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I want to evaluate an integral $I_1$ defined in $Eq.(1)$ as
\begin{align}
I_1=\int_{0}^{\infty}\frac{x\exp(-\beta x)K_1(\alpha x)}{1+x}dx\tag{1}
\end{align}
Where $K_1(.)$ is modified first order Bessel's function of second kind
In order to evaluate it, I went through transforming $\frac{x}{1+x}$ and $K_1(\alpha x)$ in... |
Let $X$ is poisson distribution.
$$f_{X}(n;\lambda)=\frac{\lambda^{n}}{n!}e^{-\lambda}$$
And there is some positive constant $\alpha$.
I like to know pmf(probability mass function) of $Z=\alpha X$.
I have searched this topic and i could find almost similar post.
This post deal with distribution of $Y=\frac{X}{m}$ ($X$ ... |
How to prove $I(X_1 \cap X_2)=\sqrt{I(X_1)+I(X_2)}$?
Clearly $\sqrt{I(X_1)+I(X_2)} \subseteq I(X_1 \cap X_2)$
But for $f \in I(X_1 \cap X_2)$ $f(x)=0 \forall x\in X_1 \cap X_2$.
how to show $f \in \sqrt{I(X_1)+I(X_2)}$?
|
Suppose $q$ is any nonzero rational number and $p$ is a fixed prime. If $q = p^k\frac{n}{m}$ for integers $n$ and $m$, neither of which has $p$ as a factor, then we define $|q|_p := p^{−k}$.
We can then, of course, define a metric $d_p(q, q') := |q − q'|_p$.
Why is it that for any rational numbers $a < b$, the interval... |
I have a large matrix (~1000 by ~2000), whose entries are purely rational numbers, typically involving large fractions, that is numbers (much) larger than 10^8 in denominator/numerator. The original problem strongly suggests that by multiplying, adding and subtracting rows one can reach a form which is "simple". By thi... |
I am trying to look for an analytic solution to the following equation
$\frac{1}{4D} (y')^2 -\frac{1}{2}y'' = -n(n+1)A\operatorname{sech}^2(\frac{x}{b})$
with $A>0$, $D>0$ and $b>0$ and $n>1$ an integer. This equation is obtained from trying to find the equivalent Fokker-Planck equation for the Poschl-Teller Schroding... |
How can this question be geometric and also negative binomial.
In order to join a group Peter needs 9 invitations.The probability that he receives an invitation on any day is 0.8, independent of other days.
He joins as soon as he receives his 9 th invitation. Given that he joined on the 14 th day, find the probability ... |
Let $V$ be the set of all polynomial $f(x)$ in $P_2$ s.t. $\int_{0}^3 f(x) dx =3f(1) $
If $V$ is a subspace of $P_2$ find a basis of $V$.
Can somebody help me get started? The integral condition kind of scares me.
|
If $ A \neq \emptyset $ and is bounded above and $ s = \sup A $ then $ s \in \bar A $ and $ s \notin \operatorname{int}A$
I've got the first part, showing that $s$ is a limit point of $A$. Thus, $ s \in \bar A$. But, how to go about showing the second part?
|
Construct a continuous function of bounded variation on $[0,1]$ which is not monotone in any subinterval.
We can follow the pattern of the Cantor-Lebesgue function (somewhat). For example, at the first stage, let the function increase to $1/2 + \epsilon$, then decrease by $2\epsilon$, and then increase again by $1/2 +... |
In Liu's answer to this MO question there is a characterization of smooth affine varieties which are étale over affine space. I was wondering if one can give a similar characterization for projective varieties.
Let $X$ be a smooth projective variety of dimension $n$. When is $X$ étale over $\mathbb A^n$? How can one c... |
I started thinking on the following problem. I am interested in finding complex functions of a complex variable such that
$\phi(z)=\phi(z^{-1})$
So far, all I could come up with was a family of functions of
$z^{a}+z^{-a}$
where $a$ is a complex parameter. Is there a way to find other functions?
In general, a way to say... |
$$3\sin x + 4\cos x = 2$$
To solve an equation like the one above, we were taught to use the double angle identity formula to get two equations in the form of $R\cos\alpha = y$ where $R$ is a coefficient and $\alpha$ is the second angle being added to $x$ when using the double angle identity.
