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I know that this answer makes a lot of assumptions, but it at least generalizes your algorithm:
Suppose that $\{A_n\}$, $\{B_n\}$, and the seeding matrix, $V_N$, all form a commuting family of normal matrices, where the eigenvalue decompositions of $\{A_n\}$ and $\{B_n\}$ are known a priori, say $U' V_N U = \Lambda_N$,... |
1d diffusion equation
Integrating the diffusion equation,
$$ \frac{\partial u}{\partial t} = D \frac{\partial^2 u}{\partial x^2}, $$
with a constant diffusion coefficient D using forward Euler for time and the finite difference approximation for space,
$$ u_i^{t+1} = u_i^t + D\frac{\Delta t}{\Delta x^2} ( u_{i+1}^{t} +... |
Introduction
Theoretically speaking, an optical fiber with a circular core has no birefringence, and the polarization state in such an optical fiber does not change during propagation. In reality, however, a small amount of birefringence is always present in an optical fiber due to external perturbations (load, bend, e... |
Hilmar's answer is of course perfectly correct, but I think there there are several points that Lyons did not address in the statement quoted by the OP (or maybe he talked about them previously and chose not to repeat himself in the paragraph quoted by the OP).
The Discrete Fourier Transform (DFT) is commonly described... |
This question already has an answer here:
Why is the Earth so fat? 7 answers
My dynamics lecture notes repeat the Earth's equatorial bulge can be approximated as: $$ \approx \frac{\Omega^2R}{g} \approx \frac{1}{300} $$ (Do they mean R/300?)
They also include statements like:
"The Earth's oblateness $ \frac{I_3-I_1}{I_1... |
In many papers the contact resistance of a metal in contact with a semiconductor is given in units of $\Omega~\mu m$, for example in the paper by Li et al. (Appl. Phys. Lett. 102 (2013), p. 183110):
$R_C$ for contacts formed to epitaxial graphene on SiC have been reported to be less than 100 $\Omega~\mu m$ and with spe... |
Consider the following plot, courtesy of this page:
Regarding the $y$-axis, how does this "expected return" relate to the "instantaneous expected return" in a geometric Brownian motion (GBM)?
E.g., assume each stock price follows $dS(t) = \mu S(t) dt + \sigma S(t) dW(t)$, and so $S(t) = S(0)\exp\left(\left(\mu - \frac{... |
i need to get a Givens-Rotation, which zeros a matrix entry when multiplied from the right side. I did already look at this Topic https://math.stackexchange.com/questions/945736/givens-rotation-from-right-side but i could not really understand the process of getting there.
Some more details: I have to matrices, A and B... |
Edit: According to comments of Eric Wofsy and Yemon Choi I edit the question.
For a (compact) topological space $X$, we put $A=\{f:X\to \mathbb{C}\mid f\text{ is bounded}\}$. We define a semi-norm on $A$ with $\parallel f \parallel=\parallel \omega_{f}\parallel_{\infty}$ where $\omega_{f}(x)$ is the oscilation of $f$ a... |
Tasks
The tasks of the challenge are two-fold:
To segment vertebrae from the given spine images that include fractured and non-fractured cases, and provide vertebra segmentation results in the form of corresponding masks.
