SYLLABUS MASTER OF MATHEMATICS

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SYLLABUS

MASTER OF MATHEMATICS

Departement of

Mathematics

Faculty of Science and Data Analytics

Institut Teknologi Sepuluh Nopember

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PROGRAMME LEARNING OUTCOMES (PLO)

PLO-1 [C3] Students are able to solve mathematical problems by applying fundamental mathematical statements, methods, and computations

PLO-2

[C4] Students are able to analyze mathematical problems in one of the fields: analysis, algebra, modeling, system, optimization or computing sciences

PLO-3

[C5] Students are able to work and research collaboratively on mathematical problems within either the area of pure mathematics or applied mathematics or computing sciences

PLO-4 Students are able to communicate and present mathematical ideas with clarity and coherence, both written and verbally

PLO-5 Students are able to make use of the principles of long life learning to improve knowledge and current issues on mathematics

PLO-6 Students are able to demonstrate religious attitude and tolerance

PLO-7

Students are able to demonstrate an attitude of responsibility and commitment to law enforcement, ethics, norms for community and environmental sustainability

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SYLLABUS

List of Master Program Courses SEMESTER 1

No. Course Code Course Name Credit

1. KM185101 Module Theory 3

2. KM185102 Functional Analysis 3

3. KM185103 Mathematical Modeling 3

4. KM185104 Numerical Computing 2

Total credits 11

SEMESTER 2

1. KM1852xx Compulsary Courses 6

2. KM1852xx Elective Courses 3

Total credits 9

SEMESTER 3

1. KM1853xx Elective Courses 8

Total credits 8

SEMESTER 4

1. KM185401 Thesis 8

Total credits 8

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List of Compulsary Courses

SEMESTER 2

No. Code Compulsary Courses Credits

1. KM185211 Approximation Theory 3

2. KM185212 Max-Plus Algebra 3

3. KM185221 Dynamical Systems 3

4. KM185222 Stochastics Calculus 3

5. KM185231 Computational Algorithm 3

6. KM185232 Mathematics of Machine Learning 3

List of Elective Courses

SEMESTER 2

No. Code Elective Courses Credits

1. KM185271 Discrete Transformation 3

2. KM185272 Formal Verification 3

3. KM185273 Systems and Controls 3

4. KM185274 Computational Fluid Dynamics 3

5. KM185275 Dynamical Optimization 3

6. KM185276 Financial Mathematics 3

7. KM185277 Digital Image Processing and Analysis 3

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SEMESTER 3

No. Code Elective Courses Credits

1. KM185372 Mathematical Biology 3

2. KM185373 Data Assimilation 3

3. KM185374 Computational Biology 3

4. KM185375 Mathematics of Derivatives 3

5. KM185376 Risk Analysis 3

6. KM185377 Graph Algebra 3

7. KM185378 Theory of Computing 3

8. KM185379 Wavelet and Applications 3

9. KM185380 Advanced Partial Differential Equations 2

10. KM185381 Inverse Problems 2

11. KM185382 Fuzzy Systems 2

12. KM185383 Graph and Applications 2

13. KM185384 Topics of Applied Analysis 2

14. KM185385 Topics of Computing 2

15. KM185386 Topics of Mathematical Modeling 2

16. KM185387 Topics of Applied Algebra 2

17. KM185388 Topics of Optimization 2

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Detail of Courses

COURSES

Name Subjects : Module Theory CS Code : KM185101

Credit : 3

Semester : 1

COURSE DESCRIPTION

This course presents an advanced study of a fundamental concept of Linear Algebra. The discussion is emphasized on the aspects of Algebra that is commutative group, ring and module theory. Furthermore, some materials will be theory Module Provided for future understanding for students who will have special abilities in the field of Algebra and other related fields or applications that need them. Assessment of learning outcomes is done through written evaluations, classroom discussions and student presentations and releases them in paper format.

ACHIEVEMENTS GRADUATES CHARGED LEARNING COURSE PLO 1 Students are able to solve mathematical problems by applying

fundamental mathematical statements, methods, and computations ACHIEVEMENT OF LEARNING COURSES

 A mature student is able to develop math and writing mathematical proof by default.

 Students are able to develop an understanding of the concept and be able to draw conclusions pituitary and in particular the theory of linear algebra ideas for module theory and computational problems.

 Students are able to appreciate the importance of understanding the structure of algebra to a higher-level concepts.

 Students can create awareness kususnya symbolic thinking within the framework of the theory of modules

 Students have the capability to use its understanding and analyzing models of mathematics, science and technology and other disciplines related fields.

 Students are able to develop an understanding matematematika framework that supports science and technology, and mathematics as

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well as communicate the results of the development of oral and written comprehension.