Why can't we use the ide... |
Say I have a size $k$ set called $S_k$ with elements that are natural numbers (repetitions are allowed). For instance $\{2, 8, 6, 6, 1, 3\}$ is a valid set for $k = 6.$ I am trying to find the least number of strict partial orders such that each partial order uses a unique element. For instance, in the above set I have... |
Suppose I have a function $f(x,y,z)$. I need to know when one can write it as $$f(x,y,z)=a(x)\cdot b(y) \cdot c(z)$$ where $a, b, c$ are functions. I don't want to know what they are, but just whether it's possible to separate.
Motivation of the question was that many times, to solve a p.d.e we use separation of variab... |
I'm trying to understand the following equation:
$\prod_{i:y_i=1} p(x_i) \prod_{i:y_i=0} (1 - p(x_i))$
The part I don't get is the subscript below the product operator. Does the $i:y_i=1$ under the product operator mean "for every i where $y_i = 1$"?
|
Given two operators or non-zero matrices $A$ and $B$, where $A\neq B$, tr$(A)=1$ and tr$(B)=1$ and tr$(A-B)=0$, what is a lower bound of the Schatten p-norm ($p=1$) $\|A-B\|_1$? Any helpful references?
|
I have a polynomial ring $R = K[x_1,\dots,x_n]$ and $N \subset \{x_1,\dots,x_n\}$. Define $R_N = K[x_i:x_i\notin N]$. If $f\not= 0$, show $(f) \cap R_N = (0) \iff f \notin R_N$.
So my entire argument hinges that $f \in R_{N^c}$ and clearly $f\in (f)$, but $f \not= 0$, so the intersection must be $(0)$ since $0 \in R_N$... |
I was solving an A Level paper when I came across this question. I tried substitution, but I'm not getting the answer with that. Would appreciate it if someone would help me.
|
A useful system in some contexts where are you counting things is writing
1 as |
2 as ||
3 as |||
.
.
.
11 as |||| |||| |
Example of usefulness is keeping track of the number of cars that pass over a bridge.
What is the system called?
|
Let $X=(X_{1},X_{2})$ have joint pdf.$$f(x_{1},x_{2})=\begin{cases}\frac{e^{-\frac{x_{2}^2}{2}}}{x_{2}\sqrt{2\pi}},\ &\text{if}\ 0<|x_{1}|\le x_{2}<\infty.\\0,\ &\text{otherwise} \end{cases}$$ Then the variance of $X_1$, is for finding out the variance I must need marginal dist. of $X_1$ $f(x_{1})$? but I am stuck in ... |
$$(4y+2x-5)dx+(6y+4x-1)dy=0,y(-1)=2$$
First method:
$$\frac{dy}{dx}=-\frac{4y+2x-5}{6y+4x-1}$$
let $Y=y-\frac{9}{2}$, $dY=dy$ and $X=x+\frac{13}{2}$, $dX=dx$;
$$\frac{dY}{dX}=-\frac{4Y+2X}{6Y+4X}$$
let $u=\frac{Y}{X}$ , $Y'=u'X+u$;
$$u'X+u=-\frac{4u+2}{6u+4}$$
$$u'X=\frac{-6u^2-8u-2}{6u+4}$$
$$\int\frac{(6u+4)du}{6u^... |
In $\mathbb R$ for a derivative to exist (or a limit generally) it is necessary that the limit be the same in both directions (from below and above) and this is the same in $\mathbb C$ where for a function to be differentiable at a point the difference quotient limit must be the same independent of the way $h$ approac... |
Disprove the following claim:
For any events $A,B$, $P(A) = P(A|B) + P(A|B^C)$
I've fiddled with the algebra of the conditionals, but didn't get to anything solvable. I'm not terribly interested in proving this by counterexample -- I want to understand why this is not the case. It looks deceptively reasonable to me.... |
Use the dot/scalar product to solve the problem
Line 1 has vector equation $(2\mathrm{i}-\mathrm{j}) + \lambda(3\mathrm{i} + 2\mathrm{j})$
Find the vector equation of the line perpendicular to Line 1 and passing through the point with position vector $(4\mathrm{i} + 3\mathrm{j})$.
I can solve this problem by converting... |
I have a question. What is
$$\int_{|z|=2}\frac{z}{(z-3)^2}dz?$$
In my optinion it must be zero, because the singularity $3$ is outside $\{z\in\mathbb{C}:|z|<2\}$, is it correct?
