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$$\alpha\,\mathrm{A}+\beta\,\mathrm{B}+\cdots\rightleftharpoons \rho\,\mathrm{R}+\sigma\,\mathrm{S}+\cdots$$ The equilibrium constant for this reaction can be calculated using the following formula:
$$\mathrm K_c = \frac{{{[\mathrm{R}]}}^\rho {{[\mathrm{S}]}}^\sigma ... } {{{[\... |
The root locus is a parametric plot. It shows how zeros and poles move in the complex plain depending on the value of a gain $k$. For all points on the plot both the phase condition
1 and the amplitude condition 2 have to be true.
$$G_0(s) = k \cdot \frac{\prod^q_{i=1}(s-s_{0i})}{\prod^n_{i=1}(s-s_{i})}$$
In your case,... |
Abbreviation:
Frm
A
is a structure $\mathbf{A}=\langle A, \bigvee, \wedge, e, 0\rangle$ of type $\langle\infty, 2, 0, 0\rangle$ such that frame
$\langle A, \bigvee, 0\rangle$ is a complete semilattice with $0=\bigvee\emptyset$,
$\langle A, \wedge, e\rangle$ is a meet semilattice with identity, and
$\wedge$ distributes ... |
Question
The angle between the two diagonals of a cube is
(a) 30°
(b) 45°
(c) \[\cos^{- 1} \left( \frac{1}{\sqrt{3}} \right)\]
(d) \[\cos^{- 1} \left( \frac{1}{3} \right)\]
Solution
\[\cos^{- 1} \left( \frac{1}{3} \right)\]
\[\text { Let a be the length of an edge of the cube and let one corner be at the origin as show... |
Z-Scores
The standard normal distribution is a normal distribution of standardized values called
z-scores. A z-score is measured in units of the standard deviation.
Definition: Z-Score
If \(X\) is a normally distributed random variable and \(X \sim N(\mu, \sigma)\), then the
z-score is:
\[z = \dfrac{x=\mu}{\sigma} \lab... |
How to show the sequence $x_n = (1 + \frac{x}{n})^{n}$ is bounded above by $e^x$?
Note: I'm not supposed to be able to use any differentiation techniques if possible. Since we techincally "don't know" it yet.
As can be deduced I am trying to show that the sequence $x_n = (1 + \frac{x}{n})^{n}$ is convergent. I have to ... |
The Annals of Probability Ann. Probab. Volume 43, Number 2 (2015), 639-681. Critical two-point functions for long-range statistical-mechanical models in high dimensions Abstract
We consider long-range self-avoiding walk, percolation and the Ising model on $\mathbb{Z}^{d}$ that are defined by power-law decaying pair pot... |
In general, a Hilbert Schmidt Operator $M$ is one where \begin{equation} \sum_{i=1}^{\infty} ||M \varphi_{i}||^2 \end{equation} is bounded for any orthonormal system $\{\varphi_i: i\in \mathbb{N}\}$.
Now, let $M_k$ be a sequence of Hilbert Schmidt operators. We call it a uniform sequence of Hilbert Schmidt operators, i... |
Find a Basis of the Subspace Spanned by Four Polynomials of Degree 3 or Less Problem 607
Let $\calP_3$ be the vector space of all polynomials of degree $3$ or less.
Let \[S=\{p_1(x), p_2(x), p_3(x), p_4(x)\},\] where \begin{align*} p_1(x)&=1+3x+2x^2-x^3 & p_2(x)&=x+x^3\\ p_3(x)&=x+x^2-x^3 & p_4(x)&=3+8x+8x^3. \end{alig... |
NOTE: This is a working document that is regularly being edited and added to.