SUBJECT

 Commutative groups and subgroups

 Commutative additive group Homomorpisma

 Ring, homomorpisma ring, subring and ideal

 Ideal Prima and Ideal Maximum

 Quasi field

 Single factorization area

 Module and submodule

 The set expander

 Non Linear Element Torque and Annihilator

 Modules and Module Quasi Homomorpisma

 Free modules and modules Noetherian

 Modules on the Main Ideal Regions PRECONDITION

-

REFERENCES

1. Subiono., "Lecture Notes: Module Theory", Mathematics Department, FMKSD-ITS, 2018.

2. Adnan Tercan and Canan C. Yücel, "Module Theory, Extending Modules and its generalizations", Birkhäuser, 2016

3. Ernest Shult and David Surowski, "Algebra, A Teaching and Source Book", Spriger, (2015)

4. Paul E. Bland, "Ring and Their Modules", Walter de Gryter GmbH &

Co., Berlin / Newyork, (2011)

5. Steven Roman, "Avanced Linear Algebra, Third Edition", SPRINGER, (2008).

6. WA Adkins and SH Weintraub, "Algebra An Approach via Module Theory", SPRINGER-Verlag, (1999)

7. DG Northcott, FRS, "Lessons on Rings, Modules and multiplicities", Cambridge at the University Press, (1968)

LIBRARY SUPPORT

Paul A. Furmann, "A polynomial Approach to Linear Algebra, Second Edition", SPRINGER, (2012)

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COURSES

Name Subjects : Functional Analysis CS Code : KM185102

Credit : 3 credits Semester : 1

In this course, it is studied about concept of metric space, topology, norm space and inner product such that the student can analyze the convergence of series function, bounded and continuity. It is also studied how to prove some theorem in those spaces. Bounded and continuity operator in those spaces are studied.

ACHIEVEMENTS GRADUATES CHARGED LEARNING COURSE PLO-1 Students are able to solve mathematical problems by applying

fundamental mathematical statements, methods, and computations

ACHIEVEMENT OF LEARNING COURSES

 The student able to explain the characteristic of vector space, metric space, norm space and inner product space.

 The student able to explain and analyze the convergence of sequences, open set and function continuity.

 The student able to prove the relevant theorems on those spaces.

 The student able to define operator and analyze the bounded and continuity of operator

SUBJECT

 Metric space

 Norm space

 Inner product space

 Linear operator PRECONDITION

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REFERENCES

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1. Yunus, M., Textbook of Functional Analysis, Department of Mathematics ITS, 2014

2. Zeidler, E., Applied Functional Analysis, Springer Verlag, 1995 LIBRARY SUPPORT

-

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COURSES

Name Subjects : Mathematical Modeling CS Code : KM185103

PLO-2 Students are able to analyze mathematical problems in one of the fields: analysis, algebra, modeling, system, optimization or computing sciences

PLO-3 Students are able to work and research collaboratively on

mathematical problems within either the area of pure mathematics or applied mathematics or computing sciences



SUBJECT



PRECONDITION

REFERENCES

LIBRARY SUPPORT

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COURSES

Name Subjects : Numerical Computing CS Code : KM185104

Credit : 2

The course is a numerical computing that gives an opportunity to the students to be able to solve the problems of numerical mathematics. This course discusses about the error, interpolation, turnan andNumerical of integration, ordinary differential equations (initial value problems), and partial differential equations.

PLO-2 Students are able to analyze mathematical problems in one of the fields: analysis, algebra, modeling, system, optimization or computing sciences

 Students are able to analyze errors and kekovergenannya of a numerical solution.

 Students are able to actively construct mathematical problem solving algorithms with numerical approach

 students can implement numerical approach to the programming language MATLAB to solve the problems of mathematics.

 Students are able to apply numerical approach to various multidisciplinary applications of science and technology.

SUBJECT

 Error Analysis: analyzing error and kekonvergenannya

 Interpolation: polynomial Newton, Newton divided difference method, Lagrange polynomial, linear and quadratic spline

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 Numerical derivative: Difference Method Forward / Backward / Center, Newton-Cotes Rules, Richardson Extrapolation, derivatives High Level

 Numerical Integral: Rule Simpson, Simpson 3/8, Romberg method, quadrature Gauss - Legendre

 Numerical GDP: Euler method, Heun method, Runge-Kutta methods, Methods Predictor - Corrector

 Numerical PDP: implicit and explicit methods PRECONDITION

-

REFERENCES

1. RL Burden and JD Faires, Numerical Analysis, 9th edition, Brooks- Cole,

2. Atkinson Kendall and Weimin Han, Elementary Numerical Analysis, 2nd edition, John Wiley & Sons, Inc.

3. Steven Chapra and Canale, Numerical methods for engineering, 4th edition, McGraw-Hill, 2002

LIBRARY SUPPORT -

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COURSES

Name Subjects : Approximation Theory CS Code : KM185211

Credit : 3

This course discusses the main frame of approximation theory, with an emphasis on classical topics related to polynomial and rational functions, along with computational approaches. The main discussion begins from Weierstass Approximation Theorem, which includes a discussion interpolan Chebyshev, polynomials and Chebyshev series. Then on the best approximation that includes the convergence function convergence diferensiabel and analytic functions. While the last part will discuss topics relating to spectral methods and accelerated convergence.

ACHIEVEMENTS GRADUATES CHARGED LEARNING COURSE PLO-2 Students are able to analyze mathematical problems in one of the

fields: analysis, algebra, modeling, system, optimization or computing sciences

 Being able to understand the main points of the classical approximation theory as a basis approximation method development and application.

 Being able to explain the advantages of some of the best approximation method

 Being able to apply some approximation methods in solving problems related approximation.