Regards
|
I'm working on creating a homography 3x3 matrix to do a perspective correction of a photograph 2D piece of paper. The paper contains 3 markers (like the 3 corner markers of a QR code) in its corners, which already have been detected in photograph coordinates.
With the detection I also have the area of each marker, whic... |
Consider the interval $ S_k = (1 − 1/k, 2 + 1/k] $ and I want to find $ \cup^\infty_{k=1} S_k $ with proof
How do I go about this?
What I was thinking... as $ k \rightarrow \infty $ $ 1/k \rightarrow 0 $ so the biggest interval will be (0, 3]. And every interval past $ k = 1 $ will be smaller than this interval hence t... |
If my $x$ belongs to non-negative real numbers then what is the expansion of $\cos \sqrt x$ . Is it sufficient to substitute $\sqrt x$ in place of $x$ in $\cos x$ expansion ? But I see differentiability issues at $x=0$ in case of substitution.
|
$a_1=2$, and $a_{n+1}=\frac{2a_n +3}{a_n +2}$. I'm not sure on how to prove this by mathematical induction it's my weak point.
|
I have this nice question:
$\int_{[0,1]^3} \min (x,y,z)\,dx\,dy\,dz .$
I think it equals
$6\int_0^1\int_0^z \int_0^y x \,dx\,dy\,dz .$
Then using my formula I got the result to be $\frac {1}{4} $
But when I used the hint below I got $\frac {17}{8} $
So what goes wrong?
|
Let $X$ and $Y$ be simply connected, locally finite CW-complexes and let $(X_i)_i$ and $(Y_i)_i$ be their Postnikov towers respectively.
Is the Postnikov tower of $X\times Y$ given by the products $(X_i \times Y_i)_i$ with the obvious maps?
|
Two random vectors $\mathbf a$ and $\mathbf b$. Vector $\mathbf a$ has uncorrelated entries satisfying $\mathbb E [\mathbf a \mathbf a^{\rm H}]=\sigma^2{\mathbf I}$. Now I need to calculate ${\mathbb E} [|\mathbf b^{\rm H} \mathbf a|^2]$.
For independent $\mathbf a$ and $\mathbf b$, it can be calculated as
\begin{alig... |
I am wondering which relation(s) there are between triangulations of combinatorially equivalent polytopes that use no new points:
Let $P,Q$ be a $n$-polytopes such that their face lattices are isomorphic (say via $\phi$), that is $P,Q$ are combinatorially equivalent. Suppose we have an ordering on the vertices of $P,Q$... |
For a matrix:
$
$[A]$ = \begin{bmatrix}
5 & 4 \\
1 & 2 \\
\end{bmatrix}
$
I have the eigenvalues: $\lambda = 6, 1$
Now for each value I need to find eigenvectors:
When $\lambda = 6$
$$
\begin{bmatrix}
5-6 & 4 \\
1 & 2-6 \\
\end{bmatrix} \begin{bmatrix}
x \\ y
\end{bmatri... |
The wikipedia entry for bounded operators shows that for the space $X$ of trigonometric polynomials on $[-\pi,\pi]$ with norm
$$\lVert P\rVert = \int_{-\pi}^{\pi}\lvert P(x)\rvert \operatorname{d}\!x,$$
the operator $L: X\rightarrow X$ by $P\mapsto P^{\prime}$ is not a bounded operator.
It is not clear to me why there ... |
I have recently been reading about the Large Mathieu Groups in "The Finite Simple Groups" by Robert Wilson and I have not come across any explanation of this isomorphism as of yet. In this context, $M_{21}$ is described as the point stabiliser of $M_{22}$, which is described as the point stabiliser of $M_{23}$, and so ... |
Let $G=\langle S\rangle$.
Now let $H=\{x_1x_2\cdots x_m\mid x_i\in S \cup S^{-1},\ i\le m\in\Bbb N,\ m\text{ is even}\}$
I'm trying to prove that $[G:H]=1$ or $2$. I started doing this by proving that $H$ is a subgroup of $G$. I assumed that $H$ was empty. Which would imply that $S \cup S^{-1}$ is empty which would imp... |
Let $f(n) = \sum_{i=1}^n i^{-1}$.
