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Some More Advanced Mathematics¶ Algebraic Geometry¶
You can define arbitrary algebraic varieties in Sage, but sometimes nontrivial functionality is limited to rings over \(\QQ\) or finite fields. For example, we compute the union of two affine plane curves, then recover the curves as the irreducible components of the u... |
I'm currently developing a program to solve 2D transient state heat conduction on a square plate using the V-cycle multigrid. Althought my program is able to reach the steady state solution, it's computational time is longer then just running the problem using Gauss-seidel method.
Problem case: 0.1 m by 0.1 m square pl... |
Question
As a test, I transform a uniform distribution over the unit square.
But when I check the transformed distribution with Monte Carlo, it is wrong.
What went wrong?
Thanks.
Problem
Random variables $\vec{X} = (X_1, X_2)$ follows the uniform distribution over the unit square.
In other words, $X_{1} \sim U[0, 1]$ a... |
And I think people said that reading first chapter of Do Carmo mostly fixed the problems in that regard. The only person I asked about the second pset said that his main difficulty was in solving the ODEs
Yeah here there's the double whammy in grad school that every grad student has to take the full year of algebra/ana... |
The time measured by the clock you carry with you on your journey is called the proper time, and the proper time is just the length of your world line (give or take a factor of $c$). So if we calculate the length of your world line as you accelerate away then back, that gives us your elapsed time. Better still, the pro... |
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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 ... |
You cannot eliminate the dependence of a solution on the risk aversion parameter (which this author confusingly calls $\lambda$).
Perhaps a source of confusion? Typically $\lambda$ is used to denote a Lagrange multiplier in Lagrangian optimization, but the author is using $\lambda$ as a risk tolerance parameter. (In yo... |
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Up a level 56. Article
Aad, G, Abbott, B, Abdallah, J et al. (2822 more authors) (2017)
Performance of algorithms that reconstruct missing transverse momentum in root s=8 TeV proton-proton collisions in the ATLAS detector. The European Physical Journal C, 77 (4). 241. ISSN 1434-6044
Aad, G, Abbott, B, ... |
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Change to browse by: Bookmark(what is this?) Mathematical Physics Title: Superradiance initiated inside the ergoregion
(Submitted on 10 Sep 2015 (v1), last revised 18 Nov 2016 (this version, v3))
Abstract: We consider the stationary metrics that have both the black hole and the ergoregio... |
To the extent you can assume your process is a white stationary ergodic process, the variance of the mean is the variance of the process divided by N where N is the number of samples.
Assuming that you can estimate the variance from your sample using
$$\sigma_x = \frac{1}{N-1}\sum_{i=1}^N(x_i-\mu_x)^2$$
NOTE: This is t... |
LaTeX typesetting is made by using special tags or commands that provide a handful of ways to format your document. Sometimes standard commands are not enough to fulfil some specific needs, in such cases new commands can be defined and this article explains how.
Contents
Most of the LaTeX commands are simple words prec... |
From a purely theoretical standpoint, the radius of the wheel doesn't matter, but heavy wheels are slow.
In an idealized scenario, the skateboard conserves energy. This means that its total energy when it gets to the bottom of the hill is the same regardless of how it goes down.
As a wheel rolls down the hill, it picks... |
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Change to browse by: Bookmark(what is this?) Nuclear Theory Title: $S$-factor and scattering-parameter extractions from ${}^{3}\mathrm{He} +{}^{4}\mathrm{He} \rightarrow {}^{7}\mathrm{Be} + γ$
(Submitted on 16 Sep 2019)
Abstract: Previous studies of the reaction ${}^{3}\mathrm{He} +{}^{4... |
I am a beginner in ergodic theory. I have read some lecture notes(such as this and this) about it in hope that I could find something which helps to prove the ergodicity of some Markov chain taking values in a general state space(say a Polish space).
Although I've learned many interesting things, such as the applicatio... |
One of the axioms of ZF set theory is the axiom of union: $$(\forall x)(\exists y)(\forall z)(z\in y \iff (\exists t)(z\in t\ \&\ t\in x)).$$ The axiom of union (together with the axiom of extensionality) guarantees that the operation $$x\mapsto\bigcup x$$ can be defined for all sets and indeed captures our intuition o... |
In machine learning, we are often dealing with high-dimension data. For convenience, we often use matrix to represent data. Numerical optimization in machine learning often involves matrix transformation and computation. To make matrix computation more efficiently, we always factorize a matrix into several special matr... |
I think the main issue is you are jumping ahead of yourself. You probably remember or read somewhere that the Fourier transform of a $rect$ function is a $sinc$ function. This is true; however, no where in this section does he mention Fourier transform! In fact, what he is doing is not Fourier transform.
