Consider the problem of a sphere of material that starts at a non-uniform temperature, $T = r^{2}$ and is covered with insulation on the outer surface so that no heat gets out. We take the coordinate $r$ to measure position radially from the centre of the sphere with the outer surface given at $ r = 1.$ We take t as time and the variable $ T (r, t)$ as the temperature. The equation governing the heat flow, is the heat equation $$\frac{\delta T}{\delta t}=\frac{1}{r^2}\frac{\delta}{\delta r}(r^2\frac{\delta T}{\delta r}), 0\leq r\leq 1, t>0$$ We insist that T remains finite as $ r → 0$ and the outer surface boundary condition is $$\frac{\delta T}{\delta t}(1,t)=0, t>0$$ with the initial condition $$T(r,0) = r^2, 0 < r < 1.$$ (a) Find the solution T (t, r) using separation of variables. Note: you can leave the answer in a form where eigenvalues are given by the roots of an equation. By exploiting the orthogonality of the eigenfunctions you should give the integral formulas necessary to compute the coefficients in the solutions. You do not need to evaluate the integrals. (b) Find numerically, to five decimal places accuracy, the value of the smallest of the eigenvalues in (a), corresponding to the first non-constant term in the solution. This is what I have obtained so far: $$\frac{\delta T}{\delta t}=\frac{1}{r^2}\frac{\delta}{\delta r}(r^2\frac{\delta T}{\delta r})$$ $=\frac{\delta^2 T}{\delta r^2}+ \frac{2}{r} \frac{\delta T}{\delta r}$ Let $T=R(r)Q(t)$ $\frac{\delta T}{\delta t}= R(r)Q'(t)$ $\frac{\delta T}{\delta r}= R'(r)Q(t)$ $\frac{\delta^2 T}{\delta r^2}= R''(r)Q(t)$ Then putting back into the equation we get $$\frac{Q'(t)}{Q(t)}=\frac{R''(r)}{R(r)} + \frac{2}{r}\frac{R'(r)}{R(r)}= \mu$$ which gives the two equations: $$Q'(t)-\mu Q(t)=0$$ and $$rR''(r)+2R'(r)-\mu rR(r)=0$$ Then I have done Let $o(r)=rR(r)$ $o'(r)=R(r)+rR'(r)$ $o''(r)=2R'(r)+rR''(r)$ Putting this back into the equation we get $o''(r)-\mu o(r)=0$ Now how do I proceed? using the boundary conditions how do I implement this into the equation.

Here is the question: The 10 members of a school board are having a meeting around a circular table. 3 members, Anna, Bob, and Carl dislike each others ideas, and refuse to sit next to each other. In how many ways can the school board members gather around the table? Rotations are considered the same, but reflections are considered distinct. I decided to use complementary counting. I found that there are $30240$ ways for A, B, and C to sit together because of $7! \times 6$. Then I broke up the pairs AB, AC, and BC into three identical cases. AB: $8! \times 2 = 80640 - 30240 = 50400$ AC: $50400$ BC: $50400$ $50400 \times 3 + 30240 = 181440 \Longrightarrow 362880-181440 \Longrightarrow 181440$ However, $181440$ isn't listed the correct answer. It is $151200$. I don't understand this answer because I feel like that would be overcounting the complementary cases. Which answer is correct, and why? Thanks!

I understand what a vector can represent, but I still don't understand what a vector space represents. I understand that you can add two vectors and that becomes a vector space. What else can you do with them? What can you apply it to? I am taking a linear algebra course right now and understand the calculation steps, but not really what these tools are for. I hope that adds some context to my strange and probably vague question. Thanks!

