q-Hypergeometric solutions of q-difference equations

AbstractWe present algorithm qHyper for finding all solutions y(x) of a linear homogeneous q-difference equation, such that y(qx)=r(x)y(x) where r(x) is a rational function of x. Applications include construction of basic hypergeometric series solutions, and definite q-hypergeometric summation in closed form

Converging to Gosper's Algorithm

Given two polynomials, we find a convergence property of the GCD of the rising factorial and the falling factorial. Based on this property, we present a unified approach to computing the universal denominators as given by Gosper's algorithm and Abramov's algorithm for finding rational solutions to linear difference equations with polynomial coefficients.Comment: 13 page

Denominator Bounds and Polynomial Solutions for Systems of q-Recurrences over K(t) for Constant K

We consider systems A_\ell(t) y(q^\ell t) + ... + A_0(t) y(t) = b(t) of higher order q-recurrence equations with rational coefficients. We extend a method for finding a bound on the maximal power of t in the denominator of arbitrary rational solutions y(t) as well as a method for bounding the degree of polynomial solutions from the scalar case to the systems case. The approach is direct and does not rely on uncoupling or reduction to a first order system. Unlike in the scalar case this usually requires an initial transformation of the system.Comment: 8 page

Error bounds for the asymptotic expansion of the partition function

Asymptotic study on the partition function $p(n)$ began with the work of Hardy and Ramanujan. Later Rademacher obtained a convergent series for $p(n)$ and an error bound was given by Lehmer. Despite having this, a full asymptotic expansion for $p(n)$ with an explicit error bound is not known. Recently O'Sullivan studied the asymptotic expansion of $p^{k}(n)$-partitions into $k$th powers, initiated by Wright, and consequently obtained an asymptotic expansion for $p(n)$ along with a concise description of the coefficients involved in the expansion but without any estimation of the error term. Here we consider a detailed and comprehensive analysis on an estimation of the error term obtained by truncating the asymptotic expansion for $p(n)$ at any positive integer $n$. This gives rise to an infinite family of inequalities for $p(n)$ which finally answers to a question proposed by Chen. Our error term estimation predominantly relies on applications of algorithmic methods from symbolic summation

Reduced risk of hypoglycemia with once-daily glargine versus twice-daily NPH and number needed to harm with NPH to demonstrate the risk of one additional hypoglycemic event in type 2 diabetes: Evidence from a long-term controlled trial

AbstractAimsThis analysis evaluated HbA1c-adjusted hypoglycemia risk with glargine versus neutral protamine Hagedorn (NPH) over a 5-year study in patients with Type 2 diabetes mellitus (T2DM). Clinical significance was assessed using number needed to harm (NNH) to demonstrate the risk of one additional patient experiencing at least one hypoglycemic event.MethodsIndividual patient-level data for symptomatic documented hypoglycemia and HbA1c values from a 5-year randomized study comparing once-daily glargine (n=513) with twice-daily NPH (n=504) were analyzed. Symptomatic hypoglycemia was categorized according to concurrent self-monitoring blood glucose levels and need for assistance. Hypoglycemic events per patient-year as a function of HbA1c were fitted by negative binomial regression using treatment and HbA1c at endpoint as independent variables. An estimate of NNH was derived from logistic regression models.ResultsThe cumulative number of symptomatic hypoglycemia events was consistently lower with glargine compared with NPH over 5years. Compared with twice-daily NPH, once-daily glargine treatment resulted in significantly lower adjusted odds ratios (OR) for all daytime hypoglycemia (OR 0.74; p=0.030) and any severe event (OR 0.64; p=0.035), representing a 26% and 36% reduction in the odds of daytime and severe hypoglycemia, respectively. Our model predicts that, if 25 patients were treated with NPH instead of glargine, then one additional patient would experience at least one severe hypoglycemic event.ConclusionsThis analysis of long-term insulin treatment confirms findings from short-term studies and demonstrates that glargine provides sustained, clinically meaningful reductions in risk of hypoglycemia compared with NPH in patients with T2DM

Hemodynamic monitoring by intracardiac impedance measured by cardiac resynchronization defibrillators:Evaluation in a controlled clinical setting (BIO.Detect HF II study)

