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In
category theory, a branch of mathematics, profunctors are a generalization of relations and also of bimodules. Definition [ edit ]
A
profunctor (also named by the French school and distributor module by the Sydney school) from a ϕ {\displaystyle \,\phi } category to a category C {\displaystyle C} , written D {\displ... |
I too was unable to derive a direct proof of the formula you have given; the following proof therefore is somewhat indirect. It rests on the following claim.
Claim. Given functions $H(t), \, f(t)$, the renewal equation $g(t) = H(t) + \int_{0}^t g(t-x) f(x) \mathrm d x$ has a unique solution.
Henceforth I will use the n... |
Let $\mathbb{X}$ be a normed space that is complete and $\mathbb{Y}$ be another normed space which is not complete. Then can a bounded linear map $A:\mathbb{X} \to \mathbb{Y}$ be bijective or not?
Edit: In fact there's a very simple theorem here that gives the whole truth: Given a bounded linear bijection $T:X\to Y$, w... |
Here is a sketch of proof. Since the eigenvalues of a linear operator are independent of the choice of basis, for ease of presentation, when we mention $A,G,Q$ or $M$ below, they are viewed as linear operators rather than matrices.
Consider an ordered orthonormal basis $\mathcal{U}=\left\{u_1,\ u_2,\ \ldots,\ \right\}$... |
Even before quantization, charged bosonic fields exhibit a certain "self-interaction". The body of this post demonstrates this fact, and the last paragraph asks the question.
Notation/ Lagrangians
Let me first provide the respective Lagrangians and elucidate the notation.
I am talking about complex scalar QED with the ... |
Consider $\mathcal{N}=1,d=4$ SUSY with $n$ chiral superfields $\Phi^i,$ Kaehler potential $K,$ superpotential $W$ and action ($\overline{\Phi}_i$ is complex conjugate of $\Phi^i$) $$ S= \int d^4x \left[ \int d^2\theta d^2\overline{\theta} K(\Phi,\overline{\Phi})+ \int d^2\theta W(\Phi) + \int d^2\overline{\theta} \over... |
Webster's Revised Unabridged Dictionary
AS. rest, ræst, rest; akin to D. rust, G. rast,. OHG. rasta, Dan. & Sw. rast, rest, repose, Icel. röst, the distance between two resting places, a mole, Goth. rasta, a mile, also to Goth. razn, house, Icel. rann, and perhaps to G. ruhe, rest, repose, AS. rōw, Gr. 'erwh`. Cf. Rans... |
For students of maths stream, the 2nd puc karnataka board is a very critical year as its time for them to acquire more in depth knowledge in their stream. If a student is able to pass successfully this year, then they can go for further studies to professional colleges or other as per their preference. The marks that t... |
How can I prove that $$2^n=2\left({n \choose 0}+{n \choose 2}+{n \choose 4}+\dots\right)$$ using the binomial theorem. I've tried expanding $(x-y)^n$ with multiple different values of $x$ and $y$ but nothing pans out. I can see why the above is true when looking at Pascal's triangle as the sum of the $n-1$ row gives $2... |
Brief sketch of ARE for one-sample $t$-test, signed test and the signed-rank test
I expect the long version of @Glen_b's answer includes detailed analysis for two-sample signed rank test along with the intuitive explanation of the ARE. So I'll skip most of the derivation. (one-sample case, you can find the missing deta... |
A particle moves along the x-axis so that at time t its position is given by $x(t) = t^3-6t^2+9t+11$ during what time intervals is the particle moving to the left? so I know that we need the velocity for that and we can get that after taking the derivative but I don't know what to do after that the velocity would than ... |
We got an argument that 3-coloring bounded degree graphs is subexponential with complexity $O(\exp{(\sqrt{n}\log^2{n})})$.
The treewidth of a planar graphs on $n$ vertices is $O(\sqrt{n})$ and 3-coloring it is $O(\exp{\sqrt{n}})$ as shown here p. 8 of the pdf.
This paper gives reduction from 3-coloring to 3-coloring pl... |
Sandbox
this is a test sandbox. feel free to edit how you like although no changes are guaranteed to stay for long.
test edit
Contents Basic text formatting[edit]
You can format this page using Wikitext special characters.
