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Quadratic Formula (deterministic---no guess and check about it)
The QF yields that $-{\frac13}$ and $5$ are roots. So $$3x^2-14x-5=c\left(x+\frac13\right)(x-5)$$ Comparing leading coefficients, $c$ must be $3$: $$\begin{align}3x^2-14x-5&=3\left(x+\frac13\right)(x-5)\\&=(3x+1)(x-5)\end{align}$$
Use Parabola Vertex Form ... |
I have recently read that the dimensional regularization scheme is "special" because power law divergences are absent. It was argued that power law divergences were unphysical and that there was no fine-tuning problem. I was immediately suspicious.
Let us take $\lambda\phi^4$ theory. For the renormalized mass $m$ (not ... |
The Annals of Probability Ann. Probab. Volume 26, Number 4 (1998), 1703-1726. The standard additive coalescent Abstract
Regard an element of the set $$\Delta := {(x_1, x_2, \dots): x_1 \geq x_2 \geq \dots \geq 0, \Sigma_i x_i = 1}$$ as a fragmentation of unit mass into clusters of masses $x_i$. The additive coalescent ... |
56 6 Homework Statement A boy decides to hang its bag from a tree branch, with the purpose of raising it. The boy walks with constant velocity in the x axis, and the bag moves in the y axis only. Taking into account that the lenght of the rope is constant and that we know the height h of the branch respect to the floor... |
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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... |
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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... |
16.1. List of Main Symbols¶
The main symbols used in this book are listed below.
16.1.1. Numbers¶
Symbol Type \(x\) Scalar \(\mathbf{x}\) Vector \(\mathbf{X}\) Matrix \(\mathsf{X}\) Tensor 16.1.2. Sets¶
Symbol Type \(\mathcal{X}\) Set \(\mathbb{R}\) Real numbers \(\mathbb{R}^n\) Vectors of real numbers in \(n\) dimensi... |
For example, consider a \(11 \times 11\) grid, and choose from the following building blocks:
Five different polyominos
Fail to cover complete grid
\[x_{i,j,k} = \begin{cases} 1 & \text{if we place polyomino $k$ at location $(i,j)$}\\ 0 & \text{otherwise} \end{cases}\]
I used as rule that the left-upper corner of each ... |
Let $L =\{ <M> | $ the amount of words $w\in\Sigma^*$ that $M$ does
not halt on is finite $\}$.
I would like to prove that $L\notin RE$.
I can show that $\overline{L}\notin RE $ that is $L\notin CoRE$.
I do this with showing a reduction: $\overline{L_{acc}} \leq \overline{L} $ and since $\overline{L_{acc}}\notin RE$ th... |
After I posted this question, a couple of months ago, and got from MO-users several good hints, I think i'm ready, after some study, to ask another related question (or rather, to focus on the main point of my previous question, after I got aware of all the necessary background).
First let me describe the setting:
WEAK... |
Quantum natural gradient¶
This example demonstrates the quantum natural gradient optimization technique for variational quantum circuits, originally proposed in Stokes et al. (2019).
Background¶
The most successful class of quantum algorithms for use on near-term noisy quantum hardwareis the so-called variational quant... |
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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 ... |
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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 ... |
2019-09-04 12:06
Soft QCD and Central Exclusive Production at LHCb / Kucharczyk, Marcin (Polish Academy of Sciences (PL)) The LHCb detector, owing to its unique acceptance coverage $(2 < \eta < 5)$ and a precise track and vertex reconstruction, is a universal tool allowing the study of various aspects of electroweak an... |
I think the main issue is that you are being confused by the Dirac notation, which works fine when it does but can be occasionally confusing when you are worried about that kind of thing. I'll try and give an accessible explanation that also addresses the issues raised by Wouter.