In the late 1950's and early 1960's, Wolfgang Paul and collaborators were developing novel methodolgies for storing and manipulating ions through the use of alternating radio frequency electric fields. Their methodologies utilized quadrupolar... |
Because the particle apparently doesn't exist, the most likely prediction for the 2016 dataset was that it disappears completely. It did. And even though CMS was against the publication of its new diphoton paper after it was for it ;-), we saw the paper with the new graphs in time, as I discussed in the previous blog p... |
skills to develop
To see a complete linear correlation and regression analysis, in a practical setting, as a cohesive whole
In the preceding sections numerous concepts were introduced and illustrated, but the analysis was broken into disjoint pieces by sections. In this section we will go through a complete example of ... |
Two matrices $A$ and $B$ are similar if there exists a nonsingular (invertible) matrix $S$ such that\[S^{-1}BS=A.\]
A matrix $A$ is diagonalizable if $A$ is similar to a diagonal matrix. Namely, $A$ is diagonalizable if there exist a nonsingular matrix $S$ and a diagonal matrix $D$ such that\[S^{-1}AS=D.\]
Some useful ... |
Electrostatic Potential and Capacitance Electric Potential Electric potential is the amount of work done in bringing a unit positive charge from infinity to given point. \tt V=\frac{W}{Q} Potential at a point \tt V=\frac{1}{4\pi\ e_{0}}.\frac{Q}{R} Electric potential is a scalar quantity. The relation between "V" and "... |
I'm trying to catch up on Algebraic Geometry for a potential Masters project. I'm just now trying to get my head around determining the projective closure, and wanted to check my reasoning.
So in the exercise underway at the moment I've been given a set of zeroes $Z(yt-x^2, xy-zt, y^2-xz)$ in $\mathbb{P}^3$ - the quest... |
I don't understand how to do this considering that the contrapositive of $x^3$ is irrational. For example $2$ to the cube root is irrational, but I am trying to prove that is is rational
Suffices to show the following:
Suppose $x$ is rational. Then $f(x)=x^3+3x+3$ is also rational.
We claim that $A(x)=x^3$ is rational ... |
The Hessian matrix $\{\partial_i \partial_j f \}$ of a function $f:\mathbb{R}^n \to \mathbb{R}$ depends on the coordinate system you choose. If $x_1,\cdots,x_n$ and $y_1,\cdots,y_n$ are two sets of coordinates (say, in some open neighborhood of a manifold), then $\frac{\partial f(y(x))}{\partial x_i} = \sum_{k} \frac{\... |
ISSN:
1556-1801
eISSN:
1556-181X
All Issues
Networks & Heterogeneous Media
March 2013 , Volume 8 , Issue 1
Special issue
dedicated to Hiroshi Matano on the occasion of his 60th birthday: Part II
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Abstract:
Professor Hiroshi Matano was born in Kyoto, Japan, on July 28th, 1952. He stu... |
The Infona portal uses cookies, i.e. strings of text saved by a browser on the user's device. The portal can access those files and use them to remember the user's data, such as their chosen settings (screen view, interface language, etc.), or their login data. By using the Infona portal the user accepts automatic savi... |
So here is the last of this series explaining the expressions on Newton’s clock:
We are now up to 8. Let’s look at\[
\mathop{\prod}\limits_{{k}{=}{0}}\limits^{1}{{(}{2}{k}{+}{2}{)}}
\]
This is another excellent example of how concise the words in maths can be. The symbol “𝚷” is the capital version of 𝜋 which correspo... |
Beloved community,
Does the following series converge?
$$\sum_{n=0}^\infty \left(\sqrt[n]{n} - \sqrt[n+1]{n+1}\right)$$
According to Wolfram Alpha, it does by the Comparison Test. However, after thinking about it long and hard, I still haven't found any series to compare it to.
Many thanks in advance! :) |
I would like to apply the known version of the conjectural formula (11) page !0 of the paper Number theory and dynamical Lefschetz trace formula.
Disclaimer: I do not have a complete understanding of this formula but I can get an sketch of it. I just know that both sides of the formula are not number but distribution. ... |
Basically 2 strings, $a>b$, which go into the first box and do division to output $b,r$ such that $a = bq + r$ and $r<b$, then you have to check for $r=0$ which returns $b$ if we are done, otherwise inputs $r,q$ into the division box..
