1
vote
0answers
13 views

Level sets volume

Suppose that $f:\mathbb{R}^d\to\mathbb{R}$ is a nice function (whatever nice should mean), non-negative, with a compact support. Fix $v >0$ and define $$ A_{\epsilon} := \{x\in \mathbb{R}^d : v \le ...
1
vote
1answer
15 views

understanding phrasing of taylor polynomial question

Show that $|\sin x - x + \frac{1}{6}x^3| < 0.08$ for $|x| \le \frac{\pi}2$. How large do you have to take $k$ so that the $k$th order Taylor polynomial of $ \sin x$ about $a=0 $ approximates $\sin ...
0
votes
1answer
23 views

Random variable related to binomial

The number of successes $X$ in $n$ independent trials with the probability of a success is $p$ for each trial is binomially-distributed. I am interested in a scenario that adds dependence to the ...
3
votes
1answer
29 views

Integration by parts (Differential Geometry)

I am studying the proof of a theorem and I am stuck. It says that by integration by parts we get that: For $g(t)$ a variation of Riemannian metrics wih $g'(0)=h,$ $$\int_{M} (-\Delta (tr h) + ...
0
votes
0answers
15 views

How to find expected number of games between two players?

I couldn't understand an answer to this question, so I'm asking it again. Can someone explain the answer or solve it by another method? The one think I didn't understood in answer is why ...
0
votes
1answer
14 views

Missing something about second derivative tests

I'm studying second derivative tests, concavity and inflection points in khan academy ...
0
votes
1answer
20 views

Checking continuity looking whether image set is interval or not

Let $A(\neq \phi)\subseteq\mathbb{R}$. Suppose $f : A \to \mathbb{R}$ is a monotone function such that the image $f (A)$ is an interval. Then prove that $f$ is a continuous function. And if $f(A)$ is ...
0
votes
0answers
15 views

A basic analysis/O.D.E/perturbation theory question

Consider a system of equations $$x'=f(x,y,\epsilon)$$ $$y'=\epsilon g(x,y,\epsilon)$$ I have seen in the book to claim the following: As $\epsilon -> 0$ the limit is $$x'=f(x,y,0)$$ $$y'=0$$ I ...
-1
votes
2answers
65 views

What is the distribution of Z=min(X,Y) [on hold]

Let X and Y be independent geometric random variables. What is the distribution of Z=min(X,Y)?
1
vote
0answers
32 views

Which math class next

I just finished and Algebra for Calculus class this semester. I'm trying to work up to taking calculus (have to do up through calc 3). One person told me I should take trig next, and another calculus. ...
4
votes
1answer
49 views

algebra with topology homework problem

Hello Everyone, I have this homework problem, I'm going to share what i have so far, not sure if Im in the right path. First, I have: $$f \sim g \, \Leftrightarrow \,x_0 \in \mathbb{R^n}, \exists ...
0
votes
0answers
9 views

Question about a example of Schwarz-Christoffel Transformation

My question is 1.What the initial point mean for the first integration, and can it be $-\infty$? 2.For the integration of $A \int_{ 1 }^{ \infty } {\frac{dt}{(t^2-1)^\frac{2}{3}}} $,why it ...
3
votes
0answers
19 views

Range of Influence of the Wave Equation?

Suppose $u$ is a solution of the two-dimensional wave equation and that at $t=0, u=u_t=0$ outside the disc $x_1^2+x_2^2 \le 1$. Up to what time can you be sure that $u=0$ at the point ...
0
votes
0answers
21 views

Did I do this big-Omega proof correctly?

Prove or disprove: 6n^3 – 4n^2 + 3n +2 is in Ω (5n^3 – n^2 + n +1). So I'm not sure if I did this right or not, any pointers or the correct steps would be helpful Ǝc ∈ ℝ+, ƎB ∈ ℕ, ∀n ∈ ℕ, n ≥ B ⇒ ...
0
votes
1answer
12 views

Show that the σ-algebras generated by the collection of all intervals are equivalent

Show that the σ-algebras generated by the collection of all intervals of the form [a,b]⊂R and by the collection of all the intervals of the form (−∞,b]⊂R are equivalent. i am having trouble with ...
0
votes
1answer
28 views

Sum of two random variable X and -X

Let $X$ and $Y$ be two random variables such that $X$ and $-Y$ have the same distribution. Prove that $P(X+Y<-t)=P(X+Y>t)$. I know how to prove this when $X$ and $Y$ are independent but how do I ...
4
votes
2answers
143 views

Graphical sense of infinite power series

Can someone please give me a graphical sense of infinite power series? I can't really picture how the graph looks like. Some holomorphic function such as exponentials, sine and cosines are infinite ...
5
votes
0answers
36 views

Assume that $f$ is $2\pi$ continuous and $C^1$ such that $\int_{-\pi}^{\pi} f(x) dx=0$.

