scholarly journals Surface energies emerging in a microscopic, two-dimensional two-well problem

2017 ◽  
Vol 147 (5) ◽  
pp. 1041-1089 ◽  
Author(s):  
Georgy Kitavtsev ◽  
Stephan Luckhaus ◽  
Angkana Rüland
Keyword(s):  
Surface Energy ◽  
Two Dimensional ◽  
Direct Control ◽  
First Order ◽  
Surface Energies ◽  
Energy Scaling ◽  
Continuous Analogue ◽  
Dimensional Shape ◽  
Set Up ◽  
Microscopic Modelling

In this paper we are interested in the microscopic modelling of a two-dimensional two-well problem that arises from the square-to-rectangular transformation in (two-dimensional) shape-memory materials. In this discrete set-up, we focus on the surface energy scaling regime and further analyse the Hamiltonian that was introduced by Kitavtsev et al. in 2015. It turns out that this class of Hamiltonians allows for a direct control of the discrete second-order gradients and for a one-sided comparison with a two-dimensional spin system. Using this and relying on the ideas of Conti and Schweizer, which were developed for a continuous analogue of the model under consideration, we derive a (first-order) continuum limit. This shows the emergence of surface energy in the form of a sharp-interface limiting model as well the explicit structure of the minimizers to the latter.

scholarly journals Development of Android-Based Appy Pie Learning Media on Mathematics in Elementary School

2020 ◽  
Vol 8 (2) ◽  
pp. 81
Author(s):  
Inas Sayyida Latifa ◽  
Aan Subhan Pamungkas ◽  
Trian Pamungkas Alamsyah ◽  
Indhira Asih Vivi Yandari
Keyword(s):  
Elementary School ◽  
3D Model ◽  
Fourth Grade ◽  
Average Score ◽  
Two Dimensional ◽  
Design And Development ◽  
Fourth Grade Students ◽  
Dimensional Shape ◽  
Expert Validation

This research aimed to develop Android-based Appy Pie learning media in mathematics subjects, especially two-dimensional shape material. Moreover, to determine the validity level of the android-based Appy Pie learning media developed and to determine the students' responses after using android-based Appy Pie learning media. This research uses the 3D model (define, design, and development) as the modification result of the 4D model by Thiagarajan. The subjects of this research were 45 fourth-grade students of SDN Rawu. The result of this research is the average score of media experts validation is 91.11% which included in the “very feasible” category, the average score of material expert validation is 98.33% which included in the “very feasible” category. The average score of the students response is 91.11% that included in the “very good” category, so it can be concluded that the Android-based Appy Pie learning media is feasible to use in the two-dimensional shape material of fourth-grade.

Applications of PNC in Numerical Methods

2017 ◽  
pp. 235-268
Keyword(s):  
Probability Distribution ◽  
Shape Representation ◽  
Mathematical Problem ◽  
Distribution Functions ◽  
Two Dimensional ◽  
Key Points ◽  
Characteristic Points ◽  
Curve Modeling ◽  
Curve Contour ◽  
Dimensional Object

Nodes are treated as characteristic points of data for modeling and analyzing. The model of data can be built by choice of probability distribution function and nodes combination. Two-dimensional object is extrapolated and interpolated via nodes combination and different functions as discrete or continuous probability distribution functions: polynomial, sine, cosine, tangent, cotangent, logarithm, exponent, arc sin, arc cos, arc tan, arc cot or power function. Curve interpolation represents one of the most important problems in mathematics and computer science: how to model the curve via discrete set of two-dimensional points? Also the matter of shape representation (as closed curve - contour) and curve parameterization is still opened. For example pattern recognition, signature verification or handwriting identification problems are based on curve modeling via the choice of key points. So interpolation is not only a pure mathematical problem but important task in computer vision and artificial intelligence.

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