# All Questions

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### Solving a probability equation.

One group of 30 people won a contest and as a reward they got a free vacation to Hawaii. The hotel they will be in has 10 rooms with 3-bed (bed for 3 people). The question: How many ways can we ...
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### How to construct an $n$ equations in $n$ unknowns so that there is only a unique solution and all are equal

Suppose we are required to create $n$ equations in $n$ unknowns such that there is only 1 unique nontrivial solution and the solutions are equal to the others i.e. $x_1 = x_2 = \dots x_n$ What are ...
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### integrate the following expression again

Question: $x - x^2 + 1$ My answer: $\frac{x^2}{2} - \frac{x^3}{3} + {x} + C$ Correct answer: $\frac{x^2}{2} - \frac{x^2}{3} - \frac{x^3}{3} + x + C$ What am I doing wrong? thanks
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### Calculate $i ^ {i+1}$ and also $i^{i^{i^{\dots}}}$

I just wanted to ask the following questions please. The first I have is calculate $i^{(i+1)}$ and also $i^i$ I was just wondering if anyone can nudge me in the right direction to solve these ...
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### Limit of maximizer not equal to maximizer of limit

I am looking for functions $f_n,f$ defined on a subset of $\mathbb{R}$ with unique maximizers $\alpha_n, \alpha$, such that $f_n$ converges to $f$ pointwise, but the $\alpha_n$ do not converge to ...
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### Theorem : If the moment of order t exists for an RV X, moments of order 0 < s < t exist.

An Introduction to probability and statistics - Rohatgi Pg. No. 74 Theorem 2 Theorem : If the moment of order $t$ exists for an RV $X$, moments of order $0 < s < t$ exist. The proof is given ...
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### Is the empty function always a bijection?

Let $f_A:\emptyset\to A$ be the empty function with range $A$. The definition of a bijection as applied to this function is: $$\forall x,y \in \emptyset (x=y \implies f_A(x)=f_A(y))$$ negating you ...
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### Family of sets which intersect isn't connected

Find any family of sets $A_n$ such that $A_n$ are connected sets and $A_{n+1} \subset A_n$ and $$\bigcap_{n=1}^{\infty}A_n$$ is not connected. I tried find family of sets such that ...
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### Fit a function for these values.

This was asked to me in a interview its not a homework problem. I really wish my school would have given me such quality homework ;) Find out f(x) such that f(1) = 3 f(2) = 6 f(3) = 10
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### find nested closed balls of polynomials s.t. intersection is empty (in a metric space)

the space is the set of all polynomials on $[0,1]$ with metric sup. the space is not complete. we need to find explicitly a nested family of closed balls with radius (of each closed ball) goes to ...
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### How to apply perspective transform to Bezier curve?

I found that both Bezier curves and B-splines are described with a formula $p(t)=\sum\limits_{i=0}^d B^i_m p_i$ but in the case of B-splines $B^i_m$ are B-spline blending functions, while for Bezier ...
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### integrate the following expression

Question: integrate $\sqrt{t}$ My answer: $t^{\frac{1}{2}} = \dfrac{t^{\frac{3}{2}}}{\frac{3}{2}} +c$ correct answer: $\frac{2}{3}.t^{\frac{3}{2}} +c$ what am I doing wrong? thank's
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### An event that is very possible to happen, hasn't happened. Why? [closed]

I showed a friend an example of conditional probability, using the formula P(A|B) = P(A intersection B)/P(B). She asked an interesting question (a random question that didn't really have to do with My ...
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### How to proof that more than half binary algebraic operations on a finite set are non-commutative?

We know that if $S$ is a set and $|S|=n$, then there are $n^{n^{2}}$ binary algebraic operations, right? The cardinality of $|S|=n$ and cardinality of $|S\times S|=n^{2}$. Also, the number of all ...
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### Explaining Hypercomplex numbers to Children.

