0
votes
0answers
11 views

Singular Perturbation Theory problem for slightly unstable pde

I'm learning about singular perturbation theory for solving an approximate solution to a pde and I'm a little confused on how to apply it. The problem I'm trying to work on is $\frac{\partial ...
0
votes
2answers
17 views

Diverging Integrals and pointwise convergence?

I am looking for a $f$ and $f_n$ such that   $(f_n) \rightarrow$ $f$ pointwise on $[0,1]$ $f$ and every $f_n$ integrable but the sequence of integrals $\int_0^1 f_n$ does not converge   ...
0
votes
1answer
38 views

What would the equation to this graph look like?

I have an equation built from obscure (step) functions, which I'd like to approximate in standard mathematical functions; it is quite lengthy and produces an odd geometric shape, which I cannot ...
1
vote
0answers
14 views

Mathematics for Guidance, navigation and control

I'm finishing my math degree this week and have been looking for some subject to practice and study on my own while I'm doing some work as a programmer. I'm interested in getting my master's later but ...
0
votes
3answers
25 views

Factoring a 4th degree trinomial

I am trying to factor $3x^4-8x^3+16$, but I have no idea how to even start. I put into Wolfram Alpha, and it said that the answer was $(x-2)^2 (3 x^2+4 x+4)$. How would you factor something like this ...
1
vote
1answer
16 views

Question about assumptions in proof of continuity of $x^2$

When proving that $x^2$ is continuous we usually show that $\lim_{x \to\ a} x^2 = a^2$ for any x. So we show that for any $\epsilon > 0$, there is a $\delta > 0$ such that $0< | x-a | ...
0
votes
0answers
5 views

primitive recursive conditional

I am confronted with the assertion that the following expression describes the conditional: $\text{Cond}\left[ t, f, g \right] = \text{Pr} \left[pr^2_1, pr^4_2 \right] \circ(f,g,t)$. This is meant ...
0
votes
3answers
36 views

finding the sum of this series $(2n-1)^2(1/2)^n$

$\sum\limits_{n=1}^{\infty}(2n-1)^2(\frac{1}{2})^n$ I know via Wolfram that the sum is 17, but I'm not sure I've ever found the sum of such a series before. Any help is appreciated.
0
votes
0answers
6 views

Scale of Oscillations

I'm reading an article which claims the following result : (paragraph 2.2) for a function $f = \sin (N g(x)) h(x) $ where $g$ and $h$ are $C^{\infty}$ scalar functions non oscilattory and $N$ a large ...
0
votes
1answer
21 views

Riemann Upper and Lower Sums

Suppose $f$ is defined on $I = [0,10]$ as follows: $f(x) = N$ if $N-1 \leq x < N$, $N$ is an integer. Let $P = \{0,1,2,3,4,5,6,7,8,9,10\} \in \prod(P)$ Find $S[f,P]$ and $s[f,P]$ If $P^* = ...
3
votes
0answers
31 views

ALL Orthogonality preserving linear maps from $\mathbb R^n$ to $\mathbb R^n$?

That is we have a linear transformation, i.e. an $ n\times n $ matrix $A$, such that for every pair of vectors $ v $ and $ w $ we have $$ \langle v,w\rangle=0 \ \ \ \implies \ \ \ \ \ \langle ...
2
votes
1answer
12 views

How are Gauss sums the analogue of the gamma function for finite fields?

I've seen this statement on the internet in a few places and I don't really get the connection. Would anyone mind fleshing out the details? Thanks.
-2
votes
1answer
43 views

A clown, a robot, a cowboy, and a mathematician are traveling through the woods when they come to a river they need to cross. [on hold]

A clown, a robot, a cowboy, and a mathematician are traveling through the woods when they come to a river they need to cross. There is a boat at shore that can only hold two people or one robot. ...
1
vote
1answer
10 views

Sequences of random variables converging in probability to the same limit a.s.

