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Determining the Derivatives of the Inverse Trigonometric Functions
Now let's determine the derivatives of the inverse trigonometric functions, \(y = \arcsin x,\) \(y = \arccos x,\) \(y = \arctan x,\) \( y = \text{arccot}\, x,\) \(y = \text{arcsec}\, x,\) and \(y = \text{arccsc}\, x.\)
One way to do this that is particu... |
Difference between revisions of "Inaccessible"
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A cardinal $\kappa$ is ''hyperinaccessible'' if it is $\kappa$-inaccessible. One may similarly define that $\kappa$ is $\alpha$-hyperinaccessible if it is hyperinaccessible and for every $\beta\lt\alpha$, it is a limit of $\beta$-hyperinaccessible car... |
Decide if the series $$\sum_{n=1}^\infty\frac{4^{n+1}}{3^{n}-2}$$ converges or diverges and, if it converges, find its sum.
Is this how you would show divergence attempt:
For $n \in [1,\infty), a_n = \frac{4^{n+1}}{3^n -2} \geq 0$
For $n \in [1,\infty), a_n = \frac{4^{n+1}}{3^n-2} \geq \frac{4^{n+1}}{3^n} = b_n$
Since ... |
There's not a
lot to say here, but we can say a couple of things. We'll assume $H\le G\le S_n$ and $H$ and $G$ are both transitive. A block system is defined by a choice of a single block $\Delta\ni 1$, so let $B_H$ be the set of blocks for $H$ containing $1$ and $B_G$ be the set of blocks for $G$ containing $1$.
Immed... |
I am trying to solve a mean-variance problem with a non-linear market impact cost term in there. This is the problem I am trying to solve
$$ \max_x \left ( \alpha x - \gamma x' \Sigma x - a\sqrt{|x-x_0|} \right ) \quad s.t. \quad \text{unconstrained}.$$
where, $x_0$ is the current portfolio holding.
I am using MATLAB's... |
I think you are interpreting too much into the matter. The $-\frac12\sigma^2$ is just a correction term that comes from Jensen's inequality.You need this when switching from supposedly symmetric returns (normal distribution) to the skewed price process (log-normal distribution).I think there are no deeper truths to be ... |
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Kaon femtoscopy in Pb-Pb collisions at $\sqrt{s_{\rm{NN}}}$ = 2.76 TeV
(Elsevier, 2017-12-21)
We present the results of three-dimensional femtoscopic analyses for charged and neutral kaons recorded by ALICE in Pb-Pb collisions at $\sqrt{s_{\rm{NN}}}$ = 2.76 TeV. Femtoscopy is used to... |
Exact Differential Equations
We will now look at another type of first order differential equation that we can solve known as exact differential equations which we define below.
Definition: Let $M(x, y)$ and $N(x, y)$ be functions, and suppose we have a differential equation in the form $M(x, y) + N(x, y) \frac{dy}{dx}... |
Towards Understanding the Origin of Cosmic-Ray Electrons
We present the precision measurement of the electron flux with a particular emphasis on the behavior at high energies. The measurement is based on 28.1 million electron events collected by AMS from May 19, 2011 to November 12, 2017. This corresponds to a factor o... |
Assuming we do not know the Lorenz curve function,
If $(X_k, Y_k)$ are the known points on the Lorenz curve, with the $X_k$ indexed in increasing order $(X_{k – 1} < X_k)$, so that:
$X_k$ is the cumulated proportion of the population variable, for $k = 0,...,n$, with $X_0 = 0, X_n = 1$. $Y_k$ is the cumulated proportio... |
A new construction of rotation symmetric bent functions with maximal algebraic degree
School of Mathematics and Statistics, Henan University, Kaifeng 475004, China
$ n = 2m\ge4 $
$ n $
$ m $
$ f(x_0,x_1\cdots,x_{n-1}) = \bigoplus\limits_{i = 0}^{m-1}(x_ix_{m+i})\oplus \bigoplus\limits_{i = 0}^{n-1}(x_ix_{i+1}\cdots x_{... |
You may want to save the results of these exercises, since we will revisit in the next two sections. In Exercises [exer:3.1.1}– [exer:3.1.5} use Euler’s method to find approximate values of the solution of the given initial value problem at the points \(x_i=x_0+ih\), where \(x_0\) is the point where the initial conditi... |
If you sent all the gifts in the Twelve Days of Christmas, then how many would you send?