SUBJECT

 Approximation Theorem Weierstass

 Best approximation

 Spectral method

 Convergence acceleration PRECONDITION

- Functional analysis - Numerical computing

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REFERENCES

Trefethen, LN, Approximation Approximation Theory and Practice, SIAM, 2013

LIBRARY SUPPORT

Christensen, O. and Christensen, KL, Approximation Theory, Birkhauser, 2005

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COURSES

Name Subjects : Max-Plus Algebra CS Code : KM185212

Credit : 3

This course is presented on a study of a fundamental concept Algebra Max Plus and development that is supertropical algebra. The discussion focused on aspects of Theory and Applications. Furthermore, given the understanding Petri net in general, especially the relationship with the max plus algebra and given the ability to perform numerical computation in any discussion of using Scilab Max Plus Algebra Toolbox. Problem-based discussion is an integrated part in the study. Assessment of learning outcomes is done through an evaluation board, presentations and discussion of learners in the classroom.

 A mature student is able to develop math and writing mathematical proofs by default

 Students are able to appreciate the importance of understanding the structure of algebra to a higher - level concepts.

 Students can create awareness kususnya symbolic thinking within the framework of algebra supertropical

 Students are able to develop an understanding of the concept and be able to draw conclusions and theories particularly pituitary max plus algebra idea to issue a large scale computing system

 Students have the understanding and the ability to use mathematical models to analyze issues, particularly the issue of scheduling and other disciplines related fields.

 Students are able to develop an understanding matematika framework that supports science and technology, and mathematics as well as

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communicate the results of the development of understanding orally in the form of presentations and writing standard in mathematics

SUBJECT

 semiring

 Petri Net

 Algebra Super Tropical PRECONDITION

Module theory

REFERENCES

1. Subiono. "Lecture Notes: Ajabar Max Plus and Applications", Department of Mathematics FMKSD-ITS, 2018.

2. Subionoand Kistosil Fahim, On Computing Supply Chain Scheduling Using Max Plus Algebra, Applied Mathematical Science, Journal for Theory and Applications, vol. 10, no. 10, 477-486, 2016 DOI 10.12988 / ams.2016.618.

3. Kistosil Fahim, Subiono and Jacob van der Woude, On a generalization of power algorithms over max-plus algebra, DEDS, Discrete Event Dyn Syst (2017) 27: 181-203, DOI 10.1007 / s10626-016-0235-4, Springer Science + Business Media New York in 2017.

4. Subiono, "On Classes of Min Max Plus Systems and Their Applications", PhD. Thesis, TU Delft, The Netherlans, (2000)

5. Olsder Gj, Heidegott B. and JW van der Woude, Maxplus at Work, Modeling and Analysis of Synchronized System: A Course on Max-Plus Algebra and ITS Applications, Princeton University Press, 2006 6. Subiono, and JW van Wounde, "Power Algorithms for (mas, +) - and

Bipartite (min, max, +) - Systems", Discreate Event Dynamic Systems:

Theory and Applications, Volume 10, pp 369-389, 2002

7. CG Cassandras and Stephane LaFortune, Introduction to Discrete Event Systems, Second Edition, Springer, 2008

8. Peter Butkovic, "Max-Linear Systems: Theory and Algorithms", Spriger 2010

9. Michel Gondran and Michel Minoux, "Graph, Dioids and Semirings, New Models and Algorithms", Springer, 2008

10. Christos G. Cassandras and Stephane LaFortune, "Introduction to Discrete Event Systems, Second Edition", Spriger 2008

11. James L. Peterson, "Petri Net Theory and the Modeling of Systems", Printice Hall, Inc., 1981

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LIBRARY SUPPORT

1. Dieky Adzkiya, "Building Petri Net Model of Traffic Lights and simulation", Thesis Department of Mathematics ITS, (2008)

2. Peter Fendiyanto " Supervisory Control on Traffic Management Systems at Airports Using Petri Net ", Thesis Department of Mathematics ITS, (2016)

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COURSES

Name Subjects : Dynamical Systems CS Code : KM185221

Credit : 3

This course on study about the dynamic behavior of a system of ordinary differential equations in the form of both linear and nonlinear in a way to stability and bifurcation analysis system

PLO-4 Students are able to communicate and present mathematical ideas with clarity and coherence, both written and verbally

 Students are able to analyze the stability of linear dynamic systems and nonlinear

 Students are able to simplify the system by way of normalization and establishment of centers manifold

 Students are able to understand and prove the theorem to determine the occurrence of bifurcation and the types

 Students are able to analyze the stability of the system with delay

 Students are able to identify the real problems in the form of a dynamical system

SUBJECT

 Stability

 Bifurcation PRECONDITION

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-

REFERENCES

1. Wiggins, S. 2009, "Introduction to Applied Non-Linear Dynamical System and Chaos- second edition", Springer-Verlag

2. Xiaoxin Liao, Wang, L. And Pei Yu, 2007, "Stability of System Dynamics", Elsivier

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COURSES

Name Subjects : Stochastic Calculus CS Code : KM185222

Credit : 3

This course provides the concept of stochastics process to learn the modern financial theory. The topics include basic concept of probability, random variables, discrete and continuous distributions, and Markov chain.

Subsequently, the course introduces the concept of martingale, Brownian motion, and Ito calculus.

