10. Let C : N × N → N by, for all (x, y) EN x N, (x + y)² – 3z – y+2 C(t, y) = (i 1) +y = Ei 1) + y- 2 Show that C is a bijection. (HINT: Use statement (5.1), rather than state- ment (5.2). For (*, yı), (x2, Y2) € N², consider the casesI tyi =2 + y2 and

Q: 2. (i) Show that for a discrete dis.ribution B2> 1. (ii) Show that B2> B1

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Q: (c) Compute of using the Chain Rule. of дf дх дf ду дf dz. + дя дх дя ду дя дz дя (Use symbolic…

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A: Natural Cubic Spline is a piece-wise cubic polynomial that is twice continuously differentiable.…

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A: Here, one matrix function, A=1-223 is given Let the X=a1b1a2b2 So, X¯=AX=a1b1a2b2.1-223…

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A: Consider the given equation.

Q: 5)Let w-f(u, v), where u = X -y andv = a) Using the chain rule, express x-y in terms of x, y, of and…

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Q: Consider thế iñifial a. 2y"- 30y" + 150y' – 234y = 0, b. y"+ 6ay' + (9a² + 1)y = 0, y(0) = -3, y(0)…

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Q: (b) Let X - N4 (u, E) with 2] 4 2 -3 4 4 2 4 1 and Σ- -1 -3 4 - 2 3 4 1 -2 8. Let Y and Z be…

A: i) X ~ N4(μ,∑ ) E(Y|Z) = ? Y|Z ~ N2 with mean μ1+∑12∑22-1(Z-μ2) Here,  μ1 = 24μ2= -13 ∑12 = 4224∑22…

Q: 6.13 Show that t: = B,/[s//E=, (x;- %3D VI- r in (6.20). |

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Q: Q1. Compute xdx x'+10 using trapezoidal and Simpson's 1/3 rules with the number of points 3 and 5.

A: The given integral is, I=∫01xdxx3+10 Dividing the interval (0,1)  into 2 intervals (for 3 points)…

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Q: dz Let z-ay" – 4r, * = uv and y-v - 3u. Compute Ən (u,r)-(1,1) -

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Q: Prove that T2 is chaotic on 1.

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A: The objective of the question check for inner product.

Q: 1) Prove that Ym,n EN such thert m¢ nt = n<m using the princdple of in duction!

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Q: Prove or disprove that 0 ≤ x2 - xy + y2 for all x and y in Z.

A: Consider the given inequality 0≤x2-xy+y2  for all x and y in  ℤ Now,…

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Q: The Wronskian of y,(t)=e 2, y,()=e" is-e St. Select one: True False

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Q: 3. Use Green's Theorem to evaluate G³ï+xj)•dr around the perimeter of the rectangle 0<x<2,0<y<3.…

A: Given ∫C(y2i+xj).dr We know that dr = dxi+dyj Thus ∫C(y2i+xj).dr = ∫C(y2i+xj).(dxi+dyj)  =…

Q: Prove that F(z) = 81 8z + 1 is chaotic on (-1,1).

A: To prove: Fx=8x4-8x2+1 is chaotic on -1,1. First plot graph of given function with -1,1 as follows:

Q: 2. Solve the following Cauchy problem: xyu, + yu, + yu = y – x with 1 u = -- y = x². on

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Q: @ Let A -a Find Cenditiens on a,b.c and d that AX =o has infailely many Solution. what must there…

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Q: 0, 2 is defined by A natural cubic spline S on 5. So()12x - 0 if 1 (1) S(r)2b(x1)c(x - 1)2d(x - 1)…

A: 5.Note that, the natural cubic spline must satisfy the conditions

Q: that y? etcos 2t and Gecos 2t knowing are solühions of. yo _uys +luy'- 20y'+ 25y=0 what is the…

A: For the solution follow the next step.

Q: Compute the Wronskian, W[y1, y2](t).

A: Let y1(t) and y2(t) be two differentiable functions in the independent variable t. Then the…

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Q: 1) Aşağıdaki u(x, y) fonksiyonu harmonik midir? Neden? Cevap olumlu ise f(z) = u(x, y) + iv(x, y)…

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Q: Show that Z, dv=2010 %3D if B= 2) : -1 Śn s3, ISySY,e

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Q: Q1. Compute xdx x' +10 using trapezoidal and Simpson's 1/3 rules with the number of points 3 and 5.

A: we have  I=∫01x dxx3+10dx  n=3h=b-an=1-03=13…

Q: 1. If x, y, z E R, prove that (a) \x+y\ < \x| + \yl- (b) |x– y| < |x – z|+|z– y|, and (c) ||지| -…

A: Since you have posted a question with multiple sub-parts, we will solve the first three subparts for…

Q: If u=u(y-x/xy,z-x/xz) prove tha x^delu/delx+y^delu/dely+z^delu/delz=0

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Q: 22. (a) Prove that [0, x] = [0, y] for all x, y E Z, x + 0, y # 0. (b) Prove that [ax, ay] = [x, y]…

A: (a) we have to prove that  0, x = 0, y for all x, y∈Z, x≠ 0, y≠0 proof as x, y∈Z, x≠ 0, y≠0 Z is the…

Q: - Let 21, 22, 23 be solutions of tx"" + 2x" + ex = 0 for t > 0 Note: W stands for the Wronskian of…

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A: Given that, X~U0,1 Yn=n-1nX

Q: Let x, y, z ∈  ℕ. Suppose gcd(x, y) = 1. Prove that if x | yz, then x | z

A: In the given question we have to prove that if x|yz , then x|z and g.c.d(x,y)=1 and x,y,z∈ℕ. we will…

Q: 7. Find T, N, ar, an at t=n. r(1) = 4cos(1)i–4sin(r)j-3rk

A: Tt = r'(t)r'(t) and N(t) = T'(t)T'(t) aT = r'(t) .r''(t)r'(t) and aN = r'(t)×r''(t)r'(t)

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Q: 5. If 4=2xy į +3x°y -3pyzk is solenoidal at(1,1.1). Find p.

A: DivA=0

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Q: Show that Z/mnZ is isomphoric to (Z/mZ)×(Z/nZ) where gcd(m,n) = 1

A: Concept Used:( The First Isomorphism Theorem) If ϕ be the  ring homomorphism from G to G then…

Q: 4: a) Let X.X., X -N (Hi,6f). Let Y,, Y2 Ym-N (u2,63). .... d (1-a) % CIfor oi when ai. a known.

A: