Consider a 5x5 chessboard. Prove that no matter how the 25 cells are colored in red and blue (each cell is either red or blue), there exist 4 cells of the same color whose centers determine a rectangle with sides parallel to the sides of the board. Is the statement true for a 4x4 chessboard? What about 4x6 chessboard?

Use the Frobenius method to solve:xy"+xy^'+3y=0. Find index r and recurrence formulas. Compute the first 5 terms using the recurrence formula for each solution and index r.

The other answer on this website has the poorest handwriting, cant tell between r y x or n. Please make sure it is legible

Suppose that Ω ⊆ R n is bounded, and path-connected, and u ∈ C² (Ω) ∩ C(∂Ω) satisfies ( −∆u = 0 in Ω, u = g on ∂Ω. Prove that if g ∈ C(∂Ω) with g(x) = ( ≥ 0 for all x ∈ ∂Ω, > 0 for some x ∈ ∂Ω, then u(x) > 0 for all x ∈ Ω

prove the intermediate value theoerom using

a) Nested interval property

b) Axiom of completeness

1. An 8-inch diameter (I.D.) pipe is filled to a depth equal to one-third of its diameter. What is the area in flow?

2. Find the area of a washer formed by two concentric circles whose chord outside the small circle is 10 cm.

3. A goat is tied to a corner of a 30 ft by 35 ft building. If the rope is 40 ft. long and the goat can reach 1 ft. farther than the rope length, what is the maximum area of the goat can cover?

I need solution pleaseeee

Solve by separation variables (2x-5y-2)dx+(5x-y-5)dy=0

An experiment is given together with an event. Find the (modeled) probability of each event, assuming that the coins are distinguishable and fair, and that what is observed are the faces uppermost.

Three coins are tossed; the result is at most one tail.

An experiment is given together with an event. Find the (modeled) probability of each event, assuming that the dice are distinguishable and fair, and that what is observed are the numbers uppermost.

Two dice are rolled; the numbers add to 3.

An experiment is given together with an event. Find the (modeled) probability of each event, assuming that the dice are distinguishable and fair, and that what is observed are the numbers uppermost.

Two dice are rolled; the numbers add to 11.

An experiment is given together with an event. Find the (modeled) probability of each event, assuming that the dice are distinguishable and fair, and that what is observed are the numbers uppermost.

Two dice are rolled; the numbers add to 13.

An experiment is given together with an event. Find the (modeled) probability of each event, assuming that the dice are distinguishable and fair, and that what is observed are the numbers uppermost.

Two dice are rolled; both numbers are prime. (A positive integer is prime if it is neither 1 nor a product of smaller integers.)

Use the given information to find the indicated probability.

P(A ∪ B) = .8, P(B) = .7, P(A ∩ B) = .4.

Find P(A).

Use the given information to find the indicated probability.

P(A) = .78.

Find P(A').

P(A') =

6. Let V be the vector space above. Consider the maps T : V → V And S : V → V

defined by T(a1,a2,a3,...) = (a2,a3,a4,...) and S(a1,a2,a3,...) = (0,a1,a2,...).

(a) [optional] Show that T and S are linear.
(b) Show that T is surjective but not injective.
(c) Show that S is injective but not surjective.

(d) Show that V = im(T) + ker(T) but im(T) ∩ ker(T) ̸= {0}.

(e) Show that im(S) ∩ ker(S) = {0} but V ̸= im(S) + ker(S).

4. Consider bit strings with length l and weight k (so strings of l 0’s and 1’s, including k 1’s). We know how to count the number of these for a fixed l and k. Now, we will count the number of strings for which the sum of the length and the weight is fixed. For example, let’s count all the bit strings for which l + k = 11.

(a) Find examples of these strings of different lengths. What is the longest string possible? What is the shortest?

(b) How many strings are there of each of these lengths. Use this to count the total number of strings (with sum 11).

(c) The other approach: Let n = l + p vary. How many strings have sum n = 1? How many have sum n = 2? And so on. Find and explain a recurrence relation for the sequence (an) which gives the number of strings with sum n.

(d) Describe what you have found above in terms of Pascal’s Triangle. What patter have you discovered?

I'd really appreciate the help on solving this problem as there were no similar example problems in the book to even help me start this problem. Thank you!

Also some background information on this problem to hopefully help someone at least start answering this problem. For this problem we previously went over sequences such as recursive and closed. Also if they were arithmetic or geometric. The last chapter did cover binomial coefficients if that has some prevalence here.

A researcher believes that female stray cats are more cautious when approaching people than male cats. He knows that the stray cat population is 50% male and 50% female. In order to test this, visits 15 different areas where stray cats live and records the gender of the first cat to approach him. He finds that 12 of the 15 cats that approach him are male. Using a sign test:

D) What is the critical value? (In number of female cats)

1. Explain how loss to follow-up could bias findings in a cohort study of physical activity in college and subsequent diagnosis of diabetes.
2. Explain how recall bias could affect a case-control study of type 2 diabetes diagnosed in adults age 45 to 64 years and physical activity during young adulthood (while age 18 to 24 years)
3. Explain how selection bias could affect a case-control study of type 2 diabetes diagnosed in adults age 45 to 64 years and physical activity during young adulthood. Assume cases are identified as adult diabetic patients attending a diabetes management seminar at a local hospital and controls are selected from non-diabetic patients admitted to the hospital during the week the seminar was offered. Further assume that cases and controls are matched on age and sex.

solve the differential equation

a) ? ′′ − ? ′ = 1 1 + ? −?

b) ? ′′ + 3? ′ + 2? = cos(?^x )

c) 10) ? ′′ − 2? ′ + ? = e^xsqrtx

Use the Cauchy Criterion to prove the Bolzano–Weierstrass Theorem, and find the point in the argument where the Archimedean Property is implictly required. This establishes the final link in the equivalence of the five characterizations of completeness discussed at the end of Section 2.6.

Consider the following subsets of the set of all students:

A = set of all science majors
B = set of all art majors
C = set of all math majors

D = set of all female students

Using set operations, describe each of the following sets in terms of A, B, C, and D:

a) set of all female physics majors

b) set of all students majoring in both science and art