The problem I was given: Calculate the value of the following determinant: $\left \begin{array}{ccc} \alpha & 1 & \alpha^2 & -\alpha\\ 1 & \alpha & 1 & 1\\ 1 & \alpha^2 & 2\alpha & 2\alpha\\ 1 & 1 & \alpha^2 & -\alpha \end{array} \right $ For which values of $\alpha \in \mathbb R$ is the following system of linear equations solvable? $\begin{array}{lcl} \alpha x_1 & + & x_2 & + & \alpha^2 x_3 & = & -\alpha\\ x_1 & + & \alpha x_2 & + & x_3 & = & 1\\ x_1 & + & \alpha^2 x_2 & + & 2\alpha x_3 & = & 2\alpha\\ x_1 & + & x_2 & + & \alpha^2 x_3 & = & -\alpha\\ \end{array}$ I got as far as finding the determinant, and then I got stuck. So I solve the determinant like this: $\left \begin{array}{ccc} \alpha & 1 & \alpha^2 & -\alpha\\ 1 & \alpha & 1 & 1\\ 1 & \alpha^2 & 2\alpha & 2\alpha\\ 1 & 1 & \alpha^2 & -\alpha \end{array} \right $ = $\left \begin{array}{ccc} \alpha - 1 & 0 & 0 & 0\\ 1 & \alpha & 1 & 1\\ 1 & \alpha^2 & 2\alpha & 2\alpha\\ 1 & 1 & \alpha^2 & -\alpha \end{array} \right $ = $(\alpha - 1)\left \begin{array}{ccc} \alpha & 1 & 1\\ \alpha^2 & 2\alpha & 2\alpha \\ 1 & \alpha^2 & -\alpha \end{array} \right $ = $(\alpha - 1)\left \begin{array}{ccc} \alpha & 1 & 0\\ \alpha^2 & 2\alpha & 0 \\ 1 & \alpha^2 & -\alpha - \alpha^2 \end{array} \right $ = $-\alpha^3(\alpha - 1) (1 + \alpha)$ However, now I haven't got a clue on solving the system of linear equations... It's got to do with the fact that the equations look like the determinant I calculated before, but I don't know how to connect those two. Thanks in advance for any help. (:

This question is in reference to Exercise 4.3 in the 'learning from data' book. Here is the question where H is the hypothesis set and f is the target function. Deterministic noise depends on H, as some models approximate f better than others. (a) Assume H is fixed and we increase the complexity of f. Will deterministic noise in general go up or down? Is there a higher or lower tendency to overfit? (b) Assume f is fixed and we decrease the complexity of H. Will deterministic noise in general go up or down? Is there a higher or lower tendency to overfit? [Hint: There is a race between two factors that affect overfitting in opposite ways, but one wins.] This is what I think: a) By making the target function more complex and keeping H fixed then deterministic noise will increase. Also, H will overfit less so than before since H is even more general on f than it previously was. b) By decreasing the complexity of H while keeping f fixed then again deterministic noise will increase and again H will overfit less than before since H is even more general on f that it previously was. The hint to me implies that there may be a more nuanced answer than what I have. Would you please let me know how to go about thinking about this? The reading was pretty informal with regards to this topic and in the AML book forum there aren't any posts covering this topic. Any insights are much appreciated.

As the health and safety adviser, you want to improve manual handling safety. To do this they need to proactively monitor the activities. Justify FOUR leading indicators that would help ‘Bricks to Homes’ achieve this? . *Nebosh diploma ID1

How many ways are there to distribute seven distinct apples and six distinct pears to three distinct people such that each person has at least one pear. Note: The first part of the problem was the same but the word distinct was replaced with identical apples and pears. That answer is $360$ ways. What has to be done differently when the word identical is replaced with distinct ?

