Variational Problems with Fractional Derivatives: Euler-Lagrange Equations

We generalize the fractional variational problem by allowing the possibility that the lower bound in the fractional derivative does not coincide with the lower bound of the integral that is minimized. Also, for the standard case when these two bounds coincide, we derive a new form of Euler-Lagrange equations. We use approximations for fractional derivatives in the Lagrangian and obtain the Euler-Lagrange equations which approximate the initial Euler-Lagrange equations in a weak sense

Collisional kinetics of non-uniform electric field, low-pressure, direct-current discharges in H2_{2}

A model of the collisional kinetics of energetic hydrogen atoms, molecules, and ions in pure H$_2$ discharges is used to predict H$_\alpha$ emission profiles and spatial distributions of emission from the cathode regions of low-pressure, weakly-ionized discharges for comparison with a wide variety of experiments. Positive and negative ion energy distributions are also predicted. The model developed for spatially uniform electric fields and current densities less than $10^{-3}$ A/m$^2$ is extended to non-uniform electric fields, current densities of $10^{3}$ A/m$^2$, and electric field to gas density ratios $E/N = 1.3$ MTd at 0.002 to 5 Torr pressure. (1 Td = $10^{-21}$ V m$^2$ and 1 Torr = 133 Pa) The observed far-wing Doppler broadening and spatial distribution of the H$_\alpha$ emission is consistent with reactions among H$^+$, H$_2^+$, H$_3^+$, and H$^-H$ ions, fast H atoms, and fast H$_2$ molecules, and with reflection, excitation, and attachment to fast H atoms at surfaces. The H$_\alpha$ excitation and H$^-$ formation occur principally by collisions of fast H, fast H$_2$, and H$^+$ with H$_2$. Simplifications include using a one-dimensional geometry, a multi-beam transport model, and the average cathode-fall electric field. The H$_\alpha$ emission is linear with current density over eight orders of magnitude. The calculated ion energy distributions agree satisfactorily with experiment for H$_2^+$ and H$_3^+$, but are only in qualitative agreement for H$^+$ and H$^-$. The experiments successfully modeled range from short-gap, parallel-plane glow discharges to beam-like, electrostatic-confinement discharges.Comment: Submitted to Plasmas Sources Science and Technology 8/18/201

Fractional variational calculus of variable order

We study the fundamental problem of the calculus of variations with variable order fractional operators. Fractional integrals are considered in the sense of Riemann-Liouville while derivatives are of Caputo type.Comment: Submitted 26-Sept-2011; accepted 18-Oct-2011; withdrawn by the authors 21-Dec-2011; resubmitted 27-Dec-2011; revised 20-March-2012; accepted 13-April-2012; to 'Advances in Harmonic Analysis and Operator Theory', The Stefan Samko Anniversary Volume (Eds: A. Almeida, L. Castro, F.-O. Speck), Operator Theory: Advances and Applications, Birkh\"auser Verlag (http://www.springer.com/series/4850

Progress in Classical and Quantum Variational Principles

We review the development and practical uses of a generalized Maupertuis least action principle in classical mechanics, in which the action is varied under the constraint of fixed mean energy for the trial trajectory. The original Maupertuis (Euler-Lagrange) principle constrains the energy at every point along the trajectory. The generalized Maupertuis principle is equivalent to Hamilton's principle. Reciprocal principles are also derived for both the generalized Maupertuis and the Hamilton principles. The Reciprocal Maupertuis Principle is the classical limit of Schr\"{o}dinger's variational principle of wave mechanics, and is also very useful to solve practical problems in both classical and semiclassical mechanics, in complete analogy with the quantum Rayleigh-Ritz method. Classical, semiclassical and quantum variational calculations are carried out for a number of systems, and the results are compared. Pedagogical as well as research problems are used as examples, which include nonconservative as well as relativistic systems

A retrospective claims analysis of combination therapy in the treatment of adult attention-deficit/hyperactivity disorder (ADHD)