What he does i... |
A subordinate Brownian motion is a Lévy process which can obtained by replacing the time of Brownian motion by an independent increasing Lévy process. The infinitesimal generator of a subordinate Brownian motion is
\(-\phi(-\Delta)\), where
\(\phi\) is the Laplace exponent of the subordinator. When
\(\phi(\lambda)=\lam... |
Implementing such a method is not a trivial task. Just FYI, there are multiple programs purpose-built to perform Hartree-Fock calculations.
In terms of a good introduction to the theory of Hartree-Fock calculations, I found this pdf extremely helpful.
First of all, you have given the expression for the
psuedo-exact (ps... |
Total Internal Reflection (TIR) is a phenomenon in optics, by which light experiences complete reflection at an interface between two media. Most optical fibers use TIR as the guiding principle. Figure 1 explains schematically how TIR takes place at an interface between two media. When an incident light ray (red) hits ... |
When studying some gauge theories approach to problems in Mechanics, I've found the following integral
$$P\exp\left[\oint A \ dt\right]=1+\dfrac{1}{2}\oint_{\partial D}\sum_{\mu,\nu}F_{\mu\nu}\gamma^{\mu}(t) \dot{\gamma}^{\nu}(t)dt,$$
where $A$ is the gauge potential, and $F$ is the field strength tensor (i.e. the pull... |
Algebraic Geometry Seminar Fall 2016
The seminar meets on Fridays at 2:25 pm in Van Vleck B305.
Contents Algebraic Geometry Mailing List Please join the AGS Mailing List to hear about upcoming seminars, lunches, and other algebraic geometry events in the department (it is possible you must be on a math department compu... |
A forum where anything goes. Introduce yourselves to other members of the forums, discuss how your name evolves when written out in the Game of Life, or just tell us how you found it. This is the forum for "non-academic" content.
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Hello everyone!
I proposed a new Stack Exchange Q&A site ... |
How to use algebra in operations that involve asymptotic notation? Like for example:
$C_1(n)= O(n)$ $C_2(n) = O(n^2)$ Is $C_2(n)/C_1(n) = O(n)$?
Computer Science Stack Exchange is a question and answer site for students, researchers and practitioners of computer science. It only takes a minute to sign up.Sign up to joi... |
As has been discussed in many questions around here (e.g. here), relativity tells us only about local properties and behavior of a space-time. There are some exceptions when we make global assumptions - if we have a space of
globally and strictly constant positive curvature, non-trivial topology is imminent because the... |
I've done a little bit of research and it seems Millikan was able to measure the ratio between the charge of the electron and its mass. But how can one measure one of the two constants to get the value of the other?
The mass-to-charge ratio $m/e$ of the electron was first measured by J.J. Thomson, the discoverer of the... |
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Change to browse by: References & Citations Bookmark(what is this?) Mathematics > Optimization and Control Title: On the Kähler Geometry of Certain Optimal Transport Problems
(Submitted on 30 Nov 2018 (v1), last revised 22 Aug 2019 (this version, v4))
Abstract: Let $X$ and $Y$ be domains... |
12:45 --- Lunch --- 14:00 BSM III - James Gainer (University of Florida (US)) (until 16:00) (G31)
14:00 Search for new physics in the low MET monophoton channel with the CMS Detector - Toyoko Orimoto (Northeastern University (US)) (G31) 14:15 Characterizing Invisible Electroweak Particles through Single-Photon Processe... |
Your confusion stems basically from comparing results in different conventions.
Basically, there is always a phase difference of $\pi$ between the two output ports of a beam splitter, but this can only ever be 'morally' true, because that statement talks about the phase of the EM field at different points, and thus the... |
I've always been confused with Delta hedging. It is well-known that for a (smooth enough) function of $(S,t)$ we have, due to Ito's lemma, that:\begin{eqnarray*}dC = \left(\frac{\partial C}{\partial ...
Let's assume the usual Black Scholes assumptions hold. My question is related to an answer on this question. There, t... |
Your problem can be reduced to the Partition problem (which is weakly NP-complete)
without an exponential blowup of the numeric values; so your problem is weakly NP-complete, too.
