David Singman
George Mason University
Department of Mathematics
Professor Emeritus
Published papers and posted videos
1. Published papers
19) Minimal fine limits on trees, with K.
GowriSankaran, Illinois.
J. Math. 48 (2004), no. 2, 359-389.
pdf
file
18) Distributions and Measures on the boundary of
a tree, with Joel Cohen and Flavia Colonna, Journal of
Mathematical Analysis and Applications, 293 (2004),
no. 1, 89-107.
pdf file
17) Trees as Brelot spaces, with Ibtesam
Bajunaid, Joel Cohen, Flavia Colonna, Advances in
Applied Mathematics, 30 (2003), no 4, 706-745.
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file
Corrigendum to
"Trees as Brelot spaces", Advances in Applied
Mathematics, 47 (2011), no 2, 401-402.
pdf file
15) Polyharmonic functions on trees, with Joel
Cohen, Flavia Colonna, K. GowriSankaran, American Journal of
Mathematics, 124 (2002), no 5, 999-1043.
pdf
file
14) A projection theorem and tangential boundary
behavior of potentials, with K. GowriSankaran, Proceedings
of the American Mathematical Society, 129 (2001),
no 2, 397-405.
pdf
file
11) A generalized Littlewood theorem for
Weinstein potentials on a halfspace, with K. GowriSankaran, Illinois
J. Math, 41 (1997), no. 4,
630-647.
pdf
file
10) Intermittent oscillation and tangential
growth of functions with respect to Nagel-Stein regions on a
half-space, with Robert Berman, Illinois J. Math., 38
(1994), no. 1, 19-46.
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file
9) Lower derivatives of functions of finite
variation and generalized BCH sets, J. Math. Ann. and
App., 173 (1993), no. 2, 483-496.
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file
8) Boundary behavior of positive solutions
of the Helmholtz equation and Helmholtz potentials, with Robert
Berman, Michigan Math. J., 38 (1991), no. 3,
381-393.
pdf
file
7) Generalized local Fatou theorems and
area integrals, with B. A. Mair and Stanton Philipp, Trans.
Amer. Math. Soc., 321 (1990),
no. 1, 401-413.
pdf file
6) A converse Fatou theorem on homogeneous
spaces, with B. A. Mair and Stanton Philipp, Illinois J.
Math., 33 (1989), no. 4, 643-656.
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file
5) A converse Fatou theorem, with B. A. Mair and
Stanton Philipp, Michigan Math. J.,36 (1989),
no. 1, 3-9.
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file
4) Boundary behavior of invariant Green's
potentials on the unit ball of C^n, with K.T. Hahn, Trans.
Amer. Math. Soc.,309 (1988), no. 1, 339-354.
pdf file
3) A generalized Fatou theorem, with B. A.
Mair, Trans. Amer. Math. Soc., 300 (1987), no. 2,
705-719.
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file
2) Removable singularities for n-harmonic
functions and Hardy classes in polydiscs, Proc. Amer. Math.
Soc., 90 (1984), no. 2, 299-302.
pdf
file
1) Exceptional sets in a product of harmonic
spaces, Math. Annalen, 262
(1983), no. 1, 29-43.
pdf
file
2. Math 290 videos posted on
YouTube
Section 1.1 Propositions and Connectives
part 1
part 2 Connectives, Propositional Forms, and Truth Tables.
part 3 Negation and Useful Denials.
Section 1.2 Conditonals and Biconditionals
part 1 Definition of a conditional.
part 2 Useful denial of a conditional.
part 3 Contrapositive, converse, and biconditional.
Section 1.3 Quantifiers
part 1 Open sentences and their truth sets.
part 2 Examples of the use of quantifiers.
part 3 Useful denials of: for all x in U, P(x); there exists x in U such that P(x).
part 4 A few exercises from 1.3.
Section 1.4 Basic Proof Methods I
part 1 Structure of a direct proof of: for all x in U, P(x) implies Q(x).Chapter 2. Set Theory
part 2 First working definitions.
part 3 First example of a direct proof.
part 4 Second example of a direct proof.
part 5 A direct proof involving inequalities.
part 6 A direct proof involving cases.
part 7 Another direct proof involving cases.
Section 1.5 Basic Proof Methods II
part 1 Structure of proofs by contraposition and contradiction.
part 2 Example of a proof by contraposition.
part 3 Second example of a proof by contraposition.
part 4 Third example of a proof by contraposition.
part 5 Fourth example of a proof by contraposition.
part 6 First proof by contradiction.
part 7 Second proof by contradiction.
part 8 Alternate ways to express conditionals; biconditional.
part 9 Primes and The Fundamental Theorem of Arithmetic.
part 10 Example of a proof of a biconditional proposition.
