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It's my understanding that General Relativity abstracts away the concept of gravity as a force, and instead describes it as a feature of spacetime by which massive objects cause curvature. Then it follows that what we experience as a force is simply the difference between a geodesic on this curved surface and our perce... |
I have packet streams $1...k$ and, streams with Prob(err) $p1...pk$. The $p$'s are consts $>0$. I'd like to maximize the probability all make it simultaneously while I'm allowing at most $N$ packets to pass through, hence, every stream should get some portion of $N$. i.e.
max $f=(1-p_1^{n_1})(1-p_2^{n_2})...(1-p_2^{n_2... |
Kumari, M and Nath, G (2004)
Transient MHD rotating flow over a rotating sphere in the vicinity of the equator. In: International Journal of Engineering Science, 42 (17-18). pp. 1817-1829.
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MHD.pdf
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Abstract
Transient rotating flow of a laminar inc... |
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In the following section, we presuppose Set Theory and Nonstandard Analysis. The exponential simplex and the polynomial intex method (
inter-/ extrapolation) solve linear programmes (LPs).
Diameter theorem for polytopes: The diameter of an \(n\)-dimensional polytope defined by \(m\) constraints with \(m... |
I know that all compact Riemann surfaces with the same genus are topologically equivalent. Moreover they are diffeomorphic. But are they biholomorphic, too? In other words, is the complex structure conserved?
Some magic words for this question are "moduli space" or "moduli stack". In the early days, one was interested ... |
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In the following section, the Set Theory is presupposed.
Definition: A family of sets \(\mathbb{Y} \subseteq \mathcal{P}(X)\) is called
topology on \(X \subseteq R\) if every intersection and union of sets of \(\mathbb{Y}\) belongs apart from \(\emptyset\) and \(X\) to \(\mathbb{Y}\). The pair \((X, \ma... |
Does fixing the reparameterization invariance of the string action, for example by choosing the light-cone gauge
$$ X^{+} = \beta\alpha' p^{+}\tau $$
$$ p^{+} = \frac{2\pi}{\beta} P^{\tau +} $$
correspond to some kind of orbifolding?
This answer explains that gauge systems are orbifolds after removing the gauge redunda... |
The generating function approach:
$$P(x)=(1+x+x^2+x^3+x^4+x^5)^6=\sum a_i x^i$$
Then $a_i$ counts the number of ways of getting a total of $i+6$ from $6$ dice.
Now, to find the even terms, you can compute $$\frac{P(1)+P(-1)}{2}=\sum_i a_{2i}.$$
But $P(1)=6^6$ and $P(-1)=0$. So $$\frac{P(1)+P(-1)}{2}=\frac{6^6}{2},$$ or... |
This answer is the proof given by Ashutosh, but formulated in terms of the splitting number.
PropositionIf the splitting number $s$ is $\aleph_{1}$, then every nonseparable metric space contains a sequence of subsets with no convergent subsequence.
Proof: Following Sierpinski, since the metric space $M$ is non-separabl... |
DG - MP - PDE Seminar: Nassif Ghoussoub (UBC) Date: 09/13/2011 Time: 15:30
University of British Columbia
A self-dual polar factorization for vector fields
Abstract
We show that any non-degenerate vector field u in L^{\infty}(\Omega, \R^N), where \Omega is a bounded domain in \R^N, can be written as {equation} \hbox{u(... |
While much more can be said about sequences, we now turn to ourprincipal interest, series. Recall that a series, roughly speaking, isthe sum of a sequence: if $\ds\{a_n\}_{n=0}^\infty$ is a sequence then theassociated series is$$\sum_{i=0}^\infty a_n=a_0+a_1+a_2+\cdots$$Associated with a series is a second sequence, ca... |
J. D. Hamkins and J. Reitz, “The set-theoretic universe $V$ is not necessarily a class-forcing extension of HOD,” ArXiv e-prints, 2017. (manuscript under review)
@ARTICLE{HamkinsReitz:The-set-theoretic-universe-is-not-necessarily-a-forcing-extension-of-HOD, author = {Joel David Hamkins and Jonas Reitz}, title = {The se... |
What is the simplest ay to describe the difference between these two concepts, that often go by the same name?