What it looks like What you type
You can
3 apostrophes will bold
5 apostrophes will bold and ital... |
Pressure coefficient, abbreviated as \(C_p\), is a dimensionless number about the relative pressures throughout a flow field in fluid mechanics.
Pressure Coefficient FORMULA
\(\large{ C_p = \frac { p \;-\; p_{\infty} } { \frac {1}{2} \; \rho_{\infty} \; v_{\infty}^2 } }\)
Where:
\(\large{ C_p }\) = pressure coefficient... |
@egreg It does this "I just need to make use of the standard hyphenation function of LaTeX, except "behind the scenes", without actually typesetting anything." (if not typesetting includes typesetting in a hidden box) it doesn't address the use case that he said he wanted that for
@JosephWright ah yes, unlike the hyphe... |
Crossing the Line
This post continues our series on developing statistical models to explore the arcane relationship between UFO sightings and population. The previous post is available here: Bayes vs. the Invaders! Part One: The 37th Parallel.
The simple linear model developed in the previous post is far from satisfyi... |
A particle moves along the x-axis so that at time t its position is given by $x(t) = t^3-6t^2+9t+11$ during what time intervals is the particle moving to the left? so I know that we need the velocity for that and we can get that after taking the derivative but I don't know what to do after that the velocity would than ... |
Going with Patrick87's hash idea, here's a practical construction that almost meets your requirements — the probability of falsely mistaking a new value for an old one is not quite zero, but can be easily made negligibly small.Choose the parameters $n$ and $k$; practical values might be, say, $n = 128$ and $k = 16$. Le... |
In
mathematics, the Fictitious domain method is a method to find the solution of a partial differential equations on a complicated domain , by substituting a given problemposed on a domain D {\displaystyle D} , with a new problem posed on a simple domain D {\displaystyle D} containing Ω {\displaystyle \Omega } . D {\di... |
I have a problem using
\Aboxed in the following
align*-environment:
\begin{align*}\omega(\sigma_{ij}) &\equiv 1 - \left(\frac{27J_3}{2\sigma_e^3}\right)^2 \\& = \omega(L) = 1 - \frac{\left( 9L - L^3\right)^2}{\left(L^2+3\right)^3}. \end{align*}
I want to use
\Aboxed to put the last part of the final equation in a box, ... |
We talked about LaTeX before.
This Full Guide on LaTeX in WordPress Explains the Js, Self Hosted & Rendered Solutions for the Users Seeking For Math, 2D or 3D Graphs. To self host the LaTeX original package, you need minimum a virtual server or cloud server. Depending on the amount of workload, that self hosted LaTeX o... |
This exercise is inspired by exercises 83 and 100 of Chapter 10 in Giancoli's book
A uniform disk ($R = 0.85 m$; $M =21.0 kg$) has a rope wrapped around it. You apply a constant force $F = 35 N$ to unwrap it (at the point of contact ground-disk) while walking 5.5 m. Ignore friction.
a) How much has the center of mass o... |
I'm trying to show:
$A\subset \mathbb{R}^n$ is open if and only if $A$ is the union of a collection of open balls accounting.
"$\Leftarrow$" Let $A=\bigcup_{i=1}^{\infty} \mathbb{B_{\varepsilon_i} (x_i)}$
If $y\in A $ then $y\in \mathbb{B}_{\varepsilon_i}(x_i)$ (for some $i$)
But note that $\mathbb{B}_{||x-y||}(y)\subs... |
As the name suggest, when two distinct points are directed from one place to another then it is done by a vector. It can also be seen as differences between velocity and speed. We get no clue about in which direction the object is moving. Therefore, we use this formula that will enable us to know in which direction the... |
Assume that we have a general one-period market model consisting of d+1 assets and N states.