 Students are able to learn the concept of probability, discrete stochastic process and martingale,

 Markov process and its applications, Brownian motion, and continuous martingale.

 Students are able to learn the concept of Ito calculus and its applications in finance and other reas.

SUBJECT

 Probability

 Stochastic integral

 Stochastic differential equations PRECONDITION

Probability Theory

REFERENCES

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1. Syamsuddin, "Financial Mathematics", Lecturer Notes

2. Brzezniak and Zastawniak, "Basic Stochastic Processes", Springer, 1999 3. Shreve, Steven, "Stochastic Calculus for Finance, a Continuous Time

Model", Springer, 2004

4. Medina and Merino, "Mathematical Finance and Probability, A Discrete Introduction", Birkhauser Verlag, 2003

5. Kelbaner, FC, "Introduction to Stochastic Calculus with Applications", Imperial College Press, 2005

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COURSES

Name Subjects : Computational Algorithm CS Code : KM185231

These courses provide the ability to formulate and solve the problems of mathematics and its applications to computational algorithms approach. In addition, students will be able to implement it with Matlab and use the concept given to reveal the back and / or communicate ideas related to the field of mathematics either in writing or orally with individual and group performance in teamwork.

The topics covered include basic concepts of design and analysis of algorithms, the basic principles of matrix computation and optimization algorithms. The learning model is done through the tutorial and discussion in the classroom / lab. In addition to self-directed learning through tasks, learners are directed to cooperate in group work. Assessment of learning outcomes is done through an evaluation board, independent tasks, and the ability to write and present a given task.

 College student be able to formulate and solve the problems of mathematics and its application with the approach of computational algorithms and implement it with Matlab and use the concept given to reveal the back and / or communicate ideas related to the field of mathematics either in writing or orally to the performance of individuals and in groups in teamwork.

 Students are able to explain the concept of the design and analysis of algorithms

 Students are able to explain and implement the basic principles of computational matrix

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 Students are able to explain and implement some optimization algorithms

SUBJECT

 Computing Matrix

 Optimization algorithm PRECONDITION

-

REFERENCES

1. Matrix Computation, 4th ed, Gene H. Golub and Charles F. Van Loan, The Johns Hopkins University Press, 2012

2. Introduction to Algorithms, 3^rd Edition, Thomas H. Cormen, CE Leiserson, RL Rivest, MIT Press, 2009

LIBRARY SUPPORT

1. Computer Algorithms: Introduction to Design and Analysis, 3^rd Edition, Sara Baase and Allan Van Gelder, 2000.

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COURSES

Name Subjects : Mathematics of Machine Learning

CS Code : KM185232

Credit : 3

SUBJECT

 Theory Mat / Stat for Machine Learning

 Convexity algorithm

 Learning algorithm PRECONDITION

REFERENCES

LIBRARY SUPPORT

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COURSES

Name Subjects : Discrete Transformation CS Code : KM185271

Credit : 3

SUBJECT

 Linear transformations

 Fourier transformation

 Wavelet transformation PRECONDITION

REFERENCES

LIBRARY SUPPORT

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COURSES

Name Subjects : Formal Verification CS Code : KM185272

Credit : 3

In this course will be given an insight to students about the background and verification processes on the system transition. In addition to theoretical studies, students are also introduced to some of the software for the verification of the model, such as SPIN or NuSMV. Study paper / paper on the topic is presented in the form of discussions and presentations.

1. Students are able to explain the formal verification methods and models system where formal verification methods can be applied.

2. Students are able to explain some of the methods of verification systems and the development of a system of verification methods.

3. Students can apply the model checking system model transitions, both theoretically and using software

4. Students are able to explain and apply various algorithms on system verification.

SUBJECT

Understanding verification system: Why it is needed, the difference with the simulation, the advantages of the methods of verification systems, the boundaries of the verification system, the models used in the verification of the system: the system transition, a few specifications that are commonly used: linear-time property, linear temporal logic, computation tree logic, some software for system verification: SPIN, NuSMV, case studies verify the application of the system

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PRECONDITION -

REFERENCES

1. Baier, C. and Katoen, J-.P, 2008, Principles of Model Checking, The MIT Press

2. Ben-Ari, M., 2008, Principles of the SPIN model of checkers, Springer LIBRARY SUPPORT

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COURSES

Name Subjects : Systems and Controls CS Code : KM185273

Credit : 3

Systems and Control consist of Definition of the system, Principles of Modeling, Linear Systems and Properties System, Input/Output Feedback Control, Input/Output Representation, Optimal Control (LQR), and the methods of control growing recently. In the process of learning in class students will be given an understanding of problem identification and reduction of mathematical models and represent into the form of the system, then determine the appropriate controls with these problems. In addition to self-directed learning through tasks, students are directed to cooperate in group work. Assessment of learning outcomes is done through an evaluation board, tasks and discussions in class activities.

 Students are able to follow developments and apply linear systems and optimum control and be able to communicate actively and properly either oral or written.

 Students are able to explain the basic principles and theory understood further from dealing specifically with linear system and capable of designing an appropriate control system.

 Students are able to explain intelligently and creatively about the significant role Optimum Linear Systems and Control in the field of knowledge related clumps or other fields.

