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Let $X$ be a topological space and $Y \subseteq X$.
Consider the following statements:
(i) Every net in $Y$ has a cluster point in $X$.
(ii) Every infinite subset of $Y$ has a complete accumulation point in $X$.
These are two possible definitions for relative compactness of $Y$ in $X$.
Question 1: Is there some relatio... |
I am given this function:
$$f(x) = e^{- \sqrt{|x|}}$$
and I want to find $k\in \mathbb{N}, \ p \ge 1$ such that $f \in W^{kp} (\mathbb{R})= \{ f \in L^p (\mathbb{R}) \ | \ \forall \alpha \le k: \ D^{\alpha} f_w \in L^p (\mathbb{R}) \}$
Here $D^{\alpha} f_w$ is the $\alpha$-th weak derivative of $f$.
$f'_w = h$ such tha... |
Let $C = 0.1234567891011121314…$ the Champernowne constant. My question is :
Does the real number $2 \cdot C \simeq 0.24691357820222426283032343638404244464850525456586062646668707274...$ contain every positive integer in its digits?
For instance, $2022$ appears here : $0.246913578\underbrace{2022}...$
Obviously this... |
Consider the 4-by-4 matrix $\boldsymbol M = \boldsymbol M_0 + \boldsymbol M_1$, where
$\boldsymbol M_0 = \alpha \left( \begin{matrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & -1 & 0 \\
0 & 0 & 0 & -1 \\
\end{matrix} \right)$ and $\boldsymbol M_1 = \beta \left( \begin{matrix}
0 & \gamma & 0 & -\gamma^{*} \\... |
Suppose that the distances between houses and the center of a city are distributed with the density function: $f(x)=2-bx$, where $x$ denotes distance. If this is a proper density function, what can we infer about the value of $b$ and the range of values for $x$?
My attempt:
We know that $f(x)>0$ and $ \int_0^\infty f(x... |
If we have such relation that for $\forall x$ $f(x)\ne x$ , how is it called in one word? I can come up with only "graph of this function is not a straight line:)"
Thank you
|
My book defines a continuous time Markov chain like this.
Let $\{X_t\}, t \in T$ be a stochastic process on $(\Omega, \mathcal{A}, P)$, with a countable state space $S$. The process is a Markov chain, if for all
$n>0, t_1<t_2<\ldots<t_n <t_{n+1} \in T$ and $i_1,i_2,\ldots i_{n+1} \in S$
With
$P[X_{t_1} =i_1 \ldots X... |
Let $X$ be an $H-space$, and let a multiplication $,\cdot,$ be given, associative up to homotopy.
Let $\Omega X$ be the loopspace of $X$ based at the identity and let the multiplication $ \circ $ on the loopspace be given by concatenating loops.
We also have a multiplication on the loopspace given by pointwise multipli... |
I recently discovered a problem similar to this one in a book about Markov chains:
Assume you can buy $n-$ different set of cards in a store, but you do not know which one you'll buy: What is the average number of sets you have to buy until have all $n-$ sets together?
I was really puzzled how this problem can be viewe... |
Hi everyone I encountered a problem that looks simple enough but I have no idea where to start.
Find Fourier transform of $e^{-\langle Ax,x\rangle}$ when $A$ is a positive definite symmetric $n \times n$ matrix.
Since $A$ can be any symmetric positive definite matrix, I have no idea how to apply Fourier transform fo... |
Let $H_s$ be a predictable and bounded process. How can I show that $$M_t = \int_0^t H_s \, dW_s$$ is a martingale?
Clearly since $H_s \in L^2_\text{loc} (W)$ we have that $M_t$ is a local martingale, but then how to proceed?
I have tried to use Lebesgue dominated convergence theorem, writing
$$\lim_{k \to \infty} E[M... |
Is this operator: $$T: \mathcal{C}^{\infty}_0 \ni \varphi \to \lim_{x \to \infty} x^2 e^{-x} \varphi'(x) \in \mathbb{R}$$
a distribution (generalized function)?
I need to check two things: whether it is linear and continuous.