The gifts for each day are the triangular numbers :
\(\begin{array}{c | cc}
\text{Day}& \text{Count} & \text {Total}\\ \hline 1 & 1 & 1\\ 2 & 1 + 2 & 3\\ 3 & 1 + 2 + 3 & 6\\ 4 & 1 + 2 + 3 + 4 & 10\\ n & 1 + 2 + 3 + 4 +… +n & \frac ... |
The question says to solve this equation: $(z+1)^5 = z^5$
I did. Just want to find out if I did it properly and if my run-around logic makes sense.
First I begin my writing the equations as:
$$ (z+1)^5 = z^5$$ $$ \mathbf{e}^{5 \mathbf{Log}(z+1)} = \mathbf{e}^{5 \mathbf{Log}(z)} $$ So $$ \mathbf{Log}(z+1) = \ln|z+1| + \... |
Well, I'm amazed that this works at all as it is in reality not supported input. The syntax for lists is
\begin{enumerate} \item ...
and nothing in between. So putting
\begin{multicols} in that space is at best adventurous.
However, if I run your document with the very latest
multicols from CTAN 1.8g everything works a... |
Eigenvalues and Eigenvectors Examples 1
Recall from the Eigenvalues and Eigenvectors page that the number $\lambda \in \mathbb{F}$ is said to be an eigenvalue of the linear operator $T \in \mathcal L (V)$ if $T(u) = \lambda u$ for some nonzero vector $u \in V$. The vectors $u$ such that $T(u) = \lambda u$ are called ei... |
Calculating the Probability of a Sample Containing Bad Parts
Received a question from a reader this morning that will make a nice tutorial.
A box contains 27 black and 3 red balls. A random sample of 5 balls is drawn without replacement. What is the probability that the sample contains one red ball?
So here’s my thinki... |
The product converges because $\sum n^{-2}$ does. Can you see why? What is the greatest sum that can possibly appear when we expand the product?
In fact, suppose that $a_n\geq 0$ for each $n$. Set $$p_n=\prod_{k=1}^n a_k$$
Then $\log p_n=\sum_{k=1}^n\log a_k$
If $\sum a_k$ converges, then $a_k\to 0$, then since $$\lim_... |
This post has been cross-posted on the Quansight LabsBlog.
As of November, 2018, I have been working at Quansight. Quansight is a new startup founded by the same people who started Anaconda, which aims to connect companies and open source communities, and offers consulting, training, support and mentoring services. I w... |
Eigenvalues and Eigenvectors Examples 2
Recall from the Eigenvalues and Eigenvectors page that the number $\lambda \in \mathbb{F}$ is said to be an eigenvalue of the linear operator $T \in \mathcal L (V)$ if $T(u) = \lambda u$ for some nonzero vector $u \in V$. The nonzero vectors $u$ such that $T(u) = \lambda u$ are c... |
@Secret et al hows this for a video game? OE Cake! fluid dynamics simulator! have been looking for something like this for yrs! just discovered it wanna try it out! anyone heard of it? anyone else wanna do some serious research on it? think it could be used to experiment with solitons=D
OE-Cake, OE-CAKE! or OE Cake is ... |
So my book proves the convergence of $\Gamma(z) = \int_0^{\infty}t^{z-1}e^{-t}dt$ in the right half plane $Re(z) > 0$, and then goes on to prove the initial recurrence relation $\Gamma(z+1)=z\Gamma(z)$ by applying integration by parts to $\Gamma(z+1)$:
$$\int_0^{\infty}t^{z}e^{-t}dt = -t^ze^{-t}|_0^{\infty} + z\int_0^{... |
Problem at hand (not homework): Let $N, K$ be subgroups of a group $G$, with $N$ normal in $G$. If $N$ and $K$ are abelian groups and $G= NK$, is $G$ the direct product of $N$ and $K$?