Show that $\int_{-\pi}^{\pi} (f(x))^2 dx \leq \int_{-\pi}^{\pi} (f'(x))^2 dx$. So here's my approach to this question: Assume that $f$ was $2\pi$ continuous and $C^1$. Therefore, we have that ...
0
votes
1answer
20 views

Martingale $X_n \to \infty$ a.s.

Construct a martingale $X_n$ such that $X_n \to \infty $ a.s. I have trouble coming up with such an example and prove it. Can someone provide an example?
0
votes
0answers
19 views

Calculus Partial derivative computation

I have the following systems of equation $$x^5v^2+2y^3u = 3$$ $$3yu - xuv^3 = 2$$ I need to show that this system defines v and u implicitly as functions of x and y near the point (x,y,u,v) = ...
3
votes
0answers
13 views

Function with divergence, curl and normal trace on boundary equals zero is zero

Let $u\in H^1(\Omega)$ with $\nabla\times u=0$ in $\Omega\subset\mathbb{R}^3$ (open bounded domain), $u\times n=0$ on $\partial\Omega$ (where $n$ is a a normal vector to $\partial\Omega$), ...
0
votes
3answers
27 views

Need explanation how we simplified expression for variance

I cannot really understand how we did simplification for our variance. Like how we got E[X^2] on the second line. Probably some algebra gaps.. but I cannot really make sense of it. Need help! Thank ...
0
votes
1answer
29 views

Integrand for a set of points

I need help finding what I should be integrating when the question asks to find the double integral to find the volume of the tetrahedron given the points $(0,0,0),(3,0,0),(2,1,0),(3,0,4)$. Would the ...
1
vote
2answers
45 views

Finding a solution for $x\frac{\partial u}{\partial x}+2y\frac{\partial u}{\partial y}=x^2$

To find a solution for $x\frac{\partial u}{\partial x}+2y\frac{\partial u}{\partial y}=x^2$ knowing that $u(x,y)=1$ if $xy=1.$ I thought it may be useful to do the change $v=\log x, w=\frac{1}{2}\log ...
1
vote
1answer
37 views

How find this the expected number of games played, if A won $n$ consecutive games?

Question 1: A and B two people play a game, where the odds of winning for one per game is $\dfrac{1}{2}$. If someone first win $n$ games consecutive,then the games end.Find this expect with the ...
0
votes
0answers
21 views

Zero eigenvalue of an Operator [on hold]

In my functional analysis notes, there is a claim with proof(which I seem to understand, but don't get the point) is the following. Consider the bounded linear operator L: H$ \rightarrow $ H, where H ...
3
votes
1answer
18 views

If the probability density on a random vector is symmetric, then each variable is identically distributed?

Let $X$ be a random vector with joint distribution $F$ and density $f$. If $f$ is symmetric, is this equivalent to each random variable being identically distributed? We say $f$ is symmetric if it is ...
0
votes
2answers
33 views

If two sets are separated, then any two subsets of those sets are also separated?

I want to prove that if two sets X and Y are separated, then subsets of those sets are also separated. The definition is that if X intersect Y closure is empty and X closure intersect Y is empty, the ...
1
vote
1answer
29 views

Question on wikipedia's proof of rolles theorem

Here is this proof. It basically says that if a function has a maximum, then this quotient, for $h>0$ is $$\frac{f(c+h)-f(c)}{h}\le0$$ That's ok to me, because if $c$ is the maximum point, then ...
0
votes
1answer
20 views

Finding connected componets of a set of continuous functions

In the metric space (C[0,1], d∞) consider the set: U= {f in C[0,1]: f(x)≠0 for all x in [0,1]} Prove that U is open and find its connected components. Proving that U is open is easy, but I don't ...
0
votes
3answers
41 views

Limit to infinity of:

I have this limit that I tried and failed to solve: $$ \lim_{n \to \infty}\frac{\sqrt{n^2-n+2}-n}{\sqrt{1+\frac1n}}=\lim_{n \to ...
1
vote
1answer
20 views

Derivation of Simple Projectile Motion with Drag

Given the initial velocity $v_0$ and angle $\theta$ of a projectile on the ground, using Newton's second law and the acceleration due to gravity $\mathbf g=\left\langle0,-g\right\rangle$, I was able ...
2
votes
2answers
20 views

Visualizing Balls in Ultrametric Spaces

I've been reading about ultrametric spaces, and I find that a lot of the results about balls in ultrametric spaces are very counter-intuitive. For example: if two balls share a common point, then one ...
0
votes
0answers
10 views

Adapted but not progressively measurable?

