# 7.8.E: Problems on Lebesgue Measure - Mathematics

Exercise (PageIndex{1})

Fill in all details in the proof of Theorems 3 and 4.

Exercise (PageIndex{1'})

Prove Note 2.

Exercise (PageIndex{2})

From Theorem 3 deduce that
[left(forall A subseteq E^{n} ight)left(exists B in mathcal{G}_{delta} ight) quad A subseteq B ext { and } m^{*} A=m B.]
[Hint: See the hint to Problem 7 in §5.]

Exercise (PageIndex{3})

Review Problem 3 in §5.

Exercise (PageIndex{4})

Consider all translates
of
[R=left{ ext {rationals in } E^{1} ight}.]
Prove the following.
(i) Any two such translates are either disjoint or identical.
(ii) Each (R+p) contains at least one element of ([0,1]).
[Hint for (ii): Fix a rational (y in(-p, 1-p),) so (0

Exercise (PageIndex{5})

Continuing Problem 4, choose one element (q in[0,1]) from each (R+p.) Let (Q) be the set of all (q) so chosen.
Call a translate of (Q, Q+r,) "good" iff (r in R) and (|r|<1.) Let (U) be the union of all "good" translates of (Q.)
Prove the following.
(a) There are only countably many "good" (Q+r).
(b) All of them lie in ([-1,2]).
(c) Any two of them are either disjoint or identical.
(d) ([0,1] subseteq U subseteq[-1,2] ;) hence (1 leq m^{*} U leq 3).
[Hint for (c): Suppose
[y in(Q+r) capleft(Q+r^{prime} ight).]
Then
[y=q+r=q^{prime}+r^{prime} quadleft(q, q^{prime} in Q, r, r^{prime} in R ight);]
so (q=q^{prime}+left(r^{prime}-r ight),) with (left(r^{prime}-r ight) in R).
Thus (q in R+q^{prime}) and (q^{prime}=0+q^{prime} in R+q^{prime}.) Deduce that (q=q^{prime}) and (r=r^{prime} =;) hence (Q+r=Q+r^{prime}).]

Exercise (PageIndex{6})

Show that (Q) in Problem 5 is not L-measurable.
[Hint: Otherwise, by Theorem 4, each (Q+r) is L-measurable, with (m(Q+r)=m Q.) By 5(a)(c), (U) is a countable disjoint union of "good" translates.
Deduce that (m U=0) if (m Q=0,) or (m U=infty,) contrary to 5(d).]

Exercise (PageIndex{7})

Show that if (f : S ightarrow T) is continuous, then (f^{-1}[X]) is a Borel set in (S) whenever (X in mathcal{B}) in (T).
[Hint: Using Note 1 in §7, show that
[mathcal{R}=left{X subseteq T | f^{-1}[X] in mathcal{B} ext { in } S ight}]
is a (sigma)-ring in (T.) As (mathcal{B}) is the least (sigma)-ring (supseteq mathcal{G}, mathcal{R} supseteq mathcal{B}) (the Borel field in (T).]

Exercise (PageIndex{8})

Prove that every degenerate interval in (E^{n}) has Lebesgue measure (0,) even if it is uncountable. Give an example in (E^{2}.) Prove uncountability.
[Hint: Take (overline{a}=(0,0), overline{b}=(0,1).) Define (f : E^{1} ightarrow E^{2}) by (f(x)=(0, x).) Show that (f) is one-to-one and that ([overline{a}, overline{b}]) is the (f)-image of ([0,1].) Use Problem 2 of Chapter 1, §9.]

Exercise (PageIndex{9})

Show that not all L-measurable sets are Borel sets in (E^{n}).
[Hint for (E^{2}:) With ([overline{a}, overline{b}]) and (f) as in Problem 8, show that (f) is continuous (use the sequential criterion). As (m[overline{a}, overline{b}]=0,) all subsets of ([overline{a}, overline{b}]) are in (mathcal{M}^{*}) (Theorem 2(i)), hence in (mathcal{B}) if we assume (mathcal{M}^{*}=mathcal{B}). But then by Problem 7 , the same would apply to subsets of ([0,1],) contrary to Problem 6.
Give a similar proof for (E^{n}(n>1)).
Note: In (E^{1},) too, (mathcal{B} eq mathcal{M}^{*},) but a different proof is necessary. We omit it.]

Exercise (PageIndex{10})

Show that Cantor's set (P) (Problem 17 in Chapter 3, 14 ) has Lebesgue measure zero, even though it is uncountable.
[Outline: Let
[U=[0,1]-P;]
so (U) is the union of open intervals removed from ([0,1].) Show that
[m U=frac{1}{2} sum_{n=1}^{infty}left(frac{2}{3} ight)^{n}=1]
and use Lemma 1 in §4.]