Using a replicating portfolio $\phi$, determine $\Pi(0;X)$, the price of a European call option, with payoff $X$, on the asset $S_1^2$ with strike price $K = 1$ given that
$$S_0 =\begin{bmatrix} 2 \\ 3\\ 1 \end{bmatrix}, S_1 = ... |
Verify Simulations with the Method of Manufactured Solutions
How do we check if a simulation tool works correctly? One approach is the Method of Manufactured Solutions. The process involves assuming a solution, obtaining source terms and other auxiliary conditions consistent with the assumption, solving the problem wit... |
Netchaiev gave you a straight-forward answer. Here is a second way to get the amount of operations.
This way of thinking might be helpful for computer science oriented people, who see algorithms as computer code and not as maths.Also this way is useful, if you have written code to analyse.
The pseudo-code for this algo... |
I want to evaluate the integral $$\frac{1}{2\pi} \int_0^{2\pi} \frac{1}{1 - 2r \cos \theta + r^2} d\theta. $$ I thought first to substitute $\cos(\theta)$ for $\frac{1}{2} (e^{i\theta} + e^{-i\theta} ) $, reducing the problem to a complex integral over the unit circle, then using the $ z = re^{i\theta}$ to change the p... |
Here is a funny exercise $$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$ (If you prove it don't publish it here please). Do you have similar examples?
closed as primarily opinion-based by Najib Idrissi, user98602, user147263, apnorton, user2345215 Jan 31 '15 at 18:16
Many good questions generate some ... |
Exercise
If $(x_{\alpha})_{\alpha \in \Lambda}$ is a net, then $x$ is an accumulation point of the net if for every $A \in \mathcal F_x$, the set $\{\alpha \in \Lambda: x_{\alpha}\in A\}$ is cofinal in $\Lambda$. Prove that $x$ is an accumulation point of the net if and only if there is a subnet of $(x_{\alpha})_{\alph... |
$V(r)=\frac{1}{r}$ means for any two electrons at position $r_1$ and $r_2$, the electric potential is given by $\frac{1}{|r_1-r_2|}$
The Fourier transform of $\frac{1}{r}$ is $\frac{1}{q^2}$.
How can we interpret $V(q)=\frac{1}{q^2}$ physically?
This is what I'm thinking about:
(1)
$V(q)=\frac{1}{|q_1-q_2|^2}$ does not... |
INTRODUCTION
Ion traps are a robust and promising platform for quantum information processing and for the implementation of a quantum computer. However, major challenges exist in scaling these systems to the level required for full-scale quantum computing. I will describe work concerned with addressing two of these cha... |
Let $X_n(\Bbb{Z})$ be the simplicial complex whose vertex set is $\Bbb{Z}$ and such that the vertices $v_0,...,v_k$ span a $k$-simplex if and only if $|v_i-v_j| \le n$ for every $i,j$. Prove that $X_n(\Bbb{Z})$ is $n$-dimensional...
no kidding, my maths is foundations (basic logic but not pedantic), calc 1 which I'm pr... |
I think that gauge invariance of a Lagrangian is not a sufficient condition for the Ward identity to be valid. So why does the Ward identity happen to hold in Yang-Mills theory, and maybe in many other gauge theories which I'm not familiar with? Or why do physical states with equivalent polarizations have the following... |
15 minute read
In this summary, I assume you are familiar with Markov Decision Processes. In a Markov Decision Process (MDP), an agent interacts with the environment, by taking actions that induce a change in the state of the environment.
MDP: The robot knows the position
of the fuze bottle.
Image courtesy of the Perso... |
I'm reading Matsuzaka Topology and it says a bijective function $f$ is continuous iff it is open. Then how can I prove this? The book also says it is obvious but it's not obvious to me. I'm so confused, how can I prove this?
This isn't true: consider the function $f:[0,1) \rightarrow S^1$ given by $f(x) = e^{2\pi ix},$... |
Okay. After 12 days, your (hypothetic) homework is past due. Let's look at a couple of solutions.
"when you're lost, draw what you know."
Undergrads' wisdom. Let's see:
Without loss of generality, set up a framework placong the fence center in $O=(0,0)$ and the tether anchor in $A=(r_0,0)$. In this framework a point $(... |
A minimization problem with free boundary related to a cooperative system.