This is the idea: you can view the $2N$ integers as nodes of a graph $G$, the pairs in $P$ indentify the edges between the nodes.
Clearly $G... |
$R$ is taken to be the ring of upper $3 \times 3$ matrices with entries in $\mathbb{R}$.
If I view $R$ as a module over itself, are any of its submodules free?
And how can I prove that its submodules are pojective $R$-modules?
Thanks!
Mathematics Stack Exchange is a question and answer site for people studying math at ... |
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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 ... |
Assume we'd like to be able to encode variables $x_1,x_2,\cdots,x_r\in \mathbb{N}$, such that $\forall i\in[r]:1\leq x_i\leq N$ and $$\sum_{i=1}^{r}x_i=M$$
It's easy to store the variables using $r\log N$ bits, but this doesn't take advantage of the fact that their sum is bounded.
Furthermore, in my application $r$ is ... |
This is the problem called Dirisibly Yours
:
We wonder who can find and explain the shortest and neatest proofthat $$5^{2n+1} + 11^{2n+1 } + 17^{2n+1}$$ is divisible by $33$ forevery non negative integer $n$.
One way to solve this problem involves modulo arithmetic. You mayhave encountered this in the form of clock ari... |
What exactly happens in the case of equi-spaced points?
Why does increase in polynomial order cause the error to rise after a certain point?
This is similar to the Runge's phenomenon where, with equi-spaced nodes, the interpolation error goes to infinity with the increase of the polynomial degree, i.e. the number of po... |
Ask yourself the following:
First, how does the integration by parts affect the solvability of the problem, and the space of solutions?
Second, for which space of functions can you build a series of subspaces (the ansatz functions) that you can implement?
Let us regard the Poisson problem $u'' = f$ for $f \in L^2$, say... |
Estimation in Sequential Analysis
In the previous post I introduced the Sequential Generalized Likelihood Ratio test, which is a sequential A/B-test that in most cases require much smaller sample sizes than the classical fixed sample-size test. In this post I’m going to explain some problems regarding estimation in seq... |
Word Representation NLP, Natural Language Processing Word embeddings, which is a way of representing words. that let your algorithms automatically understand analogies like that, man is to woman, as king is to queen, and many other examples.
Representing words using a vocabulary of words.
One of the weaknesses of this ... |
2019-09-27 09:59
Higgs boson pair production at colliders: status and perspectives / Di Micco, Biagio (Universita e INFN Roma Tre (IT)) ; Gouzevitch, Maxime (Centre National de la Recherche Scientifique (FR)) ; Mazzitelli, Javier (University of Zurich) ; Vernieri, Caterina (SLAC National Accelerator Laboratory (US)) ; ... |
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In fluid dynamics, gravity waves are waves generated in a fluid m... |
We have the following generalized eigenvalue (set of) problem(s)
$$[K_R(\kappa)]\{u_R\} = \omega^2[M_R(\kappa)]\{u_R\}\quad \forall \kappa \in [\kappa_0, \kappa_1]$$
with
\begin{align} &K_R(\kappa) = T^H(\kappa) K T(\kappa)\, ,\\ &M_R(\kappa) = T^H(\kappa) M T(\kappa)\, ,\\ &u = T(\kappa) u_R\, . \end{align}
Where $K$ ... |
The color burst is also an indicator that there is a color signal.This is for compatibility with black and white signals. No color burst means B&W signal, so only decode the luminance signal (no croma).No signal, no color burst, so the decoder falls back to B&W mode.Same idea goes to FM stereo/mono. If there is no 19 k... |
The main problem with your circuit is that the time constant of 330 ohms and 1\$\mu\$F is only \$330\mu sec\$, which is not all that long, and the LED will not be especially bright for that brief time. You have to consider the physiological response of the human eye.