Exercise (PageIndex{11})

Let (mu : mathcal{B} ightarrow E^{*}) be the Borel restriction of Lebesgue measure (m) in (E^{n}) (§7). Prove that
(i) (mu) in incomplete;
(ii) (m) is the Lebesgue extension (* and completion, as in Problem 15 of §6) of (mu.)
[Hints: (i) By Problem 9, some (mu)-null sets are not in (mathcal{B}.) (ii) See the proof (end) of Theorem 2 in §9 (the next section).]

Exercise (PageIndex{12})

Prove the following.
(i) All intervals in (E^{n}) are Borel sets.
(ii) The (sigma)-ring generated by any one of the families (mathcal{C}) or (mathcal{C}^{prime}) in Problem 3 of §5 coincides with the Borel field in (E^{n}.)
[Hints: (i) Any interval arises from a closed one by dropping some "faces" (degenerate closed intervals). (ii) Use Lemma 2 from §2 and Problem 7 of §3.]

Exercise (PageIndex{13*})

Show that if a measure (m^{prime}: mathcal{M}^{prime} ightarrow E^{*}) in (E^{n}) agrees on intervals with Lebesgue measure (m: mathcal{M}^{*} ightarrow E^{*},) then the following are true.
(i) (m^{prime}=m) on (mathcal{B},) the Borel field in (E^{n}).
(ii) If (m^{prime}) is also complete, then (m^{prime}=m) on (mathcal{M}^{*}).
[Hint: (i) Use Problem 13 of §5 and Problem 12 above.]

Exercise (PageIndex{14})

Show that globes of equal radius have the same Lebesgue measure.
[Hint: Use Theorem 4.]

Exercise (PageIndex{15})

Let (f : E^{n} ightarrow E^{n},) with
(i) ((forall A subseteq E^{n}) m^{*} f[A]=c^{n} m^{*} A) ((m^{*}=)Lebesgue outer measure).
(ii) (A in mathcal{M}^{*}) iff (f[A] in mathcal{M}^{*}).
[Hint: If, say, (A=(overline{a}, overline{b}],) then (f[A]=(c overline{a}, c overline{b}].) (Why?) Proceed as in Theorem 4, using (f^{-1}) also.]

Exercise (PageIndex{16})

From Problems 14 and 15 show that
(i) (m G_{overline{p}}(c r)=c^{n} cdot m G_{overline{p}}(r));
(ii) (m G_{overline{p}}(r)=m overline{G}_{overline{p}}(r));
(iii) (m G_{overline{p}}(r)=a cdot m I,) where (I) is the cube inscribed in (G_{overline{p}}(r)) and
[a=left(frac{1}{2} sqrt{n} ight)^{n} cdot m G_{overline{0}}(1).]
[Hints: (i) (fleft[G_{overline{0}}(r) ight]=G_{overline{0}}(c r).) (ii) Prove that
[m G_{overline{p}} leq m overline{G}_{overline{p}} leq c^{n} m G_{overline{p}}]
if (c>1.) Let (c ightarrow 1).]

Exercise (PageIndex{17})

Given (aSet ((forall n))
[delta_{n}=frac{b-a}{2^{n+1}}]
and
[G_{n}=left(a_{n}, b_{n} ight)=(a, b) capleft(r_{n}-frac{1}{2} delta_{n}, r_{n}+frac{1}{2} delta_{n} ight).]
Let
[P=A-igcup_{n=1}^{infty} G_{n}.]
Prove the following.
(i) (sum_{n=1}^{infty} delta_{n}=frac{1}{2}(b-a)=frac{1}{2} m A).
(ii) (P) is closed; (P^{o}=emptyset,) yet (m P>0).
(iii) The (G_{n}) can be made disjoint (see Problem 3 in §2), with (m P) still (>0.)
(iv) Construct such a (P subseteq Aleft(P=overline{P}, P^{o}=emptyset ight)) of prescribed measure (m P=varepsilon>0).

Exercise (PageIndex{18})

Find an open set (G subset E^{1},) with (m G[Hint: (G=cup_{n=1}^{infty} G_{n}) with (G_{n}) as in Problem 17.]

Exercise (PageIndex{19*})

If (A subseteq E^{n}) is open and convex, then (m A=m overline{A}).
[Hint: Let first (overline{0} in A.) Argue as in Problem 16.]

## Problem regarding Lebesgue measure in $mathbb^2$

Let $P=A_1 imes A_2,$ where $A_1,A_2subset mathbb$ are set of positive Lebesgue measure, and $Zsubset mathbb^2,$ be a set of zero Lebesgue measure. Can we always find positive Lebesgue measure sets $B_1,B_2subset mathbb$ such that $B_1 imes B_2 subset overline?$ What extra conditions ensure that the above is true?(I can show that the above is true if $Psetminus overline$ is of positive measure then the above is true)

In this question https://math.stackexchange.com/q/3767758/641816, it was shown that the result is true if $A_1=A_2=[0,1]$ .