Take a Hilbert space $\mathcal{H}$, and... |
I need to use
\ldbrack and
\rdbrack symbols from the package
\mathabx, but when I use the
\mathabx with the
\amssymb package (previously installed) it is causing me an issue with symbols:
\subseteq,
\cap,
\cup, etc
\documentclass{book} \usepackage{amssymb} \usepackage{mathabx}\begin{document}This a example, $$A\subsete... |
A
convex optimization problem is any problem of minimizing a convex function over a convex set.An optimization problem is quasi-convexif its feasible set is convex and its objective function $f_0$ is quasi-convex.Given convex functions $f_i(x), i=0,\dots,m$, the standard form of a convex optimization problem is:$$\begi... |
So, matrix A * its inverse gives you the identity matrix correct?
Also, if you have AB=BA, what does that tell you about the matrices? Is this only true when B is the inverse of A?
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only... |
A co-$A_n$ space is a based space $Y$ equipped with a co-action by the Stasheff associahedron operad $K_\bullet$. This means that $Y$ is comes with certain maps $c_n: Y \times K_n \to Y^{\vee n}$, $n = 2,3,\dots$ that are inductively described (the definition of $c_n$ uses $c_{n-1}$ as input; the map $c_2$ is a co-$H$ ... |
The Diesel cycle is the thermodynamic cycle which approximates the pressure and volume of the combustion chamber in a Diesel engine. In the Diesel cycle, the working medium in the combustion chamber is
compressed adiabatically from V 1, p 1to V 2< V 1, p 2> p 1, expanded isobarically at p 2from V 2to V 3> V 2, expanded... |
This question is on the same topic as this one but more complex. So, let's consider the cell with $\ce{Zn}$ and $\ce{Cu}$ electrodes inside the $\ce{NaCl}$ solution. On the $\ce{Zn}$ electrode we have $$\ce{Zn + 2Cl^- -> ZnCl2 + 2e^-},$$ and on the copper one $$\ce{H3+O + e^- -> 1/2 H2 + H2O}.$$
For both reactions we c... |
Value of windfall at \(t=\) 0 Value of windfall over time Value of steady saving over time Savings per time Rate of return on invested capital Time when the windfall and consistent savings are equal
Usually, we do not have a choice in the matter of windfalls, whether they come from wealthy relatives passing, insurance ... |
This question is about homomorphism and it's definition, which still feels kind of weird and unreliable to me.
To be more obvious, let's take graph $G = (Vertices(G), Edges(G))$. Then homomorphism of $G$ to somewhat graph $G'$
consists of two independent functions: $vrt : Vertices(G) \mapsto Vertices(G')$, $edg : Edges... |
I like using
amsthm for my theorems, etc, but to highlight their importance, I like exercises to be in
tcolorboxes. I want the exercises in the boxes to share the same counter as my theorems.
I managed to create an
exercise environment which works as desired, harmoniously with
cleveref:
But then, it took me a while to ... |
Current browse context:
stat.ML
Change to browse by: References & Citations Bookmark(what is this?) Computer Science > Machine Learning Title: Finite Precision Stochastic Optimization -- Accounting for the Bias
(Submitted on 22 Aug 2019 (v1), last revised 26 Aug 2019 (this version, v2))
Abstract: We consider first orde... |
\[
\int_{-\infty}^{\infty} \mathrm{d}x \exp\left(-\frac{(x-\mu)^4}{2}\right) \]
was given. Using RcppNumerical is straight forward. One defines a class that extends
Numer::Func for the function and an interface function that calls
Numer::integrate on it:
// [[Rcpp::depends(RcppEigen)]]// [[Rcpp::depends(RcppNumerical)]... |
in this case, you might consider a different approach, since the numerator isquite simple:
\documentclass[12pt]{article}
\usepackage{amsmath}
\begin{document}
This is the reaction equation:
\begin{equation}
\label{eq: rate}
r = P_{A_b} \bigg/ \! \biggl( \frac{1}{\frac{k_G a_{GL}}{\rho_b \epsilon_s}}
+ \frac{H}{\frac{k_... |
Source of the problem
2012 AMC(American Mathematical Contest) 8 Problem 24
Do you really need a hint? Try it first!
