# All Questions

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### Determine the distribution of $\int_0^t (W_s-\frac{s}{t}W_t) ds$, where $(W_s)_{s\geq 0}$ is a brownian motion

I have to find the distribution of $X_t:=\int_0^t (W_s-\frac{s}{t}W_t) ds$ where $(W_s)_{s\geq 0}$ is a brownian motion. I already showed the first integral $\int_0^t W_s ds$ is ...
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### There are no field structures on $\mathbb{R}^3$, but what of $\mathbb{R}^n$ for $n\geq 4$?

Has it been proved that there do not exist nice field structures on $\mathbb{R}^n$ for $n\geq 4$? The quaternions $\mathbb{H}$ fail due to lack of commutativity and the bicomplex numbers ...
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### Uncountable subset $S$ of $(0,1) \implies (0,1)$ has subinterval of limit points of $S$?

If $S$ is an uncountable subset of $(0,1)$, is there an interval $(a,b) \subseteq (0,1)$ such that every point in $(a,b)$ is a limit point of $S$?
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### Solve the diophantine equation $ax+by=xyc$

Let $a,b,c$ be non-zero co-prime integers such that $a+b \neq c$, and $x.y\neq 0$, solve the diophantine equation $ax+by=xyc$.
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### Sum of Catalan numbers

What is $C_1 +C_2 + C_3 +... + C_n$, where each $C_i$ is Catalan number? I want to know if we can bound this sum by some function of $n$. I am looking for an upper bound. For sure it is less than ...
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### When do two configurations of points belong to the same Euler Equivalence Class?

When can I say, of two or more configurations of points in a plane, that they belong to the same Euler Equivalence Class? From Euler's rotation theorem, I gather that two configurations of points are ...
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### Is this a $\sigma$-algebra(closed under contable union)?

Could I say that this $$M=\{X\subseteq\Omega=[0,1):x\in X\iff y\in X\}$$ is an $\sigma$-algebra? I don't see whether it is closed under countable union. x,y are two singetons of $\Omega$ For ...
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### If $f$ is increasing toward $1$, then $\sup\{f(x)\sin x \}=1$

Suppose $f$ is an increasing monotone function in $(0,\infty)$. If $$\lim_{x \to \infty} f(x)=1$$ then $$\sup\{f(x)\sin x\mid x>0\}=1$$ I am not really sure how to approach this, any help will ...
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### Comparison theorem for ODE

Here is something I'm trying to prove: Conjecture: Suppose $f'(x) \leq \phi(f(x), x)$ and $f(a)=\alpha$. Suppose $g'(x)=\phi(g(x),x)$ and $g(a)\geq \alpha$. Then $f(x)\leq g(x)\,\,\forall x$. ...
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### System of linear equation matrix? [duplicate]

How would I do this question. Determine the value(s) of $h$ such that the matrix is augmented of a consistent linear system. My matrix \begin{bmatrix} 1&h&4\\ 3&6&8 \end{bmatrix} I ...
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### Determine if projection of 3D point onto plane is within a triangle

In 3D, given three points $P_1$, $P_2$, and $P_3$ spanning a non-degenerate triangle $T$. How to determine if the projection of a point $P$ onto the plane of $T$ lies within $T$?
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### Closed sets in product topology

I have an assignment, I have to proof that arbitrary product of close sets is closed in the product topology, I think I have to use complements and treat with opens, what do you think?
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### Finding Transformation matrix between two 2D coordinate frames [Pixel Plane to World Coordinate Plane]

The question I'm trying to figure out states that I have N points (Pa1x,Pa1y) , (Pa2x,Pa2y)...(PaNx,PaNx) which correspond to a Pixel plane xy of a camera, and other N points (Pb1w,Pb1z) , ...
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### Alternative Monty Hall Problem

So the typical set up for Monty Hall problem, I there are 3 doors where 2 have goats and 1 has a car. I, the contestant, get to randomly guess a door looking to get the one with the car, after this ...
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### If $(\sqrt{y^2-x}-x)(\sqrt{x^2+y}-y)=y$ then $x+y=0$.

