3
votes
1answer
46 views

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 ...
2
votes
2answers
55 views

Log normal distribution - Where am I wrong?

Let $X$ be a R.V whose pdf is given by $$f(x)=p\frac{1}{\sqrt{2\pi\sigma_1^2}}\exp\left(-\frac{(x-\mu_1)^2}{2\sigma_1^2}\right)+ ...
2
votes
1answer
55 views

Is there a theory of “rings” with partially defined multiplication?

Consider the abelian group $R [[\mathbb{Z}^d]]$ of all formal Laurent series over a commutative ring $R$ (a typical element has the form $\sum_{v \in \mathbb{Z}^d} \lambda_v \cdot X_1^{v_1} \dots ...
1
vote
0answers
12 views

Distance histogram within cylinder

Suppose I randomly pick a pair of points $x$ and $y$ from inside a cylinder of radius $R$ and length $L$. What is the probability that they are a distance $d$ apart? In other words, I wish to evaluate ...
0
votes
1answer
73 views

Paradox in ring theory — what am I missing?

I saw somewhere the following exercise: Give example of prime ideal in a ring which is not maximal the answer was this: Let $R$ be our Ring and $I$ ideal such $$ R = {Z}[{X}] $$ $$ I = (x) $$ ...
0
votes
3answers
42 views

Horse Betting Odds - But Guaranteed Win!

Suppose four horses - $A, B, C$, and $D$ - are entered in a race and the odds on them, respectively, are $6$ to $1$, $5$ to $1$, $4$ to $1$, and $3$ to $1.$ If you bet $\$1$ on $A$, then you receive ...
1
vote
0answers
13 views

How many ways are there to color the $H$-shaped tree with $3$ colors such that each color is used exactly twice?

How many ways are there to color this graph with the following constraints? We have three colors: blue, red, green, and we require that the number of nodes of color green is 2, and blue 2, and red ...
1
vote
0answers
15 views

Integral of the Square of the Elliptic Integral

Someone must know a good technique for $$ \int E^{2}(x)dx $$ Where $E$ is the complete elliptic integral of the second kind: $$ ...
0
votes
1answer
60 views

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 ...
0
votes
1answer
20 views

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 $S$?
-1
votes
3answers
44 views

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$.
3
votes
1answer
107 views

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 ...
0
votes
0answers
10 views

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 ...
0
votes
1answer
54 views

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 ...
1
vote
2answers
67 views

For a complex number $c$, how does a plot of $c^{-4},c^{-3}, c^{-2}, c^{-1},c^0, c^1,\dots, c^4$ look like?

I am on the road so can't test it for myself: what would happen if I took a complex number $C = a + bi$ and plotted the following in the complex plane; $$C^{-4}, C^{-3}, C^{-2}, C^{-1}, C^0, ...
3
votes
2answers
37 views

Vocal group no two singer stand next to each other?

A vocal group consisting of alf,bill,cal,deb,eve, and fay are deciding how to arrange themselves from left to right on a stage How many way to do this if Alf and Fay are the least skilled singer and ...
4
votes
3answers
76 views

2000 Olympiad in Informatics Question on Box

I have an old Olympiad question on informatics. There are 31 boxes. In each box there is one number. We know the number if and only if we open the box. We want to calculate the minimum number of ...
1
vote
3answers
33 views

Complex number equations

I cannot solve two problems regarding complex equations. 1)Let $z^2+w^2=0$, prove that $$z^{4n+2}+w^{4n+2}=0, n \in \mathbb{N^{*}}$$ What I tried; $$z^2 \cdot z^{4n}+w^2 \cdot w^{4n}=0 \iff ...
-1
votes
1answer
40 views

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 ...
0
votes
2answers
24 views

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 $f(x)\leq g(x)\,\,\forall x$. ...
-2
votes
2answers
19 views

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 ...
0
votes
1answer
356 views

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 $T$?
1
vote
2answers
26 views

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?
5
votes
2answers
8k views

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) , ...
3
votes
2answers
34 views

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 ...
22
votes
6answers
737 views

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 ...
0
votes
2answers
20 views

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 ...
4
votes
2answers
194 views

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 ...
3
votes
1answer
26 views

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 ...
1
vote
1answer
32 views

$\|x+y\|$ 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 $x=(x-y)+y$ ...
1
vote
0answers
8 views

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 ...
0
votes
2answers
34 views

ordinal numbers addition property

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 ...
0
votes
0answers
10 views

Coherence in braided monoidal categories

Let ($\mathcal{C}$,c) be a braided monoidal (tensor) category. Then c is compatible with the morphisms l,r associated with the unit object 1 of $\mathcal{C}$, in the sense that: $l_X \circ ...
2
votes
1answer
32 views

Book on advanced Hodge Theory

I'm looking for a book on advanced real Hodge Theory. I finished working through Frank Warner's Foundations of Differentiable Manifolds and Lie Groups, which ends with the Hodge Decomposition,the ...
0
votes
0answers
21 views

Bernoulli measure

Does anyone know an elementary proof (or somewhere I can find it) of the construction of Bernoulli measure on the set of infinite binary sequences? I am having trouble to show that the measure defined ...
1
vote
6answers
93 views

Find $\int_{-1}^1 (8x^3 + 14x^2 + 6x + 3)dx$. [on hold]

Does anyone know the answer to this integration? $$\int_{-1}^1 (8x^3 + 14x^2 + 6x + 3)dx$$
4
votes
3answers
91 views

Models of set theory

How can one talk of a semantics or model of set theory (lets say ZF or ZFC) when the definition of a structure (and potential model) needs a carrier set in the first place (by its definition)?
-1
votes
3answers
36 views

Elements in $F=\bigcup \limits_{n=1}^\infty \left(\bigcap \limits_{k=n}^\infty E_k\right)$

let S be set with subsets $ E_1,E_2,E_3,....$ Show that: a) $F=\bigcup \limits_{n=1}^\infty \left(\bigcap \limits_{k=n}^\infty E_k\right)$ consists of all elements of S each of which belongs to all ...
0
votes
0answers
13 views

Smallest grammar problem on a single character.

Let the alphabet be $\Sigma = \{a\}$. Say $s = a^6 = aaa aaa$. If the repeated variable $A = aa$ appears $k$ times in the expanded starting rule of a smallest grammar $G_s$ for $s$. Then that ...
0
votes
0answers
14 views

Finding commutator of $\mathbb D_n$ and $\mathbb S_4$

I have to find $[G,G]$ (the commutator of $G$) for the the dihedral group, $\mathbb D_n$ ; and the symmetric group, $\mathbb S_4$. What tools, theorems could I use in order to find these two ...
7
votes
3answers
123 views

Hard Definite integral involving the Zeta function

Prove that: $$\displaystyle \int_{0}^{1}\frac{1-x}{1-x^{6}}{\ln^4{x}} \ {dx} = \frac{16{{\pi}^{5}}}{243\sqrt[]{{3}}}+\frac{605\zeta(5)}{54} $$ I was able to simplify it a bit by substituting ${y = ...
5
votes
1answer
58 views

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 ...
4
votes
2answers
89 views

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 ...
0
votes
0answers
46 views

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 ...
3
votes
1answer
100 views

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 ...
0
votes
1answer
34 views

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 $x>0$?
3
votes
1answer
86 views

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 ...
2
votes
2answers
62 views

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 ...
2
votes
1answer
47 views

Combinatorics of a game

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 ...
-1
votes
3answers
75 views

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 $n^2$

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