The Wilsonian effective action is an action with a given scale, where all short wavelength fluctuations (up to the scale) are integrated out. Thus the theory describes the effective dynamics of the long wavele... |
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Production of light nuclei and anti-nuclei in $pp$ and Pb-Pb collisions at energies available at the CERN Large Hadron Collider
(American Physical Society, 2016-02)
The production of (anti-)deuteron and (anti-)$^{3}$He nuclei in Pb-Pb collisions at $\sqrt{s_{\rm NN}}$ = 2.76 TeV has ... |
How can I prove that the Cartesian product of two countable sets is also countable?
closed as off-topic by Xander Henderson, Xam, Arnaud Mortier, Ethan Bolker, David Hill Mar 22 '18 at 20:28
This question appears to be off-topic. The users who voted to close gave this specific reason:
" This question is missing context... |
Set Difference with Empty Set is Self
Jump to navigation Jump to search
Theorem $S \setminus \O = S$ Proof
From Set Difference is Subset:
$S \setminus \O \subseteq S$
From the definition of the empty set:
$\forall x \in S: x \notin \O$ Let $x \in S$.
Thus:
\(\displaystyle x \in S\) \(\leadsto\) \(\displaystyle x \in S ... |
If a random variable is discrete, and we are interested in its quantile value, how to define a proper back testing procedure?
For example, the
underlying variable with a discrete value is
$$ d(\mbox{account}) = \mbox{PaymentDate} - \mbox{BillingDate} $$
the
observing variable:
$$ y = \mbox{percentile}(d, 95\%, \mbox{mo... |
There are already some good answers, but I still feel like adding yet another explanation, because I consider this topic extremely important for the understanding of many aspects of digital signal processing.
First of all it is important to understand that the DFT does not 'assume' periodicity of the signal to be trans... |
I am currently working with Pesaran & Timmermann test version from year 2009. Since I could not find any R package that contains function to calculate it (
rugarch has 1992 version in
DACTest). I recently posted thread in cross validated with similar question where kind user mlofton advised me to look into this list. S... |
I am trying to do something very simple in Mathematica 9. I want to play around with option pricing and for that I thought it best to use the new stochastic process functionality.
So, first of all I simulate one instance of a geometric brownian motion:
$$ \frac{dX_t}{X_t} = \mu dt + \sigma dW_t\\ dW_t \sim N(0, 1) $$
W... |
DIFFICULT EUCLEDIAN GEOMETRY QUESTION ON CIRCLES
Including results for difficult euclidean geometry question on circles.Do you want results only for difficult eucledian geometry question on circles?
Geometry Problems with Solutions and Answers for Grade 12 Geometry problems with solutions and answers for grade 12. Free... |
In a
pure diffusion setting, you can equivalently write no calendar arbitrage constraints:
In terms of
implied volatility: total implied variance should be non decreasing in time, and that, for any given forward moneyness level, see Gatheral top of page 4.
In terms of
European option prices: see Gatheral end of page 3.... |
Ground state homoclinic solutions for a second-order Hamiltonian system
Department of Mathematics, Xiangnan University, Chenzhou, Hunan 42300, China
$ \ddot{u}-L(t)u+\nabla W(t, u) = 0, $
$ t\in {\mathbb{R}}, u\in {\mathbb{R}}^{N} $
$ L: \mathbb{R}\rightarrow {\mathbb{R}}^{N\times N} $
$ W: {\mathbb{R}}\times {\mathbb{... |
Summer Term 2016
Please note that this term the seminars will be held on Thursdays, from 2pm to 3pm.
José Miguel Manzano, Compact stable surfaces with constant mean curvature in Killing submersions . Thursday 5th May, 2-3pm, Huxley 139.
Abstract: A Killing submersion is a Riemannian submersion from an orientable 3-mani... |
[The following includes a) two specific questions (at the end), b) an attempt to capture a dispositional concept (excitability) in geometric and physical terms.]
