Counting Complex Disordered States by Efficient Pattern Matching: Chromatic Polynomials and Potts Partition Functions

Counting problems, determining the number of possible states of a large system under certain constraints, play an important role in many areas of science. They naturally arise for complex disordered systems in physics and chemistry, in mathematical graph theory, and in computer science. Counting problems, however, are among the hardest problems to access computationally. Here, we suggest a novel method to access a benchmark counting problem, finding chromatic polynomials of graphs. We develop a vertex-oriented symbolic pattern matching algorithm that exploits the equivalence between the chromatic polynomial and the zero-temperature partition function of the Potts antiferromagnet on the same graph. Implementing this bottom-up algorithm using appropriate computer algebra, the new method outperforms standard top-down methods by several orders of magnitude, already for moderately sized graphs. As a first application, we compute chromatic polynomials of samples of the simple cubic lattice, for the first time computationally accessing three-dimensional lattices of physical relevance. The method offers straightforward generalizations to several other counting problems.Comment: 7 pages, 4 figure

Rapid Mixing for Lattice Colorings with Fewer Colors

We provide an optimally mixing Markov chain for 6-colorings of the square lattice on rectangular regions with free, fixed, or toroidal boundary conditions. This implies that the uniform distribution on the set of such colorings has strong spatial mixing, so that the 6-state Potts antiferromagnet has a finite correlation length and a unique Gibbs measure at zero temperature. Four and five are now the only remaining values of q for which it is not known whether there exists a rapidly mixing Markov chain for q-colorings of the square lattice.Comment: Appeared in Proc. LATIN 2004, to appear in JSTA

Effect of Gait Imagery Tasks on Lower Limb Muscle Activity With Respect to Body Posture

The objective of this study was to evaluate the effect of gait imagery tasks on lowerlimb muscle activity with respect to body posture. The sitting and standing position and lower limb muscle activity were evaluated in 27 healthy female students (24.4±1.3 years, 167.2±5.2 cm, 60.10±6.4 kg). Surface electromyography was assessed during rest and in three different experimental conditions using mental imagery. These included a rhythmic gait, rhythmic gait simultaneously with observation of a model, and rhythmic gait after performing rhythmic gait. The normalized root mean square EMG values with respect to corresponding rest position were compared using non-parametric statistics. Standing gait imagery tasks had facilitatory effect on proximal lower limb muscle activity. However, electromyography activity of distal leg muscles decreased for all gait imagery tasks in the sitting position, when the proprioceptive feedback was less appropriate. For subsequent gait motor imagery tasks, the muscle activity decreased, probably as result of habituation. In conclusion, the effect of motor imagery on muscle activity appears to depend on relative strength of facilitatory and inhibitory inputs

Physician decision making in selection of second-line treatments in immune thrombocytopenia in children.

Immune thrombocytopenia (ITP) is an acquired autoimmune bleeding disorder which presents with isolated thrombocytopenia and risk of hemorrhage. While most children with ITP promptly recover with or without drug therapy, ITP is persistent or chronic in others. When needed, how to select second-line therapies is not clear. ICON1, conducted within the Pediatric ITP Consortium of North America (ICON), is a prospective, observational, longitudinal cohort study of 120 children from 21 centers starting second-line treatments for ITP which examined treatment decisions. Treating physicians reported reasons for selecting therapies, ranking the top three. In a propensity weighted model, the most important factors were patient/parental preference (53%) and treatment-related factors: side effect profile (58%), long-term toxicity (54%), ease of administration (46%), possibility of remission (45%), and perceived efficacy (30%). Physician, health system, and clinical factors rarely influenced decision-making. Patient/parent preferences were selected as reasons more often in chronic ITP (85.7%) than in newly diagnosed (0%) or persistent ITP (14.3%, P = .003). Splenectomy and rituximab were chosen for the possibility of inducing long-term remission (P < .001). Oral agents, such as eltrombopag and immunosuppressants, were chosen for ease of administration and expected adherence (P < .001). Physicians chose rituximab in patients with lower expected adherence (P = .017). Treatment choice showed some physician and treatment center bias. This study illustrates the complexity and many factors involved in decision-making in selecting second-line ITP treatments, given the absence of comparative trials. It highlights shared decision-making and the need for well-conducted, comparative effectiveness studies to allow for informed discussion between patients and clinicians

Diagnosis of immune thrombocytopenia, including secondary forms, and selection of second-line treatment

This article summarizes our approach to the diagnosis of immune thrombocytopenia (ITP), its secondary forms, and choice of second-line treatment options. We very briefly summarize first-line treatment and then utilize a case-based approach. We first explore persistent, chronic ITP in a younger female. We consider many possibilities beyond primary ITP e.g., hypogammaglobulinemia, chronic infection, and anemia, and how to approach their diagnosis and management. The journey continues throughout pregnancy and the post-partum period and eventually includes fourth-line treatment after a late relapse. We then consider an older male, emphasizing differences in diagnostic considerations and management. The focus is on initiation and continuation of second-line treatment, the pros and cons of each option, and briefly the impact of treatment choices related to the endemic presence of severe acute respiratory syndrome coronavirus 2. During the review of potential second-line treatment options, we also briefly touch upon novel treatments. Finally, there is a short section on refractory disease drawn from our previous extensive review published in February 2020.1 The clinical nature of the discussions, replete with figures and tables and with the interspersion of pearls regarding efficacy and toxicity at different ages and genders, will serve the reader in the management of “typical” adult patients who develop persistent and chronic ITP

Thrombopoietin receptor agonist therapy in primary immune thrombocytopenia is associated with bone marrow hypercellularity and mild reticulin fibrosis but not other stromal abnormalities

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Generating nested quadrature rules with positive weights based on arbitrary sample sets

For the purpose of uncertainty propagation a new quadrature rule technique is proposed that has positive weights, has high degree, and is constructed using onl

Bayesian model calibration with interpolating polynomials based on adaptively weighted Leja nodes

An efficient algorithm is proposed for Bayesian model calibration, which is commonly used to estimate the model parameters of non-linear, computationally expensive models using measurement data. The approach is based on Bayesian statistics: using a prior distribution and a likelihood, the posterior distribution is obtained through application of Bayes' law. Our novel algorithm to accurately determine this posterior requires significantly fewer discrete model evaluations than traditional Monte Carlo methods. The key idea is to replace the expensive model by an interpolating surrogate model and to construct the interpolating nodal set maximizing the accuracy of the posterior. To determine such a nodal set an extension to weighted Leja nodes is introduced, based on a new weighting function. We prove that the convergence of the posterior has the same rate as the convergence of the model. If the convergence of the posterior is measured in the Kullback-Leibler divergence, the rate doubles. The algorithm and its theoretical properties are verified in three different test cases: analytical cases that confirm the correctness of the theoretical findings, Burgers' equation to show its applicability in implicit problems, and finally the calibration of the closure parameters of a turbulence model to show the effectiveness for computationally expensive problems