To classify vertebrae from the given spine images into fractured and non-fractured cases along wit... |
Consider the preliminary question of getting a sequence of $N$ heads out of $k$ throws, with probability $p(N,k)$. This is given by the recurrence formula $$ p(N,k) = \begin{cases} 0 &\text{if } k<N\\ \dfrac{1}{2^N} + \sum_{m=0}^{\min(N-1,k-N-1)} \dfrac{1}{2^N}\dfrac{1}{2} + \sum_{m=N}^{k-N-1} \{1-p(N,m)\} \dfrac{1}{2^... |
Siril processing tutorial Convert your images in the FITS format Siril uses (image import) Work on a sequence of converted images Pre-processing images Registration (Global star alignment) → Stacking Stacking
The final step to do with Siril is to stack the images. Go to the "stacking" tab, indicate if you want to stack... |
For each of the functions you mention, the answer to "why it's useful" will be somewhat different. In general, if a function appears in many unrelated contexts, that's a sure sign that it's useful. Think of other mathematical structures that appear in many unrelated contexts: the natural numbers, the real numbers, the ... |
A density matrix view of the problem:
When $p_\mu$ is near zero, one is considering a momentum near zero and the discrete lattice works fine. It's when $p_\mu$ is near $\pm \pi/a$, or more generally, near $n\pi/a$ for $n\neq 0$ an integer, that one finds problems. Instead of having very large momenta, these values of $... |
Let's consider a rigid body B and an arbitrary reference point P ${\it fixed}$ with respect to the body (P can be a particle of the body or it can be a "mathematical point" outside the body but solidary to B, it doesn't matter). As B moves, the point P has a velocity ${\bf v}_P$ that, of course, can change over time. T... |
I am reading a paper on stability of CG, and I came across the following statement:
\begin{equation} \frac{\|A\|\,\|p\|^2}{\langle p,Ap\rangle} \leq \kappa(A) \end{equation} where $\kappa(\cdot)$ is the condition number and $\langle \cdot, \cdot \rangle$ the inner product.
Can you, please, help me understand this bound... |
Let
\(\gamma(G)\) and
\(\iota(G)\) be the domination and independent domination numbers of a graph
\(G\), respectively.In this paper, we define the Price of Independence of a graph
\(G\) as the ratio
\(\frac{\iota(G)}{\gamma(G)}\). Firstly, we bound the Price of Independence by values depending on the number of vertice... |
Journal of Mathematics of Kyoto University J. Math. Kyoto Univ. Volume 39, Number 4 (1999), 649-673. Perturbation theorems for supercontractive semigroups Abstract
Let $\mu$ be a probability measure on a Riemannian manifold. It is known that if the semigroup $e^{-t\nabla *\nabla}$ is hypercontractive, then any function... |
State
A
state is something that a system is in. The system is where I perform my measurement; the state is the result of that measurement.
When I perform one kind of a measurement, and then I perform another kind of a measurement, the two results are correlated. In particular, the first result may determine the probabi... |
Mandelbrot polynomial is special kind of quadratic polynomial, written in form\( P_c(z)=z^2+c\)whrere \(c\) is parameter. Usially, it is assumed to be a complex number.
Mandelbrot set
The Mandelbrot polynomial is used to define the Mandelbrot set
\( M = \left\{c\in \mathbb C : \exists s\in \mathbb R, \forall n\in \math... |
To solve exponential problems with different bases we t ake the common logarithm or natural logarithm of each side. Use the properties of logarithms to rewrite the problem. Use Property 5 to move the exponent out front which turns this into a multiplication problem. Divide each s ide by log 3. Multiplying logarithms of... |
Probability Seminar Spring 2015 Thursdays in 901 Van Vleck Hall at 2:25 PM, unless otherwise noted. If you would like to sign up for the email list to receive seminar announcements then please send an email to join-probsem@lists.wisc.edu.
Thursday, January 15, Miklos Racz, UC-Berkeley Stats
Title: Testing for high-dime... |
L-functions of signature (0,0,0;)
These L-functions satisfy a functional equation with \(\Gamma\)-factors \[\begin{aligned}\Gamma_\R(s + i \mu_1)\Gamma_\R(s + i \mu_2)\Gamma_\R(s + i \mu_3)\end{aligned}\] with \(\mu_j\in \R\) and \(\mu_1 + \mu_2 + \mu_3 = 0\). By permuting and possibly taking the complex conjugate, we ... |
It is a question vector calculus and Maxwell's laws. I put it this way. Let's say, we are working in a $3$-Dimensional space ( e.g $x\cdot y\cdot z = 4\cdot3\cdot2$, a certain room/class of that size ) .
