0
votes
1answer
15 views

Shifting a smooth function of compact support

Let $\varphi : \mathbb{R} \rightarrow \mathbb{R}$ be an infinitely differentiable function of compact support. Define $$\psi (x) := \begin{cases} \frac{\varphi (x) - \varphi (0)}{x}, & ...
0
votes
3answers
21 views

Finding poles of $\frac{1}{1 + z^4}$?

$$f(z) = \frac{1}{1 + z^4}$$ has poles when $z^4 + 1 = 0$ $\implies (z^2 + i)(z^2 - i) = 0$ $\implies (z^2 - i) = 0$ when $z = \sqrt{i}$ or $z = -\sqrt{i}$. But how do I solve the equation $(z^2 + ...
1
vote
0answers
25 views

Lesbegue Outer Measure

Consider the unit interval $I=[0,1]$ and let $\mathcal{M}$ be the $\sigma$-algebra of all Lebesgue measurable subsets of $I$. Denote by $m_*$ the Lebesgue outer measure on $\mathcal{M}$. Suppose that ...
1
vote
2answers
27 views

Finding volumes with double integrals

I got some troubles with this problem: A swimming pool is circular with $20\,ft$ diameter. The depth is constant along east-west lines and increases linearly from $4\,ft$ at the south end to $9\,ft$ ...
0
votes
2answers
32 views

Is the group $U(8)$ cyclic?

Referring to the group of units. My first thought is yes, since $1$ would be the generator. Although I think I'm getting confused between the generator and the identity element in this case. $1$ is ...
0
votes
1answer
19 views

Prove that $(A(K), ||$ $||_{\infty})$ is a Banach space. [duplicate]

Define $A(K) = \{f : K \rightarrow \mathbb{R}$ $| f$ is continuous$\}$. $K$ is compact. Prove that $(A(K), ||$ $||_{\infty})$ is a Banach space. Since a Banach space is complete then every Cauchy ...
0
votes
0answers
9 views

Convergence of a monotonically increasing sequence

Given $a_n$ monotonically increasing and $a_n>0$. Which of the following converges? (1) $\frac{1}{a_n^2}$ (2) $e^{-a_n}$. I could not see any reason why both (1) and (2) will not converge. If ...
2
votes
2answers
25 views

If L = lim_x→a f(x) exists, then |f(x)| → |L| as x → a .

Suppose that f is a real function. a) Prove that if $L = \lim_{x\to a} f(x)$ exists, then $|f(x)|\to |L|$ as $x \to a$ . Proof: Suppose that f is a real function. And suppose $L = lim_{x\to a} ...
0
votes
1answer
18 views

An equality in SDE.

I read an example in Shreve: How to get the equality in the last line?
0
votes
0answers
16 views

Wychoff's Game Combinatorics Problem

There are two piles of checkers on a table. A takes any number of checkers from one pile, or the same number from both piles. B does the same. The winner is the last one to take the checker. Positions ...
-2
votes
0answers
15 views

what value to make this function continuous? [on hold]

(1 pt) The function f is given by the formula $f(x)= \frac{ 3 x^3 - 14 x^2+ 4 x - 45 }{ x - 5 }$ when $x < 5$ and by the formula $f(x)=-4 x^2 - 1 x + a$ when $5 \le x$. What value must be ...
2
votes
1answer
107 views

What Notation Do I Use To Fix Ambiguity Writing Chain Rule

I'm a calculus noob learning over the internet. I think the best way to ask my question is just to put up a little diagram I made in paint. Now this is my attempt to write the chain rule using d/dx ...
0
votes
2answers
30 views

Investigation: For what values of $k$ does $y=k\sqrt{x^2-1}$ intersect $y=\ln x$

I'm doing a bit of investigating to try to see where two curves intersect. More specifically, I want to see for what values of $k$ does $y=k\sqrt{x^2-1}$ intersect $y= \ln x$ When you draw the two ...
0
votes
1answer
14 views

Inverse optimization problem

This may seem like a weird question, but it's something which has been intriguing me for quite a while. In the Calculus of Variations we are told to find the extrema of a functional defined over a ...
0
votes
1answer
19 views

At what point is this piecewise function continuous?

let $$f(z) = \begin{cases} z & |z|\leq 1 \\ |z|^{2} & |z|> 1 \end{cases}\ \text{where}\ z\in \mathbb{C}$$Does anyone could help me ?Thanks!
1
vote
5answers
206 views

Limit to Infinity question?

$$\lim_{x\to\infty}\left(-\sqrt{-2x+x^2}+\sqrt{2x+x^2}\right)=2$$ I'm not sure how to go about solving this problem.
1
vote
4answers
26 views

Rotate y=e^(-x^2) about the y-axis to find the volume.