Would the simplest $g(n)$ be $1$?
If we let $g(n) = 1$, then for all $n > 1$, $g(n) \le f(n)$.
To construct an upper bound, observe that since $n$ is finite, $f(n) = C$ for some $C \in \mathbb R$. (Here $C$ is the $n$th harmonic number). Then by the Archimedean Property for real numbe... |
$$A_n=\frac{cos(n)}{n}\to 0$$ I've been asked to find the limits of this sequence however using the sandwich theorem I have just found that the limit is $0$ is this the only limit if not what else is there?
|
Let $D = \operatorname{diag}(\lambda_1, \ldots, \lambda_n)$ be a diagonal matrix with positive entries $\lambda_i > 0$ (some of them might coincide). If we have the matrix equation $A D A^t = D$, does it follow that $A$ is invertible with $A^t = A^{-1}$?
It is obviously true if $A$ and $D$ commute. For $(2\times 2)$-... |
Hi i have question i have something like this:
$2 \sin 5t - 5 \cos 5t = A \sin(5t + \varphi)$
The main question is :
Is there any formulas or something to change left side of "$=$" to look similar to right?
The main goal is to make left side similar to right not to solve equation
$A = \frac{e^{2x}}{29}\cdot e^{-2x} = ... |
$$\lim_{x \to \infty} \frac{(\frac x n)^x e^{-x}}{(x-2)!}$$
where $x$ is $\mathbb N$-valued and $n$ is some nonzero real number.
Wolfram seems to give $0$ for different values of $n$ that I tried.
Is it really $0$ for any nonzero real $n$? I tried Stolz–Cesàro, but that doesn't seem to get me anywhere.
|
EDIT: I notice that the link is hidden, but this post is made with reference to
THIS PAPER
I'm trying to solve quite an old problem (once again) - to find the distance between a point (in 3d space) and the ellipse. - Or actually the furthest distance on said ellipse (but the algorithm is similar so...).
The first step... |
Let $Y$ be any topological space. In my notes I found the exercise to show that: $I \times Y \approx S^n $ via a homeomorphism is not possible, where $S^n$ denotes the $n$-sphere and $I$ the unit interval.
It is used in the proof of the Jordan curve theorem, so maybe a proof without using this theorem would be appropri... |
I posted question about distribution of poisson distribution multiplied by constant.
Here!
From this post, i can obtain what i want.
$$P(X=x)=\frac{\lambda^{n}}{n!}e^{-\lambda}$$
$$Z=\alpha X (\alpha>0)$$
$$P(Z=z)=\frac{\lambda^{z/\alpha}}{(z/\alpha)!}e^{-\lambda}$$
And i tried to calculate pgf of this distribution lik... |
A single-celled spherical organism contains $70$% water by volume. If it loses $10$% of its water content, how much would its surface area change by approximately?
$3\text{%}$
$5\text{%}$
$6\text{%}$
$7\text{%}$
|
This is a continuation of my previous quesion concerned with finding $ \cup^\infty_{k=1} S_k $ for $ S_k = (1 − 1/k, 2 + 1/k], k \in N $.
And the answer intuitively makes sense, but what about for a relative complement with the real numbers?
$ \cap_{k=1}^\infty R$\ $S_k $
The largest interval of $S_k $ is (0, 3]. So ... |
The k-cube is the graph whose vertices are the ordered k-tuples of 0's and 1's, two vertices being joined if and only if they differ in exactly one coordinate.
Show that the k-cube has $2^k$ vertices, $k$$2^k$$^-$$^1$ edges and is bipartite.
Note : This question is from the book "Graph Theory With Applications" written... |
Let $F : \mathbf{Sets} \to \mathbf{Cat}$ be the free functor that takes the elements of a set to the objects of a discrete category.
Does it has a left adjoint? Awodey (2010 p.249 ex.8) says it does, and suggests we can construct it considering the “connected components” of a category.
I am not sure what this means tho... |
Hello friends of maths,
I've given an arbitrary polygonal cross section (in cartesian coordinates $y$ and $z$). On this cross section, there acts an arbitrary stress-field $\sigma = f(y,z)$ as integrable function of $y$ and $z$.
Now I need to numerically determine the momentum ($M_y$ and $M_z$) as well as the axial for... |
This is a question from a list.
