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1,111,828 questions
250 views

A doubt in Hatcher's Algebraic Topology.

I refer to pg. 27 of Hatcher's Algebraic Topology. I refer to the part where Hatcher proves that $f.(g.h)\cong (f.g).h$ For the life of me, I cannot figure out how the diagram on the right proves ...
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Rotate $xyz$ by use of pitch and yaw around origin

I have a project for a game which uses pitch/yaw for the direction of a players head. The pitch ranges from $0$ to $180$ and the yaw is $0$ to $360$. Yaw modifies $X$ and $Z$, pitch modifies the $Y$, ...
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Solutions to functional equation $f(f(x))=x$

Is there any more solutions to this functional equation $f(f(x))=x$? I have found: $f(x)=C-x$ and $f(x)=\frac{C}{x}$.
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The meaning of Inverse Matrix

I am studying Linear Algebra, I have 3 questions in my mind What does an inverse matrix mean. I am trying to have a meaning of it, but I don't really understand. When a matrix does not have an ...
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Linearize a first order differential equation

The system described by $x'=2x^2-8$ is linearized about the equilibrium point -2. What is the resulting linearized equation? Answer is $x'=-8x-16$. How? I have no idea how it went from the first ...
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Is it possible to have a inflection on a vertical asymptote?

I found the derivative of a function to be f'(x)=8/x^3 and thus its second derivative as f''(x)=0/3x^2. After setting the second derivative to zero and doing the substitution into the parent function, ...
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Minkowski Distance Metric

Given compact sets $A$, $B$, define the Minkowski distance between the two sets as: $$\delta(A,B):= \inf \{ r: B \subseteq \mathscr{N}_r (A) \, \, \text{and} \, \, A \subseteq \mathscr{N}_r (B) \}$$ ...
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Weak nullstellansatz in Atiyah-Macdonald 5.17

$\newcommand{\fm}{\mathfrak{m}}$ Problem 17 in the exercises after the 5th chapter of Atiyah-Macdonald is the following (with some references and hints omitted): Let $X$ be an affine algebraic ...
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Let $f: [0,1] \rightarrow \mathbb{R}$ be analytic. Let $r_f(x)$ be the radius of convergence of $f$ at $x$. Is $r_x(f)$ continuous? Alternatively, is there an $r_{min}$ I can choose so ...
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Flat connection with non-trivial holonomy? I cannot get it

maybe this is a dumb question, but I cannot understand how a principal $G$-bundle can have non-trivial holonomy with a flat connection. Maybe I'm missing something, but doesn't Ambrose-Singer theorem ...
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Distribution of random variable

I need help with this problem: Let $(X_n)_{n\in \mathbb{N}}$ a sequence of i.i.d$\sim$Uniform$(\{0,\dots,9\})$ random variables. What is the distribution of $$X= \sum_{n=1}^{\infty} X_n 10^{-n}$$ ...
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Can an inflection exist if there's no max/min?

Very quick question: if a function doesn't have a maximum nor minimum, can it still have a point of inflection? I believe that these two go hand in hand and without one you can't have the other but ...
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number of solution to the given equation.

a,b,c, are all non-negative integers such that a + b + c=100 and 1000a + 300b + 50c = 10000 How many such triplets are possible? i have tried to reduce ...
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A closed form of $\sum_{k=1}^\infty\frac{(-1)^{k+1}}{k!}\Gamma^2\left(\frac{k}{2}\right)$

I am looking for a closed form of the following series $$\mathcal{I}=\sum_{k=1}^\infty\frac{(-1)^{k+1}}{k!}\Gamma^2\left(\frac{k}{2}\right)$$ I have no idea how to ...
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Show that two spaces are not homeomorphic

Let $H=[-1,1]\times \{0\}$ and $V=\{0\}\times [-1,0)$ in the plane. Let $T=H \cup V$. Show that $T$ is not homeomorphic to the unit interval $I=[0,1]$. My idea for this problem is that , if we remove ...
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Finding whether a sum of numbers in a set generate another number

I have a set of numbers $\{a_1,\dots,a_n\}$ and another number $k$. I need to find whether sum of any combination of numbers in the set produces $k$. It can be $a_1 + a_2$ or $a_1 + a_2 + a_3 + a_7$. ...
643 views

Proof of convergence of Kaprekar's Constant

I've tried googling this one a bit but nothing seems to come up, even though its considered to be a well known fact. Why does the kaprekar process of taking a 4 digit number: L, generating L' and L'' ...
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Alternative function definitions

If you go to the wikipedia page on the sine function or the log function you'll find a number of different definitions of these functions. I know that what defines a function are it's values, for ...
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Books that integrate physical reasoning with mathematical reasoning? mathematicians?

As the title says, can anyone help me to find any book that shows how physical reasoning using concepts from classical/quantum mechanics and physics in general can enlighten us about mathematical ...
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Lebesgue integral of $\chi_{\mathbb{Q}}: \mathbb{R} \rightarrow \mathbb{R}$

Suppose $(X, \mathfrak{A}, \mu)$ is a measure space. Let $\phi$ be a simple function with canonical representation $\sum^{k}_{n=1} a_{n} \chi_{E_{n}}$. I know we define the Lebesgue integral of $\phi$ ...
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Is there a way to calculate the area of this intersection of four disks without using an integral?

Is there anyway to calculate this area without using integral ?
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Finding the area of a triangle from vertices? Linear Algebra

I pretty much did this problem, but I failed to get the few last blanks where they ask the area. Its confusing, they say its half the volume of matrix (u v w) in the start of the question. which means ...