Within this room, the heat obey a certain equation ( for e.g. $T = 25 + 5z$ ) .We know that heat flows from higher t... |
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An Introduction to MimeTex
For those who wish to format math questions on the web using super and subscripting, try using the Tex tags. Tom has been kind enough to import MimeTeX to this site. One thing I cannot yet do with it is make those clever angle brackets for terms certain.
You may see some good samples of how t... |
This is a quick advert for the excellent online course “Reinforcement Learning” by David Silver.
David Silver’s ‘teaching’ home page includes links to the slides (which are particularly helpful for Lecture 7, where there’s no video available - just sound).
These notes are mainly for my own consumption…
Musings : Eligib... |
Understanding light propagation in an optical fiber by ray optics (see “optical fiber“) gives an intuitive understanding and, in fact, is a good approximation if the size of the core is much larger than the wavelength guided in the optical fiber. The electromagnetic properties of an optical fiber, however, become essen... |
To evaluate detection performance, we plot the
miss-rate $mr(c) = \frac{fn(c)}{tp(c) + fn(c)}$ against the number of false positives per image $fppi(c)=\frac{fp(c)}{\text{#img}}$ in log-log plots. $tp(c)$ is the number of true positives, $fp(c)$ is the number of false positives, and $fn(c)$ is the number of false negat... |
If you want a numerical method to satisfy certain properties, you have to prove (or construct the method in such a way) that these properties hold.
Examples:
conservative finite-volume formulations for conservation laws Runge-Kutta methods preserve linear invariants symplectic integrators preserve symplecticity, which ... |
2018-08-25 06:58
Recent developments of the CERN RD50 collaboration / Menichelli, David (U. Florence (main) ; INFN, Florence)/CERN RD50 The objective of the RD50 collaboration is to develop radiation hard semiconductor detectors for very high luminosity colliders, particularly to face the requirements of the possible u... |
As Geoff Oxberry points out in his comment, if you know a priori that $x_k=0$ for $k\leq i$ and $k\geq j$, then there's nothing to optimize with respect to these variables and you can just solve the (smaller) reduced unconstrained problem $\min_{x\in\mathbb{R}^{j-i-1}}\hat f(x)$, where $\hat f$ takes only the nonzero v... |
This is essentially an addition to the list of @4tnemele
I'd like to add some earlier work to this list, namely Discrete Gauge Theory.
Discrete gauge theory in 2+1 dimensions arises by breaking a gauge symmetry with gauge group $G$ to some lower
discrete subgroup $H$, via a Higgs mechanism. The force carriers ('photons... |
Help:Editing Math Equations using TeX This is how you edit math equations using the TeX syntax to make nice looking equations. Please use TeX when writing math. Trying to put equations directly into the text doesn't look very nice and TeX is very easy to learn.
If you already use TeX, then all you need to know is that ... |
№ 9
All Issues Volume 68, № 8, 2016
Ukr. Mat. Zh. - 2016. - 68, № 8. - pp. 1011-1020
We prove the existence of wave operators for the multidimensional electromagnetic Schr¨odinger operator in divergent form by the Cook method. Moreover, under certain conditions on the coefficients of the given operator, we establish th... |
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... |
I'm not sure where I could pose a challenge to find best $f(n)$ so people will join in. $n\ge 5$ will never probably be proven optimal, but some lucky computations or out of the box analysis might give nice results.
(Given $n$ fixed digits and operations $(+,-,\times,\div)$, whats the highest $N\in\mathbb N$, such that... |
The only way in which your original grammar produces a string of the form $a^nb^n$ is if the production $S\to aSbb$ is never used. Similarly, the only way in which it produces $a^nb^{2n}$ is if $S \to aSb$ is never used. We thus want to force both of them to be used. Since all these productions "commute" (it doesn't ma... |
Background:
We have so far taken the bond B to be deterministic for simplicity, but some reflection shows that this is not in any way necessary. Everything works out the same way with a stochastic bond $B_1(u) \neq B_1(d)$ (except the algebra takes a little more work), as we now describe. The equations defining the hed... |
Thanks in advance for the help.
