All Questions

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Question about proving a real number

If we know that for any, $\alpha \in \{0, 2\}^\mathbb{N}$ that $0 \le \sum\limits_{k = 0}^\infty\frac{\alpha(k)}{3^k} \le 3$, then what property of real numbers do we have to use to prove that ...
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Applications of Geometry to Computer Graphics

How is differential geometry (or any type of theoretical math) related to computer graphics and/or computer programming? A friend of a friend of mine has only a bachelors degree in pure math and got ...
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Is a ruled surface of degree>2 always singular?

Let $X=\mathbb{C}\mathbb{P}^3$ and let $V\subset X$ be a closed algebraic sub variety. By V-is ruled, I mean that for every point in $V$ there is a line passing through it which also lies in V. ...
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Strong epimorphic counit iff conservative right adjoint?

On page 13 of Lack and Street's Combinatorial Categorical Equivalences, it is written (but not proven) that: A right adjoint is conservative if and only if the components of the counit are strong ...
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Almost pointwise inner automorphism of free products of groups.

Let $A,B$ be groups, let $G = A\ast B$ be their free product and let $\phi \in \text{Aut}(G)$ be a automorphism of $G$. We say that $\phi$ is pointwise inner if $\phi(g) \sim_G g$ (there is $w \in G$ ...
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How to find the minimum value of the expression?

Let $x$, $y$, $z$ be three nonegative real numbers and $x^2 + y^2 + z^2 = 5.$ Find the minimum of the expression $$E=\dfrac{1}{2}(x^2 y^2 + y^2 z^2 + z^2 x^2) + \dfrac{96}{x + y + z + 1}.$$ What ...
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Composing a smooth even function and square root

Let $f:\mathbb{R}\to\mathbb{R}$ be smooth and satisfy $f(-x)=f(x)$ for all x. Define $g:[0,\infty)\to\mathbb{R}$ by $g(x)=f(\sqrt{x})$. Is $g$ necessarily smooth at $0$? I guess the answer is ...
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Decision and the Uncountable Spectrum

In 2000, Hart, Hrushovski, and Laskowski classified all complete first order theories in a countable language up to their uncountable spectra. However, does this also imply that given a $any$ ...
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Calculate the supremum of $\frac{e^{|\gamma_n+1|}-1}{|\gamma_n+1|}$

If $\{\gamma_n\}$ is a sequence of real number and $\exists M>0$, finite, such that $|\gamma_n|\leq M$, find the supremum of the following sequence: $$\frac{e^{|\gamma_n+1|}-1}{|\gamma_n+1|}$$
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Probability Question: Who's right, me or the book?

I'll be giving some classes on probability theory later this year, and so I've been going through the textbook to check that I'm up to speed. I came across the following question: The discrete random ...
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Linear Transformation from V to W (bijective) Show that T(v) is a basis of W if B is a basis of V.

$V, W$ two vector spaces and $T: V \to W$ is a bijective linear transformation. $B$ is a basis of $V$. Prove that $\{T(\mathbf{v}) | \mathbf{v} \in B\}$ is a basis of $W$. I started by doing ...
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Chain rule application in fundamental Theorem of Calculus

I have attached a question that I came across in trying to understand the fundamental theorem of calculus. The solution (highlighted with an arrow). I have difficulty understanding why the chain rule ...
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Example of first order logic without equality.

Most logic texts say that = is a special symbol which is always part of our language. It is my understanding, though, that it is perfectly acceptable to consider = to be an ordinary binary relation ...
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$H_I^n(R)=0$ and $H_I^n(M)\neq 0$ [on hold]

Find R and M as an R-module such that $H_I^n(M)\neq 0$ and $H_I^n(R)=0$, where I an ideal of R and $n\in N$. I found it in Cohen Macaulay rings. there`s nothing to find.
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If $f_n(t):=f(t^n)$ converges uniformly to continuous function then $f$ constant

Let $f \colon [0,1] \to \mathbb R$ be continuous and let $f_n$ be defined by $$f_n(x):=f(x^n), \qquad x \in [0,1], \,\, n \in \mathbb N.$$ Suppose there exists a continuous function $g$ on ...
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How to show if A is denumerable and $x\in A$ then $A-\{x\}$ is denumerable

My thoughts: If $A$ is denumerable then it has a bijection with $\mathbb{N}$ So therefore $A\rightarrow \mathbb{N}$. Then x is a single object in A and A is infinite. So if a single object is ...
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Quick question about $\epsilon -\delta$ proofs

There is one step in $\epsilon - \delta$ proofs that I hope somebody could bring clarity to for me. Say we wanted to show $\displaystyle \lim_{x \to 2} x^2 = 4$. Somewhere along the proof we would ...
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Why are $L^p$-spaces so ubiquitous?

It always baffled me why $L^p$-spaces are the spaces of choice in almost any area (sometimes with some added regularity (Sobolev/Besov/...)). I understand that the exponenent allows for convenient ...
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