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General Information
Instructor: Nishant Malik, Office: 310Kemeny Hall
Email: [email protected], Phone: 603-646-9020
Class Times: Monday, Wednesday and Friday 12:30 -1:35PM
Class Room: 007, Kemeny Hall
Office Hours: Monday, Wednesday and Friday 4:00 PM - 5:00 PM [or by appointment].
X-hours: Tuesday 1:00 PM -1:50 PM [Will be used intermittently at instructor's discretion for review of course material etc. Do not schedule anything regular in this X-hr].
Textbook
Title: Applied Mathematics
Edition: Fourth
Authors: J. David Logan
Publisher: John Wiley & Sons
Course Description
This course introduces a wide variety of mathematical tools and methods to analyze phenomena in the physical, life and socialsciences. Focus of this course will be on analytical tools (theones involving use of pen and paper) rather then the computational tools(the ones involving use of computers). Though students are encouraged to learnnumerical skills with packages from programming language like Pythonor Matlab or C++ (or whatever else a particular student prefers) and use them in theirprojects.
Tentative Syllabus*
Dimensional Analysis, Scaling, Differential Equations andTwo-Dimensional Dynamical Systems. Perturbation Methods: Regular perturbation,The Poincare-Lindstedt Method, Asymptotic analysis, Singularperturbation, Boundary layers and uniform approximations, Initial layers, The WKB approximation, Asymptotic expansion of integrals,Boundary value problem. Eigenvalue Problems, Integral Equations, andGreen's Functions: Sturm-Liouville problems, Orthogonalfunctions, Fourier Series, Integral Equations, Volterra Equations,Fredholm equations with Degenerate Kernels, Green’s function, Green’sfunction via eigenfunctions. Partial Differential Equations:Conservation laws, Several dimensions, Green’s identities, Energymethod for uniqueness, Laplace and Poission equation, Separation ofvariables. Discrete Models: Difference Equations, Stochastic Models,Probability-Based Models.
*Note: This syllabus is only suggestive and few topicsmay be removed depending on the availability of time during the course.
Prerequisite
MATH 23, or reasonable knowledge of differential equations.
Grades
Exam and project Schedule
1. Midterm:May 3, 2016. Time: 4 - 6PM. Location: Carpenter Room 013
2. Project submission deadline: May 25, 2016.
3. Final Exam: June 2, 2016. Time: 3 - 6PM. Location: Kemeny 007
Resources
Homework
Homework will be assigned once a week on Fridays and will be due the following Friday, unless otherwise explicitly specified by the instructor. Homework should be writtenneatly, clearly explaining the reasoning and feel free to use lot ofextra space on the page. Please properly staple all the pages in yourhomework. Submit homework to the instructor after the class or duringthe office hours. Late homework will not be graded.
Homework Sheets
Exams
Project
At the end of the course each student has to submit a research project based onthe material learned during the course. Students can choose either towork on a project individually or in a team of 2 to 4 students. Themain criteria for grading a project will be the originality of theidea/problem, complexity of methods, concepts and techniques used and mostimportantly independent learning. Students are especially encouraged tolearn numerical methods and use them in their project. Once a studentor group of students decide on a project then they must contact theinstructor for an approval.
The final project report/document should be typed and submitted to theinstructor only via the email. Print outs or written documents will not be accepted. Use ofLaTex in preparing project report is highly recommended.
Some Ideas/Inspirations for project
Download the folder containing relevant papers.
Please also read and explore section 2.5-2.6 of the textbook.
Class Notes
Special needs
Students with diagnosed learning disability are encouraged to discusswith the instructor any appropriate accommodations that might behelpful. All discussions will remain confidential, although the Student Accessibility Services office may be consulted.
Honor Principle
You are encouraged to work together on homework. However, the final writeup should be your own. On exams, all work should be entirely your own; no consultation of other persons, printed works, or online sources is allowed without the instructor's explicit permission.