This is my attempt: Since $A_1,A_2$ are positive Lebesgue measure set we can find $a_1in A_1, a_2in A_2$ such that for any $r>0$ we have $B(a_1,r)cap A_1, B(a_2,r)cap A_2$ are sets of positive measure(in fact this phenomenon is true for almost every $a_1in A_1,a_2in A_2$ ). Consider $B_1^r=overline,quad B_2^r=overline$ Then I think somehow one can show that there exits some $s,t>0$ such that $B_1^s imes B_2^tsubset overline.$

## Some Elements of the Classical Measure Theory

### EXAMPLES

Lebesgue measure on ℝ r is a Radon measure.

Let ( t n ) n ∈ ℕ be any sequence in ℝ r , and ( a n ) n ∈ ℕ any summable sequence in [0, ∞[. For every E ⊆ ℝ set

Then v is a (totally finite) Radon measure on ℝ r .

Cantor measure. Recall that the Cantor set is a closed negligible subset of [0, 1], and that the Cantor function is a non-decreasing continuous function f: [0, 1] → [0, 1] such that f(0) = 0, f(l) = 1 and f is constant on each of the intervals composing [0,1]/C. It follows that if we set g ( x ) = 1 2 + f ( x ) for x ∈ [ 0 , 1 ] , then g:[0,1] → [0,1] is a continuous bijection such that the Lebesgue measure of g(C) is 1 2 consequently g −1 : [0, 1] → [0, 1] is continuous. Now extend g to a bijection h: ℝ → ℝ by setting h(x) = x for x ∈ ℝ [0, 1]. Then h and h −1 are continuous. Note that h(C) = g(C) has Lebesgue measure 1 2 .

Let v1 be the Radon measure on ℝ obtained by applying the method in the last Theorem to Lebesgue measure λ on ℝ and the function 2 χ ( h ( C ) ) . Then v 1 ( h ( C ) ) = v 1 ( ℝ ) = 1 . Let v be the measure v1h, that is, v(E) = v1(h(E)) for just those E ⊆ ℝ such that h(E) ∈ Dom v1. Then v is a Radon probability measure on ℝ, and v(C) = 1, v(ℝ C) = μ.(C) = 0.

## Measure, Integral, Derivative

This classroom-tested text is intended for a one-semester course in Lebesgue’s theory. With over 180 exercises, the text takes an elementary approach, making it easily accessible to both upper-undergraduate- and lower-graduate-level students. The three main topics presented are measure, integration, and differentiation, and the only prerequisite is a course in elementary real analysis.

In order to keep the book self-contained, an introductory chapter is included with the intent to fill the gap between what the student may have learned before and what is required to fully understand the consequent text. Proofs of difficult results, such as the differentiability property of functions of bounded variations, are dissected into small steps in order to be accessible to students. With the exception of a few simple statements, all results are proven in the text. The presentation is elementary, where σ-algebras are not used in the text on measure theory and Dini’s derivatives are not used in the chapter on differentiation. However, all the main results of Lebesgue’s theory are found in the book.

Sergei Ovchinnikov is currently Professor of Mathematics at San Francisco State University.

“It is accessible to upper-undergraduate and lower graduate level students, and the only prerequisite is a course in elementary real analysis. … The book proposes 187 exercises where almost always the reader is proposed to prove a statement. … this book is a very helpful tool to get into Lebesgue’s theory in an easy manner.” (Daniel Cárdenas-Morales, zbMATH, Vol. 1277, 2014)

“This is a brief … but enjoyable book on Lebesgue measure and Lebesgue integration at the advanced undergraduate level. … The presentation is clear, and detailed proofs of all results are given. … The book is certainly well suited for a one-semester undergraduate course in Lebesgue measure and Lebesgue integration. In addition, the long list of exercises provides the instructor with a useful collection of homework problems. Alternatively, the book could be used for self-study by the serious undergraduate student.” (Lars Olsen, Mathematical Reviews, December, 2013)

## Math 172 Homepage, Winter 2014-2015

Tentative office hours: MW 3:15-3:45, Th2-3, TW 10:30-11:30.

### On Thursday, March 12, office hour is extended to 1:30-4pm.

E-mail: rchlch "at" math.stanford.edu

Class location: MWF 2:15-3:05 pm, 380-380D. Due to an emergency, the last lecture of the quarter, on Friday, March 13, will be given by another instructor.

### There were two make-up classes, Thursday, Feb 12, and Tuesday, March 3, to replace the March 9, 11 classes (when there will be no lectures). These classes were both in GESB131 (Green Earth Sciences Building), 1:15-2:05pm.

Required textbook: Stein and Shakarchi: Real Analysis.

Recommended textbook: Stein and Shakarchi: Fourier Analysis

For topics covered in the recommended textbook, the instructor will provide his own lecture notes.