Continuous in the sense that for $\varphi_i \in \mathcal{C}^{\infty}_0$ such that $\forall i: ... |
How to show that the matrix
$$B= \left(\begin{array}(\lambda_1 & a & b \\ 0 & \lambda_1 & c\\ 0 & 0 & \lambda_2\end{array}\right)$$
is diagonalizable when $a\neq0$, when $\lambda_1\neq \lambda_2$. How should I work this out? I tried comparing the algebraic and geometric multiplicities of the eigenvalues of $B$ but I di... |
I recently started to look into restricted compositions and I found a formula for a problem that I was trying to solve, the Formula E at page 441 of this document.
In my case I have n =8, k=3, t=1 and w=4.
If I expand the summation I get negative integers in the polynomial and I don't know how to handle them in a bin... |
I need to find a book in which basic $S_n$ theory is covered, mainly the part about Bruhat order, length of an element $w \in S_n$ and invariance modulo braid relations of the expressions $w=s_{i_1}\dots s_{i_k}$ (Matsumoto's theorem)
Are you aware of any good reference?
|
Let $B = {n \in \mathbb{Z} : n = 3j + 2; j \in \mathbb{Z}}, D = {n \in Z : n = 3j − 1; j \in \mathbb{Z}}$. Is $B = D$?
How do I prove this?
To me it looks to be true. But I don't know how to put it to words.
I have a doubt on this. Here the "$j$" they are referring to is different for both sets right? Or are they refer... |
The question is quite simple: for a $N \times p$ matrix $\mathbf{X}$ with real entries, when is $\mathbf{X}^{T}\mathbf{X}+\lambda\mathbf{I}$ invertible (where $\mathbf{I}$ is the $p \times p$ identity matrix and $\lambda > 0$)?
This comes up in ridge regression. In Elements of Statistical Learning (Hastie et al.),
[T... |
I'm looking for the Fourier transform of $\nabla^2f(\vec{r}-\vec{a})$
I can assume that the 3D Fourier transform of $f(\vec{r})$ is $\tilde{f}(\vec{q})$ and the vector $\vec{a}$ is a const vector.
Thanks.
|
I'm trying to prove the following result which seems correct to me, but I'm not sure how to proceed. The proposition is:
Let $M$ be a structure, $A\subseteq M$, $(a_1,\dots,a_n)$ be a sequence from $M$ such that for all $i\le n$ the type $tp(a_i\,\vert\,A\cup\{a_1,\dots,a_{i-1}\})$ is isolated by $\varphi_i$, then $tp... |
A ball of mass 100 grams is hanging from a fixed point by a string 2.5 metres long. It is struck with a bat so that it starts to describe a circle in a vertical plane. When the ball reaches a height of 4 metres the string becomes slack. Find the speed of the ball immediately after it was struck and the tension in the s... |
I've been doing an example of changing the order of a double integral and I'm not sure if I did it right. I would really appreciate if someone would check if my solution is right and correct any mistakes.
The example is $\int_{-5}^{5}dx\int_{-7}^{\sqrt{25-x^2}}f(x,y)dy$
The area set by the integral is:
$$-5\leq x \leq... |
Let an event $e$ have probability of happening $\frac{1}{n}$.
Let us assume we have $m$ independent possibilities for similar events to happen. With $m>>n$. What is the average number of times the event will happen?
The question came out of a discussion on facebook respect to the meteorite and someone claiming that the... |
Q: Is there a normal, real matrix, which it's characteristic polynomial is $t(t-1)(t^2+1)$ ?
I think that there isn't such matrix, and I prove it by: if a matrix is real than it's diagonalizable, but the characteristic polynomial doesn't have real factors, meaning it is not diagonalizable.
am I right?
|
Let $f:(a,b)\to R$ be a continuous function such that $\limsup\limits_{n\to \infty}\frac{f(x_n)-f(x_0)}{|x_n-x_0|}\leq 0$ for every $x_0\in(a,b)$ and sequence $x_n$ converging to $x_0$ such that $x_n\neq x_0, n\geq 1$. Does this imply that f is a constant on $(a,b)$?
If this is a well-known fact, I would like to have a... |
Let $X$ be a normed linear space which is separable. Then I know that there exists a compact subset $K$ of $X$ such that $\operatorname{span} K$ is dense in $X$ (in fact we can also find compact and countable subset $K$). My question is, is the converse true? i.e. if $X$ is a normed linear space such that there exist a... |
I take a exercise from apostol's book. I was trying the next exercise and do it, but the answer (from the book) is different, and I don't know what part of my procedure it's wrong?. So I want to know what's wrong with my procedure .