Since we are rotating around the y-axis, my intuition is that I need to change my original function in terms of y instead of x. So to change y=e^(-x^2), I should end with x=(-ln(y))^(1/2). At first ...
2
votes
1answer
13 views

How to Find optimal number

Say I am tracking my sleep for 1 week and these are the number of hours I sleep each night: (5, 6, 9, 4, 8, 9, 6) everyday that I track my sleep I am also taking a test that measures how well my ...
1
vote
1answer
19 views

Train distances leaving at certain times

A train leaves Boston to Fort Lauderdale traveling at $125$ mph. An hour later, another train leaves Fort Lauderdale traveling to Boston at a rate of $140$ mph. When the two trains meet each other, ...
1
vote
4answers
66 views

$4^{x}+2^{x+1}=18$ Please help me solve?

I tried using logs on both sides or tried treating it as a quadratic but didn't manage to simplify it, Help?:D
0
votes
0answers
13 views

Difference Equation Application [on hold]

I'm currently taking an upper level mathematics course in differential equations. I'm currently looking for an interesting project topic that applies difference equations and n-cycles. Any ...
1
vote
2answers
45 views

Limit approaching negative infinity question?

I'm not sure how to solve this problem.. How do I get rid of the square root?
2
votes
0answers
39 views

Real Analysis texts: Royden versus Stein & Shakarchi. Which is better? (and other suggestions welcome)

I am taking an introductory "graduate" analysis class and am comparing Analysis books that cover measure theory. I have had an "advanced calculus" class that covered the standard topics. I am having ...
0
votes
1answer
19 views

Quick question cardinalities and onto mappings

If $f: A \to B$, and if $|B| \geq |A|$, does this mean that $f$ can never be surjective or is it the other way around? I used to remember a simple argument that can help me deduce these cases, but I ...
0
votes
1answer
17 views

What would the Big oh be of (1/2)^n

Is it just (1/2)^n? The function itself gets closer and closer to 0 as x > infinity but I don't know what its classification would be in terms of big oh. O(1)?
1
vote
0answers
16 views

Hypergeometric Distribution2

From a class of 10 boys and 15 girls, prizes are randomly awarded to 3 children. Let N be the number of boys who win prizes. Construct the probability tables for the random variable N assuming that ...
0
votes
0answers
26 views

Lang Fiber Products

STATEMENT: Let $\mathcal{C}$ be a category.A product in $\mathcal{C}_z$ is called the fiber product of $f$ and $g$ in $\mathcal{C}$ and is denoted by $X\times_zY$, together with its natural morphisms ...
0
votes
0answers
10 views

gradient of an integral with variable in the upper limit

Is the value of the following computation true? Let $x,y,z\in R^{2n}$ and the continuous vector-valued function $F:\mathbb{R^{2n}}\to\mathbb{R^{2n}}$. Then $\nabla_y \left(\int_{0}^{y}F(x)\cdot ...
0
votes
1answer
14 views

How to prove the existence of a second complex derivative…

I'm trying to prove that if $h(t)=|\gamma(t)|^2$, then $h''(t)$ exists at a point $t_0$. I know that $\gamma(t_0)=0$, $\gamma'(t_0)$ is nonzero and that $\gamma(t)=x(t)+iy(t)$ on $I=[a,b]$. I know I ...
1
vote
2answers
26 views

Separation of variables in heat equation with decay

I just want to see if I completed this problem right. Here is the problem: Consider $\frac{\partial T}{\partial t} = k \frac{\partial^{2} T}{\partial x^2} -\alpha T$ where $k,\alpha >0$ are ...
0
votes
1answer
39 views

Showing that a map, $R:\mathbb{R}^n\rightarrow\mathbb{R}^n$ can be represented by an orthogonal matrix.

Note: This is a homework question. After pages of attempts and failures, here I am. First, I will present the question then state what I have tried. The question: Let $u$ be a non-zero vector in ...
1
vote
3answers
20 views

How to find the greatest integer, j, such that j * ( j - 1 ) / 2 < k?