I'm working on a problem that would be greatly simplified if a solution exists for the following optimization problems for some $N \in \mathbb{N}^+$
$$min \underset{i \in N}{\prod} x_i \;\; s.t. \;\; y = \underset{i \in N}{\sum} x_i \; \wedge \; y,x_i \in [0,1]$$
$$max \underset{i \in N}... |
The notation is mostly taken from the book "Markov chains and mixing times" by Levin, Peres, and Wilmer.
Consider an irreducible, aperiodic, time-reversible, discrete-time Markov chain on a finite state space. The relaxation time $t_{\mathrm{rel}}$ is the reciprocal of the absolute spectral gap. The $\epsilon$-mixing t... |
Scientific posters present technical information and are intended for congress or presentations with colleagues. Since LaTeX is the most natural choice to typeset scientific documents, one should be able to create posters with it. This article explains how to create posters with latex
Contents
The two main options when... |
Given an isolated $N$-particle dynamical system with only two body interaction, that is
$$H=\sum_{i=1}^N\frac{\mathbf{p}_i^2}{2m}+\sum_{i<j}V(\mathbf{r}_i-\mathbf{r}_j)$$
In the thermodynamic limit, that is $N\gg 1$ and $N/V=$constant, it seems that not all two body interaction can make system approach thermal equilibr... |
Let
$M_d$
be a $d$-manifold generator of a subgroup of bordism group $$ \Omega_d^{G}, $$ or further generalization
$$ \Omega_d^{G}(K(\mathcal{G},n+1)), $$
which $G$ is the given structure including the tangent bundle structure (such as the SO or Spin) and the internal gauge bundle structure (such as an additional compa... |
Cost-Complexity Pruning
Post-pruning algorithm for Decision Trees
by Breiman, Olshen, Stone (1984)
Cost-Complexity Function need to optimize the cost-complexity function $R_\alpha (T) = R(T) + \alpha \cdot | f(T) |$ where $R(T)$ is the training/learning error $f(T)$ a function that returns the set of leaves of tree $T$... |
I wanted to better understand dfa. I wanted to build upon a previous question:Creating a DFA that only accepts number of a's that are multiples of 3But I wanted to go a bit further. Is there any way we can have a DFA that accepts number of a's that are multiples of 3 but does NOT have the sub...
Let $X$ be a measurable... |
I recently came across this in a textbook (NCERT class 12 , chapter: wave optics , pg:367 , example 10.4(d)) of mine while studying the Young's double slit experiment. It says a condition for the formation of interference pattern is$$\frac{s}{S} < \frac{\lambda}{d}$$Where $s$ is the size of ...
The accepted answer is c... |
2019-10-14 17:21
Performance of VELO clustering and VELO pattern recognition on FPGA/LHCb Collaboration This document contains plots and tables showing the performance obtained on VELO clustering and VELO pattern recognition using algorithms implementable on FPGA. The data used are simulated with LHCb Upgrade condition... |
I'm working on a 4-layer PCB with a U-Blox module and I'm trying to calculate the space between the fencing vias next to the Antenna trace and for the stitching vias.
According to the datasheet we have the following possible frequencies:
The case which would cause the closest vias is the last one.
