In a graph G, a vertex dominates itself and its neighbors. A subset S of vertices of G is a double dominating set if every vertex in G is dominated at least twice by the vertices of S. The minimum cardinality of a double dominating set of G is the double domination number. We determine sharp lower and upper bounds on the double domination number of a nontrivial tree, and characterize the trees attaining the bounds. As a consequence of these upper bounds, we are able to improve known bounds on the independent domination number of a tree.

The Sierpiński fractal or Sierpiński gasket ∈ is a familiar object studied by specialists in dynamical systems and probability. In this paper, we consider a graph Sn derived from the first n iterations of the process that leads to ∈, and study some of its properties, including its cycle structure, domination number and pebbling number. Various open questions are posed.

A paired-dominating set of a graph G is a dominating set of vertices whose induced subgraph has a perfect matching. The paired-domination number of G is the minimum cardinality of a paired-dominating set of G, and is obviously bounded below by the domination number of G. We give a constructive characterization of the trees with equal domination and paired-domination numbers.

A set S ⊆ V is a dominating set in a graph G = (V, E) if each vertex in V-S is adjacent to at least one vertex in S. The domination number γ(G) is the minimum cardinality of a dominating set in G. We find improved bounds for γ(G) + γ(G) for graphs which have a given minimum degree.

An in-depth look at real analysis and its applications, including an introduction to wavelet analysis, a popular topic in "applied real analysis". This text makes a very natural connection between the classic pure analysis and the applied topics, including measure theory, Lebesgue Integral, harmonic analysis and wavelet theory with many associated applications. *The text is relatively elementary at the start, but the level of difficulty steadily increases *The book contains many clear, detailed examples, case studies and exercises *Many real world applications relating to measure theory and pure analysis *Introduction to wavelet analysis.

Over a decade ago Major and colleagues obtained analytical solutions for arbitrarily branching cables (Major, 1993; Major et al., 1993a, 1993b; Major and Evans, 1994). The importance of arbitrarily branching cable models is that the full dendritic arbor can be modeled rather than a collapsed version of the tree, the latter of which is known as an equivalent model. These solutions for transients in arbitrarily branching passive dendrites represented the most advanced breakthrough obtained in the history of neuron modeling, but they did not incorporate two important physical characteristics.

An arc in the plane is convex if it is simple (i.e., one-one except that its endpoints may coincide) and lies on the boundary of its convex hull. We describe a compact convex plane set of area about 0.2466 that contains a congruent copy of each convex plane arc of unit length, a reduction of about 1.1% from the smallest such set previously known.

A set S of vertices in a graph G is a total dominating set of G if every vertex of G is adjacent to some vertex in S. The total domination number γt(G) is the minimum cardinality of a total dominating set of G. A vertex that is contained in some minimum total dominating set of a graph G is a good vertex, otherwise it is a bad vertex. We determine for which triples (x, y, z) there exists a connected graph G with γt(G) = x and with y good vertices and z bad vertices, and we give graphs realizing these triples.

Testing statistical hypotheses introduces new vocabulary, concepts and a way of thinking that some students might initially find difficult. We provide a simple case that can be used in class as a gentle introduction to the ideas and procedures of hypothesis testing.

A set S of vertices of a graph G is a total dominating set if every vertex of V(G) is adjacent to some vertex in S. The total domination number γt(G) is the minimum cardinality of a total dominating set of G. Let G be a spanning subgraph of Ks,s and let H be the complement of G relative to Ks,s; that is, Ks,s = G ⊕ H is a factorization of Ks,s. The graph G is kt-critical relative to Ks,s if γt(G) = k and γ t(G + e) < k for all e ∈ E(H). We study k t-critical graphs relative to Ks,s for small values of k. In particular, we characterize the 3t-critical and 4 t-critical graphs.