There was a guy at my university who was convinced he had proven the Collatz Conject... |
ISSN:
1937-5093
eISSN:
1937-5077 Kinetic & Related Models
June 2011 , Volume 4 , Issue 2
Issue on thermomechanics and phase change
Guest Editors: Alain Miranville and Ulisse Stefanelli
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Abstract:
A relativistic kinetic Fokker-Planck equation that has been recently proposed in the ph... |
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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 ... |
Let $E$ be a set of finite outer measure and $F$ a collection of closed, bounded intervals that cover $E$ in the sense of Vitali. Show that there is a countable disjoint collection {${I_k}$}$_{k=1}^\infty$ of intervals in $F$ for which
$m$*$[E$~$\bigcup_{k=1}^{\infty}I_k] = 0$.
This is the same as the Vitali Covering L... |
I'm trying to follow the proof in Wikipedia that the PNT is equivalent to the assertion $\psi(x)\sim x$, by proving that $\psi(x)\sim\pi(x)\log x$, which it claims is a very simple proof. One direction of inequality is an actual bound, $\psi(x)\le\pi(x)\log x$, but the other inequality has a fuzz factor:
$$\psi(x) \ge ... |
I tried to use the commands defined here: Can I change all math output to use monospaced text? to change the font in the math environment. However, the following code does not compile:
\everymath{\mathtt{\xdef\tmp{\fam\the\fam\relax}\aftergroup\tmp}}\everydisplay{\mathtt{\xdef\tmp{\fam\the\fam\relax}\aftergroup\tmp}} \... |
ISSN:
1937-5093
eISSN:
1937-5077 Kinetic & Related Models
September 2011 , Volume 4 , Issue 3
Issue on new trends in direct, inverse, and control problems for evolution equations
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Abstract:
A Gaussian beam method is presented for the analysis of the energy of the high frequency solu... |
I need to evaluate $\sum_{n = -\infty}^{\infty} J_0(\alpha n) z^{-n}$ in closed form, where $z$ is complex variable and $J_0()$ is the zeroth order Bessel function of the first kind. How do I evaluate this summation?
This is not a full answer, but maybe it will get you somewhere.
You can prove (or maybe already know) t... |
Here's an inelegant but straightforward, elementary solution. It begins with the case $n=1$.
Generally, a non-central chi-squared distribution arises as the sum of squares of independent Normal distributions, each with unit variance, but with varying and possibly non-zero means. (When all means are zero, we obtain the ... |
Given the system:
$$ \dot{r} = -\mu r + r^3, \\ \dot{\theta} = r $$
There is clearly one single node at $r=0$.
The Jacobian is then: $$ \begin{pmatrix} -\mu + 3r^2 & 0 \\ 1 & 0 \end{pmatrix}$$ Setting $r=0$ and finding the eigenvalues I get: $\lambda = 0 , \lambda = -\mu $. The problem statement says "show that a subcr... |
So one thing that I find really interesting is that if I have a vector $\vec V = V_x \hat i + V_y \hat j$ its length is just:
$$ |V|^2 = V_x^2 + V_y^2 $$
That is all well and good, but then if I transform the vector into a new basis, I can rewrite the vector in terms of a covariant basis as:
$$ \vec V = V^1 \vec b_1 + ... |
To motivate Sobolev spaces, let me pose a motivating problem.
Let $\Omega$ be a smooth, bounded domain in ${\Bbb R}^n$ and let $f$ be a $C^\infty$ function on $\Omega$. Prove that there exists a $C^2$ function $u$ satisfying $-\Delta u = f$ in $\Omega$ and $u = 0$ on the boundary of $\Omega$.
As far as PDE's go, this i... |
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Search for new resonances in $W\gamma$ and $Z\gamma$ Final States in $pp$ Collisions at $\sqrt{s}=8\,\mathrm{TeV}$ with the ATLAS Detector
(Elsevier, 2014-11-10)
This letter presents a search for new resonances decaying to final states with a vector boson produced in association with a... |
Let $X_1, X_2,\ldots$ be a random sample from the probability density function for $0<\mu<\infty$, $0<\alpha < 1$
$$f(x;\mu,\alpha)= \begin{cases}\frac{1}{\Gamma(\alpha)}(x-\mu)^{\alpha-1}e^{-(x-\mu)}, & x>\mu \\ 0, & \text{otherwise} \end{cases}$$
Does Maximum Likelihood Estimator exist for parameters $\alpha$ and $\m... |
I'm currently trying to learn Bayesian Statistics but I keep losing time trying to figure out what exactly is meant by notation. Could someone answer the following for me?