Hint figure :
Assume that the radius of the circle is \( r \) unit and proceed to compare .
Note that side length of the square is \( 2r \) unit .
Clearly , $$ area(red \ square ) – area(circle ) = area(s... |
Heart Rate Variability Logger is an app I developed to record, plot and export time and frequency domain Heart Rate Variability (HRV) features. The app is available for
iPhone and Android, however features on Android are limited. This post is a sort of user guide where I cover in more detail the different features and ... |
Background:
In Subhash Khot's original UGC paper (PDF), he proves the UG-hardness of deciding whether a given CSP instance with constraints all of the form Not-all-equal(a, b, c) over a ternary alphabet admits an assignment satisfying 1-$\epsilon$ of the constraints or whether there exist no assignments satisying $\fra... |
I'm whirling a ball tying it by a rope. Then at any instant the centrifugal force (outward force) equalizes the centripetal force (inward force) . Then both should cancel each other and there should be only tension of the rope to balance the weight of the ball. Then why do we count the centripetal force as resultant fo... |
By Murray Bourne, 28 Nov 2010
This is fun. And the best part - it's accurate!
If you don't get it, see Graphs of tan, cot, sec and csc.
Image source: LukeSurl.com
He's got some other math-flavored comics (search "math").
See the 1 Comment below.
Posted in Mathematics category - 28 Nov 2010 [Permalink]
Tags: Graphs
very... |
To motivate this question, I'm going to try and explain some background notions. This won't be absolutely necessary for experts, but I want to be vaguely honest about where this question comes from. I also want to say that this is the type of question that I would ask at a coffee break during a conference about this su... |
Some mathematical elements change their style depending on the context, whether they are in line with the text or in an equation-type environment. This article explains how to manually adjust the display style.
Let's see an example
Depending on the value of $x$ the equation \( f(x) = \sum_{i=0}^{n} \frac{a_i}{1+x} \) m... |
[−][src]Module curve25519_dalek:: backend:: vector:: avx2 only.
feature="simd_backend" and (target feature
avx2 or target feature
avx512ifma) and target feature
avx2 and non-target feature
avx512ifma
An AVX2 implementation of the vectorized point operation strategy.
Our strategy is to implement 4-wide multiplication an... |
The problem of counting such "imperfect" matchings in bipartite graphs is #P-complete.This has been proved by Les Valiant himself, on page 415 of the paperLeslie G. ValiantThe Complexity of Enumeration and Reliability ProblemsSIAM J. Comput., 8(3), 410–421
It's NP-complete by a reduction from cliques in graphs. Given a... |
When we are comparing two models against some data, will we obtain the same (set of) posterior odds for the models both when we use the Bayes' factor and when we use the discriminant rule?
If not, which one is considered more "accurate" and why is the other being used?
I have found that the Bayes' factor is described a... |
Current browse context:
cond-mat.mes-hall
Change to browse by: References & Citations Bookmark(what is this?) Condensed Matter > Mesoscale and Nanoscale Physics Title: Theory of the strongly nonlinear electrodynamic response of graphene: A hot electron model
(Submitted on 13 Aug 2019)
Abstract: An electrodynamic respon... |
As dkaeae indicated in his comment, a right-moving or staying Turing machine (TM) is essentially a finite deterministic automaton (FDA). Here's a proof.
Let $M$ be such a machine, whose transition rules is in the form of $\delta(q, \gamma)=(t, \beta, d)$ where $q$ is the current state, $\gamma$ is the contents of the c... |
Clearly, if $g$ is a deterministic function then for two random variables $X, Y$:
$X \rightarrow Y \rightarrow g(Y)$ i.e. they form a Markov Chain.
If $g$ is a one-to-one mapping we can further state: $X \rightarrow g(Y) \rightarrow Y$ as $g(Y)$ is a sufficient statistics.