### Lecture notes:

The running syllabus may change somewhat, but should give an indication of the scope and speed of the course.

This course is similar to 205A, but designed for undergraduate students, and for graduate students in other departments. It also includes basic Fourier analysis. It is the continuation of the honors analysis course 171, emphasizing rigorous (i.e. logically careful) proofs, in the spirit of 171.

Grading policy: The grade will be based on the weekly homework (25%), on the in-class (expected in the usual classroom, at the usual class time) midterm exam (30%) and on the in-class (but of course not in the usual classroom, or at usual class time) final exam (45%).

## The final exam is on Monday, March 16, 12:15-3:15pm. It will be supervised by Prof. Soundararajan.

#### A practice exam with Solutions is available.

The exam covers all of the measure theory material, Chapter 1-2 of the text, as well as the Fourier analysis, as in the 5 handouts available for the course web page. There will be little emphasis on the last topic, distributions, but even when not explicitly asked, thinking about them might help your understanding of the material in a way relevant to the exam (for instance, it places the Fourier transform on L^2 into a better context). There will be an emphasis on Fubini and Tonelli since these were not covered on the midterm. In particular, you should always use careful arguments to check the hypotheses of Fubini's theorem this often involves a use of Tonelli's theorem (checking the hypotheses again).

The midterm is on Friday, February 6, in 380D, 2:15-3:30pm. Please come a few minutes early so that we can start on time.

#### Solutions are now available!

It is a closed book, closed notes, no calculators/computers, etc. exam.

In the additivity part of the original version of 2(i) solutions,

The homework will be due either in class or by 9pm in the instructor's mailbox on the designated day, usually Wednesdays. You are allowed to discuss the homework with others in the class, but you must write up your homework solution by yourself. Thus, you should understand the solution, and be able to reproduce it yourself. This ensures that, apart from satisfying a requirement for this class, you can solve the similar problems that are likely to arise on the exams.

## Henri Léon Lebesgue

Henri Lebesgue's father was a printer. Henri began his studies at the Collège de Beauvais, then he went to Paris where he studied first at the Lycée Saint Louis and then at the Lycée Louis-le-Grand.

Lebesgue entered the École Normale Supérieure in Paris in 1894 and was awarded his teaching diploma in mathematics in 1897 . For the next two years he studied in its library where he read Baire's papers on discontinuous functions and realised that much more could be achieved in this area. Later there would be considerable rivalry between Baire and Lebesgue which we refer to below. He was appointed professor at the Lycée Centrale at Nancy where he taught from 1899 to 1902 . Building on the work of others, including that of Émile Borel and Camille Jordan, Lebesgue formulated the theory of measure in 1901 and in his famous paper Sur une généralisation de l'intégrale définie Ⓣ , which appeared in the Comptes Rendus on 29 April 1901 , he gave the definition of the Lebesgue integral that generalises the notion of the Riemann integral by extending the concept of the area below a curve to include many discontinuous functions. This generalisation of the Riemann integral revolutionised the integral calculus. Up to the end of the 19 th century, mathematical analysis was limited to continuous functions, based largely on the Riemann method of integration.

His contribution is one of the achievements of modern analysis which greatly expands the scope of Fourier analysis. This outstanding piece of work appears in Lebesgue's doctoral dissertation, Intégrale, longueur, aire Ⓣ , presented to the Faculty of Science in Paris in 1902 , and the 130 page work was published in Milan in the Annali di Matematica in the same year. Having graduated with his doctorate, Lebesgue obtained his first university appointment when in 1902 he became mâitre de conférences in mathematics at the Faculty of Science in Rennes. This was in keeping with the standard French tradition of a young academic first having appointments in the provinces, then later gaining recognition in being appointed to a more junior post in Paris. On 3 December 1903 he married Louise-Marguerite Vallet and they had two children. However the marriage only lasted until 1916 when they were divorced.

One honour which Lebesgue received at an early stage in his career was an invitation to give the Cours Peccot at the Collège de France. He did so in 1903 and then received an invitation to present the Cours Peccot two years later in 1905 . Lebesgue first fell out with Baire in 1904 , when Baire gave the Cours Peccot at the Collège de France, over who had the most right to teach such a course. Their rivalry turned into a more serious argument later in their lives. Lebesgue wrote two monographs Leçons sur l'intégration et la recherche des fonctions primitives Ⓣ (1904) and Leçons sur les séries trigonométriques Ⓣ (1906) which arose from these two lecture courses and served to make his important ideas more widely known. However, his work received a hostile reception from classical analysts, especially in France. In 1906 he was appointed to the Faculty of Science in Poitiers and in the following year he was named professor of mechanics there.