<p>Abstract</p> <p>Background</p> <p>Combination therapy in managing psychiatric disorders is not uncommon. While combination therapy has been documented for depression and schizophrenia, little is known about combination therapy practices in managing attention-deficit/hyperactivity disorder (ADHD). This study seeks to quantify the combination use of ADHD medications and to understand predictors of combination therapy.</p> <p>Methods</p> <p>Prescription dispensing events were drawn from a U.S. national claims database including over 80 managed-care plans. Patients studied were age 18 or over with at least 1 medical claim with a diagnosis of ADHD (International Classification of Diseases, Ninth Revision, Clinical Modification [ICD-9-CM] code 314.0), a pharmacy claim for ADHD medication during the study period July2003 to June2004, and continuous enrollment 6 months prior to and throughout the study period. Dispensing events were grouped into 6 categories: atomoxetine (ATX), long-acting stimulants (LAS), intermediate-acting stimulants (IAS), short-acting stimulants (SAS), bupropion (BUP), and Alpha-2 Adrenergic Agonists (A2A). Events were assigned to calendar months, and months with combined use from multiple categories within patient were identified. Predictors of combination therapy for LAS and for ATX were modeled for patients covered by commercial plans using logistic regression in a generalized estimating equations framework to adjust for within-patient correlation between months of observation. Factors included age, gender, presence of the hyperactive component of ADHD, prior diagnoses for psychiatric disorders, claims history of recent psychiatric visit, insurance plan type, and geographic region.</p> <p>Results</p> <p>There were 18,609 patients identified representing a total of 11,886 months of therapy with ATX; 40,949 months with LAS; 13,622 months with IAS; 38,141 months with SAS; 22,087 months with BUP; and 1,916 months with A2A. Combination therapy was present in 19.7% of continuing months (months after the first month of therapy) for ATX, 21.0% for LAS, 27.4% for IAS, 23.1% for SAS, 36.9% for BUP, and 53.0% for A2A.</p> <p>For patients receiving LAS, being age 25–44 or age 45 and older versus being 18–24 years old, seeing a psychiatrist, having comorbid depression, or having point-of-service coverage versus a Health Maintenance Organization (HMO) resulted in odds ratios significantly greater than 1, representing increased likelihood for combination therapy in managing adult ADHD.</p> <p>For patients receiving ATX, being age 25–44 or age 45 and older versus being 18–24 years old, seeing a psychiatrist, having a hyperactive component to ADHD, or having comorbid depression resulted in odds ratios significantly greater than 1, representing increased likelihood for combination therapy in managing adult ADHD.</p> <p>Conclusion</p> <p>ATX and LAS are the most likely drugs to be used as monotherapy. Factors predicting combination use were similar for months in which ATX was used and for months in which LAS was used except that a hyperactive component to ADHD predicted increased combination use for ATX but not for LAS.</p

The Hahn Quantum Variational Calculus

We introduce the Hahn quantum variational calculus. Necessary and sufficient optimality conditions for the basic, isoperimetric, and Hahn quantum Lagrange problems, are studied. We also show the validity of Leitmann's direct method for the Hahn quantum variational calculus, and give explicit solutions to some concrete problems. To illustrate the results, we provide several examples and discuss a quantum version of the well known Ramsey model of economics.Comment: Submitted: 3/March/2010; 4th revision: 9/June/2010; accepted: 18/June/2010; for publication in Journal of Optimization Theory and Application

Analysis of the archetypal functional equation in the non-critical case

We study the archetypal functional equation of the form $y(x)=\iint_{R^2} y(a(x-b))\,\mu(da,db)$ ($x\in R$), where $\mu$ is a probability measure on $R^2$; equivalently, $y(x)=E\{y(\alpha (x-\beta))\}$, where $E$ is expectation with respect to the distribution $\mu$ of random coefficients $(\alpha,\beta)$. Existence of non-trivial (i.e. non-constant) bounded continuous solutions is governed by the value $K:=\iint_{R^2}\ln |a| \mu(da,db) =E \{\ln |\alpha|\}$; namely, under mild technical conditions no such solutions exist whenever $K0$ (and $\alpha>0$) then there is a non-trivial solution constructed as the distribution function of a certain random series representing a self-similar measure associated with $(\alpha,\beta)$. Further results are obtained in the supercritical case $K>0$, including existence, uniqueness and a maximum principle. The case with $P(\alpha0$ is drastically different from that with $\alpha>0$; in particular, we prove that a bounded solution $y(\cdot)$ possessing limits at $\pm\infty$ must be constant. The proofs employ martingale techniques applied to the martingale $y(X_n)$, where $(X_n)$ is an associated Markov chain with jumps of the form $x\rightsquigarrow\alpha (x-\beta)$

