Small Volume Fraction Limit of the Diblock Copolymer Problem: II. Diffuse-Interface Functional

We present the second of two articles on the small volume fraction limit of a nonlocal Cahn-Hilliard functional introduced to model microphase separation of diblock copolymers. After having established the results for the sharp-interface version of the functional (arXiv:0907.2224), we consider here the full diffuse-interface functional and address the limit in which epsilon and the volume fraction tend to zero but the number of minority phases (called particles) remains O(1). Using the language of Gamma-convergence, we focus on two levels of this convergence, and derive first- and second-order effective energies, whose energy landscapes are simpler and more transparent. These limiting energies are only finite on weighted sums of delta functions, corresponding to the concentration of mass into `point particles'. At the highest level, the effective energy is entirely local and contains information about the size of each particle but no information about their spatial distribution. At the next level we encounter a Coulomb-like interaction between the particles, which is responsible for the pattern formation. We present the results in three dimensions and comment on their two-dimensional analogues

Small Volume Fraction Limit of the Diblock Copolymer Problem: I. Sharp Interface Functional

We present the first of two articles on the small volume fraction limit of a nonlocal Cahn-Hilliard functional introduced to model microphase separation of diblock copolymers. Here we focus attention on the sharp-interface version of the functional and consider a limit in which the volume fraction tends to zero but the number of minority phases (called particles) remains O(1). Using the language of Gamma-convergence, we focus on two levels of this convergence, and derive first and second order effective energies, whose energy landscapes are simpler and more transparent. These limiting energies are only finite on weighted sums of delta functions, corresponding to the concentration of mass into `point particles'. At the highest level, the effective energy is entirely local and contains information about the structure of each particle but no information about their spatial distribution. At the next level we encounter a Coulomb-like interaction between the particles, which is responsible for the pattern formation. We present the results here in both three and two dimensions.Comment: 37 pages, 1 figur

Domain structure of bulk ferromagnetic crystals in applied fields near saturation

We investigate the ground state of a uniaxial ferromagnetic plate with perpendicular easy axis and subject to an applied magnetic field normal to the plate. Our interest is the asymptotic behavior of the energy in macroscopically large samples near the saturation field. We establish the scaling of the critical value of the applied field strength below saturation at which the ground state changes from the uniform to a branched domain magnetization pattern and the leading order scaling behavior of the minimal energy. Furthermore, we derive a reduced sharp-interface energy giving the precise asymptotic behavior of the minimal energy in macroscopically large plates under a physically reasonable assumption of small deviations of the magnetization from the easy axis away from domain walls. On the basis of the reduced energy, and by a formal asymptotic analysis near the transition, we derive the precise asymptotic values of the critical field strength at which non-trivial minimizers (either local or global) emerge. The non-trivial minimal energy scaling is achieved by magnetization patterns consisting of long slender needle-like domains of magnetization opposing the applied fieldComment: 38 pages, 7 figures, submitted to J. Nonlin. Sci

Solve Non-Linear Parabolic Partial Differential Equation by Spline Collocation Method

This paper provides an overview of the formulation, analysis and implementation of Spline collocation method for the numerical solution of partial differential equation with two space variable which is of parabolic type. The method includes the solution of non-linear equation which can be expressed as in matrix form. The use of spline collocation methods in the solution of initial-boundary value problems for parabolic-type system id described, with emphasis on alternating direction implicit methods. Problem of vertical groundwater recharge solve by spline collocation method. Finally, recent applications of spline collocation method are outlined

Striped periodic minimizers of a two-dimensional model for martensitic phase transitions

In this paper we consider a simplified two-dimensional scalar model for the formation of mesoscopic domain patterns in martensitic shape-memory alloys at the interface between a region occupied by the parent (austenite) phase and a region occupied by the product (martensite) phase, which can occur in two variants (twins). The model, first proposed by Kohn and Mueller, is defined by the following functional: $${\cal E}(u)=\beta||u(0,\cdot)||^2_{H^{1/2}([0,h])}+ \int_{0}^{L} dx \int_0^h dy \big(|u_x|^2 + \epsilon |u_{yy}| \big)$$ where $u:[0,L]\times[0,h]\to R$ is periodic in $y$ and $u_y=\pm 1$ almost everywhere. Conti proved that if $\beta\gtrsim\epsilon L/h^2$ then the minimal specific energy scales like $\sim \min\{(\epsilon\beta/L)^{1/2}, (\epsilon/L)^{2/3}\}$, as $(\epsilon/L)\to 0$. In the regime $(\epsilon\beta/L)^{1/2}\ll (\epsilon/L)^{2/3}$, we improve Conti's results, by computing exactly the minimal energy and by proving that minimizers are periodic one-dimensional sawtooth functions.Comment: 29 pages, 3 figure

Are We That Map?