Speaker
Karen Yeressian
Date
Thursday, June 14, 2018
Time
15:00
Location
Conference room
Subject
Free Boundary Problems
Abstract
I will talk about the minimum problem for the functional
$$\int_{\Omega}\left( \vert \nabla u \vert^{2} + Q^{2}\chi_... |
Neural Stack API
Warning
This API is deprecated since v1.5.2, users are advised to use the new Tensorflow API.
Summary
In v1.4.2, the neural stack API was introduced. It defines some primitives for modular construction of neural networks. The main idioms will be computational layers & stacks.
The neural stack API exten... |
E C G Sudarshan
Articles written in Pramana – Journal of Physics
Volume 61 Issue 4 October 2003 pp 645-653 Reasearch Articles
We examine a number of recent proofs of the spin-statistics theorem. All, of course, get the target result of Bose-Einstein statistics for identical integral spin particles and Fermi-Dirac stati... |
A particle moves along the x-axis so that at time t its position is given by $x(t) = t^3-6t^2+9t+11$ during what time intervals is the particle moving to the left? so I know that we need the velocity for that and we can get that after taking the derivative but I don't know what to do after that the velocity would than ... |
Let $X_n(\Bbb{Z})$ be the simplicial complex whose vertex set is $\Bbb{Z}$ and such that the vertices $v_0,...,v_k$ span a $k$-simplex if and only if $|v_i-v_j| \le n$ for every $i,j$. Prove that $X_n(\Bbb{Z})$ is $n$-dimensional...
no kidding, my maths is foundations (basic logic but not pedantic), calc 1 which I'm pr... |
Here's a problem restated from Ross Starr's
General Equilibrium Theory.
Consider a two-commodity economy with an excess demand function $Z(p)=(Z_1(p),Z_2(p))$. The price space is $p \in P = \left \{ p | p \in \mathbb{R}^{2}, p \geq 0, p_1 + p_2 =1 \right \}$. Let $Z(p)$ be continuous, bounded and fulfill Walras' Law as... |
Regression is a very fundamental concept in statistics, machine learning and in Neural Network. Imagine plotting the correlation between rainfall frequency and agriculture production in high school. Increase in rainfall generally increases agriculture production. Fitting a line to those points enables us to predict the... |
Suppose $X$ and $Y$ are convex sets in $\mathbb{R}^d$ such that the origin is in each of their interiors. Then the dual of $X$, $X'$ is defined as the set of linear functionals $\alpha$ such that $\alpha(x) \leq 1$ for all $x \in X$. Is there a way to derive $(X+Y)'$ form $X'$ and $Y'$? I was thinking $(X+Y)' = X' \cap... |
$\forall n\in\mathbb{N}^*, u_n = \sqrt{n+u_{n-1}}$ with $u_1 = 1$.
I've already shown that $ u_n \sim \sqrt{n}$ and that $\underset{n\rightarrow \infty}{\lim} u_n - \sqrt{n} = \dfrac{1}{2}$
How to show that $n\rightarrow \infty, u_n - \sqrt{n} - \dfrac{1}{2} \sim \dfrac{1}{8\sqrt{n}}$ ? (Or it might not be this, maybe ... |
Visualization for 2D Axisymmetric Electromagnetics Models
Today we’ll look at how to make 3D plots of vector fields that are computed using the 2D axisymmetric formulation found in the
Electromagnetic Waves, Frequency Domain interface within the RF and Wave Optics modules.
