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
In: Advanced Math
prove the intermediate value theoerom using
a) Nested interval property
b) Axiom of completeness
In: Advanced Math
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
In: Advanced Math
Solve by separation variables (2x-5y-2)dx+(5x-y-5)dy=0
In: Advanced Math
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') =
In: Advanced Math
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).
In: Advanced Math
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.
In: Advanced Math
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)
In: Advanced Math
In: Advanced Math
solve the differential equation
a) ? ′′ − ? ′ = 1 1 + ? −?
b) ? ′′ + 3? ′ + 2? = cos(?x )
c) 10) ? ′′ − 2? ′ + ? = exsqrtx
In: Advanced Math
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.
In: Advanced Math
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
In: Advanced Math
In: Advanced Math
Derive the Catmull-Rom Spline blending function in your own words step by step.
In: Advanced Math
A company manufactures Products A, B, and C. Each product is processed in three departments: I, II, and III. The total available labor-hours per week for Departments I, II, and III are 900, 1080, and 840, respectively. The time requirements (in hours per unit) and profit per unit for each product are as follows. (For example, to make 1 unit of product A requires 2 hours of work from Dept. I, 3 hours of work from Dept. II, and 2 hours of work from Dept. III.)
Product A | Product B | Product C | |
---|---|---|---|
Dept. I | 2 | 1 | 2 |
Dept. II | 3 | 1 | 2 |
Dept. III | 2 | 2 | 1 |
Profit | $18 | $12 | $15 |
How many units of each product should the company produce in order to maximize its profit?
Product A | units |
Product B | units |
Product C | units |
What is the largest profit the company can realize?
$
Are there any resources left over? (If so, enter the amount
remaining. If not, enter 0.)
labor in Dept. I | labor-hours |
labor in Dept. II | labor-hours |
labor in Dept. III | labor-hours |
In: Advanced Math