Exercise: Let $c$ be the number such that $\sinh (c) = \frac{3}{4}$. (Do not attempt to... |
For $A\subset \mathbb{R}$ and $\lambda \in \mathbb{R}$ let's define:
$$
\lambda A = \{\lambda a: a\in A\}
$$
I have to prove that for $\lambda<0$ and bounded $A$ we have $\inf(\lambda A)=\lambda \sup(A)$ and I'm not sure if my proof is correct.
First, let's notice that $\forall a\in A$ we have:
$$
a\leq \sup(A) \iff \l... |
The elementary probability problem is as follows.
Let $(X_k)_{k\in\mathbb{N}}$ be a sequence of i.i.d. random variables such that $X_k \sim U(0,1)$ for each $k$. Define $\tau := \inf\{n\geq 0: \sum_{i=1}^n X_i > 1\}$. What is $E[\tau]$?
There are plenty of clever solutions to this problem. However, I would like to ... |
As a homework assignment I'm supposed to prove or disprove Borel measurability is closed under a.e convergence. I think this is not true because the Borel $\sigma$-field is not complete. However, I'm not sure how to construct or describe a counterexample.
The proof in case of a complete domain $X$ goes by observing tha... |
$ABCD$ is a parallelogram in which $P$ and $Q$ are the mid points of the sides $AD$ and $BC$ respectively. If $BP$ & $QD$ intersect the diagonal $AC$ at $X$ and $Y$ respectively then prove that $BP$ and $QD$ trisect the diagonal $AC$ at $X$ and $Y$.
Efforts ;
I tried to prove that $DPBQ$ is a parallelogram as follows:
... |
I was watching this video but at 8:05 I don't get why to solve for the function
$p(t) = at^2 + bt + c \geq 0$, Sal decides to input $t= \frac{b}{2a}$. Someone made this explanation:
$\frac{b}{2a}$ is something that comes up a lot when solving quadratic equations. All quadratic equations form a parabola which is given b... |
Prove or disprove that $$\left|a_1\right|+\left|a_2\right|+\ldots+\left|a_n\right|\leq n\sqrt{a_1^2+\ldots+a_n^2}$$
Where $a_1,\ldots,a_n\in\mathbb{R}$ and $n\in\mathbb{N}$.
EDIT: I was hoping there is a way without using a known inequality, ie to prove that $RHS-LHS\geq 0$
|
Let $\Sigma$ be any Riemann surface, and let $L \rightarrow \Sigma$ be a complex line bundle (which is classified according to its degree). Then the vector bundle $L \oplus L^{-1} \rightarrow \Sigma$ is a rank $2$ complex vector bundle over $\Sigma$, and an easy computation shows that it has trivial Chern class. But, i... |
In the above diagram, $ABCD$ is a Trapizium with $AD || BC$ and $BC \perp AB$
$AB = 20, \; AD = 6,\; BC = 30$
$M$ is a point on $DC$ such that $[ADMB] = [BMC]$, where $[x]$ denotes the area of $x$.
The length of $MC$ can be expressed as $\dfrac{a}{b} \sqrt{c}$.
Find the length of $MC$.
I can't find a way to start. A... |
I want to propose a problem, it's a version of Poincare' Recurrence Theorem, it's very similar to another problem proposed in this forum, but a bit different:
Another version of the Poincaré Recurrence Theorem (Proof)
The problem is this: Let $(X,\mathcal{A},\mu)$ a measure space such that $\mu(X)<\infty$. Let $T:X\to ... |
Let $f$ be a continuous function in $ (0,1)$.
Determine of the following is right or wrong:
If $ (x_n) $ is a Cauchy sequence such that $0< (x_n) <1$ then $f(x_n) $ is a Cauchy sequence.
Actually... I don't know where to start. How to approach this question?
Thanks!
|
Let be $D_n$ the number of derangements of a set of $n$ elements, by convention we have $D_0=1$
Ifound that $D_n=n!\sum\limits_{k=0}^{n}\frac{(-1)^k}{k!}$
For all $n\in \mathbb{N*}$, we write $p_n$ the derangements proportion within the permutations of the set $[1,n]$
Find $\lim\limits_{n\rightarrow+\infty}p_n$
Fr... |
$\newcommand{\R}{\mathbf R}$ For a cell $e$ in a CW complex, we write $\partial e$ to denote $\bar e-e$. Note that $\partial e$ may not be the topological boundary of $e$ in $X$.
