Given the set A = {(x, y) ∈ R² | x² + y² < 1 and y ≥ 0}. Draw sketches of cl A, int A, ∂A, (cl(A^c))^c, the limit points of A, and the isolated points of A. Try to be clear about what the sketch is describing. (The answer does not depend on whether one uses the Euclidean distance or the taxi distance on R².)

find the integer solutions to A^2 + 2*B^2 = C^2?

Which of the following are groups? + And · denote the usual addition and multiplication of real numbers.

(G, +) with G = {2 n | n ∈ Z},
(G, ·) with G = {2 n | n ∈ Z}.
Determine all subgroups of the following cyclic group
G = {e, a, a2, a3, a4, a5}.
Which of these subgroups is a normal divisor of G?

Consider the vector space P2 of all polynomials of degree less than or equal to 2 i.e. P = p(x) = ax + bx + c | a,b,c €.R Determine whether each of the parts a) and b) defines a subspace in P2 ? Explain your answer. a) ( 10 pts. ) p(0) + p(1) = 1 b) ( 10 pts.) p(1) = − p(−1)

Use the Pohlig-Hellman algorithm to solve 19x ≡ 184 (mod 337) for x. Write out at least one successive squaring in detail, and at least one instance of the Chinese Remainder Theorem.

Find the indicated probabilities using the geometric​ distribution, the Poisson​ distribution, or the binomial distribution. Then determine if the events are unusual. If​ convenient, use the appropriate probability table or technology to find the probabilities.

The mean number of heart transplants performed per day in a country is about

eight

Find the probability that the number of heart transplants performed on any given day is​ (a) exactly

six

​(b) at least

seven

and​ (c) no more than

four

​(a)

​P(6​)=

​(Round to three decimal places as​ needed.)

National Survey of Family Growth 1982-2010

The National Survey of Family Growth (NSFG) provides nationally representative estimates and trends for infertility, surgical sterilization, and fertility among U.S. women and men aged 15-44. The NSFG survey has been administered since 1973, and its latest round in 2006-2010 consisted of 22,682 interviews. Infertility was defined as a lack of pregnancy in the 12 months prior to the survey despite having had unprotected sexual intercourse in each of those months with the same husband or partner. Women were classified as surgically sterile if they had an unreversed sterilizing operation, for example, a tubal sterilization or hysterectomy. Presumed fertile women were based on the residual category of those who did not meet the definitions for surgically sterile or infertile.

1. Using these data and your knowledge, make a testable hypothesis about some aspect of infertility, fertility, or surgical sterilization using the above mentioned data. Identify and count the cases of infertility depending upon age, education, percent of poverty level in the year 2006-2010. Also compare the groups like surgical sterile, infertile and presume fertile. Clearly defined exposure groups, a specific outcome and a stated direction between the exposure and outcome in the years mentioned.

Find the general solution of the following ODE: y''x^2 − 3xy' + 3y = 2/x , for x > 0.

True or False. If true, quote a relevant theorem or reason, or give a proof. If false, give a counterexample or other justification.

There is a one-to-one and onto map between the open interval (0, 1) and the open interval (4, 8).

Prove this betweenness proposition with justification for each step.

If C * A * B and l is the line through A, B, and C, then for every point P lying on l, P either lies on the ray AB or on the opposite ray AC.

Show that strong Markov property implies Markov property.

Expand the function f(z) = (z − 1) / z^ 2 (z + 1)(z − 3) as a Laurent series about the origin z = 0 in all annular regions whose boundaries are the circles containing the singularities of this function.

Let S = {-3, -2, -1, 0, 1, 2, 3}. Define a relation R on S by: xRy if and only if x = y + 4n for some integer n.

a) Prove that R is an equivalence relation.

b) Find all the distinct equivalence classes of R.

Let T: V →W be a linear transformation from V to W.

a) show that if T is injective and S is a linearly independent set of vectors in V, then T（S） is linearly independent.

b) Show that if T is surjective and S spans V,then T（S） spans W.

Please do clear handwriting!

What are the Aspects, Application, and Solutions for a Stochastic Differential Equation?