According to the calc... |
Let $G$ be a group (for now discrete). A subgroup $H$ of $G$ is called a commensurated subgroup of $G$, if $H\cap xHx^{-1}$ is a finite index subgroup of $H$ for all $x\in G$. These subgroups are also called Hecke subgroups or almost normal subgroups. My question is:
Is there any non-elementary, closed, discrete and co... |
Some background on (compact) Belyi surfaces
$\newcommand{\Ch}{\hat{\mathbb{C}}}$A compact Riemann surface $X$ is called a
Belyi surface if there exists a branched covering map $f:X\to \Ch$ such that $f$ is branched over at most three points of $\Ch$. Here $\Ch$ denotes the Riemann sphere; we can and will take the three... |
This article is cited in scientific papers (total in 3 3 papers) Estimation of Solutions of Boundary-Value Problems in Domains with Concentrated Masses Located Periodically along the Boundary: Case of Light Masses G. A. Chechkin M. V. Lomonosov Moscow State University Abstract: We study the asymptotic behavior of solut... |
"Early in his life, Schrödinger experimented in the fields of electrical engineering, atmospheric electricity, and atmospheric radioactivity, but he usually worked with his former teacher Franz Exner. He also studied vibrational theory, the theory of Brownian movement, and mathematical statistics."
Schrödinger's cat is... |
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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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Now showing items 1-10 of 26
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 ... |
$\def\pd{\partial}$ $\def\l{\left}\def\r{\right}$ $\def\mdot{{\dot{m}}}$ $\def\eps{\varepsilon}$ Consider a tube with longitudinal coordinate $x$ from $0$ to $l$ and varying cross-section $A(x)$. Derek S. Bale (a former student of Prof. Randall J. LeVeque) gives in his PhD (http://faculty.washington.edu/rjl/students/db... |
SolidsWW Flash Applet Sample Problem 2 Flash Applets embedded in WeBWorK questions solidsWW Example Sample Problem 2 with solidsWW.swf embedded
A standard WeBWorK PG file with an embedded applet has six sections:
A tagging and description section, that describes the problem for future users and authors, An initializati... |
Inductive Logic Programming (ILP) Algorithm AKA:ILP, Inductive Logic Programming. Context: It can (typically) be used to solve a Knowledge-based Supervised Binary Classification Task. It can (typically) find a Hypothesis [math]H[/math], such that [math]B \cup H \vDash e, \forall e \in E^+[/math] [math]B \cup H \nvDash ... |
As you have already figured out that we have to use the ideal gas law $PV= nRT$, which is you have modified as $$\frac{n}{V}= \frac{P}{RT}$$
To find the density, we can further modify the equation as,\begin{align}\frac{m}{MV} &= \frac{P}{RT}\\\frac{\rho}{M}&= \frac{P}{RT}\\\rho &= \frac{PM}{RT}\\\end{align}
Now we just... |
Probability Seminar Spring 2019 Thursdays in 901 Van Vleck Hall at 2:25 PM, unless otherwise noted. We usually end for questions at 3:15 PM.
If you would like to sign up for the email list to receive seminar announcements then please send an email to join-probsem@lists.wisc.edu
January 31, Oanh Nguyen, Princeton
Title:... |
I have just started learning Riemann surfaces and I am using the book by Rick Miranda:
Algebraic curves and Riemann Surfaces. #F in section 1.3 asks to determine the genus of the curve in $\mathbb{P}^3$ defined by the two equations $x_0x_3=2x_1x_2$ and $x_0^2 + x_1^2 +x_2^2 +x_3^2 = 0$. #G also has a similar question i... |
Given that the BVP is a second-order inhomogeneous ODE, we find the characteristic equation to be in the form $$r^2+a^2=0\implies r=\pm ai.$$
Thus, the homogeneous equation is $$u_h=C_1\cos(ax)+C_2\sin(ax).$$
The particular equation is $$u_p=C_3x\sin(\pi x)+C_4x\cos(\pi x),$$ $$u_p'=-C_3\pi x\cos(\pi x)+C_3\sin(\pi x)-... |
For a simple and undirected graph $G$, is there a known upper bound on the number of edges it has, given number of vertices $n$, girth $g$ and maximum degree $\Delta$?
No, there is no bound. For instance, consider an $n\times n$ square grid. It has girth $g=4$ and maximum degree $\Delta=4$ but there are roughly $2n(n+1... |
I have a question concerning computational geometry which arises in the simulation of fields with topological defects, and I'd like to know whether there's an efficient algorithm (or any algorithm) to solve it.
