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[Updated in light of some of the comments and answers below]
This is a question about the relationship between higher-order logic and topoi. It's well-known that every topos gives a model of higher-order logic, in which the type in which sentences inhabit is taken to be the subobject classifier $\Omega$.
As far as I ca... |
I'm not quite sure that I understand how the generalized likelihood ratio test works for composite hypotheses; observe the example below:
Let $X_1,...,X_n$ be a random sample from an exponential distribution, $X_i\sim EXP(\theta) \implies E(X_i)=\theta$. Derive the generalized likelihood ratio test of $H_0:\theta=\thet... |
This question already has an answer here:
Suppose I have a $d \times n$ matrix $\mathbf X$ (each entry point has $d$ dimensions) and after some manipulation of data (i.e. summarizing the data $\mathbf X$) I get its $d \times d$ symmetric, quadratic correlation matrix $\rho$ (defined by Pearson).
Then by
SVD definition ... |
Seminar der Arbeitsgruppe Algebra: Wintersemester 2018/19 Relations between Fourier coefficients of Siegel modular forms
Dienstag, der 16. Oktober 2018, 15:15 – 16:15 Uhr, Raum S2|15 401
Referentin: Jolanta Marzec (TU Darmstadt) Abstract: Fourier coefficients of Siegel modular forms have played a prominent role in numb... |
I think I'm missing something quite basic here but consider the process:
$$ e^- + e^+ \rightarrow 2\gamma$$
Fermions have opposite parity to antifermions so the parity quantum number before the process is $P=-1 \times (-1)^L$ where $L$ is the relative orbital angular momentum, which should vanish in the zero momentum f... |
Write out the simple equations
$$\begin{align}Y_j &= a_0 Z_j + a_1 Z_{j-1} + a_2 Z_{j-2}\\Y_{j-1} &= a_0 Z_{j-1} + a_1 Z_{j-2} + a_2 Z_{j-3}\end{align}$$
There are some very simple cases that make $Y_j \perp Y_{j-1}$ due to the independence assumption of the random variables $\{Z_i\}_{i\in\mathbb{Z}}$. An example is $a... |
Current browse context:
astro-ph.HE
Change to browse by: Bookmark(what is this?) Astrophysics > High Energy Astrophysical Phenomena Title: Amplifying magnetic fields of a newly born neutron star by stochastic angular momentum accretion in core collapse supernovae
(Submitted on 20 Mar 2019 (v1), last revised 22 Aug 2019... |
Research Open Access Published: Global existence and uniqueness of solutions to the three-dimensional Boussinesq equations Boundary Value Problems volume 2016, Article number: 85 (2016) Article metrics
1127 Accesses
1 Citations
Abstract
In this paper, we study the three-dimensional Boussinesq equations and obtain the g... |
What is the relationship between $G_\infty$ (homotopy Gerstenhaber) and $B_\infty$ algebras?
In Getzler & Jones "Operads, homotopy algebra, and iterated integrals for double loop spaces" (a paper I don't well understand) a $B_\infty$ algebra is defined to be a graded vector space $V$ together with a dg-bialgebra struct... |
Surely there are many: these are all polynomials in one variable, so every two of them are algebraically dependent because of the transcendence degree argument :-)
However, I am sure that this is not what you wanted to hear, so here you are a nice argument showing how to guess your formula and obtain other formulas som... |
How to Couple a Full-Wave Simulation to a Ray Tracing Simulation
Welcome back to our discussion on multiscale modeling in high-frequency electromagnetics. Multiscale modeling is a simulation challenge that arises when there are vastly different scales in a single simulation, such as the size of an antenna compared to t... |
I have peacefully derived geometric Brownian motion by applying Ito's formula to the process $Y(t) = e^{\alpha B(t) - \frac{t}{2}}$ and letting $\alpha = 1$.
The differential form of $dY$ is clearly equal to
$$dY = \alpha YdB + \left(\frac{\alpha^2}{2} - \frac{1}{2}\right)Ydt \qquad (1)$$
and I understand this to be th... |
When we write complex numbers we write in $a+ib$ form why we don't write it as we write in Cartesian coordinate system like $(x,y)$ and how the idea of a complex plane emerge
There is a book called "A History of Vector Analysis" by Crowe which addresses this and many more topics.