 Students are able to present an understanding of science in the field of Linear Systems and Control Optimum independently or in teamwork.

SUBJECT

 The state space

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 MIMO systems

 design Control PRECONDITION

-

REFERENCES

1. Subiono., "Linear Systems and Optimal Control", Department of Mathematics-ITS, 2014.

2. Frank L. Lewis, Draguna LV, Vassilis LS, "Optimal Control and Estimation", Wiley and Son, New Jersey, Canada, Inc., (2012)

3. Olsder, GJ, "Mathematical System Theory", Fourth Edition, VSDD, Delft in The Netherlands (2011)

LIBRARY SUPPORT

1. M. Gopal, "Modern Control System Theory", New Age International (P) Limited, Publishers, (1993).

2. CT Chen, "Linear System Theory and Design", Fourth Edition, Oxford University Press. (2012)

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COURSES

Name Subjects : Computational Fluid Dynamics

CS Code : KM185274

Credit : 3

Course computational fluid dynamics is about the computational aspects of fluid dynamics.

 Students understand, control and understanding of the fluid flow equations.

 Students are able to develop the transport scalar equations and

momentum.

 Students are able to understand the basic concepts of turbulence.

SUBJECT

 Fluid flow

 flow modeling

 Numerical solution of fluid flow problems PRECONDITION

-

REFERENCES

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Anderson, JDJ, 1995, Computational Fluid Dynamics (The Basics with Applications), International Edition, Mc Graw-Hill, New York, USA.

LIBRARY SUPPORT

1. Anderson, JDJ, 1995, "Computational Fluid Dynamics (The Basics with Applications) '', International Edition, Mc Graw-Hill, New York, USA.

2. Hoffmann, KA and Chiang, ST, 1995, "Computational Fluid Dynamics For Engineers, Engineering Education System", Wichita, USA.

3. Shames, IH, 1992, "Mechanics of Fluid, 3rd Edition", Mc Graw-Hill, New York, USA.

4. Welty, JR, et al., 1995, '' Fundamentals of Momentum, Heat and Mass Transfer, 3rd Edition ", John Wiley & Sons, Inc., New York, USA.

5. Wilkes, DJF, et al., 1995, "Fluid Mechanics, 3rd Edition", Longman Publishers Singapore, Singapore.

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COURSES

Name Subjects : Dynamical Optimization CS Code : KM185275

Discussion subjects include an assessment of dynamic optimization basics of calculus of variations, optimal control, modeling, application, simulation and computing. In the learning process in the classroom learners will learn to identify the real problems, modeling, and finish it. In addition to self-directed learning through tasks, learners are directed to cooperate in group work and write scientific papers in the form of paper.

 Students are able to follow developments and apply mathematics and be able to communicate actively and properly either oral or written

 Students are able to explain the basic principles and further from the theory that understands particularly with regard to dynamic optimization

 Students are able to explain intelligently and creatively about the significant role of optimization in the areas of knowledge related clumps or other fields

SUBJECT

 calculus of Variations

 Optimal control PRECONDITION

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-

REFERENCES

A. Naidu, DS, "Optimal Control Systems '', CRC Press, 2002.

B. Subchan, S and Zbikowski, R., "Computational Optimal Control: Tools and Practice", Wiley, 2009.

C. Lewis, F. and Syrmos Vassilis, "Optimal Control", John Wiley & Sons, Singapore, 1995.

D. Suzanne Lenhart, John T. Workman, "Optimal Control Applied to Biological Models", CRC Press, 2007.

E. Krasnov, ML, Makarenko, GI, and Kiselev, AI, Problems and Exercises in the Calculus of Variations, MIR Publishers Moscow, 1975.

F. Bryson and Yu-Chi Ho, Applied Optimal Control: Optimization, Estimation and Control, Taylor and Francis Group, 1975.

LIBRARY SUPPORT

1. Kamien, ML and Schwartz, NL, "Dynamic Optimization", North- Holland, Amsterdam, 1993.

2. Lewis F., "Optimal Estimation", John Wiley & Sons, Singapore, 1986.

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COURSES

Name Subjects : Financial Mathematics CS Code : KM185276

Credit : 3

This course provides theories and models of annuity, interest rate, and portfolio investment. The modelling of annuity for various payment schemes with related various interest rate models is presented. Then the development of investment portfolio based on the annuity models is assigned for the applications.

fundamental mathematical statements, methods, and computations PLO-2 Students are able to analyze mathematical problems in one of the

 Students are able to understand and apply their mathematics ability to build annuity models.

 Students are able to understand and develop the loan repayment scheme

 Students are able to learn and determine the bond value

 Students are able to learn and develop the analysis of rate of return in investments.

SUBJECT

 Annuity

 Loan repayment

 Investment Portfolio

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PRECONDITION Calculus II

REFERENCES

1. Garrett, SJ, "An Introduction to the Mathematics of Finance '', Second Edition, Elsevier, 2013

2. Broverman, Samuel, "Mathematics of Investment and Credit", 5th Edition, ACTEX Publication 2010

3. Brigham, EF and Ehrhardt, MC, "Financial Management", Thomson Southwestern

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COURSES

Name Subjects : Digital Image Processing and Analysis

CS Code : KM185277

Credit : 3

Digital Image Analysis is a subject which contains the basic concepts of applied mathematics for image processing and algorithms for image processing. Basic math concepts covered include, namely of transformation Fourier, wavelet transform and mathematical morphological. Image processing techniques include enhancement, restoration, segmentation and image compression.