I am wondering about the converse of the above... |
I am looking for an efficient method to compute $$\sum_{i=1}^\left|B\right|\left|Ax_i-b_i\right|^2\rightarrow min$$ under the condition $$\forall i, x_i\ge 0,$$ where $A$ is an n-by-m matrix and $B$ is a set of n-dimensional vectors. Only $B$, $n$ and $m$ are given. In other words the goal is to find $A$ so that the su... |
We define the following problem as:
Let $M$ be a TM with alphabet $\Gamma$, with $\{a,b,$ #$\} \subset \Gamma$.
We define, for every natural number $n$ the graph $G_{M,n}$ by:
$V_{M,n} = \{a,b\}^n$, the vertices
$E_{M,n} = \{(x,y) : x,y \in V_{M,n} $ and $M$ accepts the string $x$#$y$ with a run that requires at most $... |
Note: There is a short summary at the bottom.
This is actually also described in Nielsen&Chuang: You don't learn about general measurements, because they are
completely equivalent to projective measurements + unitary time evolution + ancillary systems, all of which is described in your usual QM formalism. The Measureme... |
A system can be in any one of N states. Using the method of undetermined multipliers to show that for the maximum entropy, $S = -k \sum_i p_i \ln p_i$ where $p_i = 1/N$, $$S = k \ln N\,,$$
the solution is as follows:
Using an undetermined multiplier $\alpha$, write $$f = -k \sum_i(p_i \ln p_i - \alpha p_i)$$ $$\frac{\p... |
4.6. Dropout¶
Just now, we introduced the classical approach of regularizing statistical models by penalyzing the \(\ell_2\) norm of the weights. In probabilistic terms, we could justify this technique by arguing that we have assumed a prior belief that weights take values from a Gaussian distribution with mean \(0\). ... |
argmax(w) sum(sign(Aw) == sign(b))
This is a strange objective. Basically: find \(w\), with \(v=Aw\) such that we maximize the number of \(v_i\) having the same sign as \(b_i\). I have never seen such an objective.
As \(A\) and \(b\) are constants, we can precompute \[ \beta_i = \mathrm{sign}(b_i)\] This simplifies the... |
Of the top of my head...
Definitely not this salt. There will be little free ammonia from the solvation of the $\ce{NH4^+}$ cation. The $\ce{Cl^-}$ is a spectator anion.
The $\ce{Na^+}$ is a spectator cation. The $\ce{CO3^{2-}}$ will be mostly $\ce{CO3^{2-}}$ with a tiny bit of $\ce{HCO3^-}$. So the ratio of $\frac{\ce... |
Note to the following well known theorem:
Theorem (1): If $\kappa$ be a "measurable" cardinal and $\mathcal{F}$ be a "non-principal $\kappa$-complete normal" ultrafilter on it then: $\langle V_{\kappa +1},\in \rangle \cong \prod_{\mathcal{F}}\lbrace \langle V_{\alpha +1}, \in \rangle~|~\alpha\in \kappa \rbrace$ Proof: ... |
Please tell me where I've gone wrong (if I did in fact make a mistake). I'm pricing a long forward on a stock. The usual setup applies:
This has payoff $S(T) - K$ at time $T$. We are at $t$ now. $S(T) = S(t)e^{(r-\frac12 \sigma^2)(T-t)+\sigma(W(T)-W(t))}$. $W(t)$ is a Wiener process. $K \in \mathbb{R}_+$. $Q$ is the ri... |
Current browse context:
astro-ph
Change to browse by: Bookmark(what is this?) Astrophysics > Cosmology and Nongalactic Astrophysics Title: Observational constraints to a unified cosmological model
(Submitted on 29 Nov 2014 (v1), last revised 27 Aug 2015 (this version, v2))
Abstract: We propose a phenomenological unifie... |
There are all sorts of things to consider here and I doubt there can be a definitive answer.
First: how many neutron stars are there - or more pertinently, what is the density in the solar neighbourhood.