A CW complex structure on a topological space $X$ is said to be regular if each cell in $X$ admits a characteristic map which is a homeomorph... |
I was wondering about the possible values that the length of a chord of a circle can take. The Length of a chord is always greater than or equal to 0 and smaller than or equal to the diameter (sayd). Is it possible to draw a chord of length 'x', where x could be any real number? In other words, does the chord of circle... |
I need a hint to solve prove $\sin^2(6x)-\sin^2(4x) = \sin(2x)\sin(10x)$
I tried several solutions, including taking $(\sin(6x)+\sin(4x))(\sin(6x)-\sin(4x))$ but every time I ended up with a ridiculously large equation, I'm sure there is a more elegant way to solve this.
Note: I'm looking to go from the L.H.S to the R.... |
From "Lecture Notes in Computer Science" by Christoph M. Hoffmann ,
Assume that both $X$ and $X'$ have $n$ vertices. We plan to code the
graph labels as suitable subgraphs which we attach to the vertices of
$X$ and of $X'$. In time polynomial in the length of the input we can rename
the labels and may assume, t... |
Giv an example of a Lebesgue unmeasurable function $f:[0,1]\rightarrow \mathbb{R}$ such that $f^2$ is Lebesgue measurable.
|
I need to find the image of the upper half disc $|z|<1$, $Im\, z >0$ under the inverse transformation $w = 1/z$.
Now, since $|z|<1$, $|z|^{2}<1$. Rewriting this as $z\overline{z}<1$, we make the substitution $z = 1/w$ to obtain $\frac{1}{u^{2}+v^{2}}<1$, or $1 < u^{2}+v^{2}$, which is the exterior of the unit circle in... |
Let $a_{1}=2$, $a_{n+1}=2^{a_{n}}$ for $n \geq 1$
Let $b_{1}=3$, $b_{n+1}=3^{b_{n}}$ for $n \geq 1$
Is is true that $a_{n+2}>b_{n}$ for all $n \geq 1$?
If so, is the proof elementary? (Use only Mathematical techniques covered in an undergraduate degree)
|
Solve my matrix method:
$$3y+4x=2xy$$ and
$$9y-2x=\frac{5xy}{2}$$
My solution
Here:
$$3y+4x=2xy$$
Dividing both sides by $xy$
$$\frac{3}{x}+\frac{4}{y}=2$$----(1)
Again,
$$9y-2x=\frac{5xy}{2}$$
$$18y-4x=5xy$$
Dividing both sides by $xy$
$$\frac{18}{x}-\frac{4}{y}=5$$-------(2).
My solution ends here, now, whats next?
|
I.e. is the embedding $C[0,1]\hookrightarrow \left( C[0,1] \right)^{**}$ surjective?
I am having a hard time answering that question. It would be enough to find a closed subspace of $C[0,1]$ which is not reflexive since closed subspaces in reflexive spaces are again reflexive. Is there a canonical such subspace? Or al... |
Let $\mathsf{nCob}$ be the category of $n$-cobordisms, whose objects are $(n-1)$-dimensional closed manifolds and morphisms are bordisms.
Is this category bicomplete, or even finitely bicomplete?
Once again, I know this is a very simple question, but I couldn't find a single reference on it.
|
Say if I have a model where I impose the restriction that $\hat{\theta} \in (0,1)$, and I calculate the MLE to $\not\in (0,1)$, does this mean my model is incorrect, for this parameter restriction?
|
A question reads: "The weights of $910$ young deer tagged and weighed in a research study are normally distributed with a mean of $86$ pounds and a standard deviation of $2.5$ pounds."
Approximately how many deer weigh more than $91$ pounds?
Since $34.1$% fall within the first standard deviation and $13.6$% fall within... |
I found that Graham's number is :
So, can we say that it is equal to $3^x$ with $x$ is a power tower of 63 3's?
|
Which inequalities can I use with stochastic integration?