I assume that "excitability of a neuron" is a reasonable concept (and measure) to distinguish neurons: there are (as will be seen) neurons that are more or le... |
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Description
Given a set of $n$ points in $\ell_{1}$, how many dimensions are needed to represent all pairwise distances within a specific distortion ? This dimension-distortion tradeoff question is well understood ... |
ISO Sensitivity (or ISO speed) is a measure of how strongly an image sensor and/or camera responds to light. The higher the sensitivity, the less light (smaller aperture and/or shorter exposure time) required to capture a good quality image. Unfortunately there are several measures of Sensitivity, and they are not cons... |
Hi, Can someone provide me some self reading material for Condensed matter theory? I've done QFT previously for which I could happily read Peskin supplemented with David Tong. Can you please suggest some references along those lines? Thanks
@skullpatrol The second one was in my MSc and covered considerably less than my... |
Learning Objectives Calculate labor force percentages and the unemployment rate Calculating the Unemployment Rate
Remember that the unemployed are those who are out of work and who are actively looking for a job. We can calculate the unemployment rate by dividing the number of unemployed people by the total number in t... |
2007 Senior Project Archive Section Navigation
Titles are hyperlinked to pdf copies of the final project write-up. Course coordinator: Barry Balof
TITLE: Baire One Functions
AUTHOR: Johnny Hu ABSTRACT: This paper gives a general overview of Baire one functions, including examples as well as several interesting properti... |
A divide and conquer algorithm's work at a specific level can be simplified into the equation:
$\qquad \displaystyle O\left(n^d\right) \cdot \left(\frac{a}{b^d}\right)^k$
where $n$ is the size of the problem, $a$ is the number of sub problems, $b$ is the factor the size of the problem is broken down by at each recursio... |
I'm very confused about this bound, please give me any suggestions on how to prove it. (Note: $a \ll b$ is just a neater way to write $a = O(b)$)
I am starting with the bound $$f(n) \ll \frac{n}{\log(n)^2}\prod_{p|n}\left(1-\frac{1}{p}\right)^{-1}$$
then I don't see where $\phi$ comes from in $$\frac{n}{\log(n)^2}\prod... |
Here is another geometric application of the integral: find the length of a portion of a curve. As usual, we need to think about how we might approximate the length, and turn the approximation into an integral.
We already know how to compute one simple arc length, that of a line segment. If the endpoints are $\ds P_0(x... |
I'm using tex4ht for conversion of heavily maths loaded LaTeX files into HTML to be able to serve them in a web app. Can successfully convert all equations to MathML and jsMath. But as equations are not web optimized, some of them are rendered wrongly or dont even get rendered as they are in original PDF's. So I decide... |
I don’t understand why the following question
The decomposition of nitrosyl bromide $(\ce{NOBr})$ proceeds by the following reaction:
$$\ce{2 NOBr(g) <=> 2 NO(g) + Br2(g)} \qquad K = 0.0142$$
Calculate the $[\ce{NOBr}],$ $[\ce{NO}],$ and $[\ce{Br2}]$ when $\pu{10.0 mol}$ of nitrosyl bromine is placed in a $\pu{5.00 L}$... |
I have two scenarios where I use arrows with a super-scripted asterisk: math mode and
tikz-cd diagrams. I would like to be able to show such an arrow in both scenarios such that the arrows look the same, i.e., with respect to positioning of the asterisk.
Consider the below MWE. This is how I would like the arrow to loo... |
In some reactions, the rate is
apparently independent of the reactant concentration. The rates of these zero-order reactions do not vary with increasing nor decreasing reactants concentrations. This means that the rate of the reaction is equal to the rate constant, \(k\), of that reaction. This property differs from bo... |
You have to distinguish between the distance the man swims, relative to the water around him, and the total distance the man travels, relative to an observer on the river bank. The total distance relative to an observer on the river bank is the distance the man swims measured relative to the water around him combined w... |
As metioned in Wikipedia's biography, Shanks used Machin's formula$$ \pi = 16\arctan(\frac15) - 4\arctan(\frac1{239}) $$
The standard way to use that (and the various Machin-
like formulas found later) is to compute the arctangents using the power series
$$ \arctan x = x - \frac{x^3}3 + \frac{x^5}5 - \frac{x^7}7 + \fra... |
I have a question on definition/motivation of Virasoro algebra. Recall that Virasoro algebra is an infinite Lie algebra generated by elements $L_n$ $(n\in \mathbb{Z})$ and $c$ over $\mathbb{C}$ with relations $$ [L_m,L_n]=(m-n)L_{m+n}+\frac{c}{12}(m^3-m)\delta_{m+n,0}. $$ A typical explanation of this definition is the... |
Here is the question:
"A car travels round a bend which has radius $100~\text{m}$ and is banked at an angle of $20°$ to the horizontal. The car is travelling at a speed of $30 ~\text{m}\text{s}^{-1}$. What is the least possible value of the coefficient of friction if the car does not slip up the slope?"