Creating 3D Plots from 2D Axisymmetric Solutio... |
Yeah, this software cannot be too easy to install, my installer is very professional looking, currently not tied into that code, but directs the user how to search for their MikTeX and or install it and does a test LaTeX rendering
Some body like Zeta (on codereview) might be able to help a lot... I'm not sure if he doe... |
Short introduction
I created a function that interpolates the IES luminious intensities (candelas) using Hermite interpolation, so in my code all light sources have $I(\theta, \phi)$ function - but for simplicity lets assume we are dealing with a hemispherical light source with constant intensity $I = 100cd$ and we are... |
I'm interested in testing independence of two groups (e.g. case and control) in a $2\times 2$ table: i.e. $H_0: \theta=1$ against the
two-sided alternative $H_1:\theta\neq 1$, where $\theta$ is the odds-ratio. Suppose that the margins of the table are fixed, then the random variable for the number of hits among cases i... |
A particle moves along the x-axis so that at time t its position is given by $x(t) = t^3-6t^2+9t+11$ during what time intervals is the particle moving to the left? so I know that we need the velocity for that and we can get that after taking the derivative but I don't know what to do after that the velocity would than ... |
On Math SE, I've seen several questions which relate to the following. By abusing the laws of exponents for rational exponents, one can come up with any number of apparent paradoxes, in which a number seems to be shown as equal to its opposite (negative). Possibly the most concise example:
$-1 = (-1)^1 = (-1)^\frac{2}{... |
My question is about the rigorous proof of the fact that the heat goes from hot body to the cold one. There is a part of the proof I don't understand.
We consider 2 systems : one is at temperature $T_1$ and the other at $T_2$. The ensemble of those two systems is isolated. I assume they only exchange heat (no work).
I ... |
August 14th, 2014, 03:48 AM
# 1
Senior Member
Joined: Aug 2014
From: India
Posts: 470
Thanks: 1
how to find max shear stress at a given point?
The state of plane-stress at a point is given by σx = -200 MPa, σy = 100 MPa, σxy = 100 MPa
The Maximum shear stress (in MPa) is:
A) 111.8
B) 150.1
C) 180.3
D) 223.6
Explain Pro... |
Alright, I have this group $\langle x_i, i\in\mathbb{Z}\mid x_i^2=x_{i-1}x_{i+1}\rangle$ and I'm trying to determine whether $x_ix_j=x_jx_i$ or not. I'm unsure there is enough information to decide this, to be honest.
Nah, I have a pretty garbage question. Let me spell it out.
I have a fiber bundle $p : E \to M$ where ... |
Meshing Considerations for Linear Static Problems
In this blog entry, we introduce meshing considerations for linear static finite element problems. This is the first in a series of postings on meshing techniques that is meant to provide guidance on how to approach the meshing of your finite element model with confiden... |
So, I started writing my second raytracer - this time focusing on photometric rendering that uses IES lights and standard photometric units. I've got the basic raytracing up and running (using pure geometry), I've read any article I could found on radio- and photometry (I think I developed an intuition for it) but the ... |
Exercise :
Find a maximum likelihood estimator of $\theta$ for : $f(x) = \theta x^{-2}, \; \; 0< \theta \leq x < \infty$.
Attempt :
$$L(x;\theta) = \prod_{i=1}^n \theta x^{-2} \mathbb{I}_{[\theta, + \infty)}(x_i) = \theta^n \mathbb{I}_{[\theta, + \infty)}(\min x_i)$$
How should one proceed from now on to find a MLE ?
I... |
Hi! I am trying to understand the problem above, and was wondering if someone could help me with the last question. I think I am fine with all other questions.
Here is my attempt:
(i) The fiscal multiplier is $\frac{1}{1-c} \Delta G$, as it is the infinite sum $ \Delta G + c\Delta G + c^2\Delta G + ... $.
Depends on c ... |
Why in the following proof $$\sum_nA_n(X_n,X_m)=A_m(X_m,X_m)$$ ?
The author says it's because orthogonality but orthogonality means $(f,g)=\int_a^bfgdx=0$. So how come orthogonality helps to prove it ?
Could someone explain this please?
Thanks
Mathematics Stack Exchange is a question and answer site for people studying... |
To perform the gluing, we need:
two metric spaces $X$ and $Y$ a set $A\subset X$, this is the part of $X$ covered in glue an isometric embedding $f:A\to Y$, which is a way to put the glue-covered part of $X$ over $Y$.