The aim of Are We That Map? was to co-develop participatory map-making workshops that actively includes and engages children and young people with a lived experience of learning disabilities/difficulties (LD/D). The project was a partnership between Newham based Rosetta Arts, purpleSTARS, RIX Centre, University of East London (UEL) and Living Maps Network. Employing a creative, iterative approach we developed workshops that would be explorative and happy to embrace the unknown. We aimed to develop a series of exercises that could ‘wake us up’ to notice how our bodies' sensory experiences create our own unique experience of the world and how we could share this making a map. This work feeds into Living Maps Young Citizens Atlas. Working together in a unique collaboration, we devised, delivered, reviewed and refined a Map Making workshop package in which people with the lived experience of Learning Difficulties and Differences LD/D developed and facilitated, working to their strengths alongside other expert artists and educators. A co-production approach was used, that included people with a lived experience of a LD/D during the development workshops and as facilitators during workshops in school. Following a robust series of development workshops, we carefully designed, prepared and identified possible activities to include in the Map Making workshop package. We invited members from the Tower Project that supports people with LD/D in Tower Hamlets London to help develop and test our Map Making workshop activities to see what would work and what would not

The challenges of diagnosing osteoporosis and the limitations of currently available tools

Abstract Dual-energy X-ray absorptiometry (DXA) was the first imaging tool widely utilized by clinicians to assess fracture risk, especially in postmenopausal women. The development of DXA nearly coincided with the availability of effective osteoporosis medications. Although osteoporosis in adults is diagnosed based on a T-score equal to or below − 2.5 SD, most individuals who sustain fragility fractures are above this arbitrary cutoff. This incongruity poses a challenge to clinicians to identify patients who may benefit from osteoporosis treatments. DXA scanners generate 2 dimensional images of complex 3 dimensional structures, and report bone density as the quotient of the bone mineral content divided by the bone area. An obvious pitfall of this method is that a larger bone will convey superior strength, but may in fact have the same bone density as a smaller bone. Other imaging modalities are available such as peripheral quantitative CT, but are largely research tools. Current osteoporosis medications increase bone density and reduce fracture risk but the mechanisms of these actions vary. Anti-resorptive medications (bisphosphonates and denosumab) primarily increase endocortical bone by bolstering mineralization of endosteal resorption pits and thereby increase cortical thickness and reduce cortical porosity. Anabolic medications (teriparatide and abaloparatide) increase the periosteal and endosteal perimeters without large changes in cortical thickness resulting in a larger more structurally sound bone. Because of the differences in the mechanisms of the various drugs, there are likely benefits of selecting a treatment based on a patient’s unique bone structure and pattern of bone loss. This review retreats to basic principles in order to advance clinical management of fragility fractures by examining how skeletal biomechanics, size, shape, and ultra-structural properties are the ultimate predictors of bone strength. Accurate measurement of these skeletal parameters through the development of better imaging scanners is critical to advancing fracture risk assessment and informing clinicians on the best treatment strategy. With this information, a “treat to target” approach could be employed to tailor current and future therapies to each patient’s unique skeletal characteristics.https://deepblue.lib.umich.edu/bitstream/2027.42/143867/1/40842_2018_Article_62.pd

Digital Financial Inclusion in a Cashless Society

Quarriers commissioned the Rix Inclusive Research team to carry out an evaluation study to explore how people with learning disabilities manage and use their money, what works and what is difficult. This report details the activities undertaken by the research team as part of this phase. It provides an account of the aims and objectives, methodology, points of discussion, conclusion, and recommendations for Phase 2 of the project, which will consider possible practical solutions to support people with learning disabilities to move from cash to cashless (digital) payments, and towards digital finance overall, in order to fully participate in the cashless society

The existence of an inverse limit of inverse system of measure spaces - a purely measurable case

The existence of an inverse limit of an inverse system of (probability) measure spaces has been investigated since the very beginning of the birth of the modern probability theory. Results from Kolmogorov [10], Bochner [2], Choksi [5], Metivier [14], Bourbaki [3] among others have paved the way of the deep understanding of the problem under consideration. All the above results, however, call for some topological concepts, or at least ones which are closely related topological ones. In this paper we investigate purely measurable inverse systems of (probability) measure spaces, and give a sucient condition for the existence of a unique inverse limit. An example for the considered purely measurable inverse systems of (probability) measure spaces is also given