The way my text... |
Fill in each blank unshaded cell in the diagram below with a positive integer less than 100, such that every consecutive group of unshaded cells within a row or column is an arithmetic sequence.
This problem is from the USAMTS Round 3 problem set.
Puzzling Stack Exchange is a question and answer site for those who crea... |
I know that you asked for an intuitive explanation, but I'm afraid that if one wants to go beyond ron's comment, the technical aspects are somewhat necessary. It isn't incredibly important to understand the maths, but I have included it as I want to enable interested readers to go further, and it provides a basis for t... |
let $x=\sum_{i=1}^{\infty}\delta_i2^{-i},\ \delta_i\in\{0,1\}$.
Is there an algorithm that converts the sequence $(\delta_0,\ \delta_1,\ ...)$ of the binary digits of $x$ to the sequence $[a_0;a_1,\ ...]$ of its continued fraction representation?
MathOverflow is a question and answer site for professional mathematician... |
Hello, all!
I have a polynomial non-singular square matrix over $\mathbf{F} _q[x]$, $$\underset{l \times l}{G(x)} = \left( \begin{matrix} g _{0,0}(x) & g _{0,1}(x) & \ldots & g _{0,l-1}(x) \\\ \vdots & \vdots & \vdots & \vdots \\\ g _{l-1,0}(x) & g _{l-1,1}(x) & \ldots & g _{l-1,l-1}(x) \end{matrix} \right).$$ I call a... |
CryptoDB Paper: Abelian varieties with prescribed embedding degree
Authors: David Freeman Peter Stevenhagen Marco Streng Download: URL: http://eprint.iacr.org/2008/061 Search ePrint Search Google Abstract: We present an algorithm that, on input of a CM-field $K$, an integer $k \ge 1$, and a prime $r \equiv 1 \bmod k$, ... |
We have seen that some functions can be represented as series, which may give valuable information about the function. So far, we have seen only those examples that result from manipulation of our one fundamental example, the geometric series. We would like to start with a given function and produce a series to represe... |
The trigonometric functions frequently arise in problems, and often it is necessary to invert the functions, for example, to find an angle with a specified sine. Of course, there are many angles with the same sine, so the sine function doesn't actually have an inverse that reliably "undoes'' the sine function. If you k... |
Heat Transfer in Deformed Solids
In a previous blog post, we presented the applications of conjugate heat transfer involving immobile solids. The case of immobile solids simplifies the heat equation to be solved and is often a good approximation to the temperature field. Today, we will complete the description of the p... |
How to Model Residual Stresses Using COMSOL Multiphysics
Today, we will introduce the concept of residual stresses in structural mechanics and find out how to compute them by taking the example of a deep metal drawing process. First, we will explain how they can be computed and interpreted in a bending beam example wit... |
I read that 'Euclidean distance is not a good distance in high dimensions'. I guess this statement has something to do with the curse of dimensionality, but what exactly? Besides, what is 'high dimensions'? I have been applying hierarchical clustering using Euclidean distance with 100 features. Up to how many features ... |
Ahlskog, M and Mukherjee, AK and Menon, Reghu (2001)
Low temperature conductivity of metallic conducting polymers. In: Synthetic Metals, 119 (1-3). pp. 457-458.
