Applications by Dr. Co Tran


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FRACTURE PROPAGATION IN THE INTERNAL PRESSURIZED VESSEL .

FRACTURE PROPAGATION IN THE INTERNAL PRESSURIZED VESSEL .
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THE  SOLUTION   STABILITY OF  VAN DER POL’S  EQUATION .

THE SOLUTION STABILITY OF VAN DER POL’S EQUATION .
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107.778
The Runge-Kutta Method for Solving Non-linear System of Differential Equations

The Runge-Kutta Method for Solving Non-linear System of Differential Equations
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57.345
THE SOLUTION OF A VARIABLE BOUNDARY PROBLEM

THE SOLUTION OF A VARIABLE BOUNDARY PROBLEM
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22.713
Finite Difference Method and The Lame's Equation in Hereditary Solid Mechanics

Finite Difference Method and The Lame's Equation in Hereditary Solid Mechanics
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20.998
Numerical and Graphical Solutions of Duffing Equation

Numerical and Graphical Solutions of Duffing Equation
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The Relaxation function problem of an orthotropic cylinder

The Relaxation function problem of an orthotropic cylinder
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19.265

Investigation of the Power Spectral Density of Duffing's Equation By Equivalent Linearization Method

Investigation of the Power Spectral Density of Duffing's Equation By Equivalent Linearization Method
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16.696
Solving the Viscous Composite Cylinder Problem by Sokolov's Method

Solving the Viscous Composite Cylinder Problem by Sokolov's Method
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The Average Approximating Method on Functional Adjustment Quantity

The Average Approximating Method on Functional Adjustment Quantity
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Online Help

asympt - asymptotic expansion


Calling Sequence


asympt(f, x)

asympt(f, x, n)


Parameters


f

-

algebraic expression in x

x

-

name

n

-

positive integer (truncation order)


Description


• 

The function asympt computes the asymptotic expansion of f with respect to the variable x (as x approaches infinity).

• 

The third argument n specifies the truncation order of the series expansion.  If no third argument is given, the value of the global variable Order (default Order = 6) is used.

• 

Specifically, asympt is defined in terms of the series function

subs(x = 1/x, series(subs(x = 1/x, f), x = 0, n))

  

However, the expression returned will be in sum-of-products form rather than in the series form.


Examples


asympt(x/(1-x-x^2), x)

-1/x+1/x^2-2/x^3+3/x^4-5/x^5+O(1/x^6)

(4.1)

asympt(factorial(n)/sqrt(2*Pi), n, 3)

(1/(1/n)^(1/2)+(1/12)*(1/n)^(1/2)+(1/288)*(1/n)^(3/2)+O((1/n)^(5/2)))/((1/n)^n*exp(n))

(4.2)

asympt(exp(x^2)*(1-erf(x)), x)

1/(Pi^(1/2)*x)-(1/2)/(Pi^(1/2)*x^3)+(3/4)/(Pi^(1/2)*x^5)+O(1/x^7)

(4.3)

asympt(sqrt((1/2)*Pi)*BesselJ(0, x), x, 3)

sin(x+(1/4)*Pi)*(1/x)^(1/2)-(1/8)*cos(x+(1/4)*Pi)*(1/x)^(3/2)-(9/128)*sin(x+(1/4)*Pi)*(1/x)^(5/2)+O((1/x)^(7/2))

(4.4)

series(ln(x)/(x-1), x, 8)

series(-ln(x)-ln(x)*x-ln(x)*x^2-ln(x)*x^3-ln(x)*x^4-ln(x)*x^5-ln(x)*x^6-ln(x)*x^7+O(x^8),x,8)

(4.5)

asympt(ln(x)/(x-1), x, 8)

ln(x)/x+ln(x)/x^2+ln(x)/x^3+ln(x)/x^4+ln(x)/x^5+ln(x)/x^6+ln(x)/x^7+O(1/x^8)

(4.6)

See Also


limit, series


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Matlab Resources: Tutorials and Beyond


Introductions to Matlab

Tutorial Sessions at UBC:

These tutorials will discuss why we use Matlab for coursework (and not Java, C, C++, Fortran, Perl, Python, Maple, Mathematica, etc...), how to get started, and where to find resources for further exploration. They are aimed at students who have not previously used Matlab.

Times:

  • Wednesday, September 16 2009, 5pm - 7pm, location DMP 101.

Material:

Alternative Workshop at UBC:

Mathworks (the company that owns Matlab) is holding a longer introductory seminar Thursday September 17, 9:30 - 12:30, as well as an afternoon session on parallel programming in Matlab. To a large extent, the material in my seminar and the Mathworks' seminar are orthogonal: my seminar will focus on why we use Matlab and exactly how to use it in the Computer Science department, while Mathworks will include more depth on the basic language and the many new features of very recent versions. Register for the Mathworks seminar.

From the Mathworks: These resources are from the company that develops and distributes Matlab.

Other Free Sources: These sources were found using either Google or Matlab Central (see below). While the tutorials cover the basics of working with Matlab, many refer to much older versions of Matlab: The current version on the UBC CPSC machines is 7.6 (R2008a), and these tutorials often refer to versions 5.3 or 6.5. The basic command methods have not changed, but the newer versions have lots of graphical interfaces to make learning easier.