After we firmly press the spaces together and let them sit for a while, a point $x\in A$ becomes ident... |
Suppose you have two equtaions: $$2xy + y^2 = 0$$ $$x^2 + 2xy + 1 = 0$$ Subtracting the second from the first yields $y^2 - x^2 - 1 = 0$. Isolating y, we discover that $y = \pm\sqrt{x^2 + 1}$. However, by inspection we can wee that the $2xy$ term in both equations must be negative, which means that a single value of x ... |
I am working through Susan Lea's
Mathematics for Physicists however I am stuck on problem 31)b):
31) Prove
b)
$\int_{S}(\hat{n}\times\vec{\nabla})\times\vec{u}\hspace{0.5mm}dA = \unicode{x222E}_{C}d\vec{l}\times\vec{u} $
I have looked at the solution and understand that first a differential rectangle is constructed in ... |
Q4. Suppose that $P$ is an $n \times n$ matrix such that $P^{2} = P$. Show that $\mathbb{R}^{n}$ is the direct sum of the range $R(P)$ and the nullspace $N(P)$ of $P$. Show also that $P$ represents the projection from $\mathbb{R}^{n}$ onto $R(P)$.
A4. Again – not sure how to start. Its idemptotent... so???
Q5. Prove th... |
Let $(X,\mathcal{M},\mu)$ be a measure space. Let $A_1, A_2, \ldots \in \mathcal{M}$.
Then, I want to show that: $$\mu\left(\bigcup_N \bigcap_{n=N}^{\infty} A_n \right) \leq \lim \inf \mu(A_n)$$
There is a solution in lecture notes:
Let $B_N = \bigcap_{n=N}^{\infty} A_n$. $B_N$ form an increasing sequence of elements, ... |
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Measurement of electrons from beauty hadron decays in pp collisions at root √s=7 TeV
(Elsevier, 2013-04-10)
The production cross section of electrons from semileptonic decays of beauty hadrons was measured at mid-rapidity (|y| < 0.8) in the transverse momentum range 1 < pT <8 GeV/c w... |
Definition:Ordering on Natural Numbers Contents Informal Definition
Let $\N$ denote the natural numbers.
The ordering on $\N$ is the relation $\le$ everyone is familiar with.
For example, we use it when we say:
James has $6$ apples, which is more than Mary, who has $4$.
which can be symbolised as:
$6 \ge 4$
The same ho... |
How can water be absorbed into a paper towel such that it continues to be absorbed up into the material against the force of gravity? What property of paper towels/water causes this? I can understand that cohesive forces cause water molecules to follow other water molecules, but why do the molecules begin this climb in... |
An unknown function $y=f(x)$ is infinitely differentiable everywhere on $\mathbb{R}$.
Known the values of every points $ (x,y) $ where $0<x<1 $ ,
Could you please show whether there exist a unique solution of $y=f(x)$ ?
If not, could you please show that there exist infinite number of solution of $y=f(x)$ ?
My approach... |
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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 ... |
PCTeX Talk Discussions on TeX, LaTeX, fonts, and typesetting
Author Message murray Joined: 07 Feb 2006 Posts: 47 Location: Amherst, MA, USA
Posted: Wed Mar 15, 2006 10:53 pm Post subject: Y&Y TeX vs. MiKTeX with MathTimePro2 fonts I used updated fonts just prior to posting of 0.98.
Initially I had different page breaks... |
2017-04-19 08:09
Flavour anomalies after the $R_{K^*}$ measurement / D'Amico, Guido (CERN) ; Nardecchia, Marco (CERN) ; Panci, Paolo (CERN) ; Sannino, Francesco (CERN ; Southern Denmark U., CP3-Origins ; U. Southern Denmark, Odense, DIAS) ; Strumia, Alessandro (CERN ; Pisa U. ; INFN, Pisa) ; Torre, Riccardo (EPFL, Laus... |
The only time we need to know where a force acts is when we are calculating a torque. For contact forces, it is clear that the force acts at the point of contact. But for a force like gravity, that acts at a distance, it is less clear.
In reality, a rigid object is made up of many particles, and there is a small gravit... |
Let $G$ be a finite group and $H \le G$ with $g \in G$ such that all elements of the coset $Hg$ are conjugate in $G$. Let $\chi$ be a $\mathbb C$-character of $G$ such that $[\chi_H, 1_H] = 0$. Show that $\chi(g) = 0$.
