Exhibiting cross-diffusion-induced patterns for reaction-diffusion systems on evolving domains and surfaces

The aim of this manuscript is to present for the first time the application of the finite element method for solving reaction-diffusion systems with cross-diffusion on continuously evolving domains and surfaces. Furthermore we present pattern formation generated by the reaction-diffusion systemwith cross-diffusion on evolving domains and surfaces. A two-component reaction-diffusion system with linear cross-diffusion in both u and v is presented. The finite element method is based on the approximation of the domain or surface by a triangulated domain or surface consisting of a union of triangles. For surfaces, the vertices of the triangulation lie on the continuous surface. A finite element space of functions is then defined by taking the continuous functions which are linear affine on each simplex of the triangulated domain or surface. To demonstrate the role of cross-diffusion to the theory of pattern formation, we compute patterns with model kinetic parameter values that belong only to the cross-diffusion parameter space; these do not belong to the standard parameter space for classical reaction-diffusion systems. Numerical results exhibited show the robustness, flexibility, versatility, and generality of our methodology; the methodology can deal with complicated evolution laws of the domain and surface, and these include uniform isotropic and anisotropic growth profiles as well as those profiles driven by chemical concentrations residing in the domain or on the surface

Coherent states associated to the wavefunctions and the spectrum of the isotonic oscillator

Classes of coherent states are presented by replacing the labeling parameter $z$ of Klauder-Perelomov type coherent states by confluent hypergeometric functions with specific parameters. Temporally stable coherent states for the isotonic oscillator Hamiltonian are presented and these states are identified as a particular case of the so-called Mittag-Leffler coherent states.Comment: 12 page

Spectra generated by a confined softcore Coulomb potential

Analytic and approximate solutions for the energy eigenvalues generated by a confined softcore Coulomb potentials of the form a/(r+\beta) in d>1 dimensions are constructed. The confinement is effected by linear and harmonic-oscillator potential terms, and also through `hard confinement' by means of an impenetrable spherical box. A byproduct of this work is the construction of polynomial solutions for a number of linear differential equations with polynomial coefficients, along with the necessary and sufficient conditions for the existence of such solutions. Very accurate approximate solutions for the general problem with arbitrary potential parameters are found by use of the asymptotic iteration method.Comment: 17 pages, 2 figure

Analysis of ensemble learning using simple perceptrons based on online learning theory

Ensemble learning of $K$ nonlinear perceptrons, which determine their outputs by sign functions, is discussed within the framework of online learning and statistical mechanics. One purpose of statistical learning theory is to theoretically obtain the generalization error. This paper shows that ensemble generalization error can be calculated by using two order parameters, that is, the similarity between a teacher and a student, and the similarity among students. The differential equations that describe the dynamical behaviors of these order parameters are derived in the case of general learning rules. The concrete forms of these differential equations are derived analytically in the cases of three well-known rules: Hebbian learning, perceptron learning and AdaTron learning. Ensemble generalization errors of these three rules are calculated by using the results determined by solving their differential equations. As a result, these three rules show different characteristics in their affinity for ensemble learning, that is ``maintaining variety among students." Results show that AdaTron learning is superior to the other two rules with respect to that affinity.Comment: 30 pages, 17 figure

Quantum Circulant Preconditioner for Linear System of Equations

We consider the quantum linear solver for $Ax=b$ with the circulant preconditioner $C$. The main technique is the singular value estimation (SVE) introduced in [I. Kerenidis and A. Prakash, Quantum recommendation system, in ITCS 2017]. However, some modifications of SVE should be made to solve the preconditioned linear system $C^{-1} Ax = C^{-1} b$. Moreover, different from the preconditioned linear system considered in [B. D. Clader, B. C. Jacobs, C. R. Sprouse, Preconditioned quantum linear system algorithm, Phys. Rev. Lett., 2013], the circulant preconditioner is easy to construct and can be directly applied to general dense non-Hermitian cases. The time complexity depends on the condition numbers of $C$ and $C^{-1} A$, as well as the Frobenius norm $\|A\|_F$

Ultralight, reusable biopolymer aerogels: Formation mechanisms to applications in selective fluid sorption and oil spill remediation

