David  Singman
George Mason University
Department of Mathematics
Professor Emeritus 

Published papers and posted videos

1. Published papers

34) The distribution of radial eigenvalues of the Euclidean Laplacian on homogeneous isotropic trees, with Joel M. Cohen, and Flavia Colonna. Complex Anal. Synerg.7 (2021), no. 2, Paper No. 21, 25 pp.
pdf file  https://rdcu.be/cDD4j (Sharedit link to the article)

33) Carleson measures for non-negative subharmonic functions on homogeneous trees, with Joel M. Cohen, Flavia Colonna, and Massimo A. Picardello,
Potential Analysis, 52 (2020), no.1, 41-67.
pdf file  https://rdcu.be/cDH9Z (Sharedit link to the article)

32) A global right inverse of the Laplace operator on trees with a Green function, with Joel M. Cohen and Flavia Colonna,
The first NEAM, 39-46,Theta Ser. Adv. Math., 22, Editura Fundaţiei Theta, Bucharest, 2018.
pdf file

31) Fractal functions with no radial limits in Bergman spaces on trees, with Joel M. Cohen, Flavia Colonna, and Massimo A. Picardello, Hokkaido Mathematical Journal, 47 (2018), 269-289.
pdf file

30) Bergman spaces and Carleson measures on homogeneous isotropic trees, with Joel M. Cohen, Flavia Colonna,  and Massimo A. Picardello,  Potential Analysis, 44 (2016), no. 4, 745-766.
pdf file   https://rdcu.be/cDIcI (Sharedit link to the article)

29)  Carleson and vanishing Carleson measures on radial trees, with Joel Cohen and Flavia Colonna,
Mediterranean Journal of Mathematics, 3 (2013) no. 10, 1233-1256.
pdf file  https://rdcu.be/cDIdj (Sharedit link to the article)

28)  Carleson measures on a homogeneous tree,
with Joel Cohen and Flavia Colonna, Journal of Mathematical Analysis and Applications, 395 (2012), no. 1, 403-412.
pdf file

27)  Potential theory on trees and multiplication operators,  Complex Analysis and Potential Theory, CRM Proceedings and Lecture Notes, American Mathematical Society, Providence, RI, 
55 (2012) , 255-281.
pdf file

26) Norm of the multiplication operators from H infinity  to the Bloch space of a bounded symmetric domain, with Flavia Colonna and Glenn Easley,  Journal of Mathematical Analysis and Applications, 382 (2011)  621-630.

 pdf file

25) Biharmonic extensions on trees without positive potentials, with Ibtesam Bajunaid, Joel Cohen and Flavia Colonna,   Journal of Mathematical Analysis and Applications
, 378 (2011) 710-722.
 pdf file

24) Biharmonic Green functions on homogeneous trees, with Joel Cohen and Flavia Colonna, Mediterranean Journal of Mathematics,
6  (2009), no. 4, 249 - 271.
 pdf file  https://rdcu.be/cDIeC (Sharedit link to the article)

23) Classification of harmonic structures on graphs, with Ibtesam Bajunaid, Joel Cohen, Flavia Colonna, Advances in Applied Mathematics 43 (2009), no. 2,  113-136
.
 pdf file

22) A Riesz decomposition theorem on harmonic spaces without positive potentials, with Ibtesam Bajunaid, Joel Cohen, Flavia Colonna, Hiroshima Math. J.
, 38 (2008), no. 1,  37-50.
 pdf file

21) A global Riesz decomposition theorem on trees without positive potentials, with Joel Cohen, Flavia Colonna,  J.  London Math. Soc. (2), 75  (2007),  no. 1, 1-17.
 pdf file
       Corrigendum to "A global Riesz decomposition theorem on trees without positive potentials",   J.  London Math. Soc. (2), 83  (2011),  no. 3, 810.
corrigendum

20) Function series, Catalan numbers and random walks on trees,  with
Ibtesam Bajunaid, Joel Cohen and Flavia Colonna The American Mathematical Monthly, 112  (2005), 765-785.
This paper was awarded a Lester R. Ford award by the Mathematical Association of America.
pdf file

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.
 pdf file
      Corrigendum to "Trees as Brelot spaces",   Advances in Applied Mathematics, 47 (2011), no 2, 401-402.
pdf file