It is also published as a book by Dover... |
Let $K = \mathbb Q(\mu_m)$ and $\zeta_K$ it's Dedekind zeta function. We know from the class number formula that, around $0$: $$\zeta_K(s) \sim s^{r_1+r_2-1}h(K)R(K)/w(K) $$ where $h,R,w$ stand for the size of the class group, the regulator and the size of the subgroup of roots of unity.
On the other hand, we have the ... |
The biggest point is of course that all motion is relative, as many have pointed out. Even barring that, pretending that we live in Copernican times and assuming that the Sun is the center of the universe, the question posits that the force of gravity might be proportional to mass and velocity. This would mean somethin... |
RC timing
Ignoring the BJT and LED portions of the circuit, the following shows you how to approximate the RC charging process. The left side is from your schematic. The right side is the Thevenin equivalent (same thing, just slightly simplified):
simulate this circuit – Schematic created using CircuitLab
Looking at th... |
I am trying to figure out whether the following is true: a function $f$ has limit $L$ at the point $p$ if and only if for every $\epsilon > 0$ there exists a $\delta > 0$ such that $$ s,t \in \mathbb{B}_{\delta}(p) \implies |f(s) - f(t)| < \epsilon. $$ I believe that it is true because by definition of limit $$ |t - p|... |
For Discrete Time
There are two quantities you should be careful not to mix up.
One is the number of individuals who will die during a given interval: $d_x = N_x - N_{x+1}$. One is the fraction, out of those alive at the beginning of a given interval, who will die during the interval: $q_x = \frac{d_x}{N_x}$
($N_x$ bei... |
Shoka function
Shoka function is holomorphic function,
\( \mathrm{Shoka}(z)=z+\ln\!\Big( \exp(-z) +\mathrm e -1\Big)\) Contents Range of holomorphizm
The Shoka function is holomorphic at the complex plane with cuts
\( x+ (1\!+\!2n) \mathrm i \pi ~\) at real \(x\) and integer \(n\).
In such a way, the countable set of c... |
A $54.0\ \mathrm{mL}$ sample of oxygen is collected over water at $20\ \mathrm{^\circ C}$ and $770\ \mathrm{Torr}$ pressure. What is the volume of the dry gas at STP?
By using the combined gas law, I came up with the answer $\pu{51.2 mL}$.
$770 - 21.1 = 748.9\ \mathrm{Torr}$ pressure for oxygen
$21.1$ = vapor pressure ... |
Dear MO community,
let $F$ be a totally real field and $K$ be an imaginary quadratic field of class number one, and $M=FK$ be the quadratic CM extension of $F$ obtained by adjoining $K$ to $F$. Let $p$ be a prime that splits in $K$ and fix an embedding of $K$ into $\mathbb{C}$. It will determine a $p$-ordinary CM type ... |
One disadvantage of the fact that you have posted 5 identical answers (1, 2, 3, 4, 5) is that if other users have some comments about the website you created, they will post them in all these place. If you have some place online where you would like to receive feedback, you should probably also add link to that. — Mart... |
(@Gijs is right that this is just algebra, but perhaps it will help you to see it worked out.)
Consider that "$\exp(b_0+b_1RS)$" is an odds of some event, and that "${\rm Pr}(RW_i|RS_i,b_0,b_1)$" is the probability of the same event (cf., Interpretation of simple predictions to odds ratios in logistic regression). To m... |
Tweedie Distributions: mle estimation of p
Maximum likelihood estimation of the Tweedie index parameter \(p\).
Keywords models Usage
tweedie.profile(formula, p.vec=NULL, xi.vec=NULL, link.power=0, data, weights, offset, fit.glm=FALSE, do.smooth=TRUE, do.plot=FALSE, do.ci=do.smooth, eps=1/6, control=list( epsilon=1e-09,... |
(Notes added by Enrico Scalas on 19.08.2009)
What is the common notion of equilibrium in economics?
The concept of equilibrium referred to in General Equilibrium Theory is taken from Physics. It coincides with mechanical equilibrium.