Let us attempt to indicate the way that the Lebesgue integral enabled many of the problems associated with integration to be solved. Fourier had assumed that for bounded functions term by term integration of an infinite series representing the function was possible. From this he was able to prove that if a function was representable by a trigonometric series then this series is necessarily its Fourier series. There is a problem here, namely that a function which is not Riemann integrable may be represented as a uniformly bounded series of Riemann integrable functions. This shows that Fourier's assumption for bounded functions does not hold.

In 1905 Lebesgue gave a deep discussion of the various conditions Lipschitz and Jordan had used in order to ensure that a function f ( x ) f (x) f ( x ) is the sum of its Fourier series. What Lebesgue was able to show was that term by term integration of a uniformly bounded series of Lebesgue integrable functions was always valid. This now meant that Fourier's proof that if a function was representable by a trigonometric series then this series is necessarily its Fourier series became valid, since it could now be founded on a correct result regarding term by term integration of series. As Hawkins writes in [ 1 ] :-

He was appointed mâitre de conférences in mathematical analysis at the Sorbonne in 1910 . During the first world war he worked for the defence of France, and at this time he fell out with Borel who was doing a similar task. Lebesgue held his post at the Sorbonne until 1918 when he was promoted to Professor of the Application of Geometry to Analysis. In 1921 he was named as Professor of Mathematics at the Collège de France, a position he held until his death in 1941 . He also taught at the École Supérieure de Physique et de Chimie Industrielles de la Ville de Paris between 1927 and 1937 and at the École Normale Supérieure in Sèvres.

It is interesting that Lebesgue did not concentrate throughout his career on the field which he had himself started. This was because his work was a striking generalisation, yet Lebesgue himself was fearful of generalisations. He wrote:-

Although future developments showed his fears to be groundless, they do allow us to understand the course his own work followed.

He also made major contributions in other areas of mathematics, including topology, potential theory, the Dirichlet problem, the calculus of variations, set theory, the theory of surface area and dimension theory. By 1922 when he published Notice sur les travaux scientifique de M Henri Lebesgue he had written nearly 90 books and papers. This ninety-two page work also provides an analysis of the contents of Lebesgue's papers. After 1922 he remained active, but his contributions were directed towards pedagogical issues, historical work, and elementary geometry.

Lebesgue was honoured with election to many academies. He was elected to the Academy of Sciences on 29 May 1922 , to the Royal Society, the Royal Academy of Science and Letters of Belgium (6 June 1931) , the Academy of Bologna, the Accademia dei Lincei, the Royal Danish Academy of Sciences, the Romanian Academy of Sciences, and the Kraków Academy of Science and Letters. He was also awarded honorary doctorates from many universities. He also received a number of prizes including the Prix Houllevigue (1912) , the Prix Poncelet (1914) , the Prix Saintour (1917) and the Prix Petit d'Ormoy (1919) .

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## Proof of Lemma

We follow exercises #45-47 of ch. 2 in Royden's Real Analysis (4ed). Let $f$ be any strictly increasing function defined on some interval. By our analysis above, we know that such a function is a homeomorphism. This fact enables us to show that $f$ maps Borel sets to Borel sets. To do so, it suffices to show that for any continuous function $g$ the set $mathscr=(E) ext < is Borel>>$ is a $sigma$-algebra containing the open sets. Once we show this, we can conclude $mathscr$ contains all the Borel sets and therefore, taking $g$ to be $f^<-1>$ (which we know is continuous!), we'll have $(f^<-1>)^<-1>(E)=f(E)$ is Borel for any Borel set $E$, which is what we want.

## Victor Beresnevich

Metric Number Theory and Diophantine approximation. Other research interests: geometry of numbers, uniform distribution, measure and probability theory, fractal geometry, ergodic theory, dynamical systems, applications of Diophantine approximation (in PDEs, signal processing, etc.).

### Available PhD research projects

Victor Beresnevich, Jason Levesley and Sanju Velani work on a variety of problems in metric number theory and Diophantine approximation that involve a range of techniques from Diophantine approximation, analytic number theory, the geometry of numbers, probability theory, fractals and ergodic theory. Some examples include the Duffin-Schaeffer problem on rational approximations to real numbers, problems on approximation by algebraic numbers, problems on badly approximable vectors, problems on Diophantine approximation on manifolds, etc. The Diophantine approximation problems have natural `dynamical' analogues in terms of shrinking targets problems associated with the phase space of a given dynamical system. Victor Beresnevich and Sanju Velani are currently running a large scale research programme and any PhD student would become an integral part of the larger research group. If interested, please, contact either of them for possible PhD research projects.

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(Prerequisite: ACT Math subscore 24, or grade of C or better in MA 1313). Two hours lecture. Two hours laboratory. Basic concepts and methods of statistics, including descriptive statistics, probability, random variables, sampling distribution, estimation, hypothesis testing, introduction to analysis of variance, simple linear regression. (Same as ST 3123).

MA 3163. Introduction to Modern Algebra. (3)

(Prerequisite: MA 3113 and MA 3053). Three hours lecture. Rings, integral domains, and fields with special emphasis on the integers, rational numbers, real numbers and complex numbers theory of polynomials.

