Real Analysis Conception Question

1. Lemma: if P is a subset of Q, then L(f,p)<=L(f,Q) and U(f,P)>=U(f,Q)

why??? for P is a subset of Q, however, why the upper sum of U(f,P) is even bigger than or equal to U(f,Q) it doesn't make any sense. Please

draw the geometric description to help with clear hand written

2. Let's assume the partition={x0,x1...xn} of the interval [a,b]

what is the diference between m=inf{f(x):x is a element of [a,b]} and mk=inf{f(x):x is a element of [xk-1,xk]}

A force of 720 Newton stretches a spring 4 meters. A mass of 45 Kilograms is attached to the spring and is initially released from the equilibrium position with an upward velocity of 6 meters per second. Find an equation of the motion.

What is the number of ways to distribute 12 identical balls in 8 different urns, so that two of the urns will contain together at least 10 balls?

Problem 2. Suppose that R is a commutative ring, and that

R[X] := { (a_0, a_1, a_2, ...)^T | a_i is in R, a_i not equal to 0 for only finitely many i}

is the set of polynomials over R, where we have named one particular element

X := (0, 1, 0, 0, . . .)T .

Show that R[X] forms a commutative ring with a suitably-chosen addition and
multiplication on R[X]. This will involve specifying a “zero” element of R[X]
that is the identity element with respect to addition, and a “unit” element of
R[X] that is the identity element with respect to multiplication. For example,
the usual addition we use for vectors in Rn should extend nicely even though
the entries here are only in a ring and not a field (and there are infinitely many
of them).
Your operations should act like how polynomial addition and multiplication
normally act. That is: to each element p = (a_0, a_1, a_2, . . . , a_k, 0, 0, . . . ,)^T is in R[X],

we can associate a polynomial function

p hat = x mapped to a_0 + a_1*x + a_2*x^2 + · · · + a_k*x^k.

Show that the mapping p mapped to p hat is a ring homomorphism from R[X] to the
“ring of polynomial functions” (you don’t need to show that the latter is a ring)
which has the following two operations defined for two functions f, g : R implies R:

f + g := x mapped to f (x) + g (x)
fg := x mapped to f (x) g (x) .

Really all you want to do here is think of two polynomials like
p = (a_0, a_1, a_2)^T is equivalent to p hat = x mapped to a_0 + a_1*x + a_2*x^2
and
q = (b_0, b_1, b_2)^T is equivalent to q hat = x mapped to b_0 + b_1*x + b_2*x^2

and then figure out what the the coordinates of p·q should be in R[X] by looking
at the coefficients of the powers in p hat · q hat. When you have it right, (p · q) hat = p hat · q hat
should hold.

Problem 1. Suppose that R is a commutative ring with addition “+” and
multiplication “·”, and that I a subset of R is an ideal in R. In other words, suppose
that I is a subring of R such that

(x is in I and y is in R) implies x · y is in I.

Define the relation “~” on R by y ~ x if and only if y − x is in I, and assume for the moment
that it is an equivalence relation. Thus we can talk about the equivalence
classes [x] := {y is in R | y ~ x}. Define R/I to be the set of equivalence classes
R/I := {[x] | x is in R}.
First show that ~ is an equivalence relation so that this all makes sense, and
then prove that R/I forms a ring under a suitable addition and multiplication
operation inherited from R. It should have both an additive and a multiplicative
identity element. The resulting ring R/I is called a “quotient ring.”
Specifically, we will want [x] + [y] = [x + y] and [x] · [y] = [x · y]. Here I
have used the same symbols “+” and “·” to mean two different operations (one
operation in R and one in R/I), but unfortunately you’ll have to get used to
that—it’s standard practice to use the same two symbols in every ring.
In any case, showing that these operations satisfy the definition of a ring
comes down to showing that multiplication is well-defined, and you should need
to use the fact that i · r is in I whenever i is in I to do so. The problem is that [x] is a
single set that can be written multiple ways: if z is in [x], then from the definition
of an equivalence class we can write either [x] or [z] to indicate the same set.
You have to prove that your definition of addition and multiplication are the
same regardless of how you write [x] and [y].

If there are functional limit to the size of primes we can use, but how and why is there a functional limit? more detail on how primes are used in RSA.please type

Find an expression for the temperature u(x,t) in a rod of length π, if the diffusivity (k) is 1, the ends x= 0 and x=π are both thermally insulated and the initial temperature is given by

u(x,0) ={2x/π , 0 < x < π/2}

u(x,0) ={1 , π/2 < x < π}

Let XOR be the eXclusive OR connective 
   Prove that the associative law applies to XOR  I.e., prove that
    (P_1 XOR (P_2 XOR P_3)) LEQV ((P_1 XOR P_2) XOR P_3).
   - Work out a simple rule to determine exactly when
      (P_1 XOR P_2 XOR ... XOR P_n)
     is satisfied.
     Prove your rule by induction on n.
 * Prove that the associative law applies to <->.  I.e., prove that
    (P_1 <-> (P_2 <-> P_3)) LEQV ((P_1 <-> P_2) <-> P_3).
   - Work out a simple rule to determine exactly when
      (P_1 <-> P_2 <-> ... <-> P_n)
     is satisfied.
     Prove your rule by induction on n.

* Prove that the associative law does not apply to ->.  I.e., prove that
    (P_1 -> (P_2 -> P_3)) notLEQV ((P_1 -> P_2) -> P_3).
 * Right associating ...
   - Work out a simple rule to determine exactly when
      (P_1 -> (P_2 -> ( ... (P_{n-1} -> P_n)...)))
     is satisfied.
     Prove your rule by induction on n.
 * Left associating ...
   - Work out a simple rule to determine exactly when
      (((...(P_1 -> P_2) ...) -> P_{n-1}) -> P_n)
     is satisfied.
     Prove your rule by induction on n.

. An automobile company is ready to introduce a new line of hybrid cars through a national sales campaign. After test marketing the line in a carefully selected city, the marketing research department estimates that sales (in millions of Ghana Cedis) will increase at the monthly rate of S'(t) = 20t − 15e−0.15t 0 ≤ t ≤ 36 t months after the campaign has started. (a) What will be the total sales S(t) in t months after the beginning of the national campaign if we assume no sales at the beginning of the campaign? (b) What are the estimated total sales for the first 12 months of the campaign? (c) When will the estimated total sales reach 100 million Cedis?

Suppose that f(x)=x^n+a_(n-1) x^(n-1)+⋯+a_0∈Z[x]. If r is rational and x-r divides f(x), prove that r is an integer.

a. tan ^ -1(y/x) Show that the function u(x,y)define classical solution to the 2-dimentional Laplace equation Uxx+Uyy =0

b. e ^ -(x-2t)^2 Show that the function u(t,x) is a  solution to wave equation

What is the order and type of the ff PDE's.

1. (1+X^2)U_xxyy-2xy^3U_xyyy+(1+u^2)U_YY=0

2.(1+U^2)U_xx-2U^2U_xy+(1+u^2)U_YY=0

3.(1+X^2_Y)U_xx-2U_XUyUxy+(1+u^2_X)U_YY=0

The least common multiple of nonzero integers a and b is the smallest positive integer m such that a|m and b|m. It is denoted [a, b], or sometimes [a, b] for short. Prove the following:

1) If a|k and b|k, then [a, b]|k.

2) If gcd(a, b) = 1, then [a, b] =ab

3) If c >0,then [ca, cb] =c·[a, b].

4) If a >0 and b >0,then [a, b] =ab / gcd(a, b).