2
votes
0answers
131 views

Derivative of measure-valued function

I have a measure $\mu^x$ which is the law of a random variable and depends on $x$. The specific situation I am thinking of is $\mu^x$ is the law of $X_t$, the solution of an SDE with $X_0=x$. If I ...
1
vote
3answers
243 views

Finding any point on a line if you know the slope and $y$-intercept.

I am wondering if there is a way to determine where a point is if I only know the slope and $y$-intercept. For example, say I am told that the line has a slope of $3$ and a $y$-intercept of $-3$. ...
1
vote
2answers
67 views

Giving variables in a coordinate ring different weights

I am reading a book and I am curious about a certain notion. Consider $R = k[x_1,x_2,x_3,x_4,t]$ and let $G = \{\underbrace{x_1 x_3-x_2^2 + t x_3^2}_{f_1}, \underbrace{x_1 x_4-x_2 x_3 +t ...
2
votes
1answer
91 views

Sub $C^*$-algebra + Ideal of $C^*$ algebra is complete space

Let $A$ be a $C^*$ Algebra. Let $J$ be a closed ideal in A . Let $B$ be $C^*$-sub-algebra of $A$. Prove $B+J$ is complete space (i.e. every cauchy sequence in $B+J$ converges to an element of $B+J$). ...
0
votes
1answer
77 views

Weighted quadrature formula

I need to calculate a quadrature rule with maximum degree of accuracy that looks like this: $$ \int_0^\infty e^{-x}f(x)dx = A_1f(x_1) + A_2f(x_2) + R(f) $$ where $f(x) = cox(x)$, presumably. ...
3
votes
0answers
138 views

Motivation for ray class groups

Almost all books on class field theory define ray class groups out of nowhere and proceed to prove highly nontrivial theorems on them. One naturally wonders; Where do they come from? Of course, it's ...
0
votes
1answer
573 views

Limit Superior of Random Variables

Suppose $X_n$ are iid random variables with $\mathbb{P}(X_n\le x)=1-e^{-x}$. By using the Borel Cantelli Lemmas it's fairly easy to show that $\mathbb{P}(\lim\sup X_n/\log n=1)=1$. My lecture notes go ...
0
votes
1answer
275 views

Fundamental Theorem of Contour Integration question (anti derivative)

This is the definition of the fundamental theorem of contour integration that I have: If $f:D\subseteq\mathbb{C}\rightarrow \mathbb{C}$ is a continuous function on a domain $D \subseteq ...
16
votes
13answers
1k views

Creative Thinking Questions?

Math is often intimidating to the average man due to its complex appearance. To show that math requires creative thinking, not just memorization, I was wondering if anyone had any math problems that ...
0
votes
1answer
33 views

Equivalent definition of valuation

Let $R$ be a ring and given a power series $f \in R[[x]]$ , $f = \sum a_n x^n$: Define $r(f)= \textrm{max} \{i : a_i \neq 0 \ \textrm{and} \ a_j=0 \ \textrm{for all} \ j \leq i\}$ Is it always true ...
1
vote
2answers
202 views

Understanding Less Frequent Form of Induction? (Putnam and Beyond)

I won't paste the question here since my problem is not a technical one but a conceptual one. Book is here: (Page 22 of the pdf) I do not understand why it is necessarily to induct $2^{k}$ to show ...
7
votes
5answers
230 views

Notation for infinite product in reverse order

This question is related to notation of infinite product. We know that, $$ \prod_{i=1}^{\infty}x_{i}=x_{1}x_{2}x_{3}\cdots $$ How do I denote $$ \cdots x_{3}x_{2}x_{1} ? $$ One approach ...
6
votes
4answers
635 views

continuous invertible map discontinuous inverse

Is there an example of a continuous invertible map $f:X\to Y$ between topological spaces $(X,T_X)$ and $(Y,T_Y)$ such that $f$ is continuous, but its inverse $f^{-1}$ is not continuous?
2
votes
0answers
159 views

Confusion! Power series and integration

Consider the below power series: $\sum\limits_{n=1}^\infty \dfrac{x^{n}}{n^{2}}$ I know that it converges for $x\in [-1,1]$ and the sum $s(x)$ of the series is given by: $s(x) = - ...
0
votes
1answer
174 views

Find double limits

$f:\mathbb{R}^2\rightarrow\mathbb{R}, \ f(x,y)=x\cdot\mathbb{D}(y)$, where $\mathbb{D}$ is Dirichlet function (nowhere continuous function). Find all the limits: $\lim_{x\to 0}\lim_{y\to 0}f(x,y)$, ...
1
vote
1answer
349 views

A basis for the cut space C* of a graph

I'm reading Diestel's Graph Theory right now and came across an interesting exercise that I cannot solve. It has to do with the basis for the cut space $C^*$: Given a graph $G$, find among all cuts ...
0
votes
1answer
313 views

How to calculate accuracy of the sample approximate of expected value

I have independent random values $x_1 < ... < x_n$ with the same distribution function. I want to find expected value of this distribution by my . Of course it will be $X = \frac{1}{n} \sum_i ...
1
vote
1answer
153 views