When looking for mechanical equilibrium one minimizes a potential function
subject to ... |
As part of an optional assignment, we asked our students to create a pendulum, measure its frequency with phyphox and submit the results via a web form. I just picked up the data and got thrilled as the result is amazing:
I just had to clean out some obvious cases in which students did not use the correct units (no, th... |
We can prove this if you dont understand the formula.
Suppose $M$ is a point object at an actual depth $MA$ below the free surface of water $XY$ in a tank. A ray of light incident on $XY$ normally along $MA$ passes straight along $MAA'$.Another ray of light from $M$ incident at $\angle i$ on $XY$,along $MB$ gets deviat... |
How do you construct a Bayes classifier for a binary target where it is assumed: $p(y=1)=\alpha$ and $p(x|y)$ both multivariate gaussian?
Assumption:I assume that this question is asking for the Bayes classifier given that $Y$ is a Bernoullirandom variable with parameter $\alpha$ and the conditionaldistributions of the... |
Motivation:
Many interesting
irrational numbers (or numbers believed to be irrational) appear as answers to natural questions in mathematics. Famous examples are $e$, $\pi$, $\log 2$, $\zeta(3)$ etc. Many more such numbers are described for example in the wonderful book "Mathematical Constants" by Steven R. Finch. The ... |
Equations Learn easily with Video Lessons and Interactive Practice Problems Solving Equations Equations containing one or more variables are algebra equations. Variables represent unknown amounts, and any letter or symbol can be used as a variable.
Algebra equations may need
one-step, two-steps, or multiple-steps to so... |
In this tutorial you will learn how to set up and execute Purpose:
calculations using the
exciting
interface, which allows to obtain full stress tensors of crystal systems for
STRESS-exciting anycrystal structure. In addition, the application of
to the determination of stress tensor for the
STRESS-exciting
is explicitl... |
Research talks;Partial Differential Equations;Mathematical Physics
In this talk we present recent results on the Hall-MHD system. We consider the incompressible MHD-Hall equations in $\mathbb{R}^3$.
$\partial_tu +u \cdot u + \nabla u+\nabla p = \left ( \nabla \times B \right )\times B +\nu \nabla u,$ $\nabla \cdot u =0... |
Volume by rings, also known as volume by disks or volume by washers (if the area between two functions is being rotated around an axis), is a method of finding the volume of a solid of revolution. This method involves splitting the shape into infinitely small circular rings and summing them up. The formula for the volu... |
deterrant
Every day one sees politicians on TV assuring us that nuclear deterrence works because there no nuclear weapon has been exploded in anger since 1945. They clearly have no understanding of statistics.
With a few plausible assumptions, we can easily calculate that the time until the next bomb explodes could be ... |
Let $X$ be a connected, based CW complex. Then the James splitting of $\Sigma\Omega\Sigma X$ gives, in particular, a weak equivalence of spectra $$ \Sigma^{\infty} \Omega\Sigma X_+ \quad \simeq \quad \Sigma^{\infty} (S^0 \vee X \vee X^{[2]} \vee X^{[3]} \vee \cdots ) , $$ where $X^{[n]}$ denotes the $n$-fold smash prod... |
The electric circuits are closed loop or path which forms a network of electrical components, where electrons are able to flow. This path is made using electrical wires and is powered by a source, like a battery. The start of the point from where the electrons start flowing is called the source whereas the point where ... |
@ACuriousMind Something I was working on: Let $B$ be $\tilde g$-bounded. Fix $x\in B$ and let $(y_n)$ be a sequence. Connect $x$ to each $y_n$ with a curve $\gamma_n$. Each curve can be made to have finite length, in fact, one can bound them above. Then the sequence $L_{\tilde g}(\gamma_n)$ has a convergent subsequence... |
I realise, that this question is a stretch, but I was wondering, how would a bonding orbital be called if it was formed from two $f_{x(x^2−3y^2)}$ or $f_{y(3x^2−y^2)}$ orbitals. Have there been any suggestions on this, was it anywhere proposed or discussed? I am not arguing about the existence of such a thing, but as a... |