Section 1.6 More proofs involving quantifiers
part 1 Constructive and nonconstructive proofs of ``there exists x such that P(x)''.
part 2 Fundamental Theorem of Algebra, Intermediate Value Theorem, and an application to a nonconstructive existence proof.
part 3 General comments about existentially quantified statements.
part 4 Example involving the limit of a function.
part 5 Working definition of limit of a function; the Triangle Inequality and the Reverse Triangle Inequality.
part 6 Exercises involving the Triangle Inequality and the Reverse Triangle Inequality.
part 7 Second example involving the limit of a function.
part 8 Third example involving the limit of a function.
part 9 Fourth example involving the limit of a function.
part 10 Working definition of limit of an infinite sequence and an example.
part 11 A proof that a given sequence does not converge to a given real number.
part 12 Working definition that an infinite sequence diverges to infinity.
part 13 How should you respond to a statement written in symbols? It depends on whether you wish to merely read it, or to write a proof of it.
part 14 Completion of the exercise started in the previous video.
part 15 Limit proof where you need to calculate both an upper bound and a positive lower bound.
part 1 Basic set theory.Chapter 3. Relations and Partitions
part 2 Subset of a set.
part 3 The power set of a set.
part 4 A few examples of proofs with sets.
Section 2.2 Set Operations
part 1 Basic set operations.
part 2 Example of a set-theoretic proof.
part 3 Using Venn diagrams to provide examples of various set relationships.
part 4 Use of a Venn diagram to suggest a counterexample.
part 5 Proof of a distributive law in set theory.
part 6 Cartesian product of sets.
part 7 Proofs involving Cartesian product of sets.
part 8 More proofs involving Cartesian products of sets.
Section 2.3 Extended Set Operations and Indexed Families of Sets
part 1 Intersection and union of indexed families of sets.
part 2 Distributive laws for indexed families of sets.
part 3 De Morgan laws for indexed families of sets.
part 4 A simple but useful result.
part 5 A proof involving a denumerable union of intervals.
part 6 A proof involving a denumerable intersection of intervals.
Section 2.4 The Principle of Mathematical Induction (PMI)
part 1 Statement of the PMI and formats of proofs involving PMI.
part 2 Use of the PMI to define functions on the set of natural numbers.
part 3 First example of a proof using the PMI.
part 4 Second example of a proof using the PMI.
part 5 Third example of a proof using the PMI.
part 6 Fourth example of a proof using the PMI (use of Pascal's triangle).
part 7 Generalized PMI.
part 8 An incorrect proof using PMI.
Section 2.5 Other Forms of Induction
part 1 Two Additional Properties of the set of natural numbers - The Principle of Complete Induction (PCI) and the Well-Ordering Property of the natural numbers..
part 2 An incorrect proof using PCI.
part 3 First example of a proof using PCI - The Fibonacci numbers.
part 4 Second example of a proof using PCI.
part 5 Third example of a proof using PCI.
part 6 An application of PCI - Proof of the existence part of the Fundamental Theorem of Arithmetic.
part 7 Proof of the Division Algorithm using PCI.
part 8 An application of the Well-Ordering Property - proof that the square root of 2 is irrational.
part 9 An application of the Well-Ordering Property and the Division Algorithm - greatest common divisors.
part 10 More on gcd's; relatively prime integers; Euclid's Lemma.
part 11 Another application of the Division Algorithm - modular arithmetic.
part 12 Logical equivalence of PMI, PCI, and the Well-Ordering Property of the natural numbers.
part 13 Use of PMI to prove that every nonempty, upper bounded subset of the set of natural numbers has a biggest element.
part 1 Definition of relation; the identity relation.
part 2 Inverse of a relation.
part 3 Composition of relations.
part 4 Some general theorems involving relations.
part 1 Definition of equivalence relation; a few examples.Chapter 4. Functions
part 2 Example of an equivalence relation.
part 3 Equivalence classes of an equivalence relation.
part 4 Example of equivalence classes - construction of the integers from the natural numbers (intro).
part 5 The equivalence classes of modular arithmetic.
Section 3.3 Partitions and the Equivalence Class Theorem
part 1 Definition of partition of a set and some examples.
part 2 The Equivalence Class Theorem: statement and proof.
part 3 Example of the Equivalence Class Theorem.
part 4 Application of the Equivalence Class Theorem: Completion of the construction of the integers from the natural numbers.
Section 3.4 Ordering Relations
part 1 Definition of partial orders and linear orders, and a few examples.
part 2 Definition of least upper bound (supremum) and greatest lower bound (infimum), and a few examples.
part 3 Statement of the Completeness Property of the set of real numbers.
part 4 An application of the Completeness Property of the set of real numbers: proof of the existence of the square root of 2.
part 1 Functions: Basic definitions.Chapter 5. Cardinality
part 2 Some special types of functions.
Section 4.2 Construction of Functions
part 1 Inverse of functions.
part 2 Composition of functions, and the associative law of composition for functions.
part 3 Inverse functions and composition.
part 4 Restriction and extension of functions.
Section 4.3 and 4.4 Onto Functions; One-to-One Functions; One-to-One Correspondences and Inverse Functions
part 1 Definition of onto (surjective) functions and some examples.
part 2 An example involving surjectivity.
part 3 Another example involving surjectivity.
part 4 Yet another example involving surjectivity.
part 5 Definition of one-to-one (injective) functions and some examples.
part 6 An example involving injectivity.
part 7 Bijections: working definition of bijection and a theorem involving injections, surjections, and composition of functions.
part 8 Bijections: A theorem about bijections and inverse.
Section 4.5 Images and Inverse Images of Sets
part 1 Working definitions of image and inverse image of sets, and a simple example..
part 2 An example involving the image of a set.
part 3 An example involving the inverse image of a set..
part 4 A general theorem concerning inverse images, unions of sets, and intersections of sets.
part 5 A general theorem concerning images, unions of sets, and intersections of sets.