Let's say $X \sim N(\mu,\sigma^2)$
(1) I'm trying to calculate a posterior distribution $p(\mu\mid X) \propto p(X\mid\mu)p(\mu)$. So my understandin... |
Suppose that the matrix $A$ is diagonalizable by an orthogonal matrix $Q$.The orthogonality of the matrix $Q$ means that we have\[Q^{\trans}Q=QQ^{\trans}=I, \tag{*}\]where $Q^{\trans}$ is the transpose matrix of $Q$ and $I$ is the $n\times n$ identity matrix.
Since $Q$ diagonalizes the matrix $A$, we have\[Q^{-1}AQ=D,\... |
Standard finite difference formulas are usable to numerically compute a derivative under the expectation that you have function values $f(x_k)$ at evenly spaced points, so that $h \equiv x_{k+1} - x_k$ is a constant. What if I have unevenly spaced points, so that $h$ now varies from one pair of adjacent points to the n... |
If the Quotient by the Center is Cyclic, then the Group is AbelianLet $Z(G)$ be the center of a group $G$.Show that if $G/Z(G)$ is a cyclic group, then $G$ is abelian.Steps.Write $G/Z(G)=\langle \bar{g} \rangle$ for some $g \in G$.Any element $x\in G$ can be written as $x=g^a z$ for some $z \in Z(G)$ and $a \in \Z$.Usi... |
There are $2^4=16$ total possible outcomes of which only one outcome gives rise to all heads.
Thus the probability that all coins land heads is $1/16$.
Solution (2)
Consider the event that the first coin is heads.
In this case, there are total $2^3=8$ possible outcomes for the rest of coins (2nd, 3rd, and 4th).
Hence, ... |
ISSN:
1930-5311
eISSN:
1930-532X
All Issues
Journal of Modern Dynamics
July 2008 , Volume 2 , Issue 3
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Abstract:
Professor Michael Brin of the University of Maryland endowed an international prize for outstanding work in the theory of dynamical systems and related areas. The prize i... |
This week¶
This week's paper is Active Perception in Adversarial Scenarios using Maximum Entropy Deep Reinforcement Learning. The idea is that an agent interacting with another agent can learn to assess the threat it may pose. It does this by actively testing the opponent agent's behavior, and does not assume the oppon... |
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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... |
So now I’m up to “4” on Newton’s clock:
So the expression\[
{\left({2\sin\frac{\mathit{\pi}}{2}}\right)}^{2}
\]
uses the sine function which has been talked about many posts before. Only this time, it is using radian measure of angles instead of degrees. If your calculator is in degree mode, you can substitute 90° in p... |
Hubble's Law, when written in this form,$$v = H_0D,$$means: if $D$ is the
current distance of a galaxy, and $H_0$ the Hubble constant, then $v$ is the current recession velocity of the galaxy. So it tells you what the recession velocity of a galaxy is right now, not what it was in the past.
Basically, the Hubble Law is... |
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Higher harmonic flow coefficients of identified hadrons in Pb-Pb collisions at $\sqrt{s_{\rm NN}}$ = 2.76 TeV
(Springer, 2016-09)
The elliptic, triangular, quadrangular and pentagonal anisotropic flow coefficients for $\pi^{\pm}$, $\mathrm{K}^{\pm}$ and p+$\overline{\mathrm{p}}$ in Pb-... |
I'm looking for something synonymous to
tabbing, but for math mode.
I read that
alignat* should do the trick, but this behaves differently it seems. For example:
\documentclass{article}\usepackage{amsmath}\begin{document}\begin{alignat*}{2}S_4^{t} &= \{&\langle rdi, r12, rax' \rangle \mid \langle rdi, r12, rax \rangle ... |
Determine the Quotient Ring $\Z[\sqrt{10}]/(2, \sqrt{10})$
Problem 487
Let\[P=(2, \sqrt{10})=\{a+b\sqrt{10} \mid a, b \in \Z, 2|a\}\]be an ideal of the ring\[\Z[\sqrt{10}]=\{a+b\sqrt{10} \mid a, b \in \Z\}.\]Then determine the quotient ring $\Z[\sqrt{10}]/P$.Is $P$ a prime ideal? Is $P$ a maximal ideal?