$\displaystyle \sum_{n=0}^{\infty}\frac{7^n}{n!}x^n$
I'm still trying to get the hang of these and feel like I've done something wrong here. After applying the ratio test I end up with:
$\left|7x\right|\lim \limits_{n \to \infty}\left|\frac{1}{n+1}\right|$
That limit is $0$, so does this mean my radius of convergence i... |
In this post, I discuss our recent paper, Categorical Reparameterization with Gumbel-Softmax, which introduces a simple technique for training neural networks with discrete latent variables. I'm really excited to share this because (1) I believe it will be quite useful for a variety of Machine Learning research problem... |
Consider a multiset of natural numbers:
$$A_n=[a_j]_{j=1..n}$$
in the cases:
$a_j \not=a_k$ for all $j,k \in {\{1,2,3,...,n}\} \land n \gt 1\quad\quad\quad\quad\,\,\,\,\,\,\,\,\quad\quad\quad\quad\quad\quad\quad(\operatorname{i})$
$a_j \not=a_k$ for
some $j,k \in {\{1,2,3,...,n}\} \land n \gt 1$ and $a_j \not= 1$ $\qua... |
Group Presentations
We defined a lot of terms on the Words Over a Set page. Recall that if $A$ is a nonempty set and $\mathcal R$ is a set of words over $A \cup A^{-}$ then we defined an equivalence relation $\sim$ on the set of words over $A \cup A^{-}$ for words $u$ and $v$ as follows. We say that $u \sim v$ if there... |
To do it for a particular number of variables is very easy to follow. Consider what you do when you integrate a function of x and y over some region. Basically, you chop up the region into boxes of area ${\rm d}x{~\rm d} y$, evaluate the function at a point in each box, multiply it by the area of the box. This can be n... |
It's clearly a norm: If $\|\nabla u\|_{L^2}=0$ then $u$ is constant, and the only constant in $L^2$ is $0$. On the other hand, if $H^1=W^{1,2}$ were complete under this norm we would have, by the bounded inverse theorem, that both norms are equivalent, which we know to be false (the scaling below should give you an ide... |
Suppose that I want to check how good OLS works in some specific environment using Monte Carlo. I can simulate $Y=X\beta+\epsilon$. What should I do in Monte Carlo simulations, do I simulate the whole model on each replication, or do I simulate only $\epsilon$ in each replication, while $X$ is the same across all repli... |
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Now showing items 1-10 of 24
Production of Σ(1385)± and Ξ(1530)0 in proton–proton collisions at √s = 7 TeV
(Springer, 2015-01-10)
The production of the strange and double-strange baryon resonances ((1385)±, Ξ(1530)0) has been measured at mid-rapidity (|y|< 0.5) in proton–proton collisions at √s = 7 TeV with the ... |
Inaccessible
Inaccessible cardinals are the traditional entry-point to the large cardinal hierarchy (although there are some weaker large cardinal notions, such as universe cardinals).
If $\kappa$ is inaccessible, then $V_\kappa$ is a model of ZFC, but this is not an equivalence, since the weaker notion of universe car... |
Given the logarithmic spiral $$\alpha(t) = e^{-t}(\cos(t),\sin(t))$$ I take a ray from the origin given by $\lambda(\cos \theta, \sin \theta)$ and I have to prove that in $\alpha(\mathbb{R}) \cap R_{\theta}$ the tangents form a constant angle with the vector $(\cos \theta,\sin \theta)$ (constant in the sense that it do... |
The Robin's inequality says - If the Riemann hypothesis is true then - $$\sigma(n) < e^{\gamma}n \log(\log(n))$$ holds true for all $n \in \mathbb{N}$
Now it is proved for all $5-$ free integers .And thus for infinely many integers. So my question is does this proof also prove That there are infinitely many no - trivia... |
It is said that $p \left( \theta | y _ { 1 : N } \right) \propto _ { \theta } p \left( y _ { 1 : N } | \theta \right) p ( \theta )$.
And $p \left( \theta | y _ { 1 : N } \right)$ is the posterior, $ p \left( y _ { 1 : N } | \theta \right)$ is the likelihood, and $p ( \theta )$ is the prior.
Suppose we have a model M, a... |
Why does a 95% CI not imply a 95% chance of containing the mean?
There are many issues to be clarified in this question and in the majority of the given responses. I shall confine myself only to two of them.
a.
