear extensions of Lie algebras.Comment: 12 pages, 2 figure
Graded extension of SO(2,1) Lie algebra and the search for exact solutions of Dirac equation by point canonical transformations
SO(2,1) is the symmetry algebra for a class of three-parameter problems that
includes the oscillator, Coulomb and Morse potentials as well as other problems
at zero energy. All of the potentials in this class can be mapped into the
oscillator potential by point canonical transformations. We call this class the
"oscillator class". A nontrivial graded extension of SO(2,1) is defined and its
realization by two-dimensional matrices of differential operators acting in
spinor space is given. It turns out that this graded algebra is the
supersymmetry algebra for a class of relativistic potentials that includes the
Dirac-Oscillator, Dirac-Coulomb and Dirac-Morse potentials. This class is, in
fact, the relativistic extension of the oscillator class. A new point canonical
transformation, which is compatible with the relativistic problem, is
formulated. It maps all of these relativistic potentials into the
Dirac-Oscillator potential.Comment: Replaced with a more potrable PDF versio
A Group-Theoretical Method for Natanzon Potentials in Position-Dependent Mass Background
A new manner for deriving the exact potentials is presented. By making use of
conformal mappings, the general expression of the effective potentials deduced
under su(1,1) algebra can be brought back to the general Natanzon
hypergeometric potentials
Deformed algebras, position-dependent effective masses and curved spaces: An exactly solvable Coulomb problem
We show that there exist some intimate connections between three
unconventional Schr\"odinger equations based on the use of deformed canonical
commutation relations, of a position-dependent effective mass or of a curved
space, respectively. This occurs whenever a specific relation between the
deforming function, the position-dependent mass and the (diagonal) metric
tensor holds true. We illustrate these three equivalent approaches by
considering a new Coulomb problem and solving it by means of supersymmetric
quantum mechanical and shape invariance techniques. We show that in contrast
with the conventional Coulomb problem, the new one gives rise to only a finite
number of bound states.Comment: 22 pages, no figure. Archive version is already official. Published
by JPA at http://stacks.iop.org/0305-4470/37/426
BGWM as Second Constituent of Complex Matrix Model
Earlier we explained that partition functions of various matrix models can be
constructed from that of the cubic Kontsevich model, which, therefore, becomes
a basic elementary building block in "M-theory" of matrix models. However, the
less topical complex matrix model appeared to be an exception: its
decomposition involved not only the Kontsevich tau-function but also another
constituent, which we now identify as the Brezin-Gross-Witten (BGW) partition
function. The BGW tau-function can be represented either as a generating
function of all unitary-matrix integrals or as a Kontsevich-Penner model with
potential 1/X (instead of X^3 in the cubic Kontsevich model).Comment: 42 page
Questionnaire of chronic illness care in primary care-psychometric properties and test-retest reliability
<p>Abstract</p> <p>Background</p> <p>The Chronic Care Model (CCM) is an evidence-based approach to improving the structure of care for chronically ill patients with multimorbidity. The Assessment of Chronic Illness Care (ACIC), an instrument commonly used in international research, includes all aspects of the CCM, but cannot be easily extended to the German context. A new instrument called the "Questionnaire of Chronic Illness Care in Primary Care" (QCPC) was developed for use in Germany for this reason. Here, we present the results of the psychometric properties and test-retest reliability of QCPC.</p> <p>Methods</p> <p>A total of 109 family doctors from different German states participated in the validation study. Participating physicians completed the QCPC, which includes items concerning the CCM and practice structure, at baseline (T0) and 3 weeks later (T1). Internal consistency reliability and test-retest reliability were evaluated using Cronbach's alpha and Pearson's r, respectively.</p> <p>Results</p> <p>The QCPC contains five elements of the CCM (decision support, delivery system design, self-management support, clinical information systems, and community linkages). All subscales demonstrated moderate internal consistency and moderate test-retest reliability over a three-week interval.</p> <p>Conclusions</p> <p>The QCPC is an appropriate instrument to assess the structure of chronic illness care. Unlike the ACIC, the QCPC can be used by health care providers without CCM training. The QCPC can detect the actual state of care as well as areas for improvement of care according to the CCM.</p
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