MA 3253. Differential Equations I. (3)

(Prerequisite: MA 2743 or co-registration in MA 2743). Origin and solution of differential equations series solutions Laplace Transform methods applications.

MA 3353. Differential Equations II. (3)

(Prerequisite: MA 3253). Three hours lecture. Systems of differential equations matrix representations infinite series solution of ordinary differential equations selected special functions boundary-value problems orthogonal functions: Fourier series.

MA 3463. Foundations of Geometry. (3)

(Prerequisite: MA 1723 and MA 3053). Three hours lecture. The structural nature of geometry modern methods in geometry: finite geometrics.

MA 3513. History of Mathematics. (3)

(Prerequisite: MA 2733 or co-registration in MA 2733). Three hours lecture. A historical development of mathematicians and their most important contributions will be emphasized.

MA 4133/6133. Discrete Mathematics. (3)

(Prerequisites: MA 3163 or consent of instructor). Three hours lecture. Sets, relations, functions, combinatorics, review of group and ring theory, Burnside’s theorem, Polya’s counting theory, group codes, finite fields, cyclic codes, and error-correcting codes.

MA 4143/6143. Graph Theory. (3)

(Prerequisites: MA 3113 or consent of instructor). Three hours lecture. Basic concepts, graphs, and matrices, algebraic graph theory, planarity and nonplanarity, Hamiltonian graphs, digraphs, network flows, and applications.

MA 4153/6153. Matrices and Linear Algebra. (3)

(Prerequisites: MA 3113 and MA 3253). Three hours lecture. Linear transformations and matrices eigenvalues and similarity transformations linear functionals, bilinear and quadratic forms orthogonal and unitary transformations normal matrices applications of linear algebra.

MA 4163/6163. Group Theory. (3)

(Prerequisite: MA 3163 or consent of the instructor). Three hours lecture. Elementary properties: normal subgroups factor groups homomorphisms and isomorphisms Abelian groups Sylow theorems composition series solvable groups.

MA 4173/6173. Number Theory. (3)

(Prerequisite: MA 3113). Three hours lecture. Divisibility: congruences quadratic reciprocity Diophantine equations continued fractions.

MA 4213. Senior Seminar in Mathematics. (3)

(Prerequisites: MA 3163 and MA 3253 and MA 4633). Three hours lecture. Students explore topics in current mathematical research, write expository articles, and give oral presentations. Refinement of specialized writing skills needed for effective mathematical communication.

MA 4243/6243 Data Analysis I. (3)

(Prerequisite: MA 2743. Co-requisite: MA 3113). Three hours lecture. Data description and descriptive statistics, probability and probability distributions, parametric one-sample and two-sample inference procedures, simple linear regressions, one-way ANOVA. Use of SAS. (Same as ST 4243/6243.)

MA 4253/6253 Data Analysis II. (3)

(Prerequisites: MA 4243/6243 and MA 3113). Three hours lecture. Multiple linear regression fixed, mixed and random effect models block designs two-factor analysis of variance three-factor analysis of variance analysis of covariance. Use of SAS. (Same as ST 4253/6253.)

MA 4313/6313. Numerical Analysis I. (3)

(Prerequisites: CSE 1213, MA 3113, and MA 2743). Three hours lecture. Matrix operations error analysis norms of vectors and matrices transformations matrix functions numerical solutions of systems of linear equations stability matrix inversion eigenvalue problems approximations.

MA 4323/6323. Numerical Analysis II. (3)

(Prerequisites: CSE 1213 or equivalent. MA 3113 and MA 3253). Three hours lecture. Numerical solution of equations error analysis finite difference methods numerical differentiation and integration series expansions difference equations numerical solution of differential equations.

MA 4373/6373. Introduction to Partial Differential Equations. (3)

(Prerequisite: MA 3253). Three hours lecture. Linear operators: linear first order equations the wave equation Green’s function and Sturm-Liouville problems Fourier series the heat equation Laplace’s equation.

MA 4523/6523. Introduction to Probability. (3)

(Prerequisite: MA 2733). Three hours lecture. Basic concepts of probability, conditional probability, independence, random variables, discrete and continuous probability distributions, moment generating function, moments, special distributions, central limit theorem. (Same as ST 4523/6523).

MA 4533/6533. Introductory Probability and Random Processes. (3)

(Prerequisites: MA 3113 and MA 2743). Three hours lecture. Probability, law of large numbers, central limit theorem, sampling distributions, confidence intervals, hypothesis testing, linear regression, random processes, correlation functions, frequency and time domain analysis. (Credit can not be earned for this course and MA/ST 4523/6523.)