So, here's the criteria that I know:
Given normal subgroups $N_1, \cdots N_k$ of $G$ such that any element $g\in G$ can be written uniq... |
Eigenvalues and Eigenvectors Examples 3
Recall from the Eigenvalues and Eigenvectors page that the number $\lambda \in \mathbb{F}$ is said to be an eigenvalue of the linear operator $T \in \mathcal L (V)$ if $T(u) = \lambda u$ for some nonzero vector $u \in V$. The nonzero vectors $u$ such that $T(u) = \lambda u$ are c... |
Note to Self: Getting in Touch with My Inner Austrian: Memo to Self: A Start of a Model...: How can you be an Austrian, and yet have big swings in the desired capital stock—and thus bigger swings in the desired rate of investment—without having to have massive shocks to the current level of technology? How can you get ... |
You only need to consider the case $\mathfrak{h}_{s}^\ast(A) \lt \infty$, but you need to be a bit careful in choosing the outer approximations since swapping $\inf$ and $\sup$ certainly isn't allowed without some thinking. If you knew that you can always take the same set $E$ in the $\inf$ (which I will show in \eqref... |
Consider the integral operator $T : C([0,1])\to C([0,1])$ given by
$$Tf(t)=\int_0^1 K(t,\tau)f(\tau)d\tau.$$
I'm solving one exercise which is to show this operator is bounded. The exercise is from a mathematical physics course.
Also, I want to show this
without any results from Lebesgue integration. So I'm disconsider... |
Given that $(ab)^2=(bc)^4=(ca)^x=abc$ Then what is the value of $x$?
$2(\log a+\log b)=4(\log b+\log c)=x(\log c+\log a)=\log a+\log b+\log c$
Then I am lost, any other easier way to solve?
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields... |
A sine sweep is a sine function that gradually changes frequency over time.
The sine sweep can also be called "sinusoidal sweep," "frequency sweep", or "chirp".
The functions
$$x(t)=\sin(2\,\pi(f_0\, t+\frac{f_1-f_0}{2\,T}t^2))$$
and
$$x(t)=\sin(2\,\pi\,f_0 \,T \frac{(\frac{f_1}{f_0})^{\frac{t}{T}}-1}{\ln(\frac{f_1}{f_... |
I have an unknown $n$-dimensional vector $x$ whose analytical expression depends on the following sum $x = z + Ba$ where the vector $z$ and the matrix $B\in \mathbb{R}^{n\times s}$ are given. So the $s$-dimensional vector $a$ is to be computed to find $x$.
The only assumption that we have is $x=0$ when we project $x$ o... |
I've been working through Gelman's Bayesian Data Analysis 3 text and have been trying to understand one of the hierarchical models revolving around rat tumors (Chapter 5). He uses a binomial model with p assigned a beta distribution. The Beta distribution has parameters $\alpha$ and $\beta$ which need a distribution fo... |
Note: This begins [with DS] January 1, 2017.
Note: This resumes with NS January 30, 2019 Note: This starts with DA February 4, 2019 NA complete, Feb 26, 2019 to Sep 24, 2019. 100-digit eigenvalues - the regular pentagon Unit-edged Regular Pentagon, both Dirichlet and Neumann boundary conditionsBob Jones Winter 2017
Thi... |
GolfScript (23 chars)
{:^((1${\.**2^?%}+*}:f;
The sentinel result for a non-existent inverse is
0.
This is a simple application of Euler's theorem. \$x^{\varphi(2^n)} \equiv 1 \pmod {2^n}\$, so \$x^{-1} \equiv x^{2^{n-1}-1} \pmod {2^n}\$
Unfortunately that's rather too big an exponential to compute directly, so we have... |
I am trying to solve the following trigonometric equation algebraically, where $0\leq \displaystyle x \leq2 \pi$
$$\sin2x = -\frac{1}{2}.$$
My answer must be an exact solution.
Here is what I have tried: If $\displaystyle \sin(30°) = \frac{1}{2}$, then $\displaystyle \sin (2\times 15°) = \frac{1}{2}$.
Sin is negative i... |
Will be glad for a little hint: let x and n be positive integer such that $1+x+x^2+\dots+x^{n-1}$ is a prime number then show that n is prime
Hint $\ $ The sequence $\rm\:f_n = (x^n-1)/(x-1)\:$ is a divisibility sequence, i.e. $\rm\:m\:|\:n\:$ $\Rightarrow$ $\rm\:f_m\:|\:f_n.\:$
In fact it is a strong divisibility sequ... |
I'm stuck on something in the derivation of the Principal Components.