Let $(X_n)_{n \geq 1}$ and $(Y_n)_{n \geq 1}$ be two sequences of random variables s.t. $X_n$ converges to X and $Y_n$ to $Y$ both in probability. Furthemore, $X$ = $Y$ a.s. How can I prove that, ...
0
votes
2answers
23 views

Given a Matrix A, prove that 1/9A is an orthogonal matrix.

$$Let A= \begin{pmatrix} 4 & -7 & 4 \\ -1 & 4 & 8 \\ -8 & -4 & 1 \\ \end{pmatrix} $$ The problem is to prove that $1/9A$ is an ...
0
votes
2answers
16 views

Simple Random Walk - Why are these two events the same?

Let $S = (S_n)_{n \geq 1}$ be a simple random walk. We denote the hitting time of a point $b$ by $\tau_b = \min \{i \geq 1 : S_i \geq b\}$. My text says that the events $\displaystyle\{\max_{k \leq ...
1
vote
0answers
16 views

Non-atomic, ergodic measure which is left and right shift invariant.

Given a one-sided shift space, say $X = \prod\limits_{n=1}^\infty \mathbb Z_2$. Denote the left shift by $T$: $T(x_1 x_2 x_3\cdots) = x_2x_3 \cdots$. There are lots of examples of $T$-invariant ...
2
votes
1answer
26 views

Uniform convergence on compact subset

Let there be two functional squences $$a_n(x)=\sqrt[n]{x} \quad \textrm{ for $x\in(0,\infty)$}$$ $$b_n(x)=\sum_{k=0}^{n}x^k(1-x)^k=\frac{1-x^{n+1}(1-x)^{n+1}}{x^2-x+1} \quad \textrm{ for $x\in ...
2
votes
0answers
32 views

Simplify and Denest $\sqrt[3]{2+\sqrt{3}+\sqrt[3]{4}}$

I'm not too sure how to begin in denesting $\sqrt[3]{2+\sqrt{3}+\sqrt[3]{4}}$. I thought about setting it equal to $2a+b\sqrt{3}+c\sqrt[3]{4}$, but that sounds a bit sketchy.
0
votes
1answer
15 views

Parametric equations in rectangular form

$y=t^2+4, x=2t-5$ Graph and state any restrictions on the domain. I need to state any restrictions on the domain but don't know how. I added 5 to both sides of the x equation and then I divided by ...
1
vote
1answer
12 views

Why is Frobenius norm related to the inner product of characters?

This is a continuation of my question asked here. I am going through Normal Subgroup Reconstruction and Quantum Computation Using Group Representations. In Definition 2, the authors start with the ...
2
votes
4answers
47 views

How to prove that $\lim_{x \to 0^{+}} \frac{e^{-1/x}}{x} = 0$

Today someone asked me how to calculate $\lim_{x \to 0^{+}} \frac{e^{-1/x}}{x}$. At first sight that limit is $0$, because the exponential decreases faster than the lineal term in the denominator. ...
0
votes
1answer
10 views

Notation: square brackets with a unique scalar?

my question is purely about notation. I am reading papers in computer science and I see that people use the following notation $[x]$ to denote $\{1,2,\ldots,x\}$. Is that correct? Or does it mean ...
1
vote
0answers
11 views

Characteristic functions proof… $\mu\{x:|x|>r\}\le\tfrac r2\int_{-2/r}^{2/r}(1-\varphi(t))\,dt$

I am trying to prove Lemma 4.1 (1) from Olav Kallenberg's Foundations of Modern Probability: $\mu\{x:|x|>r\}\le\tfrac r2\int_{-2/r}^{2/r}(1-\varphi(t))\,dt$ From context, I believe that ...
1
vote
0answers
13 views

Show that every polynomial of degree $1,2,$ or $4$ in $Z_2[x]$ has a root in $Z_2[x]/(x^4+x+1)$.

