12,571 research outputs found
The number of independent sets in a graph with small maximum degree
Let be the number of independent sets in a graph . We show
that if has maximum degree at most then
(where is vertex degree, is the number of isolated
vertices in and is the complete bipartite graph with vertices
in one partition class and in the other), with equality if and only if each
connected component of is either a complete bipartite graph or a single
vertex. This bound (for all ) was conjectured by Kahn.
A corollary of our result is that if is -regular with then with
equality if and only if is a disjoint union of copies of
. This bound (for all ) was conjectured by Alon and Kahn and
recently proved for all by the second author, without the characterization
of the extreme cases.
Our proof involves a reduction to a finite search. For graphs with maximum
degree at most the search could be done by hand, but for the case of
maximum degree or , a computer is needed.Comment: Article will appear in {\em Graphs and Combinatorics
Spin and exchange coupling for Ti embedded in a surface dipolar network
We have studied the spin and exchange coupling of Ti atoms on a
CuN/Cu(100) surface using density functional theory. We find that
individual Ti have a spin of 1.0 (i.e., 2 Bohr Magneton) on the CuN/Cu(100)
surface instead of spin-1/2 as found by Scanning Tunneling Microscope. We
suggest an explanation for this difference, a two-stage Kondo effect, which can
be verified by experiments. By calculating the exchange coupling for Ti dimers
on the CuN/Cu(100) surface, we find that the exchange coupling across a
`void' of 3.6\AA\ is antiferromagnetic, whereas indirect (superexchange)
coupling through a N atom is ferromagnetic. We confirm the existence of
superexchange interactions by varying the Ti-N angle in a model trimer
calculation. For a square lattice of Ti on CuN/Cu(100), we find a novel
spin striped phase
Multi-Period Trading via Convex Optimization
We consider a basic model of multi-period trading, which can be used to
evaluate the performance of a trading strategy. We describe a framework for
single-period optimization, where the trades in each period are found by
solving a convex optimization problem that trades off expected return, risk,
transaction cost and holding cost such as the borrowing cost for shorting
assets. We then describe a multi-period version of the trading method, where
optimization is used to plan a sequence of trades, with only the first one
executed, using estimates of future quantities that are unknown when the trades
are chosen. The single-period method traces back to Markowitz; the multi-period
methods trace back to model predictive control. Our contribution is to describe
the single-period and multi-period methods in one simple framework, giving a
clear description of the development and the approximations made. In this paper
we do not address a critical component in a trading algorithm, the predictions
or forecasts of future quantities. The methods we describe in this paper can be
thought of as good ways to exploit predictions, no matter how they are made. We
have also developed a companion open-source software library that implements
many of the ideas and methods described in the paper
An invitation to grieve: reconsidering critical incident responses by support teams in the school setting
This paper proposes that consideration could be given to an invitational intervention rather than an expectational intervention when support personnel respond to a critical incident in schools. Intuitively many practitioners know that it is necessary for guidance/counselling personnel to intervene in schools in and following times of trauma. Most educational authorities in Australia have mandated the formulation of a critical incident intervention plan. This paper defines the term critical incident and then outlines current intervention processes, discussing the efficacy of debriefing interventions. Recent literature suggests that even though it is accepted that a planned intervention is necessary, there is scant evidence as to the effectiveness of debriefing interventions in stemming later symptoms of post traumatic stress disorder. The authors of this paper advocate for an expressive therapy intervention that is invitational rather than expectational, arguing that not all people respond to trauma in the same way and to expect that they will need to recall and retell what has happened is most likely a dangerous assumption. A model of invitation using Howard Gardner’s (1983) multiple intelligences is proposed so that students are invited to grieve and understand emotionally what is happening to them following a critical incident
Interobserver Reliability in Describing Radiographic Lung Changes After Stereotactic Body Radiation Therapy
Purpose Radiographic lung changes after stereotactic body radiation therapy (SBRT) vary widely between patients. Standardized descriptions of acute (≤6 months after treatment) and late (\u3e6 months after treatment) benign lung changes have been proposed but the reliable application of these classification systems has not been demonstrated. Herein, we examine the interobserver reliability of classifying acute and late lung changes after SBRT.
Methods and materials A total of 280 follow-up computed tomography scans at 3, 6, and 12 months post-treatment were analyzed in 100 patients undergoing thoracic SBRT. Standardized descriptions of acute lung changes (3- and 6-month scans) include diffuse consolidation, patchy consolidation and ground glass opacity (GGO), diffuse GGO, patchy GGO, and no change. Late lung change classifications (12-month scans) include modified conventional pattern, mass-like pattern, scar-like pattern, and no change. Five physicians scored the images independently in a blinded fashion. Fleiss\u27 kappa scores quantified the interobserver agreement.
Results The Kappa scores were 0.30 at 3 months, 0.20 at 6 months, and 0.25 at 12 months. The proportion of patients in each category at 3 and 6 months was as follows: Diffuse consolidation 11% and 21%; patchy consolidation and GGO 15% and 28%; diffuse GGO 10% and 11%; patchy GGO 15% and 15%; and no change 49% and 25%, respectively. The percentage of patients in each category at 12 months was as follows: Modified conventional 46%; mass-like 16%; scar-like 26%; and no change 12%. Uniform scoring between the observers occurred in 26, 8, and 14 cases at 3, 6, and 12 months, respectively.
Conclusions Interobserver reliability scores indicate a fair agreement to classify radiographic lung changes after SBRT. Qualitative descriptions are insufficient to categorize these findings because most patient scans do not fit clearly into a single classification. Categorization at 6 months may be the most difficult because late and acute lung changes can arise at that time
Phase diagram of a frustrated mixed-spin ladder with diagonal exchange bonds
Using exact numerical diagonalization and the conformal field theory
approach, we study the effect of magnetic frustrations due to diagonal exchange
bonds in a system of two coupled mixed-spin Heisenberg chains. It
is established that relatively moderate frustrations are able to destroy the
ferrimagnetic state and to stabilize the critical spin-liquid phase typical for
half-integer-spin antiferromagnetic Heisenberg chains. Both phases are
separated by a narrow but finite region occupied by a critical
partially-polarized ferromagnetic phase.Comment: 5 PRB pages, 7 eps figures, to appear in Phys. Rev.
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