16) Tangential limits of potentials on homogeneous trees, with K. GowriSankaran, Potential analysis, 18 (2003), no 1, 79-96.
 pdf file  https://rdcu.be/cDIfy  (Sharedit link to the article)

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

13) Minimal fine  limits for a class of potentials, with K. GowriSankaran, Potential Analysis, 13 (2000), no. 2, 103-114.
 pdf file  https://rdcu.be/cDIfD (Sharedit link to the article)

12) Thin sets and boundary behavior of solutions of the Helmholtz equation, with K. GowriSankaran, Potential Analysis, 9 (1998), no. 4, 383-398.
 pdf file  https://rdcu.be/cDIgi (Sharedit link to the article)

11) A generalized Littlewood theorem for Weinstein potentials on a halfspace, with K. GowriSankaran, Illinois J. Math41 (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.
 pdf file

9)  Lower derivatives of functions of finite variation and generalized BCH sets,  J. Math. Ann. and App.173 (1993), no. 2, 483-496.
 pdf 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.
pdf file

5) A converse Fatou theorem, with B. A. Mair and Stanton Philipp, Michigan Math. J.,36 (1989), no. 1, 3-9.
pdf 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.
 pdf 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

Chapter 1. Logic and Proofs
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).
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.
Chapter 2. Set Theory  
               
        Section 2.1  Basic Concepts of Set Theory

part 1    Basic set theory.
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.
Chapter 3. Relations and Partitions  
               
        Section 3.1  Cartesian Products and Relations

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.

        Section 3.2  Equivalence Relations
part 1    Definition of equivalence relation; a few examples.
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.
Chapter 4. Functions  
               
        Section 4.1  Functions as Relations

part 1    Functions: Basic definitions.
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.

Chapter 5.  Cardinality
               
        Section 5.1(a)  Equivalent Sets
(This is the first series of lectures on Section 5.1)
        part 1
    Fundamental definition that two sets have the same cardinality.
        part 2    A few examples involving cardinality of sets.
        part 3    Another example involving cardinality of sets.
        part 4    Cardinality as an equivalence relation.
        part 5
   Some general results involving cardinality (valid for all sets).

        Section 5.1(b)  Finite Sets
(This is the second series of lectures on Section 5.1)
        part 1
    Definition of finite sets and some theorems.
        part 2    Theorems converning finite sets: unions, subsets, differences, Cartesian products.

        Section 5.2-5.5  Infinite Sets
        part 1
    Definition of  infinite, denumerable, countable, and uncountable  sets, and a few simple examples of denumerable sets.
        part 2
   A surprising result- The Cartesian product of denumerable sets is denumerable.
        part 3
   Proof that the set of integers is denumerable.
        part 4
   Proof that any infinite subset of  the natural numbers is denumerable.
        part 5
   Any infinite subset of a denumerable set is denumerable.
        part 6
   The denumerable union of denumerable sets is denumerable.
        part 7
   The set of rational numbers is denumerable.
        part 8
   Cantor's diagonlization argument that (0,1) is uncountable.
        part 9
  The Cantor-Schroder-Bernstein Theorem and a few examples.
        part 10   R x R and R have the same cardinality.
        part 11   The comparison of cardinalities: what does it mean to say a set has strictly greater cardinality than another set?
        part 12   Cantor's Theorem: Given any nonempty set, there exists a set of strictly bigger cardinality.
        part 13   Proof of the Cantor-Schroder-Bernstein Theorem.
        part 14
  Algebraic and transcendental numbers.
        part 15
  The Axiom of Choice and Zorn's Lemma.
        part 16
  Proof of the Comparability Theorem :  Application of Zorn's Lemma.

          Section 5.1(c)  Proofs of Theorems Concerning Finite sets. With these lectures I fill in the proofs of results concerning finite sets stated in Section 5.1(b).
        part 1
  Proof of the Pigeonhole Principle.
        part 2   Definition of Finite Set: The main result of this lecture together with the Pigeonhole Principle allow us to define precisely what it means to say that a set is finite.
        part 3
  Cardinality of the disjoint union of finitely many finite sets. The results in this lecture play a key role in the remaining lectures.  
        part 4
  Cardinality of subsets of finite sets.
        part 5
  Cardinality of the difference of finite sets.
        part 6   Cardinality of the union of two finite sets.
        part 7   Cardinality of the Cartesian product of two finite sets.