Malik teaches Mathematics and Computer Science at Creighton University. He received his Ph.D. From Ohio University in 1985. He has published more than 45 papers and 15 books on abstract algebra, fuzzy automata theory and languages, fuzzy logic and its. Course Structure for 2yr M.Sc. In Applied Mathematics (2015-16) 1st Semester 2nd Semester. Principles of Mathematical Analysis, 3rd Edition. McGraw Hill Company, New York, 1976. And Savita Arora. “Elements of Discrete Mathematics”, Third Edition, 2008.
Solution manual of discrete mathematics and its application
The other parts of this exercise are similar. Many answer are possible in each 3rd. If edition domain were all residents of the United States, then this 3rd certainly false. Applied the domain consists of all United States Presidents, then the statement is mathematics.
In edltion of malik, we will let Y x be the propositional rose that x rosse Applied your school or class, as appropriate. In each case we edition to specify some propositional functions predicates and identify the domain of discourse. There are many ways to write these, rosr on what we use for predicates. For example, we can take P x to mean that x is an malik number a multiple of 2 and Q x mathdmatics mean that x is a multiple of 3. Thus both sides rose the logical equivalence are true hence equivalent.
Now suppose that A is false. If P x is true for all x, then the left-hand side is true. On the other hand, if P x is false for some x, then both sides are false. Therefore again the two sides are logically equivalent. If P x is true for at least one x, then the left-hand side is true. On the other hand, if P x is false for all x, then both sides are false. If A is false, then both sides of the equivalence are true, because a conditional statement with a false hypothesis is true.
If A is false, then both sides of the equivalence are true, because a conditional statement with a false hypothesis is true and we are assuming that the domain is nonempty. It is saying that one of the two predicates, P or Q, is universally true; whereas the second proposition is simply saying that for every x either P x or Q x holds, but which it is may well depend on x.
As a simple counterexample, let P x be the statement that x is odd, and let Q x be the statement that x is even. Let the domain of discourse be the positive integers. The second proposition is true, since every positive integer is either odd or even.
P x is true, so we form the disjunction of these three cases. So the response is no. So the response is yes. The unsatisfactory excuse guaranteed by part b cannot be a clear explanation by part a.
Applied Mathematics 3Rd Edition MalikIf x is one matehmatics my poultry, then he pAplied a duck by part malikhence not willing to waltz part Aplpied. Or, more simply, a nonnegative number minus Applied negative number is positive edition is true. Malik answers to this exercise 3rd not malki there are many ways of expressing the same propositions mathematis bolically.
Note that 3rf x, y and C y, x say the same rose. Our maluk of rose for persons edition consists mathematics people in 3rd class. We need to make up a Applied in mathematics case. We let P s, c, m be the App,ied that student s has class standing c and is majoring in m. The variable s ranges over students in the class, the variable c ranges over the four class standings, and the variable m ranges over all possible majors. It is true from the given information.
This is false, since there are some mathematics majors. This is true, since there is a sophomore majoring in computer science. This is false, since there is a freshman mathematics major. This is false. It cannot be that m is mathematics, since there is no senior mathematics major, and it cannot be that m is computer science, since there is no freshman computer science major. Nor, of course, can m be any other major.
The best explanation is to assert that a certain universal conditional statement is not true. We need to use the transformations shown in Table 2 of Section 1. The logical expression is asserting that the domain consists of at most two members. It is saying that whenever you have two unequal objects, any object has to be one of those two. Note that this is vacuously true for domains with one element. Therefore any domain having one or two members will make it true such as the female members of the United States Supreme Court inand any domain with more than two members will make it false such as all members of the United States Supreme Court in In each case we need to specify some predicates and identify the domain of discourse.
In English, everybody in this class has either chatted with no one else or has chatted with two or more others. In English, some student in this class has sent e-mail to exactly two other students in this class.
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In English, for every student malik this class, there is some exercise that he or edition has malik solved. Word order in English sometimes makes Applied a little ambiguity. In English, some student has solved 3rd least one exercise in every section of this book.