Continuity of function of three variables

Check continuity of a function $f:\mathbb{R}^3 \rightarrow \mathbb{R}$ given by formula: $$f(x,y,z)=\begin{cases} \frac{xz + yz}{x^2+y^2+z^2} \text{ for }(x,y,z)\neq (0,0,0) \\ 0 \text{ for } ...
0
votes
1answer
49 views

Function of a RV

Let $x$ be a R.V. that is normally distributed with mean 0 (for simplicity we can even assume its a standard normal). Let $F$ be a monotonic function such that $F(0)=0$ Does it follow that y=F(x) is ...
5
votes
1answer
364 views

Excessive use of the Yoneda lemma

In a MathOverflow thread on "nuking mosquitos", Andrej Bauer offered the following proof: If two elements in a poset have the same lower bounds then they are equal by Yoneda lemma. I understand ...
10
votes
3answers
263 views

Does the fact that $\sum_{n=1}^\infty 1/2^n$ converges to $1$ mean that it equals $1$?

I have a clueless friend who believes that $$ \sum_{n=1}^\infty \frac{1}{2^n} $$ doesn't equal $1$ in the 'normal arithmetical sense'. He doesn't believe that this series flat out equals $$1$$ Is ...
1
vote
1answer
240 views

The sgn function and permutations

Let $P=\{(i, j)|1\leq i<j\leq n\}. $For $\sigma\in S_n$, define $\operatorname{sgn}\colon S_{n}\rightarrow \{\pm 1\}$ by $$ \operatorname{sgn}(\sigma)=\prod_{(i, j)\in ...
2
votes
0answers
281 views

Does this approach to the Collatz Conjecture make any sense.

I was playing around with the Collatz Conjecture and came up with the following: Take any positive integer. If it's odd, multiply by three and add one. If it's even, divide by two. Repeat ...
2
votes
1answer
191 views

Is every semi-simple ring a product of simple rings

I was wondering if the following statements were true; 1) Every semi-simple ring is a product of simple rings. 2) Every module over a division ring $R$ is free. I think both of these statements are ...
1
vote
2answers
71 views

Reduction of $f:\mathbb{C}^n\to \mathbb{C}$ into sum of $f_{ij}:\mathbb{C}^2\to\mathbb{C}$

I was browsing wikipedia the other day when I came across the following (paraphrased) claim: $$ \exists f_{ij}:\mathbb{C}^2\to \mathbb{C} \mbox{ s.t. } f(x_1,\dots,x_n)=\sum_{i,j} f_{ij}(x_i,x_j) $$ ...
1
vote
1answer
761 views

Integration from 0 to 0 - why does my calculator say “undefined” in one case, and “0” in another?

I have the following question: Using my calculator, $\displaystyle\int_0^{0} \frac{1}{x}dx$ is "undefined". But when I type $\displaystyle\int_0^{0} - \frac{\ln(1-t)}{t} dt$, the result is 0. What ...
4
votes
4answers
352 views

Density of a set. Exercise from Spivak.

I'm trying to do a series of exercises from Spivak's Calculus, in chapter 8, Least Upper Bounds. I'm trying to tackle these two exercises, $5.$ and $^*.6$ From $5.$ I have proven the first claim ...
1
vote
3answers
131 views

Simplified form of $x^{10/3}$

I'm in a intermediate algebra class and am confused about how to get the simplified form of $\sqrt[3]{x^{10}}$ I tend to want to write it as $x^{10/3}$ creating a mixed fraction then simplifying that ...
1
vote
2answers
24 views

Functions Preserve “ultra-ness” of Prefilters

I'm trying to prove the following proposition: Let $f:X\to Y$ be a map of topological spaces. Let $F$ be a prefilter on $X$. Then $(a)$ $f(F)$ is a prefilter on $Y$. $(b)$ If $F$ is an ultra ...
4
votes
1answer
94 views

Compression of equations and coincidence?

I stumbled across an interesting paper last night. Basically, it tries to see if mathematical equations have meaning by determining how well they "compress" the results. For instance, he says the ...
1
vote
3answers
211 views

Limit points of sets

Find all limit points of given sets: $A = \left\{ (x,y)\in\mathbb{R}^2 : x\in \mathbb{Z}\right\}$ $B = \left\{ (x,y)\in\mathbb{R}^2 : x^2+y^2 >1 \right\}$ I don't know how to do that. Are ...
4
votes
1answer
83 views

What does Mumford mean by “an extension” here?

From Mumford's Red Book, Chapter 2, Example K: Take $X = Y = \mathbb{P}^2$, and let $x_0, x_1, x_2$ and $y_0, y_1, y_2$ be homogeneous coordinates on $X$ and $Y$. Let $U_0 \subset X$ and ...
2
votes
2answers
65 views

Distribution of sum of two directions on hemisphere?