Your eye acts as a kind of integrator over a period ... |
The local standard of rest is the restframe circular orbit of a star at the position of the Sun in the azimuthally averaged Galactic potential. The Sun moves with 3d Galactic velocity coordinates of (11, 12, 7) km/s with respect to the LSR (Schonrich et al. 2009. i.e The Sun has a velocity component (wrt the LSR) of 11... |
For a universe described by the RW metric, a relation between the scale factor at the time of emission of light and the redshift can be derived, and yields $$ a(t_e) = \frac{1}{1+z} $$ The above equation depends only on the time of emission and the redshift. The above relation implies that the redshift of any light fro... |
K Mazumdar
Articles written in Pramana – Journal of Physics
Volume 63 Issue 6 December 2004 pp 1359-1365
Amol Dighe Anirban Kundu K Agashe B Anantanarayan A Chandra A Datta P K Das S P Das A Dighe R Forty D K Ghosh Y -Y Keum A Kundu N Mahajan S Majhi G Mazumdar K Mazumdar P Mehta Y Nir J P Saha R Singh N Sinha R Sinha ... |
The key to the efficiency of an antenna (whether for transmitting or receiving - the two processes are essentially reciprocal) is resonance, and impedance matching with the source / receiver. The size also matters in terms of the relationship between power and current.
A nice analysis of the impact of size of an antenn... |
I am reading through documentation related to Funding Valuation Adjustments (FVA) which discuss risk free rate and funding matters and the following question came to my mind: in risk neutral valuation theory, why do we require the risk free rate to be risk free?
Indeed, let's assume a Black-Scholes framework but with a... |
Motivation: I'm using 2D regular grid (it's actually a quadtree but I can still treat it as a finite difference thing if I weight-average the solution over smaller scale cells for the purpose of estimating Laplacian in the neighbor points) to discretize my Poisson equation and I'd like to solve it iteratively. The disc... |
I would like to discuss the stability argument in a bit more detail. Since it is correct that static longitudinal stability is the main reason why these aircraft are not often developed.
However the reasoning given in the other posts is incomplete/not completely correct.
First of all, a flying wing indeed has a very sm... |
Recently I'm studying PSG and I felt very puzzled about two statements appeared in Wen's paper. To present the questions clearly, imagine that we use the Shwinger-fermion $\mathbf{S}_i=\frac{1}{2}f_i^\dagger\mathbf{\sigma}f_i$ mean-field method to study the 2D spin-1/2 system, and get a mean-field Hamiltonian $H(\psi_i... |
NGC 6397 :: new paper Abstract: see this previous post
URL: ApJ Letters website
NGC 6397 :: new paper Abstract: see this previous post
URL: ApJ Letters website
Authors:
Brad M. S. Hansen, Jay Anderson, James Brewer, Aaron Dotter, Greg. G. Fahlman, Jarrod Hurley, Jason Kalirai, Ivan King, David Reitzel, Harvey B. Richer... |
I know that we should express complex numbers generally in the standard form
$$a+bi:a,b\in\mathbb{R}$$
Like $4+5i-2=2+5i$.
But how do I express complex numbers like $e^{-i\pi/2}$ or $i+e^{2\pi i}$?
Thank you!
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professional... |
Accepted Manuscripts
There are 83 manuscripts.
1.
title: Generalized $k$--Fibonacci numbers authors: Sergio Falcon code: MMN-1233
2.
title: Krasnoselskii Type Theorems for Multivalued Operators in Generalized Metric Spaces with Applications authors: Cristina Urs code: MMN-1787
3.
title: Normal direction curves and thei... |
Guiding AMR with adjoint flagging¶
A new approach to flagging cells for refinement was introduced in Clawpack 5.6.0 – using the solution to an adjoint problem to determine what cells in the forward solution should be refined because these cells may have an impact on the some specified functional of interest. This appro... |
To get an action of $S_3$ in which $\langle(1,2,3)\rangle$ is the stabilizer of a point, take $X=\{1,-1\}$, and let $S_3$ act by parity: even permutations fix both $1$ and $-1$, odd permutations swap $1$ and $-1$. Then the stabilizer of a single point, $1$, is $A_3 = \langle (1,2,3)\rangle$.
In general, if $H$ is a sub... |
Yes, the same holds for infinite graphs. I'll provide here a probably overlong proof of this.
Let $G = (V,E)$ be an infinite graph of size $\kappa$, and let $\chi(G) = \lambda$. If $\lambda = \kappa$, then any bijection is chromatic, so we may assume that $\lambda < \kappa$.
Let $c:V \rightarrow \lambda$ be a chromatic... |
I would like to know how to compute the statistics of the discrete Fourier transform of a noise signal. To illustrate what I mean, I will first explain in detail a computation I have managed to do myself.