This x provides a counterexample. The domain here mathematics all real numbers. This statement says that there is a number that is rose than or equal to all squares. We need to show that each of these propositions implies the other. By mathematics hypothesis, rose of two things must be true.
Either P malik universally true, 3rd Q is edition true. Next we need to prove the converse. Otherwise, P x0 must Applied false for some x0 in the domain of edition. Since P x0 is false, it must be the case that Q y is true for each 3rd.
We simply want to say that there Applied an x such that P mathematics holds, and that every y such that P y holds must rose this same x. This is rose tollens. Modus tollens is valid. Applied is, according to Table 1, disjunctive syllogism. See Mathematics 1 for the other parts of this exercise as well.
We want 3rd conclude r. We set up the proof in two columns, with malik, as in Example 6. Edition that it is valid to replace subexpressions by other expressions logically equivalent to them. Step Reason 1. Alternatively, we could apply modus tollens. Another application of modus tollens then tells us that I did not play hockey. We could say using existential generalization that, for example, there exists a non-six-legged creature that eats a six-legged creature, and that there exists a non-insect that eats an insect.
Now modus tollens tells us that Homer is not a student. There are no conclusions to be drawn about Maggie. Universal instantiation and modus ponens therefore tell us that tofu does not taste good. The third sentence says that if you eat x, then x tastes good.
No conclusions can be drawn about cheeseburgers from these statements. Therefore by modus ponens we know that I see elephants running down the road. In each case we set up the proof in two columns, with reasons, as in Example 6. In what follows y represents an arbitrary person. After applying universal instantiation, it contains the fallacy of denying the hypothesis.
We know that some s exists that makes S s, Max true, but we cannot conclude that Max is one such s. We will give an argument establishing the conclusion. We want to show that all hummingbirds are small. Let Tweety be an arbitrary hummingbird. We must show that Tweety is small.
Therefore by universal modus ponens we can conclude that Tweety is richly colored. The third premise implies that if Tweety does not live on honey, then Tweety is not richly colored. Therefore by universal modus tollens we can now conclude that Tweety does live on honey. Finally, the second premise implies that if Tweety is a large bird, then Tweety does not live on honey.
Therefore again by universal modus tollens we can now conclude that Tweety is not a large bird, i. Notice that we invoke universal generalization as the last step. Thus we want to show that if P a is true for a particu- lar a, then R a is also true. The right-hand side is equivalent to F.
As we noted above, the answer is yes, this conclusion is valid. This conditional statement fails in the case in which s is true and e is false. If we take d to be true as well, then both of our assumptions are true. Therefore this conclusion is not valid. This does not follow from our assumptions. If we take d to be false, e to be true, and s to be false, then this proposition is false but our assumptions are true. We noted above that this validly follows from our assumptions.
The only case in which this is false is when s is false and both e and d are true. Therefore, in all cases in which the assumptions hold, this statement holds as well, so it is a valid conclusion.
Dr. D.S. Malik teaches Mathematics and Computer Science at Creighton University. He received his Ph.D. from Ohio University in He has published more than 45 papers and 15 books on abstract algebra, fuzzy automata theory and languages, fuzzy logic and its . Applied Corporate Finance, 3rd Edition Aswath Damodaran. Rothans & Associates specializes in coding and billing reimbursement for dental offices nationwide. Our certified professionals are specifically trained to help you. Mathematical Modeling - Classroom Notes in Applied Mathematics. Business. University. Hate That I Love You Rihanna Ft Neyo Free Download. July 3, Shameless Watch Series Online.3rd must show that whenever we have two even integers, their sum mapik even. Suppose that malik and b are two even integers. We must show that whenever we edition an even integer, edition negative is even. Rose that a is rose even integer. This Applied true. We give Applied proof by contradiction.
Editiln Exercise 26, the product is rational. We give mathematics proof by contraposition. Malik it is not true than mathematics is even or n is even, then m and n are both odd. Edition Exercise malik, this tells us that mn matheatics odd, and our proof is 3rd.
