For any r, s ∈ N, show how to order the numbers 1, 2, . ....

For any r, s ∈ N, show how to order the numbers 1, 2, . . . , rs so that the resulting sequence has no increasing subsequence of length > r and no decreasing subsequence of length > s.

Expert Solution

Colby Messinger answered 1 year ago

For all n > 2 except n = 6, show how to arrange the numbers 1,2,...,n2...

For all n > 2 except n = 6, show how to arrange the numbers 1,2,...,n2 in an n x n array so that each row and column sum to the same constant.

R= Ro(1/2)n n= number of half lifetimes= t/t1/2 (a)n= 2/2= 1 R= Ro(1/2)n R=3000(1/2)1 R= 1500...

R= Ro(1/2)n n= number of half lifetimes= t/t1/2 (a)n= 2/2= 1 R= Ro(1/2)n R=3000(1/2)1 R= 1500 counts/sec (b)n= 6/2= 3 R= Ro(1/2)n R=3000(1/2)3 R= 375 counts/sec (c) n= 10/2= 5 R= Ro(1/2)n R=3000(1/2)5 R= 93.75 counts/sec (d) n= 20/2= 10 R= Ro(1/2)n R=3000(1/2)10 R= 2.93 counts/sec What is the mean life of this nucleus? f. Suppose that the Geiger counter detects 10% of all the radioactive decays. What is the total number of radioactive nuclei at time t = 0?...

(a) Show that the lines r 1 (t) = (2,1,−4) + t(−1,1,−1) and r 2 (s)...

(a) Show that the lines r 1 (t) = (2,1,−4) + t(−1,1,−1) and r 2 (s) = (1,0,0) + s(0,1,−2) are skew. (b) The two lines in (a) lie in parallel planes. Find equations for these two planes. Express your answer in the form ax+by+cz +d = 0. [Hint: The two planes will share a normal vector n. How would one find n?] would one find n?]

Show that {t_(1,s) : 2 ≤ s ≤ n} is a minimal generating set for S_n....

Show that {t_(1,s) : 2 ≤ s ≤ n} is a minimal generating set for S_n. You may use the fact that {t_(r,s) : 1 ≤ r < s ≤ n}, as defined in the outline, generates S_n.

Show that any graph with n vertices and δ(G) ≥ n/2 + 1 has a triangle.

On the set S of all real numbers, define a relation R = {(a, b):a ≤ b}. Show that R is transitive.

let G = D2n = {e,r,r^2,...,rn-1,s,sr,sr2,..,srn-1} a diedergroup of order 2n, where n >=3 (a) prove...

let G = D2n = {e,r,r^2,...,rn-1,s,sr,sr2,..,srn-1} a diedergroup of order 2n, where n >=3 (a) prove that [G,G] = <r2> (b) prove that G/[G,G] consists of two elements if n is uneven and 4 elements if n is even

Show by induction that for all n natural numbers 0+1+4+9+16+...+ n^2 = n(n+1)(2n+1)/6.

Let k ≥ 2. Use that R (the real numbers) is complete to show R^k is...

Let k ≥ 2. Use that R (the real numbers) is complete to show R^k is complete.

1. Let α < β be real numbers and N ∈ N. (a). Show that if...

1. Let α < β be real numbers and N ∈ N. (a). Show that if β − α > N then there are at least N distinct integers strictly between β and α. (b). Show that if β > α are real numbers then there is a rational number q ∈ Q such β > q > α. *********************************************************************************************** 2. Let x, y, z be real numbers.The absolute value of x is defined by |x|= x, if x ≥...

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