We have random vectors $X$ of dimension $p$. We want to find linear combinations of $X$, $a'X,$ where $a \in \mathbb{R}^{p}$ that satisfy the constraints:
(1) $\max_{a_{1}\in\mathbb{R}^{p}} \big(Var(a_1' X) \big) $ subject to $a_1'a_1 =1$
(2) $\max_{... |
Vertex Colouring and Chromatic Numbers
Imagine that we could take the vertices of a graph and colour or label them such that the vertices of any edge are coloured (or labelled) differently. This is what we recognize as vertex colouring
Definition: A Good Vertex $k$-Colouring for $k \in \mathbb{Z^+}$ is a function $f: V... |
Is there any practical difference as to how a unity gain buffer using an op-amp might be configured? For example, the input signal can be connected to the noninverting input and the feedback to the inverting input; but it can also be configured the other way, with the inverting input as the signal and the noninverting ... |
Hints will display for most wrong answers; explanations for most right answers. You can attempt a question multiple times; it will only be scored correct if you get it right the first time. To see a new question, reload the page.
I used the official objectives and sample test to construct these questions, but cannot pr... |
Recall that the exponential function \(f(x)=2^{x}\) produces this table of values
\(x\) -3 -2 -1 0 1 2 3 \(f(x)\) \(\frac{1}{8}\) \(\frac{1}{4}\) \(\frac{1}{2}\) 1 2 4 8
Since the logarithmic function is an inverse of the exponential, \(g(x)=\log _{2} (x)\) produces the table of values
\(x\) \(\frac{1}{8}\) \(\frac{1}{... |
If I have two normally distributed independent random variables $X$ and $Y$ with means $\mu_X$ and $\mu_Y$ and standard deviations $\sigma_X$ and $\sigma_Y$ and I discover that $X+Y=c$, then (assuming I have not made any errors) the conditional distribution of $X$ and $Y$ given $c$ are also normally distributed with me... |
In this section, we demonstrate which integers have primitive roots. We start by showing that every power of an odd prime has a primitive root and to do this we start by showing that every square of an odd prime has a primitive root.
If \(p\) is an odd prime with primitive root \(r\), then one can have either \(r\) or ... |
Disclaimer. This is not an answer! But I hope the following transformations and numerical experiments can be useful in finding the solution.
First, we will get rid of trigonometric functions. Setting $t=\tan \phi$ we obtain:
$$\int_0^\infty \frac{t^a [1+t^2+2(1-t^2) \ln t]}{(1+t^b)(1+t^2)^2}dt$$
Here $a=\frac{1}{2}\sqr... |
There are two major faults in this argument.
The first is the assumption that over an infinitesimal time with an infinitesimal additional force (hence doing approximately 0 work to the object) that the object will be moving with a non-infinitesimal velocity.
The second is the assumption that because the object is movin... |
Actually the $\partial_\mu$ is a "total derivative", otherwise it wouldn't be a total divergence and we would not be able to get rid of it during the derivation of Lagrange equations
$$\partial_\mu \mathcal{L} \equiv \frac{\partial \mathcal{L}}{\partial \phi} \partial_\mu \phi + \frac{\partial \mathcal{ L}}{\partial ( ... |
Let $A = \begin{pmatrix} \cos \delta & -\sin \delta & 0 \\ \sin \delta & \cos \delta & 0 \\ 0 & 0 & 1 \end{pmatrix}$, and let $B = \begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos \delta & -\sin \delta \\ 0 & \sin \delta & \cos \delta \end{pmatrix}$ be rotation by $\delta$ along the $z$ and $x$ axes, respectively. In suitable coo... |
Theorem: Let $q:X\rightarrow\mathbb{R}$ be a sublinear functional on a real linear space $X$. Let $M$ be a linear subspace of $X$ and suppose that $f:M\rightarrow\mathbb{R}$ is a linear functional such that $f(x)\leqslant q(x)$ for all $x\in M$. Then there exists a linear functional $F:X\rightarrow\mathbb{R}$ such that... |
My brother asked me to calculate the following integral before we had dinner and I have been working to calculate it since then ($\pm\, 4$ hours). He said, it has a beautiful closed form but I doubt it and I guess he has tried to trick me again (as usual). So I am curious, what is the closed form (
if any) of the follo... |
We present a new approach to the eigensystem multiscale analysis (EMSA) for the Anderson model that relies on the Wegner estimate. The EMSA treats all energies of the finite volume operator in an energy interval at the same time, simultaneously establishing localization of all eigenfunctions with eigenvalues in the ene... |
Difference between revisions of "Vopenka"
(→Variants: gVP so?)