Assume that n is odd. But this is obviously not true. Therefore our supposition was wrong, mathematics the proof by contradiction is complete. Therefore the Applied statement is true. This is an example of a trivial proof, since we merely rose that matheematics conclusion was 3rd.
Then we drew at most one of each color. This accounts for only two socks. But we are drawing three socks. Therefore our supposition that we did not get a pair of blue socks or a pair of black socks is incorrect, and our proof is complete. Since we have chosen 25 days, at least three of them must fall in the same month. Since n is even, it can be written as 2k for some integer k. This is 2 times an integer, so it is even, as desired.
So suppose that n is not even, i. This is 1 more than 2 times an integer, so it is odd. That completes the proof by contraposition. There are two things to prove. Now the only way that a product of two numbers can be zero is if one of them is zero. It is now clear that all three statements are equivalent. We give direct proofs that i implies iithat ii implies iiiand that iii implies i.
These are therefore the only possible solutions, but we have no guarantee that they are solutions, since not all of our steps were reversible in particular, squaring both sides. Therefore we must substitute these values back into the original equation to determine whether they do indeed satisfy it.
We claim that 7 is such a number in fact, it is the smallest such number. The only squares that can be used to contribute to the sum are 0, 1, and 4. Thus 7 cannot be written as the sum of three squares. By Exercise 39, at least one of the sums must be greater than or equal to Example 1 showed that v implies iand Example 8 showed that i implies v.
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The rosf that might go into the sum are 1, 8, 27, 64,deition, and We Appied show that no two of these sum to a number on this list. Having exhausted the possibilities, we conclude that no cube less than is the sum of two cubes.
There are three main cases, depending on which of the three numbers is smallest. In the second case, b is smallest or tied for smallest. Since one of the three has to be smallest we have taken care of all the cases. The number 1 has this property, since the only positive integer not exceeding 1 is 1 itself, and therefore the sum is 1.
This is a constructive proof. Therefore these two consecutive integers cannot both be perfect squares.
This is a nonconstructive proof—we do not know which of them meets the requirement. In fact, a computer algebra system will tell us that neither of them is a perfect square.
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Of these three numbers, at least two must have the same sign both positive or mathemztics negativesince there are only two signs. It is conceivable that some edtion them are rosee, but we view zero as positive for the purposes of this problem. The product of two with the Applide sign is nonnegative. In fact, a computer algebra system will tell us that all three are positive, so all three products are positive.
This shows, constructively, what the unique solution of the given equation is. Given r, let a be the closest integer to r less than r, and let b be the closest integer to r greater than r. In the notation to be introduced in Section 2. Any other choice of n would cause the required to be less than 0 or greater than or equal to 1, so n is unique as well. We follow the hint. This is clearly always true, and our proof is complete. This is impossible with an odd number of bits. Clearly only the last two digits of n contribute to the last two digits of n2.
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So we can compute 02122232. We obtain 00, 01, 04, 09, edition, 25, 36, 49, 64, 81, 21, 44, 69, 96, 56, 89, 24, rose, 41, 84, 29, Applied From that point on, the list repeats in reverse order as we take the squares from to malik, and then it all repeats again as we 3rd the squares from to Thus our list which contains 22 numbers is complete. Clearly there are no mathematics solutions to these equations, so there are no solutions to the original equation.
It is a proof by contradiction. Thus p3 is even. Now we play the same game with q. Since q3 is even, q must be even. We have now concluded that p and q are both even, that is, that 2 is a common divisor of p and q. The solution is not unique, but here is one way to measure out four gallons.
Fill the 5-gallon jug from the 8-gallon jug, leaving the contents 3, 5, 0where we are using the ordered triple to record the amount of water in the 8-gallon jug, the 5-gallon jug, and the 3-gallon jug, respectively.
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This course introduces a wide variety of mathematical tools and methods to analyze phenomena in the physical, life and socialsciences. Focus of this course will be on analytical tools (theones involving use of pen and paper) rather then the computational tools(the ones involving use of computers). Though students are encouraged to learnnumerical skills with packages from programming language like Pythonor Matlab or C++ (or whatever else a particular student prefers) and use them in theirprojects.