We prove that t... |
I'm doing the following exercise:
Consider $$ f(x)= \begin{cases} \displaystyle\frac{1-(\cos(x))^3}{x^2}, \quad if\ x\neq 0\\ \displaystyle\frac{3}{2}, \quad if\ x= 0\\ \end{cases} $$
Calculate $f(x_0)$, with $x_0=0.000011$, with a C program (simple and double precision) and with a pocket calculator.
Identify the cause... |
I face some trouble solving Maxwell's equations inside a cylinder with perfect conductor boundaries (in 3D) ? We work with cylindrical coordinates $(r, \phi, z)$ and we make the assumption that fields have a sinusoidal "$e^{i\omega t}$" time dependence. Note that we have a $\phi$ symmetry. First, and in any coordinates... |
Under the assumption that
The probability of getting a girl or a boy is equal (50%) A male can only put his thang on his partner female (no multiple wives)
Neither of them will get significantly more boys than the other (in a long period of time) A Priori Proof : Barbarian King
In regex:
(G*)B
...
G
G B
Z B
B
The tree ... |
This week¶
This week's paper, Large-scale traffic signal control using machine learning: some traffic flow considerations, caught my eye for several reasons. First, traffic signal control is relevant to my own group's work involving microservice and network traffic management. Second, the authors use cellular automaton... |
I have a gaussian noise $\nu(t)$ with variance $\sigma^2$. After a FFT I get $X(\omega)$. If now I do the IFFT on the $X(\omega)$ can I say that the result is still a gaussian noise of variance $\sigma^2$? What is the effect of zero padding on FFT? How the zero padding affects the statistics of the noise after FFT and ... |
I think it can be solved with a simple algorithm:
Take two $x_i$,$x_j$, if they exist, such that $x_i-x_j>1$ Replace them, respectively, with $x_i-1$ and $x_j+1$ Repeat from point 1
We'll show that the algorithm is finite and that it makes $S=\sum_{k=1}^n x_k^{\alpha}$ smaller at each run. Let's demonstrate the latter:... |
K p And K c
K
p And K c are the equilibrium constant of a ideal gaseous mixture. K p is equilibrium constant used when equilibrium concentrations are expressed in atmospheric pressure and K c is equilibrium constant used when equilibrium concentrations are expressed in molarity.
For many general chemical reactions aA +... |
I actually have to prove the following :
$\mathbf{NL} \subseteq \mathbf{NC_2}$
I have the following approach :
I will prove that $\mathbf{PATH} = \{〈D, s, t〉 | \text{D is a directed graph with a path from vertex s to t}\} \in \mathbf{NC_2}$.
I will show that $\mathbf{NC_2}$ is closed under log-space reductions i.e:
$$(... |
fskilnik wrote:
GMATH practice exercise (Quant Class 18)
Last Monday N female executives (N>1) received M male managers (M>1) for a business meeting. If every person shook hands exactly once with every other person in the meeting, what is the difference between the total number of shaking hands and the number of shakin... |
Here is a short tale about
Linear Discriminant Analysis (LDA) as a reply to the question.
When we have one variable and $k$ groups (classes) to discriminate by it, this is ANOVA. The discrimination power of the variable is $SS_\text{between groups} / SS_\text{within groups}$, or $B/W$.
When we have $p$ variables, this ... |
14 0 Homework Statement A yo-yo is placed on a conveyor belt accelerating ##a_C = 1 m/s^2## to the left. The end of the rope of the yo-yo is fixed to a wall on the right. The moment of inertia is ##I = 200 kg \cdot m^2##. Its mass is ##m = 100kg##. The radius of the outer circle is ##R = 2m## and the radius of the inne... |
Once we have identified two variables that are correlated, we would like to model this relationship. We want to use one variable as a
predictor or explanatory variable to explain the other variable, the response or dependent variable. In order to do this, we need a good relationship between our two variables. The model... |
Below I expand a little bit on the point in Peter's answer by trying to carry out the quantifier removal for more than constant number of steps to see where it fails and if anything can be salvaged from such an attempt.