(One intermediate revision by the same user not shown) Line 76: Line 76:
* For every $Γ$, $VP(κ, Γ)$ for some $κ$ implies $VP(Γ)$.
* For every $Γ$, $VP(κ, Γ)$ for some $κ$ implies $VP(Γ)$.
* $VP(κ, \mathbf{Σ_1})$ holds for every uncountable cardinal $κ$.
* ... |
The Gluing Lemma
The Gluing Lemma
Lemma 1: Let $X$ and $Y$ be topological spaces and let $A, B \subset X$ be closed subsets of $X$ such that $X = A \cup B$. Furthermore, let $f : A \to Y$ and $g : B \to Y$ be continuous maps such that $f(x) = g(x)$ for all $x \in A \cap B$. Then the function $h : X \to Y$ given by $\le... |
Let $g_1, g_2, h_1, h_2 : \mathbb{R} \rightarrow \mathbb{R}$ be non-decreasing and right-continuous. Define $$ \begin{align} f_1 & := g_1 - h_1 \\ f_2 & := g_2 - h_2 \end{align} $$ and suppose $f_1 = f_2$. In other words, for every $a, b \in \mathbb{R}$ with $a < b$, the restriction of $g_1 - h_1$ and $g_2 - h_2$ to $[... |
Let $X_1,...,X_n$ i.i.d random variables, square integrable, and with $E[X_1]=0$.
Let $Y_n = \frac{|X_1 +...+X_n|}{\sqrt{n}}$
I am trying to show that $(Y_n)$ is uniformly integrable, i.e
$\sup_n\mathbb E[Y_n\mathbb 1\{Y_n\gt K\} ] \to 0$ when $K \to +\infty$
I have tried to use Cauchy-Schwarz inequality:
$\mathbb E[Y_... |
Exercise
Prove the statement using the $\epsilon$, $\delta$ definition of a limit: $$\lim \limits_{x \to 3}{(x^2+x-4)} = 8$$
The Precise Definition of a Limit In case you're not familiar with the definition of "The Precise Definition of a Limit", here it is.
Let $f$ be a function defined on some open interval that cont... |
How does one go about showing $ A \cup B \cup C $ is countable if $ A, B $ are countable and $C$ is finite?
I understand most of the confusion for resolving set theory questions online seem to be the definition. For my course we consider the following definitions:
countable: Finite or $A \sim\mathbb{N}$ uncountable: no... |
Finally, I would like to introduce
reaction-diffusion systems, a particular class of continuous field models that have been studied extensively. They are continuous field models whose equations are made of only reaction terms and diffusion terms, as shown below:
\[\dfrac{\partial{f_1}}{\partial{t}} =R_1(f_1,f_2,...,f_n) ... |
nth Order Ordinary Differential Equations
Definition: Let $D \subseteq \mathbb{R}^{n+1}$ be a domain (a nonempty, open, connected subset of $\mathbb{R}^{n+1}$) and let $h \in C(D, \mathbb{R})$. An n is of the form $y^{(n)} = h(t, y, y^{(1)}, ..., y^{(n-1)})$. th Order Ordinary Differential Equation The notation $y^{(i)... |
How can I calculate the Total Variation Distance of a transition Matrix? is there any built in function? I've searched all documentation and haven;t found anything.
** More information:
Let me try to explain it better. let's say we have a transition matrix ($P$), $4\times4$ that describes the probability of going from ... |
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Published on Oct 3, 2014
For germs of holomorphic functions $f : \mathbf{C}^{m+1} \to \mathbf{C}, g : \mathbf{C}^{n+1} \to \mathbf{C}$ having an isolated critical point at 0 with value 0, the classical T... |
I understand that during phase transition nucleation must occur. I'm wondering, once phase equilibrium is established, does nucleation still occur? For instance, in liquid-vapour equilibrium, does independent water molecules just simply enter and leave the liquid-vapour interface, or does some kind of nucleation (say l... |
The point is that the ball gets a tangential hit by the ground. This changes the angular momentum of the ball.
