# All Questions

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### Infinite Set Proof (Countable and Uncountable )

I can't figure out this problem, I have to prove that Q × Q is denumerable, but I have no idea how to do it. Thanks
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### Mathematical Induction.

We know that because of commutative law we have, $a+b=b+a$ and similarly we have associative property and likewise distributive. But isn't there a need to prove following laws by mathematical ...
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### probability of the empty set for arbitrary probability measures

I have a probability space $(\Omega, \mathcal{P}(\Omega), P)$. I want to know the probability of the empty set $\{\}$. Intuitively, I would say this probability is zero. It certainly is for the ...
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### Group actions: Why do we place the condition that $S$ be finite in the following theorem?

Theorem. Let $G$ be a group, $S$ be a $G$-set, and $S$ be finite, then $$|S|= \sum_{a \in A} [G : G_a],$$ where $A$ is a subset of $S$containing exactly one element from each orbit $[a]$. Here, $G_a$ ...
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### Tychonoff spaces with small weight

Let $\kappa$ be an infinite cardinal. Is there a Tychonoff space $(X,\tau)$ such that $|X| = 2^\kappa$ and $(X,\tau)$ has weight $\kappa$ (= a basis consisting of $\kappa$ elements)?
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### Is split-complex $j=i+2\epsilon$?

In matrix representation imaginary unit $$i=\begin{pmatrix}0 & -1 \\ 1 & 0 \end{pmatrix}$$ dual numbers unit $$\epsilon=\begin{pmatrix}0 & 1 \\ 0 & 0 \end{pmatrix}$$ ...
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### Neat Diophantine Equation Question

After some fairly tedious work including studying multiple different cases separately, I have found all the solutions to $$a^n+1=b^2$$ where $a$, $b$, $n$ can take on the value of any integer, be it ...
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### Construct a procedure which determines the location of the shadow of a rectangluar box.

I drew a 3d rectangular box on a coordinate plan consisting of x, y, and z. A procedure is to be created that will determine the location of the shadow of the box on one of the coordinate planes. I ...
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### Direct sum of two spaces

Let $\alpha_1=[1,1,0,1]$, $\alpha_2=[1,0,1,1], \alpha_3=[1,1,1,1],\alpha_4=[0,1,1,1]$ be a vectors from $\mathbb{R}^4$ let $U=span(\alpha_1, \alpha_2) \ and \ W=span(\alpha_3, \alpha_4)$ Check that ...
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### Am I upper bounding or lower bounding my probability distribution function?

I came across a probability distribution function in my work, it is however difficult to find in closed form, therefore I am looking to either upper bound or lower bound it. Assuming $a,b,T$ are ...
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### Existence & uniqueness of a second order ODE

Details on the model can be found here under III titled "Will The Valve Hold?". ...
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### Does there exist a 4D torus with a spherical cross-section, analogous to a circle for the 3D case?

I don't mean to be a bother. It seems as though the answer may be obvious, but then, seemingly simple math questions can have surprising answers. I should also like any pointers re: the general ...
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### Determinant: Continuity?

Building on the previous thread: Determinant: Definitions? Presumptions: Differential Geometry, Functional Analysis Given a vector space $V$. Consider an endomorphism $T:V\to V$. Define its ...
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### A sum involving binomial coefficients and a simple fraction

Let $a_1$ and $a_2$ be real numbers. Let $n_1$ and $n_2$ be positive integers. Finally let $\theta$ be a real number which is different from a negative integer. By the generalizing the result from ...
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### What can we say about the set X?

We have a certain set $X$ for which is valid: $\forall U\subset X:[ U\neq X ]\rightarrow U\nsim X$. What can we say about $X$? I think we've got to use the axiom of choice here. My first guess would ...
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### Statistics question involving exponential distribution and (maybe) gamma function

This is from a past stat exam that I am studying for my final tomorrow (lol). I believe this might have to do with gamma function. Could someone guide me through step by step of how to do this? An ...
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### how to find out intial direction and angle of collision

i have a problem in my game. I have a wall where a ball hit to a wall from anywhere. i need to give it to the direction according to collision law. Let suppose if a ball thrown from x == 0 and y == 0 ...
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### C*-algebras: States?