Obtain two $\mathcal{C}^\infty$ functions $f,g:\mathbb{R}\to\mathbb{R}$ satisfying these properties:
$f(x)=0 \Leftrightarrow 0\leq x\leq 1$;
$g(x)=x$ if $|x|\leq 1$, and $|g(x)|<|x|$ if $|x| >1$.
Some functions that I tried are not twice derivable. I'm searching for one evolving exp, b... |
I am a physicist so I am sorry if the following is not written in a rigorous (or even completely right) way.
As Quantum Mechanics is formed in Hilbert space, I would like to know what the so-called weight function $w(t)$ stands for, which is defined via the inner product:
$$
\langle f, g \rangle = \int_0^1 f(t) \overl... |
Assumptions: Consider a sample of i.i.d random variables $X_i$ $i=1,...,n$, where each $X_i$ is defined on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$, $X_i:\Omega\rightarrow \mathbb{R}$.
Let $\theta \in \mathbb{R}$ be the parameter underlying the probability distribution of $X_i$, $P_{\theta}$.
(1) Let... |
As a deeply enthusiastic prospective undergraduate student, there are is a fact that i'm still to completely understand about the big $O$ notation, namely:
Let $f(x), g(x) \neq x$ be nonconstant differentiable functions with $f(x), g(x) = O(x)$. Does it necessarily follow that
$\dfrac{f(x) - x}{f(x) - g(x)} = \dfrac{O... |
I'm learning Trigonometry right now with myself and at current about General solution. I have a question in my book which I don't understand how to proof. The question is Show that the two angles are indicated by the two following formulae - (1) $(2n-1)\pi/2 + (-1)^n\pi/3$ and (2) $2n\pi±\pi/6$
I don't know how to pro... |
I have been trying to prove that for a random variable that is uniformly bounded, i.e. $|X_n - c| <M$ for all $n$, convergence in probability to $c$ implies that
$$E\left(X_n \right) \to c$$
as well. This is quite intuitive as all probability mass is bounded away from infinity in this case, but I am having a hard tim... |
I am learning about Green's function, and the different ways of obtaining it, through variation of parameters and through eigenfunction expansion.
Variation of parameters gives it via one equation for a < x < t and another for t< x< b, in this form: Variation of Parameters
Eigenfunction expansion gives it in the form ... |
Integrate $$\frac{dx}{\sqrt{6x-4x^2}}$$
While completing the square of $6x-4x^2$, I want to know where did $9/16$ come from in the following after taking $4$ out as the common factor $$-4\bigg(x^2 -\frac{3x}{2} +\frac{9}{16}\bigg) +\frac{9}{4}$$. I want to know the steps of completing the square.
|
I got $2$ non zero matrices: $A$ and $B$.
$A^t=-A$ and $B^t=B$.
I try to understand if there is such matrix $B$ so the statment $AB + BA = O$ is true. I know that for $B=I$ it's not becuase $A$ will be $O$.
How can i find such matrix $B$ so it will be true? Both matrices are $3\times3$.
Thank you
|
T is defined as $$T:P_2(\mathbb R) \to M_2 (\mathbb R) \ \text{where} \ T(ax^2 +bx+c)=\begin{pmatrix}-2a +c & b+c\\-3b-3c&6a-3c\\ \end{pmatrix}$$ and I need t find bases for $ Im(T) $ and $ker(T)$
I started with $im(T)$ and got
$$Im(T) = \begin{pmatrix}-2a +c & b+c\\-3b-3c&6a-3c\\ \end{pmatrix} \\
= a\begin{pmatrix}-2... |
Let $F$ be a free group with $r$ generators. Then for every $n \in \mathbb{N}$ there are exactly $H_n = \left(\frac{n!}{2}\right)^r - 1$ nontrivial distinct homomorphisms from $F$ to $A_n$, the alternating group on $n$ elements. There is a canonical map $$\iota \colon F \to \mathcal{A} = \prod_{n = 1}^\infty A_n^{H_n},... |
I want to prove this. Say if $A-B$ is a positive definite matrix then can we find a relation between $\det(A)$ and $\det(B)$?
e.g. is $|A| < |B|$.
|
I have tried possibly all approaches.