Let's try to amplify $\mathsf{P}=\mathsf{NP}$ for more than constant number times.
Assume that $\mat... |
"Abstract: We prove, once and for all, that people who don't use superspace are really out of it. This includes QCDers, who always either wave their hands or gamble with lettuce (Monte Zuma calculations). Besides, all nonsupersymmetric theories have divergences which lead to problems with things like renormalons, insta... |
I want to draw bode plot of this transfer function:
$$G(p) = {K \over p \space (1+0.1p) \space (1+0.05p)}$$
But I don't know what to do with that K (static gain) -- I've only drawn TF with known gain.
Electrical Engineering Stack Exchange is a question and answer site for electronics and electrical engineering professi... |
If two rays are not parallel in the start, how can they become parallel at the instant when they strike the lens of a telescope? If they don't become parallel, why do we consider them to be, in the ray diagrams of telescopes?
The answer is because our ken (field of view) subtends an extremely small angle at the source.... |
Does there exist a definition for matrix exponentiation? If we have, say, an integer, one can define $A^B$ as follows:
$$\prod_{n = 1}^B A$$
We can define exponentials of fractions as a power of a radical, and we even have the following definition of the exponential:
$$e^z = \sum_{n = 0}^\infty \frac{z^n}{n!}$$
which c... |
The Annals of Mathematical Statistics Ann. Math. Statist. Volume 33, Number 4 (1962), 1272-1280. A Characterization of the Wishart Distribution Abstract
It is known that if $X$ and $Y$ are independent random variables having a Gamma distribution with parameters $(\theta, n)$ and $(\theta, m)$, i.e., with density functi... |
I don't know how your book defines the ring of fractions, but the
right (i.e. the most general as possible) way of doing things with rings of fractions is this :
If $M$ is an $R$-module and $1 \in D \subseteq R$ is a subset of $R$ which is multiplicatively closed, then we define $D^{-1} R$ as follows. Define the follow... |
Let $f\colon[0,1]\to \mathbb{R^2}$ be continuous such that $f(0)=f(1)$.
If want to find a 1-Lipschitz function $g : [0,a]\to f([0,1])$ such that $g(0)=g(a)$ and $g$ is surjective ($a>0$).
I had the following idea using the total variation of $f$: Denote $V_T(f) = \sup \left\lbrace\sum_{i=1}^n \lVert f(x_i) - f(x_{i-1})... |
Establishing the type of distribution, sample size, and known or unknown standard deviation can help you figure out how to go about a hypothesis test. However, there are several other factors you should consider when working out a hypothesis test.
Rare Events
Suppose you make an assumption about a property of the popul... |
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Search for new resonances in $W\gamma$ and $Z\gamma$ Final States in $pp$ Collisions at $\sqrt{s}=8\,\mathrm{TeV}$ with the ATLAS Detector
(Elsevier, 2014-11-10)
This letter presents a search for new resonances decaying to final states with a vector boson produced in association with a... |
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Now showing items 1-10 of 167
J/Ψ production and nuclear effects in p-Pb collisions at √sNN=5.02 TeV
(Springer, 2014-02)
Inclusive J/ψ production has been studied with the ALICE detector in p-Pb collisions at the nucleon–nucleon center of mass energy √sNN = 5.02TeV at the CERN LHC. The measurement is performed i... |
This is the students’ version of the page. Log in above for the teachers’ version. Part 1 – Adding and subtracting fractions
Between each question and the next, only one aspect is changed. Can you see how this affects the answer in each case? Click the “New questions” button for a new set of randomly generated question... |
Introduction
Drawing inferences from A/B tests is an integral job to many data scientists. Often, we hear about the frequentist (classical) approach, where we specify the alpha and beta rates and see if we can reject the null hypothesis in favor of the alternate hypothesis. On the other hand, Bayesian inference uses Ba... |
Parentheses and brackets are very common in mathematical formulas. You can easily control the size and style of brackets in LaTeX; this article explains how.
