On R², consider the function f(x, y) = ( .5y, .5sinx). Show that f is a strict contraction on R². Is the Banach contraction principle applicable here? If so, how many fixed points are there? Can you guess the fixed point?

y"+y=-1 , y(0)=0 , y(Pi/2)=0
please repeat the answer by details (by properties of green function only)

and the answer will be : sin(x)+cos(x)-1

Hello i dont get this problem

determine the center of mass of the solid limited by the graphs of Y=x2, Y=x, Z=y+2 Z=0 if the density at point P is directly proportional to the distance from the XY plane

Write 10 - 15 applications of each.
( note: formulas and diagrams are not required just applications)
1- Numerical differentiations
2-Numerical integration
3- Composite Numerical integration.

As a spring is heated, its spring “constant” decreases in value. Suppose a spring is heated so that the spring “constant” at time t is k(t) = 6 − t. If the unforced mass-spring system has mass m = 2 kg and a damping constant b = 1 N-sec/m with initial conditions x(0) = 3 m and x' (0) = 0 m/sec, then the displacement x(t) is governed by the initial value problem 2x'' + x' − (6 − t)x = 0; x(0) = 3, x' (0) = 0.

Find the first four nonzero terms in a power series solution centered at t = 0.

Compare and explain “random error of a regression” with “residuals of regression.”

a) Suppose f:[a, b] → R is continuous on [a, b] and differentiable on (a, b) and f ' < -1 on (a, b). Prove that f is strictly decreasing on [a, b].
b) Suppose f:[a, b] → R is continuous on [a, b] and differentiable on (a, b) and
f ' ≠ -1 on (a, b).   Why must it be true that either f ' > -1 on all of (a, b) or f ' < -1 on all of (a, b)?

a) Suppose f:R → R is differentiable on R. Prove that if f ' is bounded on R then f is uniformly continuous on R.
b) Show that g(x) = (sin(x⁴))/(1 + x²) is uniformly continuous on R.
c) Show that the derivative g'(x) is not bounded on R.

Solve the differential equation. y'' − 8y' + 20y = te^t, y(0) = 0, y'(0) = 0

(Answer using fractions)

(i) There are two non-isomorphic groups of order 4: C4 and C2xC2. Let C3 be a cyclic group of order 3. For each group G of order 4, determine all possible homomorphisms f an element of Hom(C3, Aut(G)).

(ii) For C3 and each G of order 4 as above, determine all possible homomorphisms phi an element of Hom(G, Aut(C3)).

How do you use strong induction to show that the coefficient of x^2 in the expansion of (1+x+x^2+...x^n)^n is (1+2+...n)?

In linear regression , why can we see the lack of fit test as the comparison between the model we choose and the saturated model?

Let T be a linear operator on a finite-dimensional complex vector space V . Prove that T is diagonalizable if and only if for every λ ∈ C, we have N(T − λI_V ) = N((T − λI_V )²).