Other Sources: These ones will cost you some money. I have not personally tried any of them, so I do not endorse or make any representation about their suitability to your needs.


Additional Matlab Resources


Matlab resource web page
created and maintained by Ian Mitchell
Created 28 August 2006.
Last updated 09 August 2009.

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Maths A Students Survival Guide 2nd
How to Help Your Child Excel in Math - An A-Z Survival Guide
Mathematical Olympiad in China Problems and Solutions
The Book of Numbers
Number: The Language of Science
Math Wonders to Inspire Teachers and Students
Instant Revision: AS Maths
QuickStudy - Calculus 1
Instant Revision: GCSE Mathematics
QuickStudy - Algebra 1
QuickStudy - Algebra 2
QuickStudy - Statistics
SAT Prep Course eBook
Mathematics for Business, Science, and Technology
Master Math Solving Word Problems
QuickStudy - Finance
More Damned Lies and Statistics: How Numbers Confuse Public Issues
Formula Card
Why Learn Math?
Foundations of Infinitesimal Calculus
Concrete Mathematics: A Foundation for Computer Science (2nd Edition)
Algebraic Topology


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By Cambridge University Press
Subject: Math
Description: [ More info... ]
Date Added: 2008-04-26


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By Career Press
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Mathematical Olympiad in China Problems and Solutions
By
Subject: Math
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The Book of Numbers
By Tim Glynne-Jones
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Number: The Language of Science
By Tobias Dantzig
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Math Wonders to Inspire Teachers and Students
By Alfred S. Posamentier
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By Jenny Sharp, Stewart Townend
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By BarCharts, Inc.
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QuickStudy - Algebra 1
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Master Math Solving Word Problems
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Subject: Math
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QuickStudy - Finance
By
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More Damned Lies and Statistics: How Numbers Confuse Public Issues
By Joel Best
Subject: Math
Description: [ More info... ]
Date Added: 2005-08-22


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Formula Card
By ''
Subject: Math
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Date Added: 2005-08-20



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Why Learn Math?
By ''
Subject: Math
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Date Added: 2005-08-20


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Foundations of Infinitesimal Calculus
By Keith Duncan Stroyan
Subject: Math
Description: [ More info... ]
Date Added: 2003-12-01



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Concrete Mathematics: A Foundation for Computer Science (2nd Edition)
By Ronald L. Graham, Donald E. Knuth, Oren Patashnik
Subject: Math
Description: [ More info... ]
Date Added: 2003-12-01


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Algebraic Topology
By Allen Hatcher
Subject: Math
Description: [ More info... ]
Date Added: 2003-12-01

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Only Problems, Not Solutions 4th Ed. (F. Smarandache)

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  ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------

Jim Herod's Home Page

James V. Herod, Professor Emeritus

School of Mathematics

Georgia Institute of Technology

Atlanta GA

USmail: P O Box 1038, Grove Hill, AL 36451

email: jherod@tds.net

Phone: (251) 275 2244

 

First note that I have retired. I no longer maintain an office at Georgia Tech. Look at my address above! I do answer email. This is the best way to contact me when I am at home. When I travel, my email is different. 

What follows is a collection of things that please me. I list them in no particular order, give a brief description of what they are about, and provide a contact point to access more details.

1. A New Interest: ��It is true that I have published a work of fiction titled Gathering Moss. Publication date was January, 2008. Some of the action in the book takes place on the campus of Georgia Tech, some at the University of Alabama, and some at the University of Montana. There are both science and mathematics in the book. The technical parts are written to be accessible to the general reading public. Is it a surprise that a person who has spent a life in mathematics would write a book of fiction? I say no. One can look at the process of engaging a person in the pleasure of science, mathematics, or fiction as a continuation of the old Southern tradition of story-telling. After all, a good lecture tells a story; and a good story-teller creates the narrative to fit the intended audience.

2. No More Same Old Calculus: Before I retired, Professor George Cain and I made a common observation: most students entering Georgia Tech have already had the calculus. Because they think they already know the calculus, they often fail to note that the college course expects a deeper understanding of the concepts. Or, the students are bored because their high school calculus was such a good course. Cain and I wanted to make the materials fresh and exciting. Our decision was to present the notions of calculus in a multidimensional setting at the outset. This would take more time than typical in a second year calculus course because we would constantly go back to re-look at the one dimensional setting from this multidimensional perspective. Where could we find a text? Can you guess?

3. Asynchronous Studies in Undergraduate Partial Differential Equations: Georgia Tech now offers graduate programs in engineering where the materials stream over the web for students scattered all over the country. I participated in this program for several summers after I retired. The last time was during summer, 2006. There were streaming lectures, class notes, detailed calculations, and a bulletin board for class discussions. Some of these materials are accessible from my web site.

4. Notes for a course in Partial Differential Equations: In the Winter of 2003, I was commissioned to write a set of notes for studies in classical linear partial differential equations and to publish these notes on the web site for Maple Waterloo Software, Inc. These notes are available at this site with minor modifications from what appears at the Maple site. The modifications occur as small changes are made to the text or to the models, and as additions are created to amplify these ideas.