There are about 1000 stars within 15pc of the Sun down to about $0.2M_{\odot}$. Most of these are main sequence star... |
Projections for HH measurements in the bbZZ(4l) final state with the CMS experiment at the HL-LHC
Pre-published on: 2019 August 18
Published on: —
Abstract
Prospects for the study of Higgs boson pair (HH) production in the $\mathrm{HH \to b\bar{b}4l}$ (l = e, $\mu$) channel are studied in the context of the High Lumino... |
Suppose we have a triangular mesh of a two dimensional shape $\Omega$, and on this mesh we define a P1 finite element structure. I know that given $u,v$ by their values at the vertices of the triangles, we can compute $\int_\Omega \nabla u \cdot \nabla v$ using the stiffness matrix $A = (\int_\Omega \nabla \varphi_i\cd... |
$\DeclareMathOperator{\diag}{diag}$ Consider the generalized eigenproblem $A\mathbf{x}=\lambda B\mathbf{x}$. When solving PDEs numerically (specifically, I am interested on finding the Dirichlet eigenmodes of the 2D laplacian in orthogonal coordinates, i.e. $\nabla^2 u=\lambda u$) this problem takes the form
$$ A_1X + ... |
In addition to the different situations we have seen above, which involve modular functions and forms, there are also many ways to create new modular forms from given ones.
Vector space or graded ring structureThe most trivial way to construct a modular form from given ones is to use the vector space structure: if $f$ ... |
65 4 Homework Statement In a double slit experiment let d=5.00 D=30.0λ. Estimate the ratio of the intensity of the third order maximum with that of the zero-order maximum. Homework Equations interference diffraction Homework Statement:In a double slit experiment let d=5.00 D=30.0λ. Estimate the ratio of the intensity o... |
I'm going to try to design an algorithm to find all the rational roots of a polynomial equation in range [a, b]. Can someone please tell me which algorithm currently solves the problem with lowest worst-case complexity? This algorithm will be for a general purpose computer(Turing Machine).
The paper Computing Real Root... |
Why is there 10k resistor at DAC output? Is it used as current limiter? Why is current limiter needed if op-amp sinks very little current at its input?
Why is there a capacitor at the same + input?
The 10k and 470p combination form a low-pass filter. Cut-off frequency is given by \$ f = \frac {1}{2 \pi R C} = \frac {1}... |
If you reversed the directions of the two spins, the spin part of the wavefunction is a
singlet and it is anti-symmetric.
The total wavefunction (spatial * spin) needs to be overall antisymmetric for fermions, which means in your case the spatial part of the wavefunction is symmetric - there is the highest probability ... |
Monte Carlo is most useful when you lack analytic tractability or when you have a highly multidimensional problem.
For example, even using simple lognormal and poisson models, there exist path-dependent payoffs or multi-asset computations such that no analytic solution exists and such that any PDE finite difference sol... |
In a previous question, link, I asked about how I could most effectively do a Fourier Transform of a radial function given at certain values and which we knew the asymptotical behaviour of. The Fourier transform reading$$\frac{4\pi}{q}\int^{\infty}_0 dr\, r \sin(qr) f(r).$$I tried several ways and ended up choosing FFT... |
I am new to Stack Exchange, so please excuse the few mistakes that might have happened.
I like the large square roots, parantheses and braces, the wide accents, underbraces, etc that one gets with
mtpro2. Also, the TeX Gyre Pagella Math font is very much to my liking, especially since it has optical sizes inbuilt.
Howe... |
I believe that you will want to process each channel separately over all of the images. Otherwise a mean and variance will have very little meaning. And if you convert to grayscale before training, then the network can only retrieve grayscale images, which doesn't seem to match with what the paper describes.
I've troub... |
Here's how I think of your result:
Let's look for integers $n$, such that the beginning of the decimal expansion of $n^{\log n}$ agrees with that of $\pi$ (up to some point). Using a for loop, I found the following approximations for $n<100,000$:
$$ \pi \approx \frac{11^{\log 11}}{10^2},\frac{53599^{\log{53599}}}{10^{5... |
Using currently available fuels and technology and an unlimited budget. How big would a rocket without stages have to be to make it to a stable orbit? and how big would a rocket without staged have to be to escape Earth's gravity? "edited" - or for the easier question how big would the rocket have to be if it was massl... |
Jan A. Sanders Vrije Universiteit Amsterdam
Under construction.
