4.89. Let a E Z. Prove that if 3 5a, then 3 a.

Q: 12. Define def M e-$dWg• s=0 For each t > 0, compute (a) E[M¡]. (b) E[M?]

A: Given  Mt=∫s=0te-s dWs      , where W is a Brownian motion.

Q: A. let w-e te Find W2

A: We can solve this using u/v rule

Q: Show tH In k CHint dbtain o formdar for Sn Hint dotau forunlar

A:

Q: Compute for the value of tanh (1 – i) Iny + 2.13i In 2 + 2.13i 2 In 5 - 4.25i 2 In v2 + 2.13i

A: To Compute: tanh-11-i Concept: 1. tanh-1z=12ln1+z-ln1-z2. lnz=lnr+iθ+2nπwhere,…

Q: Suppose that Y ~ Bin(n, p) with 0<p< 1, and that Z = aY where a is a non-zero constant. If E(Z) =…

A:

Q: Let n e N, s > 0, a > 1. Show that n! 1"e" dr sn+1 Inz dr 1 (a – 1)2

A:

Q: 2.2 If fe R(a) on [a, b] and if Saf da prove that a must be constant on [a, b]. = 0 for every f…

A:

Q: 30) Let n 2 3 and k e Zn. Prove that in Dn. p u = up"-k, %3D

A: Note: There are more than one definition for dihedral group. The question is proved using a common…

Q: If f(z, y) – e . In(9y) , then fz is given by: O - 9e In(9y) o - 27 e In(9y) O 9 e In(9y) O 27 e…

A:

Q: Let A = d e Assume that det (A) = -13, compute the following.

A: Properties of Determinants :If we given A , nxn matrix then i) The value of det(cA) = cn det(A)…

Q: 1. Compute P(8), and find the exact values of the six circular functions given that e =- 135°. %3D

A:

Q: What is the value of k such that P(z>k)=0.0630? Previous

A:

Q: 4. Work either (a) or (b), NOT BOTH (a) Let In = for n E N. Prove that nIn = 0. 0, n n%=D1

A: a. Given that    In=0,1n  for  n∈ℕ. And we have to prove that…

Q: .3 T Evaluate In In e²+j²4

A:

Q: Prove that 12) d SIAX 13)let y=x, find Ay/Ax

A: Derivative of function

Q: 1. Prove with the usual notations that: (a) EV = VE = A (b) A +V =- (c) A= SE Apply the suitable…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: 2. Let P: C(0, 1]C(0, 1] be given by Pp "- e. Compute dP(x). Be sure to justify your calculations.

A: Given: The given function is  P:C20,1→0,1 given by pφ=φ''-eφ

Q: -2 Let f(t,y) = y + 3t?e'. Then f satisfies a Lipschitz condition in y with L = 2, given that 1<t<2.…

A:

Q: 1 1 Prove that: L' a exp(a't) erfe( avt), where erfc is complementary a+v error function.

A: Formula Used:- L1t=πsLeat=1s-aLerf(at)=as(s+a)Leatf(t)=F(s-a)                   where…

Q: Suppose that C> 1. Prove that if t > Vm ln C+In C, then ()/F) >C.

A: We will use the following Strining's approximation :  n! ≈ √(2πn) e-n nn To prove the given…

Q: prove that var (XY) == 0} 03+ 2} · o3+ 13.03

A:

Q: Assume that |Xn] 0. (Hint: first, prove that P(|X|>c+1) = 0.)

A: Answer:

Q: eb Let A E M₁ (TR) be given If AAt = On HOME JATOT prove that A = On - Does this hold if A EMn (C)?

A:

Q: Differentiate the following with respect to x: b) 3*Inx

A:

Q: Show that for every e>0 ther e exists P suuch that U (P, f) < U (f) + e

A: We will ise definition of U(f) and Infimum.

Q: 28. (a) Prove that for x e R" and s < t, Ds(x) C D;(x).

A: The notation Dsx represent  an open ball of radius s, centered at x∈ℝn. Let x=x1, x2, …, xn,…

Q: ), P{|Zn – c| > ɛ} → 0 as n → o. Show that for any b on g, E[g(Z„)] → g(c) as n→o

A: *Answer:

Q: Show for Re(z) > 1 ex – 1 dr = T(z)5(2). et – 1

A:

Q: Suppose a E Z. Prove that for any n E N, 5|2"a implies 5|a.

A:

Q: S 1.Prove that L (coswt) = %3D s2+w2

A:

Q: 2. Let P:C0, 1] → C|0, 1] be given by Pp = " – e°. Compute dP(y). Be sure to justify your…

A: We have Pφ=φ''-eφ We need to find dPφ If we have  y=fx⇒dydx=f'x⇒dy=f'x dx Similarly, we find dPφ for…

Q: If m` E = 0, then E is measurable.

A:

Q: 3. ncnt) use PMI to prove that E

A:

Q: 5.) Which of the following illustrates 4D E F inscribed in O M?

A: To determine which  of the following illustrates ∠DEF  inscribed in ⊙M

Q: 1. Prove that L(t)=. 1 s2

A:

Q: 4. Show that In sin 6 d0 = – n 2 by applying the mean-value principle to In |1+ z| for |2| <r < 1,…

A: ∫0π2logsin(θ)dθ=12∫0πlogsinθdθ                         =12∫0π2logsin2θdθ…

Q: Let Xi. ---- Xnd Glle e) and Tet TX = Exi , a) show tha T) is unbiased estimator for WcO) =nG: E…

A:

Q: 1. Prove that L(t)=. s2

A:

Q: b. Let x, y,p E Z, where p is prime. Prove that x2 - p²y - p + 0. (Hint: you may use without proof…

A:

Q: Show that |Re(z)|+|Im(z)| < v2|z| for any z e C.

A: The solution is given as

Q: 9. Let S- {xeZ| 3r.seZ,x=9r+6s) and T={xeZ| 3x} b) Prove that T Cs. a) Prove that SCT. Hint for (b):…

A: Part(a) We have S=x∈Z/ ∃r,s∈ℤ, x=9r+6s and T=x∈ℤ 3/x  Now, let a=9r+6s be an element in S. Then…

Q: that H(m m). Show

A: Given function is: Hm=3m-2 We have to find the inverse function H-1m. Let Hm=y Then we get,…

Q: 10.) For n>Liestablish that the in teger nc an't 5) is of the form ok. or n21, establish the in nz1…

A: To establish given relation we will use first principle of mathematical induction. For this we will…

Q: 5. Let TE B(H). Show that (ker T)- = ran T.

A: Given: T∈BH and T† is an adjoint of an operator T. To show: ker T⊥=ranT†

Q: Let x, y,p E Z, where p is prime. Prove that x2 – p²y – p + 0. (Hint: you may use without proof that…

A: Solution :-

Q: 13. If w(x,y) = In(1+xy²), show that 2wxx + y³Wyx = 0. %3|

A: Given: w(x, y)=ln(1+xy2)  To show: 2wxx+y3wyx=0 Chain rule : ddx(f(g(x))=f'(g(x))g'(x) Quotient…

Q: Prove that e“ p.w. on R. n! n=0 M8

A:

Q: Suppose / (s) - (÷)*" valuate: Js(x) dr +C 3- In (5)

A: Given      f(x) = 1512-3x

Q: 2) Evaluate each of the following. Show your work. pIn(3) e2t +5 dt et a)

A: Given:        (a): ∫0ln(3)e2t+5etdt

Q: Prove that (A')' = A for each A E Mmxn(F).

A:

Family of Curves

A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.

XZ Plane

In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.  

Euclidean Geometry

Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.

Lines and Angles

In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.