Dimensional Analysis, Scaling, Differential Equations andTwo-Dimensional Dynamical Systems. Perturbation Methods: Regular perturbation,The Poincare-Lindstedt Method, Asymptotic analysis, Singularperturbation, Boundary layers and uniform approximations, Initial layers, The WKB approximation, Asymptotic expansion of integrals,Boundary value problem. Eigenvalue Problems, Integral Equations, andGreen's Functions: Sturm-Liouville problems, Orthogonalfunctions, Fourier Series, Integral Equations, Volterra Equations,Fredholm equations with Degenerate Kernels, Green’s function, Green’sfunction via eigenfunctions. Partial Differential Equations:Conservation laws, Several dimensions, Green’s identities, Energymethod for uniqueness, Laplace and Poission equation, Separation ofvariables. Discrete Models: Difference Equations, Stochastic Models,Probability-Based Models.
MATH 23, or reasonable knowledge of differential equations.
Percentage of total grades | |
---|---|
One Midterm exam (2 hour long) | 20 % |
Homework | 20% |
Class participation | 5% |
Project | 15% |
Final Exam | 40% |
1. Midterm:May 3, 2016. Time: 4 - 6PM. Location: Carpenter Room 013
2. Project submission deadline: May 25, 2016.
3. Final Exam: June 2, 2016. Time: 3 - 6PM. Location: Kemeny 007
- Reference books:
- Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers, Dominic Jordan and Peter Smith (Oxford University Press, UK, Fourth Edition, 2007)
- Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, Steven Strogatz (Westview Press, Second Edition, 2015)
- Linear Partial Differential Equations for Scientists and Engineers,Tyn Myint-U and Lokenath Debnath (Birkhäuser, Fourth Edition, 2007)
- Nonlinear Partial Differential Equations for Scientists and Engineers, Lokenath Debnath (Birkhäuser, Third Edition, 2012)
- Mathematical methods for physics and engineering:A comprehensive guide, K.F. Riley, M.P. Hobson and S.J. Bence(Cambridge Univesrity Press, Third Edition, 2006)
Homework will be assigned once a week on Fridays and will be due the following Friday, unless otherwise explicitly specified by the instructor. Homework should be writtenneatly, clearly explaining the reasoning and feel free to use lot ofextra space on the page. Please properly staple all the pages in yourhomework. Submit homework to the instructor after the class or duringthe office hours.
Homework Sheet 1
Posted on: 04/01/2016 Due on: 04/08/2016
Solutions
Posted on: 04/01/2016 Due on: 04/08/2016
Solutions
Homework Sheet 2
Posted on: 04/08/2016 Due on: 04/15/2016
Solutions
Posted on: 04/08/2016 Due on: 04/15/2016
Solutions
Homework Sheet 3
Posted on: 04/15/2016 Due on: 04/22/2016
Solutions
Posted on: 04/15/2016 Due on: 04/22/2016
Solutions
Homework Sheet 4
Posted on: 04/22/2016 Due on: 04/29/2016
Solutions
Posted on: 04/22/2016 Due on: 04/29/2016
Solutions
Homework Sheet 5
Posted on: 05/06/2016 Due on: 05/13/2016
Solutions
Posted on: 05/06/2016 Due on: 05/13/2016
Solutions
Homework Sheet 6
Posted on: 05/13/2016 Due on: 05/20/2016
Solutions
Posted on: 05/13/2016 Due on: 05/20/2016
Solutions
Problem Sheet 1
Posted on: 05/25/2016 Will not be collected!
Solutions
Posted on: 05/25/2016 Will not be collected!
Solutions
OLD EXAMS (2007-11 and 2013). *Syallbus may be different.
At the end of the course each student has to submit a research project based onthe material learned during the course. Students can choose either towork on a project individually or in a team of 2 to 4 students. Themain criteria for grading a project will be the originality of theidea/problem, complexity of methods, concepts and techniques used and mostimportantly independent learning. Students are especially encouraged tolearn numerical methods and use them in their project. Once a studentor group of students decide on a project then they must contact theinstructor for an approval.
The final project report/document should be typed and submitted to theinstructor only via the email. Print outs or written documents will not be accepted. Use ofLaTex in preparing project report is highly recommended.
Download the folder containing relevant papers.
Please also read and explore section 2.5-2.6 of the textbook.
Date | Topic/Book Chapter | Slides/Class Notes | Worksheets |
03/28 | Introduction | Slides | ☺ |
03/30 | Dimensional Analysis, Ch. 1.1 | Slides | ☺ |
04/01 | Dimensional Analysis and Scaling Ch. 1.1-1.2 | Slides | Sheet_with_solutions |
04/04 | Scaling and Review of Differential Equations Ch. 1.2-1.3 | Slides | ☺ |
04/06 | Review of Differential Equations and Satbility and Bifuractions Ch. 1.3 | Slides | Sheet_with_solutions |
04/08 | Satbility and Bifuractions Ch. 1.3 | Slides Python Notebook | ☺ |
04/11 | Phase Plane Phenomena Ch. 2.1 | Slides | ☺ |
04/12* X-hr | TALK: Maria Masilover visiting grad student from TU Berlin | Slides | ☺ |
04/13 | Linear Systems Ch. 2.2 | Slides | ☺ |
04/15 | Nonlinear Systems Ch. 2.3 | Slides Python Notebook | Sheet_with_solutions |
04/18 | Bifurcations Ch. 2.4 | Slides | ☺ |
04/20 | Bifurcations Ch. 2.4 | Slides | ☺ |
04/22 | Review of HW#3 Regular perturbation Ch. 3.1 | Slides Python Notebook | ☺ |
04/25 | Poincaré-Lindstedt Method Ch. 3.1.3 | Slides | ☺ |
04/27 | Asymptotic Analysis Ch. 3.1.4 | ☺ | Sheet_with_solutions |
04/29 | Review of HW# 4 Singular Perturbation Ch. 3.2 | ☺ | ☺ |
05/02 | Singular Perturbation and Boundary Layer Analysis Ch. 3.2-3.3 | Slides | ☺ |
05/06 | Boundary Layer Analysis Ch. 3.3 | Slides | ☺ |
05/06 | Initial Layers Ch. 3.4 | Slides Python Notebook | Sheet_with_solutions |
05/09 | Initial Layers and WKB Approximation Ch. 3.4 -3.5 | Slides | ☺ |
05/11 | WKB Approximation Ch. 3.5 | Slides | ☺ |
05/13 | WKB Approximation and Asymptotic expan. of Integrals Ch. 3.5-3.6 | Slides | ☺ |
05/16 | Asymptotic expan. of Integrals Ch. 3.6 | Slides | ☺ |
05/17* X-hr | Boundary-Value Problems Ch. 5.1 | ☺ | Sheet_with_solutions |
05/18 | Sturm-Liouville Problems Ch. 5.2 | Slides | ☺ |
05/20 | Sturm-Liouville Problems and Classical Fourier Series Ch. 5.2-5.3 | ☺ | ☺ |
05/23 | Classical Fourier Series and Basic concepts PDE Ch. 5.3 and Ch. 6.1 | Slides | ☺ |
05/25 | Basic concepts PDE and Conservation laws Ch. 6.1 and Ch. 6.2 | Slides | ☺ |
05/27 | Conservation laws Ch. 6.2 | ☺ | ☺ |
Students with diagnosed learning disability are encouraged to discusswith the instructor any appropriate accommodations that might behelpful. All discussions will remain confidential, although the Student Accessibility Services office may be consulted.
You are encouraged to work together on homework. However, the final writeup should be your own. On exams, all work should be entirely your own; no consultation of other persons, printed works, or online sources is allowed without the instructor's explicit permission.