Prove that for all k E N, k ose [coske -sink0j cose Lsinke cosko

Q: Suppose that j and k are positive integers such that j< k. Give a combinatorial proof of t identity:…

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Q: Show that the Stein operator for Poisson P(Y = k) = is Tf(r) = Af(k + 1) – kf(k).

A: Given information: The Stein operator for Poisson P(Y = k) = λk/k! is Tµf(x) = λf(k + 1) – kf(k).

Q: In the domain of Gaussian integers Z[i], the element 23 is * Irreducible Reducible Unit None of the…

A: (i) Every prime is irreducible in Z[i].

Q: Prove that det(At) = det(A) for any A ∈ M2× 2(F).

A: Let us consider a 2*2 matrix A whose entry belongs to F, which is given below

Q: 2. Show that A is Hermitian if and only if (|A|) ER for all |t).

A: The given question is related with functional analysis. We have to show that…

Q: shor w that To- {0} abelian 7oup. where p an is prime.

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Q: Prove that Z3 x Zz is isomorphic to Zg.

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Q: Find the value of k such that A is singular. 1 0 3 4 -1 0 2 4 k k =

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Q: (a) Compute the following residue: Res 0 z°(1+2+ 22)*

A: (a) To Compute the residue Resz=0ezz5(1+z+z2)   Note: As multiple questions are asked, so only first…

Q: 1. Prove that if A is infinite and a E A, then A – {a} is also infinite.

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Q: Prove that gcd(m + 1, n + 1)|mn - 1 for all m, n E Z \ {-1}.

A: Let, gcd(m+1, n+1)=x From the properties of GCD, we can say that, x divides m+1 and n+1 Now,…

Q: Prove that for all positive integers n.

A: To prove for all positive integer n ∫01(lnx)ndx=(-1)nn!

Q: 173. Let |G| = p" and assume that |Z(G)| = p"-1 for a prime number p. Prove that G is Abelian. %3D

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Q: 2. Let R = {(x, y) : x, y E R}. Prove or disprove that R is reflexive, symmetric and/or transitive…

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Q: For allne z, gcdin,nt)=l.

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Q: Let a and b belong to Sn. Prove that ßαß-1 and a are both even orboth odd.

A: Let, a and b belong to Sn.   Consider α is the product of a 2-cycles, and β is the product of b…

Q: 1. Prove that if A is infinite and a e A, then A – {a} is also - infinite.

A:

Q: In the domain of Gaussian integers Zli), the element 19 is * O Reducible O Unit Irreducible

A: Answer is irreducible. Proof is given below.

Q: Let x E Q'. Show that for any n EN, there exists Xn E Q such that |x- Xal <

A: Given x∈ℚ', that x is a fixed irrational number. We know that between any two real  number, there…

Q: Prove that for a E N, a always invertible in Z,2+3a+1·

A: We know that an element a∈ℤn is called invertible if there is an element b∈ℤn such that a+b≡0  mod…

Q: 32. Prove statement d of Theorem 3.9: If G is abelian, (xy)" = x"y" for all integers n. bns liD to…

A: I have proved using mathematical induction

Q: Find the missing parts of the Wronskian: x Inx ? W = inx - and 4 O x2Inx and x3Inx O 1 and Inx O 1…

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Q: Prove that 223 a E lo,1) く 22 + 1영 ( 1-계 3(1-x2)

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Q: Prove that for each n E N, (d) (n) e (d) dn

A: To prove, for each n∈ℕ, nφn=∑d|nμ2dφd. We know that the mobius function μn is defined as: μn=1if…

Q: Prove that F2 + F = F2n+2 where Fn is the nth Fibonacci number. %3D n+1

A: Given, Fn2+Fn+12=F2n+2 As we know that, F1=1,F2=1,F3=2,F4=2,F5=4,F6=5

Q: Prove that L(G) is Eulerian if G is Eulerian.

A: Let G be Eulerian. Therefore, there exists an Eulerian tour C in G. Let us take…

Q: 3. In each of the following examples, determine (with careful arguments) the bound- ary of A in X,…

A: Given  X = R  and  A = (-1,1]

Q: Let gilx) i for n z2 i define gncr) -gn1(g,(), = 2ー× Deter mine the Intervai Such that g2avu CA) is…

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Q: Prove that ( n2 + 1) / (n + 1) diverges to +∞

A: Let

Q: (b) There exists LE R such that L4 Size(O(n)).

A: Given below clear explanation

Q: Let H(x):=k for x=, (keN) and H(x):=0 elsewhere on [0,1] Show that H(x) 4ER[0,1].

A: In this question, we use the definition of Riemann integral to check the function H(x) is Riemann…

Q: Prove that there exist c1, c2 ER such that for all ne N C1 E1 sin (i)<c2 =D1

A: We will use the range of the sine function to determine the real constants c1 and c2

Q: Suppose that {A;};eI and {B;};ej are families of nonempty dasses with the same index dass I. Prove…

A: Given that Aii∈I and Bjj∈J are families of non empty classes with the same index class I.

Q: In the domain of Gaussian integers Z[i], the element 15 is 青 O Reducible O Unit O Irreducible O None…

A: Since 3∈ Z[i] and 5∈ Z[i] And 15 = 3 × 5 Hence 15 is reducible element in Z[i]

Q: p be an irreducible reprsentation of G

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Q: Show that cl(a) = {a} if and only if a ∈ Z(G).

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Q: Suppose  there exist transposition f_i , 1<=I<=t  such that

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Q: Prove that ||ư + B|| = ||T|| + ||B|| if and only if i = kủ. %3D

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Q: Prove that G and H IS sube GroP form CX H

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Q: Let z = x + yi, z2 = 1– 0.5i , and z, + z2 = 3 – 1.5i. Find In(z,) – In(z2) %3D A) In(25) ± 2niN D)…

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Q: What is Wronskian of e' and et? 1. -2 -2t O000

A: Given that  e^t and e^(-t)  To find the wronskian of e^t and e^-t

Q: Prove that where e, > (1,0)"

A: Prove that     conve1, e2, -e1, -e2=x∈ℝ2 : x1+x2≤1 where e1=1, 0T , e2=0, 1T.

Q: Use Euler-Maclaurin's formula to prove that E n² = "(n+ 1)(n +2) 6. %3D 1

A:

Q: Theorem 26. If n ∈ N then (1^n) = 1. Theorem 27. If x, n, m ∈ N then (x^m)^n = x^(mn) . Exercise…

A: Given 1n =1 where n belongs to N(natural numbers)   Prove .. 1×1×1×1.........×1 n Times.........(1)…

Q: Prove that f(n) = 1000n5 + 20000n2 + 32 is O(n6 ); Is this a tight upper bound? Why or why not?

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Q: Let fons =loag, Cln). Find t'ot at

A: Given, f(x)=logx(ln(x))

Q: Prove that : a"+ (a+ 1)° =1 +a(1+い) (mod a la+s) for all a, m,n E Z.

A:

Q: Find the value of k such that A is singular. 1 0 6 4 -1 0 6 2 k

A: Given. A is a 3X3 matrix. We need to find the value of k for which the matrix is singular. Singular…

Q: Prove that (aA + bB)t = aAt + bBt for any A , B E Mmxn(F) and any a,bϵ F.

A: Consider, aA+bBt=aAt+bBt    ∵A+Bt=At+Bt=aAt+bBt     ∵kAt=kAt for any scalar k.=aAt+bBt

Q: Prove that for any k > 0, nk e o(2v™).

A: The complete solutions are given below