0
votes
0answers
4 views

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.
0
votes
0answers
9 views

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 ...
0
votes
0answers
3 views

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))
0
votes
0answers
4 views

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 ...
0
votes
0answers
8 views

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 ...
0
votes
0answers
3 views

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 ...
0
votes
0answers
18 views

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 ...
0
votes
0answers
6 views

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 ...
1
vote
0answers
4 views

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 ...
1
vote
0answers
14 views

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 ...
0
votes
2answers
8 views

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 $R^n$
2
votes
0answers
4 views

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, ...
0
votes
0answers
10 views

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 ...
0
votes
0answers
7 views

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: ...
0
votes
1answer
24 views

Showing a Lipschitz function is differentiable

Let $f : \mathbb{R} \to \mathbb{R}$ be a Lipschitz function. Suppose $$ \lim_{n\to \infty} n[f(x + \frac{1}{n}) - f(x)] = 0.$$ Prove that $f$ is differentiable. I am tempted to just let $n = ...
3
votes
1answer
13 views

Find the inverse of $2$ modulo $17$ using the Euclidean algorithm

The question states "find the inverse of a modulo m for each of these pairs of relatively prime integers" ATTEMPT AT SOLUTION \begin{align*} 17 & = 2 \cdot 8 + 1\\ 2 & = 1 \cdot 2 ...
1
vote
0answers
10 views

Picard rank of K3s — How can it be small?

I must be missing something. How can the Picard rank of a compact Kahler manifold drop below its Hodge number $H^{1,1}(X)$? For simplicity, let $X$ be a K3 surface. After all, we have the standard ...
0
votes
1answer
8 views

How do I proove that function is solution of the Laplace equation?

How do I proove that for $\vec{r}=(x,y,z)\in \mathbb{R}^3,\vec{r}\neq 0$, function is $u(x,y,z):=1/(-ln\left \| \vec{r} \right \|)$ a solution of the Laplace equation $\Delta u=\frac{\partial^2 ...
0
votes
1answer
17 views

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~
0
votes
0answers
4 views

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 ...
-1
votes
0answers
21 views

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 ...
1
vote
1answer
21 views

Is it true that $\lim\limits_{r \rightarrow 1}f(r x) = f(x)$ in $L^1$?

Suppose $f \in L^1(\mathbb{R})$ with Lebesgue measure and $r > 0$. Does $f(rx)$ converges to $f$ in $L^1$ as $r \rightarrow 1$ ? Put differently, does $$ \| f(rx) - f(x)\|_1 \rightarrow 0$$ as $r ...
2
votes
3answers
31 views

For which values of $x$ does $\sum_{n=1}^{\infty}\frac{ x^n}{x^{2n} -1}$ converge?

Of course it does not converge for $x = \pm 1$. I tried to use the limit form of the comparison test, using a divisor "b" series of $x^n$, and $\frac{1}{x^{2n} -1}$ goes to $0$ as $n$ goes to $\infty$ ...
0
votes
1answer
6 views

Help with normal vectors to linear vector functions.

I am studying basic vector calculus and am on tangential and normal vectors. I understand why the derivative of a vector is tangential, and I also understand why the second derivative of a vector ...
0
votes
1answer
22 views

Galois subfields and subgroups

Find the minimal Galois extension $L$ of $\mathbb{Q}$ containing $\mathbb{Q}(\sqrt[4]{5})$ $L=\mathbb{Q}(\sqrt[4]{5}, i)$ is a splitting field of $X^4-5$ over $\mathbb{Q}$ Describe the ...
0
votes
1answer
15 views

Calculating variance upon removing a value from the sample

We are given that a sample of $N=7$ has mean $\bar{X} = 5$, and a sample variance of $s^2 = 9$. One person, with a score of $11$ is removed from the data. What is the mean and variance for the new set ...
0
votes
0answers
20 views

Why are all Cauchyfilters on $\mathbb{N}$ ultrafilters?

One can place an equivalence relation on the set of all Cauchyfilters in a uniform space: $\mathcal{U}\sim\mathcal{V}$ iff $\mathcal{U}\cap\mathcal{V}$ is a Cauchyfilter. In my topology course, we ...
1
vote
3answers
24 views

Stuck in integration problem

Could someone give me some hint or show me how to calculate this integration? Thanks in advance.
1
vote
1answer
10 views

How is it true that for large $t$, $(1+O(1/t))e^{-2\ln t O(1/t)}=1+O(\ln t/t)$?

The title pretty much says it all. At some point in large time analysis, the following claim popped out but I don't see how it is true: For sufficiently large $t>0$, $$ \frac{2\ln ...
0
votes
1answer
17 views

Integration by part for an integral

I have the following integral that I try to solve with integration by parts; $$\int_{0}^{L}z^{\frac{1}{\phi}-1}p\left(z\right)dz$$ What I have tried is as follows ; I know from my calculations ...
0
votes
1answer
13 views

Singular part of the pole at $z=0$ of $f(z)= \pi^2 / \sin^2(\pi z)$

In $z=0$, $f(z)$ has a pole of order 2, and I know that the singular part in this case is $\frac{1}{z^2}$, but I dont know how to obtain this result. Any help.
1
vote
0answers
15 views

Prove by induction that $ F_{2n}=F_nL_n $

In the following exercise from George E. Andrews' Number Theory, we are given that $F_n$ and $L_n$ represent the $nth$ Fibonacci and Lucas numbers respectfully, and we need to prove by induction (i.e. ...
0
votes
0answers
5 views

A simple, explained example of Lagrange inversion for a polynomial…

I have an bivariate polynomial, $\beta^{10} \alpha^2 + \beta^9 \alpha + \alpha \beta - 2\beta + 1 =0$ I want to develop an expression $\beta(\alpha) = \# + \# \alpha + \# \alpha^2 + \cdots $ at ...
1
vote
0answers
10 views

How to analyze ODE equilibrium stability with complex equilibria

Take this example: $y'=y^2+1$. There's no "real equilibrium", but is that right to say it has two "complex equilibria"? If so, what should be the conclusion of the derivative test? $y'' = 2y \implies ...
0
votes
0answers
5 views

Size of the vocabulary in Laplace smoothing for a trigram language model

Let's say we have a text document with $N$ unique words making up a vocabulary $V$, $|V| = N$. For a bigram language model with add-one smoothing, we define a conditional probability of any word ...
1
vote
3answers
34 views

Is there any example of a set, together with the “sum” operation which is non conmutative?