ACHIEVEMENTS GRADUATES CHARGED LEARNING COURSE PLO-2 Students are able to analyze mathematical problems in one of the

fields: analysis, algebra, modeling, system, optimization or computing sciences

 Able to understand and develop concepts and basic techniques of image processing

 Able to understand and implement image processing algorithms with the programming language.

 Able to apply image processing techniques for image processing applications more complex individually or in groups in the form of presentations or papers.

SUBJECT

 Image processing: image enhancement spatial and frequency domain, image restoration

 Image Segmentation: edge detection, segmentation methods

 Image Analysis: feature extraction and classification

 image compression

 wavelet PRECONDITION

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REFERENCES

1. RC Gonzalez and RE Woods, "Digital Image Processing, Third Edition", Pearson, 2008

2. John C. Russ, "The Image Processing Handbook, Sixth Edition", CRC Press, 2011.

LIBRARY SUPPORT

1. Bhabatosh, Majumder, Dwijesh Dutta, "Digital Image Processing And Analysis", Prentice Hall, 2006

2. Gonzalez, Woods, and Eddins, "" Digital Image Processing Using MATLAB (DIPUM) ", Prentice Hall, 1st edition, 2004.

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COURSES

Name Subjects : Mathematical Biology CS Code : KM185372

Credit : 3

1. Able to understand the problem in the form of a continuous population models -diffusi reaction and analyze the behavior of the system 2. Able and mastered the meaning pupolasi interaction as a function of the

transmission in the dispersion model

3. Being able to construct models of the phenomena discrete object of observation.

4. Being able to make project-related research and to publish reaction model –diffuse

SUBJECT

 Continuous Population Model

 Discrete Population Model

 Population Interaction Model PRECONDITION

system dynamics

REFERENCES

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1. Marco Di Francesco 2010. "Mathematical models in life science"

2. Eduardo D. Sontag 2006, "Lecture Notes in Mathematical Biology"

Rutgers University.

3. DW Hughes, JH Merkin, R. Sturman, 2004, "Lecture Notes in Analytic Solutions of Partial Differential Equations" School of Mathematics, University of Leeds.

4. F Brauer C. -Chavez, 2012. "Mathematical Models in Population Biology and Epidemiology", Texts in Applied Mathematics, Springer Science + Business Media

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COURSES

Name Subjects : Assimilation Data CS Code : KM185373 Credit : 3 credits Semester : 3

In this course is studied about definition of data assimilation, comparing between classical estimation and data assimilation, the application of data assimilation to estimate the stochastic dynamical system.

 The student able to explain data assimilation method and the models that can data assimilation applied.

 The student able to explain some methods of estimation and the data assimilation development

 The student able to apply data assimilation method on stochastic dynamical system and deterministic dynamical system.

 The student able to explain and apply some development of Kalman filter algorithm as one of data assimilation method.

SUBJECT

 Classical Estimates

 Estimation of Stochastic Models

 Development of Data Assimilation Methods

 Applied methods of data assimilation PRECONDITION

-

REFERENCES

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1. Lewis, JM, Lakshmivarahan, Dhall, SK 2006, "Dynamic Data Assimilation: A Least Squares Approach", Cambride

2. Kalnay 2003, "Atmospheric Modeling, Data Assimilation And Predictability", Cambridge

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COURSES

Name Subjects : Computational Biology CS Code : KM185374

Credit : 3

Computer applications increasingly important issues in the field of bioinformatics and offers a lot of challenges from the perspective of the computing process. In this course, students will gain the ability to formulate problems of bioinformatics, particularly sequence analysis into the form of a computational model and solve it with the help of software. In addition, students will learn some of the alternative settlement in sequence analysis. To deepen understanding, students will implement it with Matlab and use the concept given to reveal the back and / or communicate ideas related to the field of mathematics either in writing or orally with individual and group performance in teamwork.

The topics covered include sequence alignment problem solving, stochastic modeling for the analysis of mutations, super pairwise alignment and multiple alignment and phylogenetic tree reconstruction. The learning model is done through the tutorial and discussion in the classroom / lab. In addition to self- directed learning through tasks, learners are directed to cooperate in group work. Assessment of learning outcomes is done through an evaluation board, independent tasks, and the ability to write and present a given task.

 Be able to formulate problems of bioinformatics in the form of a computational model and solve it with the help of software.

