Representation of differential operators in wavelet basis

AbstractExisting work on the representation of operators in one-dimensional, compactly-supported, orthonormal wavelet bases is extended to two dimensions. The nonstandard form of the representation of operators is given in separable two-dimensional, periodic, orthonormal wavelet bases. The matrix representation of the partial-differential operators ∂x and ∂y are constructed and a closed form formula for the matrix representation of a general partial-differential operator g(∂x, ∂y) is derived, where g is an analytic function

Remarkable properties of the sincp,q functions and related integrals

This is the final version. Available on open access from Elsevier via the DOI in this recordDifferent types of sinc integrals are investigated when the standard sine function is replaced by the generalised sinp,q in two parameters. A striking generalisation of the improper Dirichlet integral is achieved. A second surprising but interesting generalisation of the identity between the Dirichlet integral and that of the integrand sinc2 is discovered. Moreover, an asymptotically sharpened form of Ball's integral inequality is obtained in terms of two parameters.JSP

Symptom improvement and remission in untreated adults seeking treatment for obsessive-compulsive disorder: A systematic review and meta-analysis.

Obsessive-compulsive disorder (OCD) is a common psychiatric condition that results in significant distress and impairment, and high societal costs. OCD is widely considered to be a chronic condition, however, our understanding of the chronicity of the disorder, and the incidence of spontaneous remission, has largely relied on longitudinal studies of individuals who have received treatment. The aim of the current study is to examine symptom improvement and rate of spontaneous remission in individuals with OCD who were assigned to a no-treatment control group within a randomized controlled trial using a meta-analytic approach. Twelve studies (n = 282; mean age = 35.52; 60.03 % female) were included in the meta-analysis. The pooled within-group effect size was negligible (g = -0.14; 95 % CI [-0.25, -0.04]) and only 4 % of participants demonstrated spontaneous remission across an average of 10.92 weeks (event rate = 0.04; [95 % CI: 0.01, 0.11]). Sample size and duration of OCD symptoms significantly moderated the effect size for symptom change. No moderators were found for symptom remission. The findings add to the small body of literature demonstrating that OCD has a chronic and unremitting course without treatment

Psi-series solutions of the cubic H\'{e}non-Heiles system and their convergence

The cubic H\'enon-Heiles system contains parameters, for most values of which, the system is not integrable. In such parameter regimes, the general solution is expressible in formal expansions about arbitrary movable branch points, the so-called psi-series expansions. In this paper, the convergence of known, as well as new, psi-series solutions on real time intervals is proved, thereby establishing that the formal solutions are actual solutions

Thermal maps of gases in heterogeneous reactions.

More than 85 per cent of all chemical industry products are made using catalysts1,2, the overwhelming majority of which are heterogeneous catalysts2 that function at the gas–solid interface3. Consequently, much effort is invested in optimizing the design of catalytic reactors, usually by modelling4 the coupling between heat transfer, fluid dynamics and surface reaction kinetics. The complexity involved requires a calibration of model approximations against experimental observations5,6, with temperature maps being particularly valuable because temperature control is often essential for optimal operation and because temperature gradients contain information about the energetics of a reaction. However, it is challenging to probe the behaviour of a gas inside a reactor without disturbing its flow, particularly when trying also to map the physical parameters and gradients that dictate heat and mass flow and catalytic efficiency1,2,3,4,5,6,7,8,9. Although optical techniques10,11,12 and sensors13,14 have been used for that purpose, the former perform poorly in opaque media and the latter perturb the flow. NMR thermometry can measure temperature non-invasively, but traditional approaches applied to gases produce signals that depend only weakly on temperature15,16 are rapidly attenuated by diffusion16,17 or require contrast agents18 that may interfere with reactions. Here we present a new NMR thermometry technique that circumvents these problems by exploiting the inverse relationship between NMR linewidths and temperature caused by motional averaging in a weak magnetic field gradient. We demonstrate the concept by non-invasively mapping gas temperatures during the hydrogenation of propylene in reactors packed with metal nanoparticles and metal–organic framework catalysts, with measurement errors of less than four per cent of the absolute temperature. These results establish our technique as a non-invasive tool for locating hot and cold spots in catalyst-packed gas–solid reactors, with unprecedented capabilities for testing the approximations used in reactor modelling

Having Fun in Learning Formal Specifications

There are many benefits in providing formal specifications for our software. However, teaching students to do this is not always easy as courses on formal methods are often experienced as dry by students. This paper presents a game called FormalZ that teachers can use to introduce some variation in their class. Students can have some fun in playing the game and, while doing so, also learn the basics of writing formal specifications in the form of pre- and post-conditions. Unlike existing software engineering themed education games such as Pex and Code Defenders, FormalZ takes the deep gamification approach where playing gets a more central role in order to generate more engagement. This short paper presents our work in progress: the first implementation of FormalZ along with the result of a preliminary users' evaluation. This implementation is functionally complete and tested, but the polishing of its user interface is still future work