MA 4543/6543. Introduction to Mathematical Statistics I. (3)

(Prerequisite: MA 2743.) Three hours lecture. Combinatorics probability, random variables, discrete and continuous distributions, generating functions, moments, special distributions, multivariate distributions, independence, distributions of functions of random variables. (Same as ST 4543/6543.)

MA 4573/6573. Introduction to Mathematical Statistics II. (3)

(Prerequisite: MA 4543/6543.) Three hours lecture. Continuation of MA-ST 4543/6543. Transformations, sampling distributions, limiting distributions, point estimation, interval estimation, hypothesis testing, likelihood ratio tests, analysis of variance, regression, chi-square tests. (Same as ST 4573/6573.)

MA 4633/6633. Advanced Calculus I. (3)

(Prerequisite: MA 2743 and MA 3053). Three hours lecture. Theoretical investigation of functions limits differentiability and related topics in calculus.

MA 4643/6643. Advanced Calculus II. (3)

(Prerequisite: MA 4633/6633). Three hours lecture. Rigorous development of the definite integral sequences and series of functions convergence criteria improper integrals.

MA 4733/6733. Linear Programming. (3)

(Prerequisites: MA 3113). Three hours lecture. Theory and application of linear programming simplex algorithm, revised simplex algorithm, duality and sensitivity analysis, transportation and assignment problem algorithms, integer and goal programming. (Same as IE 4733/6733).

MA 4753/6753. Applied Complex Variables. (3)

(Prerequisite: MA 2743). Three hours lecture. Analytic functions: Taylor and Laurent expansions Cauchy theorems and integrals residues contour integration introduction to conformal mapping.

MA 4933/6933. Mathematical Analysis I. (3)

(Prerequisite: MA 4633/6633 or equivalent). Three hours lecture. Metric and topological spaces functions of bounded variation and differentiability in normed spaces.

MA 4943/6943. Mathematical Analysis II. (3)

(Prerequisite: MA 4933/6933). Three hours lecture. Riemann-Stieltjes integration, sequences and series of functions implicit function theorem multiple integration.

MA 4953/6953. Elementary Topology. (3)

(Prerequisite: MA 4633/6633). Three hours lecture. Definition of a topological space, metric space, continuity in metric spaces and topological spaces sequences accumulation points.

MA 6990 Special Topics in Mathematics. (1-9)

Credit and title to be arranged. This course is to be used on a limited basis to offer developing subject matter areas not covered in existing courses. (Courses limited to two offerings under one title within two academic years.)

MA 7000 Directed Individual Study in Mathematics. (1-6)

Hours and credits to be arranged.

MA 8000 Thesis Research/ Thesis in Mathematics: (1-13)

Hours and credits to be arranged.

MA 8113. Modern Higher Algebra I. (3)

(Prerequisite: MA 4163/6163). Three hours lecture. A study of the basic mathematical systems with emphasis on rings, fields, and vector spaces.

MA 8123. Modern Higher Algebra II. (3)

(Prerequisite: MA 8113). Three hours lecture. A continuation of the topics introduced in MA 8113.

MA 8203. Foundations of Applied Mathematics I. (3)

(Prerequisites: MA 3113, MA 3253 or consent of instructor.) Three hours lecture. Principles of applied mathematics including topics from perturbation theory, calculus of variations, and partial differential equations. Emphasis of applications from heat transfer, mechanics, fluids.

MA 8213. Foundations of Applied Mathematics II. (3)

(Prerequisite: MA 8203). Three hours lecture. A continuation of MA 8203 including topics from wave propagation, stability, and similarity methods.

MA 8253. Operational Mathematics. (3)

(Prerequisite: MA 4753/6753). Three hours lecture. Theory and applications of Laplace, Fourier, and other integral transformations: introduction to the theory of generalized functions.

*Courses numbered MA 8273, 8283, 8293 and 8313 have as prerequisites at least one of the courses MA 4633/6633, MA 4153/6153, 4753/6753.

MA 8273. Special Functions. (3)

Three hours lecture. Infinite products: asymptotic series origin and properties of the special functions of mathematical physics.

MA 8283. Calculus of Variations. (3)

Three hours lecture. Functionals: weak and strong extrema necessary conditions for extrema sufficient conditions for extrema constrained extrema direct methods applications.

MA 8293. Integral Equations. (3)

Three hours lecture. Equations of Fredholm type: symmetric kernels Hilbert-Schmidt theory singular integral equations applications selected topics.

MA 8313. Ordinary Differential Equations I. (3)

Three hours lecture. Linear systems of differential equations existence and uniqueness second order systems systems with constant coefficients periodic systems matrix comparison theorems applications and selected topics.

MA 8323. Ordinary Differential Equations II. (3)

(Prerequisite: MA 8313). Three hours lecture. Existence, uniqueness, continuation of solutions of nonlinear systems properties of solutions of linear and nonlinear equations including boundedness, oscillation, asymptotic behavior, stability, and periodicity application.