For example, with the standard lebesgue integral we have $$\left|\int_\Omega f(x) dx\right| \le M |\Omega|$$
(where $M$ is the maximum of $|f|$ on $\Omega$ is exists)
Also, $$\left|\int_\Omega f(x) dx\right| \le \int_\Omega |f(x)| dx$$
Plus the ever-useful Hold... |
Let $(1,x,x^2,x^3)$ be a basis for $\mathscr{P_3}(\mathbb{R})$ and let $(1,x,x^2,x^3,x^4)$ be a basis for $\mathscr{P_4}(\mathbb{R})$. Suppose $R \in \mathscr{L}(\mathscr{P_4}(\mathbb{R}),\mathscr{P_3}(\mathbb{R}))$, $$A=\begin{bmatrix}
1 & 2 & 0 & 0 & 0\\
2 & 1 & 2 & 0 & 0\\
0 & 2 & 1 & 2 & 0\\
0 & 0 & 2 & 1 & 2
\... |
I've been reading this paper and in remark 2.1. it is mentioned that the estimate (9) from Def 2.1. $$\|\nabla \varphi \|_{L^\infty (K)}\leq \frac{C}{diam \Omega}$$ follows from the regularity of the mesh, in the case of piecewise linear hat function on triangles.
I am looking for a proper proof of this! Can anyone h... |
I'm taking a course in stochastic differential equations, and in order to solve $dX_{t} = X_{t}\,dt + \,dB_{t}$, the book gives a hint: to multiply both sides of this equation by $e^{-t}$. (But, as explained below, "multiplication" doesn't really mean usual multiplication because this is an integral equation.)
But the... |
I have a theory that you should be able to construct a category of categories and embeddings that is complete. Someone already pointed out that if the functors are full and faithful, your cat is not complete. Can we choose a type of embedding that ensures that the cat is complete? My intuition is coming from "just a... |
I'm not sure how to tackle this question. Assume a box containing $1000$ balls, $3$ of them are black and the rest are white. $2$ players $A_1$ & $A_2$ take turns and draw from the box without replacement. $A_1$ starts first. What's the probability that player $A_2$ draws a black ball given that player $A_1$ has alread... |
Show that $f(x)= x/(e^x -1)+x/2$ is even.
So an even function is such that $f(-x)=f(x)$.
So I need to show that $f(x)= (-x)/(e^{-x} -1)+(-x)/2=x/(e^x -1)+x/2=f(x)$.
I also know that $\frac{x}{e^x-1}=\sum\limits_{n=0}^\infty \frac{B_nx^n}{n!}$ where $B_n$ are bernouli numbers.
Though I don't feel that gets me any close... |
I realize this is pretty basic but recently became unsure of how to justify proof by contradiction. Is it that case that I can show $A\Rightarrow B$ by assuming $A$ and NOT $B$, and showing this leads to a contradiction? How can this be justified?
Also, let's say I want to show $A$ and $B$ imply $C$. Is it sufficient t... |
I know speed = |velocity|
Why is speed of parametric defined as
$$speed = \sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}$$
How is this derived? What is the principle here? Is this Pythag? Thinking of this in terms of vectors?
|
If $A+B+C=π$, prove that
$$\cos A \sin B \sin C + \cos B \sin C \sin A + \cos C \sin A \sin B=1+\cos A \cos B\cos C$$
ATTEMPT:
Here,
$$A+B+C=π$$
Now,
\begin{align*}
\text{L.H.S} &= \cos A \sin B \sin C + \cos B \sin C \sin A + \cos C \sin A \sin B \\
&=\sin C(\sin A \cos B + \cos A \sin B) + \cos C \sin A \sin B \\
&= ... |
$p(n)=(1-r)^2nr^{n-1},n=1,2,...$
$f(z)=1/(1-z)$ has derivative $f'(z)$ with convergent power series $f'(z)=1/(1-z)^2=1+2z+3z^2+...$
the answer I have got is $(1-r)^2e^{it}(1-re^{it})^{-2}$ , I am not sure if it's correct or not.
|
I'm stuck on a few questions relating to Index Notation. I'm trying to grasp it however help on these questions would further my understanding of the concepts.