I first expressed $80$ as $60+20$ and $40$ as $60-20$ and then used trig identities.I later used conditional identities expressing $\tan 20^\circ+\tan40^\circ+\tan120^\circ$ as $\tan 20^\circ \tan40^\circ \tan120^\circ$. But I really can't get to the end of it .
Please help.
|
I have seen lots of quadrature formulas where we have definite limits or one of the limits is infinity and the other is zero. But what about the following case
$$f(x) = \int_a^\infty e^{\frac{x}{t}} dt$$
How would I integrate this numerically when there is a non-zero lower limit?
|
Is there a way of using derivatives to get some trigonometric identities in a straight-forward fashion? I use to forget them, so that would help me a lot...
For example, since when we get the derivative of the sine function we use the angle sum identity, maybe there is a way to get the angle sum identity from the deriv... |
Does it have some closed form or approximation ? I tried on my own but i am not getting any idea regarding this.
$$\sum_{k_1=k}^{M}\sum_{k_2=k}^{M}\frac{k_1^{-\gamma} k_2^{-\gamma} {{M-k}\choose{k_1-k}}{{M-k_1}\choose{k_2-k}}}{{M\choose k_1}{M \choose k_2}}$$
One implied relation is "$k_1+k_2 \leq M+k$".
Please sha... |
How many permutations of $1,2,\cdots, n$ contain exactly $n-2$ digits that are smaller than the digit immediately to their right?
My solution proceeded with recursion. It has some chance of being wrong.
We call all such permutations good permutations. We call a digit good if it is are smaller than the next digit. Le... |
$$f(x) =
\begin{cases}
-1 &x \in [-1,0],\\
+1 &x \in (0,1]\\
\end{cases}$$
the formula for calculation of coefficients in terms of Legendre polynomial $L_k(x)$ is:
$f_k= (2k+1)/2\int_{-1}^{1}f(x)L_k(x)dx$
can you please help me in calculating the answer of its coefficients which is:
$f_k= (-1)^k \frac {(4k+3)(2k)!... |
How do I solve the following equation:
$(3x)^{ln3}=(4x)^{ln4}$
Thanks in advance!
|
I have a conceptual question about numerical methods for second-order elliptic partial differential equations.
What is the difference among finite element, continuous finite element, discontinuous finite element, continuous galerkin and discontinuous galerkin methods? Are some of them just the same thing?
|
Let us first define some terms.
Definition of Pre-pseudometric
Let $X\ne\emptyset$ and a function $\varphi:X\times X\to\mathbb{R}$ will be called a pre-pseudometric on $X$ if,
$x=y\implies \varphi(x,y)=0$
$\varphi(x,y)=-\varphi(y,x)$ for all $x,y\in X$
$\varphi(x,y)=\varphi(x,z)+\varphi(z,y)$ for all $x,y,z\in X$
... |
Find the condition that the lines represented by $$ax^2+2hxy+by^2=0$$ are such that the slope of one line is $k$ times that of the other.
I calculated the two represented by
$ax^2+2hxy+by^2=0$ by multiplying by a on both sides and adding $h^2y^2$ to both sides as :
$(ax+by)^2=y^2(h^2-ab)
(ax+by)=\pm\sqrt{h^2-ab}$. A... |
I am currently working on to derive the following form of the stock price dynamics:
$$dS_t = S_t[(r_t + \psi\sigma_S)dt + \rho \sigma_S dz_{1t} + \sqrt{1-\rho^2}\sigma_S dz_{2t}$$
where the parameter $\rho$ is the correlation between bond market returns and stock market returns, $\sigma_S$ is the volatility of the stoc... |
Find the basis for $ker(T)$ where $T: \mathbb R^2 \to \mathbb R^2 \ \ \ (x,y,z) \to (y,y,y)$
$$
\begin{pmatrix}0 & 1& 0\\
0 & 1& 0\\
0 & 1& 0\\
\end{pmatrix} $$is reduced to$$
\begin{pmatrix}0 & 1& 0\\
0 & 0& 0\\
0 & 0& 0\\
\end{pmatrix} \implies y=0 $$
$$\therefore ker(T)=(x,0,z) = x(1,0,0)+z(0,0,1) \\
\therefo... |
$\newcommand{\inv}{\text{inv}}$
Let $G$ be a Lie group. Assume $d$ is a metric on $G$ (in the sense of metric spaces) which is bi-invariant. Is it true that the inverse automorphism must be an isometry of $(G,d)$?