Identify the correct steps involved in the proof of the statement "If f(x) and g(x) are functions from the set of real numbers to the set of real numbers, then f(x) is Θ(g(x)) if and only if there are positive constants k, C1, and C2 such that C1|g(x)| ≤ |f (x)| ≤ C2|g(x)| whenever x > k." (Check all that apply.) Check All That Apply Suppose $f(x)$ is $\Theta(g(x))$. Then, $f(x)$ is both $Q(g(x))$ and $\Omega(g(x))$. There exists positive constants $C_{1}, k_{1}, C_{2}$, and $k_{2}$ such that $|f(x)| \leq C_{2} \mid g\left(x||\right.$ for all $x>k_{2}$ and $|f(x)| \geq C_{1}|g(x)|$ for all $x>k_{1}$. Let $k=\max \left\{k_{1}, k_{2}\right\}$. Then, for $x>k, C_{1}|g(x)| \leq|f(x)| \leq C_{2}|g(x)|$. Let $k=\min \left\{k_{1}, k_{2}\right\}$. Then, for $x>k, C_{1}|g(x)| \leq|f(x)| \leq C_{2}|g(x)|$. Suppose there exists positive constants $C_{1}, C_{2}$, and $k$ such that $C_{1}|g(x)| \leq|f(x)| \leq C_{2}|g(x)|$ whenever $x>k$. Let $k=k_{1}=k_{2}$. Then, $|f(x)| \leq C_{2}|g(x)|$ for all $x>k_{2}$ and $|f(x)| \geq C_{1}|g(x)|$ for all $x>k_{1}$. Let $k=k_{1}=k_{2}$. Then, $|f(x)| \leq C_{2}|g(x)|$ for all $x>k$ and $|f(x)| \leq C_{1} \mid g(x||$ for all $x>k$.

Question 5: Debt payments of $\$ 700$ in two months and $\$ 800$ in five months are scheduled to be due. If interest at $8.3 \%$ p.a. Is allowed, what single payment today is required to settle the two scheduled payments? Answer 5. Question 5 (8 marks): An investor purchased a $\$ 100,000$ 182-day T-bill on its date of issue to yield $5.5 \%$. When he sold it 30 days later, yields had dropped to $5.0 \%$. a. How much did the investor earn? b. Calculate the rate of return actually realized by the investor.

Three clocks exist in a room: The round clock is 10 minutes behind the actual time The rectangular clock is 5 minutes after the actual time The squared clock is 5 minutes slower than the round clock The time now on the squared clock is $08: 05$, what is the time now on the rectangular clock? 08:25 $08: 15$ $08: 30$ $08: 20$

Questions 19 and 20 The following matrix is a generator matrix for the Reed-Muller code $\mathcal{R}(3)$ : \[ \left[\begin{array}{llllllll} 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \\ 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 \\ 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 \end{array}\right] \] Q20 Which bit of the received word is in error? Options for Question 20 A Bit 1 B Bit 2 C Bit 3 D Bit 4 E Bit 5 F $\quad$ Bit 6 G $\quad$ Bit 7 H $\quad$ Bit 8

MATHEMATICS IN THE MODERN WORLD 5. (7 points). Find the missing value. Part II. Solve the problem using the indicated problem solving strategy. 6. (Use representation and equation). A pet shop owner just bought her supply of birds and dogs. She doesn't really remember how many she bought but she has a system. She knows she bought a total of 22 animals, a number exactly equal to her age. She also recalls that the animals had a total of 56 legs, her mother's age. How many birds and dogs did she buy? (10 pts.) 7. (Use Venn Diagram). All the members of a group of 30 students belong to at least one of these clubs: Academic, Music and Sports. 6 of the students belong to only the Sports Club 5 of the students belong to all 3 clubs 2 of the students belong to the Academic and Sports Clubs but not to the Music Club. 15 of the students belong to the Sports Club 2 of the students belong only to the Academic Club 3 of the students belong only to the Music Club. A. (7 points) Draw the Venn diagram representing the given problem. B. (2 points each) Find the number of students who belong to: B1. Academic and Music but not Sports B2. Academic Club B3. Exactly one of the three clubs B4. At least two clubs

A particle is moving on a hilly landscape where the height at (x, y) is given by h(x, y) = xy 2 +x 3 y . Find a direction v , at the point (1, 2) such that the height of the particle does not change for a small movement along v ? Can you nd another direction w that is orthogonal to v such that the height of the particle does not change for a small movement along w?

