# All Questions

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### Finding a general solution of recurrences

I am unsure how to even start the questions :S I need to learn this stuff for the final exam of my subject and its hard to find a tutorial on how to answer this type of question.
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### Proof using the product lemma

Let $S$ be the set of all finite subsets of $\mathbb N = \{1,2,3,...\}.$ We define a weight function $w$ where for a subset $X$ of $\mathbb N, w(X)$ is the sum of all the elements in $X$, with ...
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### Expectation calculation for stochastic process

How to find expectation of: e^-r(T-t)[S(T)|Ft] where S(T) = S0 * e^((mu-1/2 sigma^2)t+sigma*W(t))
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### Solving for the particular solution of a system of differential equations

Consider the IVP $\vec{y}(t)= \begin{bmatrix}0 & -1\\-1 & 0\end{bmatrix} + \begin{bmatrix}t \\e^{2t}\end{bmatrix}$ $\vec{y}(0) = \begin{bmatrix}1 \\1\end{bmatrix}$ The general solution of ...
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### Vector diagram: forces

Using a vector diagram,explain why it is easier to do chin ups when your hands are 30cm apart instead of 90 cm apart.(Assume that force exerted by your arms is the same in both cases). If someone ...
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### Finding range fo variables when doing parametrizations

I have been working on some homework problems where I am asked to convert a standard equation of a surface into cylindrical and/or spherical. I understand the conversions for the most part but I ...
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### Prove $\ \sin(x) < x \ \ \ \forall x \in(0, 2\pi)$

Problem : Prove $\sin(x) < x \ \ \ \forall x \in(0, 2\pi)$ Now I have a possible solution for this, using limits and the first derivatives of $\sin(x)$ and $x$, but I don't feel it's a very ...
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### Equation of a perpendicular plane without dot product

How can i find the equation of the plane that passes through $(6,4,-2)$ and is perpendicular to the line that passes through points $(1,4,-5)$ and $(7,-2,3)$ without dot product? I tried findind the ...
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### If $x^3+px+q$ is irreducible over a finite field then $-4p^3-27q^2$ is a square

Suppose that $x^3+px+q$ is irreducible over a finite field $F$ with characteristic not equal to $2$ or $3$. Show that $-4p^3-27q^2$ is a square in $F$. I noticed that the determinant of ...
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### What is a composition in category theory?

I'm just beginning to learn category theory. So far, the basic examples (like Set) are making sense. But I'm having a little trouble getting my head around the fundamentals. Suppose I try to define a ...
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### Prove/Disprove question on matrix vector multiplication and linear independence

If {$Bv_1$, ... ,$Bv_k$} is a linearly independent set in $R^k$ where $B$ is a $k$ x $n$ matrix in, then {$v_1$, ... ,$v_k$} is a linearly independent set in $R^n$
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### How to Prove the Stochastic Fubini Theorem? (Exercise 2.19 in Chapter IV of Revuz and Yor)

Here is the theorem statement: Let $B$ and $C$ be two independent standard Brownian motions. If $\phi$ is square integrable on the unit square ($\phi \in L^2([0,1]^2)$ ), by suitable filtrations, ...
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### Prove $\frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y} \geqslant \frac32+ \frac{27}{16}\frac{(y-z)^2}{(x+y+z)^2}$

$x,y,z >0$, prove $$\frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y} \geqslant \frac32+ \frac{27}{16}\frac{(y-z)^2}{(x+y+z)^2}$$ This inequality is easier compared to the other one. Previously, I ...
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### Affine function? Asymptote?

I'm searching what are the keywords or the good links to expand my researches. I would like to get the equation ( for programming purposes ) of more or less this curve: ...
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### problem of conditional expectation

If I already get the $E(X|Y)=y$, how could I derive $E(XY|Y)$ from $E(X|Y)$? Is $E(XY|Y)$ = $yE(X|Y)=y^2$? And then $E(XY)=E(E(XY|Y))$ How could I use the $E(XY|Y)$ to get the $E(XY)$? Thanks~
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### Showing any bounded sequence in Holder space $C^{1/2}$ has a convergent subsequence in Holder space $C^{1/3}.$

Prove that any bounded sequence in $C^{1/2}([0,1])$ admits a convergent subsequence in $C^{1/3}([0,1]),$ where we say that $f \in C^{\alpha}([0,1])$ if $f$ is Holder continuous of order $\alpha.$ The ...
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### How can I know this problem talk about conditional probability , permutation, or combination ,Do you have any techniques for this .

How can I know this problem talk about conditional probability , permutation, or combination ,Do you have any techniques for this . always my teacher gives us problem , and it does not have any ...
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