I do not assume $d$ is induced by a Riemannian metric*.
*If $d$ is induced by a Riemannian metric $g$, t... |
The definition of a topological space is a set with a collection of subsets (the topology) satisfying various conditions. A metric topology is given as the set of open subsets with respect to the metric. But if I take an arbitrary topology for a metric space, will this set coincide with the metric topology?
I'm trying... |
Consider nonconstant functions $f(x), g(x) \neq x$. Suppose there exist positive constants $k_1$ and $k_2$ such that
$k_{1} x \leq f(x) \leq k_{2} x$ and $\frac{1}{2}k_{1} x \leq g(x) \leq k_{2} x$. Does it follow that $\ \frac{f(x) + x}{\mid f(x) - g(x)\mid} < \infty$ ?
My attempt: Since $f(x) + x \leq (k_{2} + 1)x$... |
"Let $\mathcal S$={$\mathbf x \in \mathbb{R}^3 : ||\mathbf x||=1$}
Prove that the area of the part of $\mathcal S$ that lies between the two parallel planes given, say, by $x_3=a$ and $x_3=b$, is the same as the area of the part of the circumscribing cylinder (given by $x_1^2+x_2^2=1$) that lies between these two plane... |
Apologies for the somewhat cryptic title.
For any first-order formula X, let ssm(X) be the size of the smallest finite model of X. By size I mean number of individuals. So, for example, ssm('Fa & ~Fb') = 2 because the smallest models of 'Fa & ~Fb' have two individuals. If X has no model, or only infinite models, f(X) ... |
I'm attempting to prove the convergence (or divergence, though I strongly suspect it converges) of a sequence defined as $a_{n+1}=\sqrt{2-a_n}$ with $a_1=\sqrt{2}$.
I cannot use the monotonic sequence theorem as the sequence is not monotonically increasing. In fact, the first few values of the sequence are:
$a_1 =\sqr... |
We have function $f:\mathbb{R}\rightarrow \mathbb{R}$ with $$f\left(x\right)=\frac{1}{3x+1}\:$$ $$x\in \left(-\frac{1}{3},\infty \right)$$
Write the Maclaurin series for this function.
Alright so from what I learned in class, the Maclaurin series is basically the Taylor series for when we have $x_o=0$ and we write the ... |
Let $k$ be an algebraically closed field, $T$ be a integral, regular, projective $k$-scheme and $X$ another projective, integral $k$-scheme. Let $\mathcal{F}$ be a coherent (pure) sheaf on $X \times_k T$, which is flat over $T$. Is the dual sheaf $\mathcal{Hom}_{\mathcal{O}_{X \times_k T}}(\mathcal{F},\mathcal{O}_{X \t... |
An urn contains white and black balls with $p_w = p$ and $p_b = 1-p$. If we made $n$ extractions with replacement what is the probability that the number of white balls extracted is between $a$ and $b$?
I thought this:
$P=P(b) - P(a)$
$P(b) =
\binom{n}{b}\cdot\left(p\right)^{b}\cdot\left(1-p\right)^{n-b}$
$P(a)
... |
If X(n) and Y(n) converges to X and Y in distribution respectively then does X/Y(n) also converge to X/Y in distribution? Prove or disprove.
I feel that this is correct but have not been able to proceed at all with the proof. If I move to distribution of X/Y then it becomes too complicated and I am not able to proceed ... |
I'm not very good at PDEs but this particular problem seems... Strange. It requires that the answer be "continuous (!!)" all in bold.