Consider a ball thrown with a horizontal speed v. It should also not rotate.
Right before hitting the ground, the ball has an angular momentum of
$$L=mvr$$
This is a result of $\vec{L}=\vec{v}\times\vec{p}$, w... |
Taken straight-out from Prof. L. Braile's collection of earthquake hazard information, that explains it quite nicely:
The magnitude scale is really comparing amplitudes of waves on a seismogram, not the STRENGTH (energy) of the earthquakes.
While one unit of magnitude is 10 times the amplitude on a seismogram, one unit... |
Kalid Azad uses color for equations [1] Chris Olah uses tables for equations [2]
Here I’m using both color and tables (using the table tag instead of css grid), with labels for the symbols:
\(\frac{dR}{dt}\) = \(\alpha\) \(\cdot\) \(R\) \(-\) \(\beta\) \(\cdot\) \(R\) \(\cdot\) \(F\) rate of change
of rabbits
is birth ... |
ASU Electronic Theses and Dissertations
This collection includes most of the ASU Theses and Dissertations from 2011 to present. ASU Theses and Dissertations are available in downloadable PDF format; however, a small percentage of items are under embargo. Information about the dissertations/theses includes degree inform... |
Dear all experts
I have had a problem since many years ago, but I've never tried to find the solution, but now I need it urgently.
Many of friends and classmates would ask me if they could use "condition on variables" in a summation in GAMS. Obviously for me, it was not possible. But all the time my answer was one thin... |
This question already has an answer here:
Series in Real Analysis 2 answers
Prove$\sum_{n=1}^\infty a_n \in \mathbb R \mathbb \leftrightarrow \sum_{n=1}^\infty\frac{a_n}{1+a_n}\in \mathbb R$
The hint says that for one direction argue that $b_n = \frac{a_n}{\frac{a_n}{1+a_n}}$ is bounded after some fixed cutoff point. I... |
The Group Isomorphism problem for finitely presentable groups is undecidable. Restricting to finite groups, the problem is certainly decidable, with the trivial algorithm of enumerating all $n!$ permutations (assuming of course, the two groups have the same order).
If the input groups $G, H$ are given by their Cayley t... |
Orthonormal Vectors Review
Orthonormal Vectors Review
We will now review some of the recent content regarding orthonormal vectors.
Recall from the Orthonormal Bases of Vector Spaces page that if $V$ is a finite-dimensional inner product space then an Orthonormal Basisof $V$ is a basis $\{ e_1, e_2, ..., e_n \}$ such th... |
As with the sine, we do not know anything about derivatives that allows us to compute the derivatives of the exponential and logarithmic functions without going back to basics. Let's do a little work with the definition again:
\[\eqalign{ {d\over dx}a^x&=\lim_{\Delta x\to 0} {a^{x+\Delta x}-a^x\over \Delta x}\cr& =\lim... |
A. Enayat, J. D. Hamkins, and B. Wcisło, “Topological models of arithmetic,” ArXiv e-prints, 2018. (under review)
@ARTICLE{EnayatHamkinsWcislo2018:Topological-models-of-arithmetic,
author = {Ali Enayat and Joel David Hamkins and Bartosz Wcisło},
title = {Topological models of arithmetic},
journal = {ArXiv e-prints},
ye... |
I’m not sure to whom the image or the idea is due. Please comment if you have information. (See comments below for current information.)
The rules will naturally generalize those in Connect-Four. Namely, starting from an empty board, the players take turns placing their coins into the $\omega\times 4$ grid. When a coin... |
Given that $\alpha$ is a root (in the field extension) of the irreducible polynomial $X^4+X^3-X+2\in\mathbb{Q}[X]$, I have to find the minimal polynomial of $\alpha^2$. I am thinking about this for a while, but I can't find it. I need some hints. Thank you.
You have $\alpha^4 + 2 = \alpha - \alpha^3$, and squaring both... |
Apart from images of representations of subgroups of SU(2), what are the Lie subgroups of SU(3)? Where should I look for a reference?
A first approximation to an answer is to determine the Lie subalgebras of $\mathfrak{su}(3)$. Here is their Hasse diagram with edges denoting inclusions. One can work this out iterativel... |
Commutative Rings
Recall that if $(G, *)$ is a group with the additional property that for all $x, y \in G$ we have that $x * y = y * x$ then $(G, *)$ is said to be an abelian group or a commutative group.
