# All Questions

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### 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 ...
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### 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 $-3$. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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?
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### 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 ...
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### 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 ...
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### 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: $$\exists f_{ij}:\mathbb{C}^2\to \mathbb{C} \mbox{ s.t. } f(x_1,\dots,x_n)=\sum_{i,j} f_{ij}(x_i,x_j)$$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...