Contents introduction
The plan is to give an introduction to Lie algebra cohomology that can be followed on different levels. The development of the cohomological theory will require nothing beyond the basic rules for Lie algebras and representations. To m... |
This question pops up in my head every few weeks and I'm struggling to really understand the concept / theory behind it.
We all know there are different kind of measures of dependencies out there. From Pearson's $\rho$ to Kendall's $\tau$ to more sophisticated tail dependency measures from copula:
$$\lim_{q\downarrow0}... |
Preliminary note: You have not stated the support of your density, so I am going to assume it is $x \geqslant \mu$, since this gives a valid density function. Under this interpretation, this means that you have a shifted exponential distribution with shift parameter $\mu$ and scale parameter $\sigma$.
Firstly, the func... |
Theoretically it's possible to do this, although it will often not be practical.
Let's consider this in polynomial space. For a filter of order N you have 2*N+1 independent variables (N for the denominator and N+1 for the numerator). Let's look an arbitrary point $z_{k}$ in the z-plane and let's say the value of the tr... |
How to evaluate this limit $$\sum_{n=1}^\infty\frac{1}{n\sqrt[n]{n}}$$ and its convergence?
I tried ratio test, root test, Raabe's test. However, I'm not getting anywhere. Can you please help me? Thank you
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals i... |
I) Recall that the probability current 1D QM is
$$\tag{1} J(x)~:=~\frac{i\hbar}{2m}W(\psi,\psi^{\ast}) (x), $$
where
$$\tag{2} W(\psi,\psi^{\ast}) (x)~:=~\psi(x) \psi^{\prime}(x)^{\ast}- \psi^{\prime}(x)\psi(x)^{\ast}$$
is the Wronskian. Unitarity of the $S$-matrix is equivalent to the statement that
$$ \lim_{x\to -\in... |
Center manifold
Jack Carr (2006), Scholarpedia, 1(12):1826. doi:10.4249/scholarpedia.1826 revision #126955 [link to/cite this article]
One of the main methods of simplifying dynamical systems is to reducethe dimension of the system.
Centre manifold theory is a rigorous mathematical technique that makes this reduction p... |
I am just working through an argument from Halliday Resnick to derive the Lorentz contraction (see quote below).
Some paragraphs before this, the authors note that:
If the rod is moving, however, you must note the positions of the end points
simultaneously(in your reference frame) or your measurement cannot be called a... |
The spacetime geometry around a rotating uncharged black hole is described by the Kerr metric. I'll give this below, and it will look terrifying, but bear with me because there's only one small bit of the equation we need to see why the horizon disappears. Anyhow, the Kerr metric is:
$$\begin{align} ds^2 &= -(1 - \frac... |
I roughly understand the concept of the Lagrangian $L = T - V$ as well as the idea of stationary action $\delta \mathcal{S} =0$. However, I am confused what the Euler-Lagrange equation actually says.
Consider the Euler-Lagrange equation: \begin{equation} \frac{\partial L}{\partial q} - \frac{d}{dt} \left(\frac{\partial... |
Let's pretend we have a spatially discretized PDE of the following form:
\begin{align} \frac{\partial^2 \boldsymbol{u}^k}{\partial t^2} = D\boldsymbol{u}^k \end{align}
where $D$ can be any form for now and $k$ refers to this being the discretization at time $t_k$. Then let's suppose, for example, we use a central diffe... |
It is stated e.g. on Hackage, that the
ArrowApply and the
Monad typeclasses are equivalent. I have my doubts about this.
It is obvious that there are morphisms from
ArrowApply to
Monad and back, and one can check that one can prove the
Monad laws and the
ArrowApply laws in each case. What I doubt is that these morphism... |
Using Kjetil's answer answer, with process91's comment, we arrive at the following procedure.
