Let G be a group of order mn where gcd(m,n)=1

Let a and b be elements in G such that o(a)=m and 0(b)=n

Prove that G is cyclic if and only if ab=ba

4. Gradient descent. Gradient descent is one of the most popular algorithms in data science and by far the most common way to optimise neural networks. A function is minimised by iteratively moving a little bit in the direction of negative gradient. For the two-dimensional case, the step of iteration is given by the formula xn+1 , yn+1 = xn, yn − ε ∇f(xn, yn). In general, ε does not have to be a constant, but in this question, for demonstrative purposes, we set ε = 0.1. Let f(x, y) = 3.5x 2 − 4xy + 6.5y 2 and x0 and y0 be any real numbers. (a) For all x, y ∈ R compute ∇f(x, y) and find a matrix A such that [3] A x y = x y − ε ∇f(x, y). Write an expression for xn yn in terms of x0 and y0 and powers of A. (b) Find the eigenvalues of A. [1] (c) Find one eigenvector corresponding to each eigenvalue. [2] (d) Find matrices P and D such that D is diagonal and A = P DP −1 . [1] (e) Find matrices Dn , P −1 and An . Find formulas for xn and yn. [4] (f) Suppose x0 = y0 = 1. Find the smallest N ∈ N such that xN yN ≤ 0.05. [3] (g) Sketch the region R consisting of those (x0, y0) such that xN ≥ 0, yN ≥ 0 and [4] xN yN ≤ 0.05, xN−1 yN−1 > 0.05, where N is the number found in part (f). Write an equation for the boundary of R. Which points of the boundary belongs to R and which do not?

1) When a mass of 3 kilograms is attached to a spring whose constant is 48 N/m, it comes to rest in the equilibrium position. Starting at

t = 0, a force equal to f(t) = 180e^−4t cos(4t) is applied to the system. Find the equation of motion in the absence of damping.

x(t) =

2) Solve the given initial-value problem. d^(2)x/dt^2 + 9x = 5 sin(3t), x(0) = 6,  x'(0) = 0

x(t) =

Use Stokes' Theorem to find the circulation of F⃗ =2yi⃗ +2zj⃗ +4xk⃗  around the triangle obtained by tracing out the path (3,0,0) to (3,0,4), to (3,4,4) back to (3,0,0).

Circulation = ∫CF⃗ ⋅dr⃗  =

Second time I've asked this question because chegg cant solve this problem correct. 24sqrt(2) is the wrong answer

Use the divergence theorem to calculate the flux of the vector field F⃗ (x,y,z)=−5xyi⃗ +2yzj⃗ +4xzk⃗  through the sphere S of radius 2 centered at the origin and oriented outward.
∬SF⃗ ⋅dA⃗ =

For the differential equation dy/dx=ye^x check all that apply

A. PDE
B. ODE
C. homogeneous
D. linear
E. constant coefficients
F. autonomous
G. system

this is a differential equation problem. i was thinking it was A,C,F but thats not correct. any suggestions would help.