Now suppose that we have a ring, $(R, +, *)$ Then $(R, +)$ by definition is an abelian group, but there's no requi... |
Here's a proof of Ben's claim--I don't know where in Matsumura Ben is referring, but it's not too hard if you take two well known theorems for granted.
To se things up, let us first make a definition. If $f:X\to Y$ is a morphism, and $\mathscr{F}$ a quasicoherent sheaf on $X$ then define the
flat locus of $f$ (with res... |
In Riemannian geometry, a
Jacobi field is a vector field along a geodesic \gamma in a Riemannian manifold describing the difference between the geodesic and an "infinitesimally close" geodesic. In other words, the Jacobi fields along a geodesic form the tangent space to the geodesic in the space of all geodesics. They ... |
Performing Topology Optimization with the Density Method
Engineers are given significant freedom in their pursuit of lightweight structural components in airplanes and space applications, so it makes sense to use methods that can exploit this freedom, making topology optimization a popular choice in the early design ph... |
Even though the title is quite a mouthful, this post is about two really cool ideas:
A solution to the "chicken-and-egg" problem (known as the Expectation-Maximization method, described by A. Dempster, N. Laird and D. Rubin in 1977), and An application of this solution to automatic image clustering by similarity, using... |
The equation of motion for a fluid parcel in the atmosphere (in Cartesian space) is
$$\dfrac{D\mathbf u}{Dt} = -\dfrac{1}{\rho}\nabla p-2 \mathbf \Omega \times \mathbf u + \mathbf g + \mathbf F,$$
where $\mathbf u$ is the wind, $\rho$ is density, $p$ is pressure, $\mathbf\Omega$ is the angular velocity of the Earth, $\... |
Compactness Review
Compactness Review
We will now review some of the recent material regarding compactness in topological spaces.
Recall from the Covers of Sets in a Topological Spacepage that if $X$ is a topological space then an a Coverof $X$ is a collection of sets $\mathcal F$ such that:
\begin{align} \quad X \subs... |
Let $G$ be a compact connected $n$-dimensional Lie group, and let $1 \le k < n$. Do there exist a non-zero
bi-invariantdifferential $k$-form on $G$?
I know that the answer is positive for $k=n$ (top forms)-but the proof I know does not adapt to the case $k<n$. (That proof shows that every left-invariant $n$-form is in ... |
Theorems Regarding Linear Independence and Dependence
Table of Contents
Theorems Regarding Linear Independence and Dependence
Recall from the Linear Independence and Dependence page the definition of linear independence/dependence:
Definition: A set of vectors $V = \{ \mathbf{x_1}, \mathbf{x_2}, ..., \mathbf{x_n} \}$ a... |
For a discrete random variable \(X\) (a random variable whose range \(\mathcal{X}\) is countable) with probability mass function \(p(x) = P(X = x)\), we can define the (Shannon or discrete) entropy of the random variable as \[ H[X] = -E[\log_{2} p(X)] = - \sum_{x \in \mathcal{X}} p(x) \log_{2} p(x).\] That is, the entr... |
If $X$ is a discrete random variable, its entropy $H(X)$ is usually defined as something along the lines of $-\sum \def\P{\mathbb{P}}\P(x) \log_2( \P(x))$, where the sum ranges over all the possible values $x$ of $X$.
I have seen a few expositions of the problem of extending this definition to continuous random variabl... |
Chapter Review Exercises Exercise \({1}\)
Let \(f(x+y)=f(x)+f(y)+x^2y+xy^2\) and \(\lim_{x \rightarrow 0} \dfrac{f(x)}{x}=1\). Then find \( f^'(x) \).
Answer
\(1+x^2\)
Exercise \({2}\)
Use the limit definition of the derivative to find \(f'(x)\), where \(f(x)=3x^2\).
(You may use differentiation rules to check your ans... |
The solution of the model contains constant: $k = \alpha K$, it relates to: (i) probability of getting a fill ($\alpha$) and (ii) market impact ($K$).
Estimating (i). The author proposes that the market order sizes follow a power law distribution:
$$f(x)^Q \propto x^{-1-\alpha}$$
So no problem estimating this.