Let $X(t,\omega)$ be a stochastic process: $$ X: \mathbb{R}^+ \times \Omega \rightarrow \mathbb{R}, $$ where $(\Omega, \mathcal{F}_t, \mathcal{F}, \mu)$ is a stochastic basis. Some definitions: ...
0
votes
1answer
19 views

Graph Theory into

Let M and N be matchings in a graph G with (cardinality of M) > (cardinality of N). Prove that there exists matchings M' and N' such that (Cardinality of M') = ((cardinality of M)-1), and (Cardinality ...
0
votes
0answers
5 views

Use copula to find the joint distribution of two random variables

Assuming that there are two random variables $x$, $y$ both having an Arcsine marginal distribution $F$, i.e. $F(x)=\frac{2}{\pi}\arcsin{\sqrt{x}}$ The density of their Gaussian copula can thus be ...
0
votes
1answer
21 views

Finding range for h(x)

What is the range of $\frac{1}{e^{x^2}+3}$? I know that the answer is $\frac{1}{4}$>=h(x)>0, but how do I show it
3
votes
12answers
347 views

Interesting piece of math for high school students?

I'm giving an hour long lecture to high school math students with a fairly high aptitude in math. I want to present something a little advanced for them (undergrad level) that they have to struggle ...
0
votes
0answers
9 views

extension of trigonometric functions as basis functions to higher dimensions

Trigonometric functions forms an orthonormal basis functions for $L^2[a,b]$, with corresponding normalization coefficients. I want to know if this result can be extended to higher dimensions. For ...
0
votes
2answers
11 views

iid Gaussian random matrix $A\in M_n$ has full rank with probability 1?

I want to prove that: iid Gaussian random matrix $A\in M_n$(I mean whose elements are iid Gaussian) has full rank with probability 1 Below is my consideration: $$1-P(\text{full ...
1
vote
2answers
23 views

Stone's Theorem and Functional Calculus

I've asked a few questions on here before regarding functional calculus but I am still having a bit of trouble. I have been reading up on Stone's theorem for unitary groups, and going through the ...
1
vote
0answers
10 views

Probability and search Tree

I need some help with the following question. Given the random permutations of $ n > 2 $ numbers. Now, creating a binary search tree and puting it the organs one by one. Denote the input organs ...
0
votes
2answers
10 views

Does Orthogonal matrix have complex eigenvectors with the same absolute value? If it is true, how can I prove it?

Does Orthogonal matrix have complex eigenvectors with the same absolute value (or modulus or magnitude)? If it is true, how can I prove it?
0
votes
0answers
14 views

Trouble matching two shapes at an angle [on hold]

I want to connect two pieces of wood of the same width at an angle, but the ends are never the same width.
0
votes
3answers
16 views

Diagonalize Matrix A

I am told to diagonalize Matrix A i solved for P and P inverse ...
2
votes
2answers
44 views

How to prove/disprove proof on limits (delta-epsilon)

Prove or disprove: $$ \forall \epsilon > 0, \exists \delta>0, \forall x, y \in \mathbb{R}^+, |x - y| > \delta ⇒ |x + y| > \epsilon $$ I've been trying this for some time now but can't ...
1
vote
2answers
20 views

Pointwise convergence and Sobolev bounded

If $\{f_n\}$ is a sequence of function on $\Omega$, and $\lim_{n\to\infty}f_n=f$ on $\Omega$. $\|f_n\|_{W^{k,p}(\Omega)}$ is bounded (or uniformly bounded), then whether $f$ is in $W^{k,p}(\Omega)$. ...
1
vote
2answers
34 views

let $D_n$ be the number of permutations of $\{1,2,3,…n\}$ which leave no element fixed.

Let $n\geq2$ and let $D_n$ be the number of permutations of $\{1,2,3,\dots,n\}$ which leave no element fixed. How to write an expression for $D_n$ in terms of $D_k$? I don't know how to start. Please ...
1
vote
1answer
12 views

Minor Proof on a maximal planar graph

Let $G$ be a maximal planar graph of order at least $6$. Let $x$, $y$ be two non-adjacent vertices in $G$. Then $G + xy$ contains both $K_5$ and $K_{3, 3}$ as a topological minor. I am lost on this ...
2
votes
1answer
28 views

Simple questions about the Jacobson Radical

Questions: [See below] $\rm\color{#c00}{(1)}$ What structure has $R/I$ when $I$ is a maximal left ideal? I know that if $I$ is a bilateral ideal then $R/I$ is a ring and, moreover, if $I$ is ...

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