Highly porous (99.7 %), ultra-light (4.3 mg/ml) and mechanically robust cellulose ester aerogels with tailored hydrophobicity are synthesized. The aerogels achieve maximum compression strain of 92 % without failure and reach a compressive stress of 350 kPa, which is 100 times higher than that reported for cellulosic aerogels. In its native, unmodified state, the aerogels are hydrophilic and display unprecedented water uptake (45-90 g/g) while affording wet strength. Further adjustment of the aerogels towards hydrophobicity and oleophilicity via chemical vapor deposition with an organo-silane species reveal them to exhibit high oil retention (20-30 g/g aerogel) while maintaining mechanical integrity for fast oil cleanup from aqueous media under marine conditions. The modified aerogels are reusable and durable as they retain their hydrophobicity for months under ambient conditions. The Zisman and Fowkes theoretical frameworks are used to identify the selectiveness of the aerogel and establish a criterion for separation of various non-polar fluids from water media

Double-blind randomized controlled trial of letrozole versus clomiphene citrate in subfertile women with polycystic ovarian syndrome

STUDY QUESTION: Would letrozole as a primary ovulation induction agent generate better pregnancy rates than clomiphene citrate (CC) in subfertile women with anovulatory polycystic ovarian syndrome (PCOS)? SUMMARY ANSWER: Participants receiving letrozole as a primary treatment achieved a significantly (P = 0.022) higher clinical pregnancy rate per patient (61.2%) compared to CC (43.0%). WHAT IS KNOWN ALREADY: According to a recent Cochrane systematic review (2014), letrozole appears to improve live-birth (LB) and pregnancy rates in anovulatory women with PCOS, compared to CC. However, the review concluded that the quality of evidence was low due to poor reporting of study methods and possible publication bias. STUDY DESIGN, SIZE, DURATION: This double-blind randomized controlled trial (RCT) included 159 participants between April 2007 and June 2014. Subjects were randomly allocated to either CC (n = 79) or letrozole (n = 80) in a 1:1 ratio. Both drugs were encapsulated to look identical. Randomization was performed in mixed blocks and stratified by patients’ BMI (<30 and 30–35 kg/m2). PARTICIPANTS/MATERIALS, SETTING, METHODS: The trial included subfertile women diagnosed with PCOS. Treatment started with one tablet (CC 50 mg, letrozole 2.5 mg) increasing to two in non-responders and continuing until pregnancy or for up to six ovulatory cycles. Non-responders were crossed over to the other treatment after a 6-week break. Cycles were initially monitored with ultrasound follicle tracking then mid-luteal serum progesterone measurement in subsequent cycles. MAIN RESULTS AND THE ROLE OF CHANCE: Amongst the 159 participants included in the intention-to-treat analysis, four women conceived before treatment and six were lost-to-follow-up. The remaining 149 participants (74 on CC and 75 on letrozole) completed at least the first treatment. Women receiving letrozole achieved a significantly (P = 0.022; absolute difference [95% confidence interval] 18% [3–33%]) higher pregnancy rate (61.%) than those on CC (43%). The median number of treatment cycles received until pregnancy was significantly (log rank P = 0.038) smaller with letrozole (4[3–5] cycles) compared to CC (6[4–7] cycles). LB rates were not statistically (P = 0.089) different between the two groups, although there was a trend towards higher rates on letrozole (48.8%) compared to CC (35.4%). After the crossover, pregnancy and LB rates on letrozole (n = 45; 28.9 and 24.4%, respectively) were not statistically (P = 0.539 and P = 0.601) different from CC (n = 31; 22.6 and 19.4%). LIMITATIONS, REASONS FOR CAUTION: One possible limitation of this trial may be the exclusion of PCOS women with BMI > 35 kg/m2, which would limit the applicability of the results in this subgroup of PCOS. However, this group of women are generally excluded from treatment in the majority of fertility centres, especially in Europe, due to the associated challenges and risks. WIDER IMPLICATIONS OF THE FINDINGS: The results of this trial are consistent with the recent Cochrane systematic review. However, with its robust design, the current RCT provides more valid and compelling evidence for the superiority of letrozole over CC as a primary ovulation induction agent in PCOS women with 40% increase in pregnancy rates and with a shorter time-to-pregnancy. Furthermore, the participants in this RCT are a good representation of subfertile PCOS population receiving fertility treatment in Europe and worldwide. The results are therefore globally generalizable for clinical practice. STUDY FUNDING/COMPETING INTEREST(S): This RCT was mainly funded by the R&D Funding Scheme of Derby Hospitals NHS Foundation Trust. The study also received funds from School of Medicine, University of Nottingham. The Trust R&D department was involved in the development of the protocol and the running of the trial. The trial was sponsored and monitored by the University of Nottingham. The authors have no conflicts of interest. TRIAL REGISTRATION NUMBER: www.Clinicaltrials.gov: NCT00478504. TRIAL REGISTRATION DATE: Registration was verified on 23/05/2007. DATE OF FIRST PATIENT’S ENROLMENT: 25/04/2007