The problem is basically the following: consider a grid cell $\mathcal{C}$ in a 3-D cubical grid. On each of ... |
Consider the strictly convex unconstrained optimization problem $\mathcal{O} := \min_{x \in \mathbb{R}^n} f(x).$ Let $x_\text{opt}$ denote its unique minima and $x_0$ be a given initial approximation to $x_\text{opt}.$We will call a vector $x$ an $\epsilon-$ close solution of $\mathcal{O}$ if \begin{equation} \frac{||x... |
Answer
If you assume your returns are independent (yes your models might loosen this assumption) then the two models, $Q_1$ and $Q_2$ assign probability distributions to the returns on any given day, $i$: $q_1^i(r^i)$ and $q_2^i(r^i)$.
Presumably you are interested in the model that can more accurately predict the stat... |
It seems that the sample linear correlation coefficient $\hat{\rho}$ of samples generated by a copula that is parametrized by $\rho$ is unequal to $\rho$. For example, I construct a Normal copula with parameter $\rho = 0.9$. I generated a bunch of samples from this copula, and calculate their sample correlation coeffic... |
Calculators¶ ClusterExpansionCalculator¶ class
mchammer.calculators.
ClusterExpansionCalculator(
structure, cluster_expansion, name='Cluster Expansion Calculator', scaling=None, use_local_energy_calculator=True)¶
A ClusterExpansionCalculator object enables the efficient calculation of properties described by a cluster ... |
Let's say we are given a function $f(x)$, which is not defined at the point $x_0$. How do we find linear approximation of $f$ near $x_0$? P.S. I wrote "linear" just to make things simpler, I came across this problem while trying to approximate the following function near zero: $\frac{lnx}{x*e^x}$. My problem is that to... |
Well firstly, how do you define $[a,b]$ aka $\text{lcm}(a,b)$? I'll define it by the converse of a proposition in Artin Algebra (Prop 2.3.8)
Namely, we'll prove the converse of Prop 2.3.8 where $m:=\text{lcm}(a,b)$ is defined by the integer s.t.
(a) $m$ is divisible by both $a$ and $b$
(b) If $n$ is divisible by $a$ an... |
I am trying to show the following two statements:
1) Let $U$ be a connected open subset of $\mathbb{C}^n$ and let $L$ be a closed subset of $U$. Show that $$accum(U\setminus L)\cap L\neq \emptyset,$$ where accum() denotes the set of accumulation points.
2) Let $L$ be a closed subset of $\mathbb{C}^n$ and let $x\in L$. ... |
In that case, the problem becomes a non-trivial
stopping time problem.
Consider a filtered probability space $(\Omega, \mathcal{F}, \mathbb{P})$ equipped with the natural filtration of a standard Brownian motion $W_t^\mathbb{P}$.
Assuming a geometric Brownian motion for the underlying asset, one gets$$ S_t = S_0 \exp\l... |
I would like to generate an isotropic gaussian random field described by a power spectrum $P(k)$ on a 3D grid which represents spherical polar coordinates (i.e., the angular separation between pixels at each slice in the radial direction is equal and each pixel represents a solid angle).
I am currently generating a ran... |
I am really confused on how to go about this problem. I know the formula, but I get confused with all the steps and how to use what I got to get to what I need.
A student is examining a bacterium under the microscope. The E. coli bacterial cell has a mass of $m = 0.100~\mathrm{fg}$ (where a femtogram, $\mathrm{fg}$, is... |
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Now showing items 1-1 of 1
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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Now showing items 1-10 of 18
J/Ψ production and nuclear effects in p-Pb collisions at √sNN=5.02 TeV
(Springer, 2014-02)
Inclusive J/ψ production has been studied with the ALICE detector in p-Pb collisions at the nucleon–nucleon center of mass energy √sNN = 5.02TeV at the CERN LHC. The measurement is performed in... |
I am trying to fit a restricted cubic spline (natural cubic spline) with 4 knots to toy data, attempting to follow Hastie, Tibshirani, Friedman 2nd ed. 5.2.1 p.144-146, Eqs 5.4 and 5.5. Data: Is basically a transposed ‘S’ shape. R-code is:
n <© 100x <- (1:n)/ntrue <- ((exp(1.2*x)+1.5*sin(7*x))-1)/3noise <- rnorm(n, 0, ... |
As per the suggestion by Christian in the comments here, as part of my continuing quest to understand the Raviart-Thomas (RT) elements I'd like to know how exactly the RT elements are defined globally, and in particular how they have compact support.
