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1. Show that the tree-order associated with a rooted tree T is indeed a partial order on V(T), and verify the claims made about this partial order in the text.
Hint: Theorem 1.5.1.
2. Do the partition classes of a regular bipartite graph always have the same size?
Hint: Count the edges.
3. Show that a graph is bipartite if and only if every induced cycle has even length.
Hint: Show that if a graph contains any odd cycle at all it also contains an induced one.
4. Find a function f : N → N such that, for all k ∈ N, every graph of average degree at least f(k) has a bipartite subgraph of minimum degree at least k.
Hint: Given a graph G, how would you split its vertex set into two parts A and B so that the bipartite graph H defined by the A-B edges of G has minimum degree as large as possible? To find f, apply this method to a suitable subgraph G of a given graph G’, and determine how large d(G’) must be to ensure that δ(H) ≥ k.
Consider a flea of mass 4.5×10-4 at 1.0 m/s midway through its jump. What is its de Broglie wavelength?
What is a probability that 2 females and 1 male will be in a group? What is a probability of 3 females only?
John Doe borrowed $5,000 and agreed to repay the principal and interest at 15% compounded monthly in 25 equal monthly payments. The first payment is due in l½ years. How large is each payment?
Show how the application was accomplished citing hypotheses
State the null hypothesis and the alternative hypothesis that we would use for this test.
the population of a certain species of fish introduced into a lake is described by the logistic equation G(t)=10000/(1+24e^(-1.2t)). Where G(t) is the population after t years. Find the point at which the growth rate of this population begins to d..
If local mortgage rates are 7.5%/year compounded monthly for a conventional 30-yr mortgage, what is the price range of houses they should consider?
A steel company produces two types of machine dies, part A and part B. the company makes a $3.00 profit on each part A that it produces and a $6.00 profit on each part B that it produces.
there is a liquid motion due to a doublet of strength μ at the point (0,3α) with its axis along the Y axis.find the velocity potential
y" + (1/x2)y' - (1/x3)y = 0 a) Show that x = 0 is an irregular singular point b) Use the fact that y1 = x is a solution to find a second independent solution y2 by the method discussed earlier in the book. c) Show that the second solution y2 that we ..
Suppose the radius of convergence of ∑k≥0 ckzk is R. What is the radius of convergence of each of the following? (a) ∑ k≥0 ckz2k.
1. what constitutes a rational expression? how would you explain this concept to someone unfamiliar with it?2. how
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