# All Questions

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### How to identify surfaces of revolution

Given a surface $f(x,y,z)=0$, how would you determine whether or not it's a surface of revolution, and find the axis of rotation? The special case where $f$ is a polynomial is also of interest. A ...
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### Is a set closed under finite intersections? (about filters)

In my research I was faced with the problem (as a special example and a pattern for more general problems) whether the family $\operatorname{GR} ( \Delta \times^{\mathsf{FCD}} \Delta)$ of sets is ...
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### Proving a certain map on the closed unit disc must be the identity

Prove: Let $f$ be a continuous function on the closed unit disc with two properties: 1. $f$ is the identity on the boundary, i.e., on the unit circle. That is, if $|z| = 1$, then $f(z) = z$. 2. ...
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### Tangent sheaf of the Picard scheme

Let $X\overset{f}\to S$ be a flat morphism of schemes, and $S=\operatorname{Spec}(k)$ for an algebraically closed field $k=\bar{k}$. I'm just interested, for the moment, with schemes $X$ corresponding ...
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### Errata for the first edition of Robert Strichartz's “ The Way of Analysis”

I've recently acquired a copy of the very popular analysis text, The Way of Analysis by Robert Strichartz. It is the 1996 first edition, which was the only way I could afford a copy. Unfortunately, ...
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### Help in this proof

I need help to understand this proof: Theorem Let $X$ be an affine algebraic set and take $f\in k[X]$. Consider the open principal subset $X_f=\{x\in X;f(x)\neq 0\}$. Then ...
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### Commutative ID does not finitely generate its field of fractions

I want to prove that if we have a commutative integral domain $D$ with field of fractions $F\neq D$ then $F$ is not finitely generated as a $D$-module. (In this question it may be the case that ...
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### What is an awesome book as an introduction to hyper groups

I'm a grad studen and i'm choosing an area to follow on my doctorate (in?) and I've been thinking about extension of topological group theory results to topological hypergroups, but for that i need to ...
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### Unit diameter pentagons with maximum area

In the euclidean plane, if one considers the set of quadrilaterals having unit diameter (maximum distance between two points in the convex envelope), it is quite easy to give a description of the ...
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### Monotonic version of Weierstrass approximation theorem

Let $f\in\mathcal{C}^1([0,1])$ be an increasing function over $[0,1]$. Prove or disprove the existence of a sequence of real polynomials $\{p_n(x)\}_{n\in\mathbb{N}}$ with the properties: $p_n(x)$ ...
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### Variance With Martingales Problem - Answered; Ignore the Bounty

Let $(X_{j})_{j \geq 1}$ be random variables such that $X_{j}$ is $\mathcal{F}$-measurable for each $j$, where $(F_{j})_{j\geq 1}$ is an increasing sequence of $\sigma$-algebras. Assume ...
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### Banach-Stone Theorem

In here, Banach first proved a lemma which used directional derivative to identity peak point of functions. Then he used the lemma in the proof of Banach-Stone theorem. After several years, Stone ...
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### Calculating circle in taxicab geometry

I struggle with the problem of calculating radius and center of a circle when being in taxicab geometry. I need the case for two and three points including degenerate cases (collinear in the three ...
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### Consider such a homogeneous linear recurrence

$a_n$=3$a_{n-1}$+18$a_{n-2}$ with initial conditions $a_0$=1 and $a_1$=9. a). Use the algorithm to find a solution. b). Use induction to show that the solution that you found in a) is correct, be ...