Background: In patients with cardiac resynchronization therapy defibrillators (CRT-Ds), intracardiac impedance measured by dedicated CRT-D software may be used to monitor hemodynamic changes. We investigated the relationship of hemodynamic parameters assessed by intracardiac impedance and by echocardiography in a controlled clinical setting. Methods: The study enrolled 68 patients (mean age, 66 ± 9 years; 74% males) at 12 investigational sites. The patients had an indication for CRT-D implantation, New York Heart Association class II/III symptoms, left ventricular ejection fraction 15%–35%, and a QRS duration ≥150 ms. Two months after a CRT-D implantation, hemodynamic changes were provoked by overdrive pacing. Intracardiac impedance was recorded at rest and at four pacing rates ranging from 10 to 40 beats/min above the resting rate. In parallel, echocardiography measurements were performed. We hypothesized that a mean intra-individual correlation coefficient (rmean) between stroke impedance (difference between end-systolic and end-diastolic intracardiac impedance) measured by CRT-D and the aortic velocity time integral (i.e., stroke volume) determined by echocardiography would be significantly larger than 0.65. Results: The hypothesis was evaluated in 40 patients with complete data sets. The rmean was 0.797, with a lower confidence interval bound of 0.709. The study hypothesis was met (p = 0.007). A stepwise reduction of stroke impedance and stroke volume was observed with increasing heart rate. Conclusions: Intracardiac impedance measured by implanted CRT-Ds correlated well with the aortic velocity time integral (stroke volume) determined by echocardiography. The impedance measurements bear potential and are readily available technically, not requiring implantation of additional material beyond standard CRT-D system

Mutable elastic models for sculpting structured shapes

Special Issue: Proc. Eurographics, May 2013, Girona, Spain.International audienceIn this paper, we propose a new paradigm for free-form shape deformation. Standard deformable models minimize an energy measuring the distance to a single target shape. We propose a new, "mutable" elastic model. It represents complex geometry by a collection of parts and measures the distance of each part measures to a larger set of alternative rest configurations. By detecting and reacting to local switches between best-matching rest states, we build a 3D sculpting system: It takes a structured shape consisting of parts and replacement rules as input. The shape can subsequently be elongated, compressed, bent, cut, and merged within a constraints-based free-form editing interface, where alternative rest-states model to such changes. In practical experiments, we show that the approach yields a surprisingly intuitive and easy to implement interface for interactively designing objects described by such discrete shape grammars, for which direct shape control mechanisms were typically lacking

Computer-Assisted Proofs of Some Identities for Bessel Functions of Fractional Order

Abstract We employ computer algebra algorithms to prove a collection of identities involving Bessel functions with half-integer orders and other special functions. These identities appear in the famous Handbook of Mathematical Functions, as well as in its successor, the DLMF, but their proofs were lost. We use generating functions and symbolic summation techniques to produce new proofs for them

Use of a basal-plus insulin regimen in persons with type 2 diabetes stratified by age and body mass index: A pooled analysis of four clinical trials

AIMS: To evaluate the efficacy and safety of adding a single bolus dose of insulin glulisine to basal insulin ('basal-plus') in persons with type 2 diabetes. METHODS: Data from patients with poor glycemic control on oral antihyperglycemic drugs who were initiated on a 'basal-plus' regimen for up to 6 months were pooled from four randomized, multicenter studies. Glycated hemoglobin (HbA1c), fasting blood glucose, postprandial glucose (PPG), insulin dose and demographics were measured at baseline and end of study. RESULTS: 711 patients with a mean age of 59.9 years and a mean duration of diabetes of 11.0 years were included in the analysis population. A 'basal-plus' regimen was associated with significant decreases in HbA1c and PPG at 6 months, an increase in glargine and glulisine doses and small, but statistically significant, changes in body weight and BMI in all patient subsets. The proportion of patients with HbA1c<7% also increased in all populations studied, while the prevalence of severe hypoglycemia was low and did not significantly differ across patient groups. CONCLUSIONS: These results suggest that the use of 'basal-plus' can achieve a good therapeutic response with a low risk of hypoglycemia and weight gain, regardless of a patient's age or BMI. Copyright © 2015 Primary Care Diabetes Europe. Published by Elsevier Ltd. All rights reserved