1)How do I show that
$$\nabla(\mathbf{a} \cdot \mathbf{b})=\mathbf{a} \times (\nabla \times \mathbf{b})+\mathbf{b} \times (\nabla \times \mathbf{a})+ \mathbf{a... |
So I had to implement the distance formula earlier in programming, and had a thought that I should make it work for $N$ dimensions. I then smacked myself and realized that I'm programming for a maximum of three dimensions, but that got me thinking. Back in Math classes my professors alluded to the idea that the distanc... |
Let M be a von Neumann algebra with faithful normal normalized trace tr. Let $\{ e_i | i=1,2,\dots \}$ be projections in M such that:
$e_ie_{i \pm 1}e_i=\tau e_i $ for some $\tau \leq 1$
$e_ie_j=e_je_i $ for $|i-j| \geq 2$
$\text{tr}(we_i)=\tau \text{tr}(w)$ if $w$ is a word on $1,e_1,\dots,e_{i-1}$.
Let $A_n$ be the a... |
I am facing a problem in robotics where a robot is localized in 3D-space to build up a map simultaneously (see SLAM, e.g. [1]). One approach is to build up a graph of poses $x_i$ and transforms $z_{ij}$ which connect two poses $x_i$ and $x_j$. In my case both poses and transforms consist of a translation vector and a r... |
This question was asked in GATE 2016: Please help me to figure out the right answer.
Let $T$ ∶ $ℓ_2$ → $ℓ_2$ be defined by
$T((x_1,x_2,...,x_n...))$=$(X_2-X_1, X_3-X_2,...,X_{n+1}-x_n,...)$
Then
(A) $||T|| = 1$
(B) $||T||$ > 2 but bounded
(C)$ 1 < ||T|| < 2$
(D)$ ||T||$ is unbounded
I think answer is (A).
|
I'm having difficulty coming up with actual sequences that have the properties below. I've included my thoughts on the questions below.
Assume that $(a_n)\rightarrow a$.
If every $a_n$ is an upper bound for a set $B$, then $a$ is also an upper bound for $B$
Is it possible to construct a strictly decreasing sequence, ... |
I would like to know what books people currently like in Markov Chains (with syllabus comprising discrete MC, stationary distributions, etc.), that contain many good exercises. Some such book on Stochastic Processes will also suffice.
I have recently come to notice that there are some new books (read: "non-classics") t... |
There are 5 candidates for presidency and 11 people that can vote at most one of them (so they can decide not to vote). How many combinations of votes are there if no candidate can recieve more than 5 votes?
To explain better I make an example:
if the candidates are $A,B,C,D,E$ then
the votes $A-2,B-3,C-1$ are diffe... |
The exercise is to construct a sequence of continuous functions $f_n:\mathbb{R}\rightarrow \mathbb{R}, n\in \mathbb{N}$ , which is pointwise convergent to $f(x)=0 , x\in \mathbb{R}$ and not uniformly convergent on any interval $(a,b)$.
I posted this question once and I got the answer, but I had forgotten to mention tha... |
I need to show that $1 \over x^2$ is not uniformly continuous on the interval $(0,2]$ using the definition of uniform continuity.
Definition of Uniform Continuity on a set A:
Let $A \subseteq \Bbb R$, let $f:A \rightarrow \Bbb R$.
$f$ is uniformly continuous if $ \forall \epsilon > 0$ $\exists \delta > 0$ such that if ... |
I am looking for different representations of the inverse of a matrix as a power series. One obvious candidate is the Von Neumann series which is given
$$A^{-1} = \sum_{k=0}^{\infty} (I-A)^k$$
However the series above converges only of A is Positive Definite and $\|A\| \leq 1$. Are there other series that do not expli... |
I've tried to solve the equation $\cot \theta = 2\cot 2\theta$ with the command 'Reduce' of Mathematica and obtained $\theta = n\pi$ as the solution with n an integer. But $\theta=n\pi$ is clearly a singularity in a cotangent function so this is puzzling.
I've realized that the above equation can be simplified to:
$\t... |
If $f$ is a real function defined in a convex open set $E\subset \mathbb{R}^n$, such that $(D_1f)(\mathbf{x})=0$ for every $\mathbf{x}\in E$, prove that $f(\mathbf{x})$ depends only on $x_2, \dots, x_n$.
Proof: Let $\overline{a}\in E$ and $\hat{a}\in E$ two points which differs only by first coordinate, i.e. $\overlin... |
Calculate $$I=\int_0^{1}\frac{1+x}{x^2+x+1}\log\left({\frac{x}{1-x}}\right)\,\mathrm dx$$
without using complex analysis.