What is a population mean? Does exist a true population mean?
The concept of population mean is model-depende... |
23 people. In a room of just 23 people there’s a 50-50 chance of at least two people having the same birthday. In a room of 75 there’s a 99.9% chance of at least two people matching.
Put down the calculator and pitchfork, I don’t speak heresy. The birthday paradox is strange, counter-intuitive, and
completely true. It’... |
I'm trying to understand the proof of the zero-one-law for first order logic as provided in (Ebbinghaus-Flum, 1995). It goes as follows:
Let $\tau$ be a relational signature. Let $r\in\mathbb{N}$, then $\Delta_{r+1}=\{\phi(v_1,\ldots,v_r,v_{r+1})\mid\phi\text{ has the form }R\bar x\text{, where }R\in\tau\text{ and wher... |
I'm looking for help translate these posts into different languages! Please email me at <myfirstname><mylastname>2004<at>gmail.com if you are interested. Xiaoyi Yin (尹肖贻) has kindly translated this post into Chinese (中文).
In my previous blog post, I described how simple distributions like Gaussians can be “deformed” to... |
If you have a list, e.g.
{1, 2, 3}
then you can extract the $k$th part using
Part (
list[[k]]):
In[1]:= {1, 2, 3}[[2]]Out[1]= 2
The problem is that if you provide a symbolic expression in the place of the list,
Part will try to decompose it:
In[1]:= list[[2]]Part::partd: Part specification list[[2]] is longer than dept... |
Exercise \(\PageIndex{1}\)
Write each of the following as a complex number in standard form.
\[(4 + i) + (3 - 3i)\] \[5(2 - i) + i(3 - 2i)\] \[(4 + 2i)(5 - 3i)\] \[(2 + 3i)(1 + i) + (4 - 3i)\] Answer \((4 + i) + (3 - 3i) = 7 - 2i\) \(5(2 - i) + i(3 -2i) = 12 - 2i\) \((4 + 2i)(5 - 3i) = 26 - 2i\) \((2 + 3i)(1 + i) + (4 ... |
In school, I have learnt to plot simple graphs such as $y=x^2$ followed by $y=x^3$.
A grade or two later, I learnt to plot other interesting graphs such as $y=1/x$, $y=\ln x$, $y=e^x$. I have also recently learnt about trigonometric graphs and circle equations. In the internet, I have seen users posting graphs of diffe... |
I'm trying to calculate confidence intervals for a neural network (rather than prediction intervals). I'm following this paper, which treats them in the same framework as any parametric (parameter-involving?) nonlinear model (which a neural net basically is). Calculating these CI's involves computing the Jacobian -- th... |
After the excellent post by JD Long in this thread, I looked for a simple example, and the R code necessary to produce the PCA and then go back to the original data. It gave me some first-hand geometric intuition, and I want to share what I got. The dataset and code can be directly copied and pasted into R form Github.... |
It is true that K-means clustering and PCA appear to have very different goals and at first sight do not seem to be related. However, as explained in the Ding & He 2004 paper K-means Clustering via Principal Component Analysis, there is a deep connection between them.
The intuition is that PCA seeks to represent all $n... |
Consider this question,
Let $(X_1,Y_1),(X_2,Y_2),...,(X_n,Y_n)$ be independent and identically distributed pairs of random variables with $E(X_1) = E(Y_1), Var(X_1)= Var(Y_1) = 1$, and $Cov(X_1,Y_1) = \rho \in (-1,1).$
Given $\alpha \in (0,1)$, obtain a statistic $L_n$ which is a function of $(X_1,Y_1),(X_2,Y_2),...,(X... |
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.
I used the official objectives and sample test to construct these questions, but cannot promise that they accurately reflect what’... |
I am trying to decompose a time series of $n$ observations $\bf{\mathrm{v_c}}$ into the $n \times n$ variance-covariance structure $\sum$ and a random series $\bf{\mathrm{v}}$.