This is exercise 2.1 (b) from M. Isaacs, Character Theory of Finite Groups; it is given the hint to l... |
I was trying to find the residue of the function
$$f(z) = \frac{z^2 + \sin z}{\cos z - 1}.$$
Here is the my attempt:
The given function has a pole of order two at $z = 2n\pi$. So, we use the following formula to calculate residue of a function when it has a pole of order
m at $z=z_0$.
$$\mathrm{Res}(f(z))_{z=z_0}=\frac... |
In the literature on social welfare functionals, the only example I've seen of a functional which meets all of Arrow's conditions–––or at least utility analogues of Arrow's conditions–––plus invariance regarding ordinal level comparability is Rawls' maximin. E.g. Sen in
On Weights and Measures (1977, p. 1544) cites max... |
Edit: As noted by @Branimir, I can remove the conjugations from the earlier draft. $T^*$ is the Banach-space adjoint conjugated by a conjugate-linear isomorphism, so constants end up being preserved.
As this is a homework problem, I will attempt to give a hint rather than the complete solution (which is elegant). I her... |
This module demonstrates how to convert the lower triangular covariance table from the Excel Analysis ToolPak to a full covariance matrix for use with the
MMULT function.
First, a brief review of square matrices.
A
matrix is a rectangular array of numbers of the form $$A=\begin{bmatrix} a_{11} & \cdots & a_{1n} \\ \vdo... |
When discussing ordinary correlations we looked at tests for the null hypothesis that the ordinary correlation is equal to zero, against the alternative that it is not equal to zero. If that null hypothesis is rejected, then we look at confidence intervals for the ordinary correlation. Similar objectives can be conside... |
Disclaimer
My following answer is the "traditional" explanation of Hund's first rule, which is based on a smaller value of $V_\mathrm{ee}$ (electron-electron repulsions) in the triplet state arising from Fermi holes. According to Levine's
Quantum Chemistry 7th ed.:
This traditional explanation turns out to be wrong in ... |
Let $X_n(\Bbb{Z})$ be the simplicial complex whose vertex set is $\Bbb{Z}$ and such that the vertices $v_0,...,v_k$ span a $k$-simplex if and only if $|v_i-v_j| \le n$ for every $i,j$. Prove that $X_n(\Bbb{Z})$ is $n$-dimensional...
no kidding, my maths is foundations (basic logic but not pedantic), calc 1 which I'm pr... |
As far as I understand it, at least in scalar QFT, the canonical variables are the field operator $\hat{\phi}(x)$ and its conjugate momentum $\hat{\pi}_{\phi}(x)=\frac{\partial\mathcal{L}}{\partial\dot{\hat{\phi}}}$ (where $x=(t,\mathbf{x})$ and $c=\hbar =1$).
Someone has recently told me that it is usually assumed tha... |
Mooring¶
A mooring system can be composed of several components, including :
mooring lines, mooring buoys, clumpweights, fairlead, anchor, etc…
Its general purpose is to restrict the motions of a floating structure (ship, platform, etc.) in a specified area. Mooring systems can be more complex than a single line connec... |
Market
1. INTRODUCTION
Market is arguably the most known and widely used factor - think about values of broad-based equity indices such as the S&P 500 reported in news, index-tracking ETFs, and the whole universe of derivatives with indices' values as underlying assets. The market is also the most empirically studied f... |
Take the Hamiltonian you write at end of your question as the example(where your statement about the eigenvector is not right), the procedure to obtain the eigenvalue and eigenvectors of other Hamiltonian is nearly the same.
$$H=\sum_{ij}A_{ij}a_i^\dagger a_j$$
Writing in matrix form you have(suppose it is a $n\times n... |
Show that the first excited state of 8-Be nucleus fits the experimental value of the rotational band $E(2^+) = 92 keV$ (this is the first excited state, which is 92 keV above the ground state). To do so model 8-Be to be made of 2 alpha particles $4.5 fm$ apart rotating about their center of mass.