PDF
5.pdf
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Abstract
In several metallic conducting polymers, both positive and negative te... |
This is a question I feel too stupid asking my professor about. I'm having a mental block remembering how this works even though I think I understood it at one point:
I know the following properties:
$$x(t) {\longrightarrow}\boxed{\textrm{LTI System}}{\longrightarrow} y(t) = x(t) \star h(t) \longleftrightarrow X(j\omeg... |
Can I be a pedant and say that if the question states that $\langle \alpha \vert A \vert \alpha \rangle = 0$ for every vector $\lvert \alpha \rangle$, that means that $A$ is everywhere defined, so there are no domain issues?
Gravitational optics is very different from quantum optics, if by the latter you mean the quant... |
The diagram shows a British 50 pence coin.
The seven arcs $AB$, $BC$, . . . , $FG$, $GA$ are of equal length and each arc is formed from the circle of radius a having its centre at the vertex diametrically opposite the mid-point of the arc. Show that the area of the face of the coin is
$$\frac{a^2}{2}(\pi-7\tan\frac{\p... |
Suppose $p \in A \cap B$. To show connectedness of $A \cup B$, write $A \cup B=U \cup V$, where $U$ and $V$ are open disjoint subsets of $A \cup B$. It suffices to show $U$ or $V$ is empty...
So where is $p$? It must be in $U$ or in $V$, say $p\in U$ for definiteness; by symmetry it doesn't matter (or rename letters in... |
Let $1\leq p < \infty$ and consider a sequence $(x_{n})_{n}\subseteq L^{p}[0,1]$. Show the equivalence of:
$1.$ $x_{n} \xrightarrow{ w} 0$
$2.$ $\sup\limits_{n \in \mathbb N} \vert \vert x_{n}\vert\vert_{p}<\infty $ and $\int_{A}x_{n}(t)dt\xrightarrow{n \to \infty} 0$ for any borel sets $A$ on $[0,1]$.
for $1. \Rightar... |
Change Points Posted on (Update: ) Introduction Page’s (1954, 1955) classical formulation Shiryaev (1963) and Lorden (1971) then developed
One is concerned with sequential detection of a change-point, which represents a disruption in a continuous production process.
problems in fixed samples return to sequential detect... |
Definition:Division Contents Definition
Let $\struct {F, +, \times}$ be a field.
Let the zero of $F$ be $0_F$.
The operation of division is defined as: $\forall a, b \in F \setminus \set {0_F}: a / b := a \times b^{-1}$
where $b^{-1}$ is the multiplicative inverse of $b$.
The concept is usually seen in the context of t... |
Subset Equivalences
Jump to navigation Jump to search
Contents Definitions
In the following:
$S \subseteq T$ denotes that $S$ is a subset of $T$ $S \cup T$ denotes the union of $S$ and $T$ $S \cap T$ denotes the intersection of $S$ and $T$ $S \setminus T$ denotes the set difference between $S$ and $T$ $\varnothing$ den... |
Fitting Measured Data to Different Hyperelastic Material Models
Previously on the blog, we have discussed the need for appropriate measured data to fit the material parameters that correspond to a material model. We have also looked at typical experimental tests, considerations for operating conditions when choosing a ... |
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Now showing items 1-5 of 5
Forward-backward multiplicity correlations in pp collisions at √s = 0.9, 2.76 and 7 TeV
(Springer, 2015-05-20)
The strength of forward-backward (FB) multiplicity correlations is measured by the ALICE detector in proton-proton (pp) collisions at s√ = 0.9, 2.76 and 7 TeV. The measurement... |
Definition:Integer Division
Jump to navigation Jump to search
Definition
Let $a, b \in \Z$ be integers such that $b \ne 0$..
From the Division Theorem:
$\exists_1 q, r \in \Z: a = q b + r, 0 \le r < \left|{b}\right|$ The process of finding $q$ and $r$ is known as (integer) division. $29 \div 8 = 3 \rem 5$
\(\displaysty... |
Problem: The series $\sum_{n=1}^{\infty} a_n$ diverge and positive. What can be said on the series $\sum_{n=1}^{\infty} \frac{a_n}{1+n^{2}a_n}$ ?
Approach:
First, I tried separating it into two cases: $a_n \to \infty$, $a_n \to A$ (where $A$ is some const value).
Tried the ratio test and got nowhere with that.