MA 8333. Partial Differential Equations I. (3)

(Prerequisite: MA 4373/6373 or consent of instructor). Three hours lecture. Solution techniques existence and uniqueness of solutions to elliptic, parabolic, and hyperbolic equations Green’s functions.

MA 8343. Partial Differential Equations II. (3)

(Prerequisite: MA 8333). Three hours lecture. A continuation of the topics introduced in MA 8333.

MA 8363. Numerical Solution of Systems of Nonlinear Equations. (3)

(Prerequisites: MA 4313/6313 and MA 4323/6323). Three hours lecture. Basic concepts in the numerical solution of systems of nonlinear equations with applications to unconstrained optimization.

MA 8383. Numerical Solution of Ordinary Differential Equations I. (3)

(Prerequisites: MA 4313/6313 and MA 4323/6323). Three hours lecture. General single-step, multistep, multivalue, and extrapolation methods for systems of nonlinear equations convergence error bounds error estimates stability methods for stiff systems current literature.

MA 8443. Numerical Solution of Partial Differential Equations I. (3)

(Prerequisites: MA 4313/6313, MA 4323/6323, and MA 4373/6373 or consent of instructor). Three hours lecture. Basic concepts in the fi nite difference and fi nite element methods methods for parabolic, hyperbolic and elliptic equations analysis of stability and convergence.

MA 8453. Numerical Solution of Partial Differential Equations II. (3)

(Prerequisite: MA 8443). Three hours lecture. Methods for elliptic equations iterative procedures integral equation methods methods for hyperbolic equations stability dissipation and dispersion.

MA 8463. Numerical Linear Algebra. (3)

(Prerequisite: MA 4323/6323). Three hours lecture. Basic concepts of numerical linear algebra.

MA 8633. Real Analysis I. (3)

(Prerequisite: MA 4943/6943). Three hours lecture. Lebesgue measure and Lebesgue integrals convergence theorems, differentiation and L spaces.

MA 8643. Real Analysis II. (3)

(Prerequisite: MA 8633). Three hours lecture. General measures the Radon-Nikodym theorem and other topics.

MA 8663. Functional Analysis I. (3)

(Prerequisite: MA 8643). Three hours lecture. Hilbert spaces Banach spaces locally convex spaces Hahn-Banach and closed graph theorems principle of uniform boundedness weak topologies.

MA 8673. Functional Analysis II. (3)

(Prerequisite: MA 8663). Three hours lecture. Continuation of topics introduced in MA 8663.

MA 8713. Complex Analysis I. (3)

(Prerequisite MA 4943/6943 or consent of instructor). Three hours lecture. Complex numbers: functions of a complex variable continuity differentiation and integration of complex functions transformations in the complex plane.

MA 8723. Complex Analysis II. (3)

(Prerequisite: MA 8713). Three hours lecture. Series analytic continuation Riemann surfaces theory of residues.

MA 8913. Introduction to Topology I. (3)

(Prerequisite: MA 4643/6643 or MA 4953/6953). Three hours lecture. Basic general topology introduction of homotopy and homology groups.

MA 8923. Introduction to Topology II. (3)

(Prerequisite: MA 8913). Three hours lecture. Continuation of topics introduced in MA 8913.

MA 8981. Teaching Seminar. (1)

One hour lecture. Preparation for service as instructors in mathematics and statistics courses includes practice lectures and exam preparation. (May be taken for credit more than once.)

MA 8990 Special Topics in Mathematics: (1-9)

Credit and title to be arranged. This course is to be used on a limited basis to offer developing subject matter areas not covered in existing courses. (Courses limited to two offerings under one title within two academic years.)

MA 9000 Dissertation Research /Dissertation in Mathematics. (1-13)

Hours and credits to be arranged.

MA 9313. Selected Topics in Ordinary Differential Equations. (3)

(Prerequisite: MA 8313 and consent of instructor). (May be taken for credit more than once). Three hours lecture. Topics to be chosen from such areas as Bifurcation Theory, Biological Modeling, Control Theory, Dynamical Systems, Functional Differential Equations, Nonlinear Oscillations, and Quantitative Behavior.

MA 9333. Selected Topics in Partial Differential Equations. (3)

(Prerequisite: MA 8333 and consent of instructor). (May be taken for credit more than once). Three hours lecture. Topics to be chosen from such areas as Bifurcation Theory, Boundary Integral Methods, Evolution Equations, Maximum and Variational Principles, and Spectral Methods.

MA 9413. Selected Topics in Numerical Analysis. (3)

(Prerequisite: Consent of instructor). (May be taken for credit more than once). Three hours lecture. Current topics in Numerical Analysis. The subject matter may vary from year to year.

MA 9633. Selected Topics in Analysis. (3)

(Prerequisite: MA 8643 and consent of instructor). (May be taken for credit more than once). Three hours lecture. Topics will be chosen from areas of analysis of current interest.