Estimati... |
In the book
Bayesian Data Analysis by Gelman et al. (3rd edition, 2014), a hierarchical model (or one-way random-effects ANOVA) is presented in section 5.4 as follows,
\begin{equation}\label{eq:lme1} y_{ij} = b_0 + \lambda_i + \varepsilon_{ij}, \end{equation}
where the data $y_{ij}$ come from the $i$th measuring entity... |
PCTeX Talk Discussions on TeX, LaTeX, fonts, and typesetting
Author Message Michael Spivak Joined: 10 Oct 2005 Posts: 52
Posted: Tue Oct 11, 2005 3:46 pm Post subject: new version of fonts We are making a new version of the MTPro fonts, which will have Times-Italic-like characters designed into them, so that there will... |
Covering Space Examples R2 \ {(0, 0)}
Recall from the Covering Spaces page that if $X$ is a topological space then a covering space of $X$ is a pair $(\tilde{X}, p)$ where $\tilde{X}$ is a path connected and locally path connected topological space and $p : \tilde{X} \to X$ is a continuous map such that for every $x \i... |
Wronskian Determinants and Linear Homogenous Differential Equations
Recall from the The Principle of Superposition page that if we have the second order linear homogenous differential equation $\frac{d^2 y}{dt^2} + p(t) \frac{dy}{dt} + q(t)y = 0$ and $y = y_1(t)$ and $y = y_2(t)$ are solutions to this differential equa... |
If a team of researchers perform multiple (hypothesis) tests on a given data set, there is a volume of literature asserting that they should use some form of correction for multiple testing (Bonferroni, etc), even if the tests are independent. My question is this: does this same logic apply to multiple teams testing hy... |
Preprints (rote Reihe) des Fachbereich Mathematik Refine Year of publication 1995 (9) (remove)
292
Symmetry properties of average densities and tangent measure distributions of measures on the line (1995)
Answering a question by Bedford and Fisher we show that for every Radon measure on the line with positive and finit... |
Speaker: Misha Tyomkyn (TAU) Title: Lagrangians of hypergraphs and the Frankl-Furedi conjecture Abstract: Frankl and Furedi conjectured in 1989 that the maximum Lagrangian of all r-uniform hypergraphs of given size m is realised by the initial segment of the colexicographic order. For r=3 this was partially solved by T... |
As Twitter and Github and some other sites have switched from square icons to circle icons, I needed to update my logo to fit. I had cropped the previous one, but I decided I should generate new ones from scratch. The main ingredients:
The red blob logo is \(r = 5 + \sin(5\theta)\). The mouth can be drawn with my face ... |
LaTeX supports many worldwide languages by means of some special packages. In this article is explained how to import and use those packages to create documents in
Portuguese.
Contents
Portuguese language has some accentuated words. For this reason the preamble of your file must be modified accordingly to support these... |
Author Message jautschbach Joined: 17 Mar 2006 Posts: 11
Posted: Fri Apr 21, 2006 7:42 am Post subject: small caps Hello
I noticed some inconsistencies between Y&Y and MikTeX on WinXP when using the small caps font with \textsc{}. With Y&Y I get the correct Times SC font in the oouput but with MikTeX it seems like some... |
Soil composition
by
V
olume and
M
ass, by phase:
a
ir,
w
ater,
v
oid (pores filled with water or air),
s
oil, and
t
otal.
Water content or moisture content is the quantity of water contained in a material, such as soil (called soil moisture), rock, ceramics, fruit, or wood. Water content is used in a wide range of scie... |
I heard a couple of times that there is no dynamics in 3D (2+1) GR, that it's something like a topological theory. I got the argument in the 2D case (the metric is conformally flat, Einstein equations trivially satisfied and the action is just a topological number) but I don't get how it is still true or partially simi... |
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Filomat, ISSN 0354-5180, 2016, Volume 30, Issue 6, pp. 1519 - 1524
Journal Article
Physics Letters B, ISSN 0370-2693, 03/2019, Volume 790, Issue C
We present the first me... |
So this is from Charles C. Pinter’s “A Book of Abstract Algebra”- specifically, it’s from the second chapter on permutations. The question is:
[Prove that] Let $ \alpha_1$ and $ \alpha_2$ be cycles of the same length. Let $ \beta_1$ and $ \beta_2$ be cycles of the same length. Let $ \alpha_1$ and $ \beta_1$ be disjoint... |
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