Imagine a highschool freshman walks up to you and asks you what hypercomplex numbers are. Explain to her, in a fair amount of detail, the different types of hypercomplex numbers in a way that any ...
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### An integral relating to Bernoulli polynomials

Show that $$\int_{0}^{1}B_{2n+1}(x)(\cot({\pi}x)-2\sin(2{\pi}x))dx{\sim}0$$ where $B_{2n+1}(x)$ is the Bernoulli polynomials.
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### Correct formal interval notation

I can't find any definitive answer on this topic, maybe that's because there isn't one, but I figured if there was a place to ask then SE was it! To describe a set in which $x$ and $y$ are in the ...
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### $\sum\limits_{k=1}^{n-1} \frac{1}{1-cos(\frac{2\cdot \pi\cdot k}{n})} = \frac{1}{6} \cdot (n^2-1)$

I want to show that $\sum\limits_{k=1}^{n-1} \frac{1}{1-\cos(\frac{2\cdot \pi\cdot k}{n})} = \frac{1}{6} \cdot (n^2-1)$ is true. I don't know how I could prove that, with induction I don't know how ...
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### Properties of the co-countable topology on $[0,1]$

I am trying to learn topology by myself. I have the following questions and my attempts of their proofs. Since I have no one to consult with, I am posting here. If anyone check my proofs that would be ...
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### How to enclose a ball more than once with a surface homeomorphic to $S^2$? In 3D.

In 3 dimensional space, how to enclose a monopole (either point-like or ball-like, both types exist.) more than once with a surface homeomorphic to $S^2$?
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### Do pushouts exist in a cartesian closed category?

If $C$ is a cartesian closed category, then is it necessary that all pushouts exist?
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### using $\sin x$ to get a function with range $[a,b]$

$\sin x$ is a nice function on $\mathbb{R}$ whose range is $[-1,1]$. can we 'adjust' it so that its range will be $[a,b]$. By 'adjusting' I mean changing the argument $x$ to some other argument which ...
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### Drawing balls from a bin with a color-specific probabilistic discard step

(I migrated this question here myself from MathOverflow since it might be too low level there.) I have a bin with $N = (k_a + b_b)$ total balls, $k_r$ of which are red and $k_b$ of which are blue. I ...
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### Rotating an $n$-dimensional hyperplane

Let $\mathcal{H}: \mathbf{x}^T\mathbf{w}+b=0$ be a hyperplane in the $n$-dimensional Euclidean space of column vectors. Is there a way of "rotating" the above hyperplane such that it coincides with ...
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### Prove that $[a,b]$ is connected space.

Prove that $[a,b]$ is connected space. I know that $\mathbb{R}$ with euclidean metric is connected space. I would like find surjective function $f: \mathbb{R} \rightarrow [a,b]$. Because $\mathbb{R}$ ...
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### a corollary of Cauchy's integral formula in several complex variables

I'm learning several complex variables. There is a corollary of Cauchy's integral formula that I don't know how to prove. Let $X\subset\mathbb{C}^n$ be a domain. For each multi-index $\nu$, for each ...
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### which one is larger $\sqrt[n]{x+\delta}-\sqrt[n]{x}$ or $\sqrt[n]{x}-\sqrt[n]{x-\delta}$?

Which is larger? $\sqrt[n]{x+\delta}-\sqrt[n]{x}$ or $\sqrt[n]{x}-\sqrt[n]{x-\delta}$? Algebraic justilation does not help.
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### Solving a first order non-linear ODE

I encountered a problem solving the following equation: $$t^2dy/dt+2ty-y^3=0$$ I have already tried the following steps, but to no avail: 1)Separation of variables 2.1)Integration Factor of 1 ...
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### Probability: the expected value in a dice game. [duplicate]

If a dice is thrown till the sum of the numbers appearing on the top face of dice exceeds or equal to 100, what is the most likely sum?
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### local gauge invariance of field's homotopy class? Every map $S^2\rightarrow \mathrm{group } G$ is homotopic to a constant map?

In a discussion of a gauge field theory with gauge group $G$, someone says we can use a celebrated result of E. Cartan to show the gauge invariance of matter field's homotopy class. And Cartan's ...
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### Let $\, g \,$ be a function defined on $(a,b)$ such that $\, a<g(x)<x$

I am stuck on the following problem that says : What I guess option (A) is not possible. But, I am not sure about the other options. Can someone explain? Thanks in advance for your time.
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### How find this sum $\sum_{k=1}^{n}[\arctan{f(k)}\arctan{g(k)}]$

Let $n$ be a positive integer. Compute ...

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