So, I can derive the variance-covariance matrix $\sum$ from the autocorrelation function of $\bf{\mathrm{v_c}}$. This will be a Toeplitz matrix... |
Answer
A=$\frac{8}{3}$$\pi$-4$\sqrt 3$
Work Step by Step
height of the triangle=.5s$\sqrt 3$ h=2$\sqrt 3$ base of the triangle=r=4 A=$\frac{1}{6}$$\pi$4$^2$-$\frac{1}{2}$(4)(2$\sqrt 3$) A=$\frac{8}{3}$$\pi$-4$\sqrt 3$
You can help us out by revising, improving and updating this answer.Update this answer
After you claim... |
In practice, claiming that $x$ being of type $T$
usually is used to describe syntax, while claiming that $x$ is in set $S$ is usually used to indicate a semantic property. I will give some examples to clarify this difference in usage of types and sets. For the difference in what types and sets actually are, I refer to ... |
Equivalences:
The non-orthogonal vectors problem (as defined above) for a set $S$ of $n$ Boolean
vectors each of length $d$ and a positive integer $k$ is equivalent
the following:
Finding a $2$ by $k$ submatrix of 1's in a given $n$ by $d$ Boolean matrix.
Finding a $\mathrm{K}_{2,k}$ complete subgraph in a given bipart... |
I am currently doing a problem in *Introduction to Quantum Mechanics, 2nd edition Griffiths *
QUESTION
In question 2.22 part D, we are asked to calculate $\langle p^2 \rangle$ for the Gaussian Wave packet $$ \Psi(x,0) = A e^{-ax^2} $$ In an earlier problem we calculate the normalization constant $ A = (\frac{2a}{\pi})^... |
I'm looking for an upright Greek font for single Greek characters (like "β-decay" or "µ-metal") which fits to the default CM/latin style, i.e. the upright version of the default italic math mode Greek letters (
\beta,
\mu).
The "default" upright Greek font (should be
cbgreek), which is used when writing with
babel or w... |
Exercise \(\PageIndex{1}\)
In the following exercises, state whether each statement is true, or give an example to show that it is false.
1. If \(\displaystyle \sum_{n=1}^∞a_nx^n\) converges, then \(\displaystyle a_nx^n→0\) as \(\displaystyle n→∞.\)
Answer
True. If a series converges then its terms tend to zero.
2. \(\... |
If you are an NLP person, chances are you know PCFGs (probabilistic context-free grammars) pretty well. These are just generative models for generating phrase-structure trees.
A CFG includes a set of nonterminals (\(N\)) with a designed start symbol \(\mathrm{S} \in N\), a set of terminal words (the “words”) and a set ... |
The Fundamental Theorem of Arithmetic is one of the most important results in this chapter. It simply says that every positive integer can be written uniquely as a product of primes. The unique factorization is needed to establish much of what comes later. There are systems where unique factorization fails to hold. Man... |
The definition of incompressible is often unclear and changes depending on which community uses it. So let's look at some common definitions:
Constant density
This means the density is constant everywhere in space and time. So:$$\frac{D\rho}{Dt} = \frac{\partial \rho}{\partial t} + \vec{u}\cdot\nabla{\rho} = 0$$Because... |
Inner Product Spaces Review
Inner Product Spaces Review
We will now review some of the recent content regarding inner product spaces.
Recall from the Inner Product Spaces page that if $V$ is a vector space over $\mathbb{R}$ or $\mathbb{C}$ then an Inner Producton $V$ is a function which takes every pair of vectors $u, ... |
Character conditionEdit
The most common character conditions found in
Fallout 3, Fallout: New Vegas, and Fallout 4 are poisons from Chems or from the environment, and damage from combat. Fallout 4 is currently the only one in the Fallout series to include illnesses. PoisonedEdit InjuriesEdit OtherEdit Fallout 4Edit Ill... |
Difference between revisions of "Inaccessible"
(Organized a bit)
(→Hyper-inaccessible: Meta-ordinal)
Line 65: Line 65:
Therefore $2$-inaccessibility is weaker than $3$-inaccessibility, which is weaker than $4$-inaccessibility... all of which are weaker than $\omega$-inaccessibility, which is weaker than $\omega+1$-inac... |
Table of Contents
Products of Paths Relative to {0, 1} in a Topological Space
Definition: Let $X$ be a topological space. A Path in $X$ is a continuous function $\alpha : [0, 1] \to X$.