Contents
Here's how to type some common math braces and parentheses in LaTeX:
Type LaTeX markup Renders as Parentheses; round brackets
(x+y)
\((x+y)\) Brackets; s... |
The solution below is based on the one sent in by Barinder of Langley Grammar School. We had quite a number of correct, well laid out solutions to this problem this month including those from Roy of Allerton High School, Dan (no address given) and Calum of Wayland High School.
Although definitely not in proportion, thi... |
Regression Intercept Confidence Interval, is a way to determine closeness of two factors and is used to check the reliability of estimation.
${R = \beta_0 \pm t(1 - \frac{\alpha}{2}, n-k-1) \times SE_{\beta_0} }$
Where −
${\beta_0}$ = Regression intercept.
${k}$ = Number of Predictors.
${n}$ = sample size.
${SE_{\beta_... |
Al-Zamil, Qusay and Montaldi, James (2010)
Generalized Dirichlet to Neumann operator on invariant differential forms and equivariant cohomology. [MIMS Preprint]
PDF
DNoperator1.pdf
Download (140kB)
Abstract
In a recent paper, Belishev and Sharafutdinov consider a compact Riemannian manifold $M$ with boundary $\partial ... |
mrtaurho
"The mathematician does not study pure mathematics because it is useful; he studies it because he delights in it and he delights in it because it is beautiful" - Georg Cantor
Contact: mrtaurho[at]gmail[dot]com
My favorite Theorem so far:
Let $f(x)$ be an analytic function with a MacLaurin Expansion of the form... |
I was reading about energy usage in batteries and don't quite understand why it is measured in different units than home electrical usage. An ampere-hour does not include a measure of volts. But my understanding though is that a battery has a constant voltage (1.5V, 9V, ...) just as much as home electrical usage (120V,... |
Based on an original page posted by Nick Grassly on the H1N1 pandemic website.
The basic reproduction number of the swine influenza epidemic, \( R_{0} \) can be estimated from its initial rate of spread. If we assume roughly exponential growth then the basic reproductive number is related to the growth rate by the so-c... |
How does spin arise in (relativistic or not) quantum mechanics?
What are
particles in the first place? And what precisely is this property that we call "spin"?
A modern way of arriving at the notion of particles, that may be more transparent than the historical version you summarize in your post, is the way they are in... |
The Annals of Applied Probability Ann. Appl. Probab. Volume 26, Number 2 (2016), 722-759. The mean Euler characteristic and excursion probability of Gaussian random fields with stationary increments Abstract
Let $X=\{X(t),t\in\mathbb{R}^{N}\}$ be a centered Gaussian random field with stationary increments and $X(0)=0$.... |
I'm trying to typeset some axiomatic logic proofs in list form. I have tried to define a new environment such that I could combine the
proof environment of the
amsthm package with a table, resulting in the ugly picture underneath.
What I would like to achieve is that the lines of the proof are automatically numbered (I... |
According to Maxwell's equations, magnetic fields are divergence-free: $\nabla \cdot \mathbf{B} = 0$. If I understand this correctly, this means that magnetic field lines do not start or end. How can we reconcile this with magnetic reconnection?
One must be very careful in making the step from $\nabla\cdot\mathbf{B}=0$... |
Another of the uses of the \(F\) distribution is testing two variances. It is often desirable to compare two variances rather than two averages. For instance, college administrators would like two college professors grading exams to have the same variation in their grading. In order for a lid to fit a container, the va... |
Trigonometric functions
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The well known trigonometric functions can be visualized on the circle of radius 1. See http://en.wikipedia.org/wiki/Trigonometric_functions for the definitions.
Tangent: [math]\tan x = \frac{\sin x}{\cos x}[/math] Cotangent: ... |
Let $A$ be a finite alphabet. For a given language $L \subseteq A^{\ast}$ the syntactic monoid $M(L)$ is a well-known notion in formal language theory. Furthermore, a monoid $M$ recognizes a language $L$ iff there exists a morphism $\varphi : A^{\ast} \to M$ such that $L = \varphi^{-1}(\varphi(L)))$.
Then we have the n... |
Wave energy converters in coastal structures Introduction
Fig 1: Construction of a coastal structure.