The problem: Show that every polynomial of degree $1,2,$ or $4$ in $\mathbb{Z}_2[x]$ has a root in $\mathbb{Z}_2[x]/(x^4+x+1)$. My attempt: I know that the polynomials $x$ and $x+1$ have ...
0
votes
2answers
26 views

If $a$ and $b$ are nonzero integers such that each is a divisor of the other, show that $a = ± b$ .

I tried many approaches to this problem. I believe that if I did $b|a=m$ and $a|b=n$ and set $m=n$, then $a$ and $b$ would be equal. Is that how it should be done? If not, please help me out. Thanks.
0
votes
1answer
18 views

Prove this series converges to a continuous function

My problem: Prove that the series $\sum\limits_{n=0}^\infty e^{n(\sin(nx)-2)}$ converges for all $x\in\mathbb{R}$ to a continuous function. By the root test it converges, but as far as the continuous ...
1
vote
0answers
23 views

What is the significance of studying the steady state behaviors of a system? What information do steady state models provide us?

For example, http://www.jameslovelock.org/page31.html In this 1983 paper by Lovelock and Watson modeling Daisyworld, in equations (10) through (14), the paper considers the non zero steady state ...
0
votes
2answers
30 views

Two decreasing, convex functions agreeing on a closed set

Fix a closed subset $B$ of $[0,\infty)$ and assume that $0\in B$. I am striving to construct two functions $f,g:[0,\infty)\to\mathbb R$ such that $f(0)=g(0)=1$; $f(x)\geq 0$ and $g(x)\geq0$ for each ...
0
votes
1answer
34 views

solving the integral $\sqrt{\sin(y)}(4-y)$

So the question asked me to find the volume of the area between $\sqrt{\sin(y)}$ and $x=0$ rotated about $y=4$. I came up with the integral $$\int_0^{\pi}(4-y)\sqrt{\sin(y)}dy$$ I can't figure out ...
-3
votes
0answers
20 views

question on linear algebra word problems. [on hold]

How would you go about answering this problem I tried this: Wayside Auto Sales discovers that when \$1000 is spent on radio advertising, weekly sales increase by \$101,000. When \$1250 is spent on ...
1
vote
4answers
65 views

$\int \frac{1}{\sqrt{x^2+1}} dx$

So I've seen some options on the internet that are fairly good, but I have this substitution: $x^2+1=t-x$, you square both sides and get $x = (t^2-1)/t$ and $x + 1 = (t^2-1)/2t + 1$. If we call that ...
1
vote
2answers
40 views

Integrating $\int\frac{1}{kx}dx$ [duplicate]

$\int\frac{1}{kx}dx$ There are two ways to integrate this... Method 1 (Separating the coefficient from the variable): $\frac{1}{k}\int\frac{1}{x}dx$ $\frac{\ln|x|}{k} + c$ Method 2 (knowing that ...
0
votes
0answers
36 views

A not so obvious corollary?

Let $X$ be a normed vector space over $\mathbb{K}$. Then there is only one completion of $X$, the banach space $\hat{X}$ such that $X$ is a dense subspace of $\hat{X}$. I am trying to prove that there ...
1
vote
1answer
18 views

Sequences of random variables converging in probability to the same limit a.s

Let $(X_n)_{n \geq 1}$ and $(Y_n)_{n \geq 1}$ be two sequences of random variables s.t. $X_n$ converges to X and $Y_n$ to $Y$ both in probability. Furthemore, $X$ = $Y$ a.s. How can I prove that, for ...
0
votes
1answer
13 views

Calculate confidence interval when only given alpha?