 Being able to choose an alternative solution sequence alignment (sequence alignment)

 Being able to apply the sequence alignment algorithms and structure of the network to identify genetic mutations

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SUBJECT

 sequence Alignment

 Protein folds

 phylogenetic trees PRECONDITION

-

REFERENCES

1. Isaev, Alexander, "Introduction to Mathematical Methods in Bioinformatics", Springer-Verlag, 2004

2. Shen, Nankai Shiyi, "Theory and Mathematical Methods for Bioinformatics", Springer-Verlag, 2008

LIBRARY SUPPORT

Ian Korf, Mark Yandell, Joseph Bedell, "Basic Local Alignment Search Tools" Oreilly 2003

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COURSES

Name Subjects : Mathematics of Derivatives CS Code : KM185375

Credit : 3

This course provides mathematical models to solve practical problem in three basic aspect of financial market : pricing the financial assets, pricing of financial derivative products, and risk management. The discussions are focused on arbitrage principles, stochastics models of stock and interest rate, Ito’s lemma, modelling of financial derivative product, analytic and numerical methods to solve the financial derivative differential equations. The solutions are used to design the risk management of financial derivative investments.

fundamental mathematical statements, methods, and computations PLO-2 Students are able to analyze mathematical problems in one of the

 Students are able to learn 3 basic aspects of financial market : price of financial assets, financial derivative products, adn risk management.

 Students are able to learn and use the basic principles of mathematical models development of financial assets and the financial derivatives products, i.e arbitrage principles.

 Students are able to learn the development of mathematical models of financial product and its derivatives and their solutions analytically and numerically, and to provide the analysis.

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 Students are able to extend the mathematical model of financial product and its derivatives analytically and numerically.

SUBJECT

 Financial derivatives products

 Stochastic and partial differential equations

 Numerical solutions PRECONDITION

1. Numerical methods 2. Statistical methods 3. Multivariable calculus REFERENCES

1. Jiang, Lishang, Mathematical Modeling and Methods of Option Pricing, World Scientific, 2005

2. Willmot, Paul, et al, The Mathematics of Financial Derivatives, Cambridge Press, 1995

3. Higham, Desmond J, An Introduction to Financial Option Valuation:

Mathematics, Stochastics and Computation 1st Edition, Cmabridge 2004.

4. Hull, JC, Options, "Futures and Other Derivatives", Prentice Hall 2005 5. Seydel, Rüdiger, Tools for Computational Finance, Springer, 2002 LIBRARY SUPPORT

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COURSES

Name Subjects : Risk Analysis CS Code : KM185376

Credit : 3

This course provides the concepts and methodologies in risk analysis theory, risk models with uncertainty to analyze risks, optimization concepts in risk analysis. Subsequently, some the applications of optimization concepts in risk analysis are presented in some areas such as insurance, project risks, and product assesment.

PLO-5 Students are able to make use of the principles of long life learning to improve knowledge and current issues on mathematics

1. Students are able to explain the concepts and methodologies in risk analysis theories.

2. Students are able to use the risk models to analyze risk in insurance and other fields.

3. Students are able to explain the concept of optimization in risk analysis 4. Students are able to apply the concept of optimization in risk analysis

for some fields such as insurance, project risk, and product assesment.

SUBJECT

 Risk modelling: time series, Markov chain, birth and death model, copula

 Risk optimization

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PRECONDITION Probability theory

REFERENCES

1. Quantitative Risk Analysis, David Vose, Wiley, 2009

2. Probability and Risk Analysis, Igor Rychlik and Jesper Ryden, Springer, 2006

LIBRARY SUPPORT

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COURSES

Name Subjects : Graph Algebra CS Code : KM185377 Credit : 3 credits Semester : 3

SUBJECT

 Linear Algebra in Graf

 Spectral Graph Theory

 Graf Partitions PRECONDITION

REFERENCES

LIBRARY SUPPORT

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COURSES

Name Subjects : Theory of Computing CS Code : KM185378

Credit : 3

SUBJECT

 Automato

 Language theory

 complexity theory PRECONDITION

REFERENCES

LIBRARY SUPPORT

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COURSES

Name Subjects : Wavelets and Applications CS Code : KM185379

Credit : 3 Semester : 3

SUBJECT

 multiresolution analysis

 Orthogonal wavelet

 filter Bank PRECONDITION

REFERENCES

LIBRARY SUPPORT

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COURSES

Name Subjects : Advanced Partial Differential Equations CS Code : KM185380

Credit : 2

SUBJECT

 PDP Linear and Non-Linear

 Variational methods

 Free Boundary Value Problems PRECONDITION

REFERENCES

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LIBRARY SUPPORT

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COURSES

Name Subjects : Inverse Problems CS Code : KM185381 Credit : 2

In this course is studied about invers problem, some methods to solve inver problem, regulation method and convergence of linear and non linear regulation

 The student able to understand about invers problem, can formulate the problem and solve it.

 The student able to analyze the convergence of regulation method, apply to solve invers problem.

 The student able to determine the exact method for invers problem.

SUBJECT

 Linear Inverse problem

 Linear Regulation Method

 Convergence Analyze of Regulation Method

 Non Linear Regulation Method PRECONDITION

Functional analysis

REFERENCES

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1. Isakov, V, 2006, Inverse Problems for Partial Differential Equations, Springer Science Business Media, Inc.

2. Tarantola,A , 2008, Inverse Problem Theory and Methods for Model Parameter Estimation, Library of Congress Cataloging-in-Publication Data, SIAM

3. Kaipio, J dan Somersalo, E. 2005, Statistical and Computational Inverse Problems, Springer Science Business Media, Inc.