Derivation
We are given two unit column vectors, $A$ and $B$ ($\|A\|=1$ and $\|B\|=1$). The $\|\circ\|$ denotes the L-2 norm of $\circ$.
First, note that the rotation from $A$ to $B$ is just a 2D rotation on a plane with the n... |
Find the degree of the splitting field of $x^6+1$ over $\mathbb{Q}$
First I tried to find the roots of $x^6+1$ over $\mathbb{C}$ so I can create the splitting field:
$$x^6+1 = (x^3-i)(x^3+i)$$
but this strategy showed some complications like in Find the degree of the splitting field of $x^4 + 1$ over $\mathbb{Q}$
So I ... |
I want to prove that the implicit function theorem implies the inverse function theorem. I saw in another post what's written below as the proof for this but I don't understand what they've done.
$$ \text{ For } f : \mathbb{R}^n \to \mathbb{R}^n \text{, consider } F:\mathbb{R}^n\times\mathbb{R}^n \to \mathbb{R}^n \text... |
In 1988, IMO presented a problem, to prove that $k$ must be a square if $a^2+b^2=k(1+ab)$, for positive integers $a$, $b$ and $k$. I am wondering about the solutions, not obvious from the proof. Beside the trivial solutions a or $b=0$ or 1 with $k=0$ or $1$, an obvious solution is $a=b^3$ so that the equation becomes $... |
For the hydrogen atom the quantised energy levels are:
$$E_n = \frac{- 13.6 eV}{n^2}\quad\text{with}\quad n = 1,2,3...$$
One peculiar property of this quantisation is that for large $n$ the energy levels are ever closer together and for $E \geq 0$ (that is $n = \infty$) the energy spectrum becomes a continuum. The elec... |
Solutions2tma@gmail.com
Whatsapp: 08155572788 Question This combined force law \(\widetilde{F}= q(\widetilde{E}+\frac{\widetilde V}{C} \times \widetilde{B})\) is known as Answer Lorentz Question How would you describein continous medium mechanics,the response of a linear elastic medium to strain. Answer Hookes law Ques... |
I am currently studying uniform and pointwise convergence and I am stuck at a somewhat basic distinction. Until now our lecturer has been talking about sequences of functions. For example:
$$f_n(x)=\frac{n}{x} \space \space \text{or} \space \space f_n(x)=x^n$$
However, in our problem set I am supposed to test the follo... |
Scalar and vector potentials in electromagnetism. The scalar potential is potential energy per unit charge. For potential energy, use the potential-energy tag.
In electromagnetic theory, there are two kinds of potential. First, and most common, is the scalar potential, denoted $\varphi$ or $\phi$, defined as the potent... |
This problem can be expressed as the original Merton's portfolio problem.
Consider wealth process defined by SDE
$$d X _ { t } = \frac { X _ { t } \alpha _ { t } } { S _ { t } } d S _ { t } + \frac { X _ { t } \left( 1 - \alpha _ { t } \right) } { S _ { t } ^ { 0 } } d S _ { t } ^ { 0 }$$
where $\alpha_t$ is proportion... |
I'm having great difficulty with the following problem:
This question concerns the integral $\int_{0}^{2}\int_{0}^{\sqrt{4-y^2}}\int_{\sqrt{x^2+y^2}}^{\sqrt{8-x^2-y^2}}\!z\ \mathrm{d}z\ \mathrm{d}x\ \mathrm{d}y$. Sketch or describe in words the domain of integration. Rewrite the integral in both cylindrical and spheric... |
Some history: In the pre-Wilsonian interpretation of quantum field theory, renormalisability of theories was considered an essential requirement. That is, you should be able to remove all ultraviolet divergences of the theory (that occur in perturbation theory) by absorbing them in a finite number of parameters in the ... |
Markdown Cells¶
Text can be added to Jupyter Notebooks using Markdown cells. You can change the cell type to Markdown by using the
Cell menu, the toolbar, or the key shortcut
m. Markdown is a popular markup language that is a superset of HTML. Its specification can be found here:
Markdown basics¶
You can make text
ital... |
Let $T_0$ be the set theory axiomatized by $ZFC^-$ (that is $ZFC$ without powerset) + every set is countable + $\mathbb{V}=\mathbb{L}$.