Suppose we have a discrete time series of values $x_n$ with $n$ from 0 to $N-1$. Each $x_n$ is a random variable, u... |
And I think people said that reading first chapter of Do Carmo mostly fixed the problems in that regard. The only person I asked about the second pset said that his main difficulty was in solving the ODEs
Yeah here there's the double whammy in grad school that every grad student has to take the full year of algebra/ana... |
In order to simulate the following equation using FDM $$u_t(t,x)-u_{xx}(t,x)=0, \quad (t,x) \in (0,1)\times (0,1)$$ $$(u_t(t,x)-u_{x}(t,x))\rvert_{x=0}=0, \quad t \in (0,1)$$ $$(u_t(t,x)+u_{x}(t,x))\rvert_{x=1}=0, \quad t \in (0,1)$$ $$u(0,x)=u_0, \quad x \in (0,1)$$ $$u(0,0) =\mu_0, \quad u(0,1)=\mu_1,$$ where $f,g,u_... |
My plan is to solve the heat equation in the right half portion of the domain, while having the left half completely isolated with constant temperature. To do so, I model the left half with a very low conductivity. I then apply a heat flux to the top side of the right half and a Dirichlet BC to the bottom side. For the... |
Hello one and all! Is anyone here familiar with planar prolate spheroidal coordinates? I am reading a book on dynamics and the author states If we introduce planar prolate spheroidal coordinates $(R, \sigma)$ based on the distance parameter $b$, then, in terms of the Cartesian coordinates $(x, z)$ and also of the plane... |
Let's solve the exercise. It's asking for your awareness of periodicity in DFT and some trigonometry.
First, beacuse of the question content, let's state the following :$$\delta[n-d] \overset{DTFT} \longleftrightarrow e^{-j \omega d } \overset{DFT} \longleftrightarrow e^{-j \frac{2 \pi}{N} d k } $$
Then given our DFT $... |
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Change to browse by: References & Citations Bookmark(what is this?) Mathematics > Algebraic Geometry Title: On Landau-Ginzburg systems and $\mathcal{D}^b(X)$ of various toric Fano manifolds with small picard group
(Submitted on 1 May 2016 (v1), last revised 19 May 2017 (this version, v2)... |
Written, in Ex.8 Ch.9.1 of the book Advanced Calculus by P. M. Fitzpatrick :
Suppose that $\sum\limits_{k=1}^\infty a_k$ and $\sum\limits_{k=1}^\infty b_k$ are series of positive numbers such that $$\lim_{n \to \infty} \frac{a_n}{b_n}=l \ \ \ \text{and} \ l>0.$$ Prove that the series $\sum\limits_{k=1}^\infty a_k$ conv... |
For simplicity, in the following we set the electric charge $e=1$ and consider a lattice spinless free electron system in an external static magnetic field $\mathbf{B}=\nabla\times\mathbf{A}$ described by the Hamiltonian $H=\sum_{ij}t_{ij}c_i^\dagger c_j$, where $t_{ij}=\left | t_{ij} \right |e^{iA_{ij}}$ with the corr... |
In a paper by Joos and Zeh, Z Phys B 59 (1985) 223, they say:This 'coming into being of classical properties' appears related to what Heisenberg may have meant by his famous remark [7]: 'Die "Bahn" entsteht erst dadurch, dass wir sie beobachten.'Google Translate says this means something ...
@EmilioPisanty Tough call. ... |
Let me first run through the setting of my question in an example I understand well; that of modular curves. If $Y_1(N)$ denotes the usual modular curve over the complexes, the quotient of the upper half plane by the congruence subgroup $\Gamma=\Gamma_1(N)$, then there are two kinds of sheaves that one often sees showi... |
I understand the principle of less airflow, less control, but why is that the case?
Because moments of inertia don't change with speed
Control effectiveness means that the controls effect a change in the balance of moments which results in the desired attitude change. The smaller the control deflection for the same cha... |
LambertW function LambertW, called also ProductLog [1], is solution \(W=W(z)\) of equations \( \displaystyle \!\!\!\!\!\!\!\!\!\!\!\!\!\!\! (0\mathrm{a}) ~ ~ ~ W'= \frac{W}{(1+W)~z}\) \( \displaystyle \!\!\!\!\!\!\!\!\!\!\!\!\!\!\! (0\mathrm{b}) ~ ~ ~ W(0)=0\)
The cut line is drawn from the singularity at \(-1/\mathrm ... |
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Kaon femtoscopy in Pb-Pb collisions at $\sqrt{s_{\rm{NN}}}$ = 2.76 TeV
(Elsevier, 2017-12-21)
We present the results of three-dimensional femtoscopic analyses for charged and neutral kaons recorded by ALICE in Pb-Pb collisions at $\sqrt{s_{\rm{NN}}}$ = 2.76 TeV. Femtoscopy is used to... |
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The
Cool Solutions Wiki is a Cool Solutions Best Practices Compendium written collaboratively by its readers. You're welcome to add to articles, or start new ones. You are even free to copy, change or distribute articles. This model is called a wiki.