The method I used is:
... |
Let $G$ be a Lie group and $L(G)$ it's Lie algebra. We know that every left-invariant vector field $X$ in $G$ is complete, and so one can consider the integral curve defined for all $t\in \mathbb{R}$ by $t\mapsto \exp(tX)$. If $M$ is a smooth manifold on which $G$ acts on the left, then if $a\in M$ we can consider the ... |
Help:Formatting
You can format your text using the formatting toolbar or wiki markup. Wiki markup can be thought of as a simplified version of html and it consists of normal characters like asterisks, single quotes or equation marks which have a special function in the wiki. For example, to format a word in
italic, you... |
How can I show that every monoid $M$ admits a surjection from a free monoid $F(X) \rightarrow M$ ?
The set $X$ is called an alphabet. You can take $M$ itself as alphabet and then the morphism $\mu : F(M)\rightarrow M$ is just the multiplication.
To be more specific, the structure of monoid is the data of a triplet $(M,... |
Trying to make my AI hit people. So I need a formula to know the time until my projectile will hit the target and also the direction the projectile should be shot at.
Here is an example scenario.
The projectile shoots from the origin $(0,0)$ Both projectile and target will move in a straight line. Speed of projectile: ... |
I have to calculate this : $$ \lim_{x\to 0}\frac{2-x}{x^3}e^{(x-1)/x^2} $$ Can somebody help me?
Hint: It may be fruitful to substitute $\alpha = 1/x$, in which case you obtain the limit
$$ \lim_{ \alpha \rightarrow \infty} \left(2 - \frac{1}{\alpha} \right) \alpha^3 e^{\alpha - \alpha^2} $$
I should note that, here, I... |
Prandtl number, abbreviated as Pr, is a dimensionless number used in fluid dynamics is used to calculate force by the ratio of momentum diffusivity (kinematic viscosity) and thermal diffusivities. This number helps characterize the relative thickness of the velocity boundary layer to the thermal boundary layer.
\(\larg... |
There are a lot of websites and forums, which explain that there is a bijection between $\mathbb{R}$ and $\mathbb{R}^2$, and even give some bijections. (By the way: Can you generalize it? since it works with the natural Numbers and the real numbers, does there exist a bijection between any infinite set $X$ and $X^2$) O... |
Background: Bombieri and Lagarias showed that a function $f$ with roots $\rho=x+iy$ satisfies has all its roots lying on $x=\frac12$ if and only if
$$\lambda_n :=\sum_\rho 1-\left(1-\frac{1}{\rho}\right)^n$$
is nonnegative for all $n$.
When $f$ is the Riemann zeta restricted to the critical strip, this gives an equival... |
For a particle in an infinite square-well potential in an energy eigenstate, the probability distribution relating to outcomes of position measurements vanishes outside the square well and takes a sinusoidal form inside the well, approaching zero at the edges of the well.
Suppose a particle trapped in an infinite squar... |
Let $X_n(\Bbb{Z})$ be the simplicial complex whose vertex set is $\Bbb{Z}$ and such that the vertices $v_0,...,v_k$ span a $k$-simplex if and only if $|v_i-v_j| \le n$ for every $i,j$. Prove that $X_n(\Bbb{Z})$ is $n$-dimensional...
no kidding, my maths is foundations (basic logic but not pedantic), calc 1 which I'm pr... |
How do these models work? I know they have to take in various data points - what are these, and how does the model use it? Also, how do they come up with a forecast or a map of what will happen in the future?
All numerical atmospheric models are built around calculations derived from primitive equations that describe a... |
In a 2-person economy, the owner of a firm pays 20% of the total output to the worker.
What is the Gini coefficient in this economy?
The answer is supposed to be 3/5 but I end up getting 3/10 instead.
Economics Stack Exchange is a question and answer site for those who study, teach, research and apply economics and eco... |
1. Discrete returns
Let \(P_t\) be the asset price at time \(t\) and \(P_{t-1}\) be the price in the prior period – day, week, month, … For daily returns on stocks, \(P_{t-1}\) is the closing price of the stock on the previous trading day.
Example calculations are provided in figure 1 – Excel Web App #1 – Sheet1, where... |
It means the theory is expressed/discussed in the language of tensor fields, where the tensors are quantities that transform between mutually moving inertial frames according to
the Lorentz transformation. All differential equations are expressed as relations between tensor fields and their derivatives. For example, th... |
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