\begin{align}
u_{tt}&=9u_{xx},\quad x>0,\, t>0, \\
u(x,t)&=e^{-x} \\
u_t(x,0)&=\cos(5x)\\
u_x(0,t)&=u(0,t)+t\\
\end{align}
That last one is giving me trouble, and I'm certain how to ... |
$$\lim_{n \to \infty} \int_{0}^{n}\left(1-\frac{3x}{n}\right)^ne^{\frac{x}{2}}dx$$
I thought about using the theorem of monotonic convergence and had $$f_n{(x)}=\left(1-\frac{3x}{n}\right)^ne^{\frac{x}{2}} \lambda_{[0,n]}(x)$$ it is rising. but I don't know where it is valid to write:
$$\lim_{n \to \infty} \int_{0}^{n... |
A bit of background: I'm writing a report in a piece of software and have to use XSLT1.0 (no extensions supported). What this means is that I only have basic arithmetic functions and a few simple functions like "mod"
My maths is pretty rusty, but here's the problem (for 2 variables, but there could be more)
We are a ma... |
Context:
Be a vector $V \in \mathbb{R}^p$, a matrix $X\in \mathbb{R}^{(p\times p)}$ and a second vector $B\in \mathbb{R}^p$ such that
$V =X B$
I'm looking for a way to find a partition of the type
$\begin{pmatrix}
V_a\\
V_b
\end{pmatrix}
=
\begin{pmatrix}
X_{a} & X_{ab}\\
X_{ba} & X_b
\end{pmatrix}
\begin{pmatrix}
B_a\... |
Let $X$ be a topological space, $(x_\alpha)$ a net in $X$ and $A \subseteq X$ an arbitrary subset.
The point $x \in X$ is
a cluster point of $x_\alpha$ if for every neighborhood $U$ of $x$ the net $x_\alpha$ is frequently in $U$ or equivalently if $x_\alpha$ has a subnet that converges to $x$.
an accumulation point of... |
I have the following position function for a point:
$x(t) := C_x - (S_x-C_x) \cdot \cos(t\cdot\theta) + (S_y-C_y) \cdot \sin(t\cdot\theta) + t \cdot v_x$
$y(t) := C_y - (S_x-C_x) \cdot \sin(t\cdot\theta) - (S_y-C_y) \cdot \cos(t\cdot\theta) + t \cdot v_y$
These functions calculate the x and y coordinates for a point $S... |
I was studying the Inverse function theorem when I came across the following problems :
(Let the closed set $V$ i.e the range have non-empty interior)
Does there exist a continuous onto function from an open set $U$ in $\mathbb{R}^n $ to a closed set $V$ in $\mathbb{R}^m$ such that some points in the interior of $... |
Can anyone help me to solve this? Thanks.
|
I am trying to show that $(-1,1)$ and $\{(x,y)\in\mathbb{R}^2:y=x^3\}$ are homeomorphic, with the standard metrics. I cant see how to define a bijection.
|
Find the number of values of $k$ for which the system of equations has no solution:
$$(k+1)x+8y=4k$$
$$kx+(k+3)y=3k-1$$
This is the augmented matrix:
$$\begin{bmatrix}
k+1 & 8 & 4k \\
k & k+3 & 3k-1
\end{bmatrix}$$
$R_2 \rightarrow R_2-\frac{k}{k+1}R_1$
$$\begin{bmatrix}
k+1 & 8 & 4k \\
0 & k+3-\frac{8k}{k+1} &... |
I used the Fundamental Theorem of Calculus to show that if $g\in C[a,b]$ then we can define $g(x)=\int_a^x g(x)dx$, $x\in[a,b]$ and thus $g$ is differentiable and $g'(x)=g(x)$. Then since we now know that the function is differentiable and continuous then I used the MVT to get that $g'(x)=\frac{g(b)-g(a)}{b-a}$.
But ho... |
What is the domain and range of
$f(x)=\arcsin[e^{-x}]+ \arcsin [e^x]$ where $[x]$ denotes greatest integer function?
|
I think that I do not understand something in the definition of the line bundles. Line bundles have fibers of rank 1, that is isomorphic to $\mathbb{C}$. But I do not know how to connect this vector space with elements like $dz_i$, $\,dz_{i_1}\wedge dz_{i_{2}},$ etc. Here, $dz_i$ are locally defined one-forms for the c... |
Let $f(x)=(x+a_1)(x+a_2)...(x+a_n)$, where $a_1,a_2, ..., a_n -$ non-negative numbers. Prove that
$$f'(1)\ge n \left ( 1+ \sqrt[n]{f(0)} \right)^{n-1}$$
|
Let $\mathcal{B}^n$ be the borel sigma algebra generated by the rectangles in $\mathbb{R}^n$. I can write $f(\mathbf{x})=g_1(x_1)\cdots g_n(x_n)$.
Let $\mu=\mu_1\times \cdots \times \mu_n$ be the Lebesgue product measure.
Is it possible to write $\int_Af(\mathbf{x})\ d\mu(\mathbf{x})$, where $A \in \mathcal{B}^n$, as a... |
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