How to calculate without using the residue theorem?
The correct answer is
$$-\frac{1}{8}\ln^23+\frac{\pi^2}{72}-\frac{\pi\sqrt{3}}{36}\ln3$$
This integral is solved using complex analysis in th... |
I'm trying to understand a calculation made in a paper (section 2 from the supplementary contents of Likelihood based observability analysis and confidence intervals for predictions of dynamic models. Kreutz C, Raue A and Timmer J. BMC Systems Biology 6, 120, 2012) for analytically determining a maximum likelihood esti... |
Let $A$ be a subset of $B = \{1, 2,\ldots,n\}$. Suppose that $F$ is a family of subsets of $B$, each of which is a translation of $A$ and no two of which intersect more than once. What is the maximum size of $F$?
This question is similar to the following question:
Choosing subsets of a set with a specified amount of ma... |
Let $f\in L^1([0,1])$, $g_n$ a sequence of continuous functions that converges in $L^1$ to some $g\in L^1([0,1])$. Now my question is: Does $\int_0^1 f(t)e^{g_n(t)} dt$ converge to $\int_0^1 f(t)e^{g(t)} dt$?
edit: I forgot to mention that we may assume that $g$ is bounded, but this bound is not a bound for the $g_n$.
|
My question is based on one of Scott Aaronson blog post which states that a God-like being could convinced the villagers, to any degree of confidence, that she has solved chess by answering a few questions about the zeroes of some polynomials over finite fields. I know it has to do with PSPACE = IP, but I am wondering ... |
Let $M$ be a module over a ring $A$ and $R=Hom_A(M,M)$ its endomorphism ring (with respect to the composition). I need to show these following conditions are equivalent:
$\alpha = \alpha \beta \alpha$, for some $\beta \in R$
$\ker(\alpha)$ and im$(\alpha)$ are direct summands of $A$
the right ideal $\alpha R$ is a dir... |
Say we have a stochastic process described by a stochastic differential equation (in the Itô sense), and maybe we are able to find an explicit solution of it in terms of deterministic and Itô integrals (and maybe not).
In any of these cases, how can we compute the power spectral density of the process?
For instance, f... |
Assuming arithmetic operations on $\overline{\mathbb{C}}$ (that's the extended complex plane) are defined via arithmetic operations on the corresponding sequences, I need to give examples showing why $0\cdot \infty$, $\infty/\infty$, and $0/0$ are meaningless.
I also had to show the same thing for $\infty - \infty$, bu... |
Prove that one of the lines represented by $ax^2+2hxy+by^2=0$ will bisect the angle between the coordinate axes if $(a+b)^2=4h^2$.
Solution
I calculated the two lines represented by $ax^2+2hxy+by^2=0$ as follows;
here.
$$ax^2+2hxy+by^2=0$$
Multiplying by $a$ on both sides and adding $h^2y^2$ to both sides :
$$ax+hy=\p... |
Evaluation of $\displaystyle \lim_{n\rightarrow \infty}\sin \left(\frac{n}{n^2+1}\right)+\sin \left(\frac{n}{n^2+2^2}\right)+\cdots+\sin \left(\frac{n}{n^2+n^2}\right)$
$\bf{My Try::}$ We can write the Sum as $$\lim_{n\rightarrow \infty}\sum^{n}_{r=1}\sin\left(\frac{n}{n^2+r^2}\right)$$
Now how can I convert into Rie... |
I have this problem statement:
Use the cross product to find the sine angle $\phi$ between the vectors $\vec{u}=2i+j-k$ and $\vec{v}=-3i-2j+4k$. Then use the dot product to find the cosine angle $\phi$ for the same vectors. Finally show that $\sin^2{\phi}+\cos^2{\phi}=1$ for the values calculated.
My solution:
First pa... |
How do I convert a negative decimal number into other systems(binary, octal)?
I got the decimal numbers: -22,5 , -60 and 166.