Agreement Between Magnetic Resonance Imaging Proton Density Fat Fraction Measurements and Pathologist-assigned Steatosis Grades of Liver Biopsies from Adults with Nonalcoholic Steatohepatitis

Background & Aims We assessed the diagnostic performance of magnetic resonance imaging (MRI) proton density fat fraction (PDFF) in grading hepatic steatosis and change in hepatic steatosis in adults with nonalcoholic steatohepatitis (NASH) in a multi-center study, using central histology as reference. Methods We collected data from 113 adults with NASH participating in a multi-center, randomized, double-masked, placebo-controlled, phase 2b trial to compare the efficacy cross-sectionally and longitudinally of obeticholic acid vs placebo. Hepatic steatosis was assessed at baseline and after 72 weeks of obeticholic acid or placebo by liver biopsy and MRI (scanners from different manufacturers, at 1.5T or 3T). We compared steatosis estimates by PDFF vs histology. Histologic steatosis grade was scored in consensus by a pathology committee. Cross-validated receiver operating characteristic (ROC) analyses were performed. Results At baseline, 34% of subjects had steatosis grade 0 or 1, 39% had steatosis grade 2, and 27% had steatosis grade 3; corresponding mean PDFF values were 9.8%±3.7%, 18.1%±4.3%, and 30.1%±8.1%. PDFF classified steatosis grade 0–1 vs 2–3 with an area under the ROC curve (AUROC) of 0.95 (95% CI, 0.91–0.98), and grade 0–2 vs grade 3 steatosis with an AUROC of 0.96 (95% CI, 0.93–0.99). PDFF cut-off values at 90% specificity were 16.3% for grades 2–3 and 21.7% for grade 3, with corresponding sensitivities of 83% and 84%. After 72 weeks' of obeticholic vs placebo, 42% of subjects had a reduced steatosis grade (mean reduction in PDFF from baseline of 7.4%±8.7%), 49% had no change in steatosis grade (mean increase in PDFF from baseline of 0.3%±6.3%), and 9% had an increased steatosis grade (mean increase in PDFF from baseline of 7.7%±6.0%). PDFF change identified subjects with reduced steatosis grade with an AUROC of 0.81 (95% CI, 0.71–0.91) and increased steatosis grade with an AUROC of 0.81 (95% CI, 0.63–0.99). A PDFF reduction of 5.15% identified subjects with reduced steatosis grade with 90% specificity and 58% sensitivity, whereas a PDFF increase of 5.6% identified those with increased steatosis grade with 90% specificity and 57% sensitivity. Conclusions Based on data from a phase 2 randomized controlled trial of adults with NASH, PDFF estimated by MRI scanners of different field strength and at different sites, accurately classifies grades and changes in hepatic steatosis when histologic analysis of biopsies is used as a reference

Direct and Inverse Variational Problems on Time Scales: A Survey

We deal with direct and inverse problems of the calculus of variations on arbitrary time scales. Firstly, using the Euler-Lagrange equation and the strengthened Legendre condition, we give a general form for a variational functional to attain a local minimum at a given point of the vector space. Furthermore, we provide a necessary condition for a dynamic integro-differential equation to be an Euler-Lagrange equation (Helmholtz's problem of the calculus of variations on time scales). New and interesting results for the discrete and quantum settings are obtained as particular cases. Finally, we consider very general problems of the calculus of variations given by the composition of a certain scalar function with delta and nabla integrals of a vector valued field.Comment: This is a preprint of a paper whose final and definite form will be published in the Springer Volume 'Modeling, Dynamics, Optimization and Bioeconomics II', Edited by A. A. Pinto and D. Zilberman (Eds.), Springer Proceedings in Mathematics & Statistics. Submitted 03/Sept/2014; Accepted, after a revision, 19/Jan/201