A Wiki or wiki (pronounced "wicky", "weekee" or "... |
For a lattice $\mathbb Z^d$, denote by
lattice line any line that contains two (and thus infinitely many) lattice points.
For $2\le k<n$, define a
$(n,k)$-coloring, or $C_d(n,k)$ for short, as a $n$-coloring $\mathbb Z^d\to \{0,1,\dots,n-1\}$ such that in each lattice line, exactly $k$ colors occur, each color is used ... |
Hartogs Theorem says every function whose undefined locus is of codim 2 can be extend to the whole domain. I saw people saying this corresponds to the (S2) property of a ring. But I can't see why this is true. Can anybody explain this or give a heuristic argument?
Let $\mathscr F$ be a coherent sheaf on a noetherian sc... |
I am using backward Euler in a FEM scheme for a convection-diffusion problem. On a given mesh, I can take arbitrarily large time steps, as expected. But if I decrease time step, at some point it will generate oscillations in the solution (spikes). Is this a known behavior? It's not mentioned in text books, at least the... |
There are lots of statements that have been conditionally proved on the assumption that the Riemann Hypothesis is true.
What other conjectures have a large number of proven consequences?
MathOverflow is a question and answer site for professional mathematicians. It only takes a minute to sign up.Sign up to join this co... |
I have the following homework problem:
A line of charge $\lambda$ is located on the z-axis. Determine the electric flux for a rectangular surface with corners at coordinates: $(0, R, 0)$, $(w, R, 0)$, $(0,R, L)$, and $(w, R, L)$.
This is what I have come up with so far:
The line of charge, is located on the $z$-axis. W... |
We all know that \begin{equation}\exp(x) = \sum_{n=0}^\infty \frac{x^n}{n!} = 1 + x + \frac12 x^2 + \dots\end{equation}implies that for $|x| \ll 1$, we have $\exp(x) \approx 1 + x$. This means that if we have to evaluate in floating point $\exp(x) -1$, for $|x| \ll 1$ catastrophic cancellation can occur.
This can be ea... |
In some of my computations I calculate a scalar value $\lambda_h$ (in my case an eigenvalue) depending on a finite element discretization of the domain. Usually we can manage to find estimates of the form $$ |\lambda - \lambda_h| \leq C h^k $$ where $k$ depends on the problem, the degree of the approximation, etc.
Ques... |
If $G$ is a finite group with normal subgroups $M$ and $N$, then $MN$ is a subgroup, called the
normal product of $M$ and $N$.
If $\mathcal{F}$ is a set of finite groups closed under isomorphism and normal products, then there is a subgroup $O_\mathcal{F}(G)$ called the
$\mathcal{F}$-radical which is the normal product... |
I am an analyst struggling through some geometry used in physics.
Some background: For some Lie group $G$, let $P$ be a principal $G$-bundle over the smooth manifold $M$. Let $\omega$ be a connection 1-form on $P$ (a "principal connection"). This is a Lie algebra-valued 1-form.