I was wondering if there is any mathematical structure (Im not even sure this is the correct way to name what I have in mind) but basically, any set, together with the operation sum, (I don´t say just ...
0
votes
1answer
17 views

The degree of an algebraic element over a field extension

Let $ L/K $ be a field extension and let $ \alpha $ be an algebraic element of prime degree over $ K $, i.e $ [K(\alpha) : K] = p $ for some prime $ p $. Is it always the case that we have $ ...
1
vote
0answers
14 views

An analogue of the Cauchy formula for radius of convergence for power series with arbitrary (non-integer) exponents

By Cauchy formula, the radius of convergence of the series $\sum_{n=0}^{\infty}a_nr^{n}$ is $\rho=1/\limsup\limits_{n\rightarrow +\infty}\sqrt[n]{|a_n|}$. Let $\{\lambda_n\}_{n=0}^{\infty}$ be an ...
1
vote
1answer
15 views

If a normal subgroup shares elements with a conjugacy class, then it contains it entirely?

One of my group theory review problems seems to follow directly from definitions, but I'm not sure. The problem is: Let $G$ be a group and $C$ a conjugacy class of $G$. Let $N$ be a normal subgroup ...
2
votes
2answers
28 views

Why is $\beta(\mathbb{N}\times\mathbb{N})\neq\beta(\mathbb{N})\times\beta(\mathbb{N})$?

In my course on topology, we proved that the Cech-Stone compactification of $\mathbb{N}$ is homeomorphic to $(U(\mathbb{N}), \mathcal{T})$ where $U(\mathbb{N})$ are the ultrafilters on $\mathbb{N}$ ...
0
votes
0answers
11 views

hypothesis testing question.

Sorry for any English mistakes, Looking for a push in the right direction with this problem. ...... Intro: 2 researchers are testing a hypothesis separately.. $$H_0 : \mu = \mu_0, H_1 : \mu ...
0
votes
0answers
14 views

Finite module over Noetherian ring faithfully flat?

If I have Noetherian rings $B=A^G\subset A$ (for some action of finite group $G$, maybe not relevant) and $A$ is finite as $B$-module. Is it always true that A is faithfully flat over $B$? EDIT: ...
1
vote
2answers
19 views

General structure of a 3 by 3 persymmetric matrix with zero eigenvalues

Here is an interesting problem that might be extended to higher dimensions, I am looking for different simple ways of describing a 3 by 3 matrix with one or two zero eigenvalues. This persymmetric ...
0
votes
1answer
8 views

Rayleigh quotient $Q=(\frac{||\triangledown w||}{||w||})^2$ in using the eigenfunction $\sin(x)$ on the segment $(0,\pi)$

I would like to well understanding the Rayleigh quotient $Q=(\frac{\|\nabla w\|}{\|w\|})^2$. Does anyone could explain to me why we divide the norm of the gradient $\| \nabla w \|$ by $\| w \|$, and ...
1
vote
2answers
27 views

Show that $\int_{-\infty}^\infty f(w)e^{-\pi \delta w^2} dw \to \int_{-\infty}^\infty f(w) dw$ as $\delta \to 0$

Show that $\int_{-\infty}^\infty f(w)e^{-\pi \delta w^2} dw \to \int_{-\infty}^\infty f(w) dw$ as $\delta \to 0$. $f(w)$ is a Schwartz function. This is a part of the proof of Fourier inversion ...
2
votes
2answers
16 views

Showing that $f$ is not measurable

I'm learning about measure theory, specifically measurable functions, and need help with the following problem: Let $N$ be a non-measurable subset of $[0, 1]$ and define $$f(x) = ...
3
votes
1answer
39 views

Prove $\int_{0}^{\infty}\frac{4x^2}{(x^4+2x^2+2)^2}dx=\frac{\pi}{15}$

$$\int_{0}^{\infty}\frac{4x^2}{(x^4+2x^2+2)^2}dx=\frac{\pi}{15}$$ $$\int_{0}^{\infty}\frac{4x^2}{[(1+(1+x^2)^2]^2}dx=\frac{\pi}{15}$$ $u=\tan(z)$ $\rightarrow$ $du=\sec^2(z)$ $u$ $\rightarrow ...
1
vote
1answer
18 views

Proving that an equation has a unique stable limit cycle

I'm preparing for my exam and I stumbled upon a question and I am a bit lost on how to write the correct solution. The question goes as follows: Prove that the equation $\ddot{x} + ...
0
votes
1answer
10 views

“All phase plane solution points remain stationary as $t$ increases”?

Consider the linear system $y′(t)=A\vec{y}(t)$, where $A$ is a real $2\times2$ constant matrix with repeated eigenvalues. All phase plane solution points remain stationary as $t$ increases. I ...
0
votes
1answer
24 views

Probability of drawing two cards

If I have five different cards, valued at 0 - 4 with a probability of drawing each card at 1/4 (for values 0, 1 and 2), 3/20 (for value 3) and 1/10 (for value 4). I draw 2 cards at random without ...

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