Let $x,y$ be real numbers such that $$\left(\sqrt{y^{2} - x\,\,}\, - x\right)\left(\sqrt{x^{2} + y\,\,}\, - y\right)=y$$ Show that $x+y=0$. My try: Let ...
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### Need verification - Given a Hermitian matrix and two eigenvectors corresponding to distinct eigenvalues, show x and y are orthogonal.

Claim: Let $A \in \mathbb{C}^{mxm}$ be hermitian ($A = A^*)$. If $x$ and $y$ are eigenvectors corresponding to distinct eigenvalues, then x and y are orthogonal. Proof: Let $x$ and $y$ correspond to ...
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### Reference request for “Hodge Theorem”

I have been told about a theorem (it was called Hodge Theorem), which states the following isomorphism: $H^q(X, E) \simeq Ker(\Delta^q).$ Where $X$ is a Kähler Manifold, $E$ an Hermitian vector ...
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### proof involving a triangle with a point inside it. [duplicate]

Suppose we have a triangle, call it triangle $XYZ$, and a point $W$ inside triangle $XYZ$. How would I prove that $XY + YZ > XW + WZ$? So the way I labeled everything, point $X$ is the bottom left ...
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### $\|x+y\|$ vs $\|x-y\|$ for reverse triangle inequality

So I am using the text "Elementary Functional Analysis" by MacCluer. In it, for Exercise 1.1, it asks us to prove the Reverse Triangle Inequality (which I have done in the past, using the $x=(x-y)+y$ ...
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### Equation for Circle in 3D Space Given Center, Radius, and Point

I'm looking for how to derive the equation of a circle, in 3D space, given the following information: The Center Point The Radius One point on the circle I understand that this is functionally the ...
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I'm having trouble proving the following property of ordinal numbers. If $a, b, c$ are ordinal numbers such that $b \lt c$, then $b+a \le c+a$. I first started by assuming $g$ as an order ...
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### Second Stiefel-Whitney Class of a Five Manifold

There is a unique rank 4 nontrivial orientable vector bundle over the 2-torus, denote this by $p:E\rightarrow T^2$. Denote the associated sphere bundle by $S(E)$. Then since $S(E)$ is orientable, the ...
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### Inverse of $f(x)=x^n+x$ on $[0,\infty)$

Fix integer $n > 1$. The function $f_n(x) = x^n + x$ is monotone increasing on $[0,\infty)$, and so has an inverse $f_n^{-1}(x)$ that is also monotone increasing on $[0,\infty)$. I'm interested ...
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### simple equations with deep implications [on hold]

the twitter limit of $140$ characters can stimulate the mind. in the same vein i'm wondering what are the most semantically rich mathematical statements that can be made with restricted means. i was ...
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### When do counital coalgebras have a basis of grouplike elements?

Question. Under what conditions do counital coalgebras have bases consisting entirely of grouplike elements? At least in the case of finite-dimensional coalgebras, or for bialgebras (or Hopf ...
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### Solve the equation $x^2+bx+c+t\log(x)=0$

There is a explicit formula to solve the equation $x^2+bx+c+t\log(x)=0$ with the constraint $x>0$?
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### Prove that the ring of rational numbers $\Bbb Q$ is not isomorphic to the ring of real numbers $\Bbb R$

just wondering if my reasoning is correct. I said assume there is such a homomorphism f, then f(1)=1 since it is a ring homomorphism. But f(\sqrt 2)= f(1\cdot\sqrt 2)= f(1) \cdot \sqrt 2= \sqrt ...
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### One to one mapping from $A(S_1)$ into $A(S_2)$

Let $S_1$ and $S_2$ be two sets. Suppose there exists a one to one mapping $\phi$ of $S_1$ into $S_2$. Show that there exists an one to one mapping from $A(S_1)$ into $A(S_2)$, where $A(S)$ means ...
Suppose there are $n$ people sitting in a circle, with $n$ odd. The game is played in rounds until one player is left. Each round the remaining players point either to the person on their right or ...
### The sum of the $n$ smallest odd numbers is equal to $n^2$ [duplicate]
if given a positive integer $n$ is the sum of the $n$ smallest odd numbers equal to $n^2$