How to find the greatest integer, j, such that j * ( j - 1 ) / 2 < k ? Is there a way to find a formula for j in terms of k ? Thanks in advance.
0
votes
0answers
17 views

$k$-connected graphs containing trees

I've encountered the following problem in the book "Graphs and Digraphs" and I'm not sure how to do it. Show that every $k$-connected graph contains any tree of order $k+1$ as a subgraph. Any proofs ...
2
votes
2answers
17 views

Decomposing Countable Union of Measurable Sets

Why can every set $E$ in the real numbers with $\mu^{*}(E)=\infty$ be realized as the disjoint union of countably many measurable sets, each of which has finite outer measure? I'm trying to see this ...
0
votes
0answers
19 views

how write these in interval notation

how write these in interval notation 1- all real numbers except $-3/4$ I assume it will be: $(-\infty , -3/4)\cup(-3/4 , \infty)$ 2- all real numbers except $1/4$ I assume it will be: ...
0
votes
1answer
5 views

Variance changes proof

I have, say a vector of data, A of size(N*1). How do I prove or disprove: if some of the data in A, say A(1:N/2), their variance becomes large, the whole vector A's variance becomes larger? That is ...
0
votes
0answers
3 views

Find a denormal Binary32 number for a decimal number

How do I find the denormal binary32 number for a decimal number? For example if the number is $-0.625*2^{-146}$ what would be the process to go about getting the denormal number? So far I have gotten ...
0
votes
1answer
23 views

Showing a point is a saddle point

Show that if $x'=(x,y) \ \ $ is a critical point of a $\mathcal{C}^3$ function $f$ such that: $$f_{xx}(x')f_{yy}(x')-(f_{xy}(x'))^2<0$$ Then there are points $x$ and $z$ near $x'$ such that ...
4
votes
1answer
75 views

Is chaos theory really a theory? Why not just call it non-linear dynamics?

This may just be semantics, but it's always confused me. What is the thesis of Chaos Theory? I have read an entire book about it, and as far as I can tell, its just a bunch of analytical techniques, ...
0
votes
0answers
8 views

Show equalities of random variables

In the text we showed that a geometrically distributed random variable W has the lack of memory property. Now assume that the range of W is {1,2,3...} and that P(W = j + 1|W>j} = p for j = ...
0
votes
2answers
12 views

Notation for summation while skipping elements

Suppose I have a summation like so: $\sum_{i =0}^n l^i$ Except I don't want to compute for all $0 \leq i \leq n$. I just want to compute it for the arithmetic sequence: $1, 3, 5, 7, 9...$ How do I ...
0
votes
1answer
16 views

Converting $f_{2}$ to $O(f_{2})$ that isn't $f_{1}$

So the actual problem I'm trying to figure out is Find functions $f_{1}$ and $f_{2}$ such that both $f_{1}(n)$ and $f_{2}(n)$ are $O(g(n))$, but $f_{1}(n)$ is not $O(f_{2})$ I know that if I had ...
2
votes
0answers
9 views

Computing a uniformizer in a totally ramified extension of $\mathbb{Q}_p$.

Do you know how to compute a uniformizer of $\mathbb{Q}_p(\zeta_{p^n},p^\frac{1}{p})$? Where $\zeta_{p^n}$ is a primitive $p^n$-th root of 1.
-2
votes
0answers
11 views

Can someone help me with these complex variable problems? [on hold]

I was having trouble solving some of these problems, if someone can help that be great! https://www.flickr.com/photos/127187109@N04/15387754276/
0
votes
1answer
34 views

Find the limit of $3^{x+2}/5^x$ as $x$ approaches infinity.

I'm not sure how to approach this limit problem, I've attempted to solve by inspection, but I don't think this is correct. Since $5^x > 3^{x+2}$ we know that substituting a large number for $x$ ...
1
vote
0answers
27 views

Irrational solutions to some equations in two variables

The next statement is a conjecture of mine, so I dont know if it's true (though quite sure): Let $x,y$ be irrational numbers such that $x^4+y^4=1$. Prove (or disprove) that $x^5+y^5$ is irrational or ...
3
votes
3answers
143 views

Why do the concepts of linear algebra apply to differential equations?

A lot of the stuff we do to solve diff equations are taken word for word from linear algebra. The concept of linear independence, determinant of the Wronskian used to determine independence, adding a ...
0
votes
0answers
13 views

Lagrange Multiplier Method

How do I show the following? Show by Lagrange multiplier method that the maximum value of $\frac{d\phi}{ds}$ is $|\nabla\phi|.$ So $s$ is the distance and $\frac{d\phi}{ds}=\nabla\phi\cdot ...
0
votes
0answers
8 views

Is Dirichlets approximation theorem a method for finding a and b or just used to prove stuff?

I am taking an elementary number theory course and there is a problem that asks to find five rational numbers p/q with abs( (5^(1/3)) - p/q ) <= 1/(q^2). I made a spreadsheet to solve the problem, ...
-1
votes
0answers
18 views

How to prove the existence and uniqueness of a certain function? [on hold]

I would like to prove the existence and uniqueness of a function $f:R\rightarrow R^+$, such that $$ f(x) = a^x, a>0, x\in \mathbb{R}. $$ Thank you for your help.

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