4. Hohage, T., 2002, lecture notes on Inverse Problems, University of G¨ottingen

LIBRARY SUPPORT

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COURSES

Name Subjects : Fuzzy Systems CS Code : KM185382

Credit : 2

This course aims to give basic concepts and to further increase the structure of fuzzy theory and its application, this lecture consists of two parts: theory and application part. The first part (part theory) covers the basic concepts and operations of fuzzy sets, fuzzy set of multi-dimensional expansion of fuzzy theory to the number and function, development properties and the probability to fuzzy logic theory. The second part is an application that consists of a fuzzy inference techniques, application of fuzzy logic inference, decision-making in fuzzy environment

 Being able to develop mathematical concepts, especially in the form of fuzzy

 Able to formulate a common problem in the form of fuzzy mathematics models and get a settlement

 Being able to apply the frame of mathematics and computational principles to solve the problems of the development of intelligent systems

 Being able to identify problems and develop mathematical models and analyze the relevant fuzzy behavior

 Being able to communicate the results of research in a scientific forum at the national or international level.

 Able to develop contemporary science and technology by mastering and understanding, approach, method, scientific principles along with their application skills in the field of optimization of the system, or computer science

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SUBJECT

 Fuzzy Set Theory

 Fuzzy logic

 fuzzy Decision PRECONDITION

REFERENCES

1. Buckley J, and E. Eslami, "An Introduction to Fuzzy Logic and Fuzzy Sets", Physica Heidelberg, 2001

2. Klir, GJ and B. Juan, "Fuzzy Sets and Fuzzy Logic", Prentice Hall, New Jersey, 2001

3. Zimmerman H. J, "Fuzzy Set Theory and Its Applications", Kluwer Academic Publishers, 1996

4. Zadeh, LA., "Fuzzy Sets, Fuzzy Logic, and Fuzzy Systems: Selected Papers", Kluwer Academic Publishers, 1996

LIBRARY SUPPORT

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COURSES

Name Subjects : Graf and Applications CS Code : KM185383

Credit : 2

SUBJECT

 Graph Theory

 Application of graphs in Mechanical Problems PRECONDITION

REFERENCES

LIBRARY SUPPORT

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COURSES

Name Subjects : Topics of Applied Analysis CS Code : KM185384

Credit : 2

On this subject, topic-topic presented the latest in the field of analysis, algebra and its application study of paper and paper terkaitan presented the topic for the next student in the form of presentation. From this study are expected to emerge thesis topics

1. Students are able to assess the new topics of analysis, algebra and its application

2. Students are able to assess the paper / paper relating on the topic 3. Students are able to present a role in the form of presentations and

writing SUBJECT

 The items you just about the analysis and its application

 Recent Developments Analysis PRECONDITION

-

REFERENCES

Text books and related Paper

LIBRARY SUPPORT __

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COURSES

Name Subjects : Topics of Computing CS Code : KM185385

Credit : 2

On this subject, topic-topic presented the latest in the field of computer science and computing. Study of paper and paper terkaitan presented the topic for the next student in the form of presentation. From this study are expected to emerge thesis topics

1. Students are able to assess the new topics of computer science and computing

2. Students are able to assess the paper / paper relating on the topic 3. Students are able to present a role in the form of presentations and

writing SUBJECT

 The items you new to computer science and computing

 Recent Development of Computer Science and computing PRECONDITION

-

REFERENCES

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COURSES

Name Subjects : Topics of Mathematical Modeling

CS Code : KM185386

Credit : 2

On this subject, topic-topic presented the latest in the field of mathematical modeling. Study of paper and paper terkaitan presented the topic for the next student in the form of presentation. From this study are expected to emerge thesis topics

PLO-5 Students are able to make use of the principles of long life learning to improve knowledge and current issues on mathematics ACHIEVEMENT OF LEARNING COURSES

1. Students are able to assess the new topics of mathematical modeling 2. Students are able to assess the paper / paper relating on the topic 3. Students are able to present a role in the form of presentations and

writing SUBJECT

1. Pemodelanan the items you just about math 2. Mathematical modeling Recent Developments PRECONDITION

-

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REFERENCES

Text books and papers related

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COURSES

Name Subjects : Topics of Applied Algebra CS Code : KM185387

Credit : 2

Of this course-topic presented the latest topics in the field of algebra and its application. Study of paper and paper terkaitan presented the topic for the next student in the form of presentation. From this study are expected to emerge thesis topics

1. Students are able to assess new topics algebra and its application 2. Students are able to assess the paper / papers relating about the topic

mentioned

3. Students are able to present a role in the form of presentations and writing

SUBJECT

 The items you just about analysis, algebra and its application

 Recent Developments Algebra PRECONDITION

-

REFERENCES

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COURSES

Name Subjects : Topics of Optimization CS Code : KM185388

Credit : 2

On this subject, topic-topic presented the latest in the field of optimization.

Study of paper and paper terkaitan presented the topic for the next student in the form of presentation. From this study are expected to emerge thesis topics ACHIEVEMENTS GRADUATES CHARGED LEARNING COURSE PLO-2 Students are able to analyze mathematical problems in one of the

1. Students are able to assess the new topics of optimization 2. Students are able to assess the paper / paper relating on the topic 3. Students are able to present a role in the form of presentations and

writing SUBJECT

1. The items you just about pemodelanan optimization 2. Recent Developments Optimization

PRECONDITION -

REFERENCES

Text books and papers related