(1 point) (Note: you have only 4 attempts on this question.) Complete the sentences below to obtain true statements. Use the following words: increasing ", decreasing ", concave up", concave down", positive", negative ". If a particular answer cannot be determined from the given information, type in the word undetermined". (Type the words BUT DO NOT TYPE THE QUOTATION MARKS.) 1. If $f^{\prime}$ is negative on an interval, then $f(x)$ is decreasing III on that interval and $f^{\prime \prime}(x)$ is undetermined Iil on that interval. 2. If $f^{\prime \prime}(x)$ is negative on an interval, then $f^{\prime}(x)$ is undetermined III on that interval and $f(x)$ is undetermined III on that interval. 3. If $f^{\prime}(x)$ is increasing on an interval, then $f(x)$ is Iif on that interval and $f^{\prime \prime}(x)$ is III on that interval. 4. If $f(x)$ is increasing on an interval, then $f^{\prime}(x)$ is Iif on that interval and $f^{\prime \prime}(x)$ is III on that interval.

Alice and Bob play the following game. There is one pile of $N$ stones. Alice and Bob take turns to pick stones from the pile. Alice always begins by picking at least one, but less than $N$ stones. Thereafter, in each turn a player must pick at least one stone, but no more stones than were picked in the immediately preceding turn. The player who takes the last stone wins. With what property of $N$, will Alice win? When will Bob win? For odd $N$ the outcome is quite clear, as Alice will start by picking one stone and will enforce the win. But what then?

2. (10 points.) Suppose that B is the infinite set of all strings made from one or more 0’s and 1’s. B = { "0", "1", "00", "01", "10", "11", "000", "001", "010" ... } Prove that B is countable. Hint: Some strings in B represent the same base-2 number. For example, "0", "00", and "000" all represent zero.

Prove that if x^3 is irrational, then x is irrational.

Exercise 4. Consider a monotonically strictly increasing function (i.e-a function $f(x)$ whose graph is going up, that is, if $x<y$ then $f(x)<f(y)$.) Let $a<b$. Order the following three quanities in increasing magnitude: $\int_{a}^{b} f(x) d x$, $R_{100}, L_{100}$ and explain your reasoning geometrically.

For a permutation p: X → X, let $p^{k}$ denote the permutation arising by a k-fold composition of p, i.e. $p^{1} = p \ and \ p^{k} = p◦p^{k−1}$. Define a relation ≈ on the set X as follows: i ≈ j if and only if there exists a k ≥ 1 such that $p^{k}(i) = j$. Prove that ≈ is an equivalence relation on X, and that its classes are the cycles of p. I am new to this, what does it mean by its classes? does it mean its elements in the set X? I do see the connection between the k-fold composition and the permutation of p. I think I just do not understand the language. Please give me some hints.

Use the Laplace Transform to solve the given initial value problem. a) y ′′ + 3y ′ + 2y = 0, y(0) = 1, y′ (0) = 0. b) y ′′ − 2y ′ + 2y = 0, y(0) = 0, y′ (0) = 1. c) y ′′ + 2y ′ + 5y = 0, y(0) = 2, y′ (0) = −1.

Question 5 $2.5 \mathrm{pts}$ One side of a rectangular yard is bounded by the side of a bouse. The other three sides are to be fenced with $86 \mathrm{ft}$ of fencing. The length of fence opposite the house is $13 \mathrm{ft}$ less than either of the other two sides. Find its length and width. The opposite sade is $18 \mathrm{ft}$, and the other aides are $31 \mathrm{ft}$ each The opposite side is $20 \mathrm{ft}$, and the other sides are $33 \mathrm{ft}$ each. The opposite side is $21 \mathrm{ft}$, and the other sides are $34 \mathrm{ft}$ each. The opposite side is $23 \mathrm{ft}$, and the other sides are $36 \mathrm{ft}$ each The opposite side is $25 \mathrm{ft}$, and the other sides are $38 \mathrm{ft}$ each.