As for the curvature two-form, either you... |
Work done bringing charge from infinity to the origin doesn’t imply that work done is zero, because the origin is itself is a part of coordinate system but will it requires infinite amount of work that is need to done for bringing the charge from infinity to the origin where $+q$ charge is already there, we can easily ... |
The best way to do this is (as you said) to just use the definition of periodic boundary conditions and set up your equations correctly from the start using the fact that $u(0)=u(1)$. In fact, even more strongly,
periodic boundary conditions identify $x=0$ with $x=1$. For this reason, you should only have one of these ... |
I'am trying to implement a path following algorithm based on MPC (Model Predictive Control), found in this paper : Path Following Mobile Robot in the Presence of Velocity Constraints
Principle: Using the robot model and the path, the algorithm predict the behavior of the robot over N future steps to compute a sequence ... |
There is nothing wrong with effective quantum general relativity as a QFT. It works fine and we can get effective answers for calculations. That said, most are not interested in calculating scattering cross sections and amplitudes with quantum GR in a relatively low energy limit, they are interested in the character of... |
This is analogous to the definition of an empty product in mathematics. For a finite non-empty set $S=\{s_1,\ldots,s_n\}$, the product over $S$ can be defined as$$\prod_{s\in S}s=s_1\times \cdots\times s_n.$$For such a product you'd want disjoint unions to map into products: if $R\cap S=\emptyset$, then you want $\prod... |
I’m currently having some issues with my routine for the linear advection-diffusion problem. The model problem is as follows: $$ \nabla\cdot(\mathbf{s} u) - \nabla\cdot(\kappa\nabla u) = f, \;\;\;\text{in}\;\Omega \subseteq \mathbb{R}^{2} $$ $$ u = g_{D} \;\;\;\text{on}\;\Gamma_{D} $$
The boundary is purely Dirichlet i... |
In the wave equation:
$$c^2 \nabla \cdot \nabla u(x,t) - \frac{\partial^2 u(x,t)}{\partial t^2} = f(x,t)$$
Why do we first multiply by a test function $v(x,t)$ before integrating?
Computational Science Stack Exchange is a question and answer site for scientists using computers to solve scientific problems. It only take... |
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{x=y2−116+z2−116y=z2−125+x2−125z=x2−136+y2−136 \begin{cases}x = \sqrt{y^2 - \frac1{16} } + \sqrt{z^2 - \frac1{16}} \\y = \sqrt{z^2 - \frac1{25} } + \sqrt{x^2 - \frac1{25}} \\z = \sqrt{x^2 - \frac1{36} } + \sqrt{y^2 - \frac1{36}} \\\end{cases}⎩⎪⎪⎪⎨⎪⎪⎪⎧x=y2−161+z2−... |
Assuming you are sampling at a fixed time step, called \$\Delta\$, and \$t_n\$ denote your sample times (i.e. \$t_n = n \Delta\$) then a traditional estimate of the velocity is$$v(t_{n}) \sim \sum_{k=0}^{n-1} a(t_k) \Delta$$where \$a(t_k)\$ is the acceleration value at time \$t_k\$ as sampled by your accelerometer. It ... |
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If $A$ is a connected set and $\{A_i : i \in I\}$, $I$ an arbitrary set (can be countable or not) of connected sets.
How to show that if $A \cap A_i \neq \emptyset$ for all $i \in I$ then $A \cup (\cup_{i\in I} A_i)$ is connected?
I am trying to show that if $A \cap A_u \neq \e... |
I am trying to figure out what is the largest possible order that an element of the multiplicative group $\bmod{n}$ can have if $n=p_1^{k_1} \cdot p_2^{k_2} \cdot \dots \cdot p_m^{k_m}$.
then I can prove that
$$U(n) \cong U(p_1^{k_1}) \times U(p_2^{k_2}) \times \dots \times U(p_m^{k_m})$$
Now I know from my number theo... |
Consider the following integral equation: $$ \int_0^r f(t) \arcsin \left( \frac{t}{r} \right) \, \mathrm{d}t + \frac{\pi}{2} \int_r^R f(t) \, \mathrm{d} t = r \, \qquad (0<r<R) \, , $$ where $f(t)$ is the unknown function. By differentiating both sides of this equation with respect to $r$, one obtains $$ -\frac{1}{r} \... |
closed as no longer relevant by Robin Chapman, Akhil Mathew, Yemon Choi, Qiaochu Yuan, Pete L. Clark Aug 22 '10 at 9:00
This question is unlikely to help any future visitors; it is only relevant to a small geographic area, a specific moment in time, or an extraordinarily narrow situation that is not generally applicabl... |
Discounted price process of an american put (perpetual) has a $dt$ part in it, which is negative if the price at time $t$ is less than the optimal exercise price. This is the only thing that drags the discounted process down as the time goes on and makes the whole process a supermartingale. So when you don't exercise t... |
As the name suggests,
supervised learning takes place under the supervision of a teacher. This learning process is dependent. During the training of ANN under supervised learning, the input vector is presented to the network, which will produce an output vector. This output vector is compared with the desired/target ou... |
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 ... |
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