1) An observer fails to check the bubble of a level and it is off two divisions on a 100-m sight. If the sensitivity of the bubble is 20-sec, what error would there be in the reading? 2 ) The following azimuths are from the north: 329°20’, 180°35’, 48°32’, 170°30’, 145°25’, 319°35’, 350°45’, 95°49’, 11°30’, 235°45’. Express these directions as (a) azimuth from the south, (b) back azimuths, (c) bearings. 8) 6.7 The following azimuths are reckoned from the north: FE = 4°25’, ED = 90°15’, DC = 271°32’, CB = 320°21’, and BA = 190°45’. What are the corresponding bearings? What are the deflections angles between consecutive lines? 9) 6.8 The interior angles of a five-sided closed polygon ABCDE are as follows: A, 120°24’; B, 80°15’; C, 132°24’; D, 142°20’. The angle E is not measured. Compute the angle at E, assuming the given values to be correct.’ 10) 6.11 In an old survey made when the declination was 4°15’E, the magnetic bearing of a given line was N35°15’E. The declination in same locality is now 1°10’W. What are the true bearing and the present magnetic bearing that would be used in retracing the line? 11) The following bearings taken on a closed compass and fill up the table. Lines Bearings (deg-min) Azimuth N Deflection Angles Angle to the Right Interior Angles AB S 37-30 E ? 83°45’ 163°45’ ? BC S 43-30 W ? ? ? ? CD N 73-30 W ? ? ? ? DE N 11-45 E ? ? ? ? EF ? ? ? ? ? 12) ILLUSTRATE THE TRAVERSE AND SHOW THE SOLUTIONS. Transform graphically into a figure with sides containing unknown quantities are made adjoining. Determine the unknown quantities. Course Length Bearing AB 492.98 N 05°30’ E BC UNKNOWN S 12°17’ E CD 845.85 N 46°03’ E DE 852.18 S 67°24’ E EF 1210.50 UNKNOWN FA 661.26 N 55°27’ W AB BC CD DE EF FA 13) ILLUSTRATE THE TRAVERSE AND SHOW THE SOLUTIONS. Transform graphically into a figure with sides containing unknown quantities are made adjoining. Determine the unknown quantities. Course Length Bearing AB 249.18 S 19°32’ E BC UNKNOWN N 74°09’ E CD 445.10 S 36°40’ E DE 668.27 S 51°14’ W EF 866.79 N 73°25’ W FG UNKNOWN N 26°00’ W GA 560.15 N 64°32’ E 14) Given Below is the technical description of lot 2061, Cebu Cadastre. Line Length (m) Azimuth from South 1-2 22.04 S 32°17’ W 2-3 10.00 N 36°35’ W 3-4 5.00 N 15°47’ W 4-1 19.95 N 73°07’ E 1. Find the area of the lot by DMD method. 2. Find the area of the lot by DPD method. 15) Compute for the following and show the tabulated solutions: (Notes: Use 2 decimal places for all corrections and lengths, degrees and minutes for angles) Line Length (m) Azimuth from South AB 1020.87 168°35’ BC 1117.26 263°30’ CD 660.08 304°25’ DE 495.85 5°30’ EF 850.62 46°15’ FA 855.45 112°30’ 1. Using Compass Rule, solve for adjusted latitudes, departures, distances and bearings for all the traverse lines 2. Using Transit Rule, solve for adjusted latitudes, departures, distances and bearings for all the traverse lines 3. Using coordinate method and using the result of balancing traverse in # 2 (transit rule) with station A (N,E) coordinates = (20000, 20000), solve for the coordinates (N, E) of all traverse stations and the area of the traverse 4. Using DMD method and using the result of balancing traverse in # 2 (transit rule), solve for area of the traverse 5. Using DPD method and using the result of balancing traverse in # 2 (transit rule), solve for area of the traverse 16) At regular interval of 5 meters along line AB, the measured offset distances from the line AB to the edge of the stream are: 6.5, 7.3, 12.1, 12.7, 14.5, 16.0, 18.6, 18.5, 16.4, 15.9, 14.7, 13.5, 12.75, 12.5, 9.7 and 2.0. What is the area between line AB and the stream?

Does every linear transformation from a complex vector space to itself have an eigenvector?

Show that (x-2)² -lnx = 0, [1,2] and [e,4] have at least one solution in the given intervals.

Use the Fourier transform to find the solution of the following initial boundaryvalue Laplace equations

uxx + uyy = 0, −∞ < x < ∞ 0 < y < a,

u(x, 0) = f(x), u(x, a) = 0, −∞ < x < ∞

u(x, y) → 0 uniformlyiny as|x| → ∞.

Find the Fourier series of the following functions

f(x) = sinx+x^2+x,

−2 < x < 2

1.Find all rational zeros of the polynomial, and then find the irrational zeros, if any. Whenever appropriate, use the Rational Zeros Theorem, the Upper and Lower Bounds Theorem, Descartes' Rule of Signs, the Quadratic Formula, or other factoring techniques. (Enter your answers as comma-separated lists. Enter all answers including repetitions. If an answer does not exist, enter DNE.)

P(x) = 12x⁴ − 11x³ − 18x² + 5x

2.

Find all rational zeros of the polynomial. (Enter your answers as a comma-separated list. Enter all answers including repetitions.)

P(x) = 6x⁴ − 7x³ − 13x² + 4x + 4

x =

3.

Two polynomials P and D are given. Use either synthetic or long division to divide P(x) by D(x),

and express P in the form

P(x) = D(x) · Q(x) + R(x).

P(x) = 4x³ + 9x + 9,    D(x) = 2x + 1

P(x) =

4.

P(x) = 12x⁴ − 11x³ − 18x² + 5x

Among the tools (e.g., graphics tools, knowledge-based tools, etc.), determine the type of tool that you would use for process improvement framework. Next, determine the type of tool you would use for problem solving framework. Justify your response.

Suppose T ∈ L(V)

a) Show that V = ImT⁰ ⊃ ImT¹ ⊃ ImT² ⊃ ...

b) Show that if m ≥ 0 is such that ImT^m = ImT^{m + 1}, then ImT^{m + k} =ImT^{m + k + 1} for all k ≥ 0.

c) Show that if n = dim V, then ImTⁿ = ImT^{n + 1} = ImT^{n + 2} = ....

let f be analytic and not constant on a domain G. show that the area of f(G)is strictly positive

(complex variables)