Is there a formula to properly calculate the confidence interval when you're only given alpha and sample data? Thanks!
-1
votes
0answers
21 views

6 people like the Lions five people like monkeys how many fewer people like monkeys [on hold]

I know this is a simple problem I'm just not sure and wondering what the answer is
0
votes
1answer
14 views

Computing the “Mean Value” of a Point Sample From an Arbitrary Manifold

A friend of mine noticed that taking the "mean" of two points on the circle isn't as easy as just computing the arithmetic mean of their arguments: If one point has argument $-3.13$ radians and one ...
-1
votes
0answers
24 views

Gauss and Stokes Theorem Problem!! Help!! [on hold]

In $(x, y, z)$ space is considered the vector field $V(x,y,z)=(y^2 z, yx^2, ye^y)$ solid spatial region $Ω$ is given by the parameterization: $\left[ \begin{matrix} x \\ y \\ z \end{matrix} \right] ...
0
votes
0answers
19 views

Poison's equation for charge distributed on a ring

Let's consider the problem of determining the electrostatic potential given a charge distribution $\rho : U\subset \mathbb{R}^3\to \mathbb{R}$. In that case, the potential satisfies the Poisson ...
2
votes
0answers
16 views

Hypothesis Testing on Renewal Processes

We have time $[0,T]$ to observe a renewal point process, where the inter-renewal timings are i.i.d, and then decide whether the observation is according to a renewal process in which the pdf of ...
1
vote
2answers
23 views

Evaluate Distribution

Evaluate the distributions p(a), p(b|c), and p(c|a) corresponding to the joint distribution given in Table 8.2. Hence show by direct evaluation that p(a, b, c) = p(a)p(c|a)p(b|c). Table 8.2 ...
2
votes
1answer
35 views

Prove that if $f$ and $g$ are integrable on $[a,b]$, then so are $\max{(f,g)}$ and $\min{(f,g)}$

Prove that if $f$ and $g$ are integrable on $[a,b]$, then so are $\max{(f,g)}$ and $\min{(f,g)}$. Since $f$ and $g$ are integrable, we know that $U(f,\mathcal{P})-L(f,\mathcal{P}) < \epsilon$ ...
0
votes
5answers
59 views

Solving equations.

How would you solve these equations and show that they do not intersect each other? $$x^2+y^2=2x-2y$$ $$x^2+y^2=4(x^2+y^2)^{1/2} +y$$ It's isolating a term which I am struggling with. General ...
0
votes
1answer
37 views

Integrate $\frac{\sin x^3}{x^3}$ as a power series

Today, I tried to do this by taking the MacLaurin of Sin to 4 terms, putting in $x^3$ in place of $x$, multiplying the terms by $x^{-3}$, and integrating. I came out with a sum that had $x^{6n+1}$ as ...
1
vote
1answer
10 views

Convex subsets of a topological vector space

I'm trying to prove: Let $X$ be a topological vector space and $A \subseteq X$. $A$ is convex if and only if $$\forall s,t \in \mathbb{R}_{+}, (s+t)A = sA + tA $$ Let $sx + ty \in sA + tA$. How ...
3
votes
0answers
29 views

What do you call the category of groups with surjections (injections)?

Take the category $Grp$ and throw out all morphisms that are not surjections. I think what's left is still a category. The composition of two surjections is surjective, and identities exist ...
0
votes
2answers
32 views

Complex integration problem via Cauchy's integral formula

I want to integrate the following :$$\int_{|z|=2} \frac{dz}{z^{2}-1}$$ in the positive direction. So my idea is two split the integral into a sum of two integral , something like $$\int_{|z|=2} ...
1
vote
1answer
30 views

Problem in understanding the proof of lemma $7.2.5$ in Liu's book

Let's analyze the proof of the following lemma: Lemma: Let $X$ be an integral, Noetherian scheme and let $f\in K(X)^\ast$, then for all but finitely many points $x\in X$ of codimension $1$ we have ...
1
vote
1answer
21 views

$P($partial sums of $\sum X_k$ are bounded$)>0 \to \sum_{k=1}^{\infty} X_k < \infty\ \text{a.s.}$

From Williams' Probability with Martingales How is the remark deduced from the proof of $b$? I really don't see it.

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