5. An Introduction to the Mathematics of Biology: Unsuspecting students often are surprised that mathematics could be so important in Biology. Even professionals are likely to think that the mathematics of biology is statistics. Yes, statistics is important in biology. There is also modeling. A good mathematical model for a biological phenomenon can illustrate what is observed in life, can predict outcomes, and can give directions for biological research. During the years before Professor Edward Yeargers and I retired, we joined with Professor Ron Shonkwiler in teaching an undergraduate course in mathematical biology. A text came out of our joint efforts. You will find that I keep all the Maple syntax current, as well as provide a list of errors we have found after printing.

6. Studies in the First Two Years of Undergraduate Mathematics: As students are introduced to the concepts of undergraduate mathematics, they often wish to see applications of the ideas. The models presented in this section have evolved through teaching, as well as through discussions with faculty at Georgia Tech and other universities. These studies are made available with the hope that students will find the models give insight, and that other faculty will use them in methods which seem appropriate.

7. An Introduction to the Linear Algebra of Infinite Dimensional Spaces: Graduate students in Mathematics and in several engineering disciplines include in their studies the notions of Hilbert Spaces. It was a course that I taught often at Georgia Tech and, consequently, developed a set of notes. The notes are pretty compact. They serve as an introduction to studies in more general infinite dimensional spaces.

8. An Introduction to the Use of Maple for Applications: I am asked fairly often about how to accomplish different type tasks with Maple. I created some materials that can be used to get an introduction to the use of Maple for classroom instruction and for analyzing applications from a variety of sources.

9. First Order Differential Equations with Periodic Forcing Functions: In this Maple worksheet, we consider solutions Y for the differential equation Y '(t) = A Y(t) + F(t). In case A is a number and F is a real valued function, students in an introductory calculus course often can make a solution for the equation. If A is a matrix, students in an introductory ordinary differential equations course likely can formulate a solution if F is vector valued. The interest here is a view for how to handle this problem for A in a class of differential operators. We review all three situations in this expository discussion. Our principal interest is in periodic solutions in case F is periodic. The pleasure is in putting the three settings into one context.

10. MAA Short Course, March 2002: At a joint regional meeting of the American Mathematical Society and the Mathematics Association of America, Professor Ron Shonkwiler, Professor Edward Yeargers, and I presented a short course on An Introduction to the Mathematics of Biology. Notes for the course are posted on this web site.

11. Maple Conference, Summer 2006: The Maple document located at this site was the basis of a presentation at the Maple Conference in July, 2006. The paper was co-authored with Don Jones .Using the diffusion equation with periodic endpoint boundary conditions, the paper illustrates how to compute both the periodic solution and the particular solution for a partial differential equation with specified initial conditions. With three dimensional graphics and with animations, the coalescence of the particular solution to the periodic solution can be observed. A similar analysis is made for the wave equation with periodic endpoint boundary conditions. Also, techniques are shown for using Maple to compute numerical solutions for these equations.

12. Connections: Often I get email inquiries from both sides of the Atlantic asking from where my family comes. Usually, the author of the message wants to know if we are related. I am fortunate to have several cousins interested in making these connections. I am the second son of Joseph Wheeler Herod.

THIS PAGE IS NOT A PUBLICATION OF THE GEORGIA INSTITUTE OF TECHNOLOGY AND THE GEORGIA INSTITUTE OF TECHNOLOGY HAS NOT EDITED OR EXAMINEDTHE CONTENT. THE AUTHOR OF THE PAGE IS SOLELY RESPONSIBLE FOR THE CONTENT.

  ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------

Evans Harrell


Professor
School of Mathematics
&
Associate Dean
College of Sciences
Georgia Institute of Technology
Atlanta GA 30332-0365

email:h a r r e l l (at) math.gatech.edu
Phone: (404) 894 3300


Links and current events

Past and ongoing activities, such as courses from previous semesters

  • The home page for Math 2401, Vector Calculus, Fall, 2008. which was also accessible through T-Square.
  • The course page for Math 2401, Vector Calculus, Fall, 2007.

Other educational links


Scientific organizations


Links for Georgia and Georgia Tech


* The Skiles building ( pictures) shows as a rectangle with a hole in it and bears the number 2. To see it in relation to the whole campus, go to "Map Home". My office location is in the southwest corner of the building.
The opinions on this and any of my pages at locations in
      math.gatech.edu/~harrell/     or       mathphysics.com
are my own, or else satirical, or honest errors. They are not
necessarily those of Georgia Tech, although most of them should be.
As we have been instructed to say:
    THIS PAGE IS NOT A PUBLICATION OF THE GEORGIA
INSTITUTE OF TECHNOLOGY AND THE GEORGIA INSTITUTE
OF TECHNOLOGY HAS NOT EDITED OR EXAMINED
THE CONTENT. THE AUTHOR(S) OF THE PAGE ARE SOLELY
RESPONSIBLE FOR THE CONTENT.

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