I think ... |
Let $V_\mathbb{R}$ denote the $\mathbb{R}$-vector space of binary quartic forms. The group $\operatorname{GL}_2(\mathbb{R})$ acts on $V_\mathbb{R}$ via the standard substitution action. That is, if $F(x,y) \in V_\mathbb{R}$ and $T = \begin{pmatrix} t_1 & t_2 \\ t_3 & t_4 \end{pmatrix} \in \operatorname{GL}_2(\mathbb{R}... |
Major Edit: The last attempt had a fatal flaw.
Let $J$ be the set of odd integers greater than $1$, and let $p_n(x)=x^n-nx+(n-1)$ for $n\in J$. Calculating derivatives $$p_n^{\prime}(x)=n(x^{n-1}-1)\text{ and }p^{\prime\prime}_n(x)=n(n-1)x^{n-2}$$ tells us that $p_n(-1)=2(n-1)$ is a local max, $p_n(1)=0$ is a local min... |
(I apologize in advance if this question is unsuitable for MO. If so, please let me know and I will migrate it to MSE.)
Let $\sigma(M)$ be the sum of the divisors of the positive integer $M$. For example, $\sigma(6)=1+2+3+6=12$.
A number $N$ is called
perfect if $\sigma(N)=2N$.
Euler proved that an
odd perfect number $... |
The probability of finding a particle at a point is always zero.
Recall that $\rho(x) = \lvert\psi(x)\rvert^2$ is a
probability density, not a probability, and so the probability to find the particle somewhere inside the interval $[a,b]$ is given by$$ P([a,b]) = \int_a^b \rho(x)\mathrm{d}x.$$Since points have zero meas... |
What is the motivation for including the compactness and semi-simplicity assumptions on the groups that one gauges to obtain Yang-Mills theories? I'd think that these hypotheses lead to physically "nice" theories in some way, but I've never, even from a computational perspective. really given these assumptions much tho... |
Question:
Why is it important to make sure your entire unknown has vaporized? Why is it important to put a pinhole in the aluminum "cap"? If 0.750 g of a gas occupies 265 mL at 25 degree C and 680 mm Hg, what is the molar mass of the gas?
Molar mass:
Molar mass is an important parameter in stoichiometric calculations. ... |
In a question i found the recurrence relation $a_n=a_{n-2}+3^{(n-2)/2}$ then i should solve this relation with using generating factor but $3^{(n-2)/2}$ makes difficult to solve . Is there a solution for such a relations ? Or is my answer wrong ?
Expand the equation:
\begin{align*} a_n&=a_{n-2}+3^{\frac{n-2}{2}} \\ &=a... |
It might help to think of an example. A simple is example is dust, specifically a collection of particles of a fixed rest mass, all at rest with respect to each other, and we can consider uniform dust, so they are equally spaced.
If that's the only thing in our universe, then there is no momentum or stress in the frame... |
Or should we input $[1 \ 0]$ in each H gate, because we are applying H
gates to just qubit of state $|0\rangle$ each time?
Yes, when you have a two-qubit state (say you label the two qubits as $A$ and $B$ respectively), you need to apply the two Hadamard gates separately on each qubit's state. The final state will be t... |
The Sharpe ratio tells us the amount of excess return we get for taking on each additional unit of portfolio standard deviation. $$\frac{\mu_p - r_f}{\sigma_p }$$
We are looking for the combination of the two risky assets with the highest Sharpe ratio ($P^*$). Once we do that, we can take linear combinations of that po... |
I have reduced this problem (thanks @Mhenni) to the following (which needs to be proved):
$$\prod_{k=1}^n\frac{\Gamma(3k)\Gamma\left(\frac{k}{2}\right)}{2^k\Gamma\left(\frac{3k}{2}\right)\Gamma(2k)}=\prod_{k=1}^n\frac{2^k(1+k)\Gamma(k)\Gamma\left(\frac{3(1+k)}{2}\right)}{(1+3k)\Gamma(2k)\Gamma\left(\frac{3+k}{2}\right)... |
Let $G \left(X, Y, E\right)$ be a bipartite graph with two equal-sized parts (that is, $|X|=|Y|=n$).