Coastal works along European coasts are composed of very diverse structures. Many coastal structures are ageing and facing problems of stability, sustainability and erosion. Moreover climate change and especially sea l... |
Note that, by definition, the projections $P_n$ converge strongly to $I$.
Let $r\in\mathbb N$ (to be determined later), and define$$Q_n=P_{n+r}-P_n. $$The projections $Q_n$ are finite-rank, and pairwise orthogonal. Let $$S=\sum_n Q_nTQ_n,\ \ \ \ K=T-S.$$Let us check first that $SP_n=P_nS$ for all $n$. It is obvious tha... |
The key to my questions was the wonderful book "Linear Algebraic Groups" by Armand Borel (specifically, page 57).
First, a lemma (1 of 2): if $M$ is a (not necessarily algebraic) subgroup of an algebraic $G$, then $G(M) = \overline{M}$ (the closure is in the Zariski topology). Proof of lemma 1: $G(M)$ is closed so it c... |
Hint $\ {\rm mod}\ 13\!:\ \dfrac{41}7 \equiv \dfrac{28}7 = 4\ \ $ by $\ \ 41\equiv 41\!-\!13 = 28$
Alternatively $\ \dfrac{41}{7}\equiv\dfrac{(-2)(-1)}{-6}\equiv \dfrac{-2}{-2}\dfrac{12}3\equiv 4\ \ $ by $\ \ \begin{eqnarray}41&&\equiv\ \ 2\\ 7 &&\equiv -6\end{eqnarray}$
Alternatively $\ \dfrac{41}{7}\equiv \dfrac{2}7\... |
PGFplots allows you to use all the usual functions from Ti
kZ within its
{axis} environment. You have access to the coordinate system through
axis cs so that
\node at (axis cs: 3, 4) {}; places a node at the
x- y coordinate
(3, 4). In version 1.11,
axis cs became the default coordinate system used by Ti
kZ within the
{... |
"Abstract: We prove, once and for all, that people who don't use superspace are really out of it. This includes QCDers, who always either wave their hands or gamble with lettuce (Monte Zuma calculations). Besides, all nonsupersymmetric theories have divergences which lead to problems with things like renormalons, insta... |
vignettes/ctstm.Rmd
ctstm.Rmd
Continuous time state transition models (CTSTMs) simulate trajectories for patients between mutually exclusive health states. These models can be parameterized using multi-state models, which are generalizations of survival models with more than two states. Transitions between health state... |
Well done Robert of Madras College, St Andrew's, Scotland andAndrei of School No. 205, Bucharest, Romania for your solutions tothis problem.
In both parts of this question we consider the limiting case of aprocess which is repeated infinitely often and things are not whatthey might seem to be.
(a) In a square $ABCD$ wi... |
With $\hat i$ a unit vector, a definition of simple harmonic motion might go like this.
The motion of a particle is simple harmonic if
the acceleration of the particle, $\vec a$, is proportional to the displacement of the particle from a fixed point, $\vec x$ the acceleration of the particle is always directed towards ... |
Let $B$ be the $3\times 3$ matrix whose columns are the vectors $\mathbf{x},\mathbf{y}, \mathbf{z}$, that is,\[B=[\mathbf{x} \mathbf{y} \mathbf{z}].\]
Then we have\[AB=\begin{bmatrix}1 & 0 & 1 \\0 &1 &1 \\1 & 0 & 1\end{bmatrix}.\]
Then we have\[\det(A)\det(B)=\det(AB)=\begin{vmatrix}1 & 0 & 1 \\0 &1 &1 \\1 & 0 & 1\end{... |
With the set of parameters available to you, you cannot do this.
If you have the
actual track instead of the desired track, you will be able to calculate the wind.
The simplest way to do this is using vector math.
There are three vectors to consider:
ground speed vector $\vec{V_{gs}}$ air speed vector $\vec{V_{as}} $ w... |
Abbreviation:
All
An
is an expanded category $\mathbf{M}=\langle M,\circ,\text{dom},\text{rng},\text{id},\vee,\wedge,^\smile\rangle$ such that allegory
$...$ is …: $...$
$...$ is …: $...$
Remark: This is a template.
It is not unusual to give several (equivalent) definitions. Ideally, one of the definitions would give a... |
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