Definition: Let $X$ be a topological space and let $\alpha, \beta : [0, 1] \to X$ be paths such that $\alpha(1) = \beta (0)$. The Prod... |
I would like to ask that at what distance from the Earth's surface the curvature of the Earth is visible. What layer of the atmosphere is this?
I've noticed that at the height of 9-12 Km (the view from from aeroplanes) it is not visible.
Earth Science Stack Exchange is a question and answer site for those interested in... |
Linear Maps Examples 4
Recall from the Linear Maps page that a linear map or linear transformation from the vector space $V$ to the vector space $W$ is a function $T : V \to W$ such that for all $u, v \in V$ and for all $a \in \mathbb{F}$ we have that $T(u + v) = T(u) + T(v)$ (additivity property) and $T(av) = aT(v)$ (... |
This story actually starts with Einstein's paper on the photoelectric effect. Einstein proposed that for light waves, $E \propto f$, with a proportionality constant that eventually became known as $h$. Using the relation $E = pc$ from special relativity, you can derive that $pc = hf$, and with $\lambda f = c$ you get $... |
Preservation of Connectivity under Continuous Maps
Table of Contents
Preservation of Connectivity under Continuous Maps
One particularly nice property of connected topological spaces $X$ is that if $f : X \to Y$ is a continuous map then the range $f(X)$ is also connected. We prove this result below.
Theorem 1: Let $X$ ... |
Just curious: This monday, I had an exam in Knowledge Processing. They asked what's the problem with FOL (compared to propositional), and I gave the textbook answer that iterating functions gives infinitely many ground terms and makes it undecidable. And since I'm a bigmouth, lavedida, I added that I strongly suspect t... |
Here are the conditions for making a substitution:
$\int_{y_0}^{y_1}f(y) dy = \int_{x_0}^{x_1}f(h(x))h'(x) dx$
if:1. $f : [y_0, y_1] → R$ is continuous on $[y_0, y_1]$,
$h : [x_0, x_1] → [y_0, y_1]$ is differentiable on $[x_0, x_1]$, with $h′$ continuous on $[x_0, x_1]$, $h(x_0) = y_0$ and $h(x_1) = y_1$
note that this... |
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Now showing items 1-1 of 1
Anisotropic flow of inclusive and identified particles in Pb–Pb collisions at $\sqrt{{s}_{NN}}=$ 5.02 TeV with ALICE
(Elsevier, 2017-11)
Anisotropic flow measurements constrain the shear $(\eta/s)$ and bulk ($\zeta/s$) viscosity of the quark-gluon plasma created in heavy-ion collisions... |
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Now showing items 1-1 of 1
Production of charged pions, kaons and protons at large transverse momenta in pp and Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV
(Elsevier, 2014-09)
Transverse momentum spectra of $\pi^{\pm}, K^{\pm}$ and $p(\bar{p})$ up to $p_T$ = 20 GeV/c at mid-rapidity, |y| $\le$ 0.8, in pp and ... |
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Now showing items 1-1 of 1
Production of charged pions, kaons and protons at large transverse momenta in pp and Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV
(Elsevier, 2014-09)
Transverse momentum spectra of $\pi^{\pm}, K^{\pm}$ and $p(\bar{p})$ up to $p_T$ = 20 GeV/c at mid-rapidity, |y| $\le$ 0.8, in pp and ... |
Take $f:[0,1]\to [0,1]^n$ a continuous tour around $[0,1]^n,$ say, some iteration of a Hilbert curve. For $\varepsilon \in (0,1)$ what is the following thing called and are there any nontrivial upper bounds?
\begin{equation} \max_{|a-b|<\varepsilon} \|f(a)-f(b)\|. \end{equation}
Or if not a maximum, then the typical va... |
Instrumental Cutsets Efficient Identification in Linear SCM
We’re going to be presenting our paper, “Efficient Identification in Linear Structural Causal Models with Instrumental Cutsets”, at NeurIPS 2019. The work is extremely technical, requiring a huge amount of specialized background knowledge, so I’m offering this... |
Focus Questions
The following questions are meant to guide our study of the material in this section. After studying this section, we should understand the concepts motivated by these questions and be able to write precise, coherent answers to these questions. What is a complex number? What does it mean for two complex... |
I am trying to prove or disprove the following statement:
$$\sum_{r|n} d(r^2) = d^2(n),$$
where $d$ is the number of divisors function.