For RT0 on the reference square, one of the basis functions is $\math... |
The action which describes a string propagating in a $D$ dimensional spacetime, with given metric $g_{\mu\nu}$, is given by the Polyakov action $$S_{\text{p}}=-\frac{T}{2}\int \mathrm{d}\sigma\mathrm{d}\tau\sqrt{-h}\eta^{\alpha\beta}\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}g_{\mu\nu}\tag{1}$$ where the symbols ha... |
Octopress has become a popular blogging framework among programmers since its 2.0 release. We can observe this growing from the reading of google keyword tool. It did serve the needs of most programmers comparing to other existing blogging frameworks. To the author, I give my full compliement and respect.
However, as w... |
Given a tree with $$$n$$$ nodes numbered from $$$1$$$ to $$$n$$$. Each node $$$i$$$ has an associated value $$$V_i$$$.
If the simple path from $$$u_1$$$ to $$$u_m$$$ consists of $$$m$$$ nodes namely $$$u_1 \rightarrow u_2 \rightarrow u_3 \rightarrow \dots u_{m-1} \rightarrow u_{m}$$$, then its alternating function $$$A... |
The basic reason that it is justified is that the time integral is a linear operator.
So let's step way back into the abstract, what you
really have is some configuration space $\mathcal C$ of possible configurations of some system, and some space of paths $\mathcal P$ through that, which is some subset of the function... |
Problem: Let $ A_1,A_2,...,A_n$ be sets such that $X=\bigcup_{i=1}^n A_i$.
Prove that there exists a sequence of sets $ B_1,B_2,...,B_n$ such that
a) $B_i \subseteq A_i$ for each $i=1,2,...,n$
b) $B_i \cap B_j = \emptyset$ for $i \neq j$
c) $X=\bigcup_{i=1}^n B_i$
My observation: I look at the "new things" added to $X$... |
In an action-packed three pages of Lurie's DAG-XIII: Rational and p-adic Homotopy Theory, section 2.2: Power Operations on $\mathbb{E}_{\infty}$-algebras, one finds a construction of the power operation $P^0$ following a few observations on the $p$-power Tate construction in the category of $k$-module spectra: $\hat{T}... |
I am wondering how the confidence interval for the Area under the Curve statistic (ROC curves) is derived. I have heard that the AUC can be assumed to be normally distributed, but I am looking for a proof of this statement or a derivation of the confidence intervals
AUC can be viewed as Wilcoxon-Mann-Whitney Test. And ... |
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Anisotropic flow of inclusive and identified particles in Pb–Pb collisions at $\sqrt{{s}_{NN}}=$ 5.02 TeV with ALICE
(Elsevier, 2017-11)
Anisotropic flow measurements constrain the shear $(\eta/s)$ and bulk ($\zeta/s$) viscosity of the quark-gluon plasma created in heavy-ion collisions... |
Bulletin of the American Physical Society 65th Annual Meeting of the APS Division of Fluid Dynamics Volume 57, Number 17 Sunday–Tuesday, November 18–20, 2012; San Diego, California
Session L11: Bubbles IV and Drug Delivery Hide Abstracts Chair: Kausik Sarkar, George Washington University
Room:
26A
Monday, November 19, ... |
I am doing a molecular dynamics simulation. I need to assign initial velocities to the atoms. I want to assign the initial velocities which follow the Maxwell-Boltzmann distribution. How do I calculate such initial velocities using a uniform random number generator with range [0,1)?