Question 1: Suppose $\phi$ is a sentence of set theory. Must there be a large cardinal axiom $A$ such that $\phi$ is decided by $T_0$ + "there are transitive set models of $A$ of arbit... |
What's the relationship between Kahler differentials and ordinary differential forms?
Let $M$ be a differentiable manifold, $A=C^\infty (M)$ its ring of global differentiable functions and $\Omega^1 (M)$ the A-module of global differential forms of class $C^\infty$. The A-module of Kähler differentials $\Omega_k(A)$ is... |
Consider a ring with radius $R$ and charge density $\lambda=\lambda_0\cos\phi$, where $\phi$ is the angular coordinate in the cylindrical coordinate. If I want to find the dipole moment of this charge distribution, then I put it into $$\vec{p}=\int{\mathrm d^3r~\rho(\vec{r})\vec{r}},$$ where I tried $$\rho(\vec{r})=\la... |
I want to calculate the VaR for a long position (S) in stockprices after one year. Therefore i tried two methods:
analytical solution: $VaR = S\cdot p_0\cdot \sigma_d \cdot \Phi^{-1}(1-\alpha)\cdot \sqrt{252}$
MC with geometric brownian motion:
I. model stock price (assuming $\mu = 0$): $p_{t+1} = p_t + p_t\cdot \sigma... |
8.11. Bidirectional Recurrent Neural Networks¶
So far we assumed that our goal is to model the next word given what we’ve seen so far, e.g. in the context of a time series or in the context of a language model. While this is a typical scenario, it is not the only one we might encounter. To illustrate the issue, conside... |
For the elastic scattering of identical particles, A + A → A + A, what are the Mandelstam variables?
Solution
The Mandelstam variables are defined, for a process 1 + 2 → 3 + 4, as
\begin{eqnarray}
s &=& (p_1 + p_2)^2 \nonumber \\ t &=& (p_1 − p_3)^2 \nonumber \\ u &=& (p_1 − p_4)^2 \nonumber \end{eqnarray}
where the \(... |
statsmodels.multivariate.factor_rotation.rotate_factors¶
statsmodels.multivariate.factor_rotation.
rotate_factors(
A, method, *method_args, **algorithm_kwargs)¶
Subroutine for orthogonal and oblique rotation of the matrix \(A\). For orthogonal rotations \(A\) is rotated to \(L\) according to\[L = AT,\]
where \(T\) is a... |
Lieb-Liniger model of a Bose Gas
Elliott H. Lieb (2008), Scholarpedia, 3(12):8712. doi:10.4249/scholarpedia.8712 revision #91428 [link to/cite this article] The Lieb-Liniger model describes a gas of particles moving in one-dimension and satisfying Bose-Einstein statistics.
Contents Introduction
A model of a gas of part... |
In this post of Prof Tao, it shows a lemma about the decomposition of bounded linear functional and a related exercise. The content is
Lemma 9(Jordan decomposition for functions) Let $I \in C_c(X \rightarrow \bf{R})^*$ be a (real) continuous linear functional. Then there exist positive linear functions $I^+, I^- \in C_... |
What is a good, single, "periodic table" of all the particles of the Standard Model?
I thought Particle Data Group would have a single-page PDF of this, but I couldn't find a single table listing all the particles.
Physics Stack Exchange is a question and answer site for active researchers, academics and students of ph... |
updated: Added the MathML code in source form at the bottom of this, as requested.
I am now configuring tex4ht to generate MathML instead of .png for the math. Then MathJax is loaded for final display. I found 3 problems (so far) with the MathML code generated by tex4ht.
The first one,
\pm (plus or minus sign) generate... |
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