• Title/Summary/Keyword: graphs of function and its derivatives

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COMPUTATIONS OF NATURAL CONVECTION FLOW WITHIN A SQUARE CAVITY BY HERMITE STREAM FUNCTION METHOD (Hermite 유동함수법에 의한 정사각형 공동 내부의 자연대류 유동계산)

  • Kim, J.W.
    • Journal of computational fluids engineering
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    • v.14 no.4
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    • pp.67-77
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    • 2009
  • This paper is a continuation of a recent development on the Hermite-based divergence-free element method and deals with a non-isothermal fluid flow driven by the buoyancy force in a square cavity with temperature difference across the two sides. Two Hermite functions are considered for numerical computations in this paper. One is a cubic function and the other is a quartic function. The degrees-of-freedom of the cubic Hermite function are stream function and its first and second derivatives for the velocity field, and temperature and its first derivatives for the temperature field. The degrees-of-freedom of the quartic Hermite function include two second derivatives and one cross derivative of the stream function in addition to the degrees-of-freedom of the cubic stream function. This paper presents a brief review on the Hermite based divergence-free basis functions and its finite element formulations for the buoyancy driven flow. The present algorithm does not employ any upwinding or a stabilization term. However, numerical values and contour graphs for major flow variables showed good agreements with those by De Vahl Davis[6].

On the students' thinking of the properties of derivatives (도함수의 성질에 관련한 학생들의 사고에 대하여)

  • Choi, Young Ju;Hong, Jin Kon
    • The Mathematical Education
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    • v.53 no.1
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    • pp.25-40
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    • 2014
  • Mathematical concept exists in the structural form, not in the independent form. The purpose of this study is to consider the network which students actually have for the mathematical concept structure related to the properties of derivatives. First, we analyzed the properties of derivatives in 'Mathematics II' and showed the mathematical concept structure of the relations among derivatives, functions, and primitive functions as a network. Also, we investigated the understanding of high school students for the mathematical concept structure between derivatives and functions, and the structure between functions and second order derivatives when the functional formula is not given, and only the graph is given. The results showed that students mainly focus on the relation of 'function-derivatives', the thinking process for direction of derivative and the thinking style for algebra. On this basis, we suggest the educational implication that is necessary for students to build the network properly.

A STUDY OF THE RIGHT LOCAL GENERAL TRUNCATED M-FRACTIONAL DERIVATIVE

  • Chauhan, Rajendrakumar B.;Chudasama, Meera H.
    • Communications of the Korean Mathematical Society
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    • v.37 no.2
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    • pp.503-520
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    • 2022
  • We introduce a new type of fractional derivative, which we call as the right local general truncated M-fractional derivative for α-differentiable functions that generalizes the fractional derivative type introduced by Anastassiou. This newly defined operator generalizes the standard properties and results of the integer order calculus viz. the Rolle's theorem, the mean value theorem and its extension, inverse property, the fundamental theorem of calculus and the theorem of integration by parts. Then we represent a relation of the newly defined fractional derivative with known fractional derivative and in context with this derivative a physical problem, Kirchoff's voltage law, is generalized. Also, the importance of this newly defined operator with respect to the flexibility in the parametric values is described via the comparison of the solutions in the graphs using MATLAB software.

Comparison of Deep Learning Frameworks: About Theano, Tensorflow, and Cognitive Toolkit (딥러닝 프레임워크의 비교: 티아노, 텐서플로, CNTK를 중심으로)

  • Chung, Yeojin;Ahn, SungMahn;Yang, Jiheon;Lee, Jaejoon
    • Journal of Intelligence and Information Systems
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    • v.23 no.2
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    • pp.1-17
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    • 2017
  • The deep learning framework is software designed to help develop deep learning models. Some of its important functions include "automatic differentiation" and "utilization of GPU". The list of popular deep learning framework includes Caffe (BVLC) and Theano (University of Montreal). And recently, Microsoft's deep learning framework, Microsoft Cognitive Toolkit, was released as open-source license, following Google's Tensorflow a year earlier. The early deep learning frameworks have been developed mainly for research at universities. Beginning with the inception of Tensorflow, however, it seems that companies such as Microsoft and Facebook have started to join the competition of framework development. Given the trend, Google and other companies are expected to continue investing in the deep learning framework to bring forward the initiative in the artificial intelligence business. From this point of view, we think it is a good time to compare some of deep learning frameworks. So we compare three deep learning frameworks which can be used as a Python library. Those are Google's Tensorflow, Microsoft's CNTK, and Theano which is sort of a predecessor of the preceding two. The most common and important function of deep learning frameworks is the ability to perform automatic differentiation. Basically all the mathematical expressions of deep learning models can be represented as computational graphs, which consist of nodes and edges. Partial derivatives on each edge of a computational graph can then be obtained. With the partial derivatives, we can let software compute differentiation of any node with respect to any variable by utilizing chain rule of Calculus. First of all, the convenience of coding is in the order of CNTK, Tensorflow, and Theano. The criterion is simply based on the lengths of the codes and the learning curve and the ease of coding are not the main concern. According to the criteria, Theano was the most difficult to implement with, and CNTK and Tensorflow were somewhat easier. With Tensorflow, we need to define weight variables and biases explicitly. The reason that CNTK and Tensorflow are easier to implement with is that those frameworks provide us with more abstraction than Theano. We, however, need to mention that low-level coding is not always bad. It gives us flexibility of coding. With the low-level coding such as in Theano, we can implement and test any new deep learning models or any new search methods that we can think of. The assessment of the execution speed of each framework is that there is not meaningful difference. According to the experiment, execution speeds of Theano and Tensorflow are very similar, although the experiment was limited to a CNN model. In the case of CNTK, the experimental environment was not maintained as the same. The code written in CNTK has to be run in PC environment without GPU where codes execute as much as 50 times slower than with GPU. But we concluded that the difference of execution speed was within the range of variation caused by the different hardware setup. In this study, we compared three types of deep learning framework: Theano, Tensorflow, and CNTK. According to Wikipedia, there are 12 available deep learning frameworks. And 15 different attributes differentiate each framework. Some of the important attributes would include interface language (Python, C ++, Java, etc.) and the availability of libraries on various deep learning models such as CNN, RNN, DBN, and etc. And if a user implements a large scale deep learning model, it will also be important to support multiple GPU or multiple servers. Also, if you are learning the deep learning model, it would also be important if there are enough examples and references.