Computing it for small numbers yields equality, so I at least think it's true. But I can't see how to prove it formally. I have tried working with prime factorisations, but finding equ... |
Answer
$S=2$
Work Step by Step
$\displaystyle \sum_{i=0}^{∞}(0.5)^i=(0.5)^0+(0.5)^1+(0.5)^2+...$ $a_1=(0.5)^0=1$ $r=\frac{a_2}{a_1}=\frac{(0.5)^1}{(0.5)^0}=0.5$ $S=\frac{a_1}{1-r}=\frac{1}{1-0.5}=\frac{1}{0.5}=2$
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The Neighbourhood of a Vertex
Definition: The Neighbourhood of the vertex $x \in V(G)$ is the set of all vertices which are adjacent to $x$. For the graph $G = (V(G), E(G))$ we have that the neighbourhood of $x$ denoted $N_{G} (x)$ is $\displaystyle{N_{G}(x) = \left\{ {y \in V(G) : {x, y} \in E(G)}\right\}}$. When unde... |
The Handshaking Lemma
We will now look at a very important and well known lemma in graph theory.
Lemma 1 (The Handshaking Lemma): In any graph $G = (V(G), E(G))$, the sum of the degrees in the degree sequence of $G$ is equal to one half the number of edges in the graph, that is $\displaystyle{\sum_{v \in V(G)} \deg (v)... |
Bedforms and roughness
Bed forms are relief features initiated by the fluid motions generated downstream of small local obstacles at the bottom consisting of movable (alluvial) sediment materials.
Contents Introduction
Many types of bed forms can be observed in nature. The bed form regimes for steady flow over a sand b... |
Section 7.4 Exercises
Find a possible formula for the trigonometric function whose values are given in the following tables.
1.
\(x\) 0 3 6 9 12 15 18 \(y\) -4 -1 2 -1 -4 -1 2
2.
\(x\) 0 2 4 6 8 10 12 \(y\) 5 1 -3 1 5 1 -3
3. The displacement \(h(t)\), in centimeters, of a mass suspended by a spring is modeled by the f... |
The suggestion in the comments to use brute force is a little unsatisfactory; what if $15$ were $15000$? Here is a general method.
Suppose$$m = \sigma(n) = \prod_{p^k ||\: n} \frac{p^{k+1}-1}{p-1}.\tag{1}$$Each factor $d$ in the product
is an integer (a finite geometric series). The prime factorization of $m$ will seve... |
A first order ODE is an equation of the form
\[\dfrac{dy}{dx}=f(x,y)\]
or just
\[y'=f(x,y)\]
In general, there is no simple formula or procedure one can follow to find solutions. In the next few lectures we will look at special cases where solutions are not difficult to obtain. In this section, let us assume that \(f\)... |
If $1_K \in K$ and $1_F \in F$ are the multiplicative neutral elements of $K$ and $F$, notice that $\Bbb Q$ naturally embeds in $K$ by the map $\Bbb Q \ni \frac p q \mapsto (p \cdot 1_K) \cdot (q \cdot 1_K)^{-1} \in K$. This is where we use the fact that the characteristic is $0$: this guarantees that $q \cdot 1_K \ne ... |
In the Hamilton-Jacobi equation, we take the partial time derivative of the action. But the action comes from integrating the Lagrangian over time, so time seems to just be a dummy variable here and hence I do not understand how we can partial differentiate $S$ with respect to time? A simple example would also be helpf... |
Question: As we know, (1) the macroscopic spatial dimension of our universe is 3 dimension, and (2) gravity attracts massive objects together and the gravitational force is isotropic without directional preferences. Why do we have the spiral 2D plane-like Galaxy(galaxies), instead of spherical or elliptic-like galaxies... |
I'm struggling with some anomalous behavior in an analysis I'm running and was hoping for some advice/insights. I'm attempting to extract the implied funding/borrow costs from ETF option prices (say ...
Trying to fit variants of SVI (Zeliade method, SSVI etc) to options on futures price data. One of the core ideas of t... |
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