I have to convert them to binary(16 bit) and octal(by hand). I know the system for decimal numbers that are not negative and without decimal point.
|
Let $\zeta(s)$ be the Riemann zeta function. Then we know it satisfies
the Euler product for $\text{Re}(s) > 1$,
$$
\zeta(s) = \prod_{p} (1 - p^{-s})^{-1}.
$$
The proof I read, if I recall correctly, was by considering the partial product of the right hand side, bound $\zeta(s)$ from above and below and take the limit,... |
Are there (simple) examples of martingales which aren't square integrable?
|
$w(n)$ denotes the number of distinct prime factors of $n$. I am wondering if any such result is known.
|
An urn contains white and black balls with $p_w=p$ and $p_b=1−p$. If I made some extractions with replacement, what are the support and the probability function of $X_a$, where $X_a$ is the random variable representing the number
of extraction made for get the a-th white ball? How can I get the conditional probabili... |
Let $x\colon\mathcal{H}\to\mathcal{H}$ be a self-adjoint operator,
the support $s(x)$ of $x$ is defined as the smallest projection $e\in B(\mathcal{H})$ such that $ex=xe=x$.
Let $x=\int\lambda \, dE_x$ be the spectral decomposition.
Then $s(x)=E_x(\mathbb{R}\setminus\{0\})$ und $E_x(\{0\})=\ker x$.
How can these proper... |
I was wondering about the following.
Let $M$ be a (smooth, closed, connected and oriented) manifold endowed with a regular foliation (i.e. such that all the leaves are smooth submanifolds of the same dimension and such that their tangent bundles form a smooth distribution in $TM$). Also assume all leafs are closed (tha... |
I was reading from the book Apostol calculus, There is this theorem
Possibility of Subtraction : Given $a$ and $b$, there is exactly one $x$ such that $a+x=b$.
The proof is - Given $a$ and $b$, choose $y$ so that $a + y = 0$ and let $x = y + b$. Then $a+x = a+(y+b) = (a+y) +b = 0+b = b$. Therefore there is at least ... |
I am trying to understand the Beilinson-Bernstein localization theorem (following the book by Hotta, Takeuchi and Tanisaki). I got stuck at the following two steps. Any help will be greatly appreciated.
For an integral weight $\lambda$, I will write $\mathcal L(\lambda)$ for the sheaf associated to the line bundle $G... |
Let $K$ be a field, and $K(x)$ the field of rational functions over $K$. Consider the degree valuation $v$ on $K(x)$, That is
$v\left(\frac{f(x)}{g(x)}\right)=\deg(g)-\deg(f)$. So for every $f(x)\in K[x]$, $v(f(x))\leq 0$.
Now let $F$ be a finite extension of $K(x)$, and $R$ the integral closure of $K[x]$ in $F$. Let ... |
I've been given these relations and I've been told to determine whether they're reflexive and I know the definition of reflexive but I don't really understand it.
$R=\{(x,y)\ \in\ \mathbb Z^2\ |\ x^2+y^2\ \text{is odd}\}$
$R=\{(x,y)\ \in\ \mathbb Q^2\ |\ xy \ge 0\}$
|
For a homework assignment I was told to prove that given $f\in L^1(\mathbb R)$, the following function is continuous $$F(x)=\int_{(-\infty,x]}fd\lambda.$$
I thought to use DCT and show sequential continuity. So given $x_n\rightarrow x$, we look at the sequence of indicators $\mathbf 1_{(-\infty,x_n]}f$. These are domin... |
I'm trying to determine if the matrix:
\begin{bmatrix}
0 & i \\
1 & 0
\end{bmatrix}
is a unitary matrix.
Therefore, the first step I'm taking is to figure out if both $\langle 0, 1\rangle$ and $\langle i, 0\rangle$ are unit vectors. $\langle 0, 1\rangle$ is obviously a unit vector, however, I'm not sure exactly what t... |
If $F$ is a formally real field, then there exists a total order relation on $F$ which is compatible with its sum and product, but it needs not be unique.
a) How different can be two (compatible with the field) order relations over the same formally real field, in the sense of not sharing important properties (e.g. bei... |
$z=x_1^2+x_2^2$
$u=(u_1,u_2)$
$v=(v_1,v_2)$
Height of arc: $f[\theta u+(1-\theta )v]= f[\theta u_1+(1-\theta)v_1,\theta u_2 + (1-\theta)v_2 ]$
$= [\theta u_1+(1-\theta)v_1]^2 + [\theta u_2 + (1-\theta)v_2 ]^2$
Height of line segment: $\theta f(u) + (1-\theta)f(v) =\theta(u_1^2+u_2^2)+(1-\theta)(v_1^2+v_2^2)$
My bo... |
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