# All Questions

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### How many functions f : $\{0,1,2,3\}^n$ -> $\{1,2,3\}$ are there, that take the value 1 exactly once?

I know the answer to this question is $4^n \cdot 2^{4^n-1}$ but i don´t understand at all how to arrive at this result.
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### Solve differential equation: x * dy/dx = 1 − y 2 to find y in terms of x

Solve differential equation : $$x \frac{dy}{dx} = 1- y^2$$ Find $y$ in terms of $x$. I've got upto here : $3y-y^3 = \ln \bigg|\dfrac{x}{2}\bigg|$, if it's correct. How do I get $y$ in terms of ...
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### Percentage of class 8

Population of a town increases by 5 % annually. If population now is 185220. Find the population last year. Solve by taking x or 100. Is answer 175959. If yes verify or if no prove.
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### Are these subgraphs of equal size?

Let $G$ be a (finite) connected graph. Let $H_1,\ldots, H_n$ be distinct subgraphs of $G$ of the same order (=number of vertices). Suppose that there exists $k$ such that every edge of $G$ belongs ...
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### induced connections

I am studying "Spin Geometry" and i have the following question. In page 106 after the connection 1-form is defined there is a discussion about induced connections.Supposing we have a connection on ...
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### Minimize the sum of the lengths of the cevians

This question is inspired by this other one. $ABC$ is an acute-angled triangle. Given a point $P$ inside $ABC$, we take $P_A,P_B,P_C$ as $AP\cap BC,BP\cap AC,CP\cap AB$ respectively. For which ...
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### Matrix inverse using Sherman–Morrison formula

For a square matrix $(\mathbf{D^TD})$, how to compute its inverse $(\mathbf{D^TD})^{-1}$ where $\mathbf{D=[X\;a]}$ i.e., a column vector added to $\mathbf{X}$ and we know the inverse ...
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### The meaning of dependent and independent variables in ODEs

This thought has occured to me a few days ago, and now I am puzzled about some fundamental properties /definitions in the theory of differential equations. Suppose I have an ode ...
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### Simple questions (linear and non linear graphs)

1.A line has a gradient of $\frac23$ and passes through the points $(4,7)$. Do the points $(7,9)$ and $(-4,5)$ lie on the line? $MN$ and $OP$ are parallel. What is the $y$-coordinate for $N$? ...
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### Statistics problem:independence of order statistics

Let $X_1,...,X_n$ be the independent and identically distributed samples taken from an uniform distribution on $(a,b)$. $-\infty<a<b<\infty$. $X_{(1)},...,X_{(n)}$ are the order statistics of ...
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### Creating an increasing function out of a continuous function

Given a continuous function $f$ over an interval, must there exist a continuous, increasing function $g$ such that for all $x,y$ $$|f(x)-f(y)| \leq |g(x)-g(y)|$$ I've tried assuming the opposite, ...
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### Continuation of uniformly continuous function on a metric space.

$(Y, \rho_Y)$ - complete metric space, $(X, \rho_X)$ - metric space, $(X^*, \rho^*)$ - completion $(X, \rho_X)$. $f \in C(X,Y)$ - uniformly continuous function from $X$ to $Y$. It's necessary to ...
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### Limit of complete Elliptic integral of the first kind times 1 minus modulus squared

I am stuck trying to prove that $\lim_{k\rightarrow 1} (k^2-1) \int_{0}^{\pi/2} \frac{d\theta}{1-k^2 \sin^2(\theta)}=0$ Can anybody help me out here? ;-) thanks!!
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### What part of digraphs are posets?

Let $D_n$ denote the collection of all directed graphs on vertex set $[n]=\{1,\ldots, n\}$. Let $P_n$ denote the subcollection of all digraphs in $D_n$, which have the structure of a partial order. ...
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### How to show that the limit exists?

the limit of $$f(x,y) = \frac{x}{\sqrt y}$$ as $(x,y) \rightarrow (0,0)$? I have approached the origin via different paths and have come to the conclusion that the limit exists. I just cant prove it. ...
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### Does a field extension equal $F(\alpha) = a_0+a_1\alpha+a_2\alpha^2+a_3\alpha^3+\cdots$

Let $F$ be a field, does $F(\alpha)=a_0+a_1\alpha+a_2\alpha^2+a_3\alpha^3+\cdots$? If it does, this was never explicitly stated, and I have gained confusion from this, since in the case that: ...
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### Positive definite and continuous function

I'm reading a proof with the following statement. Let $\nabla^{2} f(x)$ be the hessian matrix and continous. Assume it is positive definite in $x'$. Now there exists an open ball around $x'$ such ...
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### Find the interval for which $2\arctan x + \arcsin \frac{2x}{1+x^{2}}$ is an independent of x?

I used formula and simplified the expression to $\Rightarrow 4\tan^{-1} x$
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### Proof of Central Limit Theorem via MaxEnt principle

Let $X_i$'s be i.i.d. with mean $0$ and variance $\sigma^2$. After reading Jaynes' book: Probability the Logic of Science, I decided to try out and actually prove CLT via the following steps: a) ...
Let $X_1,...,X_n$ be the independent and identically distributed samples taken from an uniform distribution on $(a,b)$. $-\infty<a<b<\infty$. $X_{(1)},...,X_{(n)}$ are the order statistics of ...
Let $a_1, \ldots ,a_k,b,n$ be integers with $n > 0$. Show that the congruence $a_1z_1+\cdots+a_kz_k \equiv b (\text{mod n})$ has a solution $z_1, \ldots , z_k \in \mathbb{Z}$ if and only if $d|b$, ...