An
envy-free matching is a perfect matching between two subsets $X_1 \subseteq X$ and $Y_1 \subseteq Y$ such that no unmatched $x$ (that is, $x \in X \setminus X_1$) wants (i.e., is connected to) any matched $y$ (that i... |
I began to study Field Theory from Lang's book and I would be happy to discuss some questions which I am going to write down below:
Question 1. Suppose that $\mathbb{k}$-field, $E$ extension field of $\mathbb{k}$ and $\alpha_1,\alpha_2\in E$.
Am I right that $\mathbb{k}(\alpha_1,\alpha_2)=\mathbb{k}(\alpha_1)(\alpha_2)... |
Suppose that $f:\mathbb{R}^n\to \mathbb{R}^m$ is of class $C^1$ and $Df(x_0)$ has rank $m$. Then show there is a whole neighborhood of $f(x_0)$ lying in the image of $f$.
My attempt: if $Df(x_0)$ is onto (rank $m$) and $n\leq m$, then I can use the Rank Theorem and justify that exists open sets $V,W\subset \mathbb{R}^m... |
In Classical Mechanics one usually considers the Lagrangian as $L = K - U$ where $K$ is the kinetic energy of the system and $U$ is the potential energy. One then gets the Euler-Lagrange equations and everything is fine: if we have a system we can plug in the kinetic energy and potential and find the Lagrangian for it.... |
Assume we have $n$ double elements $a_1 \dots a_n$. We want to find out if two of the elements of the array are identical. And we have a hash function $h(x)$ which assigns each double value an integer between $1$ and $n$ and which can be calculate in $O(1)$ time. Let $m := \{(i,j) : a_j \neq a_i \text{ and } h(a_j) = h... |
Suppose we are interested in a more detailed inventory of the colorings of an object, namely, instead of the total number of colorings we seek the number of colorings with a given number of each color.
Example 6.3.1 How many distinct ways are there to color the vertices of a regular pentagon modulo $D_5$ so that one ve... |
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D-meson nuclear modification factor and elliptic flow measurements in Pb–Pb collisions at $\sqrt {s_{NN}}$ = 5.02TeV with ALICE at the LHC
(Elsevier, 2017-11)
ALICE measured the nuclear modification factor ($R_{AA}$) and elliptic flow ($\nu_{2}$) of D mesons ($D^{0}$, $D^{+}$, $D^{⁎+}$... |
The exact answer depends on the exact kind of superposition you want. The answers by pyramids and Niel both give you something like
$$A\sum_{t=1}^n |\,\,f_t (x)\,\,\rangle \otimes |F_t\rangle$$
Here I've followed Niel in labelling the different functions $f_1$, $f_2$, etc, with $n$ as the total number of functions you ... |
\[\def\bigtimes{\mathop{\vcenter{\huge\times}}}\]
Introduction
In a recent post, principles of Dynamic Programming were used to derive a recursive control algorithm for Deterministic Linear Control systems. The challenges with the approach used in that blog post is that it is only readily useful for Linear Control Syst... |
Apps for Teaching Mathematical Modeling of Tubular Reactors
The Tubular Reactor application is a tool where students can model a nonideal tubular reactor, including radial and axial variations in temperature and composition, and investigate the impact of different operating conditions. It also exemplifies how teachers ... |
Show that $\mathbb{Z}[x]=\lbrace \sum_{i=0}^{n}{a_ix^i}:a_i \in \mathbb{Z}, n \geq 0 \rbrace$ is not a principal ideal ring. I know the definition of principal ideal ring is that every ideal is generated by a single element. So my aim here is to find an ideal which is not generated by a single element. But I fail to lo... |
No long explanation is needed,
What would happen if I were to allow one end of a rope to fall past the event horizon of a black hole while I held the other end?
Would I be able to pull it out? Would the rope feel extremely (infinitely?) heavy?
Physics Stack Exchange is a question and answer site for active researchers,... |
A second order differential equation is one containing the second derivative. These are in general quite complicated, but one fairly simple type is useful: the second order linear equation with constant coefficients.