I previously asked the following question: Is average of two random directions also a random direction? Apparently the answer is no, so as a follow up question I would like to find an expression for ...
0
votes
3answers
2k views

Calculate third point with two given point.

I have a point for example A (0,0) and B(10,10). Now I want to calculate a third point which lies in the same direction. I want to calculate point (x3,y3). I need a formula to calculate the new point. ...
2
votes
3answers
106 views

Lesson : Solving System of Equations using matrices

I have a matrix $$ A = \begin{pmatrix} a & 0 & 0 \\ 2 & b & 5 \\ -3 & 1 & b \end{pmatrix} $$ in my try, I came up with $$ bx1 = 0,\quad x2 + 5/a x3 = 0,\quad x2 + a x3 ...
2
votes
1answer
1k views

Finding extremas of a three variable function

Find all points on he portion of the plane $x+y+z=5$ in the first octant at which $f(x,y,z)=xy^2z^2$ has a maximum value. Attempt; Since $x+y+z=5$; $x=5-y-z$. I plug this into the $f(x,y,z)$: ...
0
votes
1answer
422 views

Conditional Poisson process

I am having difficulty with the following problem: A store promises to give a small gift to every thirteenth customer to arrive. If the arrivals of customers form a Poisson process with rate ...
1
vote
1answer
226 views

Integrating rational functions of sine using residues

Can anyone point me to a book that discusses integrals of the following type? $$\int_{\mathbb{R}}^{}\frac{\cos(ax)dx}{1+x^2}$$
10
votes
2answers
2k views

trigonometry with alternative parametrizations of the circle

In trigonometry it is conventional when defining $\cos\theta$, $\sin\theta$, etc., to parametrize the circle by arc length $\theta$. Some trigonometric identities don't depend at all on which ...
6
votes
3answers
242 views

What's known about the number of primes in the range $(n..2n)$?

What asymptotic lower bounds are known on the number of primes between $n$ and $2n$, call it $\Pi(n) = \pi(2n)-\pi(n)$? Obviously Bertrand's Theorem is the statement that $\Pi(n) \geq 1$ for all $n$, ...
1
vote
2answers
129 views

Does the decimal portion of $n\log_3 2$ cover all irrational numbers between 0 and 1 repeatedly, where n is all positive integers?

Does the decimal portion of $n\log_3 2$ include all irrational numbers between 0 and 1 repeatedly, where $n$ is all positive integers? Oddly enough I ran into this question while fooling around with ...
4
votes
3answers
179 views

Is average of two random directions also a random direction?

Given two uniformly random directions on a hemisphere, n0 and n1, is the normalized sum of these vectors also a uniformly random direction on the same hemisphere?
0
votes
1answer
641 views

Marginal PDF from a joint PDF with an integral that does not converge

I have a joint PDF that has gone through some transformations of $f(x,y) = 12x\displaystyle\frac{1-y}{y^3}$,$0<x<y^2$, $0<y<1$ It definitely is a valid PDF as it has a double ...
2
votes
1answer
180 views

I have a question from the Book of Leithold.

I have solved it but i need a review for this problem. This is the question: Find an equation of the tangent to the curve $y=\frac{1}{3}x^2+2$ which is perpendicular to the line $x-y=0$. The ...
2
votes
1answer
151 views

criteria if given irrational number is or not transcedental

It is known that constants $\pi$ and $e$ are irrational numbers but also transcedental. Where consist difference between irrationality and transcedentality. How we know that given irrational number is ...
4
votes
1answer
658 views

Software to render formulas to ASCII art

I guess all computer algebra systems have command line interfaces which render formula as 2D monospace "ascii art". But the only standalone tool to render TeX to 2D I know is ...
0
votes
2answers
167 views

Why is special about the reverse inclusion in the following example ?

Given a point $x$ in a topological space, let $N_x$ denote the set of all neighbourhoods containing $x$. Then $N_x$ is a directed set, where the direction is given by reverse inclusion, so that $S ≥ ...
2
votes
3answers
194 views

Is there an operational isomorphism from $(\mathbb{Z},+)$ to $(\mathbb{Q}^{+},\cdot)$?

Let $\left(\mathbb{Z},+\right)$ and $\left(\mathbb{Q}^{+},\cdot\right)$ be groups (let the integers be a group with respect to addition and the positive rationals be a group with respect to ...
1
vote
2answers
379 views

Finding the second-degree polynomial that is the best approximation for cos(x)

So, I need to find the second-degree polynomial that is the best approximation for $f(x) = cos(x)$ in $L^2_w[a, b]$, where $w(x) = e^{-x}$, $a=0$, $b=\infty$. "Best approximation" for f is a function ...
2
votes
0answers
59 views

gamma funtion and estimates-typo or mistake?

In one of the lecture notes I've found that $C_n$ $$ C_n= \begin{cases} \frac{n!}{\sqrt 2 \Gamma((n/2+1)}\pi^{-1/42^{-n/2}(n!)^{-1/2}} & n\text{ even} \\[4mm] \frac{2(n!)}{(\sqrt2n+1/(\sqrt2 ...

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