The initial velocities are drawn fro... |
One thing that really helped me understand poles and zeros is to visualize them as amplitude surfaces. Several of these plots can be found in A Filter Primer. Some notes:It's probably easier to learn the analog S plane first, and after you understand it, then learn how the digital Z plane works.A zero is a point at whi... |
A distribution $\mathcal{D}$ is said to $\epsilon$-fool a function $f$ if $|E_{x\in U}(f(x)) - E_{x\in \mathcal{D}}(f(x))| \leq \epsilon$. And it is said to fool a class of functions if it fools every function in that class.
It is known that $\epsilon$-biased spaces fool the class of parities over subsets. (see Alon-Go... |
Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $E$ be a complete locally compact separable metric space, $(X^n_t)_{t\ge0}$ be an $E$-valued càdlàg process on $(\Omega,\mathcal A,\operatorname P)$ for $n\in\mathbb N$ and $(X_t)_{t\ge0}$ be an $E$-valued continuous process on $(\Omega,\mathcal A,\opera... |
I am trying to find the minimum of the so called Beale’s function given by
$f(x_1,x_2) = (1.5-x_1+x_1x_2)^2 + (2.25-x_1+x_1x_2^2)^2 + (2.625-x_1+x_1x_2^3)^2$
Using Newton iteration
$x^{(k+1)} = x^{(k)} - \alpha^{(k)} \eta^{(k)} $ with $ \eta^{(k)} = \left( \frac{\partial ^2f}{\partial x^2}(x^{(k)}) \right)^{-1}\frac{\p... |
If you converge, you would expect the steps to get small. Ideally, a step $\delta x_k$ in an optimization algorithm would go from the current iterate $x_k$ to the exact solution $x^\ast$, so $\|\delta x_k\| \approx \|x_k-x^\ast\|$ which is, as you approach the solution, going to be small.
Now you say that $\delta x_k$ ... |
In the last lesson, we derived the functions that help us descend down cost functions efficiently. Remember that this technique is not so different from what we saw when using the derivative to tell us the next step size and direction in two dimensions.
When descending down our cost curve in two dimensions, we used the... |
I don't know much about the deformation of compact complex manifolds, I've only read chapter 6 of Huybrechts' book
Complex Geometry: An Introduction. There are two parts to this chapter. The second goes through the standard approach, that is, considering a family of compact complex manifolds as a proper holomorphic sub... |
I'm trying to understand tensor notation and working with indices in special relativity. I use a book for this purpose in which $\eta_{\mu\nu}=\eta^{\mu\nu}$ is used for the metric tensor and a vector is transformed according to the rule $$x'^\mu= \Lambda^\mu{}_{\alpha}x^\alpha$$ (Lorentz-transformation).
I think I und... |
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J/Ψ production and nuclear effects in p-Pb collisions at √sNN=5.02 TeV
(Springer, 2014-02)
Inclusive J/ψ production has been studied with the ALICE detector in p-Pb collisions at the nucleon–nucleon center of mass energy √sNN = 5.02TeV at the CERN LHC. The measurement is performed in... |
Why do we care about eigenvalues, eigenvectors, and singular values? Intuitively, what do they tell us about a matrix? When I first studied eigenvalues in college, I regarded it as yet another theoretical math trick that is hardly applicable to my life. Once I passed the final exam, I shelved all my eigen-knowledge to ... |
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"""
Definition of the semi-grand canonical ensemble class.
"""
import numpy as np
from ase import Atoms
from ase.data import atomic_numbers, chemical_symbols
from ase.units import kB
f... |
I have two questions. First, given a finite dimensional complex vector space $V$ and a finite group representation $\rho:G \to GL(V)$, Maschke's theorem tells us that we may decompose $V$ into a direct sum
$$V= \bigoplus_{\lambda}m_{\lambda}V^{\lambda}$$
of irreducible representations $V^{\lambda}$ of multiplicity $m_{... |
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