Solve the puzzle: Two words = "202315 4129238" But three words $=" 2212071992222$ 231518419" Can you decipher these words? "9 4261372015221372292085141811 9142051814198916"

If $\|\mathbf{v}\|=3$ and $\|\mathbf{w}\|=5$, what are the largest and smallest values possible for $\|\mathbf{v}-\mathbf{w}\|$ ? Give a geometric explanation of your results. The largest value possible for $\|\mathbf{v}-\mathbf{w}\|$ is The smallest value possible for $\|\mathbf{v}-\mathbf{w}\|$ is

Your grandparents have an annuity. The value of the annuity increases each month as $1 \%$ interest on the previous month's balance is deposited. Your grandparents withdraw $\$ 1000$ each month for living expenses. Currently, they have $\$ 50,000$ in the annuity. Model the annuity with a dynamical system. Find the equilibrium value. What does the equilibrium value represent for this problem? Build a numerical solution to determine when the annuity is depleted.

Solve the IVP, y'=(x^2-y^2)/(xy) with y(2)=2

Consider the following valid argument: If Mark likes Algebra, then he is a mathematician. Mark does not like Geometry. Mark is a mathematician or Mark is smart. If Mark is a mathematician, then he likes Geometry. Therefore, Mark is smart, or if Mark likes Algebra, then Mark likes Geometry. A. Symbolize the given argument using A for "Mark likes Algebra", M

Find the principal value of the following. a. $(2 i)^{(1-2 i)}$ b. $(\mathbf{1}-i)^{(\mathbf{1}+i)}$ c. $(-2)^{-3 i}$ Solve for $z$. Express the value of $z$ in the form $x+i y$. a. $\ln z=-\frac{\pi i}{2}$ b. $\ln z=4-3 i$ c. $\ln z=e-\pi i$ d. $\ln z=0.6+0.4 i$ Show that $\arccos z=-i \ln \left(z+\sqrt{z^{2}-1}\right)$. Find all the values of $z$ that would satisfy the equation $z^{4}=\mathbf{1}-\boldsymbol{i}$. Show your complete solution.

Let $\mathbf{u}, \mathbf{v}$, and $\mathbf{w}$ be vectors in $\mathbb{R}^{n}$ and let $c$ and $d$ be scalars. Then a. $\mathbf{u}+\mathbf{v}=\mathbf{v}+\mathbf{u}$ b. $(\mathbf{u}+\mathbf{v})+\mathbf{w}=\mathbf{u}+(\mathbf{v}+\mathbf{w})$ c. $\mathbf{u}+\mathbf{0}=\mathbf{u}$ d. $\mathbf{u}+(-\mathbf{u})=\mathbf{0}$ e. $c(\mathbf{u}+\mathbf{v})=c \mathbf{u}+c \mathbf{v}$ f. $(c+d) \mathbf{u}=c \mathbf{u}+d \mathbf{u}$ g. $c(d \mathbf{u})=(c d) \mathbf{u}$ h. $1 \mathbf{u}=\mathbf{u}$ Commutativity Associativity Distributivity Distributivity Simplify the given vector expression. Indicate which properties in the theorem above \[ \begin{array}{rlrl} 2(\mathbf{a}-5 \mathbf{b})+5(2 \mathbf{b}+\mathbf{a}) & \\ 2(\mathbf{a}-5 \mathbf{b})+5(2 \mathbf{b}+\mathbf{a}) & =(2 \mathbf{a}-10 \mathbf{b})+(10 \mathbf{b}+5 \mathbf{a}) & \text { properties ? } \\ & =(2 \mathbf{a}+5 \mathbf{a})+(10 \mathbf{b}-10 \mathbf{b}) \text { properties ? } \end{array} \]

How would/could understanding essay structure and the writing process help you be a better writer? Why is it important to understand Academic Writing Style andaudience awareness? What are some ideas to keep in mind as you begin writing for this class--and your other classes too?

help Question 27 3.33 pts People are entering a building at a rate modeled by $f(t)$ people per hour and exiting the building at a rate modeled by $g(t)$ people per hour, where $t$ is measured in hours. The functions $f$ and $g$ are nonnegative and differentiable for all times $t$. Which of the following inequalities indicates that the rate of change of the number of people in the building is increasing at time $t$ ? $f(t)-g(t)>0$ $f^{\prime}(t)>0$ $f^{\prime}(t)-g^{\prime}(t)>0$ $f(t)>0$