Example 19.5.1 Consider the intial value problem $\ddot y-\dot y-2y=0$, $y(0)=5$, $\dot y(0)=0$. We mak... |
I understand how to get the proper maclaurin series representation for $\cos x$, but I'm having trouble understanding the following part conceptually:
I get $\cos x$ as $\sum_{n=0}^\infty (-1)^n\frac{x^{2n}}{2n!}$ but,
Can the maclaurin series of $\cos x$ also be $\sum_{n=0}^\infty (-1)^n\frac{x^n}{n!}$?
I'm confused b... |
Functional PCA Posted on PCA and eigenanalysis
Define the covariance function $v(s,t)$ by
Each of those principal component weight functions $\xi_j(s)$ satisfies the equation
for an appropriate eigenvalue $\rho$. The left side is an
integral transform $V$ of the weight function $\xi$ defined by
The integral transform i... |
Answer
The radius of the curve is approximately 73.3 feet.
Work Step by Step
We can convert the angle to radians: $\theta = (42.0^{\circ})(\frac{\pi~rad}{180^{\circ}}) = 0.733~rad$ Since the chord is approximately equal to the arc length, the we can use the chord length $d$ to find the approximate value of the radius: ... |
Let $\Omega\subset\mathbb{R}^n$ be a bounded open set. Let us say it has a Lipschitz boundary.
Consider the Laplacian $\Delta$ in the classical sense. Suppose $\Delta u=\frac{\partial^2}{\partial x_1^2}u+\dotsb+\frac{\partial^2}{\partial x_n^2}u$ is bounded.
Q: Can we say $u\in C^1(\Omega)$? Does it depend on the dimen... |
Image Denoising and Other Multidimensional Variational Problems
We previously discussed how to solve 1D variational problems with the COMSOL Multiphysics® software and implement complex domain and boundary conditions using a unified constraint enforcement framework. Here, we extend the discussion to multiple dimensions... |
Title: On u-deformed Kottwitz's involution modules
报告人：胡峻 教授（北京理工大学）
地点：玉泉校区工商楼105
摘要: Let (W,S) be a Coxeter system and $\ast$ an automorphism of W with order $\leq 2$ and $S^{\ast}=S$.
Lusztig and Vogan have introduced a u-deformed version $M_u$ of Kottwitz's involution module over the Iwahori--Hecke algebra $H_{u}(W... |
There isn't a single way in which one can approach a discrete optimization problem using Differential Evolution (DE).
Widespread techniques listed under the Discrete Differential Evolution label aren't DE-specific.
You can allow variables to take values in a continuous range and use penalty functions to enforce integer... |
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Calculus Calculus is a branch of mathematics which helps us understand changes between values that are related by a function. For example, if you had one formula telling how much money you got every day, calculus would help you understand related formulas like how much ... |
In this section, we define what is arguably the single most important function in all of mathematics. We have already noted that the function $\ln x $ is injective, and therefore it has an inverse.
Definition 9.3.1 The inverse functionof $\ln(x) $ is$y=\exp(x)$, called the
natural exponential function.
The domain of $\... |
Erdős cardinals
The $\alpha$-Erdős cardinals were introduced by Erdős and Hajnal in [1] and arose out of their study of partition relations. A cardinal $\kappa$ is $\alpha$-Erdős for an infinite limit ordinal $\alpha$ if it is the least cardinal $\kappa$ such that $\kappa\rightarrow (\alpha)^{\lt\omega}_2$ (if any such... |
I have $d$ Normal Distributions, $N_1(\mu_1, \sigma_1^2) \cdots N_d(\mu_d, \sigma_d^2)$. We pick one of the $d$ distributions with each distribution having a probability of $\frac{1}{d}$ of being picked and generate a sample, $s_0$. What is the probability that it was generated from the distribution $N_1$?
I think this... |
For a constant,
N, what value of
x will maximize the cosine (or any trig) function?
\begin{equation} 1 = \cos{(Nx)} \end{equation}
I am looking for the exact form, not the approximation because, \begin{equation} \frac{\arccos{(1)}}{N} = x = 0 \end{equation}
For example, WolframAlpha.com states that if
N = 19.013, then,... |
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