I'd like to better understand states on C*-algebras. I suppose basic facts about functional analysis. (C*-algebras, spectral theory, functional calculus, Banach-Alaoglu, etc.) What properties ...
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### My question pair about diagonalization

Let A = $\begin{bmatrix}1 & 1 & 4\\0 & 3 & -4\\0&0&-1\end{bmatrix}$. Is the matrix $A$ diagonalizable? If so find a matrix $P$ that diagonalizes $A$. Can you write $A$ as a ...
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### What is my maths score as a percentage?

In my maths module, there are 2 phases. In the 1st phase, there are 2 tests. In the 2nd phase there is 1 test. Phase 1's total weight is 60% and phase 2's total weight is 40%. This is what I ...
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### How to show that the following relation is not an equivalence relation?

We have the relation $\sim$ in $\mathbb{R}^n$: $x\sim y \leftrightarrow d(x,y)\in \mathbb{Q}$, where $d(x,y)=\sqrt{\sum^n_{i=1}(x_i-y_i)^2}$. How do you prove that this isn't an equivalence relation ...
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### CDF of RVs taking infinite values

How can we define the CDF of a RV that takes positive infinite values with a tagged probability? Thanks in advance
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### If $X$ is Hausdorff, then so is $E$

Let $q:E \to X$ be a covering map. If $X$ is Hausdorff, then so is $E$. OK, suppose $X$ is Hausdorff and let $x,y \in E$ with $x\neq y$. Let $V$ denote the evenly covered neighbourhood for $q(x)$, ...
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### Inverse Laplace Transformation

I was solving a problem but I am stuck at it. Here is the question : $\frac{7s^2+9s+3}{(s^2-12s+40)(s^2+9)}$ Find inverse Laplace transform. I performed these operation : ...
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### Show that $\{e^{in}: n\in\Bbb N\}$ is Dense in the Unit Circle

This problem gave me fits when I was in grad school. Looking back at it now, it still escapes me. The problem is from Conway's Functions of One Complex Variable. I'm looking for a proof from basic ...
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### Odd coefficient in $M\in \mathcal{M}_n(\Bbb{Z})$ satisfies $n\le m\le n²-n+1$.

Let $M\in \mathcal{M}_n(\Bbb{Z})$ I would like to prove that all odd coefficient of $M$ satisfies $n\le m\le n²-n+1$. In fact I don't see why $m$ is necessary bigger than $n$. I can only prove ...
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### Fourier transform, quadratic function

I'm trying to compute this convolution: $\frac{2 \alpha}{\alpha ^2 + 4 \pi ^2 x^2} * \frac{2 \beta}{\beta ^2 + 4 \pi ^2 x^2}$ I know that the Fourier transform of a convolution of two functions is ...
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### Prove existence of (Nash) equilibrium

My question is about proving the existence of Nash equilibrium for a game involving two players. $x$ is player 1's strategy and $y$ is player 2's strategy; both strategies are continuous. For each ...
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### Reference for work on abelian divisible groups $G$ such that for every $n \in \mathbb N , g \in G , \exists$ unique $x \in G$ such that $g=x^n$

Is there any work or reference in the literature about those abelian divisible groups $G$ such that for every $n \in \mathbb N , g \in G , \exists$ unique $x \in G$ such that $g=x^n$ ; I think then I ...
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Why $$\left\|\sum_{i=1}^nx_i\right\|^2\leq n\sum_{i=1}^n\|x_i\|^2$$ for arbitrary norm on the inner product space over the real field? My attempt $$\left\|\sum_{i=1}^nx_i\right\|^2 = ... 1answer 20 views ### Mechanics question involving integration. a particle P moves so that its position vector r satisfies the differential equation$$\frac{dr}{dt}= c \times r,$$where c is a constant vector. Show that P moves with constant speed on a circular ... 2answers 34 views ### Prove \lim_{x \to 0} \frac{e^{\sin(x)} - e^{\tan (x)}}{e^{\sin (2x)}-e^{\tan (2x)}} = \frac{1}{8} Here's a nice little problem.$$\lim_{x \to 0} \frac{e^{\sin(x)} - e^{\tan (x)}}{e^{\sin (2x)}-e^{\tan (2x)}} What's the quickest way to do this? One line solutions will be applauded :D Cheers, my ...

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